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1,314,259,996,963 | arxiv | \section*{Acknowledgements}
Authors partially supported by the grant PID2020-113192GB-I00 (Mathematical Visualization: Foundations, Algorithms and Applications) from the Spanish MICINN.
Part of this work was developed during a research visit of the first author to CUNEF University in Madrid.
\section{Introduction}
The study and analysis of the behavior of algebraic or algebraic-geometric objects under specializations is of great interest from a theoretical, computational or applied point of view. For instance, some techniques for computing resultants, gcds, or polynomial factorizations, rely on Hensel's lemma or the Chinese remainder theorem (see e.g. \cite{Geddes}, \cite{winkler}). From a more theoretical point of view, also computational, it is important to control, for instance, when a resultant, or more generally a Gr\"obner basis with parameters, specializes properly (see e.g. \cite{cox}, \cite{montes}).
The question whether a given irreducible polynomial over $\mathbb{K}(a_1,\ldots,a_n)$ remains irreducible when the parameters are replaced by values in a field $\mathbb{K}$ was studied intensively by Hilbert~\cite{hilbert1892} and Serre~\cite{Serre} and is the defining property of ``Hilbertian fields''.
The work of Serre can be seen in a more general context.
With respect to applications, there is a vast amount of applications of algebraic curves involving parameters: level curves of surfaces~\cite{alcazar2007}, linear homotopy deformation of curves (see Section \ref{sec-application}), curve recognition~\cite{torrente2017}, geometric constructions in computer aided design, like offsets, conchoids, cissoids etc, where the final object depends on the distance, the focus, etc.; see e.g. \cite{arrondo1}, \cite{arrondo2}, \cite{lu}, \cite{cissoids}, \cite{conchoids}.
Another type of applications is the computation with meromorphic functions in linear algebra (see \cite{penrose}) or the rational solutions of functional algebraic equations (see Section \ref{sec-application}).
In this work we study algebraic curves $\mathcal{C}(F)$ given as the zero-set of a polynomial
\begin{equation}\label{eq-parametric}
F(x,y) = 0 ~\text{ with }~ F \in \mathbb{K}(a_1,\ldots,a_n)[x,y]
\end{equation}
where $\mathbb{K}$ is a computable field of characteristic zero, $a_1,\ldots,a_n$ are a set of parameters, and $F$ is irreducible over the algebraic closure of the coefficient field $\mathbb{K}(a_1,\ldots,a_n)$, denoted by $\overline{\mathbb{K}(a_1,\ldots,a_n)}$.
In this paper, we focus on the problem that for certain values of the parameters $a_1,\ldots,a_n$ the algebraic properties of the resulting curve do not coincide with the generic properties of $\mathcal{C}(F)$.
More precisely, we define several Zariski-closed sets in the space of parameter values where non-generic behavior may appear.
Of particular interest are the singularities, their multiplicities and their character. This leads to a partition of the affine space, where the parameters take values, so that in each subset of the partition the specialized curve is either reducible or its (geometric) genus is invariant.
When the generic curve has genus zero, for a given rational parametrization can be given a better description.
In particular, the set of parameters where Hilbert's irreducibility theorem does not hold can be isolated. Moreover, the proper specialization of the rational parametrization is guaranteed.
In~\cite{myBook,walker} and references therein are studied algebraic curves and their rationality.
The problem of finding rational parametrizations of plane curves is a classical problem and has already been studied by Hilbert~\cite{HH}, and more recently in \cite{hoeij}, \cite{schicho}, \cite{SW91},\cite{SW97}.
In addition,
for evaluating the parameters, it is important to control field extensions which might be necessary for computing parametrizations.
Optimal fields of parametrizations have been studied in~\cite{hoeij} and \cite{SW97}.
When introducing parameters in the coefficients, new phenomena have to be considered and lead to Tsen's study of finding solutions in a minimal field~\cite{tsen}.
The structure of the paper is as follows.
In Section~\ref{sec-prel} we present notations, preliminaries on algebraic curves and rational parametrizations.
Of particular interest is the computation of the genus and a rational parametrization, if it exists.
Some of the details are attached in the appendix\footnote{There exist different methods to deal computationally with the genus: the adjoint curve based method (see e.g. \cite{walker} and \cite{myBook}), the method based on the anticanonical divisor (see \cite{hoeij}) or the method based on Puiseux expansions (see \cite{PW}), among others. In this paper we will follow the adjoint curve based method which is described, for completeness, in the appendix.}~\ref{App}.
In Section~\ref{sec-specialization}, we introduce the unspecified parameters and their specialization.
The computation of the genus and rational parametrizations is followed to define several computable Zariski-open subsets $\Omega$ where the specialized curve behaves, up to irreducibility, as in the generic case.
The actual computation of the genus is presented in Section~\ref{sec-genus}.
In Theorem~\ref{theorem:genus-weak} is shown that the genus of the specialized curve, where the parameters take values in $\Omega_{\mathrm{singOrd}}$, is less or equal to the generic genus or the defining polynomial is reducible.
A direct corollary of that is that specialized curves of rational curves are also rational or reducible (Corollary~\ref{cor:genus-zero}).
For values in a smaller set $\Omega_{\mathrm{genusOrd}}$, it is shown that the genus of the curve remains exactly the same, again up to irreducibibility, see Theorem~\ref{theorem:genus-strong}.
Section~\ref{sec-parametricCurves} is devoted to the case where the generic curve is rational; in this frame the irreducibility can be guaranteed.
For some of the parameter values the genus may remain the same but an evaluation of the parametrization is not possible.
In Theorem~\ref{thrm:genus0general}, however, is presented an open set where the specialization is possible and results in a parametrization of the specialized curve.
These open sets can be recursively used for decomposing the whole parameter space as it is explained in Section~\ref{sec:decomposition}.
Applications as described above are presented by using illustrative examples in Section~\ref{sec-application}.
This manuscript is a self-contained work on the computation of the genus and rational parametrizations of algebraic curves involving parameters.
Results from various mathematical disciplines are combined for this purpose and presented in a coherent way.
A rigorous construction of such computable Zariski-open sets were, up to our knowledge, missing in the literature.
The theorems mentioned in the previous paragraph are novel and can be directly applied in several interesting problems involving parametric curves.
\section{Preliminaries and notation}\label{sec-prel}
Throughout this paper, the following notation will be used. $\mathbb{K}$ is a computable field of characteristic zero. We denote by $\mathbf{a}$ a tuple of parameters, and we represent by $\mathbb{L}$ the field extension $\mathbb{L}:=\mathbb{K}(\mathbf{a})$. In addition, we consider an algebraic element $\gamma$ over $\mathbb{L}$. Let $\mathbb{F}$ be the field $\mathbb{F}:=\mathbb{L}(\gamma)$. Furthermore, $K$ represents any field extension of $\mathbb{K}$. We denote by $\overline{K}$ the algebraic closure of $K$, similarly for any field appearing in the paper. $\mathbb{S}$ is the affine space
\begin{equation}\label{eq-S}
\mathbb{S}=\overline{\mathbb{K}}^{\,\#(\mathbf{a})}
\end{equation}
where $\mathbf{a}$ will take values.
For $G\in K[x,y]\setminus K$, we denote by $\mathcal{C}(G)$ the plane affine algebraic curve
\[ \mathcal{C}(G)=\{ p\in \overline{K}^2\,|\, G(p)=0\}. \]
We denote by $G^h(x,y,z)$ the homogenization of $G$, and by $G_x,G_y$ (similarly for $G^{h}_{x}, G^{h}_{y}, G^{h}_{z}$) the partial derivative of $G$ w.r.t. $x$ and $y$ respectively. For a homogeneous polynomial $M\in K[x,y,z]\setminus \{0\}$, $\mathcal{C}(M)$ denotes the projective plane curve
\[ \mathcal{C}(M)=\{ p\in \mathbb{P}^2(K)\,|\, M(p)=0\}. \]
For polynomials $f,g$ in the variable $t$, and coefficients in an integral domain, we denote by $\mathrm{res}_t(f,g)$ the resultant of $f$ and $g$ w.r.t. $t$.
Let $\{f_1,\ldots,f_k\}\subset K[\overline{v}]$, where $\overline{v}$ is a tuple of variables. We denote by $\mathbb{V}(f_1,\ldots,f_k)$ the zero set, over $\overline{K}$, of the polynomials $\{f_1,\ldots,f_k\}$; similarly for $\mathbb{V}(\mathrm{I})$ where $\mathrm{I}$ is an ideal in $K[\overline{v}]$.
\subsection{Rational Curves}\label{sec-pre}
Throughout this section, let $G\in K[x,y]\setminus K$ be irreducible over $\overline{K}$. A \textit{rational (affine) parametrization} of the irreducible affine plane curve $\mathcal{C}(G)$ is a pair of rational functions $\mathcal{P}(t)\in \overline{K}(t)^2 \setminus \overline{K}^2$ such that $G(\mathcal{P}(t))=0$. A rational (projective) parametrization of $\mathcal{C}(G^h)$ is of form $\mathcal{Q}(h,t)=(p_1(h,t):p_2(h,t):p_3(h,t))$ where $p_i$ are homogeneous co-prime polynomials of the same degree over $\overline{K}$, not all zero, such that $G^h(\mathcal{Q})=0$. We observe that the degree, the irreducibility and the rationality of $\mathcal{C}(G)$ and $\mathcal{C}(G^h)$ are equivalent. Moreover the parametrizations of $\mathcal{C}(G)$ and $\mathcal{C}(G^h)$ relate each other by means of homogenizing and dehomogenizing. So, in the following we will focus on affine parametrizations.
The parametrization $\mathcal{P}(t)$ is called \textit{birational} or \textit{proper} if the map $\overline{K} \dashrightarrow \mathcal{C}(G); t\mapsto \mathcal{P}(t)$ is injective in a non--empty open Zariski subset of $\overline{K}$ (see e.g. \cite{myBook} for further details). Curves admitting a rational parametrization are called \textit{rational}, and they correspond to those of genus zero; note that the genus of $\mathcal{C}(G)$ is defined as the genus of $\mathcal{C}(G^h)$. There exist algorithmic methods to compute the genus of an algebraic curves and to determine, when the genus is zero, a rational parametrization of the curve (see e.g. \cite{hoeij}, \cite{SW91}, \cite{SW97}, \cite{myBook}). In Appendix \ref{App} we summarize the adjoint curves based method for parametrizing curves. Some of the ideas in this paper will use those methods.
In general, if one computes a parametrization $\mathcal{P}(t)$ of $\mathcal{C}(G)$, the ground field $K$ has to be extended (see e.g. Sections 4.7. and 4.8. in \cite{myBook}). A subfield $\mathbb{E}$ of $\overline{K}$ is called a \textit{parametrizing field} or \textit{field of parametrization} of $\mathcal{C}(G)$ if there exists a parametrization of $\mathcal{C}(G)$ with coefficients in $\mathbb{E}$.
\subsection{Fields of Parametrization}\label{sec-FieldOfparametrization}
In this section, we work with the field $\mathbb{L}:=\mathbb{K}(\mathbf{a})$. Let $G\in \mathbb{L}[x,y]$ be an irreducible (over $\overline{\mathbb{L}}$) non-constant polynomial, and let us assume $\mathcal{C}(G)$ is a rational curve. We analyze the fields of parametrization of $\mathcal{C}(G)$.
$\overline{\mathbb{L}}$ is always a field of parametrization of $\mathcal{C}(G)$. Nevertheless, in \cite{SW97} (see also Chapter 5 in \cite{myBook}), the optimality of the fields of parametrization is analyzed and, as a consequence of Hilbert-Hurwitz Theorem (see \cite{HH}), there always exists a field extension of $\mathbb{L}$, of degree at most $2$, being a field of parametrization of $\mathcal{C}(G)$. Indeed, this field extension, of degree at most two, is the field extension used in Step (4), of the parametrization computation (see Subsection \ref{subsec:param}), to express the simple point utilized in the parametrization of either the conic or the line.
We observe that if the two degree field extension is $\mathbb{L}(\alpha)$, with minimal polynomial $t^2+bt+c\in \mathbb{L}[t]$, then $\mathbb{L}(\alpha)=\mathbb{L}(\beta)$ where $\beta=\alpha+b/2$ which minimal polynomial is $t^2+c-b^2/4$. Therefore, the following holds.
\begin{theorem}\label{theorem:HHext} \
\begin{enumerate}
\item If $\deg(\mathcal{C}(G))$ is odd then $\mathbb{L}$ is a field of parametrization.
\item If $\deg(\mathcal{C}(G))$ is even then either $\mathbb{L}$ is a field of parametrization or there exists $\delta\in \overline{\mathbb{L}}$ algebraic over $\mathbb{L}$, with minimal polynomial $t^2-\alpha\in \mathbb{L}[t]$, such that $\mathbb{L}(\delta)$ is a field of parametrization of $\mathcal{C}(G)$.
\end{enumerate}
\end{theorem}
\begin{remark}
Observe that the previous result is valid taking $\mathbb{L}$ as any field extension of $\mathbb{K}$.
\end{remark}
The case where $\mathbf{a}$ contains a single element admits a particular treatment because of Tsen's Theorem; we refer to \cite{tsen} for this topic.
\begin{corollary}\label{cor:TsenN2}
If $\#(\mathbf{a})=1$, then $\mathbb{L}$ is a field of parametrization of $\mathcal{C}(G)$.
\end{corollary}
\begin{proof}
By Hilbert-Hurwitz Theorem (see e.g. Theorem 5.8. in \cite{myBook} or Subsection \ref{subsec:param}), $\mathcal{C}(G)$ is $\mathbb{L}$--birationally equivalent to either a line or a conic. So, fields of parametrization are preserved.
In the line case, the result is clear. In the conic case, the result follows from Tsen's Theorem (see e.g. Corollary 1.11. in \cite{shafa}).
\end{proof}
\begin{remark}
The proof of Tsen's Theorem provides a method for computing an $\mathbb{L}$-simple point on the conic. An alternative approach for computing this point can be found in \cite{HW} and \cite{vo}.
\end{remark}
\begin{remark}\label{rem:TsenGeneralization}
In the following section we will work with $G\in \mathbb{K}[\mathbf{a},\gamma][x,y]$ where $\gamma$ is algebraic over $\mathbb{K}(\mathbf{a})$.
In the case where $\#(\mathbf{a})=1$, we can view $\gamma$ as the only parameter and write $\mathbf{a}$ in terms of $\gamma$.
More precisely, let $M(\mathbf{a},c) \in \mathbb{K}(\mathbf{a})[c]$ be the minimal polynomial of $\gamma$. We can view $M$ as rational expression in $\mathbf{a}$ and consider $H(a):=\num(M)(\mathbf{a}=a,c=\gamma) \in \mathbb{K}(\gamma)[a]$ as polynomial in $a$ with the root $\mathbf{a}$.
Thus, $\mathbb{K}(\gamma,\mathbf{a}):\mathbb{K}(\gamma)$ is a field extension of degree $d \le \deg_c(M)$.
If $d=1$, by Corollary~\ref{cor:TsenN2}, $\mathbb{K}(\gamma)$ is a field of parametrization of $\mathcal{C}(G)$.
\end{remark}
\begin{remark}
In Corollary \ref{cor:TsenN2}, we have seen that if $\#(\mathbf{a})=1$ then $\mathbb{L}$ is a field of parametrization. The following example shows that if $\#(a)>1$, in general, $\mathbb{L}$ is not a field of parametrization. We consider the conic defined by $F:=a_1 x^2+a_2 y^2-1$, and we see that it does not have a parametrization over $\mathbb{C}(a_1,a_2)$. Let us assume that $\mathbb{C}(a_1,a_2)$ is a field of parametrization of $\mathcal{C}(F)$, then $\mathcal{C}(F)$ has infinitely many point in $\mathbb{C}(a_1,a_2)^2$. $\mathcal{C}(F)$ can be properly parametrized by
\[ \mathcal{P}:=\left( \sqrt{a_1}\, \frac{1-t^{2}}{t^{2}+1}, \sqrt{a_2} \,\frac{2 t}{t^{2}+1}
\right), \]
which inverse is
\[ \mathcal{P}^{-1}(x,y)=\frac{\sqrt{a_{2}}\, \left(\sqrt{a_{1}}+x \right)}{\sqrt{a_{1}}\, y}.\]
So, there are infinitely many points in $\mathcal{C}(F)\cap \mathbb{C}(a_1,a_2)^2$ that are injectively reachable, via $\mathcal{P}$, for $t\in \mathbb{C}(\sqrt{a_1},\sqrt{a_2})$. Indeed, note that all points of $\mathcal{C}(F)$, with the exception of $(-\sqrt{a_1},0)$, are reachable by $\mathcal{P}$. Let $t_0\in \mathbb{C}(\sqrt{a_1},\sqrt{a_2}) \setminus \{ 0,\pm \mathrm{i}\}$ be one of these parameter values; say $\mathcal{P}(t_0)=(x_0,y_0)\in \mathbb{C}(a_1,a_2)^2$.
Then $t_{0}^2 = (\sqrt{a_1}+x_0)/(\sqrt{a_1}-x_0) \in \mathbb{C}(\sqrt{a_1},a_2).$ For $x_0 \ne 0$, it holds that $\sqrt{a_1}+x_0$, $\sqrt{a_1}-x_0$ are coprime (seen as polynomials in $\sqrt{a_1}$) and $t_0 = \pm \sqrt{\tfrac{\sqrt{a_1}+x_0}{\sqrt{a_1}-x_0}} \notin \mathbb{C}(\sqrt{a_1},\sqrt{a_2})$, a contradiction.
For $x_0=0$ we have the curve-points $(x_0,y_0)=(0,\pm 1/\sqrt{a_2})$ which are not in $\mathbb{C}(a_1,a_2)$.
\end{remark}
\section{Specializations}\label{sec-specialization}
Throughout the paper, we will specialize the tuple of parameters $\mathbf{a}$ taking values in $\mathbb{S}$ (see \eqref{eq-S}). We will write $\mathbf{a}^0$ to emphasize that the parameters in $\mathbf{a}$ have been substituted by elements in $\overline{\mathbb{K}}$. In the following we discuss different aspects on the specializations.
\subsection{General statements}
The elements in $\overline{\mathbb{K}}(\mathbf{a})$ are assumed to be represented in reduced form; that is, the numerator and denominator are assumed to be coprime. Then, for $f:=p/q\in \overline{\mathbb{K}}(\mathbf{a})$, where by assumption $\gcd(p,q)=1$, and for $\mathbf{a}^0\in \mathbb{S}$ (see \eqref{eq-S}) such that $q(\mathbf{a}^0)\neq 0$, we denote by $f(\mathbf{a}^0)$ the $\overline{\mathbb{K}}$--element $p(\mathbf{a}^0)/q(\mathbf{a}^0)$.
We may need to work in the finite field extension $\mathbb{F}=\mathbb{L}(\gamma)$. Let $p(\mathbf{a},t) \in \mathbb{K}(\mathbf{a})[t]$, of degree $k$ in $t$, be the minimal polynomial of $\gamma$. We might simply write $p(t)$ instead of $p(\mathbf{a},t)$ and express it as
\begin{equation}\label{eq-alpha}
p(t)= t^k+\frac{N_{k-1}(\mathbf{a})}{D_{k-1}(\mathbf{a})}\,t^{k-1} + \cdots + \frac{N_{0}(\mathbf{a})}{D_{0}(\mathbf{a})}, \,\,N_i,D_i \in \mathbb{K}[\mathbf{a}]
\end{equation}
where $\gcd(N_i,D_i)=1$.
Then, for $\mathbf{a}^0\in \mathbb{S}$ such that all $D_i(\mathbf{a}^0)\neq 0$, we denote by $\gamma^0$ the algebraic element, over $\mathbb{K}(\mathbf{a}^0)$, defined by an irreducible factor of
$$p(\mathbf{a}^0,t)= t^k+\frac{N_{k-1}(\mathbf{a}^0)}{D_{k-1}(\mathbf{a}^0)} \,t^{k-1}+ \cdots + \frac{N_{0}(\mathbf{a}^0)}{D_{0}(\mathbf{a}^0)} \in \mathbb{K}(\mathbf{a}^0)[t]\subset \overline{\mathbb{K}}[t].$$
For an element $f \in \mathbb{F}$, specialized at $\mathbf{a}^0\in \mathbb{S}$, we might simply write $f(\mathbf{a}^0)$ instead of $f(\mathbf{a}^0,\gamma^0)$.
\begin{definition}\label{def-gamma}
We define the open subset $\Omega_{\gamma}:=\mathbb{S} \setminus \mathbb{V}(D)$ where $D:=\lcm(D_0,\ldots,D_{n-1})$.
\end{definition}
Clearly for $\mathbf{a}^0\in \Omega_\gamma$, $\gamma(\mathbf{a}^0)$ is well--defined. The elements in $\mathbb{F}$ are assumed to be expressed in canonical form; that is, $f\in \mathbb{F}$ is expressed as
\begin{equation}\label{eq-f}
f=\sum_{i=0}^{k-1}\dfrac{U_i(\mathbf{a})}{W(\mathbf{a})}\gamma^i
\end{equation}
where $U_i,W\in \mathbb{K}[\mathbf{a}]$ and $\gcd(U_1,\ldots,U_{k-1},W)=1$. In addition, the coefficients of polynomials in $\mathbb{F}[\overline{v}]$, w.r.t. to the tuple of variables $\overline{v}$, are also supposed to be written in canonical form.
Moreover, for $f$ as in \eqref{eq-f}, we denote by $\mathrm{Norm}(f)$ the $\mathbb{L}$--field element
$$\mathrm{Norm}(f):= \prod f(\mathbf{a},\gamma_i) \in \mathbb{L}$$
where the product is taken over all roots $\gamma_i$ in $\overline{\mathbb{L}}$ of $p(\mathbf{a},t)$ (see \eqref{eq-alpha}). Note that $\mathrm{Norm}(f)=\mathrm{res}_t(f(\mathbf{a},t),p(\mathbf{a},t))$.
\begin{lemma}\label{lem:N}
Let $f$ be as in \eqref{eq-f}.
Let $\mathbf{a}^0\in \mathbb{S}$ be such that $D(\mathbf{a}^0)W(\mathbf{a}^0)\neq 0$ (see Def. \ref{def-gamma}). If $f(\mathbf{a}^0)=0$, then $\mathrm{Norm}(f)(\mathbf{a}^0)=0$.
\end{lemma}
\begin{proof}
Since $D(\mathbf{a}^0)W(\mathbf{a}^0)\neq 0$, then $\gamma(\mathbf{a}^0)$, $f(\mathbf{a}^0)$
Then $\mathrm{Norm}(f)(\mathbf{a}^0)= \prod f(\mathbf{a}^0,\gamma_i(\mathbf{a}^0))$ is well-defined and, since one of the factors on the right hand side is equal to $f(\mathbf{a}^0,\gamma(\mathbf{a}^0))=0$, we obtain $\mathrm{Norm}(f)(\mathbf{a}^0)=0$.
\end{proof}
\begin{definition}\label{def-open1}
Let $H\in \mathbb{F} [\overline{v}]$, where $\overline{v}$ is a tuple of variables. Let $S$ be the set of all non-zero coefficients of $H$ w.r.t. $\overline{v}$. Let
$$\mathcal{D}(H):=\lcm(\{\mathrm{denom}(C)\,|\, C\in S\})\in \overline{\mathbb{K}}[\mathbf{a}],$$
and let
$$\mathcal{V}(H ):=\{ \mathrm{Norm}(\mathrm{numer}(C))\,|\, C\in S \} \subset \overline{\mathbb{K}}[\mathbf{a}].$$
We associate to $H$ the following open subsets
\begin{enumerate}
\item $\Omega_{\mathrm{def}(H)}:=\Omega_\gamma \cap\left( \mathbb{S}\setminus \mathbb{V}(\mathcal{D}(H))\right).$
\item $\Omega_{\mathrm{nonZ}(H)}:=\Omega_{\mathrm{def}(H)} \cap \left(\mathbb{S}\setminus \mathbb{V}(\mathcal{V}(H ))\right).$
\end{enumerate}
\end{definition}
\begin{remark}
Throughout the paper, we will define several open subsets of $\mathbb{S}$. All these open subsets will be included in $\Omega_{\gamma}$ (for the corresponding algebraic element $\gamma$). So, we observe that $\gamma^0$ will be always well--defined.
\end{remark}
The next lemma justifies the previous definitions.
\begin{lemma}\label{lem:defVan}
Let $H\in \mathbb{F}[\overline{v}]$, where $\overline{v}$ is a tuple of variables. It holds that
\begin{enumerate}
\item If $\mathbf{a}^0\in \Omega_{\mathrm{def}(H)}$ then $H(\mathbf{a}^0,\gamma^0,\overline{v})$ is well-defined.
\item If $\mathbf{a}^0\in \Omega_{\mathrm{nonZ}(H)}$ then $H(\mathbf{a}^0,\gamma^0,\overline{v})\neq 0$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) Let $\mathbf{a}^0\in \Omega_{\mathrm{def}(H)}\subset \Omega_\gamma$. Then, $\gamma^0=\gamma(\mathbf{a}^0)$ is well--defined, and the result follows from the definition of $\mathcal{D}$.\\
(2) Let $\mathbf{a}^0\in \Omega_{\mathrm{nonZ}(H)}\subset \Omega_{\mathrm{def}(H)}$. Then, by (1), $H(\mathbf{a}^0,\gamma^0,\overline{v})$ is well--defined. Furthermore, there exists a coefficient of $H$ w.r.t. $\overline{v}$, say $C(\mathbf{a},\gamma)$, such that $\mathrm{Norm}(\mathrm{numer}(C))(\mathbf{a}^0)\neq 0$. Since $D(\mathbf{a}^0)\neq 0$ and the denominator of $C$ does not vanish at $\mathbf{a}^0$, by Lemma \ref{lem:N}, we get that $C(\mathbf{a}^0,\gamma^0)\neq 0$. So, $H(\mathbf{a}^0,\gamma^0,\overline{v})\neq 0$.
\end{proof}
The following lemma is an adaptation of Lemma 3 in \cite{SW01} to our case, and will be used to control the birationality of a curve parametrization $\mathcal{P}(\mathbf{a},t)$ under specializations of $\mathbf{a}$.
\begin{definition}\label{def-gcd}
Let $f_1,f_2\in \mathbb{F}[u][v]\setminus\{0\}$ for $i\in \{1,2\}$, where $u,v$ are variables. Let $f_i=f_{i}^{*} \, g$, for $i\in \{1,2\}$, where $g=\gcd(f_1,f_2)$. Let $A_i\in \mathbb{F}[u]$ be the leading coefficient of $f_i$ w.r.t. $v$ for $i\in \{1,2\}$ and $B\in \mathbb{F}[u]$ the leading coefficient of $g$ w.r.t. $v$. Let $R=\mathrm{res}_{v}(f_{1}^{*},f_{2}^{*})\in \mathbb{F}[u]$.
Let
\[ \begin{array}{l}
\Omega_1:= \Omega_{\mathrm{def}(f_1)} \cap \Omega_{\mathrm{def}(f_2)} \cap \Omega_{\mathrm{def}(f_{1}^{*})} \cap \Omega_{\mathrm{def}(f_{2}^{*})}\cap \Omega_{\mathrm{def}(g)} \cap \Omega_{\mathrm{def}(R)},\\
\Omega_2:=\Omega_{\mathrm{nonZ}(A_1)} \cap \Omega_{\mathrm{nonZ}(A_2)} \cap \Omega_{\mathrm{nonZ}(B)}\cap \Omega_{\mathrm{nonZ}(R)}.
\end{array}
\]
We define the set
\[ \Omega_{\mathrm{gcd}(f_1,f_2)}:= \Omega_1\cap \Omega_2. \]
\end{definition}
\begin{lemma}\label{lem-gcd}
Let $f_1,f_2,f_{1}^{*},f_{2}^{*},g$ be as in Def. \ref{def-gcd}.
For
$\mathbf{a}^0\in \Omega_{\mathrm{gcd}(f_1,f_2)},$ it holds that
\[ g(\mathbf{a}^0,\gamma^0,u,v)= \lambda(u) \gcd(f_1(\mathbf{a}^0,\gamma^0,u,v),f_2(\mathbf{a}^0,\gamma^0,u,v)), \]
with $\lambda(u)\in \overline{\mathbb{K}}[u]\setminus \{0\}$. Moreover, $\deg_v(g(\mathbf{a}^0,\gamma^0,u,v))=\deg_v(g(\mathbf{a},\gamma,u,v)).$
\end{lemma}
\begin{proof}
Let $A_i,B,R$ be as in Def.~\ref{def-gcd}.
Since $\mathbf{a}^0\in \Omega_{\mathrm{gcd}(f_1,f_2)}\subset \Omega_1$ (see Def. \ref{def-gcd}), by Lemma~\ref{lem:defVan}~(1), the specializations of $f_i,f_{i}^{*},g,R$ at $\mathbf{a}^0$ are well-defined. So,
\begin{equation}\label{eq-gcd1}
f_i(\mathbf{a}^0,\gamma^0,u,v)=f_{i}^{*}(\mathbf{a}^0,\gamma^0,u,v)g(\mathbf{a}^0,\gamma^0,u,v).
\end{equation}
Moreover, since $\mathbf{a}^0\in \Omega_{\mathrm{gcd}(f_1,f_2)}\subset \Omega_2$ (see Def.~\ref{def-gcd}), the specializations of $f_i,g,R$ at $\mathbf{a}^0$ preserve the degree in $v$ and, in particular, are non--zero.
This implies that $f_{i}^{*}(\mathbf{a}^0,\gamma^0,u,v)$ are non--zero too. From \eqref{eq-gcd1}, one has that there exists $\lambda\in \overline{\mathbb{K}}[u]\setminus\{0\}$ such that
\[ \begin{array}{c}\gcd(f_1(\mathbf{a}^0,\gamma^0,u,v),f_2(\mathbf{a}^0,\gamma^0,u,v))
\lambda(u) \, \gcd(f_{1}^{*}(\mathbf{a}^0,\gamma^0,u,v),f_{2}^{*}(\mathbf{a}^0,\gamma^0,u,v)) g(\mathbf{a}^0,\gamma^0,u,v). \end{array}\]
Let us assume that $\gcd(f_{1}^{*}(\mathbf{a}^0,\gamma^0,u,v),f_{2}^{*}(\mathbf{a}^0,\gamma^0,u,v))$ has positive degree in $v$. Then, if $\tilde{R}(u)$ is the resultant w.r.t. $v$ of $f_{1}^{*}(\mathbf{a}^0,\gamma^0,u,v)$ and $f_{2}^{*}(\mathbf{a}^0,\gamma^0,u,v)$, we get that $\tilde{R}$ is zero (see e.g. Corollary page 288 in \cite{Geddes}). However, since $\mathbf{a}^0\in \Omega_{\mathrm{gcd}(f_1,f_2)}\subset \Omega_2$ (see Def.~\ref{def-gcd}), by Lemma~\ref{lem:defVan}(2), $A_1,A_2$ do not vanish at $\mathbf{a}^0$ and, hence, the leading coefficients of $f_{i}^{*}$ do not vanish either at $\mathbf{a}^0$. Therefore, by Lemma 4.3.1 in~\cite{winkler}, $R(\mathbf{a}^0,\gamma^0,u)=\tilde{R}(u)$. Nevertheless, since $\mathbf{a}^0\in\Omega_2$ by Lemma~\ref{lem:defVan}~(2), $R(\mathbf{a}^0,\gamma^0,u)\neq 0$ which is a contradiction. So, $g(\mathbf{a}^0,\gamma^0,u,v)$ and $\gcd(f_1(\mathbf{a}^0,\gamma^0,u,v),f_2(\mathbf{a}^0,\gamma^0,u,v))$ are associated. In addition, since $\mathbf{a}^0\in \Omega_2$, by Lemma~\ref{lem:defVan}~(2), $B(\mathbf{a}^0,\gamma^0,u)\neq 0$ and, hence, $\deg_v(g(\mathbf{a}^0,\gamma^0,u,v))=\deg_v(g(\mathbf{a},\gamma,u,v)).$
\end{proof}
If in Lemma \ref{lem-gcd} all coefficients are assumed to be in a field, the statement can be simplified as follows.
\begin{corollary}\label{cor-lemma-gcd}
Let $f_1,f_2\in \mathbb{F}[v]\setminus\{0\}$ for $i\in \{1,2\}$. Let $f_i=f_{i}^{*} \, g$, for $i\in \{1,2\}$, where $g=\gcd(f_1,f_2)$. For $\mathbf{a}^0\in \Omega_{\mathrm{gcd}(f_1,f_2)},$ it holds that
\[ g(\mathbf{a}^0,\gamma^0,t)=\gcd(f_1(\mathbf{a}^0,\gamma^0,v),f_2(\mathbf{a}^0,\gamma^0,v)). \]
Moreover, $\deg_v(g(\mathbf{a}^0,\gamma^0,v))=\deg_v(g(\mathbf{a},\gamma,v)).$
\end{corollary}
Let us now generalize the previous statement to several univariate polynomials with coefficients in $\mathbb{F}$.
\begin{definition}\label{def:gcd-several}
Let $f_1,\ldots,f_r\in \mathbb{F}[v]\setminus\{0\}$. Let $f_i=f_{i}^{*} \, g$, for $i\in \{1,\ldots,r\}$, where $g=\gcd(f_1,\ldots,f_r)$. We consider the polynomial $f_Z:=f_2+f_3Z+\cdots +f_{r} Z^{r-2}\in \mathbb{F}(Z)[v]$ where $Z$ is a new variable. We define
\[ \Omega_{\mathrm{gcd}(f_1,\ldots,f_r)}= \Omega_{\mathrm{gcd}(f_1,f_Z)}\cap \Omega_{\mathrm{def}(g)}\cap \Omega_{\mathrm{nonZ}(A)}\]
where $A$ is the leading coefficient of $g$ w.r.t. $v$.
\end{definition}
\begin{remark}
Observe that if $r=2$ in Def.~ \ref{def:gcd-several}, then Def.~\ref{def-gcd} and~\ref{def:gcd-several} coincide.
\end{remark}
\begin{theorem}\label{theorem-lemma-gcd-several-pol}
Let $f_1,\ldots,f_r,f_{1}^{*},\ldots,f_{r}^{*},g$ be as in Def.~\ref{def:gcd-several}. For $\mathbf{a}^0\in \Omega_{\mathrm{gcd}(f_1,\ldots,f_r)},$ it holds that
\[ g(\mathbf{a}^0,\gamma^0,v)=\gcd(f_1(\mathbf{a}^0,\gamma^0,v),\ldots,f_r(\mathbf{a}^0,\gamma^0,v)). \]
Moreover, $\deg_v(g(\mathbf{a}^0,\gamma^0,v))=\deg_v(g(\mathbf{a},\gamma,v)).$
\end{theorem}
\begin{proof}
Let $g^*:=\gcd(f_1,f_Z)\in \mathbb{F}(Z)[v]$. Since $f_1$ does not depend on $Z$, $g^*\in \mathbb{F}[v]$. This implies
that $g=\lambda g^*$ with $\lambda\in \mathbb{F}$.
By Corollary~\ref{cor-lemma-gcd}, we know that $g^*(\mathbf{a}^0,\gamma^0,t)=\gcd(f_1(\mathbf{a}^0,\gamma^0,v),f_Z(\mathbf{a}^0,\gamma^0,Z,v))$ and that $\deg_v(g^*(\mathbf{a}^0,\gamma^0,v))=\deg_v(g^*(\mathbf{a},\gamma,v)).$ Note that $\deg_v(g^*(\mathbf{a},\gamma,v))=\deg_v(g(\mathbf{a},\gamma,v))$. Moreover, since
$\mathbf{a}^0\in \Omega_{\mathrm{def}(g)}\cap \Omega_{\mathrm{nonZ}(A)}$, then $g(\mathbf{a}^0,\gamma^0,v)$ is well--defined. Furthermore, since both leading coefficients of $g$ and $g^*$ w.r.t. $v$ do not vanish at $\mathbf{a}^0$, then $\lambda(\mathbf{a}^0,\gamma^0)$ is well--defined and non--zero. Thus, $\deg_v(g^*(\mathbf{a}^0,\gamma^0,v))=\deg_v(g(\mathbf{a}^0,\gamma^0,v))$.
Summarizing, $\deg_v(g(\mathbf{a}^0,\gamma^0,v))=\deg_v(g(\mathbf{a},\gamma,v)).$ On the other hand, $$g(\mathbf{a}^0,\gamma^0,v)=\lambda(\mathbf{a}^0,\gamma^0)\, g^*(\mathbf{a}^0,\gamma^0,v) = \lambda(\mathbf{a}^0,\gamma^0)\,\gcd(f_1(\mathbf{a}^0,\gamma^0,v),f_Z(\mathbf{a}^0,\gamma^0,Z,v))$$
and, since $\lambda(\mathbf{a}^0,\gamma^0)\neq 0$, this implies
that $g(\mathbf{a}^0,\gamma^0,v)=\gcd(f_1(\mathbf{a}^0,\gamma^0,v),\ldots,f_r(\mathbf{a}^0,\gamma^0,v))$.
\end{proof}
Our next step is to analyze the squarefreeness.
\begin{definition}\label{def:sqfree}
Let $f\in \mathbb{F}[v]\setminus \mathbb{F}$ be squarefree. Let $R$ be the discriminant of $f$ w.r.t. $v$ and let $A$ be the leading coefficient of $f$ w.r.t. $v$. We define the open subset
\[ \Omega_{\mathrm{sqfree}(f)}:=\Omega_{\mathrm{def}(f)} \cap \Omega_{\mathrm{nonZ}(R)}\cap \Omega_{\mathrm{NonZ}(A))}.\]
\end{definition}
\begin{lemma}\label{lem:sqfree}
Let $f\in \mathbb{F}[v]\setminus \mathbb{F}$ be squarefree. If $\mathbf{a}^0\in \Omega_{\mathrm{sqfree}(f)}$, then $\deg_{v}(f(\mathbf{a},\gamma,v))= \deg_{v}(f(\mathbf{a}^0,\gamma^0,v))$ and $f(\mathbf{a}^0,\gamma^0,v)$ is squareefree.
\end{lemma}
\begin{proof}
Since $\mathbf{a}^0\in\Omega_{\mathrm{def}(f)}$, by Lemma \ref{lem:defVan}, $f(\mathbf{a}^0,\gamma^0,v)$ is well--defined and, since $\mathbf{a}\in\Omega_{\mathrm{NonZ}(A))}$, $f(\mathbf{a}^0,\gamma^0,v)\neq 0$. Furthermore, one has the equality of the degrees.
Moreover, since $\mathbf{a}^0\in \Omega_{\mathrm{nonZ}(R)}$, also by Lemma \ref{lem:defVan}, $R(\mathbf{a}^0,\gamma^0,v)$ is well-defined and non--zero. Since $A(\mathbf{a}^0,\gamma^0,v)\neq 0$, by \cite[Lemma 4.3.1]{winkler}, the discrimininant of $f(\mathbf{a}^0,\gamma^0,v)$ is not zero. Then, by \cite[Theorem 4.4.1]{winkler}, $f(\mathbf{a}^0,\gamma^0,v)$ is squarefree.
\end{proof}
\subsection{Specialization of the curve defining polynomial}\label{subsec:defpol}
In this subsection, we deal with the specialization of defining polynomials of irreducible plane curves. Let $G\in \mathbb{F}[x,y]\setminus \mathbb{F}$ be irreducible over $\overline{\mathbb{F}}$ of total degree $d$ and let $G^h\in \mathbb{F}[x,y,z]$ be its homogenization. In the following, let $G$ be written as
\begin{equation}\label{eq-G}
G=g_d(x,y)+\cdots+g_0(x,y)
\end{equation}
where $g_i$ is either the zero polynomial or a form of degree $i$.
\begin{definition}\label{def:omegaG}
We associate to $G$ the open subset (see \eqref{eq-G})
$\Omega_G:=\Omega_{\mathrm{def}(G)}\cap \Omega_{\mathrm{nonZ}(g_d)}.$
\end{definition}
\begin{lemma}\label{lem:omegaG} Let $G$ be as above and let $\mathbf{a}^0\in \Omega_G$. Then
\begin{enumerate}
\item $G(\mathbf{a}^0,x,t)$ is well--defined and $\deg(G(\mathbf{a}^0, x,y))=\deg(G)$;
\item $G(\mathbf{a}^0, x,y)^{h}=G^{h}(\mathbf{a}^0, x,y,z)$;
\item the partial derivatives of $G^h$, of any order, specialize properly.
\end{enumerate}
\end{lemma}
\begin{proof}
Since $\mathbf{a}^0\in \Omega_{\mathrm{def}(G)}$, by Lemma \ref{lem:defVan}, $G(\mathbf{a}^0,x,y)$ is well--defined and, since $\mathbf{a}^0\in \Omega_{\mathrm{nonZ}(g_d)}$, the equality on the degree holds. Since $G(\mathbf{a}^0, x,y)$ is well--defined, the other statements directly follow.
\end{proof}
\subsection{Specialization of families of points}\label{subsec-spec-families}
Let us now deal with the specialization of conjugate families of points associated to a curve. More precisely, let $G$ and $G^h$ be as in Subsection \ref{subsec:defpol}. We will study the specialization of families in the standard decomposition of the singular locus of $\mathcal{C}(G^h)$. For this purpose, we observe that, for each $\mathbf{a}^0\in \mathbb{S}$ such that $G(\mathbf{a}^0,x,y)\not\in \overline{\mathbb{K}}$, $G(\mathbf{a}^0,x,y)$ defines an affine plane curve over $\overline{\mathbb{K}}$. Let us denote by $\mathcal{C}(G)$ the first curve and by $\mathcal{C}(G,\mathbf{a}^0)$ the second.
The conjugate families of $\mathcal{C}(G^h)$ will be over $\mathbb{F}$. When we specialize $\mathbf{a}$ we need to have a reference field where the conjugation of the points is defined. This motivates the following definition.
\begin{definition}\label{def-Ka} For $\mathbf{a}^0\in \mathbb{S}$, we define $\mathbb{K}_{\mathbf{a}^0}$ as the smallest subfield of $\overline{\mathbb{K}}$ containing the coefficients of $G(\mathbf{a}^0,\gamma^0,x,y)$. Moreover, if $\mathcal{F}=\{(f_1:f_2:f_3)\}_{m(t)}$ is an $\mathbb{F}$--conjugate family, and $\mathbf{a}^0\in \mathbb{S}$ is such that $\gamma^0, f_1(\mathbf{a}^0,t), f_2(\mathbf{a}^0,t), f_3(\mathbf{a}^0,t), m(\mathbf{a}^0,t)$ are well--defined, we denote by $\mathcal{F}(\mathbf{a}^0)$ the specialization of $\mathcal{F}$ at $\mathbf{a}^0$ (and $\gamma^0$).
\end{definition}
Let $\mathscr{F}(G^h)$ be an $\mathbb{F}$--standard decomposition of the singular locus of $\mathcal{C}(G^h)$ obtained using the process described in Subsection \ref{subsec-genus}. Let $\mathscr{F}(G^h)$ decompose as
\begin{equation}\label{eq-SingularLocus}
\mathscr{F}(G^h)=\bigcup_{m(t)\in \mathcal{A}_a} \{(f_{1,m}(t):f_{2,m}(t):1)\}_{m(t)} \, \cup \, \bigcup_{m(t)\in \mathcal{A}_{\infty}} \{(L_{1,m}(t):L_{2,m}(t):0)\}_{m(t)}
\end{equation}
where $f_{i,m},m(t)\in \mathbb{F}[t]$, $L_{i,m}\in \mathbb{K}[t]$ (recall that the transformation $\mathcal{L}$, in the standard decomposition process in Subsection \ref{subsec-genus}, can be taken over $\mathbb{K}$) with $\deg(L_{i,m})\leq 1$, $\gcd(L_{1,m},L_{2,m})=1$, and $m(t)$ irreducible over $\mathbb{F}$, and where $\mathcal{A}_a$ and $\mathcal{A}_{\infty}$ are finite sets of irreducible polynomials in $\mathbb{F}[t]$. By abuse of notation, we will write $\mathcal{F}\in \mathscr{F}(G^h)$ for such a component $\mathcal{F}$ of $\mathscr{F}(G^h)$.
\begin{definition}\label{def-defFam} Let $\mathcal{F}:=\{(f_{1,m}(t):f_{2,m}(t):1)\}_{m(t)}\in \mathscr{F}(G^h)$ be an irreducible $\mathbb{F}$-conjugate family of affine singularities of $\mathcal{C}(G^h)$ (see \eqref{eq-SingularLocus}).
Let $A$ be the product of the leading coefficient of $m$ w.r.t. $t$ and the leading coefficients w.r.t. $t$ of $f_{1,m}, f_{2,m}$. We associate to $\mathcal{F}$ the open set
\[ \Omega_{\mathrm{def}(\mathcal{F})}:= \displaystyle{ \Omega_{\mathrm{def}(f_{1,m})}\cap \Omega_{\mathrm{def}(f_{2,m})}\cap \Omega_{\mathrm{sqfree}(m)} \cap \Omega_{\mathrm{NonZ}(A))} \cap \, \Omega_{G}}. \]
Let $\mathcal{F}:=\{(L_{1,m}(t):L_{2,m}(t):0)\}_{m(t)}\in \mathscr{F}(G^h)$ be an irreducible $\mathbb{F}$-conjugate family of singularities of $\mathcal{C}(G^h)$ at infinity (see \eqref{eq-SingularLocus}).
Let $A$ be the leading coefficient of $m$ w.r.t. $t$. We associate to $\mathcal{F}$ the open set
\[ \Omega_{\mathrm{def}(\mathcal{F})}:= \displaystyle{\Omega_{\mathrm{sqfree}(m)} \cap \Omega_{\mathrm{NonZ}(A))} \cap \, \Omega_{G}}. \]
\end{definition}
We start our analysis with a technical lemma.
\begin{lemma}\label{lem:spec-mod} Let $H,m\in \mathbb{F}[t]$. Let $H=R$ mod $m$, and let $A$ be the leading coefficient of $m$ w.r.t. $t$. If $H(\mathbf{a}^0,t), m(\mathbf{a}^0,t)$ are well--defined and $A(\mathbf{a}^0)\neq 0$, then $H(\mathbf{a}^0,t)=R(\mathbf{a}^0,t)$ mod $m(\mathbf{a}^0,t)$.
\end{lemma}
\begin{proof} Let $Q$ be the quotient of $H$ by $m$ w.r.t. $t$. So, $H=Q \cdot m +R$ with $\deg_t(R)<\deg_t(m)$. Since $H(\mathbf{a}^0), m(\mathbf{a}^0,t)$ is well--defined, $\gamma^0$ is well--defined or all polynomials are independent of $\gamma$. Since $A(\mathbf{a}^0)\neq 0$, then $Q(\mathbf{a}^0,t),R(\mathbf{a}^0,t)$ are well--defined. Moreover, $\deg_t(m(\mathbf{a},t))=\deg_t(m(\mathbf{a}^0,t))$. Then $H(\mathbf{a}^0,t)=Q(\mathbf{a}^0,t) m(\mathbf{a}^0,t) +R(\mathbf{a}^0,t)$ with $$\deg_t(R(\mathbf{a}^0,t))\leq \deg_t(R)<\deg_t(m)=\deg_t(m(\mathbf{a}^0,t)).$$ This concludes the proof.
\end{proof}
\begin{lemma}\label{lem:defF}
Let $\mathcal{F}\in \mathscr{F}(G^h)$ be an irreducible $\mathbb{F}$--family of $\mathcal{C}(G^h)$ (see~\eqref{eq-SingularLocus}). If $\mathbf{a}^0\in \Omega_{\mathrm{def}(\mathcal{F})}$, then $\mathcal{F}(\mathbf{a}^0)$ is a $\mathbb{K}_{\mathbf{a}^0}$-conjugate family of points of $\mathcal{C}(G^h,\mathbf{a}^0)$ and $\#(\mathcal{F})=\#(\mathcal{F}(\mathbf{a}^0))$.
\end{lemma}
\begin{proof}
Let $\mathcal{F}=\{(f_{1,m}(t):f_{2,m}(t):1)\}_{m(t)}$.
Let us first show that $\mathcal{F}(\mathbf{a}^0)$ is a $\mathbb{K}_{\mathbf{a}^0}$-conjugate family of points of $\mathcal{C}(G^h,\mathbf{a}^0)$. Since $\mathbf{a}^0\in \Omega_{\mathrm{def}(f_{i,m})}$ and $\mathbf{a}^0\in \Omega_{\mathrm{sqrfree}(m)}\subset \Omega_{\mathrm{def}(m)}$, we have that $\gamma^0$, $f_{i,m}(\mathbf{a}^0,t)$ and $m(\mathbf{a}^0,t)$ are well-defined.
Furthermore, since $\mathbf{a}^0\in \Omega_{\mathrm{NonZ}(A))}$, the degree of all non-constant polynomials $f_{i,m}$ and $m$ is preserved under the specialization. In addition, since $\mathbf{a}^0\in \Omega_{\mathrm{sqfree}(m)}$, it holds that $m(\mathbf{a}^0,t)$ is squarefree (see Lemma \ref{lem:sqfree}) and condition (3) in Def. \ref{def-family} holds; note that conditions (1) and (2) in Def. \ref{def-family} hold trivially. Furthermore, note that, after specialization, all polynomials are over $\mathbb{K}_{\mathbf{a}^0}$. So, $\mathcal{F}(\mathbf{a}^0)$ is a family over $\mathbb{K}_{\mathbf{a}^0}$. It remains to prove that the points in $\mathcal{F}(\mathbf{a}^0)$ are in the specialized curve.
Since $\mathbf{a}^0\in \Omega_{G}$, by Lemma \ref{lem:omegaG}, it holds that $G(\mathbf{a}^0, x,y)^h=G^{h}(\mathbf{a}^0, x,y,z)$.
Let $T(\mathbf{a},t)=G^h(\mathbf{a},f_{1,m},f_{2,m},1)$. Since $\mathcal{F}$ is a family of points in $\mathcal{C}(G^h)$, it holds that $T=0$ mod $m$. Since $G^h(\mathbf{a}^0,x,y,z)$ and $f_{i,m}(\mathbf{a}^0,t)$ are well--defined, then $T(\mathbf{a}^0,t)$ is well--defined too. We know that $m(\mathbf{a}^0,t)$ is well--defined and that the leading coefficient of $m$ in $t$ does not vanish after specialization. Therefore, by Lemma \ref{lem:spec-mod}, $G^h(\mathbf{a}^0,f_{1,m}(\mathbf{a}^0,t),f_{2,m}(\mathbf{a}^0,t),1)=0$ modulo $m(\mathbf{a}^0,t)$. Hence, $\mathcal{F}(\mathbf{a}^0)$ is a $\mathbb{K}_{\mathbf{a}^0}$-conjugate family of points of $\mathcal{C}(G^h,\mathbf{a}^0)$.
If $\mathcal{F}:=\{(L_{1,m}(t):L_{2,m}(t):0)\}_{m(t)}$ all the arguments above apply and conditions (1) and (2) in Def. \ref{def-family} also hold because $L_{i,m}$ do not depend on $\mathbf{a}$ or $\gamma$.
We already know that $\mathcal{F}(\mathbf{a}^0)$ is a $\mathbb{K}_{\mathbf{a}^0}$-conjugate family of points of $\mathcal{C}(G^h,\mathbf{a}^0)$, and $\deg_t(m)=\deg_t(m(\mathbf{a}^0, t))$, where $m$ is the defining polynomial of $\mathcal{F}$. It remains to prove that $\#(\mathcal{F}(\mathbf{a}^0))=\#(\mathcal{F})$. Let $\mathcal{L}$ be the $\mathbb{K}$--linear change of coordinates transforming $\mathcal{C}(G)$ in regular position; see Step (1) in the standard decomposition process described in Subsection \ref{subsec-genus}. Then, $\#(\mathcal{F})=\#(\mathcal{L}^{-1}(\mathcal{F}))$. Furthermore, $\mathcal{L}^{-1}(\mathcal{F})$ is in the form appearing either in \eqref{eq-sing-infinity} or in \eqref{eq-sing-affine}. Therefore, $\#(\mathcal{F})=\deg_t(m)$. Since $\mathcal{L}$ is over $\mathbb{K}$, we may apply it to $\mathcal{F}(\mathbf{a}^0)$ and $\mathcal{L}^{-1}(\mathcal{F}(\mathbf{a}^0))$ will be of the form either $\{(t:B(\mathbf{a}^0,t):1)\}_{m(\mathbf{a}^0,t)}$ or $\{(1:t:0)\}_{m(\mathbf{a}^0,t)}$. In both cases, $\#(\mathcal{F}(\mathbf{a}^0))=\#(\mathcal{L}^{-1}(\mathcal{F}(\mathbf{a}^0)))=\deg_t(m(\mathbf{a}^0,t))$. Now, the result follows using that $\deg_t(m)=\deg_t(m(\mathbf{a}^0,t))$.
\end{proof}
\begin{remark}\label{rem-Fa}
Given an $\mathbb{F}$--conjugate family $\mathcal{F}\in \mathscr{F}(G^h)$ (see equation \eqref{eq-SingularLocus}), and $\mathbf{a}^0\in \Omega_{\mathrm{def}(\mathcal{F})}$ (see Def. \ref{def-defFam}), we observe that, even though $\mathcal{F}$ is irreducible, $\mathcal{F}(\mathbf{a}^0)$ may be reducible. We will be interested in working with irreducible specialized families. So, factoring over $\mathbb{K}_{\mathbf{a}^0}$ the defining polynomial of $\mathcal{F}(\mathbf{a}^0)$, the family will be decomposed as
\[ \mathcal{F}(\mathbf{a}^0)=\bigcup_{i\in I} \mathcal{F}_i \]
where $\mathcal{F}_i$ is an irreducible $\mathbb{K}_{\mathbf{a}^0}$--family. We will refer to $\mathcal{F}_i$ as the \textit{irreducible subfamilies of $\mathcal{F}(\mathbf{a}^0)$.}
\end{remark}
In the sequel we analyze the multiplicity of families of singularities under specializations.
\begin{definition}\label{def-multFam} Let $\mathcal{F}\in \mathscr{F}(G^h)$ (see \eqref{eq-SingularLocus}) be an irreducible $\mathbb{F}$-conjugate family of $r$-fold points of $\mathcal{C}(G^h)$ with defining polynomial $m(t)$, and let $H^*$ be one of the order $r$ derivatives of $G^h$ such that $H^*(\mathcal{F})\neq 0$ modulo $m(t)$. Let $H(\mathbf{a},t)$ be the reduction of $H^*(\mathcal{F})$ modulo $m(t)$. Let $R(\mathbf{a}):=\mathrm{res}_t(H(t),m(t))$. We define the open subset
\[ \Omega_{\mathrm{mult}(\mathcal{F})}:=\Omega_{\mathrm{def}( \mathcal{F})}\cap \Omega_{\mathrm{nonZ}(H)} \cap \,\Omega_{\mathrm{nonZ}(R)}.
\]
\end{definition}
\begin{remark}
We observe that in Def. \ref{def-multFam}, $m$ is irreducible (note that $\mathcal{F}$ belongs to a standard decomposition of the singular locus) over $\mathbb{F}$, $H\in \mathbb{F}[t]\setminus\{0\}$ and $\deg_t(H)<\deg_t(m)$. Therefore, $\gcd(m,H)=1$ and hence $R\neq 0$. \end{remark}
\begin{lemma}\label{lem:r-fold}
Let $\mathcal{F}\in \mathscr{F}(G^h)$ (see \eqref{eq-SingularLocus}) be an irreducible $\mathbb{F}$-conjugate family of $r$-fold points of $\mathcal{C}(G^h)$.
If $\mathbf{a}^0\in \Omega_{\mathrm{mult}(\mathcal{F})}$, then every irreducible subfamily of $\mathcal{F}(\mathbf{a}^0)$ (see Remark \ref{rem-Fa}) is a $\mathbb{K}_{\mathbf{a}^0}$--conjugate family of $r$-fold points of $\mathcal{C}(G^h,\mathbf{a}^0)$.
\end{lemma}
\begin{proof} Let $\mathcal{F}$ be expressed as $\mathcal{F}=\{(f_1:f_2:\lambda)\}_{m(t)}$ where $f_1,f_2,m\in \mathbb{F}[t]$, $\lambda\in \{0,1\}$,
$m$ irreducible over $\mathbb{F}$, and such that, for the case $\lambda=0$, $\deg_{t}(f_i)\leq 1$ and $\gcd(f_1,f_2)=1$ (see \eqref{eq-SingularLocus}).
Let $H^*$ and $H$ be as in Def. \ref{def-multFam}, and let $\mathcal{F}_i$, with defining polynomial $m_i$, be an irreducible subfamily of $\mathcal{F}(\mathbf{a}^0)$.
Since $\mathbf{a}^0\in \Omega_{\mathrm{def}(\mathcal{F})}$, by Lemma \ref{lem:defF}, $\mathcal{F}(\mathbf{a}^0)$ is a $\mathbb{K}_{\mathbf{a}^0}$--conjugate family of points of $\mathcal{C}(G^h,\mathbf{a}^0)$; in particular $f_i(\mathbf{a}^0,t)$ and $m(\mathbf{a}^0,t)$ are well--defined and the leading coefficient of $m$ w.r.t. $t$ does not vanish at $\mathbf{a}^0$. Moreover, by Lemma \ref{lem:omegaG}, since $\mathbf{a}^0\in \Omega_G$, all partial derivatives specialize properly; note that $\Omega_{\mathrm{def}(\mathcal{F})} \subset \Omega_G$.
Therefore, by Lemma \ref{lem:spec-mod}, the multiplicity of $\mathcal{F}_i$ is at least $r$. Moreover, since $\mathbf{a}^0\in \Omega_{G}$, $H^*(\mathbf{a}^0,f_1(\mathbf{a}^0,t),f_2(\mathbf{a}^0,t),\lambda)$ is well--defined. So, again by Lemma \ref{lem:spec-mod}, $H^*(\mathbf{a}^0,f_1(\mathbf{a}^0,t),f_2(\mathbf{a}^0,t),\lambda)=H(\mathbf{a}^0,t)$ modulo $m(\mathbf{a}^0,t)$. Furthermore, since $\mathbf{a}^0\in \Omega_{\mathrm{nonZ}(H)}$, then $H(\mathbf{a}^0,t)\neq 0$. Moreover, since the leading coefficient of $m$ w.r.t. $t$ does not vanish at $\mathbf{a}^0$, we have that
$\mathrm{res}_{t}(H(\mathbf{a}^0,t),m(\mathbf{a}^0,t))=\mu \,R(\mathbf{a}^0)$ for some non-zero constant $\mu$ (see Lemma 4.3.1 in \cite{winkler}). So, since $\mathbf{a}^0\in \Omega_{\mathrm{nonZ}(R)}$, $\mathrm{res}_{t}(H(\mathbf{a}^0,t),m(\mathbf{a}^0,t))\neq 0$. Therefore, $\gcd(H(\mathbf{a}^0,t),m(\mathbf{a}^0,t))=1$ and hence $H(\mathbf{a}^0,t)\neq 0$ mod $m_i$.
Summarizing, the multiplicity of $\mathcal{F}_i$ is $r$.
\end{proof}
In the last part of this subsection, we deal with the tangents to $\mathcal{C}(G^h)$ at an irreducible $\mathbb{F}$-conjugate family; since the family $\mathcal{F}$ is assumed to be irreducible, one may think on the tangents at $\mathcal{F}$ to the curve $\mathcal{C}(G^{h}_{\mathcal{F}})$ (see Remark \ref{rem-fam}(3) in Subsection \ref{subsec-genus}).
\begin{definition}\label{def:tangent} Let $\mathcal{F}$, with defining polynomial $m(t)$, be an irreducible $\mathbb{F}$-conjugate family of $r$-fold points of $\mathcal{C}(G^h)$.
Let $\mathbb{F}_m$ be the quotient field of $\mathbb{F}[t]/\!\!<\!\!m(t)\!\!>$.
The \textit{defining tangent polynomial} of $\mathcal{C}(G^h)$ at $\mathcal{F}$ is defined as the homogenous polynomial $T\in\mathbb{F}_m[x,y,z]$ of degree $r$ which factors over the algebraic closure of $\mathbb{F}_m$ into the tangents, with the according multiplicities, of $\mathcal{C}(G^h)$ at $\mathcal{F}$. Similarly, we introduce the \textit{defining tangent polynomial} to an specialized curve.
\end{definition}
\begin{remark}\label{rem-character} \
\begin{enumerate}
\item Let us assume that $\mathcal{F}=\{(f_1:f_2:1)\}_{m(t)} \in \mathscr{F}(G^h)$ (see equation \eqref{eq-SingularLocus}) is a family of $r$--fold points with irreducible $m$; similarly if the family is at infinity. Then $T$ is the reduction of
\begin{equation}\label{eq-Tstar}
T^*(\mathbf{a},t,x,y,z)=\sum_{i=0}^{r} \binom{r}{i} \dfrac{\partial^r G}{\partial^i x\,\partial^{r-i} y}(f_1,f_2) (x-f_1z)^i (y-f_2 z)^{r-i}
\end{equation}
modulo $m(t)$.
\item Let $\mathcal{F}$ be as in Def. \ref{def:tangent}. We observe that $\mathcal{F}$ is a family of ordinary $r$--fold points if and only if $T$ is squarefree over $\mathbb{F}_m$. In the sequel, we assume w.l.o.g. that there is no tangent of $\mathcal{C}(G^{h}_{\mathcal{F}})$ at $\mathcal{F}$ independent of $x$ and for two different tangents $T_1(x,y,z), T_2(x,y,z)$ it holds that $T_1(x,1,1)\neq T_2(x,1,1)$. Note that, if this is not the case, one can apply a linear change over $\mathbb{K}$ (and thus invariant under specializations of the parameters $\mathbf{a}$). Then, the ordinary character of the family is readable from the squarefreeness of $T(\mathbf{a},t,x,1,1)$ over $\mathbb{F}_m$.
\end{enumerate}
\end{remark}
\begin{definition}\label{def:ordSet}
Let $\mathcal{F}$, with defining polynomial $m(t)$, be an irreducible $\mathbb{F}$-conjugate family of ordinary $r$-fold points of $\mathcal{C}(G^h)$. Let $T$ be the defining tangent polynomial of $\mathcal{F}$, where we assume w.l.o.g. that the hypotheses in Remark~\ref{rem-character}~(2) are satisfied. Let $D(\mathbf{a},t)$ be the reduction modulo $m(t)$ of the discriminant w.r.t. $x$ of $T(\mathbf{a},t,x,1,1)$. Let $N(\mathbf{a})=\mathrm{res}_t(D,m)$. Let $A(\mathbf{a},t)$ be the leading coefficient of $T(\mathbf{a},t,x,1,1)$ w.r.t. $x$ and let $R(\mathbf{a})=\mathrm{res}_{t}(A,m)$. Let $S(\mathbf{a},x)=\mathrm{res}_t(T(\mathbf{a},t,x,0,0),m)$.
We define the set
\[ \Omega_{\mathrm{ord}(\mathcal{F})}=\Omega_{\mathrm{mult}(\mathcal{F})}\cap \Omega_{\mathrm{nonZ}(R)}\cap \Omega_{\mathrm{nonZ}(N)} \cap \Omega_{\mathrm{nonZ}(S)} . \]
\end{definition}
\begin{remark} In relation to Def. \ref{def:ordSet}, we observe the following.
\begin{enumerate}
\item By construction, $\deg_t(A)<\deg_t(m)$ and clearly $A$ is not zero. Since $m$ is irreducible, $\gcd(A, m)=1$. Hence, $R\neq 0$.
\item Since $\mathcal{F}$ is ordinary and the two hypotheses in Remark \ref{rem-character}~(2) are satisfied, $D\neq 0$.
Because $\deg_t(D)<\deg_t(m)$ and $m$ is irreducible, $\gcd(m,H)=1$ and hence, $N\neq 0$.
\item $T(\mathbf{a},t,x,y,z)$ has a factor in $\mathbb{F}_{m}[y,z]$ if and only if $T(\mathbf{a},t,x,0,0)=0$.
This follows from the fact that the tangents are of degree one and thus, $T(\mathbf{a},t,x,y,z)=\prod (A_i(\mathbf{a},t)x+B_i(\mathbf{a},t)y+C_i(\mathbf{a},t)z)$ for some $A_i,B_i,C_i \in \mathbb{F}_{m}$.
Thus, under our assumption that $T(\mathbf{a},t,x,y,z)$ does not have a factor independent of $x$, $T(\mathbf{a},t,x,0,0) \ne 0$.
Since $m$ is irreducible, and $\deg_t(T(\mathbf{a},t,x,0,0))<\deg_t(m)$, it follows that $S \ne 0$.
\end{enumerate}
\end{remark}
\begin{lemma}\label{lem:ord-fold}
Let $\mathcal{F}\in \mathscr{F}(G^h)$ (see equation \eqref{eq-SingularLocus}) be an irreducible $\mathbb{F}$-conjugate family of ordinary $r$-fold points of $\mathcal{C}(G^h)$.
If $\mathbf{a}^0\in \Omega_{\mathrm{ord}(\mathcal{F})}$, then every irreducible subfamily of $\mathcal{F}(\mathbf{a}^0)$ (see Remark \ref{rem-Fa}) is a $\mathbb{K}_{\mathbf{a}^0}$--conjugate family of ordinary $r$-fold points of $\mathcal{C}(G^h,\mathbf{a}^0)$.
\end{lemma}
\begin{proof}
Since $\mathbf{a}^0\in \Omega_{\mathrm{ord}(\mathcal{F})}\subset \Omega_{\mathrm{mult}(\mathcal{F})}\subset \Omega_{\mathrm{def}(\mathcal{F})}$, then $\deg_t(m(\mathbf{a},t))=\deg_{t}(m(\mathbf{a}^0,t))$ and, by Lemma \ref{lem:r-fold}, every irreducible subfamily of $\mathcal{F}(\mathbf{a}^0)$ is a $\mathbb{K}_{\mathbf{a}^0}$--conjugate family of $r$-fold points of $\mathcal{C}(G^h,\mathbf{a}^0)$. Let us prove that all points in $\mathcal{F}(\mathbf{a}^0)$ are ordinary.
Let $T^*$ be as in \eqref{eq-Tstar}, or similarly if the family is at infinity, and let $T$ be the reduction of $T^*$ modulo $m$. Since $\mathbf{a}^0\in \Omega_{\mathrm{ord}(\mathcal{F})}\subset \Omega_{\mathrm{mult}(\mathcal{F})}\subset \Omega_{\mathrm{def}(\mathcal{F})}\subset \Omega_G$, by Lemma \ref{lem:omegaG}, $T^*$ specializes properly at $\mathbf{a}^0$, and since $\deg_t(m(\mathbf{a},t))=\deg_t(m(\mathbf{a}^0,t))$, by Lemma \ref{lem:spec-mod}, $T$ also specializes properly at $\mathbf{a}^0$.
Now, let $P$ be a point in $\mathcal{F}(\mathbf{a}^0)$. Then, there exists a root $t_0$ of $m(\mathbf{a}^0,t)$ such that $P$ is obtained by specializing $\mathcal{F}$ at $\mathbf{a}^0$ and $t_0$. Since $P$ belongs to one of the irreducible subfamilies, $P$ is an $r$--fold point of the curve $\mathcal{C}(G^h,\mathbf{a}^0)$. Because of the discussion above, $E(x,y,z):=T(\mathbf{a}^0,t_0,x,y,z)$ is the defining tangent polynomial of $\mathcal{C}(G^h,\mathbf{a}^0)$ at $P$. It remains to prove that $E$ is squareefree. First, let us see that there is no factor of $E$ independent of $x$. Assume that $e(y,z)$ is a factor of $E$. Then $E(x,0,0)=0$, and $(\mathbf{a}^0,t_0,x,0,0)$ is a common zero of $T(\mathbf{a},t,x,y,z)$ and $m(\mathbf{a},t)$. Therefore, see e.g. Theorem 4.3.3 in \cite{winkler}, $S(\mathbf{a}^0,x)=0$ which contradicts that $\mathbf{a}^0\in \Omega_{\mathrm{nonZ}(S)}$.
Thus, it is sufficient to prove the squarefreeness of $E(x,1,1)$. Let us assume that it is not squarefree, then its discriminant is zero. That is, the discriminant of $T(\mathbf{a}^0,t_0,x,1,1)$ is zero. On the other hand, $\mathbf{a}^0\in \Omega_{\mathrm{nonZ}(R)}$, $R(\mathbf{a}^0)\neq 0$ and thus, $A(\mathbf{a}^0,t_0)\neq 0$.
By \cite[Lemma 4.1.3]{winkler} and the fact that $m(\mathbf{a}^0,t_0)=0$, it follows that $D(\mathbf{a}^0,t_0)=0$ and consequently, $N(\mathbf{a}^0)=0$, in contradiction to $\mathbf{a}^0 \in \Omega_{\mathrm{nonZ}(N)}$.
\end{proof}
As a consequence of Lemmas \ref{lem:defF}, \ref{lem:r-fold}, \ref{lem:ord-fold}, and taking into account that $\Omega_{\mathrm{ord}(\mathcal{F})}\subset\Omega_{\mathrm{mult}(\mathcal{F})}\subset \Omega_{\mathrm{def}(\mathcal{F})}$, we get the following corollary.
\begin{corollary}\label{cor:mul+ord}
Let $\mathcal{F}\in \mathscr{F}(G^h)$ (see equation \eqref{eq-SingularLocus}) be an irreducible $\mathbb{F}$-conjugate family of ordinary $r$-fold points of $\mathcal{C}(G^h)$.
If $\mathbf{a}^0\in \Omega_{\mathrm{ord}(\mathcal{F})}$, then all points in $\mathcal{F}(\mathbf{a}^0)$ are ordinary $r$-fold points of $\mathcal{C}(G^h,\mathbf{a}^0)$ and $\#(\mathcal{F})=\#(\mathcal{F}(\mathbf{a}^0))$.
\end{corollary}
\section{Preservation of the Genus}\label{sec-genus}
We consider a polynomial
\begin{equation}\label{eq-F} F(\mathbf{a},\gamma,x,y)\in \mathbb{K}[\mathbf{a},\gamma][x,y] \setminus \mathbb{K}[\mathbf{a},\gamma].
\end{equation}
$F$, as a non--constant polymomial in $\mathbb{F}[x,y]$, defines an affine plane curve over $\overline{\mathbb{F}}$ that we assume irreducible.
As introduced in Subsection \ref{subsec-spec-families}, for each $\mathbf{a}^0\in \mathbb{S}$ such that $F(\mathbf{a}^0,\gamma^0,x,y)\not\in \overline{\mathbb{K}}$, we denote by $\mathcal{C}(F,\mathbf{a}^0)$ the curve $\mathcal{C}(F(\mathbf{a}^0,\gamma^0,x,y))$; similarly for $\mathcal{C}(F^h,\mathbf{a}^0)$. Also, we denote by $\mathbb{K}_{\mathbf{a}^0}$ the ground field of $\mathcal{C}(F,\mathbf{a}^0)$ (see Def. \ref{def-Ka}). Our goal is to analyze the relation between the genus of $\mathcal{C}(F)$ and the genus of $\mathcal{C}(F,\mathbf{a}^0)$ under the assumption that $\mathcal{C}(F,\mathbf{a}^0)$ is irreducible.
\subsection{Ordinary singular locus case}\label{subsec-ord-locus}
We start our analysis assuming that $\mathcal{C}(F^h)$ has only ordinary singularities.
Let $\mathscr{F}(F^h)$ be an $\mathbb{F}$--standard decomposition of the singular locus of $\mathcal{C}(F^h)$ obtained by using the process described in Subsection \ref{subsec-genus}. Let $\mathscr{F}(F^h)$ decompose as in \eqref{eq-SingularLocus}
\begin{equation}\label{eq-SingularLocusF}
\mathscr{F}(F^h)=\bigcup_{m(t)\in \mathcal{A}_a} \{(f_{1,m}(t):f_{2,m}(t):1)\}_{m(t)} \, \cup \, \bigcup_{m(t)\in \mathcal{A}_{\infty}} \{(L_{1,m}(t):L_{2,m}(t):0)\}_{m(t)}
\end{equation}
where $f_{i,m}\in \mathbb{F}[t]$, $L_{i,m}\in \mathbb{K}[t]$ with $\gcd(L_{1,m},L_{2,m})=1$ and $\deg(L_{i,m})\leq 1$, and $\mathcal{A}_a$, $\mathcal{A}_{\infty}$ are finite sets of irreducible polynomials in $\mathbb{F}[t]$. We start with the following definition, where $\mathrm{sing}(\mathcal{C}(F^h))$ denotes the singular locus of $\mathcal{C}(F^h)$.
\begin{definition}\label{def-singOrd} We define the open subset (see \eqref{eq-SingularLocusF} and Def. \ref{def:omegaG} and Def. \ref{def:ordSet})
$$\Omega_{\mathrm{singOrd}(F^h)}:=\left\{ \begin{array}{cc} \displaystyle{\bigcap_{ \mathcal{F}\in \mathscr{F}(F^h)} \Omega_{\mathrm{ord}(\mathcal{F})}} & \text{if $\mathrm{sing}(\mathcal{C}(F^h))\neq \emptyset$}\\
\Omega_{F} & \text{if $\mathrm{sing}(\mathcal{C}(F^h))=\emptyset$} \end{array} \right.$$
\end{definition}
Then, the following result holds.
\begin{theorem}\label{theorem:genus-weak}
Let $\mathbf{a}^0\in \Omega_{\mathrm{singOrd}(F^h)}$. If $\mathcal{C}(F,\mathbf{a}^0)$ is irreducible, then
\[ \mathrm{genus}(\mathcal{C}(F))\geq \mathrm{genus}(\mathcal{C}(F,\mathbf{a}^0)). \]
\end{theorem}
\begin{proof} Let $d$ be the degree of $\mathcal{C}(F)$. If $\mathrm{sing}(\mathcal{C}(F^h))=\emptyset$, then $\mathbf{a}^0\in \Omega_F$. So, by Lemma \ref{lem:omegaG}, $\deg(F(\mathbf{a}^0,\gamma^0,x,y))=d$. Since $\mathcal{C}(F,\mathbf{a}^0)$ is irreducible, by \eqref{eq-genus}, one has that
\[ \mathrm{genus}(\mathcal{C}(F))=\dfrac{(d-1)(d-2)}{2} \geq \mathrm{genus}(\mathcal{C}(F,\mathbf{a}^0)). \]
If $\mathrm{sing}(\mathcal{C}(F^h))\neq \emptyset$, then $\mathbf{a}^0\in \Omega_{\mathrm{singOrd}(F^h)}\subset \Omega_F$ and $\deg(F(\mathbf{a}^0,\gamma^0,x,y))=d$. By Corollary \ref{cor:mul+ord}, all elements in $\mathrm{sing}(\mathcal{C}(F^h))$ have the same multiplicity and character as their corresponding elements in $\mathrm{sing}(\mathcal{C}(F^h,\mathbf{a}^0))$ after specialization.
New singularities, however, may appear in $\mathrm{sing}(\mathcal{C}(F^h,\mathbf{a}^0))$. So, reasoning as above with the genus formula in \eqref{eq-genus}, or \eqref{eq-genus2}, we get the result.
\end{proof}
The next result is a direct consequence of the previous theorem.
\begin{corollary}\label{cor:genus-zero}
Let $\mathcal{C}(F)$ be a rational curve. Let $\mathbf{a}^0\in \Omega_{\mathrm{SingOrd}(F^h)}$. If $\mathcal{C}(F,\mathbf{a}^0)$ is irreducible, then $\mathcal{C}(F,\mathbf{a}^0)$ is rational.
\end{corollary}
The inequality in Theorem \ref{theorem:genus-weak} comes from the fact that, using $\Omega_{\mathrm{SingOrd}(F^h)}$, we cannot ensure that $\mathrm{sing}(\mathcal{C}(F^h,\mathbf{a}^0))$ does not include new singularities apart from those coming from the specialization of the singular locus of $\mathcal{C}(F^h)$. To control this phenomenon, we will ensure that certain Gr\"obner bases behave properly under specializations. By, exercises 7, 8, pages 315--316 in \cite{cox}, or by Proposition 1, page 308 in \cite{cox}, we know that there exists an open Zariski set such the Gr\"obner basis specializes properly; in fact, a description of this open subset is also available. For a more general analysis of Gr\"obner bases with parametric coefficients we refer to \cite{montes} and \cite{weispfennin}. On the other hand, since we are working with bivariate polynomials in $\mathbb{F}[x,y]$, the open subset above can be determined by using resultants.
This motivates the next definition.
\begin{definition}\label{def-OpenGB}
Let $\mathrm{I}$ be an ideal in $\mathbb{F}[\overline{v}]$, where $\overline{v}$ is tuple of variables, generated by $\mathscr{G}\subset \mathbb{F}[\overline{v}]$. Let $\mathcal{G}$ be a Gr\"obner basis of $\mathscr{G}$ w.r.t. some order. We define $\Omega_{\mathrm{spGB}(\mathcal{G})}\subset \mathbb{S}$ as a non-empty open subset such that for every $\mathbf{a}^0\in \Omega_{\mathrm{spGB}(\mathcal{G})}$ it holds that
$\{ g(\mathbf{a}^0,\gamma^0,\overline{v}) \,|\, g\in \mathcal{G}\}$ is a Gr\"obner basis, w.r.t. the same order, of the ideal generated by $\{ g(\mathbf{a}^0,\gamma^0,\overline{v}) \,|\, g\in \mathscr{G}\}$ in $\mathbb{K}_{\mathbf{a}^0}[\overline{v}]$.
\end{definition}
Now, we focus our attention on the standard decomposition of the singular locus of $\mathcal{C}(F^h)$ described in Subsection \ref{subsec-genus}.
In the first step, if necessary, we apply a $\mathbb{K}$ linear change of coordinates to ensure that the curve is in regular position. Hence, this linear transformation it is not affected by the specializations of $\mathbf{a}$. Therefore, for our reasonings, we may assume w.l.o.g. that $F$ is already in regular position. Next, let $\mathcal{G}_1$ be a Gr\"obner basis of $<\!\!F,F_{x},F_{y}\!\!>$ w.r.t. the lexicographic order with $x<y$, and let $\mathcal{G}_2$ be a Gr\"obner basis of the same ideal w.r.t. the lexicographic order with $y<x$. Let $\{f(\mathbf{a},\gamma,x)\}=\mathcal{G}_1\cap \mathbb{F}[x]$, $\{g(\mathbf{a},\gamma,y)\}=\mathcal{G}_2\cap \mathbb{F}[y]$, $\tilde{f}=f/\gcd(f,\frac{\partial f}{\partial x})$ and $\tilde{g}=g/\gcd(g,\gcd(g,\frac{\partial g}{\partial x})$. Finally, let $\mathcal{G}_3:=\{A(\mathbf{a},x),y-B(\mathbf{a},x)\}$, with $A$ square-free and $\deg(B)<\deg(A)$, be the normed reduced Gr\"obner basis w.r.t. the lexicographic order with $x<y$ of $<\!\!F, F_{x}, F_{y}, \tilde{f}, \tilde{g}\!\!>$.
Then, we introduce the following definition.
\begin{definition}\label{def:genusOrd}
With the notation introduced above, let (see also Def. \ref{def-open1}, \ref{def-gcd}, and \ref{def:sqfree})
\[ \Omega_1=\Omega_{\mathrm{nonZ}(U)} \cap \Omega_{\mathrm{gcd}(f,\frac{\partial f}{\partial x})}\cap \Omega_{\mathrm{sqfree}(\tilde{f})}, \,\, \Omega_2=\Omega_{\mathrm{nonZ}(V)} \cap \Omega_{\mathrm{gcd}(g,\frac{\partial g}{\partial y})}\cap \Omega_{\mathrm{sqfree}(\tilde{g})}
\]
where $U$ and $V$ are the leading coefficients of $f$ and $g$ w.r.t. $x$ and $y$, respectively.
We define the open subset
\[ \Omega_{\mathrm{sing}_a(F)}:=\bigcap_{i=1}^{3} \Omega_{\mathrm{spGB}(\mathcal{G}_i)} {\bigcap_{q\in \mathcal{G}_1\setminus \{f\}}
\Omega_{\mathrm{nonZ}(W_{q,y})}} {\bigcap_{q\in \mathcal{G}_2\setminus \{g\}}
\Omega_{\mathrm{nonZ}(W_{q,x})}} \bigcap_{i=1}^{2} \Omega_i \cap \Omega_{\mathrm{sqfree}(A)}\]
where $W_{q,y}$ denotes the leading coefficient of $q$ w.r.t. $y$; similarly with $W_{q,x}$.
In addition, let $G(\mathbf{a},\gamma,y,z):=F^h(\mathbf{a},\gamma,1,y,z)$, let $U(\mathbf{a},\gamma,t)=\gcd(G(\mathbf{a},\gamma,t,0),G_y(\mathbf{a},\gamma,t,0),G_z(\mathbf{a},\gamma,t,0))$ and $\tilde{U}(\mathbf{a},\gamma,t):=U/\gcd(U,U')$, where $U'$ is the derivative of $U$ w.r.t. $t$. Let $\Omega_{(0:1:0)}:=\mathbb{S}$ if $(0:1:0)\in \mathrm{sing}(\mathcal{C}(F^h))$ and else $\Omega_{(0:1:0)}:=\Omega_{\mathrm{nonZ}(J(\mathbf{a},\gamma,0,1,0))}$ where $J$ is one the first derivatives of $F^h$ not vanishing at $(0:1:0)$.
We define the open subset (see Definitions \ref{def:gcd-several} and \ref{def:sqfree})
\[ \Omega_{\mathrm{sing}_\infty(F)}:=\Omega_{\mathrm{gcd}(G(\mathbf{a},\gamma,t,0),G_y(\mathbf{a},\gamma,t,0),G_z(\mathbf{a},\gamma,t,0))} \cap \Omega_{\mathrm{gcd}(U,U')} \cap \Omega_{(0:1:0)}. \]
Then, we define (see Def. \ref{def-singOrd})
\[ \Omega_{\mathrm{genusOrd}(F^h)}:= \Omega_{\mathrm{singOrd}(F^h)} \cap \Omega_{\mathrm{sing}_a(F)}\cap \Omega_{\mathrm{sing}_\infty(F)}. \]
\end{definition}
\begin{remark}
Note that $\mathcal{G}_1\cap \mathbb{F}[x]=\{f\}$. So all polynomials in $\mathcal{G}_1\setminus \{f\}$ do depend on $y$; similarly for
$\mathcal{G}_2\setminus \{g\}$. The idea of controlling the coefficients $W_{q,x}$ and $W_{q,y}$ in Def. \ref{def:genusOrd} is to ensure that the elimination ideal of the specialized Gr\"obner basis does not include additional generators.
\end{remark}
In the following lemma, we see that the cardinality of the singular locus, as a set, is preserved under specializations.
\begin{lemma}\label{lem:card-sing}
Let $\mathbf{a}^0\in \Omega_{\mathrm{sing}_a(F)}\cap \Omega_{\mathrm{sing}_\infty(F)}$. Then
\[ \#(\mathrm{sing}(\mathcal{C}(F^h))=\#(\mathrm{sing}(\mathcal{C}(F^h,\mathbf{a}^0)). \]
\end{lemma}
\begin{proof}
Let $F^{0}(x,y):=F(\mathbf{a}^0,\gamma^0,x,y)$. Let $\mathcal{G}^{0}_{1}$ be a Gr\"obner basis of $<\!\!F^0,F^{0}_{x},F^{0}_{y}\!\!>$ w.r.t. the lexicographic order $x<y$, and let $\mathcal{G}^{0}_2$ be a Gr\"obner basis of the same ideal w.r.t. the lexicographic order $y<x$. Since $$\mathbf{a}^0\in \Omega_{\mathrm{spGB}(\mathcal{G}_1)} \cap \bigcap_{q\in \mathcal{G}_1\setminus \{f\}}\Omega_{\mathrm{nonZ}(W_{q,y})}\cap \Omega_1,$$ then $\{f(\mathbf{a}^0,\gamma^0,x)\}=\mathcal{G}_{1}^{0}\cap \mathbb{K}_{\mathbf{a}^0}[x]$ and $\deg_x(f(\mathbf{a},\gamma,x))=\deg_x(f(\mathbf{a}^0,\gamma^0,x))$.
Similarly, $\{g(\mathbf{a}^0,\gamma^0,x)\}=\mathcal{G}_{2}^{0}\cap \mathbb{K}_{\mathbf{a}^0}[y]$ and $\deg_y(f(\mathbf{a},\gamma,y))=\deg_y(f(\mathbf{a}^0,\gamma^0,y))$.
Since $\mathbf{a}^0 \in \Omega_1$, by Corollary~\ref{cor-lemma-gcd}, $\gcd(f,\frac{\partial f}{\partial x})(\mathbf{a}^0,\gamma^0,x)=\gcd(f(\mathbf{a}^0,\gamma^0,x),\frac{\partial f(\mathbf{a}^0,\gamma^0,x)}{\partial x})$ and $\deg_x(\gcd(f,\frac{\partial f}{\partial x})(\mathbf{a}^0,\gamma^0,x))=\deg_x(\gcd(f,\frac{\partial f}{\partial x})(\mathbf{a},\gamma,x))$.
Thus, $$\tilde{f}(\mathbf{a}^0,\gamma^0,x)=\frac{f(\mathbf{a}^0,\gamma^0,x)}{\gcd(f(\mathbf{a}^0,\gamma^0,x),\frac{\partial f(\mathbf{a}^0,\gamma^0,x)}{\partial x})}.$$
By Lemma~\ref{lem:sqfree}, it holds that $\deg_x(\tilde{f}(\mathbf{a},\gamma,x))=\deg_x(\tilde{f}(\mathbf{a}^0,\gamma^0,x))$ and $\tilde{f}(\mathbf{a}^0,\gamma^0,x)$ is squarefree.
Similarly for $g$ and $\tilde{g}$ since $\mathbf{a}^0\in \Omega_2$. In addition, by Lemma \ref{lem:omegaG}, $F^{0}_{x}(x,y)=F_x(\mathbf{a}^0,\gamma^0,x,y)$ and $F^{0}_{y}(x,y)=F_y(\mathbf{a}^0,\gamma^0,x,y)$. Thus,
\[ \sqrt{<\!\! F^0,F^{0}_{x},F^{0}_{y} \!>}=<\!\! F(\mathbf{a}^0,\gamma^0,x,y),F_{x}(\mathbf{a}^0,\gamma^0,x,y),F_{y}(\mathbf{a}^0,\gamma^0,x,y),\tilde{f}(\mathbf{a}^0,\gamma^0,x),\tilde{g}(\mathbf{a}^0,\gamma^0,y)\!\!>.\]
Since $\mathbf{a}^0\in \Omega_{\mathrm{spGB}(\mathcal{G}_3)}$, $\{A(\mathbf{a}^0,\gamma^0,x),y-B(\mathbf{a}^0,\gamma^0,x)\}$ is a Gr\"obner basis of $\sqrt{<\!\!F^0,F^{0}_{x},F^{0}_{y} \!>}$.
Since $\mathbf{a}^0\in \Omega_{\mathrm{sqfree}(A)}$, by Lemma~\ref{lem:sqfree}, $A(\mathbf{a}^0,\gamma^0,x)$ is squarefree. Therefore, the number of affine singularities of $\mathcal{C}(F,\mathbf{a}^0)$ is $\deg_x(A(\mathbf{a}^0,\gamma^0,x))$ and $\deg_x(A(\mathbf{a},\gamma,x))$ is the number of affine singularities of $\mathcal{C}(F)$. By Lemma~\ref{lem:sqfree}, we get that $\deg_x(A(\mathbf{a}^0,\gamma^0,x))=\deg_x(A(\mathbf{a},\gamma,x))$ and, hence, $\mathcal{C}(F)$ and $\mathcal{C}(F,\mathbf{a}^0)$ have the same number of affine singularities.
It remains to prove that the number of singularities at infinity is also the same. First we observe
that, if $(0:1:0)\in \mathrm{sing}(\mathcal{C}(F^h))$, then $(0:1:0)\in \mathrm{sing}(\mathcal{C}(F^h,\mathbf{a}^0))$. Moreover, since $\mathbf{a}^0\in \Omega_{(0:1:0)}$,
if $(0:1:0)\not\in \mathrm{sing}(\mathcal{C}(F^h))$, then $(0:1:0)\not\in \mathrm{sing}(\mathcal{C}(F^h,\mathbf{a}^0))$.
For the remaining singularities at infinity, denote by $\Sigma$ the set of the singularities of the form $(1:\mu:0)\in \mathrm{sing}(\mathcal{C}(F^h))$; similarly let $\Sigma^0$ be the set of singularities of this type in $\mathrm{sing}(\mathcal{C}(F^h,\mathbf{a}^0))$. Let $G^0(y,z):=G(\mathbf{a}^0,\gamma^0,y,z)$ (see Def. \ref{def:genusOrd}), $U^0(t):=\gcd(G^0(t,0),G^{0}_{y}(t,0),G^{0}_{z}(t,0))$ and $\tilde{U}^0(t):=U^0/\gcd(U^0,(U^0)')$. Then,
\begin{equation}\label{eq-uo}
\#(\Sigma)=\deg_t(\tilde{U}) \,\,\, \text{and}\,\,
\#(\Sigma^0)=\deg_t(\tilde{U}^0).
\end{equation}
Since $\mathbf{a}^0\in \Omega_{\mathrm{gcd}(G(\mathbf{a},\gamma,t,0),G_y(\mathbf{a},\gamma,t,0),G_z(\mathbf{a},\gamma,t,0))}$, by Theorem~\ref{theorem-lemma-gcd-several-pol}, it holds that
\begin{equation}\label{eq-u1}
U^0(t)=U(\mathbf{a}^0,\gamma^0,t) \,\,\, \text{and}\,\, \deg_t(U^0(t))=\deg_t(U(\mathbf{a},\gamma,t)).
\end{equation}
Let $D(\mathbf{a},\gamma,t)=\gcd(U,U')$ and $D^0(t)=\gcd(U^0,(U^0)')$. Since $\mathbf{a}^0\in \Omega_{\mathrm{gcd}(U,U')}$, by Corollary~\ref{cor-lemma-gcd}, one has that
\begin{equation}\label{eq-u2}
D^0(t)=D(\mathbf{a}^0,\gamma^0,t) \,\,\, \text{and}\,\, \deg_t(D^0(t))=\deg_t(D(\mathbf{a},\gamma,t)).
\end{equation}
Now, by \eqref{eq-uo}, \eqref{eq-u1} and \eqref{eq-u2}, we get that $\#(\Sigma)=\#(\Sigma^0)$ and this concludes the proof.
\end{proof}
\begin{theorem}\label{theorem:genus-strong}
Let $\mathbf{a}^0\in \Omega_{\mathrm{genusOrd}(F^h)}$. If $\mathcal{C}(F,\mathbf{a}^0)$ is irreducible, then
\[ \mathrm{genus}(\mathcal{C}(F))= \mathrm{genus}(\mathcal{C}(F,\mathbf{a}^0)). \]
\end{theorem}
\begin{proof}
Since $\mathbf{a}^0\in \Omega_{\mathrm{sing}_a(F)}\cap \Omega_{\mathrm{sing}_\infty(F)}$, by Lemma~\ref{lem:card-sing}, it holds that $\#(\mathrm{sing}(\mathcal{C}(F^h))=\#(\mathrm{sing}(\mathcal{C}(F^h,\mathbf{a}^0)).$ On the other hand,
since $\mathbf{a}^0\in \Omega_{\mathrm{SingOrd}(F^h)}$, by Corollary~\ref{cor:mul+ord}, we know that each $r$--fold in $\mathrm{sing}(\mathcal{C}(F^h))$ generates an ordinary $r$-fold in $\mathrm{sing}(\mathcal{C}(F^h,\mathbf{a}^0))$.
Therefore, applying \eqref{eq-genus}, we conclude the proof.
\end{proof}
\subsection{General case}
Let $F$ be as in \eqref{eq-F}. But now, differently to the case of Subsection \ref{subsec-ord-locus}, we do not introduce any assumption on the singular locus of the irreducible curve $\mathcal{C}(F^h)$. The key of our analysis is to reduce the general case to the case studied in Subsection \ref{subsec-ord-locus}. For this purpose, we recall that any irreducible curve is birationally equivalent to a curve having only ordinary singularities; see e.g. \cite[Theorem 7.4.]{walker} or \cite[Section 3.2.]{myBook} for a more computational description. This transformation, say $\varphi$, can be seen as a finite sequence of blowups of the irreducible families of non-ordinary singularities and, hence, as a finite sequence of compositions of quadratic Cremona transformations and linear transformations.
Now, our goal is to find an open subset $\Omega_{\mathrm{blowup}}$ of $\mathbb{S}$ such that, when $\mathbf{a}$ is specialized in $\Omega_{\mathrm{blowup}}$, the birationality of $\varphi$ is preserved. For this purpose, let $F_0(x,y)=F(x,y)$ and let $\mathcal{F}(F_0^h)$ be a $\mathbb{F}$-conjugate irreducible family of non-ordinary singularities of $\mathcal{C}(F_{0}^{h})$ with defining polynomial $m_1(t_1) \in \mathbb{F}[t_1]$. Let $\mathbb{F}_{m_1}$ be the quotient field of $\mathbb{F}[t_1]/\!\!<\!m_1(t_1)\!>$, that is, $\mathbb{F}_{m_1}=\mathbb{F}(t_1)$ with $m_1(t_1)$ as minimal polynomial of $t_1$.
Then we apply a linear transformation $\mathcal{L}_1$, given by a matrix $M_1 \in \text{M}_{3 \times 3}(\mathbb{F}_{m_1})$, and the Cremona transformation $\mathcal{Q}_1=(yz:xz:xy)$ as described in the blow up basic step in Subsection~\ref{subsec-genus} of the appendix.
Denote by $\Delta_1$ the determinant $\det(M_1)$ and let $\mathcal{C}(F_1^h)$ be the curve over $\overline{\mathbb{F}_{m_1}}$ obtained after the quadratic transformation $\mathcal{Q}_1 \circ \mathcal{L}_1$. Note that $F_{1}^{h}$, the quadratic transformation of $F_0^h$, is the cofactor of $F_{0}^h(\mathcal{Q}_1(\mathcal{L}_1))$ not being divisible by neither $x$, $y$ nor $z$.
We repeat the above process for $F_1^h(t_1,x,y,z)$, $F_2^h(t_1,t_2,x,y,z), \ldots, F_r^h(t_1,\ldots,t_r,x,y,z)$ until all singularities of $\mathcal{C}(F_r^h)$ are ordinary.
Then \[ \varphi(\mathbf{a},\gamma,t_1,\ldots,t_r,x,y,z) = (\mathcal{Q}_r \circ \mathcal{L}_r) \circ \cdots \circ (\mathcal{Q}_1 \circ \mathcal{L}_1) .\]
Note that $F_r^h$ is defined over $\mathbb{F}(t_1,\ldots,t_r)$ and $\mathcal{C}(F_r^h)$ over the algebraic closure of $\mathbb{F}(t_1,\ldots,t_r)$.
In addition, $\mathbb{F}(t_1,\ldots,t_r)=\mathbb{K}(\mathbf{a},\gamma,t_1,\ldots,t_r)=\mathbb{L}(\gamma,t_1,\ldots,t_r)$. So, we consider a primitive element of the extension over $\mathbb{L}$, say $\gamma^*$, and we work over $\mathbb{L}(\gamma^*)=\mathbb{L}(\gamma,t_1,\ldots,t_r)$; note that results in Section \ref{sec-specialization} apply to this new frame. In this situation, let us denote by $\Delta=\Delta_1 \cdots \Delta_r \in \mathbb{L}(\gamma^*)$ the product of the determinants of the linear transformations $\mathcal{L}_1,\ldots,\mathcal{L}_m$. In addition, let
$$\mathcal{M}:=\{ \text{all entries of $M_i\in \mathrm{M}_{3\times 3}(\mathbb{L}(\gamma^*))\}_{i\in \{1,\ldots,r\}}$}.$$
and let
$$
\begin{array}{lc}
\mathcal{B}:=&\{ F_{i}^h(\mathcal{Q}_{i+1}(\mathcal{L}_{i+1}))(\mathbf{a},\gamma^*,0,y,z), F_{i}^h(\mathcal{Q}_{i+1}(\mathcal{L}_{i+1}))(\mathbf{a},\gamma^*,x,0,z), \\
\noalign{\vspace*{1mm}}
& F_{i}^h(\mathcal{Q}_{i+1}(\mathcal{L}_{i+1}))(\mathbf{a},\gamma^*,x,y,0)\}_{i\in \{0,\ldots,r-1\}}.
\end{array}$$
\begin{definition}\label{def-openBlup}
With the notation introduced above, we define the set $$\Omega_{\mathrm{blowup}(F)}:= \Omega_F \cap \bigcap_{h\in \mathcal{M}} \Omega_{\mathrm{def}(h)} \cap \Omega_{\mathrm{nonZ}({\Delta})} \cap \bigcap_{h\in \mathcal{B}} \Omega_{\mathrm{nonZ}(h)}.$$
\end{definition}
The previous observations lead to the following result.
\begin{lemma}\label{lem:specializationBirational}
Let $\mathbf{a}^0 \in \Omega_{\mathrm{blowup}}(F)$. Then $\varphi(\mathbf{a}^0,(\gamma^*)^0,x,y,z)$ is birational.
\end{lemma}
\begin{proof}
Since $\mathbf{a}^0 \in \Omega_{\mathrm{blowup}}(F) \subset \bigcap_{h\in \mathcal{M}} \Omega_{\mathrm{def}(h)} \cap \Omega_{\mathrm{nonZ}({\Delta})}$, all $\mathcal{Q}_i \circ \mathcal{L}_i$ are well--defined, and birational, when $\mathbf{a}$ is specialized as $\mathbf{a}^0$. So $\varphi(\mathbf{a}^0,(\gamma^*)^0,x,y,z)$ is birational.
\end{proof}
\begin{lemma}\label{lemm:biratR}
Let $\mathbf{a}^0 \in \Omega_{\mathrm{blowup}}(F)$ and let $\varphi^0:=\varphi(\mathbf{a}^0,(\gamma^*)^0,x,y,z)$. If $F_0(\mathbf{a}^0,\gamma^0,x,y,z)$ is irreducible, then $F_r(\mathbf{a}^0,(\gamma^*)^0,x,y,z)$ is the quadratic transformation of $F_0(\mathbf{a}^0,\gamma^0,x,y,z)$.
\end{lemma}
\begin{proof}
First we observe that because of (the proof of) Lemma \ref{lem:specializationBirational} each $\varphi_i:=\mathcal{Q}_i \circ \mathcal{L}_i$ is well defined at $\mathbf{a}^0$ and it is birational. Let us prove the result by induction. By hypothesis $F_{0}^{h}(\mathbf{a}^0,\gamma^0,x,y,z)$ is irreducible. We have the equality
$F_{0}^{h}(\varphi_i)=x^{n_1}y^{n_2}z^{n_3} F_{1}^{h}$ for some $n_i\in \mathbb{N}$ and that neither $x$, $y$ nor $z$ divides $F_{1}^{h}$. So, $F_{0}(\varphi_i)(\mathbf{a}^0,(\gamma^*)^0,x,y,z)=x^{n_1}y^{n_2}z^{n_3} F_{1}^{h}(\mathbf{a}^0,(\gamma^*)^0,x,y,z)$. Moreover, since $\mathbf{a}^0\in \bigcap_{h\in \mathcal{B}} \Omega_{\mathrm{nonZ}(h)}$, we know that neither $x$, $y$ nor $z$ divides $F_1(\mathbf{a}^0,(\gamma^*)^0,x,y,z)$. Furthermore, since $\varphi_i$ is birational when specialized at $\mathbf{a}^0$ and $F_{0}^{h}(\mathbf{a}^0,\gamma^0,x,y,z)$ is irreducible, we have that $F_{1}^{h}(\mathbf{a}^0,(\gamma^*)^0,x,y,z)$ is also irreducible.
Thus, $F_{1}^{h}(\mathbf{a}^0,(\gamma^*)^0,x,y,z)$ is the quadratic transformation of $F_{0}^{h}$. Now, the $i$-induction step is reasoned analogously using that, by induction, $F_{i}^{h}(\mathbf{a}^0,(\gamma^*)^0,x,y,z)$ is irreducible.
\end{proof}
With this, we can now give an open set where the genus is preserved.
\begin{definition}\label{def-genus}
Let $F \in \mathbb{F}[x,y]$ be as in~\eqref{eq-F}, and let $G \in \mathbb{F}(\gamma^*)[x,y]$ be the polynomial obtained after the blowup process of $F^h$. We define the set $$\Omega_{\mathrm{genus}(F)}:= \Omega_{\mathrm{blowup}(F)} \cap \Omega_{\mathrm{genusOrd}(G^h)}.$$
\end{definition}
\begin{theorem}\label{thm:genusPreservation}
Let $F \in \mathbb{F}[x,y]$ be as in~\eqref{eq-F}, and let $\mathbf{a}^0 \in \Omega_{\mathrm{genus}(F)}$. If $\mathcal{C}(F,\mathbf{a}^0)$ is irreducible, then
\[ \mathrm{genus}(\mathcal{C}(F))= \mathrm{genus}(\mathcal{C}(F,\mathbf{a}^0)). \]
\end{theorem}
\begin{proof} Since $G^h$ is the quadratic transformation of $F^h$, we have that
\begin{equation}\label{eqq-genus1}
\mathrm{genus}(\mathcal{C}(F^h(\mathbf{a},x,y,z)))=\mathrm{genus}(\mathcal{C}(G^h(\mathbf{a},\gamma^*,x,y,z))).
\end{equation}
Let $\varphi^0$ denote the map $\varphi(\mathbf{a}^0,(\gamma^*)^0,x,y,z)$. Since $\mathbf{a}^0 \in \Omega_{\mathrm{blowup}(F)}$, by Lemma \ref{lem:specializationBirational}, $\varphi^0$ is birational and, by Lemma \ref{lemm:biratR}, $G^h(\mathbf{a}^0,(\gamma^*)^0,x,y,z)$ is the quadratic transformation of $F^h(\mathbf{a}^0,\gamma^0,x,y,z)$ via $\varphi^0$. Therefore,
\begin{equation}\label{eqq-genus2}
\mathrm{genus}(\mathcal{C}(F^h(\mathbf{a}^0,\gamma^0,x,y,z)))=\mathrm{genus}(\mathcal{C}(G^h(\mathbf{a}^0,(\gamma^*)^0,x,y,z))).
\end{equation}
Moreover, since $\mathbf{a}^0 \in \Omega_{\mathrm{genusOrd}(G^h)}$, by Theorem \ref{theorem:genus-strong}, it holds that
\begin{equation}\label{eqq-genus3}
\mathrm{genus}(\mathcal{C}(G^h(\mathbf{a},\gamma^*x,y,z))) = \mathrm{genus}(\mathcal{C}(G^h(\mathbf{a}^0,(\gamma^*)^0,x,y,z))) .
\end{equation}
Now, the proof follows from \eqref{eqq-genus1}, \eqref{eqq-genus2} and \eqref{eqq-genus3}.
\end{proof}
\section{Birational Parametrization of Parametric Rational Curves}\label{sec-parametricCurves}
In Section \ref{sec-genus}, and more precisely in Theorem \ref{theorem:genus-strong} and \ref{thm:genusPreservation}, we have described open subsets of $\mathbb{S}$ where the genus of the curve is preserved under specializations; even in Corollary \ref{cor:genus-zero} the particular case of genus zero was treated. Nevertheless, in all these results the additional condition that the specialized polynomial is irreducible over $\overline{\mathbb{K}}$ was required. Avoiding the irreducibility is in general a difficult problem related to the Hilbert irreducibility problem (see e.g. \cite{Serre}). More precisely, there is no algorithm known that finds for given irreducible $F \in \mathbb{F}[x,y]$ the specializations $\mathbf{a}^0 \in \mathbb{S}$ such that $F(\mathbf{a}^0,x,y)$ is reducible.
Nevertheless, in this section, we see how in the case of genus zero the problem can be solved. Furthermore, we described an open subset where the specialized parametrization parametrizes the specialized curve.
For this purpose, throughout this section, let assume that $F \in \mathbb{F}[x,y]$ is as in \eqref{eq-F} and additionally assume that $\mathcal{C}(F)$ is rational. Moreover, let us assume that $\mathcal{P}$ is a proper (i.e. birational) parametrization of $\mathcal{C}(F)$ which can be computed, for instance, by the algorithm described in Subsection \ref{subsec:param} in the appendix. Note that, in general, one may need to extend $\mathbb{F}$ with an algebraic element $\delta$ of degree two.
If $\#(\mathbf{a})=1$ and $\deg(\gamma)=1$, or $\deg_{x,y}(F)$ is odd, then no extension of $\mathbb{F}$ is required (see Remark~\ref{rem:TsenGeneralization} and Theorem~\ref{theorem:HHext}).
So, we may consider a primitive element of $\mathbb{L}(\gamma,\delta)$, say $\gamma^*$, and express our parametrization in $\mathbb{L}(\gamma^*)$. Throughout this section, by abuse of notation, let $\mathbb{F}$ denote the field $\mathbb{L}(\gamma^*)$. Let us write the proper parametrization $\mathcal{P}$ of $\mathcal{C}(F)$ as
\begin{equation}\label{eq-Pa} \mathcal{P}(\mathbf{a},\gamma,t)=\left(\dfrac{p_1}{q_1},\dfrac{p_2}{q_2} \right) \in \mathbb{F}(t)^2 \setminus \mathbb{F}^2
\end{equation}
where we assume that $\mathcal{P}$ is in reduced form, that is $\gcd(p_1,q_1)=\gcd(p_2,q_2)=1$.
Let us start with the simple case of degree one.
\begin{definition}\label{def:deg1}
Let $F=A_2(\mathbf{a},\gamma)x+A_1(\mathbf{a},\gamma) y+A_0(\mathbf{a},\gamma)\in \mathbb{F}[x,y] \setminus \mathbb{F}$, and let $\mathcal{P}$ be expressed as $\mathcal{P}(\mathbf{a},\gamma,t)=(\lambda_1 t+\lambda_0, \mu_1 t+ \mu_0)\in \mathbb{F}(t)^2\setminus \mathbb{F}^2$ be a proper polynomial parametrization of $\mathcal{C}(F)$. We define the set (see Def. \ref{def-open1} and \ref{def:omegaG})
\[ \Omega_{\mathrm{proper}(\mathcal{P})}:= \Omega_{F} \cap \Omega_{\mathrm{def}(\lambda_1 t+\lambda_0)} \cap \Omega_{\mathrm{def}(\mu_1 t+\mu_0)} \cap \left( \Omega_{\mathrm{nonZ}(\lambda_1)} \cup \Omega_{\mathrm{nonZ}(\mu_1)}\right).\]
\end{definition}
\begin{proposition}\label{prop:deg1}
Let $F$ and $\mathcal{P}$ be as in Def. \ref{def:deg1}. Then, for every $\mathbf{a}^0\in \Omega_{\mathrm{proper}(\mathcal{P})}$, it holds that $\mathcal{P}(\mathbf{a}^0,t)$ is a proper polynomial parametrization of $\mathcal{C}(F,\mathbf{a}^0)$.
\end{proposition}
\begin{proof}
Since $\mathbf{a}^0\in \Omega_{F}$, by Lemma~\ref{lem:omegaG}, $F(\mathbf{a}^0,\gamma^0,x,y)$ is well defined and $\mathcal{C}(F,\mathbf{a}^0)$ is an affine line, obviously irreducible. Since $\mathbf{a}^0\in \Omega_{\mathrm{def}(\lambda_1 t+\lambda_0)} \cap \Omega_{\mathrm{def}(\mu_1 t+\mu_0)}$ then $\mathcal{P}(\mathbf{a}^0,\gamma^0,t)$ is well--defined. Moreover, since $\mathbf{a}^0\in \left( \Omega_{\mathrm{nonZ}(\lambda_1)} \cup \Omega_{\mathrm{nonZ}(\mu_1)}\right)$, $\mathcal{P}(\mathbf{a}^0,\gamma^0,t)$ is a polynomial parametrization, clearly proper. Furthermore, $F(\mathbf{a}^0,\gamma^0,\mathcal{P}(\mathbf{a}^0,\gamma^0,t))=0$. Thus, since $F(\mathbf{a}^0,\gamma^0,x,y)$ is irreducible, we conclude that $\mathcal{P}(\mathbf{a}^0,\gamma^0,t)$ is a proper parametrization of $\mathcal{C}(F,\mathbf{a}^0)$.
\end{proof}
\begin{remark}\label{rem-caseslineal}
Let us use the notation in Proposition \ref{prop:deg1}. If $\mathbf{a}^0\in \mathbb{S}\setminus \Omega_{\mathrm{proper}(\mathcal{P})}$ then:
\begin{enumerate}
\item If $\mathbf{a}^0\not\in \Omega_F\setminus \Omega_{\mathrm{def}(F)}$, it holds that $ F(\mathbf{a}^0,\gamma^0,x,y)=A_0(\mathbf{a}^0,\gamma^0)\in \overline{\mathbb{K}}$, and hence $\mathcal{C}(F,\mathbf{a}^0)$ does not define an affine curve.
\item If $\mathbf{a}^0\not\in \Omega_{\mathrm{def}(\lambda_1 t+\lambda_0)}$ but $\mathbf{a}^0\in \Omega_F$ (similarly if $\mathbf{a}^0\not\in\Omega_{\mathrm{def}(\mu_1 t+\mu_0)}$), the specialization $\mathcal{P}(\mathbf{a}^0,\gamma^0,t)$ is not well--defined even though $\mathcal{C}(F,\mathbf{a}^0)$ is a line. Clearly, in this case, one has that $\mathcal{C}(F,\mathbf{a}^0)$ is rational and a proper parametrization of the specialized line can be provided.
\item If $\mathbf{a}^0\not\in \left( \Omega_{\mathrm{nonZ}(\lambda_1)} \cup \Omega_{\mathrm{nonZ}(\mu_1)}\right)$ but $\mathbf{a}^0\in \Omega_F\cap \Omega_{\mathrm{def}(\lambda_1 t+\lambda_0)} \cap \Omega_{\mathrm{def}(\mu_1 t+\mu_0)}$, then $\mathcal{P}(\mathbf{a}^0,\gamma^0,t)\in \overline{\mathbb{K}}^2$ and hence it is not a parametrization although $\mathcal{C}(F,\mathbf{a}^0)$ is a line. Again, as in the previous case, one can easily provide a polynomial parametrization of the specialized line.
\end{enumerate}
\end{remark}
In the sequel, we assume that $\mathcal{C}(F)$ is not a line. Then, we generalize the open subset in Def. \ref{def:deg1} as follows.
\begin{definition}\label{def-omega}\
\begin{enumerate}
\item Let $\Omega_1:=\Omega_{F}$ (see Def. \ref{def:omegaG}).
\item $\Omega_2:=\Omega_{\mathrm{def}(p_1)}\cap \Omega_{\mathrm{def}(p_2)}\cap \Omega_{\mathrm{nonZ}(q_1)}\cap \Omega_{\mathrm{nonZ}(q_2)}$.
\item We consider the polynomials $G_i=p_i(h)q_i(t)-p_i(t)q_i(h)\in \mathbb{F}[h][t]\setminus \{0\}$ for $i\in \{1,2\}$. Let $\Omega_{3}:=\Omega_{\mathrm{gcd}(G_1,G_2)}$ (see Def. \ref{def-gcd}).
\item Let $\Omega_{4}:=\Omega_{\mathrm{gcd}(p_1,q_1)} \cap \Omega_{\mathrm{gcd}(p_2,q_2)}$; note that $p_i,q_j\in \mathbb{F}[t]\subset \mathbb{F}[h,t]$ and, since $\mathcal{C}(F)$ is not a line, the $p_i$ and $g_i$ are are non--zero (see Def. \ref{def-gcd}).
\end{enumerate}
We define $\Omega_{\mathrm{proper}(\mathcal{P})}$ as
\[ \Omega_{\mathrm{proper}(\mathcal{P})}= \bigcap_{i=1}^{4} \Omega_i. \]
\end{definition}
The following theorem generalizes Prop.~\ref{prop:deg1}.
\begin{theorem}\label{thrm:genus0general}
Let $\mathbf{a}^0\in \Omega_{\mathrm{proper}(\mathcal{P})}$. Then $\mathcal{C}(F,\mathbf{a}^0)$ is a rational affine curve in $\overline{\mathbb{K}}^2$ properly parametrized by $\mathcal{P}(\mathbf{a}^0,\delta^0,t)$.
\end{theorem}
\begin{proof}
If $\deg(F)=1$, the result follows from Prop. \ref{prop:deg1}. Let $\deg(F)>1$. Since $\mathbf{a}^0\in \Omega_1$, then $F(\mathbf{a}^0,\gamma^0,x,y)$ is well--defined and $\deg(\mathcal{C}(F))=\deg(\mathcal{C}(F,\mathbf{a}^0))$. In particular $\mathcal{C}(F,\mathbf{a}^0)$ is an affine curve. On the other hand, since $\mathbf{a}^0\in \Omega_2$, by Lemma~\ref{lem:defVan}~(1), we have that $\mathcal{P}(\mathbf{a}^0,\gamma^0,t)$ is well--defined.
In addition, since $\mathbf{a}^0\in \Omega_4$, the leading coefficients of $p_1,p_2,q_1,q_2$ do not vanish at $\mathbf{a}^0$ (see Def.~\ref{def-gcd}). Consequently the degree of all numerators and denominators of $\mathcal{P}$ after specialization are preserved.
Furthermore, by Corollary~\ref{cor-lemma-gcd}, and using that $p_i/q_i$ are in reduced form, we get that $p_i(\mathbf{a}^0,\gamma^0,t)/q_i(\mathbf{a}^0,\gamma^0,t)$ are also in reduced form. Therefore,
\begin{equation}\label{eq-deg}
\deg_t\left(\dfrac{p_i(\mathbf{a}^0,\gamma^0,t)}{q_i(\mathbf{a}^0,\gamma^0,t)}\right)= \deg_t\left(\dfrac{p_i(\mathbf{a},\gamma,t)}{q_i(\mathbf{a},\gamma,t)}\right) \,\, \text{for $i\in\{1,2\}$.}
\end{equation}
In particular, $\mathcal{P}(\mathbf{a}^0,\gamma^0,t)\not\in \overline{\mathbb{K}}^2$, and hence $\mathcal{P}(\mathbf{a}^0,\gamma^0,t)$ is a parametrization.
Moreover, $F(\mathbf{a}^0,\gamma^0,\mathcal{P}(\mathbf{a}^0,\delta^0,t))=0$. Thus, $\mathcal{P}(\mathbf{a}^0,\gamma^0,t)$ parametrizes the curve defined by one of the factors, say $H(x,y)$, of $F(\mathbf{a}^0,\gamma^0,x,y)$. Let us see that indeed $\mathcal{C}(F,\mathbf{a}^0)=\mathcal{C}(H)$.
Let $G_i(\mathbf{a},\gamma,h,t)$ as in Def.~\ref{def-omega}~(3), and let $G:=\gcd(G_1(\mathbf{a},\gamma,h,t),G_2(\mathbf{a},\gamma,h,t))$. Let $\tilde{G}_i(h,t)$ be the corresponding polynomials associated, as in Def.~\ref{def-omega}~(3), to $\mathcal{P}(\mathbf{a}^0,\gamma^0,t)$. Let $\tilde{G}:=\gcd(\tilde{G}_1,\tilde{G}_2)$. Since $p_i(\mathbf{a}^0,\gamma^0,t)/q_i(\mathbf{a}^0,\gamma^0,t)$ are in reduced form, no simplification of the rational functions have been required, and therefore $\tilde{G}_i(h,t)=G_i(\mathbf{a}^0,\gamma^0,h,t)$. Moreover, since $\mathbf{a}^0\in \Omega_3$, by Lemma~\ref{lem-gcd}, it holds that
\begin{equation}\label{eq:gcd}
\deg_t(G(\mathbf{a}^0,\gamma^0,h,t))=\deg_t(\tilde{G}(h,t)),
\end{equation}
and $\deg_t(G(\mathbf{a}^0,\gamma^0,h,t))=\deg_t(G(\mathbf{a},\gamma,h,t))$. By \cite[Theorem 3]{SW01}, since $\mathcal{P}(\mathbf{a},\gamma,t)$ is proper, we have that $\deg_t(G(\mathbf{a},\gamma,h,t))=1$. Therefore, it holds that $\deg_t(\tilde{G}(h,t))=\deg_t(G(\mathbf{a}^0,\gamma^0,h,t))=1$. Again by \cite[Theorem 3]{SW01}, $\mathcal{P}(\mathbf{a}^0,\gamma^0,t)$ is proper.
On the other hand, by \eqref{eq-deg}, $\deg_t(\mathcal{P}(\mathbf{a},\gamma,t))=\deg_t(\mathcal{P}(\mathbf{a}^0,\gamma^0,t))$. Therefore, by Theorem 4.21 in \cite{myBook}, we have that
\[ \begin{array}{rl} \max\{\deg_{x}(F(\mathbf{a},\gamma,x,y)),\deg_{y}(F(\mathbf{a},\gamma,x,y)\}=&\deg_t(\mathcal{P}(\mathbf{a},\gamma,t))
\\
=&\deg_t(\mathcal{P}(\mathbf{a}^0,\gamma^0,t))\\
=& \max\{\deg_{x}(H),\deg_{y}(H)\}.
\end{array} \]
Moreover, since $F$ is not linear, by \eqref{eq-deg}, no component of $\mathcal{P}(\mathbf{a}^0,\gamma^0,t)$ is constant. Applying again \cite[Theorem 4.21.]{myBook}, we have that
\[ \begin{array}{l} \deg_{x}(H)=\deg_t\left(\dfrac{p_2(\mathbf{a}^0,\gamma^0,t)}{q_2(\mathbf{a}^0,\gamma^0,t)}\right)=\deg_t\left(\dfrac{p_2(\mathbf{a},\gamma,t)}{q_2(\mathbf{a},\gamma,t)}\right)=\deg_{x}(F) \\
\noalign{\vspace*{1mm}}
\deg_{y}(H)=\deg_t\left(\dfrac{p_1(\mathbf{a}^0,\gamma^0,t)}{q_1(\mathbf{a}^0,\gamma^0,t)}\right)=\deg_t\left(\dfrac{p_1(\mathbf{a},\gamma,t)}{q_1(\mathbf{a},\gamma,t)}\right)=\deg_{y}(F). \end{array} \]
Finally, since $H(x,y)$ divides $F(\mathbf{a}^0,\gamma^0,x,y)$, one
has that $\mathcal{C}(F,\mathbf{a}^0)=\mathcal{C}(H)$, which concludes the proof.
\end{proof}
\begin{remark}\label{cases} Let us analyze the behavior of $F$ and/or $\mathcal{P}$ when specializing in $\mathbb{S}\setminus \Omega_{\mathrm{proper}(\mathcal{P})}$.
\begin{enumerate}
\item If $\mathbf{a}^0\in \mathbb{S}\setminus \Omega_{1}$, since $F\in \mathbb{K}[\mathbf{a}][x,y]$ (see \eqref{eq-F}), then $F(\mathbf{a}^0,x,y)$ is always well--defined, and hence, $\deg_{\{x,y\}}(F(\mathbf{a}^0,x,y))< \deg_{\{x,y\}}(F(\mathbf{a},x,y))$. So, it can happen that either $0<\deg_{\{x,y\}}(F(\mathbf{a}^0,x,y))< \deg_{\{x,y\}}(F(\mathbf{a},x,y))$, in which case $\mathcal{C}(F,\mathbf{a}^0)$ is an affine curve; or $0=\deg_{\{x,y\}}(F(\mathbf{a}^0,x,y))$, which implies that $\mathcal{C}(F,\mathbf{a}^0)$ is the empty set or $\overline{\mathbb{K}}^2$.
\item If $\mathbf{a}^0\in \mathbb{S}\setminus \Omega_{2}$, then $\mathcal{P}(\mathbf{a}^0,\delta^0,t)$ is not defined, and hence the specialization fails.
\item If $\mathbf{a}^0\in \left(\mathbb{S}\setminus (\Omega_{3}\cap \Omega_{4})\right)\cap \Omega_1\cap \Omega_2$, at least one of the following assertions hold.
\begin{enumerate}
\item $\mathcal{P}(\mathbf{a}^0,t)$ is not proper.
\item $\mathcal{P}(\mathbf{a}^0,t)\in \overline{\mathbb{K}}^2$ and hence, $\mathcal{P}(\mathbf{a}^0,t)$ is not a parametrization.
\item$\mathcal{P}(\mathbf{a}^0,t)$ parametrizes a proper factor of $F(\mathbf{a}^0,x,y)$, that is, $\mathcal{C}(F,\mathbf{a}^0)$ decomposes and one of its components is rational and parametrized by $\mathcal{P}(\mathbf{a}^0,t)$.
\end{enumerate}
\end{enumerate}
\end{remark}
The next result follows from Theorem \ref{thrm:genus0general} and emphasizes the polynomiality of the parametrizaion.
\begin{corollary}
If $\mathcal{P}$ is proper and polynomial and $\mathbf{a}^0\in \Omega_{\mathrm{proper}(\mathcal{P})}$, then $\mathcal{P}(\mathbf{a}^0,\gamma^0,t)$ parametrizes properly and polynomially $\mathcal{C}(F,\mathbf{a}^0)$.
\end{corollary}
We now analyze the normality (i.e. the surjectivity, see~\cite{myBook}) of the parametrization. We recall that any parametrization can be reparametrized surjectively (see~\cite[Theorem 6.26]{myBook}). This reparametrization requires, in our case, a new algebraic extension of $\mathbb{F}$ via a new algebraic element. Alternatively, one may reparametrize normally the specialized parametrizations. In the following we deal with the case where $\mathcal{P}$ is already normal and we want to preserve this property through the specializations.
For this purpose, we first introduce a new definition.
\begin{definition}
Let $\mathcal{P}$ be as in \ref{eq-Pa}. If $\mathcal{P}$ is normal, we define the set
\[ \Omega_{\mathrm{normal}(\mathcal{P})}:=\left\{
\begin{array}{cl}
\mathbb{S} & \text{if $\deg_t(p_1)>\deg_t(q_1)$ or $\deg_t(p_2)>\deg_t(q_2)$,} \\
\Omega_{\mathrm{gcd}(N_1,N_2)} & \text{if $\deg_t(p_1)\leq \deg_t(q_1)$ and $\deg_t(p_2)\leq \deg_t(q_2)$}
\end{array}
\right.
\]
where $(\alpha_1/\beta_1, \alpha_2/\beta_2)\in \mathbb{F}^2$ is the critical point of $\mathcal{P}$ (see~\cite[Def. 6.24]{myBook}) and, for $i\in \{1,2\}$, $N_i= \alpha_{i} q_i -\beta_{i} p_i$,
\end{definition}
\begin{corollary}
Let $\mathcal{P}$ be proper and normal. For $\mathbf{a}^0\in \Omega_{\mathrm{proper}(\mathcal{P})}\cap \Omega_{\mathrm{normal}(\mathcal{P})}$, $\mathcal{P}(\mathbf{a}^0,\gamma^0,t)$ parametrizes properly and normally $\mathcal{C}(F,\mathbf{a}^0)$.
\end{corollary}
\begin{proof}
In the proof of Theorem \ref{thrm:genus0general} we have seen that, for $\mathbf{a}^0\in \Omega_{\mathrm{proper}(\mathcal{P})}$, $\deg_t(p_i(\mathbf{a},\gamma,t))=\deg_t(p_i(\mathbf{a}^0,\gamma^0,t))$, similarly for $q_i$, and that the rational functions in $\mathcal{P}(\mathbf{a}^0,\gamma^0,t)$ are in reduced form.
Now, if $\deg_t(p_1)>\deg_t(q_1)$ or $\deg_t(p_2)>\deg_t(q_2)$, the result follows from Theorem \ref{thrm:genus0general} and~\cite[Theorem 6.22]{myBook}. If $\deg_t(p_1)\leq \deg_t(q_1)$ and $\deg_t(p_2)\leq \deg_t(q_2)$, since $\mathbf{a}^0\in \Omega_2$ in Def. \ref{def-omega}, we get that $$C:=\left(\frac{\alpha_1(\mathbf{a}^0,\gamma^0)}{\beta_1(\mathbf{a}^0,\gamma^0)}, \frac{\alpha_2(\mathbf{a}^0,\gamma^0)}{\beta_2(\mathbf{a}^0,\gamma^0)}\right)$$ is well defined and, by the above remark on the degrees, $C$ is the critical point of $\mathcal{P}(\mathbf{a}^0,\gamma^0,t)$. Let $$\tilde{N}_i:= \alpha_{i}(\mathbf{a}^0,\gamma^0) q_i(\mathbf{a}^0,\gamma^0,t) -\beta_{i}(\mathbf{a}^0,\gamma^0) p_i(\mathbf{a}^0,\gamma^0)=N_i(\mathbf{a}^0,\gamma^0,t).$$
Now, since $\mathbf{a}^0\in\Omega_{\mathrm{gcd}(N_1,N_2)}$, by Corollary \ref{cor-lemma-gcd}, one has that $\deg_t(\gcd(\tilde{N}_1,\tilde{N}_2))=\deg_t(\gcd(N_1,N_2))>0$; recall that $\mathcal{P}$ is normal. Now, the result follows from Theorem \ref{thrm:genus0general} and~\cite[Theorem 6.22]{myBook}.
\end{proof}
Let us illustrate these ideas in an example.
\begin{example}\label{ex-1}
Let us consider $\mathbb{K}=\mathbb{Q}$ and $\mathbb{F}=\mathbb{L}:=\mathbb{Q}(a_1,a_2)$. Let
\vspace*{1mm}
\noindent
{\Small
\begin{align*}
F(\mathbf{a},x,y) =\, &((a_1^5 + a_2^5 + 3a_1^2a_2^2 - a_1a_2)y^2 + (2a_1^3a_2 + (-9a_2 - 1)a_1^2 + 3a_1 - 6a_2^4 + a_2^3)y + a_2^2(a_1 + 9a_2 - 3))x^3 \\ &+ ((-3a_1^3a_2^2 - 6a_1^4 + 3a_1^2a_2 - 6a_1a_2^2)y^2 + 9((a_2 + 2/9)a_1^2 + (-(8a_2)/9 - 1)a_1 - a_2^3/9 + 2a_2 + 2/9)a_1y \\ &+ 3(a_1 - 2/3)a_2^2)x^2 - 3(((a_2 - 4)a_1 - 2a_2^2)a_1y + a_1^3/3 - 3a_1^2 + (6a_2 + 4/3)a_1 - (8a_2)/3)a_1xy \\ &+ ((a_1^2a_2 - 8)y - 3a_1^2 + 2a_1)a_1^2y
\end{align*}
}
\vspace*{1mm}
\noindent $\mathcal{C}(F)$ is a rational quintic that can be properly parametrized as
\begin{equation}\label{eq-exP} \mathcal{P}(\mathbf{a},t)= \left(\dfrac{t a_{1}+2}{t^{2} a_{2}+t +a_{1}},
\dfrac{t +3}{t^{3} a_{1}+a_{2}}\right).
\end{equation}
The field of parametrization is $\mathbb{L}$. We determine the open subset $\Omega_{\mathrm{proper}(\mathcal{P})}$ (see Def. \eqref{def-omega}). Let us deal with $\Omega_1$. Clearly $\Omega_{\mathrm{def}(F)}=\mathbb{C}^2$. The homogeneous component of $F$ of maximum degree is
\[ \left(a_{1}^{5}+a_{2}^{5}+3 a_{1}^{2} a_{2}^{2}-a_{1} a_{2}\right) x^{3} y^{2} \]
So,
\[ \Omega_{1}:= \mathbb{C}^2 \setminus \mathbb{V}(a_{1}^{5}+a_{2}^{5}+3 a_{1}^{2} a_{2}^{2}-a_{1} a_{2}). \]
One has that
\[\Omega_2:=\mathbb{C}^2\setminus \{(0,0)\}\]
Note that, if $\mathbf{a}^0\not\in \Omega_2$, the second component of $\mathcal{P}$ is not well-defined.
Let us deal with $\Omega_3$. The polynomials $G_i$, $G_{i}^{*}$, $R$ are
\[ \begin{array}{lcl} G_1&=& h^{2} t a_{1} a_{2}-h \,t^{2} a_{1} a_{2}+2 h^{2} a_{2}-h a_{1}^{2}-2 t^{2} a_{2}+t a_{1}^{2}+2 h -2 t \\
G_2& =& h^{3} t a_{1}-h \,t^{3} a_{1}+3 h^{3} a_{1}-3 t^{3} a_{1}-h a_{2}+t a_{2}\\
G&= & h-t \\
G_{1}^{*}&=& \left(h t a_{1}+2 h +2 t \right) a_{2}-a_{1}^{2}+2 \\
G_{2}^{*}&=& \left(\left(t +3\right) h^{2}+\left(t^{2}+3 t \right) h +3 t^{2}\right) a_{1}-a_{2}
\\
R&= & 3 h^{4} a_{1}^{3} a_{2}^{2}-2 h^{4} a_{1}^{2} a_{2}^{2}+h^{3} a_{1}^{4} a_{2}+6 h^{3} a_{1}^{2} a_{2}^{2}+3 h^{2} a_{1}^{4} a_{2}-h^{2} a_{1}^{2} a_{2}^{3}-2 h^{3} a_{1}^{2} a_{2}-2 h^{2} a_{1}^{3} a_{2}\\
&& +h a_{1}^{5}-6 h^{2} a_{1}^{2} a_{2}+12 h^{2} a_{1} a_{2}^{2}-6 h a_{1}^{3} a_{2}-4 h a_{1} a_{2}^{3}+3 a_{1}^{5}+4 h^{2} a_{1} a_{2}-4 h a_{1}^{3}+12 h a_{1} a_{2}\\&&-12 a_{1}^{3}-4 a_{2}^{3}+4 h a_{1}+12 a_{1}
\end{array}
\]
Moreover, $A_1=-h a_{1} a_{2}-2 a_{2},A_2=-h a_{1}-3 a_{1},B=-1$ (see Def. \eqref{def-gcd}).
Hence, $\Omega_1=\mathbb{C}^2$. Moreover, $\Omega_{\mathrm{nonZ}(A_1)}=\mathbb{C}^2\setminus \mathbb{V}(a_2)$, $\Omega_{\mathrm{nonZ}(A_2)}=\mathbb{C}^2\setminus \mathbb{V}(a_1)$ and $\Omega_{\mathrm{nonZ}(B)}=\mathbb{C}^2$.
On the other hand, $\Omega_{\mathrm{nonZ}(R)}$ can be expressed as
\[ \mathbb{C}^2 \setminus \{ (0,0), (\pm\sqrt{2},0)\}. \]
Therefore,
$$\Omega_3=\Omega_{\mathrm{gcd}(G_1,G_2)}=\mathbb{C}^2 \cap \left( \mathbb{C}^2 \setminus \mathbb{V}(a_1)\right)
\cap \left( \mathbb{C}^2 \setminus \mathbb{V}(a_2)\right) \cap \left( \mathbb{C}^2 \setminus \{ (0,0), (\pm\sqrt{2},0)\}\right)= \mathbb{C}^2 \setminus \mathbb{V}(a_1 a_2).$$
Finally, we deal with $\Omega_4$. We have
\[ \begin{array}{lcl}
p_1=t a_{1}+2 & & p_2=t +3 \\
q_1= t^{2} a_{2}+t +a_{1} & & q_2=t^{3} a_{1}+a_{2} \\
\gcd(p_1,q_1)=1 & &\gcd(p_2,q_2) =1 \\
\mathrm{res}_t(p_1,q_1)= a_{1}^{3}-2 a_{1}+4 a_{2} && \mathrm{res}_t(p_2,q_2)=a_{2}-27 a_{1}
\end{array}
\]
Therefore,
\begin{equation*}
\Omega_4=\mathbb{C}^2 \setminus \mathbb{V}(a_1 a_2 (a_{1}^{3}-2 a_{1}+4 a_{2})(a_{2}-27 a_{1})).
\end{equation*}
Summarizing (see Fig. \ref{fig-1}, left)
\begin{equation}\label{eq-omegaEx1} \Omega_{\mathrm{proper}(\mathcal{P})}=\mathbb{C}^2\setminus \mathbb{V}(a_1 a_2 (a_{1}^{5}+a_{2}^{5}+3 a_{1}^{2} a_{2}^{2}-a_{1} a_{2})(a_{1}^{3}-2 a_{1}+4 a_{2})(a_{2}-27 a_{1})).\
\end{equation}
\begin{center}
\begin{figure}[h]
\includegraphics[width=6cm]{plot1.pdf} \includegraphics[width=7cm]{plot2.jpg} \caption{Left: Plot of the real part of the closed set defining $\Omega_{\mathrm{proper}(\mathcal{P})}$ in Example \ref{ex-1}. Right: Plot of the real part of the closed set defining $\Omega_{\mathrm{genus}(F)}$ in Example \ref{ex-cubic}}
\label{fig-1}
\end{figure}
\end{center}
\end{example}
\section{Decomposition of $\mathbb{S}$}\label{sec:decomposition}
The goal in this section is to provide an algorithm decomposing the space $\mathbb{S}$ so that in each subset of the decomposition we can give information on the genus of the corresponding specialized curve.
Let $F$ be as in \eqref{eq-F}, irreducible over $\overline{\mathbb{F}}$. We first compute the genus of $\mathcal{C}(F^h)$. Let $\mathfrak{g}:=\mathrm{genus}(\mathcal{C}(F^h))$. Furthermore, if $\mathfrak{g}=0$, let $\mathcal{P}(\mathbf{a},\gamma,t)$ be, as in \eqref{eq-Pa}, a proper parametrization of $\mathcal{C}(F^h)$. We consider the open subset
\begin{equation}\label{eq-Sigma1}
\Sigma:=\left\{ \begin{array}{cl}
\Omega_{\mathrm{genus}(F)} & \text{if $\mathfrak{g}>0$ (see Def. \eqref{def-genus})} \\
\noalign{\vspace*{1mm}}
\Omega_{\mathrm{proper}(\mathcal{P})} & \text{if $\mathfrak{g}=0$ (see Def. \eqref{def-omega})}
\end{array}\right.
\end{equation}
At this level of the process we know that (see Theorems \eqref{thm:genusPreservation} and \eqref{thrm:genus0general})
\begin{enumerate}
\item If $\mathfrak{g}>0$, then for $\mathbf{a}^0\in \Sigma$ it holds that $\mathcal{C}(F,\mathbf{a}^0)$ is either reducible or its genus is $\mathfrak{g}$.
\item If $\mathfrak{g}=0$, then for $\mathbf{a}^0\in \Sigma$ it holds that $\mathcal{C}(F,\mathbf{a}^0)$ is rational and $\mathcal{P}(\mathbf{a}^0,\gamma^0,t)$ parametrizes properly $\mathcal{C}(F,\mathbf{a}^0)$.
\end{enumerate}
In the following, we analyze the specializations when working in the closed set
\begin{equation}\label{eq-Z}
\mathcal{Z}:=\mathbb{S}\setminus \Sigma.
\end{equation} First, let us discuss the computational issues that may appear.
Let $\mathcal{A}\subset \mathbb{K}[\mathbf{a}]$ be a set of generators of $\mathcal{Z}$, and let $\mathrm{I}$ be the ideal generated by $\mathcal{A}$ in $\mathbb{K}[\mathbf{a}]$.
We consider the prime decomposition of $\mathrm{I}$
$$\mathrm{I}=\bigcup_{j=1}^{\ell} \mathrm{I}_j.$$
Now, for each prime ideal $\mathrm{J}\in \{\mathrm{I}_1,\ldots,\mathrm{I}_\ell\}
$ we consider the quotient field of $\mathbb{K}[\mathbf{a}]/\mathrm{J}$; we denote it by $\mathbb{L}_{\mathrm{J}}$. Elements in $\mathbb{L}_{\mathrm{J}}$ are quotients of equivalence classes of $\overline{\mathbb{K}}[\mathbf{a}]/\mathrm{J}$. We will assume that elements in $\overline{\mathbb{K}}[\mathbf{a}]/\mathrm{J}$ are always expressed by means of a canonical representative of the class in the following sense. We fix a Gr\"obner basis $\mathcal{G}$ of $\mathrm{J}$ w.r.t. some fixed order. Then, the elements in $\overline{\mathbb{K}}[\mathbf{a}]/\mathrm{J}$ are uniquely represented by their normal form w.r.t. $\mathcal{G}$ (see e.g. Prop. 1 and Ex. 13, Chap. 2, Sect. 6. in \cite{cox}) and, hence, elements in $\mathbb{L}_{\mathrm{J}}$ are represented as the quotient of the canonical representatives of their numerators and denominators. So, by {abuse of notation}, we will identify, via the canonical representation, the elements in $\mathbb{L}_{\mathrm{J}}$ with elements in $\overline{\mathbb{K}}(\mathbf{a})$. In addition, we consider an algebraic element $\gamma_\mathrm{J}$ over $\mathbb{L}_{\mathrm{J}}$ and we denote by $\mathbb{F}_\mathrm{J}$ the field $\mathbb{F}_{\mathrm{J}}:=\mathbb{L}_{\mathrm{J}}(\gamma_{\mathrm{J}})$.
We observe that $\mathbb{F}_{\mathrm{J}}$ is a computable field with a polynomial factorization algorithm available; zero test and basic arithmetic (addition, multiplication and inverse computation) can be carried out e.g. by taking the normal forms w.r.t. a Gr\"obner basis of $\mathrm{J}$. For the polynomial factorization we refer to (see Section 10.2 and Appendix B in \cite{wang2}, see also \cite{wang1}). As a particular case, as in Example \ref{ex-1} and \ref{ex-2}, if $\mathbb{V}(\mathrm{J})$ is a rational variety, one may work over $\mathbb{K}(\mathcal{Q}(\lambda_1,\ldots,\lambda_m))$ instead of $\mathbb{F}_{\mathrm{J}}$, where $\mathcal{Q}(\lambda_1,\ldots,\lambda_m)$ is a parametrization of $\mathbb{V}(\mathrm{J})$.
Concerning specializations, instead of working in $\mathbb{S}$ (see \eqref{eq-S}), we take the parameter values in the irreducible variety $\mathbb{V}(\mathrm{J})$. Then, for $\mathbf{a}^0\in \mathbb{V}(\mathrm{J})\subset \mathbb{S}$, and $f\in\overline{\mathbb{K}}[\mathbf{a}]/\mathrm{J}$, we denote by $f(\mathbf{a}^0)$ the specialization at $\mathbf{a}^0$ of the equivalence class of $f$; note that since $\mathbf{a}^0\in \mathbb{V}(\mathrm{J})$ the specialization does not depend on the representative. Similarly, if $f:=p/q\in\mathbb{F}_{\mathrm{J}}$ and $q(\mathbf{a}^0)\neq 0$, then $f(\mathbf{a}^0):=p(\mathbf{a}^0)/q(\mathbf{a}^0)\in \overline{\mathbb{K}}$.
In this situation, for each prime ideal $\mathrm{J}\in \{\mathrm{I}_1,\ldots,\mathrm{I}_\ell\}$ we consider the polynomial $F$ in \eqref{eq-F} as a polynomial in $\mathbb{F}_{\mathrm{J}}[x,y]$. To emphasize this fact, we write $F_\mathrm{J}$. First we check the irreducibility of $F_\mathrm{J}$ over the algebraic closure of $\mathbb{F}_\mathrm{J}$. If $F_\mathrm{J}$ is reducible we can either stop the decomposition over this closed subset, and claim that the specialization over $\mathbb{V}(\mathrm{J})$ is reducible, or continue the process with each irreducible factor of $F_\mathrm{J}$. For irreducible $F_\mathrm{J}$, the process continues, as in the initial step, by computing the genus of $\mathcal{C}(F_\mathrm{J})$. Since in each iteration of the process the dimension of the variety $\mathbb{V}(\mathrm{J})$ decreases, we, at the end, reach the zero--dimensional case, and the decomposition ends.
Let us say that a specialization \textit{degenerates} if either $F(\mathbf{a}^0,\gamma^0,x,y)$ is not well--defined or $F(\mathbf{a}^0,\gamma^0,x,y)\in \overline{\mathbb{K}}$.
As a result of the process described above, we find a disjoint decomposition
\begin{equation}\label{eq-decomp}
\mathbb{S}=\dot{\bigcup}_{i\in I} \mathbb{S}_i
\end{equation}
such that, for every specialization $\mathbf{a}^0\in \mathbb{S}_i$, one of the following holds
\begin{enumerate}
\item the specialization degenerates;
\item the genus is positive and preserved, or the specialized curve is reducible;
\item the genus is zero and a proper parametrization of $\mathcal{C}(F,\mathbf{a}^0)$ is provided.
\end{enumerate}
\begin{remark}\label{rem:decomposition}
Let us remark that in the decomposition~\eqref{eq-decomp} we can take the union of those $\mathbb{S}_i$ corresponding to each of the three items above; say $\mathbb{S}_1, \mathbb{S}_2, \mathbb{S}_3$ representing the corresponding item.
In this way, we can achieve a unique decomposition of the parameter space $\mathbb{S}$.
The $\mathbb{S}_i$ obtained in this way are constructible sets of $\mathbb{S}$, and $\mathbb{S}_2,\mathbb{S}_3$ are a finite union of subsets $\Sigma$ as in~\eqref{eq-Sigma1} and $\mathbb{S}_1$ is a closed subset directly defined from the implicit equation $F$.
Moreover, $\mathbb{S}_3$ can further be decomposed into a finite union of subsets $\mathbb{S}_{3,j}$ such that for every $j$, there is a proper parametrization $\mathcal{P}_j$ which is well--defined for every $\mathbf{a}^0 \in \mathbb{S}_{3,j}$ and specialized properly.
Finally, since $\Sigma$ of $F$ as in~\eqref{eq-Sigma1} is open on non-empty, depending on the genus of $F$, either $\mathbb{S}_2$ or $\mathbb{S}_3$ is a dense subset of $\mathbb{S}$.
\end{remark}
We illustrate the previous ideas by continuing the analysis of Example \ref{ex-1}.
\begin{example}\label{ex-2} (Continuation of Example \ref{ex-1})
Taking into account \eqref{eq-omegaEx1}, the closed set $\mathcal{Z}$ (see \eqref{eq-Z}) decomposes as
\[ \mathcal{Z}= \mathbb{V}(a_1) \cup \mathbb{V}(a_2) \cup \mathbb{V}(a_{2}-27 a_{1}) \cup \mathbb{V}(a_{1}^{3}-2 a_{1}+4 a_{2})\cup \mathbb{V}(a_{1}^{5}+a_{2}^{5}+3 a_{1}^{2} a_{2}^{2}-a_{1} a_{2})\subset \overline{\mathbb{Q}}^2.
\]
We start with $\mathrm{J}_1:= <\!\!a_1\!\!>$ and $\mathbb{V}_1:=\mathbb{V}(\mathrm{J}_1)$. Since $\mathbb{V}_1$ is rational, surjectively parametrized by $\mathcal{Q}_1:=(0,\lambda)$, we work over the field $\mathbb{Q}(\lambda)[x,y]$. We have that
\[ F_{\mathrm{J}_1}:=\lambda^{2} x^{2} \left(\lambda^{3} x \,y^{2}-6 \lambda^{2} x y +\lambda x y +9 \lambda x -3 x -2\right)
\]
and therefore all specializations in $\mathbb{V}(a_1)$ lead to a reducible curve. Additionally, one may distinguish the cases $\lambda=0$, that corresponds to the point $(0,0)$, where the specialization degenerates, and $\lambda\neq 0$ where $\mathcal{C}(F,\mathbf{a}^0)$ decomposes to the union of a double line and a rational cubic.
The analysis for $\mathrm{J}_2:=<\!\!a_2\!\!>$, and $\mathbb{V}_2:=\mathbb{V}(\mathrm{J}_2)$ looks similar. Since $\mathbb{V}_2$ is rational, parametrized by $\mathcal{Q}_2:=(\lambda,0)$, we work over the field
$\mathbb{Q}(\lambda)[x,y]$. We have that
\vspace*{2mm}
\noindent
{\small
\begin{align*}
F_{\mathrm{J}_2} := \, &\lambda y (\lambda^{4} x^{3} y -6 \lambda^{3} x^{2} y -\lambda^{3} x +2 \lambda^{2} x^{2}+12 \lambda^{2} x y -\lambda \,x^{3}-3 \lambda^{3}+9 \lambda^{2} x -9 \lambda \,x^{2}+3 x^{3}+2 \lambda^{2}-4 \lambda x -8 \lambda y +2 x^{2}).
\end{align*}
}
\vspace*{2mm}
\noindent Thus, all specializations in $\mathbb{V}_2$ lead to a reducible curve; note that $\mathcal{Q}_2$ is surjective. The case $\lambda=0$ is covered above, and for $\lambda\neq 0$, the specialization $\mathcal{C}(F,\mathbf{a}^0)$ decomposes to the union of a line and a rational quartic.
Let us study $\mathrm{J}_3:=<\!\! a_{2}-27 a_{1} \!\!>$ and $\mathbb{V}_3:=\mathbb{V}(\mathrm{J}_3)$. Again, $\mathbb{V}_3$ is rational, parametrized by $\mathcal{Q}_3:=(\lambda,27\lambda)$, and we work over the field
$\mathbb{Q}(\lambda)[x,y]$. We have that
\vspace*{2mm}
\noindent
{\small
\begin{align*}
F_{\mathrm{J}_3} := \, &\lambda (244 \lambda x +3 \lambda -3 x -2 ) (58807 \lambda^{3} x^{2} y^{2}-732 \lambda^{3} x \,y^{2}+732 \lambda^{2} x^{2} y^{2}+9 \lambda^{3} y^{2}-13068 \lambda^{2} x^{2} y \\ &+464 \lambda^{2} x \,y^{2}+9 \lambda \,x^{2} y^{2}+81 \lambda^{2} x y +6 \lambda^{2} y^{2}-81 \lambda \,x^{2} y -6 \lambda x \,y^{2}-\lambda^{2} y +729 \lambda \,x^{2}-106 \lambda x y +4 \lambda \,y^{2}-x^{2} y ).
\end{align*}
}
\vspace*{2mm}
\noindent The analysis of $\mathbb{V}_3$ is identical to $\mathbb{V}_2$.
Let us study $\mathrm{J}_4:=<\!\! a_{1}^{3}-2 a_{1}+4 a_{2} \!\!>$ and $\mathbb{V}_4:=\mathbb{V}(\mathrm{J_4})$ that is again rational and it is properly and surjectively parametrized by $\mathcal{Q}_4:=(\lambda, -\frac{1}{4} \lambda^{3}+\frac{1}{2} \lambda)$. We work over the field $\mathbb{Q}(\lambda)[x,y]$. We have that
\vspace*{2mm}
\noindent
{\small
\begin{align*}
F_{\mathrm{J}_4} := \, & \lambda (\lambda^{5} y -2 \lambda^{3} y +12 \lambda^{2}-8 \lambda +32 y) (\lambda^{9} x^{3} y -8 \lambda^{7} x^{3} y +12 \lambda^{6} x^{3}+24 \lambda^{5} x^{3} y +8 \lambda^{5} x^{3}-32 \lambda^{4} x^{3} y \\ &-72 \lambda^{4} x^{3}-32 \lambda^{3} x^{3} y -16 \lambda^{4} x^{2}-32 \lambda^{3} x^{3}+192 \lambda^{3} x^{2} y +144 \lambda^{2} x^{3}+16 \lambda \,x^{3} y +64 \lambda^{2} x^{2}-384 \lambda^{2} x y \\ &+32 \lambda \,x^{3}-96 x^{3}+256 \lambda y -64 x^{2} ).
\end{align*}
}
\vspace*{2mm}
\noindent Again $\mathbb{V}_4$ behaves as $\mathbb{V}_2$ and $\mathbb{V}_3$.
Finally, let us analyze $\mathrm{J}_5:=<\!\! a_{1}^{5}+a_{2}^{5}+3 a_{1}^{2} a_{2}^{2}-a_{1} a_{2}\!\!>$ and $\mathbb{V}_5:=\mathbb{V}(\mathrm{J}_5)$ which is a rational quintic and properly and surjectively parametrized as
{\small
\[ \mathcal{Q}_5(\lambda):=\left({\frac { \left( 5\,\lambda-1 \right) \left( 5\,\lambda-2 \right) ^{4
}}{3125\,{\lambda}^{4}-3750\,{\lambda}^{3}+1750\,{\lambda}^{2}-375\,
\lambda+31}},-{\frac { \left( 5\,\lambda-2 \right) \left( 5\,\lambda-
1 \right) ^{4}}{3125\,{\lambda}^{4}-3750\,{\lambda}^{3}+1750\,{\lambda
}^{2}-375\,\lambda+31}}
\right). \]
}
Those values for which the parametrization is not defined, i.e. the poles, play no role in this analysis.
The polynomial $F_{\mathrm{J}_5}$ is $F_{\mathrm{J}_5}(\lambda,x,y)=F(\mathcal{Q}_5(\lambda),x,y)\in \mathbb{Q}(\lambda)[x,y]$.
It holds that $\deg(\mathcal{C}(F_{\mathrm{J}_5}))=4$, $\mathrm{genus}(\mathcal{C}(F_{\mathrm{J}_5}))=0$ and a proper surjective parametrization is $\mathcal{P}_{\mathrm{J}_5}(\lambda,t):=\mathcal{P}(\mathcal{Q}(\lambda),t)$ (see \eqref{eq-exP}). So, we get (see Def. \ref{def-omega})
\[ \Omega_{\mathrm{proper}(\mathcal{P}_{\mathrm{J}_5})}:=\mathbb{V}_5 \setminus \{ \mathcal{P}_{\mathrm{J}_5}(\lambda_0)\,|\, f(\lambda_0)=0\} \]
where $f:=p_1\, p_2\,p_3\, p_4$ and
\[ \begin{array}{ll}p_1:=& 5\,\lambda-1,\,\,\,
p_2:= 5\,
\lambda-2,\,\,\,
p_3:= 25\,{\lambda}^{2}-15\,\lambda+3,
\\
p_4:= & \text{\small{$
390625\,{\lambda}^{8}-937500\,{\lambda}^{7}+1343750\,{\lambda}
^{6}-1237500\,{\lambda}^{5}+711875\,{\lambda}^{4}-253500\,{\lambda}^{3
}$}}\\ &\text{\small{$+54475\,{\lambda}^{2}-6495\,\lambda+331.$}} \end{array}
\]
By Theorem \ref{thrm:genus0general}, for every $\mathbf{a}^0\in \Omega_{\mathrm{proper}(\mathcal{P}_{\mathrm{J}_5})}$ it holds that $\mathcal{C}(F,\mathbf{a}^0)$ is rationally parametrized by $\mathcal{P}(\mathbf{a}^0,t)$ (see \eqref{eq-exP}) or, equivalently, by $\mathcal{P}_{\mathrm{J}_5}(\mathcal{Q}^{-1}(\mathbf{a}^0),t)$. Now, let us analyze the curve in $\mathcal{Z}_5:= \mathbb{V}_5\setminus \Omega_{\mathrm{proper}(\mathcal{P}_{\mathrm{J}_5})}=\{ \mathcal{P}_{\mathrm{J}_5}(\lambda_0)\,|\, f(\lambda_0)=0\}$ (see \eqref{eq-Z}).
We define $\mathbf{a}^0_1:=(0,0)=\mathcal{P}_{\mathrm{J}_5}(\lambda_0)$, where $\lambda_{0}$ is a root of $p_1 \,p_2$; $\mathbf{a}^0_2:=(-1,-1)=\mathcal{P}_{\mathrm{J}_5}(\lambda_0)$, where $\lambda_{0}$ is a root of $p_3$; and observe that $p_4$ generates 8 points on the curve that we denote by $\mathbf{a}^0_{i}, i\in \{3,\ldots,10\}$ and which correspond to $\mathcal{P}_{\mathrm{J}_5}(\lambda_0)$ where $\lambda_{0}$ is one of the roots of $p_4$. Thus, $\mathcal{Z}=\{\mathbf{a}^0_1,\ldots,\mathbf{a}^0_{10}\}$, $\mathcal{C}(F,\mathbf{a}^0_1)=\mathbb{C}^2$, and, for $i\in \{2,\ldots,10\}$, $\mathcal{C}(F,\mathbf{a}^0_i)$ are rational cubics parametrized by $\mathcal{P}_{\mathrm{J}_5}(\mathbf{a}^0_i,t)$.
Summarizing, $\mathbb{S}$ decomposes as (see \eqref{eq-decomp} and Remark~\ref{rem:decomposition})
\begin{align*}
\mathbb{S} = &\left(\mathbb{S}_1:=\{(0,0)\}\right) \cup \left(\mathbb{S}_2:=\cup_{i=1}^{4}\mathbb{V}_i \setminus \{(0,0)\} \right) \\ & \cup \left(\mathbb{S}_{3,1}:=\Omega_{\mathrm{proper}(\mathcal{P})}\right) \cup \left(\mathbb{S}_{3,2}:=\Omega_{\mathrm{proper}(\mathcal{P}_{\mathrm{J}_5})}\right) \cup \left(\mathbb{S}_{3,3}:=\{\mathbf{a}^0_2,\ldots,\mathbf{a}^0_{10}\}\right).
\end{align*}
For $\mathbf{a}^0 \in \mathbb{S}_1$, $\mathcal{C}(F,\mathbf{a}^0)$ degenerates.
For $\mathbf{a}^0 \in \mathbb{S}_2$, $\mathcal{C}(F,\mathbf{a}^0)$ is reducible (note that $\mathbf{a}^0_1\in \mathbb{S}_2$).
For $\mathbf{a}^0\in \mathbb{S}_{3,1}$, the specialized curve $\mathcal{C}(F,\mathbf{a}^0)$ is a quintic parametrized by $\mathcal{P}(\mathbf{a}^0,t)$; for $\mathbf{a}^0 \in \mathbb{S}_{3,2}$, $\mathcal{C}(F,\mathbf{a}^0)$ is a quartic parametrized by $\mathcal{P}_{\mathrm{J}_5}(\mathbf{a}^0,t)$; and for $\mathbf{a}^0 \in \mathbb{S}_{3,3}$, $\mathcal{C}(F,\mathbf{a}^0)$ is a cubic parametrized by $\mathcal{P}_{\mathrm{J}_5}(\mathbf{a}^0,t)$.
\end{example}
\begin{example}\label{ex-cubic}
Let $\mathbb{K}=\mathbb{Q}$ and $\mathbb{F}=\mathbb{L}=\mathbb{Q}(a_1,a_2,a_3)$. We consider
\[ F={x}^{3}+{x}^{2}a_{{1}}+{y}^{3}+a_{{2}}a_{{3}}\in \mathbb{F}[x,y]. \]
One has that $\mathrm{genus}(\mathcal{C}(F))=1$. Using the ideas in Section \ref{sec-genus}, we compute $\Omega_{\mathrm{genus}(F)}$ (see Def. \ref{def-genus}). We observe that since $\mathcal{C}(F)$ is an elliptic cubic, no blowup is required and, hence, $\Omega_{\mathrm{genus}(F)}=
\Omega_{\mathrm{genusOrd}(F)}$. Indeed, one gets that (see Fig. \ref{fig-1}, right)
\[ \Omega_{\mathrm{genus}(F)}:=\mathbb{C}^3 \setminus \mathbb{V}\left(-a_{{2}}a_{{3}} \left( -4\,{a_{{1}}}^{3}-27\,a_{{2}}a_{{3}} \right)\right).\]
Thus, by Theorem \ref{theorem:genus-strong}, for every $\mathbf{a}^0\in \Omega_{\mathrm{genus}(F)}$, $\mathcal{C}(F,\mathbf{a}^0)$ is either reducible or it is a genus 1 cubic curve. Let us analyze the specializations in $\mathcal{Z}:=\mathbb{C}^3\setminus \Omega_{\mathrm{genus}(F)}=\mathbb{V}\left(-a_{{2}}a_{{3}} \left( -4\,{a_{{1}}}^{3}-27\,a_{{2}}a_{{3}} \right)\right)$.
Let $\mathrm{J}_1:=<\!\!a_2\!\!>$ and $\mathbb{V}_1:=\mathbb{V}(\mathrm{J}_1)$. Then, $F_{\mathrm{J}_1}:={x}^{3}+a_{{1}}{x}^{2}+{y}^{3}$. We observe that $\mathrm{genus}(\mathcal{C}(F_{\mathrm{J}_1}))=0$ and
\[\mathcal{P}_{\mathrm{J}_1}:=\left(-{\frac {{t}^{3}a_{{1}}}{{t}^{3}+1}},-{\frac {{t}^{2}a_{{1}}}{{t}^{3}
+1}}
\right) \]
is a proper parametrization of $\mathcal{C}(F_{\mathrm{J}_1})$. Applying Def. \ref{def-omega} to $F_{\mathrm{J}_1}$ and $\mathcal{P}_{\mathrm{J}_1}$ we get that $\Omega_{\mathrm{proper}(\mathcal{P}_{\mathrm{J}_1})}:=\mathbb{V}_1\setminus \mathbb{V}(a_1)$. Therefore, by Theorem \ref{thrm:genus0general}, for all $\mathbf{a}^0\in \Omega_{\mathrm{proper}(\mathcal{P}_{\mathrm{J}_1})}$ it holds that $\mathcal{C}(F,\mathbf{a}^0)$ is a rational curve parametrized by $\mathcal{P}_{\mathrm{J}_1}(\mathbf{a}^0,t)$. However, for the remaining case, namely the points $(0,0,\mu)$ for $\mu\in\mathbb{C}$, $\mathcal{C}(F,(0,0,\mu))$ decomposes as the product of three lines.
Let $\mathrm{J}_2:=<\!\!a_3\!\!>$ and $\mathbb{V}_2:=\mathbb{V}(\mathrm{J}_2)$. Now, the situation is identical to the previous case. Let $\mathrm{J}_3:=<\!\!-4\,{a_{{1}}}^{3}-27\,a_{{2}}a_{{3}}\!\!>$ and $\mathbb{V}_3:=\mathbb{V}(\mathrm{J}_3)$. The surface $\mathbb{V}_3$ can be properly parametrized as
\[ \mathcal{Q}(\lambda_1,\lambda_2)=\left(\lambda_{{1}},\lambda_{{2}},-{\frac {4\,{\lambda_{{1}}}^{3}}{27\,\lambda_{{2}}}}\right). \]
We observe that $\mathcal{Q}$ is not surjective. Indeed, $\mathcal{Q}(\mathbb{C}^2)=\mathbb{V}_3\setminus \{(0,0,\mu)\,|\, \mu\in \mathbb{C}\}$ (see e.g. Remark 3 in \cite{SSV}).
However, the specializations in $\{(0,0,\mu)\,|\, \mu\in \mathbb{C}\}$ have already been analyzed.
So, we treat the case $\mathcal{Q}(\mathbb{C}^2)$. Then, $F_{\mathrm{J}_3}$ can be taken as
$$F_{\mathrm{J}_3}:=F(\mathcal{Q}(\lambda_1,\lambda_2),t)= {x}^{3}+{x}^{2}\lambda_{{1}}+{y}^{3}-{\frac {4\,{\lambda_{{1}}}^{3}}{27}},$$ where $(\lambda_1,\lambda_2)\in \mathbb{C}^2 \setminus \{(0,0)\}$. It holds that $\mathrm{genus}(\mathcal{C}(F_{\mathrm{J}_3}))=0$ and
\[ \mathcal{P}_{\mathrm{J}_3}:=\left({\frac {{t}^{3}\lambda_{{1}}-2\,\lambda_{{1}}}{3\,{t}^{3}+3}},{\frac
{{t}^{2}\lambda_{{1}}}{{t}^{3}+1}}
\right) \]
is a proper parametrization.
Applying Def. \ref{def-omega} to $F_{\mathrm{J}_3}$ and $\mathcal{P}_{\mathrm{J}_3}$, we get that $\Omega_{\mathrm{proper}(\mathcal{P}_{\mathrm{J}_3})}=\mathbb{V}_3\setminus \{(0,0,\mu)\,|\, \mu\in \mathbb{C}\}$. Therefore, by Theorem \ref{thrm:genus0general}, for all $\mathbf{a}^0\in \Omega_{\mathrm{proper}(\mathcal{P}_{\mathrm{J}_3})}$ it holds that $\mathcal{C}(F,\mathbf{a}^0)$ is a rational curve parametrized by $\mathcal{P}_{\mathrm{J}_3}(\mathbf{a}^0,t)$.
Summarizing, $\mathbb{S}$ decomposes as (see \eqref{eq-decomp})
\[ \mathbb{S}=\left(\mathbb{S}_2:=\Omega_{\mathrm{genus}(F)} \cup \{(0,0,\mu)\,|\, \mu\in \mathbb{C}\}\right) \cup \left(\mathbb{S}_{3}:=\Omega_{\mathrm{proper}(\mathcal{P}_{\mathrm{J}_1})}\right). \]
For $\mathbf{a}^0\in \mathbb{S}_2, \mathcal{C}(F,\mathbf{a}^0)$ is either reducible or an elliptic curve; and for $\mathbf{a}^0\in \mathbb{S}_3, \mathcal{C}(F,\mathbf{a}^0)$ is a rational cubic parametrized by $\mathcal{P}_{\mathrm{J}_3}(\mathbf{a}^0,t)$.
\end{example}
\section{Some Illustrating Applications}\label{sec-application}
In this section, we illustrate by examples some possible applications of the theory developed in the paper.
In the first example, given a surface, we consider the problem of determining its rational level curves, if any.
\begin{example}\label{ex-app1} \textsf{Level curves of a surface.} \\
Let $\mathcal{S}$ be the surface defined over $\mathbb{C}$ by the polynomial
\[ F=x^{6}-5 x^{4} y +3 x^{4} z -y^{5}+2 y^{4} z -y^{3} z^{2}-x^{3} z +5 x^{2} y^{2}-7 x^{2} y z +3 x^{2} z^{2}+y^{2} z -2 y z^{2}+z^{3}-x^{2} \]
where $F \in \mathbb{Q}(z)[x,y]$. So, with the terminology of the paper, $\mathbf{a}=z$, $\mathbb{K}=\mathbb{Q}$ and $\mathbb{L}=\mathbb{F}=\mathbb{Q}(z)$. With this interpretation, the idea is to analyze the genus of $\mathcal{C}(F)\subset \overline{\mathbb{Q}(z)}^2$ under specializations in $\mathbb{S}:=\mathbb{C}$. For this purpose, we first compute a standard decomposition of the singular locus as in~\eqref{eq-SingularLocusF}. Following the steps described in Subsection~\ref{subsec-genus}, we get
\[ \mathscr{F}(F^h)=\left\{\left(0:z: 1\right) \right\}_{m_1=t}. \]
Moreover, one can easily check that the family consists in one ordinary double point. Therefore, one gets that $\mathrm{genus}(\mathcal{C}(F))=9$. Since the singularities are all ordinary, we get that (see Def. \ref{def-genus}) $\Omega_{\mathrm{genus}(F)}=\Omega_{\mathrm{genusOrd}(F)}.$
Moreover (see Def. \ref{def:genusOrd}),
\[ \begin{array}{ll} \Omega_{\mathrm{genusOrd}(F)}= &\mathbb{C}\setminus \left\{ -1, 0, 1\right\}. \end{array}\]
Using Theorem \ref{thm:genusPreservation}, for $z_0\in \Omega_{\mathrm{genusOrd}(F)}$, $\mathcal{C}(F,(x,y,z_0))$ is either reducible or its genus is 9. In any case, no rational level curve appears. For the elements in $\mathcal{Z}:=\mathbb{C} \setminus \Omega_{\mathrm{genusOrd}(F)}$, we get that $\mathcal{C}(F,(x,y,\pm 1))$ are irreducible of genus 7 and $\mathcal{C}(F,(x,y,0))$ is irreducible of genus 0. Indeed, $\mathcal{C}(F,(x,y,0))$ can be parametrized by $(t^{5}, t^{6}-t^{2})$.
\end{example}
In the second example, we consider the linear homotopy deformation of two curves and we analyze the genus of each instance curve.
\begin{example}\label{ex-app2} \textsf{Linear homotopy deformation of curves.} \\
Let us consider the linear homotopy between the Fermat cubic curve and the unit circle. That is we consider the polynomial
\[ F=\left(1-\lambda \right) \left(x^{3}+y^{3}-1\right)+\lambda \left(x^{2}+y^{2}-1\right). \]
We consider $F$ as a polynomial in $\mathbb{Q}(\lambda)[x,y]$ and we analyze the genus behavior through the deformation.
So, with the terminology of the paper, $\mathbf{a}=\lambda$, $\mathbb{K}=\mathbb{Q}$ and $\mathbb{L}=\mathbb{F}=\mathbb{Q}(\lambda)$. Now, the idea is to study the genus of $\mathcal{C}(F)$ under specializations in $\mathbb{S}:=\mathbb{C}$; or, in particular, in the real interval $[0,1]$. For this purpose, we first observe that $\mathrm{genus}(\mathcal{C}(F))=1$ and hence (see Def. \ref{def-genus}), $\Omega_{\mathrm{genus}(F)}=\Omega_{\mathrm{genusOrd}(F)}.$
We get that $\Omega_{\mathrm{genusOrd}(F)}=\mathbb{C}\setminus \mathbb{V}(g)$
where
\noindent {\small
\begin{align*}
g(\lambda) = &-\left(2 \lambda^{3}-5 \lambda^{2}+7 \lambda -3\right) \left(\lambda^{6}-4 \lambda^{5}+15 \lambda^{4}-29 \lambda^{3}+43 \lambda^{2}-33 \lambda +9\right) \left(4 \lambda -3\right) \left(-1+\lambda \right) \left(\lambda -3\right) \\ & \left(2 \lambda^{9}+27 \lambda^{8}+5049 \lambda^{7}-40068 \lambda^{6}+148716 \lambda^{5}-315657 \lambda^{4}+398763 \lambda^{3}-295245 \lambda^{2}+118098 \lambda -19683\right) \\ &\left(8 \lambda^{3}-27 \lambda^{2}+54 \lambda -27\right).
\end{align*}
}
\vspace*{1mm}
Using Theorem \ref{thm:genusPreservation}, for $\lambda_0\in \Omega_{\mathrm{genusOrd}(F)}$, $\mathcal{C}(F,(x,y,\lambda_0))$ is either reducible or its genus is 1. In any case, no rational deformation instance appears. For the elements in $\mathcal{Z}=\mathbb{C}\setminus \Omega_{\mathrm{genusOrd}(F)}$, we get that $\mathcal{C}(F,(x,y,1))$ is rational, the specialized cubics $\mathcal{C}(F,(x,y,3/4))$ and $\mathcal{C}(F,(x,y,3))$ factor as a union of a line and a conic, and for all the other cases the genus remains one (see Fig~\ref{fig-2}).
\begin{center}
\begin{figure}[h]
\includegraphics[width=5cm]{plot3.jpg} \includegraphics[width=5cm]{plot4.jpg} \caption{Left: Plot of the real part of different instances of the deformation in Example~\ref{ex-app2}. Right: Plot of the real part of $\mathcal{C}(F,(x,y,3/4))$ and $\mathcal{C}(F,(x,y,3))$ in Example~\ref{ex-app2}.}
\label{fig-2}
\end{figure}
\end{center}
\end{example}
Let $\mathcal{O}$ be a connected open subset of $\mathbb{C}$ and let $\mathrm{Mer}(\mathcal{O})$ be the field of meromorphic functions in $\mathcal{O}$ (see \cite{meromorfas}).
We consider a polynomial equation of the form
\begin{equation}\label{eq-fun}
\sum_{i,j\in {I}} f_{i,j}(t) x^i y^j=0
\end{equation}
where $I$ is a finite subset of $\mathbb{N}^2$, and where $f_{i,j}\in \mathrm{Mer}(\mathcal{O})$. Let $\mathbf{f}$ be the tuple with all the functions $f_{i,j}$ appearing in \eqref{eq-fun}. The question now is to decide, and indeed compute, whether there exists rational solutions of the equation; that is $p,q\in \overline{\mathbb{C}(\mathbf{f})}$ such that $\sum_{i,j\in {I}} f_{i,j}(t) p^i q^j=0$.
We may proceed as follows. We consider the polynomial $F(\mathbf{a},x,y)$ resulting from the formal replacement in \eqref{eq-fun} of each function $f_{i,j}$ by a parameter $a_{k}$. Now, with the terminology of the paper, we take $\mathbb{K}=\mathbb{C}(\mathbf{f})$ and $\mathbb{L}=\mathbb{F}=\mathbb{K}(\mathbf{a})$.
Then, we decompose $\mathbb{S}$ (see \eqref{eq-S}) as described in Section \ref{sec:decomposition}; note that the computations can be carried out over $\mathbb{C}(\mathbf{a})$ instead of over $\mathbb{F}$. Then, if any subset in the decomposition has genus zero, and the functions $\mathbf{f}$ belong to it, we obtain a (family of) rational solutions. Let us see a particular example.
\begin{example}\label{ex-app3} \textsf{Rational solution of functional algebraic equations.} \\
We consider the functional algebraic equation
\begin{equation}\label{eq-funEx}
x^{2} y^{3} f_{1}^{4}-x^{2} y^{2} f_{2}^{4} f_{3}-2 x \,y^{3} f_{1}^{2} f_{2}+2 x y f_{1} f_{2}^{2} f_{3}^{2}+y^{3} f_{2}^{2}-f_{1}^{2} f_{3}^{3}
=0, \end{equation}
where $\mathbf{f}=(f_1,f_2,f_3):=(\sin(t),\cos(t),\mathrm{e}^{t})$. We associate to \eqref{eq-funEx} the curve $\mathcal{C}(F)$ where
\[ F=x^{2} y^{3} a_{1}^{4}-x^{2} y^{2} a_{2}^{4} a_{3}-2 x \,y^{3} a_{1}^{2} a_{2}+2 x y a_{1} a_{2}^{2} a_{3}^{2}+y^{3} a_{2}^{2}-a_{1}^{2} a_{3}^{3}. \]
It holds that $\mathrm{genus}(\mathcal{C}(F))=0$ and that a proper parametrization is
\begin{equation}\label{eq-P-ex}
\mathcal{P}(\mathbf{a},W)=\left(\frac{W^{3} a_{1}+a_{2}}{W a_{2}^{2}+a_{1}^{2}},
\frac{a_{3}}{W^{2}}\right).
\end{equation}
The open subset in Def. \ref{def-omega} turns to be
\[ \Omega_{\mathrm{proper}(\mathcal{P})}= \overline{\mathbb{C}(\mathbf{f})}^3\setminus \mathbb{V}_{\overline{\mathbb{C}(\mathbf{f})}}\left(a_{1} a_{2} \left(a_{1}-a_{2}\right) \left(a_{1}^{6}+a_{1}^{5} a_{2}+a_{1}^{4} a_{2}^{2}+a_{1}^{3} a_{2}^{3}+a_{1}^{2} a_{2}^{4}+a_{1} a_{2}^{5}+a_{2}^{6}\right) a_{3} \right). \]
Since $\mathbf{f}\in \Omega_{\mathrm{proper}(\mathcal{P})}$, by Theorem \ref{thrm:genus0general}, we have that
\begin{equation}\label{eq-solApp} \left\{ x=\frac{W^{3} \sin \! \left(t \right)+\cos \! \left(t \right)}{W \left(\cos^{2}\left(t \right)\right)+\sin^{2}\left(t \right)}
, y=\frac{{\mathrm e}^{t}}{W^{2}}
\right\},
\end{equation}
for every $W\in \overline{\mathbb{C}(\mathbf{f})}$ such that~\eqref{eq-solApp} is well--defined, is a rational solution of~\eqref{eq-funEx}. In fact, the last statement holds more generally for every $W\in \overline{\mathrm{Mer}(\mathbb{C})}$.
\end{example}
\noindent
|
1,314,259,996,964 | arxiv | \section{CR Geometry}
\label{sec:bg}
Throughout this article, we will follow the conventions used by Gover and Graham~\cite{GoverGraham2005} for describing CR and pseudohermitian invariants and performing local computations using a choice of contact form. These conventions are identical to the the conventions used by Lee in his work on pseudo-Einstein structures~\cite{Lee1988}, except that we will sometimes describe invariants as densities rather than functions. This has the effect that exponential factors will generally not appear in our formulae for how these invariants transform under a change of contact form. Both for the convenience of the reader and to hopefully avoid any confusion caused by the many different notations used in the literature, we use this section to make precise these conventions as necessary for this article.
\subsection{CR and pseudohermitian manifolds}
\label{sec:bg/cr}
A \emph{CR manifold} is a pair $(M^{2n+1},J)$ of a smooth oriented (real) $(2n+1)$-dimensional manifold together with a formally integrable complex structure $J\colon H\to H$ on a maximally nonintegrable codimension one subbundle $H\subset TM$. In particular, the bundle $E=H^\perp\subset T^\ast M$ is orientable and any nonvanishing section $\theta$ of $E$ is a \emph{contact form}; i.e.\ $\theta\wedge (d\theta)^n$ is nonvanishing. We will assume further that $(M^{2n+1},J)$ is \emph{strictly pseudoconvex}, meaning that the symmetric tensor $d\theta(\cdot,J\cdot)$ on $H^\ast\otimes H^\ast$ is positive definite; since $E$ is one-dimensional, this is independent of the choice of contact form $\theta$.
Given a CR manifold $(M^{2n+1},J)$, we can define the subbundle $T^{1,0}$ of the complexified tangent bundle $T_{\mathbb{C}}M$ as the $+i$-eigenspace of $J$, and $T^{0,1}$ as its conjugate. We likewise denote by $\Lambda^{1,0}$ the space of $(1,0)$-forms --- that is, the subbundle of $T_{\mathbb{C}}^\ast M$ which annihilates $T^{0,1}$ --- and by $\Lambda^{0,1}$ its conjugate. The \emph{canonical bundle $K$} is the complex line-bundle $K=\Lambda^{n+1}\left(\Lambda^{1,0}\right)$.
A \emph{pseudohermitian manifold} is a triple $(M^{2n+1},J,\theta)$ of a CR manifold $(M^{2n+1},J)$ together with a choice of contact form $\theta$. The assumption that $d\theta(\cdot,J\cdot)$ is positive definite implies that the \emph{Levi form $L_\theta(U\wedge\bar V)=-2id\theta(U\wedge\bar V)$} defined on $T^{1,0}$ is a positive-definite Hermitian form. Since another choice of contact form $\hat\theta$ is equivalent to a choice of (real-valued) function $\sigma\in C^\infty(M)$ such that $\hat\theta=e^\sigma\theta$, and the Levi forms of $\hat\theta$ and $\theta$ are related by $L_{\hat\theta}=e^\sigma L_\theta$, we see that the analogy between CR geometry and conformal geometry begins through the similarity of choosing a contact form or a metric in a conformal class (cf.\ \cite{JerisonLee1987}).
Given a pseudohermitian manifold $(M^{2n+1},J,\theta)$, the \emph{Reeb vector field $T$} is the unique vector field such that $\theta(T)=1$ and $T\contr d\theta=0$. An \emph{admissible coframe} is a set of $(1,0)$-forms $\{\theta^\alpha\}_{\alpha=1}^n$ whose restriction to $T^{1,0}$ forms a basis for $\left(T^{1,0}\right)^\ast$ and such that $\theta^\alpha(T)=0$ for all $\alpha$. Denote by $\theta^{\bar\alpha}=\overline{\theta^\alpha}$ the conjugate of $\theta^\alpha$. Then $d\theta=ih_{\alpha\bar\beta}\theta^\alpha\wedge\theta^{\bar\beta}$ for some positive definite Hermitian matrix $h_{\alpha\bar\beta}$. Denote by $\{T,Z_\alpha,Z_{\bar\alpha}\}$ the frame for $T_{\mathbb{C}}M$ dual to $\{\theta,\theta^\alpha,\theta^{\bar\alpha}\}$, so that the Levi form is
\[ L_\theta\left(U^\alpha Z_\alpha,V^{\bar\alpha}Z_{\bar\alpha}\right) = h_{\alpha\bar\beta}U^\alpha V^{\bar\beta} . \]
Tanaka~\cite{Tanaka1975} and Webster~\cite{Webster1977} have defined a canonical connection on a pseudohermitian manifold $(M^{2n+1},J,\theta)$ as follows: Given an admissible coframe $\{\theta^\alpha\}$, define the \emph{connection forms $\omega_\alpha{}^\beta$} and the \emph{torsion form $\tau_\alpha=A_{\alpha\beta}\theta^\beta$} by the relations
\begin{align*}
d\theta^\beta & = \theta^\alpha\wedge\omega_\alpha{}^\beta + \theta\wedge\tau^\beta, \\
\omega_{\alpha\bar\beta}+\omega_{\bar\beta\alpha} & = dh_{\alpha\bar\beta}, \\
A_{\alpha\beta} & = A_{\beta\alpha},
\end{align*}
where we use the metric $h_{\alpha\bar\beta}$ to raise and lower indices; e.g.\ $\omega_{\alpha\bar\beta}=h_{\gamma\bar\beta}\omega_\alpha{}^\gamma$. In particular, the connection forms are pure imaginary. The connection forms define \emph{the pseudohermitian connection} on $T^{1,0}$ by $\nabla Z_\alpha=\omega_\alpha{}^\beta\otimes Z_\beta$, which is the unique connection preserving $T^{1,0}$, $T$, and the Levi form.
The curvature form $\Pi_\alpha{}^\beta:=d\omega_\alpha{}^\beta-\omega_\alpha{}^\gamma\wedge\omega_\gamma{}^\beta$ can be written
\[ \Pi_\alpha{}^\beta=R_\alpha{}^\beta{}_{\gamma\bar\delta}\theta^\gamma\wedge\theta^{\bar\delta} \mod \theta , \]
defining the curvature of $M$. The \emph{pseudohermitian Ricci tensor} is the contraction $R_{\alpha\bar\beta}:=R_\gamma{}^\gamma{}_{\alpha\bar\beta}$ and the \emph{pseudohermitian scalar curvature} is the contraction $R:=R_\alpha{}^\alpha$. As shown by Webster~\cite{Webster1977}, the contraction $\Pi_\gamma{}^\gamma$ is given by
\begin{equation}
\label{eqn:domegacontraction}
\Pi_\gamma{}^\gamma = d\omega_\gamma{}^\gamma = R_{\alpha\bar\beta}\theta^\alpha\wedge\theta^{\bar\beta} + \nabla^\beta A_{\alpha\beta} \theta^\alpha\wedge\theta - \nabla^{\bar\beta} A_{\bar\alpha\bar\beta} \theta^{\bar\alpha}\wedge\theta .
\end{equation}
For computational and notational efficiency, it will usually be more useful to work with the \emph{pseudohermitian Schouten tensor}
\[ P_{\alpha\bar\beta} := \frac{1}{n+2}\left( R_{\alpha\bar\beta} - \frac{1}{2(n+1)}Rh_{\alpha\bar\beta}\right) \]
and its trace $P:=P_\alpha{}^\alpha=\frac{R}{2(n+1)}$. The following higher order derivatives
\begin{align*}
T_\alpha & = \frac{1}{n+2}\left(\nabla_\alpha P - i\nabla^\beta A_{\alpha\beta}\right) \\
S & = -\frac{1}{n}\left(\nabla^\alpha T_\alpha + \nabla^{\bar\alpha}T_{\bar\alpha} + P_{\alpha\bar\beta}P^{\alpha\bar\beta} - A_{\alpha\beta}A^{\alpha\beta}\right)
\end{align*}
will also appear frequently (cf.\ \cite{GoverGraham2005,Lee1986}).
In performing computations, we will usually use abstract index notation, so for example $\tau_\alpha$ will denote a $(1,0)$-form and $\nabla_\alpha\nabla_\beta f$ will denote the $(2,0)$-part of the Hessian of a function. Of course, given an admissible coframe, these expressions give the components of the equivalent tensor. The following commutator formulae established by Lee~\cite[Lemma~2.3]{Lee1988} will be useful.
\begin{lem}
\label{lem:lee_commutator}
Let $(M^{2n+1},J,\theta)$ be a pseudohermitian manifold. Then
\begin{align*}
\nabla_\alpha\nabla_\beta f - \nabla_\beta\nabla_\alpha f & = 0, & \nabla_{\bar\beta}\nabla_\alpha f - \nabla_\alpha\nabla_{\bar\beta}f & = ih_{\alpha\bar\beta}\nabla_0f, \\
\nabla_\alpha\nabla_0f - \nabla_0\nabla_\alpha f & = A_{\alpha\gamma}\nabla^\gamma f, & \nabla^\beta\nabla_0\tau_\alpha - \nabla_0\nabla^\beta\tau_\alpha & = A^{\gamma\beta}\nabla_\gamma\tau_\alpha + \tau_\gamma\nabla_\alpha A^{\gamma\beta} ,
\end{align*}
where $\nabla_0$ denotes the derivative in the direction $T$.
\end{lem}
The following consequences of the Bianchi identities established in~\cite[Lemma~2.2]{Lee1988} will also be useful.
\begin{lem}
\label{lem:lee_bianchi}
Let $(M^{2n+1},J,\theta)$ be a pseudohermitian manifold. Then
\begin{align}
\label{eqn:schouten_bianchi} \nabla^\alpha P_{\alpha\bar\beta} & = \nabla_{\bar\beta}P + (n-1)T_{\bar\beta} \\
\label{eqn:nabla_0r} \nabla_0 R & = \nabla^\alpha\nabla^\beta A_{\alpha\beta} + \nabla_\alpha\nabla_\beta A^{\alpha\beta} .
\end{align}
\end{lem}
In particular, combining the results of Lemma~\ref{lem:lee_commutator} and Lemma~\ref{lem:lee_bianchi} yields
\begin{equation}
\label{eqn:subplacian_R}
\Delta_b R - 2n\Imaginary\nabla^\alpha\nabla^\beta A_{\alpha\beta} = -2\nabla^\alpha\left(\nabla_\alpha R - in\nabla^\beta A_{\alpha\beta}\right) .
\end{equation}
An important operator in the study of pseudohermitian manifolds is the \emph{sublaplacian}
\[ \Delta_b := -\left(\nabla^\alpha\nabla_\alpha + \nabla_\alpha\nabla^\alpha\right) . \]
Defining the \emph{subgradient $\nabla_bu$} as the projection of $du$ onto $H^\ast\otimes\mathbb{C}$ --- that is, $\nabla_bf=\nabla_\alpha f+\nabla_{\bar\alpha}f$ --- it is easy to show that
\[ \int_M u\Delta_b v\,\theta\wedge d\theta^n = \int_M \langle\nabla_b u,\nabla_b v\rangle \theta\wedge d\theta^n \]
for any $u,v\in C^\infty(M)$, at least one of which is compactly supported, and where $\langle\cdot,\cdot\rangle$ denotes the Levi form.
One important consequence of Lemma~\ref{lem:lee_commutator} is that the operator $C$ has the following two equivalent forms:
\begin{equation}
\label{eqn:subplacian_squared}
\begin{split}
Cf & := \Delta_b^2f + n^2\nabla_0^2f - 2in\nabla_\beta\left(A^{\alpha\beta}\nabla_\alpha f\right) + 2in\nabla^\beta\left(A_{\alpha\beta}\nabla^\alpha f\right) \\
& = 4\nabla^\alpha\left(\nabla_\alpha\nabla_\beta\nabla^\beta f + in A_{\alpha\beta}\nabla^\beta f\right) .
\end{split}
\end{equation}
In dimension $n=1$, the operator $C$ is the compatibility operator found by Graham and Lee~\cite{GrahamLee1988}. Hirachi~\cite{Hirachi1990} later observed that in this dimension $C$ is a CR covariant operator, in the sense that it satisfies a particularly simple transformation formula under a change of contact form. Thus, in this dimension $C$ is the CR Paneitz operator $P_4$; for further discussion, see Section~\ref{sec:defn}.
\subsection{CR pluriharmonic functions}
\label{sec:bg/pluriharmonic}
Given a CR manifold $(M^{2n+1},J)$, a \emph{CR pluriharmonic function} is a function $u\in C^\infty(M)$ which is locally the real part of a \emph{CR function} $v\in C^\infty(M;\mathbb{C})$; i.e.\ $u=\Real(v)$ for $v$ satisfying $\overline{\partial} v:=\nabla_{\bar\alpha}v=0$. We will denote by $\mathcal{P}$ the space of pluriharmonic functions on $M$, which is usually an infinite-dimensional vector space. When additionally a choice of contact form $\theta$ is given, Lee~\cite{Lee1988} proved the following alternative characterization of CR pluriharmonic functions which does not require solving $\overline{\partial} v=0$.
\begin{prop}
\label{prop:pluriharmonic}
Let $(M^{2n+1},J,\theta)$ be a pseudohermitian manifold. A function $u\in C^\infty(M)$ is CR pluriharmonic if and only if
\begin{align*}
B_{\alpha\bar\beta}u := \nabla_{\bar\beta}\nabla_\alpha u - \frac{1}{n}\nabla^\gamma\nabla_\gamma u\,h_{\alpha\bar\beta} & = 0, \qquad \text{if $n\geq 2$} \\
P_\alpha u := \nabla_\alpha\nabla_\beta\nabla^\beta u + inA_{\alpha\beta}\nabla^\beta u & = 0, \qquad \text{if $n=1$} .
\end{align*}
\end{prop}
Using Lemma~\ref{lem:lee_commutator}, it is straightforward to check that (cf.\ \cite{GrahamLee1988})
\begin{equation}
\label{eqn:grahamlee}
\nabla^{\bar\beta}\left(B_{\alpha\bar\beta}u\right) = \frac{n-1}{n}P_\alpha u .
\end{equation}
In particular, we see that the vanishing of $B_{\alpha\bar\beta}u$ implies the vanishing of $P_\alpha u$ when $n>1$. Moreover, the condition $B_{\alpha\bar\beta}u=0$ is vacuous when $n=1$, and by~\eqref{eqn:grahamlee}, we can consider the condition $P_\alpha u=0$ from Proposition~\ref{prop:pluriharmonic} as the ``residue'' of the condition $B_{\alpha\bar\beta}u=0$ (cf.\ Section~\ref{sec:lee} and Section~\ref{sec:defn}).
Note also that, using the second expression in~\eqref{eqn:subplacian_squared}, we have that $C=4\nabla^\alpha P_\alpha$. In particular, it follows that $\mathcal{P}\subset\ker P_4$ for three-dimensional CR manifolds $(M^3,J)$. It is easy to see that this is an equality when $(M^3,J)$ admits a torsion-free contact form (cf.\ \cite{GrahamLee1988}), but a good characterization of when equality holds is not yet known.
\subsection{CR density bundles}
\label{sec:bg/density}
One generally wants to study CR geometry using \emph{CR invariants}; i.e.\ using invariants of a CR manifold $(M^{2n+1},J)$. However, it is frequently easier to do geometry by making a choice of contact form $\theta$ so as to make use of the Levi form and the associated pseudohermitian connection. If one takes this point of view, it then becomes important to know how the pseudohermitian connection and the pseudohermitian curvatures transform under a change of contact form, and also to have a convenient way to describe objects which transform in a simple way with a change of contact form. This goal is met using CR density bundles.
Given a CR manifold $(M^{2n+1},J)$, choose a $(n+2)$-nd root of the canonical bundle $K$ and denote it by $E(1,0)$; this can always be done locally, and since we are entirely concerned with local invariants, this poses no problems. Given any $w,w^\prime\in\mathbb{R}$ with $w-w^\prime\in\mathbb{Z}$, the \emph{$(w,w^\prime)$-density bundle $E(w,w^\prime)$} is the complex line bundle
\[ E(w,w^\prime) = E(1,0)^{\otimes w} \otimes \overline{E(1,0)}^{\otimes w^\prime} . \]
For our purposes, the important property of $E(w,w^\prime)$ is that a choice of contact form $\theta$ induces an isomorphism between the space $\mathcal{E}(w,w^\prime)$ of smooth sections of $E(w,w^\prime)$ and $C^\infty(M;\mathbb{C})$,
\[ \mathcal{E}(w,w^\prime) \ni u \cong u_\theta \in C^\infty(M;\mathbb{C}), \]
with the property that if $\hat\theta=e^\sigma\theta$ is another choice of contact form, then $u_{\hat\theta}$ is related to $u_\theta$ by
\begin{equation}
\label{eqn:complex_transformation}
u_{\hat\theta} = e^{\frac{w}{2}\sigma}e^{\frac{w^\prime}{2}\bar\sigma} u_\theta ;
\end{equation}
for details, see~\cite{GoverGraham2005}. We will also consider density-valued tensor bundles; for example, we will denote by $E^\alpha(w,w^\prime)$ and $E^{\bar\alpha}(w,w^\prime)$ the tensor products $T^{1,0}\otimes E(w,w^\prime)$ and $T^{0,1}\otimes E(w,w^\prime)$, respectively, and by $\mathcal{E}^\alpha(w,w^\prime)$ and $\mathcal{E}^{\bar\alpha}(w,w^\prime)$ their respective spaces of smooth sections. In this way, we may regard the Levi form as the density $h_{\alpha\bar\beta}\in\mathcal{E}_{\alpha\bar\beta}(1,1)$, thereby suppressing the exponential factor which normally appears when writing how it transforms under a change of contact structure.
Since we will be primarily interested in real-valued functions and tensors, our primary interest will be in the $(w,w)$-density bundles and tensor products thereof. In particular, if $u\in\mathcal{E}(w,w)$ is real-valued and we restrict ourselves to real-valued contact forms, the transformation rule~\eqref{eqn:complex_transformation} becomes $u_{\hat\theta}=e^{w\sigma}u_\theta$.
In~\cite{Lee1988} (see also~\cite{GoverGraham2005}), the transformation formulae for the pseudohermitian connection and its torsion and curvatures under a change of contact form are given, which we record below:
\begin{lem}
\label{lem:cr_change}
Let $(M^{2n+1},J,\theta)$ be a pseudohermitian manifold and regard the torsion $A_{\alpha\beta}\in\mathcal{E}_{\alpha\beta}(0,0)$, the pseudohermitian Schouten tensor $P_{\alpha\bar\beta}\in\mathcal{E}_{\alpha\bar\beta}(0,0)$, and its trace $P=P_\alpha{}^\alpha\in\mathcal{E}(-1,-1)$. Additionally, let $f\in\mathcal{E}(w,w)$ and $\tau_\alpha\in\mathcal{E}_\alpha(w,w)$. If $\hat\theta=e^\sigma\theta$ is another choice of contact form and $\hat A_{\alpha\beta},\hat P_{\alpha\bar\beta},\hat P$ are its torsion, pseudohermitian Schouten tensor, and its trace, respectively, then
\begin{align*}
\hat A_{\alpha\beta} & = A_{\alpha\beta} + i\nabla_\beta\nabla_\alpha\sigma - i(\nabla_\alpha\sigma)(\nabla_\beta\sigma) \\
\hat P_{\alpha\bar\beta} & = P_{\alpha\bar\beta} - \frac{1}{2}\left(\nabla_{\bar\beta}\nabla_\alpha\sigma+\nabla_\alpha\nabla_{\bar\beta}\sigma\right) - \frac{1}{2}\lvert\nabla_\gamma\sigma\rvert^2 h_{\alpha\bar\beta} \\
\hat P & = P + \frac{1}{2}\Delta_b\sigma - \frac{n}{2}\lvert\nabla_\gamma\sigma\rvert^2 \\
\hat\nabla_\alpha f & = \nabla_\alpha f + wf\nabla_\alpha\sigma \\
\hat\nabla_0 f & = \nabla_0f + i(\nabla_\alpha\sigma)(\nabla^\alpha f) - i(\nabla^\alpha\sigma)(\nabla_\alpha f) + wf\nabla_0\sigma \\
\hat\nabla_\alpha\tau_\beta & = \nabla_\alpha\tau_\beta + (w-1)\tau_\beta\nabla_\alpha\sigma - \tau_\alpha\nabla_\beta\sigma \\
\hat\nabla_{\bar\beta}\tau_\alpha & = \nabla_{\bar\beta}\tau_\alpha + w\tau_\alpha\nabla_{\bar\beta}\sigma + \tau_\gamma\nabla^\gamma\sigma h_{\alpha\bar\beta} .
\end{align*}
\end{lem}
There are a few technical comments necessary to properly interpret Lemma~\ref{lem:cr_change}.
First, we define the norm $\lvert\nabla_\gamma\sigma\rvert^2:=(\nabla_\gamma\sigma)(\nabla^\gamma\sigma)$. In particular, $\lvert\nabla_\gamma\sigma\rvert^2=\frac{1}{2}\langle\nabla_b\sigma,\nabla_b\sigma\rangle$. We define norms on all (density-valued) tensors in a similar way; for example, $\lvert A_{\alpha\beta}\rvert^2=A_{\alpha\beta}A^{\alpha\beta}$ and $\lvert P_{\alpha\bar\beta}\rvert^2=P_{\alpha\bar\beta}P^{\alpha\bar\beta}$.
Second, for these formulae to be valid component-wise, one also needs to change the admissible frame in which one computes the components of the torsion and CR Schouten tensor. Explicitly, if $\{\theta,\theta^\alpha,\theta^{\bar\alpha}\}$ is an admissible coframe for the contact form $\theta$, one defines
\[ \hat\theta^\alpha = \theta^\alpha + i(\nabla^\alpha\sigma)\theta \]
and $\hat\theta^{\bar\alpha}$ by conjugation, ensuring that $\{\hat\theta,\hat\theta^\alpha,\hat\theta^{\bar\alpha}\}$ is an admissible coframe for the contact form $\hat\theta$. In the above formulae, this frame is used to compute the components of $\hat\nabla_\alpha$ and $\hat\nabla_{\bar\alpha}$, while the coframe $\{\theta,\theta^\alpha,\theta^{\bar\alpha}\}$ is used to compute the components of $\nabla_\alpha$ and $\nabla_{\bar\alpha}$.
Third, to regard $P\in\mathcal{E}(-1,-1)$ means to extend the \emph{function} $P$ to a \emph{density} $\rho\in\mathcal{E}(-1,-1)$ by requiring $\rho_\theta=P$, and we use $h_{\alpha\bar\beta}\in\mathcal{E}_{\alpha\bar\beta}(1,1)$ to raise and lower indices. This has the effect that, at the level of functions, Lemma~\ref{lem:cr_change} states that
\[ \hat P = e^{-\sigma}\left( P + \frac{1}{2}\Delta_b\sigma - \frac{n}{2}\lvert\nabla_\gamma\sigma\rvert^2\right) , \]
which is the transformation formula proven in~\cite{Lee1988}. It also means that we can quickly compute how $\nabla_\alpha P$ transforms under a change of contact form: Using Lemma~\ref{lem:cr_change} with $P\in\mathcal{E}(-1,-1)$, it follows immediately that
\[ \widehat{\nabla_\alpha} \widehat{P} = \nabla_\alpha\left(P+\frac{1}{2}\Delta_b\sigma-\frac{n}{2}\lvert\nabla_\gamma\sigma\rvert^2\right) - \left(P+\frac{1}{2}\Delta_b\sigma-\frac{n}{2}\lvert\nabla_\gamma\sigma\rvert^2\right)\nabla_\alpha\sigma . \]
These conventions will be exploited heavily in Section~\ref{sec:covariance}.
\section{CR covariance of the $P^\prime$-operator}
\label{sec:covariance}
In this section we give a direct computational proof of transformation formula~\eqref{eqn:q_operator_covariant} of the $P^\prime$-operator after a conformal change of contact form. Indeed, we will compute the transformation formula for the operator $P_4^\prime$ as defined by~\eqref{eqn:q_crit} acting on functions --- rather than only pluriharmonic functions --- and thus establish that one cannot hope to find an invariant operator acting instead on the kernel of the CR Paneitz operator.
To begin, we recall from Lemma~\ref{lem:cr_change} and~\eqref{eqn:Walpha} that given a three-dimensional pseudohermitian manifold $(M^3,J,\theta)$ and an arbitrary one-form $\tau_\alpha\in\mathcal{E}_\alpha(-1,-1)$, if $\hat\theta=e^\sigma\theta$ then
\begin{equation}
\label{eqn:hirachi}
\widehat{\nabla^\alpha}\widehat{\tau_\alpha} = \nabla^\alpha\tau_\alpha \quad\text{and}\quad \widehat{W_\alpha} = W_\alpha - 3P_\alpha\sigma .
\end{equation}
Another useful computation in preparation for our identification of the transformation law of $P_4^\prime$ is the following expression for the CR Paneitz operator applied to a product of two functions.
\begin{lem}
\label{lem:cr_prod}
Let $(M^3,J,\theta)$ be a pseudohermitian three-manifold and let $P_4$ be the CR Paneitz operator. Given $f,\sigma\in\mathcal{E}(0,0)$, it holds that
\begin{align*}
\frac{1}{4}P_4(f\sigma) & = \frac{1}{4}fP_4(\sigma) + \frac{1}{4}\sigma P_4(f) + 4\Real\left(\nabla^\alpha\sigma\nabla_\alpha\nabla^\beta\nabla_\beta f\right) + 4\Real\left(\nabla^\alpha f\nabla_\alpha\nabla^\beta\nabla_\beta\sigma\right) \\
& \quad + 2\Real\left(R\nabla^\alpha\sigma\nabla_\alpha f\right) + 2\Real\left(\nabla^\alpha\nabla^\beta\sigma\nabla_\alpha\nabla_\beta f\right) + 4\Real\left(\nabla^\alpha\nabla_\alpha\sigma\nabla_\beta\nabla^\beta f\right) .
\end{align*}
Alternatively,
\begin{align*}
P_4(\sigma f) & = \sigma P_4(f) + f P_4(\sigma) + 4\nabla^\alpha\sigma P_\alpha f + 4\nabla^\alpha f P_\alpha\sigma \\
& \quad + 4\nabla^\alpha\left(2\nabla_\alpha\sigma\nabla_\beta\nabla^\beta f + \nabla^\beta\sigma \nabla_\alpha\nabla_\beta f + 2\nabla_\alpha f\nabla_\beta\nabla^\beta\sigma + \nabla^\beta f \nabla_\alpha\nabla_\beta\sigma\right) .
\end{align*}
\end{lem}
\begin{proof}
The second expression follows by a direct expansion using the second formula for $P_4$ in~\eqref{eqn:subplacian_squared}. To establish the first expression, observe that the term involving $\nabla^\alpha\sigma$ in the second expression of the lemma is
\begin{equation}
\label{eqn:intermediate}
\nabla^\alpha\sigma\left(\nabla_\alpha\nabla_\beta\nabla^\beta f + iA_{\alpha\beta}\nabla^\beta f + \nabla^\beta\nabla_\beta\nabla_\alpha f\right) .
\end{equation}
Using the assumption that $M$ is three-dimensional and the commutator formula
\[ \nabla_{\bar\gamma}\nabla_\beta\nabla_\alpha f - \nabla_\beta\nabla_{\bar\gamma}\nabla_\alpha f = i\nabla_0\nabla_\alpha f\, h_{\beta\bar\gamma} + R_\alpha{}^\rho{}_{\beta\bar\gamma}\nabla_\rho f \]
due to Lee~\cite[Lemma~2.3]{Lee1988}, it holds that
\begin{equation}
\label{eqn:lee_third_commutator}
\nabla^\beta\nabla_\beta\nabla_\alpha f - \nabla_\alpha\nabla^\beta\nabla_\beta f = i\nabla_0\nabla_\alpha f + R\nabla_\alpha f .
\end{equation}
It then follows from Lemma~\ref{lem:lee_commutator} that~\eqref{eqn:intermediate} can be rewritten
\[ \nabla^\alpha\sigma\left(2\nabla_\alpha\nabla^\beta\nabla_\beta f + R\nabla_\alpha f\right), \]
from which the desired expression immediately follows.
\end{proof}
The transformation formulae~\eqref{eqn:hirachi} immediately yield the transformation formulae for the zeroth and first order terms of $P_4^\prime$. Thus it remains to compute the transformation formulae for the higher order terms.
\begin{prop}
\label{prop:D_covariant}
Let $(M^3,J,\theta)$ be a pseudohermitian three-manifold and define the operator $D\colon\mathcal{E}(0,0)\to\mathcal{E}(-2,-2)$ by
\[ Df = 4\Delta_b^2 f - 8\Imaginary\left(\nabla^\alpha(A_{\alpha\beta}\nabla^\beta f)\right) - 4\Real\left(\nabla^\alpha(R\nabla_\alpha f)\right) . \]
If $\hat\theta=e^\sigma\theta$ is another choice of contact form, then
\begin{align*}
\widehat{Df} & = Df + 8\Real\nabla^\alpha\left(2\nabla_\beta\nabla^\beta\sigma\nabla_\alpha f + \nabla^\beta\nabla_\beta\sigma\nabla_\alpha f + \nabla_\alpha\nabla_\beta f \nabla^\beta\sigma - \nabla^\beta\nabla_\beta f \nabla_\alpha\sigma\right) .
\end{align*}
\end{prop}
\begin{proof}
To begin, consider how each summand of $D$ transforms. By a straightforward application of Lemma~\ref{lem:cr_change}, we compute that
\begin{align*}
\widehat{\Delta_b^2} \widehat{f} & = \Delta_b\left(\Delta_b f - \langle\nabla f,\nabla\sigma\rangle\right) + 2\Real\nabla^\alpha\left((\Delta_b f - \langle\nabla f,\nabla\sigma\rangle)\nabla_\alpha\sigma\right) \\
\widehat{\nabla^\alpha}\left(\widehat{A_{\alpha\beta}}\widehat{\nabla^\beta} \widehat{f}\right) & = \nabla^\alpha\left((A_{\alpha\beta} + i\nabla_\alpha\nabla_\beta\sigma - i\nabla_\alpha\sigma\nabla_\beta\sigma)\nabla^\beta f\right) \\
\widehat{\nabla^\alpha}\left(\widehat{R}\widehat{\nabla_\alpha}\widehat{f}\right) & = \nabla^\alpha\left((R+2\Delta_b\sigma - 2\lvert\nabla_\gamma\sigma\rvert^2)\nabla_\alpha f\right) .
\end{align*}
The conclusion of the proposition then follows immediately from a straightforward computation.
\end{proof}
Together, \eqref{eqn:hirachi}, Lemma~\ref{lem:cr_prod}, and Proposition~\ref{prop:D_covariant} yield another proof of the transformation law~\eqref{eqn:q_operator_covariant} of the $P^\prime$-operator. In fact, the computations above allow us to compute the transformation rule for $P_4^\prime$ under a change of contact form when the local formula~\eqref{eqn:q_crit} is extended to all of $C^\infty(M)$.
\begin{prop}
Let $(M^3,J,\theta)$ be a pseudohermitian three-manifold and let $\sigma\in C^\infty(M)$. Set $\hat\theta=e^\sigma\theta$ and denote by $\widehat{P_4^\prime}$ and $P_4^\prime$ the operator~\eqref{eqn:q_crit} defined in terms of $\hat\theta$ and $\theta$, respectively. Then
\begin{equation}
\label{eqn:p4prime_crit_genl_transformation}
e^{2\sigma}\widehat{P_4^\prime}(f) = P_4^\prime(f) + P_4(f\sigma) - \sigma P_4(f) - 8\Real\left(P_\alpha f \nabla^\alpha \sigma\right)
\end{equation}
for all $f\in C^\infty(M)$. In particular,
\[ e^{2\sigma}\widehat{P_4^\prime}(f) = P_4^\prime(f) + P_4(f\sigma) \]
for all $f\in\mathcal{P}$.
\end{prop}
\begin{remark}
The transformation rule~\eqref{eqn:p4prime_crit_genl_transformation} obviously remains true when one adds a multiple of the CR Paneitz operator to $P_4^\prime$.
\end{remark}
\begin{proof}
It follows from~\eqref{eqn:hirachi} and Proposition~\ref{prop:D_covariant} that
\begin{align*}
e^{2\sigma}\widehat{P_4^\prime}(f) & = P_4^\prime(f) + f P_4\sigma - 8\Real\left(P_\alpha\sigma \nabla^\alpha f\right) \\
& \quad + 8\Real\nabla^\alpha\left(2(\nabla_\beta\nabla^\beta\sigma)(\nabla_\alpha f) + (\nabla^\beta\nabla_\beta\sigma)(\nabla_\alpha f) + (\nabla_\alpha\nabla_\beta f)(\nabla^\beta\sigma) - (\nabla^\beta\nabla_\beta f)(\nabla_\alpha\sigma)\right) .
\end{align*}
Using Lemma~\ref{lem:cr_prod} to write $fP_4\sigma$ in terms of $P_4(f\sigma)$, we find that
\begin{align*}
e^{2\sigma}\widehat{P_4^\prime}(f) & = P_4^\prime(f) + P_4(\sigma f) - \sigma P_4f - 4\Real\left(P_\alpha f\nabla^\alpha\sigma\right) \\
& \quad + 4\Real\left((2\nabla^\alpha\nabla_\beta\nabla^\beta\sigma - \nabla^\alpha\nabla^\beta\nabla_\beta\sigma - \nabla_\beta\nabla^\beta\nabla^\alpha\sigma + 3iA^{\alpha\beta}\nabla_\beta\sigma)\nabla_\alpha f\right) \\
& \quad - 4\Real\left((2\nabla^\alpha\nabla_\beta\nabla^\beta f + 2\nabla^\alpha\nabla^\beta\nabla_\beta f - \nabla_\beta\nabla^\beta\nabla^\alpha f)\nabla_\alpha\sigma\right) .
\end{align*}
The result then follows by using~\eqref{eqn:lee_third_commutator} to commute derivatives in the last two lines and the definition of the third order operator $P_\alpha$.
\end{proof}
\section*{Acknowledgments}
\input{bg}
\input{lee}
\input{defn}
\input{covariance}
\input{qprime}
\section{The CR Paneitz and $Q$-Curvature Operators}
\label{sec:defn}
The CR Paneitz operator in dimension three is well-known and given by $P_4=C$. However, in higher dimensions the operator $C$ is not CR covariant. The correct definition, in that $P_4$ is CR covariant, is as follows.
\begin{defn}
\label{defn:cr_paneitz}
Let $(M^{2n+1},J,\theta)$ be a CR manifold. The \emph{CR Paneitz operator $P_4$} is the operator
\begin{align*}
P_4f & := \Delta_b^2 f + \nabla_0^2 f - 4\Imaginary\left(\nabla^\alpha(A_{\alpha\beta}\nabla^\beta f)\right) + 4\Real\left(\nabla_{\bar\beta}(\tracefree{P}{}^{\alpha\bar\beta}\nabla_\alpha f)\right) \\
& \quad - \frac{4(n^2-1)}{n}\Real\left(\nabla^\beta(P\nabla_\beta f)\right) + \frac{n-1}{2}Q_4f .
\end{align*}
where
\begin{align*}
Q_4 & = \frac{2(n+1)^2}{n(n+2)}\Delta_b P - \frac{4}{n(n+2)}\Imaginary\left(\nabla^\alpha\nabla^\beta A_{\alpha\beta}\right) - \frac{2(n-1)}{n}\lvert A_{\alpha\beta}\rvert^2 \\
& \quad - \frac{2(n+1)}{n}\lvert \tracefree{P_{\alpha\bar\beta}}\rvert^2 + \frac{2(n-1)(n+1)^2}{n^2}P^2
\end{align*}
and $\tracefree{P_{\alpha\bar\beta}}=P_{\alpha\bar\beta} - \frac{P}{n}h_{\alpha\bar\beta}$ is the tracefree part of the CR Schouten tensor.
\end{defn}
The above expression for the CR Paneitz operator in general dimension does not seem to appear anywhere in the literature, though its existence and two different methods to derive the formula have been established by Gover and Graham~\cite{GoverGraham2005}. In particular, their construction immediately implies that the CR Paneitz operator is CR covariant,
\begin{equation}
\label{eqn:cr_covariant}
P_4 \colon \mathcal{E}\left(-\frac{n-1}{2},-\frac{n-1}{2}\right) \to \mathcal{E}\left(-\frac{n+3}{2},-\frac{n+3}{2}\right) .
\end{equation}
By inspection, it is clear that $P_4$ is a real, (formally) self-adjoint fourth order operator of the form $\Delta_b^2+T^2$ plus lower order terms, and thus has the form one expects of a ``Paneitz operator'' (cf.\ \cite{GoverGraham2005}). For convenience, we derive in the appendices the above expression for the CR Paneitz operator using both methods described in~\cite{GoverGraham2005}, namely the CR tractor calculus and restriction from the Fefferman bundle.
As mentioned in Section~\ref{sec:bg}, in the critical case $n=1$ we have that $\mathcal{P}\subset\ker P_4$. Motivated by~\cite{BransonFontanaMorpurgo2007,BransonGover2005}, we define the $P^\prime$-operator corresponding to the CR Paneitz operator as a renormalization of the part of $P_4$ which doesn't annihilate pluriharmonic functions.
\begin{defn}
\label{defn:q}
Let $(M^{2n+1},J,\theta)$ be a CR manifold. The \emph{$P^\prime$-operator $P_4^\prime\colon\mathcal{P}\to C^\infty(M)$} is defined by
\[ P_4^\prime f = \frac{2}{n-1}P_4f . \]
When $n=1$, we define $P_4^\prime$ by the formal limit
\[ P_4^\prime f = \lim_{n\to1} \frac{2}{n-1}P_4f . \]
\end{defn}
The key property of the $P^\prime$-operator, which we check explicitly below, is that the expression for $P_4^\prime$ as defined in Definition~\ref{defn:q} is rational in the dimension and does not have a pole at $n=1$; in particular, it is meaningful to discuss the $P^\prime$-operator on three-dimensional CR manifolds.
\begin{lem}
\label{lem:q}
Let $(M^{2n+1},J,\theta)$ be a CR manifold. Then the $P^\prime$-operator is given by
\begin{equation}
\label{eqn:q}
\begin{split}
P_4^\prime f & = \frac{2(n+1)}{n^2}\Delta_b^2f - \frac{8}{n}\Imaginary\left(\nabla^\alpha(A_{\alpha\beta}\nabla^\beta f)\right) - \frac{8(n+1)}{n}\Real\left(\nabla^\alpha(P\nabla_\alpha f)\right) \\
& \quad + \frac{16(n+1)}{n(n+2)}\Real\left((\nabla_\alpha P-\frac{in}{2(n+1)}\nabla^\beta A_{\alpha\beta})\nabla^\alpha f\right) \\
& \quad + \bigg[\frac{2(n+1)^2}{n(n+2)}\Delta_b P - \frac{4}{n(n+2)}\Imaginary\left(\nabla^\alpha\nabla^\beta A_{\alpha\beta}\right) \\
& \qquad - \frac{2(n-1)}{n}\lvert A_{\alpha\beta}\rvert^2 - \frac{2(n+1)}{n}\lvert\tracefree{P_{\alpha\bar\beta}}\rvert^2 + \frac{2(n-1)(n+1)^2}{n^2}P^2\bigg] f .
\end{split}
\end{equation}
In particular, if $n=1$, the \emph{critical $P^\prime$-operator} is given by
\begin{equation}
\label{eqn:q_crit}
\begin{split}
P_4^\prime f & = 4\Delta_b^2 f - 8\Imaginary\left(\nabla^\alpha(A_{\alpha\beta}\nabla^\beta f)\right) - 4\Real\left(\nabla^\alpha(R\nabla_\alpha f)\right) \\
& \quad + \frac{8}{3}\Real\left((\nabla_\alpha R - i\nabla^\beta A_{\alpha\beta})\nabla^\alpha f\right) + \frac{2}{3}\left(\Delta_b R - \frac{1}{2}\Imaginary\nabla^\alpha\nabla^\beta A_{\alpha\beta}\right) f .
\end{split}
\end{equation}
\end{lem}
\begin{proof}
When $n>1$, this follows directly from the definition of the CR Paneitz operator and the fact that $f\in\mathcal{P}$ if and only if
\[ \nabla_{\bar\beta}\nabla_\alpha f = \mu\,h_{\alpha\bar\beta} \]
for some $\mu\in C^\infty(M)$, which in turn implies, using~\eqref{eqn:subplacian_squared}, that
\[ \Delta_b^2 f + n^2\nabla_0^2 f - 4n\Imaginary\left(\nabla^\alpha(A_{\alpha\beta}\nabla^\beta f)\right) = 0 . \]
Letting $n\to1$ then yields the case $n=1$.
\end{proof}
Note that, as an operator on $C^\infty(M)$, the $P^\prime$-operator is only determined uniquely up to the addition of operators which annihilate $\mathcal{P}$. We have chosen the expression~\eqref{eqn:q_crit} so that our expression does not involve $T$-derivatives. In particular, this allows us to readily connect the $P^\prime$-operator to similar objects already appearing in the literature.
\begin{enumerate}
\item On a general CR manifold $(M^3,J,\theta)$,
\[ P_4^\prime(1) = \frac{2}{3}\left(\Delta_b R - 2\Imaginary\nabla^\alpha\nabla^\beta A_{\alpha\beta}\right) , \]
which is, using~\eqref{eqn:subplacian_R}, Hirachi's $Q$-curvature~\eqref{eqn:Q_divergence}.
\item On $(S^3,J,\theta)$ with its standard CR structure, the $P^\prime$-operator is given by
\[ P_4^\prime = 4\Delta_b^2 + 2\Delta_b, \]
which is the operator introduced by Branson, Fontana and Morpurgo~\cite{BransonFontanaMorpurgo2007}.
\end{enumerate}
The means by which we defined the $P^\prime$-operator, and which we will further employ to establish its CR covariance, is called ``analytic continuation in the dimension'' (cf.\ \cite{Branson1995}). However, due to the relatively simple form of the expression for $P_4^\prime$, we can also check its CR covariance directly, as is carried out in Section~\ref{sec:covariance}.
In the case that $(M^3,J,\theta)$ is a pseudo-Einstein manifold, the $P^\prime$-operator takes the simple form
\begin{equation}
\label{eqn:p4prime_invariant_contact_form}
P_4^\prime = 4\Delta_b^2 -8\Imaginary\nabla^\alpha\left(A_{\alpha\beta}\nabla^\beta\right) - 4\Real\nabla^\alpha\left(R\nabla_\alpha\right) .
\end{equation}
In particular, we see that $P_4^\prime$ annihilates constants, leading us to consider the ``$Q$-curvature of the $P^\prime$-operator,'' which we shall simply call the $Q^\prime$-curvature.
\begin{defn}
Let $(M^{2n+1},J,\theta)$ be a pseudo-Einstein manifold. The \emph{$Q^\prime$-curvature $Q_4^\prime\in C^\infty(M)$} is the local invariant defined by
\[ Q_4^\prime = \frac{2}{n-1}P_4^\prime(1) = \frac{4}{(n-1)^2}P_4(1) . \]
When $n=1$, we define $Q_4^\prime$ as the formal limit
\[ Q_4^\prime = \lim_{n\to 1}\frac{4}{(n-1)^2}P_4(1) . \]
\end{defn}
Again, it is straightforward to give an explicit formula for $Q_4^\prime$.
\begin{lem}
\label{lem:qprime}
Let $(M^{2n+1},J,\theta)$ be a pseudo-Einstein manifold. Then the $Q^\prime$-curvature is given by
\begin{equation}
\label{eqn:q4prime}
Q_4^\prime = \frac{2}{n^2}\Delta_b R - \frac{4}{n}\lvert A_{\alpha\beta}\rvert^2 + \frac{1}{n^2}R^2 .
\end{equation}
In particular, when $n=1$ the $Q^\prime$-curvature is
\begin{equation}
\label{eqn:q4prime_crit}
Q_4^\prime = 2\Delta_b R - 4\lvert A_{\alpha\beta}\rvert^2 + R^2 .
\end{equation}
\end{lem}
\begin{proof}
When $n>1$, it follows from~\eqref{eqn:divtrfreep} and the pseudo-Einstein assumption that $\lvert\tracefree{P_{\alpha\bar\beta}}\rvert^2=0$ and
\[ \Delta_b R - 2n\Imaginary\left(\nabla^\alpha\nabla^\beta A_{\alpha\beta}\right) = 0 . \]
Plugging in to~\eqref{eqn:q}, we see that
\[ P_4^\prime(1) = \frac{n-1}{n^2}\Delta_b R - \frac{2(n-1)}{n}\lvert A_{\alpha\beta}\rvert^2 + \frac{(n-1)}{2n^2}R^2 . \]
Multiplying by $\frac{2}{n-1}$ then yields the desired result.
\end{proof}
Let us now verify some basic properties of the $P^\prime$-operator and the $Q^\prime$-curvature. These objects are best behaved and the most interesting in the critical dimension $n=1$, so we shall make our statements only in this dimension.
\begin{prop}
\label{prop:q_covariant}
Let $(M^3,J,\theta)$ be a pseudohermitian manifold with $P^\prime$-operator $P_4^\prime\colon\mathcal{P}\to C^\infty(M)$. Then the following properties hold.
\begin{enumerate}
\item $P_4^\prime$ is formally self-adjoint.
\item Given another choice of contact form $\hat\theta=e^\sigma\theta$ with $\sigma\in C^\infty(M)$, it holds that
\begin{equation}
\label{eqn:q_operator_covariant}
e^{2\sigma}\hat P_4^\prime(f) = P_4^\prime(f) + P_4(f\sigma)
\end{equation}
for all $f\in\mathcal{P}$, where $\hat P_4^\prime$ denotes the $P_4^\prime$-operator defined in terms of $\hat\theta$.
\end{enumerate}
\end{prop}
\begin{proof}
On a general pseudohermitian manifold $(M^{2n+1},J,\theta)$, it follows from Definition~\ref{defn:q} and the self-adjointness of $P_4$ that, given $u,v\in\mathcal{P}$,
\[ \frac{n-1}{2}\int_M u\,P_4^\prime v = \int_M u\,P_4v = \int_M v\,P_4u = \frac{n-1}{2}\int_M v\,P_4^\prime u, \]
establishing the self-adjointness of $P_4^\prime$. Likewise, the covariance~\eqref{eqn:cr_covariant} of the CR Paneitz operator implies that for all $u\in\mathcal{P}$,
\[ \frac{n-1}{2}e^{\frac{n+3}{2}\sigma}\hat P_4^\prime(u) = P_4\left(e^{\frac{n-1}{2}\sigma}u\right) = \frac{n-1}{2}P_4^\prime(u) + P_4\left(\left(e^{\frac{n-1}{2}\sigma}-1\right)u\right) . \]
Multiplying both sides by $\frac{2}{n-1}$ and taking the limit $n\to 1$ yields~\eqref{eqn:q_operator_covariant}.
\end{proof}
\begin{remark}
It would be nice to have a formula for the critical $P_4^\prime$-operator which is manifestly formally self-adjoint on \emph{all} functions. At present, we have not been able to find such a formula without the assumption that $\theta$ is a pseudo-Einstein contact form, in which case~\eqref{eqn:p4prime_invariant_contact_form} gives such a formula.
\end{remark}
Using the same argument with $Q_4^\prime$ in place of $P_4^\prime$ and $P_4^\prime$ in place of $P_4$, we get a similar result for transformation law of the $Q^\prime$-curvature when the contact form is changed by a CR pluriharmonic function.
\begin{prop}
\label{prop:qprime_covariant}
Let $(M^3,J,\theta)$ be a pseudo-Einstein manifold. Given $\sigma\in\mathcal{P}$, denote $\hat\theta=e^\sigma\theta$. Then
\begin{equation}
\label{eqn:qprime_operator_covariant}
e^{2\sigma}\hat Q_4^\prime = Q_4^\prime + P_4^\prime(\sigma) + \frac{1}{2}P_4(\sigma^2) .
\end{equation}
In particular, if $M$ is compact then
\begin{equation}
\label{eqn:qprime_integral}
\int_M \hat Q_4^\prime\,\hat\theta\wedge d\hat\theta = \int_M Q_4^\prime\,\theta\wedge d\theta .
\end{equation}
\end{prop}
\begin{proof}
For $n>1$ and $\sigma\in\mathcal{P}$, we have that
\begin{align*}
\left(\frac{n-1}{2}\right)^2 e^{\frac{n+3}{2}\sigma}\hat Q_4^\prime & = \left(\frac{n-1}{2}\right)^2 Q_4^\prime + P_4\left(e^{\frac{n-1}{2}\sigma} - 1 \right) \\
& = \left(\frac{n-1}{2}\right)^2 Q_4^\prime + \frac{n-1}{2}P_4\left(\sigma + \frac{n-1}{4}\sigma^2 + O\big((n-1)^2\big)\right) .
\end{align*}
Multiplying by $\frac{4}{(n-1)^2}$ and taking the limit $n\to 1$ yields~\eqref{eqn:qprime_operator_covariant}. The invariance~\eqref{eqn:qprime_integral} then follows from the self-adjointness of $P_4^\prime$ and $P_4$ on their respective domains and the facts that $P_4(1)=0$ for any contact form and $P_4^\prime(1)=0$ for any pseudo-Einstein contact form.
\end{proof}
We conclude this section with a useful observation about the sign of the $P^\prime$-operator, which can be regarded as a CR analogue of a result of Gursky~\cite{Gursky1999} for the Paneitz operator in conformal geometry.
\begin{prop}
\label{prop:gursky}
Let $(M^3,J)$ be a compact CR manifold which admits a pseudo-Einstein contact form $\theta$ with nonnegative scalar curvature. Then $P_4^\prime\geq 0$ and the kernel of $P^\prime$ consists of the constants.
\end{prop}
\begin{proof}
It follows from~\eqref{eqn:q_operator_covariant} that the conclusion $P_4^\prime\geq0$ with $\ker P_4^\prime=\mathbb{R}$ is CR invariant, so we may compute in the scale $\theta$. From the definition of the sublaplacian we see that
\[ \Delta_b^2 - 2\Imaginary\nabla^\beta\left(A_{\alpha\beta}\nabla^\alpha\right) = 2\Real\nabla^\alpha\left(\nabla_\alpha\nabla^\beta\nabla_\beta + P_\alpha\right) . \]
It thus follows that the $P^\prime$-operator is equivalently defined via the formula
\begin{equation}
\label{eqn:signed_version}
P_4^\prime u = 4\Real\nabla^\alpha\left(2\nabla_\alpha\nabla^\beta\nabla_\beta u - R\nabla_\alpha u\right)
\end{equation}
for all $u\in\mathcal{P}$. Multiplying~\eqref{eqn:signed_version} by $u$ and integrating yields
\[ \int_M u\,P_4^\prime u = 4\int_M \left(2\left|\nabla^\beta\nabla_\beta u\right|^2 + 2R\left|\nabla_\beta u\right|^2\right) . \]
Since $R\geq0$, this is clearly nonnegative, showing that $P_4^\prime\geq0$. Moreover, if equality holds, then $\nabla^\beta\nabla_\beta u=0$, which is easily seen to imply that $u$ is constant, as desired.
\end{proof}
It would be preferable for Proposition~\ref{prop:gursky} to require checking only CR invariant assumptions. For instance, one might hope to prove the same result assuming that $(M^3,J)$ has nonnegative CR Yamabe constant and admits a pseudo-Einstein contact form. However, it is at present unclear whether these assumptions imply that one can choose a contact form as in the statement of Proposition~\ref{prop:gursky}.
\section{Checking Via the Fefferman Metric}
\label{sec:fefferman}
In this appendix, we follow the other perspective of Gover and Graham~\cite{GoverGraham2005} and give the formula for the CR Paneitz operator using the Fefferman metric. To arrive at the formula given in Definition~\ref{defn:cr_paneitz}, we use Lee's intrinsic formulation~\cite{Lee1986} of the Fefferman metric.
To begin, let $(M^{2n+1},J,\theta)$ be a pseudohermitian manifold and let $(\tilde M^{2n+2},g)$ be the Fefferman bundle, which is an $S^1$-bundle over $M$ with $g$ a particular Lorentzian metric. The Paneitz operator $L_4$ on a pseudo-Riemannian manifold is defined by
\[ L_4u = \Delta^2 u + 4P^{ij}\nabla_i\nabla_j u - (N-2)P_i^i\Delta u - (N-6)(\nabla^jP_i^i)(\nabla_ju) + \frac{N-4}{2}Qu, \]
where $N=2n+2$ is the dimension of $\tilde M$, $P_{ij}=\frac{1}{N-2}\left(R_{ij} - \frac{1}{2(N-1)}R_k^kg_{ij}\right)$ is the Schouten tensor of $g$, $\Delta=\nabla^i\nabla_i$ is the Laplacian (with nonpositive spectrum), and
\[ Q = -\Delta P_i^i - 2P_{ij}P^{ij} + \frac{N}{2}\left(P_i^i\right)^2 \]
is the (conformal) $Q$-curvature. The key facts about the Paneitz operator on the Fefferman bundle are that it is conformally invariant and that its restriction to functions which are invariant under the circle action is itself invariant under the circle action. In particular, these facts together imply that $L_4$ descends to a CR covariant operator on $(M^{2n+1},J,\theta)$. Explicitly, the operator $P_4$ defined by
\begin{equation}
\label{eqn:fefferman_paneitz}
P_4u = \frac{1}{4}\pi_\ast\left(L_4(\pi^\ast u)\right)
\end{equation}
will necessarily be a CR covariant operator of the form $\Delta_b^2+T^2$ plus lower order terms. Its explicit form can be computed using the following sequence of lemmas which are a consequence of Lee's intrinsic characterization~\cite{Lee1986} of the Fefferman bundle. First, we have the following simple expressions for the scalar curvature, the Laplacian, and the inner product of two gradients on both manifolds.
\begin{lem}
Let $(M^{2n+1},J,\theta)$ be a pseudohermitian manifold and let $(\tilde M^{2n+2},g)$ denote the associated Fefferman bundle. Then, given any $u,v\in C^\infty(M)$, it holds that
\[ \pi_\ast\left(\Delta (\pi^\ast u)\right) = -2\Delta_b u , \quad \pi_\ast J = 2P, \quad \pi_\ast\langle\nabla(\pi^\ast u),\nabla(\pi^\ast v)\rangle = 4\Real\left(\nabla^\alpha u\nabla_\alpha v\right) . \]
\end{lem}
Next, we have the relationship between the norms of the Schouten tensor on both manifolds.
\begin{lem}
Let $(M^{2n+1},J,\theta)$ be a pseudohermitian manifold and let $(\tilde M^{2n+2},g)$ denote the associated Fefferman bundle. It holds that
\begin{align*}
\pi_\ast\left(P_{ij}P^{ij}\right) & = \frac{2(n+1)}{n}P_{\alpha\bar\beta}P^{\alpha\bar\beta} + \frac{2(n-1)}{n}A_{\alpha\beta}A^{\alpha\beta} \\
& \quad + \frac{4}{n(n+2)}\Imaginary\left(\nabla^\alpha\nabla^\beta A_{\alpha\beta}\right) + \frac{4}{n(n+2)}\Real\left(\nabla^\alpha\nabla_\alpha P\right) .
\end{align*}
\end{lem}
The last ingredient from~\cite{Lee1986} is the inner product of the Schouten tensor with a Hessian, which follows from the formulae for the Ricci tensor and the connection on both manifolds.
\begin{lem}
Let $(M^{2n+1},J,\theta)$ be a pseudohermitian manifold and let $(\tilde M^{2n+2},g)$ denote the associated Fefferman bundle. Then, given any $u\in C^\infty(M)$, it holds that
\[ \pi_\ast\left(P^{ij}\nabla_i\nabla_ju\right) = 4\nabla_0^2 u - 16\Imaginary\left(A_{\alpha\beta}\nabla^\alpha\nabla^\beta u\right) + 16\Real\left(P^{\alpha\bar\beta}\nabla_{\bar\beta\alpha}^2u\right) - 48\Real\left(T_\alpha\nabla^\alpha u\right) . \]
\end{lem}
Putting these together, we provide another derivation for the formula given in Definition~\ref{defn:cr_paneitz} for the CR Paneitz operator.
\begin{prop}
Let $(M^{2n+1},J,\theta)$ be a pseudohermitian manifold and let $(\tilde M^{2n+2},g)$ denote the associated Fefferman bundle. Denote by $F$ and $Q$ the operators
\begin{align*}
F(u) & = 4P^{ij}\nabla_i\nabla_ju - (N-2)P_k^k\Delta u - (N-6)(\nabla^jP_i^i)(\nabla_ju) \\
Q & = -\Delta P_i^i - 2P_{ij}P^{ij} + \frac{N}{2}(P_i^i)^2
\end{align*}
on $\tilde M^N$. Then, given any $u\in C^\infty(M)$, it holds that
\begin{align*}
\frac{1}{4}\pi_\ast\left(\Delta^2 (\pi^\ast u)\right) & = \Delta_b^2 u \\
\frac{1}{4}\pi_\ast\left(F(\pi^\ast u)\right) & = \nabla_0^2 u - 4\Imaginary\nabla^\alpha\left(A_{\alpha\beta}\nabla^\beta u\right) + 4\tracefree{P}{}^{\alpha\bar\beta}\nabla_{\bar\beta\alpha}^2u - \frac{4(n^2-1)}{n}\Real\left(\nabla^\alpha(P\nabla_\alpha u)\right) \\
& \quad - \frac{32(n^2-1)}{n(n+2)}\Real\left((\nabla_\alpha P - \frac{in}{2(n+1)}\nabla^\beta A_{\alpha\beta})\nabla^\alpha u\right) \\
\frac{1}{4}\pi_\ast Q & = \frac{(n+1)^2}{n(n+2)}\Delta_b P - \frac{2}{n(n+2)}\Imaginary\left(\nabla^\alpha\nabla^\beta A_{\alpha\beta}\right) \\
& \quad - \frac{n+1}{n}\lvert \tracefree{P}{}^{\alpha\bar\beta}\rvert^2 - \frac{n-1}{n}\lvert A\rvert^2 + \frac{(n-1)(n+1)^2}{n^2} P^2 .
\end{align*}
In particular, Definition~\ref{defn:cr_paneitz} for the CR Paneitz operator agrees with the definition via~\eqref{eqn:fefferman_paneitz}.
\end{prop}
\section{Introduction}
\label{sec:intro}
It is well-known that there is a deep analogy between the study of three-dimensional CR manifolds and of four-dimensional conformal manifolds. Two important ingredients in the study of the latter are the Paneitz operator $P_4$ and the $Q$-curvature $Q_4$. Given a metric $g$, the Paneitz operator is a formally self-adjoint fourth-order differential operator of the form $\Delta^2$ plus lower-order terms, while the $Q$-curvature is a scalar invariant of the form $\Delta R$ plus lower-order terms, where $R$ is the scalar curvature of $g$ and ``order'' is measured according to the number of derivatives taken of $g$. The pair $(P_4,Q_4)$ generalizes to four-dimensions many important properties of the pair $(-\Delta,K)$ of the Laplacian and the Gauss curvature of a two-manifold. For example, if $(M^4,g)$ is a Riemannian manifold and $\hat g=e^{2\sigma}g$ is another choice of metric, then
\begin{align}
\label{eqn:conformal_p_trans} e^{4\sigma}\hat P_4(f) & = P_4(f) \\
\label{eqn:conformal_q_trans} e^{4\sigma}\hat Q_4 & = Q_4 + P_4(\sigma)
\end{align}
for all $f\in C^\infty(M)$. Since also $P_4(1)=0$, the transformation formula~\eqref{eqn:conformal_q_trans} implies that on a compact conformal manifold $(M^4,[g])$, the integral of the $Q$-curvature is a conformal invariant; indeed, the Gauss--Bonnet--Chern formula states that this integral is a linear combination of the Euler characteristic of $M^4$ and the integral of a pointwise conformal invariant, namely the norm of the Weyl tensor. The pair $(P_4,Q_4)$ also appears in the linearization of the Moser--Trudinger inequality. Denoting by $(S^4,g_0)$ the standard four-sphere with $g_0$ a metric of constant sectional curvature one, it was proven by Beckner~\cite{Beckner1993}, and later by Chang and the second author~\cite{ChangYang1995} using a different technique, that
\begin{equation}
\label{eqn:conformal_mt}
\int_{S^4} u\,P_4u + 2\int_{S^4} Q_4u - \frac{1}{2}\left(\int_{S^4} Q_4\right)\log\left(\fint_{S^4} e^{4u}\right) \geq 0
\end{equation}
for all $u\in C^\infty(S^4)$, and that equality holds if and only if $e^{2u}g_0$ is an Einstein metric on $(S^4,g_0)$.
A natural question is whether there exist analogues of $P_4$ and $Q_4$ defined for a three-dimensional pseudohermitian manifold $(M^3,J,\theta)$. In a certain sense this is already known; the compatibility operator studied by Graham and Lee~\cite{GrahamLee1988} is a fourth-order CR invariant operator with leading order term $\Delta_b^2+T^2$ and Hirachi~\cite{Hirachi1990} has identified a scalar invariant $Q_4$ which is related to $P_4$ through a change of contact form in a manner analogous to~\eqref{eqn:conformal_q_trans}. However, while the total $Q$-curvature of a compact three-dimensional CR manifold is indeed a CR invariant, it is always equal to zero. Moreover, the $Q$-curvature of the standard CR three-sphere vanishes identically; indeed, this is true for the boundary of any strictly pseudoconvex domain~\cite{FeffermanHirachi2003}, as is explained in Section~\ref{sec:defn}. In particular, while~\eqref{eqn:conformal_mt} is true on the CR three-sphere, it is trivial, as it only states that the Paneitz operator is nonnegative.
Using spectral methods, Branson, Fontana and Morpurgo~\cite{BransonFontanaMorpurgo2007} have recently identified a new operator $P_4^\prime$ on the standard CR three-sphere $(S^3,J,\theta_0)$ such that $P_4^\prime$ is of the form $\Delta_b^2$ plus lower-order terms, $P_4^\prime$ is invariant under the action of the CR automorphism group of $S^3$, and $P_4^\prime$ appears in an analogue of~\eqref{eqn:conformal_mt} in which the exponential term is present. There is, however, a catch: the operator $P_4^\prime$ acts only on the space $\mathcal{P}$ of CR pluriharmonic functions on $S^3$, namely those functions which are the boundary values of pluriharmonic functions in the ball $\{(z,w)\colon \lvert z\rvert^2+\lvert w\rvert^2<1\}\subset\mathbb{C}^2$. The space of CR pluriharmonic functions on $S^3$ is itself invariant under the action of the CR automorphism group, so it makes sense to discuss the invariance of $P_4^\prime$. Using this operator, Branson, Fontana and Morpurgo~\cite{BransonFontanaMorpurgo2007} showed that
\begin{equation}
\label{eqn:cr_mt}
\int_{S^3} u\,P_4^\prime u + 2\int_{S^3} Q_4^\prime u - \left(\int_{S^3} Q_4^\prime\right)\log\left(\fint_{S^3} e^{2u}\right) \geq 0
\end{equation}
for all $u\in\mathcal{P}$, where $Q_4^\prime=1$ and equality holds in~\eqref{eqn:cr_mt} if and only if $e^u\theta_0$ is a torsion-free contact form with constant Webster scalar curvature.
Formally, the operator $P_4^\prime$ is constructed using Branson's principle of analytic continuation in the dimension~\cite{Branson1995}. More precisely, there exists in general dimensions a fourth-order CR invariant operator with leading order term $\Delta_b^2+T^2$, which we shall also refer to as the Paneitz operator. On the CR spheres, this is an intertwining operator, and techniques from representation theory allow one to quickly compute the spectrum of this operator. By carrying out this program, one observes that the Paneitz operator on the standard CR three-sphere kills CR pluriharmonic functions, and moreover, the Paneitz operator $P_{4,n}$ on the standard CR $(2n+1)$-sphere acts on CR pluriharmonic functions as $\frac{n-1}{2}$ times a well-defined operator, called $P_4^\prime$. One observation in~\cite{BransonFontanaMorpurgo2007} is that this operator is in fact a fourth-order differential operator acting on CR pluriharmonic functions which is, in a suitable sense, CR invariant.
The purpose of this article is to show that there is a meaningful definition of the ``$P^\prime$-operator'' on general three-dimensional CR manifolds enjoying the same algebraic properties as the operator $P_4^\prime$ defined in~\cite{BransonFontanaMorpurgo2007}, and also to investigate the possibility of defining a scalar invariant $Q_4^\prime$ which is related to $P_4^\prime$ in a manner analogous to the way in which the $Q$-curvature is related to the Paneitz operator. It turns out that one cannot define $Q_4^\prime$ in a meaningful way for a general choice of contact form on a CR three-manifold, though one can for a distinguished class of contact forms, namely the so-called pseudo-Einstein contact forms. These are precisely those contact forms which are locally volume-normalized with respect to a closed section of the canonical bundle, which is a meaningful consideration in dimension three (cf.\ \cite{Lee1988} and Section~\ref{sec:lee}). Having made these definitions, we will also begin to investigate the geometric meaning of these invariants.
To describe our results, let us begin by discussing in more detail the ideas which give rise to the definitions of $P_4^\prime$ and $Q_4^\prime$. To define $P_4^\prime$, we follow the same strategy of Branson, Fontana, and Morpurgo~\cite{BransonFontanaMorpurgo2007}. First, Gover and Graham~\cite{GoverGraham2005} have shown that on a general CR manifold $(M^{2n+1},J)$, one can associate to each choice of contact form $\theta$ a formally-self adjoint real fourth-order operator $P_{4,n}$ which has leading order term $\Delta_b^2+T^2$, and that this operator is CR covariant. On three-dimensional CR manifolds, this reduces to the well-known operator
\[ P_4 := P_{4,1} = \Delta_b^2 + T^2 - 4\Imaginary \nabla^\alpha A_{\alpha\beta}\nabla^\beta \]
which, through the work of Graham and Lee~\cite{GrahamLee1988} and Hirachi~\cite{Hirachi1990}, is known to serve as a good analogue of the Paneitz operator of a four-dimensional conformal manifold. As pointed out by Graham and Lee~\cite{GrahamLee1988}, the kernel of $P_4$ (as an operator on a three-dimensional CR manifold) contains the space $\mathcal{P}$ of CR pluriharmonic functions, and thus one can ask whether the operator
\[ P_4^\prime := \lim_{n\to1} \frac{2}{n-1}P_{4,n} \rvert_{\mathcal{P}} \]
is well-defined. As we verify in Section~\ref{sec:defn}, this is the case. It then follows from standard arguments (cf.\ \cite{BransonGover2005}) that if $\hat\theta=e^\sigma\theta$ is any other choice of contact form, then the corresponding operator $\widehat{P_4^\prime}$ is related to $P_4^\prime$ by
\begin{equation}
\label{eqn:cr_p_trans}
e^{2\sigma}\widehat{P_4^\prime}(f) = P_4^\prime(f) + P_4(\sigma f)
\end{equation}
for any $f\in\mathcal{P}$. Thus the relation between $P_4^\prime$ and $P_4$ is analogous to the relation~\eqref{eqn:conformal_q_trans} between the $Q$-curvature and the Paneitz operator; more precisely, the $P^\prime$-operator can be regarded as a $Q$-curvature operator in the sense of Branson and Gover~\cite{BransonGover2005}. Moreover, since the Paneitz operator is self-adjoint and kills pluriharmonic functions, the transformation formula~\eqref{eqn:cr_p_trans} implies that
\[ e^{2\sigma}\widehat{P_4^\prime}(f) = P_4^\prime(f) \mod \mathcal{P}^\perp \]
for any $f\in\mathcal{P}$, returning $P_4^\prime$ to the status of a Paneitz-type operator. This is the sense in which the $P^\prime$-operator is CR invariant, and is the way that it is studied in~\eqref{eqn:cr_mt}.
From its construction, one easily sees that $P_4^\prime(1)$ is exactly Hirachi's $Q$-curvature. Thus, unlike the Paneitz operator, the $P^\prime$-operator does not necessarily kill constants. However, there is a large and natural class of contact forms for which the $P^\prime$-operator does kill constants, namely the pseudo-Einstein contact forms; see Section~\ref{sec:lee} for their definition. It turns out that two pseudo-Einstein contact forms $\hat\theta$ and $\theta$ must be related by a CR pluriharmonic function, $\log\hat\theta/\theta\in\mathcal{P}$ (cf.\ \cite{Lee1988}). If $(M^3,J)$ is the boundary of a domain in $\mathbb{C}^2$, such contact forms exist in profusion, arising as solutions to Fefferman's Monge-Amp\`ere equation (cf.\ \cite{Fefferman1976,FeffermanHirachi2003}). In this setting, it is natural to ask whether there is a scalar invariant $Q_4^\prime$ such that $P_4^\prime(1)=\frac{n-1}{2}Q_4^\prime$. This is true; we will show that if $(M^3,J,\theta)$ is a pseudo-Einstein manifold, then the scalar invariant
\[ Q_4^\prime := \lim_{n\to1} \frac{4}{(n-1)^2}P_{4,n}(1) \]
is well-defined. As a consequence, if $\hat\theta=e^\sigma\theta$ is another pseudo-Einstein contact form (in particular, $\sigma\in\mathcal{P}$), then
\begin{equation}
\label{eqn:cr_q_trans}
e^{2\sigma}\widehat{Q_4^\prime} = Q_4^\prime + P_4^\prime(\sigma) + \frac{1}{2}P_4(\sigma^2) .
\end{equation}
Taking the point of view that $P_4^\prime$ is a Paneitz-type operator, we may also write
\[ e^{2\sigma}\widehat{Q_4^\prime} = Q_4^\prime + P_4^\prime(\sigma) \mod \mathcal{P}^\perp . \]
The upshot is that, on the standard CR three-sphere, $Q_4^\prime=1$, so that this indeed recovers the interpretation of the Beckner--Onofri-type inequality~\eqref{eqn:cr_mt} of Branson--Fontana--Morpurgo~\cite{BransonFontanaMorpurgo2007} as an estimate involving some sort of Paneitz-type operator and $Q$-type curvature. Additionally, we also see from~\eqref{eqn:cr_q_trans} that the integral of $Q_4^\prime$ is a CR invariant; more precisely, if $(M^3,J)$ is a compact CR three-manifold and $\theta,\hat\theta$ are two pseudo-Einstein contact forms, then
\[ \int_M \widehat{Q_4^\prime}\,\hat\theta\wedge d\hat\theta = \int_M Q_4^\prime\, \theta\wedge d\theta . \]
In conformal geometry, the total $Q$-curvature plays an important role in controlling the topology of the underlying manifold. For instance, the total $Q$-curvature can be used to prove sphere theorems (e.g.\ \cite[Theorem~B]{Gursky1999} and~\cite[Theorem~A]{ChangGurskyYang2003}). We will prove the following CR analogue of Gursky's theorem~\cite[Theorem~B]{Gursky1999}.
\begin{thm}
\label{thm:qprime_upper_bound}
Let $(M^3,J,\theta)$ be a compact three-dimensional pseudo-Einstein manifold with nonnegative Paneitz operator and nonnegative CR Yamabe constant. Then
\[ \int_M Q_4^\prime \, \theta\wedge d\theta \leq \int_{S^3} Q_0^\prime\,\theta_0\wedge d\theta_0, \]
with equality if and only if $(M^3,J)$ is CR equivalent to the standard CR three sphere.
\end{thm}
Here, the CR Yamabe constant of a CR manifold $(M^3,J)$ is the infimum of the total Webster scalar curvature over all contact forms $\theta$ such that $\int\theta\wedge d\theta=1$ (cf.\ \cite{JerisonLee1987}). The proof of Theorem~\ref{thm:qprime_upper_bound} relies upon the existence of a CR Yamabe contact form --- that is, the existence of a smooth unit-volume contact form with constant Webster scalar curvature equal to the CR Yamabe constant~\cite{ChengMalchiodiYang2013,JerisonLee1987}. In particular, it relies on the CR Positive Mass Theorem~\cite{ChengMalchiodiYang2013}. One complication which does not arise in the conformal case~\cite{Gursky1999} is the possibility that the CR Yamabe contact form may not be pseudo-Einstein. We overcome this difficulty by computing how the local formula~\eqref{eqn:q4prime_crit} for $Q_4^\prime$ transforms with a general change of contact form; i.e.\ without imposing the pseudo-Einstein assumption. For details, see Section~\ref{sec:qprime}.
In conformal geometry, the total $Q$-curvature also arises when considering the Euler characteristic of the underlying manifold. Burns and Epstein~\cite{BurnsEpstein1988} have shown that there is a biholomorphic invariant, now known as the Burns--Epstein invariant, of the boundary of a strictly pseudoconvex domain which is related to the Euler characteristic of the domain in a similar way. It turns out that the Burns--Epstein invariant is a constant multiple of the total $Q^\prime$-curvature, and thus there is a nice relationship between the total $Q^\prime$-curvature and the Euler characteristic.
\begin{thm}
\label{thm:burns_epstein}
Let $(M^3,J)$ be a compact CR manifold which admits a pseudo-Einstein contact form $\theta$, and denote by $\mu(M)$ the Burns--Epstein invariant of $(M^3,J)$. Then
\[ \mu(M) = -16\pi^2\int_M Q^\prime\,\theta\wedge d\theta . \]
In particular, if $(M^3,J)$ is the boundary of a strictly pseudoconvex domain $X$, then
\[ \int_X\left(c_2 - \frac{1}{3}c_1^2\right) = \chi(X) - \frac{1}{16\pi^2}\int_M Q^\prime\,\theta\wedge d\theta , \]
where $c_1$ and $c_2$ are the first and second Chern forms of the K\"ahler--Einstein metric in $X$ obtained by solving Fefferman's equation and $\chi(X)$ is the Euler characteristic of $X$.
\end{thm}
While we were discussing a preliminary version of this work at Banff in Summer 2012, it was suggested to us by Kengo Hirachi that a version of Theorem~\ref{thm:burns_epstein} should be true. It was then pointed out to us by Jih-Hsin Cheng that Theorem~\ref{thm:burns_epstein} can be proved by using the formula given by Burns and Epstein~\cite{BurnsEpstein1988} (see also~\cite{ChengLee1990}) for their invariant. This fact has since been independently verified by Hirachi~\cite{Hirachi2013}, to which we refer the reader for the details of the verification of Theorem~\ref{thm:burns_epstein}.
Finally, we point out that much of the background described above generalizes to higher dimensions. On any even-dimensional Riemannian manifold $(M^{2n},g)$ there exists a pair $(P_{2n},Q_{2n})$ of a conformally-invariant differential operator $P_{2n}$ of the form $(-\Delta)^n$ plus lower order terms, the so-called GJMS operators~\cite{GJMS1992}, and scalar invariants $Q_{2n}$ of the form $(-\Delta)^{n-1}R$ plus lower-order terms, the so-called (critical) $Q$-curvatures~\cite{Branson1995}, which satisfy transformation rules analogous to~\eqref{eqn:conformal_p_trans} and~\eqref{eqn:conformal_q_trans}. On the standard $2n$-sphere, Beckner~\cite{Beckner1993} and Chang--Yang~\cite{ChangYang1995} showed that the analogue of~\eqref{eqn:conformal_mt} still holds, including the characterization of equality. Likewise, Branson, Fontana and Morpurgo~\cite{BransonFontanaMorpurgo2007} defined operators $P_{2n+2}^\prime$ on the standard CR $(2n+1)$-sphere which are CR invariant operators of order $2n+2$ and for which an analogue of~\eqref{eqn:cr_mt} holds, including the characterization of equality, where again $Q_{2n+2}^\prime$ are only identified as explicit constants. After a preliminary version of this article was presented at Banff in Summer 2012, Hirachi~\cite{Hirachi2013} showed how to use the ambient calculus to extend the $P^\prime$-curvature and $Q^\prime$-curvature to higher dimensions in such a way that the transformation formulae~\eqref{eqn:cr_p_trans} and~\eqref{eqn:cr_q_trans} hold. In a forthcoming work with Rod Gover, we produce tractor formulae for the $P^\prime$-operator and the $Q^\prime$-curvature. This allows us to produce for pseudo-Einstein manifolds with vanishing torsion a product formula for the $P^\prime$-operator and an explicit formula for the $Q^\prime$-curvature, giving a geometric derivation of the formulae given by Branson, Fontana and Morpurgo~\cite{BransonFontanaMorpurgo2007}.
This article is organized as follows. In Section~\ref{sec:bg}, we recall some basic definitions and facts in CR geometry, and in particular recall the depth of the analogy between aspects of conformal and CR geometry. In Section~\ref{sec:lee}, we introduce the notion of a pseudo-Einstein contact form on a three-dimensional CR manifold, and explore some basic properties of such forms. In Section~\ref{sec:defn}, we give a general formula for the Paneitz operator on a CR manifold $(M^{2n+1},J,\theta)$. We then use this formula to give the definitions of the $P^\prime$-operator and the $Q^\prime$-curvature, and establish some of their basic properties. In Section~\ref{sec:covariance}, we check by direct computation that the $P^\prime$-operator satisfies the correct transformation law. Indeed, this computation shows that $P^\prime$ no longer satisfies this rule if it is considered on a space strictly larger than the space of CR pluriharmonic functions. In Section~\ref{sec:qprime}, we check by direct computation that the $Q^\prime$-curvature satisfies the correct transformation law, and use this computation to prove Theorem~\ref{thm:qprime_upper_bound}. In the appendices, we will derive in two different ways the local formula for the CR Paneitz operator in general dimension. First, Appendix~\ref{sec:tractor} gives the derivation using the CR tractor calculus~\cite{GoverGraham2005}. Second, Appendix~\ref{sec:fefferman} gives the derivation using Lee's construction~\cite{Lee1986} of the Fefferman bundle.
\section{Pseudo-Einstein contact forms in three dimensions}
\label{sec:lee}
In~\cite{Lee1988}, Lee defined pseudo-Einstein manifolds as pseudohermitian manifolds $(M^{2n+1},J,\theta)$ such that $P_{\alpha\bar\beta}-\frac{1}{n}Ph_{\alpha\bar\beta}=0$ and studied their existence when $n\geq 2$. In particular, he showed in this case that $\theta$ is pseudo-Einstein if and only if it is locally volume-normalized with respect to a closed nonvanishing section of $K$; that is, using the terminology of Fefferman and Hirachi~\cite{FeffermanHirachi2003}, $\theta$ is pseudo-Einstein if and only if it is an invariant contact form. While Lee's definition of pseudo-Einstein contact forms is vacuous in dimension three, the notion of an invariant contact form is not. It turns out that, analogous to Proposition~\ref{prop:pluriharmonic}, there is a meaningful way to extend the notion of pseudo-Einstein contact forms to the case $n=1$ as a higher order condition on $\theta$ which retains the equivalence with invariant contact forms. As this notion will be essential to our discussion of the $Q^\prime$-curvature, and because it did not appear elsewhere in the literature at the time the work of this paper was being completed, we devote this section to explaining this three-dimensional notion of pseudo-Einstein contact forms.
\begin{defn}
\label{defn:pseudo_einstein}
A pseudohermitian manifold $(M^{2n+1},J,\theta)$ is said to be \emph{pseudo-Einstein} if
\begin{align*}
R_{\alpha\bar\beta} - \frac{1}{n}Rh_{\alpha\bar\beta} & = 0, \qquad \text{if $n\geq 2$}, \\
\nabla_\alpha R - i\nabla^\beta A_{\alpha\beta} & = 0, \qquad \text{if $n=1$}.
\end{align*}
\end{defn}
One way to regard this definition is as an analogue of Lee's characterization~\cite{Lee1988} of CR pluriharmonic functions from Proposition~\ref{prop:pluriharmonic}. Indeed, a straightforward computation using Lemma~\ref{lem:lee_bianchi} shows that
\begin{equation}
\label{eqn:divtrfreep}
\nabla^{\bar\beta}\left( R_{\alpha\bar\beta} - \frac{1}{n}Rh_{\alpha\bar\beta}\right) = \frac{n-1}{n}\left(\nabla_\alpha R - in\nabla^\beta A_{\alpha\beta}\right)
\end{equation}
holds for any CR manifold $(M^{2n+1},J,\theta)$. In particular, our definition of a three-dimensional pseudo-Einstein manifold can be regarded as the ``residue'' of the usual definition when $n\geq2$.
The characterization of pseudo-Einstein contact forms as invariant contact forms persists in the case $n=1$. To see this, let us first recall what it means for a contact form to be volume-normalized with respect to a section of $K$.
\begin{defn}
Given a CR manifold $(M^{2n+1},J)$ and a nonvanishing section $\omega$ of the canonical bundle $K$, we say that a contact form $\theta$ is \emph{volume-normalized with respect to $\omega$} if
\[ \theta\wedge(d\theta)^n = i^{n^2}n! \theta\wedge(T\contr\omega)\wedge(T\contr\overline{\omega}) . \]
\end{defn}
By considering all the terms in $d\omega_\alpha{}^\alpha$, Lee's argument~\cite[Theorem~4.2]{Lee1988} establishing the equivalence between pseudo-Einstein contact forms and invariant contact forms can be extended to the case $n=1$.
\begin{thm}
\label{thm:lee}
Let $(M^3,J)$ be a three-dimensional CR manifold. A contact form $\theta$ on $M$ is pseudo-Einstein if and only if for each point $p\in M$, there exists a neighborhood of $p$ in which there is a closed section of the canonical bundle with respect to which $\theta$ is volume-normalized.
\end{thm}
The main step in the proof of Theorem~\ref{thm:lee} is the following analogue of~\cite[Lemma~4.1]{Lee1988}.
\begin{lem}
\label{lem:lee}
Let $(M^3,J)$ be a three-dimensional CR manifold. A contact form $\theta$ on $M$ is pseudo-Einstein if and only if with respect to any admissible coframe $\{\theta,\theta^\alpha,\theta^{\bar\alpha}\}$ the one-form $\omega_\alpha{}^\alpha+iR\theta$ is closed.
\end{lem}
\begin{proof}
Using~\eqref{eqn:domegacontraction} and the assumption $n=1$, it holds in general that
\begin{equation}
\label{eqn:domega}
d\omega_\alpha{}^\alpha = Rh_{\alpha\bar\beta}\theta^\alpha\wedge\theta^{\bar\beta} + \nabla^\beta A_{\alpha\beta} \theta^\alpha\wedge\theta - \nabla^{\bar\beta}A_{\bar\alpha\bar\beta}\theta^{\bar\alpha}\wedge\theta .
\end{equation}
It thus follows that
\[ d\left(\omega_\alpha{}^\alpha+iR\theta\right) = 2i\Real\left((\nabla_\alpha R-i\nabla^\beta A_{\alpha\beta})\theta^\alpha\wedge\theta\right), \]
from which the conclusion follows immediately.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:lee}]
First suppose that $\theta$ is volume-normalized with respect to a closed section $\xi\in K$ on a neighborhood $U$ of $p$, and choose an admissible coframe $\{\theta,\theta^\alpha,\theta^{\bar\alpha}\}$ such that $d\theta=i\theta^\alpha\wedge\theta^{\bar\alpha}$. Since $\xi\in K$, there is a function $\lambda\in C^\infty(M,\mathbb{C})$ such that $\xi=\lambda\theta\wedge\theta^\alpha$. On the other hand, since $\theta$ is volume-normalized with respect to $\xi$, it must hold that $\lvert\lambda\rvert=1$. Thus, upon replacing $\theta^\alpha$ by $\lambda^{-1}\theta^\alpha$, we have that $\xi=\theta\wedge\theta^\alpha$.
Now, using the definition of the connection one-form $\omega_\alpha{}^\alpha$, it holds in general that
\begin{equation}
\label{eqn:dxi}
d\xi = -\omega_\alpha{}^\alpha\wedge\xi .
\end{equation}
Since $\xi$ is closed, this shows that $\omega_\alpha{}^\alpha$ is a $(1,0)$-form. But $\omega_\alpha{}^\alpha$ is also pure imaginary, hence $\omega_\alpha{}^\alpha=iu\theta$ for some $u\in C^\infty(M)$. Differentiating, we see that
\[ d\omega_\alpha{}^\alpha = -u\theta^\alpha\wedge\theta^{\bar\alpha} + i\nabla_\alpha u\theta^\alpha\wedge\theta + i\nabla_{\bar\alpha}u\theta^{\bar\alpha}\wedge\theta . \]
It thus follows from~\eqref{eqn:domega} that $R=-u$ and $\nabla^\beta A_{\alpha\beta}=i\nabla_\alpha u$. In particular, $\theta$ is pseudo-Einstein.
Conversely, suppose that $\theta$ is pseudo-Einstein. In a neighborhood of $p\in M$, let $\{\theta,\theta^\alpha,\theta^{\bar\alpha}\}$ be an admissible coframe such that $d\theta=i\theta^\alpha\wedge\theta^{\bar\alpha}$, and define $\xi_0=\theta\wedge\theta^\alpha\in K$. By~\eqref{eqn:dxi} it holds that $d\xi_0=-\omega_\alpha{}^\alpha\wedge\xi_0$, while by Lemma~\ref{lem:lee} there exists a function $\phi$ such that
\[ \omega_\alpha{}^\alpha + iR\theta = id\phi . \]
Since $\omega_\alpha{}^\alpha$ is pure imaginary, we can take $\phi$ to be real, whence $d\left(e^{i\phi}\xi_0\right)=0$. Since $\theta$ is volume-normalized with respect to $e^{i\phi}\xi_0$, this gives the desired section of $K$.
\end{proof}
Another nice property of pseudo-Einstein contact forms when $n\geq2$ is that, when they exist, they can be characterized in terms of CR pluriharmonic functions. This too persists in the case $n=1$, which is crucial to making sense of the $Q^\prime$-curvature. To see this, let $(M^3,J,\theta)$ be a pseudohermitian manifold and define the $(1,0)$-form $W_\alpha$ by
\[ W_\alpha := \nabla_\alpha R - i\nabla^\beta A_{\alpha\beta} . \]
Observe that $W_\alpha$ vanishes if and only if $\theta$ is pseudo-Einstein. As first observed by Hirachi~\cite{Hirachi1990}, $W_\alpha$ satisfies a simple transformation formula; given another contact form $\hat\theta=e^\sigma\theta$, a straightforward computation using Lemma~\ref{lem:cr_change} shows that
\begin{equation}
\label{eqn:Walpha}
\hat W_\alpha = W_\alpha - 3P_\alpha\sigma,
\end{equation}
where here we regard $W_\alpha\in\mathcal{E}_\alpha(-1,-1)$. An immediate consequence of~\eqref{eqn:Walpha} is the following correspondence between pseudo-Einstein contact forms and CR pluriharmonic functions.
\begin{prop}
\label{prop:pluriharmonic_psie}
Let $(M^3,J,\theta)$ be a pseudo-Einstein three-manifold. Then the set of pseudo-Einstein contact forms on $(M^3,J)$ is given by
\[ \left\{ e^u\theta \colon u \text{ is a CR pluriharmonic function} \right\} . \]
\end{prop}
Following~\cite{Lee1988}, there are topological obstructions to the existence of an invariant contact form $\theta$ on a three-dimensional CR manifold $(M^3,J)$. However, if $(M^3,J)$ is the boundary of a strictly pseudoconvex domain in $\mathbb{C}^2$, then there always exists a closed section of $K$, and hence a pseudo-Einstein contact form. This is a slight refinement of the observation by Fefferman and Hirachi~\cite{FeffermanHirachi2003} that any such CR manifold admits a contact form $\theta$ such that the $Q$-curvature
\begin{equation}
\label{eqn:Q_divergence}
Q := -\frac{4}{3}\nabla^\alpha W_\alpha
\end{equation}
vanishes.
\section{CR transformation property of the $Q^\prime$-curvature}
\label{sec:qprime}
In this section we give a direct computational proof of the transformation formula~\eqref{eqn:qprime_operator_covariant} relating the $Q^\prime$-curvatures of two pseudo-Einstein contact forms on the same CR manifold. As in Section~\ref{sec:covariance}, we will in fact compute how the scalar~\eqref{eqn:q4prime_crit} transforms under a conformal change of contact form without assuming either contact form is pseudo-Einstein. This has two benefits. First, it makes clear that the $Q^\prime$-curvature only transforms as in~\eqref{eqn:qprime_operator_covariant} when both contact forms are pseudo-Einstein, as opposed to having vanishing $Q$-curvature. Second, it will allow us to prove Theorem~\ref{thm:qprime_upper_bound} by appealing to the resolution of the CR Yamabe Problem~\cite{ChengMalchiodiYang2013,JerisonLee1987,JerisonLee1988}.
First, as a consequence of Lemma~\ref{lem:cr_change}, we see that if $\hat\theta=e^\sigma\theta$, then
\begin{align*}
\widehat{R}^2 & = R^2 + 4R\Delta_b\sigma + 4(\Delta_b\sigma)^2 - 4R\lvert\nabla_\gamma\sigma\rvert^2 - 8\lvert\nabla_\gamma\sigma\rvert^2\Delta_b\sigma + 4\lvert\nabla_\gamma\sigma\rvert^4 \\
4\lvert\widehat{A_{\alpha\beta}}\rvert^2 & = 4\lvert A_{\alpha\beta}\rvert + 8\Imaginary\left(A_{\alpha\beta}\nabla^\alpha\nabla^\beta\sigma\right) + 4\lvert\nabla_\alpha\nabla_\beta\sigma\rvert^2 \\
& \quad - 8\Imaginary\left(A_{\alpha\beta}\nabla^\alpha\sigma\nabla^\beta\sigma\right) - 8\Real\left((\nabla_{\alpha\beta}\sigma)\nabla^\alpha\sigma\nabla^\beta\sigma\right) + 4\lvert\nabla_\gamma\sigma\rvert^4 \\
2\widehat{\Delta_b} \widehat{R} & = 2\Delta_b\left(P+2\Delta_b\sigma-2\lvert\nabla_\gamma\sigma\rvert^2\right) + 4\Real\nabla^\beta\left(\left(R+2\Delta_b\sigma-2\lvert\nabla_\gamma\sigma\rvert^2\right)\nabla_\beta\sigma\right) .
\end{align*}
It is immediately clear that the transformation law for $Q_4^\prime$ depends at most quadratically on $\sigma$. Using the three-dimensional Bochner formula (cf.\ \cite{ChangChiu2007,ChangTieWu2010,ChanilloChiuYang2010,Chiu2006})
\begin{align*}
-\Delta_b\lvert\nabla_\gamma\sigma\rvert^2 & = 2\nabla_\alpha\nabla_\beta\sigma\nabla^\alpha\nabla^\beta\sigma + 2\nabla_\alpha\nabla^\alpha\sigma\nabla^\beta\nabla_\beta\sigma - \langle\nabla_b\sigma,\nabla_b\Delta_b\sigma\rangle \\
& \quad - 2\Real\left(\nabla^\alpha\sigma(\nabla_\alpha\nabla_\beta\nabla^\beta\sigma-\nabla^\beta\nabla_\beta\nabla_\alpha\sigma)\right)
\end{align*}
together with the consequence
\begin{align*}
\frac{1}{2}P_4(\sigma^2) - \sigma P_4(\sigma) & = 8\Real\left(\nabla^\alpha\sigma P_\alpha\sigma\right) + 8\Real\left(\nabla^\alpha\sigma\nabla^\beta\nabla_\beta\nabla_\alpha\sigma\right) \\
& \quad + 4\nabla^\alpha\nabla^\beta\nabla_\alpha\nabla_\beta\sigma + 8\nabla_\alpha\nabla^\alpha\sigma\nabla^\beta\nabla_\beta\sigma - R\lvert\nabla_\gamma\sigma\rvert^2
\end{align*}
of Lemma~\ref{lem:cr_prod}, it follows immediately that the term of $\hat Q_4^\prime$ which is quadratic in $\sigma$ is given by
\[ U(\sigma) := \frac{1}{2}P_4(\sigma^2) - \sigma P_4(\sigma) - 16\Real\left(\nabla^\alpha\sigma P_\alpha\sigma\right) . \]
In particular, if $\sigma\in\mathcal{P}$, then
\[ U(\sigma) = \frac{1}{2}P_4(\sigma^2) , \]
as expected.
On the other hand, the term of $\hat Q_4^\prime$ which is linear in $\sigma$ is given by
\begin{align*}
V(\sigma) & := 4\Delta_b^2\sigma - 8\Imaginary\left(\nabla^\alpha(A_{\alpha\beta}\nabla^\beta\sigma)\right) - 4\Real\left(\nabla^\alpha(R\nabla_\alpha\sigma)\right) + 8\Real\left(W_\alpha\nabla^\alpha\sigma\right) \\
& = P_4^\prime(\sigma) + \frac{16}{3}\Real\left(W_\alpha\nabla^\alpha\sigma\right) - Q\sigma \\
& = P_4^\prime(\sigma) + \frac{16}{3}\Real\nabla^\alpha\left(\sigma W_\alpha\right) + 3Q\sigma .
\end{align*}
In particular, if $\theta$ is a pseudo-Einstein contact form, then
\[ V(\sigma) = P_4^\prime(\sigma), \]
as expected. In fact, we have computed the general transformation formula for the scalar invariant
\begin{equation}
\label{eqn:general_q4prime_defn}
Q_4^\prime = 2\Delta_b R - 4\lvert A_{\alpha\beta}\rvert^2 + R^2 .
\end{equation}
\begin{prop}
Let $(M^3,J,\theta)$ be a three-dimensional pseudohermitian manifold, regard $P_4^\prime$ as an operator $P_4^\prime\colon C^\infty(M)\to C^\infty(M)$, and define $Q_4^\prime$ by~\eqref{eqn:general_q4prime_defn}. Given any $\sigma\in C^\infty(M)$, the scalars $Q_4^\prime$ and $\hat Q_4^\prime$ defined in terms of the contact forms $\theta$ and $\hat\theta=e^\sigma\theta$, respectively, are related by
\begin{equation}
\label{eqn:general_q4prime}
\begin{split}
e^{2\sigma}\hat Q_4^\prime & = Q_4^\prime + P_4^\prime(\sigma) + \frac{16}{3}\Real\nabla^\alpha\left(\sigma W_\alpha\right) + 3Q\sigma \\
& \quad + \frac{1}{2}P_4(\sigma^2) - \sigma P_4(\sigma) - 16\Real\left((\nabla^\alpha\sigma)(P_\alpha\sigma)\right) .
\end{split}
\end{equation}
In particular, if $M$ is compact, then
\begin{equation}
\label{eqn:general_integral_q4prime}
\int_M \hat Q_4^\prime\,\hat\theta\wedge d\hat\theta = \int_M Q_4^\prime\,\theta\wedge d\theta + 3\int_M \left(\sigma P_4\sigma + 2Q\sigma\right)\theta\wedge d\theta
\end{equation}
for $Q=P_4^\prime(1)$ Hirachi's $Q$-curvature~\eqref{eqn:Q_divergence}.
\end{prop}
\begin{proof}
\eqref{eqn:general_q4prime} follows from the computations given above. \eqref{eqn:general_integral_q4prime} follows by integration by parts.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:qprime_upper_bound}]
Let $\hat\theta$ be a CR Yamabe contact form; that is, suppose that $\Vol_{\hat\theta}(M)=1$ and $R_{\hat\theta}=\Lambda[\theta]$ for $\Lambda[\theta]$ the CR Yamabe constant of $(M^3,J,\theta)$. Then
\begin{equation}
\label{eqn:basic_sphere_step1}
\int_M \hat R^2\,\hat\theta\wedge d\hat\theta = \Lambda[\theta]^2 \leq \Lambda[S^3]^2
\end{equation}
for $\Lambda[S^3]=\Vol(S^3)$ the CR Yamabe constant of the standard CR three-sphere. Moreover, by the CR Positive Mass Theorem~\cite{ChengMalchiodiYang2013}, equality holds in~\eqref{eqn:basic_sphere_step1} if and only if $(M^3,J,\theta)$ is CR equivalent to the standard CR three-sphere. On the other hand, the nonnegativity of the CR Paneitz operator combined with~\eqref{eqn:general_integral_q4prime} yield
\begin{equation}
\label{eqn:basic_sphere_step2}
\int_M Q_4^\prime\,\theta\wedge d\theta \leq \int_M \hat Q_4^\prime\,\hat\theta\wedge d\hat\theta,
\end{equation}
while the expression~\eqref{eqn:general_q4prime_defn} yields
\begin{equation}
\label{eqn:basic_sphere_step3}
\int_M \hat Q_4^\prime\,\hat\theta\wedge d\hat\theta \leq \int_M \hat R^2\,\hat\theta\wedge d\hat\theta .
\end{equation}
The result then follows from~\eqref{eqn:basic_sphere_step1}, \eqref{eqn:basic_sphere_step2}, and~\eqref{eqn:basic_sphere_step3}.
\end{proof}
\section{CR tractor bundles and the CR Paneitz operator}
\label{sec:tractor}
In this appendix we give the derivation of the CR Paneitz operator in general dimension using tractor bundles in CR geometry, as outlined by Gover and Graham~\cite{GoverGraham2005}. In the interests of brevity, we compute in a fixed scale $\theta$ and only state the necessary tractor formulae, and refer the reader to~\cite{GoverGraham2005} for definitions of the tractor bundles and operators we use here.
The main objects we are concerned with are the CR tractor bundle $\mathcal{E}_A\cong\mathcal{E}(1,0)\oplus\mathcal{E}_\alpha(1,0)\oplus\mathcal{E}(0,-1)$, its canonical connection, and the tractor-$D$ operator $\mathbb{D}\colon\mathcal{E}^\ast(w,w^\prime)\to\mathcal{E}_A\otimes\mathcal{E}^\ast(w-1,w^\prime)$, which are given by
\begin{align*}
\nabla_\beta\begin{pmatrix}\sigma\\\tau_\alpha\\\rho\end{pmatrix} & = \begin{pmatrix}\nabla_\beta\sigma-\tau_\beta\\\nabla_\beta\tau_\alpha+i\sigma A_{\alpha\beta}\\\nabla_\beta\rho - P_\beta{}^\alpha\tau_\alpha + \sigma T_\beta\end{pmatrix} \\
\nabla_{\bar\beta}\begin{pmatrix}\sigma\\\tau_\alpha\\\rho\end{pmatrix} & = \begin{pmatrix}\nabla_{\bar\beta}\sigma\\\nabla_{\bar\beta}\tau_\alpha+\sigma P_{\alpha\bar\beta}+\rho h_{\alpha\bar\beta}\\\nabla_{\bar\beta}\rho+iA^\alpha{}_{\bar\beta}\tau_\alpha-\sigma T_{\bar\beta}\end{pmatrix} \\
\nabla_0\begin{pmatrix}\sigma\\\tau_\alpha\\\rho\end{pmatrix} & = \begin{pmatrix}\nabla_0\sigma+\frac{i}{n+2}P\sigma-i\rho\\\nabla_0\tau_\alpha-iP_\alpha{}^\beta\tau_\beta+\frac{i}{n+2}P\tau_\alpha+2i\sigma T_\alpha \\\nabla_0\rho + \frac{i}{n+2}P\rho + 2iT^\alpha\tau_\alpha+iS\sigma\end{pmatrix} \\
\mathbb{D}_A f & = \begin{pmatrix} w(n+w+w^\prime)f\\(n+w+w^\prime)\nabla_\alpha f\\-\left(\nabla^\beta\nabla_\beta f + iw\nabla_0 f + w(1+\frac{w^\prime-w}{n+2})Pf\right)\end{pmatrix} ,
\end{align*}
where $\mathcal{E}^\ast(w,w^\prime)$ denotes any (weighted) tractor bundle. As always, the topmost nonvanishing slot is CR invariant. In particular, we see that the bottom row of $\mathbb{D}_A\mathbb{D}_Bf$ is the topmost nonvanishing row if $n+w+w^\prime=1$; as is straightforward to check, if we assume that $f\in\mathcal{E}(w,w^\prime)$ for $n+w+w^\prime=1$, then the ``bottom left'' spot is the only nonvanishing term, and hence is CR invariant. More precisely, the operator $P_4$ defined by
\begin{equation}
\label{eqn:tractor_paneitz}
-\left(\nabla^\beta\nabla_\beta+i(w-1)\nabla_0+(w-1)(1+\frac{w^\prime-w+1}{n+2})P\right)\mathbb{D}_A f = \begin{pmatrix} 0\\0\\\frac{1}{4}P_4f\end{pmatrix}
\end{equation}
will necessarily be a CR covariant operator which, as we shall see, has leading order term $\Delta_b^2+T^2$ (this is the reason for the factor of $4$). To get the usual CR Paneitz operator, we need to assume further that $w=w^\prime$; in particular, $w=-\frac{n-1}{2}$.
In order to evaluate~\eqref{eqn:tractor_paneitz} to determine $P_4$, we need to know the bottom components of both $\nabla_0\mathbb{D}_A f$ and $\nabla^\beta\nabla_\beta\mathbb{D}_A f$. The latter is the most involved computation: Marking irrelevant terms by asterisks, we see that
\begin{align*}
\nabla^\beta\nabla_\beta\begin{pmatrix}\sigma\\\tau_\alpha\\\rho\end{pmatrix} & = h^{\beta\bar\gamma}\nabla_{\bar\gamma}\nabla_\beta\begin{pmatrix}\sigma\\\tau_\alpha\\\rho\end{pmatrix} \\
& = h^{\beta\bar\gamma}\nabla_{\bar\gamma}\begin{pmatrix}\nabla_\beta\sigma-\tau_\beta\\\nabla_\beta\tau_\alpha+i\sigma A_{\alpha\beta}\\\nabla_\beta\rho - P_\beta^\alpha\tau_\alpha + \sigma T_\beta\end{pmatrix} \\
& = \begin{pmatrix}\ast\\\ast\\\nabla^\beta\nabla_\beta\rho - \nabla^\beta(P_\beta^\alpha\tau_\alpha - \sigma T_\beta) + iA^{\alpha\beta}\nabla_\beta\tau_\alpha - \sigma A^{\alpha\beta}A_{\alpha\beta}-(\nabla_\beta\sigma-\tau_\beta) T^\beta\end{pmatrix}
\end{align*}
In particular, we see that the bottom component of $\nabla^\beta\nabla_\beta\mathbb{D}_Af$ is given by
\begin{equation}
\label{eqn:tractor_laplacian_special}
\begin{split}
& -\nabla^\beta\nabla_\beta(\nabla^\alpha\nabla_\alpha f+iw\nabla_0f+wPf)-\nabla^\beta\left(P_\beta^\alpha\nabla_\alpha f-wfT_\beta\right) \\
& \quad + iA^{\alpha\beta}\nabla_\beta\nabla_\alpha f - wf A^{\alpha\beta}A_{\alpha\beta} - (w-1)T^\beta\nabla_\beta f .
\end{split}
\end{equation}
The other derivative we must compute is $\nabla_0\mathbb{D}_A f$; it is straightforward to check that the bottom component is given by
\begin{equation}
\label{eqn:tractor_0_special}
\begin{split}
& -\nabla_0\left(\nabla^\beta\nabla_\beta f + iw\nabla_0f + wPf\right) - \frac{i}{n+2}P\left(\nabla^\beta\nabla_\beta f + iw\nabla_0f + wPf\right) \\
& \quad + 2iT^\alpha\nabla_\alpha f+iwSf .
\end{split}
\end{equation}
Evaluating~\eqref{eqn:tractor_paneitz} using~\eqref{eqn:tractor_laplacian_special} and~\eqref{eqn:tractor_0_special}, we thus find that (after identifying tractor terms with their bottom components)
\begin{align*}
\frac{1}{4}P_4^\prime f & = -\nabla^\beta\nabla_\beta\mathbb{D}_A f - i(w-1)\nabla_0\mathbb{D}_A f - \frac{(w-1)(n+3)}{n+2}P\mathbb{D}_A f \\
& = \nabla^\beta\nabla_\beta(\nabla^\alpha\nabla_\alpha f+iw\nabla_0f+wPf) + \nabla^\beta\left(P_\beta^\alpha\nabla_\alpha f-wfT_\beta\right) \\
& \quad - iA^{\alpha\beta}\nabla_\beta\nabla_\alpha f + wf A^{\alpha\beta}A_{\alpha\beta} + (w-1)T^\beta\nabla_\beta f \\
& \quad + i(w-1)\nabla_0\left(\nabla^\beta\nabla_\beta f + iw\nabla_0f + wPf\right) + (w-1)P\left(\nabla^\beta\nabla_\beta f + iw\nabla_0f + wPf\right) \\
& \quad + 2(w-1)T^\alpha\nabla_\alpha f + w(w-1)Sf .
\end{align*}
Our goal is now to simplify this so that we can identify the CR Paneitz operator. Towards that end, let us regroup terms into those with a $w$ coefficient and those without; in other words, write
\begin{equation}
\label{eqn:lf_split}
P_4^\prime f = Af + wBf
\end{equation}
for
\begin{align*}
\frac{1}{4}Af & = \nabla^\beta\nabla_\beta\nabla^\alpha\nabla_\alpha f - i\nabla_0\nabla^\beta\nabla_\beta f \\
& \quad + \nabla^\beta\left(P_\beta^\alpha\nabla_\alpha f\right) - iA^{\alpha\beta}\nabla_\beta\nabla_\alpha f - 3T^\beta\nabla_\beta f - P\nabla^\beta\nabla_\beta f \\
\frac{1}{4}Bf & = -(w-1)\nabla_0\nabla_0 f + i\nabla_0\nabla^\beta\nabla_\beta f + i\nabla^\beta\nabla_\beta\nabla_0f \\
& \quad + \nabla^\beta\nabla_\beta(Pf) - \nabla^\beta(T_\beta f) + A^{\alpha\beta}A_{\alpha\beta} f + 3T^\beta\nabla_\beta f + i(w-1)\nabla_0(Pf) \\
& \quad + P\nabla^\beta\nabla_\beta f + i(w-1)P\nabla_0f + (w-1)P^2f + (w-1)Sf .
\end{align*}
First, let us rewrite $Af$ in a more familiar way. Using~\eqref{eqn:schouten_bianchi} and~\eqref{eqn:subplacian_squared}, it is straightforward to check that
\begin{align*}
\frac{1}{4}Af & = \frac{1}{4}Cf + i(n-1)\nabla_0\nabla^\beta\nabla_\beta f + i(n-1)A^{\alpha\beta}\nabla_\beta\nabla_\alpha f + in\left(\nabla_\beta A^{\alpha\beta}\right)\nabla_\alpha f \\
& \quad + (\nabla^\beta P+(n-1)T^\beta)\nabla_\beta f + (P^{\alpha\bar\beta}-Ph^{\alpha\bar\beta})\nabla_{\bar\beta}\nabla_\alpha f - 3T^\beta\nabla_\beta f \\
& = \frac{1}{4}Cf + \tracefree{P}{}^{\alpha\bar\beta}\nabla_{\bar\beta}\nabla_\alpha f \\
& \quad + \frac{n-1}{2}\bigg[2i\nabla_0\nabla^\beta\nabla_\beta f +2i\nabla_\beta(A^{\alpha\beta}\nabla_\alpha f) - \frac{2}{n}P\nabla^\beta\nabla_\beta f + 4T^\beta\nabla_\beta f\bigg] .
\end{align*}
Since $\tracefree{P_{\alpha\bar\beta}}=0$ and $w=0$ when $n=1$, we check in particular that $P_4f=Cf$ in this dimension.
Second, recalling that $w=-\frac{n-1}{2}$, we see from the above that
\begin{align*}
Af & = Cf + 4\tracefree{P}{}^{\alpha\bar\beta}\nabla_{\bar\beta}\nabla_\alpha f + wEf \\
\frac{1}{4}Ef & := -2i\nabla_0\nabla^\beta\nabla_\beta f - 2i\nabla_\beta(A^{\alpha\beta}\nabla_\alpha f) + \frac{2}{n}P\nabla^\beta\nabla_\beta f - 4T^\beta\nabla_\beta f .
\end{align*}
In particular, the operator $F$ defined by $F=B+E$ is such that $P_4f=Cf+4\tracefree{P}{}^{\alpha\bar\beta}\nabla_{\bar\beta}\nabla_\alpha f + wFf$, and is given by
\begin{align*}
\frac{1}{4}Ff & = (1-w)\nabla_0\nabla_0 f + i\nabla^\beta\nabla_\beta\nabla_0 f - i\nabla_0\nabla^\beta\nabla_\beta f - 2i\nabla_\beta(A^{\alpha\beta}\nabla_\alpha f) \\
& \quad + \frac{2(n+1)}{n}P\nabla^\beta\nabla_\beta f + 2i(w-1)P\nabla_0 f + \nabla^\beta P \nabla_\beta f + \nabla_\beta P \nabla^\beta f - T^\beta\nabla_\beta f - T_\beta\nabla^\beta f \\
& \quad + \left(\nabla^\beta\nabla_\beta P - \nabla^\beta T_\beta + i(w-1)\nabla_0 P + A^{\alpha\beta}A_{\alpha\beta} + (w-1)P^2 + (w-1)S\right)f \\
& = (1-w)\nabla_0\nabla_0f + i\nabla^\beta(A_{\alpha\beta}\nabla^\alpha f) - i\nabla_\beta(A^{\alpha\beta}\nabla_\alpha f) \\
& \quad + \frac{2(n+1)}{n}P\nabla^\beta\nabla_\beta f + 2i(w-1)P\nabla_0 f + \nabla^\beta P \nabla_\beta f + \nabla_\beta P \nabla^\beta f - T^\beta\nabla_\beta f - T_\beta\nabla^\beta f \\
& \quad + \left(\nabla^\beta\nabla_\beta P - \nabla^\beta T_\beta + i(w-1)\nabla_0 P + A^{\alpha\beta}A_{\alpha\beta} + (w-1)P^2 + (w-1)S\right)f .
\end{align*}
Writing this entirely in terms of $n$, $P_{\alpha\bar\beta}$, $P$, and $A_{\alpha\beta}$ then yields the desired form. |
3,212,635,537,408 | arxiv | \section{Introduction}
Cen X--3 is an X-ray pulsar with an O-type supergiant companion. The
orbital period is $\sim 2.1$ days and eclipses are observed. Out of
eclipse, an iron emission line was detected at $6.5 \pm 0.1$ keV
(Nagase et al., 1992; Burderi et al., 2000). Nagase et al. (1992)
suggested that the feature at 6.5 keV, observed in the Ginga spectrum,
could be fitted by two Gaussian lines centered at 6.4 keV and 6.67
keV, respectively, with the latter stronger than the former during the
eclipse. The 6.5 keV line was found to be pulsating, supporting the
fluorescence origin (Day et al., 1993) and implying that the
fluorescence region does not uniformly surround the neutron star.
Because of the large X-ray luminosity ($10^{37}-10^{38}$ erg s$^{-1}$)
and a strong stellar wind from the companion, Day \& Stevens (1993)
proposed that photoionization of the circumstellar wind by X-ray
irradiation will be significant in Cen X--3 system. Therefore, we
expect the presence of emission lines due to recombination in highly
ionized plasma. Ebisawa et al. (1996), using ASCA data taken at
different orbital phases, identified the presence of three emission
lines centered at around 6.40 keV (\ion{Fe}{1}), 6.67 keV
(\ion{Fe}{25}), and 6.95 keV (\ion{Fe}{26}), respectively. The
intrinsic width of each line was fixed at 0.01 keV, which was much
smaller than the instrumental resolution. The equivalent widths (EWs)
associated to the three lines were 105 eV, 78 eV, and 43 eV,
respectively, at orbital phases between 0.14 and 0.18. Ebisawa et al.
(1996) suggested that the line at 6.65 keV could be a blend of three
lines at 6.63 keV, 6.67 keV, and 6.70 keV. The simultaneous presence
of the \ion{Fe}{25} and \ion{Fe}{26} lines in the spectrum implied a
ionization parameter of the photoionized plasma of $\xi \sim
10^{3.4}$.
The common idea is that the K$_\alpha$ neutral iron line is produced
in a low ionized region near the neutron star surface, because during
the eclipse this line is weaker than out of the eclipse. On the other
hand the \ion{Fe}{25} and \ion{Fe}{26} lines are produced in a
region far from the neutron star, probably in the photoionized wind of
the companion star, because the intensities of these lines do not
change with the orbital phase.
Recently Wojdowski et al. (2003), using Chandra data, analysed the
spectrum of Cen X--3 during the eclipse. They resolved the
\ion{Si}{13} triplet and partially the \ion{Fe}{25} triplet,
concluding that the helium-like triplet component flux ratios outside
of eclipse are consistent with emission from recombination and
subsequent cascades (recombination radiation) from a photoionized
plasma with a temperature of 100 eV. The best-fit velocity shifts and
(Gaussian $\sigma$) velocity widths are generally less than 500 km
s$^{-1}$. These velocities are significantly smaller than terminal
velocities of isolated O star winds [$(1-2) \times 10^3$ km s$^{-1}$;
e.g., Lamers et al. 1999].
In this work we present a spectral analysis of Cen X--3 in the 6--7
keV energy range from a 45 ks Chandra observation. The observation
covers the orbital-phase interval 0.13--0.40. We detect the presence
of the K$_\alpha $ neutral iron line at 6.4 keV and, for the first
time, we resolve the triplet associated to the He-like ion of
\ion{Fe}{25}.
\section{Observation}
Cen X--3 was observed with the Chandra observatory on 2000 Dec 30 from
00:13:30 to 13:31:53 UT using the HETGS. The observation has a total
integration time of 45.3 ks, and was performed in timed graded mode.
The HETGS consists of two types of transmission gratings, the Medium
Energy Grating (MEG) and the High Energy Grating (HEG). The HETGS
affords high-resolution spectroscopy from 1.2 to 31 \AA\ (0.4--10 keV)
with a peak spectral resolution of $\lambda/\Delta \lambda \sim 1000$
at 12 \AA\ for HEG first order. The dispersed spectra were recorded
with an array of six charge-coupled devices (CCDs) which are part of
the Advanced CCD Imaging Spectrometer-S (Garmire et al.,
2003)\footnote{See http://asc.harvard.edu/cdo/about\_chandra for more
details.}. The current relative accuracy of the overall wavelength
calibration is on the order of 0.05\%, leading to a worst-case
uncertainty of 0.004 \AA\ in the 1st-order MEG and 0.006 \AA\ in the
1st-order HEG. We processed the event list using available software
(FTOOLS and CIAO v3.2 packages). We computed aspect-corrected
exposure maps for each spectrum, allowing us to correct for effects
from the effective area of the CCD spectrometer.
The brightness of the source required additional efforts to mitigate
``photon pileup'' effects. A 349 row ``subarray'' (with the first row
= 1) was applied during the observation that reduced the CCD frame
time to 1.3 s. The zeroth-order image is affected by heavy pileup: the
event rate is so high that two or more events are detected in the CCD
during the 1.3 s frame exposure. Pileup distorts the count spectrum
because detected events overlap and their deposited charges are
collected into single, apparently more energetic, events. Moreover,
many events ($\sim 90 \%$) are lost as the grades of the piled up
events overlap those of highly energetic background particles and are
thus rejected by the on board software. We therefore will ignore the
zeroth-order events in the subsequent analysis. On the other hand,
the grating spectra are not, or only moderately (less than 10 \%),
affected by pileup. In this work we utilize the HEG 1st-order
spectrum in order to study the 6--7 keV energy range.
To determine the zero-point position in the image as precisely as
possible, we calculated the mean crossing point of the zeroth-order
readout trace and the tracks of the dispersed HEG and MEG arms. This
results in the following source coordinates: R.A.= $11^h21^m15^s.095$,
DEC=-60$^{\circ}$37\arcmin 25\arcsec.53 (J2000.0, with a 90\%
uncertainty circle of the absolute position of 0.6\arcsec\footnote{See
http://cxc.harvard.edu/cal/ASPECT/celmon/ for more details.}). Note
that the Chandra position of Cen X--3 is distant $\sim$1.6\arcsec\ from,
and fully compatible with, the coordinates previously reported (Bradt \&
McClintock, 1983) based on the measure of the
optical counterpart.
Finally we used the ephemeris of Nagase et al. (1992) to determine
which orbital phase interval was covered by our observation. The
observation covers the phases between 0.13 and 0.40; it was taken just
after the egress from the eclipse, in the high, post-egress phase (see
Nagase et al., 1992).
\section{Spectral Analysis}
We selected the 1st-order spectra from the HEG. Data were extracted
from regions around the grating arms; to avoid overlapping between HEG
and MEG data, we used a region size of 26 pixels for the HEG along the
cross-dispersion direction. The background spectra were computed, as
usual, by extracting data above and below the dispersed flux. The
contribution from the background is $0.4 \%$ of the total count rate.
We used the standard CIAO tools to create detector response files
(Davis 2001) for the HEG -1 and HEG +1 order (background-subtracted)
spectra. After verifying that these two spectra were compatible with
each other in the whole energy range we coadded them using the script
{\it add\_grating\_spectra} in the CIAO software; the resulting
spectrum was rebinned to 0.0075 \AA. Initially we fitted the 6--8 keV
energy spectrum corresponding to the whole observation (orbital phases
$\phi_{orb}=0.13-0.40$) using a power-law component as a continuum.
We fixed the equivalent hydrogen column to N$_H = 1.95 \times 10^{22}$
cm$^{-2}$, and the photon index to 1.2 as obtained by Burderi et al.
(2000) from a BeppoSAX observation taken at similar orbital phases.
The absorbed flux was $\sim 6.5 \times 10^{-9}$ ergs cm$^{-2}$
s$^{-1}$ in the 2--10 keV energy band, similar to $5.7 \times 10^{-9}$
ergs cm$^{-2}$ s$^{-1}$ observed by Burderi et al. (2000) and one
order of magnitude larger than that measured by ASCA at the orbital
phase 0.14--0.18 ($\sim 8.4 \times 10^{-10}$ ergs cm$^{-2}$ s$^{-1}$;
see Ebisawa et al., 1996). In Fig.~\ref{fig2} (left panel) we show
the spectrum and the residuals with respect to this model in the
6--7.6 keV energy range. It is evident the presence of a K$_\alpha$
neutral iron emission line at 6.4 keV with a width of 0.012 keV, and
the presence of an absorption edge at 7.19 keV associated to low
ionized iron (\ion{Fe}{2}--\ion{Fe}{6}). We note that near 6.65 keV a
more complex structure is present in the residuals; three peaks are
evident. For this reason we used three Gaussian lines to fit these
residuals. The energies of the lines are 6.61 keV, 6.67 keV, and 6.72
keV, the corresponding widths are $<0.018$ keV, $<0.023$ keV, and
$<0.037$ keV; finally, the corresponding EWs are 6 eV, 8.8 eV, and 4.8
eV. In Fig.~\ref{fig2} (right panel) we show the data and the
residuals in the 6--7.6 keV energy range after adding the emission
lines described above.
In Fig.~\ref{fig3} (left panel) we note the presence of a feature at 2
keV. We fitted this feature using two Gaussian lines centered at 2
keV and 2.006 keV (corresponding to Ly$_{\alpha_{2}}$ \ion{Si}{14} and
Ly$_{\alpha_{1}}$ \ion{Si}{14}, respectively). The widths of these
two lines are around 2 eV. In Table \ref{tab1} we report the
parameters of the emission lines and of the absorption edge.
The EW of the neutral iron K$_\alpha$ line is $\sim 13$ eV, that is a
factor 8 and 14 lower than the EW of the neutral iron K$_\alpha$ line
obtained during the ASCA (Ebisawa et al., 1996) and the Ginga
observation (Nagase et al., 1992) taken in the egress and post-egress
phases, respectively, and in which the unabsorbed flux of the
continuum emission, in the 2--10 keV energy range, was $\sim 1 \times
10^{-9}$ and $\sim 4 \times 10^{-9}$ ergs cm$^{-2}$ s$^{-1}$,
respectively. It is a common idea that the line at 6.4 keV is
produced in the inner region of the system and that during dips and
eclipses its flux decreases proportionally to the flux of the
continuum, leaving unchanged the EW of the line at $\sim 100$ eV.
At the light of these results we have reanalyzed the previous Chandra
observation (start time: 2000 Mar 5) of Cen X--3 during the
pre-eclipse phases (between -0.16 and -0.12), already studied by
Wojdowski et al. (2003) and corresponding to the "interval b" in
their work (see Fig. 1 in Wojdowski et al., 2003). We fitted the
1st-order MEG and HEG data using a power-law component absorbed by an
equivalent hydrogen column density fixed to $ 1.95 \times 10^{22}$
cm$^{-2}$ and by a partial covering component with an equivalent
hydrogen column density of $(1.2 \pm 0.4) \times 10^{23}$ cm$^{-2}$
and a covered fraction of the emitting region of $85 \pm 6$ \%. The
photon index of the power law was 0.43 and the unabsorbed flux in the
2--10 keV energy band was $\sim 1.8 \times 10^{-9}$ ergs cm$^{-2}$
s$^{-1}$. As already discussed by Wojdowski et al. (2003), this
spectrum shows a neutral iron K$_\alpha$ emission line at $ 6.384 \pm
0.013$ keV with a width of $\sigma < 39$ eV, a normalization of $9.4
\times 10^{-4}$ ph cm$^{-2}$ s$^{-1}$, and a EW of $42 \pm 17$ eV,
respectively. As done by Ebisawa et al. (1996) for the spectrum
during the pre-eclipse state, we also added the lines centered at 6.67
and 6.97 keV associated to \ion{Fe}{25} and \ion{Fe}{26}, fixing the
centroids at the expected values and the widths to 10 eV. We found
upper limits on the line intensities of $\sim 4.2 \times 10^{-4}$ and
$\sim 2.1 \times 10^{-4}$ ph cm$^{-2}$ s$^{-1}$; moreover the inferred
EWs are $<22$ and $<11$ eV, respectively for the \ion{Fe}{25} and
\ion{Fe}{26} emission lines. We conclude that these results are in
absolute agreement with those reported by Ebisawa et al. (1996).
However, the relatively low value (42 eV) of the EW of the neutral
iron K$_\alpha$ emission line is indeed intermediate between the value
measured by ASCA (75 eV) and the one we measure with Chandra (13 eV)
and might indicate a trend of decreasing EW in 2000.
\section{Discussion}
We analyzed a 45 ks Chandra observation of the high mass X-ray binary
Cen X--3. The position of the zeroth-order image of the source
provides improved X-ray coordinates for Cen X--3 (R.A.=
$11^h21^m15^s.095$, DEC=-60$^{\circ}$37\arcmin 25\arcsec.53),
compatible with the optical coordinates previously reported for this
source (see Bradt \& McClintock, 1983). We performed a spectral
analysis of the HEG 1st-order spectra of Cen X--3. The continuum
emission was well fitted by a power-law component with a photon index
of 1.2 absorbed by an equivalent hydrogen column density fixed at $
1.95 \times 10^{22}$ cm$^{-2}$. The inferred unabsorbed flux was
$\sim 7.4 \times 10^{-9}$ ergs cm$^{-2}$ s$^{-1}$ in the 2--10 keV
energy band, corresponding to a luminosity of $5.6 \times 10^{37}$
ergs s$^{-1}$ in the 2--10 keV energy band assuming a distance to the
source of 8 kpc (Krzeminski, 1974). We detected a complex structure
in the X-ray spectrum at 6--7.6 keV. In particular, a K$_\alpha$
neutral iron emission line at 6.4 keV with a width of 12 eV,
significantly different from zero and corresponding to a velocity
dispersion of 1270 km s$^{-1}$. We also resolved the triplet of
\ion{Fe}{25} at about 6.6--6.7 keV. Furthermore we detected an
absorption edge associated to \ion{Fe}{2}--\ion{Fe}{6}.
We can explain the broad width of the K$_\alpha$ neutral iron line
assuming that it was produced in an accretion disk. In fact, supposing
that the width of the line was produced by a thermal velocity
dispersion, then $T_4 \sim v^2/2.89$ K, where $T_4$ was the
temperature in units of $10^4$ K and $v$ was the velocity dispersion
in units of km s$^{-1}$. Since $v \sim 1270$ km s$^{-1}$ for the
K$_\alpha$ neutral iron line then the corresponding temperature should
be $T \sim 5.6 \times 10^9$ K, physically not acceptable. On the
other hand, assuming that the broadening is produced by the Keplerian
motion of the accretion disk, we can infer the radius $r=GM/c^2
(\Delta E/E)^{-2}$ where the line is produced; we find the inner
radius of the reflecting region between $8.3 \times 10^{9}$ cm and
$5.5 \times 10^{10}$ cm, assuming for the source an inclination angle
of $75^{+12}_{-13}$ degrees (see Nagase 1989). This radius is
compatible with the upper limit of $3.4 \times 10^{10}$ cm for the
emission region of the K$_\alpha$ neutral iron line given by Nagase et
al. (1992) and Day et al. (1993). Therefore, we conclude that the
most probable origin of the 6.4 keV line is reflection from the outer
accretion disk.
As noted in section 3 the low value of the EW of the 6.4 keV line
measured in the Chandra observations of March and December 2000 might
indicate that there is a trend of decreasing EW of the 6.4 keV line
component in 2000.
A possible explanation of the low EW observed during our observation
might involve changes in the geometry of the reflector. Perhaps the
solid angle subtended by the reflector with respect to the
illuminating source changes with time (from 2$\pi$ to less), maybe due
to a precession of the accretion disk.
For the first time, thanks to the high energy resolution of Chandra,
we were able to resolve the 6.6 keV line in the Cen X--3 spectrum as
three lines centered at 6.61 keV, 6.67 keV, and 6.72 keV with an
EW of 6 eV, 9 eV, and 5 eV. The \ion{Fe}{25} He-like
triplet is potentially a powerful diagnostic tool of density and
temperature. We obtain that the intensities of the intercombination
(i), resonance (r), and forbidden (f) lines are $7.6 \times 10^{-4}$
photons cm$^{-2}$ s$^{-1}$, $4.2 \times 10^{-4}$ photons cm$^{-2}$
s$^{-1}$, and $5.2 \times 10^{-4}$ photons cm$^{-2}$ s$^{-1}$,
respectively. We find that $R \equiv f/i$ and $G \equiv (f+i)/r$ are
$0.68 \pm 0.39$ and $3.05 \pm 2.25$ (where the uncertainties are at
90\% confidence level for a single parameter). As Bautista \& Kallman
(2000) point out, caution should be exercised with the use of $R$ and
$G$ as density and temperature diagnostics, respectively, since they
are sensitive to whether the plasma is collisionally ionized or
photoionized. Nevertheless the temperature curves in Bautista \&
Kallman (2000) indicate that $T$ is between either $6.3 \times 10^5$ K
and $1.5 \times 10^7$ K or $ 10^7$ K and $4 \times 10^7$ K for
collisional and photoionized gas, respectively. Because in the case
of Cen X--3 (a bright X-ray source emitting near the Eddington limit)
we are probably in the case of photoionized gas, we can conclude that
the temperature of the emitting region is $(1-4) \times 10^7$ K.
Furthermore the density curves are consistent with $n_e < 10^{17}$
cm$^{-3}$. Although we obtain a weak upper limit on the electron
density this is compatible with the typical stellar wind density of
$n_e=10^{10}-10^{11}$ cm$^{-3}$.
Finally we note that we do not observe the \ion{Fe}{26} line at 6.9
keV that was instead observed in the previous ASCA observation
(Ebisawa et al., 1996). This implies that the stellar wind has, during
our observation, a lower ionization parameter with respect to $\xi
\sim 10^{3.4}$ estimated by Ebisawa et al. (1996). We estimate that
$\xi$ should be $ \sim 10^{2.6}-10^{2.8}$ (see fig. 7 in Ebisawa et
al., 1996). This is also compatible with the detection of the
Ly$_{\alpha}$ \ion{Si}{14} at 2 keV. From $\xi=L_x/n_{e}^2 d $ (see
Krolik et al., 1981), assuming a source luminosity of $L_x \sim 1.3
\times 10^{38}$ erg s$^{-1}$ in the 0.1--200 keV energy band (see
Burderi et al., 2000), a ionization parameter $ \xi \simeq
10^{2.6}-10^{2.8}$ and a separation between the neutron star, and the
companion star of $d \simeq 10^{12}$ cm (Nagase et al., 1992), we find
an electron density of the emitting region of $2 \times 10^{11}$
cm$^{-3}$, in agreement with the previous results and compatible with
the upper limit obtained above.
\acknowledgements We thank the referee for the useful suggestions. This
work was partially supported by the Italian Space Agency (ASI) and the
Ministero della Istruzione, della Universit\'a e della Ricerca (MIUR).
|
3,212,635,537,409 | arxiv | \section{Introduction}\label{sec:intro}
Our daily lives surround us with practical applications of assembled supramolecular structures. %
Their applications vary in complexity and size ranging from micelle formation using soap\cite{klevens1946critical} over neurotransmitter transport via vesicles\cite{del1956biophysical} to specific DNA-Origami\cite{rothemund2006folding} in cancer therapy\cite{zhao2012dna,zhang2014dna}.
These structures, such as complexes, enzymes and even cell membranes, are artificially constructed and serve a particular purpose, e.g. drug delivery\cite{sharma1997liposomes,tian2014doxorubicin}. %
As the desire for methods and structural shapes grew especially in surfactant and lipid systems, the increasing demand for designable building blocks \cite{bayburt2002self,zhang2017design} and especially DNA scaffolds\cite{yang2016self,franquelim2018membrane} followed. %
In \citeyear{dong2014frame} \citet{dong2014frame} published a new concept with the goal of generating vesicles of programmable size and shape based on a predefined frame, namely, the frame-guided assembly. %
The basic frame was generated by single strand DNA, of which many were attached onto the surface of a gold nanoparticle. %
Via DNA-hybridization a second, complementary DNA strand was bound to the first one and extended the strand by a head group. %
These DNA strands and their respective head groups form a frame-guide around the gold nanoparticle. %
The head group was designed in a way to allow for free monomers in bulk to be accumulated between the head groups of these guiding elements. %
As a prerequisite for the frame-guided assembly it is essential to chose a monomer concentration below the point of self aggregation. %
This implies, that the frame-guided assembly facilitates aggregation below the critical micelle concentration (cmc) of the monomers. %
By shaping the surface on which the single strand DNA is attached, the authors were able to generate many shapes of vesicles resembling the structure of the underlying base of the frame-guide. %
This building block yields a high flexibility and was also be combined with DNA-origami scaffolds\cite{dong2017cuboid} to create even more complex structures. %
In computer simulations, it is of great importance to reach time scales that are long enough to observe the phenomena of interest. %
This often is achieved by the usage of coarse-graining methods that reduce the complexity of the algorithm are used. %
Coarse graining is a widely used method in biochemical computer simulations of membranes\cite{shelley2001simulations,lenz2005,marrink2007,hakobyan2013phase}, nucleic acids\cite{sim2012modeling,maciejczyk2010coarse} and proteins\cite{baaden2013coarse,poulain2008insights}. %
The unnecessary complexity is hidden and one is able to study effects and dynamics in systems, that would otherwise be beyond the reach of contemporary computational capabilities. %
In this work, we take a detailed look into the mechanism of the frame-guided assembly method with the use of a simplistic, highly coarse-grained theoretical model and investigate the assembly of planar frame-guided structures. %
In order achieve the long time scales that are needed due to the slowdown in dynamics around cmc we take a recently developed single bead surfactant model \cite{raschke2019non} and extend it in a way to allow us to study this new assembly strategy. %
Our goal is threefold. Firstly, for that coarse-grained surfactant model we characterize the frame-guided assembly in dependence of the relevant parameters such as the density of guiding elements. Secondly, we present a microscopic picture of the underlying processes of the assembly process. Thirdly, in parallel we study a 2D lattice gas model. Beyond analogous simulations as for the surfactant model it is also possible to formulate analytical predictions based on a mean-field approximation of the effect of the guiding elements. They are compared with the outcome of the simulations and can explain some of the key numerical results of the frame-guided assembly. Thus, the lattice gas model may serve as a kind of theoretical reference which allows us to gain additional information about the underlying behavior, e.g., with respect to the relevance of a critical temperature.
\section{Models and Methods}\label{sec:method}
\subsection{Definition continuous model}
Here, we present an application of the coarse grained model by \citet{raschke2019non}, which was developed for modelling micelle formation of surfactants in bulk. %
Due to the high flexibility of the model, it can be mapped onto various types of surfactants. %
Molecules are modelled as point-like particles with an orientation vector, which interact via a modified Lennard-Jones(12,6) potential. %
Via the orientation vector an anisotropy was introduced into the interaction. %
This made it possible to fit the model to critical packing parameters\cite{israelachvili1976theory} of various surfactant systems and carry out simulations. %
By adjusting an optimum angle of interaction \(\gamma\), the anisotropy was controlled so that structures of different size and shape could be sampled. %
In principle the application ranges from micelles to inverted micelles and is also suitable for planar structures like membranes. %
In this work, the model is extended upon for the application of the frame-guided assembly strategy. %
A frame-guide is an object of multiple structure defining elements, which facilitate cluster growth in the predefined frame. %
The structure defining elements are called the guiding elements. %
These elements can be placed in arbitrary formations of which vesicle\cite{dong2015using,dong2015frame} and planar\cite{zhou2016precisely} shapes were discussed in literature. %
Therefore, we introduce a new particle type, the guiding element, into the model. %
We define these guiding elements to interact with the free particles in the same way free particles interact with each other. The 3D density of the free particles is denoted \(\rho_\text{3D}\).
Guiding elements, however, are constrained in their translational and rotational degrees of freedom and are bound loosely to their initial position and orientation. %
The magnitude of these constraints are a compromise between the rigidity of the frame-guide and the ability to adapt to free particles moving in and out of the frame-guide. %
\begin{figure}
\centering
\subfloat[\label{sfig:fga_setup3d}]{\includegraphics[width=.44\linewidth]{planar3setup.pdf}}\hfill
\subfloat[\label{sfig:fga_setup2d}]{\includegraphics[width=.52\linewidth]{lattice_setup.pdf}}
\caption{Construction of a planar frame-guide with \(N_\text{GE}=9\) guiding elements. %
The dashed line denoted setup plane marks the virtual plane with an edge length of \(12\) Lennard-Jones units \(\sigma_\text{LJ}\) in which the setup takes place. %
The grey area in between the guiding elements is the activity region. %
The setup is shown for \protect\subref{sfig:fga_setup3d} the continuous model and \protect\subref{sfig:fga_setup2d} the lattice gas model. %
}
\label{fig:fga_setup}
\end{figure}
The setup of the planar frame-guide was realized via placing guiding elements on a grid with the edge length of \(12\sigma_\text{LJ}\) in Lennard Jones units in the center of the simulation box. %
The limits of the box far exceed the size of the frame-guide in avoidance of large clusters spanning the complete box and thereby becoming artificially very stable due to not experiencing border effects when a cluster stabilizes itself across the periodic boundary conditions. %
On the grid, guiding elements were placed equidistant so that individual particles had a distance of the edge length divided by the number of guiding elements per dimension \(12\sigma_\text{LJ} / N_\text{GE} \), where \(N_\text{GE}\) is the total number of guiding elements. %
\Cref{sfig:fga_setup3d} exemplarily shows the setup for \(N_\text{GE}{=}9\). %
The activity region is spanned by the outer guiding elements. %
It has the area \(A_\text{act}=144 \sigma^2 \qty(\sqrt{N_\text{GE}} - 1)^2/N_\text{GE}\). %
In analogy to previous work on this model \cite{raschke2019non} simulations were carried out via the Metropolis\cite{metropolis1953equation} Monte Carlo\cite{metropolis1949monte} sampling method in the NVT ensemble. %
Frame-guided systems were set up with an optimum angle of \(\gamma{=}0^\circ\) in order to generate planar structures. %
Guiding elements were set up in the center of the box and free particles were distributed uniformly in a random way across the simulation box. %
The exact number of free particles is adjusted to match a predefined particle density \(\rho_\text{3D}\). %
All production simulation runs sampled \(4\cdot10^7\) translational and rotational Monte Carlo trial moves each per particle in the system. %
The step width of a Monte Carlo step was fixed to a maximum of \(0.2\sigma_\text{LJ}\) for translation and a maximum \(0.2\;\text{rad}\) for rotation in order to allow for a time comparison between different sets of simulation parameters. %
The temperatures $T$, used in this work, are expressed in units of the Lennard-Jones energy of the pair-interaction potential.
\subsection{Coverage continuous model}
For a given configuration of particles within the plane, spanned by the guiding elements, we want to calculate the coverage. %
For the number of particles in the activity region there are two contributions. %
We start with the effective number of guiding elements. %
Here we have to take into account that guiding elements at the edge only contribute with 50\% and those at the corner with 25\%. %
A straightforward yields for their effective number $\qty(\sqrt{N_\text{GE}}{-}1)^2$. %
A second contribution is the number of free particles $N_\text{p}$. %
The sum of these values has to be related to the maximum possible number of particles in the activity region when all particles are arranged on a triangular lattice. Then the area \(A_\text{p}\), occupied by a single particle, is given by the area of a regular hexagon with its edge points given by the midpoints between two neighbor particles. The distance of opposite midpoints just equals the distance of two particles in the minimum of the Lennard Jones potential. Thus, the incircle radius \(r\) is exactly half of the Lennard Jones optimum distance $ 2^{1/6}\sigma_\text{LJ}$ and we can write
\begin{equation}
A_\text{p} = 2\sqrt{3} \; r^2 = 1.091\sigma_\text{LJ}^2\;,
\end{equation}
From this, we can derive the maximum surface density of particles as %
\begin{equation}
\rho_\text{2D}^\text{max} = \frac{1}{A_\text{p}} = \frac{1}{1.091\sigma_\text{LJ}^2} = 0.9166\; \sigma_\text{LJ}^{-2}\;.
\end{equation}
Now we are in a position to express the plane coverage \(\mathcal{C}\) as %
\begin{equation}
\mathcal{C} = \frac{N_\text{p}{+}\qty(\sqrt{N_\text{GE}}{-}1)^2}{A_\text{act}\rho_\text{2D}^\text{max}}\;.
\end{equation}
For \(N_\text{p}{=}0\) the coverage is the coverage due to the presence of the guiding elements \(\cov_\text{GE}=\mathcal{C}\qty(N_\text{p}{=}0)\). %
\subsection{Lattice gas model}
We consider a triangular lattice with $24 \cdot 24$ sites and periodic boundary conditions in 2D. %
As shown in \cref{sfig:fga_setup2d} we choose, in close analogy to the particle-based simulations, a local arrangement of \(N_\text{GE}\) guiding elements such that the number of rows/columns one has to add from one guiding element to the next amounts to $12/\sqrt{N_\text{GE}}$. %
If two adjacent sites are occupied the interaction energy is $J$.
The Boltzmann constant $k_\text{B}$ will be set to unity.%
During our Metropolis Monte Carlo procedure we attempt to move a lattice particle to an occupied and neighboring empty lattice site. %
The guiding elements are fixed in space. %
Beyond the rearrangement of lattice particle we also attempt to either generate a lattice particle on an empty site or remove a lattice particle from an occupied site. %
For the generation of a lattice particle one calculates the (possible) gain in energy $E_\text{gain}{<}0$ and adds the constant value $\mu {>} 0$, representing the negative chemical potential in the dilute limit. %
The acceptance is checked with a Metropolis Monte Carlo criterion. %
In analogy, the possible removal of a particle is realized by subtracting $\mu$ from the (possible) loss in energy $E_\text{loss}{>}0$ and then taking the same criterion. %
In case of no interactions among the particles, i.e. $J{=}0$, the resulting equilibrium concentration is %
\begin{equation}
\rho_\text{LG} = \frac{1}{1 + e^{\mu/T}}\;. %
\label{eq:rhoLG}
\end{equation}
The relation to the bulk density \(\rho_\text{3D}\) of the continuous model will be discussed further below. %
When starting the simulations we have started with an lattice without free particles. In all cases we have averaged over at least 100 independent runs, each with $10^6$ Monte Carlo steps.
\subsection{Mapping to Ising model}
There is a straightforward mapping from the lattice gas model to the Ising model. %
The Hamilton of the lattice gas model reads %
\begin{equation}
\mathcal{H}_\text{latticegas} = -\frac{J}{2}\sum_{\langle \text{i,j} \rangle} \sigma_\text{i} \sigma_\text{j} + \mu \sum_\text{i} \sigma_\text{i}. %
\end{equation}
where the first sum for a given index $i$ runs over all neighbors $j$. %
Here $\sigma_\text{i}$ expresses whether site $i$ is populated ($\sigma_\text{i} {=}1$) or not ($\sigma_\text{i} {=}0$). %
The mapping to the Ising model involves the transformation to spin variables $s_\text{i} {=} 2\sigma_\text{i} {-} 1$. %
This yields for a triangular lattice %
\begin{equation}
\label{eq:Hising}
\mathcal{H}_\text{Ising} = -\frac{J}{8}\sum_{\langle \text{i,j} \rangle} s_\text{i} s_\text{j} +\qty( \frac{1}{2} \mu -\frac{3}{2} J ) \sum_\text{i} s_\text{i} + \text{const}\;. %
\end{equation}
Thus, naturally the thermodynamic properties of the Ising model directly translate to those of the lattice gas model. %
In particular, the critical temperature of the lattice gas model $T_\text{c}$ is one fourth the critical temperature of the triangular Ising model, i.e. $\qty(T/J)_\text{c} =3.6403 /4 \approx 0.91$ \cite{zhi2009critical} The inverse reads $\qty(J/T)_\text{c} \approx 1.10$.
Furthermore, from \cref{eq:Hising} the effective magnetic field can be identified as
\begin{equation}
B_\text{eff} = (1/2)(\mu - 3J).
\end{equation}
Thus, both the chemical potential and the interaction strength contribute to $B_\text{eff}$. %
At cmc the system needs to be symmetric with respect to the spin direction since the lattice gas model with high coverage has the same free energy as the system with low coverage. %
As a consequence at cmc one exactly has $B_\text{eff}{=}0$. %
In the Ising representation a guiding element can be described as a spin which is in a permanent up state $s_\text{i} {=} 1$. %
Thus, a neighbor j of a guiding element experiences an additional energy contribution, related to the presence of the guiding element, given by $-(J/4)(1 - \langle s \rangle ) s_\text{j}$. Here $\langle s \rangle$ denotes the average magnetization. Since we are mainly interested in the behaviour around cmc, one has $\langle s \rangle=0$.
If the coverage of guiding elements is denoted \(\cov_\text{GE}\) and taking into account that a guiding element has 6 neighbors one gets %
\begin{equation}
B_\text{eff} = (1/2)(\mu - 3J - 3J \cov_\text{GE}).
\end{equation}
This relation implies that the impact of a guiding element is equally distributed over all sites, which corresponds to a mean-field picture.
Since at cmc one has $B_\text{eff}{=}0$ one can calculate the required chemical potential via
\begin{equation}
\mu =3J \cdot (1+\cov_\text{GE}).
\label{eq:mutheo}
\end{equation}
Together with \cref{eq:rhoLG} this yields (using $\mu/T \gg 1$ which will always be the case)
\begin{equation}
\text{cmc} = \exp[-3(J/T)] \exp[-3(J/T)\cov_\text{GE}].
\label{eq:cmc_exact}
\end{equation}
For the coverage of the lattice gas model we just count the occupied sites in the activity region and compare them with the total number of sites. Again, the effective number of guiding elements, to be taken into account in the activity region, is $(\sqrt{N_\text{GE}}-1)^2$ .
\subsection{Quality of mean-field approximation}
\begin{figure}
\centering
\subfloat[\(J T^{-1}{=}1.2\).\label{sfig:mu_theory_actual_J12}]{\includegraphics[width=.5\linewidth]{mu_theory_actual_J12.pdf}}\hfill
\subfloat[\(J T^{-1}{=}1.8\).\label{sfig:mu_theory_actual_J18}]{\includegraphics[width=.5\linewidth]{mu_theory_actual_J18.pdf}}
\caption{Chemical potential divided by the temperature \(\mu T^{-1}\) as a function of the guiding element coverage \(\cov_\text{GE}\) under periodic boundary conditions (PBC), in the activity region and for the mean-field theory for \protect\subref{sfig:mu_theory_actual_J12} \(J T^{-1}{=}1.2\) and \protect\subref{sfig:mu_theory_actual_J18} \(J T^{-1}{=}1.8\). For the case of PBC, $N_\text{GE}$ is chosen to be $0^2, 1^2, ..., 4^2$ whereas for the analysis of the activity region data are shown for $2^2, 3^2, 4^2$. %
}
\label{fig:mu_theory_actual}
\end{figure}
We checked the quality of the mean-field approximation for two different temperatures, $J/T{=}1.2$ and $J/T{=}1.8$. Specifically, we determined via systematic variation of the chemical potential, based on interval bisectioning, when the total system is at cmc, i.e. on average half of the available sites are populated. %
This value can be compared with the mean-field result in \cref{eq:mutheo}.
Two different scenarios were evaluated. %
First, we used a system with $12 \cdot 12$ sites together with periodic boundary conditions. %
In this way the boundary effects at the active plane are removed. %
As seen in \cref{fig:mu_theory_actual}, the agreement between simulation and mean-field approximation is nearly perfect for both temperatures. %
In the second scenario we just considered the activity region in the larger system. %
For extrapolation to low coverage \(\cov_\text{GE}\) the results, obtained from analysis of the activity region, agree with the simulations, using periodic boundary conditions. This is a necessary consequence of the condition $B_\text{eff}=0$ at cmc. Interestingly, the dependence on \(\cov_\text{GE}\) is somewhat weaker which holds in particular for the lower temperature. Thus, application of \cref{eq:cmc_exact} should be particularly suited for higher temperatures.
\section{Results}\label{sec:Results}
\begin{figure*}
\centering
\subfloat[Setup.\label{sfig:snap_setup}]{
\begin{tabular}{@{}c@{}}
{\frame{\includegraphics[width=.23\linewidth, height=4cm]{local_setup.pdf}}}
\end{tabular}
}\hfill
\subfloat[Onset of agglomeration.\label{sfig:snap_first}]{
\begin{tabular}{@{}c@{}}
{\frame{\includegraphics[width=.23\linewidth, height=4cm]{local_first.pdf}}}\\[2ex]
{\frame{\includegraphics[width=.23\linewidth, height=4cm]{general_first.pdf}}}
\end{tabular}
}\hfill
\subfloat[Growing structures.\label{sfig:snap_continued}]{
\begin{tabular}{@{}c@{}}
{\frame{\includegraphics[width=.23\linewidth, height=4cm]{local_connected.pdf}}}\\[2ex]
{\frame{\includegraphics[width=.23\linewidth, height=4cm]{general_growing.pdf}}}
\end{tabular}
}\hfill
\subfloat[Fully assembled plane.\label{sfig:snap_complete}]{
\begin{tabular}{@{}c@{}}
{\frame{\includegraphics[width=.23\linewidth, height=4cm]{local_populated.pdf}}}
\end{tabular}
}
\caption{%
Extracts from snapshots of the assembly process of a planar frame-guided system with 16 guiding elements \protect\subref{sfig:snap_setup} in its initial setup, \protect\subref{sfig:snap_first} while the first agglomeration is formed, \protect\subref{sfig:snap_continued} as the cluster grows and, \protect\subref{sfig:snap_complete} as a completely assembled plane. %
The above row shots the assembly process as it takes place in a localized fashion from a single nucleation site. %
The bottom row presents a less localized more general assembly process. %
Particles are represented as sticks, where the stick denotes the center of mass between the endpoints and the orientation vector from endpoint to endpoint. %
Guiding elements are depicted in red, while free particles are shown in black. %
}
\label{fig:assembly_snapshots}
\end{figure*}
\subsection{Qualitative behavior}
An example of cluster formation is shown in \cref{fig:assembly_snapshots} at different stages of the assembly process of a planar frame-guided cluster for the continuous model. The concentration is chosen such that in equilibrium the plane is (nearly) fully covered.
Prior to the assembly process, free particles interact with guiding elements randomly and dissociate after some simulation steps. %
Once a certain amount of free particles is located between guiding elements, these particles are stabilized in the guiding element structure as one observes in \cref{sfig:snap_first}. %
The assembled particles function as a nucleation site for further cluster assembly, which is facilitated further via the next guiding elements in proximity. %
This is depicted in \cref{sfig:snap_connected}, where the frame-guide is approximately half-filled with free particles. %
Note that the assembly process can be more localized as seen in the upper figures or more delocalized as shown in the lower figures. This will be quantified further below.
For long times the activity region is fully covered by particles, extending even beyond the activity region, see \cref{sfig:snap_complete}. As expected the particles basically arrange in a triangular lattice.
\subsection{Coverage}
\begin{figure*}
\centering%
\subfloat[Continuous model.\label{sfig:plane2d_coverage3d}]{\includegraphics{plane2d_coverage_t023_logx.pdf}}\hspace{1cm}%
\subfloat[Lattice gas model.\label{sfig:plane2d_coverage2d}] {\includegraphics{coverage_rho_NGEsorted_new.pdf}}%
\caption{%
Coverage \(\mathcal{C}\) of the guiding element plane with particles \protect\subref{sfig:plane2d_coverage3d} as a function of the overall particle density \(\rho_\text{3D}\) at temperature \(T{=}0.23\) in the continuous 3D model and \protect\subref{sfig:plane2d_coverage2d} as a function of the density \(\rho\) at temperature \(J{=}1.4\) the 2D lattice model with 4, 9 and 16 guiding elements \(N_\text{GE}\). %
}%
\label{fig:coverage}
\end{figure*}
We start by studying the coverage \(\mathcal{C}\) of the activity region which can hold values between \(\cov_\text{GE}\) and 1. It is shown in \cref{fig:coverage} as a function of \(\rho_\text{3D}\) at various \(N_\text{GE}\). %
In general, one observes a steep increase in plane coverage at certain \(\rho_\text{3D}\) values.
One can state that the cluster formation process is very sensitive to changes in particle density. Small changes shift the system from no cluster formation to a fully populated activity region. %
Most importantly, with the addition of guiding elements to the system, cluster formation was observed below the cmc of the non-guided system, i.e. the value of cmc decreases with increasing \(N_\text{GE}\). %
This reflects the impact of the frame-guide to provide nucleation sites for free particles. Since we work below the cmc of the cluster formation without a frame-guide, cluster formation is not observed outside of the frame-guide. Thus, all processes of interest are expected to occur in the activity region. %
A fully analogous picture we obtain for the lattice gas model. Again, the presence of guiding elements reduces the density where the coverage strongly starts to increase.
\subsection{Efficiency of guiding elements}\label{subsec:efficiency}
\begin{figure}
\centering
\includegraphics{plane_zdistFWHM_T022_ge16_d010_fit.pdf}
\caption{Probability density of particles in z-direction of the simulation box centered around the plane of the frame-guide \(z{=}0\). The histogram normalized to 1 and an ensemble average of 10 systems at \(t=4\cdot10^7\) simulation steps with a temperature of \(T{=}0.22\) and \(N_\text{GE}{=}16\) guiding elements. The red line represents the Gaussian fit of the histogram, and the dashed lines indicate the full width at half maximum FWHM of the Gaussian. }%
\label{fig:zfit}
\end{figure}
Now the dependence of cmc on $N_\text{GE}$ is discussed in more detail.
We define cmc as the lowest particle density in the activity region at which \(50\%\) of the trajectories showed a density greater or equal to half of the maximum density \(\rho_\text{2D}^\text{max}\). %
Note that cmc is a specific value of \(\rho_\text{3D}\). Thus, we need to convert \(\rho_\text{3D}\) into a coverage representing the 3D particle density in a planar 2D structure with layer thickness $d$. In this way it is possible to define cmc via the coverage criterion and to compare the resulting 2D density at cmc with the corresponding values of the lattice gas model.
The thickness of the particle layer in the frame-guide is depicted in \cref{fig:zfit} as the full width at half maximum FWHM of the gaussian fit of the \(z\)-distribution of particles in the box. %
Here, with \(N_\text{GE}{=}16\), the cluster formation showed no significant bending of the membrane. Therefore, we take the resulting value of \(d{=}\text{FWHM}{=}0.86\sigma_\text{LJ}\) as a reasonable estimate of the layer thickness and will use it as a constant for all \(N_\text{GE}\) values in further calculations. We checked that in the range of relevant temperatures this value is insensitive to temperature.%
Accordingly, we express the 2D coverage at the 3D critical micelle concentration as %
\begin{equation}
\cov^\text{cmc}}%_\text{bulk} = d \; A_\text{p} \; \left.\rho_\text{3D}\right|_\text{cmc}\;.
\end{equation}
Here\(\left.\rho_\text{3D}\right|_\text{cmc}\) is the overall particle density at cmc. %
The term \(d \; A_\text{p}\) expresses the volume one particle occupies in the triangular lattice. %
\begin{figure*}
\centering%
\subfloat[Continuous model.\label{sfig:cmc_plane_3d}]{\includegraphics{plane_cmc_paper.pdf}}\hspace{1cm}%
\subfloat[Lattice gas model.\label{sfig:cmc_plane_2d}]{\includegraphics{cmc_ge_new.pdf}}%
\caption{%
Bulk coverage at the critical micelle concentration \(\cov^\text{cmc}}%_\text{bulk}\) as a function of guiding elements \(N_\text{GE}\) at various temperatures \(T\). The top axis shows the complementary coverage of guiding elements \(\cov_\text{GE}\) \protect\subref{sfig:plane2d_coverage3d} in the activity region of the continuous 3D model and \protect\subref{sfig:plane2d_coverage2d} \(\mathcal{C}^\text{cmc}\) in the 2D lattice model. The broken line is the prediction of the superposition hypothesis (see text). The offset in y-direction is chosen arbitrarily.%
}
\label{fig:cmc_plane}
\end{figure*}
\(\cov^\text{cmc}}%_\text{bulk}\) was calculated at \(T{=}0.22\), \(0.23\), \(0.24\) and \(0.25\) and \(N_\text{GE}{=}4\), \(9\) and \(16\) guiding elements as depicted in \cref{fig:cmc_plane}. %
We observed a decrease in cmc with the increase of guiding elements in the system at all sampled temperatures in agreement with the results in \cref{fig:coverage}. Furthermore, as expected, the value of cmc gets smaller at lower temperatures. The slope of the logarithm of \(\cov^\text{cmc}}%_\text{bulk}\) turns out to be approximately temperature independent.
To quantify the efficiency of guiding elements for the cluster formation process we started from the {\it superposition hypothesis}, that a certain coverage of guiding elements \(\cov_\text{GE}\) is equivalent to the same coverage with free particles. %
In this limiting case, the sum
\begin{equation}\label{eq:superpos}
\cov_\text{GE} + \qty(1-\cov_\text{GE})\;\cov^\text{cmc}}%_\text{bulk}
\end{equation}
should have no dependence on the bulk density. Here \(\qty(1-\cov_\text{GE})\) represents the area already occupied by guiding elements and \(\cov^\text{cmc}}%_\text{bulk}\) is the 2D representation of particle density at cmc in bulk. %
Since \mbox{\(\cov_\text{GE}{\ll}1\)}, from \cref{eq:superpos} we may conclude %
\begin{equation}
\cov^\text{cmc}}%_\text{bulk} = \text{const} - \covg
\end{equation}
The numerical relation between $\cov_\text{GE}$ and $\cov^\text{cmc}}%_\text{bulk}$ is shown in \cref{fig:cmc_plane} for different temperatures together with the prediction of the superposition hypothesis (formulated for the data at $T=0.25$). The bending results from the logarithmic representation of the y-axis.
Obviously the superposition hypothesis predicts a much stronger dependence on the number of guiding elements as compared to the actually observed dependence and, furthermore has a different functional dependence on $N_\text{GE}$. %
We conclude, that guiding elements are capable of allowing cluster formation below the cmc-value of the non guided system, but are far less efficient in doing so when compared to a corresponding increase of the bulk density. %
This is due to the conceptual difference between frame-guided particles and free particles. %
In the latter case they represent the overall 3D concentration, i.e. the ability to feed additional particles for further growth. %
In contrast, the frame-guided particles just represent a localized 2D density.
It is also straightforward to argue why the superposition hypothesis has to fail when starting from \cref{eq:cmc_exact}. This relation yields $\text{cmc} \approx \exp\qty(-3J/T) {-} \qty(3J/T) \exp\qty(-3J/T) \cov_\text{GE} $. The prefactor of $\cov_\text{GE} $ is significantly smaller than one (since $x \exp(-x) \ll 1$ for $x \gg 1$). As mentioned above this again reflects the fact that the guiding elements just modify the chemical potential but do not change the effective density of particles which is connected with the exponential of the chemical potential.
Obviously, to a good approximation the data display an exponential dependence on $N_\text{GE}$. Indeed, whereas this was expected for the lattice model, this behavior, at least on a qualitative level, also holds for the continuous model.
\subsection{Locality of cluster formation}\label{subsec:locality}
\begin{figure*}
\centering%
\subfloat[Continuous model.\label{sfig:localization3d}]{\includegraphics{locality_ge9.pdf}}\hfill%
\subfloat[Lattice gas model.\label{sfig:localization2d}] {\includegraphics{coop.pdf}}%
\caption{%
Frame-guide coverage and locality of cluster formation \(\mathcal{L}\) as a function of simulation time at various temperatures \(T\) and particle densities \(\rho_\text{3D}\) for 9 guiding elements above cmc for the continuous model and the lattice gas model. %
The time scale is shifted such that just at \(t{=}0\) a threshold of \(\mathcal{C}{=}50\%\) coverage was exceeded for an individual run. %
In \protect\subref{sfig:localization3d} every curve shows an ensemble average across multiple trajectories and a rolling mean a \(5\cdot10^3\) steps time frame. %
The insets show the ratio of locality peak heights and the plateau values before cluster growth.
Here, the intersection of the linear regression curves (black line) extrapolate to a temperature, where the cluster growth would occur in a delocalized way. %
}
\label{fig:localization}
\end{figure*}
The behavior, shown in \cref{fig:assembly_snapshots}, indicates that the complete assembly can be preceded by localized agglomeration of the particles. %
Here we want to quantify the relevance of localized behavior. %
We took the activity region and divided it into specific domains each spanning the area between 4 guiding elements. %
In this way a grid of 9 guiding elements, to be analysed in this context, results in 4 distinct domains (see \cref{fig:assembly_snapshots}).
Within these domains the number of particles, populating each domain, was tracked. Since every guiding element is contributing to 4 domains, the number of particles per domain contain one guiding element.%
We calculated the locality as the ratio of the number of particles in the domain with the highest population\(N_\text{max}\) and the average number of particles per domain \(\mean{N}\) in the activity region %
\begin{equation}
\mathcal{L}=\frac{N_\text{max}}{\mean{N}}\;.
\end{equation}
When averaging over different clustering events, we shift the relative times such that at time 0 the coverage is half between minimum and maximum coverage, i.e. close to 50\%. %
Furthermore, we have chosen the density approx. 10\% above the cmc, as given in \cref{fig:cmc_plane}. %
For the lattice gas model we adjusted the chemical potential accordingly. %
This just guarantees that at some stage the system manages to form a large cluster in the activity region.
The subsequent results are very insensitive to the exact value of that increase. %
\Cref{fig:localization} shows the results for the plane coverage in the upper half as well as the locality of cluster formation as a function of time. %
Data are presented for both the continuous model and the lattice gas model. %
For early times one has a few scattered particles in the different domains which gives rise to a time-independent value of the $\mathcal{L}$, reflecting the stochastic fluctuations of particle number per domain. %
In the long-time limit every domain is nearly fully covered. By definition, this gives rise to a $\mathcal{L}$ value of unity.
Of key interest is the peak of $\mathcal{L}$ which at the lower temperatures appears for slightly negative time scales, i.e. shortly before the half coverage is reached. %
On this qualitative level the observations are fully analogous for the two models.
At the point where the assembly process begins we observed a steep increase in the locality of the process.
The magnitude of this effect is strongly dependent on system temperature, showing stronger localization at lower temperatures.
To characterize the height of the localization peak we divided the maximum value of $\mathcal{L}$ by the average $\mathcal{L}$ value before cluster formation; see the inset of \cref{sfig:localization3d}. This value is denoted as $\mathcal{L}_\text{rel}$.%
We observed that the vanishing of the peak, corresponding to $\mathcal{L}_\text{rel}$, occurs at a well-defined temperature $T_\text{loc}= 0.265$ for the continuous model and $\qty(T/J)_\text{loc} = 0.83$ for the lattice gas model. %
This temperature is slightly below the critical temperature $\qty(T/J)_\text{c} = 0.91$.
In the next section a physical picture is presented which relates these observations to the phenomenology of the Ising model.
For the subsequent discussion we explored that $T_\text{loc}$ can be used as a characteristic energy scale which, in principle, would allow to compare both models more quantitatively.
\subsection{Gibbs free energy}\label{subsec:gibbs}
Finally, we analyse how the value of cmc depends on temperature for fixed number of guiding elements. The results are shown in \cref{fig:detlaGfit}, where temperatures are scaled by \(T_\text{loc} \).
\begin{figure*}
\centering%
\subfloat[Continuous model.\label{sfig:deltaGfit3d}] {\includegraphics{plane_deltaG_fit_all.pdf}}%
\subfloat[Lattice gas model.\label{sfig:deltaGfit2d}]{\includegraphics{deltaG_fit_2d_new.pdf}}%
\caption{%
Critical micelle concentration \(\cov^\text{cmc}}%_\text{bulk}\) as a function of inverse temperature with respect to temperature of localization \(T_\text{loc}\;T^{-1}\) for various number of guiding elements \(N_\text{GE}\) in the \protect\subref{sfig:deltaGfit3d} continuous model and \protect\subref{sfig:deltaGfit2d} the lattice model as comparison. %
Solid lines indicate a fit in the form of \mbox{\(\cov^\text{cmc}}%_\text{bulk} \propto \exp\qty(\frac{\Delta G_\text{m}}{RT})\)}. %
Temperatures are scaled to the temperature of localization \(T_\text{loc}\). %
}%
\label{fig:detlaGfit}
\end{figure*}
\begin{figure}
\centering
\includegraphics{deltaG_new.pdf}
\caption{%
Gibbs free energy change of micellization with respect to the temperature of localization \(\Delta G_\text{m}\;T_\text{loc}^{-1}\), derived from the exponential fits in \cref{fig:detlaGfit}, as a function of guiding elements \(N_\text{GE}\) comparing the continuous 3D and the lattice 2D models.%
}%
\label{fig:deltaG}
\end{figure}
To analyse the temperature dependence one may start with the law of mass-action regarding micellization as described by \citet{olesen2015determination}. It turns out that the Gibbs free energy change of micellization can be expressed as %
\begin{equation}
\text{cmc} = a_T e^{\frac{\Delta G_\text{m}}{RT}}
\end{equation}
with a prefactor $a_T=1$.
This is also compatible with \cref{eq:cmc_exact}.
In order to reduce the impact of the presence of guiding elements our later discussion will mainly focus on the case of lowest number of guiding elements, i.e. $N_\text{GE}=4$. For reasons, to be discussed below, $a_T$ is treated as an additional adjustable parameter when performing the corresponding Arrhenius fit.
The results for the normalized value \(\Delta G_\text{m}/T_\text{loc} \) are shown in \cref{fig:deltaG}. %
One notices the significant difference in \(\Delta G_\text{m}\) in this scaled representation. For $N_\text{GE}=4$ we obtain \(\Delta G_\text{m}/T_\text{loc} = -8.55\) for the continuous model and \(\Delta G_\text{m}/T_\text{loc} = -3.57\) for the lattice gas model. The corresponding values of $a_T$ are 360 for the continuous model and 0.96 for the lattice gas model. Thus, the continuous model displays a major deviation from $a_T=1$.
\section*{Discussion and Summary}
We observed that the presence of guiding elements has two main effects. %
First, the assembly process may already start below the cmc value which one would find without guiding elements. %
Second, the presence of guiding elements determines the spatial region where the assembly takes place. %
In this work, we mainly concentrated on the theoretical understanding of the variation of the cmc value upon addition of guiding elements as well as the nature of the assembly process.
Additionally, we performed simulations of a lattice gas model. We observed an increased locality index $\mathcal{L}$ with decreasing temperature. %
It allowed us to define the localization temperature $T_\text{loc}$. %
For the Ising model the value of $T_\text{loc} $ is 10\% lower than the theoretical critical temperature. Indeed, one expects that for a finite system the observed crossover behavior occurs at slightly lower temperatures due to the additional fluctuations. Thus, it is reasonable to relate $T_\text{loc}$ to the critical temperature. This relation is not a pure coincidence but is consistent with the observed phenomenology of the assembly process. For the Ising model it is known that the correlation length of the fluctuations is very large close to the critical temperature. Since we are dealing with concentrations close to cmc, corresponding to $B_{eff} \approx 0$, indeed we expect a highly correlated and thus delocalized behavior. With decreasing temperature the correlation length decreases. Thus, one expects the assembly process to be more localized as also seen from the behavior of the locality index. A second key prediction for the lattice model is the exponential dependence of cmc on the density of guiding elements. To a good approximation, this was seen for the lattice model. In particular, the alternative scenario, based on the superposition hypothesis, could be discarded. This supports the notion that the guiding elements serve as an additional field which shift the chemical potential as derived within a mean-field approximation.
Importantly, both key results, namely the localization properties and the exponential dependence of cmc on the density of guiding elements, was also observed for the continuous model. This strongly suggests that the lattice model captures essential properties of the particle-based continuous model. Thus, via this analogy the simulations of the continuous model have gained an additional theoretical basis. In particular, the values of $T_\text{loc}$ may serve as a measure of the respective energy scale.
From a more quantitative perspective, however, deviations between both models are present. (1) The value of $a_T$, used for fitting the temperature dependence of cmc, significantly deviates from unity for the continuous model. In contrast, the lattice model fulfills the expectation $a_T \approx 1$. This is basically equivalent to the observation that the Gibbs free energy relative to $T_\text{loc}$, i.e. \(\Delta G_\text{m}/T_\text{loc} \) is higher for the continuous model. (2) The localization effects display a stronger temperature dependence for the continuous model. (3) The sensitivity of cmc on the number of guiding elements, expressed by the dimensionless slope $a_{GE}$, is higher for the continuous case although in both cases temperatures close $T_\text{loc}$ are taken, i.e. one works in comparable regions of parameter space.
Here we argue that on a qualitative level these deviations may be related to the additional complexity of the particle-based continuous model as compared to the simple lattice model.
First, the interaction between two particles in the continuous model strongly depends on the orientation. By construction of the potential, only for parallel alignment in optimum distance the full Lennard-Jones attraction is possible. When just considering a pair of particles, due to entropy effects the effective interaction strength $J$ is lower than the maximum possible interaction strength. This effect becomes smaller for lower temperatures. Naturally, entropy effects are less pronounced in this limit. This was already shown in \cite{raschke2019non}. Furthermore, at lower temperatures the growth mode is more localized as discussed above. A locally higher density naturally gives rise to a stronger parallel alignment as shown in \cref{fig:assembly_snapshots} and thus to a stronger interaction. In summary, due to both reasons with decreasing temperature the effective interaction and thus temperature-dependent effects become stronger. This naturally rationalizes (1) and (2).
Second, the guiding elements in the continuous model can locally translate and thus adjacent guiding elements can come together to accelerate the local assembly process. In contrast, in the lattice model the guiding elements are strictly localized. This may be one reason for the observation (3). Also other differences should be mentioned. For example the actual growth of the assembly in the continuous case may be slowed down due to the finite diffusive transport of particles to the active region. In contrast, in the lattice model the availability of additional particles is just governed by the chemical potential.
Finally, we note that the reduction in cmc upon addition of guiding elements can be further enhanced if the interaction between guiding elements and surfactant molecules is very strong. Based on our theoretical understanding it would be possible to predict the reduction in cmc in dependence on that interaction strength.
\section*{Acknowledgement}
We gratefully acknowledge financial support from the DFG (HE 2570/3-1 / TRR 61 / SFB 858) and helpful discussions with D. Liu.
|
3,212,635,537,410 | arxiv | \section{Introduction}
QCD simplifies greatly in the chiral \cite{chiral}, heavy-quark \cite{heavy},
and large-$N_c$ \cite{bign} limits (where $N_c$ is the number of colors). The
Skyrme \cite{sky1} model of heavy baryons \cite{wit,skyrme,jmw} incorporates
all of these, and makes powerful predictions \cite{chow} about the baryonic
Isgur-Wise functions $\eta$, $\eta_1$, and $\eta_2$. Since it has been
conjectured that all parameter-independent Skyrme model predictions can be
derived just from large-$N_c$ unitarity constraints, we investigate what these
constraints alone can tell us about Isgur-Wise functions.
Our model-independent large-$N_c$ results turn out to be less powerful than
Chow's Skyrme model predictions. Using the exponential form $\eta \sim
\exp\{-\lambda N_c^{3/2} (w-1)\}$ given by Jenkins, Manohar and Wise
\cite{jmw}, unitarity requires only that the usual normalization conditions
hold at $w=1$, and that $\eta=\eta_1$ near threshold.
\subsection{Baryon Isgur-Wise Functions}
The weak transition $\Lambda_b \to \Lambda_c\, W$ is characterized by a single
Isgur-Wise function, which represents the overlap of the spin-0 light degrees
of freedom (``brown muck''):
\begin{equation}} \def\End{\end{equation} \langle L(v') | L(v) \rangle = \eta(w) \End
where $w\equiv v\cdot v'$. $\Lambda_Q$ is an isospin singlet ($I \!=\! J \!=\!
0$).
The weak transition $\Sigma^{(*)}_b \to \Sigma^{(*)}_c\, W$ is characterized by
two other functions, since in this system the brown muck has spin 1:
\begin{equation}} \def\End{\end{equation} \langle L_\nu(v') | L_\mu(v) \rangle = -\eta_1(w) g_{\mu\nu} + \eta_2(w)
v'_\mu v_\nu \End
The $\Sigma_Q$ and $\Sigma^*_Q$ can be treated together in the heavy quark
limit, as a single ``superfield'' $\Sigma$ \cite{cho}, since they differ only
in the relative spin orientation of the heavy quark and the brown muck.
$\Sigma$ is an isospin triplet ($I \!=\! J \!=\! 1$).
The normalization of these functions at $w=1$ (``threshold'') is:
\begin{equation}} \def\End{\end{equation} \eta(1)=\eta_1(1)=1 \Endl{norm}
Heavy quark symmetry makes no prediction for the value of $\eta_2(1)$.
Chow \cite{chow} found the following relations among baryon Isgur-Wise
functions using the Skyrme model:\footnote{Chow writes $(\zeta_1,\zeta_2)$ for
$(\eta_1,\eta_2)$ and uses ``east coast'' metric $g_{\mu\nu} = {\rm
diag}(-1,1,1,1)$. Chow has recently derived the same relations from $SU(4)$
symmetry. \cite{chow2}}
\begin{equation}} \def\End{\end{equation} \eta_1(w) = -(1+w)\eta_2(w) = \eta(w) \Endl{chowrel}
These relations are consistent with the normalizations in \puteq{norm},
and additionally predict that $\eta_2(1)=-1/2$.
\section{Diagrams}
\subsection{1-Loop Renormalization of $\eta(w)$}
In Fig.~1 we show the 1-loop renormalization (vertex and wavefunction) of
$\Lambda_b(v) \to \Lambda_c(v') W$ ({\it i.e.}} \def\ord#1{{\cal O}({#1})\ of $\eta$), which is calculated by Cho
\cite[eq.~(3.4)]{cho}. Since $(g_{\Sigma\Lambda}} \def\eps{\epsilon/f)^2 \sim N_c$, the term that it multiplies
must vanish at least as fast as $1/N_c$. The relevant piece is
\begin{equation}} \def\End{\end{equation} \left[ 3\eta - (2r+w)\eta_1 + (w^2-1)\eta_2 \right] = \ord{1\over N_c},
\qquad r \equiv {\ln\left(w + \sqrt{w^2-1}\right) \over \sqrt{w^2-1}}
\Endl{1lp}
\INSERTFIG{6.49}{bign_1.eps}{Fig.~1: 1-loop renormalization of $\Lambda_b(v)
\to \Lambda_c(v') W$}
One might be tempted to use the renormalization of $\Sigma\to \Sigma' W$ ({\it i.e.}} \def\ord#1{{\cal O}({#1})\
of $\eta_1$ and $\eta_2$), also calculated by Cho, to derive more relations.
However, in the large-$N_c$ limit there exists an $I \!=\! J$ tower of states
above the $\Lambda$ and $\Sigma$. In particular, the state with $I \!=\! J
\!=\! 2$ contributes to the 1-loop renormalization of $\Sigma\to \Sigma' W$.
It introduces 3 new Isgur-Wise functions \cite[eq.~(2.26)]{falk}, only one of
which is normalized at $w=1$. Thus no useful new information is obtained.
\subsection{Single Pion Emission}
In Fig.~2, we look at weak decay accompanied by single pion emission:
$\Lambda_b(v) \to \Sigma_c(v') \pi_l(q) W$. The sum of the two diagrams gives
an invariant amplitude
\begin{equation}} \def\End{\end{equation} {\cal M} = {g_{\Sigma\Lambda}} \def\eps{\epsilon\over 2f}{g_2 V_{cb}\over 2\sqrt2} \left[ \bar
\Sigma_{ij}^\mu \slash\epsilon^* (1-\gamma_5) \Lambda \right] \left[
\epsilon^{ki} (T_l)^j_k + \epsilon^{kj} (T_l)^i_k \right] (F q_\mu + G
v_\mu) \Endl{calM}
where
\begin{equation}} \def\End{\end{equation} F \equiv {\eta\over v'\cdot q} - {\eta_1\over v\cdot q}, \qquad
G \equiv \eta_2 {v'\cdot q \over v\cdot q} + \eta_1 - w \eta_2 \End
and the $T_l$'s are flavor SU(2) generators. We used Cho's \cite{cho} Feynman
rules restricted to SU(2), so $\{i,j,k\}\in\{1,2\}$, and $l\in\{1,2,3\}$; the
group theory factor is just the Clebsch-Gordan coefficient $\langle 1,\alpha;
1,\alpha'|0,0\rangle$. These rules automatically obey unitarity constraints
for $\Lambda\pi\to\Lambda\pi$ analogous to those derived elsewhere \cite{unit}
for $N\pi\to N\pi$.
\INSERTFIG{4.5}{bign_2.eps}{Fig.~2: Single pion emission $\Lambda_b(v) \to
\Sigma_c(v') \pi(q) W$.}
Since $(g_{\Sigma\Lambda}} \def\eps{\epsilon/f) \sim \sqrt{N_c}$, the last factor of \puteq{calM} must vanish
at least as fast as $1/\sqrt{N_c}$ when contracted with any final state
$\Sigma^\mu$, which is in turn constrained only by $v'_\mu \Sigma^\mu = 0$:
\begin{equation}} \def\End{\end{equation} \Sigma^\mu (F q_\mu + G v_\mu) = \ord{1\over\sqrt{N_c}} \Endl{1pi}
for any $q$ satisfying $q^2 = m_\pi^2 \approx 0$ (in the chiral limit) and
kinematic constraints.
\subsection{Double Pion Emission}
Double pion emission, $\Lambda_b(v) \to \Lambda_c(v') \pi_l(p) \pi_m(q) W$,
arises from the 3 diagrams of Fig.~3, plus 3 ``crossed'' diagrams related by
$\{l,p\} \leftrightarrow\{m,q\}$. As long as we restrict our indices to SU(2)
as before, the group theory factor of the crossed diagrams equals that of the
uncrossed diagrams. Since $(g_{\Sigma\Lambda}} \def\eps{\epsilon/f)^2 \sim N_c$, the remaining term must
vanish at least as fast as $1/N_c$:
\begin{eqnarray}} \def\Enda{\end{eqnarray} 2\eta
&+& (\eta_1 - w \eta_2) \left[ 2w - {v'\cdot p \over v\cdot p} - {v\cdot p
\over v'\cdot p} - {v'\cdot q \over v\cdot q} - {v\cdot q \over v'\cdot q}
\right]
- \eta_2 \left[ {(v'\cdot p)(v\cdot q) \over (v'\cdot q)(v\cdot p)} +
{(v'\cdot q)(v\cdot p) \over (v'\cdot p)(v\cdot q)} \right] \nonumber\\
&+& \eta_1 \left[ {p\cdot q \over (v'\cdot q) (v\cdot p)} + {p\cdot q \over
(v'\cdot p) (v\cdot q)} \right]
- \eta \left[ {p\cdot q \over (v\cdot p) (v\cdot q)} + {p\cdot q \over
(v'\cdot p) (v'\cdot q)} \right]
= \ord{1\over N_c} \Endla{2pi}
Again, we cannot continue with $n$-pion emission because higher states in the
$I\!=\!J$ tower come into play for $n>2$.
\INSERTFIG{6.49}{bign_3.eps}{Fig.~3: Two-pion emission, $\Lambda_b(v) \to
\Lambda_c(v') \pi_l(p) \pi_m(q) W$ (3 crossed diagrams not shown).}
\section{Analysis}
\subsection{Taylor Expansion}
Let $\eps^2 = w-1$. {\it Assume\/} the Isgur-Wise functions can be expanded in
$\eps$; then to $\ord{\eps^2}$,
\begin{equation}} \def\End{\end{equation} \eta(w) = 1 + \eps \et01 + \eps^2 \et02, \qquad
\eta_1(w) = 1 + \eps \et11 + \eps^2 \et12, \qquad
\eta_2(w) = \et20 + \eps \et21 + \eps^2 \et22 \End
\Puteq{1lp} becomes
\begin{equation}} \def\End{\end{equation} \eps \left[ 3\et01 - 3\et11 \right] + \eps^2 \left[ 3\et02
- 3\et12 + 2\et20 - \frac13 \right] + \ord{\eps^3} =
\ord{1\over N_c} \End
Over different ranges for $\eps$, different terms are constrained:
\begin{equation}} \def\End{\end{equation} \renewcommand{\arraycolsep}{20pt} \begin{array}{rr}
\hbox{[With $\eps = \ord{N_c^{-3/4}}$]} & \et01 - \et11 =
\ord{N_c^{-1/4}} \\
\hbox{[With $\eps = \ord{N_c^{-1/2}}$]} & \et01 - \et11 =
\ord{N_c^{-1/2}} \\
\hbox{[With $\eps = \ord{N_c^{-1/4}}$]} & 3\et02 - 3\et12 +
2\et20 - \frac13 = \ord{N_c^{-1/4}} \end{array} \Endl{tay1}
The latter relation is {\it inconsistent\/} with Chow's result.
Turning to single-pion emission, we go to the $\Sigma_c$ rest frame (where
$\Sigma^0=0$):
\begin{equation}} \def\End{\end{equation} v'=(1,0,0,0), \quad v=(1+\eps^2,0,0,\sqrt2\eps), \quad q\sim (1,
\sin\theta,0,\cos\theta) \End
and we use the result $\et01 - \et11 = \ord{N_c^{-1/2}}$ from \puteq{tay1}.
Then \puteq{1pi} becomes
\begin{equation}} \def\End{\end{equation} \eps \, (\sqrt2 \sin\theta) \; \vec\Sigma \cdot (\cos\theta,0,-\sin\theta)
+ \ord{\eps^2} = \ord{1\over\sqrt{N_c}} \End
With $\eps = \ord{N_c^{-1/4}}$, this gives a constraint on $\Sigma^\mu$,
representing angular momentum conservation among the light degrees of freedom.
We obtain no information about the $\eta$'s.
We analyze 2-pion emission in the $\Lambda_c$ rest frame, with $p \sim
(1, \sin\bar\theta\cos\bar\phi, \sin\bar\theta\sin\bar\phi, \cos\bar\theta)$
(the normalization of $p$ and $q$ drop out). Then \puteq{2pi} becomes
\begin{eqnarray}} \def\Enda{\end{eqnarray} && \quad \eps \left[ 2(1-p\cdot q)(\et01-\et11) \right] \nonumber\\
&+& 2\eps^2 \;\Bigl[ \bigl\{ 1 - \cos^2\theta - \cos^2 \bar\theta +
(\et02-\et12) + 2 \cos\theta \cos\bar\theta \,\et20 \bigr\} \nonumber\\
&& \quad + \; (p\cdot q) \bigl\{ -\cos\theta \cos\bar\theta -
\frac{1}{\sqrt2} (\cos\theta + \cos\bar\theta) (\et01-\et11) -
(\et02-\et12) \bigr\} \Bigr] \nonumber\\
&+& \ord{\eps^3} = \ord{1\over N_c} \Enda
Again using $\et01 - \et11 = \ord{N_c^{-1/2}}$ from \puteq{tay1}, and taking
$\eps = \ord{N_c^{-1/4}}$, we find
\begin{equation}} \def\End{\end{equation} 1 - \cos^2\theta - \cos^2 \bar\theta + (1-p\cdot q)(\et02-\et12) +
\cos\theta \cos\bar\theta \, (2\et20 - p\cdot q) = \ord{N_c^{-1/4}}
\Endl{nons}
\subsection{The Exponential Dropoff}
\Puteq{nons}, derived for $\eps = \ord{N_c^{-1/4}}$, cannot be generally true.
We conclude that our assumption of analyticity must be invalid this far from
threshold. [Nothing significant changes if we try expanding in some other
power of $(w-1)$.] The Isgur-Wise functions must vanish, {\it e.g.}\ exponentially,
for $\eps \ge N_c^{-1/4}$, in which case \puteq{2pi} is trivially satisfied.
Indeed, Jenkins, Manohar and Wise \cite{jmw} showed that $\eta \sim
\exp\{-\lambda N_c^{3/2} (w-1)\}$. So in fact, the Isgur-Wise functions vanish
exponentially fast for $\eps > N_c^{-3/4}$, which is a stronger statement than
ours.
Unfortunately, the only relation we can then retain is the first line of
\puteq{tay1}. There is no inconsistency with Chow or with kinematics, but
neither can we verify Chow's prediction for $\eta_2(1)$.
\section{Conclusions}
We have analyzed three weak-decay processes ($\Lambda_b \to \Lambda_c W$ at one
loop, $\Lambda_b \to \Sigma_c \pi W$, and $\Lambda_b \to \Lambda_c \pi\pi W$)
in the chiral/heavy/large-$N_c$ limits. These are the {\it only\/} processes
that do not involve higher states in the $I\!=\!J$ tower. In this diagrammatic
approach, unitarity requires certain constraints on the baryonic Isgur-Wise
functions. At threshold, $\eta(1)=\eta_1(1)$ by heavy quark symmetry. For
$w-1 \approx N_c^{-3/2}$, we still find $\eta=\eta_1$, in agreement with Chow.
The functions vanish exponentially beyond that, and we can derive no further
information.
These unitarity constraints are consistent with, but not as powerful as, Chow's
Skyrme model [or $SU(4)$] relations \cite{chow,chow2}. In particular,
unitarity constraints give no prediction for $\eta_2(1)$, whereas the Skyrme
model analysis predicts $\eta_2(1)=-1/2$.
We emphasize that we do {\it not\/} disagree with Chow's results. Rather, we
have shown that perturbative unitarity is incapable of reproducing them. Here
is an explicit counterexample to the widely-held belief that all large-$N_c$
predictions can be derived from unitarity constraints.
\newpage
\leftline{\Large\bf Acknowledgments} \bigskip
The authors thank C.K. Chow, Ming Lu, Martin Savage, and Mark Wise for helpful
discussions. This work was partially supported by the U.S. Dept.\ of Energy
under Contract DE-FG02-91-ER40682.
\def\ap#1{{Ann.\ Phys.} {\bf #1}}
\def\pl#1{{Phys.\ Lett.} {\bf #1}}
\def\np#1{{Nucl.\ Phys.} {\bf #1}}
\def\pr#1{{Phys.\ Rev.} {\bf #1}}
\def\prl#1{{Phys.\ Rev.\ Lett.} {\bf #1}}
\def\prs#1{{Proc.\ Roy.\ Soc.} {\bf #1}}
\def\zp#1{{Z.\ Phys.} {\bf #1}}
\def{ibid.\ }{{ibid.\ }}
|
3,212,635,537,411 | arxiv | \section{Introduction}
\label{sec:intro}
Learning semantic representations for words is a fundamental task in NLP
that is required in numerous higher-level NLP applications~\cite{Collobert:2011}.
Distributed word representations have gained much popularity lately because of their accuracy
as semantic representations for words~\cite{Milkov:2013,Pennington:EMNLP:2014}.
However, the meaning of a word often varies from one domain to another.
For example, the phrase \emph{lightweight} is often used in a positive sentiment in the
portable electronics domain because a lightweight device is easier to carry around,
which is a positive attribute for a portable electronic device.
However, the same phrase has a negative sentiment assocition in the \emph{movie}
domain because movies that do not invoke deep thoughts in viewers are considered to be lightweight~\cite{Bollegala:ACL:2014}.
However, existing word representation learning methods are agnostic to such domain-specific semantic variations
of words, and capture semantics of words only within a single domain.
To overcome this problem and capture domain-specific semantic orientations of words,
we propose a method that learns separate distributed representations for each domain in which a word occurs.
Despite the successful applications of distributed word representation
learning methods~\cite{Pennington:EMNLP:2014,Collobert:2011,Milkov:2013}
most existing approaches are limited to learning only a single representation for a given word~\cite{Reisinger:NAACL:2010}.
Although there have been some work on learning multiple \emph{prototype}
representations~\cite{Huang:ACL:2012,neelakantan-EtAl:2014:EMNLP2014}
for a word considering its multiple senses, such methods do not consider the semantics of the domain
in which the word is being used.
If we can learn separate representations for a word for each
domain in which it occurs, we can use the learnt representations for domain adaptation tasks
such as cross-domain sentiment classification~\cite{Bollegala:ACL:2011},
cross-domain POS tagging~\cite{schnabel-schutze:2013:IJCNLP},
cross-domain dependency parsing~\cite{McClosky:NAACL:2010}, and
domain adaptation of
relation extractors~\cite{Bollegala:IJCAI:2013,Bollegala:TKDE:RA:2013,Bollegala:IJCAI:2011,Jiang:ACL:2007,Jiang:CIKM:2007}.
We introduce the \emph{cross-domain word representation learning} task, where given two domains,
(referred to as the \emph{source} ($\cS$) and the \emph{target} ($\cT$))
the goal is to learn two separate representations $\vec{w}_\cS$ and $\vec{w}_\cT$ for a word
$w$ respectively from the source and the target domain that capture \emph{domain-specific} semantic variations of $w$.
In this paper, we use the term \emph{domain} to represent a collection of documents related to
a particular topic such as user-reviews in Amazon for a product category (e.g. \textit{books}, \textit{dvds}, \textit{movies}, etc.).
However, a domain in general can be a field of study (e.g. \textit{biology}, \textit{computer science}, \textit{law}, etc.)
or even an entire source of information (e.g. \textit{twitter}, \textit{blogs}, \textit{news articles}, etc.).
In particular, we do not assume the availability of any labeled data for learning word representations.
This problem setting is closely related to unsupervised domain adaptation~\cite{Blitzer:EMNLP:2006},
which has found numerous useful applications such as, sentiment classification and POS tagging.
For example, in unsupervised cross-domain sentiment classification~\cite{Blitzer:EMNLP:2006,Blitzer:ACL:2007},
we train a binary sentiment classifier using
positive and negative labeled user reviews in the source domain, and apply the trained classifier to predict
sentiment of the target domain's user reviews.
Although the distinction between the source and the target domains is not important for the word representation learning
step, it is important for the domain adaptation tasks in which we subsequently evaluate the learnt word representations.
Following prior work on domain adaptation~\cite{Blitzer:EMNLP:2006},
high-frequent features (unigrams/bigrams) common to both domains are referred to as \emph{domain-independent}
features or \emph{pivots}.
In contrast, we use \emph{non-pivots} to refer to features that are specific to a single domain.
We propose an unsupervised cross-domain word representation learning method that jointly optimizes two criteria:
(a) given a document $d$ from the source or the target domain, we must accurately
predict the non-pivots that occur in $d$ using the pivots that occur in $d$, and
(b) the source and target domain representations we learn for pivots must be similar.
The main challenge in domain adaptation is \emph{feature mismatch}, where the features that we use for training
a classifier in the source domain do not necessarily occur in the target domain.
Consequently, prior work on domain adaptation~\cite{Blitzer:EMNLP:2006,Pan:WWW:2010}
learn lower-dimensional mappings from non-pivots to pivots, thereby overcoming the feature mismatch problem.
Criteria (a) ensures that word representations for domain-specific non-pivots in each domain are related to
the word representations for domain-independent pivots. This relationship enables us to discover
pivots that are similar to target domain-specific non-pivots, thereby overcoming the feature mismatch problem.
On the other hand, criteria (b) captures the prior knowledge that high-frequent words
common to two domains often represent domain-independent
semantics. For example, in sentiment classification, words such as \emph{excellent} or \emph{terrible} would express
similar sentiment about a product irrespective of the domain.
However, if a pivot expresses different semantics in source and the target domains,
then it will be surrounded by dissimilar sets of non-pivots, and reflected in the first criteria.
Criteria (b) can also be seen as a regularization constraint imposed on word representations to prevent overfitting by
reducing the number of free parameters in the model.
Our contributions in this paper can be summarized as follows.
\begin{itemize}
\item We propose a distributed word representation learning method that learns separate representations for a word
for each domain in which it occurs.
To the best of our knowledge, ours is the first-ever \emph{domain-sensitive distributed} word representation learning method.
\item Given domain-specific word representations, we propose a method to learn a cross-domain sentiment classifier.
Although word representation learning methods have been used for various related tasks in NLP such as
similarity measurement~\cite{Mikolov:NAACL:2013}, POS tagging~\cite{Collobert:2011},
dependency parsing~\cite{Socher:ICML:2011}, machine translation~\cite{Zou:EMNLP:2013},
sentiment classification~\cite{socher-EtAl:2011:EMNLP},
and semantic role labeling~\cite{roth-woodsend:2014:EMNLP2014}, to the best of our knowledge,
word representations methods have not yet been used for cross-domain sentiment classification.
\end{itemize}
Experimental results for cross-domain sentiment classification on a benchmark
dataset show that the word representations learnt using the proposed method
statistically significantly outperform a state-of-the-art domain-insensitive word representation learning method~\cite{Pennington:EMNLP:2014},
and several competitive baselines. In particular, our proposed cross-domain word representation learning method
is not specific to a particular task such as sentiment classification, and in principle, can be in applied to a wide-range
of domain adaptation tasks.
Despite this task-independent nature of the proposed method, it achieves the best sentiment classification accuracies
on all domain-pairs, reporting statistically comparable results to the
current state-of-the-art unsupervised cross-domain sentiment classification methods~\cite{Pan:WWW:2010,Blitzer:EMNLP:2006}.
\section{Related Work}
\label{sec:related}
Representing the semantics of a word using some algebraic structure such as a vector (more generally a tensor)
is a common first step in many NLP tasks~\cite{Turney:JAIR:2010}. By applying algebraic operations on the word representations,
we can perform numerous tasks in NLP, such as composing representations for larger textual units beyond individual words such
as phrases~\cite{Mitchell:ACL:2008}. Moreover, word representations are found to be useful for measuring semantic similarity,
and for solving proportional analogies~\cite{Mikolov:NAACL:2013}.
Two main approaches for computing word representations can be identified
in prior work~\cite{baroni-dinu-kruszewski:2014:P14-1}: \emph{counting-based} and \emph{prediction-based}.
In counting-based approaches~\cite{Baroni:DM}, a word $w$ is represented by a vector $\vec{w}$ that contains other words that
co-occur with $w$ in a corpus. Numerous methods for selecting co-occurrence contexts
such as proximity or dependency relations have been proposed~\cite{Turney:JAIR:2010}.
Despite the numerous successful applications of co-occurrence counting-based distributional word representations,
their high dimensionality and sparsity are often problematic in practice.
Consequently, further post-processing steps such as dimensionality reduction,
and feature selection are often required when using counting-based word representations.
On the other hand, prediction-based approaches first assign each word, for example, with a $d$-dimensional
real-vector, and learn the elements of those vectors by applying them in an auxiliary task such as language modeling,
where the goal is to predict the next word in a given sequence.
The dimensionality $d$ is fixed for all the words in the vocabulary, and,
unlike counting-based word representations, is much smaller (e.g. $d \in [10, 1000]$ in practice) compared to the vocabulary size.
The neural network language model (NNLM)~\cite{Bengio:JMLR:2003}
uses a multi-layer feed-forward neural network to predict the next word in a sequence,
and uses backpropagation to update the word vectors such that the prediction error is minimized.
Although NNLMs learn word representations as a by-product, the main focus on language modeling is
to predict the next word in a sentence given the previous words, and not learning word representations that capture semantics.
Moreover, training multi-layer neural networks using large text corpora is time consuming.
To overcome those limitations, methods that specifically focus on learning word
representations that model word co-occurrences in large corpora
have been proposed~\cite{Milkov:2013,Mnih:2013,Huang:ACL:2012,Pennington:EMNLP:2014}.
Unlike the NNLM, these methods use \emph{all} the words in a contextual window in the prediction task.
Methods that use one or no hidden layers are proposed to improve the scalability of the learning algorithms.
For example, the skip-gram model~\cite{Mikolov:NIPS:2013} predicts the words $c$ that appear in the local context of a word $w$, whereas
the continuous bag-of-words model (CBOW) predicts a word $w$ conditioned on all the words $c$ that appear in
$w$'s local context~\cite{Milkov:2013}.
Methods that use global co-occurrences in the entire corpus to learn word
representations have shown to outperform methods that use only local co-occurrences~\cite{Huang:ACL:2012,Pennington:EMNLP:2014}.
Overall, prediction-based methods have shown to outperform counting-based methods~\cite{baroni-dinu-kruszewski:2014:P14-1}.
Despite their impressive performance, existing methods for word representation learning
do not consider the semantic variation of words across different domains.
However, as described in Section~\ref{sec:intro},
the meaning of a word vary from one domain to another, and must be considered.
To the best of our knowledge,
the only prior work studying the problem of word representation variation across domains is due to
Bollegala et al.~\shortcite{Bollegala:ACL:2014}. Given a source and a target domain,
they first select a set of pivots using pointwise mutual information, and create two distributional representations for each pivot using
their co-occurrence contexts in a particular domain. Next, a projection matrix from the source to the target domain feature spaces
is learnt using partial least squares regression. Finally, the learnt projection matrix is used to
find the nearest neighbors in the source domain for each target domain-specific features.
However, unlike our proposed method, their method \emph{does not} learn domain-specific word representations,
but simply uses co-occurrence counting when creating in-domain word representations.
Faralli et al. \shortcite{Faralli:EMNLP:2012} proposed a domain-driven word sense disambiguation (WSD) method where they
construct glossaries for several domain using a pattern-based bootstrapping technique.
This work demonstrates the importance of considering the domain specificity of word senses.
However, the focus of their work is not to learn representations for words or their senses in a domain, but to
construct glossaries. It would be an interesting future research direction to explore the possibility
of using such domain-specific glossaries for learning domain-specific word representations.
Neelakantan et al.~\shortcite{neelakantan-EtAl:2014:EMNLP2014} proposed a method that jointly performs WSD and word embedding learning, thereby
learning multiple embeddings per word type. In particular, the number of senses per word type is
automatically estimated. However, their method is limited to a single domain, and does not consider how the representations vary across domains.
On the other hand, our proposed method learns a single representation for a particular
word for each domain in which it occurs.
Although in this paper we focus on the monolingual setting where source and target domains belong to the same language,
the related setting where learning representations for words that are translational pairs across languages
has been studied~\cite{Moritz:ICLR:2014,Klementiev:COLING:2012,Gouws:ICML:2015}.
Such representations are particularly useful for cross-lingual information retrieval~\cite{Duc:WI:2010}.
It will be an interesting future research direction to extend our proposed method to learn such cross-lingual word representations.
\section{Cross-Domain Representation Learning}
We propose a method for learning word representations that are sensitive to the semantic variations of words
across domains. We call this problem \emph{cross-domain word representation learning}, and provide a definition
in Section~\ref{sec:definition}. Next, in Section~\ref{sec:model}, given a set of pivots that occurs in both a source
and a target domain, we propose a method for learning cross-domain word representations.
We defer the discussion of pivot selection methods to Section~\ref{sec:pivots}.
In Section~\ref{sec:DA}, we propose a method for using the learnt word representations to train
a cross-domain sentiment classifier.
\subsection{Problem Definition}
\label{sec:definition}
Let us assume that we are given two sets of documents $\cD_\cS$ and $\cD_\cT$ respectively for a source ($\cS$)
and a target ($\cT$) domain.
We do not consider the problem of retrieving documents for a domain, and assume such a collection of documents to be given.
Then, given a particular word $w$, we define cross-domain representation learning as the task of learning two separate representations
$\vec{w}_\cS$ and $\vec{w}_\cT$ capturing $w$'s semantics in respectively the source $\cS$ and the target $\cT$ domains.
Unlike in domain adaptation, where there is a clear distinction between the source (i.e. the domain on which we train)
vs. the target (i.e. the domain on which we test) domains, for representation learning purposes we do not make a distinction
between the two domains.
In the \emph{unsupervised} setting of the cross-domain representation learning that we
study in this paper, we do not assume the availability of labeled data for any domain for the purpose of learning word representations.
As an extrinsic evaluation task, we apply the trained word representations
for classifying sentiment related to user-reviews (Section~\ref{sec:DA}).
However, for this evaluation task we require sentiment-labeled user-reviews from the
source domain.
Decoupling of the word representation learning from any tasks in which those representations
are subsequently used, simplifies the problem as well as enables us to learn \emph{task-independent} word representations
with potential generic applicability.
Although we limit the discussion to a pair of domains for simplicity,
the proposed method can be easily extended to jointly learn word representations for more than two domains.
In fact, prior work on cross-domain sentiment analysis show that
incorporating multiple source domains improves sentiment classification
accuracy on a target domain~\cite{Bollegala:ACL:2011,Glorot:ICML:2011}.
\subsection{Proposed Method}
\label{sec:model}
To describe our proposed method, let us denote a pivot and a non-pivot feature respectively by $c$ and $w$.
Our proposed method does not depend on a specific pivot selection method,
and can be used with all previously proposed methods for selecting pivots as explained later in Section~\ref{sec:pivots}.
A pivot $c$ is represented in the source and target domains respectively
by vectors $\vec{c}_\cS \in \R^n$ and $\vec{c}_\cT \in \R^n$.
Likewise, a source specific non-pivot $w$ is represented by $\vec{w}_\cS$ in the source domain,
whereas a target specific non-pivot $w$ is represented by $\vec{w}_\cT$ in the target domain.
By definition, a non-pivot occurs only in a single domain. For notational convenience we use $w$ to
denote non-pivots in both domains when the domain is clear from the context.
We use $\cC_\cS$, $\cW_\cS, \cC_\cT$, and $\cW_\cT$ to denote the sets of word representation vectors
respectively for the source pivots, source non-pivots, target pivots, and target non-pivots.
Let us denote the set of documents in the source and the target domains respectively by $\cD_\cS$ and $\cD_\cT$.
Following the bag-of-features model, we assume that a document $D$ is represented by the set of pivots and non-pivots that occur in $D$
($w \in d$ and $c \in d$).
We consider the co-occurrences of a pivot $c$ and a non-pivot $w$ within a fixed-size contextual window in a document.
Following prior work on representation learning~\cite{Milkov:2013}, in our experiments,
we set the window size to $10$ tokens, without crossing sentence boundaries.
The notation $(c, w) \in d$ denotes the co-occurrence of a pivot $c$ and a non-pivot $w$ in a document $d$.
We learn domain-specific word representations by maximizing the
prediction accuracy of the non-pivots $w$ that occur in the local context of a pivot $c$.
The hinge loss, $L(\cC_\cS, \cW_\cS)$, associated with predicting a non-pivot $w$ in a source document $d \in \cD_\cS$
that co-occurs with pivots $c$ is given by:
\begin{equation}
\small
\label{eq:source-loss}
\sum_{d \in \cD_\cS} \sum_{(c, w) \in d} \sum_{w^*\!\by\!p(w)} \max \left(0, 1 -{ \vec{c}_\cS}\T \vec{w}_\cS + {\vec{c}_\cS}\T \vec{w}^{*}_\cS \right)
\end{equation}
Here, $\vec{w}^{*}_\cS$ is the source domain representation of a non-pivot $w^*$ that \emph{does not occur} in $d$.
The loss function given by Eq.~\ref{eq:source-loss} requires that a non-pivot $w$ that co-occurs with a pivot $c$
in the document $d$ is assigned a higher ranking score as measured by the inner-product between $\vec{c}_\cS$
and $\vec{w}_\cS$ than a non-pivot $w^*$ that does not occur in $d$. We randomly sample $k$
non-pivots from the set of all source domain non-pivots that do not occur in $d$ as $w^*$.
Specifically, we use the marginal distribution of non-pivots $p(w)$, estimated from the corpus counts, as the sampling distribution.
We raise $p(w)$ to the $3/4$-th power as proposed by Mikolov et al.~\shortcite{Milkov:2013},
and normalize it to unit probability mass prior to sampling $k$ non-pivots $w^*$ per each co-occurrence of $(c,w) \in d$.
Because non-occurring non-pivots $w^*$ are randomly sampled, prior work on noise contrastive estimation
has found that it requires more negative samples than positive samples to accurately learn a prediction model~\cite{Mnih:2013}.
We experimentally found $k = 5$ to be an acceptable trade-off between the prediction accuracy and the number of training instances.
Likewise, the loss function $L(\cC_\cT, \cW_\cT)$ for predicting non-pivots using pivots in the target domain is given by:
\begin{equation}
\small
\label{eq:target-loss}
\sum_{d \in \cD_\cT} \sum_{(c,w) \in d} \sum_{w^*\!\by\!p(w)} \max \left(0, 1 -{ \vec{c}_\cT}\T \vec{w}_\cT + {\vec{c}_\cT}\T \vec{w}^{*}_\cT \right)
\end{equation}
Here, $w^*$ denotes target domain non-pivots that \emph{do not occur} in $d$, and are randomly sampled from $p(w)$ following the same procedure as in the source domain.
The source and target loss functions given respectively by Eqs.~\ref{eq:source-loss} and \ref{eq:target-loss}
can be used on their own to independently learn source and target domain word representations.
However, by definition, pivots are common to both domains.
We use this property to relate the source and target word representations via a \emph{pivot-regularizer}, $R(\cC_\cS, \cC_\cT)$, defined as:
\begin{equation}
\label{eq:pivot-regularizer}
R(\cC_\cS, \cC_\cT) = \frac{1}{2}\sum_{i=1}^{K} {||\vec{c}^{(i)}_\cS - \vec{c}^{(i)}_\cT||}^2
\end{equation}
Here, $||\vec{x}||$ represents the $l_2$ norm of a vector $\vec{x}$, and $c^{(i)}$ is the $i$-th pivot in a total collection of $K$ pivots.
Word representations for non-pivots in the source and target domains are linked via the pivot regularizer because,
the non-pivots in each domain are predicted using the word representations for the pivots in each domain,
which in turn are regularized by Eq.~\ref{eq:pivot-regularizer}.
The overall objective function, $L(\cC_\cS, \cW_\cS, \cC_\cT, \cW_\cT)$, we minimize is the sum\footnote{Weighting the source and
target loss functions by the respective dataset sizes did not result in any significant increase in performance. We believe that this is
because the benchmark dataset contains approximately equal numbers of documents for each domain.} of the source and target loss functions, regularized via Eq.~\ref{eq:pivot-regularizer} with coefficient $\lambda$, and is given by:
\begin{equation}
\label{eq:overall}
L(\cC_\cS, \cW_\cS,) + L(\cC_\cT, \cW_\cT) + \lambda R(\cC_\cS, \cC_\cT)
\end{equation}
\subsection{Training}
\label{sec:train}
Word representations of pivots $c$ and non-pivots $w$
in the source ($\vec{c}_\cS$, $\vec{w}_\cS$) and the target ($\vec{c}_\cT$, $\vec{w}_\cT$) domains are
parameters to be learnt in the proposed method.
To derive parameter updates, we compute the gradients of the overall loss function in Eq.~\ref{eq:overall}
w.r.t. to each parameter as follows:
{\small
\begin{align}
&\frac{\partial L}{\partial \vec{w}_\cS} =
\begin{cases}
0 & \text{if \ \ } \vec{c}_\cS\T (\vec{w}_\cS - \vec{w}^*_\cS) \geq 1 \\
-\vec{c}_\cS & \text{otherwise}
\end{cases} \\
&\frac{\partial L}{\partial \vec{w}^*_\cS} =
\begin{cases}
0 & \text{if \ \ } \vec{c}_\cS\T (\vec{w}_\cS - \vec{w}^*_\cS) \geq 1 \\
\vec{c}_\cS & \text{otheriwse}
\end{cases} \\
&\frac{\partial L}{\partial \vec{w}_\cT} =
\begin{cases}
0 & \text{ if \ \ } \vec{c}_\cT\T (\vec{w}_\cT - \vec{w}^*_\cT) \geq 1 \\
-\vec{c}_\cT & \text{otherwise}
\end{cases} \\
& \frac{\partial L}{\partial \vec{w}^*_\cT} =
\begin{cases}
0 & \text{ if \ \ } \vec{c}_\cT\T (\vec{w}_\cT - \vec{w}^*_\cT) \geq 1 \\
\vec{c}_\cT & \text{otherwise}
\end{cases} \\
&\frac{\partial L}{\partial \vec{c}_\cS} =
\begin{cases}
\lambda (\vec{c}_\cS - \vec{c}_\cT) & \text{\!\!\!\!\!\!\!\!\!\!if \ \ } \vec{c}_\cS\T (\vec{w}_\cS - \vec{w}^*_\cS) \geq 1 \\
\vec{w}^*_\cS - \vec{w}_\cS + \lambda(\vec{c}_\cS - \vec{c}_\cT) & \text{otherwise}
\end{cases} \\
&\frac{\partial L}{\partial \vec{c}_\cT} =
\begin{cases}
\lambda(\vec{c}_\cT - \vec{c}_\cS) & \text{\!\!\!\!\!\!\!\!\!\!if \ \ } \vec{c}_\cT\T (\vec{w}_\cT - \vec{w}^*_\cT) \geq 1 \\
\vec{w}^*_\cT- \vec{w}_\cT + \lambda (\vec{c}_\cT - \vec{c}_\cS) & \text{otherwise}
\end{cases}
\end{align}
}%
Here, for simplicity, we drop the arguments inside the loss function and write it as $L$.
We use mini batch stochastic gradient descent with a batch size of $50$ instances.
AdaGrad~\cite{Duchi:JMLR:2011} is used to schedule the learning rate.
All word representations are initialized with $n$ dimensional random vectors sampled from
a zero mean and unit variance Gaussian. Although the objective in Eq.~\ref{eq:overall} is not jointly convex in
all four representations, it is convex w.r.t. the representation of a particular feature (pivot or non-pivot)
when the representations for all the other features are held fixed. In our experiments, the training
converged in all cases with less than $100$ epochs over the dataset.
The rank-based predictive hinge loss (Eq.~\ref{eq:source-loss}) is inspired by the prior work on word representation learning
for a single domain~\cite{Collobert:2011}.
However, unlike the multilayer neural network in \newcite{Collobert:2011}, the proposed method uses
a computationally efficient single layer to reduce the number of parameters that must be learnt, thereby scaling to large datasets.
Similar to the skip-gram model~\cite{Milkov:2013}, the proposed method predicts occurrences of contexts (non-pivots) $w$
within a fixed-size contextual window of a target word (pivot) $c$.
Scoring the co-occurrences of two words $c$ and $w$ by the bilinear form given by the
inner-product is similar to prior work on domain-insensitive word-representation
learning~\cite{Mnih:HLBL:NIPS:2008,Milkov:2013}.
However, unlike those methods that use the softmax function to convert inner-products to probabilities,
we directly use the inner-products without any further transformations, thereby avoiding
computationally expensive distribution normalizations over the entire vocabulary.
\subsection{Pivot Selection}
\label{sec:pivots}
Given two sets of documents $\cD_\cS$, $\cD_\cT$ respectively for the source and the target domains,
we use the following procedure to select pivots and non-pivots.
First, we tokenize and lemmatize each document using the
Stanford CoreNLP toolkit\footnote{\url{http://nlp.stanford.edu/software/corenlp.shtml}}.
Next, we extract unigrams and bigrams as features for representing a document.
We remove features listed as stop words using a standard stop words list.
Stop word removal increases the effective co-occurrence window size for a pivot.
Finally, we remove features that occur less than $50$ times in the entire set of documents.
Several methods have been proposed in the prior work on domain adaptation for selecting
a set of pivots from a given pair of domains such as the minimum frequency of occurrence of a feature
in the two domains, mutual information (MI), and the entropy of the feature distribution over the documents~\cite{Pan:WWW:2010}.
In our preliminary experiments, we discovered that a normalized version of the PMI (NPMI)~\cite{Bouma:2009}
to work consistently well for selecting pivots from different pairs of domains.
NPMI between two features $x$ and $y$ is given by:
\begin{equation}
\label{eq:NPMI}
\mathrm{NPMI}(x, y) = \log\left(\frac{p(x,y)}{p(x)p(y)}\right) \frac{1}{-\log(p(x,y))}
\end{equation}
Here, the joint probability $p(x,y)$, and the marginal probabilities $p(x)$ and $p(y)$ are estimated
using the number of co-occurrences of $x$ and $y$ in the sentences in the documents.
Eq.~\ref{eq:NPMI} normalizes both the upper and lower bounds of the PMI.
We measure the appropriateness of a feature as a pivot according to the score given by:
\begin{equation}
\label{eq:pivot-score}
\mathrm{score}(x) = \min \left( \mathrm{NPMI}(x, \cS), \mathrm{NPMI}(x, \cT) \right) .
\end{equation}
We rank features that are common to both domains in the descending order of
their scores as given by Eq.~\ref{eq:pivot-score}, and select the top $N_\cP$ features as pivots.
We rank features $x$ that occur only in the source domain by $\mathrm{NPMI}(x, \cS)$,
and select the top ranked $N_\cS$ features as source-specific non-pivots.
Likewise, we rank the features $x$ that occur only in the target domain by $\mathrm{NPMI}(x, \cT)$,
and select the top ranked $N_\cT$ features as target-specific non-pivots.
The pivot selection criterion described here differs from that of
Blitzer et al.~\shortcite{Blitzer:EMNLP:2006,Blitzer:ACL:2007},
where pivots are defined as features that behave similarly both in the source and the target domains.
They compute the mutual information between a feature (i.e. unigrams or bigrams) and the sentiment labels
using source domain labeled reviews. This method is useful when selecting pivots that are closely
associated with positive or negative sentiment in the source domain. However, in unsupervised domain adaptation
we do not have labeled data for the target domain. Therefore, the pivots selected using this approach are not
guaranteed to demonstrate the same sentiment in the target domain as in the source domain.
On the other hand, the pivot selection method proposed in this paper focuses on identifying a subset of
features that are closely associated with both domains.
It is noteworthy that our proposed cross-domain word representation learning method (Section~\ref{sec:model})
\emph{does not} assume any specific pivot/non-pivot selection method.
Therefore, in principle, our proposed word representation learning method could be used with any of the
previously proposed pivot selection methods.
We defer a comprehensive evaluation of possible combinations of pivot selection methods and their effect on
the proposed word representation learning method to future work.
\subsection{Cross-Domain Sentiment Classification}
\label{sec:DA}
As a concrete application of cross-domain word representations, we describe a method for learning a cross-domain sentiment
classifier using the word representations learnt by the proposed method.
Existing word representation learning methods that learn from only a single domain are typically evaluated for their
accuracy in measuring semantic similarity between words, or by solving word analogy problems.
Unfortunately, such gold standard datasets capturing cross-domain semantic variations of words are unavailable.
Therefore, by applying the learnt word representations in a cross-domain sentiment classification task,
we can conduct an indirect extrinsic evaluation.
The train data available for unsupervised cross-domain sentiment classification
consists of unlabeled data for both the source and the target domains
as well as labeled data for the source domain. We train a binary sentiment classifier using those train data,
and apply it to classify sentiment of the target test data.
Unsupervised cross-domain sentiment classification is challenging due to two reasons: \emph{feature-mismatch},
and \emph{semantic variation}.
First, the sets of features that occur in source and target domain documents are different.
Therefore, a sentiment classifier trained using source domain labeled data is likely to encounter unseen features during
test time. We refer to this as the feature-mismatch problem. Second, some of the features that occur in both domains
will have different sentiments associated with them (e.g. \emph{lightweight}). Therefore, a sentiment classifier trained
using source domain labeled data is likely to incorrectly predict similar sentiment (as in the source) for such features.
We call this the semantic variation problem. Next, we propose a method to overcome both problems using cross-domain word representations.
Let us assume that we are given a set $\{(\vec{x}^{(i)}_{\cS}, y^{(i)})\}_{i=1}^{n}$ of $n$ labeled reviews $\vec{x}^{(i)}_{\cS}$
for the source domain $\cS$. For simplicity, let us consider binary sentiment classification where each review $\vec{x}^{(i)}$ is
labeled either as positive (i.e.\ $y^{(i)} = 1$) or negative (i.e.\ $y^{(i)} = -1$).
Our cross-domain binary sentiment classification method can be easily extended to multi-class classification.
First, we lemmatize each word in a source domain labeled review $\vec{x}^{(i)}_{\cS}$,
and extract unigrams and bigrams as features to represent $\vec{x}^{(i)}_{\cS}$ by a binary-valued feature vector.
Next, we train a binary linear classifier, $\vec{\theta}$, using those feature vectors.
Any binary classification algorithm can be used for this purpose.
We use $\vec{\theta}(z)$ to denote the weight learnt by the classifier for a feature $z$.
In our experiments, we used $l_2$ regularized logistic regression.
At test time, we represent a test target review by a binary-valued vector $\vec{h}$ using a the set of unigrams and bigrams
extracted from that review. Then, the activation score, $\psi(\vec{h})$, of $\vec{h}$ is defined by:
\begin{equation}
\small
\psi(\vec{h}) = \sum_{c \in \vec{h}} \sum_{c' \in \vec{\theta}} \vec{\theta}(c') f(\vec{c}'_\cS, \vec{c}_\cS) +
\sum_{w \in \vec{h}} \sum_{w' \in \vec{\theta}} \vec{\theta}(w') f(\vec{w}'_\cS, \vec{w}_\cT)
\label{eq:score}
\end{equation}
Here, $f$ is a similarity measure between two vectors. If $\psi(\vec{h}) > 0$, we classify $\vec{h}$ as positive,
and negative otherwise. Eq.~\ref{eq:score} measures the similarity between each feature in $\vec{h}$ against the
features in the classification model $\vec{\theta}$.
For pivots $c \in \vec{h}$, we use the the source domain representations to measure
similarity, whereas for the (target-specific) non-pivots $w \in \vec{h}$, we use their target domain representations.
We experimented with several popular similarity measures for $f$ and found cosine similarity to perform consistently well.
We can interpret Eq.~\ref{eq:score} as a method for \emph{expanding} a test target document using
nearest neighbor features from the source domain labeled data.
It is analogous to query expansion used in information retrieval to improve document recall~\cite{Fang:ACL:2008}.
Alternatively, Eq.~\ref{eq:score} can be seen as a linearly-weighted additive kernel function over two feature spaces.
\section{Experiments and Results}
\label{sec:exp}
For train and evaluation purposes, we use the Amazon product reviews collected by \newcite{Blitzer:ACL:2007}
for the four product categories:
books (\textbf{B}), DVDs (\textbf{D}), electronic items (\textbf{E}), and kitchen appliances (\textbf{K}).
There are $1000$ positive and $1000$ negative sentiment labeled reviews for each domain.
Moreover, each domain has on average $17,547$ unlabeled reviews.
We use the standard split of $800$ positive and $800$ negative labeled reviews from each domain
as training data, and the rest (200+200) for testing.
For validation purposes we use \textit{movie} (source) and
\textit{computer} (target) domains, which were also collected by \newcite{Blitzer:ACL:2007}, but not part of the train/test domains.
Experiments conducted using this validation dataset revealed that the performance of the proposed method
is relatively insensitive to the value of the regularization parameter $\lambda \in [10^{-3}, 10^{3}]$.
For the non-pivot prediction task we generate positive and negative instances using the procedure described in Section~\ref{sec:model}.
As a typical example, we have $88,494$ train instances from the books source domain and $141,756$ train instances
from the target domain (1:5 ratio between positive and negative instances in each domain).
The number of pivots and non-pivots are set to $N_\cP = N_\cS = N_\cT = 500$.
\begin{figure*}[t]
\centering
\includegraphics[width=17cm]{overall.pdf}
\caption{Accuracies obtained by different methods for each source-target pair in cross-domain sentiment classification.}
\label{fig:overall}
\end{figure*}
In Figure~\ref{fig:overall}, we compare the proposed method against two baselines (\textbf{NA}, \textbf{InDomain}),
current state-of-the-art methods for unsupervised cross-domain sentiment classification (\textbf{SFA}, \textbf{SCL}),
word representation learning (\textbf{GloVe}), and cross-domain similarity prediction (\textbf{CS}).
The \textbf{NA} (no-adapt) lower baseline uses a classifier trained on source labeled data to classify target test data without any
domain adaptation. The \textbf{InDomain} baseline is trained using the labeled data for the target domain,
and simulates the performance we can expect to obtain if target domain labeled data were available.
Spectral Feature Alignment (\textbf{SFA})~\cite{Pan:WWW:2010} and
Structural Correspondence Learning (\textbf{SCL})~\cite{Blitzer:ACL:2007} are
the state-of-the-art methods for cross-domain sentiment classification. However, those methods do not learn word representations.
We use Global Vector Prediction (\textbf{GloVe})~\cite{Pennington:EMNLP:2014}, the current state-of-the-art word representation
learning method, to learn word representations separately from the source and target domain unlabeled data, and use the learnt representations
in Eq.~\ref{eq:score} for sentiment classification. In contrast to the \textit{joint} word representations learnt by the proposed method,
\textbf{GloVe} simulates the level of performance we would obtain by learning representations \textit{independently}.
\textbf{CS} denotes the cross-domain vector prediction method proposed by \newcite{Bollegala:ACL:2014}.
Although \textbf{CS} can be used to learn a vector-space translation matrix, it \emph{does not} learn word representations.
Vertical bars represent the classification accuracies (i.e. percentage of the correctly classified test instances)
obtained by a particular method on target domain's test data, and Clopper-Pearson $95\%$ binomial confidence intervals are superimposed.
Differences in data pre-processing (tokenization/lemmatization), selection (train/test splits), feature representation (unigram/bigram),
pivot selection (MI/frequency), and the binary classification algorithms used to train the final classifier
make it difficult to directly compare results published in prior work. Therefore, we re-run the original algorithms
on the same processed dataset under the same conditions such that any differences reported in Figure~\ref{fig:overall}
can be directly attributable to the domain adaptation, or word-representation learning methods compared.
All methods use $l_2$ regularized logistic regression
as the binary sentiment classifier, and the regularization coefficients are set to their optimal values on the validation dataset.
\textbf{SFA}, \textbf{SCL}, and \textbf{CS} use the same set of $500$ pivots as used by the proposed method
selected using NPMI (Section~\ref{sec:pivots}).
Dimensionality $n$ of the representation is set to $300$ for both \textbf{GloVe} and the proposed method.
From Fig.~\ref{fig:overall} we see that the proposed method reports the highest classification accuracies in all $12$ domain pairs.
Overall, the improvements of the proposed method over \textbf{NA}, \textbf{GloVe}, and \textbf{CS} are statistically significant,
and is comparable with \textbf{SFA}, and \textbf{SCL}. The proposed method's improvement over \textbf{CS} shows the importance
of \emph{predicting} word representations instead of \emph{counting}.
The improvement over \textbf{GloVe} shows that it is inadequate to simply apply existing word representation learning methods
to learn independent word representations for the source and target domains.
We must consider the correspondences between the two domains as expressed by the pivots
to jointly learn word representations. As shown in Fig.~\ref{fig:dims}, the proposed method reports superior accuracies over
\textbf{GloVe} across different dimensionalities. Moreover, we see that when the dimensionality of the representations increases,
initially accuracies increase in both methods and saturates after $200-600$ dimensions. However, further increasing the dimensionality
results in unstable and some what poor accuracies due to overfitting when training high-dimensional representations.
Although our word representations learnt by the proposed method are not specific to sentiment classification,
the fact that it clearly outperforms \textbf{SFA} and \textbf{SCL} in all domain pairs is encouraging, and implies the wider-applicability
of the proposed method for domain adaptation tasks beyond sentiment classification.
\begin{figure}[t]
\centering
\includegraphics[height=50mm]{dims.pdf}
\caption{Accuracy vs. dimensionality of the representation.}
\label{fig:dims}
\end{figure}
\section{Conclusion}
We proposed an unsupervised method for learning cross-domain word representations using a given set of pivots
and non-pivots selected from a source and a target domain. Moreover, we proposed a domain adaptation
method using the learnt word representations.
Experimental results on a cross-domain sentiment classification task showed that the proposed method outperforms several
competitive baselines and achieves best sentiment classification accuracies for all domain pairs.
In future, we plan to apply the proposed method to other types of domain adaptation tasks such as cross-domain part-of-speech
tagging, named entity recognition, and relation extraction.
Source code and pre-processed data etc. for this publication are publicly available\footnote{\url{www.csc.liv.ac.uk/~danushka/prj/darep}}.
\bibliographystyle{acl}
|
3,212,635,537,412 | arxiv | \section{Introduction}
\subsection{Random geometric complexes}The subject of random geometric complexes has recently attracted a lot of attention, with a special focus on the study of expectation of topological properties of these complexes \cite{Kahle2011, Penrose, NSW, YSA, BobrowskiMukherjee, BobrowskiKahle}\footnote{This list is by no mean complete, see \cite{BobrowskiKahle} for a survey and a more complete set of references!} (e.g. number of connected components, or more generally Betti numbers). In a recent paper \cite{Antonio}, Auffinger, Lerario and Lundberg have imported methods from \cite{NazarovSodin, SarnakWigman} for the study of finer properties of these random complexes, namely the distribution of the homotopy types of the connected components of the complex. Before moving to the content of the current paper, we discuss the main ideas from \cite{Antonio} and introduce some terminology.
Let $(M,g)$ be a compact, Riemannian manifold of dimension $m$. We normalize the metric $g$ in such a way that
\begin{equation} \mathrm{vol}(M)=1.\end{equation}
We denote by $\hat{B}(x,r)\subset M$ the Riemannian ball centered at $x$ of radius $r>0$ and we construct a \emph{random $M$--geometric complex} in the \emph{thermodynamic regime} as follows. We let $\{p_1,\ldots,p_n\}$ be a set of points independently sampled from the uniform distribution on $M$, we fix a positive number $\alpha>0$, and we consider:
\begin{equation}\label{eq:alpha}
\mathcal{U}_n:=\bigcup_{k=1}^n\hat{B}(p_k,r)\quad \textrm{where}\quad r:=\alpha n^{-1/m}.\end{equation}
The choice of such $r$ is what defines the so-called \emph{critical} or \emph{thermodynamic regime}\footnote{Quoting the Introduction from \cite{Antonio}: random geometric complexes are studied \emph{within three main phases or regimes based on the relation between density of points and radius of the neighborhoods determining the complex: the subcritical regime (or ``dust phase'') where there are many connected components with little topology, the critical regime (or ``thermodynamic regime'') where topology is the richest (and where the percolation threshold appears), and the supercritical regime where the connectivity threshold appears.
The thermodynamic regime is seen to have the most intricate topology.}} and it's the regime where topology is the richest \cite{Antonio,Kahle}. We say that $\mathcal{U}_n$ is a \emph{random $M$--geometric complex} and the name is motivated by the fact that, for $n$ large enough, $\mathcal{U}_n$ is homotopy equivalent to its \v{C}ech complex, as we shall see in Lemma \ref{nervelemma} below.
Auffinger, Lerario and Lundberg \cite{Antonio} proved that, in the case when $\vol M=1$, the normalized counting measure of connected components of such complexes, counted according to homotopy type, converges in probability to a deterministic measure. That is,
\begin{equation}\label{equation:oldconvergence}
\tilde{\Theta}_n:=\frac{1}{b_0(\mathcal{U}_n)}\sum\delta_{[u]}\xrightarrow[n\rightarrow\infty]{\mathbb{P}}\tilde{\Theta},
\end{equation}where the sum is over all connected components $u$ of $\mathcal{U}_n$, $[u]$ denotes their homotopy type and $b_0$ is the zero-th Betti number, therefore $b_0(\mathcal{U}_n)$ is the number of connected components of $\mathcal{U}_n$. In \eqref{equation:oldconvergence} the measure $\tilde{\Theta}_n$ is a \emph{random} probability measure on the countable set of all possible homotopy types of connected geometric complexes and the convergence is in probability with respect to the total variation distance (see Section \ref{section:Random measures} for more precise definitions). The support of the limiting deterministic measure $\tilde{\Theta}$ equals the set of all homotopy types for Euclidean geometric complexes of dimension $m = \dim M$. Roughly speaking, \eqref{equation:oldconvergence} tells that, for every fixed homotopy type $[u]$ of connected geometric complexes, denoting by $\mathcal{N}_n([u])$ the random variable ``number of connected components of $\mathcal{U}_n$ which are in the homotopy equivalence class $[u]$'', there is a convergence of the random variable $\mathcal{N}_n([u])/b_0(\mathcal{U}_n)$ to a constant $c_{[u]}$ as $n\to \infty$ (the convergence is in $L^1$ and $c_{[u]}>0$ if and only if $[u]$ contains a $\mathbb{R}^m$--geometric complex).
\subsection{Isotopy classes of geometric complexes}
\begin{figure}
\begin{center}
\includegraphics[width=5cm]{knots4.jpg}
\end{center}
\caption{The unknot and the trefoil knot are homotopy equivalent but they are not isotopic.}\label{Figureknots}
\end{figure}We now move to the content of the current paper. Our first goal is to include the results of \cite{Antonio} into a more general framework which allows to make even more refined counts (e.g. according to the type of the embedding of the components, or on the structure of their skeleta, or on the property of containing a given motif\footnote{A \emph{motif} in a graph (or more generally in a complex) is a recurrent and statistically significant sub-graph or pattern.}). The first result that we prove is that (\ref{equation:oldconvergence}) still holds if we consider \emph{isotopy classes} instead of homotopy classes: intuitively, two complexes are isotopic if the vertices of one can be moved continuously to the vertices of the other without ever changing the combinatorics of the intersection of the corresponding balls (see Definition \ref{def:isotopy}). From now on we will always make the assumption that our complexes are \emph{nondegenerate}, i.e. that the boundaries of the balls defining them intersect transversely (see Definition \ref{def:nondeg}); our random geometric complexes will be nondegenerate with probability one, and the notion of isotopic nondegenerate complexes coincides with the one from differential topology.
In Theorem \ref{mainthm} we show that
\begin{equation}\label{equation:newconvergence}
\Theta_n\xrightarrow[n\rightarrow \infty]{\mathbb{P}}\Theta,
\end{equation}where $\Theta_n$ is defined in a similar way as $\tilde{\Theta}_n$ above, with isotopy classes instead of homotopy classes Interestingly, the limiting measure depends only on $\alpha$ on the dimension of $M$.\footnote{In the rest of the paper we will consider $\alpha$ as fixed from the very beginning and omit it from the notation; the study of the dependence of the various objects on $\alpha>0$ is an interesting problem, on which for now we cannot say much.}.
To appreciate the difference with the results from \cite{Antonio}: the unknot and the trefoil knot in $\mathbb{R}^3$ (Figure \ref{Figureknots}) are homotopy equivalent but they are not isotopic, and with positive probability there are connected $M$--geometric complexes whose embedding looks like these two knots (see Proposition \ref{propexistence} below); Theorem \ref{mainthm} is able to distinguish between them, whereas the construction from \cite{Antonio} is not.
\subsection{A cascade of measures}Theorem \ref{mainthm} contains in a sense the richest possible information on the topological structure of our geometric complexes and the convergence of many other counting measures can be deduced from it. To explain this idea, we consider the space
\begin{equation} \left(\mathcal{G}/{\cong}\right):=\{\textrm{isotopy classes of connected geometric complexes}\}\end{equation}
and we put an equivalence relation $\rho$ on $\mathcal{G}/{\cong}$ (the relation can be for example: two isotopy classes are the same if their $k$--skeleta are isomorphic, or if they contain the same number of a given motif). Then the natural map $\psi:\left(\mathcal{G}/{\cong}\right)\to \left(\mathcal{G}/{\cong}\right)/{\rho}$ defines the random pushforward measure $\psi_*\Theta_n$ on $\left(\mathcal{G}/{\cong}\right)/{\rho}$ and Theorem \ref{mainthm} implies that $\psi_*\Theta_n\to \psi_*\Theta$.
This idea can be used to produce a ``cascade'' of random relevant measures. Consider in fact the following diagram of maps and spaces:
\begin{equation}
\begin{tikzcd}
\mathcal{G}/{\cong} \arrow[r, "\varphi"] \arrow[rd, "\varphi^{(k)}"'] \arrow[rr, "\phi", bend left=49] & \mathcal{G}/{\simeq} \arrow[d] \arrow[r] & \mathcal{G}/{\sim} \\
& \mathcal{G}^{(k)}/{\simeq} & \end{tikzcd}\end{equation}
where the spaces are:
\begin{align} \left(\mathcal{G}/{\simeq}\right)&:=\{\textrm{isomorphism classes of connected geometric \v{C}ech complexes}\}\\
\left(\mathcal{G}^{(k)}/{\simeq}\right)&:=\{\textrm{isomorphism classes of components of the $k$--skeleton of \v{C}ech complexes}\}\\
\left(\mathcal{G}/{\sim}\right)&:=\{\textrm{homotopy classes of connected geometric complexes}\}\end{align}
and the maps are the natural ``forgetful'' maps. For example, the map $\varphi$ takes the isotopy class of a nondegenerate complex and associates to it its homotopy class; the map $\varphi^{(k)}$ associates to it the isomorphism class of its $k$--skeleton (it is well defined since isotopic complexs have isomorphic \v{C}ech complexes).
Then for all the pushforward measures defined by these maps we have convergence in probability with respect to the total variation distance (see Section \ref{section:Random measures}) and as $n\to \infty$
\begin{equation} \phi_*\Theta_n\to\phi_*\Theta,\quad \varphi_*\Theta_n\to\varphi_*\Theta\quad \textrm{and}\quad \varphi^{(k)}_*\Theta_n\to\varphi^{(k)}_*\Theta.\end{equation}
\subsection{Random Geometric Graphs}
Of special interest is the case of random geometric graphs: vertices of a random $M$--geometric graph $\Gamma_n$ are the points $\{p_1,\ldots,p_n\}$ and we put an edge between $p_i$ and $p_j$ if and only if $i\neq j$ and $\hat{B}(p_i, r)\cap\hat{B}(p_j, r)\neq \emptyset$. Using the above language, a random $M$--geometric graph is the $1$--skeleton of the \v{C}ech complex associated to the complex $\mathcal{U}_n$.
To every random $M$--geometric graph $\Gamma_n$ we can associate the measure:
\begin{equation} \label{eq:graphsintro}\varphi^{(1)}_{*}\Theta_n= \frac{1}{b_0(\Gamma_n)}\sum\delta_{\gamma}\to \varphi^{(1)}_*\Theta,\end{equation}
where the sum is over all connected components of $\Gamma_n$ and $\gamma$ denotes their isomorphism class (as graphs). There is an interesting fact regarding the random variable $b_0(\Gamma_n)$ appearing in \eqref{eq:graphsintro}: it is the same random variable as $b_0(\mathcal{U}_n)$ (the number of components of the random graph and of the random complex are the same), and in \cite{Antonio} it is proved that there exists a constant $\beta$ (depending on the parameter $\alpha$ in \eqref{eq:alpha}) such that:
\begin{equation}
\frac{b_0(\Gamma_n)}{n}=\frac{b_0(\mathcal{U}_n)}{n}\xrightarrow{L^1}\beta.
\end{equation}
The existence of this limit also follows from \cite{GoTrTs}, where the authors establish a limit law in the thermodynamic regime for Betti numbers
of random geometric complexes built over possibly inhomogeneous Poisson point processes in Euclidean space, including the case when the point process is supported on a submanifold.
Moreover, we note that for a related model of random graphs (the Poisson model on $\mathbb{R}^m$, see Section \ref{sec:poissonintro} below) Penrose \cite{Penrose} has proved that there exists a constant $\beta$ (depending on the parameter $\alpha$ in \eqref{eq:alpha}) such that the normalized component count converges to a constant in $L^2$. In fact, as we will see below, related to our $M$--geometric model there is a way to construct a corresponding $\mathbb{R}^m$--geometric model, which is in a sense the rescaled limit of the Riemannian one, and the limit constants for the two models are the same.
In fact the limit measure $\varphi^{(1)}_{*}\Theta$ also comes from the rescaled Euclidean limit and it is supported on connected $\mathbb{R}^m$--geometric graphs. For a given $m$, the set of such graphs is not easy to describe, but in the case $m=1$ they can be characterized by a result of Roberts \cite{unitgraphs2}, and from this result we can deduce a description of the support of the limit measure in \eqref{eq:graphsintro} (see Corollary \ref{cor:limit1} and Section \ref{section:Geometric graphs} for more details).
\begin{rmk}[Related work on random geometric graphs]
The general theory of random graphs has been founded in 1959 by Erd\"os and Rényi, who proposed a model of random graph $G(n,p)$ where the number of vertices is fixed to be $n$ and each pair of distinct vertices is joined by an edge with probability $p$, independently of other edges \cite{erdos1,erdos2,erdos3,erdos4,referee3}. Later on, other models have been proposed in the literature \cite{referee2}, as for instance the Barabási–Albert scale-free network model \cite{referee2-ref2} and the Watts-Strogatz small-world network model \cite{referee2-ref53}. For general references on random graphs, the reader is referred to \cite{referee3-ref10,Chung,chung-random,referee3-ref16,referee3-ref18,referee3-ref23,referee3-ref30,referee3-ref34,referee3-ref40}. Here we focus on the random geometric graph model and we refer to \cite{referee2-ref23,referee2-ref42,referee2-ref50} for more literature on this topic. Applications of random geometric graphs can be found, for instance, in the contexts of wireless networks, epidemic spreading, city growth, power grids, protein-protein interaction networks \cite{referee2}. \end{rmk}
\subsection{The spectrum of a random geometric graph}
When talking about a graph, a natural associated object to look at is its normalized Laplace operator, see Section \ref{section:Laplacian of a graph and its spectrum}. It is known that the spectrum of the (symmetric) normalized Laplace operator for graphs encodes important information about the graphs \cite{Chung}. For example, it tells us how many connected components a graph has; it tells whether a graph is bipartite and whether it is complete; it tells us how difficult it is to partition the vertex set of a graph into two disjoint sets $V_1$ and $V_2$ such that the number of edges between $V_1$ and $V_2$ is as small as possible and such that the \emph{volume} of both $V_1$ and $V_2$, i.e. the sum of the degrees of their vertices, is as big as possible. Therefore, the normalized Laplace operator gives a partition of graphs into families and \emph{isospectral graphs} share important common features. Since, furthermore, the computation of the eigenvalues can be performed with tools from linear algebra, such operator is a very powerful and used tool in graph theory and data analytics.
In the context of random $M$--geometric graphs, the convergence of the counting measure in \eqref{eq:graphsintro} can be used to deduce the existence of a limit measure for the spectrum of the normalized Laplace operator for random geometric graphs. More specifically, we define the \emph{empirical spectral measure} of a graph as the normalized counting measure of eigenvalues of the normalized Laplace operator and we prove that there exists a deterministic measure $\mu$ on the real line such that (Theorem \ref{teomu})
\begin{equation}\label{equation:mu}
\mu_{\Gamma_n}:=\frac{1}{n}\sum_{i=1}^{n}\delta_{\lambda_i(\Gamma_n)}\overset{*}{\underset{n\rightarrow\infty}{\rightharpoonup}}\mu.
\end{equation}Here, $\lambda_1(\Gamma_n),\ldots,\lambda_n(\Gamma_n)$ are the eigenvalues of the normalized Laplace operator of $\Gamma_n$ and the convergence in \eqref{equation:mu} means that for every continuous function $f:[0,2]\to \mathbb{R}$ we have:
\begin{equation} \lim_{n\to \infty}\mathbb{E}\int_{[0,2]}f d\mu_{\Gamma_n}=\int_{[0,2]} f d\mu.\end{equation}
The measure $\mu$ in \eqref{equation:mu} is far from trivial and we don't have yet a clear understanding of it: we know it is supported on the interval $[0,2]$, but for example it is not absolutely continuous with respect to Lebesgue measure (in fact $\mu(\{0\})=\beta>0$).
\begin{rmk}Interestingly, \cite{referee3} studies the convergence of $\mu_{\Gamma_n}$ as $n\rightarrow\infty$ in the case where $\Gamma_n$ is a $G(n,p)$ random graphs and the eigenvalues are the ones of the non-normalized Laplacian or the ones of the adjacency matrix. In particular, it is shown that in such context, under suitable conditions, $\mu_{\Gamma_n}$ converges to the semi-circle law if associated to the adjacency matrix and it converges to the free convolution of the standard normal distribution if associated to the non-normalized Laplacian.\end{rmk}
\begin{rmk}\label{rmk:JJ}In \cite{JJ}, Gu, Jost, Liu and Stadler introduce the notion of \emph{spectral class} of a family of graphs. Given a Radon measure $\rho $ on $[0,2]$ and a sequence $(\Gamma_n)_{n\in \mathbb{N}}$ of graphs with $\#(V(\Gamma_n))=n$, they say that this sequence belongs to the spectral class $\rho$ if $\mu_{\Gamma_n}\overset{*} \rightharpoonup \rho$ as $n\to \infty$. We can interpret \eqref{equation:mu} as saying that our family of random geometric graphs $(\Gamma_n)_n$ belongs to the spectral class $\mu$ (in a probabilistic sense).
\end{rmk}
\begin{rmk}[Related work on spectral theory]Similarly to the spectrum of the normalized Laplace operator, also the spectra of the non-normalized Laplacian matrix (defined in Section \ref{section:Laplacian of a graph and its spectrum}) and the one of the adjacency matrix have been widely studied. We refer the reader to \cite{Chung,referee3-ref42} for general references on spectral graph theory. We refer to \cite{referee3-ref9,Hypergraphs} for applications of spectral graph theory in chemistry and we refer to \cite{referee3-ref24,referee3-ref25,referee3-ref26,referee3-ref38,referee3-ref39,referee3-ref43,referee3-ref44} for applications in theoretical physics and quantum mechanics. For references on spectral graph theory of (not necessarily geometric) random graphs, we refer to \cite{chung-random,referee3,referee2,referee4}. In \cite{referee2}, in particular, the eigenvalues of the adjacency matrix for random gometric graphs are studied using numerical and statistical methods. Remarkably, it is shown that random geometrix graphs are statistically very similar to the other random graph models we have mentioned above: Erdős-Rényi random graphs, Barabási–Albert scale-free networks, Watts-Strogatz small-world networks. On the other hand, in \cite{referee4}, it is shown that symmetric structures abundantly occur in random geometric graphs, while the same doesn't hold for the other random graph models. Our main results on spectral graph theory for random geometric graphs, Theorem \ref{teomu} and Proposition \ref{prop:last} below, follow the same general idea as \cite{referee2} and \cite{referee4}, in the sense that we are interested on the limiting spectrum of large random geometric graphs. The main difference is that \cite{referee2} is focused on the adjacency matrix, \cite{referee4} gives a focus on the non-normalized Laplacian and we focus on the normalized Laplacian. Therefore the final implications differ very much.
\end{rmk}
\subsection{The Euclidean Poisson model}\label{sec:poissonintro} As we already observed, in \cite{Antonio}, the proof of (\ref{equation:oldconvergence}) is based on a \emph{rescaling limit} idea. Namely, one can fix $R>0$ and a point $p\in M$, and study the limit structure of the random complexes inside the ball $\hat{B}(p,Rn^{-1/d})$. The random geometric complex obtained as $n\rightarrow\infty$ can then be described as follows. Let $P:=\{p_1,p_2,\ldots\}$ be a set of points sampled from the standard spatial Poisson distribution in $\mathbb{R}^m$. For $\alpha>0$, let
\begin{equation*}
\mathcal{P}:=\bigcup_{p\in P}B(p,\alpha)
\end{equation*}and let
\begin{equation*}
\mathcal{P}_R:=\{\text{connected components of $\mathcal{P}$ entirely contained in the interior of }B(0,R)\}.
\end{equation*}For the random complex $\mathcal{P}_R$, one can define completely analogue measures, where now the parameter is $R>0$, and all the above discussion applies also to this model (this is discussed throughout the paper).
\subsection*{Structure of the paper}This paper is structured as follows. In Section \ref{section:Geometric complexes} we discuss (deterministic) $M$--geometric complexes and, in particular, we define and see some properties of the set $\mathcal{G}/{\cong}$ of isotopy classes of connected, nondegenerate $M$--geometric complexes. In Section \ref{section: Random geometric complexes} we discuss random $M$--geometric complexes; in Section \ref{section:Random measures} we prove (\ref{equation:newconvergence}). Morevorer, in Section \ref{section:Geometric graphs} we define and see some properties of geometric graphs; in Section \ref{section:Laplacian of a graph and its spectrum} we recall the definition of the normalized Laplace operator $\hat{L}$ for graphs and we prove some properties of the spectral measure in the case of geometric graphs. Finally, in Section \ref{section:Random geometric graphs} we prove (\ref{equation:mu}).
\subsection*{Acknowledgements}
We are grateful to Bernd Sturmfels, because without him this paper would not exist. We are grateful to Fabio Cavalletti, J\"urgen Jost, Matthew Kahle, Erik Lundberg, Leo Mathis and Michele Stecconi for helpful comments, discussions and forbidden graphs. We are grateful to the anonymous referees for the constructive comments.
\section{Geometric complexes}\label{section:Geometric complexes}
Throughout this paper we fix a Riemannian manifold $(M,g)$ of dimension $m$.
\begin{definition}[$M$--geometric complex and its skeleta]Let $p_1,\ldots,p_n$ be points in $M$ and fix $r\geq 0$. We define a \emph{$M$--geometric complex} as
\begin{align*}
\mathcal{U}(\{p_1,\ldots,p_n\},r):&=\bigcup_{k=1}^n\hat{B}(p_k,r)\\
&=\{x\in M: d_M(x,\{p_1,\ldots,p_n\})\leq r\}.
\end{align*}For $\mathcal{U}:=\mathcal{U}(\{p_1,\ldots,p_n\},r)$, we also let
\begin{align*}
\check{C}(\mathcal{U})&:=\check{C}(\{p_1,\ldots,p_n\},r)\\&:=\text{ nerve of the cover }\{\hat{B}(p_k,r)\}_{k=1}^n
\end{align*}and we let
\begin{align*}
\check{C}^{(k)}(\mathcal{U})&:=\check{C}^{(k)}(\{p_1,\ldots,p_n\},r)\\
&:=k-\text{skeleton of }\check{C}(\{p_1,\ldots,p_n\},r).
\end{align*}In particular, we call $\check{C}^{(1)}(\{p_1,\ldots,p_n\},r)$ a \emph{$M$--geometric graph}.
\end{definition}
\begin{rmk}\label{remark:smooth} In order to avoid unnecessary complications, in the sequel we will always assume that the injectivity radius\footnote{Recall that the injectivity radius $\textrm{inj}_p(M)$ of $M$ at one point $p$ is defined to be the largest radius of a ball in the tangent space $T_pM$ on which the exponential map $\textrm{exp}_p:T_{p}M\to M$ is a diffeomorphism and the injectivity radius of $M$ is defined as the infimum of the injectivity radii at all points:
\begin{equation} \textrm{inj}(M)=\inf_{p\in M}\textrm{inj}_p(M).
\end{equation}}
$\textrm{inj}(M)$ of $M$ is strictly positive (which is true if $M$ is compact or if $M=\mathbb{R}^m$ with the flat metric) and that \begin{equation} \label{eq:inj(M)}0<r\leq\textrm{inj}(M).\end{equation}
This requirement ensures that for every point $p\in M$ the set
\begin{equation} \partial\hat{B}(p,r)=\{x\in M:d(x,p_k)=r\}\end{equation}
is smooth (in fact it is the image of the sphere of radius $r$ in the tangent space at $p$ under the exponential map, which is a diffeomorphism on $B_{T_pM}(0, \textrm{inj}(M))$).
Observe also that for $r\leq \textrm{inj}(M)$ the ball $\widehat{B}(p,r)$ is contractible, but not necessarily geodesically convex.
\end{rmk}
\begin{definition}\label{def:nondeg}
We say that $\mathcal{U}(\{p_1,\ldots,p_n\},r)$ is \emph{nondegenerate} if for each $J=\{j_1,\ldots,j_l\}\in\genfrac{\lbrace}{\rbrace}{0pt}{}{n}{l}$ the intersection \begin{equation}\label{eq:transv} \bigcap_{k=1}^l\hat{B}(p_{j_k},r)\quad \textrm{is transversal}.\end{equation}
\end{definition}
The next result is classical and relates the homotopy of a geometric complex to the one of its associated \v{C}ech complex.
\begin{lem}[Nerve Lemma]\label{nervelemma}
If $M$ is compact, there exists $\rho>0$ such that, for each $r\leq \rho$,
\begin{equation*}
\mathcal{U}(\{p_1,\ldots,p_n\},r)\sim \check{C}(\{p_1,\ldots,p_n\},r),
\end{equation*}i.e. they are homotopy equivalent.
\end{lem}
\begin{proof}For the proof in this setting, see \cite[Lemma 6.1]{Antonio}.
\end{proof}
\begin{definition}[Isotopy classes of connected geometric complexes]\label{def:isotopy}Let $p_1,\ldots,p_n$ and $q_1,\ldots,q_n$ be points in $M$ and let $r_0,r_1\geq 0$ such that
\begin{equation*}
\mathcal{U}_0:=\mathcal{U}(\{p_1,\ldots,p_n\},r_0)\quad\textrm{and}\quad
\mathcal{U}_1:=\mathcal{U}(\{q_1,\ldots,q_n\},r_1)
\end{equation*}are nondegenerate $M$--geometric complexes. We say that $\mathcal{U}_0$ and $\mathcal{U}_1$ are \emph{(rigidly) isotopic} and we write $\mathcal{U}_0\cong\mathcal{U}_1$ if there exists an isotopy of diffeomorphisms $\varphi_t:M\rightarrow M$ with $\varphi_0=\id_M$ and a continuous function $r(t)>0$, for $t\in[0,1]$, such that:
\begin{itemize}
\item For each $t\in[0,1]$, $\mathcal{U}(\{\varphi_t(p_1),\ldots,\varphi_t(p_n)\},r(t))$ is nondegenerate,
\item $r(0)=r_0$,
\item $r(1)=r_1$ and
\item $\varphi_1(p_1)=q_1,\ldots,\varphi_1(p_n)=q_n$.
\end{itemize}
\end{definition}
\begin{rmk}[Isotopy classes and discriminants]\label{rmkdiscriminant}The definition of two complexes being (rigidly) isotopic is very reminiscent of the notion of rigid isotopy from algebraic geometry, where the ``regular'' deformations are those which do not intersect some discriminant. We can make this analogy more precise. For every $n\in\mathbb{N}$ consider the smooth manifold
\begin{equation}\label{eq:H} H_{n}:=\underbrace{M\times \cdots\times M}_{\textrm{$n$ many times}}\times (0, \textrm{inj}(M)),\end{equation}
together with the \emph{discriminant}
\begin{equation}\label{eq:Sigma} \Sigma_{n}:=\{(p_1, \ldots, p_n, r)\,|\,\textrm{ $\mathcal{U}(\{p_1, \ldots, p_n\}, r)$ is degenerate}\}.
\end{equation}
The set $\Sigma_{n}$ is closed since its complement $R_{n}$ is defined by the (finitely many) transversality conditions \eqref{eq:transv}. Adopting this point of view, isotopy classes of nondegenerate $M$--geometric complexes built using $n$ many balls are labeled by the connected components of $R_{n}:=H_{n}\backslash \Sigma_{n}$ (the complement of a discriminant).
In fact, given a nondegenerate complex $\mathcal{U}(\{p_1, \ldots, p_n\}, r)$, then $(p_1, \ldots, p_n, r)\in R_{n}$ (because it is nondegenerate) and viceversa every point in $R_{n}$ correspond to a nondegenerate complex. Moreover, a nondegenerate isotopy of two nondegenerate complexes defines a curve between the corresponding points of $R_{n}$, and this curve is entirely contained in $R_{n}$; the two corresponding parameters must therefore lie in the same connected component of $R_{n}$; viceversa, because for an open set of a manifold connected and path connected are equivalent, every two points in the same component $R_{n}$ can be joined by an arc all contained in $R_{n}$ and give therefore rise to isotopic complexes.
\end{rmk}
\begin{definition}
We define the set
\begin{equation*}
\mathcal{G}(M):=\{\text{connected, nondegenerate $M$--geometric complexes}\}
\end{equation*}and use the notation $\mathcal{G}:=\mathcal{G}(M)$ when $M$ is given. We also let
\begin{equation*}
\left(\mathcal{G}/{\cong}\right):=\{\text{isotopy classes of connected, nondegenerate $M$--geometric complexes}\}.
\end{equation*}
\end{definition}
\begin{rmk}
Observe that, by definition, each class $[\mathcal{U}]\in\mathcal{G}/{\cong}$ keeps also the information on the way $\mathcal{U}$ is embedded in $M$. In particular, it might be that to two nondegenerate $M$--geometric complexes $\mathcal{U}_1$ and $\mathcal{U}_2$ there correspond isomorphic \v{C}ech complexes $\check{C}(\mathcal{U}_1)\simeq \check{C}(\mathcal{U}_2)$, but at the same time the complexes $\mathcal{U}_1$ and $\mathcal{U}_2$ themselves are not rigidly isotopic (see Figure \ref{fig:complexes}).
\end{rmk}
\begin{rmk}For each $M$ of dimension $m$,
$\mathcal{G}(M)/{\cong}\supset\mathcal{G}(\mathbb{R}^m)/{\cong}$.
\end{rmk}
\begin{theorem}\label{teocountable}
$\mathcal{G}/{\cong}$ is a countable set.
\end{theorem}
\begin{proof}
We first partition $\mathcal{G}/{\cong}$ into countably many sets. For every $n\in \mathbb{N}$ we consider the set:
\begin{equation}
\left(\mathcal{G}_{n}/{\cong}\right):=\{\textrm{classes of complexes in $\mathcal{G}/{\cong}$ which are built using $n$ many balls}\},\end{equation}
and we need to prove that this set is countable.
We have already seen (Remark \ref{rmkdiscriminant}) that isotopy classes of nondegenerate complexes which are built using $n$ many balls are in one-to-one correspondence with the connected components of $R_{n}=H_{n}\backslash \Sigma_{n}$ (these sets are defined by \eqref{eq:H} and \eqref{eq:Sigma}). The function ``number of connected components of a nondegenerate complex'' is constant on each component of $R_{n}$ and consequently the number of isotopy classes of \emph{connected} and nondegenerate complexes (i.e. the cardinality of $\mathcal{G}_{n}/{\cong}$) is smaller than the number of components of $R_{n}$:
\begin{equation} \#\bigl(\mathcal{G}_{n}/{\cong}\bigr)\leq \#\bigl(\{\textrm{connected components of $R_{n}$}\}\bigr).\end{equation}
We are therefore reduced to prove that $R_{n}$ has at most countably many components. To this end we write $R_{n}$ as the disjoint union of its components (each of which is an open set in $H_{n}$):
\begin{equation} \label{eq:Ca}R_{n}=\bigsqcup_{\alpha \in A}C_{\alpha}.\end{equation}
We cover now the manifold $H_{n}$ with countably many manifold charts $\{(V_j, \varphi_j)\}_{j\in \mathbb{N}}$ with $\varphi_j:V_j\stackrel{\sim}{\longrightarrow} \mathbb{R}^m$. For every $j\in \mathbb{N}$ we consider also the decomposition of the open set $R_{n}\cap V_j$ into its connected components:
\begin{equation} R_{n}\cap V_j=\bigsqcup_{\beta \in B_j}C_{\beta, j}.\end{equation}
Since each $C_\alpha$ from \eqref{eq:Ca} is the union of elements of the form
\begin{equation} C_\alpha=\bigcup_{j\in N_\alpha,\, \beta \in B_j}C_{\beta, j}\end{equation}
with the index set $j$ running over the countable set $N_\alpha\subset \mathbb{N}$, it is therefore enough to prove that for every $j\in \mathbb{N}$ the index set $B_{j}$ is countable, i.e. that the number of connected components of $R_{n}\cap V_j$ is countable.
Observe now that, since $\varphi_j$ is a diffeomorphism between $V_j$ and $\mathbb{R}^m$, then the number of connected components of $R_{n}\cap V_j$ is the same of the number of connected components of $\varphi_j(R_{n}\cap V_j),$ which is an open subset of $\mathbb{R}^m$. Since in each component of $\varphi_j(R_{n}\cap V_j)$ we can pick a point with rational coordinates, it follows that the number of such components is countable, and this concludes the proof.
\end{proof}
\begin{rmk}Observe that the key point of the proof of Theorem \ref{teocountable} is showing that the number of connected components of an open set in a differentiable manifold is countable.
\end{rmk}
\begin{definition}
We let
\begin{equation*}
\left(\mathcal{G}/{\sim}\right):=\{\text{homotopy classes of connected, nondegenerate $M$--geometric complexes}\}
\end{equation*}
and we let
\begin{equation*}
\left(\mathcal{G}^{(k)}/{\simeq}\right):=\{\text{isomorphism classes of $k$--skeleta of connected $M$--geometric complexes}\}.
\end{equation*}In particular,
\begin{equation*}
\mathcal{G}^{(1)}/{\simeq}=\{\text{isomorphism classes of connected $M$--geometric graphs}\}.
\end{equation*}
\end{definition}
\begin{rmk}\label{rmkpallettine}In order to appreciate the difference between the classes in $\left(\mathcal{G}/{\cong}\right)$, $\left(\mathcal{G}/{\sim}\right)$ and $\left(\mathcal{G}^{(k)}/{\simeq}\right)$, we look at Figure \ref{fig:complexes}. Here, we have three $\mathbb{R}^3$--geometric complexes, given by the union of balls in $\mathbb{R}^3$, that have three different shapes. All three complexes are homotopy equivalent to each other and they are all pairwise not isotopic. Moreover, while the first one and the second one have isomorphic $1$--skeleta, the $1$--skeleton of the third complex is not isomorphic to the other ones.
\end{rmk}
\begin{figure}
\begin{center}
\includegraphics[width=9cm]{sferette2.jpg}
\caption{The three $\mathbb{R}^3$--geometric complexes described in Remark \ref{rmkpallettine}. From left to right, we call them $\mathcal{U}_1, \mathcal{U}_2$ and $ \mathcal{U}_3$ and we assume the number of balls and their combinatorics needed to to describe the first two are the same. Then $[\mathcal{U}_1]=[\mathcal{U}_2]=[\mathcal{U}_3]$ in $\mathcal{G}/{\sim}$, because they are all homotopy equivalent, $[\mathcal{U}_1]=[\mathcal{U}_2]\neq[\mathcal{U}_3]$ in $\mathcal{G}/{\simeq}$ (the first two give rise to same \v{C}ech complexes, which is however different from the last one) and $[\mathcal{U}_1]\neq[\mathcal{U}_2]\neq[\mathcal{U}_3]\neq [\mathcal{U}_1]$ in $\mathcal{G}/{\cong}$ (they are all pairwise non-isotopic).}\label{fig:complexes}
\end{center}
\end{figure}
\begin{rmk}\label{rmk:forgetful}
There are natural forgetful maps
\begin{align*}
\phi:\mathcal{G}/{\cong}\longrightarrow&\mathcal{G}/{\sim}\\
[\mathcal{U}]\longmapsto&[\mathcal{U}]
\end{align*}and
\begin{align*}
\varphi^{(k)}:\mathcal{G}/{\cong}\longrightarrow&\mathcal{G}^{(k)}/{\simeq}\\
[\mathcal{U}]\longmapsto&[\check{C}^{(k)}(\mathcal{U})].
\end{align*}
\end{rmk}
\begin{definition}[Component counting function]
Given a nondegenerate geometric complex $\mathcal{U}\subset M$, a topological subspace $Y\subset M$ and a class $w\in\mathcal{G}/{\cong}$, we define
\begin{equation*}
\mathcal{N}(\mathcal{U};w):=\#\bigl(\text{components of $\mathcal{U}$ of type $w$}\bigr);
\end{equation*}
\begin{equation*}
\mathcal{N}(\mathcal{U},Y;w):=\#\bigl(\text{components of $\mathcal{U}$ of type $w$ entirely contained in the interior of $Y$}\bigr);
\end{equation*}
\begin{equation*}
\mathcal{N}^*(\mathcal{U},Y;w):=\#\bigl(\text{components of $\mathcal{U}$ of type $w$ intersecting $Y$}\bigr).
\end{equation*}
\end{definition}
In the paper \cite{NazarovSodin} Nazarov and Sodin have introduced a powerful tool (the ``integral geometry sandwitch'') for localizing the count of the number of components of the zero set of random waves in a Riemannian manifold. This tool has been used by Sarnak and Wigman \cite{SarnakWigman} for the study of distribution of components type of the zero set of random waves on a Riemannian manifold, and it has been adapted to geometric complexes in \cite{Antonio}. We recall here this tool, stated in the language of this paper.
\begin{theorem}[Analogue of the Integral Geometry Sandwich]\label{teosandwich}The following two estimates are true:
\begin{enumerate}
\item \emph{(The local case)} Let $\mathcal{U}$ be a generic geometric complex in $\mathbb{R}^m$ and fix $w\in \mathcal{G}(\mathbb{R}^m)/{\cong}$. Then for $0<r<R$
\begin{equation*}
\int_{B_{R-r}}\frac{\mathcal{N}(\mathcal{U},B(x,r);w)}{\vol(B_r)}\diff x\leq \mathcal{N}(\mathcal{U},B_R;w)\leq\int_{B_{R+r}}\frac{\mathcal{N}^*(\mathcal{U},B(x,r);w)}{\vol (B_r)}\diff x.
\end{equation*}
\item \emph{(The global case)} Let $\mathcal{U}$ be a generic geometric complex in a compact Riemannian manifold $M$ and fix $w\in \mathcal{G}(M)/{\cong}$. Then for every $\varepsilon>0$ there exists $\eta>0$ such that for every $r<\eta$:
\begin{equation} (1-\varepsilon)\int_{M}\frac{\mathcal{N}(\mathcal{U},B(x,r);w)}{\vol(B_r)}\diff x\leq \mathcal{N}(\mathcal{U},M;w)\leq(1+\varepsilon)\int_{M}\frac{\mathcal{N}^*(\mathcal{U},B(x,r);w)}{\vol (B_r)}\diff x.
\end{equation}
\end{enumerate}
\end{theorem}
\begin{proof}The proof of both statements is exactly the same as in \cite{SarnakWigman}, after noticing that the only property needed on the counting functions is that the considered topological spaces only have finitely many components, and these components are counted according to a specific type (here are selected according to isotopy type, but we could consider instead any function that partitions the set of components and count only the components belonging to a given class).
\end{proof}
\section{Random geometric complexes (thermodynamic regime)}\label{section: Random geometric complexes}
\begin{definition}[Riemannian case]Let $M$ be a compact Riemannian manifold and consider a set of points $\{p_1,\ldots,p_n\}$ independently sampled from the uniform distribution on $M$. Fix a positive number $\alpha>0$, let $r:=\alpha n^{-1/m}$ and let
\begin{equation*}
\mathcal{U}_n:=\mathcal{U}(\{p_1,\ldots,p_n\},r).
\end{equation*}We say that $\mathcal{U}_n$ is a \emph{random $M$--geometric complex}. The choice of such $r$ is what defines the so-called \emph{critical} or \emph{thermodynamic regime}.
\end{definition}
\begin{definition}[Euclidean Poisson case]
Let $P:=\{p_1,p_2,\ldots\}$ be a set of points sampled from the standard spatial Poisson distribution in $\mathbb{R}^m$. For $\alpha>0$, let
\begin{equation*}
\mathcal{P}:=\bigcup_{p\in P}B(p,\alpha)
\end{equation*}and, for $R>0$, let
\begin{equation*}
\mathcal{P}_R:=\{\text{connected components of $\mathcal{P}$ entirely contained in the interior of }B(0,R)\}.
\end{equation*}
\end{definition}
\begin{rmk}
With probability $1$, we have that $\#\bigl(P\cap B(0,R)\bigr)$ is finite. To see this, observe that
\begin{equation*}
\mathbb{P}\{\#\bigl(P\cap B(0,R)\bigr)\geq l\}=\sum_{k\geq l}\frac{\vol(B(0,R))}{k!}e^{-\vol(B(0,R))}
\end{equation*}and
\begin{equation*}
\mathbb{P}\{\#\bigl(P\cap B(0,R)\bigr)=\infty\}\leq\mathbb{P}\{\#\bigl(P\cap B(0,R)\bigr)\geq l\}\xrightarrow{l\rightarrow\infty}0.
\end{equation*}
\end{rmk}
From now on, we will only consider nondegenerate complexes, without further mentioning this assumption. This is not reductive, since our random complexes are nondegenerate with probability one.
\section{Random measures}\label{section:Random measures}
We fix the following notation. Given a set $A$ with a fixed sigma algebra (omitted from the notation), we denote by:
\begin{equation*}
\mathcal{M}(A):=\{\text{measures on }A\}
\qquad\text{and}\qquad
\mathcal{M}^1(A):=\{\text{probability measures on }A\}.
\end{equation*}
\begin{definition}
Let $\mathcal{U}\subset M$ be a finite geometric complex and let
\begin{equation*}
\mathcal{U}=\mathcal{U}^1\sqcup\ldots\sqcup\mathcal{U}^{b_0(\mathcal{U})}
\end{equation*}be its decomposition into connected components.
We define $\Theta_\mathcal{U}\in\mathcal{M}^1(\mathcal{G}/{\cong})$ as
\begin{equation*}
\Theta_\mathcal{U}:=\frac{1}{b_0(\mathcal{U})}\sum_{j=1}^{b_0(\mathcal{U})}\delta_{[\mathcal{U}^j]}.
\end{equation*}
Observe that the measure $\Theta_{\mathcal{U}}$ just defined is a probability measure.
We also endow $\mathcal{M}^1(\mathcal{G}/{\cong})$ with the \emph{total variation distance}:
\begin{equation*}
d_{\tv}(\Theta_1,\Theta_2):=\sup_{A\subset\mathcal{G}/{\cong}}|\Theta_1(A)-\Theta_2(A)|.
\end{equation*}When $\mathcal{U}$ is a random geometric complex, $\Theta_{\mathcal{U}}$ is a random variable with values in the metric space $(\mathcal{M}^1(\mathcal{G}/{\cong}),d_{\tv})$. In this context, recall the notion of \emph{convergence in probability}:
\begin{equation*}
\Theta_n\xrightarrow{\mathbb{P}}\Theta \iff \forall\varepsilon, \lim_{n\rightarrow \infty}\mathbb{P}(d_{\tv}(\Theta_n,\Theta)\geq\varepsilon)=0.
\end{equation*}Using the previous notation, we set
\begin{equation*}
\Theta_n:=\Theta_{\mathcal{U}_n} \qquad\text{and}\qquad\Theta_R:=\Theta_{\mathcal{P}_R}.
\end{equation*}
\end{definition}
\begin{theorem}\label{mainthm}
There exists a probability measure $\Theta\in\mathcal{M}^1(\mathcal{G}/{\cong})$ such that
\begin{enumerate}
\item $\Theta_n\xrightarrow[n\rightarrow \infty]{\mathbb{P}}\Theta\qquad$ and $\qquad\Theta_R\xrightarrow[R\rightarrow \infty]{\mathbb{P}}\Theta$
\item $\supp(\Theta)=\mathcal{G}(\mathbb{R}^m)/{\cong}$.
\end{enumerate}
\end{theorem}
We shall see the proof of Theorem \ref{mainthm} in Section \ref{sectionmainproof}. As a first corollary we recover the results from \cite{Antonio}.
\begin{cor}[Theorem 1.1 and Theorem 1.3 from \cite{Antonio}]
Consider the forgetful map \begin{align*}
\phi:\mathcal{G}/{\cong}\longrightarrow&\,\mathcal{G}/{\sim}\\
[\mathcal{U}]\longmapsto&[\mathcal{U}].
\end{align*}We have that
\begin{equation*}
\phi_*\Theta_n\xrightarrow{\mathbb{P}}\phi_*\Theta\qquad\text{and}\qquad\phi_*\Theta_R\xrightarrow{\mathbb{P}}\phi_*\Theta.
\end{equation*}Also, $\supp\phi_*\Theta=\mathcal{G}(\mathbb{R}^m)/{\sim}$.
\end{cor}
As a second corollary we see that, because Theorem \ref{mainthm} keeps track of fine properties of the geometric complex, we can use other forgetful maps and obtain information on the limit distribution of the components type of each $k$--skeleton.
\begin{cor}\label{corsupppihik}
For each $k\in\mathbb{N}$, consider the forgetful map from Remark \ref{rmk:forgetful}:
\begin{align*}
\varphi^{(k)}:\mathcal{G}/{\cong}\longrightarrow&\,\mathcal{G}^{(k)}/{\simeq}\\
[\mathcal{U}]\longmapsto&[\check{C}^{(k)}(\mathcal{U})].
\end{align*}We have that
\begin{equation*}
\varphi^{(k)}_*\Theta_n\xrightarrow{\mathbb{P}}\varphi^{(k)}_*\Theta\qquad\text{and}\qquad\varphi^{(k)}_*\Theta_R\xrightarrow{\mathbb{P}}\varphi^{(k)}_*\Theta.
\end{equation*}Also, $\supp\varphi^{(k)}_*\Theta=\mathcal{G}^{(k)}(\mathbb{R}^m)/{\simeq}$.
\end{cor}
\begin{prop}[Existence of all isotopy types]\label{propexistence}
Let $\mathcal{U}$ be a nondegenerate geometric complex in $\mathbb{R}^m$ and let $\alpha>0$. Let $(\mathcal{U}_n)_n$ be a sequence of random $M$--geometric complexes constructed using $\alpha$. There exist $n_0$, $R$, $c>0$ (depending on $\Gamma$ and $\alpha$ but not on $M$) such that for every $p\in M$ and for every $n\geq n_0$:
\begin{equation*}
\mathbb{P}\{\mathcal{U}_n\cap\hat{B}(p,Rn^{-1/m})\cong\mathcal{U}\}>c.
\end{equation*}
\end{prop}
\begin{proof}The proof is similar to the proof of \cite[Proposition 1.2]{Antonio}. Here we sketch this proof for the sake of completeness, pointing out what is the main difference with \cite{Antonio}.
Assume that $\mathcal{U}\subset B(0, R')$ is constructed using balls of radius $r=1$:
\begin{equation} \mathcal{U}=\bigcap_{k=1}^\ell B(y_k, 1),\end{equation}
set $R=\alpha R'$ and
consider the sequence of maps:
\begin{equation}\psi_n:\hat{B}(p, Rn^{-1/m})\stackrel{\mathrm{exp}_p^{-1}}{\longrightarrow}B_{T_pM}(0, Rn^{-1/m})\stackrel{\mathrm{dilation}}{\longrightarrow}B_{T_pM}(0, R')\simeq B(0, R')\end{equation}
For $n>$ large enough the map $\psi_n$ becomes a diffeomorphism and we denote by $\varphi_n$ its inverse. \cite[Proposition 6.2]{Antonio} implies that there exist $\epsilon_0>0$ and $n_0>0$ such that if $\|\tilde{y}_k-y_k\|\leq \epsilon_0$ for every $k=1, \ldots, \ell$, then for $n\geq n_0$ the two complexes $\bigcup_{k=1}^\ell \hat{B}(\varphi_n(\tilde y_k), \alpha n^{-1/m})$ and $\mathcal{U}$ are isomorphic. In fact, because $\mathcal{U}$ is nondegenerate, possibly choosing $\epsilon_0>0$ even smaller, we can make sure that the complex $\bigcup_{k=1}^\ell \hat{B}(\varphi_n(\tilde y_k), \alpha n^{-1/m})$ belong to the same rigid isotopy class of $\mathcal{U}$, because this is an open condition.
One then proceeds considering the event:
\begin{equation} E_{n}=\left\{\exists I_\ell\in\genfrac{\{ }{\}}{0pt}{}{n}{\ell}\,:\, \forall j\in I_\ell \quad p_{j}\in \psi_n^{-1}(B(y_j, \epsilon)), \text{ and} \quad \forall j\notin I_\ell \quad p_j\in \hat{B}(p, (R+\alpha)n^{-1/m})^c\right\}.\end{equation}
Observe that:
\begin{equation} E_n\implies \mathcal{U}_n\cap\hat{B}(p,Rn^{-1/m})\cong\mathcal{U},\end{equation}
and in particular, in order to get the conclusion, it is enough to estimate from below the probability of $E_n$. This is done in the last lines of the proof of \cite[Proposition 1.2]{Antonio}.
\end{proof}
\begin{ex}
Let $C_{137}$ be the cycle with $137$ vertices and $137$ edges. By Proposition \ref{propexistence} there exist $n_0$, $R$ and $c>0$ (depending on $C_{137}$ and $\alpha$ but not on $M$) such that for every $p\in M$ and for every $n\geq n_0$:
\begin{equation*}
\mathbb{P}\{\Gamma_n\cap\hat{B}(p,Rn^{-1/m})\cong C_{137}\}>c.
\end{equation*}
\end{ex}
\subsection{Proof of Theorem \ref{mainthm}}\label{sectionmainproof}
We split the statement of Theorem \ref{mainthm} into two parts. The first part, Theorem \ref{theo:main1}, states that the random measure $\Theta_R$ converges in probability to a deterministic measure $\Theta\in\mathcal{M}^1(\mathcal{G}/{\cong})$ supported on the set $\mathcal{G}(\mathbb{R}^m)/{\cong}$. The second part, Theorem \ref{theo:main2}, states that also the random measure $\Theta_n$ converges in probability to $\Theta$.
\subsubsection{The local model}
\begin{prop}\label{prop2.1}
For every $w\in \mathcal{G}(\mathbb{R}^m)/{\cong}$ there exists a constant $c_w>0$ such that the random variable
\begin{equation*}
c_{R,w}:=\frac{\mathcal{N}(\mathcal{P}_R,w)}{\vol(B(0,R))}
\end{equation*}converges to $c_w$ in $L^1$ and almost surely as $R\rightarrow\infty$.
\end{prop}
\begin{proof}[Proof of Proposition \ref{prop2.1}]
Following the proof of \cite[Proposition 2.1]{Antonio} with $\mathcal{N}(\mathcal{P}_R,\tau,w)$ instead of $\mathcal{N}(\mathcal{P}_R,\tau,\gamma)$ and with the application of Theorem \ref{teosandwich} instead of \cite[Theorem 6.6]{Antonio}, one proves that there exists a constant $c_w$ such that
\begin{equation*}
\frac{\mathcal{N}(\mathcal{P}_R,B(0,R);w)}{\vol(B(0,R))}\longrightarrow c_w.
\end{equation*}Since $\mathcal{P}_R\subset B(0,R)$, this implies that
\begin{equation*}
\frac{\mathcal{N}(\mathcal{P}_R,w)}{\vol(B(0,R))}\longrightarrow c_w.
\end{equation*}We now have to prove that $c_w>0$. Since $w\in \mathcal{G}(\mathbb{R}^m)/{\cong}$, given $\mathcal{U}\in w$ there exist $\beta>0$ and $y_1,\ldots,y_n\in\mathbb{R}^m$ such that
\begin{equation*}
\mathcal{U}\cong\mathcal{U}(\{y_1,\ldots,y_n\},\beta).
\end{equation*}Let $R_1$ be such that $\mathcal{U}\subset B(0,R_1)$ and choose $r$ such that $r\beta=\alpha$, where $\alpha$ is the constant that we used for constructing the random complex $\mathcal{P}$. We can then rescale $\mathcal{U}$ in $B(0,r\cdot R_1)$ so that it is constructed on radius $\alpha$. Now, since $\mathcal{U}$ is nondegenerate, there exists $\varepsilon>0$ such that, if for every $i$ we have that $\|y_i-y'_i\|<\varepsilon$, then the complex
\begin{equation*}
\mathcal{U}'(\{y_1',\ldots,y_n'\},\alpha )
\end{equation*}
is isotopic to $\mathcal{U}$.
Take $K_R=k\cdot\mathrm{vol}(B(0, R))$ disjoint balls $\{B(y_j,r\cdot R_1)\}_{j=1,\ldots,K_R}$ inside $B(0,R)$ with $c>0$. Then
\begin{equation*}
\frac{\mathcal{N}(\mathcal{P}_R,B(0,R);w)}{\vol(B(0,R))}\geq \frac{1}{\vol(B(0,R))}\cdot\sum_{j=1}^{K_R}\mathcal{N}(\mathcal{P}_R,B(y_j,r\cdot R_1);w).
\end{equation*}Therefore
\begin{align*}
\mathbb{E}\Biggl(\frac{\mathcal{N}(\mathcal{P}_R,B(0,R);w)}{\vol(B(0,R))}\Biggr)&\geq \mathbb{E}\Biggr(\frac{1}{\vol(B(0,R))}\cdot\sum_{j=1}^{K_R}\mathcal{N}(\mathcal{P}_R,B(y_j,r\cdot R_1);w)\Biggr)\\
&=\frac{K_R}{\vol(B(0,R))}\cdot\mathbb{E}\Biggl(\mathcal{N}(\mathcal{P}_R,B(y,r\cdot R_1);w)\Biggr),
\end{align*}since the random variables $\mathcal{N}(\mathcal{P}_R,B(y_j,r\cdot R_1);w)$ are identically distributed by the fact that the balls are disjoint. Now,
\begin{equation*}
\frac{K_R}{\vol(B(0,R))}\geq \frac{k\cdot \mathrm{vol}(B(0, R))}{\vol(B(0,R))}=k>0
\end{equation*}
and
\begin{equation*}
\mathbb{E}\Biggl(\mathcal{N}(\mathcal{P}_R,B(y,r\cdot R_1);w)\Biggr)>0.
\end{equation*}Therefore, $c_w>0$.
\end{proof}
Proposition \ref{prop2.1} allows us to deduce the following theorem.
\begin{theorem}\label{theo:main1}There exists $\Theta\in\mathcal{M}^1(\mathcal{G}/{\cong})$ such that
\begin{equation*}
\Theta_R\xrightarrow[R\rightarrow \infty]{\mathbb{P}}\Theta.
\end{equation*}
Also, $\supp(\Theta)=\mathcal{G}(\mathbb{R}^m)/{\cong}$.
\end{theorem}
\begin{proof}The proof is the same as the one of \cite[Theorem 1.3]{Antonio}, replacing the homotopy type counting function with the isotopy type one.
\end{proof}
\subsubsection{Riemannian case}
\begin{theorem}\label{teo3.1}
Let $p\in M$. For every $\delta>0$ and for $R>0$ sufficiently big there exists $n_0$ such that for every $w\in \mathcal{G}(\mathbb{R}^m)/{\cong}$ and for $n\geq n_0$:
\begin{equation*}
\mathbb{P}\{\mathcal{N}(\mathcal{P}_R,B(0,R);w)=\mathcal{N}(\mathcal{U}_n,\hat{B}(p,Rn^{-1/m});w)\}\geq 1-\delta.
\end{equation*}
\end{theorem}
\begin{proof}The proof is the same as the one of \cite[Theorem 3.1]{Antonio}, with the following differences:
\begin{itemize}
\item In point (3), instead of considering the homotopy equivalence between the unions of the balls, we consider the isotopy equivalences between their $k$--skeleta. This is allowed because, at the end of the proof of point (3), it is proved that the combinatorics of the covers are the same.
\item After assuming point (1), point (2) and the modified point (3), we can say that the $k$--skeleta of the two unions of balls are isotopic and also the unions of all the components entirely contained in $B(0,R)$ (respectively $\hat{B}(p,Rn^{-1/m})$) have isotopic $k$--skeleta. In particular, the number of components of a given isotopy class $w$ is the same for both sets with probability at least $1-\delta$.
\end{itemize}
\end{proof}
\begin{cor}\label{cor3.2}
For each $w\in \mathcal{G}(\mathbb{R}^m)/{\cong}$, $\alpha>0$, $x\in M$ and $\varepsilon>0$, we have
\begin{equation*}
\lim_{R\rightarrow\infty}\limsup_{n\rightarrow\infty}\mathbb{P}\biggl\{\biggl|\frac{\mathcal{N}(\mathcal{U}_n,B(x,Rn^{-1/d});w)}{\vol (B(0,R))}-c_w\biggr|>\varepsilon\biggr\}=0.
\end{equation*}
\end{cor}
\begin{proof}
The proof is the same as the one of \cite[Corollary 3.2]{Antonio}, with the application of Theorem \ref{teo3.1} and Proposition \ref{prop2.1} instead of \cite[Theorem 3.1]{Antonio} and \cite[Proposition 2.1]{Antonio}.
\end{proof}
\begin{theorem}\label{teo4.1}
For every $w\in \mathcal{G}(\mathbb{R}^m)/{\cong}$, the random variable
\begin{equation*}
c_{n,w}:=\frac{\mathcal{N}(\mathcal{U}_n;w)}{n}
\end{equation*}converges in $L^1$ to $c_w\cdot\vol(M)$, where $c_w$ is the constant appearing in Proposition \ref{prop2.1}. In particular, this implies that the random variable
\begin{equation*}
c_{n}:=\frac{\mathcal{N}(\mathcal{U}_n)}{n},
\end{equation*}i.e. when we consider all components with no restriction on their type, converges in $L^1$ to $c$, where $c=\sum_{w\in \mathcal{G}(\mathbb{R}^m)/{\cong}}c_w>0$.
\end{theorem}
\begin{proof}
The proof is the same as the one of \cite[Theorem 4.1]{Antonio}, with the application of Theorem \ref{teosandwich} instead of \cite[Theorem 6.7]{Antonio} and the application of Corollary \ref{cor3.2} instead of \cite[Corollary 3.2]{Antonio}.
\end{proof}
\begin{theorem}\label{theo:main2}The measure $\Theta\in\mathcal{M}^1(\mathcal{G}/{\cong})$ appearing in Theorem \ref{theo:main1} is such that
\begin{equation*}
\Theta_n\xrightarrow[n\rightarrow \infty]{\mathbb{P}}\Theta.
\end{equation*}
\end{theorem}
\begin{proof}
The proof is the same as the one of \cite[Theorem 1.1]{Antonio}, with the following differences:
\begin{itemize}
\item We apply Theorem \ref{teo4.1} instead of \cite[Theorem 4.1]{Antonio};
\item We use $w\in \mathcal{G}(\mathbb{R}^m)/{\cong}$ instead of $\gamma\in \mathcal{G}$, $\Theta_n$ instead of $\hat{\mu}_n$ and $\mathcal{G}(M)/{\cong}$ instead of $\hat{\mathcal{G}}$.
\end{itemize}
\end{proof}
\section{Geometric graphs}\label{section:Geometric graphs}
We specialize the previous discussion to the case $k=1$, and consider
\begin{equation*}
\mathcal{G}^{(1)}=\{\text{isomorphism classes of connected, nondegenerate $M$--geometric graphs}\}.
\end{equation*}
\begin{rmk}\label{rem:open}The set of $M$--geometric graphs defined using closed balls equals the set of $M$--geometric graphs defined using open balls. To see this, assume that a geometric graph $\Gamma=(V(\Gamma),E(\Gamma))$ is defined using closed balls of radius $r$. Then, for each pair of distinct vertices $(p_i,p_j)$,
\begin{equation*}
(p_i,p_j)\in E(\Gamma) \iff d(p_i,p_j)\leq r.
\end{equation*}Now, choose $\varepsilon\geq 0$ small enough that, for each pair of distinct vertices $(p_i,p_j)$,
\begin{equation*}
(p_i,p_j)\in E(\Gamma) \iff d(p_i,p_j)< r+\varepsilon.
\end{equation*}Therefore, $\Gamma$ can be constructed as a $M$--geometric graph using open balls of radius $r+\varepsilon$. The inverse implication is analogous.
\end{rmk}
In the case $M=\mathbb{R}^m$, the problem of describing the set $\mathcal{G}^{(1)}(\mathbb{R}^m)$ is equivalent to asking which graphs are realizable as $\mathbb{R}^m$--geometric graphs in a given dimension $m$. There is a vast literature about this problem and, commonly, geometric graphs realizable in dimension $m$ are called \emph{$m$--sphere graphs} while the minimal dimension $m$ such that a given graph is a $m$--sphere graph is called its \emph{sphericity}. In \cite{sphericity1} it is proved that every graph has finite sphericity; in \cite{sphericity2} the authors prove that the problem of deciding, given a graph $\Gamma$, whether $\Gamma$ is a $m$--sphere is NP-hard for all $m>1$. We can also observe that, for each $m>0$, there are graphs that are not $m$--sphere graphs. To see this, consider the \emph{kissing number} $k(m)$ in dimension $m$, defined as the number of non-overlapping unit spheres that can be arranged such that they each touch a common unit sphere. Consider the star graph with a central vertex connected to $n$ external vertices, where $n>k(m)$. In order to have a realization of dimension $m$ of this graph, we need a central sphere that touches $n$ spheres which do not touch each other. Since $n>k(m)$, this is not possible.
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{Kissing.jpg}
\end{center}
\caption{$7$ kissing spheres in dimension $2$. The eighth sphere doesn't know where to go.}\label{Figurekissing}
\end{figure}
\begin{ex}
In dimension $2$, the kissing number is $6$, as shown in Figure \ref{Figurekissing}. Therefore, any star graph $S_n$ on $n+1$ vertices, with $n>6$, is not realizable in dimension $2$ as a sphere graph.
\end{ex}
In the particular case of $m=1$, $1$--sphere graphs are called \emph{indifference graphs}, \emph{unit interval graphs} and there are many characterizations of such graphs \cite{unitgraphs1, unitgraphs2, unitgraphs3, unitgraphs4, unitgraphs5, unitgraphs6}. A classical characterization is due to Roberts and Wegner \cite{unitgraphs2,unitgraphs5,unitgraphs3} and it characterizes unit interval graphs by the absence of certain forbidden subgraphs, this is recalled in Theorem \ref{lekkerkerker}.
\begin{theorem}[Roberts and Wegner]\label{lekkerkerker}
A graph is a unit interval graph, i.e. it's an element of $\mathcal{G}^{(1)}(\mathbb{R})$, if and only if it does not contain any cycle of length at least four and any of the graphs shown in Figure \ref{fig:proi} as induced subgraph.
\end{theorem}
As a consequence, we get the following corollary.
\begin{cor}\label{cor:limit1}The support of the measure $\varphi^{(1)}_*\Theta$ defined in Corollary \ref{corsupppihik} is given by all graphs that do not contain any cycle of length at least four and any of the graphs shown in Figure \ref{fig:proi} as induced subgraph.
\end{cor}
\begin{figure}
\centering
\subfloat{{
\begin{tikzpicture}
\node[bluenode] (1) {};
\node[bluenode] (2) [below left = 0.6 cm and 0.3 cm of 1] {};
\node[bluenode] (3) [below right = 0.6 cm and 0.3 cm of 1] {};
\node[bluenode] (4) [below right = 0.6 cm and 0.6 cm of 3] {};
\node[bluenode] (5) [below left = 0.6 cm and 0.6 cm of 2] {};
\node[bluenode] (6) [above = 0.6 cm of 1] {};
\path[draw,thick]
(1) edge node {} (2)
(1) edge node {} (3)
(2) edge node {} (3)
(4) edge node {} (3)
(2) edge node {} (5)
(1) edge node {} (6);
\end{tikzpicture}
}}
\qquad
\subfloat{{
\begin{tikzpicture}
\node[bluenode] (1) {};
\node[bluenode] (2) [below left = 1 cm and 1 cm of 1] {};
\node[bluenode] (3) [below right = 1 cm and 1 cm of 1] {};
\node[bluenode] (4) [above = of 1] {};
\path[draw,thick]
(1) edge node {} (2)
(1) edge node {} (4)
(1) edge node {} (3);
\end{tikzpicture}
}}\qquad
\subfloat{{
\begin{tikzpicture}
\node[bluenode] (1) {};
\node[bluenode] (2) [below left = 1 cm and 0.5 cm of 1] {};
\node[bluenode] (3) [below right = 1 cm and 0.5 cm of 1] {};
\node[bluenode] (4) [below left = 1 cm and 0.5 cm of 2] {};
\node[bluenode] (5) [below right = 1 cm and 0.5 cm of 2] {};
\node[bluenode] (6) [below right = 1 cm and 0.5 cm of 3] {};
\path[draw,thick]
(1) edge node {} (2)
(3) edge node {} (2)
(1) edge node {} (3)
(3) edge node {} (6)
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(2) edge node {} (5)
(3) edge node {} (5)
(4) edge node {} (5)
(5) edge node {} (6);
\end{tikzpicture}
}}
\caption{Together with the cycles of length at least four, these are the non-admissible induced subgraphs for unit interval graphs on the line.}
\label{fig:proi}
\end{figure}
\section{Normalized Laplacian of a graph and its spectrum}\label{section:Laplacian of a graph and its spectrum}
We now fix a graph $\Gamma$ on $n$ vertices $v_1,\ldots,v_n$ and we recall the definition of the (symmetric) normalized Laplace matrix, together with other common matrices associated to graphs. We shall then define the \emph{spectrum} and the \emph{spectral measure} associated to these matrices, and show some properties.
\begin{definition}[Matrices associated to a graph]
Let $A$ be the \emph{adjacency matrix} of $\Gamma$; let $D:=\diag(\deg v_1,\ldots,\deg v_n)$ be the \emph{degree matrix}; let $L:=D-A$ be the \emph{non-normalized Laplacian matrix} and let $\hat{L}:=I_n-D^{-1/2}AD^{-1/2}$ be the \emph{symmetric normalized Laplacian matrix}.
\end{definition}
\begin{definition}[Spectrum of a matrix]
Given $Q\in \Sym(n,\mathbb{R})$ let $\spec(Q)$ be the spectrum of $Q$, i.e. the collection of its eigenvalues repeated with multiplicity,
\begin{equation*}
\lambda_1(Q)\leq\ldots\leq \lambda_n(Q).
\end{equation*}
We define the \emph{empirical spectral measure of} $Q$ as
\begin{equation*}
\mu_Q:=\frac{1}{n}\sum_{i=1}^n\delta_{\lambda_i(Q)}.
\end{equation*}
\end{definition}
\begin{definition}[Spectrum of a graph]\label{defspectrum}
We define \emph{spectrum} of $\Gamma$, $\spec(\Gamma)$, as the spectrum of $\hat{L}$ and we write it as
\begin{equation*}
\lambda_1(\Gamma)\leq\ldots\leq \lambda_n(\Gamma).
\end{equation*}
We also define
\begin{equation*}
s(\Gamma):=\sum_{i=1}^{n}\delta_{\lambda_i(\Gamma)}
\end{equation*}and the \emph{spectral measure of} $\Gamma$ as
\begin{equation*}
\mu_\Gamma:=\mu_{\hat{L}}=\frac{1}{n}\sum_{i=1}^n\delta_{\lambda_i(\Gamma)}.
\end{equation*}
\end{definition}
Recall that, for every $i=1,\ldots,n$, $\lambda_i(\Gamma)\in[0,2]$ \cite[Equation (1.1) and Lemma 1.7]{Chung}. In particular, this implies that $s(\Gamma)\in\mathcal{M}([0,2])$ and $\mu_\Gamma\in\mathcal{M}^1([0,2])$.
\begin{theorem}\label{teoo1edges}
Let $(\Gamma_{1,n})_n$ and $(\Gamma_{2,n})_n$ be two sequences of graphs such that, for every $n$, $(\Gamma_{1,n})_n$ and $(\Gamma_{2,n})_n$ are two graphs on $n$ nodes that differ at most by $c$ edges. Denote by $\mu_{1,n}$ and $\mu_{2,n}$ the spectral measures associated to one of the matrices $A$, $D$, $L$, $\hat{L}$. Then
\begin{equation*}
\mu_{1,n}-\mu_{2,n}\overset{*} \rightharpoonup 0,
\end{equation*} where $\overset{*} \rightharpoonup$ denotes the weak star convergence, i.e. for each $f\in C^{0}_c(\mathbb{R},\mathbb{R})$
\begin{equation*}
\bigg| \int_\mathbb{R}f\diff\mu_{1,n}-\int_\mathbb{R}f\diff\mu_{2,n}\bigg|\rightarrow 0.
\end{equation*}
\end{theorem}
We shall prove Theorem \ref{teoo1edges} in Section \ref{sectionweakconvergence}.
\begin{rmk}
In the case of $\hat{L}$, we have convergence in total variation distance for ``connected sum'' of complete graphs, but not for paths, as we shall see in Section \ref{sectionstrongconvergence}.
\end{rmk}
\begin{rmk} Theorem 2.8 from \cite{JJ} tells that, if two families $(\Gamma_{1,n})_n$ and $(\Gamma_{2, n})_n$ differ by at most $c$ edges \emph{and} their corresponding spectral measures have weak limits, then they belong to the same spectral class (the notion of spectral class of a family of graph is recalled in Remark \ref{rmk:JJ}). In this sense our previous Theorem \ref{teoo1edges} can be considered as an analogue of \cite[Theorem 2.8]{JJ}: the difference of the spectral measure of two families of graphs $(\Gamma_{1,n})_n$ and $(\Gamma_{2, n})_n$ differing by at most a finite number $c$ of edges, goes to zero weakly (without the assumption that the corresponding spectral measures have weak limits).
\end{rmk}
\subsection{Proof of Theorem \ref{teoo1edges}}\label{sectionweakconvergence}
\subsubsection{Preliminaries}
Given $Q\in \Sym(n,\mathbb{R})$, we define the \emph{$1$--Shatten norm of} $Q$ as
\begin{equation*}
\|Q\|_{S^1}:=\sum_{i=1}^n|\lambda_i(Q)|.
\end{equation*}The \emph{Weilandt-Hoffman inequality} \cite[Exercise 1.3.6]{Tao} holds:
\begin{equation}\label{Weilandt-Hoffman inequality}
\sum_{i=1}^n|\lambda_i(Q_1)-\lambda_i(Q_2)|\leq \|Q_1-Q_2\|_{S^1}.
\end{equation}
\begin{prop}\label{prop1proof}
Let $Q_1,Q_2\in \Sym(n,\mathbb{R})$ such that
\begin{equation*}
\|Q_1-Q_2\|_{S^1}\leq C.
\end{equation*}Then, for each $f\in C^0_c(\mathbb{R},\mathbb{R})$ and for each $\varepsilon>0$, there exists $\delta>0$ such that
\begin{equation*}
\biggl|\int_{\mathbb{R}}f\diff\mu_{Q_1}-\int_{\mathbb{R}}f\diff\mu_{Q_2}\biggr|\leq\varepsilon+\frac{2\sup |f|}{\delta n}\cdot C.
\end{equation*}
\end{prop}
\begin{proof}
Denote by $\{\lambda_i^{(1)}\}_{i=1}^n$ and $\{\lambda_i^{(2)}\}_{i=1}^n$ the eigenvalues of $Q_1$ and $Q_2$ respectively. Then
\begin{equation*}
\int_{\mathbb{R}}f\diff\mu_{Q_1}-\int_{\mathbb{R}}f\diff\mu_{Q_2}=\frac{1}{n}\sum_{i=1}^{n}f(\lambda_i^{(1)})-f(\lambda_i^{(2)}),
\end{equation*}therefore
\begin{equation*}
\biggl|\int_{\mathbb{R}}f\diff\mu_{Q_1}-\int_{\mathbb{R}}f\diff\mu_{Q_2}\biggr|\leq\frac{1}{n}\sum_{i=1}^{n}\bigl|f(\lambda_i^{(1)})-f(\lambda_i^{(2)})\bigr|.
\end{equation*}Now, since $f\in C^0_c(\mathbb{R},\mathbb{R})$, $f$ is uniformly continuous and given $\varepsilon>0$ there exists $\delta=\delta(f)$ such that
\begin{equation*}
|\lambda_1-\lambda_2|\leq\delta \qquad \Longrightarrow \qquad |f(\lambda_1)-f(\lambda_2)|\leq\varepsilon.
\end{equation*}Therefore, since by Equation \ref{Weilandt-Hoffman inequality} and by hypothesis we have that
\begin{equation*}
\sum_{i=1}^{n}|\lambda_i^{(1)}-\lambda_i^{(2)}|\leq \|Q_1-Q_2\|_{S^1}\leq C,
\end{equation*}it follows that
\begin{equation*}
|\{|\lambda_i^{(1)}-\lambda_i^{(2)}|>\delta\}|\leq\frac{C}{\delta}.
\end{equation*}Therefore,
\begin{align*}
\biggl|\int_{\mathbb{R}}f\diff\mu_{Q_1}-\int_{\mathbb{R}}f\diff\mu_{Q_2}\biggr|&\leq\frac{1}{n}\sum_{i=1}^{n}\bigl|f(\lambda_i^{(1)})-f(\lambda_i^{(2)})\bigr|\\
&= \frac{1}{n}\sum_{|\lambda_i^{(1)}-\lambda_i^{(2)}|<\delta}\bigl|f(\lambda_i^{(1)})-f(\lambda_i^{(2)})\bigr|+\frac{1}{n}\sum_{|\lambda_i^{(1)}-\lambda_i^{(2)}|\geq\delta}\bigl|f(\lambda_i^{(1)})-f(\lambda_i^{(2)})\bigr|\\
&\leq \frac{1}{n}\sum_{|\lambda_i^{(1)}-\lambda_i^{(2)}|<\delta}\varepsilon+\frac{1}{n}\sum_{|\lambda_i^{(1)}-\lambda_i^{(2)}|\geq\delta}\bigl|f(\lambda_i^{(1)})\bigr|+\bigl|f(\lambda_i^{(2)})\bigr|\\
&\leq\frac{1}{n}\cdot\varepsilon\cdot\bigl|\{|\lambda_i^{(1)}-\lambda_i^{(2)}|<\delta\}\bigr|+\frac{1}{n}\cdot 2\sup |f|\cdot\bigl|\{|\lambda_i^{(1)}-\lambda_i^{(2)}| \geq\delta\}\bigr|\\
&\leq\frac{1}{n}\cdot\varepsilon\cdot n + \frac{1}{n}\cdot\frac{2\sup |f|}{\delta}\cdot C\\
&\leq \varepsilon+\frac{2\sup |f|}{\delta n}\cdot C.
\end{align*}
\end{proof}
\subsubsection{Applications to graphs}
\begin{lem}\label{lemci}Let $\Gamma_1$, $\Gamma_2$ be two graphs with $V(\Gamma_1)=V(\Gamma_2)$ that differ by at most $C$--many edges. Then,
\begin{align*}
&\|A_1-A_2\|_{S^1}\leq 4C\\
&\|D_1-D_2\|_{S^1}\leq 4C^2\\
&\|L_1-L_2\|_{S^1}\leq 4C^2\\
&\|\hat{L}_1-\hat{L}_2\|_{S^1}\leq 2C\cdot\sqrt{2}\cdot\sqrt{n-1}.
\end{align*}
\end{lem}
\begin{proof}Observe that any of the matrices
\begin{equation*}
\Delta_1:=A_1-A_2,\qquad\Delta_2:=D_1-D_2,\qquad\Delta_3:=L_1-L_2
\end{equation*}consists of all zeros except for at most $4C$ entries, all of which entries are bounded by a constant (it is $1$ for $\Delta_1$ and $C$ for $\Delta_2$ and $\Delta_3$). Therefore, each $\Delta_i\in\Sym(n,\mathbb{R})$ for $i=1,2,3$ has rank at most $4C$ and all its eigenvalues are zero, except for at most $4C$ of them. It follows that, for $i=1,2,3$,
\begin{align*}
\|\Delta_i\|_{S^1}&=\sum_{j=n-2C+1}^n|\lambda_j(\Delta_i)|\\
&=\langle(|\lambda_{n-2C+1}|,\ldots,|\lambda_n|),(1,\ldots,1) \rangle\\
&\leq \sum_{j=n-2C+1}^n\bigl(\lambda_j(\Delta_i)^2\bigr)^{1/2}\sqrt{4C}\\
&=2C^{1/2}\|\Delta_i\|_F\\
&=2C^{1/2}\bigl(\sum_{j,k}(\Delta_i)_{jk}^2\bigr)^{1/2}\\
&=2C^{1/2}(4C\tilde{C}^2)^{1/2}\\
&\leq 4C\tilde{C}_i,
\end{align*}where
\begin{equation*}
\tilde{C}_i=\begin{cases}1 & \textrm{if }i=1\\ C & \textrm{if }i=2,3.\end{cases}
\end{equation*}
Similarly, $\Delta_4:=\hat{L}_1-\hat{L}_2$ consists of all zeros except for at most $2C(n-1)$ entries, all of which entries are bounded by $1$, and it has rank at most $4C$. Therefore,
\begin{align*}
\|\Delta_4\|_{S^1}&=\sum_{j=1}^n|\lambda_j(\Delta_4)|\\
&\leq \sum_{j=1}^n\bigl(\lambda_j(\Delta_4)^2\bigr)^{1/2}\cdot(4C)^{1/2}\\
&=\bigl(\sum_{j,k}(\Delta_4)_{jk}^2\bigr)^{1/2}\cdot2C^{1/2}\\
&\leq \bigl(\sum_{1}^{2C(n-1)}1\bigr)^{1/2}\cdot2C^{1/2}\\
&=2C\cdot\sqrt{2}\cdot\sqrt{n-1}.
\end{align*}
\end{proof}As a corollary, we can prove Theorem \ref{teoo1edges}.
\begin{proof}[Proof of Theorem \ref{teoo1edges}]We prove that, for each $f\in C^{0}_c(\mathbb{R},\mathbb{R})$ and for each $\varepsilon>0$,
\begin{equation*}
\lim_{n\rightarrow\infty}\bigg| \int_\mathbb{R}f\diff\mu_{1,n}-\int_\mathbb{R}f\diff\mu_{2,n}\bigg|\leq \varepsilon.
\end{equation*}Let $c_1:=4C$, $c_2:=c_3:=4C^2$ and $c_4:=2C\cdot\sqrt{2}$. By Lemma \ref{lemci}, we have that
\begin{equation*}
\|\Delta_i\|_{S^1}\leq c_i
\end{equation*}for each $i=1,2,3$. By Proposition \ref{prop1proof}, there exists $\delta>0$ such that
\begin{equation*}
\biggl|\int_{\mathbb{R}}f\diff\mu_{1,n}-\int_{\mathbb{R}}f\diff\mu_{2,n}\biggr|\leq\varepsilon+\frac{2\sup |f|}{\delta n}\cdot c_i.
\end{equation*}Therefore,
\begin{equation*}
\lim_{n\rightarrow\infty}\bigg| \int_\mathbb{R}f\diff\mu_{1,n}-\int_\mathbb{R}f\diff\mu_{2,n}\bigg|\leq \varepsilon.
\end{equation*}Similarly, by Lemma \ref{lemci} we have that
\begin{equation*}
\|\Delta_4\|_{S^1}\leq c_4\sqrt{n-1}.
\end{equation*}By Proposition \ref{prop1proof}, there exists $\delta>0$ such that
\begin{equation*}
\biggl|\int_{\mathbb{R}}f\diff\mu_{1,n}-\int_{\mathbb{R}}f\diff\mu_{2,n}\biggr|\leq\varepsilon+\frac{2\sup |f|}{\delta n}\cdot c_4\sqrt{n-1}.
\end{equation*}Therefore,
\begin{equation*}
\lim_{n\rightarrow\infty}\bigg| \int_\mathbb{R}f\diff\mu_{1,n}-\int_\mathbb{R}f\diff\mu_{2,n}\bigg|\leq \varepsilon.
\end{equation*}
\end{proof}
\subsection{Strong convergence for complete graphs}\label{sectionstrongconvergence}
\begin{lem}\label{lemmaKN}
Given $N\in\mathbb{N}^+$, let $K_N$ and $K'_N$ be two complete graphs on $N$ nodes. Let $K_N\sqcup K'_N$ be their disjoint union and let $\Gamma_N:=K_N\bigcup_{c\text{ edges }} K'_N$ be their union together with $c$ edges $(v_i,v'_i)$ where $v_i\in K_N$ and $v'_i\in K'_N$, for $i=1,\ldots,c$. Let also $\mu_{K_N\sqcup K'_N}$ and $\mu_{\Gamma_N}$ be the spectral measures of these two graphs. Then,
\begin{equation*}
\mu_{K_N\sqcup K'_N}=\frac{1}{N}\cdot\delta_0+\biggl(\frac{N-1}{N}\biggr)\cdot\delta_{\frac{N}{N-1}}
\end{equation*}and
\begin{equation*}
\mu_{\Gamma_N}=\frac{1}{2N}\cdot\delta_0+\frac{1}{2N}\cdot\sum_{i=1}^{2c+1}\delta_{a_i}+\biggl(\frac{N-1-c}{N}\biggr)\cdot\delta_{\frac{N}{N-1}}
\end{equation*}for some $a_i\in (0,2)$.
\end{lem}
\begin{rmk}In order to prove Lemma \ref{lemmaKN}, we make the following observation. It is easy to see that the spectrum of the symmetric normalized Laplacian matrix $\hat{L}=I_n-D^{-1/2}AD^{-1/2}$ equals the spectrum of the \emph{random walk normalized Laplacian matrix} $\tilde{L}:=I_n-D^{-1}A$. Moreover, for a graph with vertex set $V$, $\tilde{L}$ can be seen as an operator from the set $\{f:V\rightarrow\mathbb{R}\}$ to itself. We shall work on this operator for proving Lemma \ref{lemmaKN} and, for a graph $\Gamma$, we shall use the simplified notation $L^{\Gamma}$ in order to indicate the random walk normalized Laplace operator for $\Gamma$.
\end{rmk}
\begin{proof}[Proof of Lemma \ref{lemmaKN}]
Since the spectrum of $K_N\sqcup K'_N$ is given by $0$ with multiplicity $2$ and $\frac{N}{N-1}$ with multiplicity $2(N-1)$, we have that
\begin{equation*}
\mu_{K_N\sqcup K'_N}=\frac{1}{2N}\Biggl(2\cdot\delta_0+2(N-1)\cdot\delta_{\frac{N}{N-1}}\Biggr).
\end{equation*}In order to prove the second part of the lemma, we shall find $2(N-1-c)$ functions on $V(\Gamma_N)$ that are eigenfunctions for the normalized Laplace operator with eigenvalue $\frac{N}{N-1}$ and are orthogonal to each other. In particular, by the symmetry of $\Gamma_N$, it suffices to find $N-1-c$ such functions that are $0$ on the vertices of $K'_N$.
Observe that $K_{N-1-c}$ is a subgraph of $K_N\setminus\{v_1,\ldots,v_c\}$ that has $N-1-c$ eigenfunctions $f_1,\ldots,f_{N-1-c}$ for the largest eigenvalue. These are orthogonal to each other and orthogonal to the constants, therefore
\begin{equation*}
0=\sum_{v\in V(K_{N-1-c})}\deg_{K_{N-1-c}}(v)f_i(v)f_j(v)=\sum_{v\in V(K_{N-1-c})}f_i(v)f_j(v)
\end{equation*}and
\begin{equation*}
\sum_{v\in V(K_{N-1-c})}\deg_{K_{N-1-c}}(v)f_i(v)=\sum_{v\in V(K_{N-1-c})}f_i(v)=0
\end{equation*}
for each $i,j\in\{1,\ldots,N-1-c\}$. Now, for $i\in\{1,\ldots,N-1-c\}$, let $\tilde{f}_i$ be the function on $V(K_N)$ that is equal to zero on $v_1,\ldots,v_c$ and is equal to $f_i$ otherwise. Then, $\tilde{f}_1,\ldots,\tilde{f}_{N-1-c}$ are orthogonal to each other and orthogonal to the constants because
\begin{align*}
\sum_{v\in V(K_N)}\deg_{K_N}(v)\tilde{f}_i(v)\tilde{f}_j(v)&=\sum_{v\in V(K_{N-1-c})}(N-1)\tilde{f}_i(v)\tilde{f}_j(v)\\
&=\sum_{v\in V(K_{N-1-c})}\tilde{f}_i(v)\tilde{f}_j(v)\\
&=\sum_{v\in V(K_{N-1-c})}f_i(v)f_j(v)\\
&=0
\end{align*}and
\begin{equation*}
\sum_{v\in V(K_N)}\tilde{f}_i(v)=\sum_{v\in V(K_{N-1-c})}f_i(v)=0.
\end{equation*}Since for complete graphs any function that is orthogonal to the constants is an eigenfunction for $\frac{N}{N-1}$, we have that $\tilde{f}_1,\ldots,\tilde{f}_{N-1-c}$ are (pairwise orthogonal) eigenfunctions for $\frac{N}{N-1}$.
Analogously, for $i\in\{1,\ldots,N-1-c\}$, let now $\hat{f}_i$ be the function on $V(\Gamma_N)$ that is equal to zero on $K_N'\cup \{v_1,\ldots,v_c\}$ and is equal to $f_i$ otherwise. It's then easy to see that also these functions are orthogonal to each other and orthogonal to the constants. Now, for each $i$ and for each $v\in \Gamma_N$ with $\tilde{f}_i(v)=0$, we have that $v\in K_{N-1-c}$, therefore
\begin{align*}
L^{\Gamma_N}\hat{f}_i(v)&=\hat{f}_i(v)-\frac{1}{\deg_{\Gamma_N}(v)}\sum_{w\sim v}\hat{f}_i(w)\\
&=\tilde{f}_i(v)-\frac{1}{\deg_{K_N}(v)}\sum_{w\sim v\text{ in }K_N}\tilde{f}_i(w)\\
&=L^{K_N}\tilde{f}_i(v)\\
&=\frac{N}{N-1}\cdot\tilde{f}_i(v)\\
&=\frac{N}{N-1}\cdot\hat{f}_i(v).
\end{align*}
This proves that the functions $\hat{f}_i$'s are $N-1-c$ orthogonal eigenfunctions of the Laplace Operator in $\Gamma_N$ for the eigenvalue $\frac{N}{N-1}$. Since they are all $0$ on $K'_N$, by symmetry we can also get $N-1-c$ eigenfunctions for $\frac{N}{N-1}$ on $\Gamma_N$ that are $0$ on $K_N$ and therefore are orthogonal to the first $N-1-c$ functions. This implies that the multiplicity of $\frac{N}{N-1}$ for $\Gamma_N$ is at least $2(N-1-c)$. Therefore,
\begin{equation*}
\mu_{\Gamma_N}=\frac{1}{2N}\cdot\delta_0+\frac{1}{2N}\cdot\sum_{i=1}^{2c+1}\delta_{a_i}+\biggl(\frac{N-1-c}{N}\biggr)\cdot\delta_{\frac{N}{N-1}}
\end{equation*}for some $a_i\in (0,2)$.
\end{proof}
\begin{cor}
The total variation distance between the probability measures $\mu_{K_N\sqcup K'_N}$ and $\mu_{\Gamma_N}$ defined in the previous lemma is
\begin{equation*}
\sup_{A\subseteq[0,2] \text{ measurable }}\biggl|\mu_{K_N\sqcup K'_N}(A)-\mu_{\Gamma_N}(A)\biggr|=\frac{2c+1}{2N}.
\end{equation*}In particular, if $c=o(N)$, the total variation distance tends to zero for $N\rightarrow\infty$.
\end{cor}
\begin{ex}\label{counterexamplestrong}
The previous corollary doesn't hold in general. As a counterexample, take two copies of the path on $N$ vertices, $P_N$ and $P'_N$. Their union via one external edge can be for example the path on $2N$ vertices, and
\begin{equation*}
\mu_{P_{2N}}=\frac{1}{2N}\Biggl(\sum_{k=0}^{2N-1}\delta_{1-\cos\frac{\pi k}{2N-1}}\Biggr),
\end{equation*}while
\begin{equation*}
\mu_{P_N\sqcup P'_N}=\frac{1}{N}\Biggl(\sum_{k=0}^{N-1}\delta_{1-\cos\frac{\pi k}{N-1}}\Biggr).
\end{equation*}The total variation distance between these two measures does not tend to zero for $N\rightarrow\infty$.
\end{ex}
\section{Random geometric graphs}\label{section:Random geometric graphs}
Specializing Corollary \ref{corsupppihik} to the case $k=1$, we get
\begin{equation*}
\varphi^{(1)}_*\Theta_n\xrightarrow{\mathbb{P}}\varphi^{(1)}_*\Theta\qquad\text{and}\qquad\varphi^{(1)}_*\Theta_R\xrightarrow{\mathbb{P}}\varphi^{(1)}_*\Theta,
\end{equation*}with $\supp\varphi^{(1)}_*\Theta=\mathcal{G}^{(1)}(\mathbb{R}^m)$.
\begin{definition}
We define the random geometric graphs
\begin{equation*}
\Gamma_n:=\check{C}^{(1)}(\mathcal{U}_n)\qquad\text{and}\qquad\Gamma_R:=\check{C}^{(1)}(\mathcal{P}_R)
\end{equation*}and we associate the empirical spectral measures
\begin{equation*}
\mu_n:=\mu_{\Gamma_n}=\frac{1}{n}\sum_{i=1}^{n}\delta_{\lambda_i(\Gamma_n)}\qquad\text{ and }\qquad \mu_R:=\mu_{\Gamma_R}=\frac{1}{\#(V(\Gamma_R))}\sum_{i=1}^{\#(V(\Gamma_R))}\delta_{\lambda_i(\Gamma_R)}.
\end{equation*}
\end{definition}
Since the spectrum of a graph $\Gamma$ is finite, we can rewrite the two random measures $\mu_n$ and $\mu_R$ above as follows. First, the set of all possible isomorphism classes of graphs is countable and therefore there exists a sequence $\{x_\ell\}_{\ell\in \mathbb{N}}\subset [0,2]$ such that:
\begin{equation}\mu_n=\sum_{\ell=1}^\infty c_{\ell, n}\delta_{x_\ell}\quad \textrm{and}\quad \mu_R=\sum_{\ell=1}^\infty c_{\ell, R}\delta_{x_\ell},
\end{equation}
where the random variables $c_{\ell, n}$ and $c_{\ell, R}$ are defined by:
\begin{equation}
c_{\ell, n}=\frac{1}{n}\#\{\textrm{components $\Gamma_{n,j}$ of $\Gamma_n=\sqcup\Gamma_{n, j}$ such that $x_\ell$ belongs to the spectrum of $\Gamma_{n,j}$}\}
\end{equation}
and
\begin{equation}
c_{\ell, R}=\frac{1}{\#(V(\Gamma_R))}\#\{\textrm{components $\Gamma_{R,j}$ of $\Gamma_R=\sqcup\Gamma_{R, j}$ such that $x_\ell$ belongs to the spectrum of $\Gamma_{R,j}$}\}.
\end{equation}
In the above definitions of the coefficients $c_{\ell, n}$ and $c_{\ell, R}$, the components should be counted ``with multiplicity": if $x_\ell$ belongs to the spectrum of $\Gamma_{n, j}$ (respectively $\Gamma_{R, j}$) with some multiplicity $m(x_\ell)$, then the component $\Gamma_{n, j}$ (respectively $\Gamma_{R, j}$) is counted $m(x_\ell)$ times.
We will need the following Lemma.
We will need the following Lemma.
\begin{lem}\label{lemg1}For every $\delta>0$ there exists $L>0$ such that
\begin{equation} \mathbb{E}\sum_{\ell\geq L}c_{\ell,n}<\frac{\delta}{4}\quad \textrm{and}\quad \mathbb{E}\sum_{\ell\geq L}c_{\ell,R}<\frac{\delta}{4}.\end{equation}
\end{lem}
\begin{proof}Let $A\subset\{x_\ell\}_{\ell\in \mathbb{N}} $ be any subset. Then:
\begin{equation}\sum_{\ell}\mathbb{E}c_{\ell, n}=\mathbb{E}\sum_{\ell}c_{\ell,n}\leq 1\quad \textrm{and}\quad \sum_{\ell}\mathbb{E}c_{\ell, R}=\mathbb{E}\sum_{\ell}c_{\ell,R}\leq 1.
\end{equation}
In particular the two series $\mathbb{E}\sum_{\ell}c_{\ell,n}$ and $\mathbb{E}\sum_{\ell}c_{\ell,R}$ converge and therefore the existence of such $L$ is clear (as the tails of the series must be arbitrarily small).
\end{proof}
\begin{cor}For every $\ell\in \mathbb{N}$ there exists constants $c_\ell\geq 0$ such that we have the following convergence of random variables:
\begin{equation} c_{\ell, n}\stackrel{L^1}{\longrightarrow} c_\ell\quad \textrm{and}\quad c_{\ell, R}\stackrel{L^1}{\longrightarrow} c_\ell.\end{equation}
The constants $c_\ell$ are positive if and only if $x_\ell$ belongs to the spectrum of a $\mathbb{R}^m$--geometric graph. Moreover the measure
\begin{equation}\label{eq:measure} \mu:=\sum_{\ell\in \mathbb{N}}c_\ell \delta_{x_\ell}\end{equation}
is a probability measure on $\mathbb{R}$ with support contained in $[0,2]$.
\end{cor}
\begin{proof}The convergence in $L^1$ of the random variales $c_{\ell, n}$ and $c_{\ell, R}$ follows from their description as ``number of components such that $x_\ell$ belongs to the spectrum of the random geometric graph'': in fact for every $\ell\geq 0$ we can introduce the counting function:
\begin{equation} \mathcal{N}(\Gamma, \ell)=\#\{\textrm{components of $\Gamma$ for which $x_\ell$ belongs to their spectrum}\}.\end{equation}
With this notation we have:
\begin{equation} c_{\ell, n}=\frac{\mathcal{N}(\Gamma_n,\ell) }{n}\quad \textrm{and}\quad c_{\ell, R}=\frac{\mathcal{N}(\Gamma_R,\ell) }{\textrm{vol}(B(0,R))}=\frac{\mathcal{N}(\Gamma_R,\ell) }{\#(V(\Gamma_R))}\cdot \frac{\#(V(\Gamma_R))}{\textrm{vol}(B(0,R))}.\end{equation}
The convergence of $c_{\ell, n}$ follows the exact same proof of Theorem \ref{teo4.1} and the convergence of $c_{\ell, R}$ proceeds as in Proposition \ref{prop2.1}. The measure $\mu$ is well defined and the fact that it is a probability measure follows from the same proof as \cite[Proposition 6.4]{Antonio}, using the above Lemma \ref{lemg1} as a substitute of \cite[Lemma 6.3]{Antonio}.
\end{proof}
\begin{theorem}\label{teomu}
Let $\mu\in \mathcal{M}^1([0,2])$ be the measure defined in \eqref{eq:measure}. Then
\begin{equation*}
\mu_n\overset{*}{\underset{n\rightarrow\infty}{\rightharpoonup}}\mu\qquad \text{and}\qquad
\mu_R\overset{*}{\underset{R\rightarrow\infty}{\rightharpoonup}}\mu.
\end{equation*}
i.e. $\forall f\in C^0([0,2],\mathbb{R})$ we have $\mathbb{E}\int f\diff \mu_n\rightarrow\int f\diff\mu$ and $\mathbb{E}\int f\diff \mu_R\rightarrow\int f\diff\mu$
\end{theorem}
\begin{proof}
The proof of the statement for $\mu_n$ and $\mu_R$ is the same: we do it for $\mu_R$. Let $f\in C^{0}([0,2], \mathbb{R})$, and fix $\varepsilon>0$. Apply Lemma \ref{lemg1} with the choice of $\delta=\varepsilon/\sup|f|$, and get the corresponding set $F=\{\ell_1, \ldots, \ell_a\}\subset \mathbb{N}$. Also, from the convergence of the series $\sum c_\ell$ we get the existence of a finite set $F'$ such that $\sum_{\ell\notin F}c_\ell<\varepsilon/\sup|f|$. Define $F''=F\cup F''$ (this is still a finite set) and:
\begin{align}
\left|\mathbb{E}\int_{[0,2]}f d\mu_R-\int_{[0,2]}fd\mu\right|=&\left| \mathbb{E}\sum_\ell c_{\ell,R}f(x_\ell)-\sum_{\ell}c_{\ell}f(x_\ell)\right|\\
=&\left|\mathbb{E}\sum_{\ell\in F''} c_{\ell,R}f(x_\ell)+\mathbb{E}\sum_{\ell\notin F''} c_{\ell,R}f(x_\ell)-\sum_{\ell\in F''} c_{\ell}f(x_\ell)-\sum_{\ell\notin F''} c_{\ell}f(x_\ell)\right|\\
\leq &\left|\mathbb{E}\sum_{\ell\in F''} c_{\ell,R}f(x_\ell)-\sum_{\ell\in F''} c_{\ell}f(x_\ell)\right|\\
&+\left|\mathbb{E}\sum_{\ell\notin F''} c_{\ell,R}f(x_\ell)\right|+\left|\sum_{\ell\notin F''} c_{\ell}f(x_\ell)\right|\\
\leq &\left|\sum_{\ell\in F''} \mathbb{E}(c_{\ell,R}-c_\ell)f(x_\ell)\right|+\sup|f|\cdot\left| \sum_{\ell\notin F''} \mathbb{E}c_{\ell,R}\right|+\sup|f|\cdot\left| \sum_{\ell\notin F''} c_{\ell}\right|\\
\leq &\sup|f|\sum_{\ell\in F''} \mathbb{E}|c_{\ell,R}-c_\ell|+\sup|f|\frac{\delta}{4}+\sup|f|\frac{\delta}{4}\\
\label{eq:epsilon}\leq&\mathbb{E}|c_{\ell,R}-c_\ell|+\frac{\varepsilon}{2}\longrightarrow \frac{\varepsilon}{2}\quad \textrm{as $R\to \infty$},
\end{align}
where in the last line we have used the $L^1$ convergence of $c_{\ell, R}\to c_\ell$. Since this is true for every $\varepsilon>0$, it follows that:
\begin{equation} \lim_{R\to \infty}\mathbb{E}\int_{[0,2]}fd\mu_R=\int_{[0,2]}d\mu.\end{equation}
\end{proof}
\begin{prop}\label{prop:last}
The measure $\mu$ appearing in Theorem \ref{teomu} has the following properties:
\begin{enumerate}
\item $\mu$ is not absolutely continuous with respect to Lebesgue;
\item $\lim_{n\rightarrow\infty}\frac{1}{n}\cdot\mathbb{E}\#\bigl(\text{eigenvalues of $\Gamma_n$ in } [a,b]\bigr)=\mu([a,b])>0$;
\item $\mu(\{0\})=\beta$.
\end{enumerate}
\end{prop}
\begin{proof}We start with point (3): we have that $\mu(\{0\})=c_{\ell_0}$ where $0=x_{\ell_0}$ and $c_{\ell_0}$ is the $L^1$-limit of $c_{\ell_0, R}$, which is the random variable:
\begin{equation} c_{\ell_0, R}=\#\{\textrm{components of $\Gamma_R$ containing $0$ in their spectrum}\}=b_0(\Gamma_R),\end{equation}
and therefore (3) follows from the definition of $\beta$.
Point (1) also follows immediately, since $\beta_0>0$ and $\mu$ charges positively sets of Lebesgue measure zero (hence it cannot be absolutely continuous with respect to Lebesghe measure).
For the proof of point (2) we argue exactly as in the proof of Theorem \ref{teomu}, by replacing $f$ with $\chi_{[a,b]}$ (the characteristic function of the interval $[a,b]$) and observing that the only property of $f$ that we have used is its boundedness.
\end{proof}
\bibliographystyle{alpha}
|
3,212,635,537,413 | arxiv | \section{Introduction}
Although the contraction principle appears partly in the method of successive approximation in the works of
Cauchy \cite{Cau1835}, Liouville \cite{Lio1837}, and Picard \cite{Pic1890}, its abstract and powerful version
is due to Banach \cite{Ban22}. This form of the contraction principle, commonly quoted as the Banach Fixed Point
Theorem, states that any contraction of a complete metric space has exactly one fixed point. Until now, this
seminal result has been generalized in several ways and some of these generalizations initiated new branches
in the field of Iterative Fixed Point Theory. The books by Granas and Dugundji \cite{GraDug03}, and by Zeidler
\cite{Zei86} give an excellent and detailed overview of the topic.
Some generalizations of Banach's fundamental result replace contractivity by a weaker but still effective
property; for example, the self-mapping $T$ of a metric space $X$ is supposed to satisfy
\Eq{main}{
d(Tx,Ty)\le\phi\bigl(d(x,y)\bigr)\qquad(x,y\in X).}
To the best of our knowledge, the assumption above appears first in the paper by Browder \cite{Brow68} and
by Boyd and Wong \cite{BoyWon69}. One of the most important result in this setting was obtained by Matkowski
\cite{Mat75} who established the following statement.
\Thmn{Assume that $(X,d)$ is a complete metric space and $\phi\colon\mathbb R_+\to\mathbb R_+$ is a monotone increasing
function
such that the sequence of iterates $(\phi^n)$ tends to zero pointwise on the positive half-line. If $T\colon
X\to X$
is a mapping satisfying \eqref{main}, then it has a unique fixed point in $X$.}
Another branch of generalizations of Banach's principle is based on relaxing the axioms of the metric space.
For an account of such developments, see the monographs by Rus \cite{Rus01}, by Rus, Petru\c{s}el and Petru\c{s}el
\cite{RusPetPet08}, and by Berinde \cite{Ber07}. The aim of this note is to extend the contraction principle
combining these two directions: The main result is formulated in the spirit of Matkowski, while the underlying
space is a complete semimetric space that fulfills an extra regularity condition. These kind of spaces involve
standard metric, ultrametric and inframetric spaces. The stability of fixed points is also investigated in this
general setting.
\section{Conventions and Basic Notions}
Throughout this note, $\mathbb R_+$ and $\overline{\mathbb R}_+$ stand for the set of all nonnegative and extended nonnegative
reals, respectively. The \emph{iterates} of a mapping $T\colon X\to X$ are defined inductively by the
recursion
$T^1=T$ and $T^{n+1}=T\circ T^n$.
Dropping the third axiom of Fr\'echet \cite{Fre1906}, we arrive at the notion of semimetric spaces: Under a
\emph{semimetric space} we mean a pair $(X,d)$, where $X$ is a nonempty set, and $d\colon X\times X\to\mathbb R_+$
is a nonnegative and symmetric function which vanishes exactly on the diagonal of the Cartesian product
$X\times X$. In semimetric spaces, the notions of \emph{convergent} and \emph{Cauchy sequences}, as like as
(open) \emph{balls} with given center and radius, can be introduced in the usual way. For an open ball with
center $p$ and radius $r$, we use the notation $B(p,r)$. Under the \emph{diameter} of $B(p,r)$ we mean the
supremum of distances taken over the pairs of points of the ball. Under the \emph{topology} of a semimetric
space we mean the topology induced by open balls.
\begin{Def}
Consider a semimetric space $(X,d)$. We say that $\Phi\colon\overline{\mathbb R}_+^2\to\overline{\mathbb R}_+$ is a
\emph{triangle function} for $d$, if $\Phi$ is symmetric and monotone increasing in both of its arguments,
satisfies $\Phi(0,0)=0$ and, for all $x,y,z\in X$, the generalized triangle inequality
\Eq{*}{
d(x,y)\le\Phi\bigl(d(x,z),d(y,z)\bigr)}
holds.
\end{Def}
The construction below plays a key role in the further investigations. For a semimetric space $(X,d)$, define
the function
\Eq{basic}{
\Phi_d(u,v):=\sup\{d(x,y)\mid\exists p\in X: d(p,x)\le u,\,d(p,y)\le v\}\qquad(u,v\in\overline{\mathbb R}_+).}
Simple and direct calculations show, that $\Phi_d\colon\overline{\mathbb R}^2_+\to\overline{\mathbb R}_+$ is a triangle
function for $d$. This function is called the \emph{basic triangle function}. Note also, that the basic
triangle function is optimal in the following sense: If $\Phi$ is a triangle function for $d$, then
$\Phi_d\le\Phi$ holds.
Obviously, metric spaces are semimetric spaces with triangle function $\Phi(u,v):=u+v$. Ultrametric spaces are
also semimetric spaces if we choose $\Phi(u,v):=\max\{u,v\}$. Not claiming completeness, we present here
some further examples that can be interpreted in this framework:
\begin{itemize}
\item $\Phi(u,v)=c(u+v)$ ($c$-relaxed triangle inequality);
\item $\Phi(u,v)=c\max\{u,v\}$ ($c$-inframetric inequality);
\item $\Phi(u,v)=(u^p+v^p)^{1/p}$ ($p$th-order triangle inequality, where $p>0$).
\end{itemize}
Briefly, each semimetric space $(X,d)$ can be equipped with an optimal triangle function attached to $d$; this
triangle function provides an inequality that corresponds to and plays the role of the classical triangle
inequality. In this sense, semimetric spaces are closer relatives of metric spaces then the system of axioms
suggests.
Throughout the present note, we shall restrict our attention only to those semimetric spaces whose basic
triangle function is continuous at the origin. These spaces are termed \emph{regular}. Clearly,
the basic triangle function of a regular semimetric space is bounded in a neighborhood of the origin. The
importance of regular semimetric spaces is enlightened by the next important technical result.
\Lem{regsem}{The topology of a regular semimetric space is Hausdorff. A convergent sequence in a regular
semimetric space has a unique limit and possesses the Cauchy property. Moreover, a semimetric space $(X,d)$
is regular if and only if
\Eq{diameter}{
\lim_{r\to 0}\sup_{p\in X}\mathop{\hbox{\rm diam}} B(p,r)=0.}}
\begin{proof}
For the first statement, assume to the contrary that there exist distinct points $x,y\in X$ such that, for all
$r>0$, the balls $B(x,r)$ and $B(y,r)$ are not disjoint. The continuity and the separate monotonicity of the
basic triangle function guarantee the existence of $\delta>0$ such that, for all $r<\delta$, we have $\Phi_d(r,r)<d(x,y)$.
Therefore, if $p\in B(x,r)\cap B(y,r)$, we get the contradiction
\Eq{*}{
d(x,y)\le\Phi_d\bigl(d(p,x),d(p,y)\bigr)\le\Phi_d(r,r)<d(x,y).}
The Hausdorff property immediately implies that the limit of a convergent sequence is unique. Assume now that
$(x_n)$ is convergent and tends to $x\in X$. Then the generalized triangle inequality implies the estimation
$d(x_n,y_m)\le\Phi_d\bigl(d(x,x_n),d(x,x_m)\bigr)$. The regularity of the underlying space provides that the
right-hand side tends to zero if we take the limit $n\to\infty$, yielding the Cauchy property.
If $(X,d)$ is a regular semimetric space, then the basic triangle function $\Phi_d$ is continuous at the origin.
Hence, for ${\varepsilon}>0$, there exists $u_0,v_0>0$ such that $\Phi_d(u,v)<{\varepsilon}$ whenever $0<u<u_0$ and $0<v<v_0$. Let
$r_0=\min\{u_0,v_0\}$. Fix $0<r<r_0$ and $p\in X$. If $x,y\in B(p,r)$, then, using the separate monotonicity of
triangle functions,
\Eq{*}{
d(x,y)\le\Phi_d\bigl(d(p,x),d(p,y)\bigr)\le\Phi_d(r,r)<{\varepsilon}}
follows. That is, $\mathop{\hbox{\rm diam}} B(p,r)\le{\varepsilon}$ holds for all $p\in X$. Since ${\varepsilon}$ is an arbitrary positive number,
we arrive at the desired limit property.
Assume conversely that the diameters of balls with small radius are uniformly small, and take sequences of
positive numbers $(u_n)$ and $(v_n)$ tending to zero. For fixed $n\in\mathbb N$, define $r_n=\max\{u_n,v_n\}$ and
take elements $p,x,y\in X$ satisfying $d(p,x)\le u_n$ and $d(p,y)\le v_n$. Then, $x,y\in B(p,r_n)$; therefore
\Eq{*}{
d(x,y)\le\mathop{\hbox{\rm diam}} B(p,r_n)\le\sup_{p\in X}\mathop{\hbox{\rm diam}} B(p,r_n).}
Taking into account the definition of the basic triangle function $\Phi_d$ and the choice of $p,x,y$, we arrive
at the inequality
\Eq{*}{
\Phi_d(u_n,v_n)\le\sup_{p\in X}\mathop{\hbox{\rm diam}} B(p,r_n).}
Here the right-hand side tends to zero as $n\to\infty$ by hypothesis, resulting the continuity of $\Phi_d$ at
the origin.
\end{proof}
As usual, a semimetric space is termed to be \emph{complete}, if each Cauchy sequence of the space is convergent.
In view of the previous lemma, convergence and Cauchy property cannot be distinguished in complete and regular
semimetric spaces.
In order to construct a large class of complete, regular semimetric spaces, we introduce the notion of
equivalence of semimetrics. Given two semimetrics $d_1$ and $d_2$ on $X$, an increasing function
$L\colon\overline{\mathbb R}_+\to\overline{\mathbb R}_+$ such that $L(0)=0$ and
\Eq{*}{
d_1(x,y)\leq L(d_2(x,y)) \qquad(x,y\in X)
}
is called a \emph{Lipschitz modulus with respect to the pair $(d_1,d_2)$}. It is immediate to see that
the function $L_{d_1,d_2}\colon\overline{\mathbb R}_+\to\overline{\mathbb R}_+$ defined by
\Eq{*}{
L_{d_1,d_2}(t):=\sup\{d_1(x,y)\mid x,y\in X,\,d_2(x,y)\leq t\} \qquad(t\in \overline{\mathbb R}_+)
}
is the smallest Lipschitz modulus with respect to $(d_1,d_2)$. The semimetrics $d_1$ and $d_2$ are said to be
\emph{equivalent} if
\Eq{*}{
\lim_{t\to0+} L_{d_1,d_2}(t)=0\qquad\mbox{and}\qquad \lim_{t\to0+} L_{d_2,d_1}(t)=0
}
i.e., if $L_{d_1,d_2}$ and $L_{d_2,d_1}$ are continuous at zero.
It is easy to verify that, indeed, this notion of equivalence is an equivalence relation. For the proof of
the transitivity one should use the inequality $ L_{d_1,d_3}\leq L_{d_1,d_2}\circ L_{d_2,d_3}$.
Our next result establishes that the convergence, completeness and the regularity of a semimetric space is
invariant with respect to the equivalence of the semimetrics.
\Lem{equiv}{If $d_1$ and $d_2$ are semimetrics on $X$, then
\Eq{Lip}{
\Phi_{d_1}\leq L_{d_1,d_2}\circ\Phi_{d_2}\circ(L_{d_2,d_1},L_{d_2,d_1}).
}
Provided that $d_1$ and $d_2$ are equivalent semimetrics, we have that
\begin{enumerate}[(i)]
\item a sequence converges to point in $(X,d_1)$ if and only if it converges to the
same point in $(X,d_2)$;
\item a sequence is Cauchy in $(X,d_1)$ if and only if it is Cauchy in $(X,d_2)$;
\item $(X,d_1)$ is complete if and only if $(X,d_2)$ is complete;
\item $(X,d_1)$ is regular if and only if $(X,d_2)$ is regular.
\end{enumerate}}
\begin{proof} Using the monotonicity properties, for $x,y,z\in X$, we have
\Eq{*}{
d_1(x,y)\leq L_{d_1,d_2}(d_2(x,y))
&\leq L_{d_1,d_2}\Big(\Phi_{d_2}(d_2(x,z),d_2(z,y))\big)\Big) \\
&\leq L_{d_1,d_2}\Big(\Phi_{d_2}\big(L_{d_2,d_1}(d_1(x,z)),L_{d_2,d_1}(d_1(z,y))\big)\Big),
}
whence it follows that the map $L_{d_1,d_2}\circ\Phi_{d_2}\circ(L_{d_2,d_1},L_{d_2,d_1})$ is a triangle
function for $d_1$, hence \eqref{Lip} must be valid.
For (i), let $(x_n)$ be a sequence converging to $x$ in $(X,d_1)$. Then $(d_1(x_n,x))$ is a null-sequence.
By the continuity of $L_{d_2,d_1}$ at zero, the right-hand side of the inequality
\Eq{*}{
d_2(x_n,x)\leq L_{d_2,d_1}(d_1(x_n,x))}
also tends to zero, hence $(d_2(x_n,x))$ is also a null-sequence. The reversed implication holds analogously.
The proof for the equivalence of the Cauchy property is completely similar.
To prove (iii), assume that $(X,d_1)$ is complete and let $(x_n)$ be a Cauchy sequence in $(X,d_2)$. Then,
by (ii), $(x_n)$ is Cauchy in $(X,d_1)$. Hence, there exists an $x\in X$ such that $(x_n)$ converges to $x$
in $(X,d_1)$. Therefore, by (i), $(x_n)$ converges to $x$ in $(X,d_2)$. This proves the completeness of
$(X,d_2)$. The reversed implication can be verified analogously.
Finally, assume that $(X,d_2)$ is a regular semimetric space, which means that $\Phi_{d_2}$ is continuous at
$(0,0)$. Using inequality \eqref{Lip}, it follows that $\Phi_{d_1}$ is also continuous at $(0,0)$ yielding
that $(X,d_1)$ is regular, too.
\end{proof}
By the above result, if a semimetric is equivalent to a complete metric, then it is regular and complete.
With a similar argument that was followed in the above proofs, one can easily verify that equivalent
semimetrics on $X$ generate the same topology.
\medskip
In the sequel, we shall need a concept that extends the notion of classical contractions to nonlinear ones.
This extension is formulated applying comparison functions fulfilling the assumptions of Matkowski.
\begin{Def}
Under a \emph{comparison function} we mean a monotone increasing function $\phi\colon\mathbb R_+\to\mathbb R_+$ such
that the limit property $\lim_{n\to\infty}\phi^n(t)=0$ holds for all $t\ge 0$. Given a semimetric space
$(X,d)$ and a comparison function $\phi$, a mapping $T\colon X\to X$ is said to be \emph{$\phi$-contractive}
or a \emph{$\phi$-contraction} if it fulfills \eqref{main}.
\end{Def}
The statement of the next lemma is well-known, we provide its proof for the convenience of the reader.
\Lem{mainprop}{If $\phi$ is a comparison function, then $\phi(t)<t$ for all positive $t$. If $(X,d)$ is a
semimetric space and $T\colon X\to X$ is a $\phi$-contraction, then $T$ has at most one fixed point.}
\begin{proof}
For the first statement, assume at the contrary that $t\le\phi(t)$ for some $t>0$. Whence, by monotonicity and
using induction, we arrive at
\Eq{*}{
t\le\phi(t)\le\phi^2(t)\le\cdots\le\phi^n(t).}
Upon taking the limit $n\to\infty$, the right hand-side tends to zero, contradicting to the positivity of $t$.
In view of this property, the second statement follows immediately. Indeed, if $x_0,y_0$ were distinct fixed
points of a $\phi$-contraction $T$, then we would arrive at
\Eq{*}{
t=d(x_0,y_0)=d(Tx_0,Ty_0)\le\phi(d(x_0,y_0))=\phi(t)<t.}
This contradiction implies $x_0=y_0$.
\end{proof}
\section{The Main Results}
The main results of this note is presented in two theorems. The first one is an extension of the Matkowski Fixed
Point Theorem \cite{Mat75} for complete and regular semimetric spaces. The most important ingredient of the proof
is that a domain invariance property remains true for sufficiently large iterates of $\phi$-contractions.
\Thm{mainfix}{If $(X,d)$ is a complete regular semimetric space and $\phi$ is a comparison function, then every
$\phi$-contraction has a unique fixed point.}
\begin{proof}
Let $T\colon X\to X$ be a $\phi$-contraction and let $p\in X$ be fixed arbitrarily. Define the sequence
$(x_n)$ by the
standard way $x_n:=T^np$. Observe that, for all fixed $k\in\mathbb N$, the sequence $\bigl(d(x_n,x_{n+k})\bigr)$ tends
to zero. Indeed, by the asymptotic property of comparison functions,
\Eq{*}{
d(x_n,x_{n+k})=d(Tx_{n-1},Tx_{n+k-1})\le
\phi\bigl(d(x_{n-1},x_{n+k-1})\bigr)\le\cdots\le
\phi^n\bigl(d(p,T^kp)\bigr)\longrightarrow 0.}
We are going to prove that $(x_n)$ is a Cauchy sequence. Fix ${\varepsilon}>0$. The continuity of the basic triangle
function guarantees the existence of a neighborhood $U$ of the origin such that, for all $(u,v)\in U$, we
have the inequality $\Phi_d(u,v)<{\varepsilon}$. Or equivalently, applying the separate monotonicity, there exists
some $\delta({\varepsilon})>0$ such that $\Phi_d(u,v)<{\varepsilon}$ holds if $0\le u,v<\delta({\varepsilon})$. The asymptotic property of
comparison functions allows us to fix an index $n({\varepsilon})\in\mathbb N$ such that $\phi^{n({\varepsilon})}({\varepsilon})<\delta({\varepsilon})$
hold. Then, $\psi:=\phi^{n({\varepsilon})}$ is a comparison function. Hence, if $0\le u,v<\min\{{\varepsilon},\delta({\varepsilon})\}$,
\Eq{*}{
\Phi_d\bigl(u,\psi(v)\bigr)\le\Phi_d(u,v)<{\varepsilon}.}
Immediate calculations show, that the mapping $S:=T^{n({\varepsilon})}$ is a $\psi$-contraction.
Let the nonnegative integer $k$ and the points $x,y\in X$ be arbitrary. Then, applying the monotonicity
properties of comparison functions and
their iterates,
\Eq{*}{
d(T^kSx,T^kSy)\le\psi\bigl(d(T^kx,T^ky)\bigr)
\le\psi\circ\phi^k\bigl(d(x,y)\bigr)
\le\psi\bigl(d(x,y)\bigr)}
follows. This inequality immediately implies that $T^kS$ maps the ball $B(x,{\varepsilon})$ into itself if it makes small
perturbation on the center. Indeed, if $y\in B(x,{\varepsilon})$ and $d(x,T^kSx)<\delta({\varepsilon})$, the choices of ${\varepsilon}$ and
$\delta({\varepsilon})$ moreover the separate monotonicity of the basic triangle function yield
\Eq{*}{
d(x,T^kSy)
\le\Phi_d\bigl(d(x,T^kSx),d(T^kSx,T^kSy)\bigr)
\le\Phi_d\bigl(d(x,T^kSx),\psi(d(x,y)\bigr)<{\varepsilon}.}
The properties of $(x_n)$ established at the beginning of the proof ensure that, for all nonnegative $k$, there
exists some $n_k\in\mathbb N$ such that the inequalities $d(x_n,T^kSx_n)<\delta({\varepsilon})$ hold whenever $n\ge n_k$. Choose
\Eq{*}{
n_0=\max\{n_k\mid k=1,\ldots,n({\varepsilon})\}.}
Then, taking into account the previous step, $T^kS\colon B(x_{n_0},{\varepsilon})\to B(x_{n_0},{\varepsilon})$ for
$k=1,\ldots,n({\varepsilon})$.
In particular, each iterates of $S$ is a self-mapping of the ball $B(x_{n_0},{\varepsilon})$. Let $n>n_0$ be an arbitrarily
given natural number. Then $n=mn({\varepsilon})+k$, where $m\in\mathbb N$ and $k\in\{1,\ldots,n({\varepsilon})\}$; hence, the definition of
$S$ leads to
\Eq{*}{
T^nS=T^{mn({\varepsilon})+k}S=T^kT^{mn({\varepsilon})}S=T^kS^mS=T^kS^{m+1}.}
Therefore,
\Eq{*}{
T^nS\bigl(B(x_{n_0},{\varepsilon})\bigr)
=T^kS^{m+1}\bigl(B(x_{n_0},{\varepsilon})\bigr)
\subset T^kS\bigl(B(x_{n_0},{\varepsilon})\bigr)\subset B(x_{n_0},{\varepsilon}).}
In other words, due to property \eqref{diameter} of \lem{regsem}, the sequence $(T^nSx_{n_0})$ is Cauchy and hence
so is $(x_n)$. The completeness implies, that it tends to some element $x_0$ of $X$. Our claim is that $x_0$ is a
fixed point of $T$. Applying the generalized triangle inequality,
\Eq{*}{
d(x_0,Tx_0)&\le\Phi_d\bigl(d(x_0,x_{n+1}),d(Tx_0,x_{n+1})\bigr)\\
&=\Phi_d\bigl(d(x_0,x_{n+1}),d(Tx_0,Tx_n)\bigr)\\
&\le\Phi_d\bigl(d(x_0,x_{n+1}),\phi(d(x_0,x_n))\bigr)\\
&\le\Phi_d\bigl(d(x_0,x_{n+1}),d(x_0,x_n)\bigr).}
Therefore,
\Eq{*}{
d(x_0,Tx_0)\le\lim_{n\to\infty}\Phi_d\bigl(d(x_0,x_{n+1}),d(x_0,x_n)\bigr)=\Phi_d(0,0)=0}
follows. That is, $Tx_0=x_0$, as it was desired. To complete the proof, recall that a $\phi$-contraction
may have at most one fixed point.
\end{proof}
Our second main result is based on \thm{mainfix} whose proof requires some additional ideas. It asserts
the stability of fixed points of iterates of $\phi$-contractions.
\Thm{stabfix}{Let $(X,d)$ be a complete and regular semimetric space. If $(T_n)$ is a sequence of
$\phi$-contractions converging pointwise to a $\phi$-contraction $T_0\colon X\to X$, then
the sequence of the fixed points of $(T_n)$ converges to the unique fixed point of $T_0$.}
\begin{proof}
First we show that, for all $k\in\mathbb N$, $(T_n^k)$ converges to $T_0^k$ pointwise.
We prove by induction on $k$. By the assumption of the theorem, we have the statement for $k=1$.
Now assume that $T_n^k \to T_0^k$ pointwise. Then, for every $x\in X$, we have
\Eq{*}{
d(T_n^{k+1}x,T_0^{k+1}x)
&\leq \Phi_d\bigl(d(T_n^{k+1}x,T_n^{k}T_0x),d(T_n^{k}T_0x,T_0^{k+1}x)\bigr)\\
&\leq \Phi_d\bigl(\phi^k(d(T_nx,T_0x)),d(T_n^{k}T_0x,T_0^{k}T_0x)\bigr)\\
&\leq \Phi_d\bigl(d(T_nx,T_0x),d(T_n^{k}T_0x,T_0^{k}T_0x)\bigr)\to0
}
since $T_n\to T_0$, $T_n^k \to T_0^k$ pointwise and the semimetric $d$ is regular. This proves that
$(T_n^{k+1})$ converges to $T_0^{k+1}$ pointwise.
The previous theorem ensures that, for all $n\in\mathbb N$, there exists
a unique fixed point $x_n$ of $T_n$ as well as a unique fixed point $x_0$ of $T_0$.
To verify the statement of the theorem, assume indirectly that
\Eq{*}{
{\varepsilon}:=\limsup_{n\to\infty}d(x_0,x_n)>0.}
Choose $\delta>0$ such that $\Phi_d\bigl(\delta,\delta\bigr)<{\varepsilon}$ hold, and then choose $m\in\mathbb N$ such
that $\phi^m(2{\varepsilon})<\delta$, finally denote $\psi:=\phi^m$. Using induction, one can easily prove that
$S_n=T^m_n$ is a $\psi$-contraction and $x_n$ is a fixed point of $S_n$ for all $n\geq 0$. Furthermore,
in view of the previous step, the sequence $(S_nx_0)$ converges to $S_0x_0=x_0$. Hence, for large $n$,
$d(x_0,S_nx_0)<\delta$ and $d(x_n,x_0)<2{\varepsilon}$. Therefore, for large $n$, we have
\Eq{*}{
d(x_0,x_n)&\le\Phi_d\bigl(d(x_0,S_nx_0),d(S_nx_0,x_n)\bigr)\\
&=\Phi_d\bigl(d(x_0,S_nx_0),d(S_nx_0,S_nx_n)\bigr)\\
&\le\Phi_d\bigl(d(x_0,S_nx_0),\psi(d(x_n,x_0))\bigr)
\le\Phi_d\bigl(\delta,\psi(2{\varepsilon})\bigr).
}
Taking the limes superior,
${\varepsilon}\leq\Phi_d\bigl(\delta,\psi(2{\varepsilon})\bigr) \leq\Phi_d\bigl(\delta,\delta\bigr)<{\varepsilon}$ follows, which is a
contradiction. That is, the sequence of fixed points of $T_n$ tends to the fixed point of $T_0$, as it was
stated.
\end{proof}
If the semimetric $d$ is \emph{self-continuous}, that is, $d(x_n,y_n)\to d(x,y)$ whenever $x_n\to x$ and
$y_n\to y$, then the $\phi$-contractivity of the members of the sequence $(T_n)$ implies the
$\phi$-contractivity of $T_0$. Indeed, taking the limit $n\to\infty$ in the inequalities
$d(T_nx,T_ny)\le\phi\bigl(d(x,y)\bigr)$ and using the self-continuity of $d$, the inequality
$d(T_0x,T_0y)\le\phi\bigl(d(x,y)\bigr)$ follows. Note that the self-continuity of $d$ holds automatically in
metric spaces and also in ultrametric spaces.
\section{Applications and Concluding Remarks}
Not claiming completeness, let us present here some immediate consequences of \thm{mainfix} and \thm{stabfix},
respectively. For their proof, one should observe only that in each case the underlying semimetric spaces
are regular. Moreover, the extra assumption of \thm{stabfix} is obviously satisfied.
\Cor{Matkowski1}{If $(X,d)$ is a complete $c$-relaxed metric space or complete $c$-inframetric space and
$\phi$ is a comparison function, then every $\phi$-contraction has a unique fixed point.}
\Cor{Matkowski2}{Let $(X,d)$ be a complete metric space or a complete ultrametric space. If $(T_n)$ is a
sequence of $\phi$-contractions converging pointwise to $T\colon X\to X$, then $T$ is a
$\phi$-contraction, and the sequence of the fixed points of $(T_n)$ converges to the fixed point of $T$.}
Note, that the special case $c=1$ of \cor{Matkowski1} reduces to the result of Matkowski \cite{Mat75} which
is a generalization that of Browder \cite{Brow68}. Each of these results generalizes the Banach Fixed Point
Theorem under the particular choice $\phi(t)=qt$ where $q\in[0,1[$ is given. Let us also mention, that the
contraction principle was also discovered and applied independently (in a few years later after Banach) by
Caccioppoli \cite{Cac30}. Our last corollary demonstrates the efficiency of \thm{mainfix} in a particular
case when the Banach Fixed Point Theorem cannot be applied directly. For its proof, one should combine
\cor{Matkowski1} with \lem{equiv}.
\Cor{extension}{If $(X,d)$ is a semimetric space with a semimetric $d$ equivalent to a complete $c$-relaxed
metric or to a complete $c$-inframetric on $X$, then every $\phi$-contraction has a unique fixed point.}
As a final remark, let us quote here a result due to Jachymski, Matkowski, and \'Swi\k{a}tkowski, which
is a generalization of the Matkowski Fixed Point Theorem (for precise details, consult \cite{JacMatSwi95}).
\Thmn{Assume that $(X,d)$ is a complete Hausdorff semimetric space such that there is some $r>0$ for which the
diameters of balls with radius $r$ are uniformly bounded and in which the closure operator induced by $d$ is
idempotent. If $\phi$ is a comparison function, then every $\phi$-contraction has a unique fixed point.}
The proof of this theorem is based on the fundamental works of Chittenden \cite{Chi17} and Wilson \cite{Wil31}.
In view of \lem{regsem}, the first two conditions of the result above is always satisfied in regular semimetric
spaces. However, the exact connection between the properties of the basic triangle function and the idempotence
of the metric closure has not been clarified yet. The interested Readers can find further details on the topology
of semimetric spaces in the papers of Burke \cite{Bur72}, Galvin and Shore \cite{GalSho84}, and by McAuley
\cite{Mca56}.
|
3,212,635,537,414 | arxiv | \section{Introduction}
Spin-transfer torque induced by an applied electric current in ferromagnetic metals is a crucially important effect in spintronics.
The idea was first proposed theoretically by Berger \cite{Berger86} in the case of a domain wall motion and by Slonczewski \cite{Slonczewski96} and Berger \cite{Berger96} in the case of the uniform magnetization of thin films.
The spin-transfer effect arises from the transfer of spin angular momentum from conduction electrons to localized spins which induce the magnetization.
The effect is caused by the $sd$ exchange interaction, and the angular momentum transfer occurs owing to the angular momentum conservation \cite{Slonczewski96}.
The interaction Hamiltonian describing the spin-transfer effect is
\begin{align}
\mathcal{H}_{\rm st}&= \int {\rm{d}}^3r \frac{\hbar P}{2e}(1-\cos\theta) (\bm{j}\cdot\nabla)\phi,
\end{align}
where $\theta$ and $\phi$ are the polar coordinates representing the localized spin direction, $\bm{j}$ denotes the applied electric current density, $P$ is the spin polarization of the conduction electron, and $e$ is the electron charge.
The interaction is represented as a gauge coupling to a spin gauge field $\bm{A}_{\rm s}^z$
\cite{Bazaliy98,TKS_PR08},
$\mathcal{H}_{\rm st}= \int {\rm{d}}^3r (\bm{j}_{\rm s}\cdot\bm{A}_{\rm s}^z)$, where $\bm{j}_{\rm s}\equiv P\bm{j}$ and $\bm{A}_{\rm s}^z=\frac{\hbar}{2e}(1-\cos\theta) \nabla\phi$.
The interaction is thus expressed as
$\mathcal{H}_{\rm st}= \int {\rm{d}}^3r P\sigma_{\rm B}(\bm{E}\cdot\bm{A}_{\rm s}^z)$, where $\sigma_{\rm B}$ is the Boltzmann conductivity.
This expression clearly shows that the spin-transfer effect is due to a coupling of two gauge fields, the conventional electromagnetic field of the electric charge, and the gauge field acting on the electron spin.
The aim of this work is to study the coupling between the two gauge fields by calculating an effective Hamiltonian.
We shall show that the effective Hamiltonian in the case of a slowly varying magnetization is made up of three contributions, one representing the spin-transfer torque and the others describing the couplings between the electric and magnetic fields.
In the case of charge electromagnetism coupled to relativistic charged particles, the effective Lagrangian induced by the particles is always written in a relativistically invariant form as $\sum_{\mu\nu}F_{\mu\nu}F^{\mu\nu}$, where $F_{\mu\nu}$ is the field strength and $\mu$ and $\nu$ are indices representing $x,y,z$, and $t$. The only terms allowed in the relativistic case are thus proportional to either $|\bm{E}|^2$ or $|\bm{B}|^2$.
In ferromagnetic metals, conduction electrons interact with two gauge fields, $\bm{A}$ acting on the charge and $\bm{A}_{\rm s}$ acting on the spin, and the total electric and magnetic fields become $\bm {E}+\bm{E}_{\rm s}$ and $\bm {B}+\bm{B}_{\rm s}$, where $\bm{E}_{\rm s}$ and $\bm{B}_{\rm s}$ are the effective spin electric and spin magnetic fields, respectively. If the system is relativistic, we would thus expect to have interactions in the form of $\bm {E}\cdot\bm{E}_{\rm s}$ and $\bm {B}\cdot\bm{B}_{\rm s}$ arising from $(\bm {E}+\bm{E}_{\rm s})^2$ and $(\bm {B}+\bm{B}_{\rm s})^2$. In reality, there are other contributions in ferromagnetic metals since the electrons are not relativistic and they have a finite lifetime of elastic scattering.
We shall demonstrate that a coupling term proportional to $\bm{E}\cdot \bm{A}_{\rm s}^z$ arises as the dominant contribution. This term derived first in Ref. 6 represents the spin-transfer effect, as was discussed there. We also investigate other coupling terms, $\bm {E}\cdot\bm{E}_{\rm s}$ and $\bm {B}\cdot\bm{B}_{\rm s}$.
\subsection{Spin electromagnetic field}
An effective electromagnetic field arises from the $sd$ exchange interaction
described by
\begin{align}
\mathcal{H}_{sd}=-\Delta_{sd}\int {\rm{d}}^3r \bm{n}\cdot \bm{s}_{\rm e},
\end{align}
where $\Delta_{sd}$ is the exchange energy, $\bm{n}$ is a unit vector representing the direction of the localized spin, and $\bm{s}_{\rm e}$ is the direction of the conduction electron spin.
When this exchange interaction is strong, the conduction electron spin is aligned parallel to the localized spin direction, and this effect results in a quantum mechanical phase attached to the electron spin when the electron moves (see Ref. 7 for details of derivation).
The spin part of the electron wave function with the expectation value along $\bm{n}$ is
$|\bm{n}\rangle =\cos\frac{\theta}{2}|\uparrow\rangle+\sin\frac{\theta}{2}e^{i\phi}|\downarrow\rangle$,
where $\theta$ and $\phi$ are the polar coordinates of $\bm{n}$ and
$|\uparrow\rangle$ and $|\downarrow\rangle$ denote the spin states \cite{Sakurai93}.
When the electron hops over a small distance ${\rm{d}}{\bm r}$ to a nearby site
where the localized spin is along $\bm{n}'$, the overlap of the wave functions is calculated as
$\langle \bm{n}'|\bm{n}\rangle\simeq e^{\frac{i}{\hbar}e{\bm A}_{\rm s}^{z}\cdot {\rm{d}}{\bm r}}$, where
\begin{equation}
{\bm A}_{\rm s}^{z}=\frac{\hbar}{2e}(1-\cos\theta)\nabla\phi, \label{Asdef}
\end{equation}
and the factor of $\frac{1}{2}$ is due to the magnitude of the electron spin.
The field ${\bm A}_{\rm s}^{z}$ is an effective vector potential or an effective gauge field.
When the electron's path is finite, the phase becomes
$
\varphi=\frac{e}{\hbar} \int_C {\rm{d}}{\bm r}\cdot \bm{A}_{\rm s}^z
$.
The existence of the phase means that there is an effective magnetic field ${\bm B}_{\rm s}$, as seen by rewriting the integral over a closed path using the Stokes theorem as
\varphi=\frac{e}{\hbar} \int_S {\rm{d}}{\bm S}\cdot{\bm B}_{\rm s}
$,
where
${\bm B}_{\rm s}\equiv \nabla\times\bm{A}_{\rm s}^z$.
The time derivative of the phase is equivalent to a voltage, and thus, we have an effective electric field defined by
\dot{\varphi}=-\frac{e}{\hbar} \int_C{\rm{d}}{\bm r}\cdot{\bm E}_{\rm s}
$,
where ${\bm E}_{\rm s}\equiv-\dot{\bm{A}}_{\rm s}^z$.
These two fields satisfy Faraday's law,
$
\nabla\times{\bm E}_{\rm s}+\dot{{\bm B}_{\rm s}}=0
$.
We therefore have effective electromagnetic fields that couple to the conduction electron spin as a result of the $sd$ exchange interaction.
We call the field a spin electromagnetic field \cite{Tatara_smf13}.
Using the explicit form of the effective gauge field, Eq. (\ref{Asdef}), we see that the emergent spin electromagnetic fields are
\begin{eqnarray}
{{\bm E}_{\rm s}}_{,i}&=&-\frac{\hbar}{2e} \bm{n} \cdot (\dot{\bm{n}} \times \nabla_i \bm{n}), \nonumber\\
{{\bm B}_{\rm s}}_{,i}&=& \frac{\hbar}{4e}\sum_{jk}\epsilon_{ijk} \bm{n} \cdot (\nabla_j \bm{n} \times \nabla_k \bm{n}) .\label{EBtopological}
\end{eqnarray}
The magnetic component ${{\bm B}_{\rm s}}$ is the spin Berry's curvature \cite{Xiao10} or scalar chirality.
The electric component ${{\bm E}_{\rm s}}$, called the spin motive force, is a chirality in the space-time, which arises when the localized spin structure $\bm{n}$ is time-dependent.
The expression Eq. (\ref{EBtopological}) was derived by Volovik in 1987 \cite{Volovik87}.
Originally, the emergence of the effective electric field ${\bm E}_{\rm s}$ from moving magnetic structures was found in 1986 by Berger, where a voltage generated by canting a moving domain wall was calculated \cite{Berger86}.
Stern discussed the motive force in the context of the spin Berry's phase and the Aharonov-Bohm effect in a ring, and showed similarity to Faraday's law \cite{Stern92}.
The spin motive force was rederived in Ref. 13 in the case of the domain wall motion, and discussed in the context of topological pumping in Ref. 14.
Those works consider only the adiabatic limit, i.e., in the case of a strong $sd$ exchange interaction and in the absence of spin-dependent scattering.
The idea of the spin motive force has recently been extended to include the spin-orbit interaction \cite{Duine08,Lucassen11,Shibata09,Shibata11,Kim12,Tatara_smf13,Nakabayashi14}, and it was shown that the spin-orbit interaction modifies the spin electric field.
It was also shown that the spin electromagnetic field arises even in the limit of a weak $sd$ interaction \cite{Takeuchi12,Tatara12}.
The case of the Rashba spin-orbit interaction has been studied in detail recently. It was shown that the spin electric field in this case emerges even from a uniform precession of magnetization \cite{Tatara_smf13,Nakabayashi14}.
This fact suggests that the Rashba interaction at interfaces would be useful in controlling the spin-charge conversion.
The Rashba-induced spin electric field induces a voltage in the same direction as in the inverse spin Hall and inverse Edelstein effects \cite{Saitoh06,Sanchez13} driven by the spin pumping effect \cite{Tserkovnyak02}.
It was also pointed out that the spin electromagnetic fields in the presence of spin relaxation satisfy Maxwell's equations with spin magnetic monopoles that are driven dynamically \cite{Takeuchi12}.
The coupling between the spin magnetic field and the helicity of light was theoretically studied in the context of the topological inverse Faraday effect, which is a nonlinear effect with respect to the incident electric field \cite{Taguchi12}.
Experimentally, the spin magnetic field (the spin Berry's curvature) has been observed using the anomalous Hall effect in frustrated ferromagnets \cite{Lee09, Nagaosa10}.
The spin electric field has been measured in the motion of various ferromagnetic structures such as domain walls \cite{Yang09}, magnetic vortices \cite{Tanabe12}, and skyrmions \cite{Schulz12}.
\section{Derivation of Effective Hamiltonian}
The effective Hamiltonian is calculated in the imaginary-time (denoted by $\tau$) path integral formalism \cite{Sakita85}. In this section, we set $\hbar = 1$.
The system we consider is a ferromagnetic metal, where conduction electrons, represented by two-component annihilation and creation fields, ${c} (\bm{r},\tau)$ and $\bar{c} (\bm{r},\tau)$, interact with localized spins, described by the vector field $\bm{n}(\bm{r},\tau)$, via the $sd$ exchange interaction.
The Hamiltonian thus reads
\begin{align}
\mathcal{H} &= \mathcal{H}_{\text{0}} +\mathcal{H}_{sd} + \mathcal{H}_{\text{em}} , \\
\mathcal{H}_{\text{0}} &= \int {\rm{d}}^{3} r \left(\frac{1} {2m} |\bm{\nabla}{c} (\bm{r},\tau)|^2 - \mu \bar{c} (\bm{r},\tau) c (\bm{r},\tau)\right) ,\nonumber\\
\mathcal{H}_{sd} &= -\Delta_{sd} \int {\rm{d}}^{3} r \bm{n} (\bm{r},\tau) \cdot \left(\bar{c} (\bm{r},\tau) \bm{\sigma} c (\bm{r},\tau)\right),
\end{align}
where $\mu$ is the chemical potential, $m$ is the electron mass, and $\bm{\sigma}$ is the vector of Pauli matrices.
The term $\mathcal{H}_{\text{em}}$ represents the interaction between the conduction electron and the applied electromagnetic field, described by a vector potential $\bm{A}$, which reads
\begin{align}
\mathcal{H}_{\text{em}} &= - \int {\rm{d}}^{3} r \bm{A}(\bm{r},\tau) \cdot \left(\frac{ie}{2m} \bar{c} (\bm{r},\tau) \overleftrightarrow{\nabla} c (\bm{r},\tau) - \frac{e^2}{2m} \bm{A}(\bm{r},\tau) \bar{c} (\bm{r},\tau) c (\bm{r},\tau)\right),
\end{align}
where $\bar{c} \overleftrightarrow{\nabla} c $ $\equiv$ $\bar{c}\left(\bm{\nabla} c \right)-\left(\bm{\nabla} \bar{c} \right)c $ and $-e$ is the electron charge ($e>0$). The system we consider is a film thinner than the penetration depth of the electromagnetic field.
The Lagrangian of the system is
\begin{align}
\mathcal{L}&= \int {\rm{d}}^{3} r \bar{c} (\bm{r},\tau) \partial_\tau c(\bm{r},\tau)+\mathcal{H},
\end{align}
and the effective Hamiltonian describing the localized spin and the gauge field is obtained by carrying out a path integral over the conduction electrons as
$\mathcal{H}_{\rm eff}(\theta,\phi,\bm{A})\equiv -\ln Z$, where
\begin{align}
Z & = \int D\bar{c}(\bm{r},\tau)Dc(\bm{r},\tau)e^{-\int_0^\beta d\tau \mathcal{L}},
\end{align}
is the partition function and $D$ denotes the path integral.
We are interested in the case where the $sd$ exchange interaction is large and thus the conduction electron spin is aligned parallel to the localized spin direction $\bm{n}$, i.e., the adiabatic limit.
To describe this limit, the use of the spin gauge field, which characterizes the deviation from the adiabatic limit, is convenient \cite{TKS_PR08}.
The spin gauge field is introduced by diagonalizing the $sd$ interaction using a unitary transformation, $c(\bm{r},\tau)=U(\bm{r},\tau)a(\bm{r},\tau)$, where $U(\bm{r},\tau)$ is a $2\times2$ unitary matrix and $a$ is a new electron field operator.
A convenient choice of $U(\bm{r},\tau)$ is
$U(\bm{r},\tau)=\bm{m}(\bm{r},\tau) \cdot \bm{\sigma}$ with $\bm{m}(\bm{r},\tau)=\left(\sin\frac{\theta}{2} \cos\phi , \sin\frac{\theta}{2} \sin\phi , \cos\frac{\theta}{2} \right)$, where $\theta$ and $\phi$ are the polar angles of $\bm{n}$.
It is easy to confirm that $U^{\dagger}(\bm{n}\;\cdot\; \bm{\sigma})U$ = $\sigma_{z}$ is satisfied.
Because of this local unitary transformation, derivatives of the electron field become covariant derivatives
$\partial_\mu c=U(\partial_\mu+ieA_{{\rm s},\mu})a$, where
$A_{{\rm s},\mu}\equiv -\frac{i}{e}U^{-1}\partial_\mu U$ is the gauge field.
Since $U$ is a $2\times2$ matrix, the gauge field $A_{{\rm s},\mu}$ is written using Pauli matrices as $A_{{\rm s},\mu}=\sum_{\alpha}A_{{\rm s},\mu}^\alpha \sigma_{\alpha}$ ($\mu=x,y,z,\tau$ is a suffix for space and time and $\alpha=x,y,z$ is for spin).
It is thus an SU(2) gauge field, which we call the spin gauge field.
The Lagrangian in the rotated space is thus given by
\begin{align}
\mathcal{L}&\equiv \mathcal{L}_{\text 0}+\mathcal{L}_{A},\\
\mathcal{L}_{\text 0}&\equiv \int {\rm{d}}^{3} r \bar{a} \left(\partial_\tau - \frac{1} {2m} \bm{\nabla}^{2} - \mu - \Delta_{sd}\sigma_{z}\right) a, \\
\mathcal{L}_{A}&\equiv \int {\rm{d}}^{3} r \left[ i e\bar{a} A_{{\rm s},\tau}a
+\sum_{i} \left(\sum_{\alpha} A_{{\rm s},i}^\alpha j_{{\rm s},i}^\alpha + A_{i}j_{i}\right) + \frac{e^2}{m}\sum_{i,\alpha}A_{i}A_{{\rm s},i}^\alpha \bar{a} \sigma^{\alpha}a\right.\nonumber\\
&\left.+ \frac{e^2}{2m}\sum_i\left(\sum_{\alpha} (A_{{\rm s},i}^\alpha) ^{2}+ (A_i)^{2}\right)\bar{a} a
\right],
\end{align}
where
$j_{{\rm s},i}^\alpha\equiv -ie\frac{1} {2m}\bar{a}\left(\overrightarrow{\nabla}_{i} - \overleftarrow{\nabla}_{i}\right)\sigma^\alpha a$
and $j_{i}\equiv -ie\frac{1} {2m}\bar{a}\left(\overrightarrow{\nabla}_{i} - \overleftarrow{\nabla}_{i}\right)a$ are the spin current and charge current, respectively.
The electron field $a$ is strongly spin-polarized owing to the $sd$ exchange interaction (the last term of $\mathcal{L}_{\text 0}$).
We carry out the path integral with respect to the electron field and derive the effective Hamiltonian for the two gauge fields $A_{{\rm s},i}^\alpha$ and $A_{i}$ describing the spin and charge gauge fields, respectively.
The spin gauge field is written using the localized spin direction, $\theta$ and $\phi$, and thus the effective Hamiltonian can be regarded as that describing the interaction of the localized spin and the charge electromagnetic field.
Up to the second order with respect to the gauge fields, the effective Hamiltonian reads
\begin{align}
\mathcal{H}_{\rm eff} &= -\int_{0}^{\beta} {\rm{d}}\tau \int {\rm{d}}^3r \left[2ieA_{{\rm s},\tau}^{z}s_{\rm e} (\bm{r},\tau)+ \frac{2e^2}{m}\sum_{i}A_{i}A_{{\rm s},i}^{z}s_{\rm e}(\bm{r},\tau)\right.\nonumber\\
&\left.+ \frac{e^2}{2m}\sum_i\left(\sum_{\alpha} (A_{{\rm s},i}^\alpha) ^{2}+ (A_i)^{2}\right)n(\bm{r},\tau) \right]
+ \frac{1}{2}\int_{0}^{\beta} {\rm{d}}\tau\int_{0}^{\beta} {\rm{d}}\tau'\int {\rm{d}}^3r\int {\rm{d}}^3r'\sum_{ij} \nonumber\\
&\times \left[ \sum_{\alpha\beta} A_{{\rm s},i}^\alpha A_{{\rm s},j}^\beta \chi_{ij}^{\alpha\beta}(\bm{r},\bm{r}',\tau,\tau')
+ 2\sum_{\alpha} A_{{\rm s},i}^\alpha A_{j}\chi_{ij}^{\alpha}(\bm{r},\bm{r}',\tau,\tau')
+ A_{i} A_{j}\chi_{ij}(\bm{r},\bm{r}',\tau,\tau') \right],
\end{align}
where $s_{\rm e} (\bm{r},\tau)\equiv \frac{1}{2}\langle \bar{a}(\bm{r},\tau)\sigma^{z}a(\bm{r},\tau)\rangle$, $n(\bm{r},\tau)\equiv \langle \bar{a}(\bm{r},\tau)a(\bm{r},\tau)\rangle$
, $\langle\ \rangle$ denotes the thermal average, and
\begin{align}
\chi_{ij}^{\alpha\beta}(\bm{r},\bm{r}',\tau,\tau')
& \equiv \langle j_{{\rm s},i}^\alpha (\bm{r},\tau) j_{{\rm s},j}^\beta(\bm{r}',\tau') \rangle \nonumber\\
\chi_{ij}^{\alpha}(\bm{r},\bm{r}',\tau,\tau')
& \equiv \langle j_{{\rm s},i}^\alpha (\bm{r},\tau) j_{j}(\bm{r}',\tau') \rangle \nonumber\\
\chi_{ij}(\bm{r},\bm{r}',\tau,\tau')
& \equiv \langle j_{i} (\bm{r},\tau) j_{j}(\bm{r}',\tau') \rangle ,
\end{align}
are the current-current correlation functions.
The spin density $s_{\rm e}$ and the electron density $n$ are calculated as
$
s_{\rm e}=\frac{1}{2V}\sum_{\bm{k}}\sum_{\sigma=\pm}\sigma f(\epsilon_{\bm{k} \sigma})$ and
$n=\frac{1}{V}\sum_{\bm{k}}\sum_{\sigma=\pm}f(\epsilon_{\bm{k} \sigma})
$, respectively,
where $f(\epsilon_{\bm{k} \sigma})$=$(e^{\beta \epsilon_{\bm{k} \sigma}}+1)^{-1}$ is the Fermi-Dirac distribution function, $\epsilon_{\bm{k}\sigma}=\frac{k^2}{2m}-\mu-\sigma\Delta_{sd}$, and $\sigma$ = $\pm$ is the spin index.
The Fourier components of the correlation functions are
\begin{align}
\chi_{ij}^{\alpha\beta}(\bm{q},i\Omega_\ell)
& = -\frac{e^2}{m^2\beta V}\sum_{n,\bm{k}} k_i k_j {\rm tr}\left[\sigma_\alpha G_{\bm{k}-\frac{\bm{q}}{2},n} \sigma_\beta G_{\bm{k}+\frac{\bm{q}}{2},n+\ell}\right]
\nonumber\\
\chi_{ij}^{\alpha}(\bm{q},i\Omega_\ell)
& = -\frac{e^2}{m^2\beta V}\sum_{n,\bm{k}} k_i k_j {\rm tr}\left[\sigma_\alpha G_{\bm{k}-\frac{\bm{q}}{2},n} G_{\bm{k}+\frac{\bm{q}}{2},n+\ell} \right]
\nonumber\\
\chi_{ij}(\bm{q},i\Omega_\ell)
& = -\frac{e^2}{m^2\beta V}\sum_{n,\bm{k}} k_i k_j {\rm tr}\left[G_{\bm{k}-\frac{\bm{q}}{2},n} G_{\bm{k}+\frac{\bm{q}}{2},n+\ell} \right] .
\end{align}
Here $G_{\bm{k},n}$ is defined as $G_{\bm{k},n}\equiv\left[i\omega_n-\epsilon_{\bm{k}}+ \frac{i}{2\tau_e} {\rm sgn}(n)\right]^{-1}$, where ${\rm sgn}(n)$ = $1$ and $-1$ for $n > 0$ and $n < 0$, respectively,
$\epsilon_{\bm{k}}=\frac{k^2}{2m}-\mu-\Delta_{sd}\sigma_z$ is the electron energy in the matrix representation, $\tau_e$ is the electron elastic scattering lifetime,
and ${\rm tr}$ denotes the trace over spin space.
The Fermionic thermal frequency is represented by $\omega_n$ $\equiv$ $\frac{(2n+1)\pi}{\beta}$, and $\Omega_\ell$ $\equiv$ $\frac{2\pi \ell}{\beta}$ is a bosonic thermal frequency.
The correlation functions are calculated by rewriting the summation over the thermal frequency using the contour integral ($z$ $\equiv$ $i\omega_n$) as
\begin{align}
\chi_{ij}^{zz}(\bm{q},i\Omega_\ell)
& = \frac{e^2}{m^2V} \sum_{\bm{k}} k_i k_j \sum_{\sigma=\pm}\int_C \frac{{\rm{d}}z}{2\pi i} f(z) g_{\bm{k}-\frac{\bm{q}}{2},\sigma}(z) g_{\bm{k}+\frac{\bm{q}}{2},\sigma}(z+i\Omega_\ell)
\nonumber\\
\chi_{ij}^{+-}(\bm{q},i\Omega_\ell)
& = \frac{e^2}{m^2V} \sum_{\bm{k}} k_i k_j \sum_{\sigma=\pm}\int_C \frac{{\rm{d}}z}{2\pi i} f(z) g_{\bm{k}-\frac{\bm{q}}{2},\sigma}(z) g_{\bm{k}+\frac{\bm{q}}{2},-\sigma}(z+i\Omega_\ell)
\nonumber\\
\chi_{ij}^{z}(\bm{q},i\Omega_\ell)
& = \frac{e^2}{m^2V} \sum_{\bm{k}} k_i k_j\sum_{\sigma=\pm}\int_C \frac{{\rm{d}}z}{2\pi i} \sigma f(z) g_{\bm{k}-\frac{\bm{q}}{2},\sigma}(z) g_{\bm{k}+\frac{\bm{q}}{2},\sigma}(z+i\Omega_\ell)
\nonumber\\
\chi_{ij}(\bm{q},i\Omega_\ell)
& = \frac{e^2}{m^2V}\sum_{\bm{k}} k_i k_j\sum_{\sigma=\pm}\int_C \frac{{\rm{d}}z}{2\pi i} f(z) g_{\bm{k}-\frac{\bm{q}}{2},\sigma}(z) g_{\bm{k}+\frac{\bm{q}}{2},\sigma}(z+i\Omega_\ell) ,
\end{align}
where $C$ is an anticlockwise contour surrounding the imaginary axis \cite{AGD75,Altland06} and $g_{\bm{k},\sigma}(z)$ is defined as $g_{\bm{k},\sigma}(z)\equiv \left[z-\epsilon_{\bm{k}\sigma}+\frac{i}{2\tau_e}{\rm sgn}({\rm Im}z)\right]^{-1}$.
We expand the correlation functions with respect to the external wave vector $\bm{q}$ and frequency $\Omega$ after the analytical continuation to $\Omega +i0$ $\equiv$ $i\Omega_\ell$ \cite{Altland06}.
The result up to the second order in $\bm{q}$ and $\Omega$ is
\begin{align}
\chi_{ij}^{\alpha\beta}(\bm{q},\Omega )
& = \frac{e^2}{m} \left\{ (\delta_{\alpha\beta} - \delta_{\alpha z}\delta_{\beta z})\delta_{ij} b +\delta_{\alpha z}\delta_{\beta z}
\left[\delta_{ij} {n}\left(1+ i\Omega \tau_{\rm e} -(\Omega \tau_{\rm e})^2\right)
+ c (q_iq_j - q^2\delta_{ij})\right]\right\}
\nonumber\\
\chi_{ij}^{z}(\bm{q},\Omega )
&= \frac{e^2}{m} \left[\delta_{ij} 2s_{\rm e}\left(1+ i\Omega \tau_{\rm e} -(\Omega \tau_{\rm e})^2 \right) +
d (q_iq_j - q^2\delta_{ij})\right]
\nonumber\\
\chi_{ij}(\bm{q},\Omega )
& = \frac{e^2}{m} \left[\delta_{ij}n\left(1+ i\Omega \tau_{\rm e} -(\Omega \tau_{\rm e})^2\right) +
c (q_iq_j - q^2\delta_{ij})\right] ,
\end{align}
where $b\equiv \frac{1}{3mV\Delta_{sd}}\sum_{\bm{k}}\sum_{\sigma=\pm}\sigma\bm{k}^{2}f(\epsilon_{\bm{k}\sigma}) $, $c \equiv \frac{1}{12m}\sum_{\sigma=\pm}\nu_\sigma$,
$d \equiv \frac{1}{12m}\sum_{\sigma=\pm}\sigma\nu_\sigma$,
$n=\sum_{\sigma=\pm}\frac{k_{{\rm F}\sigma}^2\nu_\sigma}{3m}$,
$s_{\rm e}= \frac{1}{2}\sum_{\sigma=\pm}\sigma\frac{k_{{\rm F}\sigma}^2\nu_\sigma}{3m}$,
and $k_{{\rm F}\sigma}\equiv \sqrt{k_{\rm F}^2+2m\sigma\Delta_{sd}}$ and $\nu_\sigma\equiv\frac{m^{\frac{3}{2}}}{\sqrt{2}\pi^2}\sqrt{\epsilon_{\rm F}+\sigma\Delta_{sd}}$
are the spin-dependent Fermi wave number and the density of states per unit volume, respectively. Vertex corrections are irrelevant, since they are proportional to $\bm{\nabla} \cdot \bm{E}$, which vanishes in metals. In this work, we do not consider surface effects played by induced surface charges such as the surface plasmon effect.
We thus obtain the effective Hamiltonian up to the order of ${\bm{q}}^2$ and $\Omega^2$ as
\begin{align}
\mathcal{H}_{\text{eff}} &= - \Bigg\{
2ieA_{{\rm s},\tau}^z s_{\rm e} + \frac{1}{2m}\left(n-b\right)\sum_{i,\bm{q},\Omega} A_{{\rm s},i}^+(-\bm{q},-\Omega)A_{{\rm s},i}^-(\bm{q},\Omega) \nonumber\\
&-\frac{e^{2}}{m} 2s_{\rm e}\tau_{\rm e} \sum_{i,\bm{q},\Omega} i\Omega A_{{\rm s},i}^z(-\bm{q},-\Omega)A_{i}(\bm{q},\Omega) \nonumber\\
&+ \frac{e^2}{2m}{\tau_{e}}^2\sum_{i,\bm{q},\Omega}\Omega^{2}\left[ n( A_{{\rm s},i}^z(-\bm{q},-\Omega)A_{{\rm s},i}^z(\bm{q},\Omega)+A_{i}(-\bm{q},-\Omega)A_{i}(\bm{q},\Omega) )
+ 4s_{\rm e} A_{{\rm s},i}^z(-\bm{q},-\Omega)A_{i}(\bm{q},\Omega) \right] \nonumber\\
&- \frac{e^2}{2m}\sum_{ij,\bm{q},\Omega}(q_iq_j - q^2\delta_{ij}) \nonumber\\
&
\times \left[ c ( A_{{\rm s},i}^z(-\bm{q},-\Omega)A_{{\rm s},j}^z(\bm{q},\Omega)+A_{i}(-\bm{q},-\Omega)A_{j}(\bm{q},\Omega) )
+ d A_{{\rm s},i}^z(-\bm{q},-\Omega)A_{j}(\bm{q},\Omega) \right] \Bigg\},
\end{align}
where $A_{{\rm s},i}^{\pm}(\bm{q},\Omega)$ $\equiv$ $A_{{\rm s},i}^x(\bm{q},\Omega) \pm iA_{{\rm s},i}^y(\bm{q},\Omega)$.
The terms quadratic in the charge gauge field describe the electric permittivity and magnetic permeability of the media.
We used the fact that $\int dt A_{i}\dot{A}_{i}=0$ to drop the term proportional to $\Omega A_{i}(-\Omega)A_{i}(\Omega)$.
The terms quadratic in the spin gauge field contribute to the renormalization of the exchange interaction and dissipation as shown in Ref. 35.
In fact, the contribution of the order of $\Omega^{0}$ reduces to
\begin{align}
\frac{1}{m}\left(n-b\right)\sum_{i} A_{{\rm s},i}^{+}A_{{\rm s},i}^{-}=J_{\text{eff}}(\bm{\nabla} \bm{n})^2,
\end{align}
where $J_{\text{eff}}\equiv \frac{1}{4m}\left(n-b\right)$. We treat this renormalization as a term in the original exchange interaction and do not consider it further.
The term quadratic in the spin gauge field and linear in $\Omega$ represents a dissipation \cite{TF94_JPSJ}, but we neglect this effect since we are not interested in the spin dynamics where dissipation plays an important role.
Instead, we are interested in the coupling between the two gauge fields.
The effective Hamiltonian describing the coupling reads
\begin{align}
\mathcal{H}_{\rm int} &= \frac{e^2}{m} \int {\rm{d}}^3r \left(
2s_{\rm e} \tau_{\rm e} \bm{E}\cdot\bm{A}_{{\rm s}}^z+ 2s_{\rm e} \tau_{\rm e}^2 \bm{E}\cdot\bm{E}_{{\rm s}}
+ \frac{d}{2} \bm{B}\cdot\bm{B}_{\rm s} \right),
\label{Hcoupling}
\end{align}
where $\bm{E}\equiv -\dot{\bm{A}}$ and $\bm{B}\equiv \nabla\times \bm{A}$ are the electric and magnetic fields, respectively, and $\bm{E}_{\rm s}\equiv -\dot{\bm{A}}_{\rm s}^z$ and $\bm{B}_{\rm s}\equiv \nabla\times\bm{A}_{\rm s}^z$ are the effective spin electric and magnetic fields, respectively.
\section{Discussion}
Let us discuss the effect of the coupling terms, Eq. (\ref{Hcoupling}).
The first term indicates that $\bm{A}_{\rm s}^z$ is induced when an electric field is applied.
In fact, this term is the term describing the spin-transfer torque,
as seen by denoting
$2s_{\rm e} \frac{e^2}{m} \tau_{\rm e} \bm{E}=P\bm{j}$, where $P\equiv \frac{n_\uparrow-n_\downarrow}{n}$, $\bm{j}=\sigma_{\rm B}\bm{E}$, and $\sigma_{\rm B}\equiv \frac{e^2}{m}n \tau_{\rm e}$ is the Boltzmann conductivity.
As pointed out in Ref. 6, the effective Hamiltonian method that we used thus easily reproduces the spin-transfer effect, which is usually discussed in the context of the conservation law of angular momentum.
Although the spin gauge field $\bm{A}_{\rm s}^z$ is related to the spin electric field as
$\bm{E}_{\rm s}=-\dot{\bm{A}}_{\rm s}^z$, the generation of $\bm{A}_{\rm s}^z$ does not always imply the generation of a spin electric field.
In fact, a direct consequence of the spin-transfer torque is to drive magnetization textures \cite{TKS_PR08}.
Only when the induced magnetization dynamics creates a non-coplanarity, the spin electric field is induced.
The emergence of the spin electric field thus depends in an essential way on the dynamics of the magnetization.
\subsection{Spin electric field induced by domain wall motion}
Let us consider as an example a domain wall.
A domain wall favors a non-coplanar motion since its center of mass coordinate $X$ and the angle of the wall plane $\phi$ are canonical conjugates to each other \cite{TKS_PR08}.
When the spin-transfer torque due to an electric field is applied, the wall plane starts to tilt and its angle drives the motion of the wall \cite{TK04}.
The spin electric field generated by this wall motion is calculated as follows.
We consider a case of uniaxial anisotropy and neglect the nonadiabaticity which is represented by $\beta$ in Ref. 5.
A planar domain wall with the magnetization changing along the $x$-direction at position $x=X(t)$ is described by $\cos\theta=\tanh\frac{x-X}{\lambda}$ and
$\sin\theta=\frac{1}{\cosh\frac{x-X}{\lambda}}$ with a constant $\phi$.
The equations of motion for $X$ and $\phi$ are \cite{TTKSNF06}
\begin{align}
\dot{X}-\alpha\lambda \dot{\phi}
&=
\frac{a_0^3}{2eS}Pj \nonumber \\
\dot{\phi}+\alpha\frac{\dot{X}}{\lambda}
&= 0, \label{DWeq}
\end{align}
where $\lambda$ is the width of the wall, $\alpha$ is the Gilbert damping parameter, $a_0$ is the lattice constant,
and $S$ is the magnitude of the localized spin.
The solution for Eq. (\ref{DWeq}) is
$\dot{X}=\frac{1}{1+\alpha^2}\frac{a_0^3}{2eS}Pj$ and
$\dot{\phi}=-\frac{\alpha}{1+\alpha^2}\frac{a_0^3}{2eS\lambda}Pj$.
The spin electric field, Eq. (\ref{EBtopological}), for a moving domain wall is
calculated using
$\nabla_x{\bm{n}}=-\frac{\sin\theta}{\lambda}\bm{e}_\theta$ and
$\dot{\bm{n}}=\sin\theta\left(\dot{\phi}\bm{e}_{\phi}+\frac{\dot{X}}{\lambda}\bm{e}_\theta\right)$,
where $\bm{e}_\phi=(-\sin\phi,\cos\phi,0)$ and $\bm{e}_\theta=(\cos\theta\cos\phi,\cos\theta\sin\phi,-\sin\theta)$.
The spin electric field then arises along the $x$-direction and the magnitude is
\begin{align}
E_{\rm{s}}&=
\frac{\hbar}{2e} \frac{1}{2\lambda} \dot{\phi}= \frac{\hbar}{2e^2}\frac{\alpha}{1+\alpha^2}\frac{a_0^3P}{4S\lambda^2}\sigma_{\rm B}E.
\label{EsDW}
\end{align}
Let us estimate the magnitude choosing $\alpha \sim 10^{-2}$, $P \sim 0.8$ \cite{Ueda12}, $\sigma_{\rm B} \sim 10^{8}$ $\Omega^{-1}\text{m}^{-1}$, and $S = 1$. For a field of $E\sim 10^{4}$ $\text{V}/\text{m}$ corresponding to $j=10^{12}$ A/m$^2$, we have $|\dot{X}|$ $\sim$ $4$ $\text{m}/\text{s}$ and $|\dot{\phi}|\sim4\times10^{6}$ $\text{s}^{-1}$.
For $\lambda$ $\sim$ $10^{-8}$ $\text{m}$ \cite{Fukami08}, we thus obtain $|E_{\rm s}|$ $\sim$ $0.1$ $\text{V}/\text{m}$. In experiments, what is measured is the voltage due to $E_{\rm s}$. Since $E_{\rm s}$ is localized at the wall, the voltage generated by a single domain wall is $E_{\rm s}\lambda \sim 1 \;\text{nV}$. This value is not large, but is detectable. For instance, in the case of a domain wall driven by an external magnetic field, a voltage of $400\;\text{nV}$ was observed for a wall speed of $150\;\text{m}/\text{s}$ \cite{Yang09}.
The conversion efficiency from the electric field to the spin electric field is given by
\begin{equation}
\mu\equiv \frac{E_{\rm s}}{E}=\frac{\hbar}{2e^2}\frac{\alpha}{1+\alpha^2}\frac{a_0^3P}{4S\lambda^2}\sigma_{\rm B}
\sim \frac{P\alpha}{4(k_{\rm F}\lambda)^2} \left(\frac{\epsilon_{\rm F}\tau_{\rm e}}{\hbar}\right),
\end{equation}
where we approximated $s_{\rm e}a_0^3 \sim P/2$. A typical value of $\mu$ is $\mu \sim 10^{-4}$ for $\alpha \sim 10^{-2}$, $k_{\rm F}\lambda \sim 100$, and $(\epsilon_{\rm F}\tau_{\rm e})/\hbar \sim 100$. The conversion efficiency is larger in a thin wall, such as perpendicular anisotropy magnets and weak ferromagnets.
In contrast to the spin-transfer term, the second term in Eq. (\ref{Hcoupling}) describes a direct coupling between the electric field and the spin electric field.
The strength of the induced spin electric field is determined by solving spin dynamics as we did above, because $E_{\rm s}$ is determined using the equation of motion for the spin (the Laudau-Lifshitz equation) and not for the Lagrangian like $(\bm{E}_{\rm s})^2-(\bm{B}_{\rm s})^2$ as in the charge electromagnetism.
The effect is generally small, since the $\bm{E}\cdot\bm{E}_{\rm s}$ coupling term is smaller than the spin-transfer term by a factor of $\Omega\tau_{\rm e}$, which is small in the GHz frequency where magnetization structures can respond [$\tau_{\rm e}\gtrsim 10^{-13}$ $\text{s}$ for a metal with $(\epsilon_{\rm F}\tau_{\rm e})/\hbar = 100$].
The term may be important in a very clean metal in the $\text{THz}$ range.
\subsection{Magnetic coupling}
The third term of Eq. (\ref{Hcoupling}) is due to the external magnetic field.
It indicates that the spin magnetic field is induced when a uniform external magnetic field is applied.
This effect indicates a novel intrinsic mechanism involving frustration, since a finite $\bm{B}_{\rm s}$ denotes a non-coplanar spin structure, while a uniform magnetic field favors the uniform magnetization along the magnetic field.
When the magnetization structure has a finite non-coplanarity at the scale of $\lambda$, the magnitude of the induced spin magnetic field is ${B}_{\rm s}\sim \frac{\hbar}{e \lambda^2}$.
The energy gain per site due to the non-coplanarity is then
$\left(\frac{e\hbar}{m}\right)^{2}\frac{{B}_{\rm s}}{\epsilon_{\rm F}}B\simeq \frac{e\hbar}{m}B \frac{2}{(k_{\rm F} \lambda)^2}$, assuming that $d \sim O(\frac{1}{12m\epsilon_{\rm F}})$. [Note that $d=\frac{1}{12m}(\nu_{+}-\nu_{-})$ can be positive or negative.]
In ferromagnetic systems, a non-coplanar structure costs the exchange energy of
$\frac{J}{2}(\nabla \bm{n})^2\simeq \frac{J}{a_0^2} \frac{1}{(k_{\rm F} \lambda)^2}$, where $J$ is the exchange energy of the localized spin,
in the absence of other frustration.
The total energy cost due to the generation of $\bm{B}_{\rm s}$ is thus
\begin{align}
\Delta E\simeq \frac{1}{(k_{\rm F} \lambda)^2} \left( \frac{J}{a_0^2} - \frac{e\hbar}{m}B \right).
\end{align}
In common $3d$ ferromagnetic metals, the exchange energy $J/a_0^2$ is on the order of 1 $\text{eV}$, while the applied magnetic field of 1 $\text{T}$ corresponds to the energy of $\frac{e\hbar}{m}B=10^{-4}$ $\text{eV}$.
The spin magnetic field $\bm{B}_{\rm s}$ is therefore not induced by simply applying a uniform magnetic field.
The situation may be different in very weak ferromagnets like in molecular conducting ferromagnets.
For a system with a ferromagnetic critical temperature of 5 $\text{K}$ \cite{Coronado04}, a magnetic field of 5 $\text{T}$ may be sufficient to induce a finite $\bm{B}_{\rm s}$ if the Fermi energy is on the order of 1 $\text{eV}$. Our present study assuming a strong $sd$ exchange interaction does not directly apply to weak ferromagnets, and different approaches like in Ref. 21 are needed to study the interplay between $\bm{B}_{\rm s}$ and $\bm{B}$.
Molecular conducting ferromagnets would be unique systems in the context of an emergent spin electromagnetic field.
\subsection{Spin electric field induced by spin wave}
The spin-transfer term $\mathcal{H}_{\rm st}$ induces a spin wave excitation when the electric field is applied if the induced current exceeds a threshold value \cite{STK05}.
If the spin wave is monochromatic, a spin electric field is not induced, since a monochromatic plane wave, such as $n_x\pm i n_y=\varphi e^{\pm i(kx-\Omega t)}$, does not have a non-coplanarity because $\dot{\bm{n}}$ and $\nabla_x\bm{n}$ are parallel to each other.
It is possible to excite a spin electric field if we use two spin waves having different wave vectors or frequencies.
Let us consider the case described by
\begin{align}
s_{\pm}=\frac{1}{2}\sum_{j=1,2} \varphi_j e^{\pm i(k_j x-\Omega_j t)},
\end{align}
where $s_{\pm}\equiv \frac{1}{2}(s_x\pm i s_y)$ is a small spin fluctuation, $\varphi_j$ are the amplitudes of spin waves, and $k_j$ and $\Omega_j$ ($j=1,2$) are the wave vector and angular frequency of the two spin-wave excitations, respectively.
The spin electric field associated with the spin waves then reads
\begin{align}
E_{{\rm s},x}=-\frac{\hbar}{2e}\varphi_1 \varphi_2 (\Omega_1k_2-\Omega_2k_1)\sin[(k_1-k_2)x-(\Omega_1-\Omega_2)t].
\end{align}
Namely, a spin electric field having a wave vector $(k_1-k_2)$ and a frequency $(\Omega_1-\Omega_2)$ is induced by the two spin-wave excitations. Two spin waves having the same frequency of about 7 $\text{GHz}$ have been generated recently using a magnetic field which is induced by applying a current through antennas \cite{Sato13}.
The method would be applicable to create a spin electric field.
For spin waves with $k=0.4$ $\mu$m$^{-1}$ and an angular frequency of $2\pi \times7$ $\text{GHz}$ \cite{Sato13}, $k\Omega\sim 1.7\times 10^{15}$ 1/(ms), and the spin electric field is expected to be on the order of $\frac{\hbar}{e}k\Omega\varphi^2\simeq 1.7 \times \varphi^2$ V/m for the amplitude of spin waves $\varphi$. The spin electric field is expected to be induced generally by the spin wave excitation owing to a nonlinear effect ($E_{{\rm s},x}$ nonlinearly depend on the spin wave amplitude), if the wave is non-monochromatic.
The present method of generating a spin electric field applies to a uniform ferromagnet and would have better possibilities of applications than the conventional methods using magnetic structures such as domain walls.
\section{Summary}
We have derived the effective Hamiltonian describing the coupling between the emergent spin electromagnetic field and the charge electromagnetic field. The dominant term turns out to be the one corresponding to the spin-transfer torque. The coupling between the magnetic components suggests an interesting possibility of inducing frustration by applying a uniform external magnetic field on weak ferromagnets.
We have proposed a generation mechanism of a spin electric field using a nonlinear effect of non-monochromatic spin-wave excitations.
This mechanism is applicable to the case of a uniform magnetization, and it would have a great advantage in applications over common setups using non-coplanar structures.
Our theoretical considerations call for an experimental verification of the effect.
\section*{Acknowledgments}
The authors thank N. Nakabayashi, H. Kohno, H. Saarikoski, and H. Seo for valuable comments and discussions.
This work was supported by a Grant-in-Aid for Scientific Research (C) (Grant No. 25400344) and (A) (Grant No. 24244053) from the Japan Society for the Promotion of Science and UK-Japanese Collaboration on Current-Driven Domain Wall Dynamics from JST.
|
3,212,635,537,415 | arxiv | \section{Introduction and overview}
\label{sec:intro}
Our main goal in this paper is to study global rates of convergence of
the Maximum Likelihood Estimator (MLE) in one simple model for multivariate
interval-censored data. In section 3 we will show that under some reasonable
conditions the MLE converges in a Hellinger metric to the true distribution
function on ${\mathbb R}^d$ at a rate no worse than $n^{-1/3} (\log n)^{\gamma_d}$ for $\gamma_d = (5d - 4)/6$
for all $d\ge 2$. Thus the rate of convergence is only worse than the known rate of $n^{-1/3}$ for
the case $d=1$ by a factor involving a power of $\log n$ growing linearly with the dimension.
These new rate results rely heavily on recent bracketing entropy bounds for $d-$dimensional
distribution functions obtained by \cite{Gao:12}.
We begin in Section~\ref{sec:IntCensOnR} with a review of interval censoring problems and known results in the case
$d=1$. We introduce the multivariate interval censoring model of interest here in
Section~\ref{sec:MultIntervalCensoring}, and
obtain a rate of convergence for this model for $d\ge 2$ in Theorem~\ref{NewMultHellingerRateThm}.
Most of the proofs are given in Section~\ref{sec:proofs}, with the exception being
a key corollary of \cite{Gao:12}, the statement and proof of which are given in
the Appendix (Section~\ref{sec:Appendix}).
Finally, in Section~\ref{sec:OtherModelsFurtherProblems}
we introduce several related models and further problems.
\section{Interval Censoring (or Current Status Data) on ${\mathbb R}$}
\label{sec:IntCensOnR}
Let $Y\sim F_0$ on ${\mathbb R}^+$, and let $T \sim G_0$ on ${\mathbb R}^+$ be independent of $Y$. Suppose that we
observe $X_1, \ldots , X_n$ i.i.d. as $X = (\Delta , T)$ where $\Delta = 1_{[Y \le T]}$.
Here $Y$ is often the time until some event of interest and $T$ is an observation time.
The goal is to estimate $F_0$ nonparametrically based on observation of the $X_i$'s.
To calculate the likelihood, we first calculate the distribution of $X$
for a general distribution function $F$: note that the conditional distribution of $\Delta$ conditional on $T$ is
Bernoulli:
$$
( \Delta | T ) \sim \mbox{Bernoulli} (p(T))
$$
where $p(T) = F(T)$. If $G_0$ has density $g_0$ with respect to some measure $\mu$ on ${\mathbb R}^+$, then
$X = (\Delta , T)$ has density
$$
p_{F,g_0} (\delta, t) = F(t)^{\delta} (1-F(t))^{1-\delta} g_0 (t), \ \ \ \delta \in \{ 0,1 \}, \ \ t \in {\mathbb R}^+,
$$
with respect to the dominating measure (counting measure on $\{ 0,1 \}) \times \mu$.
The nonparametric Maximum Likelihood Estimator (MLE) $\hat{F}_n$ of $F_0$ in this interval
censoring model was first obtained by
\cite{MR0073895}. It is simply described as follows: let $T_{(1)} \le \cdots \le T_{(n)}$ denote
the order statistics corresponding to $T_1, \ldots , T_n$ and let
$\Delta_{(1)}, \ldots , \Delta_{(n)}$ denote the corresponding $\Delta$'s.
Then the part of the log-likelihood of $X_1 ,\ldots , X_n$ depending on $F$
is given by
\begin{eqnarray}
l_n (F) & = & \sum_{i=1}^n \{ \Delta_{(i)} \log F(T_{(i)}) + (1-\Delta_{(i)} ) \log (1-F(T_{(i)})) \} \nonumber\\
& \equiv & \sum_{i=1}^n \{ \Delta_{(i)} \log F_{i} + (1-\Delta_{(i)} ) \log (1-F_{i}) \} \label{LogLikelihood}
\end{eqnarray}
where
\begin{eqnarray}
0 \le F_1 \le \cdots \le F_n \le 1 .
\label{MonConstr}
\end{eqnarray}
It turns out that the maximizer $\hat{F}_n$ of (\ref{LogLikelihood}) subject to (\ref{MonConstr})
can be described as follows:
let $H^*$ be the (greatest) convex minorant of the points $\{ (i, \sum_{j\le i} \Delta_{(j)}): \ i \in \{ 1, \ldots , n \} \}$:
\begin{eqnarray*}
H^* (t) = \sup \left \{ \begin{array}{l} H(t) : \ \ H(i) \le \sum_{j\le i} \Delta_{(j)}\ \ \mbox{for each} \ \ 0 \le i \le n \\
\ \ H(0) = 0, \ \mbox{and} \ \ H \ \ \mbox{is convex}
\end{array} \right \} .
\end{eqnarray*}
Let $\hat{F}_i$ denote the left-derivative of $H^*$ at $T_{(i)}$. Then
$(\hat{F}_1, \ldots , \hat{F}_n)$ is the unique vector maximizing (\ref{LogLikelihood}) subject to
(\ref{MonConstr}), and we therefore take the MLE $\hat{F}_n$ of $F$ to be
\begin{eqnarray*}
\hat{F}_n (t) = \sum_{i=0}^n \hat{F}_i 1_{[T_{(i)}, T_{(i+1)})} (t)
\end{eqnarray*}
with the conventions $T_{(0)} \equiv 0$ and $T_{(n+1)} \equiv \infty$. See
\cite{MR0073895} or \cite{MR1180321}, pages 38-43, for details.
\cite{Groeneboom:87} initiated the study of $\hat{F}_n$ and proved the following limiting
distribution result at a fixed point $t_0$.
\begin{thm} (Groeneboom, 1987).
\label{GroeneboomPointwiseLimitDistributionRone}
Consider the current status model on ${\mathbb R}^+$.
Suppose that $0 < F_0(t_0) , G_0(t_0) < 1$ and suppose that
$F$ and $G$ are differentiable at $t_0$ with strictly positive derivatives
$f_0(t_0)$ and $g_0(t_0)$ respectively. Then
\begin{eqnarray*}
n^{1/3} ( \hat{F}_n (t_0) - F_0(t_0) ) \rightarrow_d c(F_0,G_0) {\mathbb Z}
\end{eqnarray*}
where
\begin{eqnarray*}
c(F_0,G_0) = 2 \left ( \frac{F_0(t_0) (1-F_0(t_0))f_0(t_0)}{2g_0 (t_0)} \right )^{1/3}
\end{eqnarray*}
and
\begin{eqnarray*}
{\mathbb Z} = \mbox{argmin} \{ W(t) + t^2 \}
\end{eqnarray*}
where $W$ is a standard two-sided Brownian motion starting from $0$.
\end{thm}
\medskip
The distribution of ${\mathbb Z}$ has been studied in detail by
\cite{MR981568}
and computed by
\cite{MR1939706}.
\cite{BalabdaWell:12} show that the density $f_{{\mathbb Z}}$ of ${\mathbb Z}$ is log-concave.
\cite{MR1212164} (see also \cite{MR1739079}) obtained the following
global rate result for $p_{\hat{F}_n}$.
Recall that the Hellinger distance $h(p,q)$ between two densities with respect
to a dominating measure $\mu$ is given by
$$
h^2 (p,q) = \frac{1}{2} \int \{ \sqrt{p} - \sqrt{q} \}^2 d \mu .
$$
\begin{prop} (van de Geer, 1993)
\label{VdGHellingerRateRone}
$ h( p_{\hat{F}_n} , p_{F_0}) = O_p (n^{-1/3})$.
\end{prop}
\par\noindent
Now for any distribution functions $F$ and $F_0$ the (squared) Hellinger distance $h^2 (p_F , p_{F_0})$
for the current status model is given by
\begin{eqnarray}
h^2 (p_F , p_{F_0})
&= & \frac{1}{2} \left \{ \int ( \sqrt{F} - \sqrt{F_0} )^2 dG_0 + \int ( \sqrt{1-F} - \sqrt{1-F_0} )^2 d G_0 \right \} \nonumber \\
& = & \frac{1}{2} \int \frac{\{ ( \sqrt{F} - \sqrt{F_0} )(\sqrt{F} + \sqrt{F_0}) \}^2}{ (\sqrt{F} + \sqrt{F_0})^2} dG_0 \nonumber \\
&& \ \ + \ \frac{1}{2} \int \frac{\{ ( \sqrt{1-F} - \sqrt{1-F_0} )(\sqrt{1-F} + \sqrt{1-F_0}) \}^2}{ (\sqrt{1-F} + \sqrt{1-F_0})^2} dG_0
\nonumber \\
& \ge & \frac{1}{8} \int (F - F_0 )^2 d G_0 + \frac{1}{8} \int ((1-F)- (1-F_0) )^2 d G_0 \nonumber \\
& = & \frac{1}{4} \int (F - F_0 )^2 d G_0 , \label{L2-Hellinger-Inequal}
\end{eqnarray}
and hence
Proposition~\ref{VdGHellingerRateRone} yields
\begin{eqnarray}
\int_0^\infty ( \hat{F}_n (z) - F_0(z) )^2 d G_0 (z) = O_p (n^{-2/3}) ,
\end{eqnarray}
or $\| \hat{F}_n - F_0 \|_{L_2 (G_0)} = O_p (n^{-1/3})$.
\medskip
For generalizations of these and other asymptotic results
for the current status model to more complicated interval censoring schemes
for real-valued random variables $Y$, see e.g.
\cite{MR1180321},
\cite{MR1212164},
\cite{MR1600884},
\cite{MR1739079},
\cite{MR1774042},
and \cite{MR2418648, MR2418649}.
Our main focus in this paper, however, concerns one simple generalization of the
interval censoring model for ${\mathbb R}$ introduced above to interval censoring in ${\mathbb R}^d$.
We now turn to this generalization.
\section{Multivariate interval censoring: multivariate current status data}
\label{sec:MultIntervalCensoring}
Let $\underline{Y} = (Y_1, \ldots , Y_d) \sim F_0$ on ${\mathbb R}^{+d} \equiv [0,\infty)^d$,
and let $\underline{T} = (T_1, \ldots , T_d) \sim G_0 $ on ${\mathbb R}^{+d}$ be independent of $\underline{Y}$.
We assume that $G_0$ has density $g_0$ with respect to some dominating measure $\mu$ on ${\mathbb R}^d$.
Suppose we observe $\underline{X}_1, \ldots , \underline{X}_n$ i.i.d. as
$\underline{X} = ( \underline{\Delta}, \underline{T} )$ where $\underline{\Delta} = (\Delta_1 , \ldots , \Delta_d)$
is given by $\Delta_j = 1_{[Y_j \le T_j ]}$, $j =1, \ldots , d$.
Equivalently, with a slight abuse of notation, $\underline{X} = (\underline{\Gamma}, \underline{T} )$
where $\underline{\Gamma} = ( \Gamma_1 , \ldots , \Gamma_{2^d} )$
is a vector of length $2^d$ consisting of $0$'s and $1$'s and with at most one $1$
which indicates into which of the $2^d$ orthants of ${\mathbb R}^{+d}$ determined by
$\underline{T}$ the random vector $\underline{Y}$
belongs. More explicitly, define $K\equiv 1+\sum_{j=1}^d (1-\Delta_j) 2^{j-1}$. Then set
$\Gamma_k \equiv 1\{ k =K \}$ for $k=1,\ldots , 2^d$, so that $\Gamma_K = 1$ and $\Gamma_{l} = 0$ for
$l \in \{ 1, \ldots , 2^d \} \setminus \{ K \}$. Much as for univariate current status data,
$\underline{Y}$ represents a vector of times to events, $\underline{T}$ is a vector of observation times,
and the goal is nonparametric estimation of the joint distribution function $F_0$ of $\underline{Y}$ based on observation
of the $\underline{X}_i$'s.
See \cite{MR1891046},
\cite{MR2416109},
\cite{MR2736679},
and \cite{MR2818097}
for examples of settings in which data of this type arises.
To calculate the likelihood, we first calculate the distribution of $\underline{X}$
for a general distribution function $F$: note that the conditional distribution of $\underline{\Gamma}$
conditional on $\underline{T}$ is Multinomial:
$$
( \underline{\Gamma} | \underline{T} ) \sim \mbox{Mult}_{2^d} ( 1, \underline{p}(\underline{T}; F))
$$
where $\underline{p}(\underline{T};F) = (p_1 (\underline{T};F) , \ldots , p_{2^d} (\underline{T};F))$ and the probabilities
$p_j (\underline{t};F)$, $j = 1, \ldots , 2^d$, $\underline{t} \in {\mathbb R}^{+d}$ are determined
by the $F$ measures of the corresponding sets.
Then our model ${\cal P}$ for multivariate current status data
is the collection of all densities with respect to the dominating measure
$(\mbox{counting measure on} \ \{ 0,1\}^{2^d}) \times \mu$ given by
$$
\prod_{j=1}^{2^d} p_j (\underline{t};F)^{\gamma_j} g_0 (\underline{t})
$$
for some distribution function $F$ on ${\mathbb R}^{+d}$ where
$\underline{t} \in {\mathbb R}^{+d}$ and $\gamma_j \in \{ 0, 1\}$ with $\sum_{j=1}^{2^d} \gamma_j = 1$.
Now the part of the log-likelihood that depends on $F$ is given by
\begin{eqnarray*}
l_n (F) = \sum_{i=1}^n \sum_{j=1}^{2^d} \Gamma_{i,j} \log p_j (\underline{T}_i ; F),
\end{eqnarray*}
and again the MLE $\hat{F}_n$ of the true distribution function $F_0$ is given by
\begin{eqnarray}
\hat{F}_n = \mbox{argmax} \{ l_n (F) : \ F \ \mbox{is a distribution function on} \ {\mathbb R}^{+d} \}.
\label{MultivariateIntCensMLE}
\end{eqnarray}
For example, when $d=2$, we can write
$\Gamma_1 = \Delta_1 \Delta_2$,
$\Gamma_2 = (1-\Delta_1)\Delta_2$,
$\Gamma_3 = \Delta_1 (1-\Delta_2)$, and
$\Gamma_4 = (1-\Delta_1)(1-\Delta_2)$, and then
\begin{eqnarray*}
&& p_1 (\underline{T};F) = F(T_1, T_2), \\
&& p_2 (\underline{T};F) = F(\infty, T_2) - F(T_1, T_2) , \\
&& p_3 (\underline{T};F) = F(T_1, \infty) - F(T_1 , T_2) , \\
&& p_4 (\underline{T};F) = 1- F(T_1 , \infty) - F(\infty, T_2 ) + F(T_1, T_2) .
\end{eqnarray*}
Thus
$$
P_{F} ( \underline{\Gamma} = \underline{\gamma} | \underline{T})
= \prod_{j=1}^4 p_j (\underline{T}; F)^{\gamma_j} , \ \ \mbox{for}
\ \ \underline{\gamma} = (\gamma_1 , \gamma_2 ,\gamma_3 , \gamma_4), \ \
\gamma_j \in \{ 0,1 \}, \ \sum_{j=1}^4 \gamma_j = 1 .
$$
Note that
\begin{eqnarray}
p_j (\underline{t}; F) = \int_{[0, \infty)^2} 1_{C_j (\underline{t})} (\underline{y}) dF(\underline{y}), \qquad j = 1, \ldots , 4
\label{StructureOfTheCellProbabilities}
\end{eqnarray}
where
\begin{eqnarray*}
&& C_1 (\underline{t}) = [0,t_1]\times [0,t_2], \\
&& C_2 (\underline{t}) = [0,t_1]\times (t_2, \infty) ,\\
&& C_3 (\underline{t}) = (t_1, \infty) \times [0,t_2] ,\\
&& C_4 (\underline{t}) = (t_1 , \infty) \times (t_2, \infty) .
\end{eqnarray*}
Characterizations and computation of the MLE (\ref{MultivariateIntCensMLE}), mostly
for the case $d=2$ have
been treated in
\cite{Song/PHD},
\cite{MR1964427},
and
\cite{MR2160818, MR2708977}.
Consistency of the MLE for more general interval censoring models
has been established by \cite{MR2236498}.
For an interesting application see
\cite{Betensky-Fink:99}.
This example and other examples of multivariate interval censored data
are treated in \cite{MR2287318}
and and \cite{MR2489672}.
For a comparison of the MLE with alternative estimators in the case $d=2$, see
\cite{Groeneboom:12}.
An analogue of Groeneboom's Theorem~\ref{GroeneboomPointwiseLimitDistributionRone}
has not been established in the multivariate case.
\cite{Song/PHD} established an asymptotic minimax lower bound for pointwise convergence when $d=2$: if
$F_0$ and $G_0$ have positive continuous densities at $\underline{t}_0$, then no estimator has a local minimax
rate for estimation of $F_0 (\underline{t}_0)$ faster than $n^{-1/3}$.
By making use of additional smoothness hypotheses,
\cite{Groeneboom:12} has constructed estimators which achieve the pointwise $n^{-1/3}$
rate, but it is not yet known if the MLE achieves this.
Our main goal here is to prove the following theorem concerning the global rate of convergence of the
MLE $\hat{F}_n$.
\begin{thm}
\label{NewMultHellingerRateThm}
Consider the multivariate current status model.
Suppose that $F_0$ has $\mbox{supp} (F_0) \subset [0,M]^d$ and that $F_0$ has density $f_0$
which satisfies
\begin{eqnarray}
c_1^{-1} \le f_0 (\underline{y}) \le c_1 \ \ \mbox{for all} \ \ \underline{y} \in [0,M]^d
\label{ConditionOne}
\end{eqnarray}
where $0< c_1 < \infty$. Suppose that $G_0$ has density $g_0$ which
satisfies
\begin{eqnarray}
c_2^{-1} \le g_0 (\underline{y}) \le c_2 \ \ \mbox{for all} \ \ \underline{y} \in [0,M]^d.
\label{ConditionTwo}
\end{eqnarray}
Then the MLE $\widehat{p}_n \equiv p_{\widehat{F}_n}$ of $p_0\equiv p_{F_0}$ satisfies
\begin{eqnarray*}
h( \widehat{p}_n , p_0 ) = O_p \left ( \frac{ (\log n)^{\gamma}}{n^{1/3}} \right )
\end{eqnarray*}
for $\gamma \equiv \gamma_d \equiv (5d-4)/6$.
\end{thm}
Since the inequality (\ref{L2-Hellinger-Inequal}) continues to hold in ${\mathbb R}^d$ for $d\ge 2$
(with $1/4$ replaced by $1/8$ on the right side), we obtain the
following corollary:
\begin{cor}
\label{L2-rateCorollaryMultCase}
Under the conditions of Theorem~\ref{NewMultHellingerRateThm} it follows that
\begin{eqnarray*}
\int_{{\mathbb R}^{+d}} ( \widehat{F}_n (z) - F_0 (z) )^2 d G_0 (z) = O_p ( n^{-2/3} (\log n)^{\beta} )
\end{eqnarray*}
for $\beta \equiv \beta_d = 2 \gamma_d = (5d- 4)/3$.
\end{cor}
\section{Proofs}
\label{sec:proofs}
Here we give the proof of Theorem~\ref{NewMultHellingerRateThm}.
The main tool is a method developed by
\cite{MR1739079}. We will use the following lemma in combination with
Theorem 7.6 of \cite{MR1739079} or Theorem 3.4.1 of \cite{MR1385671} (Section 3.4.2, pages 330-331).
Without loss of generality we can take $M=1$ where $M$ is the upper bound of the support of $F$
(see Theorem~\ref{NewMultHellingerRateThm}).
Let ${\cal P}$ be a collection of probability densities $p$ on a sample space ${\cal X}$ with respect to
a dominating measure $\mu$. Define
\begin{eqnarray}
&& {\cal G}^{(conv)} \equiv \left \{ \frac{2 p}{p+p_0} : \ p \in {\cal P} \right \} ,
\label{CalGDefn} \\
&& \sigma( \delta ) \equiv \sup \{ \sigma \ge 0 : \ \ \int_{\{ p_0 \le \sigma \}} p_0 d \mu \le \delta^2 \}
\qquad \mbox{for} \ \ \delta > 0 ,
\label{SigmaSubDeltaDefn}\\
&& {\cal G}_{\sigma}^{(conv)} \equiv \left \{ \frac{2 p}{p+p_0} 1_{[p_0 > \sigma]} : \ p \in {\cal P} \right \} ,
\qquad \mbox{for} \ \ \sigma > 0 .
\label{CalGSubSigmaDefn}
\end{eqnarray}
The following general result relating the bracketing entropies $\log N_{[\,]} (\cdot , {\cal G}^{(conv)} , L_2 (P_0))$,
$\log N_{[\,]} (\cdot , {\cal G}_{\sigma(\epsilon)}^{(conv)} , L_2 (P_0))$,
$\log N_{[\,]} (\cdot , {\cal P}_ , L_2 (Q_{\sigma(\epsilon)} ))$,
and
$\log N_{[\,]} (\cdot , {\cal P}_ , L_2 (\tilde{Q}_{\sigma(\epsilon)} ))$
is due to \cite{MR1739079}.
\begin{lem} (van de Geer, 2000)
\label{VdGBasicLemma}
For every $\epsilon > 0$
\begin{eqnarray}
\log N_{[\, ]} ( 3 \epsilon , {\cal G}^{(conv)} , L_2 (P_0) )
& \le & \log N_{[\, ]} ( \epsilon , {\cal G}_{\sigma(\epsilon)}^{(conv)} , L_2 (P_0))
\label{VDG_First_Inequality} \\
& \le & \log N_{[\, ]} ( \epsilon/2 , {\cal P}, L_2 ( Q_{\sigma(\epsilon)} ) )
\label{VDG_Second_Inequality} \\
& = & \log N_{[\, ]} \left ( \frac{\epsilon/2}{\sqrt{Q_{\sigma(\epsilon)} ({\cal X} )}} ,
{\cal P}, L_2 (\tilde{Q}_{\sigma(\epsilon)} ) \right )
\label{VDG_Third_Equality}
\end{eqnarray}
where $dQ_{\sigma} \equiv p_0^{-1} 1_{[p_0 > \sigma]} d \mu$ and
$\tilde{Q}_{\sigma} \equiv Q_{\sigma} / Q_{\sigma} ({\cal X} )$.
\end{lem}
\begin{proof}
We first show that (\ref{VDG_First_Inequality}) holds. Suppose that
$\{ [g_{L,j} , g_{U,j} ] , \ j = 1, \ldots , m \}$ are $\epsilon$-brackets with respect to $L_2(P_0)$
for ${\cal G}_{\sigma (\epsilon)}^{(conv)}$ with
$$
{\cal G}_{\sigma (\epsilon)}^{(conv)} \subset \bigcup_{j=1}^m [g_{L,j} , g_{U,j} ] , \qquad
m = N_{[\, ]} ( \epsilon , {\cal G}_{\sigma(\epsilon)}^{(conv)} , L_2 (P_0)) .
$$
Then for $g \in {\cal G}^{(conv)}$, let $g_{\sigma} \equiv g 1_{[p_0 > \sigma]}$ be the corresponding
element of ${\cal G}_{\sigma (\epsilon)}^{(conv)}$. Suppose that
$g_{\sigma} \in [g_{L,j} , g_{U,j} ]$ for some $j \in \{ 1, \ldots , m \}$. Then
\begin{eqnarray*}
g = g 1_{[p_0 \le \sigma]} + g_{\sigma}
\ \left \{ \begin{array}{l} \le g 1_{[p_0 \le \sigma ]} + g_{U,j} \equiv \tilde{g}_{U,j} \\
\ge 0 + g_{L,j} \equiv \tilde{g}_{L,j} ,
\end{array} \right .
\end{eqnarray*}
where, by the triangle inequality, $ 0 \le g \le 2 $ for all $g \in {\cal G}^{(conv)}$,
and the definition of $\sigma(\epsilon)$,
it follows that
\begin{eqnarray*}
\big \| \tilde{g}_{U,j} - \tilde{g}_{L,j} \big \|_{P_0,2} \le \big \|{g}_{U,j} - {g}_{L,j} \big \|_{P_0,2} + 2 \epsilon
\le 3 \epsilon .
\end{eqnarray*}
Thus $\{ [ \tilde{g}_{L,j} , \tilde{g}_{U,j}] : \ j \in \{ 1, \ldots , m\} \}$ is a collection of
$3\epsilon-$brackets for ${\cal G}^{(conv)}$ with respect to $L_2 (P_0)$
and hence (\ref{VDG_First_Inequality}) holds.
Now we show that (\ref{VDG_Second_Inequality}) holds.
Suppose that $ \{ [ p_{L,j}, p_{U,j} ] : \ j = 1, \ldots , m \}$
is a set of $\epsilon/2-$brackets with respect to $L_2 (Q_{\sigma} )$ for ${\cal P}$ with
$$
{\cal P} \subset \bigcup_{j=1}^m [ p_{L,j}, p_{U,j} ] \qquad \mbox{and} \qquad
m = N_{[\, ]} ( \epsilon/2 , {\cal P}, L_2 ( Q_{\sigma(\epsilon)} ) ) .
$$
Suppose $p \in [ p_{L,j}, p_{U,j} ] $ for some $j$. Then, since
\begin{eqnarray*}
\frac{2p}{p+p_0} 1_{[p_0 > \sigma]} \ \ \left \{
\begin{array}{l} \le \frac{2p_{U,j}}{p_{U,j} + p_0 } 1_{[p_0 > \sigma]} \equiv g_{U,j} , \\
\ \ \\
\ge \frac{2p_{L,j}}{p_{U,j} + p_0} 1_{[p_0 > \sigma ]} \equiv g_{L,j}
\end{array} \right .
\end{eqnarray*}
where
\begin{eqnarray*}
\lefteqn{| g_{U,j} - g_{L,j} | } \\
& = & \bigg | \frac{2p_{U,j}}{p_{U,j} + p_0 } 1_{[p_0 > \sigma]} - \frac{2p_{L,j}}{p_{U,j} + p_0 } 1_{[p_0 > \sigma]} \bigg | \\
& = & \frac{2(p_{U,j} - p_{L,j})}{p_{L,j} + p_0 } 1_{[p_0 > \sigma]}
\le \frac{2 | p_{U,j} - p_{L,j} |}{p_0 } 1_{[p_0 > \sigma]} .
\end{eqnarray*}
Thus
\begin{eqnarray*}
\big \| g_{U,j} - g_{L,j} \|_{P_0 , 2} \le 2 \big \| p_{U,j} - p_{L,j} \big \|_{Q_{\sigma} , 2} \le \epsilon ,
\end{eqnarray*}
and hence $\{ [ g_{L,j}, g_{U,j} ] : \ j = 1, \ldots , m \}$ is a set of $\epsilon$-brackets with respect to
$L_2 (P_0)$ for ${\cal G}_{\sigma}^{(conv)}$. This shows that (\ref{VDG_Second_Inequality}) holds.
It remains only to show that (\ref{VDG_Third_Equality}) holds. But this is easy since
$\| g \|_{Q_{\sigma},2}^2 = \| g \|_{\tilde{Q}_\sigma,2}^2 \cdot Q_{\sigma} ({\cal X} )$.
This lemma is based on
\cite{MR1739079}, pages 101 and 103.
Note that our constants differ slightly from those of van de Geer.
\end{proof}
\begin{lem}
\label{BoundsForFZeroF}
Suppose that $F_0$ has density $f_0$
which satisfies, for some $0 < c_1 < \infty$,
\begin{eqnarray}
\frac{1}{c_1} \le f_0 (\underline{y} ) \le c_1 \qquad
\mbox{for all} \ \
\underline{y} \in [0,1]^d .
\label{BoundsForFZeroDensity}
\end{eqnarray}
Then $p_0$ (which we can identify with the vector $p_0 (\cdot , F_0)$) satisfies
\begin{eqnarray*}
&& p_{0,1} (\underline{t}; F_0) \left \{ \begin{array}{l} \le c_1 \prod_{j=1}^d t_j \\ \ge c_1^{-1} \prod_{j=1}^d t_j , \end{array}
\right . \qquad \mbox{for all} \ \ \underline{t}\in [0,1]^d , \\
&& \vdots \\
&& p_{0,2^d} (\underline{t}; F_0) \left \{
\begin{array}{l} \le c_1 \prod_{j=1}^d (1-t_j) \\ \ge c_1^{-1} \prod_{j=1}^d (1-t_j ), \end{array}
\right . \qquad \mbox{for all} \ \ \underline{t} \in [0,1]^d .
\end{eqnarray*}
\end{lem}
\begin{proof}
This follows immediately from the general $d$ version of
(\ref{StructureOfTheCellProbabilities}) and the assumption on $f_0$.
\end{proof}
These inequalities can also be written in the following compact form: For $k=1+ \sum_{j=1}^d (1-\delta_j )2^{j-1}$
with $\delta_j \in \{ 0,1\}$,
\begin{eqnarray*}
p_{0,k} (\underline{t}; F_0) \left \{ \begin{array}{l} \le c_1 \prod_{j=1}^d t_j^{\delta_j} (1-t_j)^{1-\delta_j} \\
\ge c_1^{-1} \prod_{j=1}^d t_j^{\delta_j} (1-t_j)^{1-\delta_j} ,
\end{array} \right . \qquad \mbox{for all} \ \ t \in [0,1]^d .
\end{eqnarray*}
\begin{lem}
\label{BoundsForSmallValuesDensity}
Suppose that the assumption of Lemma~\ref{BoundsForFZeroF}
holds.
Suppose, moreover, that
$G_0$ has density $g_0$
which satisfies
\begin{eqnarray}
\frac{1}{c_2} \le g_0 (\underline{y} ) \le c_2 \qquad
\mbox{for all} \ \
\underline{y} \in [0,1]^d.
\label{BoundsForGZeroDensity}
\end{eqnarray}
Then
\begin{eqnarray*}
\int_{[p_0 \le \sigma ]} p_0 d \mu \le 2^d (c_1 c_2)^2 \sigma .
\end{eqnarray*}
Furthermore, with
$\sigma (\delta ) \equiv \delta^2 / (2^d (c_1 c_2)^2)$
we have
$$
\int_{[p_0 \le \sigma( \delta )]} p_0 d \mu \le \delta^2 .
$$
\end{lem}
\begin{proof}
The first inequality follows easily from Lemma~\ref{BoundsForFZeroF}: note that
\begin{eqnarray*}
\int_{[p_0 \le \sigma]} p_0 d\mu
& = & \sum_{k=1}^{2^d} \int_{[p_k (\underline{t}, F_0) \le \sigma]} p_k (\underline{t} , F_0 ) g_0 (\underline{t}) \underline{dt} \\
& \le & 2^d \int_{[F_0 (\underline{t})g_0 (\underline{t}) \le \sigma]} F_0 (\underline{t} ) g_0 (\underline{t} ) \underline{dt} \\
& \le & 2^d c_1 c_2 \int_{[ c_1^{-1} c_2^{-1} \prod_{j=1}^d t_j \le \sigma]} \prod_{j=1}^d t_j \, \underline{dt} \le 2^d (c_1 c_2)^2 \sigma .
\end{eqnarray*}
The second inequality follows from the first inequality of the lemma.
\end{proof}
\begin{lem}
\label{MassOfQsigma}
If the hypotheses of Lemmas~\ref{BoundsForFZeroF} and~\ref{BoundsForSmallValuesDensity}
hold, then the measure $Q_{\sigma}$
defined by $dQ_{\sigma} \equiv (1/p_0) 1 \{ p_0 > \sigma\}d \mu $
has total mass $Q_{\sigma} ({\cal X})$
given by
\begin{eqnarray}
\int d Q_{\sigma}
& = & \int_{\{ p_0 > \sigma\}} \frac{1}{p_0} d \mu
\nonumber \\
& = & \sum_{j=1}^{2^d} \int_{\{\underline{t} : \ p_{0,j} (\underline{t}) g_0(\underline{t}) > \sigma \}}
\frac{1}{ p_{0,j} (\underline{t}) g_0(\underline{t}) } \underline{dt} \nonumber \\
& \le & 2^d \int_{\{\underline{t} \in [0,1]^d: \ \prod_{j=1}^d t_j > \sigma/(c_1 c_2) \} }
\frac{ c_1c_2}{ \prod_{j=1}^d t_j } \underline{dt}
\label{FirstCrackAnalyticExpressionTotalMass} \\
& = & \frac{2^d c_1 c_2}{d!} ( \log (c_1 c_2 / \sigma ))^d .
\label{AsympApproxTotalMass}
\end{eqnarray}
\end{lem}
\begin{proof}
This follows from Lemma~\ref{BoundsForFZeroF},
followed by an explicit calculation.
In particular, the equality in (\ref{AsympApproxTotalMass}) follows from
\begin{eqnarray*}
\int_{[\prod_{j=1}^d t_j > b]} \frac{1}{\prod_{j=1}^d t_j} \underline{dt}
& = & \int_{[ \sum_1^d x_j \le \log (1/b)]} \underline{dx} \ \ \mbox{by the change of variables} \ t_j = e^{-x_j},\\
& = &\frac{1}{d!} \left ( \log (1/b) \right )^d \ \ \ \ \mbox{for} \ \ 0 < b \le 1
\end{eqnarray*}
where the second equality follows by induction: it holds easily for $d=1$ (and $d=2$); and then an easy calculation
shows that it holds for $d$ if it holds for $d-1$.
\end{proof}
\begin{lem}
\label{CruxEntropyBound}
If the hypotheses of Lemmas~\ref{BoundsForFZeroF} and ~\ref{BoundsForSmallValuesDensity} hold,
and $d\ge2$, then
\begin{eqnarray*}
\log N_{[\, ]} ( \epsilon , {\cal G}^{(conv)} , L_2 (P_0)) \le K \frac{ \left [ \log (1/\epsilon)\right ]^{5d/2-2}}{\epsilon}
\end{eqnarray*}
for all $0 < \epsilon < \mbox{some}\ \epsilon_0 $ and some constant $K< \infty$.
\end{lem}
\begin{proof}
This follows by combining the results of Lemmas~\ref{BoundsForSmallValuesDensity} and
\ref{MassOfQsigma} with Lemma~\ref{VdGBasicLemma}, and
then using Corollary~\ref{CorollariesOfGaosThm}
of the bracketing entropy bound of \cite{Gao:12} and stated here as Theorem~\ref{GaosThm}.
Here is the explicit calculation:
\begin{eqnarray*}
\lefteqn{\log N_{[\, ]} ( 6 \epsilon , {\cal G}^{(conv)} , L_2 (P) )} \\
& \le & \log N_{[\, ]} \left ( \frac{\epsilon}{\sqrt{Q_{\sigma(\epsilon)} ({\cal X} )}},
{\cal P}, L_2 (\tilde{Q}_{\sigma(\epsilon)} ) \right ) \qquad \mbox{by Lemma~\ref{VdGBasicLemma}} \\
& \le & \log N_{[\, ]} \left ( \frac{\epsilon}{\sqrt{\frac{2^d c_1c_2}{d!} \left [ \log ((c_1c_2)^3\cdot 2^d /(\epsilon^2 ) \right ]^d }}, {\cal P} ,
L_2 (\tilde{Q}_{\sigma(\epsilon)} ) \right )
\qquad \mbox{by \ Lemmas~\ref{BoundsForSmallValuesDensity}~and~\ref{MassOfQsigma}} \\
& \le & \log N_{[\, ]} \left ( \frac{V \epsilon}{\left [ \log (1/\epsilon ) \right ]^{d/2} }, {\cal P} ,
L_2 (\tilde{Q}_{\sigma(\epsilon)} ) \right ) \qquad \mbox{for} \ \ V = V_d (c_1,c_2) \\
& \le & K \frac{\left [ \log(1/\epsilon) \right ]^{d/2}}{V \epsilon}
\left [ \log \left ( \frac{ (\log (1/\epsilon))^{d/2}}{V\epsilon} \right ) \right ]^{2(d-1)}
\qquad \mbox{by Corollary~\ref{CorollariesOfGaosThm}(b)} \\
& \le & \tilde{K} \frac{\left [\log (1/\epsilon) \right ]^{5d/2 -2}}{\epsilon}
\end{eqnarray*}
for $\epsilon$ sufficiently small.
\end{proof}
\begin{proof} (Theorem~\ref{NewMultHellingerRateThm})
This follows from Lemma~\ref{CruxEntropyBound} and Theorem 7.6 of
\cite{MR1739079} or Theorem 3.4.1 of \cite{MR1385671} together with the
arguments given in Section 3.4.2. By Lemma~\ref{CruxEntropyBound} the bracketing entropy integrals
\begin{eqnarray*}
J_{[\,]} (\delta , {\cal G}^{(conv)} , L_2 (P_0)) \equiv \int_0^{\delta}
\sqrt { 1 + \log N_{[\, ]} (\epsilon , {\cal G}^{(conv)} , L_2 (P_0)) } \ d\epsilon
\lesssim \int_0^\delta \epsilon^{-1/2} \left \{ \log (1/\epsilon) \right \}^{3\gamma_d/2} \ d \epsilon
\end{eqnarray*}
where the bound on the right side
behaves asymptotically as a constant times $2 \delta^{1/2} ( \log (1/\delta ))^{3\gamma_d/2}$
with $3\gamma_d \equiv 5d/2 -2$,
and hence (using the notation of Theorem 3.4.1 of \cite{MR1385671}), we can take
$\phi_n (\delta ) = K 2 \delta^{1/2} ( \log (1/\delta ))^{3\gamma_d/2}$. Thus with
$r_n \equiv n^{1/3} / (\log n)^\beta $ with $\beta = \gamma_d$ we find that
$r_n^2 \phi_n (1/r_n ) \sim \tilde{K} \sqrt{n} $ and hence the claimed
order of convergence holds.
\end{proof}
\section{Some related models and further problems}
\label{sec:OtherModelsFurtherProblems}
There are several related models in which we expect to see the same basic
phenomenon as established here, namely a global convergence rate of the form
$n^{-1/3} (\log n)^{\gamma}$ in all dimensions $d\ge 2$ with only the power
$\gamma$ of the log term depending on $d$. Three such models are:\\
(a) the ``in-out model'' for interval censoring in ${\mathbb R}^d$;\\
(b) the ``case 2'' multivariate interval censoring models studied by \cite{MR2489672}; and\\
(c) the scale mixture of uniforms model for decreasing densities in ${\mathbb R}^{+d}$.\\
Here we briefly sketch why we expect the same phenomenon to hold
in these three cases, even though we do not yet know pointwise convergence rates in any of these cases.
\subsection{The ``in-out model'' for interval censoring in ${\mathbb R}^d$}
\label{subsec:InOut}
The ``in-out model'' for interval censoring in ${\mathbb R}^d$ was explored in the case $d=2$ by
\cite{Song/PHD}. In this model $\underline{Y} \sim F$ on ${\mathbb R}^2$, $R$ is a random rectangle
in ${\mathbb R}^2$ independent of $\underline{Y}$
(say $[\underline{U},\underline{V}] = \{ \underline{x} = (x_1, x_2) \in {\mathbb R}^2 : \ U_1 \le x_1 \le V_1, \ U_2 \le x_2 \le V_2 \}$
where $\underline{U}$ and $\underline{V}$ are random vectors in ${\mathbb R}^2$
with $\underline{U} \le \underline{V}$ coordinatewise).
We observe only $ (1_R (\underline{Y}), R)$, and the goal is to estimate the unknown distribution function
$F$.
\cite{Song/PHD} (page 86) produced a local asymptotic minimax lower bound for estimation of
$F$ at a fixed $\underline{t}_0 \in {\mathbb R}^2$. Under the assumption that $F$ has a positive density $f$
at $\underline{t}_0$, \cite{Song/PHD} showed that any estimator of $F(\underline{t}_0)$ can have a local-minimax
convergence rate which is at best $n^{-1/3}$. \cite{Groeneboom:12} has shown that this rate can
be achieved by estimators involving smoothing methods. Based on the results
for current status data in ${\mathbb R}^d$ obtained in Theorem~\ref{NewMultHellingerRateThm}
and the entropy results for the class of distribution functions on ${\mathbb R}^d$,
we conjecture that the global Hellinger rate of convergence of the MLE
$\hat{F}_n (\underline{t}_0)$ will be $n^{-1/3} (\log n)^{\nu}$ for all $d \ge 2$ where $\nu = \nu_d$.
\subsection{``Case 2'' multivariate interval censoring models in ${\mathbb R}^d$}
\label{subsec:MultVarCase2}
Recall that ``case 2'' interval censored data on ${\mathbb R}$ is as follows:
suppose that $\underline{Y} \sim F_0$ on ${\mathbb R}^+$, the pair of observation times
$(U,V)$ with $U \le V$ determines a random interval $(U,V]$, and we observe
$\underline{X} = ( \underline{\Delta}, U, V) = (\Delta_1, \Delta_2, \Delta_3, U,V)$
where $\Delta_1 = 1\{ Y \le U \}$, $\Delta_2 = 1\{U<Y \le V\}$, and $\Delta_3 = 1\{ V < Y \}$.
Nonparametric estimation of $F_0$ based on $\underline{X}_1, \ldots , \underline{X}_n )$
i.i.d. as $\underline{X}$ has been discussed by a number of authors, including
\cite{MR1180321}, \cite{MR1714713}, and \cite{MR1600884}.
\cite{MR2489672} studied generalizations of this model to ${\mathbb R}^d$, and obtained
rates of convergence of the MLE with respect to the Hellinger metric given by
$n^{-(1+d)/(2(1+2d)} (\log n)^{d^2/(2(2d+1)}$ in the case most comparable to
the multivariate interval censoring model studied here. While this rate reduces when $d=1$ to the
known rate $n^{-1/3} (\log n)^{1/6}$, it is slower than $n^{-1/3} (\log n)^{\nu}$ for some $\nu$
when $d>1$ due to the use of entropy bounds involving convex hulls (see \cite{MR2489672},
Proposition A.1, page 66) which are not necessarily sharp. We expect that rates
of the form $n^{-1/3} (\log n)^{\nu}$ with $\nu>0$ are possible in these models as well.
\subsection{Scale mixtures of uniform densities on ${\mathbb R}^{+d}$}
\label{subsec:ScaleMixUniform}
\cite{MR2717518}
and \cite{MR2890434}
studied the family of scale mixtures of uniform densities of the following form:
\begin{eqnarray}
f_G (\underline{x}) = \int_{{\mathbb R}^{+d}} \frac{1}{\prod_{j=1}^d y_j } 1_{(0,\underline{y}]} (\underline{x}) dG(\underline{y})\equiv
\int_{{\mathbb R}^{+d}} \frac{1}{|\underline{y}|} 1_{(0,\underline{y}]} (\underline{x}) dG(\underline{y})
\label{ScaleMixtureOfUniforms}
\end{eqnarray}
for some distribution function $G$ on $(0,\infty)^{d}$.
(Note that we have used the notation $\prod_{j=1}^d y_j = | \underline{y} | $ for
$\underline{y} = (y_1, \ldots , y_d) \in {\mathbb R}^{+d}$.)
It is not difficult to see
that such densities are decreasing in each coordinate and that they also satisfy
\begin{eqnarray*}
(\Delta_d f_G) (\underline{u},\underline{v}] =
(-1)^d \int_{(\underline{u},\underline{v}]} |\underline{y}|^{-1} 1_{(\underline{y},\underline{v}]} d G(\underline{y}) \ge 0
\end{eqnarray*}
for all $\underline{u},\underline{v} \in {\mathbb R}^{+d} $ with $\underline{u} \le \underline{v}$; here
$\Delta_d$ denotes the $d-$dimensional
difference operator. This is the same key property of
distribution functions which results in (bracketing) entropies which depend on
dimension only through a logarithmic term. The difference here is that
the density functions $f_G$ need not be bounded, and even if the true density
$f_0 $ is in this class and satisfies $f_0 (\underline{0}) < \infty$, then we do not yet
know the behavior of the MLE $\hat{f}_n $ at zero. In fact we conjecture that:
(a) If $f_0 (\underline{0}) < \infty$ and $f_0$ is a scale mixture of uniform densities on rectangles as in
(\ref{ScaleMixtureOfUniforms}), then $\hat{f}_n (\underline{0}) = O_p ((\log n)^{\beta})$ for some
$\beta = \beta_d>0$. (b) Under the same hypothesis as in (a) and the hypothesis that
$f_0$ has support contained in a compact set,
the MLE converges with respect to the Hellinger distance with a rate that is no worse than
$n^{-1/3} (\log n)^{\xi}$ where $\xi= \xi_d$. Again \cite{MR2717518} and \cite{MR2890434}
establish asymptotic minimax lower bounds for estimation of $f_0 (\underline{x}_0)$ proving that
no estimator can have a (local minimax) rate of convergence faster than $n^{-1/3}$ in all dimensions.
This is in sharp contrast to the class of block-decreasing densities on ${\mathbb R}^{+d}$ studied
by \cite{MR2911847} and by \cite{MR1994726}: \cite{MR2911847} shows that the local
asymptotic minimax rate for estimation of $f_0 (x_0)$ is no faster than $n^{-1/(d+2)}$,
while \cite{MR1994726} show that there exist (histogram type) estimators $\tilde{f}_n $
which satisfy $E_{f_0} \| \tilde{f}_n - f_0 \|_1 = O (n^{-1/(d+2)})$.
\section{Appendix}
\label{sec:Appendix}
We begin by summarizing the results of \cite{Gao:12}.
For a (probability) measure $\mu$ on $[0,1]^d$, let $F \equiv F_{\mu}$ denote the
corresponding distribution function given by
$$
F(\underline{x}) = F_{\mu} (\underline{x}) = \mu ( [0,\underline{x}] ) = \mu ( [0,x_1]\times \cdots \times [0,x_d] )
$$
for all $\underline{x} = (x_1, \ldots , x_d) \in [0,1]^d$.
Let ${\cal F}_d$ denote the collection of all distribution functions on $[0,1]^d$; i.e.
\begin{eqnarray*}
{\cal F}_d = \{ F : \ F \ \ \mbox{is a distribution function on} \ \ [0,1]^d \} .
\end{eqnarray*}
For example, if $\lambda_d$ denotes Lebesgue measure on $[0,1]^d$, then the corresponding
distribution function is
$F(\underline{x}) = F_{\lambda_d} (\underline{x}) = \prod_{j=1}^d x_j$.
\smallskip
\begin{thm} (Gao, 2012).
\label{GaosThm}
For $d \ge 2$ and $1 \le p < \infty$
\begin{eqnarray*}
\log N_{[\,]} (\epsilon , {\cal F}_d , L_p (\lambda_d) ) \lesssim \epsilon^{-1} \left ( \log (1/\epsilon) \right )^{2(d-1)}
\end{eqnarray*}
for all $0 < \epsilon \le 1$.
\end{thm}
Our goal here is to use this result to control bracketing numbers for ${\cal F}_d$ with respect to
two other measures $C_d$ and $R_{d,\sigma}$ defined as follows. Let $C_d$ denote the finite
measure on $[0,1]^d$ with density with respect to $\lambda_d$ given by
\begin{eqnarray*}
c_d (\underline{u}) = \frac{d!}{d^d} \prod_{j=1}^d \frac{1}{u_j^{1-1/d}} \cdot 1 \left \{ \sum_{j=1}^d u_j^{1/d} > d-1 \right \} .
\end{eqnarray*}
For fixed $\sigma > 0$, let $R_{d,\sigma}$ denote the (probability) measure on $(0,1]^d$ with
density with respect to $\lambda_d$ given by
\begin{eqnarray*}
r_{d,\sigma} (\underline{t}) = \frac{d!}{(\log(1/\sigma))^d} \frac{1}{\prod_{j=1}^d t_j } 1\left \{\prod_{j=1}^d t_j > \sigma \right \} .
\end{eqnarray*}
\begin{cor}
\label{CorollariesOfGaosThm}
(a) \ For each $d \ge 2$ it follows that for $\epsilon \le \epsilon_0 (d)$
\begin{eqnarray*}
\log N_{[\, ]} (2^{d/2}\epsilon , {\cal F}_d , L_2 (C_d) )
\lesssim \epsilon^{-1} \left ( \log (1/\epsilon) \right )^{2(d-1)} .
\end{eqnarray*}
(b) \ For each $d \ge 2$ and $\sigma \le \sigma_0 (d)$ it follows that for $\epsilon \le \epsilon_0 (d)/2$
\begin{eqnarray*}
\log N_{[\, ]} (2^{d/2 +1}\epsilon , {\cal F}_d , L_2 (R_{d,\sigma}) )
\lesssim \epsilon^{-1} \left ( \log (1/\epsilon) \right )^{2(d-1)} .
\end{eqnarray*}
\end{cor}
\begin{proof}
We first prove (a).
We set $p\equiv p_d = 2 r_d \equiv 2r$ where $r \equiv r_d = 2d-1$
and $s =(d-1/2)/(d-1)$ satisfy $r^{-1} + s^{-1} =1$.
Let $\{ [g_j, h_j ], j = 1, \ldots , m\}$ be a collection of $\epsilon-$brackets for ${\cal F}_d$
with respect to $L_p (\lambda_d)$. (Thus for $d=2$, $r = 3$, $s=3/2$, and $p=6$, while for
$d=4$, $r=7$, $s= (13/2)/3 = 13/6$, and $p= 14$.)
By Theorem A.1 we know that $m \lesssim \epsilon^{-1} ( \log (1/\epsilon) )^{2(d-1)} $.
Now we bound the size of the brackets
$[g_j, h_j]$ with respect to $C_d$.
Using H\"older's inequality with $1/r + 1/s =1$ as chosen above we find that
\begin{eqnarray}
\int_{[0,1]^d} ( h_j - g_j )^2 c_d (u) du
& \le & \left ( \int_{[0,1]^d} | h_j - g_j |^{2r} \underline{du} \right )^{1/r}
\cdot \left ( \int_{[0,1]^d} c_d (\underline{u})^s \underline{du} \right )^{1/s} \nonumber \\
&\le & (\epsilon^p )^{1/r} \cdot 2^{d/s} \le 2^{d} \epsilon^2 .
\label{L2EntropyWrtCd}
\end{eqnarray}
Here are some details of the computation leading to (\ref{L2EntropyWrtCd}):
\begin{eqnarray*}
\int_{[0,1]^d} c_d (\underline{u})^s \underline{du}
& = & \int_{[0,1]^d} \left ( \frac{d!}{d^d} \right )^s \prod_{j=1}^d \frac{1}{u_j^{(d-1/2)/d}}
\cdot 1 \left \{ \sum_{j=1}^d u_j^{1/d} > d-1 \right \} \underline{du} \\
& = & \left ( \frac{d!}{d^d} \right )^s \cdot (2d)^d \int_{[0,1]^d} 1 \left \{ \sum_{j=1}^d x_j^{2} > d-1 \right \} \underline{dx}\\
& \le & \left ( \frac{d!}{d^d} \right )^s \cdot (2d)^d \cdot \int_{[0,1]^d}1 \left \{ \sum_{j=1}^d x_j > d-1 \right \} \underline{dx}\\
& \le & \left ( \frac{d!}{d^d} \right )^s \cdot (2d)^d \cdot \int_{[0,1]^d}1 \left \{ \sum_{j=1}^d t_j < 1 \right \} \underline{dt}\\
&=&2^d\left ( \frac{d!}{d^d} \right )^{s-1}
\le 2^d .
\end{eqnarray*}
To prove (b) we introduce monotone transformations $t_j (u_j)$ and their inverses $u_j (t_j)$ which
relate $c_d$ and $r_{d,\sigma}$: we set
\begin{eqnarray*}
&& u_j (t_j) \equiv \left ( \frac{ \log (t_j / \sigma )}{\log (1/\sigma)} \right )^d, \\
&& t_j (u_j) \equiv \sigma \exp ( u_j^{1/d} \log (1/\sigma) )
\end{eqnarray*}
for $j = 1, \ldots , m$. These all depend on $\sigma>0$, but this dependence is suppressed in
the notation.
For the same brackets $[g_j , h_j]$ used in the proof of (a), we define new brackets
$[ \tilde{g}_j , \tilde{h}_j ]$ for $j = 1, \ldots , m$ by
\begin{eqnarray*}
&& \tilde{g}_j (\underline{t}) \equiv \tilde{g}_{j, \sigma} (\underline{t}) = g_j ( u(\underline{t})) = g_j (u_1 (t_1), \ldots , u_d (t_d) ) ,\\
&& \tilde{h}_j (\underline{t}) \equiv \tilde{h}_{j, \sigma} (\underline{t}) = h_j ( u(\underline{t})) = h_j (u_1 (t_1), \ldots , u_d (t_d))) .
\end{eqnarray*}
Then it follows easily by direct calculation using
\begin{eqnarray*}
\prod_{j=1}^d t_j &= & \sigma^d \exp \left ( \log (1/\sigma) \sum_{j=1}^d u_j^{1/d} \right ), \\
\underline{d t} & = & \prod_{j=1}^d \left \{ \sigma \exp ( \log (1/\sigma) u_j^{1/d} )
\cdot d^{-1} u_j^{1/d-1} \cdot \log (1/\sigma) (d u_j) \right \} \\
& = & \frac{\sigma^d (\log (1/\sigma))^d}{d^d} \prod_{j=1}^d t_j \cdot \prod_{j=1}^d u_j^{- (1-1/d)} \cdot \underline{du} \\
\left \{ \prod_{j=1}^d t_j > \sigma \right \}
& = & \left \{ \exp \left ( \log (1/\sigma) \sum_{j=1}^d u_j^{1/d} \right ) > \sigma^{- (d-1)} \right \} \\
& = & \left \{ \log (1/\sigma) \sum_{j=1}^d u_j^{1/d} > (d-1) \log (1/\sigma) \right \} \\
& = & \left \{ \sum_{j=1}^d u_j^{1/d} > d-1 \right \} ,
\end{eqnarray*}
that
\begin{eqnarray*}
\int_{[0,1]^d} (\tilde{h}_j (t) - \tilde{g}_j (t))^2 r_{d, \sigma} (t) \underline{dt}
& = & \int_{[0,1]^d} ( h_j (u) - g_j (u))^2 c_d (u) \underline{du} .
\end{eqnarray*}
Thus for $\sigma\le \sigma_0 (d)$ we have
\begin{eqnarray*}
\| \tilde{h}_j - \tilde{g}_j \|_{L_2 (R_{d, \sigma })} \le 2^{d/2+1} \epsilon
\end{eqnarray*}
by the arguments in (a).
Hence the brackets
$[\tilde{g}_j , \tilde{h}_j]$ yield a collection of $2^{d/2+1} \epsilon -$ brackets for ${\cal F}_d$
with respect to $L_2 (R_{d,\sigma)}$, and this
implies that (b) holds.
\end{proof}
\bigskip
\par\noindent
{\bf Acknowledgements:} We owe thanks to the referees for a number of helpful suggestions
and for pointing out the work of \cite{MR2236498} and \cite{MR2489672}.
|
3,212,635,537,416 | arxiv | \section{Introduction}
The degenerate affine Hecke algebra (dAHA) of any Coxeter group
was defined by Drinfeld and Lusztig(\cite{Dri},\cite{Lus}).
It is generated by the group algebra of the Coxeter group and by the
commuting generators $y_{i}$ with some relations.
The degenerate double affine Hecke algebra (dDAHA) of a root system
was introduced by Cherednik (see \cite{Ch}). It is generated by the the
group algebra of the Weyl group, commuting generators $y_{i}$, and
by another kind of commuting generators $X_{i}$ with some relations.
The dAHA corresponding to the Weyl group can be realized
as a subalgebra of dDAHA, generated by the Weyl group and the
elements $y_i$.
The paper \cite{AS} gives a Lie-theoretic construction of
representations of the dAHA of type $A_{n-1}$. Namely (see
\cite{CEE}, section 9), for every $\mathfrak{sl}_N$-bimodule
$M$, an action of the dAHA $\mathcal{H}$ of type $A_{n-1}$ is
constructed on the space
\begin{equation*}
F_n(M):=(M\otimes (\mathbb{C}^{N})^{\otimes n})^{\mathfrak{sl}_N},
\end{equation*}
where the invariants are taken with respect to the adjoint action
of $\mathfrak{sl}_N$ on $M$.
This construction is upgraded to a Lie-theoretic construction of
representations of dDAHA of type $A_{n-1}$
\cite{CEE}, Section 9. Namely, for any
$\mathcal{D}$-module $M$ on $SL_{N}$,
the paper \cite{CEE} constructs an action of dDAHA $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ with parameter
$k=N/n$ on the space $F_n(M)$, such that the induced action of
the dAHA $\mathcal{H}\subset \hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$
coincides with the action of \cite{AS}, obtained by regarding $M$ as an
$\mathfrak{sl}_N$-bimodule via left- and right-invariant vector
fields on $SL_{N}$.
The main result of this paper is an analog of the constructions
of \cite{AS} and \cite{CEE} for dAHA and dDAHA of type $BC_n$,
which gives a method of obtaining representations of these
algebras from Lie theory. Specifically, given a
module $M$ over the Lie algebra
$\mathfrak{g}:=\mathfrak{gl}_N$, we first construct an action of
the dAHA $\mathcal{H}$ of type $B_{n}$ on the space
$F_{n,p,\mu}(M)$ of $\mu$-invariants in $M\otimes
(\mathbb{C}^{N})^{\otimes n}$ under the subalgebra
$\mathfrak{k}_0:=(\mathfrak{gl}_p\oplus \mathfrak{gl}_q)\cap
\mathfrak{sl}_N\subset \mathfrak{g}$, where $q=N-p$, and
$\mu\in \Bbb C$ is a parameter (here by $\mu$-invariants we mean
eigenvectors of $\mathfrak{k}_0$ with eigenvalues given by
the character $\mu\chi$, where $\chi$ is a basic character of
$\mathfrak{k}_0$). In this construction, the
parameters of $\mathcal{H}$ are certain explicit functions of
$\mu$ and $p$. Thus we obtain an functor $F_{n,p,\mu}$ from the
category of $\mathfrak{gl}_N$-modules to the category of
representations of $\mathcal{H}$. It is easy to see that this
functor factors through the category of Harish-Chandra modules
for the symmetric pair $(\mathfrak{gl}_N,\mathfrak{gl}_p\oplus
\mathfrak{gl}_q)$, so it suffices to restrict our attention to
Harish-Chandra modules.
In particular,
the resulting functor after this restriction, i.e. the functor
from the category of Harish-Chandra modules to the category of
representations of $\mathcal{H}$, is exact.
\footnote{The functor $F_{n,p,\mu}$ is not exact without this restriction.
The reason is that
the functor of $\mathfrak{g}$-invariants for a semisimple Lie algebra $\mathfrak{g}$ is only exact
on the category of locally finite $\mathfrak{g}$-modules.}
Then we upgrade this construction to one giving representations
of dDAHA $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ of type $BC_n$. Namely, let $G=GL_N$, and
$K=GL_p\times GL_q\subset G$. Then for any
${\lambda}$-twisted $\mathcal{D}$-module $M$ on $G/K$ we
construct an action of the dDAHA $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ of type $BC_n$ on
the space $F_{n,p,\mu}(M)$. In this construction, the parameters
of $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ are certain explicit functions of $\lambda$, $\mu$, and
$p$. Moreover, the underlying representation of $\mathcal{H}$
coincides with the representation obtained in the previous
construction, if we regard $M$ as a $\mathfrak{gl}_N$-module
via the vector fields corresponding to the action of $G$ on
$G/K$. Thus we obtain an functor $F^\lambda_{n,p,\mu}$ from the
category of $\lambda$-twisted $\mathcal{D}$-modules on $G/K$ to
the category of representations of $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$. This functor factors
through the category of $K$-monodromic twisted
$\mathcal{D}$-modules, so it suffices to restrict our attention
to such $\mathcal{D}$-modules. In particular,
the resulting functor after this restriction, i.e. the functor
from the category of $K$-monodromic twisted
$\mathcal{D}$-modules to the category of representations of $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$, is exact.
\footnote{Similarly to the affine case, $F^\lambda_{n,p,\mu}$ is not exact without this restriction.}
Since the appearance of the first version of this paper,
the functor $F_{n,p,\mu}$ has been studied in the followup paper
\cite{M} by the third author. In this paper, it was shown that
the principal series representations of $U(p,q)$ are mapped by
the $F_{n,p,\mu}$ to certain induced modules over the dAHA.
This is analogous to the result of \cite{AS}, where it is shown that
in the type $A$ case, standard modules go to standard modules.
We expect that further careful study of the functors $F_{n,p,\mu}$ and
$F_{n,p,\mu}^\lambda$ will reveal an interesting new connection
between, on the one hand, the representation theory of
(the universal cover of) the Lie group $U(p,q)$ and the theory of
monodromic $\mathcal{D}$-modules on the (complexification of the)
corresponding symmetric space $U(p,q)/U(p)\times U(q)$, and, on
the other hand, the representation theory of dAHA and dDAHA of
type $BC_n$. We plan to discuss this in future publications.
The paper is organized as follows. In Section 2, we recall the
definitions of dAHA and dDAHA of a root system. In Section 3,
we write down the explicit definitions for dAHA and dDAHA of type
$BC_{n}$. In Section 4, we construct the
functor $F_{n,p,\mu}$. In Section 5, we construct the functor
$F_{n,p,\mu}^\lambda$. In Section 6, we study some properties of
this functor.
\section{Definitions and notations}
In this section, we recall the definitions
of degenerate affine and double affine Hecke algebras.
For more details, see \cite{Ch}.
Let $\mathfrak{h}$ be a finite dimensional real vector space with a
positive definite symmetric bilinear form $(\cdot,\cdot)$. Let
$\{\epsilon_{i}\}$ be a basis for $\mathfrak{h}$ such that
$(\epsilon_{i},\epsilon_{j})=\delta_{ij}$. Let $R$ be an irreducible root
system in $\mathfrak{h}$ (possibly non-reduced).
Let $R_{+}$ be the set of positive roots of $R$, and
let $\Pi=\{\alpha_{i}\}$ be the set of simple roots.
For any root $\alpha$, the corresponding
coroot is $\alpha^{\vee}=2\alpha/(\alpha,\alpha)$.
Let $Q$ and $Q^{\vee}$ be the root lattice and the coroot lattice. Let
$P=\text{Hom}_{\mathbb{Z}}(Q^{\vee},\mathbb{Z})$ be the weight lattice.
Let $W$ be the Weyl group of $R$.
Let $\Sigma$ be the set of reflections in $W$. Let
$S_\alpha\in \Sigma$ be the reflection corresponding to the root
$\alpha$. In particular, we write $S_i$ for the simple
reflections $S_{\alpha_i}$.
If $\alpha$ is a root, let $\nu_\alpha=1$ if $\alpha$ an indivisible root,
and $\nu_\alpha=2$ otherwise.
Let us define dAHA.
Let $\kappa: \Sigma\to \Bbb C$ be a conjugation invariant
function.
\begin{definition}
The {\em degenerate affine Hecke algebra (dAHA)}
$\mathcal{H}(\kappa)$ is the quotient of the free product
$\Bbb CW * S\mathfrak{h}$ by the
relations
$$
S_i y-y^{S_i} S_i=\kappa(S_i)\alpha_i(y),\quad y\in \mathfrak{h}.
$$
\end{definition}
For any $a\ne 0$, multiplication of $y$-generators by $a$ defines an isomorphism of $\mathcal{H}(\kappa)$ with $\mathcal{H}(a\kappa)$
(under this transformation, nothing happens to $\alpha_{i}(y)$ since they are just numbers).
Thus in the simply laced situation, there is only one
nontrivial case, $\kappa=1$, and in the non-simply laced case,
the function $\kappa$ takes
two values $\kappa_1,\kappa_2$ (the values of $\kappa$ on
the root reflections for long and short indivisible roots,
respectively), and the algebra depends only
on the ratio $\kappa_2/\kappa_1$ (unless both values are zero).
Now let us define dDAHA.
Let $k: R\to \Bbb C$, $\alpha\mapsto k_\alpha$, be a
function such that $k_{g(\alpha)}=k_{\alpha}$ for all $g\in
W$. Let $t\in \Bbb C$.
\begin{definition}
For $\epsilon\in \mathfrak{h}$,
define the Dunkl-Cherednik operator
\begin{equation*}
D_{\epsilon}(t,k)=t\partial_{\epsilon}-\sum_{\alpha\in R_{+}}
\frac{k_{\alpha}\alpha(\epsilon)}{1-e^{-\alpha}}(1-S_{\alpha})
+\rho(k)(\epsilon),
\end{equation*}
where $\partial_{\epsilon}$ is the differentiation along
$\epsilon$, and $\rho(k)=\frac{1}{2}\sum_{\alpha\in R_{+}}k_{\alpha}\alpha$.
This operator acts on the space $E$
of trigonometric polynomials on $\mathfrak{h}/Q^\vee$.
\end{definition}
An important property of the operators $D_{\epsilon}$ is that
they commute with each other.
\begin{definition}
The {\em degenerate double affine Hecke algebra (dDAHA)}
$\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}(t,k)$ is generated by $W$, the Dunkl-Cherednik operators,
and the elements $e^{\lambda}\quad(\lambda\in P)$.
\end{definition}
\begin{remark}
This is not the original definition of dDAHA. But it is
equivalent to the original definition by a theorem of Cherednik,
see \cite{Ch}.
\end{remark}
Obviously, for any $a\ne 0$, we have a natural isomorphism
between $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}(t,k)$ and $\mathcal{H}(at,ak)$;
thus, there are only two essentially different cases:
$t=0$ (the classical case) and $t=1$ (the quantum case).
The following proposition can be proved by a straightforward
computation.
\begin{proposition} (see \cite{Ch})
The subalgebra of the dDAHA $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}(t,k)$ generated by $W$ and the
Dunkl-Cherednik operators is isomorphic to the dAHA
$\mathcal{H}(\kappa)$, where $\kappa(S)=\sum_{\alpha:
S=S_\alpha}k_\alpha\nu_\alpha$.
\end{proposition}
\section{Type $BC_n$ $d$AHA and $d$DAHA}
\subsection{Definitions of the type $BC_n$ dAHA and dDAHA}
Let us now describe the dAHA and dDAHA of type $BC_n$ more
explicitly.
Let $\mathfrak{h}$ be a real vector space of dimension $n$ with
orthonormal basis $\epsilon_1, \ldots, \epsilon_n$.
We will identify $\mathfrak{h}$ with its dual by the bilinear
form, and set $X_{i}=e^{\epsilon_{i}}$, $y_{i}=D_{\epsilon_{i}}$.
The roots of
type $BC_{n}$ are
\begin{equation*}
R=\{\pm\epsilon_{i}\}\cup\{\pm2\epsilon_{i}\}
\cup\{\pm\epsilon_{i}\pm\epsilon_{j}\}_{i\neq j},
\end{equation*}
and the positive roots are
\begin{equation*}
R_{+}=\{\epsilon_{i}\}\cup\{2\epsilon_{i}\}\cup\{\epsilon_{i}\pm\epsilon_{j}\}_{i< j}.
\end{equation*}
The function $\kappa$ considered in the previous section
reduces to two parameters $\kappa=(\kappa_1,\kappa_2)$,
while the function $k$ reduces
to three parameters $k=(k_{1},k_{2},k_{3})$
corresponding to the three kinds of positive roots: those of
lengths $2,1,4$, respectively. Namely, $k_1=k_{\epsilon_i-\epsilon_j}$,
$k_2=k_{\epsilon_i}$, $k_3=2k_{2\epsilon_i}$.
Let $W=S_n\ltimes (\Bbb Z/2\Bbb Z)^n$
be the Weyl group of type $BC_{n}$.
We denote by $S_{ij}$ the reflection in this group
corresponding to the root $\epsilon_i-\epsilon_j$,
and by $\gamma_{i}$ the reflection corresponding to
$\epsilon_{i}$. Then $W$ is generated by
$S_{i}=S_{i,i+1}, i=1,\ldots, n-1 $ and $\gamma_{n}$.
The type $BC_{n}$
dAHA $\mathcal{H}(\kappa_1,\kappa_2)$
is then defined as follows:
\begin{itemize}
\item generators: $y_{1},\ldots,y_{n}$
and $\mathbb{C}W$;
\item relations:
\begin{enumerate}
\item[i)] $S_{i}$ and $\gamma_{n}$ satisfy the Coxeter relations;
\item[ii)] $S_{i}y_{i}-y_{i+1}S_{i}=\kappa_{1}$,
$[S_{i},y_{j}]=0, (j\neq i,i+1)$;
\item[iii)] $\gamma_{n}y_{n}+y_{n}\gamma_{n}=\kappa_2$,
$[\gamma_{n},y_{j}]=0, (j\neq n)$;
\item[iv)] $[y_{i},y_{j}]=0$.
\end{enumerate}
\end{itemize}
On the other hand, the type $BC_n$
dDAHA $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}(t,k_{1},k_{2},k_{3})$ is
defined as follows:
\begin{itemize}
\item generators: $X_{1}, \ldots, X_{n}$, $y_{1},\ldots,y_{n}$
and $\mathbb{C}W$;
\item relations:
\begin{enumerate}
\item[i)] $S_{i}$ and $\gamma_{n}$ satisfy the Coxeter relations;
\item[ii)] $S_{i}X_{i}-X_{i+1}S_{i}=0$, $[S_{i},X_{j}]=0, (j\neq i,i+1)$;
\item[iii)] $S_{i}y_{i}-y_{i+1}S_{i}=k_{1}$, $[S_{i},y_{j}]=0, (j\neq i,i+1)$;
\item[iv)] $\gamma_{n}y_{n}+y_{n}\gamma_{n}=k_{2}+k_{3}$,
$\gamma_{n}X_{n}=X_{n}^{-1}\gamma_{n}$,
\\$[\gamma_{n},y_{j}]=[\gamma_{n},X_{j}]=0, (j\neq n)$;
\item[v)] $[X_{i},X_{j}]=[y_{i},y_{j}]=0$;
\item[vi)]$[y_{j},X_{i}]=k_{1}X_{i}S_{ij}-k_{1}X_{i}S_{ij}\gamma_{i}
\gamma_{j}$, \\ $[y_{i},X_{j}]=k_{1}X_{i}S_{ij}-k_{1}X_{j}S_{ij}\gamma_{i}\gamma_{j},(i<j)$;
\item[vii)]
\begin{eqnarray*}
[y_{i},X_{i}]
&=&tX_{i}-k_{1}X_{i}\sum_{k>i}S_{ik}-k_{1}\sum_{k<i}S_{ik}X_{i}-k_{1}X_{i}\sum_{k\neq i}S_{ik}\gamma_{i}\gamma_{k}\\
&&\qquad- (k_{2}+k_{3})X_{i}\gamma_{i}-k_{2}\gamma_{i}.
\end{eqnarray*}
\end{enumerate}
\end{itemize}
In particular, we see that the subalgebra in the dDAHA generated
by $W$ and $y_i$ is $\mathcal{H}(\kappa_1,\kappa_2)$, where
$\kappa_1=k_1$ and $\kappa_2=k_2+k_3$.
\subsection{Another set of generators of dAHA of type $BC_{n}$}
The advantage of the generators $y_i$ is that they commute with
each other, but their disadvantage is that they do not change
according to the standard representation of the Weyl group.
It turns out that it is possible (and useful in some situations,
including one of this paper) to trade the first property for the
second one, by replacing the generators $y_i$ by their shifted
versions $\tilde y_i$, passing from Lusztig's presentation
of dAHA (\cite{Lus}) to Drinfeld's one (\cite{Dri}).
\footnote{See \cite{RS} for more details.}
Namely, for each $i=1,\ldots, n$, define
\begin{equation*}
\tilde{y}_{i}=y_{i}-\frac{\kappa_{2}}{2}\gamma_{i}-
\frac{\kappa_{1}}{2}\sum_{k>i}S_{ik}+\frac{\kappa_{1}}{2}\sum_{k<i}S_{ik}-
\frac{\kappa_{1}}{2}\sum_{i\neq k}S_{ik}\gamma_{i}\gamma_{k}.
\end{equation*}
\begin{lemma}\label{rel-tildey1}
The type $BC_{n}$ dAHA $\mathcal{H}(\kappa_{1},\kappa_{2})$ is generated by
$w\in W$ and $\tilde{y}_{i}$ with the following relations:
\begin{enumerate}
\item[i)] $S_{i}\tilde{y}_{i}-\tilde{y}_{i+1}S_{i}=0$,
\quad $S_{j}\tilde{y}_{i}-\tilde{y}_{i}S_{j}=0$, $(i\neq j)$;
\item[ii)] $\tilde{y}_{n}\gamma_{n}+\gamma_{n}\tilde{y}_{n}=0$,
\quad $\tilde{y}_{i}\gamma_{n}-\gamma_{n}\tilde{y}_{i}=0$, $(i\neq n)$;
\item[vi)]
\begin{eqnarray*}\label{ycom1}
[\tilde{y}_i, \tilde{y}_j]
&=& \frac{\kappa_{1}\kappa_{2}}{2}S_{ij}(\gamma_{j}
-\gamma_{i})
+ \frac{\kappa_{1}^{2}}{4}\sum_{k \neq i, j}S_{jk}S_{ik}
- \frac{\kappa_{1}^{2}}{4}\sum_{k \neq i, j}S_{ik}S_{jk} \\
&& \quad + \frac{\kappa_{1}^{2}}{4}
\sum_{k \neq i, j}S_{ik}S_{jk}
(-\gamma_i\gamma_k + \gamma_i\gamma_j+\gamma_j\gamma_k)\\
&&
\qquad- \frac{\kappa_{1}^{2}}{4}\sum_{k \neq i, j}
S_{jk}S_{ik}(\gamma_i\gamma_j-\gamma_j\gamma_k+\gamma_i\gamma_k).
\end{eqnarray*}
\end{enumerate}
\end{lemma}
\begin{proof}
The proof is contained in the proof of Lemma \ref{rel-tildey}
given in the next subsection.
\end{proof}
\subsection{Another set of generators of dDAHA of type $BC_{n}$}
Similarly to the previous subsection,
for each $i=1,\ldots, n$, define
\begin{equation*}
\tilde{y}_{i}=y_{i}-\frac{k_{2}+k_{3}}{2}\gamma_{i}-
\frac{k_{1}}{2}\sum_{k>i}S_{ik}+\frac{k_{1}}{2}\sum_{k<i}S_{ik}-
\frac{k_{1}}{2}\sum_{i\neq k}S_{ik}\gamma_{i}\gamma_{k}
\end{equation*}
(together with the Weyl group, the elements $\tilde y_i$
generate dAHA in Drinfeld's presentation \cite{Dri}).
\begin{lemma}\label{rel-tildey}
The type $BC_{n}$ $d$DAHA $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}(t,k_{1},k_{2},k_{3})$ is generated by
$w\in W$, $X_{i}$ and $\tilde{y}_{i}$ with the following relations:
\begin{enumerate}
\item[i)] the relations among elements of $W$ and $X_{i}$ are the same as before;
\item[ii)] $S_{i}\tilde{y}_{i}-\tilde{y}_{i+1}S_{i}=0$,
\quad $S_{j}\tilde{y}_{i}-\tilde{y}_{i}S_{j}=0$, $(i\neq j)$;
\item[iii)] $\tilde{y}_{n}\gamma_{n}+\gamma_{n}\tilde{y}_{n}=0$,
\quad $\tilde{y}_{i}\gamma_{n}-\gamma_{n}\tilde{y}_{i}=0$, $(i\neq n)$;
\item[iv)] $[\tilde{y}_{j},X_{i}]=\dfrac{k_{1}}{2}(X_{i}+X_{j})S_{ij}-
\dfrac{k_{1}}{2}(X_{i}+X_{j}^{-1})S_{ij}\gamma_{i}\gamma_{j}$,
$(i\ne j)$;
\item[v)]
\begin{eqnarray*}
[\tilde{y}_{i},X_{i}]
& = & tX_{i}-\frac{k_{2}+k_{3}}{2}(X_{i}^{-1}+X_{i})\gamma_{i}-
k_{2}\gamma_{i}\\
&&\qquad-\frac{k_{1}}{2}\sum_{k\neq i}((X_{i}+X_{k})S_{ik}+(X_{i}+X_{k}^{-1})S_{ik}\gamma_{i}\gamma_{k});
\end{eqnarray*}
\item[vi)]
\begin{eqnarray*}\label{ycom}
[\tilde{y}_i, \tilde{y}_j]
&=& \frac{k_{1}(k_{2}+k_{3})}{2}S_{ij}(\gamma_{j}
-\gamma_{i})
+ \frac{k_{1}^{2}}{4}\sum_{k \neq i, j}S_{jk}S_{ik}
- \frac{k_{1}^{2}}{4}\sum_{k \neq i, j}S_{ik}S_{jk} \\
&& \quad + \frac{k_{1}^{2}}{4}\sum_{k \neq i, j}S_{ik}S_{jk}(-\gamma_i\gamma_k + \gamma_i\gamma_j+\gamma_j\gamma_k)\\
&&
\qquad- \frac{k_{1}^{2}}{4}\sum_{k \neq i, j}S_{jk}S_{ik}(\gamma_i\gamma_j-\gamma_j\gamma_k+\gamma_i\gamma_k), (i \ne j).
\end{eqnarray*}
\end{enumerate}
\end{lemma}
\begin{proof}
Only the last relation is nontrivial. Its proof is by a direct
computation. To make formulas more compact,
set
$$
R_{i}=-\frac{k_{1}}{2}\sum_{k>i}S_{ik}
+\frac{k_{1}}{2}\sum_{k<i}S_{ik}
-\frac{k_{1}}{2}\sum_{i\neq k}S_{ik}\gamma_{i}\gamma_{k},
$$
then we have
$$
y_{i}=\tilde{y}_{i}+\frac{k_{2}+k_{3}}{2}\gamma_{i}
-R_{i}.
$$
Assume $i<j$, then we have
\begin{eqnarray*}
&&[y_{i},y_{j}]\\
&&\quad=[\tilde{y}_{i},\tilde{y}_{j}]-[\tilde{y}_{i},R_{j}]-[R_{i},\tilde{y}_{j}]-\frac{k_{2}+k_{3}}{2}([\gamma_{i},R_{j}]+[R_{i},\gamma_{j}])
+[R_{i},R_{j}].
\end{eqnarray*}
Since
\begin{eqnarray*}
&&[\tilde{y}_{i},R_{j}]
=\frac{k_{1}}{2}S_{ij}\tilde{y}_{j}
-\frac{k_{1}}{2}S_{ij}\tilde{y}_{i}
-\frac{k_{1}}{2}S_{ij}\tilde{y}_{j}\gamma_{i}\gamma_{j}
-\frac{k_{1}}{2}S_{ij}\tilde{y}_{i}\gamma_{i}\gamma_{j},\\
&&[R_{i},\tilde{y}_{j}]
=\frac{k_{1}}{2}S_{ij}\tilde{y}_{i}
-\frac{k_{1}}{2}S_{ij}\tilde{y}_{j}
+\frac{k_{1}}{2}S_{ij}\tilde{y}_{j}\gamma_{i}\gamma_{j}
+\frac{k_{1}}{2}S_{ij}\tilde{y}_{i}\gamma_{i}\gamma_{j},\\
&&[\gamma_{i},R_{j}]
=k_{1}S_{ij}\gamma_{j}
-k_{1}S_{ij}\gamma_{i},\\
&&[R_{i},\gamma_{j}]
=0,
\end{eqnarray*}
we have
$$[\tilde{y}_{i},\tilde{y}_{j}]
=\frac{k_{1}(k_{2}+k_{3})}{2}S_{ij}(\gamma_{j}
-\gamma_{i})
-[R_{i},R_{j}].
$$
By direct computation, we have $[R_i,R_j]=\frac{k_1^2}{4}R_{ij}$, where
\begin{eqnarray*}
R_{ij}
&=& \sum_{k>j}S_{ij}S_{jk}
- \sum_{k>j}S_{jk}S_{ij}
+ \sum_{k>j} S_{ik}S_{jk}
- \sum_{k>j}S_{jk}S_{ik} \\
&&-\mathop{\sum_{i<k<j}}_{\text{or }k>j}S_{ik}S_{ij}
+ \mathop{\sum_{i<k<j}}_{ \text{or }k>j}S_{ij}S_{ik}
- \mathop{\sum_{k<i\text{ or}}}_{i<k<j}S_{ij}S_{jk}
+ \mathop{\sum_{k<i\text{ or}}}_{i<k<j}S_{jk}S_{ij}\\
&&-\sum_{i<k<j}S_{ik}S_{jk}
+ \sum_{i<k<j}S_{jk}S_{ik}
+ \mathop{\sum_{i<k<j}}_{\text{or }k>j}S_{ik}S_{ij}\gamma_i\gamma_j\\
&&- \mathop{\sum_{i<k<j}}_{\text{or }k>j}S_{ij}S_{ik}\gamma_j\gamma_k
+\mathop{\mathop{\sum_{k<i\text{ or}}}_{i<k<j}}_{\text{or }k>j}S_{ij}S_{jk}\gamma_j\gamma_k
- \mathop{\mathop{\sum_{k<i \text{ or}}}_{i<k<j}}_{\text{or }k>j} S_{jk}S_{ij}\gamma_i\gamma_k\\
&&+\mathop{\sum_{i<k<j}}_{\text{or }k>j}S_{ik}S_{jk}\gamma_j\gamma_k
- \mathop{\sum_{i<k<j}}_{\text{or }k>j}S_{jk}S_{ik}\gamma_i\gamma_j\\
&&+ \sum_{k<i}S_{ik}S_{ij}
- \sum_{k<i}S_{ij}S_{ik}
+ \sum_{k<i} S_{ik}S_{jk}
- \sum_{k<i}S_{jk}S_{ik}\\
&& -\sum_{k<i}S_{ik}S_{ij}\gamma_i\gamma_j
+ \sum_{k<i}S_{ij}S_{ik}\gamma_j\gamma_k
- \sum_{k<i}S_{ik}S_{jk}\gamma_j\gamma_k
+ \sum_{k<i} S_{jk}S_{ik}\gamma_i\gamma_j\\
&&+ \sum_{k>j}S_{ij}S_{jk}\gamma_i\gamma_k
- \sum_{k>j}S_{jk}S_{ij}\gamma_i\gamma_j
+ \sum_{k>j}S_{ik}S_{jk}\gamma_i\gamma_j
- \sum_{k>j}S_{jk}S_{ik}\gamma_i\gamma_k \\
&&- \mathop{\mathop{\sum_{k<i \text{ or}}}_{i<k<j}}_{\text{or }k>j}S_{ik}S_{ij}\gamma_j\gamma_k
+ \mathop{\mathop{\sum_{k<i \text{ or}}}_{i<k<j}}_{\text{or }k>j}S_{ij}S_{ik}\gamma_i\gamma_k
- \mathop{\sum_{k<i \text{ or}}}_{i<k<j}S_{ij}S_{jk}\gamma_i\gamma_k \\
&&+ \mathop{\sum_{k<i \text{ or}}}_{i<k<j}S_{jk}S_{ij}\gamma_i\gamma_j
-\mathop{\sum_{k<i \text{ or}}}_{i<k<j}S_{ik}S_{jk}\gamma_i\gamma_j
+ \mathop{\sum_{k<i \text{ or}}}_{i<k<j}S_{jk}S_{ik}\gamma_i\gamma_k \\
&&+ \sum_{k \neq i, j}S_{ik}S_{ij}\gamma_i\gamma_k
- \sum_{k \neq i, j}S_{ij}S_{ik}\gamma_i\gamma_j
+ \sum_{k \neq i, j}S_{ij}S_{jk}\gamma_i\gamma_j \\
&&- \sum_{k \neq i, j}S_{jk}S_{ij}\gamma_j\gamma_k
+ \sum_{k \neq i, j} S_{ik}S_{jk}\gamma_i\gamma_k
- \sum_{k \neq i, j}S_{jk}S_{ik}\gamma_j\gamma_k
\end{eqnarray*}
\begin{eqnarray*}
&=&-\sum_{k \neq i, j}S_{jk}S_{ik}
+ \sum_{k \neq i, j}S_{ik}S_{jk} \\
&& - \sum_{k \neq i, j}S_{ik}S_{jk}(-\gamma_i\gamma_k - \gamma_i\gamma_j+\gamma_j\gamma_k)
+ \sum_{k \neq i, j}S_{jk}S_{ik}(\gamma_i\gamma_j-\gamma_j\gamma_k+\gamma_i\gamma_k).
\end{eqnarray*}
Thus, from the commutativity of $y_{i}$ and the above, we get
\begin{eqnarray*}
[\tilde{y}_i, \tilde{y}_j]
&=& \frac{k_{1}(k_{2}+k_{3})}{2}S_{ij}(\gamma_{j}
-\gamma_{i})
+ \frac{k_{1}^{2}}{4}\sum_{k \neq i, j}S_{jk}S_{ik}
- \frac{k_{1}^{2}}{4}\sum_{k \neq i, j}S_{ik}S_{jk} \\
&& \quad + \frac{k_{1}^{2}}{4}\sum_{k \neq i, j}S_{ik}S_{jk}(-\gamma_i\gamma_k + \gamma_i\gamma_j+\gamma_j\gamma_k)\\
&&
\qquad- \frac{k_{1}^{2}}{4}\sum_{k \neq i, j}S_{jk}S_{ik}(\gamma_i\gamma_j-\gamma_j\gamma_k+\gamma_i\gamma_k),
\end{eqnarray*}
as desired.
\end{proof}
\section{Construction of the functor $F_{n,p,\mu}$}
\subsection{Notations}
We will use the following notations. Let $p,q\in \mathbb{N}$
and let $N=p+q$.
Let $E_{ij}$ be the $N$ by $N$ matrix which has a $1$ at the
$(i,j)$-th position and 0 elsewhere. Set
$J=\left(\begin{array}{cc}I_p & \\ & -I_q\end{array}\right)$,
where $I_p$ is the identity matrix of size $p$.
Let $\mathfrak{g}=\mathfrak{gl}_N(\mathbb{C})$
be the Lie algebra of $G=GL_N(\mathbb{C})$. Let
$\mathfrak{k}=\mathfrak{gl}_p(\Bbb C)\times\mathfrak{gl}_q(\Bbb
C)$ be the Lie algebra of $K=GL_p(\mathbb{C})\times GL_q(\mathbb{C})$. Let
$\mathfrak{k}_{0}$ be the subalgebra of trace zero
matrices in $\mathfrak{k}$.
Define a character $\chi$ of $\mathfrak{k}$ by
$$\chi(\left(\begin{array}{cc}X_1 & 0 \\0 &
X_2\end{array}\right))=
q\text{tr}\, X_{1}-p\text{tr}\, X_{2}.$$
For this $\chi$, we have the following obvious lemma.
\begin{lemma}\label{chi-mu}
We have $\chi(E_{ij})=0$ for $i\neq j$, and
$$
\chi (E_{ii})=\left\{\begin{array}{cc}q, & i\leq p; \\
&\\
-p ,& i> p.\end{array}\right.
$$
In particular, $\chi(I_N)=0$.
\end{lemma}
\begin{remark}
The property $\chi(I_N)=0$ is very important for the future
discussion. In fact, this is also the reason why we choose such $\chi$.
\end{remark}
We will also use the following summation notations:
$$
\sum_{i,\ldots,j}=\sum_{i=1}^{n}\cdots \sum_{j=1}^{n},\quad
\sum_{i\ldots j}=\sum_{i=1}^{p}\cdots
\sum_{j=1}^{p}+\sum_{i=p+1}^{n}\cdots\sum_{j=p+1}^{n},
$$
$$
\sum_{i\ldots j|k\ldots
l}=\sum_{i=1}^{p}\cdots\sum_{j=1}^{p}\sum_{k=p+1}^{n}\cdots
\sum_{l=p+1}^{n}+\sum_{k=1}^{p}\cdots
\sum_{l=1}^{p}\sum_{i=p+1}^{n}\cdots\sum_{j=p+1}^{n}.
$$
Thus we have two ranges of summation ($[1,p]$ and $[p+1,N]$), and
the indices not separated by anything must be in the same range,
while indices separated by a vertical line must be in
different ranges. Indices separated by a comma are independent.
\subsection{Construction of the functor $F_{n,p,\mu}$}
Let $Y$ be a $\mathfrak{k}_0$-module.
For any $\mu\in \mathbb{C}$,
define the space of $\mu$-invariants
$Y^{\mathfrak{k}_0,\mu}$ to be the space
of those $v\in Y$ for which $xv=\mu\chi(x)v$ for all $x\in
\mathfrak{k}_0$.
Let $V=\mathbb{C}^{N}$ be the vector representation of $\mathfrak{g}$.
Let $M$ be a $\mathfrak{g}$-module. Define
\begin{equation*}
F_{n,p,\mu}(M)=(M\otimes V^{\otimes n})^{\mathfrak{k}_0,\mu}.
\end{equation*}
The Weyl group $W$ acts on $M\otimes V^{\otimes n}$
in the following way: the element $S_{ij}$ acts by exchanging
the $i$-th and $j$-th factors,
and $\gamma_{i}$ acts by multiplying the $i$-th factor by $J$
(here we regard $M$ as the 0-th factor).
Thus we have a natural action of $W$ on
$F_{n,p,\mu}(M)$.
Define elements $\tilde{y}_{k}\in \text{End}(F_{n,p,\mu}(M))$ as follows:
\begin{equation}\label{new-y}
\tilde{y}_{k} =-\sum_{i|j}E_{ij}\otimes(E_{ji})_{k},
\text{ for } k=1,\ldots, n,
\end{equation}
where the first component acts
on $M$ and the second component acts on the $k$-th factor
of the tensor product.
The main result of this section is the following theorem.
\begin{theorem}\label{affi} The above action of $W$ and
the elements $\tilde{y}_k$ given by \eqref{new-y} combine into
a representation of the degenerate affine Hecke algebra
$\mathcal{H}(\kappa_1,\kappa_2)$ (in the presentation of
Lemma \ref{rel-tildey1}) on the space $F_{n,p,\mu}(M)$,
with
\begin{equation}\label{par-rel-1}
\kappa_1=1,\ \kappa_2=p-q-\mu N.
\end{equation}
So we have an functor $F_{n,p,\mu}$
from the the category of
$\mathfrak{g}$-modules to the category of
representations of type $BC_{n}$ dAHA with such parameters.
If we restrict this functor on the category of Harish-Chandra modules,
we get an exact functor.
\end{theorem}
\begin{proof}
Our job is to show that
the elements $\tilde{y}_{k}$, $S_{k}$ and $\gamma_{n}$
satisfy the relations in Lemma \ref{rel-tildey1}.
We only need to prove the commutation
relation between the elements $\tilde{y}_{k}$, since the other
relations are trivial.
Let $a\neq b$ and $\delta_{ij}$ be the identity matrix,
then we have
\begin{eqnarray*
&&[\tilde{y}_a, \tilde{y}_b]\\
&=& \sum_{il|j}E_{il}\otimes( E_{ji})_a\otimes( E_{lj})_b
- \sum_{i|kj}E_{kj}\otimes( E_{ji})_a\otimes( E_{ik})_b\\
&=&\sum_{il|j}(E_{il}-\frac{I_N}{N}\delta_{il})\otimes(E_{ji})_{a}\otimes(E_{lj})_{b}
-\sum_{i|kj}(E_{kj}-\frac{I_N}{N}\delta_{kj})\otimes(E_{ji})_{a}\otimes(E_{ik})_{b}\\
&&\text{(By the $\mu$-invariance and lemma \ref{chi-mu})}\\
&=&((q-p)+\mu N)(\sum_{i\leq p, j>p}1\otimes(E_{ji})_{a}\otimes(E_{ij})_{b}
-\sum_{i>p, j\leq p}1\otimes(E_{ji})_{a}\otimes(E_{ij})_{b})\\
&&-\sum_{a\neq c\neq b}(\sum_{il|j}1\otimes(E_{ji})_{a}\otimes(E_{lj})_{b}\otimes(E_{il})_{c}
-\sum_{i|kj}1\otimes(E_{ji})_{a}\otimes(E_{ik})_{b}\otimes(E_{kj})_{c})
\end{eqnarray*}
\begin{eqnarray*}
&=&\frac{p-q-\mu N}{2}S_{ab}(\gamma_{b}-\gamma_{a})
-\frac{1}{4}\sum_{a\neq c\neq b}(1-\gamma_{a}\gamma_{b}-\gamma_{a}\gamma_{c}+\gamma_{b}\gamma_{c})S_{ac}S_{bc}\\
&&\qquad+\frac{1}{4}\sum_{a\neq c\neq b}(1-\gamma_{a}\gamma_{b}+\gamma_{a}\gamma_{c}-\gamma_{b}\gamma_{c})S_{bc}S_{ac}.
\end{eqnarray*}
Comparing this to the relation in Lemma \ref{rel-tildey1}, we get
the result.
\end{proof}
\subsection{Example}
Consider the example $p=q=1$, $N=2$.
Thus, $\kappa_1=1$, $\kappa_2=-\mu N$.
For the module $M$, let us take the module
${\mathcal F}_{\lambda,\nu}$ of tensor fields
$p(z)z^{\nu/2} (dz/z)^\lambda$, where $p$ is a Laurent polynomial;
the Lie algebra $\mathfrak{gl}_2$ acts in it by infinitesimal fractional
linear transformations of $z$. Then we get
$F_{n,p,\mu}({\mathcal F}_{\lambda,\mu-n})=Y_\lambda$, a representation
of $\mathcal{H}(\kappa_1,\kappa_2)$ of dimension $2^n$, which is isomorphic to
$V^{\otimes n}$ as a $W$-module. The structure of $Y_\lambda$ as
a dAHA-module is discussed in the recent paper \cite{M}.
\section{Construction of the functor $F^\lambda_{n,p,\mu}$}
\subsection{The main theorem}
Let $\lambda\in \Bbb C$.
For $x\in \mathfrak{g}$, let $L_x$ denote the vector field on
$G$ generated by the left action of $x$.
Thus, $(L_xf)(A)=\frac{d}{dt}|_{t=0}f(e^{tx}A)$ for a function
$f$. Note that $L_{[x,y]}=-[L_x,L_y]$ (the minus sign comes from the
fact that left multiplication by elements of $G$ gives rise to a
{\it right} action of $G$ on functions on $G$).
Let $\mathcal{D}^{\lambda}(G/K)$ be the sheaf of differential
operators on $G/K$, twisted by the character $\lambda\chi$.
Local sections of ${\mathcal D}^{\lambda}(G/K)$ act
naturally on $\lambda\chi$-twisted functions on $G/K$, i.e.
analytic functions $f$ on a small open set $U\subset G$
such that \linebreak $R_zf=\lambda\chi(z)f$, $z\in \mathfrak{k}$, where $R_z$ is the
left invariant vector field corresponding to the right
translation by $z$. This action is faithful. Note that we can regard elements
$L_x$ as global sections of $\mathcal{D}^{\lambda}(G/K)$,
with the same commutation law $[L_x,L_y]=-L_{[x,y]}$.
Let $M$ be a $\mathcal{D}^{\lambda}(G/K)$-module.
Then $M$ is naturally a $\mathfrak{g}$-module, via the vector
fields $L_x$. Define
$$
F^\lambda_{n,p,\mu}(M)=
(M\otimes V^{\otimes n})^{\mathfrak{k}_0,\mu}.
$$
Then $F^\lambda_{n,p,\mu}(M)$ is a $W$-module as in the previous
section.
For $k=1,\ldots, n$, define the following linear operators
on the space $F^\lambda_{n,p,\mu}(M)$:
\begin{eqnarray}\label{rep-X-y}
X_{k}=\sum_{i,j}(AJA^{-1}J)_{ij}\otimes(E_{ij})_{k},\quad
\tilde{y}_{k}=\sum_{i|j}L_{ij}\otimes(E_{ji})_{k},
\end{eqnarray}
where $(AJA^{-1}J)_{ij}$ is the function of $A\in G/K$ which takes the $ij$
-th element of $AJA^{-1}J$,
$L_{ij}=L_{E_{ij}}$, and the second component acts on the
$k$-th factor in $V^{\otimes n}$.
From now on, we write $X=AJA^{-1}J$ and $X^{-1}=JAJA^{-1}$.
Thus we have $JX=X^{-1}J$.
The main result of this section is the following theorem.
\begin{theorem}\label{daffi} The above action of $W$ and
the elements in \eqref{rep-X-y} combine into
a representation of the degenerate double affine Hecke algebra
$\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}(t,k_{1},k_{2},k_{3})$ (in the presentation of Lemma \ref{rel-tildey})
on the space $F^\lambda_{n,p,\mu}(M)$, with
\begin{equation}\label{par-rel-2}
t=\dfrac{2n}{N}+(\lambda+\mu)(q-p),\quad k_{1}=1,\quad k_{2}=p-q-\lambda N,\quad k_{3}=(\lambda-\mu)N.
\end{equation}
So we have an functor $F^\lambda_{n,p,\mu}$
from the the category of
$\mathcal{D}^{\lambda}(G/K)$-modules
to the category of representations of the type $BC_{n}$
dDAHA with such parameters.
If we restrict this functor on the category of $K$-monodromic twisted
$\mathcal{D}$-modules, we get an exact functor.
\end{theorem}
Note that the restriction of the representation
$F^\lambda_{n,p,\mu}(M)$ to the affine Hecke algebra
$\mathcal{H}$ clearly
coincides with the representation of Theorem \ref{affi}.
\vskip .05in
{\bf Remark.} This construction is parallel to the
similar construction in type $A$ performed in \cite{CEE}.
The most important difference is that here we use
the functions $(AJA^{-1}J)_{ij}$ on $G/K$ instead of the functions
$A_{ij}$ on $G$ used in \cite{CEE}. The motivation for this is that
the matrix elements of $AJA^{-1}J$ are ``the simplest''
nonconstant algebraic functions on $G/K$, similarly to
how the matrix elements of $A$ are ``the simplest'' nonconstant
algebraic functions on $G$.
\vskip .05in
The rest of this section is devoted to the proof of
Theorem \ref{daffi}.
\subsection{Proof of Theorem \ref{daffi}}
Our job is to show that
the elements $X_{k}$,
$\tilde{y}_{k}$, $S_{k}$ and $\gamma_{n}$ satisfy the relations in Lemma \ref{rel-tildey}.
First of all, the relations in Lemma \ref{rel-tildey} which
don't involve $X_i$ can be established as in the proof of Theorem \ref{affi}
(as \eqref{par-rel-1} is compatible with \eqref{par-rel-2}).
Second, there are some trivial relations:
\begin{eqnarray*}
&[X_{i},X_{j}]=0, \qquad [\gamma_{i}, X_{j}]=0,\quad(i\neq j),\\
&[S_{i},X_{j}]=0,\quad(j\neq i, i+1),\qquad S_{i}X_{i}-X_{i+1}S_{i}=0,\\
&[\gamma_{i},\tilde{y}_{j}]=0,\quad(j\neq i), \qquad \gamma_{i}\tilde{y}_{i}+\tilde{y}_{i}\gamma_{i}=0,
\end{eqnarray*}
and since $JX=X^{-1}J$, we have
$$
\gamma_{i}X_{i}=X_{i}^{-1}\gamma_{i}.
$$
Third, we have the following result.
\begin{lemma}\label{rel-Xy}
We have the following commutation relations: if $m\neq k$ then
\begin{eqnarray*}
&[\tilde{y}_{m},X_{k}] & = \frac{1}{2}(X_{k}+X_{m})S_{mk}-\frac{1}{2}(X_{k}+X_{m}^{-1})S_{mk}\gamma_{m}\gamma_{k},\\
&[\tilde{y}_{m},X_{k}^{-1}] & = -\frac{1}{2}(X_{k}^{-1}+X_{m}^{-1})S_{mk}+\frac{1}{2}(X_{k}^{-1}+X_{m})S_{mk}\gamma_{m}\gamma_{k}.
\end{eqnarray*}
So we have
\begin{equation*}
[\tilde{y}_{m},X_{k}+X_{k}^{-1}] = \frac{1}{2}(X_{k}-X_{k}^{-1}+X_{m}-X_{m}^{-1})S_{mk}+\frac{1}{2}(X_{k}^{-1}-X_{k}+X_{m}-X_{m}^{-1})S_{mk}\gamma_{m}\gamma_{k}.
\end{equation*}
\end{lemma}
\begin{proof}
The proof is by direct computation.
First, we have for $r\leq p<s$ or $s\leq p<r$
\begin{eqnarray*}
L_{rs}(X)_{ij}
=\delta_{sj}(X)_{ir}+\delta_{ri}(X)_{sj}.
\end{eqnarray*}
So
\begin{eqnarray*}
&&[\tilde{y}_{m},X_{k}]\\
&=& \sum_{r|s}\sum_{i,j}L_{rs}(X)_{ij}\otimes(E_{sr})_{m}\otimes(E_{ij})_{k}\\
&=&\sum_{r|s}\sum_{i}(X)_{ir}\otimes(E_{sr})_{m}\otimes(E_{is})_{k}
+\sum_{r|s}\sum_{j}(X)_{sj}\otimes(E_{sr})_{m}\otimes(E_{rj})_{k}\\
&= &\frac{1}{2}(X_{k}+X_{m})S_{mk}-\frac{1}{2}(X_{k}+X_{m}^{-1})S_{mk}\gamma_{m}\gamma_{k}.
\end{eqnarray*}
By a similar method, we can get the other identities.
\end{proof}
Thus, we only need to show that $X_{m}$ and $\tilde{y}_{m}$
satisfy v) in Lemma \ref{rel-tildey} if the parameters satisfy
\eqref{par-rel-2}. Instead of computing $[\tilde{y}_{m},X_{m}]$,
we will compute \linebreak $[\tilde{y}_{m},X_{m}+X_{m}^{-1}]$ and
$[\tilde{y}_{m},X_{m}-X_{m}^{-1}]$.
\subsubsection{Computing $[\tilde{y}_{m},X_{m}+X_{m}^{-1}]$}
Let us define
$$
T=\text{tr}\,(X)=\sum_{i}(X)_{ii}\otimes 1.
$$
Suppose $X=\left(\begin{array}{cc}A_1 & A_2 \\A_3 &
A_4\end{array}\right)$
where $A_{1}$ is a $p$ by $p$ matrix.
Then
\begin{equation}
\text{tr}\,(A_{1})=\text{tr}\,(X(J+1)/2),\qquad \text{tr}\,(A_{4})=\text{tr}\,(X(1-J)/2).
\end{equation}
But $\text{tr}\,(XJ)=\text{tr}\,(AJA^{-1})=\text{tr}\,(J)=p-q$, so we get
\begin{equation}\label{tr12}
\text{tr}\,(A_{1})=\frac{T+p-q}{2},\qquad \text{tr}\,(A_{4})=\frac{T-p+q}{2}.
\end{equation}
\begin{lemma}\label{lemma-T}
We have the relation
$$\sum_{m}(X_{m}+X_{m}^{-1})=(\frac{2n}{N}+(\lambda+\mu)(q-p))T+(\lambda+\mu)(p^{2}-q^{2}).$$
\end{lemma}
\begin{proof}
Since $(X)_{ij}=-(X^{-1})_{ij}$ unless $i,j\leq p$ or $i,j>p$,
and $(X)_{ij}=(X^{-1})_{ij}$ if $i,j\leq p$ or $i,j>p$,
we have
$$
X_{m}+X_{m}^{-1}=\sum_{ij}(X+X^{-1})_{ij}\otimes (E_{ij})_{m}.
$$
Thus we have
\begin{eqnarray}\label{sumx}
&& \sum_m(X_{m}+X_{m}^{-1})\\\nonumber
& = & \sum_m\sum_{ij}(X+X^{-1})_{ij}
\otimes (E_{ij}-\frac{I_N}{N}\delta_{ij})_{m}+\sum_m\sum_{i}
(X+X^{-1})_{ii}\otimes (\frac{I_N}{N})_{m}\\\nonumber
&&\text{(By the $\mu$-invariance and Lemma \ref{chi-mu})}\\\nonumber
& = &Y +\frac{2n}{N}T+\sum_{i\leq p}\mu q(X+X^{-1})_{ii}
\otimes 1-\sum_{i> p}\mu
p(X+X^{-1})_{ii}\otimes 1\\\nonumber
&&\text{(By (\ref{tr12}))}\\\nonumber
&=&Y+(\frac{2n}{N}+\mu(q-p))T+\mu(p^{2}-q^{2}),
\end{eqnarray}
where
$Y=\sum_{ij}(X+X^{-1})_{ij}L_{E_{ij}}\otimes
1$.
It remains to calculate the expression $Y$
in the algebra ${\mathcal D}^{\lambda}(G/K)$.
We can calculate $Y$ by acting with it on
$\lambda\chi$-twisted functions $f$ on $G/K$.
We have
\begin{eqnarray*}
(Yf)(A)&=&\frac{d}{dt}|_{t=0}f(A+t(X+X^{-1})A)\\
&=&
\frac{d}{dt}|_{t=0}f(A+tAJA^{-1}JA+tJAJ)\\
&=&
\frac{d}{dt}|_{t=0}f(A+tA(JA^{-1}JA+A^{-1}JAJ))\\
&=&
\frac{d}{dt}|_{t=0}f(A+tA(X_*+X_*^{-1})),
\end{eqnarray*}
where $X_*:=JA^{-1}JA$. Now, $X_*+X_*^{-1}\in \mathfrak{k}$,
so we have
$$
Yf=\lambda\chi(X_*+X_*^{-1})f=\lambda((q-p)T+(p^2-q^2))f.
$$
Combining this with formula (\ref{sumx}), we obtain the statement
of the lemma.
\end{proof}
Notice that
\begin{eqnarray*}
[\tilde{y}_{m}, T]
& = & \sum_{r|s}L_{rs}(\text{tr}\,(X))\otimes (E_{sr})_{m}\\
& = &\sum_{r|s}((X)_{sr}-(X^{-1})_{sr})\otimes (E_{sr})_{m}\\
& = & X_{m}-X_{m}^{-1}.
\end{eqnarray*}
So from Lemma \ref{rel-Xy} and Lemma \ref{lemma-T}, we have
\begin{eqnarray*}
[\tilde{y}_{m},X_{m}+X_{m}^{-1}]=-\sum_{k\neq
m}[\tilde{y}_{m},X_{k}+X_{k}^{-1}]+(\frac{2n}{N}+
(\lambda+\mu)(q-p))[\tilde{y}_{m},T].
\end{eqnarray*}
Thus, we have obtained
\begin{lemma}\label{xplusx}
\begin{eqnarray}\label{rel-y-x+x}
&&[\tilde{y}_{m},X_{m}+X_{m}^{-1}]\\\nonumber
& = &(\frac{2n}{N}+(\lambda+\mu)(q-p))(X_{m}-X_{m}^{-1})-
\frac{1}{2}\sum_{k\neq m}(X_{k}-X_{k}^{-1}+X_{m}-X_{m}^{-1})S_{mk}\\\nonumber
&&\qquad+\frac{1}{2}\sum_{k\neq m}(X_{k}-X_{k}^{-1}-X_{m}+X_{m}^{-1})S_{mk}\gamma_{m}\gamma_{k}.
\end{eqnarray}
\end{lemma}
\subsubsection{Computing $[\tilde{y}_{m},X_{m}-X_{m}^{-1}]$}
At first, we need the following lemmas for the future computation.
\begin{lemma}\label{lemma-qpsum}
We have the equality
\begin{eqnarray*}
&&q\sum_{s\leq p}\sum_{j}(X+X^{-1})_{sj}\otimes (E_{sj})_{m}+
p\sum_{s>p}\sum_{j}(X+X^{-1})_{sj}\otimes (E_{sj})_{m}\\
&=&\frac{1}{2}(N+(q-p)\gamma_{m})(X_{m}+X_{m}^{-1}).
\end{eqnarray*}
\end{lemma}
\begin{proof}
By direct computation.
\end{proof}
\begin{lemma}\label{lemma-vf-equal}
In $\mathcal{D}^{\lambda}(G/K)$, we have for $r,j\leq p$ or $r,j>p$,
$$L_{[X-X^{-1},E_{rj}]}=-L_{\{X+X^{-1},E_{rj}\}}+2\lambda\chi(Q_{rj}),$$
where $\{a,b\}=ab+ba$ and $Q_{rj}=JA^{-1}JE_{rj}A+A^{-1}E_{rj}JAJ$.
\end{lemma}
\begin{proof}
Let $f(A)$ be a $\lambda\chi$-twisted function on $G/K$,
i.e. an analytic function on a small open set $U\subset G$ such
that $R_zf=\lambda\chi(z)f$, $z\in \mathfrak{k}$.
Then we have
\begin{eqnarray*}
&&L_{[X-X^{-1},E_{rj}]}f(A)\\
&=&\frac{d}{dt}|_{t=0}f(A+t(XE_{rj}+E_{rj}X^{-1})A
-t(X^{-1}E_{rj}+E_{rj}X)A).
\end{eqnarray*}
Notice that
\begin{eqnarray*}
&&f(A+t(XE_{rj}+E_{rj}X^{-1})A
-t(X^{-1}E_{rj}+E_{rj}X)A)\\
&=&f(A+2tAQ_{rj}-t(XE_{rj}+E_{rj}X^{-1})A
-t(X^{-1}E_{rj}+E_{rj}X)A),
\end{eqnarray*}
and $Q_{rj}$ is an element of $\mathfrak{k}$.
So we have
\begin{eqnarray*}
&&\frac{d}{dt}|_{t=0}f(A+tAQ_{rj}-t(X^{-1}E_{rj}+E_{rj}X)A)\\
&=&\frac{d}{dt}|_{t=0}f(A-t(X^{-1}E_{rj}+E_{rj}X)A-t(XE_{rj}+E_{rj}X^{-1})A)+\frac{d}{dt}|_{t=0}f(A+2tAQ_{rj})\\
&=&-L_{\{X+X^{-1},E_{rj}\}}f(A)+2\lambda\chi(Q_{rj})f(A).
\end{eqnarray*}
Thus we get the lemma.
\end{proof}
Now let us compute $[\tilde{y}_{m},X_{m}-X_{m}^{-1}]$.
By the definition and Lemma \ref{lemma-vf-equal}, we have
\begin{eqnarray}\label{eqn-1}
&&[\tilde{y}_{m},X_{m}-X_{m}^{-1}]\\\nonumber
& = & \sum_{r|s}\sum_{i,j}L_{rs}((X)_{ij}-(X^{-1})_{ij})\otimes (E_{sr}E_{ij})_{m}
-\sum_{rj}L_{\{X+X^{-1},E_{jr}\}}\otimes (E_{rj})_{m}\\
&&\qquad+
2\lambda\sum_{rj}\chi(JA^{-1}JE_{rj}A+A^{-1}E_{rj}JAJ)
\otimes (E_{jr})_{m}.\nonumber
\end{eqnarray}
Since we have
\begin{eqnarray*}
L_{rs}((X)_{ij}-(X^{-1})_{ij})
= (X)_{ir}\delta_{sj}+(X)_{sj}\delta_{ir}
+(X^{-1})_{ir}\delta_{sj}+(X^{-1})_{sj}\delta_{ir},
\end{eqnarray*}
by Lemma \ref{lemma-qpsum}, the first summand of \eqref{eqn-1} is
\begin{eqnarray*}
\frac{1}{2}(N+(q-p)\gamma_{m})(X_{m}+X_{m}^{-1})+(1+\gamma_{m})\frac{T-p+q}{2}+(1-\gamma_{m})\frac{T+p-q}{2}.
\end{eqnarray*}
Now let us compute the second summand of \eqref{eqn-1}.
By definition, we have
\begin{eqnarray*}
&&-\sum_{rj}L_{\{X+X^{-1},E_{rj}\}}\otimes (E_{jr})_{m}\\\nonumber
&=&-\sum_{ijr}\left((X+X^{-1})_{ir}L_{E_{ij}}+(X+X^{-1})_{ji}L_{E_{ri}}\right)\otimes(E_{jr})_{m}\\\nonumber
&=&-\sum_{ijr}\left((X+X^{-1})_{ir}L_{E_{ij}-\frac{I_N}{N}\delta_{ij}}+(X+X^{-1})_{ji}L_{E_{ri}-\frac{I_N}{N}\delta_{ri}}\right)\otimes(E_{jr})_{m}\\
&=&-\sum_{ijr}(X+X^{-1})_{ir}\otimes(E_{jr}E_{ij})_{m}-\sum_{k\neq m}\sum_{ijr}(X+X^{-1})_{ir}\otimes(E_{ij})_{k}\otimes(E_{jr})_{m}\\
&&-\sum_{ijr}(X+X^{-1})_{ji}\otimes(E_{jr}E_{ri})_{m}-\sum_{k\neq m}\sum_{ijr}(X+X^{-1})_{ji}\otimes(E_{ri})_{k}\otimes(E_{jr})_{m}\\
&&+\sum_{ijr}(X+X^{-1})_{ir}\otimes(E_{jr}\frac{I_N}{N}\delta_{ij})_{m}+\sum_{k\neq m}\sum_{ijr}(X+X^{-1})_{ir}\otimes(\frac{I_N}{N}\delta_{ij})_{k}\otimes(E_{jr})_{m}\\
&&+\sum_{ijr}(X+X^{-1})_{ji}\otimes(E_{jr}\frac{I_N}{N}\delta_{ri})_{m}+\sum_{k\neq m}\sum_{ijr}(X+X^{-1})_{ji}\otimes(\frac{I_N}{N}\delta_{ri})_{k}\otimes(E_{jr})_{m}\\
&&+\mu((q-p)+N\gamma_{m})(X_{m}+X_{m}^{-1})
\end{eqnarray*}
\begin{eqnarray*}
&=&-T+(q-p)\gamma_{m}-\frac{1}{2}(X_{m}+X_{m}^{-1})(N+(p-q)\gamma_{m})\\
&&\quad-\frac{1}{2}\sum_{k\neq m}(X_{m}+X_{m}^{-1}+X_{k}+X_{k}^{-1})S_{km}(1+\gamma_{k}\gamma_{m})\\
&&\qquad+\frac{2n(X_{m}+X_{m}^{-1})}{N}+\mu((q-p)+N\gamma_{m})(X_{m}+X_{m}^{-1}).
\end{eqnarray*}
Now let us compute the third summand of \eqref{eqn-1}.
\begin{lemma}\label{chiS}
For $r,j\leq p$ or $r,j>p$, we have:
\begin{eqnarray*}
&&\lambda\chi(JA^{-1}JE_{rj}A+A^{-1}E_{rj}JAJ)\\
&=&\left\{\begin{array}{cc}
\dfrac{\lambda(q-p)}{2}(X+X^{-1})_{jr}+\lambda N\delta_{rj}, & r,j\leq p; \\
&\\
\dfrac{\lambda(q-p)}{2}(X+X^{-1})_{jr}-\lambda N\delta_{rj}, & r,j> p. \end{array}\right.
\end{eqnarray*}
\end{lemma}
\begin{proof}
Let us denote
$$
B=A^{-1}E_{rj}JA=\left(\begin{array}{cc}B_{1} & B_{2} \\B_{3} & B_{4}\end{array}\right), \text{ where $B_{1}$ is a $p$ by $p$ matrix.}
$$
Then
$$
JA^{-1}JE_{rj}A+A^{-1}E_{rj}JAJ
=JB+BJ
=\left(\begin{array}{cc}2B_{1} & \\ & -2B_{4}\end{array}\right).
$$
Then we have
$$
2\text{tr}\,(B_{1})-2\text{tr}\,(B_{4})
=(X+X^{-1})_{jr}.
$$
On the other hand we have
$$
\text{tr}\,(B)=\text{tr}\,(B_{1})+\text{tr}\,(B_{4})=\text{tr}\,(A^{-1}E_{rj}JA)=
\left\{\begin{array}{cc}
\delta_{rj}, & rj\leq p; \\
-\delta_{rj},& rj>p. \end{array}\right.
$$
Then we have for $j,r\leq p$,
$$\text{tr}\,(B_{1})=\frac{1}{4}(X+X^{-1})_{jr}
+\frac{1}{2}\delta_{rj},\quad \text{tr}\,(B_{4})=-\frac{1}{4}(X+X^{-1})_{jr}
+\frac{1}{2}\delta_{rj},$$
for $j,r> p$,
$$\text{tr}\,(B_{1})=\frac{1}{4}(X+X^{-1})_{jr}
-\frac{1}{2}\delta_{rj},\quad \text{tr}\,(B_{4})=-\frac{1}{4}(X+X^{-1})_{jr}
-\frac{1}{2}\delta_{rj}.$$
So we get the lemma.
\end{proof}
From Lemma \ref{chiS}, we have:
\begin{eqnarray*}
&&\sum_{rj}\chi(JA^{-1}JE_{rj}A+A^{-1}E_{rj}JAJ)\otimes (E_{jr})_{m}\\
&&\qquad
=\frac{1}{2}(q-p)(X_{m}+X_{m}^{-1})+N\gamma_{m}.
\end{eqnarray*}
Thus, combining the above formulas, we have
\begin{lemma}\label{xminx}
\begin{eqnarray}\label{rel-y-x-x}
&&[\tilde{y}_{m},X_{m}-X_{m}^{-1}]\\\nonumber
&=&(\frac{2n}{N}+(\lambda+\mu)(q-p))(X_{m}+X_{m}^{-1})-\frac{1}{2}\sum_{k\neq m}(X_{k}+X_{k}^{-1})(1+\gamma_{m}\gamma_{k})S_{km}\\\nonumber
&&\qquad-\frac{1}{2}\sum_{k\neq m}(X_{m}+X_{m}^{-1})(1+\gamma_{m}\gamma_{k})S_{km}\\\nonumber
&&\qquad+((q-p)+\mu N)
\gamma_{m}(X_{m}+X_{m}^{-1})+2((q-p)+\lambda N)\gamma_{m}.
\end{eqnarray}
\end{lemma}
\subsubsection{Conclusion}
Adding equations (\ref{rel-y-x-x}) and (\ref{rel-y-x+x}),
and comparing with Lemma \ref{rel-tildey}, we conclude the
proof of Theorem \ref{daffi}.
\section{Action of $F^\lambda_{n,p,\mu}$ on some subcategories}
As we mentioned, the functors $F_{n,p,\mu}$,
$F_{n,p,\mu}^\lambda$ factor through
the category of modules $M$ on which the action of $\mathfrak{k}$ is
locally finite; more precisely,
$F_{n,p,\mu}(M)=F_{n,p,\mu}(M_f)$,
where $M_f$ is the locally finite part of $M$ under $\mathfrak{k}$.
Now let $M$ be a $\mathcal{D}^{\lambda}(G/K)$-module,
locally finite under $\mathfrak{k}$.
The support of such a $\mathcal{D}$-module is a $K$-invariant subset of
$G/K$, i.e. a union of $K$-orbits. Recall that closed $K$-orbits
of $G/K$ are labeled by the points of the categorical quotient
$K\backslash G/K$, i.e. the spectrum of the ring
$R_{p,q}={\mathcal O}(G/K)^K$.
For every point $\psi\in K\backslash G/K$ ($\psi: R_{p,q}\to \Bbb C$), we
can define the subcategory $D^\lambda(\psi)$ of the category
of ${\mathcal D}^{\lambda}$-modules on $G/K$ which are
set-theoretically supported on the preimage of $\psi$ in $G/K$,
i.e. those on which $R_{p,q}$ acts with generalized eigenvalue
$\psi$.
On the other hand, let $\Bbb T=\Bbb C^n/\Bbb Z^n=\mathfrak{h}/Q^\vee$, and
let $\beta\in \Bbb T/W$. Then we can define the category
${\mathcal O}_\beta$ of modules over the dDAHA $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$
on which the subalgebra $\Bbb C[P]^W\subset \hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ acts with
generalized eigenvalue $\beta$.
The following theorem tells us how the functor
$F^\lambda_{n,p,\mu}$ relates $\psi$ and $\beta$.
\begin{theorem}\label{thet}
The functor $F^\lambda_{n,p,\mu}$ maps
$D^{\lambda}(\psi)$ to ${\mathcal O}_\beta$, where $\beta=\theta(\psi)$,\linebreak
and $\theta: K\backslash G/K\to\Bbb T/W$ is the regular map defined by the
formula
$$
\theta^*(\sum_{m=1}^n g(X_m))=ng(1)+
(\frac{n}{N}+\frac{1}{2}(\lambda+\mu)(q-p))\text{tr}\,(g(X)-g(1)),
$$
where $g$ is a
Laurent polynomial in one variable such that $g(Z)=g(Z^{-1})$.
\end{theorem}
\begin{proof}
The proof is obtained by generalizing the proof of
Lemma \ref{lemma-T}.
We'll need the following lemma.
\begin{lemma}
$\text{tr}\,(X^sJ)=p-q$ for any $s\in \Bbb Z$.
\end{lemma}
\begin{proof}
It's easy to see that $X^sJ$ is conjugate to $J$.
\end{proof}
The lemma implies that
$$
\text{tr}\,(X^s(J+1)/2)=\frac{\text{tr}\,(X^s)+p-q}{2},
$$
and
$$
\text{tr}\,(X^s(1-J)/2)=\frac{\text{tr}\,(X^s)-p+q}{2}.
$$
Thus we have
\begin{eqnarray}\label{sumx1}
&& \sum_m g(X_{m})\\\nonumber
& = & \sum_m\sum_{ij}g(X)_{ij}
\otimes (E_{ij}-\frac{I_N}{N}\delta_{ij})_{m}+\sum_m\sum_{i}
g(X)_{ii}\otimes (\frac{I_N}{N})_{m}\\\nonumber
&&\text{(By the $\mu$-invariance and Lemma \ref{chi-mu})}\\\nonumber
& = &Y_g +\frac{n}{N}\text{tr}\,(g(X))+
\sum_{i\leq p}\mu qg(X)_{ii}\otimes 1-\sum_{i> p}\mu pg(X)_{ii}\otimes 1\\\nonumber
&=&Y_g+(\frac{n}{N}+\frac{1}{2}\mu(q-p))\text{tr}\,(g(X))+\frac{1}{2}
\mu(p^{2}-q^{2})g(1),
\end{eqnarray}
where
$Y_g=\sum_{ij}g(X)_{ij}L_{E_{ij}}\otimes
1$.
It remains to calculate the expression $Y_g$
in the algebra ${\mathcal D}^{\lambda}(G/K)$.
We can calculate $Y_g$ by acting with it on
$\lambda\chi$-twisted functions $f$ on $G/K$.
We have
\begin{eqnarray*}
(Y_gf)(A)&=&\frac{d}{dt}|_{t=0}f(A+tg(X)A)\\
&=&\frac{d}{dt}|_{t=0}f(A+tAg(X_*)),
\end{eqnarray*}
where $X_*:=JA^{-1}JA$. Now, $X_*+X_*^{-1}\in \mathfrak{k}$,
so we have
$$
Y_gf=\lambda\chi(g(X_*))f=
\frac{1}{2}\lambda((q-p)\text{tr}\,(g(X))+\frac{1}{2}(p^2-q^2)g(1))f.
$$
Combining this with formula (\ref{sumx1}), we obtain the statement
of the theorem.
\end{proof}
\begin{remark}
In particular, Theorem \ref{thet} implies that
$\theta(1)=1$, where $1\in K\backslash G/K$ is the double coset
of $1$, and $1\in \Bbb T/W$ is the image of the unit of the group $\Bbb T$.
Thus the functor $F^\lambda_{n,p,\mu}$ maps the category $D^{\lambda}(1)$
to the category ${\mathcal O}_1$. Note that $D^{\lambda}(1)$ is the
category of twisted $\mathcal{D}$-modules supported on the ``unipotent
variety'' in $G/K$ (which is equivalent to the category
on $\mathcal{D}$-modules
on $\mathfrak{g}/\mathfrak{k}$ supported on the nilpotent cone),
and ${\mathcal O}_1$ is the category of $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$-modules on which
$X_i$ act unipotently (which is equivalent to category ${\mathcal
O}$ for the rational Cherednik algebra of type $B_n$).
\end{remark}
\begin{remark}
Another very interesting question is how the functor $F_{n,p,\mu}$
transforms the central characters, i.e. how the central character
of a Harish-Chandra module $M$ is related to the central
character of the dAHA module $F_{n,p,\mu}(M)$. This question is
discussed in the paper \cite{M} (for $n=1$).
\end{remark}
\section*{acknowledgments}
{The work of the first author was partially supported by the NSF grant
DMS-0504847. The work of the second and the third author was supported by
the Summer Program of Undergraduate Research in the Department of
Mathematics at MIT. We thank Ju-Lee Kim, David Vogan, and Ting Xue for useful discussions.}
|
3,212,635,537,417 | arxiv | \section{Introduction}
Topological concepts in condensed matter physics have led to the realization of new states of matter.\cite{bansilrev16, *kane10, *qi11, *Moore11} Ongoing generalizations of topological concepts continue to generate profound discoveries. Many of the new predicted emergent properties have been experimentally confirmed, some analogous to concepts originating in particle physics like the Dirac,\cite{wang12,wang13,xuHasan13,LiuZXShen14,xiongOngCA15,neupane14,liuNature14,jeon14,yi14,borisenkoCava14} Weyl,\cite{wan2011, *burkov11, *weng14,*huang15,*lv15,*xu15,*lvn215} and Majorana fermion.\cite{fu08,*mourik12,* nadj14}
In the condensed matter version, Dirac fermions exist in the valence and conduction bands of 3-dimensional (3D) Dirac semimetals, which touch at a pair of points and disperse linearly away from the nodes. These bands derive from 4-fold degenerate band crossings that are protected against gapping by crystal symmetry. If either crystal inversion or time reversal symmetry is broken, each Dirac node splits into a pair of opposite chirality Weyl nodes, topological objects that act as a source or sink of Berry's phase curvature. This topological band structure effect is analogous to opposite-polarity magnetic monopoles residing at the nodes in momentum space, which fundamentally alter the semiclassical equations of motion and Maxwell's constitutive relations.\cite{zhong15,*goswami13,*zyuzin12,*fujikawa79} Some of the unique properties that may be exploited in potential technological applications include Fermi-arc surface states, chiral pumping effects, and magneto-electric-like effects in plasmonics and optics in the absence of an applied field.\cite{xuHasan13,LiuZXShen14,borisenkoCava14,neupane14,xiongOngCA15,xiongOng15,Hofmann14, Hofmann15,ashby14}
Unlike surface probes like photoemission and tunneling spectroscopy, optical measurements probe bulk band structure and carrier dynamics over a broad range of energy scales. In many ways, optical measurements are ideal probes of the bulk electronic properties of 3D Dirac systems. Sensitive measurement of the free carrier response is possible due to the low carrier densities achievable in Dirac semimetals. The Dirac interband transitions extend down to zero frequency as the carrier density becomes vanishingly small.\cite{hosur12, ashby14} This behavior of the interband transitions gives rise to a logarithmic singularity in the static dielectric constant. The logarithmic divergence, analogous to the ultraviolet divergence encountered in quantum electrodynamics, leads to charge renormalization\cite{Hofmann14, Hofmann15} and screening effects.\cite{skinner14, throckmortonPRB15} Another interesting aspect of 3D Dirac systems is the strong electron-electron interactions characterized by the ratio of the Coulomb to kinetic energy equal to an effective fine structure constant $e^2/(\hbar v_F \epsilon)$ that is substantially larger than 1 for typical values of the Fermi velocity $v_F$ and dielectric constant $\epsilon$. This behavior is striking since the interaction strength is independent of carrier density, and has been predicted to give rise to plasmaron modes at finite density that could be optically accessible.\cite{Hofmann14, Hofmann15, tediosi2007} Optical probes are also sensitive to predicted signatures of the chiral anomaly as well as the underlying chiral nature of the Weyl states using magneto-optical measurement schemes in zero field.\cite{ashby14,sonspivak13, Hofmann15,goswami15, kargarian15, hofmann16}
Since Na$_3$Bi is highly reactive with air,\cite{kushwaha15} no optical measurements have previously been reported. Providing broadband optical access to samples in a cryogenic environment while protecting them from atmospheric water and oxygen presents substantial obstacles. The high mobility of Cd$_3$As$_2$,\cite{rosenberg59,liangOng15} historically known as a narrow band semiconductor with inverted bands and non-parabolic conduction band, \cite{arush92} attracted many optical studies over the last half century.\cite{turner1961,*haide66,*gelten80,houde86} Only recently have theoretical concepts been developed predicting a pair of Dirac cones\cite{wang13} and subsequent confirmation of their existence by surface probe measurements.\cite{liuNature14,jeon14,yi14,borisenkoCava14,neupane14} Therefore previous optical studies do not report optical effects unique to a Dirac cone except for two very recent optical measurements of Cd$_3$As$_2$. One of these studies reports a nearly constant $\epsilon_2$ in the mid-IR spectral region interpreted as a Dirac cone signature,\cite{neubauer16} while the other is a broadband cyclotron resonance study reporting a linear band structure.\cite{akrap2016}
In this article, the optical spectra of Cd$_3$As$_2$ and Na$_3$Bi are presented together with parallel first-principles band structure calculations. The expected optical signatures and thermal occupation effects in a Dirac cone pair is discussed in section \ref{sec:expectedOptics}. The first optical characterization of Na$_3$Bi is reported and discussed in section \ref{sec:NaBiResults}. In section \ref{sec:CdAsResults}, a peak in reflectivity in Cd$_3$As$_2$ identifies the onset of Dirac cone interband transitions. A summary of results is presented in section \ref{sec:conclusion}.
\begin{figure}
\includegraphics[width = 7 cm]{Fig1.png}
\caption{\label{fig:ExpectedOpticalSigs} (a)
The model dispersion of a Dirac cone pair ($k_\perp \equiv k_x=k_y$), which is used to numerically calculate the dielectric function ($\epsilon=\epsilon_1 + i \epsilon_2$), is graphed as solid red and blue lines. The Fermi level (blue plane), $E_F$, lies in the conduction band between the Dirac point and the saddle-point located at the $\Gamma$-point, midway between the Dirac nodes. Intersections of the red planes with the model dispersions depict the initial and final state energies of optical transitions for two cases: the onset of Dirac cone transitions at the Pauli-blocked edge $\omega=2 E_F$ , and transitions between the two saddle points at the Lifshitz gap energy $\omega=\Delta E_{LS}$. In the graph, the blue-dotted horizontal line is the expected $\epsilon_2$ in the low frequency limit with the Fermi level at the Dirac node predicted by the $k\cdot p$ dispersion\cite{wangPRB12} with anisotropic Fermi velocities, where $v_{z2}=5 \text{ eV} \AA>>v_{z1}$ (see Appendix \ref{app:kdotp}). At finite frequency in the vicinity of the Pauli-blocked edge, a highly anisotropic Fermi surface will broaden the edge as qualitatively depicted by the black-dashed curve of $\epsilon_2$. In the vicinity of the saddle points, band structure calculations show that the dipole transition matrix elements are suppressed, which reduces $\epsilon_2$ as qualitatively shown by the black dash-dotted curve. }
\end{figure}
\begin{figure}
\includegraphics[width=7 cm]{Fig2.png}
\caption{\label{fig:ExpectedOpticalThermalSigs}
A single Dirac cone is presumed to be electron-hole symmetric with a Fermi level $E_F=25 \text{ meV}$ in the conduction band. The chemical potential shown in panel (a) is numerically calculated from the dispersion (red curve) and analytically derived in the low temperature limit (black-dashed curve). The carrier density and Drude weight of thermally excited carriers are shown in panels (b) and (c), respectively, where the contributions from electrons (blue), holes (green), and the sum of the two (red), are plotted. The analytic solution of the Drude weight in the low temperature limit is shown as the black-dashed line in panel (c). }
\end{figure}
\section{expected optical signatures in Dirac cone systems}\label{sec:expectedOptics}
\subsection{Interband transitions}\label{sec:OpticsIdealDiracCones}
In an ideal 3D Dirac cone with the Fermi level at the node, interband transitions occur at all frequencies and give rise to a linear conductivity $\sigma_1 \sim \omega/v_F$ where $v_F$ is the Fermi velocity and $\omega$ is the photon energy.\cite{ashby14} At a nonzero Fermi level, the interband transitions are blocked by carrier occupation below $\omega = 2 E_F$. The lost interband spectral weight below $2 E_F$ gives rise to an equal free carrier (Drude) spectral weight thereby satisfying the f-sum rule.
Since the complex dielectric function is given by $\epsilon = (4 \pi / i \omega ) \sigma$, the Dirac cone interband transition contribution leads to $\epsilon_2=(1/6) N_d \alpha' \Theta(\omega - 2 E_F)$ that is constant above the transition onset, where $N_d$ is the degeneracy of the Dirac cone, $\alpha'=e^2/\hbar v_F$ is the effective fine structure constant, and $v_F$ is the Fermi velocity. The frequency independence of $\epsilon_2$ results from cancellations that occur in Fermi's golden rule between the joint density of states and the dipole transition matrix elements for a linear dispersion.\cite{ashby14,YuCardona}
The Kramers-Kronig transformation of the interband $\epsilon_2$ gives
$\epsilon_1\propto \log{ \frac{\Delta^2 - \omega^2}{(2 E_F)^2 - \omega^2}}$ where $\Delta$ is an energy cut-off defined by the bandwidth. Temperature broadening of the interband transition onset is taken into account by replacing the Heaviside step function in $\epsilon_2$ with a Fermi-distribution function expression, which results in $\epsilon_1\propto \textrm{Re} [\log{ \frac{\Delta^2 - \omega^2}{(2 \mu - \imath \pi T)^2 - \omega^2}}]$ where $\mu$ is the chemical potential and T is the temperature.
For the case of Na$_3$Bi, two Dirac cones are separated by $\delta k_d = \pm 0.1 {\AA}^{-1}$ along the $k_z$ direction. \cite{xuHasan13,LiuZXShen14} The conduction and valence band of the Dirac cones merge, forming two saddle points at the $\Gamma$-point midway between the nodes as depicted by the idealized dispersion in Fig. \ref{fig:ExpectedOpticalSigs}. The Fermi velocity $v_F \approx 2.5$ eV$\cdot {\AA}$ at each Dirac node is reasonably consistent with photoemission and transport measurements, and band structure calculations.\cite{xuHasan13,LiuZXShen14,xiongOngCA15,xiongOng15,kushwaha15,wangPRB12,narayan14,cheng14} For illustration purposes, the Lifshitz gap at the $\Gamma$-point is arbitrarily set at $\sim 4 E_F$.
Considering this dispersion with the Fermi level set at the Dirac node in the limit $\omega \rightarrow 0$, the two Dirac cones are well described by the ideal case, so that $\epsilon_2=(1/6) N_d \, \alpha' \approx 4$. This low frequency value is depicted by the dashed-blue line in Fig. \ref{fig:ExpectedOpticalSigs}, providing an estimated scale of the expected interband optical response. This scale also applies to the anisotropic Dirac cone case derived from band structure calculations in the low frequency limit (see Appendix \ref{app:kdotp}). At nonzero values of the Fermi level, the Pauli-blocked edge occurs at $\omega=2 E_F$, giving rise to a step in $\epsilon_2$ and a distinctive cusp-like lineshape in $\epsilon_1$ as shown in Fig. \ref{fig:ExpectedOpticalSigs}.
At higher photon energies, the nonlinearity of the bands along $k_z$ between the nodes become increasingly important. The saddle point region gives rise to a large and rapidly changing joint density of states as well as dipole transition matrix elements that strongly deviate from the linear Dirac case. Both effects should be considered for describing the optical response even though such modeling is numerically difficult. By ignoring the effects of the transition matrix elements, the effects arising from corrections to the joint density of states can be calculated.\cite{YuCardona} For this simplified case, a rendering of the features in $\epsilon$ is shown in Fig. \ref{fig:ExpectedOpticalSigs} in the vicinity of the saddle points. We will return below to consider contributions from the dipole transition matrix elements near the saddle points.
Two main features are thus expected in the optical signal from the Dirac cone interband transitions, one related to the Pauli-blocked edge and the other to the high density of states at the Lifshitz gap energy $\Delta E_{LS}$ at the $\Gamma$-point. The magnitude of the interband contributions to $\epsilon$ is therefore expected to be in the vicinity $\sim 5$ based on reasonable Fermi velocity estimates.
\subsection{Thermal occupation effects}
In Dirac systems with relatively low Fermi level, the temperature dependence of the chemical potential and carrier density can be substantial. These thermal occupation effects can therefore drive observable optical effects.\cite{sushkov2015} The Pauli-blocked edge will thermally broaden, and shift as $\omega= 2 \mu(T)$. The free carrier (Drude weight) response will also change consistent with the f-sum rule.
An analytic form of the chemical potential $\mu$ in the low $T$ limit is $\mu-E_F= - \frac{1}{6} \partial_E \ln[g(E_F)] \,(\pi T)^2 $ where $g$ is the density of states. \cite{ashcroft} Here the band dispersion information is encoded via the derivative of the density of states. An isotropic 3D conduction band where the dispersion is given by $E \propto k^\beta$ (where $\beta=1$ for a Dirac band) results in $ \partial_E \ln[g(E_F)]= E_F^{-1} (3-\beta)/\beta$. The chemical potential is therefore driven away from regions of higher density of states as temperature is increased. The Drude weight $D_W=n e^2/m$ depends on the energy dependence of both the number density $n$ and mass $m$, but for a linear dispersion the energy dependence is given by $D_W=N_d\frac{e^2}{6 \pi^2 \hbar^3}\frac{E_F^2}{v_F}$, where $N_d=4$ is the degeneracy for a Dirac cone pair, and in the low temperature limit we now obtain $\Delta D_W(T)/D_W(0) = -\frac{1}{3}(\frac{\pi T}{E_F})^2$.\cite{throckmortonPRB15}
Numerical solutions for the temperature dependence of the chemical potential, carrier density, and Drude weight are shown in Fig. \ref{fig:ExpectedOpticalThermalSigs}(a-c) for a Dirac cone (see Appendix \ref{app:chempot} and \ref{app:EggFS}). The dispersion is assumed linear with anisotropic velocities, where $v_z$ can differ from $v_\perp \equiv v_x=v_y$ (resulting in an ellipsoidal or an egg-shaped Fermi surface described in Appendix \ref{app:EggFS}), and the applied electric field is in the x-y plane. The results shown in Fig. \ref{fig:ExpectedOpticalThermalSigs}(a-c) are independent of the velocities and only depend on the Fermi energy that is set to 25 meV. When the chemical potential is within the half-width of the Fermi distribution function ($\pi T/2$) of the Dirac node, copious numbers of additional electrons and holes are thermally excited as shown in Fig. \ref{fig:ExpectedOpticalThermalSigs}(b). The Drude weight involves the sum of responses from holes and electrons, so that in the high temperature limit where the chemical potential is approximately zero and constant, $D_W \propto T^2$. In the low temperature limit, the decreasing $D_W$ with temperature is caused by the decreasing chemical potential $D_W\sim \mu^2 \sim -T^2$.\cite{throckmorton15} The temperature-dependent Drude weight is therefore nonmonotonic. The minimum demarcates the point where a substantial number of holes and electrons are thermally excited from the valence band, $\mu(T)\sim\pi T/2$.
\section{N\lowercase{a}$_3$B\lowercase{i} Results}\label{sec:NaBiResults}
\begin{figure*}
\includegraphics[width=\textwidth]{Fig3.png}
\caption{\label{fig:Refdataphonons} (a-f) Reflectivity of Na$_3$Bi for two samples are labeled Sample 1 (a,c,e) and Sample 2 (b,d,f). The black-dashed curve in panel (a) is a fit to the 8K spectra below 1400 cm$^{-1}$ using the Lorentzian oscillator parameters for $\epsilon$ given in panel (j), and shown in graph of panel (i). For Sample 1, a more sensitive detector was used in panels (c) and (e) compared with the broadband measurements reported in panel (a). A strongly temperature-dependent, screened plasma edge is observed in panels (c) and (d). The arrows in panel (c) point to absorptive features (a plasmaron excitation) that tracks the temperature-dependent plasma edge. Panel (h) shows a graph of a reflectivity model based on the parameters in panel (j) except that the free carrier response is replaced by a temperature dependent Drude weight consistent with thermal occupation effects in a Dirac cone. The plasma edge shifts are tracked along the red-dotted lines and summarized in panel (g), where the edge position shifts are normalized to the low temperature value $\omega_0$. The green-dashed model results of the plasma edge shifts are the same as those shown in panel (h) along the red-dotted line except with many more temperatures represented. Panels (e) and (f) show the temperature dependence of the interband transition spectral region.}
\end{figure*}
\subsection{Na$_3$Bi spectra, phonons, and crystal symmetry}
Single crystals of n-doped Na$_3$Bi were prepared as in Ref. \onlinecite{kushwaha15}. All manipulations were performed inside a nitrogen filled glove box to avoid air exposure, including the mounting and sealing of the sample inside a cryostat. The as-grown facets are c-axis (001) oriented.
The normal-incidence reflectivity spectra of two Na$_3$Bi crystals at a set of temperatures are reported in Figs. \ref{fig:Refdataphonons}(a) and (b). The largest two crystals are optically thick (opaque), accommodate a 2.4 mm and a 1.5 mm diameter aperture, and are labeled as Sample 1 and Sample 2, respectively. The reflection approaches unity at low frequency indicative of a metallic response, and phonon features are observable throughout the far-infrared (FIR) region. The high reflection in the range $650-1200 \text{ cm}^{-1}$ is a reststrahlen band. The screened plasma frequency is near 1300 $\text{cm}^{-1}$ indicated by a sharp edge accompanied by a reflection minimum. Features higher in frequency are due to electronic interband transitions.
The most accurately normalized spectra are from the largest and flattest crystal, so the 8K reflectivity spectrum of sample 1 is fit up to 1400 $\text{cm}^{-1}$ to determine the free carrier and phonon parameters. The model reflectance is generated from the dielectric function $\epsilon = \epsilon_{\infty}+\sum_j {\Omega^2_P}_j / ({\omega_0^2}_j-\omega^2- i \gamma \omega)$ where each Lorentzian oscillator represents a phonon mode with a center frequency $\omega_0$, characteristic width $\gamma$, and strength $\Omega_P$. The free carrier (Drude) response corresponds to $\omega_0=0$ where $2 \pi c \gamma= 1/\tau$ is the inverse lifetime of the carriers, and $\Omega_P^2= 4 \pi D_W$ where $\Omega_P$ is the bare (unscreened) plasma frequency.
The modeled reflectance that best fits the spectrum also incorporates a thin dielectric film on the Na$_3$Bi crystal. The best fit to the reflectivity data of sample 1 was found with a $2 \text{ }\mu\text{m}$ thick dielectric film with an index set to $n=1.9$. The optical path length is consistent with the faint but visibly colored interference patterns observable under magnification from the as-cleaved samples. The thin film model smoothly modifies the photometrics over a very broad range, with a weak periodic Fabry-Perot-like etalon period of $1200 \text{ cm}^{-1}$. When the thin film is removed from the model, the resulting spectrum better resembles the spectrum of sample 2 in Fig. \ref{fig:Refdataphonons}(b). The thin dielectric is therefore attributed to a surface layer on sample 1 which is inconsequential to the results presented. The fit to the reflectivity spectrum is shown by the dashed-black curve in Fig. \ref{fig:Refdataphonons}(a), and the bulk Na$_3$Bi parameters and associated dielectric function are reported in Figs. \ref{fig:Refdataphonons}(j) and (i), respectively.
The observed phonon spectrum is important since the number of IR active phonon modes relates to the crystal symmetry. The ground state of Na$_3$Bi is currently contentious.\cite{cheng14} The strongest observed phonons at 418 and 553 $\text{cm}^{-1}$, which give rise to the broad reststrahlen band, are a factor of two larger than the predicted highest phonon frequency from our \textit{ab initio} band structure calculations that agree with earlier studies.\cite{cheng14} Three candidate crystal symmetries are analyzed using point group analysis and the number of allowed acoustic, IR active, and Raman active phonons are reported in Appendix \ref{app:PGAnalysis}.
A recent x-ray study reports that Na$_3$Bi is in the hexagonal space group P6$_ 3$/mmc.\cite{kushwaha15} The unit cell consists of two formula units with a Na(1)-Bi honeycomb structure separated by interstitial Na(2) atoms. The number of expected phonons is therefore 24, of which 2 are expected to be IR active in the ab-plane. This is inconsistent with the 9 minimum observable oscillators reported in Fig. \ref{fig:Refdataphonons}(j) necessary to describe our data, which rules out the P6$_ 3$/mmc symmetry.
A recent \textit{ab initio} calculation shows that the P$\overline{\text{3}}$c1 and P6$_3$cm ground states are $\sim4$ meV lower than the P6$_3$/mmc structure. All three point group symmetries produce nearly the same x-ray diffraction pattern and similar Dirac cone bands.\cite{cheng14, wang12} The P$\overline{\text{3}}$c1 and P6$_3$cm structures, however, have a distorted Na-Bi honeycomb resulting in additional inequivalent Na Wykoff sites. The unit cell therefore increases from two formula units to six, and the number of phonon modes triples. Eleven infrared active phonons in the ab-plane are expected from point group analysis in both buckled-hexagonal-plane symmetries. The optical spectrum is fit well with the minimum of 9 phonon oscillators, but some are unusually broad which could imply multiple closely-spaced phonons. The optical data appear consistent with either the P$\overline{\text{3}}$c1 or P6$_3$cm structure.
However, the P6$_3$cm symmetry has no center of inversion and therefore cannot be a Dirac semimetal, but would rather split into a Weyl state system with four nodes. There is no evidence from surface probe measurements that this is the case. Furthermore, numerical calculations show that the P6$_3$cm symmetry (as well as the P6$_3$/mmc structure) may be unstable due to the existence of imaginary phonons.\cite{cheng14} Therefore, Na$_3$Bi likely belongs to the P$\overline{\text{3}}$c1 spacegroup.
The Drude fit parameters are determined by the low frequency response and the plasma edge feature. Some uncertainty is introduced since the zero-frequency Lorentzian is not sufficiently distinguishable from low frequency bismuth phonons. Reasonable fits to the data give a range of Drude parameters, where $\gamma < 15 \text{ cm}^{-1}$ and $500 \text{ cm}^{-1} <\Omega_P<1000 \text{ cm}^{-1}$. The Fermi level is estimated from the plasma frequency using a model dispersion. A Dirac cone model is described in Appendix \ref{app:EggFS} that produces an elongated egg-shaped Fermi surface. This shape approximates the Fermi surface produced by a more realistic dispersion derived from a $k\cdot p$ model with parameters that fit the Dirac cone bands obtained from first-principles numerical band structure calculations.\cite{wangPRB12} The Fermi level is then estimated by $E=\sqrt{3\pi\hbar^3 v_{z1}/N_d} \Omega_P$ where the degeneracy $N_d=4$ and $v_{z1}$ is the slower of the two velocity roots along the c-axis. For $v_{z1}= 0.5 \text{ eV} \AA$ as measured by photoemission (ARPES),\cite{xuHasan13} the Fermi energy ranges from $16 \text{ meV} <E_F< 34 \text{ meV}$, and is $25\text{ meV}$ for the Drude best fit parameter $\Omega_P=746 \text{ cm}^{-1}$.
The static dielectric constant is $\epsilon_0=120^{+10}_{-30}$. The uncertainty is based upon the uncertainty in $\Omega_P$ and therefore the uncertainty in the strength of the low frequency phonons.
\subsection{Pauli-blocking and Lifshitz gap}
Since Na and Bi are relatively heavy atoms, phonon features are relegated to low frequency, well below the measured plasma edge, as verified by our band structure calculations.\cite{cheng14} Considering the estimate of the Fermi level and consulting the band structure calculations in Fig. \ref{fig:bandstructure}(a-d) (our results for the three candidate symmetries verify those of references \onlinecite{wangPRB12} and \onlinecite{cheng14}), a conservative estimate of the spectral region where a Pauli-blocked edge may be found is between $300$ and $1500 \text{ cm}^{-1}$. Nearly this entire region is within the reststrahlen band where the reflectivity is extremely sensitive to small features in $\epsilon$ on the scale expected by a sharp Pauli-blocked edge $\sim 5$, as demonstrated by the phonon features in the reflectivity located at $700$ and $880 \text{ cm}^{-1}$ produced by much smaller associated $\epsilon$ features shown in Fig. \ref{fig:Refdataphonons}(i). Furthermore, the steep slope of the plasma edge and the deep minimum in the reflectivity just above the plasma edge in the vicinity of $1300 \text{ cm}^{-1}$, where $\epsilon_1\approx0$ and therefore $R_{min}\approx (\epsilon_2/4)^2$, requires $\epsilon_2<1$. An onset of Dirac cone interband transitions anywhere below $1300 \text{ cm}^{-1}$ is expected to contribute a much larger $\epsilon_2$.
No discernable features in the reflectivity spectra resemble the expected features from a Pauli-blocked edge or Lifshitz gap shown in Fig. \ref{fig:ExpectedOpticalSigs}. Band structure calculations show that the assumptions that led to these expectations must be modified. The large anisotropy of the Dirac cone, as demonstrated along the $k_z$ direction ($\Gamma$ -$A$) in Fig. \ref{fig:bandstructure}(c), gives rise to a wide range of interband transition onset frequencies for a nonzero Fermi level. The Pauli blocked edge therefore becomes broadened, as diagrammatically represented by the black-dashed line in Fig. \ref{fig:ExpectedOpticalSigs}. Furthermore, band structure calculations show that dipole transition matrix elements are strongly modified in the vicinity of the saddle points at $\Gamma$. The Dirac cone bands in Fig. \ref{fig:bandstructure}(c) have $s$ and $p$ orbital character with a strength proportional to the size of the red dots. Allowable dipole Dirac interband transitions therefore must involve $s\leftrightarrow p$ transitions. The Dirac cone bands along $\Gamma-A$ have $p$ orbital character, but only one of the Dirac bands has $s$ orbital character and it is strongly suppressed as the $\Gamma$-point is approached. The large joint density of states at the $\Gamma$-point that gave rise to the sharp increase in $\epsilon_2$ in Fig. \ref{fig:ExpectedOpticalSigs} is strongly modified by the diminution of the matrix elements (see the black dotted-dashed line in Fig. \ref{fig:ExpectedOpticalSigs}).
\subsection{Thermal occupation effects and electronic transitions in the Dirac cone}
\subsubsection{Plasma edge and Drude weight temperature dependence}
\begin{figure*}
\includegraphics[width=\textwidth]{Fig4.png}
\caption{\label{fig:bandstructure} (a) The calculated band structure of Na$_3$Bi is shown for the $P\bar{3}c1$ space group. Results are very similar for the $P6_3cm$ structure. (b) The Brillouin zone for the crystal structure in (a) is depicted with Dirac nodes marked by the two red points along $\Gamma-A$. (c,d) The projected orbital-characters of the bands are shown for $s$, $p_y$, $p_z$, and $p_x$ orbitals along the $\Gamma-A$ momentum direction as well as through the Dirac node parallel to the $\Gamma-M$ direction, denoted by $\bar{\Gamma}-\bar{M}$. Bands are plotted as blue lines, overlayed by dotted red lines with thickness proportional to the weight of the orbital character. The orbital character of the bands along the $\bar{\Gamma}-\bar{K}$ direction is very similar to that along $\bar{\Gamma}-\bar{M}$. Panel (c) shows the lack of s-orbital character of the Dirac cone heavy Bi-like band as well as the lighter Na-like band in the vicinity $\Gamma$, which causes the optical transition matrix elements associated with the Lifshitz gap region to be suppressed. (e-f) Fermi velocities of the two Dirac bands are plotted along $\Gamma-A$ and $\bar{\Gamma}-\bar{M}$ for the $P\bar{3}c1$ space group; velocities for the $P6_3cm$ structure are identical. Velocity plots along $\bar{\Gamma}-\bar{K}$ and $\bar{\Gamma}-\bar{M}$ are similar.}
\end{figure*}
Although the Pauli-blocked edge and the Lifshitz gap optical features are complicated by band structure anisotropy and transition matrix elements, the nonmonotonic temperature dependence of the plasma edge summarized in Fig. \ref{fig:Refdataphonons}(g) encodes Dirac cone information. The strength of the zero frequency oscillator in the dielectric function relates to the Drude weight, $D_W=\Omega_P^2/4\pi$. A decrease in Drude weight shifts the zero of $\epsilon_1$, and therefore the plasma edge, to lower frequency. The resemblance between the temperature dependence of the plasma edge in Fig. \ref{fig:Refdataphonons}(g) and of the Drude weight in Fig. \ref{fig:ExpectedOpticalThermalSigs}(c) suggests that the plasma edge shifts are caused by thermal occupation effects in the Dirac cone. As mentioned previously, the results of Fig. \ref{fig:ExpectedOpticalThermalSigs}(a-c) are independent of Fermi velocity for a linear dispersion, even for a Dirac cone with anisotropic velocities, and depend only on the Fermi level. The minimum frequency of the plasma edge in Fig. \ref{fig:Refdataphonons}(g) occurs at $T\approx 100$K. Assuming these shifts are caused by the temperature dependent Drude weight, the Fermi level is estimated to be $E_F=25 \text{ meV}$ since this value gives rise to a minimum in $D_W(T)$ at 100K.
This connection between thermal occupation effects in the Dirac cone that drive the Drude weight temperature dependence and the plasma edge shifts is verified by the quantitative agreement of the reflectivity model results shown in Fig. \ref{fig:Refdataphonons}(h). The temperature dependent Drude weight of Fig. \ref{fig:ExpectedOpticalThermalSigs}(c) with $\Omega_{P0}=950 \text{ cm}^{-1}$ is substituted into the complex dielectric function that includes the phonons reported in Fig. \ref{fig:Refdataphonons}(j) (with the parameter $\epsilon_\infty$ increased by 10 percent) and the reflectivity calculated. Utilizing the results of Appendix \ref{app:EggFS} that show $E_F = \sqrt{3 \pi \hbar^3 v_{z1}/N_d} \Omega_P$ and substituting this value of $\Omega_{P0}$ and $E_F=25$ meV, the slow root of the dispersion which physically corresponds to the conduction band between the nodes is found to be $v_{z1}\approx 0.3 \text{ eV}\AA $. This is a very reasonable number since $v_{z1}\sim v_\perp/10$ as shown by band structure results in Fig. \ref{fig:bandstructure}(e) and ARPES measurements.\cite{xuHasan13,LiuZXShen14} Despite some subtle differences between the measured temperature dependence of the plasma edge of the two samples in Fig. \ref{fig:Refdataphonons}(c) and (d), the model results in Fig. \ref{fig:Refdataphonons}(h) agree extremely well.
The temperature dependence of $\mu$ or $D_W$ ideally contains information associated with the large density of state region at the saddle point as well as the degree of electron-hole asymmetry of the Dirac bands, both of which the model neglects. For example, if the Fermi energy were in the vicinity of the conduction band saddle point where the density of states rapidly increases, the factor $ \partial_E \ln[g(E_F)]$ (in the expression for $\mu(T)$) would be larger than the linearly dispersing value of $2/E_F$. The increase of this factor would cause the chemical potential to decrease more quickly with temperature than a linear dispersion. As a result, a discrepancy bewteen the model rate of decrease of the plasma edge and data would be expected. Along this line, the discrepancy between the model and Sample 2 at low temperatures could be taken as evidence that the Lifshitz point is in the vicinity of 25 meV above the Dirac point. In principle, a low temperature characterization of $\mu(T)$ or $D_W(T)$ could be used to discern the temperature dependence of the $T^2$ coefficient and therefore determine the Lifshitz transition energy in the density of states in relation to the Fermi level, but the exercise requires many more than four or five low temperature data points (below 100K).
As mentioned already, the calculations leading to Fig. \ref{fig:ExpectedOpticalThermalSigs}(a-c) assume electron-hole symmetry. In a more realistic Dirac cone pair system with asymmetric saddle points such that $|E_{LS}^{CB}|<<|E_{LS}^{VB}|$, the assumption applies near the Dirac point where linear approximations are valid. In this case, a valence or conduction band Fermi pocket within $\pm|E_F|$ has the same size and shape. However, the assumption breaks down when the chemical potential and thermal half-width approach the Lifshitz energy $\mu(T) + \pi T/2\sim E_{LS}^{CB}$. The low temperature consequences were discussed in the previous paragraph. At high temperature, the chemical potential will be pushed below the Dirac node. A numerical calculation with electron-hole asymmetry such that $E_{LS}^{CB}\sim 30 \text{ meV}=(1/2) |E_{LS}^{VB}|$ and $E_F=25 \text{ meV}$ results in a chemical potential which crosses zero at about 150K reaching $-10$ meV at 300 K. This effect on $\mu(T)$ lowers the temperature of the Drude weight minimum a small amount, where $\mu(T)=\pi T/2$ gives $T=90K$, but does not significantly effect the high temperature Drude weight since the thermal width becomes substantially larger than the chemical potential. The upshot is that even fairly large asymmetries between valence and conduction bands do not appreciably modify the quantitative conclusions of the thermal analysis presented in this section.
\subsubsection{Interband transitions and thermal occupation of the Dirac cone saddle point}
A strong temperature dependence is observed over the interband transition region between $1500 $ and $3000 \text{ cm}^{-1}$. The reflectance over this entire spectral region continually decreases with temperature, but precipitously drops in the temperature range between $125$K and $150$K.
\begin{figure}
\includegraphics[width=8 cm]{Fig5.png
\caption{\label{fig:plasmaron}
The reflectance $R$ shown in Fig. \ref{fig:Refdataphonons}(c) for Sample 1 in the vicinity of the plasma edge at a set of temperatures is plotted as $\partial R/\partial\omega$. The curves are vertically offset for clarity. The highest peak of each plot is associated with the plasma edge. A dip feature in Figs. \ref{fig:Refdataphonons}(c) and \ref{fig:Refdataphonons}(d) below the plasma edge frequency is present in both samples, although it is much sharper in sample 1 as highlighted by the arrows in Fig. \ref{fig:Refdataphonons}(c). The feature manifests as a peak-dip structure in $\frac{\partial R}{\partial \omega}$, like a side lobe to the plasma edge peak, that tracks the plasma edge as it moves with temperature. The two black-dashed parallel lines are guides for the eye that show that the peak-dip plasmaron feature tracks the plasma edge up to $100$ K, and clearly persists at $200$ K.}
\end{figure}
As mentioned previously, three crystal structures considered in this study have nearly the same ground state energy to within a few meV.\cite{cheng14} This suggests that a phase change may occur as a function of temperature. However, the IR active phonons shows no anomalous behavior. Also, band structure calculations were performed for the three candidate crystal symmetries in which the lattice spacing was varied to simulate temperature changes. No discernable changes in the electronic structure or orbital characters were identified that correlated to the observed behavior.
Thermal occupation effects of a band with a large density of states within $\pi 150K/2\sim 20$ meV of the chemical potential provides a plausible explanation of the observed behavior. At these high temperatures, the chemical potential is expected to be near the Dirac point. Based on the band structure calculations in Figs. \ref{fig:bandstructure}(a-d), the only conduction band that is in the vicinity of $20$ meV of the Dirac node is the Dirac cone conduction band saddle point, which has only $p$-orbital character.
A candidate valence band with $s$-orbital character exists at the $\Gamma$-point, but lies $\sim750$ meV below the Dirac node as shown in Fig. \ref{fig:bandstructure}(c). Band structure calculations show that the energy of this band is very sensitive to the spin-orbit coupling strength. Decreasing the spin-orbit coupling by a factor of two does not significantly alter the Dirac cone bands, but pushes the s-band up in energy by about a factor of two. The optical results together with band structure calculations may therefore provide a sensitive method to determine the spin-orbit coupling strength.
In this picture, transitions at low temperature between this $s$-character valence band and the $p$-character Dirac cone conduction band give rise to allowable transitions in the vicinity of the $\Gamma$-point with a large joint density of states, provided that $E_F<E_{LS}^{CB}$. As the temperature is raised and the chemical potential lowers toward the Dirac point, these transitions remain active until the thermal broadening is large enough that a copious number of carriers occupy the conduction band saddle-point region. The thermal occupation of the final states at high temperatures will therefore suppress these interband transitions.
The temperature dependence of these interband transitions is only appreciable up to $\sim 3000 \text{ cm}^{-1}$ since, away from the $\Gamma$-point in the Dirac conduction band along the $k_\perp$ direction, the final state energy of interband transitions rapidly increases above the scale associated with thermal occupation effects.
\subsection{Dirac cone transitions above the Lifshitz energy}
The higher energy transitions above $3000 \text{ cm}^{-1}$ are larger than the Lifshitz gap energy where the Dirac cone pair merges into a single Dirac cone. Over the spectral range $\sim3000-6000 \text{ cm}^{-1}$, $\epsilon_2=1.5\pm0.2$ is frequency and temperature independent, which is derived from fitting the reflectivity using a Kramers-Kronig constrained variational dielectric function.\cite{kuzmenkoVDF2005} Since $\epsilon_2=(1/6) N_d \alpha'$ where $N_d=2$ for a single Dirac cone, a reasonable Fermi velocity of $v_F \approx 3 \text{ eV }\AA$ in the ab-plane is attained consistent with other measurements of $v_\perp$.\cite{xuHasan13,LiuZXShen14,kushwaha15,xiongOng15}
\subsubsection{Plasmaron feature}
Fig. \ref{fig:Refdataphonons}(c) shows a dip feature, indicated by the arrows, about $60 \text{ cm}^{-1}$ below the plasma edge, which tracks the temperature dependence of the plasma edge. This tracking behavior is more clearly observed by taking the derivative $\partial R/\partial \omega$ shown in Fig. \ref{fig:plasmaron}. The low temperature lineshape of the dip feature in reflectivity is reproduced by adding a very small Lorentzian absorption to the total dielectric function, which has a characteristic width $\gamma = 40 \text{ cm}^{-1}$ and strength $\Omega_P=50 \text{ cm}^{-1}$ resulting in a small peak value of only $\sim 0.05$ in $\epsilon_2$. Such a tiny absorptive feature is observable only because the total $\epsilon$ is small near the plasma edge.
Sample 2 shows similar behavior in Fig. \ref{fig:Refdataphonons}(d), but the suppression of the reflectivity just below the plasma edge is much broader (as with nearly all the features of Sample 2 in comparison with Sample 1), and appears as a broad sideband-shoulder in $\partial R/\partial \omega$ instead of a clear peak-dip feature.
The observation of an absorption feature that tracks the ab-plane plasma frequency strongly suggests a plasmon-coupled excitation that is electronic in origin. A possible excitation is a charge that couples to the plasmon density modes,\cite{lundqvist67} called a plasmaron excitation, which has recently been predicted in 3D Dirac systems: at a finite value of the Fermi level, the Coulomb interaction induces satellite quasiparticle peaks in the spectral function, which form sidelobes off the main quasiparticle branch.\cite{Hofmann14, Hofmann15}
Plasmaron modes must be excited by a longitudinal field component. A scattering processes is required that induces the longitudinal mode that can then couple to the c-axis plasmon. Such a process has been observed in similar optical measurements on bulk bismuth crystals,\cite{tediosi2007,armitageBi10} although the mechanism is far from clear: impurity scattering \cite{gerlachPlasmaron74} and an electron-hole decay scenario has been proposed without reaching a definitive conclusion.\cite{tediosi2007,armitageBi10}
Optically excited plasmaron excitations in 3D materials have rarely been observed, which makes the observation in a 3D Dirac cone system particularly interesting. In the case of elemental bismuth, a plasmaron excitation is observed at a higher energy than the plasmon mode.\cite{tediosi2007}. For Na$_3$Bi, the c-axis plasmaron excitation is observed below the ab-plane plasmon energy. Therefore, the c-axis plasmon must be lower in energy than the ab-plane plasmon.
The plasmon energy is determined by the pole in $1/\epsilon_z$ and therefore it involves a sum of many contributing terms: free carrier (Drude) response, strength and number of IR active phonon modes, and the high energy interband transitions that cumulatively determine the value of $\epsilon_\infty$. The strength of the c-axis phonons and $\epsilon_\infty$ is not currently known, but can be easily determined optically with an appropriately oriented crystal. What is known is that the Drude weight is smaller for an electric field along the c-axis since the Fermi velocity is smaller than $v_\perp$, and the number of IR active phonons along the c-axis is substantially less than in the ab-plane (see Appendix \ref{app:PGAnalysis}). Both effects would tend to decrease the c-axis plasmon frequency below the ab-plane plasma edge.
Clear evidence of a collective plasmon-electronic excitation in bismuth and now in the 3D Dirac system Na$_3$Bi has been found. Na$_3$Bi and elemental bismuth share many characteristics, such as a Dirac-like (L point) conduction band that has a high Fermi velocity and a small associated Fermi surface, carrier density, and Fermi wavevector. These observations suggest that collective plasmon-coupled excitations are perhaps more ubiquitous, and open up the possibility of further investigating such collective modes in the various types of Weyl and Dirac systems.
\begin{figure}
\includegraphics[width=\columnwidth]{Fig6.png
\caption{\label{fig:CdAs} (a) The measured mid-IR data (solid colors) is shown with the modeled reflectivity (black) that includes the fitted phonon parameters from Ref. \onlinecite{houde86}. (b) An expanded view of the peak in reflectivity due to the onset of Dirac cone interband transitions and the modeled reflectivities (offset for clarity). Model 1 includes only thermal effects. Contributions to the width in addition to the thermal effects include potential fluctuations, shown in Model 2, or continuum of interband transition onset energies, shown in Model 3. (c) The same data is shown over a broader spectral range, offset for clarity, with a temperature independent feature demarcated by the gray dotted line. (d) The temperature dependent peak positions are fit and plotted relative to the 7 K value $\omega_0$, expressed as $\Delta \omega/\omega_0$, for the data (blue dots), Model 3 in panel (b) (black dots), and the corrected data taking into account the changing slope of the background (red dots). Error bars represent $\pm \sigma$, a standard deviation, generated from fits to the derivative of the peak. Also shown are quadratic fits (solid lines) and $\pm \sigma$ confidence intervals (dotted lines) to the data and corrected data for $T \leq 150$ K.}
\end{figure}
\section{ C\lowercase{d}$_3$A\lowercase{s}$_2$ spectra and Pauli-blocked edge}\label{sec:CdAsResults}
Cd$_3$As$_2$ n-type single crystals were prepared as in Ref. \onlinecite{aliCava14}, and the facet was oriented normal to [112]. The largest crystal accommodates a 0.4 mm aperture and is opaque. A continuous scan FTIR spectrometer measured normally incident reflection.
The small size of the crystal limited throughput power, which precluded measurements in the FIR spectral region. The mid-IR data is reported in Fig. \ref{fig:CdAs}(a-c) at a set of temperatures. A strong temperature-dependent peak in the vicinity of 1650 $\text{cm}^{-1}$ is identified as the Dirac cone Pauli-blocked edge. Band structure calculations and surface probe measurements indicate that other bands do not contribute at such relatively low energies.\cite{wang13, jeon14,liuNature14,neupane14,borisenkoCava14}The peak in the low temperature data implies a Fermi level in the vicinity of $\omega/2 \sim 100$ meV.
Surface tunneling microscopy (STM) measurements and \emph{ab initio} calculations indicate that the Lifshitz gap energy is only $\sim 40$ meV.\cite{jeon14,wang13} In the scenario where the Fermi energy is much larger than the Lifshitz gap, the Dirac cone pair merges into a single Fermi pocket. ARPES, STM, and transport measurements indicate that the bands appear very linear in this regime\cite{jeon14,liangOng15,liuNature14,neupane14,borisenkoCava14} with nearly isotropic velocity\cite{jeon14,liangOng15} with a single large-Dirac-cone-like dispersion .
We consider a model of reflectivity derived from a dielectric function that includes contributions from phonons, ideal Dirac cone interband transitions (where $N_d=2$, $E_F=100$ meV, and\cite{liangOng15} $v_F = c/322$), and a Drude weight consistent with the interband transition parameters given by $\Omega_P^2 / (4 \pi)= \frac{2}{3 \pi^2 \hbar^3} \frac{E_F^2}{v_F}$. FIR reflectivity data from Ref. \onlinecite{houde86} is fit to derive the phonon parameters. The model is shown in Fig \ref{fig:CdAs}(a) with the mid-IR reflectance data superimposed. The plasma edge is below our measured frequency range due to limitations in throughput power as a result of the small size of the sample.
By modeling several lineshape broadening effects and comparing to the data, the origin of the distinctive lineshape can be determined. The results of three different models are compared with the data in Fig. \ref{fig:CdAs}(b). Model 1: Thermal effects broaden the Pauli-blocked edge step in $\epsilon_2$ via a Fermi distribution function, which modifies $\epsilon_1$ via the Kramers-Kronig relations. The resulting cusp-like peak in the reflectivity that is dominated by the logarithmic divergence in $\epsilon_1$ is much too narrow to account for the data. Model 2: Gaussian potential fluctuations are added into the step of $\epsilon_2$ in conjunction with thermal broadening. The best match to the width of the 7K reflectivity peak is given by an amplitude potential fluctuation (RMS) $\Gamma_{\text{rms}} \sim 26$ meV. The resulting characteristic lineshape is very different from the data. Notably, an estimate of the potential fluctuations in Cd$_3$As$_2$ given in Ref. \onlinecite{skinner14} is substantially smaller, $\Gamma_{\text{rms}} \sim 4$ meV (using $E_F=100$ meV, $\epsilon_0 = 70$, $v_F$=c/322, $N_d$=2, and an assumed charged impurity density equal to the carrier density). Model 3: Anisotropies between the conduction and valence bands can result in a continuum of interband transition onset frequencies. To model this, the expected step height in $\epsilon_2$ for an ideal Dirac cone is divided into a series of equal step heights separated by equal frequency spacings. Each step is thermally broadened. Model 3 used in Fig. \ref{fig:CdAs}(b) was generated with 20 steps over a frequency range of $125 \text{ cm}^{-1}$.
The strong resemblance of Model 3 to the distinctive lineshape and thermal dependence of the data indicates a continuum of onsets in the Dirac cone over an energy range $\Delta \omega_{\text{onset}}\approx15$ meV. Incrementally adding in potential fluctuation broadening effects into Model 3 gradually evolves the lineshape towards Model 2, but also tempers the rate of decrease of the low temperature peak heights to better resemble the data. The low temperature lineshapes markedly begin deviating from the data at $\Gamma_{\text{rms}}= 7$ meV, and become untenable by $\Gamma_{\text{rms}}= 10$ meV, which sets a hard upper bound. These values are in reasonable agreement with the theoretical estimate, $\Gamma_\text{rms}\sim 4$ meV.\cite{skinner14}
The spread in Dirac cone interband transitions $\Delta \omega_{\text{onset}}$ is caused by velocity anisotropy of the Dirac bands. Using the ellipsoidal Fermi surface described in Appendix \ref{app:EggFS}, this energy spread translates into a $10\%$ variation of velocity, a very small degree of anisotropy, and therefore a nearly spherical Fermi surface. This agrees with STM, SdH, and recent cyclotron resonance results that show a nearly isotropic Fermi surface at high Fermi levels well above the Lifshitz gap.\cite{jeon14,liangOng15,akrap2016}
A shift of the peak towards lower frequency with temperature is driven by the chemical potential. This relationship is derived in Appendix \ref{app:chempot} in the low temperature limit. The relative shift of the peak position normalized to the low temperature value depends only on the Fermi level, where $\delta \omega/\omega_0=-\frac{1}{3}(\frac{\pi T}{E_F})^2$ for a linear dispersion, which is measured by ARPES and tunneling microscopy.\cite{jeon14,liuNature14,neupane14,borisenkoCava14}. The peaks are fit to determine the center frequency. The results are plotted in Fig. \ref{fig:CdAs} (d) as blue dots with error bars. The temperature dependence is fit for $T\leq150$ K to the expected quadratic form (blue solid plot) with confidence intervals as dashed lines. Using the $T^2$ fit coefficient yields $E_F = 111\pm 4$ meV.
Since the experimental peaks reside on a smooth non-constant background, the peak positions are slightly skewed as a function of temperature. To estimate these corrections, the peaks of Model 3, where the center of the Pauli-blocked edge was set to a constant $\omega_0$ for all temperatures, are fit using the same procedure as the experimental data. The centers of peak positions determined in this way are plotted in Figure \ref{fig:CdAs} (d) (black dots), appearing temperature dependent as the peak thermally broadens. These relatively small corrections to the peak positions are subtracted from the experimentally determined positions and reported as red dots and error bars. This corrected dataset is fit as before yielding $E_F = 96\pm 3$ meV, and a carrier density of $n=1.3 \times 10^{17}\text{ cm}^{-3}$. This is somewhat lower than for similarly grown crystals where the carrier density corresponded to a Fermi level in the vicinity of $200$ meV.\cite{liangOng15,jeon14}
The Fermi level is about half of the interband transition onset energy, indicating that the Dirac point is about midway between the final state (conduction band) and initial state (valence band), and the valence and conduction bands are more or less symmetrical. Band structure calculations and surface probe measurements show that the valence band is notably heavier than the conduction band, but that the two bands are not strongly asymmetrical.\cite{wang13,borisenkoCava14,jeon14,neupane14}
A very weak feature present at $\sim 2900 \text{ cm}^{-1}=360$ meV does not discernably shift with temperature (see in Fig. \ref{fig:CdAs}(c)) and is too high in energy to be associated with the Lifshitz gap energy. No optical signature of the Lifshitz gap is observed over the measured spectral region. However, even if it were within the measured range, it may not be optically measurable. The transition matrix elements in the vicinity of the $\Gamma$ point are expected to be suppressed like in the Na$_3$Bi case since the Dirac band orbital-characters are very similar.\cite{wang13}
\section{Conclusion}\label{sec:conclusion}
In both Cd$_3$As$_2$ and Na$_3$Bi, thermal occupation effects in the Dirac cone pair play a crucial role in the optical response. Thermal excitation of carriers change the chemical potential and therefore the Dirac interband transition energy as well as the free carrier response.
In Cd$_3$As$_2$, the sharp Pauli-blocked edge at the onset of Dirac cone interband transitions induces a peak in the reflectivity with a very distinctive lineshape, providing a fingerprint of the underlying Dirac cone dispersion and the associated logarithmic divergence in $\epsilon_1$. The frequency of the Pauli-blocked edge is controlled by the chemical potential that depends only on the power law exponent of the dispersion and the Fermi level in the low temperature limit. Our characterization of the peak location with temperature indicates a linear Dirac cone dispersion, a number density of $n=1.3 \times 10^{17}\text{ cm}^{-3}$, and a Fermi energy much larger than the Lifshitz gap energy as measured by STM.\cite{jeon14} The low temperature spectral width of the peak is caused by Fermi velocity anisotropy that gives rise to a narrow spectral range of Dirac cone interband transition onsets. The spectral width of the reflection peak translates into a Fermi velocity anisotropy of $10\%$, indicating a nearly spherical Fermi surface. The lineshape is incompatible with large Gaussian broadening effects, giving an upper bound energy scale for potential fluctuations of $\Gamma= 7$ meV.
In Na$_3$Bi, evidence of the Dirac cone manifests in a temperature dependent plasma edge caused by changes in the free carrier response. The Drude weight temperature dependence is nonmonotonic, attaining a minimum when the chemical potential is within $\sim kT$ of the Dirac node. The minimum in the temperature dependence of the plasma edge frequency at $T=100$K is characterized only by the Fermi level for a Dirac cone, giving $E_F=25$ meV. Unlike Cd$_3$As$_2$, evidence of the Dirac cone in Na$_3$Bi is not observable from the onset of Dirac cone interband transitions. The unobservable edge presumably reflects the large Dirac cone anisotropy, which is consistent with band structure calculations. At transition energies well above the Lifshitz gap where the low energy Dirac cone pair has merged into one large Dirac cone, a frequency and temperature independent $\epsilon_2$ is observed. The constant value of $\epsilon_2$ is a fingerprint of the Dirac dispersion that only depends on Fermi velocity, and translates into an ab-plane Fermi velocity of $v_{\perp} \approx 3 \text{ eV} \AA$. The ground state of Na$_3$Bi has been reported as belonging to the $\text{P}6_3\text{/mmc}$ space group symmetry, but the number of observable IR active phonons that we observe rules this out in favor of the $\text{P}\bar{3}\text{c}1$ candidate symmetry. Finally, we have observed a plasmaron excitation near the plasma edge in Na$_3$Bi, which tracks the shifting ab-plane plasmon energy over a broad range of temperatures.
\section{Acknowledgments}
The work at UMD was supported by DOE under grant No. ER 46741-SC0005436. The research at Princeton was supported by the ARO MURI on topological insulators, Grant No. W911NF-12-1-0461 and ARO Grant No. W911NF-11-1-0379 and the MRSEC program at the Princeton Center for Complex Materials, Grant No. NSF-DMR-0819860 and Grant No. DOE DE-FG-02-05ER46200.
T.R.C. and H.T.J. are supported by the Ministry of Science and Technology,
National Tsing Hua University, and Academia Sinica, Taiwan, and they thank NCHC, CINC-NTU and NCTS, Taiwan for technical support. H.L. acknowledges the Singapore National Research Foundation for the support under NRF Award No. NRF-NRFF2013-03. The work at Northeastern University was supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences grant number DE-FG02-07ER46352, and benefited from Northeastern University's Advanced Scientific Computation Center (ASCC) and the NERSC supercomputing center through DOE grant number DE-AC02-05CH11231. We thank Rolando V. Aguilar for useful conversations.
|
3,212,635,537,418 | arxiv | \section{Introduction}
\begin{figure}[!t]
\centering
\includegraphics[width=1.0\columnwidth]{augur-overview-01}
\caption{Augur mines human activities from a large dataset of modern fiction. Its statistical associations give applications an understanding of when each activity might be appropriate.}
\label{fig:splash}
\end{figure}
Our most compelling visions of human-computer interaction depict worlds in which computers understand the breadth of human life.
Mark Weiser's first example scenario of ubiquitous computing, for instance, imagines a smart home that predicts its user may want coffee upon waking up \cite{weiser}. Apple's Knowledge Navigator similarly knows not to let the user's phone ring during a conversation \cite{knowledgenavigator}. In science fiction, technology plays us upbeat music when we are sad, adjusts our daily routines to match our goals, and alerts us when we leave the house without our wallet.
In each of these visions, computers understand the actions people take, and when.
If a broad understanding of human behavior is needed, no method yet exists that would produce it.
Today, interaction designers instead create special-case rules and single-use machine learning models. The resulting systems can, for example, teach a phone (or Knowledge Navigator) not to respond to calls during a calendar meeting. But even the most clever developer cannot encode behaviors and responses for every human activity -- we \textit{also} ignore calls while eating lunch with friends, doing focused work, or using the restroom, among many other situations.
These original HCI visions assumed such breadth of information.
To achieve this breadth, we need a knowledge base of human activities, the situations in which they occur, and the causal relationships between them.
Even the web and social media, serving as large datasets of human record, do not offer this information readily.
In this paper, we show it is possible to create a broad knowledge base of human behavior by text mining a large dataset of \emph{modern fiction}.
Fictional human lives provide surprisingly accurate accounts of real human activities.
While we tend to think about stories in terms of the dramatic and unusual events that shape their plots, stories are also filled with prosaic information about how we navigate and react to our everyday surroundings.
Over many millions of words, these mundane patterns are far more common than their dramatic counterparts. Characters in modern fiction turn on the lights after entering rooms; they react to compliments by blushing; they do not answer their phones when they are in meetings. Our knowledge base, \emph{Augur} (Figure~\ref{fig:splash}), learns these associations between activities and objects by mining \emph{1.8 billion words} of modern fiction from the online writing community Wattpad.
Our main technical contribution is a vector space model for predicting human activities, powered by a domain specific language for data mining unstructured natural langage text. The domain specific language, TC, enables simple parser scripts that recognize syntactic patterns in a corpus (e.g., a verb with a human pronoun or name as its subject). TC compiles these scripts into parser combinators, which combine into more complex parsers that can, for example, run a co-occurrence analysis to identify objects that often appear near a human activity. We represent this information via a vector space model that uses smoothed co-occurrence statistics to represent each activity. For example, Augur maps \emph{eating} onto hundreds of food items and relevant tools such as cutlery, plates and napkins, and connects disconnected objects such as a fruit and store onto activities like \emph{buying groceries}. Similar parser scripts can extract subject-verb-object patterns such as ``he sips coffee'' or ``coffee spills on the table'' to understand common object behaviors. For example, Augur learns that coffee can be \textit{sipped}, and is more likely to be \textit{spilled} when another person \textit{appears}.
To enable software development under Augur, we expose three core abstractions. \emph{Activity detection} maps a set of objects onto an activity. \emph{Object affordances} return a list of actions that can be taken on or by an object. \emph{Activity prediction} uses temporal sequences of actions in fiction to predict which activities may occur next. Augur allows applications to subscribe to events on these three APIs, firing when an activity or prediction is likely. For example, a
heads up display might register a listener onto \emph{pay money}, which fires when objects such as cash registers, bills, and credit cards are recognized by a computer vision system. The application might then react by making your bank balance available with a single tap.
Using this API, we have created a proof-of-concept application called \textit{Soundtrack for Life}, a Google Glass application that plays music to suit your current activity.
Our evaluation tests the external validity of Augur. To examine whether the nature of fiction introduces bias into Augur's system predictions, we compare the frequency counts of its mined activities to direct human estimates, finding that 84\% of its estimates are nearly indistinguishable from human estimates. Next, we perform a field deployment of \textit{Soundtrack for Life} to collect a realistic two-hour dataset of daily activities with a human subject and then manually evaulate Augur's precision and recall. Augur demonstrated 97\% recall and 76\% precision over these daily activities. Finally, we stress-test Augur on a broader set of activities by asking external raters to evaluate its predictions on widely ranging images drawn from a sample of 50 \textit{\#dailylife} posts on Instragram. Here, raters classified 94\% of Augur's predictions as sensible, even though computer vision extracted the most semantically relevant objects in only 64\% of the images.
Our work contributes infrastructure and interfaces that draw on broad information about the associations between human interactions in the world. These interfaces can query how human behavior is conditioned by the context around the user. We demonstrate that fiction can provide deep insight into the minutae of human behavior, and present an architecture based a domain specific language for text mining that can extract this information at scale.
\section{Related Work}
Our work is inspired by techniques for mining user behavior from data. For example, query-feature graphs show how to encode the relationships between high-level descriptions of user goals and underlying features of a system \cite{qfgraphs}, even when these high-level descriptions are different from an application's domain language \cite{commandspace}. Researchers have applied these techniques to applications such as AutoCAD \cite{communitycommands} and Photoshop \cite{commandspace}, where the user's description of a domain and that domain's underlying mechanics are often disjoint. With Augur, we introduce techniques that mine real-world human activities that typically occur outside of software.
Other systems have developed powerful domain-specific support by leveraging user traces. For example, in the programming community, research systems have captured emergent practice in open source code \cite{codex}, drawn on community support for debugging computer programs \cite{helpmeout}, and modeled how developers backtrack and revise their programs \cite{backtracking}. In mobile computing, the space of user actions is small enough that it is often possible to predict upcoming actions \cite{yanglipredictions}. In design, a large dataset of real-world web pages can help guide designers to find appropriate ideas \cite{Kumar2013}. Creativity-support applications can use such data to suggest backgrounds or alternatives to the current document \cite{lee2011shadowdraw,Simon2008}.
Augur complements these techniques by focusing on unstructured data such as text and modeling everyday life rather than behavior within the bounds of one program.
Ubiquitous computing research and context-aware computing aim to empower interfaces to benefit from the context in which they are being used \cite{fabryq, contextaware}. Their visions motivated the creation of our knowledge base (e.g., \cite{weiser, knowledgenavigator}). Some applications have aimed to model specific activities or contexts such as jogging and cycling (e.g., \cite{ubifit}). Augur aims to augment these models with a broader understanding of human life. For example, what objects might be nearby before someone starts jogging? What activities do people perform before they decide to go jogging? Doing so could improve the design and development of many such applications.
We draw on work in natural language processing, information extraction, and computer vision to distill human activites from fiction. Prior work discusses how to extract patterns from text by parsing sentences \cite{nlpnarrative, reverb, relgram, tokens-regex}. We adapt and extend these approaches in our text mining domain-specific language, producing an alternative that is more declarative and potentially easier to inspect and reason about.
Other work in NLP and CV has shown how vector space models can extract useful patterns from text \cite{vectorspace}, or how other machine learning algorithms can generate accurate image labels \cite{cv} and classify images given a small closed set of human actions \cite{cv-act}. Augur draws on insights from these approaches to make conditional predictions over thousands of human activities.
Our research also benefits from prior work in commonsense knowledge representation. Existing databases of linguistic and commonsense knowledge provide networks of facts that computers should know about the world \cite{conceptnet}. Augur captures a set of relations that focus more deeply on human behavior and the causal relationships between human activities. We draw on forms of commonsense knowledge, like the WordNet hierarchy of synonym sets \cite{wordnet}, to more precisely extract human activities from fiction. Parts of this vocabulary may be mineable from social media, if they are of the sort that people are likely to advertise on Twitter \cite{emreactions}. We find that fiction offers a broader set of local activities.
\section{Augur}
Augur is a knowledge base that uses fiction to connect human activities to objects and their behaviors. We begin with an overview of the basic activities, objects, and object affordances in Augur, then then explain our approach to text mining and modeling.
\subsection{Human Activities}
Augur is primarily oriented around \textit{human activities}, which we learn from verb phrases that have human subjects, for example ``he opens the fridge'' or ``we turn off the lights.'' Through co-occurrence statistics that relate objects and activities, Augur can map contextual knowledge onto human behavior.
For example, we can ask Augur for the five activities most related to the object ``facebook'' (in modern fiction, characters use social media with surprising frequency):
\vspace{.2em}
\begin{lstlisting}
activity score frequency
----------------------------------------
message 0.71 1456
get message 0.53 4837
chat 0.51 4417
close laptop 0.45 1480
open laptop 0.39 1042
\end{lstlisting}
\vspace{-.4em}
Here \textit{score} refers to the cosine similarity between a vector-embedded query and activities in the Augur knowledge base (we'll soon explain how we arrive at this measure).
Like real people, fictional characters waste plenty of time \textit{messaging} or \textit{chatting} on Facebook. They also engage in activities like \textit{post}, \textit{block}, \textit{accept}, or \textit{scroll feed}.
Similarly, we can look at relations that connect multiple objects. What activities occur around a shirt and tie? Augur captures not only the obvious sartorial applications, but notices that shirts and ties often follow specific other parts of the morning routine such as \emph{take shower}:
\vspace{.2em}
\begin{lstlisting}
activity score frequency
----------------------------------------
wear 0.05 58685
change 0.04 56936
take shower 0.04 14358
dress 0.03 16701
slip 0.03 59965
\end{lstlisting}
\vspace{-.4em}
In total, Augur relates 54,075 human activities to 13,843 objects and locations.
While the head of the distribution contributes many observed activities (e.g., extremely common activities like \textit{ask} or \textit{open door}), a more significant portion lie in the bulk of the tail. These less common activities, like \textit{reply to text message} or \textit{take shower}, make up much of the average fictional human's existence. Further out, as the tail diminishes, we find less frequent but still semantically interesting activities like \textit{throw out flowers} or \textit{file bankruptcy}.
Augur associates each of its activities with many objects, even activities that appear relatively infrequently. For example, \textit{unfold letter} occurs only 203 times in our dataset, yet Augur connects it to 1072 different objects (e.g., handwriting, envelope). A more frequent activity like \textit{take picture} occurs 10,249 times, and is connected with 5,250 objects (e.g., camera, instagram). The abundance of objects in fiction allows us to make inferences for a large number of activities.
\subsection{Object Affordances}
Augur also contains knowlege about \textit{object affordances}: actions that are strongly associated with specific objects. To mine object affordances, Augur looks for subject-verb-object sentences with objects either as their subject or direct object.
Understanding these behaviors allows Augur to reason about how humans might interact with their surroundings. For example, the ten most related affordances for a car:
\vspace{.2em}
\begin{lstlisting}
activity score frequency
-------------------------------------------
honk horn 0.38 243
buckle seat-belt 0.37 203
roll window 0.35 279
start engine 0.34 898
shut car-door 0.33 140
open car-door 0.33 1238
park 0.32 3183
rev engine 0.32 113
turn on radio 0.30 523
drive home 0.26 881
\end{lstlisting}
\vspace{-.4em}
Cars undergo basic interactions like \textit{roll window} and \textit{buckle seat-belt} surprisingly often. These are relatively mundane activities, yet abundant in fiction.
Like the distribution of human activities, the distribution of objects is heavy-tailed. The head of this distribution contains objects such as phone, bag, book, and window, which all appear more than one million times. The thick ``torso'' of the distribution is made of objects such as plate, blanket, pill, and wine, which appear between 30,000 and 100,000 times. On the fringes of the distribution are more idiosyncratic objects such as kindle (the e-book reader), heroin, mouthwash, and porno, which appear between 500 and 1,500 times.
\subsection{Connections between activities}
Augur also contains information about the connections between human activities. To mine for sequential activties, we can look at extracted activities that co-occur within a small span of words. Understanding which activities occur around each other allows Augur to make predictions about what a person might do next.
For example, we can ask Augur what happens after someone orders coffee:
\vspace{.2em}
\begin{lstlisting}
activity score frequency
--------------------------------------
eat 0.48 49347
take order 0.40 1887
take sip 0.39 11367
take bite 0.39 6914
pay 0.36 23405
\end{lstlisting}
\vspace{-.4em}
Even fictional characters, it seems, must \textit{pay} for their orders.
Likewise, Augur can use the connections between activities to determine which activities are similar to one another. For example, we can ask for activities similar to the social media photography trend of \textit{take selfie}:
\vspace{.2em}
\begin{lstlisting}
activity score frequency
------------------------------------------
snap picture 0.78 1195
post picture 0.76 718
take photo 0.67 1527
upload picture 0.58 121
take picture 0.57 10249
\end{lstlisting}
\vspace{-.4em}
By looking for activities with similar object co-occurrence patterns, we can find near-synonyms.
\subsection{A data mining DSL for natural language}
Creating Augur requires methods that can extract relevant information from large-scale text and then model it. Exploring the patterns in a large corpus of text is a difficult and time consuming process. While constructing Augur, we tested many hypotheses about the best way to capture human activties. For example, we asked: what level of noun phrase complexity is best? Some complexity is useful. The pattern \textit{run to the grocery store} is more informative for our purposes than \textit{run to the store}. But too much complexity can hurt predictions. If we capture phrases like \textit{run to the closest grocery store}, our data stream becomes too sparse. Worse, when iterating on these hypotheses, even the cleanest parser code tends not to be easily reusable or interpretable.
To help us more quickly and efficiently explore our dataset, we created TC (Text Combinator), a data mining DSL for natural language. TC allows us to build parsers that capture patterns in a stream of text data, along with aggregate statistics about these patterns, such as frequency and co-occurrence counts, or the mutual information (MI) between relations. TC's scripts can be easier to understand and reuse than hand-coded parsers, and its execution can be streamed and parallelized across a large text dataset.
TC programs can model syntactic and semantic patterns to answer questions about a corpus. For example, suppose we want to figure out what kinds of verbs often affect laptops:
\vspace{.2em}
\begin{lstlisting}[language=Haskell]
laptop = [DET]? ([ADJ]+)? "laptop"
verb_phrase = [VERB] laptop-
freq(red_vp)
\end{lstlisting}
\vspace{-.4em}
Here the \textit{laptop} parser matches phrases like ``a laptop'' or ``the old broken laptop'' and returns exactly the matched phrase. The \textit{verb\_phrase} parser matches pharses like ``throw the broken laptop'' and returns just the verb in the phrase (e.g., ``throw''). The \textit{freq} aggregator keeps a count of unique tokens in the output stream of the \textit{verb\_phrase} parser. On a small portion of our corpus, we see as output:
\vspace{.2em}
\begin{lstlisting}
open 11
close 7
shut 6
restart 4
\end{lstlisting}
\vspace{-.4em}
To clarify the syntax for this example: square brackets (e.g., \lstinline{[NOUN]}) define a parser that matches on a given part of speech, quotes (e.g., \lstinline{"laptop"}) matches on an exact string, whitespace is an implicit then-combinator (e.g., \lstinline{[NOUN] [NOUN]} matches two sequential nouns), a question mark (e.g., \lstinline{[DET]?} optionally matches an article like ``a'' or ``the'', also matching on the empty string), a plus (e.g., \lstinline{[VERB]+} matches on as many verbs as appear consecutively), and a minus (e.g., \lstinline{[NOUN]-} matches on a noun but removes it from the returned match). We describe TC's full set of operators in this paper's appendix.
We wrote the compiler for TC in Python. Behind the scenes, our compiler transforms an input program into a parser combinator, instantiates the parser as a Python generator, then runs the generator to lazily parse a stream of text data. Aggregation commands (e.g., \lstinline{freq} frequency counting and \lstinline{MI} for MI calculation) are also Python generators, which we compose with a parser at compile time. Given many input files, TC also supports parallel parsing and aggregation.
\subsection{Mining activity patterns from text}
To build the Augur knowledge base, we index more than one billion words of fiction writing from 600,000 stories written by more than 500,000 writers on the Wattpad writing community\footnote{\url{http://wattpad.com}}. Wattpad is a community where amateur writers can share their stories, oriented mostly towards writers of genre fiction. Our dataset includes work from 23 of these genres, including romance, science fiction, and urban fantasy, all of which are set in the modern world.
Before processing these stories, we normalize them using the spaCy part of speech tagger and lemmatizer\footnote{\url{https://honnibal.github.io/spaCy/})}. The tagger labels each word with its appropriate part of speech given the context of a sentence. Part of speech tagging is important for words that have multiple senses and might otherwise be ambiguous. For example, ``run'' is a noun in the phrase, ``she wants to go for a run'', but a verb in the phrase ``I run into the arms of my reviewers.'' The lemmatizer converts each word into its singular and present-tense form. For example, the plural noun ``soldiers'' can be lemmatized to the singular ``soldier'' and the past tense verb ``ran'' to the present ``runs.''
\subsubsection{Activity-Object statistics}
Activity-object statistics connect commonly co-occurring objects and human activities. These statistics will help Augur detect activities from a list of objects in a scene. We define activities as verb phrases where the subject is a human, and objects as compound noun phrases, throwing away adjectives.
To generate these edges, we run the TC script:
\vspace{.2em}
\begin{lstlisting}[language=Haskell]
human_pronoun = "he" | "she" | "i" | "we" | "they"
np = [DET]? ([ADJ]- [NOUN])+
vp = human_pronoun ([VERB] [ADP])+
MI(freq(co-occur(np, vp, 50)))
\end{lstlisting}
\vspace{-.4em}
For example, \textit{backpack} co-occurs with \textit{pack} 2413 times, and \textit{radio} co-occurs with \textit{singing} 7987 times. Given the scale of our data, Augur's statistics produce meaningful results by focusing just on pronoun-based sentences.
In this TC script, \lstinline{MI} (mutual information, as defined by \cite{mi-ref}) processes our final co-occurence statistics to calculate the mututal information of our relations:
\footnotesize
$$MI(A,B) = \log_{10}\left(\frac{AB*corpusSize}{A*B*span}/log_{10}{2}\right)$$
\normalsize
Where $A$ and $B$ are the frequencies of two relations, the term $AB$ is the frequency of collocation between $A$ and $B$, the term $corpusSize$ is the number of words in our corpus, and $span$ is the window size for the co-occurrence analysis.
MI describes how much one term of a co-occurrence tells us about the other. For example, if people \textit{type} with every kind of object in equal amounts, then knowing there is a computer in your room doesn't mean much about whether you are typing. However, if people type with computers far more often than anything else, then knowing there is a computer in your room tells us significant information, statistically, about what you might be doing.
\subsubsection{Object-affordance statistics}
The object-affordance statistic connects objects directly to their uses and behaviors, helping Augur understand how humans can interact with the objects in a scene. We define \textit{object affordances} as verb phrases where an object serves as either the subject or direct object of the phrase, and we again capture physical objects as compound noun phrases.
To generate these edges, we run the TC script:
\vspace{.2em}
\begin{lstlisting}[language=Haskell]
np = [DET]? ([ADJ]- [NOUN])+
vp = ([VERB] [ADP])+
svo = np vp np?
MI(freq(svo))
\end{lstlisting}
\vspace{-.4em}
For example, \textit{coffee} is \textit{spilled} 229 times, and \textit{facebook} is \textit{logged into} 295 times.
\subsubsection{Activity-Activity statistics}
Activity-activity statistics count the times that an activity is followed by another activity, helping Augur make predictions about what is likely to happen next.
To generate these statistics, we run the TC script:
\vspace{.2em}
\begin{lstlisting}[language=Haskell]
human_pronoun = "he" | "she" | "i" | "we" | "they"
vp = human_pronoun ([VERB] [ADP])+
MI(freq(skip-gram(vp,2,50)))
\end{lstlisting}
\vspace{-.4em}
Activity-activity statistics tend to be more sparse, but Augur can still uncover patterns. For example, \textit{wash hair} precedes \textit{blow dry hair} 64 times, and \textit{get text} (e.g., receive a text message) precedes \textit{text back} 34 times.
In this TC script, \lstinline{skip-gram(vp,2,50)} constructs skip-grams of length $n=2$ sequential $vp$ matches on a window size of 50. Unlike co-occurrence counts, skip-grams are order-dependent, helping Augur find potential causal relationships.
\subsection{Vector space model for retrieval}
Augur's three statistics are not enough by themselves to make useful predictions. These statistics represent pairwise relationships and only allow prediction based on a single element of context (e.g., activity predictions from a single object), ignoring any information we might learn from similar co-occurrences with other terms. For many applications it is important to have a more global view of the data.
To make these global relationships available, we embed Augur's statistics into a vector space model (VSM), allowing Augur to enhance its predictions using the signal of \textit{multiple} terms. Queries based on multiple terms narrow the scope of possibility in Augur's predictions, strengthing predictions common to many query terms, and weaking those that are not.
VSMs encode concepts as vectors, where each dimension of the vector conveys a feature relevant to the concept. For Augur, these dimensions are defined by $MI > 0$ with Laplace smoothing (by a constant value of 10), which in practice reduces bias towards uncommon human activities \cite{vecspace-meta}.
Augur has three VSMs. 1). \textit{Object-Activity}: each vector is a human activity and its dimensions are smoothed MI between it and every object. 2). \textit{Object-Affordance}: each vector is an affordance and its dimensions are smoothed MI between it and every object. 3). \textit{Activity-Prediction}: each vector is a activity and its dimensions are smoothed MI between it and every other activity.
To query these VSMs, we construct a new empty vector, set the indices of the terms in the query equal to 1, then find the closest vectors in the space by measuring cosine similarity.
\begin{figure}[!t]
\centering
\includegraphics[width=1.0\columnwidth]{food-act-smaller}
\vspace{-0.15in}
\caption{Augur's activity detection API translates a photo into a set of likely relevant activities. For example, the user's camera might automatically photojournal the food whenever the user may be \emph{eating food}. Here, Clarifai produced the object labels.}
\label{fig:food-act}
\end{figure}
\section{Augur API and Applications}
Applications can draw from Augur's contents to identify user activities, understand the uses of objects, and make predictions about what a user might do next. To enable software development under Augur, we present these three APIs and a proof-of-concept architecture that can augment existing applications with if-this-then-that human semantics.
We begin by introducing the three APIs individually, then demonstrate additional example applications to follow. To more robustly evaluate Augur, we have built one of these applications, \textit{Soundtrack for Life}, into Google Glass hardware.
\subsection{Identifying Activities}
\begin{figure*}[!t]
\centering
\includegraphics{trigger-overview}
\caption{Augur's APIs map input images through a deep learning object detector, then initializes the returned objects into a query vector. Augur then compares that vector to the vectors representing each activity in its database and returns those with lowest cosine distance.}
\label{fig:triggeroverview}
\end{figure*}
\textit{What are you currently doing?} If Augur can answer this question, applications can potentially help you with that activity, or determine how to behave given the context around you.
Suppose a designer wants to help people stick to their diets, and she notices that people often forget to record their meals. So the designer decides to create an automatic meal photographer. She connects the user's wearable camera to a scene-level object detection computer vision algorithm such as R-CNN \cite{girshick2014rcnn}. While she could program the system to fire a photo whenever the computer vision algorithm recognizes an object class categorized as food, this would produce a large number of false positives throughout the day, and would ignore a breadth of other signals such as silverware and dining tables that might actually indicate eating.
So, the designer connects the computer vision output to Augur (Figure \ref{fig:food-act}). Instead of programming a manual set of object classes, the designer instructs Augur to fire a notification whenever the user engages in the activity \emph{eat food}. She refers to the activity using natural language, since this is what Augur has indexed from fiction:
\vspace{.2em}
\begin{lstlisting}[language=c]
image = /* capture picture from user's wearable camera */
if(augur.detect(image, "eat food"))
augur.broadcast("take photo");
\end{lstlisting}
\vspace{-.4em}
The application takes an \lstinline{image} at regular intervals. The \lstinline{detect} function processes the latest image in that stream, pings
a deep learning computer vision server (\url{http://www.clarifai.com/}), then runs its object results through Augur's object-activity VSM to return activity predictions. The \lstinline{broadcast} function broadcasts an object affordance request keyed on the activity \textit{take photo}: in this case, the wearable camera might respond by taking a photograph.
Now, the user sits down for dinner, and the computer vision algorithm detects a plate, steak and broccoli (Figure \ref{fig:food-act}). A query to Augur returns:
\vspace{.2em}
\begin{lstlisting}
GET /detect/plate+steak+broccoli
prediction score frequency
--------------------------------------
fill plate 0.39 203
put food 0.23 1046
take plate 0.15 1321
(*@\textbf{eat food}@*) 0.14 2449
set plate 0.12 740
cook 0.10 6566
\end{lstlisting}
\vspace{-.4em}
The activity \textit{eat food} appears as a strong prediction, as is (further down) the more general activity \textit{eat}. The ensemble of objects reinforce each other: when the plate, steak and broccoli are combined to form a query, eating has 1.4 times higher cosine similarity than for any of the objects individually. The camera fires, and the meal is saved for later.
\begin{figure}[!t]
\centering
\includegraphics[width=1.0\columnwidth]{bench-smaller}
\vspace{-0.15in}
\caption{Augur's object affordance API translates a photo into a list of possible affordances. For example, Augur could help a blind user who is wearing an intelligent camera and says they want to \emph{sit}. Here, Clarifai produced the object labels.}
\label{fig:blind}
\end{figure}
\subsection{Expanding Activites with Object Affordances}
\textit{How can you interact with your environment?} If Augur knows how you can manipulate your surroundings, it can help applications facilitate that interaction.
Object affordances can be useful for creating accessible technology. For example, suppose a blind user is wearing an intelligent camera and tells the application they want to \emph{sit} (Figure \ref{fig:blind}). Many possible objects would let this person sit down, and it would take a lot of designer effort to capture them all. Instead, using Augur's object affordance VSM, an application could scan nearby objects and find something sittable:
\vspace{.2em}
\begin{lstlisting}[language=c]
image = /* capture picture from user's wearable camera */
if(augur.affordance(image, "sit"))
alert("sittable object ahead");
\end{lstlisting}
\vspace{-.4em}
The \lstinline{affordance} function will process the objects in the latest image, executing its block when Augur notices an object with the specified affordance.
Now, if the user happens to be within eyeshot of a bench:
\vspace{.2em}
\begin{lstlisting}
GET /affordance/bench
prediction score frequency
---------------------------------------
(*@\textbf{sit}@*) 0.13 600814
take seat 0.12 24257
spot 0.11 16132
slump 0.09 8985
plop 0.07 12213
\end{lstlisting}
\vspace{-.4em}
Here the programmer didn't need to stop and think about all the scenarios or objects where a user might sit. Instead, they just stated the activity and augur figured it out.
\subsection{Predicting Future Activities}
\textit{What will you do next?} If Augur can predict your next activity, applications can react in advance to better meet your needs in that situation. Activity predictions are particularly useful for helping users avoid problematic behaviors, like forgetting their keys or spending too much money.
In Apple's Knowledge Navigator \cite{knowledgenavigator}, the agent ignores a phone call when it knows that it would be an inappropriate time to answer. Could Augur support this?
\vspace{.2em}
\begin{lstlisting}[language=c]
answer = augur.predict("answer call")
ignore = augur.predict("ignore call")
if(ignore > answer)
augur.broadcast("silence phone");
else
augur.broadcast("unsilence phone");
\end{lstlisting}
\vspace{-.4em}
The \lstinline{augur.predict} fuction makes new activity predictions based on the user's activities over the past several minutes. If the current context suggests that a user is using the restroom, for example, the prediction API will know that answering a call is an unlikely next action. When provided with an activity argument, \lstinline{augur.predict} returns a cosine similarity value reflecting the possibility of that activity happening in the near future. The activity \emph{ignore call} has less cosine similarity than \emph{answer call} for most queries to Augur. But if a query ever indicates a greater cosine similarity for \textit{ignore call}, the application can silence the phone. As before, Augur broadcasts the desired activity to any listening devices (such as the phone).
\begin{comment}
The \lstinline{broadcast} command allows Trigger to command other devices connected to its network. Devices registered under the name of an object that has an affordance associated with the broadcast (e.g., a phone) will respond to the command if capable. For example:
\vspace{.2em}
\begin{lstlisting}[language=Haskell]
register({type:"phone"})
if(recieve("silence") { mute(); }
if(recieve("unsilence") { unmute(); }
\end{lstlisting}
\vspace{-.4em}
When this phone receives a silence command, \lstinline{recieve} will execute the \lstinline{mute} function, to switch off all sound features via an internal API.
\end{comment}
Suppose your phone rings while you are talking to your best friend about their relationship issues. Thoughtlessly, you \textit{curse}, and your phone stops ringing instantly:
\vspace{.2em}
\begin{lstlisting}
GET /predict/get call+curse
prediction score frequency
-------------------------------------
throw phone 0.24 3783
(*@\textbf{ignore call}@*) 0.18 567
ring 0.18 7245
answer call 0.17 4847
call back 0.17 1883
leave voicemail 0.17 146
\end{lstlisting}
\vspace{-.4em}
Many reactions besides cursing might also trigger \textit{ignore call}. In this case, adding \textit{curse} to the prediction mix shifts the odds between ignoring and answering significantly. Other results like \textit{throw phone} reflect the biases in fiction. We will investigate the impact of these biases in our Evaluation.
\subsection{Applications}
Augur allows developers to build situationally reactive applications across many activities and contexts. Here we present three more applications designed to illustrate the potential of its API. We have deployed one of these applications, \textit{A Soundtrack for Life}, as a Google Glass prototype.
\begin{comment}
\subsubsection{The Autonomous Activity Journal}
We often forget where we have gone and what we have done. Augur allows us to journal our activities passively, automatically (and probabilistically):
\vspace{.2em}
\begin{lstlisting}[language=c]
predictions = augur.predict()
for(p in predictions where p.score > 0.8)
file.write("journal" , p.activity);
\end{lstlisting}
\vspace{-.4em}
When Augur returns new predictions about our life, this program will write the most likely ones to log. We might search this log later, or use it find patterns in our daily behavior. For example, what days are we most likely to exercise? How often do we tend to go our to eat, or hang our with friends? Some of Augur's predictions will inevitably be false positives, but in aggregate they may provide useful analytics into our lives.
\end{comment}
\subsubsection{The Coffee-Aware Smart Home}
In Weiser's ubiquitous computing vision \cite{weiser}, he introduces the idea of calm computing via a scenario where a woman wakes up and her smart home asks if she wants coffee. Augur's activity prediction API can support this vision:
\vspace{.2em}
\begin{lstlisting}[language=c]
if(augur.predict("make coffee") { askAboutCoffee(); }
\end{lstlisting}
\vspace{-.4em}
Suppose that your alarm goes off, signaling to Augur that your activity is \emph{wake up}. Your smart coffeepot can start brewing when Augur predicts you want to make coffee:
\vspace{.2em}
\begin{lstlisting}
GET /predict/wake up
prediction score frrequency
----------------------------------------
want breakfast 0.38 852
throw blanket 0.38 728
shake awake 0.37 774
hear shower 0.36 971
take bath 0.35 1719
(*@\textbf{make coffee}@*) 0.34 779
check clock 0.34 2408
\end{lstlisting}
\vspace{-.4em}
After people \textit{wake up} in the morning, they are likely to make coffee. They may also \textit{want breakfast}, another task a smart home might help us with.
\subsubsection{Spending Money Wisely}
We often spend more money than we have. Augur can help us maintain a greater awarness of our spending habits, and how they affect our finances. If we are reminded of our bank balence before spending money, we may be less inclined to spend it on frivolous things:
\vspace{.2em}
\begin{lstlisting}[language=c]
if(predict("pay") {
balance = secure_bank_query();
speak("your balance is "+ balance);
}
\end{lstlisting}
\vspace{-.4em}
If Augur predicts we are likely to pay for something, it will tell us how much money we have left in our account. What might triggger this prediction?
\vspace{.2em}
\begin{lstlisting}
GET /predict/enter store
prediction score frequency
------------------------------------
scan 0.19 5319
ring 0.19 7245
(*@\textbf{pay}@*) 0.17 23405
swipe 0.17 1800
shop 0.13 3761
\end{lstlisting}
\vspace{-.4em}
For example, when you enter a store, you may be about to \textit{pay} for something. The \textit{pay} prediction also triggers on \textit{ordering} food or coffee, \textit{entering} a cafe, \textit{gambling}, and \textit{calling a taxi}.
\vspace{.2em}
\begin{lstlisting}
GET /predict/call taxi
prediction score frequency
------------------------------------
hail taxi 0.96 228
(*@\textbf{pay}@*) 0.96 181
take taxi 0.96 359
get taxi 0.96 368
tell address 0.95 463
get suitcase 0.82 586
\end{lstlisting}
\vspace{-.4em}
\begin{figure}[!t]
\centering
\includegraphics[width=1.0\columnwidth]{glass-test}
\vspace{-0.15in}
\caption{A Soundtrack for Life is a Google Glass application that plays musicians based on the user's predicted activity, for example associating \textit{working} with The Glitch Mob.}
\label{fig:glass-test}
\end{figure}
\subsubsection{A Soundtrack for Life}
Many of life's activities are accompanied by music: you might \textit{cook} to the refined arpeggios of Vivaldi, \textit{exercise} to the dark ambivalence of St.\ Vincent, and \textit{work} to the electronic pulse of the Glitch Mob. Through an activity detection system we have built into Google Glass (Figure \ref{fig:glass-test}), Augur can arrange a soundtrack for you that suits your daily preferences. We built a physical prototype for this application as it takes advantage of the full range of activities Augur can detect.
\vspace{.2em}
\begin{lstlisting}[language=c]
var act2music = {
"cook": "Vivaldi", "drive": "The Decemberists",
"surfing": "Sea Wolf", "buy": "Atlas Genius",
"work": "Glitch Mob", "exercise": "St. Vincent",
};
var act = augur.predict();
if (act in act2music){
play(act2music[act]);
}
\end{lstlisting}
\vspace{-.4em}
For example, if you are brandishing a spoon before a pot on the stove, you are likely \textit{cooking}. Augur plays Vivaldi.
\vspace{.2em}
\begin{lstlisting}
GET /predict/stove+pot+spoon
prediction score frequency
------------------------------------
(*@\textbf{cook}@*) 0.50 6566
pour 0.39 757
place 0.37 25222
stir 0.37 2610
eat 0.34 49347
\end{lstlisting}
\vspace{-.4em}
\begin{figure*}[!t]
\centering
\includegraphics[width=2.05\columnwidth]{augur-field}
\caption{We deployed an Augur-powered wearable camera in a field test over common daily activities, finding average rates of 96\% recall and 71\% precision for its classifications.}
\label{fig:glasstest}
\end{figure*}
\section{Evaluation}
Can fiction tell us what we need in order to endow our interactive systems with basic knowledge of human activities? In this section, we investigate this question through three studies. First, we compare Augur's activity predictions to human activity predictions in order to understand what forms of bias fiction may have introduced. Second, we test Augur's ability to detect common activities over a two-hour window of daily life. Third, to stress test Augur over a wider range of activities, we evaluate its activity predictions on a dataset of 50 images sampled from the Instagram hashtag \textit{\#dailylife}.
\subsection{Bias of Fiction}
If fiction were truly representative of our lives, we might be constantly drawing swords and kissing in the rain. Our first evaluation investigates the character and prevelance of fiction bias. We tested how closely a distribution of 1000 activities sampled from Augur's knowledge base compared against human-reported distributions. While these human-reported distributions may differ somewhat from the real world, they offer a strong sanity check for Augur's predictions.
\subsubsection{Method}
To sample the distribution of activities in Augur, we first randomly sampled 100 objects from the knowledge base. We then used Augur's activity identification API to select 10 human activities most related to each object by cosine similarity. In general, these selected activities tended to be relatively common (e.g., \textit{cross} and \textit{park} for the object ``street''). We normalized these sub-distributions such that the frequencies of their activities summed to 100.
Next, for each object we asked five workers on Amazon Mechanical Turk to estimate the relative likelihood of its selected activities. For example, given a piano: ``Imagine a random person is around a piano 100 times. For each action in this list, estimate how many times that action would be taken. The overall counts must sum to 100.'' We asked for integer estimates because humans tend to be more accurate when estimating frequencies \cite{cogillusion}.
Finally, we computed the estimated true human distribution (ETH) as the mean distribution across the five human estimates. We compared the mean absolute error (MAE) of Augur and the individual human estimates against the ETH.
\subsubsection{Results}
Augur's MAE when compared to the ETH is 12.46\%, which means that, on average, its predictions relative to the true human distribution are off by slightly more than 12\%. The mean MAE of the individual human distributions when compared to the ETH is 6.47\%, with a standard deviation of 3.53\%. This suggests that Augur is biased, although its estimates are not far outside the variance of individual humans.
Investigating the individual distributions of activities suggests that the vast majority of Augur's prediction error is caused by a few activities in which its predictions differ radically from the humans. In fact, for 84\% of the tested activities Augur's estimate is within 4\% of the ETH. What accounts for the these few radically different estimates?
The largest class of prediction error is caused by general activities such as \textit{look}. For example, when considering raw co-occurrence frequencies, people \textit{look} at clocks much more often than they \textit{check the time}, because \text{look} occurs far more often in general. When estimating the distribution of activities around \textit{clock}, human estimators put most of their weight on \textit{check time}, while Augur put nearly all its weight on \textit{look}. Similar mistakes involved the common but understated activities of \textit{getting into} cars or \textit{going to} stores. Human estimators favored \textit{driving} cars and \textit{shopping} at stores.
A second and smaller class of error is caused by strong connections between dramatic events that take place more often in fiction than in real life. For example, Augur put nearly all of its prediction weight for cats on \textit{hissing} while humans distributed theirs more evenly across a cat's possible activities. In practice, we saw few of these overdramaticized instances in Augur's applications and it may be possible to use paid crowdsourcing to smooth out them out. Further, this result suggests that the ways fiction deviates from real life may be more at the macro-level of plot and situation, and less at the level of micro-behaviors. Yes, fictional characters somtimes find themselves defending their freedom in court against a murder charge. However, their actions within that courtroom do tend to mirror reality --- they don't tend to leap onto the ceiling or draw swords.
\subsection{Field test of A Soundtrack for Life}
Our second study evaluates Augur through a field test of our Glass application, \textit{A Soundtrack for Life}. We recorded a two-hour sample of one user's day, in which she walked around campus, ordered coffee, drove to a shopping center, and bought groceries, among other activities (Figure \ref{fig:glasstest}).
\subsubsection{Method}
We gave a Google Glass loaded with \textit{A Soundtrack for Life} to a volunteer and asked her, over a two hour period, to to enact the following eight activities: walk, buy, eat, read, sit, work, order, and drive.
We then turned on the Glass, set the Soundtrack's sampling rate to 1 frame every 10 seconds, and recorded all data.
The Soundtrack logged its predictions and images to disk.
Blind to Augur's predictions, we annotated all image frames with a set of correct activities. Frames could consist of no labeled activities, one activity, or several. For example, a subject sitting at a table filled with food might be both \textit{sitting} and \textit{eating}. We included plausible activities among this set. For example, when the subject approaches a checkout counter, we included \textit{pay} both under circumstances in which she did ultimately purchase something, and also under others in which she did not.
Over these annotated image frames, we computed precision and recall for Augur's predictions.
\subsubsection{Results}
We find rates of 96\% recall and 71\% precision across activity predictions in the dataset (Figure \ref{tbl:soundtrack}).
When we break up these rates by activity, Augur succeeds best at activities like \textit{walk}, \textit{buy} and \textit{read}, with precision and recall score higher than 82\%. On the other hand, we see that the activities \textit{work}, \textit{drive}, and \textit{sit} cause the majority of Augur's errors. Work is triggered by a diverse set of contextual elements. People \textit{work} at cafes or grocercy stores (for their jobs), or do construction work, or work on intellectual tasks, like writing research papers on their laptops. Our image annotations did not capture all these interpretations of work, so Augur's disagreement with our labeling is not surprising. Drive is also triggered by a large number of contexuntual elements, including broad scene descriptors like ``store'' or ``cafe,'' presumably because fictional characters often drive to these places. And \textit{sit} is problematic mostly because it is triggered by the common scene element ``tree'' (real-world people probably do this less often than fictional characters). We also observe simpler mistakes: for example, our computer vision algorithm thought the bookstore our subject visited was a restaurant, causing a large precision hit to \textit{eat}.
\begin{table}[tb]\scriptsize
\renewcommand{\arraystretch}{1.4}
\begin{helvetica}
\begin{tabular}{p{5em}@{\hspace{2em}}p{11em}@{\hspace{1em}}p{5em}@{\hspace{1em}}p{5em}}
\textbf{Activity} & \textbf{Ground Truth Frames} & \textbf{Precision} & \textbf{Recall} \\
\hline
Walk & 787 & 91\% & 99\% \\
Drive & 545 & 63\% & 100\% \\
Sit & 374 & 59\% & 86\% \\
Work & 115 & 44\% & 97\% \\
Buy & 78 & 89\% & 83\% \\
Read & 33 & 82\% & 87\% \\
Eat & 12 & 53\% & 83\% \\
\hline
\textbf{Average} & & 71\% & 96\%
\end{tabular}
\end{helvetica}
\caption{We find average rates of 96\% recall and 71\% precision over common activities in the dataset. Here \textit{Ground Truth Frames} refers to the total number of frames labeled with each activity.}
\label{tbl:soundtrack}
\end{table}
\subsection{A stress test over \#dailylife}
Our third evaluation investigates whether a broad set of inputs to Augur would produce meaningful activity predictions.
We tested the quality of Augur's predictions on a dataset of 50 images sampled from the Instagram hashtag \textit{\#dailylife}. These images were taken in a variety of environments across the world, including homes, city streets, workplaces, restaurants, shopping malls and parks. First, we sought to measure whether Augur predicts meaningful activities given the objects in the image. Second, we compared Augur's predictions to the human activity that best describes each scene.
\subsubsection{Method}
To construct a dataset of images containing real daily activites, we sampled 50 scene images from the most recent posts to the Instagram \textit{\#dailylife} hashtag \footnote{\url{https://instagram.com/explore/tags/dailylife/}}, skipping 4 images that did not represent real scenes of people or objects, such as composite images and drawings.
We ran each image through an object detection service to produce a set of object tags, then removed all non-object tags with WordNet.
For each group of objects, we used Augur to generate 20 activity predictions, making 1000 in total.
We used two external evaluators to independently analyze each of these predictions as to their plausibility given the input objects, and blind to the original photo. A third external evaluator decided any disagreements. High quality predictions describe a human activity that is likely given the objects in a scene: for example, using the objects \textit{street, mannequin, mirror, clothing, store} to predict the activity \textit{buy clothes}. Low quality predictions are unlikely or nonsensical, such as connecting \textit{car, street, ford, road, motor} to the activity \textit{hop}.
Next, we showed evaluators the original image and asked them to decide: 1) whether computer vision had extracted the set of objects most important to understanding the scene 2) whether one of Augur's predictions accurately described the most important activity in each scene.
\subsubsection{Results}
The evaluators rated 94\% of Augur's predictions are high quality (Table \ref{tbl:predictions}). Among the 44 that were low quality, many can be accounted for by tagging issues (e.g., ``sink'' being mistagged as a verb). The others are largely caused by relatively uncommon objects connecting to frequent and overly-abstract activities, for example the uncommon object ``tableware'' predicts ``pour cereal''.
Augur makes activity predictions that accurately describe 82\% of the images, despite the fact that CV extracted the most important objects in only 62\%. Augur's knowledge base is able to compensate for some noise in the neural net: across those images with good CV extraction, Augur succeeded at correctly predicting the most relevant activity on 94\%.
\begin{table}[tb]\footnotesize
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{p{10em}@{\hspace{2em}}p{4em}@{\hspace{2em}}p{8em}}
\tabhead{Quality} & \tabhead{Samples} & \tabhead{Percent Success} \\
\hline
Augur VSM predictions & 1000 & 94\% \\
Augur VSM scene recall & 50 & 82\% \\
Computer vision object detection & 50 & 62\% \\
\end{tabular}
\caption{As rated by external experts, the majority of Augur's predictions are high-quality.}
\label{tbl:predictions}
\end{table}
\section{Discussion}
Augur's design presents a set of opportunities and limitations, many of which we hope to address in future work.
First, we acknowledge that data-driven approaches are not panaceas. Just because a pattern appears in data does not mean that it is interpretable. For example, ``boyfriend is responsible'' is a statistical pattern in our text, but it isn't necessarily useful. Life is full of uninterpretable correlations, and developers using Augur should be careful not to trigger unusual behaviors with such results. A crowdsourcing layer that verifies Augur's predictions in a specific topic area may help filter out any confusing artifacts.
Similarly, while fiction allows us to learn about an enormous and diverse set of activities, in some cases it may present a vocabulary that is too open ended. Activities may have similar meanings, or overly broad ones (like \textit{work} in our evaluation). How does a user know which to use? In our testing, we have found that choice of phrase is often unimportant. For example, the cosine similarity between \textit{hail taxi} and \textit{call taxi} is 0.97, which means any trigger for one is in practice equivalent to the other (or \textit{take taxi} or \textit{get taxi}). In this sense a large vocabulary is actively helpful. However, for other activities choice of phrase does matter, and to identify and collpase these activities, we again see potential for the refinement of Augur's model through crowdsourcing.
In the process of pursuing this research, we found ourselves in many data mining dead ends. Human behavior is complex, and natural language is complex. Our initial efforts included heavier-handed integration with WordNet to identify object classes such as locations and peoples' names; unfortunately, ``Virginia'' is both. This results in many false positives. Likewise, activity prediction requires an order of magnitude more data to train than the other APIs given the $N^2$ nature of its skip-grams. Our initial result was that very few scenarios lent themselves to accurate activity prediction. Our solution was to simplify our model significantly (e.g., looking at only pronouns) and gather ten times the raw data from Wattpad. In this case, more data beat more modeling intelligence.
More broadly, Augur suggests a reinterpretation of our role as designers. Until now, the designer's goal in interactive systems has been to articulate the user's goals, then fashion an interface specifically to support those goals. Augur proposes a kind of ``open-space design'' where the behaviors may be left open to the users to populate, and the designer's goal is to design reactions that enable each of these goals. To support such an open-ended design methdology, we see promise in Augur's natural language descriptions. Activities such as ``sit down'', ``order dessert'' and ``go to the movies'' are not complex activity codes but human-language descriptions. We speculate that each of Augur's activities could become a command. Suppose any device in a home could respond to a request to ``turn down the lights''. Today, Siri has tens of commands; Augur has potentially thousands.
\section{Conclusion}
Interactive systems find themselves in a double bind: they cannot sense the environment well enough to build a broad model of human behavior, and they cannot support the breadth of human needs without that model. Augur proposes that fiction --- an ostensibly false record of humankind --- provides enough information to break this stalemate. Augur captures behaviors across a broad range of activities, from drinking coffee, to starting a car, to going out to a movie. These behaviors often represent concepts that designers may have never thought to hand-author.
Moving forward, we will integrate this information as a prior into the kinds of activity trackers and machine learning models that developers already use. We aim to develop a broader suite of Augur applications and test them in the wild.
\section{Acknowledgments}
Special thanks to our reviewers and colleagues at Stanford for their helpful feedback.
This work is supported by a NSF Graduate Fellowship.
\bibliographystyle{acm-sigchi}
|
3,212,635,537,419 | arxiv | \section{Introduction}
The discovery of nucleon resonances in the first pion-nucleon scattering experiments
provided first indications for a complicated intrinsic structure of the nucleon.
With establishing the quark picture of hadrons and developments of the
constituent quark models the interest in the study of the nucleon excitation spectra was renewed.
The major question was the number of the excited states and their properties. This problem was attacked
both experimentally and theoretically.
On the theory side constituent quark (CQM) models, lattice QCD and Dyson-Schwinger approaches
have been developed to describe and predict the nucleon resonance spectra
(see e.g. \cite{Aznauryan:2009da} for a review).
The main problem remains, however, a serious disagreement between the theoretical calculations and
the experimentally observed baryon spectra. This concerns both the number and the properties of excited states.
On the experimental side pion-induced reactions have been studied
to establish resonance spectra. However, due to difficulties in detecting neutral
particles
most experiments were limited to pion-nucleon elastic scattering
with charged particles in the final state.
Being the lightest non-strange particle next to the pion the $\eta$-meson also becomes an
interesting probe to study nucleon excitations.
A few experiments have been made in the past to investigate $\eta$-production.
The first near-threshold measurements \cite{Jones:1966zza,Richards:1970cy,Bulos:1970zk} demonstrated
that the reaction proceeds through a strong $S$-wave
resonance excitation which was later identified with $S_{11}(1535)$.
An extensive study of the $ \pi^- p \to\eta n$ reaction above W$>$1.7 GeV has been
made in \cite{ Brown:1979ii,Baker:1979aw}. Both differential cross section and asymmetry
data have been obtained. However, due to possible problems with the energy-momentum
calibration \cite{Clajus:1992dh} the use of these data might lead to wrong conclusions on the
reaction mechanism. Note that these problems
are present not only in the $\eta$-measurements \cite{ Brown:1979ii,Baker:1979aw}
but also in the charge-exchange data obtained in the same experiment.
Presently the development of the high-duty electron facilities (ELSA, JLAB, MAMI, SPring) offers
new possibilities to study the $\eta$-photoproduction both on the proton ($\eta p$) and on the
neutron ($\eta n$).
The first measurement of the $\eta$-photoproduction on the neutron reported an indication
for a resonance-like structure in the reaction cross section at
W=1.68 GeV \cite{Kuznetsov:2006, Kuznetsov:2006kt}.
Independent experimental studies \cite{Jaegle:2008ux,Jaegle:2011sw} confirmed the
existence of this effect in the $\gamma n$ reaction. This phenomenon was
predicted in \cite{Azimov:2005jj} as a signal from a narrow state - a possible
non-strange partner of the pentaquark \cite{Diakonov:1997mm}. Another
explanation has been suggested in \cite{Shklyar:2006xw} where the observed effect was
described by the contributions from the $S_{11}(1650)$ or $P_{11}(1710)$ states.
Due to the lack of knowledge of the $S_{11}(1650)$ and $P_{11}(1710)$
resonance couplings to $\gamma n$ a clean separation of the relative contributions from
these states is difficult. The general conclusion made in \cite{Shklyar:2006xw}
is that both states might be good candidates to explain the
observed structure.
By fitting to the $\eta n$ cross sections and beam asymmetry the Bonn-Gatchina group
provided an explanation \cite{Anisovich:2008wd} for the second peak in terms of the $S_{11}(1650)$ state.
Another contribution to the field has been made by the authors of \cite{Doring:2009qr}.
There the peak in the $\sigma_p/\sigma_n$ cross section ratio was explained by a cusp
effect from the $K\Sigma$ and $K\Lambda$ rescattering channel. All these studies
have been done assuming
scattering on a quasi-free nucleon. At the same time a realistic analysis of
meson photoproduction on the quasi-free neutron should include the nucleon-nucleon and meson-nucleon
correlations (FSI-effect) which were shown to be very important \cite{Tarasov:2011ec} and take into
account corresponding experimental cuts applied by the extraction of the quasi-free neutron
data from $\gamma D$-scattering. The later issue might be crucial for the unambiguous identification
of the narrow resonance contribution as discussed in \cite{MartinezTorres:2010zzb}.
If it is granted that the signal observed in the $\gamma n$ scattering
\cite{Kuznetsov:2006, Kuznetsov:2006kt,Jaegle:2008ux,Jaegle:2011sw}
is due to the narrow (exotic) state one may expect to observe a similar effect in other eta-production
reactions at the same energies, e.g in gamma-proton scattering.
The experimental investigations of the
$\eta$-production on the proton made by the CLAS, GRAAL, and CB-ELSA/TAPS collaborations
\cite{Dugger:2002ft,Crede:2009zzb,Bartholomy:2007zz,Bartalini:2007fg} have found an indication of the dip
structure around W=1.68 GeV in the differential cross section but not a resonance-like structure.
This effect was also accompanied by the change in the angular distribution of the differential cross section.
However, despite of extensive theoretical studies of the $\eta$ -production
the reaction mechanism is still under discussion
\cite{PhysRevC.84.045207,Chiang:2002vq,Bartholomy:2007zz,Feuster:1998b,An:2011sb,Anisovich:2011ka,Shyam:2008fr,
Nakayama:2008tg,Choi:2007gy,Zhong:2007fx,Fix:2007st,Gasparyan:2003fp,Ruic:2011wf}.
Recently the $\eta$-photoproduction on the proton has been measured with high-precision
by the Crystal Ball collaboration at MAMI \cite{McNicoll:2010qk}.
These high-resolution data provides a new step forward
in understanding the reaction dynamics and in the search for a signal from the
'weak' resonance states.
The main result reported in \cite{McNicoll:2010qk} is a very clean signal of a dip
structure around W=1.68 GeV. It is interesting to note that the old measurements of the
$\pi N \to \eta N$ reaction \cite{Richards:1970cy} also give an indication for the second structure in the
differential cross section at W=1.7 GeV. This raises a question whether the
dip reported in the $\eta p$ reaction, the resonance-like signal observed in $\eta n$ and the possible
structure in the $\pi N \to \eta N $ cross section are originating from the same degrees of freedom
or not. The second question is whether one of these phenomena
can be attributed to the signal from a narrow (exotic) resonance state as discussed
in \cite{Azimov2,Azimov:2005jj,Azimov:2005hv}.
In our previous coupled-channel PWA study \cite{Shklyar:2006xw}
we proposed an explanation of the possible dip in the $\eta$-proton cross section in terms of
the destructive interference of the $S_{11}(1535)$ and $S_{11}(1650)$ states.
The result was based on the $\eta p$ photoproduction data taken before 2006 \cite{Dugger:2002ft,Crede:2003ax}.
The aim of the present study is to extend our previous coupled-channel analysis of the $\gamma p\to \eta p$
reaction by including the data from the high-precision measurements \cite{McNicoll:2010qk}.
The main question is whether the $\eta p$ reaction dynamics can be understood in terms of the established
resonance states. We emphasize that for reliable identification of
the resonance contributions the calculations should maintain unitarity. Another complication comes
from the fact that the most contributions to the resonance self-energy (total decay width) is driven by its hadronic
couplings. Therefore the analysis of the photoproduction data requires the knowledge of the
hadronic transition amplitudes. Hence the simultaneous analysis of all open channels (both hadronic
and electromagnetic ) is inevitable for the identification of the resonances and extraction of their properties.
In the present study we concentrate on the combined description of the $(\gamma/\pi) p \to \eta p$ scattering
taking also the $(\gamma/\pi) \to \pi N$, $2\pi N$, $\omega N$, $K \Lambda$ channels into account.
The results on the $\eta n$ reaction will be reported elsewhere.
First, we corroborate our previous findings \cite{Penner:2002a,Penner:2002b,Shklyar:2006xw}
where the important contributions from the $S_{11}(1535)$, $S_{11}(1650)$ and $P_{11}(1710)$
resonances to the $\pi N \to \eta N$ reaction have been found.
The major effect comes from the $S_{11}$ and $P_{11}$ partial waves.
The interference between the $S_{11}(1535)$ and $S_{11}(1650)$ states
produces a dip in the $S_{11}$ amplitude. The $P_{11}$ amplitude is
influenced by the contributions from the $P_{11}(1710)$ state.
The interference between the $S_{11}$ and $P_{11}$ partial waves leads to the
forward peak in the differential cross section around W=1.7 GeV.
We stress that the interference between two nearby states also includes rescattering and coupled-channel
effects which are hard to simulate by the simple sum of two Breit-Wigner forms.
We also confirm our previous finding that the interference between $S_{11}(1535)$ and $S_{11}(1650)$
is responsible for the dip seen in the $\eta p$ data. The effect from the $\omega N$ threshold is found to be
relatively small which is also in line with the conclusion of \cite{Shklyar:2006xw}. Opposite to
\cite{Anisovich:2011ka} we do not find any strong indications for a narrow state in the Crystal Ball/Taps
data around W=1.68 GeV.
We have also checked our results for
the $\eta p$ reaction above W=2 GeV where a number of new experimental data are available. Note that we do not
use Reggezied $t-$channel exchange but include all $t$-channel contributions consistently into
our unitarization procedure. Because of the normalization
problem \cite{Sibirtsev:2010yj,Dey:2011rh} between the CLAS \cite{Williams:2009yj} and
the CB-ELSA \cite{Crede:2009zzb} datasets
the simultaneous description of these data is not possible.
Above W=2 GeV our calculations are found to be in closer agreement with the
CLAS measurements \cite{Williams:2009yj}. The CB-ELSA data \cite{Crede:2009zzb}
demonstrates a step rise around W=1.925 GeV for the scattering angles $\cos\theta=0.85...0.95$.
It is not clear whether this phenomenon could be related to a threshold effect (e.g. $\phi N$, $a_0(980) N$,
$f_0(980)$, or $\eta' N$) or attributed to other reaction mechanisms.
We conclude that further progress in understanding of the $\eta$-meson production dynamics
would be hardly possible without new measurements of the $\pi N\to \eta N$ reaction.
\section{Database}
\label{data}
Here we present a short overview of the experimental database relevant for the present calculations.
The details on the
$K\Lambda$, $K\Sigma$, $\omega N$ channels will be given elsewhere.
\mbox{ $\pi N\to\eta N$}:
The thorough overview of the $\pi N\to\eta N$ experimental data (except the
recently published Crystal Ball
measurements \cite{Prakhov}), is given in \cite{Clajus:1992dh}.
As already mentioned in Introduction only few measurements of the $\eta$-production have been made
with pion beams: except for \cite{Danburg:1971} where the eta-meson was produced in $\pi^+ D$
collisions, all the data have been taken from the $\pi^- p$ scattering
\cite{Prakhov,Baker:1979aw,Brown:1979ii,Bulos:1970zk,Richards:1970cy,Jones:1966zza,Debenham,Deinet:1969cd}.
Unfortunately due to numerous problems with the experimental data from \cite{Baker:1979aw,Brown:1979ii}
(see discussion in \cite{Clajus:1992dh} and references therein) the use of these measurements in the
analysis might lead to wrong conclusions for the reaction mechanism.
Therefore, opposite to \cite{Shrestha:2012va} we do not include these data in the analysis.
Another measurement available above W=1.65 GeV is the data from Richards et al
\cite{Richards:1970cy}. In the first resonance energy region this cross section
tends to be lower than results from other experiments.
Since the old measurements quote only statistical uncertainties the reason for these
differences is unclear. In their study the authors of \cite{Batinic:1995} added systematical
errors to all differential cross sections. We do not follow this procedure and include only quoted
uncertainties in the analysis.
$\gamma p\to\eta p$:
a number of experimental studies have been performed in the resonance energy region \cite{McNicoll:2010qk,
Williams:2009yj,GRAAL:2002, Dugger:2002ft,Krusche:1995nv,Bartalini:2007fg,Nakabayashi:2006ut,Crede:2009zzb,Crede:2003ax,
Bartholomy:2007zz,Elsner:2007hm,Bock:1998rk,Ajaka:1998zi}. Most of these measurement are differential cross sections.
The target asymmetry has been studied in \cite{Bock:1998rk}. It has been observed that close to the
$\eta N$ production threshold the asymmetry changes the sign at moderate scattering angles.
The previous calculations of the
Giessen Model \cite{Penner:2002a,Shklyar:2006xw} and the Mainz group \cite{PhysRevC.60.035210} could not explain
this feature. The description of this data would require an unexpected phase shift between the $S_{11}$ and $D_{13}$
resonances as noted in \cite{PhysRevC.60.035210}. One may hope that the upcoming new measurements of the target
asymmetry at the ELSA facility will solve this puzzle \cite{Hartmann:2011uv}.
For the beam asymmetry we use the recent data from the GRAAL \cite{Bartalini:2007fg}
and CB-ELSA/TAPS \cite{Elsner:2007hm} collaborations which cover the energy region up to W=1.91 GeV.
For the differential cross section we use the recent high-quality Crystal Ball data \cite{McNicoll:2010qk}.
Above W=1.89 GeV our calculations are constrained by the amalgamated data set from experiments
\cite{Williams:2009yj,Crede:2009zzb,Crede:2003ax,Bartholomy:2007zz}.
Since the experimental uncertainties of the data \cite{Williams:2009yj,Crede:2009zzb,Crede:2003ax,Bartholomy:2007zz}
are much larger than those in \cite{McNicoll:2010qk} we reduce them by factor of 2.
In the $(\pi/\gamma) N \to \pi N$ channels our calculations are constrained by the single-energy solutions
from the GWU (former SAID) analysis \cite{Workman:2011vb,Arndt:2006bf,Arndt:2008zz}.
For the $\pi N\to 2\pi N$ transitions we follow the procedure described in
\cite{Penner:2002a,Penner:2002b,Feuster:1998a,Feuster:1998b}. We continue to parameterize the $2\pi N$ channel
in terms of the effective $\zeta N$ state, where $\zeta$ is an isovector scalar meson of two pion mass:
$m_\zeta=2m_\pi$. The final $\zeta N$ state is only allowed to couple to nucleon resonances. Therefore the decay
$N^*\to\zeta N$ stands for the sum of transitions $N^*\to \Delta \pi$, $\sigma N$, $\rho N$ etc.
This procedure allows for the good description of the $\pi N \to 2\pi N$ partial wave cross sections
extracted in \cite{Manley:1984}. However of case of the $\gamma p\to 2\pi N$ the same agreement
cannot be expected. This is because of the enhanced role of the background contributions
(due to e.g. the contact $\gamma\rho NN$ interaction in the $\gamma N\to \rho N$ transitions).
After fixing the database a $\chi^2$ minimization is performed to fix the model parameters.
\section{Giessen Model}
\label{model}
Here we briefly outline the main ingredients of the model. More details can be found in
\cite{Feuster:1998a,Feuster:1998b,Penner:2002a,Penner:2002b,shklyar:2004a,shklyar:2005c}. The Bethe-Salpeter equation
is solved in the $K$-matrix approximation to obtain multi-channel scattering $T$-matrix:
\begin{eqnarray}
T(\sqrt{s},p,p') = K(\sqrt{s}, p, p') + \int \frac{d^4q}{(2\pi)^4}
K(\sqrt{s},p,q)G_{BS}(\sqrt{s},q)T(\sqrt{s},q,p'),
\label{bse}
\end{eqnarray}
where $p$ ($k$) and $p'$ ($k'$) are the incoming and outgoing baryon (meson) four-momenta,
$T(\sqrt{s},p,p')$ is a coupled-channel scattering amplitude, $G_{BS}$ is a meson-nucleon
propagator and $K(\sqrt{s}, p, p')$ is an interaction kernel. The quantities $T(\sqrt{s},p,p')$, $G_{BS}$,
and $K(\sqrt{s}, p, p')$ are in fact multidimensional matrices where the elements of the matrix
stand for the different scattering reactions.
To solve the coupled-channel scattering problem with a large number of inelastic channels,
we apply the so-called K-matrix approximation by neglecting the real part of the BSE propagator
$G_{BS}$. After the integration over the relative
energy, Eq. \refe{bse} reduces to
\begin{eqnarray}
T^{\lambda_f\lambda_i}_{fi} = K^{\lambda_f\lambda_i}_{fi} + i\int d\Omega_n
\sum_{n}\sum_{\lambda_n} T^{\lambda_f\lambda_n}_{fn}K^{\lambda_n\lambda_i}_{ni},
\label{GBS2}
\end{eqnarray}
where $T_{fi}$ is a scattering matrix and $\lambda_i$($\lambda_f$) stands for the quantum
numbers of initial(final) states
\mbox{$f,i,n =$ $ \gamma N$}, $\pi N$, $2\pi N$, $\eta N$, $\omega N$, $K\Lambda$, $K\Sigma$.
Using the partial-wave decomposition of $T$, $K$ in terms of Wigner d-functions
the angular integration can be easily carried out and
the equation is further simplified to the algebraic form
\begin{eqnarray}
T^{J\pm,I}_{fi} = \left[\frac{ K^{J\pm,I}}{1-iK^{J\pm,I}}\right]_{fi}.
\label{GBS3}
\end{eqnarray}
The validity of this approximation
was demonstrated by Pearce and Jennings in \cite{Pearce:1990uj} by studying different approximations
to the BSE for $\pi N$ scattering. Considering different BSE propagators
they concluded that an important feature of the reduced intermediate two particle
propagator is the on-shell part of $G_{BS}$. It has been argued
that there is no much difference between physical parameters obtained using the $K$-matrix
approximation and other schemes. It has also been shown in \cite{Goudsmit:1993cp,Oset:1997it}
that for $\pi N$ and $\bar K N$ scattering the main effect from the off-shell part is
a renormalization of the couplings and the masses.
Due to the smallness of the electromagnetic coupling the dominant contributions to the self energy stem
from the hadronic part. Therefore we treat the photoproduction reactions perturbatively.
This is equivalent to neglecting $\gamma N$ in the sum over intermediate
states $n$ in Eq.~\refe{GBS2}. Thus, for a photoproduction process the equation \refe{GBS3} can be
rewritten as follows \cite{Penner:2002b,Feuster:1998b}
\begin{eqnarray}
T^{J\pm,I}_{f\gamma} = K^{J\pm,I}_{f\gamma} + i \sum_{n} T^{J\pm,I}_{f n} K^{J\pm,I}_{n \gamma},
\label{photo}
\end{eqnarray}
where the summation in Eq.\refe{photo} is done over all hadronic intermediate states. Here the matrix $T^{J\pm,I}_{f n}$
stems only from the hadronic transitions: indices $f$ and $n$ run over $\pi N$, $2\pi N$, $\eta N$, $K\Lambda$,
$K\Sigma$, $\omega N$ channels. The sum in Eq. \refe{photo} reflects the importance of the hadronic part
of the transition amplitude in the description of photoproduction reactions. In other words, the amplitudes
for the $\pi N \to \pi N$, $\eta N$, $\omega N$ etc. transitions should always be included in the calculation
of the photoproduction amplitudes.
\subsection{Interaction kernel and resonance parameters}
Here we present the main ingredients of the interaction kernel to the BSE Eq.\refe{bse} relevant for
$\eta$-production. More details on other reactions can be found in
\cite{shklyar:2004a,Penner:2002a,Penner:2002b,Feuster:1998a,Feuster:1998b,shklyar:2004b}.
The interaction potential ($K$-matrix) of the BSE is built up as a sum of
$s$-, $u$-, and $t$-channel contributions corresponding to the tree level Feynman diagrams
shown in Fig.~\refe{diag}.
\begin{figure}
\begin{center}
\includegraphics[width=14cm]{fig1.eps}
\caption{$s$-,$u$-, and $t$- channel contributions to the interaction potential. $i$ and $f$
stand for the initial and final $ \gamma N$, $\pi N$, $2\pi N$, $\eta N$,
$\omega N$, $K\Lambda$, $K\Sigma$ states. $m$ denotes intermediate $t$-channel meson.
\label{diag}}
\end{center}
\end{figure}
In the isospin $I=\frac{1}{2}$ channel we checked for the contributions from the
$S_{11}(1535)$, $S_{11}(1650)$, $P_{11}(1440)$, $P_{11}(1710)$, $P_{13}(1720)$, $P_{13}(1900)$, $D_{13}(1520)$
$D_{13}(1900)$, $D_{15}(1675)$, $F_{15}(1680)$ , $F_{15}(2000)$ resonances.
The resonance and background contributions are consistently generated from the same effective interaction.
The Lagrangian densities are given in
\cite{shklyar:2004a,Penner:2002a,Penner:2002b,Feuster:1998a,Feuster:1998b,shklyar:2004b} and respect the chiral symmetry
in low-energy regime.
The properties of the $t$-channel mesons important for $\eta$ production are
given in Table~\ref{tchannel}.
\begin{table}
\begin{center}
\begin{tabular}
{l|c|r|r|c}
\hhline{=====}
& mass [GeV] & $J^P$ & $I$ & final state \\
\hhline{=====}
$\omega$ & 0.783 & $1^-$ & $0$ & $(\gamma,\eta)$ \\
$\rho$ & 0.769 & $1^-$ & $1$ & $(\pi,\eta)(\gamma,\eta)$ \\
$a_0$ & 0.983 & $0^+$ & $1$ & $(\pi,\eta)$ \\
$\phi$ & 1.02 & $1^-$ & $0$ & $(\gamma,\eta)$ \\
\hhline{=====}
\end{tabular}
\end{center}
\caption{Properties of mesons which give contributions
to the $\eta N$ final state via the $t$-channel exchange. The notation $(\gamma,\eta)$ means
$\gamma N\to \eta N$ etc.
\label{tchannel}}
\end{table}
Using the interaction Lagrangians and values of the corresponding meson
decay widths taken from the PDG \cite{pdg} the following coupling constants are obtained:
\begin{eqnarray}
\begin{array}{lcrclcr}
g_{a_0 \eta \pi} &=& -2.100 \; , & & g_{\omega \eta \gamma} &=& -0.27\; , \\
g_{\rho \eta \gamma} &=& -0.64 \; , & & g_{\phi \eta \gamma} &=& -0.385 \; . \\
\end{array}
\label{mesdeccons}
\end{eqnarray}
All other coupling constants were allowed to be varied during the fit.
The obtained values are given in Table~\ref{BornCoupling}. For the $\eta NN$ interaction we use
pseudoscalar coupling , which has been also utilized in our previous studies
\cite{Feuster:1998b,Feuster:1998a,Penner:2002a,Penner:2002b,Shklyar:2006xw}.
The derived $g_{\eta NN}$ constant is found to be small which is in line with
our previous results \cite{Shklyar:2006xw,Penner:2002b}. To check the dependence of
our results on the choice of the $\eta NN$ interaction we have also performed
calculations with the pseudovector coupling. However also in the latter case only a
small $g_{\eta NN}$ coupling constant has been found.
Since the PDG gives only
the upper limit for the decay branching ratio R$(\rho\to\pi\eta) <6\times 10^{-3}$ we allowed this constant to
be varied during fit. However due to lack of experimental constraints this coupling cannot
be fully fixed in the present calculation. We find a small overall contribution from the $t$-channel
$\rho$-meson exchange to the $\pi^- p \to\eta n$ reaction.
The $g_{\phi NN }$ coupling is calculated from $g_{\omega NN}$
using the relation
$$
\frac{g_{\phi NN }}{g_{\omega NN }} = -\tan\Delta \theta_{\phi/\omega},
$$
where $ \Delta\theta_{\phi/\omega}$ is a deviation from the ideal $\phi$-$\omega$ mixing angle.
Taking $ \Delta\theta_{\phi/\omega}=3.7^0$ from \cite{pdg} one gets for
the ratio $g_{\phi NN }/g_{\omega NN } \approx -1/15$. Using this value a very small contribution
from the $t$- channel $\phi$-meson exchange to the $\eta$-photoproduction has been found.
\begin{table}
\begin{tabular}{cccccccc}
\hline
\hline
$g_{\pi NN}$ & 12.85 & $g_{\rho\eta\pi}$ & 0.133 & $g_{\rho NN}$ & 4.98 & $\kappa_\rho$ & 2.18 \\
$g_{\eta NN}$ & 0.31 & $g_{a_0 NN} $ & -44.37 & $g_{\omega NN}$ & 7.23 & $\kappa_\omega$ & -1.50 \\
\hline
\hline
\end{tabular}\\
\caption{Nucleon and $t$-channel couplings obtained in the present study.\label{BornCoupling}}
\end{table}
To take into account the finite size of mesons and baryons each vertex is
dressed by a corresponding form factor:
\begin{eqnarray}
F_p (q^2,m^2) &=& \frac{\Lambda^4}{\Lambda^4 +(q^2-m^2)^2},
\label{formfact}
\end{eqnarray}
where $q$ is a c.m. four-momentum of an intermediate particle and $\Lambda$ is a cutoff parameter.
The cutoffs $\Lambda$ in Eq.~\refe{formfact} are treated as free parameters being varied
during the calculation. However, we keep the same cutoffs in all channels for a
given resonance
spin $J$ : $\Lambda^{J}_{\pi N}=\Lambda^{J}_{\pi\pi N}=\Lambda^{J}_{\eta N}=...$ etc.,
($J=\frac{1}{2},~ \frac{3}{2},~ \frac{5}{2}$). This significantly reduces the number of free parameters; i.e. for all
spin-$\frac{5}{2}$ resonances there is only one cutoff $\Lambda=\Lambda_{\frac{5}{2}}$ for all decay channels.
However for the photoproduction reactions we use different cutoffs at the $s$- and $u$-channel
electromagnetic vertices. All values are given in Table \ref{tabcutoff}. Except for the spin-$\frac{3}{2}$ states,
the $s$- and $u$-channel cutoffs almost coincide.
\begin{table}[t]
\begin{center}
\begin{tabular}
{c|c|c|c|c|c|c|c}
\hhline{========}
$\Lambda_N$ [GeV] &
$\Lambda_\frac{1}{2}^h$ [GeV] &
$\Lambda_\frac{3}{2}^h$ [GeV] &
$\Lambda_\frac{5}{2}^h$ [GeV] &
$\Lambda_\frac{1}{2}^\gamma$ [GeV] &
$\Lambda_\frac{3}{2}^\gamma$ [GeV] &
$\Lambda_\frac{5}{2}^\gamma$ [GeV] &
$\Lambda_t^{h,\gamma}$[GeV] \\
\hhline{========}
0.952 & 3.0 & 0.97 & 1.13 & 1.69~(1.69) & 4.20~(2.9) & 1.17~(1.25) & 0.7 \\
\hhline{========}
\end{tabular}
\end{center}
\caption{Cutoff values for the form factors.
The lower index denotes an intermediate particle, i.e.
$N$: nucleon, $\frac{1}{2}$: spin-$\frac{1}{2}$ resonance, $\frac{3}{2}$: spin-$\frac{3}{2}$, $\frac{5}{2}$:
spin-$\frac{5}{2}$ resonance, $t$: $t$-channel meson. The upper index $h$($\gamma$)
denotes whether the value is applied to a hadronic or electromagnetic vertex.
The cutoff values used at electromagnetic $u$-channel vertices are given in brackets.
\label{tabcutoff}}
\end{table}
The use of vertex form factors requires special care for maintaining
the current conservation when the Born contributions to photoproduction reactions
are considered. Since the resonance and intermediate meson vertices are constructed from
gauge invariant Lagrangians they can be independently multiplied by the corresponding form
factors. For the nucleon contributions to meson photoproduction we apply the suggestion of Davidson
and Workman \cite{Davidson:2001rk} and use the crossing symmetric common form factor:
\begin{eqnarray}
\tilde F(s,u,t) = F(s) + F(u) + F(t) - F(s)F(u) - F(s)F(t) - F(u)F(t) + F(s)F(u)F(t).
\label{formfact2}
\end{eqnarray}
The extracted resonance parameters given in Table~\ref{res_param} are very close to the values
deduced in our previous calculations \cite{shklyar:2004b,Shklyar:2006xw} which indicates the stability
of the obtained solution. However some values changed upon inclusion of the new MAMI data
\cite{McNicoll:2010qk}. The total width of $S_{11}(1650)$ tends to be larger than that deduced in
our previous calculations \cite{shklyar:2004b}. The helicity amplitude is also modified but still is
in good agreement with the parameter range provided by PDG \cite{pdg}. The opposite effect is found
for the $P_{11}(1710)$ state where the total width is reduced once the data of \cite{McNicoll:2010qk} are included.
The remaining resonance parameters are only slightly modified as compared to our previous results.
The mass and width of the Roper resonance is found to be larger than deduced
in other analyses \cite{pdg}. However the authors of \cite{Vrana:2000}
give $490\pm120$ MeV for the total width. The large decay width $545\pm 170$\,MeV has also been
deduced by Cutkosky and Wang \cite{Cutkosky:1990}. Note that properties of this state are strongly
influenced by its decay into the $2\pi N$ final state. Arndt et al \cite{Arndt:1990bp} found a second
pole structure for the Roper resonance which might be attributed to the coupling to the $\pi \Delta$ subchannel.
Since we use a simplified prescription for the $2\pi N$ reaction this effect cannot be properly described
in the present calculations.
The recent GWU(SAID) study of the $\pi N$ data shows no evidence for the $P_{11}(1710)$
resonance. An indirect indication for the existence of this state can be concluded from the analysis
of the $\pi N$ inelasticity and $2\pi N$ cross section in the $P_{11}$ partial wave,
see discussion in Section~\ref{pe}. We find a small coupling of this resonance to the $\pi N$
final state. Since a clear signal from this state is not seen in the recent GWU solution, the
determination of the total width turns out to be difficult. In our calculations we assume that this
resonance has a large decay branching ratio to the $\eta N$. However the quality of
the $\pi^- p \to \eta n$ data does not allow for an unambiguous determination of the properties of this
state.
The mass and width of the $D_{13}(1520)$ is more close to the values obtained by Arndt et al
\cite{Arndt:1995ak}: $1516\pm10$ MeV and $106\pm4$ MeV respectively.
It is interesting to note that the mass of this resonance deduced from the pion
photoproduction tends to be 10 MeV lower that the values derived from the pion-induced
reactions \cite{pdg}. The second $D_{13}(1900)$ has a very large decay width.
We associate this state with $D_{13}(2080)$ as suggested in PDG.
This resonance is rated with two stars and its existence is still under discussion. In our updated
coupled-channel calculation of the $\omega$-production \cite{shklyar:2004b} a large $\omega N$ and
$2\pi N$ decay branching ratios have been obtained.
The properties of other resonances are very close to the values given in PDG.
Except for $S_{1}(1535)$ and $P_{11}(1710)$ we find only small resonance couplings to $\eta N$ which is
in accordance with our previous conclusions. One needs to stress that the smallness of the resonance coupling does not
necessarily mean that the contribution from the state is negligible. The $S_{11}(1650)$ state
produces for example a sizable effect in the eta-production due to overlapping with $S_{11}(1535)$.
Another example is the effect from the $D_{13}(1520)$ state in $\eta$-photoproduction on the proton.
Here the smallness of the $\eta N$ branching ratio is compensated by the strong electromagnetic coupling
of this resonance.
Therefore the effect from this state could be seen in the $E_{2-}$ and $M_{2-}$ multipoles, see Section~\ref{multipoles}.
However in most cases the resonance contributions with small
branching ratios to the eta
are hard to resolve unambiguously.
\subsection{Pole parameters}
It is interesting to compare the poles positions and elastic residues with the results from other studies, see
Table\,\ref{poles}. The calculated pole masses are very close to the values obtained in other
analyses, see \cite{pdg}. The agreement between imaginary parts and elastics residues is also good,
though some differences exist between the present values and the results from other groups.
For the $S_{11}(1535)$ state we obtain a smaller elastic residue (for definition of $|R|$ see \cite{pdg})
$|R|=15$\,MeV which is almost identical to the result of the GWU group $|R|$=16\,MeV \cite{Arndt:2006bf}.
Both values seem to be out of the range given in PDG \cite{pdg} 50$\pm$20 MeV.
It is interesting to note that the elastic residue from \cite{Arndt:2006bf} is included into
the estimation made in \cite{pdg} but still does not fit to the provided range. The value
$\Gamma_{\rm pole}=89$ MeV for the $S_{11}(1650)$ state is also comparable with the result
from \cite{Arndt:2006bf}: $\Gamma_{\rm pole}=80$ MeV which are again less than the lower
bound given in \cite{pdg}.
Though the derived pole mass of $P_{11}(1440)$ is very close to the values deduced in other calculations
we obtain a significantly larger pole width. As a result the elastic pole residue
turns out to be also large $|R|$=126 MeV. We note, that the extraction of the properties of $P_{11}(1440)$
in the complex energy plane might require a proper treatment of the
$P_{11}(1440)\to\pi\Delta(1232)\to 2\pi N$ isobar decay
channel where the overlap of the self-energies of the $P_{11}(1440)$ and $\Delta(1232)$ states might be important for
the determination of the properties of $P_{11}(1440)$. This question will be addressed in \cite{shklyar:2012}.
As we already mentioned the results for $P_{11}(1710)$ are controversial.
We find 159 MeV for the pole width. Somewhat greater value of 189 MeV has been obtained
in \cite{Tiator:2010rp,Batinic:2010zz}.
The recent issue of PDG \cite{pdg} summarizes results
for the pole parameters taken from
four different analyses.
Whereas the calculations \cite{Anisovich:2011fc,Hoehler:1993} give 200 MeV for the pole width, Cutkosky
obtains a significantly lower value $\Gamma$=80 MeV \cite{Cutkosky:1990zh,Cutkosky:1979fy}. This results in a
large spread of the resonance width given by PDG, see Table \ref{poles}.
The elastic residue is found to be small which is in accordance with the small decay
branching ratio to $\pi N$. The similar conclusion has also been drawn in \cite{Tiator:2010rp}.
Investigation of the $P_{13}$- wave inelasticity \cite{Arndt:2006bf} shows that the $P_{13}(1720)$ state could
have a strong decay flux into the $3\pi N$ channel \cite{Manley:1984}.
Therefore the calculation of its pole width might be affected by
deficiencies in description of this channel. PDG estimations are based on several studies
where $\Gamma_{\rm pole}= 120\pm 40$ by Cutkosky
\cite{Cutkosky:1979fy} is the lower limit. The upper bound $\Gamma_{\rm pole}=450\pm 100$\,MeV is given by the recent Bonn-Gatchina
analysis \cite{Anisovich:2011fc}. Neither of these calculations includes the $3\pi N$ channel explicitely.
The situation with the second $P_{13}(1900)$ state is even more complicated. This resonance is rated by two stars
in PDG and supposed to be rather broad. The latest GWU analysis \cite{Arndt:2006bf} does not find any
indication for this state. The present information about the pole parameters in PDG is based
solely on the result of the Bonn-Gatchina calculations \cite{Anisovich:2011fc}
which deduce the pole mass $1900\pm 30$\,MeV and the pole width $200^{+100}_{-60}$\,MeV.
These values are very close to those derived in the present work.
The pole width of the $D_{13}(1520)$ state ( 94 MeV) turns out to be 10 MeV less than the
lower limit given in PDG\cite{pdg}. The similar value of $\Gamma_{\rm pole}=$ 95\,MeV has also been
obtained in the J\"ulich model \cite{Doring:2009yv}. Some analyses find additional poles associated with the $D_{13}(1700)$
and $D_{13}(1875)$ states \cite{pdg}. We do not find any indication for $D_{13}(1700)$. The pole
position for the second resonance is close to the results of other calculation \cite{pdg}.
Though the elastic residues for the $D_{15}(1675)$ and $F_{15}(1680)$ states are comparable
with the values given in PDG their
pole widths are somewhat lower than those obtained in other studies \cite{pdg}. We also find an indication for the
second state N(2000) with the pole mass of 1900 MeV and the width of $123$\,MeV, see Table\,\ref{poles}.
This resonance has a small
coupling to the $\pi N$ final state what is in agreement with results from other calculations.
\begin{table}
\begin{tabular}{ccccccc|cc}
\hline
\hline
$N^*$ & mass (MeV) & $\Gamma_{tot}(MeV)$ &
R$_{\pi N}$ & R$_{2\pi N}$ & R$_{\eta N}$
& R$_{\omega N}$& $A^{p}_{\frac{1}{2}}$ & $A^{p}_{\frac{3}{2}}$
\\
\hline
\hline
$S_{11}$(1535)
& 1526(2) & 131(12) & 35(3) & $ 8(2)$ & $58(4)$ & --- & 91(4) & --- \\
& 1526 & 136 & 34.4 & $ 9.5$ & $56.1 $ & --- & 92 & --- \\
& 1536(10) & 150(25) & 45(10) & $ 5(5)$ & $42(10)$ & --- & 90(30) & --- \\
$S_{11}$(1650)
& 1665(2) & 147(14) & 74(3) & $23(2)$ & $ 1(2)$ & --- & 63(6) & --- \\
& 1664 & 131 & 72.4 & $23.1 $ & $ 1.4 $ & --- & 57 & --- \\
& 1657(13) & 150(30) & 70(20) & $15(5)$ & $ 10(5)$ & --- & 53(16) & --- \\
\hline
$P_{11}$(1440)
& 1515(15) & 605(90) & 56(2) & $44(2)$ & --- & --- & -85(3) & --- \\
& 1517 & 608 & 56.0 & $44.0$ & --- & --- & -84 & --- \\
& 1445(25) & 300(150)& 65(10) & $35(5)$ & --- & --- & -60(4) & --- \\
$P_{11}$(1710)
& 1737(17) &368(120)& 2(2) & $49(3)$ & $45(4)$ & 3(2) & -50(1) & --- \\
& 1723 & 408 & 1.7 & $49.8 $ & $43.0$ & 0.2 & -50 & --- \\
& 1710(30) &150(100)& 13(7) & $65(25)$ & $20(10)$ & 13(2)& 24(10) & --- \\
\hline
$P_{13}$(1720)
& 1700(10) & 152(2) & 17(2) & $79(2)$ & $ 0(1)$ & --- & -65(2) & 35(2) \\
& 1700 & 152 & 17.1 & $78.7$ & $ 0.2$ & --- & -65 & 35 \\
& 1725(24) & 225(125) & 11(3) & $>70$ & $ 4(1)$ & --- & 50(60) & -19(20) \\
$P_{13}$(1900)
& 1998(3) & 359(10) & 25(1) & $61(2)$ & $ 2(2)$ & 10(3)& -8(1) & 0(1) \\
& 1998 & 404 & 22.2 & $59.4 $ & $ 2.5$ & 14.9 & -8 & 0 \\
& 1900(-) & 250(-) & 10(-) & --- & $ 12(-)$ & 39(-)& 26(15) & -65(30) \\
\hline
$D_{13}$(1520)
& 1505(4) & 100(2) & 57(2) & $44(2)$ & $ 0(1)$ & --- & -15(1) & 146(1) \\
& 1505 & 100 & 56.6 & $43.4 $ & 1.2 & --- & -13 & 145 \\
& 1520(5) & 112(12) & 60(5) & $25(5)$ & 2.3$\pm 10^{-3}$ & --- & -24(8) & 150(15) \\
$D_{13}$(1875)
& 1934(10) &857(100)& 11(1) & $69(2)$ & $ 0(1)$ & 20(5) & 11(1) & 26(1)\\
& 1934 & 859 & 10.5 & $68.7 $ & $ 0.5$ & 20.1 & 11 & 26 \\
& 1875(45) &220(100)& 12(10)& $70(20)$ & $ 3.5(3.5)$ & 21(7) & 18(10) & -9(5) \\
\hline
$D_{15}$(1675)
& 1666(2) & 148(1) & 41(1) & $58(1)$ & $ 0(1)$ & --- & 9(1) & 21(1) \\
& 1666 & 148 & 41.1 & $58.5$ & $ 0.3 $ & --- & 9 & 20 \\
& 1675(5) & 150(15) & 40(5) & $55(5)$ & $ 0(1)$ & --- & 19(8) & 15(9) \\
\hline
$F_{15}$(1680)
& 1676(2) & 115(1) & 68(1) & $32(1)$ & $ 0(1)$ & --- & 3(1) & 116(1) \\
& 1676 & 115 & 68.3 & $31.6$ & $ 0.0 $ & --- & 3 & 115 \\
& 1685(5) & 130(10)& 67(3) & $35(5)$ & $ 0(1)$ & --- & -15(6) & 132(13) \\
$F_{15}$(2000)
& 1946(4) & 198(2) & 10(1) & $87(1)$ & $ 2(2)$ & 1(1) & 11(1) & 25(1) \\
& 1946 & 198 & 9.9 & $87.2 $ & $ 2.0 $ & 0.4 & 10 & 25 \\
& 2050(100) & 350(200) & 15(7) & --- & --- & --- & 35(15) & 50(14) \\
\hline
\end{tabular}\\
\caption{Resonance parameters extracted in the present study. The uncertainties are given in brackets.
Helicity decay amplitudes are given in $10^{-3}$GeV$^{-\frac{1}{2}}$.
1st line: present study; 2nd line: \cite{shklyar:2004b}, 3th line: \cite{pdg}.
(-): the validity range is not given.
\label{res_param}}
\end{table}
\begin{table}
\begin{tabular}{ccccc}
\hline
\hline
& Re\,$z_0$(GeV) & -2Im\,$z_0$(MeV) & $|$R$|$(MeV) & $\theta^0$ \\
\hline
\hline
$S_{11}(1535)$ & 1.49 & 100 & 15 & -51 \\
& 1.49-1.53 & 90-250 & 30...70 & -1...-30 \\
\hline
$S_{11}(1650)$ & 1.65 & 89 & 19 & -46 \\
& 1.64-1.67 & 100-170 & 20-50 & -50...-80 \\
\hline
\hline
$P_{11}(1440)$ & 1.386 & 277 & 126 & -60 \\
& 1.35-1.38 & 160-220 & 40-52 & -75...-100 \\
\hline
$P_{11}(1710)$ & 1.67 & 159 & 11 & 9 \\
& 1.67-1.77 & 80-380 & 2-15 & -160...+190 \\
\hline
\hline
$P_{13}(1720)$ & 1.67 & 118 & 12 & -45 \\
& 1.66-1.69 & 150-400 & 7-23 & -90...-160 \\
\hline
$P_{13}(1900)$ & 1.91 & 173 & 10 & -64 \\
& 1.870-1.93 & 140-300 & 1-5 & 45...-25 \\
\hline
\hline
$D_{13}(1520)$ & 1.492 & 94 & 27& -35 \\
& 1.505-1.515 & 105-120 & 32-38 & -5...-15 \\
\hline
$D_{13}(1875)$ & 1.81 & 98 & 3 & -76 \\
& 1.8-1.95 & 150-250 & 2-10 & 20...180 \\
\hline
\hline
$D_{15}(1675)$ & 1.64 & 108 & 20 & -49 \\
& 1.655-1.665 & 125-150 & 22-32 & -21...40 \\
\hline
\hline
$D_{15}(1680)$ & 1.66 & 98 & 33 & -32 \\
& 1.665-1.68 & 110-135 & 35-45 & 0...-30 \\
\hline
$F_{15}(2000)$ & 1.90 & 123 & 11 & -6 \\
& 1.92-2.15 & 380-580 & 20-115 & -60...-140 \\
\hline
\end{tabular}\\
\caption{Pole positions and elastic pole residues. First line: present study,
second line: values from PDG \cite{pdg}.\label{poles}}
\end{table}
\section{Results and discussion}
\label{results}
\label{pe}
The lack of the experimental data for the pion-induced reactions does not provide enough constraints on
the resonance parameters. Also the discrepancy among various measurements (see Section~\refe{data}) does
not allow for a consistent description of the data in a full kinematical region.
While the contribution from the $S_{11}(1535)$ state is well established
the reaction dynamics above W=1.6 GeV is still under discussion.
One of the early Giessen coupled-channel calculations \cite{Penner:2002a,Penner:2002b,Shklyar:2006xw}
found a destructive interference between $S_{11}(1535)$ and $S_{11}(1650)$ states.
The second suggestion is a strong contribution from the
$P_{11}(1710)$-resonance excitation above W=1.68 GeV. This resonance was established in the early
single-channel Karlsruhe-Helsinki and Carnegie Mellon-Berkeley analyses (see PDG \cite{pdg} and references
therein). The independent study of the $\pi N \to (\pi/\eta) N$ reactions by the Zagreb group
\cite{Ceci:2006ra} provides an additional evidence for the existence of $P_{11}(1710)$. The result
of \cite{Ceci:2006ra} confirm the assumption made in \cite{Penner:2002a,Penner:2002b} on
the important contribution from this state to the $\eta$-production.
However the recent analysis from the GWU group \cite{Arndt:2006bf} finds no evidence
for this state. The absence of a clear signal in the $P_{11}$ partial wave of the elastic $\pi N$ scattering
does not necessarily mean that this state does not exist. If the coupling to the
final $\pi N$ state is small, the effect from this state might not be seen in $\pi N$ scattering.
The evidence for the signal from the $P_{11}(1710)$ resonance has also been reported from the
study of the $\pi N \to K\Lambda$ reaction \cite{Ceci:2005vf}.
On the other hand the result of the Bayestian analysis performed by the
Gent group \cite{DeCruz:2012bv} demonstrates that $P_{11}(1710) $
is not needed to describe the $K\Lambda$ photoproduction. An opposite conclusion was drawn by the
Bonn-Gatchina group which finds
decay branching ratio of $23\pm 7$\% of this state to $K\Lambda$ \cite{Anisovich:2011fc}.
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{fig2.eps}
\caption{(Color online) Calculated $\pi N$ inelasticity and $\pi N\to 2\pi N$ cross section in the $P_{11}$ partial wave
in comparison with the results from \cite{Arndt:2006bf} (GWU\,2006) and \cite{Manley:1984}(Manley\,1984).
\label{2piN}}
\end{center}
\end{figure}
Another indication for this state comes from the analysis of an inelastic flux in the $P_{11}$ partial wave.
In Fig.~\refe{2piN} the total inelasticity from the GWU analysis vs. the total $2\pi$ cross section extracted
in \cite{Manley:1984} is compared. The difference between the total $\pi N$ inelasticity and
the total $2\pi N$ cross section at W=1.7 GeV in the $P_{11}$-wave
can be attributed to the sum of inelastic channels like $3\pi N$, $\eta N$, $\eta \pi N$ etc. We assume here
that the observed difference is due to the $\eta N$ production channel dominated by the $P_{11}(1710)$ state.
As $g_{\pi N N^*(1710)}$ is assumed to be small this raises the question about the magnitude of the
$P_{11}(1710)$ contribution in the $\pi N \to \eta N$ reaction.
However the situation in $\eta$-production is different from the $\pi N$ elastic scattering. Here the
contribution from $P_{11}(1710)$ is proportional to the product $g_{\pi N N^*(1710)}g_{\eta N N^*(1710)}$,
where $g_{\pi N N^*(1710)}$ is the coupling constant at the $N(1710)\to \eta N$ transition vertex.
It follows that the contribution from the $P_{11}(1710)$ can be significant provided that
$g_{\eta N N^*(1710)}$ is large enough. The interplay with background and coupled-channel rescattering would
further increase this effect.
\subsection{$\pi N \to \eta N$}
The results of our calculations are presented in Fig.~\refe{pe_dif} in comparison with the world data.
The first peak at W=1.54 GeV is related to the well established $S_{11}(1535)$ resonance contribution.
\begin{figure}
\begin{center}
\includegraphics[width=18cm]{fig3.eps}
\caption{(Color online) Calculated differential $\pi^- p \to \eta n$ cross section in comparison with the
experimental data from: Prakhov\,2005:\cite{Prakhov}, Deinet\,1969:\cite{Deinet:1969cd},
Richards\,1970:\cite{Richards:1970cy}, Morrison\,2000:\cite{Morrison:2000kx}.
\label{pe_dif}}
\end{center}
\end{figure}
Though the effect from the $S_{11}(1650)$ state is hardly visible in the differential cross section
this state plays an important role leading to the destructive interference between $S_{11}(1535)$
and $S_{11}(1650)$ as it has been pointed out in our previous calculations
\cite{Penner:2002a,Penner:2002b}.
The second rise is due to the $P_{11}(1710)$ resonance. This state has a small branching
ratio to the $\pi N$ system but due to the large $\eta$-coupling this resonance affects the
production cross section at W=1.7 GeV. The coupled-channel effects and interference
with other partial waves further enlarge the overall contribution from this state.
The total partial wave cross sections are shown in Fig.~\ref{pe_tot}. The destructive interference
between the $S_{11}(1535)$
and $S_{11}(1650)$ leads to the dip in the total $S_{11}$-partial wave cross section around W=1.64 GeV
(dotted line). The effect from the $P_{11}(1710)$ state is shown by the dashed line, Fig.~\ref{pe_tot}.
The contributions from other partial waves are found to be small. We also corroborate our previous
results \cite{shklyar:2004a} where only minor contributions from spin $J\ge\frac{3}{2}$ resonance states
were obtained. Both t-channel $a_{0}$ and $\rho$ meson exchange and $u-$channel graphs give
small effects. The inclusion of the higher spin state $D_{13}(1520)$ into the calculations
is still important to reproduce the correct shape of the cross section.
This feature is also found in many other calculations, e.g.\cite{Penner:2002a,Batinic:1995}.
It is interesting to note that importance of the $P_{11}(1710)$ resonance
contribution has recently been found in \cite{Shrestha:2012va} which is in line with our previous
results \cite{Shklyar:2006xw,Penner:2002b}.
Since the main contributions in our calculations come mainly from the $S_{11}$ and $P_{11} $ partial waves
it is interesting to trace back the interference effect between them. Neglecting the higher partial waves
the differential cross section can be written in the form
\begin{eqnarray}
\frac{d\sigma}{d\cos\,(\theta)}~\sim~ 1+ \alpha \,\sin^2\left(\frac{\theta}{2}\right),
\label{s_p_interf}
\end{eqnarray}
where $\theta$ is a scattering angle and $\alpha =\left(\frac{|S_{11} - P_{11}|^2}{ |S_{11} + P_{11}|^2}-1 \right) $
only depends
on the c.m. energy. Then the angular distribution should have a maximum
(minimum) at forward angles depending on the relative phase between the nonvanishing $S_{11}$ and $P_{11}$
amplitudes. In our calculation the interference between $S_{11}$ and $P_{11}$ partial waves produces
a peak at forward scattering angles and energies above W=1.67 GeV, see Fig.~\refe{pe_dif}.
As a result the signal from the $P_{11}(1710)$ resonance becomes more transparent for forward
scattering.
This is in line with the data of Richards et al \cite{Richards:1970cy}
confirming our guess about the production mechanism. The inclusion of higher partial waves would
modify Eq.~\refe{s_p_interf}. However these contributions are relatively
small (see Fig.\refe{pe_tot}) thus producing only minor deviations from the distribution
Eq.~\refe{s_p_interf}.
Note, that due to numerous problems with the experimental data our calculations above W=1.6 GeV
are only partly constrained by experiment. Indeed, once the data \cite{Baker:1979aw,Brown:1979ii}
are neglected there are only 30 datapoints from experiment \cite{Richards:1970cy}. This data has
relatively large error bars and seems not to be fully consistent with other measurements
\cite{Clajus:1992dh}. Therefore, the results for the differential cross section
might be regarded as a prediction rather than an outcome of the fit.
This demonstrates an urgent need for new measurements of the
$\pi^- N \to \eta N$ reactions above W=1.6 GeV. This would be a challenge for the the
upcoming pion-beams experiment carried out by the HADES collaboration at GSI.
\begin{figure}
\begin{center}
\includegraphics[width=10cm]{fig4.eps}
\caption{(Color online) Total partial wave cross section $\pi^- p \to \eta n$ vs. experimental data.
\label{pe_tot}}
\end{center}
\end{figure}
\subsection{$\eta N \to \eta N$ amplitude and $\eta N$ scattering lengths}
The result for the $\eta N \to \eta N$ transition amplitude in the $S_{11}$ partial wave is presented in
Fig.~\ref{ee_S11}. Close to threshold the elastic $\eta N$ scattering is completely determined by the
contribution of the $S_{11}(1535)$ resonance. At higher energies the excitation of $S_{11}(1650)$
also becomes important. The interference between those two $S_{11}$-states produces an excess structure in the imaginary
part of the amplitude at W=1.65 GeV.
The rapid variation of the $S_{11}$-amplitude close to threshold indicates that this energy dependence should be
taken into account when the $\eta N$ scattering length is calculated.
Here we use the definition for the effective range expansion from \cite{PhysRevC.55.R2167}:
\begin{eqnarray}
\frac{q_{\rm c.m.}}{ S_{11}^{\,\eta N}} + i q_{\rm c.m.} = \frac{1}{a_{\eta N}} + \frac{r_0}{2} q_{\rm c.m.}^2 +s\, q^4_{\rm c.m.},
\end{eqnarray}
where $S_{11}^{\,\eta N}$ is an elastic partial S-wave amplitude, and $a_{\eta N}$, $r_0$ and $s$ are scattering length, effective range,
and effective volume respectively. The results are shown in Table~\ref{tscatt_lengths} in comparison with
values deduced from other coupled-channel calculations ( results published before 1997 are discussed in
\cite{PhysRevC.55.R2167} ). The obtained value of $a_{\eta N}$ is very close to our previous results
\cite{Penner:2002a}. The values for the real part deduced in \cite{Batinic:1996me} and \cite{PhysRevC.55.R2167}
are lower than in this work. The study \cite{Batinic:1996me} gives 1.550 GeV for the mass and 204 MeV
for the width of the $S_{11}(1535)$ state which are somewhat greater than in the present calculation.
This could be one of the reasons for the differences in $Re\,a_{\eta N}$.
In \cite{PhysRevC.55.R2167} only the $S_{11}(1535)$ state is taken into account
to calculate transition amplitudes to the $\eta N$ channel.
Since the parameters of $S_{11}(1535)$ in \cite{PhysRevC.55.R2167} are close to the values obtained in the present study
the observed difference in ${\rm Re}\,a_{\eta N}$ might be attributed to the different treatment of background
contributions which have been assumed in \cite{PhysRevC.55.R2167} to be energy-independent. The second piece of uncertainty
is related to the quality of the world data of $\pi N \to \eta N$ scattering. Hence, precise measurements of this
reaction would provide an additional constraint on $\eta N$ scattering length.
The non-vanishing imaginary part of $ a_{\eta N}$ is mostly driven by rescattering in the $\pi N$ channel. Since the
largest contributions to the scattering length are produced by the $S_{11}(1535)$ state the imaginary part of $a_{\eta N}$ is
strongly influenced by the decay branching ratio of this resonance to $\pi N$. Only a minor effect is found from
the rescattering induced by background contributions and inelastic flux to the $2\pi N$ channel.
Since the $\pi NN^*(1535)$ coupling is well fixed
an agreement in ${\rm Im}( a_{\eta N})$ between various model calculations can be expected provided that unitarity
is maintained.
The obtained value of the scattering length should be taken with care when in-medium properties of the $\eta$-meson
are considered. As it has already been pointed out in \cite{PhysRevC.55.R2167} the $S_{11}$ amplitude has a strong energy
dependence - a feature which might affect the $\eta$-potential. The second reason is that properties of
the $S_{11}(1535)$ resonance might also be subjected to in-medium modifications \cite{Lehr:2003km}. Both effects should be taken into
account when $\eta$-meson properties in nuclei are studied.
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig5.eps}
\caption{Calculated $S_{11}$ partial wave amplitude of the elastic $\eta N$ scattering.
\label{ee_S11}}
\end{center}
\end{figure}
\begin{table}
\begin{center}
\begin{tabular}
{l|c|c }
\hhline{===}
Reference & $a_{\eta N}$(fm) & $r_0$(fm) \\
\hhline{===}
present work & 0.99$\pm $0.08 + i0.25$\pm$0.06 & -1.98$\pm$0.1 - i0.43$\pm$0.15 \\
\cite{Penner:2002a} & 0.99 + i0.34 & -2.08 - i0.81 \\
\cite{Batinic:1996me} & 0.734$\pm$0.026 + i0.269$\pm0.019$ & \\
\cite{PhysRevC.55.R2167}& 0.75$\pm 0.04$ + i0.27$\pm 0.03$ & -1.5$\pm$0.13 - i0.24$\pm$0.04 \\
\cite{Lutz:2001} & 0.43+ i0.21 & \\
\hhline{===}
\end{tabular}
\end{center}
\caption{ Calculated scattering length and effective range in comparison with results from other works.
\label{tscatt_lengths}}
\end{table}
\subsection{$\gamma N \to \eta N$ below 1.89 GeV}
The results of our calculation of the differential cross section in comparison with the recent
Crystal Ball/MAMI
measurements are shown in Fig.~\refe{ge_dif}. Our calculations demonstrate a nice agreement with
the experimental data in the whole kinematical region. The first peak is related to the $S_{11}(1535)$
resonance contribution. Similar to the $\pi^- p \to \eta n$ reaction the $S_{11}(1650)$ and $S_{11}(1650)$
states interfere destructively producing a dip around W=1.68 GeV.
Though the effect from the $P_{11}(1710)$
state is only minor, the contribution from this resonance produces a rapid change in the $M_{1-}$
photoproduction multipole, see Section \ref{multipoles}.
The coherent sum of all partial waves leads to the more pronounced effect from the dip
at forward angles.
Note that
the resonance contribution to the photoproduction reaction stems from two sources: the first is related
to the direct electromagnetic excitation of the nucleon resonance and the second comes from
rescattering e.g. $\gamma p \to \pi N \to \eta N$, Eq.~\refe{photo}. At this stage the hadronic
transition amplitudes e.g. $T_{\pi N \to\eta N}$ become an important part of the production mechanism.
The sum of these contributions in the $P_{11}$ wave turns out to be destructive which reduces the overall contribution
from the $P_{11}(1710)$ state.
We also corroborate our previous findings \cite{Shklyar:2006xw} where a small
effect from the $\omega N$ threshold was found.
\begin{figure}
\begin{center}
\includegraphics[width=10cm]{fig6.eps}
\caption{ Differential $\eta p$ cross section vs. recent MAMI data \cite{McNicoll:2010qk}.
\label{ge_dif}}
\end{center}
\end{figure}
We also do not find any strong indication for contributions from a hypothetic narrow $P_{11}$
state with a width of 15-20 MeV around W=1.68 GeV.
It is natural to assume that the contribution from this state would induce a strong modification
of the beam asymmetry for energies close to the mass of this state. This is because the beam asymmetry
is less sensitive to the absolute magnitude of the various partial wave contributions but strongly affected
by the relative phases between different partial waves. Thus even a small admixture of a contribution
from a narrow state might result into a strong modification of the beam asymmetry in the energy
region of W=1.68 GeV.
\begin{figure}
\begin{center}
\includegraphics[width=10cm]{fig7.eps}
\caption{(Color online) $\gamma p \to \eta p$ partial wave cross sections vs. measurements \cite{Crede:2003ax,Crede:2009zzb,Bartholomy:2007zz}.
\label{ge_tot}}
\end{center}
\end{figure}
In Fig. \refe{ge_sig} we show the calculation of the photon-beam asymmetry in comparison with the GRAAL
measurements \cite{Bartalini:2007fg}. One can see that even close to the $\eta N$ threshold where our calculations
exhibit a dominant $S_{11}$ production mechanism (see Fig.~\refe{ge_tot} ) the beam asymmetry is
nonvanishing for angles $\cos(\theta)\ge-0.2$. This shows
that this observable is very sensitive to very small contributions from higher partial waves.
At W=1.68 GeV and forward angles the GRAAL measurements show a rapid change of the asymmetry
behavior. We explain this effect by a destructive interference between the $S_{11}(1535)$ and $S_{11}(1650)$
resonances which induces the dip at W=1.68 GeV in the $S_{11}$ partial wave. The strong drop in
the $S_{11}$ partial wave modifies the interference between $S_{11}$ and other partial waves and
changes the asymmetry behavior.
Note that the interference between $S_{11}(1535)$ and $S_{11}(1650)$ and
the interference between different partial waves are of different nature.
The overlapping of the
$S_{11}(1535)$ and $S_{11}(1650)$ resonances does not simply mean a coherent sum of two independent
contributions, but also includes rescattering (coupled-channel effects). Such interplay is hard to simulate
by the simple sum of two Breit-Wigner forms since it does not take into account
rescattering due to the coupled-channel treatment.
The GRAAL collaboration finds no evidence for a narrow state around W=1.68 GeV.
We also find no strong need for the narrow $P_{11}$ resonance contribution to describe the
asymmetry data. Taking contributions from the established states into account our results are in
close agreement with the experimental data \cite{Bartalini:2007fg}.
\begin{figure}
\begin{center}
\includegraphics[width=12cm]{fig8.eps}
\caption{Calculated beam asymmetry. Experimental data are taken from \cite{Bartalini:2007fg}(GRAAL07).
\label{ge_sig}}
\end{center}
\end{figure}
\subsection{$\gamma N \to \eta N$ above 1.89 GeV}
\begin{figure}
\begin{center}
\includegraphics[width=14cm]{fig9.eps}
\caption{(Color online) Differential $\eta p$ cross section as a function of the scattering angle.
The data are taken from CLAS\,2009:\cite{Williams:2009yj} and CB-ELSA:\cite{Crede:2009zzb}. \label{ge_dif2}}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=16cm]{fig10.eps}
\caption{(Color online) Differential $\eta p$ cross section as a function of c.m. energy at fixed forward angles.
Data are taken from CLAS\,2009:\cite{Williams:2009yj}, CB-ELSA:\cite{Crede:2009zzb}, and MAMI2010:\cite{McNicoll:2010qk}.
\label{ge_d3f}}
\end{center}
\end{figure}
Since the MAMI measurements are available up to W=1.89 GeV the calculations in the region W=1.89 ...2.GeV
are constrained by the combined data set constructed out of the recent CLAS and CB-ELSA/TAPS
\cite{Williams:2009yj,Crede:2009zzb} data. Due to some inconsistencies between these
two experiments \cite{Dey:2011rh,Sibirtsev:2010yj} we did not try to fit the data above W=2.GeV but instead
extrapolate our calculation into the higher energies. In this region the $t$-channel exchange starts to
play a dominant role. One of the accepted prescriptions is to use a Reggeized $t$-channel meson exchange
as suggested in \cite{Chiang:2002vq}. We do not follow this procedure here but include all $t$-channel
exchanges into the interaction kernel. This allows for a consistent unitary treatment of resonance and background
contributions. The calculated differential cross section is presented
in Fig.~\refe{ge_dif2} as a function of the scattering angle.
Except for the energy bin W = 2.097 GeV our results are found to be in close
agreement with the CLAS measurements. The major contribution to the differential cross section at forward
angles comes from $\rho$- and $\omega$-meson exchanges. The effect from the $\phi$-meson
is small due to the weakness of the $\phi NN$ coupling as dictated by the OZI rule
\cite{Okubo:1963fa,Zweig:1964jf,Iizuka:1966fk}.
We also checked for the contributions from the Primakoff
effect which is found to be negligible at these energies.
It is interesting to compare our calculations with the data \cite{Williams:2009yj,Crede:2009zzb} at
forward angles plotted as a function of the c.m. energy, see Fig.~\refe{ge_d3f}.
The cusp due to the $\omega N$
production threshold is clearly seen in our calculations around W=1.72 GeV. The quality of the data is still
not good enough to unambiguously resolve the cusp induced by the $\omega N$ threshold in the experimental data.
Note, that the calculations
are done assuming a stable $\omega$-meson. Taking into account the final $\omega$-width would smear out this effect.
Since the $\omega N$ threshold lies 45 MeV above the dip position ( W=1.68 GeV) we conclude that this effect cannot explain
the dip in the differential cross section. This conclusion is opposite to that drawn
in \cite{Anisovich:2011ka}.
The discrepancy between the CLAS \cite{Williams:2009yj} and CB-ELSA/TAPS data is better seen at $\cos(\theta)=0.75$
whereas for $\cos(\theta)=0.85$ the measurements are found to be in better agreement.
One of the interesting features observed in the recent CB-ELSA data is a sudden rise of the differential cross
section at W=1.92 GeV. The effect is more pronounced at $\cos(\theta)=0.85...0.95$ and is absent at other scattering
angles. This phenomena might be attributed to sidefeeding from of one of the inelastic channels (e.g. $\phi N$, $a_0(980) N$,
$f_0(980)$, or $\eta' N$). However the problem with normalization inconsistencies
between the CLAS and CB-ELSA data should be solved
first before any physical interpretation can be given.
\subsection{eta-photoproduction multipoles \label{multipoles}}
The extracted $\gamma p \to \eta p$ multipoles are presented in Fig.~\ref{ge_multiploes}.
The major contribution to the $E_{0+}$ multipole comes from the $S_{11}(1535)$ resonance.
The second $S_{11}(1650)$ plays an important role in the region W=1.6...1.7 GeV.
We corroborate our previous results \cite{Shklyar:2006xw} where only a small effect from the spin-$\frac{5}{2}$
states has been found. A very small signal from the $F_{15}(1680)$ resonance is seen in
the $E_{3-}$ and $M_{3-}$ amplitudes at W=1.68 GeV.
It is interesting to note that the effect of $D_{13}(1520)$ is clearly seen in the $E_{2-}$ and $M_{2-}$ though
the overall contribution from this state turns out to be small.
The $M_{1-}$ multipole is affected by the Roper and $P_{11}(1710)$ resonances leading to the rapid change in
both real and imaginary parts of the amplitude at W=1.7 GeV. In the region W=1.48...1.6 GeV both the imaginary
and the real parts of all multipoles with $l\neq 0$ are of the order of magnitude smaller than $E_{0+}$
due to the strong dominant contribution from $S_{11}(1535)$. However for higher energies the influence of
amplitudes with $l\neq 0$ becomes also important.
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{fig11.eps}
\caption{(Color online) $\gamma p \to \eta p$ multipoles extracted in the present study. \label{ge_multiploes}}
\end{center}
\end{figure}
\section{Conclusion}
We have performed a coupled-channel analysis of pion- and photon-induced reactions including the recent
eta-photoproduction data from the Crystal Ball/MAMI collaboration. In the region W=1.89...2.0 GeV our solution is constrained by
the combined dataset built from the recent CLAS and CB-ELSA/TAPS measurements. The dip
in the differential cross-sections at W=1.68 GeV reported in \cite{McNicoll:2010qk} is described in
terms of an interference of the $S_{11}(1535)$ and $S_{11}(1650)$ states.
We stress that such an interference also includes coupled-channel effects and
rescattering which is hard to simulate by a simple sum of two
Breit-Wigner contributions.
The additional contribution at W=1.68 GeV comes from the $M_{1-}$ multipole where the excitation of the $P_{11}(1710)$
leads to a rapid change of the real and imaginary parts of the amplitude.
We conclude that the cusp due to the $\omega N$ threshold seen at 1.72 GeV is not important
for the explanation of the dip at W=1.68 GeV. However the quality
of the data is still not sufficient to resolve the threshold effect completely.
Above W=1.9 GeV the $t$-channel $\rho$- and $\omega$-exchanges start
to play a dominant role in the calculations. The effect from the $\phi$-meson exchange is less
important because of the smallness of the $\phi NN$ coupling. We have also checked for the contribution from the
Primakoff-effect which is found to be negligible. In the region W=1.9...2.2~GeV our calculations
tend to be in closer agreement with the CLAS data.
It is interesting to note that above W=1.92 GeV the cross sections of the CB-ELSA/TAPS collaboration
indicate a sudden rise from 0.2~$\mu$b up to 0.3~$\mu$b. The effect is observed only for scattering angles
$\cos(\theta)=0.85...0.95$. This phenomenon might be attributed to sidefeeding from of one of the inelastic
channels (e.g. $\phi N$, $a_0(980) N$, $f_0(980)$, or $\eta' N$). However the origin of
the normalization discrepancies between the CLAS and CB-ELSA/TAPS data should first be understood
before any physical interpretation can be given.
In the $\pi^- p \to \eta n$ reaction the main effect comes from three resonances $S_{11}(1535)$,
$S_{11}(1650)$, and $P_{11}(1710)$. Similar to eta-photoproduction on the proton the overlap
of the $S_{11}(1535)$ and $S_{11}(1650)$ states produces a dip around W=1.68 GeV. For energies $W>1.68$ GeV
the contribution from $P_{11}(1710)$ is found to be important.
The above reaction mechanism for the $(\gamma/\pi)N\to\eta N$ reaction is in line with our early
findings \cite{Shklyar:2006xw} where the resonance like-structure in $\eta$-photoproduction at W=1.68 GeV
on the neutron was explained by the excitations of the $S_{11}(1650)$, and $P_{11}(1710)$
resonances.
We conclude that further progress in understanding of $\eta$-meson production
would be hardly possible without new measurements of the $\pi N\to \eta N$ reaction.
The experimental investigation of this reaction would help to establish the resonance contributions
to the $\eta$-photoproduction above $W >1.6 $ GeV. Finally, the study of the $\eta N$-channel with
pion beams would solve the question whether the observed phenomena in $\eta$ photoproduction
have their counterparts in $\pi N \to\eta N$ scattering.
\bibliographystyle{h-physrev}
|
3,212,635,537,420 | arxiv | \section{Introduction}
An {\it $n$-dimensional crystallographic group} ({\it $n$-space group})
is a discrete group $\Gamma$ of isometries of Euclidean $n$-space $E^n$
whose orbit space $E^n/\Gamma$ is compact.
The 3-space groups are the symmetry groups of crystalline structures,
and so are of fundamental importance in the science of crystallography.
In response to Hilbert's Problem 18, L. Bieberbach \cite{B} proved that for each dimension $n$
there are only finitely many isomorphism classes of $n$-space groups.
In this paper we prove a relative version of Bieberbach's theorem.
We prove that for each dimension $n$
there are only finitely many isomorphism classes of pairs of groups $(\Gamma,{\rm N})$
such that $\Gamma$ is an $n$-space group and
${\rm N}$ is a normal subgroup of $\Gamma$ such that $\Gamma/{\rm N}$ is a space group.
Our relative Bieberbach theorem has a geometric interpretation in the theory of flat orbifolds.
By Theorems 7, 8, and 10 of \cite{R-T} the isomorphism classes of pairs of groups $(\Gamma, {\rm N})$
such that $\Gamma$ is an $n$-space group and ${\rm N}$ is a normal subgroup of $\Gamma$ such that $\Gamma/{\rm N}$ is a space group
correspond to the affine equivalence classes of geometric orbifold fibrations of compact,
connected, flat $n$-orbifolds.
Therefore, our relative Bieberbach theorem is equivalent to the theorem that for each dimension $n$ there are only finitely many
affine equivalence classes of geometric orbifold fibrations of compact, connected, flat $n$-orbifolds.
This is known for $n = 3$ by the work of Conway-Friedrichs-Huson-Thurston\cite{C-T} and Ratcliffe-Tschantz \cite{R-T}.
We now outline the proof of our relative Bieberbach theorem.
Let $m$ be a positive integer less than $n$.
Let ${\rm M}$ be an $m$-space group and
let $\Delta$ be an $(n-m)$-space group.
Let $\mathrm{Iso}(\Delta,{\rm M})$ be the set of isomorphism classes
of pairs $(\Gamma, {\rm N})$
where ${\rm N}$ is a normal subgroup of an $n$-space group $\Gamma$
such that ${\rm N}$ is isomorphic to ${\rm M}$ and $\Gamma/{\rm N}$ is isomorphic to $\Delta$.
As there are only finitely many isomorphism classes of the groups $\Delta$ and ${\rm M}$ by
Bieberbach's theorem \cite{B}, it suffices to prove that $\mathrm{Iso}(\Delta,{\rm M})$ is finite.
Next, we define a set $\mathrm{Out}(\Delta,\mathrm{M})$ in terms of $\mathrm{Out}(\Delta)$ and $\mathrm{Out}(\mathrm{M})$.
That the set $\mathrm{Out}(\Delta,\mathrm{M})$ is finite follows easily from a theorem of Baues and Grunewald \cite{B-G}
that the outer automorphism group of a crystallographic group is an arithmetic group.
We define a function $\omega: \mathrm{Iso}(\Delta,{\rm M}) \to \mathrm{Out}(\Delta,\mathrm{M})$.
We prove that $\mathrm{Iso}(\Delta,{\rm M})$ is finite by showing that the fibers of $\omega$ are finite
by a cohomology of groups argument.
\section{Normal Subgroups of Space Groups}
A map $\phi:E^n\to E^n$ is an isometry of $E^n$
if and only if there is an $a\in E^n$ and an $A\in {\rm O}(n)$ such that
$\phi(x) = a + Ax$ for each $x$ in $E^n$.
We shall write $\phi = a+ A$.
In particular, every translation $\tau = a + I$ is an isometry of $E^n$.
Let $\Gamma$ be an $n$-space group.
Define $\eta:\Gamma \to {\rm O}(n)$ by $\eta(a+A) = A$.
Then $\eta$ is a homomorphism whose kernel is the group $\mathrm{T}$
of translations in $\Gamma$.
The image of $\eta$ is a finite group $\Pi$ called the
{\it point group} of $\Gamma$.
Let ${\rm H}$ be a subgroup of an $n$-space group $\Gamma$.
Define the {\it span} of ${\rm H}$ by the formula
$${\rm Span}({\rm H}) = {\rm Span}\{a\in E^n:a+I\in {\rm H}\}.$$
Note that ${\rm Span}({\rm H})$ is a vector subspace $V$ of $E^n$.
Let $V^\perp$ denote the orthogonal complement of $V$ in $E^n$.
\begin{theorem} {\rm (Theorem 2 \cite{R-T})}
Let ${\rm N}$ be a normal subgroup of an $n$-space group $\Gamma$,
and let $V = {\rm Span}({\rm N})$.
\begin{enumerate}
\item If $b+B\in\Gamma$, then $BV=V$.
\item If $a+A\in {\rm N}$, then $a\in V$ and $ V^\perp\subseteq{\rm Fix}(A)$.
\item The group ${\rm N}$ acts effectively on each coset $V+x$ of $V$ in $E^n$
as a space group of isometries of $V+x$.
\end{enumerate}
\end{theorem}
Let $\Gamma$ be an $n$-space group.
The {\it dimension} of $\Gamma$ is $n$.
If ${\rm N}$ is a normal subgroup of $\Gamma$,
then ${\rm N}$ is a $m$-space group with $m= \mathrm{dim}(\mathrm{Span}({\rm N}))$
by Theorem 1(3).
\vspace{.15in}
\noindent{\bf Definition:}
Let ${\rm N}$ be a normal subgroup ${\rm N}$ of an $n$-space group $\Gamma$, and let $V = {\rm Span}({\rm N})$. Then ${\rm N}$ is said to be a {\it complete normal subgroup} of $\Gamma$ if
$${\rm N}= \{a+A\in \Gamma: a\in V\ \hbox{and}\ V^\perp\subseteq{\rm Fix}(A)\}.$$
\begin{lemma} {\rm (Lemma 1 \cite{R-T})}
Let ${\rm N}$ be a complete normal subgroup of an $n$-space group $\Gamma$,
and let $V={\rm Span}({\rm N})$.
Then $\Gamma/{\rm N}$ acts effectively as a space group of isometries of $E^n/V$
by the formula
$({\rm N}(b+B))(V+x) = V+ b+Bx.$
\end{lemma}
\noindent{\bf Remark 1.} A normal subgroup ${\rm N}$ of a space group $\Gamma$ is complete
precisely when $\Gamma/{\rm N}$ is a space group by Theorem 5 of \cite{R-T}.
\vspace{.15in}
Let ${\rm N}$ be a complete normal subgroup of an $n$-space group $\Gamma$,
let $V = \mathrm{Span}({\rm N})$, and let $V^\perp$ be the orthogonal complement of $V$ in $E^n$.
Let $\gamma \in \Gamma$. Then $\gamma = b+B$ with $b\in E^n$ and $B\in \mathrm{O}(n)$.
Write $b = \overline b + b'$ with $\overline b \in V$ and $b' \in V^\perp$.
Let $\overline B$ and $B'$ be the orthogonal transformations of $V$ and $V^\perp$, respectively,
obtained by restricting $B$.
Let $\overline \gamma = \overline b + \overline B$ and $\gamma' = b' + B'$.
Then $\overline \gamma$ and $\gamma'$ are isometries of $V$ and $V^\perp$, respectively.
Euclidean $n$-space $E^n$ decomposes as the Cartesian product $E^n = V \times V^\perp$.
Let $x\in E^n$. Write $x = v+w$ with $v\in V$ and $w\in V^\perp$. Then
$$(b+B)x = b+Bx = \overline{b}+b' + Bv + Bw = (\overline{b}+\overline{B}v) + (b'+B'w).$$
Hence the action of $\Gamma$ on $E^n$ corresponds to the diagonal action of $\Gamma$
on $V\times V^\perp$ defined by the formula
$$\gamma(v,w) = (\overline{\gamma}v,\gamma'w).$$
Here $\Gamma$ acts on both $V$ and $V^\perp$ via isometries.
The kernel of the corresponding homomorphism from $\Gamma$ to $\mathrm{Isom}(V)$
is the group
$${\rm K} = \{b+B\in\Gamma: b \in V^\perp\ \hbox{and}\ V \subseteq \mathrm{Fix}(B)\}.$$
We call ${\rm K}$ the {\it kernel of the action} of $\Gamma$ on $V$.
The group ${\rm K}$ is a normal subgroup of $\Gamma$.
The action of $\Gamma$ on $V$ induces an effective action of $\Gamma/{\rm K}$ on $V$ via isometries.
The group $\Gamma/{\rm K}$ acts on $V$ as a discrete group of isometries if and only if
$\Gamma/{\rm N}{\rm K}$ is a finite group by Theorem 3(4) of \cite{R-T-Cal}.
The group ${\rm N}$ is the kernel of the action of $\Gamma$ on $V^\perp$, and so the action
of $\Gamma$ on $V^\perp$ induces an effective action of $\Gamma/{\rm N}$ on $V^\perp$ via isometries.
Orthogonal projection from $E^n$ to $V^\perp$ induces an isometry from $E^n/V$ to $V^\perp$.
Hence $\Gamma/{\rm N}$ acts on $V^\perp$ as a space group of isometries by Lemma 1.
Let $\overline \Gamma = \{\overline \gamma: \gamma \in \Gamma\}$.
If $\gamma \in \Gamma$, then $(\overline{\gamma})^{-1} = \overline{\gamma^{-1}}$, and
if $\gamma_1, \gamma_2 \in \Gamma$,
then $\overline{\gamma_1}\,\overline{\gamma_2}= \overline{\gamma_1\gamma_2}$.
Hence $\overline \Gamma$ is a subgroup of $\mathrm{Isom}(V)$.
The map ${\rm B}: \Gamma \to \overline \Gamma$ defined by ${\rm B}(\gamma)= \overline\gamma$ is an epimorphism
with kernel ${\rm K}$.
The group $\overline\Gamma$ is a discrete subgroup of $\mathrm{Isom}(V)$
if and only if $\Gamma/{\rm N}{\rm K}$ is finite by Theorem 3(4) of \cite{R-T-Cal}.
Let $\Gamma' = \{\gamma' : \gamma \in \Gamma\}$.
If $\gamma \in \Gamma$, then $(\gamma')^{-1} = (\gamma^{-1})'$, and
if $\gamma_1, \gamma_2 \in \Gamma$,
then $\gamma_1'\gamma_2' = (\gamma_1\gamma_2)'$.
Hence $\Gamma'$ is a subgroup of $\mathrm{Isom}(V^\perp)$.
The map ${\rm P}' : \Gamma \to \Gamma'$
defined by ${\rm P}'(\gamma) = \gamma'$ is epimorphism with kernel ${\rm N}$,
since ${\rm N}$ is a complete normal subgroup of $\Gamma$.
Hence ${\rm P}'$ induces an isomorphism ${\rm P}: \Gamma/{\rm N} \to \Gamma'$ defined by ${\rm P}({\rm N}\gamma) = \gamma'$.
The group $\Gamma'$ is a space group of isometries of $V^\perp$
with $V^\perp/\Gamma' = V^\perp/(\Gamma/{\rm N})$.
Let $\overline {\rm N} = \{\overline \nu: \nu \in {\rm N}\}$.
Then $\overline {\rm N}$ is a subgroup of $\mathrm{Isom}(V)$.
The map ${\rm B}: {\rm N} \to \overline {\rm N}$ defined by ${\rm B}(\nu) = \overline\nu$ is an isomorphism.
The group $\overline{\rm N}$ is a space group of isometries of $V$ with $V/\overline{{\rm N}} = V/{\rm N}$.
The action of $\Gamma$ on $V$ induces an action of $\Gamma/{\rm N}$ on $V/{\rm N}$
defined by
$$({\rm N}\gamma)({\rm N} v) = {\rm N} \overline{\gamma}v.$$
The action of $\Gamma/{\rm N}$ on $V/{\rm N}$ determines a homomorphism
$$\Xi : \Gamma/{\rm N} \to \mathrm{Isom}(V/{\rm N})$$
defined by $\Xi({\rm N} \gamma) = \overline \gamma_\star$, where $\overline \gamma_\star: V/{\rm N} \to V/{\rm N}$
is defined by $\overline\gamma_\star({\rm N} v) = {\rm N} \overline\gamma(v)$.
\begin{theorem}
Let ${\rm M}$ be an $m$-space group, let $\Delta$ be an $(n-m)$-space group,
and let $\Theta:\Delta \to \mathrm{Isom}(E^m/{\rm M})$ be a homomorphism.
Identify $E^n$ with $E^m\times E^{n-m}$, and extend ${\rm M}$ to a subgroup ${\rm N}$ of $\mathrm{Isom}(E^n)$
such that the point group of ${\rm N}$ acts trivially on $(E^m)^\perp= E^{n-m}$.
Then there exists a unique $n$-space group $\Gamma$ containing ${\rm N}$ as a complete normal subgroup
such that $\Gamma' = \Delta$, and if $\Xi:\Gamma/{\rm N} \to \mathrm{Isom}(E^m/{\rm N})$ is
the homomorphism induced by the action of $\Gamma/{\rm N}$ on $E^m/{\rm N}$, then $\Xi = \Theta{\rm P}$
where ${\rm P}:\Gamma/{\rm N} \to \Gamma'$ is the isomorphism defined by ${\rm P}({\rm N}\gamma) = \gamma'$
for each $\gamma\in\Gamma$.
\end{theorem}
\begin{proof} Let $\delta \in \Delta$.
By Lemma 1 of \cite{R-T-Isom}, there exists $\tilde\delta \in N_E({\rm M})$ such that $\tilde\delta_\star = \Theta(\delta)$.
The isometry $\tilde\delta$ is unique up to multiplication by an element of ${\rm M}$.
Let $\delta = d + D$, with $d\in E^{n-m}$ and $D\in \mathrm{O}(n-m)$,
and let $\tilde\delta = \tilde d + \tilde D$, with $\tilde d \in E^m$ and $\tilde D \in \mathrm{O}(m)$.
Let $\hat d = \tilde d + d$, and let $\hat D = \tilde D \times D$.
Then $\hat D E^m = E^m$ and $\hat D E^{n-m} = E^{n-m}$.
Let $\hat \delta = \hat d + \hat D$.
Then $\hat \delta$ is an isometry of $E^n$ such that $\overline{\hat\delta} = \tilde\delta$
and $\big(\hat\delta\big)' = \delta$.
The isometry $\hat \delta$ is unique up to multiplication by an element of ${\rm N}$.
We have that
$$\hat\delta^{-1} = -\hat D^{-1}\hat d + \hat D^{-1} = -\tilde D^{-1}\tilde d -D^{-1}d + \tilde D^{-1} \times D^{-1},$$
and so $\overline{\hat\delta^{-1} }= \big(\tilde \delta\big)^{-1}$ and $(\hat\delta^{-1})' = \delta^{-1}$.
We have that
$$\overline{\hat\delta{\rm N}\hat\delta^{-1}} = \overline{\hat\delta}\overline{\rm N}\overline{\hat\delta^{-1}} = \tilde\delta{\rm M}\tilde\delta^{-1} = {\rm M} = \overline{\rm N}$$
and
$$(\hat\delta{\rm N}\hat\delta^{-1})' = \hat\delta'{\rm N}'(\hat\delta^{-1})' = \delta\{I'\}\delta^{-1} = \{I'\}.$$
Therefore $\hat \delta{\rm N}\hat\delta^{-1} = {\rm N}$.
Let $\Gamma$ be the subgroup of $\mathrm{Isom}(E^n)$
generated by ${\rm N} \cup \{\hat \delta : \delta \in \Delta\}$.
Then $\Gamma$ contains ${\rm N}$ as a normal subgroup,
and the point group of $\Gamma$ leaves $E^m$ invariant.
Suppose $\gamma \in \Gamma$. Then there exists $\nu \in {\rm N}$ and $\delta_1, \ldots, \delta_k \in \Delta$
and $\epsilon_1,\ldots, \epsilon_k \in \{\pm 1\}$
such that $\gamma = \nu \hat{\delta}_1^{\epsilon_1} \cdots \hat{\delta}_k^{\epsilon_k}$.
Then we have
$$\gamma' = \nu' (\hat{\delta}_1^{\epsilon_1})' \cdots (\hat{\delta}_k^{\epsilon_k})' = \delta_1^{\epsilon_1}\cdots \delta_k^{\epsilon_k}.$$
Hence $\Gamma' = \Delta$, and
we have an epimorphism ${\rm P}':\Gamma \to \Delta$ defined by ${\rm P}'(\gamma) = \gamma'$.
The group ${\rm N}$ is in the kernel of ${\rm P}'$, and so ${\rm P}'$ induces an epimorphism ${\rm P}: \Gamma/{\rm N} \to \Delta$
defined by ${\rm P}({\rm N}\gamma) = \gamma'$.
Suppose ${\rm P}({\rm N}\gamma) = I'$. By Lemma 1 of \cite{R-T-Isom}, we have that
\begin{eqnarray*}
{\rm P}({\rm N}\gamma) = I' \ \ \Rightarrow \ \ \gamma' = I' & \Rightarrow & \delta_1^{\epsilon_1}\cdots \delta_k^{\epsilon_k} =I' \\
& \Rightarrow & \Theta(\delta_1)^{\epsilon_1} \cdots \Theta(\delta_k)^{\epsilon_k} = \overline I_\star\\
& \Rightarrow &(\tilde \delta_1)_\star^{\epsilon_1} \cdots (\tilde \delta_k)_\star^{\epsilon_k} = \overline I_\star\\
& \Rightarrow &{\rm M}(\tilde \delta_1)^{\epsilon_1} \cdots {\rm M}(\tilde \delta_k)^{\epsilon_k} = {\rm M}\\
& \Rightarrow &{\rm M} (\tilde \delta_1)^{\epsilon_1} \cdots (\tilde \delta_k)^{\epsilon_k} = {\rm M} \\
& \Rightarrow &{\rm M} \overline{\hat \delta_1^{\epsilon_1}} \cdots \overline{\hat \delta_k^{\epsilon_k}} = {\rm M}
\ \ \Rightarrow \ \ {\rm M} \overline\gamma = {\rm M} \ \ \Rightarrow \ \ \overline\gamma \in {\rm M}.
\end{eqnarray*}
As $\gamma' = I'$ and $\overline \gamma \in \overline{\rm N}$, we have that $\gamma \in {\rm N}$.
Thus ${\rm P}$ is an isomorphism.
We next show that $\Gamma$ acts discontinuously on $E^n$.
Let $C$ be a compact subset of $E^n$.
Let $K$ and $L$ be the orthogonal projections of $C$ into $E^m$ and $E^{n-m}$, respectively.
Then $C \subseteq K \times L$.
As $\Delta$ acts discontinuously on $E^{n-m}$, there exists only finitely many elements $\delta_1, \ldots, \delta_k$ of $\Delta$
such that $L\cap \delta_iL \neq \emptyset$ for each $i$.
Let $\gamma_i = \hat \delta_i$ for $i = 1,\ldots, k$.
The set $K_i = K \cup \overline \gamma_i(K)$ is compact for each $i = 1,\ldots, k$.
As ${\rm N}$ acts discontinuously on $E^m$, there is a finite subset $F_i$ of ${\rm N}$ such that $K_i \cap \nu K_i \neq \emptyset$
only if $\nu \in F_i$ for each $i$. Hence $K \cap \nu\overline\gamma_i(K)\neq\emptyset$ only if $\nu \in F_i$ for each $i$.
The set $F = F_1\gamma_1\cup \cdots \cup F_k\gamma_k$ is a finite subset of $\Gamma$.
Suppose $\gamma \in \Gamma$ and $C\cap \gamma C \neq \emptyset$.
Then $L \cap \gamma' L \neq \emptyset$, and so $ \gamma' = \delta_i$ for some $i$.
Hence $\gamma = \nu \gamma_i$ for some $\nu \in {\rm N}$.
Now we have that $K\cap \overline \gamma K \neq \emptyset$, and so $K\cap \nu\overline\gamma_iK \neq \emptyset$.
Hence $\nu \in F_i$. Therefore $\gamma \in F$. Thus $\Gamma$ acts discontinuously on $E^n$.
Therefore $\Gamma$ is a discrete subgroup of $\mathrm{Isom}(E^n)$ by Theorem 5.3.5 of \cite{R}.
Let $D_{\rm M}$ and $D_\Delta$ be fundamental domains for ${\rm M}$ in $E^m$ and $\Delta$ in $E^{n-m}$, respectively.
Then their topological closures $\overline{D}_{\rm M}$ and $\overline{D}_\Delta$ are compact sets.
Let $x \in E^n$. Write $x = \overline x + x'$ with $\overline x \in E^m$ and $x' \in E^{n-m}$.
Then there exists $\delta \in \Delta$ such that $\delta x' \in \overline D_\Delta$,
and there exists $\nu \in {\rm N}$ such that $\nu \tilde\delta\overline x \in \overline D_{\rm M}$.
We have that
$$\nu\hat \delta x = \overline{ \nu\hat\delta} \overline x + (\nu \hat\delta)'x' = \nu\tilde \delta \overline x + \delta x'
\in \overline D_{\rm M} \times \overline D_\Delta.$$
Hence the quotient map $\pi: E^n \to E^n/\Gamma$ maps the compact set $\overline D_{\rm M} \times \overline D_\Delta$
onto $E^n/\Gamma$. Therefore $E^n/\Gamma$ is compact. Thus $\Gamma$ is an $n$-space group.
Let $\Xi: \Gamma/{\rm N} \to \mathrm{Isom}(E^n/{\rm N})$ be the homomorphism induced by the action of $\Gamma/{\rm N}$ on $E^m/{\rm N}$.
Let $\gamma \in \Gamma$. Then there exists $\delta \in \Delta$ such that ${\rm N}\gamma = {\rm N}\hat \delta$.
Then $\gamma' = (\hat\delta)' = \delta$, and we have that $\Xi = \Theta{\rm P}$, since
$$\Xi({\rm N}\gamma) = \Xi({\rm N}\hat\delta) = (\overline{\hat\delta})_\star = \tilde \delta_\star =
\Theta(\delta) = \Theta(\gamma') = \Theta{\rm P}({\rm N}\gamma).$$
Suppose $\gamma$ is an isometry of $E^n$ such that $\gamma{\rm N}\gamma^{-1} = {\rm N}$
and $\gamma' \in \Delta$ and $\overline\gamma_\star = \Theta(\gamma')$.
Then $\widehat{\gamma'} \in \Gamma$.
Now $\overline\gamma_\star = \widetilde{\gamma'}_\star$,
and so $\overline\gamma = \overline\nu\widetilde{\gamma'}$ for some $\nu\in {\rm N}$ by Lemma 1 of \cite{R-T-Isom}.
Then $\gamma = \nu\widehat{\gamma'}$. Hence $\gamma\in \Gamma$.
Thus $\Gamma$ is the unique $n$-space group that contains ${\rm N}$ as a complete normal subgroup such that $\Gamma' = \Delta$
and $\Xi = \Theta{\rm P}$.
\end{proof}
\section{Isomorphisms of Pairs of Space Groups}
An {\it affinity} $\alpha$ of $E^n$ is a map $\alpha: E^n\to E^n$
for which there is an element $a\in E^n$ and a matrix $A \in \mathrm{GL}(n,{\mathbb R})$ such that
$\alpha(x) = a + Ax$ for all $x$ in $E^n$. We write simply $\alpha = a + A$.
The set $\mathrm{Aff}(E^n)$ of all affinities of $E^n$ is a group
that contains $\mathrm{Isom}(E^n)$ as a subgroup.
Let ${\rm N}_i$ be a complete normal subgroup of an $n$-space group $\Gamma_i$ for $i=1,2$.
We want to know when $(\Gamma_1,{\rm N}_1)$ is isomorphic to $(\Gamma_2, {\rm N}_2)$,
that is, when there is an isomorphism $\zeta: \Gamma_1 \to \Gamma_2$ such that $\zeta({\rm N}_1) = {\rm N}_2$.
By a Theorem of Bieberbach an isomorphism $\zeta: \Gamma_1 \to \Gamma_2$ is equal to
conjugation by an affinity $\phi$ of $E^n$.
In this section, we determine necessary and sufficient conditions such that there exists
an affinity $\phi$ of $E^n$ such that $\phi(\Gamma_1,{\rm N}_1)\phi^{-1}=(\Gamma_2,{\rm N}_2)$.
Let $\phi$ be an affinity of $E^n$ such that $\phi(\Gamma_1,{\rm N}_1)\phi^{-1}=(\Gamma_2,{\rm N}_2)$.
Write $\phi = c+C$ with $c\in E^n$ and $C\in \mathrm{GL}(n,{\mathbb R})$.
Let $V_i = \mathrm{Span}({\rm N}_i)$, for $i=1,2$.
Let $a+I\in {\rm N}_1$. Then $\phi(a+I)\phi^{-1} = Ca+I$. Hence $CV_1 \subseteq V_2$.
Let $\overline{C}:V_1 \to V_2$ be the linear transformation obtained by restricting $C$.
Let $\overline{C'}:V_1^\perp \to V_2$ and $C': V_1^\perp \to V_2^\perp$ be the linear transformations
obtained by restricting $C$ to $V_1^\perp$ followed by the orthogonal projections to $V_2$ and $V_2^\perp$, respectively.
Write $c = \overline{c} + c'$ with $\overline{c}\in V_2$ and $c'\in V_2^\perp$.
Let $\overline{\phi}: V_1 \to V_2$ and $\phi': V_1^\perp \to V_2^\perp$ be the affine transformations
defined by $\overline{\phi} = \overline{c}+\overline{C}$ and $\phi' = c'+C'$.
\begin{lemma}
Let ${\rm N}_i$ be a complete normal subgroup of an $n$-space group $\Gamma_i$, with $V_i = \mathrm{Span}({\rm N}_i)$, for $i=1,2$.
Let $\phi = c+C$ be an affinity of $E^n$ such that $\phi(\Gamma_1,{\rm N}_1)\phi^{-1}=(\Gamma_2,{\rm N}_2)$.
Then $\overline{C}$ and $C'$ are invertible, with $\overline{C}^{-1} = \overline{C^{-1}}$ and $(C')^{-1} = (C^{-1})'$,
and $\overline{(C^{-1})'} = -\overline{C}^{-1}\overline{C'}(C')^{-1}$.
If $b+B\in \Gamma_1$, then $\overline{C'}B' = \overline C \overline B \overline C^{-1} \overline{C'}$.
Moreover $\overline{C'}V_1^\perp \subseteq \mathrm{Span}(Z({\rm N}_2))$.
\end{lemma}
\begin{proof} We have that
$$\dim V_1 = \dim {\rm N}_1 = \dim {\rm N}_2 = \dim V_2.$$
Let $y\in E^n$ and write $y = \overline{y} + y'$ with $\overline{y}\in V_2$ and $y'\in V_2^\perp$.
Then
$$\overline{y} = CC^{-1}(\overline{y}) = \overline{C}\overline{C^{-1}}(\overline{y}),$$
and so $\overline{C}$ is invertible with $\overline{C}^{-1} = \overline{C^{-1}}$.
We have that
\begin{eqnarray*}
y' & = & CC^{-1}y' \\
& = &C(\overline{(C^{-1})'}(y') + (C^{-1})'(y')) \\
& = &C\overline{(C^{-1})'}(y')+ C(C^{-1})'(y') \\
& = & \overline{C}\overline{(C^{-1})'}(y')+\overline{C'}(C^{-1})'(y')+ C'(C^{-1})'(y').
\end{eqnarray*}
Hence $C'$ is invertible, with $(C')^{-1} = (C^{-1})'$, and $\overline{C}\overline{(C^{-1})'}+\overline{C'}(C^{-1})' =0$.
Therefore $\overline{(C^{-1})'} = -\overline{C}^{-1}\overline{C'}(C')^{-1}$.
Let $\gamma = b+ B \in \Gamma_1$. Then $\phi\gamma\phi^{-1} \in \Gamma_2$. Let $w \in V_2^\perp$.
Then we have
\begin{eqnarray*}
w& = & CBC^{-1}w\\
& = &CB(\overline{(C^{-1})'}(w) + (C^{-1})'(w)) \\
& = &C(\overline B\overline{(C^{-1})'}(w)+ B'(C^{-1})'(w)) \\
& = &C\overline B\overline{(C^{-1})'}(w)+ CB'(C^{-1})'(w) \\
& = & \overline{C}\overline B\overline{(C^{-1})'}(w)+\overline{C'}B'(C^{-1})'(w)+ C'B'(C^{-1})'(w). \\
\end{eqnarray*}
As $\phi\gamma\phi^{-1}(w) \in V_2^\perp$,
we have that
$$\overline{C}\overline B\overline{(C^{-1})'}+\overline{C'}B'(C^{-1})'=0.$$
Therefore
$$\overline{C}\overline B(-\overline{C}^{-1}\overline{C'}(C')^{-1})+\overline{C'}B'(C')^{-1}=0.$$
Hence $\overline{C'}B' = \overline C \overline B\overline{C}^{-1}\overline{C'}$.
Now suppose $\gamma \in {\rm N}_1$. Then $B' = I'$ by Theorem 1(2).
Hence $\overline B\overline{C}^{-1}\overline{C'} = \overline{C}^{-1}\overline{C'}$.
By Lemma 5 of \cite{R-T-Isom}, we deduce that $\overline{C}^{-1}\overline{C'}V_1^\perp \subseteq \mathrm{Span}(Z({\rm N}_1))$.
As $\phi Z({\rm N}_1)\phi^{-1} = Z({\rm N}_2)$, we have
that $\overline{C'}V_1^\perp \subseteq \mathrm{Span}(Z({\rm N}_2))$.
\end{proof}
\begin{theorem}
Let ${\rm N}_i$ be a complete normal subgroup of an $n$-space group $\Gamma_i$, with $V_i = \mathrm{Span}({\rm N}_i)$ for $i=1,2$.
Let $\Xi_i:\Gamma_i/{\rm N}_i \to \mathrm{Isom}(V_i/{\rm N}_i)$ be the homomorphism induced
by the action of $\Gamma_i/{\rm N}_i$ on $V_i/{\rm N}_i$ for $i=1,2$.
Let $\alpha: V_1 \to V_2$, and $\beta: V_1^\perp \to V_2^\perp$ be affinities such that $\alpha\overline{\rm N}_1\alpha^{-1} = \overline{\rm N}_2$
and $\beta\Gamma'_1\beta^{-1} = \Gamma_2'$.
Let $\alpha_\star: V_1/{\rm N}_1 \to V_2/{\rm N}_2$ be the affinity defined by $\alpha_\star({\rm N}_1v) = {\rm N}_2\alpha(v)$,
and let $\alpha_\sharp: \mathrm{Aff}(V_1/{\rm N}_1) \to \mathrm{Aff}(V_2/{\rm N}_2)$ be the isomorphism
defined by $\alpha_\sharp(\chi) = \alpha_\star\chi\alpha_\star^{-1}$.
Let $\beta_\ast: \Gamma_1' \to \Gamma_2'$ be the isomorphism defined by $\beta_\ast(\gamma') = \beta\gamma'\beta^{-1}$.
Write $\alpha = \overline c + \overline C$ with $\overline c \in V_2$ and $\overline C: V_1 \to V_2$ a linear isomorphism.
Let $D: V_1^\perp \to \mathrm{Span}(Z({\rm N}_2))$ be a linear transformation
such that if $b+B\in \Gamma_1$, then $DB' = \overline C\overline B\overline C^{-1} D$.
Let ${\rm P}_i:\Gamma_i/{\rm N}_i\to \Gamma_i'$ be the isomorphism
defined by ${\rm P}_i({\rm N}_i\gamma) = \gamma'$ for each $i = 1,2$,
and let $p_1:\Gamma_1/{\rm N}_1 \to V_1^\perp$ be defined by $p_1({\rm N}_1(b+B)) = b'$.
Let $\mathcal{K}_i$ be the connected component of the identity of the Lie group $\mathrm{Isom}(V_i/{\rm N}_i)$ for each $i = 1,2$.
Note that $\mathcal{K}_i = \{(v+\overline I)_\star: v \in \mathrm{Span}(Z({\rm N}_i))\}$ by Theorem 1 of \cite{R-T-Isom}.
Then the following are equivalent:
\begin{enumerate}
\item There exists an affinity $\phi = c + C$ of $E^n$ such that $\phi(\Gamma_1,{\rm N}_1)\phi^{-1} = (\Gamma_2,{\rm N}_2)$,
with $\overline\phi = \alpha$, and $\phi' = \beta$, and $\overline{C'} = D$.
\item We have that
$$\Xi_2{\rm P}_2^{-1}\beta_\ast{\rm P}_1 = (Dp_1)_\star\alpha_\sharp\Xi_1$$
with $(Dp_1)_\star:\Gamma_1/{\rm N}_1 \to \mathcal{K}_2$ a crossed homomorphism defined by
$$(Dp_1)_\star({\rm N}_1(b+B)) = (Db'+\overline I)_\star$$
and $\Gamma_1/{\rm N}_1$ acting on $\mathcal{K}_2$ by ${\rm N}_1(b+B)(v+\overline I)_\star = (\overline C\overline B\overline C^{-1}v+\overline I)_\star$
for each $b+B \in \Gamma_1$ and $v \in \mathrm{Span}(Z({\rm N}_2))$.
\item We have that
$$\alpha_\sharp^{-1}\Xi_2{\rm P}_2^{-1} \beta_\ast{\rm P}_1 = (\overline C^{-1}Dp_1)_\star\Xi_1$$
with $(\overline C^{-1}Dp_1)_\star: \Gamma_1/{\rm N}_1 \to \mathcal{K}_1$ a crossed homomorphism
defined by
$$(\overline C^{-1}Dp_1)_\star({\rm N}_1(b+B)) = (\overline C^{-1}D(b')+\overline I)_\star$$
and $\Gamma_1/{\rm N}_1$ acting on $\mathcal{K}_1$ by ${\rm N}_1(b+B)(v + \overline I)_\star = (\overline B v+\overline I)_\star$ for each $b+B \in \Gamma_1$
and $v \in \mathrm{Span}(Z({\rm N}_1))$.
\end{enumerate}
\end{theorem}
\begin{proof}
Suppose there exists an affinity $\phi = c+ C$ of $E^n$ such that $\phi(\Gamma_1,{\rm N}_1)\phi^{-1} = (\Gamma_2,{\rm N}_2)$,
with $\overline\phi = \alpha$, and $\phi' = \beta$, and $\overline{C'} = D$.
Let $\gamma = b+ B \in \Gamma_1$.
Then we have
$$\phi\gamma\phi^{-1} = (c+C)(b+B)(c+C)^{-1} = Cb + (I-CBC^{-1})c+CBC^{-1}.$$
Hence we have
\begin{eqnarray*}
\overline{\phi\gamma\phi^{-1}} & = & \overline{C}\overline b + \overline{C'}b'+ (\overline I-\overline C\overline B{\overline C}^{-1})\overline c+\overline C\overline B{\overline C}^{-1} \\
& = & ( \overline{C'}b'+\overline I)(\overline c +{\overline C})(\overline b+\overline B)(\overline c + {\overline C})^{-1} \\
& = & (Db'+\overline I)\overline \phi\overline \gamma\overline \phi^{-1} \ \
= \ \ ( Db'+\overline I)\alpha\overline \gamma\alpha^{-1},
\end{eqnarray*}
and
\begin{eqnarray*}
(\phi\gamma\phi^{-1})' & = & C'b'+ (I'-C'B'(C')^{-1})c' + C'B'(C')^{-1} \\
& = & (c'+C')(b'+B')(c'+C')^{-1} \\
& = & \phi' \gamma' (\phi')^{-1} \ \ = \ \ \beta \gamma' \beta^{-1}.
\end{eqnarray*}
Observe that
\begin{eqnarray*} \Xi_2{\rm P}_2^{-1}\beta_\ast{\rm P}_1({\rm N}_1\gamma)
& = & \Xi_2{\rm P}_2^{-1}\beta_\ast(\gamma') \\
& = & \Xi_2{\rm P}_2^{-1}(\beta\gamma'\beta^{-1}) \\
& = & \Xi_2{\rm P}_2^{-1}(\phi\gamma\phi^{-1})' \\
& = & \Xi_2({\rm N}_2\phi\gamma\phi^{-1}) \\
& = & (\overline{\phi\gamma\phi^{-1}})_\star \\
& = & ((Db'+\overline I)\alpha\overline \gamma\alpha^{-1})_\star \\
& = & (Db'+\overline I)_\star\alpha_\star\overline \gamma_\star \alpha_\star^{-1} \\
& = & (Dp_1({\rm N}_1(b+B)))_\star \alpha_\sharp(\overline \gamma_\star) \ \,
= \ \, (Dp_1({\rm N}_1\gamma))_\star \alpha_\sharp(\Xi_1({\rm N}_1\gamma)).
\end{eqnarray*}
Hence we have that $\Xi_2{\rm P}_2^{-1}\beta_\ast{\rm P}_1 = (Dp_1)_\star\alpha_\sharp\Xi_1$.
Conversely, suppose that $\Xi_2{\rm P}_2^{-1}\beta_\ast{\rm P}_1 = (Dp_1)_\star\alpha_\sharp\Xi_1$.
Define an affine transformation $\phi: E^n \to E^n$ by
$$\phi(x) = \alpha(\overline x) + D(x')+ \beta(x')$$
for each $x\in E^n$, where $x = \overline x + x'$ with $\overline x \in V_1$ and $x'\in V_1^\perp$.
Then $\phi$ is an affinity of $E^n$, with
$$\phi^{-1}(y) = \alpha^{-1}(\overline y) -\overline C^{-1}D\beta^{-1}(y') + \beta^{-1}(y')$$
for each $y\in E^n$ where $y = \overline y + y'$ with $\overline y \in V_2$ and $y'\in V_2^\perp$.
Write $\beta = c' + C'$ with $c' \in V_2^\perp$ and $C': V_1^\perp \to V_2^\perp$ a linear isomorphism.
Write $\phi = c+ C$ with $c \in E^n$ and $C$ a linear isomorphism of $E^n$.
Then $c =\overline c + c'$ and $Cx = \overline C\overline x + Dx' + C'x'$ for each $x\in E^n$ with $\overline x \in V_1$ and $x'\in V_1^\perp$.
We have that $\overline \phi = \alpha$, and $\phi' = \beta$ and $\overline{C'} = D$.
We also have $\phi^{-1} = -C^{-1}c+C^{-1}$ and
$$C^{-1}y = \overline C^{-1}\overline y - \overline C^{-1}D(C')^{-1}y' + (C')^{-1}y'$$
for all $y \in E^n$ with $\overline y \in V_2$ and $y' \in V_2^\perp$.
Let $\gamma \in \Gamma_1$. Write $\gamma = b+ B$ with $b\in E^n$ and $B\in \mathrm{O}(n)$.
Then we have
$$ \phi\gamma\phi^{-1} = (c+C)(b+B)(c+C)^{-1} = Cb+(I-CBC^{-1})c + CBC^{-1}.$$
Let $y \in E^n$. Write $y = \overline y + y'$ with $\overline y \in V_2$ and $y' \in V_2^\perp$.
Then we have that
\begin{eqnarray*} CBC^{-1}
& = & CB(\overline C^{-1}\overline y - \overline C^{-1}D(C')^{-1}y' + (C')^{-1}y' \\
& = & C(\overline B\overline C^{-1}\overline y -\overline B\overline C^{-1}D(C')^{-1}y' +B'(C')^{-1}y' \\
& = & \overline C\overline B\overline C^{-1}\overline y -\overline C\overline B\overline C^{-1}D(C')^{-1}y' +DB'(C')^{-1}y' + C'B'(C')^{-1}y' \\
& = & \overline C\overline B\overline C^{-1}\overline y + C'B'(C')^{-1}y'.
\end{eqnarray*}
Hence $CBC^{-1} = \overline C\overline B\overline C^{-1} \times C'B'(C')^{-1}$ as a linear isomorphism
of $E^n = V_2\times V_2^\perp$.
Moreover, we have
\begin{eqnarray*}
\overline{\phi\gamma\phi^{-1}} & = & \overline{C}\overline b + \overline{C'}b'+ (\overline I-\overline C\overline B{\overline C}^{-1})\overline c+\overline C\overline B{\overline C}^{-1} \\
& = & ( \overline{C'}b'+\overline I)(\overline c +{\overline C})(\overline b+\overline B)(\overline c + {\overline C})^{-1} \\
& = & (Db'+\overline I)\overline \phi\overline \gamma\overline \phi^{-1} \ \
= \ \ ( Db'+\overline I)\alpha\overline \gamma\alpha^{-1},
\end{eqnarray*}
and
\begin{eqnarray*}
(\phi\gamma\phi^{-1})' & = & C'b'+ (I'-C'B'(C')^{-1})c' + C'B'(C')^{-1} \\
& = & (c'+C')(b'+B')(c'+C')^{-1} \\
& = & \phi' \gamma' (\phi')^{-1} \ \ = \ \ \beta \gamma' \beta^{-1}.
\end{eqnarray*}
As $\Xi_2{\rm P}_2^{-1}\beta_\ast{\rm P}_1 = (Dp_1)_\star\alpha_\sharp\Xi_1$,
we have that $(\alpha\overline\gamma\alpha^{-1})_\star$ is an isometry of $V_2/{\rm N}_2$.
By Lemmas 1 and 7 of \cite{R-T-Isom}, we have that $\alpha\overline\gamma\alpha^{-1}$ is an isometry of $V_2$.
Hence $\overline C \overline B \overline C^{-1}$ is an orthogonal transformation of $V_2$.
As $\beta\Gamma_1'\beta^{-1} = \Gamma_2'$, we have that $C'B'(C')^{-1}$ is an orthogonal transformation of $V_2^\perp$.
Hence
$CBC^{-1} = \overline C\overline B\overline C^{-1} \times C'B'(C')^{-1}$
is an orthogonal transformation of $E^n = V_2 \times V_2^\perp$.
Therefore $\phi\gamma\phi^{-1}$ is an isometry of $E^n$ for each $\gamma \in \Gamma_1$.
As $\Gamma_1$ acts discontinuously on $E^n$ and $\phi$ is a homeomorphism of $E^n$,
we have that $\phi\Gamma_1\phi^{-1}$ acts discontinuously on $E^n$.
Therefore $\phi\Gamma_1\phi^{-1}$ is a discrete subgroup of $\mathrm{Isom}(E^n)$ by Theorem 5.3.5 of \cite{R}.
Now $\phi$ induces a homeomorphism
$\phi_\star: E^n/\Gamma_1 \to E^n/\phi\Gamma_1\phi^{-1}$
defined by $\phi_\star(\Gamma_1 x) = \phi\Gamma_1\phi^{-1}\phi(x)$.
Hence $E^n/\phi\Gamma_1\phi^{-1}$ is compact.
Therefore $\phi\Gamma_1\phi^{-1}$ is a $n$-space group.
Now $\phi_\ast: \Gamma_1 \to \phi\Gamma_1\phi^{-1}$, defined by $\phi_\ast(\gamma) = \phi\gamma\phi^{-1}$,
is an isomorphism that maps the normal subgroup ${\rm N}_1$ to the normal subgroup
$\phi{\rm N}_1\phi^{-1}$ of $\phi\Gamma_1\phi^{-1}$,
and $\phi\Gamma_1\phi^{-1}/\phi{\rm N}_1\phi^{-1}$ is isomorphic to $\Gamma_1/{\rm N}_1$.
Hence $\phi\Gamma_1\phi^{-1}/\phi{\rm N}_1\phi^{-1}$ is a space group.
Therefore $\phi{\rm N}_1\phi^{-1}$ is a complete normal subgroup of $\phi\Gamma_1\phi^{-1}$ by Theorem 5 of \cite{R-T}.
Now suppose $\nu = a+A \in {\rm N}_1$. Then $a' = 0$ and $A' = I'$, and so $\nu' = I'$.
Hence $\overline{\phi\nu\phi^{-1}} = \alpha\overline \nu\alpha^{-1}$ and $(\phi\nu\phi^{-1})' = I'$.
As $\alpha\overline{\rm N}_1\alpha^{-1} = \overline{\rm N}_2$, we have that $\phi{\rm N}_1\phi^{-1} = {\rm N}_2$.
Moreover, as $\beta\Gamma_1'\beta^{-1}=\Gamma_2'$, we have that $(\phi\Gamma_1\phi^{-1})' = \Gamma_2'$.
Let $\Xi: \phi\Gamma_1\phi^{-1}/{\rm N}_2 \to \mathrm{Isom}(V_2/{\rm N}_2)$ be the homomorphism induced by the action
of $\phi\Gamma_1\phi^{-1}/{\rm N}_2$ on $V_2/{\rm N}_2$. Let $\gamma =b+B \in \Gamma_1$,
and let ${\rm P}:\phi\Gamma_1\phi^{-1}/{\rm N}_2 \to \Gamma_2'$
be the isomorphism defined by ${\rm P}({\rm N}_2\phi\gamma\phi^{-1}) = (\phi\gamma\phi^{-1})'$.
Then we have that
\begin{eqnarray*} \Xi{\rm P}^{-1}(\beta\gamma'\beta^{-1})
& = & \Xi{\rm P}^{-1}((\phi\gamma\phi^{-1})') \\
& = & \Xi({\rm N}_2\phi\gamma\phi^{-1}) \\
& = & (\overline{\phi\gamma\phi^{-1}})_\star \\
& = & ((Db'+\overline{I})\alpha\overline{\gamma}{\alpha}^{-1})_\star \\
& = & (Db'+\overline{I})_\star\alpha_\star\overline{\gamma}_\star{\alpha}^{-1}_\star \\
& = & (Dp_1({\rm N}_1(b+B)))_\star\alpha_\sharp(\overline\gamma_\star) \\
& = & (Dp_1({\rm N}_1\gamma))_\star\alpha_\sharp(\Xi_1({\rm N}_1\gamma)) \\
& = & \Xi_2{\rm P}_2^{-1}\beta_\ast{\rm P}_1({\rm N}_1\gamma) \ \
= \ \ \Xi_2{\rm P}_2^{-1}(\beta\gamma'\beta^{-1}).
\end{eqnarray*}
Hence we have that $\Xi{\rm P}^{-1} = \Xi_2{\rm P}_2^{-1}$.
Therefore $\phi\Gamma_1\phi^{-1} = \Gamma_2$ by Theorem 2.
Thus $\phi(\Gamma_1,{\rm N}_1)\phi^{-1} = (\Gamma_2,{\rm N}_2)$.
Let $\gamma = b+B$ and $\gamma_1 = b_1+B_1$ be elements of $\Gamma_1$.
Then we have that
\begin{eqnarray*} (Dp_1)_\star({\rm N}_1\gamma{\rm N}_1\gamma_1)
& = &(Dp_1)_\star({\rm N}_1(b+Bb_1+BB_1)) \\
& = & (D(b+Bb_1)'+\overline I)_\star \\
& = &(D(b'+B'b_1')+\overline I)_\star \\
& = &(Db'+DB'b_1'+\overline I)_\star\\
& = & (Db'+\overline C \overline B \overline C^{-1}Db_1'+\overline I)_\star \\
& = &(Db'+\overline I)_\star (\overline C \overline B \overline C^{-1}Db_1'+\overline I)_\star \\
& = & (Db'+\overline I)_\star ({\rm N}_1(b+B))(Db_1'+\overline I)_\star\\
& = & (Dp_1)_\star({\rm N}_1\gamma) (({\rm N}_1\gamma)(Dp_1)_\star({\rm N}_1\gamma_1)).
\end{eqnarray*}
Therefore $(Dp_1)_\star : \Gamma_1/{\rm N}_1 \to \mathcal{K}_2$ is a crossed homomorphism.
Thus statements (1) and (2) are equivalent.
The equation $\Xi_2{\rm P}_2^{-1}\beta_\ast{\rm P}_1 = (Dp_1)_\star\alpha_\sharp\Xi_1$
is equivalent to the equation
$$\alpha_\sharp^{-1} \Xi_2{\rm P}_2^{-1}\beta_\ast{\rm P}_1 = \alpha_\sharp^{-1} (Dp_1)_\star\alpha_\sharp\Xi_1.$$
Observe that
\begin{eqnarray*} \alpha_\sharp^{-1} (Dp_1)_\star\alpha_\sharp\Xi_1({\rm N}_1\gamma)
& = & \alpha_\sharp^{-1}( (Dp_1)_\star ({\rm N}_1\gamma) \alpha_\sharp \Xi_1({\rm N}_1\gamma)) \\
& = & \alpha_\sharp^{-1}( (Db'+ \overline I)_\star \alpha_\sharp \overline \gamma_\star) \\
& = & \alpha_\sharp^{-1}( (Db'+ \overline I)_\star \alpha_\star \overline \gamma_\star \alpha_\star^{-1}) \\
& = & \alpha_\star^{-1}(Db'+ \overline I)_\star \alpha_\star \overline \gamma_\star \alpha_\star^{-1} \alpha_\star \\
& = & (\alpha^{-1}(Db' + \overline I)\alpha)_\star\overline\gamma_\star \\
& = & (\overline C^{-1}Db'+\overline I)_\star \Xi_1({\rm N}_1\gamma) \\
& = & (\overline C^{-1}Dp_1)_\star({\rm N}_1\gamma)\Xi_1({\rm N}_1\gamma).
\end{eqnarray*}
Hence we have that $\alpha_\sharp^{-1} (Dp_1)_\star\alpha_\sharp\Xi_1 = (\overline C^{-1}Dp_1)_\star\Xi_1$.
By the same argument as with $(Dp_1)_\star : \Gamma_1/{\rm N}_1 \to \mathcal{K}_2$,
we have that $(\overline C^{-1}Dp_1)_\star : \Gamma_1/{\rm N}_1 \to \mathcal{K}_1$ is a crossed homomorphism.
Thus (2) and (3) are equivalent.
\end{proof}
\section{Outer Automorphism Groups of Space Groups}
Through this section, let $m$ be a positive integer less than $n$.
Let ${\rm M}$ be an $m$-space group and
let $\Delta$ be an $(n-m)$-space group.
\vspace{.15in}
\noindent{\bf Definition:} Define $\mathrm{Iso}(\Delta,{\rm M})$ to be the set of isomorphism classes
of pairs $(\Gamma, {\rm N})$
where ${\rm N}$ is a complete normal subgroup of an $n$-space group $\Gamma$
such that ${\rm N}$ is isomorphic to ${\rm M}$ and $\Gamma/{\rm N}$ is isomorphic to $\Delta$.
We denote the isomorphism class of a pair $(\Gamma,{\rm N})$ by $[\Gamma,{\rm N}]$.
\vspace{.15in}
Let ${\rm N}$ be a complete normal subgroup of an $n$-space group $\Gamma$,
and let $\mathrm{Out}_E({\rm N})$ be the Euclidean outer automorphism group of ${\rm N}$
defined in \S 4 of \cite{R-T-Isom}.
The group $\mathrm{Out}_E({\rm N})$ is finite by Theorem 2 of \cite{R-T-Isom}.
The action of $\Gamma$ on ${\rm N}$ by conjugation induces a homomorphism
$$\mathcal{O}: \Gamma/{\rm N} \to \mathrm{Out}_E({\rm N})$$
defined by $\mathcal{O}({\rm N}\gamma) = \gamma_\ast\mathrm{Inn}({\rm N})$ where $\gamma_\ast(\nu) = \gamma\nu\gamma^{-1}$
for each $\gamma \in \Gamma$ and $\nu\in{\rm N}$.
Let $\alpha: {\rm N}_1 \to {\rm N}_2$ be an isomorphism. Then $\alpha$ induces an isomorphism
$$\alpha_\#: \mathrm{Out}({\rm N}_1) \to \mathrm{Out}({\rm N}_2)$$
defined by $\alpha_\#(\zeta\mathrm{Inn}({\rm N}_1) )= \alpha\zeta\alpha^{-1}\mathrm{Inn}({\rm N}_2)$ for each $\zeta\in\mathrm{Aut}({\rm N}_1)$.
\begin{lemma}
Let ${\rm N}_i$ be a complete normal subgroup of an $n$-space group $\Gamma_i$ for $i = 1,2$.
Let $\mathcal{O}_i:\Gamma_i/{\rm N}_i \to \mathrm{Out}_E({\rm N}_i)$ be the homomorphism
induced by the action of $\Gamma_i$ on ${\rm N}_i$ by conjugation for $i = 1, 2$, and
let $\alpha:{\rm N}_1\to {\rm N}_2$ and $\phi: \Gamma_1\to \Gamma_2$ and $\beta:\Gamma_1/{\rm N}_1 \to \Gamma_2/{\rm N}_2$ be isomorphisms
such that the following diagram commutes
\[\begin{array}{ccccccccc}
1 & \to & {\rm N}_1 & \rightarrow & \Gamma_1 & \rightarrow & \Gamma_1/{\rm N}_1 & \to & 1 \\
& & \hspace{.12in} \downarrow \, \alpha & & \hspace{.12in}\downarrow\, \phi & & \hspace{.12in} \downarrow\, \beta & \\
1 & \to & {\rm N}_2 & \rightarrow & \Gamma_2 & \rightarrow & \Gamma_2/{\rm N}_2 & \to & 1,
\end{array}\]
where the horizontal maps are inclusions and projections, then $\mathcal{O}_2 = \alpha_\#\mathcal{O}_1\beta^{-1}$.
\end{lemma}
\begin{proof}
Let $\gamma\in \Gamma_1$. Then we have that
$$\mathcal{O}_2({\rm N}_2\phi(\gamma)) = \phi(\gamma)_\ast\mathrm{Inn}({\rm N}_2),$$
whereas
\begin{eqnarray*}
\alpha_\#\mathcal{O}_1\beta^{-1}({\rm N}_2\phi(\gamma)) & = & \alpha_\#\mathcal{O}_1({\rm N}_1\gamma) \\
& = & \alpha_\#(\gamma_\ast\mathrm{Inn}({\rm N}_1)) \ \ = \ \ \alpha\gamma_\ast\alpha^{-1}\mathrm{Inn}({\rm N}_2).
\end{eqnarray*}
If $\nu \in {\rm N}$, then
\begin{eqnarray*}
\alpha\gamma_\ast\alpha^{-1}(\nu) & = & \alpha\gamma_\ast\alpha^{-1}(\nu) \\
& = & \alpha(\gamma\alpha^{-1}(\nu)\gamma^{-1}) \\
& = & \phi(\gamma\phi^{-1}(\nu)\gamma^{-1}) \\
& = & \phi(\gamma)\nu\phi(\gamma)^{-1} \ \ = \ \ \phi(\gamma)_\ast(\nu).
\end{eqnarray*}
Hence $\alpha\gamma_\ast\alpha^{-1} = \phi(\gamma)_\ast$.
Therefore $\mathcal{O}_2 = \alpha_\#\mathcal{O}_1\beta^{-1}$.
\end{proof}
\noindent{\bf Definition:}
Define $\mathrm{Hom}_f(\Delta,\mathrm{Out}({\rm M}))$ to be the set of all homomorphisms from $\Delta$ to $\mathrm{Out}({\rm M})$
that have finite image.
\vspace{.15in}
The group $\mathrm{Out}({\rm M})$ acts on the left of $\mathrm{Hom}_f(\Delta,\mathrm{Out}({\rm M}))$ by conjugation,
that is, if $g\in \mathrm{Out}({\rm M})$ and $\eta\in \mathrm{Hom}_f(\Delta,\mathrm{Out}({\rm M}))$,
then $g \eta = g_\ast\eta$ where $g_\ast: \mathrm{Out}({\rm M}) \to \mathrm{Out}({\rm M})$ is defined by $g_\ast(h) = ghg^{-1}$.
Let $\mathrm{Out}({\rm M})\backslash\mathrm{Hom}_f(\Delta,\mathrm{Out}({\rm M}))$ be the set of $\mathrm{Out}({\rm M})$-orbits.
The group $\mathrm{Aut}(\Delta)$ acts on the right of $\mathrm{Hom}_f(\Delta,\mathrm{Out}({\rm M}))$
by composition of homomorphisms.
If $\beta\in \mathrm{Aut}(\Delta)$ and $\eta\in \mathrm{Hom}_f(\Delta,\mathrm{Out}({\rm M}))$ and $g\in \mathrm{Out}({\rm M})$,
then
$$(g\eta)\beta = (g_\ast\eta)\beta = g_\ast(\eta\beta) = g(\eta\beta).$$
Hence $\mathrm{Aut}(\Delta)$ acts on the right of
$\mathrm{Out}({\rm M})\backslash\mathrm{Hom}_f(\Delta,\mathrm{Out}({\rm M}))$
by
$$(\mathrm{Out}({\rm M})\eta)\beta = \mathrm{Out}({\rm M})(\eta\beta).$$
Let $\delta, \epsilon \in \Delta$ and $\eta\in \mathrm{Hom}_f(\Delta,\mathrm{Out}({\rm M}))$.
Then we have that
$$\eta\delta_\ast(\epsilon) = \eta(\delta\epsilon\delta^{-1}) =\eta(\delta)\eta(\epsilon)\eta(\delta)^{-1}= \eta(\delta)_\ast\eta(\epsilon)
= (\eta(\delta)\eta)(\epsilon). $$
Hence $\eta\delta_\ast = \eta(\delta)\eta$. Therefore $\mathrm{Inn}(\Delta)$ acts trivially on
$\mathrm{Out}({\rm M})\backslash\mathrm{Hom}_f(\Delta,\mathrm{Out}({\rm M}))$.
Hence $\mathrm{Out}(\Delta)$ acts on the right of $\mathrm{Out}({\rm M})\backslash\mathrm{Hom}_f(\Delta,\mathrm{Out}({\rm M}))$
by
$$(\mathrm{Out}({\rm M})\eta)(\beta\mathrm{Inn}(\Delta)) = \mathrm{Out}({\rm M})(\eta\beta).$$
\noindent{\bf Definition:} Define the set $\mathrm{Out}(\Delta,{\rm M})$ by the formula
$$\mathrm{Out}(\Delta,{\rm M}) = (\mathrm{Out}({\rm M})\backslash\mathrm{Hom}_f(\Delta,\mathrm{Out}({\rm M})))/\mathrm{Out}(\Delta).$$
If $\eta\in \mathrm{Hom}_f(\Delta,\mathrm{Out}({\rm M}))$, let $[\eta] = (\mathrm{Out}({\rm M})\eta)\mathrm{Out}(\Delta)$
be the element of $\mathrm{Out}(\Delta,{\rm M})$ determined by $\eta$.
\vspace{.15in}
Let $(\Gamma,{\rm N})$ be a pair such that $[\Gamma,{\rm N}]\in\mathrm{Iso}(\Delta,{\rm M})$.
Let $\mathcal{O}: \Gamma/{\rm N} \to \mathrm{Out}_E({\rm N})$
be the homomorphism induced by the action of $\Gamma$ on ${\rm N}$ by conjugation.
Let $\alpha: {\rm N} \to {\rm M}$ and $\beta:\Delta \to \Gamma/{\rm N}$ be isomorphisms.
Then $\alpha_\#\mathcal{O}\beta \in \mathrm{Hom}_f(\Delta,\mathrm{Out}({\rm M}))$.
Let $\alpha':{\rm N}\to{\rm M}$ and $\beta':\Delta\to\Gamma/{\rm N}$ are isomorphisms.
Observe that
\begin{eqnarray*}
\alpha'_\#\mathcal{O}\beta' & = & \alpha'_\#\alpha_\#^{-1}\alpha_\#\mathcal{O}\beta\beta^{-1}\beta' \\
& = & (\alpha'\alpha^{-1})_\#\alpha_\#\mathcal{O}\beta(\beta^{-1}\beta') \\
& = & (\alpha'\alpha^{-1}\mathrm{Inn}({\rm M}))_\ast(\alpha_\#\mathcal{O}\beta(\beta^{-1}\beta')) \\
& = & (\alpha'\alpha^{-1}\mathrm{Inn}({\rm M}))(\alpha_\#\mathcal{O}\beta)(\beta^{-1}\beta'). \\
\end{eqnarray*}
Hence $[\alpha_\#\mathcal{O}\beta]$ in $\mathrm{Out}(\Delta,{\rm M})$ does not depend
on the choice of $\alpha$ and $\beta$,
and so $(\Gamma,{\rm N})$ determines the element $[\alpha_\#\mathcal{O}\beta]$
of $\mathrm{Out}(\Delta,{\rm M})$ independent of the choice of $\alpha$ and $\beta$.
Suppose $[\Gamma_i,{\rm N}_i]\in \mathrm{Iso}(\Delta,{\rm M})$ for $i=1,2$,
and $\phi:(\Gamma_1,{\rm N}_1) \to (\Gamma_2,{\rm N}_2)$ is an isomorphism of pairs.
Let $\alpha:{\rm N}_1\to {\rm N}_2$ be the isomorphism obtained by restricting $\phi$,
and let $\beta: \Gamma_1/{\rm N}_1\to \Gamma_2/{\rm N}_2$ be the isomorphism induced by $\phi$.
Let $\mathcal{O}_i:\Gamma_i/{\rm N}_i \to \mathrm{Out}_E({\rm N}_i)$ be the homomorphism
induced by the action of $\Gamma_i$ on ${\rm N}_i$ by conjugation for $i=1,2$.
Then $\mathcal{O}_2 = \alpha_\#\mathcal{O}_1\beta^{-1}$ by Lemma 17.
Let $\alpha_1:{\rm N}_1 \to {\rm M}$ and $\beta_1:\Delta \to \Gamma_1/{\rm N}_1$ be isomorphisms.
Let $\alpha_2 = \alpha_1\alpha^{-1}$ and $\beta_2= \beta\beta_1$.
Then we have
$$(\alpha_2)_\#\mathcal{O}_2\beta_2
= (\alpha_1\alpha^{-1})_\#\alpha_\#\mathcal{O}_1\beta^{-1}\beta_2 \\
= (\alpha_1)_\#\mathcal{O}_1\beta_1. $$
Hence $(\Gamma_1,{\rm N}_1)$ and $(\Gamma_2,{\rm N}_2)$ determine the same element of $\mathrm{Out}(\Delta,{\rm M})$.
Therefore there is a function
$$\omega:\mathrm{Iso}(\Delta,{\rm M}) \to \mathrm{Out}(\Delta,{\rm M})$$
defined by $\omega([\Gamma,{\rm N}]) = [\alpha_\#\mathcal{O}\beta]$ for any choice of isomorphisms $\alpha:{\rm N}\to{\rm M}$
and $\beta: \Delta\to \Gamma/{\rm N}$.
\begin{lemma}
The set $\mathrm{Out}(\Delta,{\rm M})$ is finite.
\end{lemma}
\begin{proof}
The group $\mathrm{Out}({\rm M})$ is arithmetic by Theorem 1.1 of \cite{B-G}.
Hence $\mathrm{Out}({\rm M})$ has only finitely many conjugacy classes of finite subgroups, cf.\,\S 5 of \cite{Borel}.
As $\Delta$ is finitely generated there are only finitely many homomorphisms from $\Delta$ to a finite group $G$.
Therefore $\mathrm{Out}({\rm M})\backslash\mathrm{Hom}_f(\Delta,\mathrm{Out}({\rm M}))$ is finite.
Hence $\mathrm{Out}(\Delta,{\rm M}) $ is finite.
\end{proof}
\section{Fiber Cohomology Classes}
Consider $\omega:\mathrm{Iso}(\Delta,{\rm M}) \to \mathrm{Out}(\Delta,{\rm M})$.
Suppose $[\Gamma_1,{\rm N}_1]$ and $[\Gamma_2,{\rm N}_2]$ are in the same fiber of $\omega$.
We want to define a class $[\Gamma_1,{\rm N}_1;\Gamma_2,{\rm N}_2;\alpha,\beta]$ in the cohomology group $H^1(\Gamma_1/{\rm N}_1, \mathcal{K}_1)$
where $\Gamma_1/{\rm N}_1$ acts on $\mathcal{K}_1$ by ${\rm N}_1(b+B)(v+\overline I)_\star = (\overline B v + \overline I)_\star$.
Let $\alpha_i: {\rm N}_i \to {\rm M}$ and $\beta_i: \Delta \to \Gamma_i/{\rm N}_i$ be isomorphisms for $i = 1,2$.
Let $\mathcal{O}_i:\Gamma_i/{\rm N}_i \to \mathrm{Out}_E({\rm N}_i)$ be the homomorphism
induced by the action of $\Gamma_i$ on ${\rm N}_i$ by conjugation for $i=1,2$.
As $\omega([\Gamma_1,{\rm N}_1]) = \omega([\Gamma_2,{\rm N}_2])$, we have that
$[(\alpha_1)_\#\mathcal{O}_1\beta_1]=[(\alpha_2)_\#\mathcal{O}_2\beta_2]$.
Then there exists $\alpha_0$ in $\mathrm{Aut}({\rm M})$ and $\beta_0$ in $\mathrm{Aut}(\Delta)$ such that
$(\alpha_1)_\#\mathcal{O}_1\beta_1 = (\alpha_0)_\#(\alpha_2)_\#\mathcal{O}_2\beta_2\beta_0$.
We have that
$$\mathcal{O}_1 = (\alpha_1^{-1}\alpha_0\alpha_2)_\#\mathcal{O}_2\beta_2\beta_0\beta_1^{-1}.$$
Let $\alpha: {\rm N}_1 \to {\rm N}_2$ be the isomorphism $\alpha_2^{-1}\alpha_0^{-1}\alpha_1$,
and let $\beta: \Gamma_1/{\rm N}_1 \to \Gamma_2/{\rm N}_2$ be the isomorphism $\beta_2\beta_0\beta_1^{-1}$.
Then $\mathcal{O}_1 = \alpha^{-1}_\#\mathcal{O}_2\beta$.
Now $\alpha$ induces an isomorphism $\overline\alpha: \overline{\rm N}_1 \to \overline{\rm N}_2$ defined by $\overline\alpha(\overline\nu) = \overline{\alpha(\nu)}$ for each $\nu$ in ${\rm N}_1$.
Let $V_i = \mathrm{Span}({\rm N}_i)$ for $i = 1, 2$,
and let $\tilde\alpha: V_1 \to V_2$ be an affinity such that $\tilde\alpha\overline{\rm N}_1\tilde\alpha^{-1} = \overline{\rm N}_2$ and $\tilde\alpha_\ast = \overline\alpha$,
that is, $\tilde\alpha \overline\nu \tilde\alpha^{-1} = \overline\alpha(\overline\nu)$ for each $\nu$ in ${\rm N}_1$.
Let $\Xi_i: \Gamma_i/{\rm N}_i \to \mathrm{Isom}(V_i/{\rm N}_i)$ be the homomorphism induced by the action of $\Gamma_i/{\rm N}_i$ on $V_i/{\rm N}_i$
for $i = 1, 2$.
Let $\Omega_i : \mathrm{Isom}(V_i/{\rm N}_i) \to \mathrm{Out}_E({\rm N}_i)$ be defined so that $\Omega_i(\zeta) = \mathrm{Inn}({\rm N}_i)\hat\zeta_\ast$
where $\hat\zeta$ is an isometry of $V_i$ that lifts $\zeta$ and $\hat\zeta_\ast$ is the automorphism of ${\rm N}_i$ defined
by $\overline{\hat\zeta_\ast(\nu)} = \hat\zeta\overline \nu \hat\zeta^{-1}$ for $i = 1, 2$.
Then we have that $\Omega_i\Xi_i = \mathcal{O}_i$ for $i = 1, 2$.
By Lemma 10 of \cite{R-T-Isom}, we have that
$$\Omega_1\Xi_1 = \alpha_\#^{-1}\Omega_2\Xi_2\beta =
\Omega_1(\tilde\alpha_\sharp^{-1}\Xi_2\beta).$$
Let $\phi: \Gamma_1/{\rm N}_1 \to \mathrm{Aff}(V_1/{\rm N}_1)$ and $\psi: \Gamma_1/{\rm N}_1 \to \mathrm{Aff}(V_1/{\rm N}_1)$
be the homomorphisms defined by $\phi = \tilde\alpha_\sharp^{-1}\Xi_2\beta$ and $\psi = \Xi_1$.
Then we have that $\phi(g)\psi(g)^{-1}$ is in $\mathcal{K}_1$ for each $g$ in $\Gamma_1/{\rm N}_1$ by Theorem 3 of \cite{R-T-Isom}.
As $\psi$ takes values in $\mathrm{Isom}(V_1/{\rm N}_1)$ and $\mathcal{K}_1$ is a subgroup of $\mathrm{Isom}(V_1/{\rm N}_1)$,
we have that $\phi$ takes values in $\mathrm{Isom}(V_1/{\rm N}_1)$.
Let $g, h$ be in $\Gamma_1/{\rm N}_1$, then we have that
\begin{eqnarray*}
\phi(gh)\psi(gh)^{-1} & = & \phi(g)\phi(h)(\psi(g)\psi(h))^{-1} \\
& = & \phi(g)\phi(h)\psi(h)^{-1}\psi(g)^{-1} \\
& = & \phi(g)\psi(g)^{-1}\psi(g)\phi(h)\psi(h)^{-1}\psi(g)^{-1} \\
& = & (\phi(g)\psi(g)^{-1})\psi(g)_\ast(\phi(h)\psi(h)^{-1})
\end{eqnarray*}
with
$$\psi({\rm N}_1(b+B))_\ast(v+\overline I) = (\overline b+ \overline B)_\star (v+ \overline I)_\star (\overline b + \overline B)_\star^{-1} = (\overline Bv + \overline I)_\star.$$
Hence the function $\phi\psi^{-1}: \Gamma_1/{\rm N}_1 \to \mathcal{K}_1$ is a crossed homomorphism,
and so determines a class $[\Gamma_1,{\rm N}_1; \Gamma_2,{\rm N}_2;\alpha,\beta]$ in $H^1(\Gamma_1/{\rm N}_1, \mathcal{K}_1)$,
cf.\,p.\,105 of \cite{M}.
Let $[\Gamma,{\rm N}]$ be a class in $\mathrm{Isom}(\Delta, {\rm M})$,
and let $V = \mathrm{Span}({\rm N})$.
Let $\mathcal{C}$ be the centralizer of $\overline{\rm N}$ in $\mathrm{Aff}(V)$.
By Lemmas 6 and 8 of \cite{R-T-Isom}, we have that
$$\mathcal{C}=\{v+\overline I: v \in \mathrm{Span}(Z({\rm N}))\}.$$
The group $\Gamma/{\rm N}$ acts on $\mathcal{C}$ by $({\rm N}(b+B))(v+\overline I) = \overline Bv + \overline I$
and $\Gamma/{\rm N}$ acts on $Z({\rm N})$ by $({\rm N}(b+B))(u+ I) = Bu + I$.
We have a short exact sequence of $(\Gamma/{\rm N})$-modules
$$0 \to Z({\rm N})\ {\buildrel \iota \over \longrightarrow}\ \mathcal{C}\ {\buildrel \kappa \over \longrightarrow}\ \mathcal{K} \to 0$$
where $\iota(u+I ) = u+\overline I$ and $\kappa(v+ \overline I ) = (v+\overline I)_\star$.
Let ${\rm T}$ be the group of translations of $\Gamma$.
Then ${\rm T}{\rm N}/{\rm N}$ is a normal subgroup of $\Gamma/{\rm N}$ of finite index
and ${\rm T}{\rm N}/{\rm N}$ is a subgroup of the group of translations of $\Gamma/{\rm N}$ of finite index by Theorem 16 of \cite{R-T}.
The group $\Gamma/{\rm N}$ acts on the abelian group ${\rm T}{\rm N}/{\rm N}$ by $({\rm N}(b+B))(a+ I) = {\rm N}(Ba + I)$,
and so ${\rm T}{\rm N}/{\rm N}$ is a $(\Gamma/{\rm N})$-module.
Moreover the group ${\rm T}{\rm N}/{\rm N}$ acts trivially on $\mathcal{C}$.
\begin{lemma}
Let $f: \Gamma/{\rm N} \to \mathcal{C}$ be a crossed homomorphism,
and let $f_{res}: {\rm T}{\rm N}/{\rm N} \to \mathcal{C}$ be the restriction of $f$.
Then $f_{res}$ is a homomorphism of $(\Gamma/{\rm N})$-modules
and the class of $f$ in $H^1(\Gamma/{\rm N}, \mathcal{C})$ is completely determined by $f_{res}$.
\end{lemma}
\begin{proof}
According to \cite{M}, p.\,354, we have an exact sequence of homomorphisms
$$H^1(\Gamma/{\rm T}{\rm N},\mathcal{C})\ {\buildrel inf \over \longrightarrow}\ H^1(\Gamma/{\rm N},\mathcal{C})
\ {\buildrel res \over \longrightarrow}\ H^1({\rm T}{\rm N}/{\rm N},\mathcal{C})^{\Gamma/{\rm N}} \to H^2(\Gamma/{\rm T}{\rm N},\mathcal{C}).$$
The group $\Gamma/{\rm T}{\rm N}$ is finite and $\mathcal{C}$ is a torsion-free, divisible, abelian group,
and so $H^i(\Gamma/{\rm T}{\rm N},\mathcal{C})= 0$ for $i =1, 2$ by Corollary IV.5.4 of \cite{M}.
Hence
$$res: H^1(\Gamma/{\rm N},\mathcal{C}) \to H^1({\rm T}{\rm N}/{\rm N},\mathcal{C})^{\Gamma/{\rm N}}$$
is an isomorphism.
Here $res([f]) = [f_{res}]$.
By the Universal Coefficients Theorem (p.\,77 of \cite{M}),
we have that
$$H^1({\rm T}{\rm N}/{\rm N},\mathcal{C})^{\Gamma/{\rm N}} = \mathrm{Hom}({\rm T}{\rm N}/{\rm N},\mathcal{C})^{\Gamma/{\rm N}}.$$
Here $\Gamma/{\rm N}$ acts on $\mathrm{Hom}({\rm T}{\rm N}/{\rm N},\mathcal{C})$ by $(({\rm N}\gamma)h)(x) = ({\rm N}\gamma) h({\rm N}\gamma^{-1}x)$ for each $\gamma\in\Gamma$,
homomorphism $h: {\rm T}{\rm N}/{\rm N} \to \mathcal{C}$, and element $x \in {\rm T}{\rm N}/{\rm N}$.
Therefore, we have that
$$\mathrm{Hom}({\rm T}{\rm N}/{\rm N},\mathcal{C})^{\Gamma/{\rm N}} = \mathrm{Hom}_{\Gamma/{\rm N}}({\rm T}{\rm N}/{\rm N},\mathcal{C}).$$
Hence $[f_{res}] =\{f_{res}\}$ and $f_{res}$ is a homomorphism of $(\Gamma/{\rm N})$-modules.
Therefore the class of $f$ in $H^1(\Gamma/{\rm N}, \mathcal{C})$ is completely determined by $f_{res}$.
\end{proof}
\begin{lemma}
Suppose that $\omega([\Gamma_1,{\rm N}_1]) = \omega([\Gamma_2,{\rm N}_2])$ with $\mathcal{O}_1 = \alpha_\#^{-1}\mathcal{O}_2\beta$.
If the class $[\Gamma_1,{\rm N}_1; \Gamma_2,{\rm N}_2;\alpha,\beta]$ is in the image of $\kappa_\ast: H^1(\Gamma_1/{\rm N}_1,\mathcal{C}_1) \to H^1(\Gamma_1/{\rm N}_1,\mathcal{K}_1)$,
then $[\Gamma_1,{\rm N}_1] = [\Gamma_2,{\rm N}_2]$.
\end{lemma}
\begin{proof}
Suppose that $[\Gamma_1,{\rm N}_1; \Gamma_2,{\rm N}_2;\alpha,\beta]$ is in the image of
$\kappa_\ast: H^1(\Gamma_1/{\rm N}_1,\mathcal{C}_1) \to H^1(\Gamma_1/{\rm N}_1,\mathcal{K}_1)$.
Then there is a crossed homomorphism $f: \Gamma_1/{\rm N}_1 \to \mathcal{C}_1$ such that
$$\kappa_\ast([f]) = [\Gamma_1,{\rm N}_1; \Gamma_2,{\rm N}_2;\alpha,\beta].$$
By Lemma 5, the cohomology class $[f]$ is completely determined by the restriction $(\Gamma_1/{\rm N}_1)$-module homomorphism
$f_{res}: {\rm T}_1{\rm N}_1/{\rm N}_1 \to C_1$.
To simplify notation, replace $\Gamma_1/{\rm N}_1$ with $\Gamma_1'$.
Then ${\rm T}_1{\rm N}_1/{\rm N}_1$ corresponds to $T_1' = \{b'+I': b+I \in T_1\}$.
Moreover $\Gamma_1/{\rm N}_1$ acts on $T_1'$ by $({\rm N}_1(b+B))(b'+I') = B'b'+I'$
and $f_{res}$ corresponds to a homomorphism of $(\Gamma_1/{\rm N}_1)$-modules $f_{res}': T_1' \to \mathcal{C}_1$.
The group ${\rm T}_1{\rm N}_1/{\rm N}_1$ has finite index in the group of translations of $\Gamma_1/{\rm N}_1$,
and so $T_1'$ has finite index in the group of translations of $\Gamma_1'$.
Hence $\{b': b+I \in T_1\}$ contains a basis of the vector space $V_1^\perp$.
Therefore $f_{res}': T_1' \to C_1$ induces a linear transformation $L: V_1^\perp \to \mathrm{Span}(Z({\rm N}_1))$
such that $f_{res}'(b'+I') = L(b')+\overline I$ for each $b+I$ in $T_1$ and if $b+ B$ is in $\Gamma_1$, then $LB' = \overline BL$.
Consider the function $h: \Gamma_1/{\rm N}_1 \to \mathcal{C}_1$ defined by $h({\rm N}_1(b+ B)) =L(b')+\overline I$.
Then $h$ is a crossed homomorphism.
If $b+I$ is in $T_1$, then
$$h_{res}({\rm N}_1(b+I)) = L(b')+\overline I = f_{res}'(b'+I') = f_{res}({\rm N}_1(b+I)).$$
Hence $h_{res} = f_{res}$.
Therefore $[h] = [f]$ in $H^1(\Gamma_1/{\rm N}_1, \mathcal{C}_1)$ by Lemma 5.
Let $\tilde\alpha: V_1 \to V_2$ be the affinity defined above and write $\tilde\alpha = \overline c + \overline C$
with $\overline c \in V_2$ and $\overline C: V_1 \to V_2$ a linear isomorphism.
Define a linear transformation $D: V_1^\perp \to \mathrm{Span}({\rm N}_2)$ by $D = \overline C L$.
If $b+B \in \Gamma_1$, then $DB' = \overline C\overline B\overline C^{-1} D$.
Let $p_1: \Gamma_1/{\rm N}_1 \to V_1^\perp$ be the crossed homomorphism defined by $p_1({\rm N}_1(b+B)) = b'$.
Then $h({\rm N}_1\gamma) = \overline C^{-1}D p_1({\rm N}_1\gamma)+ \overline I$.
Observe that $\kappa_\ast([h]) = [h_\star]$ where $h_\star$ is defined
by $h_\star({\rm N}_1\gamma) = (h({\rm N}_1\gamma))_\star$.
Thus $h_\star = (\overline C^{-1}D p_1)_\star$ as defined in Theorem 3(3).
Let $v$ be in $\mathrm{Span}(Z({\rm N}_1))$, and let $f_v: \Gamma_1/{\rm N}_1 \to \mathcal{K}_1$ be the principal crossed homomorphism
determined by $(v+ I)_\star$.
Then we have that
\begin{eqnarray*}
f_v({\rm N}_1(b+B)) & = & ({\rm N}_1(b+B))(v+\overline I)_\star (v+\overline I)_\star^{-1} \\
& = & (\overline Bv + \overline I)_\star(-v+\overline I)_\star\ \ =\ \ (\overline Bv - v +\overline I)_\star.
\end{eqnarray*}
Now we have that $[h_\star] = [\Gamma_1,{\rm N}_1; \Gamma_2,{\rm N}_2;\alpha,\beta]$ in $H^1(\Gamma_1/{\rm N}_1,\mathcal{K}_1)$.
Hence there exists $v$ in $\mathrm{Span}(Z({\rm N}_1))$ such that
$$(\tilde\alpha_\sharp^{-1}\Xi_2\beta)\Xi_1^{-1}f_v = (\overline C^{-1}D p_1)_\star.$$
Let $\tilde \alpha_v: V_1 \to V_2$ be the affinity defined by $\tilde \alpha_v = \tilde\alpha(v+\overline I)$.
Then $\tilde\alpha_v\overline N_1\tilde\alpha_v^{-1}= \overline N_2$ and $(\tilde\alpha_v)_\ast = \tilde\alpha_\ast$ by Lemma 6 of \cite{R-T-Isom},
and so $(\tilde\alpha_v)_\ast = \overline \alpha$.
Observe that
\begin{eqnarray*}
\lefteqn{\big((\tilde\alpha_\sharp^{-1}\Xi_2\beta)\Xi_1^{-1}f_v\big)({\rm N}_1(b+B))} \\
& = & (\tilde\alpha_\sharp^{-1}\Xi_2\beta)({\rm N}_1(b+B))\Xi_1^{-1}({\rm N}_1(b+B)) f_v({\rm N}_1(b+B)) \\
& = & \tilde\alpha_\star^{-1}(\Xi_2\beta)({\rm N}_1(b+B))\tilde\alpha_\star (\overline b + \overline B)_\star^{-1}(\overline Bv - v +\overline I)_\star \\
& = & \tilde\alpha_\star^{-1}(\Xi_2\beta)({\rm N}_1(b+B))\tilde\alpha_\star (\overline b + \overline B)_\star^{-1}(v -\overline Bv +\overline I)_\star^{-1} \\
& = & \tilde\alpha_\star^{-1}(\Xi_2\beta)({\rm N}_1(b+B))\tilde\alpha_\star (v -\overline Bv +\overline b + \overline B)_\star^{-1} \\
& = & \tilde\alpha_\star^{-1}(\Xi_2\beta)({\rm N}_1(b+B))\tilde\alpha_\star ((v +\overline I)(\overline b + \overline B)(-v+I))_\star^{-1} \\
& = & \tilde\alpha_\star^{-1}(\Xi_2\beta)({\rm N}_1(b+B))\tilde\alpha_\star (v +\overline I)_\star (\overline b + \overline B)_\star^{-1}(-v+I)_\star \\
& = & (-v+I)_\star\tilde\alpha_\star^{-1}(\Xi_2\beta)({\rm N}_1(b+B))\tilde\alpha_\star (v +\overline I)_\star (\overline b + \overline B)_\star^{-1} \\
& = & (\tilde\alpha_v)_\star^{-1}(\Xi_2\beta)({\rm N}_1(b+B))(\tilde\alpha_v)_\star (\overline b + \overline B)_\star^{-1} \\
& = & \big(((\tilde\alpha_v)_\sharp^{-1}\Xi_2\beta)\Xi_1^{-1}\big)({\rm N}_1(b+B))
\end{eqnarray*}
Hence we have
$$(\tilde\alpha_\sharp^{-1}\Xi_2\beta)\Xi_1^{-1}f_v = ((\tilde\alpha_v)_\sharp^{-1}\Xi_2\beta)\Xi_1^{-1}.$$
Thus we have that
$$(\tilde\alpha_v)_\sharp^{-1}\Xi_2\beta = (\overline C^{-1}D p_1)_\star\Xi_1.$$
Let ${\rm P}_i:\Gamma_i/{\rm N}_i \to \Gamma_i'$ be the isomorphism defined by ${\rm P}_i({\rm N}_i\gamma) = \gamma'$ for each $i = 1,2$.
Let $\beta': \Gamma_1' \to \Gamma_2'$ be the isomorphism so that ${\rm P}_2^{-1}\beta'{\rm P}_1 = \beta$.
Let $\tilde\beta: V_1^\perp \to V_2^\perp$ be an affinity such that $\tilde\beta\Gamma_1'\tilde\beta^{-1} = \Gamma_2'$ and $\tilde\beta_\ast = \beta'$, that is, and $\tilde\beta\gamma'\tilde\beta^{-1}= \beta'(\gamma')$ for each $\gamma$ in $\Gamma_1$.
Then we have that
$$(\tilde\alpha_v)_\sharp^{-1}\Xi_2{\rm P}_2^{-1}\tilde\beta_\ast{\rm P}_1= (\overline C^{-1}D p_1)_\star\Xi_1.$$
Therefore there exists an affinity $\phi = c+ C$ of $E^n$ such that $\phi(\Gamma_1,{\rm N}_1)\phi^{-1} = (\Gamma_2,{\rm N}_2)$
with $\overline \phi = \tilde\alpha_v$, $\phi' = \tilde\beta$, and $\overline{C'} = D$ by Theorem 3.
Thus $[\Gamma_1,{\rm N}_1] = [\Gamma_2,{\rm N}_2]$.
\end{proof}
\begin{lemma}
Suppose that $\omega([\Gamma,{\rm N}]) = \omega([\Gamma_1,{\rm N}_1])$ with $\mathcal{O} = (\alpha_1)_\#^{-1}\mathcal{O}_1\beta_1$
and that $\omega([\Gamma,{\rm N}]) = \omega([\Gamma_2,{\rm N}_2])$ with $\mathcal{O} = (\alpha_2)_\#^{-1}\mathcal{O}_2\beta_2$.
If $[\Gamma,{\rm N}; \Gamma_i,{\rm N}_i;\alpha_i,\beta_i]$ for $i = 1, 2$
are in the same coset of the image of $\kappa_\ast: H^1(\Gamma/{\rm N},\mathcal{C}) \to H^1(\Gamma/{\rm N},\mathcal{K})$ in $H^1(\Gamma/{\rm N},\mathcal{K})$,
then $[\Gamma_1,{\rm N}_1] = [\Gamma_2,{\rm N}_2]$.
\end{lemma}
\begin{proof}
Let $b+B \in \Gamma$, and let $b_1+B_1$ be an element of $\Gamma_1$ such that
$${\rm N}_1(b_1+B_1) = \beta_1({\rm N}(b+B)).$$
As $\mathcal{O} = (\alpha_1)_\#^{-1}\mathcal{O}_1\beta_1$, we have that
\begin{eqnarray*}
(b+B)_\ast \mathrm{Inn}({\rm N})& = & (\alpha_1)_\#^{-1}\mathcal{O}_1\beta_1({\rm N}(b+B)) \\
& = & (\alpha_1)_\#^{-1}\mathcal{O}_1({\rm N}_1(b_1+B_1)) \\
& = & (\alpha_1)_\#^{-1}((b_1+B_1)_\ast\mathrm{Inn}({\rm N}_1)) \\
& = & \alpha_1^{-1}(b_1+B_1)_\ast \alpha_1\mathrm{Inn}({\rm N}) \\
& = & (\tilde\alpha_1)_\ast^{-1}(b_1+B_1)_\ast (\tilde\alpha_1)_\ast\mathrm{Inn}({\rm N}) \\
& = & (\overline c_1 + \overline C_1)_\ast^{-1}(b_1+B_1)_\ast (\overline c_1 + \overline C_1)_\ast\mathrm{Inn}({\rm N}).
\end{eqnarray*}
The action of $\Gamma/{\rm N}$ on $\mathcal{C}$ is given by ${\rm N}(b+B)(u+\overline I) = \overline Bu + \overline I$
and is determined by the action of $\Gamma/{\rm N}$ on $Z({\rm N})$ induced by conjugation.
Now $\mathrm{Inn}({\rm N})$ acts trivially on $Z({\rm N})$, and so the last computation implies
that $\overline B = \overline C_1^{-1} \overline B_1\overline C_1$ on $\mathrm{Span}(Z({\rm N}))$.
Hence the pair of isomorphisms
$$\pi = (\beta_1:\Gamma/{\rm N} \to \Gamma_1/{\rm N}_1,(\tilde\alpha_1)^{-1}_\ast:\mathcal{C}_1 \to \mathcal{C})$$
is a change of groups isomorphism in the sense of \cite{M} p.\,108.
Therefore we have an isomorphism $\pi^\ast: H^1(\Gamma_1/{\rm N}_1,\mathcal{C}_1) \to H^1(\Gamma/{\rm N},\mathcal{C})$
defined so that if $f_1:\Gamma_1/{\rm N}_1 \to \mathcal{C}_1$ is a crossed homomorphism, then $\pi^\ast[f_1] = [\pi^\ast f_1]$
where $\pi^\ast f_1: \Gamma/{\rm N} \to \mathcal{C}$ is the crossed homomorphism defined by
$\pi^\ast f_1(x) = (\tilde\alpha_1)^{-1}_\ast(f_1(\beta_1(x)))$.
Likewise the pair of isomorphisms
$$\varpi= (\beta_1:\Gamma/{\rm N} \to \Gamma_1/{\rm N}_1,(\tilde\alpha_1)^{-1}_\sharp: \mathcal{K}_1 \to \mathcal{K})$$
is a change of groups isomorphism which induces an isomorphism
$\varpi^\ast$ such that the following diagram commutes
$$\begin{array}{ccc}
H^1(\Gamma_1/{\rm N}_1,\mathcal{C}_1) & {\buildrel \pi^\ast \over \longrightarrow} & H^1(\Gamma/{\rm N},\mathcal{C}) \vspace{.05in} \\
\downarrow (\kappa_1)_\ast & & \downarrow \kappa_\ast \\
H^1(\Gamma_1/{\rm N}_1,\mathcal{K}_1) & {\buildrel \varpi^\ast \over \longrightarrow} & H^1(\Gamma/{\rm N},\mathcal{K}).
\end{array}$$
Now we have that $\omega([\Gamma_1,{\rm N}_1]) = \omega([\Gamma_2,{\rm N}_2])$
with $\mathcal{O}_1 = (\alpha_2\alpha_1^{-1})_\#\mathcal{O}_2\beta_2\beta_1^{-1}$.
Moreover, we have that
$$[\Gamma_1,{\rm N}_1;\Gamma_2,{\rm N}_2;\alpha_2\alpha_1^{-1},\beta_2\beta_1^{-1}] = [((\tilde\alpha_2\tilde\alpha_1^{-1})_\sharp^{-1}\Xi_2\beta_2\beta_1^{-1})\Xi_1^{-1}].$$
For $i = 1, 2$, we have that
$$[\Gamma,{\rm N};\Gamma_i,{\rm N}_i;\alpha_i,\beta_i] = [((\tilde\alpha_i)_\sharp^{-1}\Xi_i\beta_i)\Xi^{-1}].$$
Let $\gamma \in \Gamma$. Observe that
\begin{eqnarray*}
\lefteqn{\big(((\tilde\alpha_2)_\sharp^{-1}\Xi_2\beta_2)\Xi^{-1}\big)\big(((\tilde\alpha_1)_\sharp^{-1}\Xi_1\beta_1)\Xi^{-1}\big)^{-1}({\rm N}\gamma)} \\
& = & ((\tilde\alpha_2)_\sharp^{-1}\Xi_2\beta_2)({\rm N}\gamma)\Xi({\rm N}\gamma)^{-1}\Xi({\rm N}\gamma)\big(((\tilde\alpha_1)_\sharp^{-1}\Xi_1\beta_1)({\rm N}\gamma)\big)^{-1} \\
& = & ((\tilde\alpha_2)_\sharp^{-1}\Xi_2\beta_2)({\rm N}\gamma)(\tilde\alpha_1)_\sharp^{-1}\big(\Xi_1\beta_1({\rm N}\gamma)\big)^{-1} \\
& = & ((\tilde\alpha_2)_\star^{-1}\Xi_2\beta_2)({\rm N}\gamma)(\tilde\alpha_2)_\star(\tilde\alpha_1)_\star^{-1}\big(\Xi_1\beta_1({\rm N}\gamma)\big)^{-1}(\tilde\alpha_1)_\star \\
& = & (\tilde\alpha_1)_\sharp^{-1}((\tilde\alpha_2\tilde\alpha_1^{-1})_\sharp^{-1}\Xi_2\beta_2)({\rm N}\gamma)\big(\Xi_1\beta_1({\rm N}\gamma)\big)^{-1} \\
& = & (\tilde\alpha_1)_\sharp^{-1}((\tilde\alpha_2\tilde\alpha_1^{-1})_\sharp^{-1}\Xi_2\beta_2\beta_1^{-1}(\beta_1({\rm N}\gamma))\big(\Xi_1\beta_1({\rm N}\gamma)\big)^{-1} \\
& = & \varpi^\ast\big(((\tilde\alpha_2\tilde\alpha_1^{-1})_\sharp^{-1}\Xi_2\beta_2\beta_1^{-1})\Xi_1^{-1}\big)({\rm N}\gamma). \\
\end{eqnarray*}
Hence we have that
$$\varpi^\ast([\Gamma_1,{\rm N}_1;\Gamma_2,{\rm N}_2;\alpha_2\alpha_1^{-1},\beta_2\beta_1^{-1}] )=[\Gamma,{\rm N};\Gamma_2,{\rm N}_2;\alpha_2,\beta_2][\Gamma,{\rm N};\Gamma_1,{\rm N}_1;\alpha_1,\beta_1]^{-1}.$$
The right-hand side of the above equation is in the image of $\kappa_\ast: H^1(\Gamma/{\rm N},\mathcal{C}) \to H^1(\Gamma/{\rm N},\mathcal{K})$.
Therefore $[\Gamma_1,{\rm N}_1;\Gamma_2,{\rm N}_2;\alpha_2\alpha_1^{-1},\beta_2\beta_1^{-1}]$ is in the image of
$(\kappa_1)_\ast: H^1(\Gamma_1/{\rm N}_1,\mathcal{C}_1) \to H^1(\Gamma_1/{\rm N}_1,\mathcal{K}_1)$.
Hence $[\Gamma_1,{\rm N}_1] = [\Gamma_2,{\rm N}_2]$ by Lemma 6.
\end{proof}
\section{The relative Bieberbach Theorem}
Let $[\Gamma, {\rm N}]$ be a class in $\mathrm{Isom}(\Delta,{\rm M})$,
and let ${\rm T}$ be the group of translations of $\Gamma$.
Then $\Gamma/{\rm T}{\rm N}$ is a finite group, since $\Gamma/{\rm T}$ is finite.
The group ${\rm T}{\rm N}/{\rm N}$ acts trivially on $Z({\rm N})$, ${\rm T}{\rm N}/{\rm N}$, $\mathcal{C}$, $\mathcal{K}$,
and so the action of $\Gamma/{\rm N}$ on $Z({\rm N})$, ${\rm T}{\rm N}/{\rm N}$, $\mathcal{C}$, $\mathcal{K}$ induces
an action of $\Gamma/{\rm T}{\rm N}$ on $Z({\rm N})$, ${\rm T}{\rm N}/{\rm N}$, $\mathcal{C}$, $\mathcal{K}$ making
$Z({\rm N})$, ${\rm T}{\rm N}/{\rm N}$, $\mathcal{C}$, $\mathcal{K}$ into $(\Gamma/{\rm T}{\rm N})$-modules.
\begin{lemma}
The group $H^1(\Gamma/{\rm T}{\rm N}, \mathcal{K})$ is finite.
\end{lemma}
\begin{proof}
The short exact sequence $0 \to Z({\rm N}) \to \mathcal{C} \to \mathcal{K} \to 0$ of $(\Gamma/{\rm T}{\rm N})$-modules induces
an exact sequence of cohomology groups
$$H^1(\Gamma/{\rm T}{\rm N},\mathcal{C}) \to H^1(\Gamma/{\rm T}{\rm N},\mathcal{K}) \to H^2(\Gamma/{\rm T}{\rm N},Z({\rm N})) \to H^2(\Gamma/{\rm T}{\rm N},\mathcal{C}).$$
As explained in the proof of Lemma 5, the outside groups are trivial, and so $H^1(\Gamma/{\rm T}{\rm N},\mathcal{K})$ is isomorphic to $H^2(\Gamma/{\rm T}{\rm N},Z({\rm N}))$.
The group $H^2(\Gamma/{\rm T}{\rm N},Z({\rm N}))$ is a torsion group by Proposition IV.5.3 of \cite{M}.
As $Z({\rm N})$ is a free abelian group of finite rank, the group $H^2(\Gamma/{\rm T}{\rm N},Z({\rm N}))$ is finitely generated.
Hence $H^2(\Gamma/{\rm T}{\rm N},Z({\rm N}))$ is finite, and so $H^1(\Gamma/{\rm T}{\rm N}, \mathcal{K})$ is finite.
\end{proof}
\begin{lemma}
The cokernel of $\kappa_\ast: H^1({\rm T}{\rm N}/{\rm N}, \mathcal{C})^{\Gamma/{\rm T}{\rm N}} \to H^1({\rm T}{\rm N}/{\rm N}, \mathcal{K})^{\Gamma/{\rm T}{\rm N}}$ is finite.
\end{lemma}
\begin{proof}
By the Universal Coefficients Theorem, we have that
$$H^1({\rm T}{\rm N}/{\rm N}, \mathcal{C})^{\Gamma/{\rm T}{\rm N}} = \mathrm{Hom}({\rm T}{\rm N}/{\rm N},\mathcal{C})^{\Gamma/{\rm T}{\rm N}} = \mathrm{Hom}_{\Gamma/{\rm T}{\rm N}}({\rm T}{\rm N}/{\rm N}, \mathcal{C}),$$
$$H^1({\rm T}{\rm N}/{\rm N}, \mathcal{K})^{\Gamma/{\rm T}{\rm N}} = \mathrm{Hom}({\rm T}{\rm N}/{\rm N},\mathcal{K})^{\Gamma/{\rm T}{\rm N}} = \mathrm{Hom}_{\Gamma/{\rm T}{\rm N}}({\rm T}{\rm N}/{\rm N}, \mathcal{K}).$$
The short exact sequence $0 \to Z({\rm N}) \to \mathcal{C} \to \mathcal{K} \to 0$ induces
an exact sequence
$$0 \to \mathrm{Hom}({\rm T}{\rm N}/{\rm N},Z({\rm N})) \to\mathrm{Hom}({\rm T}{\rm N}/{\rm N},\mathcal{C}) \to \mathrm{Hom}({\rm T}{\rm N}/{\rm N},\mathcal{K}) \to \mathrm{Ext}({\rm T}{\rm N}/{\rm N},Z({\rm N}))$$
by Theorem III.3.4 of \cite{M} (with $R = {\mathbb Z}$).
We have that $\mathrm{Ext}({\rm T}{\rm N}/{\rm N},Z({\rm N})) = 0$ by Theorems I.6.3 and III.3.5 of \cite{M}, since ${\rm T}{\rm N}/{\rm N}$ is a free abelian group.
Hence we have a short exact sequence of $(\Gamma/{\rm T}{\rm N})$-modules
$$0 \to \mathrm{Hom}({\rm T}{\rm N}/{\rm N},Z({\rm N})) \to\mathrm{Hom}({\rm T}{\rm N}/{\rm N},\mathcal{C}) \to \mathrm{Hom}({\rm T}{\rm N}/{\rm N},\mathcal{K}) \to 0.$$
Hence we have an exact sequence of cohomology groups
$$H^0(\Gamma/{\rm T}{\rm N}, \mathrm{Hom}({\rm T}{\rm N}/{\rm N},\mathcal{C})) \to H^0(\Gamma/{\rm T}{\rm N}, \mathrm{Hom}({\rm T}{\rm N}/{\rm N},\mathcal{K}))$$
$$\to H^1(\Gamma/{\rm T}{\rm N}, \mathrm{Hom}({\rm T}{\rm N}/{\rm N},Z({\rm N}))),$$
which is equivalent to an exact sequence
$$\mathrm{Hom}_{\Gamma/{\rm T}{\rm N}}({\rm T}{\rm N}/{\rm N},\mathcal{C}) \to \mathrm{Hom}_{\Gamma/{\rm T}{\rm N}}({\rm T}{\rm N}/{\rm N},\mathcal{K}) \to H^1(\Gamma/{\rm T}{\rm N}, \mathrm{Hom}({\rm T}{\rm N}/{\rm N},Z({\rm N}))).$$
The group $H^1(\Gamma/{\rm T}{\rm N}, \mathrm{Hom}({\rm T}{\rm N}/{\rm N},Z({\rm N})))$ is finite, since $\mathrm{Hom}({\rm T}{\rm N}/{\rm N},Z({\rm N}))$ is a free abelian group of finite rank.
Hence the cokernel of $\mathrm{Hom}_{\Gamma/{\rm T}{\rm N}}({\rm T}{\rm N}/{\rm N},\mathcal{C})) \to \mathrm{Hom}_{\Gamma/{\rm T}{\rm N}}({\rm T}{\rm N}/{\rm N},\mathcal{K}))$ is finite.
Therefore the cokernel of $\kappa_\ast: H^1({\rm T}{\rm N}/{\rm N}, \mathcal{C})^{\Gamma/{\rm T}{\rm N}} \to H^1({\rm T}{\rm N}/{\rm N}, \mathcal{K})^{\Gamma/{\rm T}{\rm N}}$ is finite.
\end{proof}
\begin{lemma}
The cokernel of $\kappa_\ast: H^1(\Gamma/{\rm N}, \mathcal{C}) \to H^1(\Gamma/{\rm N}, \mathcal{K})$ is finite.
\end{lemma}
\begin{proof}
We have a short exact sequence $1 \to {\rm T}{\rm N}/{\rm N} \to \Gamma/{\rm N} \to \Gamma/{\rm T}{\rm N} \to 1$, and so
we have a commutative diagram with horizontal exact sequences (cf.\,p.\,354 of \cite{M})
$$\begin{array}{ccccccc}
0= H^1(\Gamma/{\rm T}{\rm N},\mathcal{C}) &\hspace{-.1in} \to \hspace{-.1in} & H^1(\Gamma/{\rm N},\mathcal{C}) & \hspace{-.1in}\to \hspace{-.1in}& H^1({\rm T}{\rm N}/{\rm N},\mathcal{C})^{\Gamma/{\rm T}{\rm N}} & \hspace{-.1in}\to \hspace{-.1in} & H^2(\Gamma/{\rm T}{\rm N},\mathcal{C}) = 0
\vspace{.05in} \\
\downarrow \alpha & & \downarrow \beta & & \downarrow \gamma & & \downarrow \vspace{.05in} \\
0\to H^1(\Gamma/{\rm T}{\rm N},\mathcal{K}) & \hspace{-.1in}\to \hspace{-.1in}& H^1(\Gamma/{\rm N},\mathcal{K}) & \hspace{-.1in}\to \hspace{-.1in}& H^1({\rm T}{\rm N}/{\rm N},\mathcal{K})^{\Gamma/{\rm T}{\rm N}} &\hspace{-.1in} \to \hspace{-.1in} & H^2(\Gamma/{\rm T}{\rm N},\mathcal{K}) \phantom{= 0}
\end{array}$$
where the homomorphism $\alpha, \beta, \gamma$ are induced by $\kappa : \mathcal{C} \to \mathcal{K}$.
By the snake lemma (Lemma III.5.1 of \cite{H-S}), we have an exact sequence
$$\mathrm{coker}(\alpha) \to \mathrm{coker}(\beta) \to \mathrm{coker(\gamma}).$$
Now $\mathrm{coker}(\alpha) = H^1(\Gamma/{\rm T}{\rm N},\mathcal{K})$ is finite by Lemma 8, and $\mathrm{coker(\gamma})$ is finite by Lemma 9.
Hence $\mathrm{coker(\beta})$ is finite.
Thus the cokernel of $\kappa_\ast: H^1(\Gamma/{\rm N}, \mathcal{C}) \to H^1(\Gamma/{\rm N}, \mathcal{K})$ is finite.
\end{proof}
\begin{theorem}
For each dimension $n$, there are only finitely many isomorphism classes of pairs of groups $(\Gamma, {\rm N})$ such that $\Gamma$
is an $n$-space group and ${\rm N}$ is a normal subgroup of $\Gamma$ such that $\Gamma/{\rm N}$ is a space group.
\end{theorem}
\begin{proof}
Let $m$ be a positive integer less than $n$.
Let ${\rm M}$ be an $m$-space group and
let $\Delta$ be an $(n-m)$-space group.
Let $\mathrm{Iso}(\Delta,{\rm M})$ be the set of isomorphism classes
of pairs $(\Gamma, {\rm N})$
where ${\rm N}$ is a normal subgroup of an $n$-space group $\Gamma$
such that ${\rm N}$ is isomorphic to ${\rm M}$ and $\Gamma/{\rm N}$ is isomorphic to $\Delta$.
As there are only finitely many isomorphism classes of the groups $\Delta$ and ${\rm M}$ by
Bieberbach's theorem \cite{B}, it suffices to prove that $\mathrm{Iso}(\Delta,{\rm M})$ is finite.
In \S 4, we defined a function $\omega: \mathrm{Iso}(\Delta,{\rm M}) \to \mathrm{Out}(\Delta,\mathrm{M})$
with $\mathrm{Out}(\Delta,\mathrm{M})$ finite by Lemma 4.
The fibers of $\omega$ are finite by Lemmas 7 and 10.
Therefore $\mathrm{Iso}(\Delta,{\rm M})$ is finite.
\end{proof}
In view of Theorem 10 of \cite{R-T}, Theorem 4 is equivalent to the following theorem.
\begin{theorem}
For each dimension $n$, there are only finitely many affine equivalence classes of geometric orbifold fibrations of compact, connected, flat $n$-orbifolds.
\end{theorem}
|
3,212,635,537,421 | arxiv | \section*{References}
\begin{document}
\maketitle
\begin{abstract}
Consider the cotangent bundle
of a closed Riemannian manifold
and an almost complex
structure close
to the one induced by
the Riemannian metric.
For Hamiltonians which
grow for instance
quadratically in the fibers
outside of a compact set,
one can define
Floer homology and show that it is naturally
isomorphic to singular homology
of the free loop space.
We review the three isomorphisms
constructed by Viterbo~\cite{V96},
Salamon-Weber~\cite{JOA3} and
Abbondandolo-Schwarz~\cite{AS04}.
The theory is illustrated
by calculating
Morse and Floer homology
in case of the euclidean
$n$-torus.
Applications include
existence of noncontractible
periodic orbits
of compactly supported
Hamiltonians on open unit
disc cotangent bundles
which are sufficiently large
over the zero section.
\end{abstract}
\section{Chain group and boundary operators}
\label{sec:intro}
Let $M$ be a closed smooth manifold and
fix a Riemannian metric. Let $\nabla$ be
the associated Levi-Civita connection.
This endows the free loop space
$\Ll M=C^\infty(S^1,M)$
with an $L^2$ and a $W^{1,2}$ metric
given by
$$
\langle\xi,\eta\rangle_{L^2}
=\int_0^1\langle\xi(t),\eta(t)\rangle
\: dt,\quad
\langle\xi,\eta\rangle_{W^{1,2}}
=\langle\xi,\eta\rangle_{L^2} +
\langle\Nabla{t}\xi,
\Nabla{t}\eta\rangle_{L^2},
$$
where $\xi$ and $\eta$ are
smooth vector fields
along $x\in\Ll M$.
Here and throughout we identify
$S^1={\mathbb{R}}/{\mathbb{Z}}$ and think of $x\in\Ll M$
as a smooth map $x:{\mathbb{R}}\to M$
which satisfies $x(t+1)=x(t)$.
Fix a time-dependent function
$V\in C^\infty(S^1\times M)$
and set $V_t(q):=V(t,q)$.
The \emph{classical action functional
on $\Ll M$} is defined by
$$
{\mathcal{S}}_V(x)
:=\int_0^1\left(\frac{1}{2}
\abs{\dot x(t)}^2-V_t(x(t))\right) dt.
$$
The set ${\mathcal{P}}(V)$ of critical points
consists of the $1$-periodic
solutions of the ODE
\begin{equation}\label{eq:crit}
\Nabla{t} {\partial}_t x=-\nabla V_t(x).
\end{equation}
Here $\nabla V_t$ denotes the gradient.
These solutions are called
\emph{perturbed closed geodesics}.
Two features make the functional ${\mathcal{S}}_V$
accessible to standard variational methods,
boundedness from below
and finiteness of the
\emph{Morse index}\footnote{The dimension
of the largest subspace
on which the Hessian is negative definite.}
$\IND_V(x)$ of every critical point.
A critical point is called \emph{nondegenerate}
if its Hessian is nondegenerate.
A function with nondegenerate critical
points only is a \emph{Morse function}.
If ${\mathcal{S}}_V$ is Morse,
the change of topology of the sublevel set
$$
\{{\mathcal{S}}_V\le a\}:=\{x\in\Ll M\mid
{\mathcal{S}}_V(x)\le a\}
$$
when $a\in{\mathbb{R}}$ passes
through a critical value
is the subject of classical
Morse theory leading
to a CW-complex homotopy
equivalent to $\{{\mathcal{S}}_V\le a\}$
(see e.g. Milnor~\cite{M64}).
In the case $V=0$
we use the notation
$\Ll^a M:=\{{\mathcal{S}}_0\le a\}$.
\subsubsection*{\boldmath$W^{1,2}$ Morse homology}
\label{subsubsec:W^{1,2}-HM}
A geometric reincarnation of
the idea of encoding the topology
of a sublevel set in terms of a Morse function
came (back) to light in 1982 through the
work of Witten~\cite{Wi82}.
Roughly speaking, the
\emph{Morse-Witten complex}
consists of chain groups
generated by the critical points of
a Morse function and a boundary
operator which counts flow lines
of the negative gradient flow
between critical points of
Morse index difference one
(for details see
e.g.~\cite{Sch93} and~\cite{JOA5}).
In recent years
Abbondandolo and Majer~\cite{AM04}
extended the theory from finite dimensions
to a class of Hilbert manifolds.
The free loop space fits into this framework
after completion with respect to the
Sobolev $W^{1,2}$ norm.
From now on we assume that ${\mathcal{S}}_V $
is a Morse function.
(A proof that this holds for a generic
potential $V$ is given in~\cite{JOA2}).
Then the set
$$
{\mathcal{P}}^a(V)
:=\{x\in{\mathcal{P}}(V)\mid {\mathcal{S}}_V(x)\le a\}
$$
is finite for every real number $a$.
Also from now on we assume that $a$
is a regular value of ${\mathcal{S}}_V$.
The chain groups
are the free abelian groups
generated by ${\mathcal{P}}^a(V)$
and graded by the
Morse index, namely
\begin{equation}\label{eq:chain-group}
C^a_k(V)
=\bigoplus_{\stackrel
{\scriptstyle x\in{\mathcal{P}}^a(V)}
{\IND_V(x)=k}}{\mathbb{Z}} x,\qquad
k\in{\mathbb{Z}}.
\end{equation}
Our convention is that the direct
sum over an empty set equals $\{0\}$.
The negative of the $W^{1,2}$ gradient
vector field induces a flow
on the ($W^{1,2}$ completion of the)
loop space whose unstable manifolds
are of finite dimension and
whose stable manifolds are of finite
codimension. Let us choose an orientation
of the unstable manifold
of every critical point.
If the Morse-Smale condition holds
(this means that all stable and
unstable manifolds intersect
transversally),
then there are only
finitely many flow lines
between critical points of index difference
one. These are called
\emph{isolated flow lines}.
Each one inherits an orientation,
because it is the intersection
of an oriented and a cooriented
submanifold.
Let the characteristic sign
of an isolated flow line
be $+1$ if the inherited
orientation coincides with
the one provided by the flow
and $-1$ else.
Counting isolated flow lines with
characteristic signs
defines the Morse-Witten boundary operator.
The associated homology groups
$\HM_*^a(\Ll M,{\mathcal{S}}_V,W^{1,2})$
are called \emph{$W^{1,2}$ Morse homology}.
By the theory of Abbondandolo and Majer
it is naturally isomorphic to
integral singular homology
$\Ho_*(\{{\mathcal{S}}_V\le a\})$.
If $\Ll M$ has several connected
components $\Ll_\alpha M$,
then there is a separate isomorphism
for each of them.
The label $\alpha$ denotes
a homotopy class
of free loops in $M$.
\subsubsection*{\boldmath$L^2$ Morse homology}
\label{subsubsec:L^2-HM}
Replacing the $W^{1,2}$ metric on
the free loop space by the $L^2$ metric
leads to a new boundary
operator on the chain
groups~(\ref{eq:chain-group}), namely
by counting negative
$L^2$ gradient 'flow lines'.
In fact the $L^2$ metric
gives rise only to a semiflow
in forward time and so we
view -- in the
spirit of Floer theory --
the negative gradient
flow equation on the loop
space as a PDE for
smooth cylinders in $M$.
Flow lines are then replaced by
solutions $u:{\mathbb{R}}\times S^1\to M$
of the \emph{heat equation}
\begin{equation}\label{eq:heat}
{\partial}_s u-\Nabla{t}{\partial}_t u-\nabla V_t(u)=0
\end{equation}
which satisfy
\begin{equation}\label{eq:heat-lim}
\lim_{s\to\pm\infty} u(s,t)
= x^\pm(t),\qquad
\lim_{s\to\pm\infty}{\partial}_su(s,t)
= 0.
\end{equation}
Here $x^\pm\in{\mathcal{P}}(V)$ and
the limits are supposed to be
uniform in $t$.
Stable manifolds can still
be defined via the forward semiflow,
whereas to define
unstable manifolds we
use the heat flow lines~(\ref{eq:heat}).
The former are of
finite codimension
and the latter of finite
dimension. Hence
characteristic signs
can be assigned to isolated
(index difference one)
flow lines just as
in the case of $W^{1,2}$
Morse homology above.
The parabolic moduli space
$\Mm^0(x^-,x^+;V)$ is the
set of solutions of~(\ref{eq:heat})
and~(\ref{eq:heat-lim}).
In this setting we say that the
\emph{Morse-Smale condition} holds,
if the linear operator obtained by
linearizing~(\ref{eq:heat}) at a
solution $u$ is onto
for all $u\in \Mm^0(x^-,x^+;V)$
and all $x^\pm\in{\mathcal{P}}(V)$.
In this case $\Mm^0(x^-,x^+;V)$
is a smooth manifold whose dimension
equals the difference of the Morse indices.
In the case of index difference one
the quotient by the free time shift
action is a finite set.
Counting its elements
with characteristic signs
defines the $L^2$
Morse-Witten boundary
operator ${\partial}^M$.
The associated homology
$\HM_*^a(\Ll M,{\mathcal{S}}_V,L^2)$ is
called $L^2$ Morse homology.
It is naturally isomorphic
to $\Ho_*(\{{\mathcal{S}}_V\le a\})$.
(We should emphasize that this
is work in progress~\cite{JOA-FUTURE}).
\subsubsection*{Floer homology}
\label{subsubsec:HF}
The critical points of ${\mathcal{S}}_V$
can be interpreted via the
Legendre transformation as the
critical points of the
\emph{symplectic action functional}
$$
\Aa_V(z)=\Aa_{H_V}(z)
:= \int_0^1 \biggl(
\inner{y(t)}{{\partial}_t x(t)}
- H_V(t,x(t),y(t))\biggr)\,dt.
$$
Here $z=(x,y)$ where
$x:S^1\to M$ is a smooth map
and $y(t)\in T_{x(t)}^*M$
depends smoothly on $t\in S^1$.
The Hamiltonian
$H_V:S^1\times T^*M\to{\mathbb{R}}$ is
given by
$$
H_V(t,q,p) = \frac12|p|^2+V_t(q).
$$
A loop
$z=(x,y)$ in $T^*M$
is a critical point of $\Aa_V$ iff $x$
is a critical point of ${\mathcal{S}}_V$
and $y(t)\in T_{x(t)}^*M$ is related to
${\partial}_t x (t)\in T_{x(t)}M$ via the isomorphism
$g:TM\to T^*M$ induced by the Riemannian metric.
For such loops $z$ the symplectic action $\Aa_V(z)$
agrees with the classical action ${\mathcal{S}}_V(x)$.
In contrast to the classical
action, the symplectic action
is in general neither bounded
below nor do the critical
points admit finite Morse indices,
and most importantly
its $L^2$ gradient does not
define a flow on the loop space.
It was a great achievement
of Floer~\cite{F89} to
nevertheless set up
a Morse-Witten type complex.
His key idea was to reinterpret
the negative $L^2$ gradient equation
as elliptic PDE for maps from the cylinder
to the symplectic manifold imposing
appropriate boundary conditions
to make the problem Fredholm.
Floer's original setup
was a \emph{closed} symplectic manifold
subject to two topological assumptions
to ensure compactness of moduli spaces
and existence of a natural grading.
A Hamiltonian function $H$
and an almost complex structure $J$
need to be chosen
to define the Floer complex.
The power of Floer theory
lies in the fact that Floer homology
is independent of these choices.
This is called the
\emph{Floer continuation principle}.
Floer showed that if
$H$ is autonomous
and a $C^2$-small
Morse function,
then the Floer chain complex
equals the Morse-Witten complex.
Hence Floer homology is naturally
isomorphic to singular integral homology
of the closed symplectic manifold itself.
For introductory reading
we refer to Salamon's lecture notes~\cite{S97}
and the recent survey by Laudenbach~\cite{L04}.
A discussion
on a more advanced level, also
including applications
of Floer theory, can be found
in Chapter~12 of~\cite{MS04}.
Now consider the
cotangent bundle $T^*M$ equipped
with its canonical symplectic structure
$\omega_0=-d\theta$, where $\theta$
is the Liouville form. Since the
symplectic form is exact and
the first Chern class of $T^*M$
with respect to the metric
induced almost complex structure
vanishes, both topological
assumptions of Floer are met.
The former excludes
existence of nonconstant
$J$-holomorphic spheres,
which is an obstruction towards
compactness of the moduli spaces,
and the latter implies that
the Conley-Zehnder index
of 1-periodic Hamiltonian
orbits is well defined.
Since the 1-periodic orbits
of the Hamiltonian flow
are precisely the critical
points of $\Aa_V$ and these
coincide with the critical points
of ${\mathcal{S}}_V$ up to
natural identification,
the Floer chain groups are again
given by the free abelian
groups~(\ref{eq:chain-group}).
(The standard Floer grading
is the negative Conley-Zehnder index,
which is proved in~\cite{JOA2} to
equal the Morse index; up to
a constant if $M$ is not orientable).
The Riemannian metric on $M$
provides the isomorphism
\begin{equation}\label{eq:splitting}
T_{(x,y)}T^*M\to T_xM\oplus T_x^*M,
\end{equation}
which takes the derivative of a curve
${\mathbb{R}}\to T^*M:t\mapsto z(t)=(x(t),y(t))$
to the derivatives
of the two components, namely
$$
{\partial}_t z(t)
\mapsto
({\partial}_t x(t),\Nabla{t}y(t)).
$$
The metric also induces
an almost complex
structure $J_g$ and a metric $G_g$ on $T^*M$.
These and the symplectic
form are represented by
$$
J_g=\begin{pmatrix}
0&-g^{-1}\\g&0\end{pmatrix},\qquad
G_g=\begin{pmatrix}
g&0\\0&g^{-1}\end{pmatrix},\qquad
\omega_0=\begin{pmatrix}
0&-{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l\\{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l&0\end{pmatrix}.
$$
These three structures
are compatible in the sense that
$\omega_0(\cdot,J_g\cdot)=G_g(\cdot,\cdot)$.
Flow lines are then
replaced by solutions
$w:{\mathbb{R}}\times S^1\to T^*M$
of Floer's equation
$$
{\partial}_s w+J_g(w){\partial}_t w-\nabla H_V(t,w)=0.
$$
It is the negative $L^2$ gradient
equation for the symplectic action
viewed as an elliptic PDE.
Its solutions are called
\emph{Floer trajectories} or
\emph{Floer cylinders}.
If we identify $T^*M$ and $TM$
via the metric isomorphism
and represent Floer's equation
with respect to the
splitting~(\ref{eq:splitting}),
we obtain the
pair of equations
\begin{equation}\label{eq:floer}
{\partial}_su-\Nabla{t}v-\nabla V_t(u)
= 0,\qquad
\Nabla{s}v+{\partial}_tu-v=0
\end{equation}
for $(u,v):{\mathbb{R}}\times S^1\to TM$.
The Floer moduli space
$\Mm^1(x^-,x^+;V)$ is the set
of solutions
of~(\ref{eq:floer}) subject
to the boundary conditions
\begin{equation}\label{eq:floer-lim}
\lim_{s\to\pm\infty}u(s,t)
=x^\pm(t),\qquad
\lim_{s\to\pm\infty}v(s,t)
={\partial}_t x^\pm(t),
\end{equation}
and ${\partial}_su$ and $\Nabla{s}v$
converge to zero
as $|s|\to\infty$, and all limits
are uniformly in $t$.
If the Morse-Smale condition
is satisfied,
then $\Mm^1(x^-,x^+;V)$ is
a smooth manifold whose dimension
is given by the difference
of Morse indices.
Since $T^*M$ is
noncompact, we
do not obtain for free
uniform apriori
$C^0$-bounds for the
Floer solutions as
in the standard case
of a \emph{closed} symplectic manifold.
Such bounds were established in 1992
by Cieliebak in his diploma thesis
(published in~\cite{C94})
and recently extended to
a class of radial Hamiltonians
by the author~\cite{JOA4}
and to another class
of not necessarily radial
Hamiltonians by Abbondandolo
and Schwarz~\cite{AS04}.
Given these bounds,
proving compactness
and setting up Floer homology
is standard.
In the case that $\Ll T^*M$ has several
connected components,
Floer homology is defined for
each of them separately and
denoted by
$\HF_*^a(T^*M,H_V,J_g;\alpha)$.
Here $\alpha$ denotes a homotopy
class of free loops in $M$.
(Throughout we identify
homotopy classes of free loops
in $M$ and in $T^*M$).
In fact Floer homology can be
defined for classes of
Hamiltonians more general
than $H_V$, for instance
those growing
quadratically in $p$
outside of a compact set
(see Section~\ref{sec:AS})
or convex radial Hamiltonians
(see Section~\ref{sec:noncontractible}).
\subsubsection*{Isomorphisms
between the theories}
\label{subsubsec:relations}
When Floer homology for
physical Hamiltonians of the
type kinetic plus potential energy
could be defined by~1992
due to Cieliebak's breakthrough,
the obvious question was
\emph{`What is it equal to?'}.
The first
answer\footnote{By then
Theorem~\ref{thm:main}
had been expected to be true by
part of the community.
For instance the problem was proposed
as a PhD project to the present author
by Helmut Hofer during
winter term 1993/94 at ETH Z\"urich.
In summer~1996
upon meeting Dietmar Salamon in Oberwolfach
we matched up and started our joint approach.
A short time later~\cite{V96} appeared.
In private communication at a Warwick conference,
around~1998, Matthias Schwarz first told me about
an alternative approach via a
mixed boundary value problem.}
in the literature
is due to Viterbo who
conjectured in his~1994
ICM talk~\cite{V95} -- based on
formal interpretation of the symplectic
action functional as a generating function --
that Floer homology of the cotangent bundle
represents singular homology of the
free loop space.
In his 1996 preprint~\cite{V96}
he gave a beautiful line of argument
(for the component $\Ll_0 M$
of contractible loops, case $V=0$,
coefficients in ${\mathbb{Z}}_2$).
The idea is to
view the time-1-map
$\varphi_1$ of the Hamiltonian flow
as an $r$-fold composition
of symplectomorphisms close
to the identity
(set $\psi:=\varphi_{1/r}$)
in order to arrive at the well known
finite dimensional approximation
of the free loop space
via broken geodesics.
While the argument
consists of numerous steps and the idea
of each one is described in detail,
not all technical details are
provided.
Also in 1996 a first example
was computed by the present
author~\cite{JOA0c}, namely
Floer homology of the cotangent
bundles of the euclidean torus
${\mathbb{T}}^n={\mathbb{R}}^n/2\pi{\mathbb{Z}}^n$
confirming the conjecture
for all connected components
of $\Ll{\mathbb{T}}^n$, for every $n\in{\mathbb{N}}$.
This is reviewed in
Section~\ref{sec:torus}.
In 2003 Salamon and the
present author~\cite{JOA3}
proved existence of a natural
isomorphism
$$
\HM_*^a(\Ll M,{\mathcal{S}}_V,L^2;{\mathbb{Z}})
\to
\HF_*^a(T^*M,H_V,J_g;{\mathbb{Z}}).
$$
(Partial results were established
in~1999 in the
PhD thesis~\cite{JOA1}).
The idea is to introduce a
real parameter ${\varepsilon}>0$ and to replace
the standard almost complex structure $J_g$
by $J_{{\varepsilon}^{-1}g}$.
Both Floer homologies are naturally isomorphic
by Floer continuation.
The key step is then
to prove that for every
sufficiently small ${\varepsilon}$
the parabolic and elliptic moduli
spaces can be identified.
This means that
the ${\varepsilon}$-Floer
and the $L^2$ Morse chain complexes
are \emph{identical}.
In 2004 Abbondandolo and
Schwarz~\cite{AS04}
proved existence of a natural isomorphism
$$
\HM_*^a(\Ll M,{\mathcal{S}}_V,W^{1,2})
\to
\HF_*^a(T^*M,H_V,J_g)
$$
by constructing
a \emph{chain isomorphism}
in the case of orientable $M$.
Their approach works
for more general Hamiltonians
and almost complex structures
(see Section~\ref{sec:AS}).
The idea is to study
a mixed boundary value problem for
Floer half cylinders
$w=(u,v):[0,\infty)\to T^*M$.
For $s\to+\infty$ the standard Floer
boundary condition~(\ref{eq:floer-lim})
is imposed, whereas at $s=0$
the base loop $u(0,\cdot)$
is required to belong to an unstable
manifold of the negative $W^{1,2}$
gradient flow of ${\mathcal{S}}_V$.
Also in~2004 the present
author~\cite{JOA4} extended
the definition and
computation of Floer homology
to the class of convex
radial Hamiltonians
(those of the form
$h=h(\abs{p})$ with $h^{\prime\prime}\ge 0$).
The main result of~\cite{V96},
\cite{JOA3} and~\cite{AS04}
as formulated in~\cite{JOA3} is
the following theorem.
\begin{theorem}\label{thm:main}
Let $M$ be a closed Riemannian
manifold.
Assume ${\mathcal{S}}_V$ is Morse and $a$
is either a regular
value of ${\mathcal{S}}_V$ or is equal to infinity.
Then there is a natural isomorphism
$$
\HF^a_*(T^*M,H_V,J_g;R)
\simeq\mathrm{H}_*(\{{\mathcal{S}}_{\mathcal{V}}\le a\};R)
$$
for every principal ideal domain $R$.
If $M$ is not simply connected,
then there is a separate
isomorphism for each component of the loop space.
The isomorphism commutes with
the homomorphisms
$\HF^a_*(T^*M,H_V,J_g)\to\HF^b_*(T^*M,H_V,J_g)$
and $\mathrm{H}_*(\{{\mathcal{S}}_{\mathcal{V}}\le a\})
\to\mathrm{H}_*(\{{\mathcal{S}}_{\mathcal{V}}\le b\})$, for $a<b$,
which are induced by inclusion.
\end{theorem}
We summarize the discussion
by the diagram below
in which arrows represent isomorphisms.
The homologies are defined as usual
by first perturbing to achieve Morse-Smale
transversality and then taking the
homology of the perturbed chain complex.
The branch on the right hand side
indicates Viterbo's
finite dimensional approximation
argument (which he actually formulated
in terms of cohomology;
see Section~\ref{sec:V}).
\begin{equation*}
\begin{split}
\xymatrix{
\HF_*^a(T^*M,H_0,J_{{\varepsilon}^{-1}g})
&
\HF_*^a(T^*M,H_0,J_g)
\ar[r]_{\textstyle\cite{V96}}
\ar[l]^{\overset{\scriptstyle\text{Floer}}
{\scriptstyle\text{continuation}}}
&
\HF^a_*(\Delta_r,\Gamma_r(\varphi^{H_0}))
\ar[d]^{\textstyle\cite{V96}}
\\
\HM_*^a(\Ll M,{\mathcal{S}}_0,L^2)
\ar[dr]_{\textstyle\cite{JOA-FUTURE}}
\ar[u]^{\textstyle\cite{JOA3}}
&
\HM_*^a(\Ll M,{\mathcal{S}}_0,W^{1,2})
\ar[d]^{\textstyle\cite{AM04}}
\ar[u]_{\textstyle\cite{AS04}}
&
\HM^a_*(U_{r,{\varepsilon}},S_r)
\ar[d]^{\textstyle\cite{V96,V97}}
\\
&
\Ho_*(\Ll^a M)
&
\Ho_*(\Lambda^a_r)
\ar[l]_{\textstyle\cite{V97}}
}
\end{split}
\end{equation*}
It would be interesting
to fill in the missing link between
$L^2$ and $W^{1,2}$ Morse
homology, i.e. construct an
isomorphism which is natural
in the sense that the
corresponding triangle
and rectangle in the diagram
are both commutative.
The remaining part of this
text is organized as follows.
We present the three appoaches
towards Theorem~\ref{thm:main}
in chronologically reverse order
in Sections~\ref{sec:AS}--\ref{sec:V}.
This way complexity increases
-- as it should be.
In Section~\ref{sec:torus} we calculate
Floer homology of the cotangent
bundle of the euclidean $n$-torus.
An application of
Theorem~\ref{thm:main} to
existence of noncontractible
periodic orbits is reviewed in
Section~\ref{sec:noncontractible}.
For an application towards
Arnold's chord conjecture
we refer to Cieliebak's
paper~\cite{C02}.
\medskip\noindent
{\small\it Acknowledgements:}
{\footnotesize
We gratefully acknowledge
partial financial support
by DFG SPP~1154
{\sl Globale Differentialgeometrie},
Scuola Normale Superiore Pisa
and \'Ecole Polytechnique Paris.
We are particularly indebted to
Alberto Abbondandolo, Kai Cieliebak,
Pietro Majer and
Claude Viterbo
for numerous helpful and pleasant
conversations on the subject.}
\section{Mixed boundary value problem}
\label{sec:AS}
In three steps we review
the approach of Abbondandolo
and Schwarz~\cite{AS04}.
They assume for simplicity that
$M$ is orientable.
First of all, the authors
set up Floer theory for a more
general class of Hamiltonians $H$
and almost complex structures $J$.
Let ${\mathcal{P}}(H)$ denote the set of
critical points of the symplectic
action $\Aa_H$. These are precisely the
1-periodic orbits of the
Hamiltonian flow on $T^*M$.
The crucial (metric independent)
assumptions on $H$
are the following.
Outside of a compact set
$H$ is supposed to satisfy
$$
dH(t,q,p) \; p{\partial}_p-H(t,q,p)
\ge c_0\abs{p}^2-c_1,
\eqno{\text{(H1)}}
$$
$$
\Abs{\Nabla{q} H(t,q,p)}
\le c_2(1+\abs{p}^2),\qquad
\Abs{\Nabla{p} H(t,q,p)}
\le c_2(1+\abs{p}),
\eqno{\text{(H2)}}
$$
for some constants $c_0>0$
and $c_1,c_2\ge 0$.
Assumptions~(H1) and~(H2)
guarantee that the set ${\mathcal{P}}^a(H)$
is finite for every real number $a$,
whenever $\Aa_H$ is Morse (which we shall
assume from now on, since
it is true for generic $H$).
More importantly, assumptions~(H1) and~(H2)
allow the authors to establish
$C^0$-bounds for Floer solutions
associated to
almost complex structures $J$
sufficiently $C^\infty$-close to $J_g$.
Then the definition of the chain complex
$(\CF_*(H),{\partial}_*(H,J))$
proceeds by standard arguments.
Any two choices of $J$
lead to isomorphic \emph{chain complexes}.
In contrast homology is independent of $(H,J)$
and denoted by $\HF_*(T^*M)$.
Secondly, the Morse-Witten complex is defined
for the Hilbert manifold $W^{1,2}(S^1,M)$
and the classical action functional
${\mathcal{S}}_L(x):=\int_0^1 L(t,x(t),\dot x(t))\:dt$.
Here the admissible Lagrangians $L$
are those for which
there exist constants
$d_0>0$ and $d_1\ge 0$ such that
$$
\Nabla{vv} L(t,q,v)\ge d_0{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l,
\eqno{\text{(L1)}}
$$
\begin{minipage}{11cm}
$$
\Abs{\Nabla{qq} L(t,q,v)}
\le d_1(1+\abs{v}^2),\qquad
\Abs{\Nabla{qv} L(t,q,v)}
\le d_1(1+\abs{v}),
$$
$$
\Abs{\Nabla{vv} L(t,q,v)}
\le d_1,
$$
\end{minipage}
\hfill
\begin{minipage}{.66cm}
(L2)
\end{minipage}
\vspace{.2cm}
\noindent
for all $(t,q,v)\in S^1\times TM$.
Perturbing $L$
if necessary we assume
from now on that ${\mathcal{S}}_L$
is Morse and denote
the set of its critical points
by ${\mathcal{P}}(L)$.
The classical action
exhibits a number of
rather useful features.
For instance, it satisfies
the Palais-Smale condition,
it is bounded from below
and its critical points have
finite Morse indices (which
equal minus the corresponding
Conley-Zehnder indices)\footnote{The
sign difference
between~\cite{JOA2,JOA3} and~\cite{AS04}
is due to the different normalizations
$\mu_{CZ}(t\mapsto e^{i\pi t})=1$
and $\mu_{CZ}(t\mapsto e^{-i\pi t})=1$
(with $t$ running through $[0,1]$),
respectively.}.
Choosing an auxiliary
Morse-Smale metric
$\Gg$ on the Hilbert manifold,
the work of Abbondandolo and
Majer~\cite{AM04} establishes
existence of the Morse complex
$(\CM_*({\mathcal{S}}_L),{\partial}_*({\mathcal{S}}_L,\Gg))$
and shows that its homology
is naturally isomorphic to the singular
homology of the free loop space.
Given $L$ and $\Gg$ as in Step~2,
the crucial third step
is to construct
a grading preserving chain complex
isomorphism
$$
\Theta_*:
(\CM_*({\mathcal{S}}_L),{\partial}_*({\mathcal{S}}_L,\Gg))
\to
(\CF_*(H_L),{\partial}_*(H_L,J))
$$
where the Hamiltonian $H_L$
arises from the Lagrangian $L$
via Legendre transformation
$(t,q,p)\mapsto (t,q,v(t,q,p))$.
More precisely, define
$$
H_L(t,q,p)
:=\max_{v\in T_qM}\left(
\langle p,v \rangle-L(t,q,v)\right).
$$
For each $(t,q,p)$ the maximum
is achieved at a unique point
$v(t,q,p)$ by condition~(L1).
The Legendre transformation
provides a natural identification
of the critical points
of $\Aa_{H_L}$ and ${\mathcal{S}}_L$, namely
${\mathcal{P}}(H_L)\to{\mathcal{P}}(L):
(x,y)\mapsto x$.
This shows that both chain groups coincide.
\\
To define a chain homomorphism
fix $q\in{\mathcal{P}}(L)$ and
$z\in{\mathcal{P}}(H_L)$. Then
consider half cylinders
$w:[0,\infty)\times S^1\to T^*M$
solving Floer's equation
$$
{\overline{\partial}}_{J,L} w
:={\partial}_sw+J(w){\partial}_t w
-\nabla H_L(t,w)=0
$$
and such that $w(s,\cdot)$
converges uniformly to $z$,
for $s\to+\infty$.
The boundary condition
at the other end is
that the loop
$w(0,\cdot)$ projects to the
unstable manifold of $q$, i.e.
$u:=\pi\circ w(0,\cdot) \in W^u(q)$
where $\pi:T^*M\to M$ is
the bundle projection.
These half cylinders are the
elements of the moduli space
\begin{equation*}
\begin{split}
\Mm^+(q,z)
:=
\{ w:[0,+\infty)\times S^1\to T^*M
\mid\:
&\text{$\pi\circ w(0,\cdot)\in W^u(q)$,
${\overline{\partial}}_{J,L} w=0$,}\\
&\text{$\lim_{s\to\infty}
w(s,\cdot)=z(\cdot)$}
\}.
\end{split}
\end{equation*}
The problem is Fredholm,
because the unstable manifolds $W^u(q)$
are finite dimensional
and the boundary conditions for
the $\overline{\partial}$-equation are Lagrangian
and nondegenerate, respectively.
For generic $J$ this
moduli space is a smooth manifold
of dimension $\IND_L(q)-\IND_L(z)$.
In the case of equal indices
it is a discrete set and $\Theta_*$
is defined by counting its elements.
Of course, compactness of
the moduli space needs to be
established first.
Here a crucial observation of Abbondandolo
and Schwarz enters, namely the inequality
\begin{equation}\label{eq:AS-inequality}
\Aa_{H_L}(x,y)\le{\mathcal{S}}_L(x)
\end{equation}
for every loop $(x,y):S^1\to T^*M$.
On critical points equality holds.
To prove compactness
a uniform action bound is needed
to start with, but this follows
immediately from~(\ref{eq:AS-inequality}):
if $w\in\Mm^+(q,z)$, then
$\Aa_{H_L}(w(s,\cdot))\le{\mathcal{S}}_L(q)$
for all $s\ge0$.
Compactness now follows from
the $C^0$ estimate discussed
in the first step.
A gluing argument proves that
$\Theta_*$ is a chain map.
Moreover,
inequality~(\ref{eq:AS-inequality})
shows that
$\Mm^+(q,z)=\emptyset$
whenever ${\mathcal{S}}_L(q)\le\Aa_{H_L}(z)$,
unless $q$ and $z$ correspond
to the same critical point.
In this case $\Mm^+(q,z)$
consists of a single element,
the constant solution
$w(s,\cdot)=z(\cdot)$.
There is a differential version
of~(\ref{eq:AS-inequality})
at $z\in{\mathcal{P}}(H_L)$, namely
$$
d^2\Aa_{H_L}(z)\:(\cdot,\cdot)
\le d^2{\mathcal{S}}_L(\pi(z))\:
(D\pi(z)\cdot,D\pi(z)\cdot).
$$
It is used to prove that
at a constant
solution $w(s,\cdot)=z(\cdot)$
the Morse-Smale
condition is automatically true.
Ordering the generators
of the chain groups
by increasing action,
it follows that $\Theta_k$
is an upper triangular matrix
with diagonal entries $\pm1$.
This proves that
$\Theta_*$ is a chain
isomorphism.
\section{Singular perturbation
and adiabatic limit}
\label{sec:SW}
This section reviews
the approach by Salamon and the
present author who
established a natural isomorphism
$\HF_*^a(T^*M,H_V,J_g)\to
\HM_*^a(\Ll M,{\mathcal{S}}_V,L^2)$
in~\cite{JOA3}.
It is work in
progress~\cite{JOA-FUTURE}
to show that the latter
is naturally isomorphic to
$\Ho_*(\Ll^a M)$.
The main idea to relate Floer and
Morse homology is to
introduce a real parameter ${\varepsilon}>0$
in Floer's equations:
replace $(J_g,G_g)$ by
$$
(J_{\varepsilon},G_{\varepsilon})
:=(J_{{\varepsilon}^{-1}g},G_{{\varepsilon}^{-1}g}).
$$
By Floer's continuation principle
this will not change Floer homology.
Let us identify $T^*M$ with $TM$
via the metric isomorphism. A map
$(\tilde u,\tilde v):{\mathbb{R}}\times S^1\to TM$
is a solution to the
$(J_{\varepsilon},G_{\varepsilon})$-Floer
equations~(\ref{eq:floer})
if and only if
$u(s,t):=\tilde u({\varepsilon}^{-1}s,t)$
and $v(s,t):=v({\varepsilon}^{-1}s,t)$
satisfy
\begin{equation}\label{eq:floer-eps}
{\partial}_su-\Nabla{t}v-\nabla V_t(u)
= 0,\qquad\Nabla{s} v
+{\varepsilon}^{-2}\left({\partial}_tu-v\right)=0.
\end{equation}
Denote the space of solutions
to~(\ref{eq:floer-eps})
and~(\ref{eq:floer-lim}) by
$\Mm^{\varepsilon}(x^-,x^+;V)$.
We shall outline how to
prove that there is
a one-to-one correspondence
between the solutions
of~(\ref{eq:floer-eps})
and~(\ref{eq:heat})
subject to boundary
conditions~(\ref{eq:floer-lim})
and~(\ref{eq:heat-lim}), respectively,
whenever the index difference
is one\footnote{The case of
arbitrary index difference
is closely related to Cohen's
conjecture~\cite{Co04}. It
is the missing link
in proving that cylindrical
Gromov-Witten invariants
of the cotangent bundle
represent string topology
of the free loop space.
A different proof relating
the particular cases of
the three point invariant
provided by the pair of pants product
in Floer homology
and the Chas-Sullivan
loop product~\cite{CS99}
is in preparation
by Abbondandolo and
Schwarz~\cite{AS04b}.
}.
This then shows that both
chain complexes are identical.
A first hint that such a
bijection between parabolic and
${\varepsilon}$-elliptic flow lines
exists is provided
by the \emph{energy identity}
\begin{equation*}
\begin{split}
E^{\varepsilon}(u,v)
&:=
\frac12\int_{-\infty}^\infty
\int_0^1\left(
|{\partial}_su|^2
+|\Nabla{t}v+\nabla V_t(u)|^2
+{\varepsilon}^2|\Nabla{s}v|^2
+{\varepsilon}^{-2}|{\partial}_tu-v|^2
\right) \\
&=
{\mathcal{S}}_V(x^-)-{\mathcal{S}}_V(x^+)
\end{split}
\end{equation*}
for the solutions
of~(\ref{eq:floer-eps})
and~(\ref{eq:floer-lim}).
It shows that ${\partial}_tu-v$ must
converge to zero in
$L^2$ as ${\varepsilon}\to0$, but if
${\partial}_tu=v$ then the first equation
in~(\ref{eq:floer-eps}) is equivalent
to~(\ref{eq:heat}).
The next step would be to prove
that ${\mathcal{S}}_V$ is Morse-Smale
for generic $V$.
Unfortunately, this remains
an \emph{open problem}.
Instead we introduce a more
general class of perturbations
for which Morse-Smale transversality
can be achieved generically
by standard methods.
These perturbations take
the form of smooth functions
${\mathcal{V}}:\Ll M\to{\mathbb{R}}$ satisfying
a list of axioms (which
contains the properties used at some
point in the proof; see~\cite{JOA3}).
Assume for the moment that $M$ was
embedded isometrically in some euclidean
space ${\mathbb{R}}^N$, fix a loop $x_0$
in $M$ and let $\rho:{\mathbb{R}}\to[0,1]$
be a smooth cutoff function.
Then a typical example of an
abstract perturbation is given by
$$
{\mathcal{V}}(x)
:=\rho\left(\left\|x
-x_0\right\|_{L^2}^2\right)
\int_0^1V_t(x(t))\,dt.
$$
With new functionals
${\mathcal{S}}_{\mathcal{V}}:={\mathcal{S}}_0+{\mathcal{V}}$
and $\Aa_{\mathcal{V}}:=\Aa_0+{\mathcal{V}}$
equation~(\ref{eq:floer-eps})
turns into
\begin{equation}\label{eq:floer-V}
{\partial}_su-\Nabla{t}v-\grad{\mathcal{V}}(u)
=0,\qquad
\Nabla{s}v+{\varepsilon}^{-2}({\partial}_tu-v)
=0
\end{equation}
and the limit equation is
of the form
\begin{equation}\label{eq:heat-V}
{\partial}_su-\Nabla{t}{\partial}_tu-\grad{\mathcal{V}}(u)
=0.
\end{equation}
Here the $L^2$ gradient
$\grad{\mathcal{V}}(x)\in{\Omega}^0(S^1,x^*TM)$
of ${\mathcal{V}}$ at $x\in\Ll M$ is defined by
$$
\int_0^1\inner{\grad{\mathcal{V}}(u)}
{{\partial}_su}\,dt
:= \frac{d}{ds}{\mathcal{V}}(u)
$$
for every smooth path
${\mathbb{R}}\to\Ll M:s\mapsto u(s,\cdot)$.
The set
${\mathcal{P}}({\mathcal{V}})$ consists of loops
$x:S^1\to M$ satisfying
$\Nabla{t}{\partial}_t x=-\grad{\mathcal{V}}(x)$.
Define $\Mm^{\varepsilon}(x^-,x^+;{\mathcal{V}})$ and
$\Mm^0(x^-,x^+;{\mathcal{V}})$
as before with~(\ref{eq:floer-V})
and~(\ref{eq:heat-V})
replacing~(\ref{eq:floer-eps})
and~(\ref{eq:heat}),
respectively.
Let $V_t$ be a potential such that
${\mathcal{S}}_V$ is a Morse function and denote
$$
{\mathcal{V}}(x):=\int_0^1V_t(x(t))\,dt.
$$
Observe that this choice reproduces
${\mathcal{S}}_V$ and $\Aa_V$, hence the
geometric equations~(\ref{eq:heat})
and~(\ref{eq:floer}).
Fix a regular value $a$ of ${\mathcal{S}}_V$ and
choose a sequence of perturbations
${{\mathcal{V}}_i:\Ll M\to{\mathbb{R}}}$
converging to ${\mathcal{V}}$ in the
$C^{\infty}$ topology and such that
${\mathcal{S}}_{{\mathcal{V}}_i}:\Ll M\to{\mathbb{R}}$
is Morse--Smale for every $i$.
We may assume without
loss of generality that the
perturbations agree with
${\mathcal{V}}$ near the critical points
and that ${\mathcal{P}}({\mathcal{V}}_i)={\mathcal{P}}(V)$
for all $i$. Assume there
is a sequence ${\varepsilon}_i>0$
converging to zero such that,
for every ${\varepsilon}_i$ and
every pair $x^\pm\in{\mathcal{P}}^a(V)$
of index difference one,
there is a ($s$-shift equivariant)
bijection
\begin{equation}\label{eq:bijection}
{\mathcal{T}}^{{\varepsilon}_i}:
\Mm^0(x^-,x^+;{\mathcal{V}}_i)\to
\Mm^{{\varepsilon}_i}(x^-,x^+;{\mathcal{V}}_i).
\end{equation}
The following diagram --
in which arrows represent isomorphisms --
shows how this implies our goal
(with ${\mathbb{Z}}_2$-coefficients).
\begin{equation*}
\begin{split}
\xymatrix{
\HF_*^a(T^*M,{\mathcal{V}}_i,J_{{\varepsilon}_i})
\ar[d
^{(\ref{eq:bijection})}
&
\HF_*^a(T^*M,H_V+W,J_{{\varepsilon}_i})
\ar[l]_{\text{Floer}\quad}
^{\text{contin.}\quad}
&
\HF_*^a(T^*M,H_V,J_{{\varepsilon}_i})
\ar[l]_{\quad\;\text{def}}
\\
\HM_*^a(\Ll M,{\mathcal{S}}_{{\mathcal{V}}_i},L^2)
\ar[d]^{\textstyle\cite{JOA-FUTURE}
&
&
\\
\Ho_*(\{{\mathcal{S}}_{{\mathcal{V}}_i}\le a\})
\ar[rr]^{\text{homotopy
equivalent spaces}}
&
&
\Ho_*(\{{\mathcal{S}}_V\le a\})
}
\end{split}
\end{equation*}
Starting at the upper right corner,
the first step is by
definition of Floer homology
for nonregular $H_V$, namely
add a small Hamiltonian
perturbation $W$ making
$H_V+W$ regular.
Floer continuation shows
independence of the choice.
Again by Floer's continuation argument
small Hamiltonians
and small abstract perturbations
lead to isomorphic homology groups.
It remains to construct the bijection
${\mathcal{T}}^{\varepsilon}$ in the case of index
difference one.
Assume throughout that
${\mathcal{S}}_{\mathcal{V}}$ is Morse-Smale and fix a regular
value $a$ and a pair $x^\pm\in{\mathcal{P}}^a(V)$
of index difference one.
Denote $\Mm^0:=\Mm^0(x^-,x^+;{\mathcal{V}})$ and
$\Mm^{\varepsilon}:=\Mm^{\varepsilon}(x^-,x^+;{\mathcal{V}})$.
\subsubsection*{Existence and uniqueness}
\label{subsubsec:exist-unique}
Given a parabolic solution we shall prove
by Picard-Newton iteration
existence and uniqueness of an elliptic
solution nearby.
The first step is to define, for smooth maps
$(u,v):{\mathbb{R}}\times S^1\to TM$ and
$p>2$, a map ${\mathcal{F}}^{\varepsilon}_{u,v}$
between the Banach spaces
of $W^{1,p}$ and $L^p$ sections of the bundle
$u^*TM\oplus u^*TM\to{\mathbb{R}}\times S^1$.
To obtain uniform estimates for
small ${\varepsilon}$ we introduce
weighted norms. The weights for $p=2$
are suggested by the energy identity.
Let ${\mathcal{F}}_{\varepsilon}(u,v)$
be given by the left hand side of
the ${\varepsilon}$-equations~(\ref{eq:floer-eps})
and denote by
$\Phi(x,\xi):T_xM\to T_{\exp_x(\xi)}M$
parallel transport along the
geodesic $\tau\mapsto\exp_x(\tau\xi)$.
For compactly supported vector fields
$\zeta=(\xi,\eta)\in
{\Omega}^0({\mathbb{R}}\times S^1,u^*TM\oplus u^*TM)$
define the weighted norms
$$
\left\|\zeta\right\|_{0,p,{\varepsilon}}
:= \left(\int_{-\infty}^\infty\int_0^1
\left(\left|\xi\right|^p
+ {\varepsilon}^p\left|\eta\right|^p\right)\,dtds
\right)^{1/p},
$$
\begin{equation*}
\begin{split}
\left\|\zeta\right\|_{1,p,{\varepsilon}}
&:=
\biggl(\int_{-\infty}^\infty\int_0^1
\bigl(\left|\xi\right|^p
+ {\varepsilon}^p\left|\eta\right|^p
+ {\varepsilon}^p\left|\Nabla{t}\xi\right|^p
+ {\varepsilon}^{2p}\left|\Nabla{t}\eta\right|^p \\
&\quad
+\, {\varepsilon}^{2p}\left|\Nabla{s}\xi\right|^p
+ {\varepsilon}^{3p}\left|\Nabla{s}\eta\right|^p
\bigr)\,dtds\biggr)^{1/p},
\end{split}
\end{equation*}
and ${\mathcal{F}}_{u,v}^{\varepsilon}:
W^{1,p}({\mathbb{R}}\times S^1,u^*TM\oplus u^*TM) \to
L^p({\mathbb{R}}\times S^1,u^*TM\oplus u^*TM)$ by
\begin{equation*
{\mathcal{F}}_{u,v}^{\varepsilon} \begin{pmatrix} \xi \\
\eta \end{pmatrix}
:=
\begin{pmatrix} \Phi(u,\xi)^{-1} & 0 \\
0 & \Phi(u,\xi)^{-1} \end{pmatrix}
{\mathcal{F}}_{\varepsilon} \begin{pmatrix} \exp_u \xi \\
\Phi(u,\xi) (v + \eta) \end{pmatrix}.
\end{equation*}
Abbreviate ${\mathcal{D}}_{u,v}^{\varepsilon}
:=d{\mathcal{F}}_{u,v}^{\varepsilon}(0,0)$ and
${\mathcal{D}}_u^{\varepsilon}:={\mathcal{D}}_{u,{\partial}_tu}^{\varepsilon}$.
Let us fix a parabolic cylinder
$u\in\Mm^0$,
viewed as an approximate
solution of the ${\varepsilon}$-elliptic
equations. Equivalently,
we view the origin as
approximate zero
of the map ${\mathcal{F}}_u^{\varepsilon}
:={\mathcal{F}}_{u,{\partial}_tu}^{\varepsilon}$
between Banach spaces.
Carrying out the Newton-Picard iteration
for this map we shall prove existence
of a true zero nearby.
The iteration method works if,
firstly, the initial value is small,
secondly, the linearized operator admits
a right inverse, and thirdly,
second derivatives
of the map can be controlled.
These conditions must be satisfied
uniformly for small ${\varepsilon}>0$.
Choosing the origin as the
initial point of the iteration
we observe that
$$
\Norm{{\mathcal{F}}_u^{\varepsilon}(0,0)}_{0,p,{\varepsilon}}
=\Norm{{\mathcal{F}}_{\varepsilon}(u,{\partial}_tu)}_{0,p,{\varepsilon}}
=\Norm{(0,\Nabla{s}{\partial}_tu)}_{0,p,{\varepsilon}}
={\varepsilon} \Norm{\Nabla{s}{\partial}_tu}_p
\le c_0{\varepsilon}.
$$
The second identity uses
the heat equation~(\ref{eq:heat})
and the final estimate is by
exponential decay of heat flow solutions
with nondegenerate boundary conditions.
Verification of the second condition relies
heavily on the fact that ${\mathcal{D}}_u^{\varepsilon}$
is Fredholm and surjective. (Fredholm
follows by nondegeneracy of the boundary
conditions~(\ref{eq:floer-lim})
and surjectivity is a consequence
of the Morse-Smale assumption
for ${\mathcal{S}}_{\mathcal{V}}$).
Let ${{\mathcal{D}}_u^{\varepsilon}}^*$
be the adjoint operator of ${\mathcal{D}}_u^{\varepsilon}$
with respect to the $L^2$ inner product
$\langle\cdot,\cdot\rangle_{\varepsilon}$ with
associated norm ${\rm norm}{\cdot}_{0,2,{\varepsilon}}$.
A right inverse of ${\mathcal{D}}_u^{\varepsilon}$
is given by
$
{\mathcal{Q}}_u^{\varepsilon}
:={{\mathcal{D}}_u^{\varepsilon}}^*
\left({\mathcal{D}}_u^{\varepsilon}{{\mathcal{D}}_u^{\varepsilon}}^*\right)^{-1}.
$
It allows to solve the equation
$0={\mathcal{D}}_u^{\varepsilon}\zeta_0+{\mathcal{F}}_u^{\varepsilon}(0,0)$
and provides the correction term
$\zeta_0:=-{\mathcal{Q}}_u^{\varepsilon}{\mathcal{F}}_u^{\varepsilon}(0,0)$.
Recursively, for $\nu\in{\mathbb{N}}$, define
the sequence of correction terms
$\zeta_\nu=(\xi_\nu,\eta_\nu)$
by
\begin{equation*}
\zeta_\nu
:=- {{\mathcal{D}}_u^{\varepsilon}}^*
({\mathcal{D}}_u^{\varepsilon}{{\mathcal{D}}_u^{\varepsilon}}^*)^{-1}
{\mathcal{F}}_u^{\varepsilon} (Z_\nu),\qquad
Z_\nu
:=\sum_{\ell=0}^{\nu-1} \zeta_\ell.
\end{equation*}
To estimate the terms $\zeta_\nu$
it is crucial to have an estimate
for ${\mathcal{D}}_u^{\varepsilon}$ on the image of its adjoint
operator. This is a consequence
of a Calderon-Zygmund estimate
for a Cauchy-Riemann type operator.
It follows that $Z_\nu$
is a Cauchy sequence.
The third condition is verified
by quadratic estimates, i.e.
estimates for
$$
{\mathcal{F}}_u^{\varepsilon}(Z_\nu+\zeta_\nu)
-{\mathcal{F}}_u^{\varepsilon}(Z_\nu)
-d{\mathcal{F}}_u^{\varepsilon}(Z_\nu)\zeta_\nu
,\qquad
d{\mathcal{F}}_u^{\varepsilon}(Z_\nu)\zeta_\nu
-{\mathcal{D}}_u^{\varepsilon}\zeta_\nu.
$$
One proves by induction that
there is a constant $c>0$ such that
$$
\left\|{\mathcal{F}}_u^{\varepsilon}(Z_{\nu+1})
\right\|_{0,p,{\varepsilon}^{3/2}}
\le\frac{c}{2^\nu}{\varepsilon}^{7/2-3/2p}.
$$
for all $\nu\in{\mathbb{N}}$ and ${\varepsilon}>0$ small.
Therefore the Cauchy sequence $Z_\nu$
converges to a zero $Z$ of ${\mathcal{F}}_u^{\varepsilon}$
with $Z\in\im {{\mathcal{D}}_u^{\varepsilon}}^*$
and $\Norm{Z}_{1,p,{\varepsilon}}\le c{\varepsilon}^2$.
Now $Z=(X,Y)$ corresponds to
a zero $(u^{\varepsilon},v^{\varepsilon})$ of ${\mathcal{F}}_{\varepsilon}$,
more precisely to an element of $\Mm^{\varepsilon}$,
and we define
$$
{\mathcal{T}}^{\varepsilon}(u)
:=(u^{\varepsilon},v^{\varepsilon})
:=(\exp_u X,\Phi(u,X)({\partial}_t u+Y)).
$$
Using again the quadratic
estimates, roughly speaking,
one can prove uniqueness
of the constructed solution
$(u^{\varepsilon},v^{\varepsilon})$ in an even
bigger neighbourhood of $(u,{\partial}_tu)$.
Here a crucial assumption is
that the 'difference' $Z$ between
the parabolic and ${\varepsilon}$-elliptic solution
is in the image of ${{\mathcal{D}}_u^{\varepsilon}}^*$.
It follows that ${\mathcal{T}}^{\varepsilon}$ is well defined.
Moreover, it is time shift equivariant.
Injectivity of ${\mathcal{T}}^{\varepsilon}$ follows,
because the quotient
$\Mm^0/{\mathbb{R}}$ by the time shift action
is a finite set and so admits
a positive smallest distance
between its elements.
Since the existence and uniqueness
range shrinks like ${\varepsilon}^2$, injectivity holds
for all sufficiently small ${\varepsilon}$.
\subsubsection*{Surjectivity}
\label{subsubsec:surjectivity}
In four steps we sketch the proof that
the map ${\mathcal{T}}^{\varepsilon}:\Mm^0\to\Mm^{\varepsilon}$
is surjective whenever
${\varepsilon}>0$ is sufficiently
small.
The first step establishes
uniform \emph{apriori $L^\infty$ bounds}
for elliptic solutions
$(u^{\varepsilon},v^{\varepsilon})\in\Mm^{\varepsilon}$ and
their first
and second derivatives.
In each case this is based
on first proving
\emph{slicewise} $L^2$ bounds,
i.e. bounds in $L^2(S^1)$
for arbitrary fixed $s\in{\mathbb{R}}$,
and then applying a mean value inequality
for the operator $L_{\varepsilon}:={\varepsilon}^2{\partial}_s^2
+{\partial}_t^2-{\partial}_s$. More precisely, we prove that
there is a constant $c>0$ such that
for all ${\varepsilon}>0$, $r\in(0,1]$, and
$\mu\ge0$ the following is true.
If $w$ is a smooth function
(for instance $w=\abs{v^{\varepsilon}}^2$)
defined on the
\emph{parabolic domain} $P_r^{\varepsilon}:=
(-r^2-{\varepsilon} r,{\varepsilon} r)\times(-r,r)$, then
$$
L_{\varepsilon} w
:=\left({\varepsilon}^2{{\partial}_s}^2+{{\partial}_t}^2
-{\partial}_s\right)w
\ge-\mu w, \quad w\ge0
$$
imply
$$
w(0)
\le\frac{2ce^{\mu r^2}}{r^3}
\int_{P_r^{\varepsilon}}w.
$$
Now use the slicewise $L^2$
estimates on the right hand side
of the inequality to obtain
the $L^\infty$ estimate.
A bubbling argument
enters the proof in the case of
first derivatives,
as is expected by comparison
with standard Floer theory.
It provides a rather weak
(in terms of powers of ${\varepsilon}$)
preliminary $L^\infty$
estimate which, however, suffices
to establish the slicewise $L^2$ bounds.
The second step proves \emph{uniform
exponential decay} of $\Abs{{\partial}_su^{\varepsilon}(s,t)}
+\Abs{\Nabla{s}v^{\varepsilon}(s,t)}$
towards the ends of the cylinder.
The standard method of proof
works uniformly in ${\varepsilon}\in(0,1]$
under the assumption that
the energy near the ends of the cylinder
is uniformly bounded by a small
constant (no energy concentration
near infinity).
In Step three we establish
\emph{local surjectivity} by means of
a time shift argument.
Given $u\in\Mm^0$
and $(u^{\varepsilon},v^{\varepsilon})\in\Mm^{\varepsilon}$
sufficiently close to $(u,{\partial}_tu)$,
define $\zeta=(\xi,\eta)$
by $u^{\varepsilon}=\exp_u(\xi)$ and
$v^{\varepsilon}=\Phi(u,\xi)({\partial}_tu+\eta)$.
The idea is to prove that
after a suitable time shift
the pair $\zeta=(\xi,\eta)$
satisfies the crucial hypothesis
$\zeta\in\im\,{{\mathcal{D}}^{\varepsilon}_u}^*$
in the uniqueness theorem
and therefore
$(u^{\varepsilon},v^{\varepsilon})={\mathcal{T}}^{\varepsilon}(u)$.
More precisely, since the Fredholm
index of ${\mathcal{D}}^{\varepsilon}_u$
is given by the Morse index
difference
and ${\mathcal{D}}^{\varepsilon}_u$ is surjective
by the Morse-Smale assumption,
it follows that $\ker {\mathcal{D}}^{\varepsilon}_u$
is generated by a nonzero vector
$Z^{\varepsilon}$.
Because $\im {{\mathcal{D}}^{\varepsilon}_u}^*$
can be identified with
the orthogonal complement
of $\ker {\mathcal{D}}^{\varepsilon}_u$ with respect to
the $L^2$ inner product
$\langle\cdot,\cdot\rangle_{\varepsilon}$,
it remains to prove that the function
$$
\theta^{\varepsilon}(\sigma)
:=-\left\langle Z^{\varepsilon}_\sigma,\zeta_\sigma
\right\rangle_{\varepsilon},\qquad
Z^{\varepsilon}
:=\begin{pmatrix}{\partial}_su\\
\Nabla{t}{\partial}_su\end{pmatrix}
-{{\mathcal{D}}^{\varepsilon}_u}^*
\left({\mathcal{D}}^{\varepsilon}_u{{\mathcal{D}}^{\varepsilon}_u}^*\right)^{-1}
{\mathcal{D}}^{\varepsilon}_u
\begin{pmatrix}{\partial}_su\\\Nabla{t}{\partial}_su
\end{pmatrix}
$$
admits a zero.
Here the $s$-shift is defined by
$\zeta_\sigma(\cdot,\cdot)
:=\zeta(\cdot+\sigma,\cdot)$.
In Step four we prove surjectivity.
Assume by contradiction that there
is a sequence $(u_i,v_i)\in\Mm^{{\varepsilon}_i}$
with ${\varepsilon}_i$ converging to zero and such
that $(u_i,v_i)\notin{\mathcal{T}}^{{\varepsilon}_i}(\Mm^0)$.
Viewing the $u_i$ as approximate zeroes
of the parabolic section ${\mathcal{F}}_0$, defined
by the left hand side of~(\ref{eq:heat}),
we construct a sequence of parabolic
solutions $u_i^0\in\Mm^0$ by
Newton-Picard iteration.
Here two conditions need to be satisfied
uniformly for large $i\in{\mathbb{N}}$.
Firstly, we need
a small initial value of ${\mathcal{F}}_0$.
This follows from the elliptic
equations~(\ref{eq:floer-eps})
and Step one:
$$
\Norm{{\mathcal{F}}_0(u_i)}_p
=\Norm{{\partial}_su_i-\Nabla{t}{\partial}_tu_i
-\grad{\mathcal{V}}(u_i)}_p
={\varepsilon}_i^2\Norm{\Nabla{t}\Nabla{s} v_i}_p
\le c{\varepsilon}_i^2.
$$
Secondly, one needs to prove
asymptotic decay of the form
$$
\Abs{{\partial}_su_i(s,t)}+\Abs{\Nabla{s}v_i(s,t)}
\le\frac{c}{1+s^2}.
$$
This follows from uniform exponential decay
proved in Step two. The iteration shows
that $(u_i^0,{\partial}_tu_i^0)$ and
the given elliptic solution
$(u_i,v_i)$ are sufficiently close
(for large $i$) such that the time shift
argument of Step three applies.
Hence $(u_i,v_i)={\mathcal{T}}^{{\varepsilon}_i}(u_i^0)$, but
this contradicts the assumption.
\section{Finite dimensional approximation}
\label{sec:V}
Reviewing Viterbo's
paper~\cite{V96}
is somewhat delicate
due to its state
of presentation.
As a way out we decided to
enlist the steps of proof
as provided by~\cite{V96}
using the original notation
and conventions.
In view of its independent
interest, we discuss
Step~1 including full details
(up to the hypothesis that
both gradings coincide)
in a separate section.
Throughout let $(M,g)$ be
a closed Riemannian manifold
of dimension $n$.
In~\cite{V96} cohomology
is considered and the main
result is stated in the form
\begin{equation}\label{eq:viterbo-main}
\HF^*(DT^*M)
\simeq\Ho^*(\Ll M)
\end{equation}
where $DT^*M$ denotes the
open unit disc bundle.
On both sides
contractible loops are
considered and the isomorphism
is claimed with rational
coefficients. A further claim is
a version of~(\ref{eq:viterbo-main})
for $S^1$-equivariant cohomologies
(see~\cite{V99} for applications).
Since orientation
of moduli space
is not discussed in~\cite{V96},
we assume
throughout Section~\ref{sec:V}
that all homologies take
coefficients in ${\mathbb{Z}}_2$.
Actually~\cite{V96} is part two
of a series of two papers.
The left hand side
of~(\ref{eq:viterbo-main})
is defined in part one~\cite{V99}
in the more general context
of symplectic manifolds with
contact type boundary (see the
excellent recent survey by
Oancea~\cite{O04}):
take the symplectic
completion of the closed
unit disc bundle
(which is symplectomorphic
to $(T^*M,\omega_0)$ itself),
fix $\delta<1$ close to $1$
and consider Hamiltonians of the form
$$
H(t,q,p)=h_\lambda(\abs{p}).
$$
Here $h_\lambda$ is a smooth
convex real function
which vanishes for $\abs{p}<\delta$
and which is linear
of slope $\lambda>0$
for $\abs{p}\ge1$.
To define Floer cohomology
associated to $h_\lambda$
it is important to assume
that $\lambda$ is not
the length of a contractible
periodic geodesic in $M$
(see Remark~\ref{rmk:radial}).
The left hand
side of~(\ref{eq:viterbo-main})
is then defined
by the direct limit
$$
\HF^*(DT^*M)
:=\underset{\lambda\to\infty}
{\underrightarrow{\lim}}
\HF^*(T^*M,h_\lambda,J_g).
$$
To prove~(\ref{eq:viterbo-main})
it suffices to show
$$
\HF^*(T^*M,h_\lambda,J_g)
\simeq
\Ho^*(\Ll^{\lambda^2/2} M).
$$
The proof rests on the idea
of Chaperon~\cite{Ch84}
to adapt the finite
dimensional approximation
of the loop space
via piecewise geodesics
to the case of Hamiltonian
flow lines. Givental~\cite{G89}
refined this idea for
Hamiltonian flows
which are periodic in time.
From now on fix
$h=h_\lambda$, denote
by $\varphi_t^h$ the time-t-map
generated by the Hamiltonian
vector field $X_h$, and
set $\varphi^h:=\varphi_1^h$.
The proof has seven steps:
\begin{equation*}
\begin{split}
\HF^*(T^*M,h,J_g)
&\stackrel{1}{\simeq}
\HF^*(\Delta_r,\Gamma_r
(\varphi^h),
(T^*M\times
\overline{T^*M})^{\times r},J_r
) \\
&\stackrel{2}{\simeq}
\HF^*(U_{r,{\varepsilon}},
\graph\: dS_\psi,
T^*U_{r,{\varepsilon}},\Psi_*J_r
) \\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
&\stackrel{3}{\simeq}
\HM^{*+rn}(U_{r,{\varepsilon}},
S_\psi)\\
&\stackrel{4}{\simeq}
\HI^{*+rn}(U_{r,{\varepsilon}},
\nabla S_\psi)\\
&\stackrel{5}{\simeq}
\HI^{*+rn}(U_{r,{\varepsilon}},
\xi_\psi)\\
&\stackrel{6}{\simeq}
\Ho^*(\Lambda^a_{r,{\varepsilon}}) \\
&\stackrel{7}{\simeq}
\Ho^*(\Ll^{\lambda^2/2} M).
\end{split}
\end{equation*}
Here $\HF^*(L_0,L_1;N)$ denotes
Floer cohomology associated to
Lagrangian submanifolds
$L_0$ and $L_1$
of a symplectic manifold $N$,
Morse cohomology
is denoted by $\HM^*$,
and the cohomological
Conley index by $\HI^*$.
The other symbols are
introduced as they appear
in our discussion of
the seven steps.
The whole proof relies
on writing $\varphi^h$
as an $r$-fold composition
$\psi^r:=\psi\circ\dots \circ\psi$
for sufficiently large $r\in{\mathbb{N}}$.
Since $h$ does
not depend on time,
we can indeed choose
$\psi:=\varphi_{1/r}^h$.
\smallbreak
\noindent
{\sc Step 1}
Let $\omega_0$ denote the
canonical symplectic structure
on $T^*M$, fix $r\in{\mathbb{N}}$, and consider
the symplectic manifold
$$
(T^*M\times\overline{T^*M})^{\times r}
:=
\Bigl((T^*M\times T^*M)^{\times r},
\omega_r:=\bigoplus_1^r
\omega_0 \oplus -\omega_0\Bigr)
$$
and the Lagrangian submanifolds
$$
\Gamma_r(\varphi^h)
:=\{(z_1,\psi z_r;
z_2,\psi z_1; \dots ;
z_r,\psi z_{r-1})\mid
z_1,\dots,z_r \in T^*M\},\quad
\Delta_r
:=\Delta^{\times r}.
$$
Here $\Delta$ denotes the diagonal
in $T^*M\times T^*M$.
Let ${\mathcal{P}}(h)$ denote
the set of contractible 1-periodic
orbits of $h$.
Since
$$
\Gamma_r(\varphi^h)
\cap\Delta_r
=\{(z,z;\psi z,\psi z; \dots ;
\psi^{r-1} z,\psi^{r-1} z) \mid
z\in\Fix \:\varphi^h\}
\simeq {\mathcal{P}}(h),
$$
it follows that the generators
of both chain groups
coincide up to natural identification.
That this identification
preserves the grading
given by the Conley-Zehnder index
and the Maslov index, respectively,
seems to be an open problem.\\
The boundary operator
of Lagrangian intersection
Floer homology counts $r$-tuples
of pairs of $J_r$-holomorphic strips
in $T^*M$, where $J_r$
is defined by~(\ref{eq:J_r}).
The boundary operator
of periodic
Floer homology counts
$J_g$-holomorphic cylinders in $T^*M$.
The idea to prove that both boundary
operators coincide is to establish
a one-to-one correspondence
by gluing together the strips
of an $r$-tuple to obtain a cylinder.
Details of this argument will be
discussed in a separate
section below.
\begin{remark}[Transversality]\rm
Working with time independent
Hamiltonians seems unrealistic
at a first glance,
since Morse-Smale
transversality requires
time dependence.
We suggest perturbing $h$
in $C^{\infty}({\mathbb{R}}/r^{-1}{\mathbb{Z}}\times DT^*M)$.
Due to the $1/r$-periodicity of
the perturbed Hamiltonian
$H$ the crucial decomposition
$\varphi^H=\psi^r$ is still available:
fix an initial time, say $t_0=0$,
and set $\psi:=\varphi^H_{0,1/r}$.
Since $H$ is still radial
outside a compact set,
Step~1 still goes through.
\end{remark}
\smallbreak
\noindent
{\sc Step 2}
In this step coordinates
are changed via
a symplectomorphism which
identifies $\Delta_r$ with
$(T^*M)^{\times r}$ and
the essential part of
$\Gamma_r(\varphi^h)$ with the
graph of the differential of a function.
Let $d$ denote
the Riemannian distance on $M$ and
consider the neighborhood
$(V_{\varepsilon})^{\times r}$ of $\Delta_r$ where
$$
V_{\varepsilon}:=\{ (q,p;Q,P)\in
T^*M\times T^*M \mid
d(q,Q)\le {\varepsilon} \}.
$$
Since $\Delta$ is diffeomorphic
to $T^*M$, clearly
$\Delta_r$ is diffeomorphic
to $(T^*M)^{\times r}$.
For sufficiently small ${\varepsilon}>0$
a proper symplectic embedding
$$
\Psi:(V_{\varepsilon})^{\times r}\to
T^*(T^*M)^{\times r}
$$
is constructed
in~\cite[Lemma~1.1]{V97}
identifying $\Delta_r$ with
the zero section
(see Figure~\ref{fig:fig-spsi}).
According to~\cite{V97}
there exists ${\varepsilon}>0$ sufficiently small
such that for every
sufficiently large $r\in{\mathbb{N}}$
there is a function $S_\psi$
(denoted by ${\mathcal{S}}_\Phi$ in~\cite{V97};
note that $\psi$ depends on $r$)
which is defined on the set
$$
U_{r,{\varepsilon}}
:=\{ (q_1,p_1;\dots;q_r,p_r)\in
(T^*M)^{\times r} \mid
d(q_j,q_{j+1})\le \frac{{\varepsilon}}{2}\,\,
\forall j\in{\mathbb{Z}}_r \}.
$$
It has the property that
the image under $\Psi$ of
the part of $\Gamma_r(\varphi^h)$
contained in $(V_{\varepsilon})^{\times r}$
equals the graph of $dS_\psi$.
\begin{figure}[ht]
\centering
\epsfig{figure=fig-spsi.eps}
\caption{Change of coordinates $\Psi$
identifying
$\Gamma_r(\varphi^h)
\cap(V_{\varepsilon})^{\times r}$
with $\graph\:dS_\psi$.}
\label{fig:fig-spsi}
\end{figure}
\noindent
To see existence of $S_\psi$
note that the Lagrangian
$\Psi\left(\Gamma_r(\id)
\cap(V_{\varepsilon})^{\times r}\right)$
is a graph over $U_{r,{\varepsilon}}$ and
$\Psi\left(\Gamma_r(\varphi^h)
\cap(V_{\varepsilon})^{\times r}\right)$
is close to this graph
whenever $r$ is sufficiently large.
Hence it is a Lagrangian
graph over $U_{r,{\varepsilon}}$ itself.
It is exact,
because the symplectomorphism
$\varphi^h$ is Hamiltonian.
To arrive at the claim of Step~2
consider the two
isomorphisms given by
\begin{equation*}
\begin{split}
&\HF^*(\Delta_r,\Gamma_r
(\varphi^h),
(T^*M\times
\overline{T^*M})^{\times r},J_r
) \\
&\simeq
\HF^*((T^*M)^{\times r},
\Psi(\Gamma_r(\varphi^h)
\cap(V_{\varepsilon})^{\times r}),
T^*(T^*M)^{\times r},\Psi_*J_r
) \\
&\simeq
\HF^*(U_{r,{\varepsilon}},
\graph\: dS_\psi,
T^*U_{r,{\varepsilon}},\Psi_*J_r
).
\end{split}
\end{equation*}
The first isomorphism
is induced by the symplectic
embedding $\Psi$.
One needs to prove
that the Floer complex
associated to
$(\Delta_r,\Gamma_r(\varphi^h);
(T^*M\times\overline{T^*M})^{\times r};
J_r)$ lives entirely inside
the neighborhood
$(V_{\varepsilon})^{\times r}$ of $\Delta_r$.
This is clear for the
generators, i.e. the
points of intersection,
but not so much for
the connecting trajectories,
i.e. the $J_r$-holomorphic strips.
The second isomorphism
follows, if again the whole Floer complex
associated to the large
space $T^*(T^*M)^{\times r}$
lives in the smaller
space $T^*U_{r,{\varepsilon}}$.
This is clear for the
generators and for
the connecting trajectories
a proof is given in~\cite{V96}.
\smallbreak
\noindent
{\sc Step 3}
The Hamiltonian
flow on $T^*U_{r,{\varepsilon}}$
generated by the Hamiltonian
$(X,Y)\mapsto -S_\psi(X)$
is given by
$$
\phi_t(X,Y)=(X,Y+tdS_\psi(X)).
$$
According to~\cite{V96}
the Floer cohomologies
associated to $\Psi_*J_r$
and $(\phi_t)_*\Psi_*J_r$,
respectively, are isomorphic
by continuation.
Furthermore, it is
shown in~\cite{V96}
that the generators and
connecting trajectories
of the Floer complex
associated to $(\phi_t)_*\Psi_*J_r$
are in one-to-one correspondence
with those of the Morse complex
associated to $(U_{r,{\varepsilon}},\nabla S_\psi)$.
Therefore the corresponding
cohomologies coincide up to
the shift in the grading.
\smallbreak
\noindent
{\sc Step 4}
Here $\HI^*$ denotes the
cohomological Conley index
and the isomorphism is refered
to~\cite{F89b}.
\smallbreak
\noindent
{\sc Step 5}
Existence of the pseudogradient vector
field $\xi_\psi$ for $S_\psi$
is established in~\cite{V97}.
Let ${\rm I}_*(U_{r,{\varepsilon}},\xi_\psi)$ be
the Conley index, i.e.
the homotopy type
of the quotient of $U_{r,{\varepsilon}}$
by the exit set with respect
to the flow generated by $\xi_\psi$.
According to~\cite{V96}
it holds
$$
{\rm I}_*(U_{r,{\varepsilon}},\nabla S_\psi)
\simeq
{\rm I}_*(U_{r,{\varepsilon}/2},\xi_\psi)
$$
and the spaces are independent
of ${\varepsilon}$ whenever $r$
is sufficiently large.
\smallbreak
\noindent
{\sc Step 6}
Let ${\varepsilon}>0$ be smaller than the injectivity
radius of $M$ and consider the finite
dimensional approximation
of the free loop space given by
$$
\Lambda_{r,{\varepsilon}}
:=\{ (q_1,\dots,q_r)\in
M^{\times r} \mid
d(q_j,q_{j+1})\le \frac{{\varepsilon}}{2}\,\,
\forall j\in{\mathbb{Z}}_r \}.
$$
Define a function on $\Lambda_{r,{\varepsilon}}$ by
$$
E_\psi(q_1,\dots,q_r)
:=
\sup_{(p_1,\dots,p_r)}
S_\psi(q_1,p_1;\dots;q_r,p_r)
$$
and set
$\Lambda_{r,{\varepsilon}}^a
:=\{E_\psi\le a\}$.
The Conley index of
$(U_{r,{\varepsilon}},\xi_\psi)$
is calculated
in~\cite[Prop.~1.7]{V97}
and the result is
the Thom space of some
vector bundle of rank $rn$
over $\Lambda_{r,{\varepsilon}}^a$,
for sufficiently large $a$.
Taking cohomology
the Thom isomorphism
leads to the shift
in the grading and proves
Step~6.
\smallbreak
\noindent
{\sc Step 7}
According to~\cite[p.~438]{V97}
the space $\Lambda_{r,{\varepsilon}}^a$ approximates
the free loop space
$\Ll^{\lambda^2/2} M$ for $a\to \infty$,
$r\to \infty$ and ${\varepsilon}\to0$.
\smallbreak
Note that
generating function homology
has not been used throughout
the seven steps -- in contrast
to what one expects after
a glimpse into~\cite{V96}.
\subsection*{Step~1 revisited}
We shall consider the
case of Floer homology
instead of Floer cohomology
and continue our discussion of
Step~1 above.
First we review
the original
approach in~\cite{V96}.
Then we propose a
proof of the final argument
along different lines.
To establish the isomorphism
between periodic and Lagrangian
Floer homology
it remains to check
that the connecting trajectories
arising in both situations
are in one-to-one correspondence.
In the case of periodic Floer homology
these are solutions
$w:{\mathbb{R}}\times S^1\to T^*M$
satisfying
\begin{equation}\label{eq:floer-H}
{\partial}_s w+J_g(w){\partial}_t w
-\nabla h(w)
=0
\end{equation}
and appropriate
boundary conditions.
In the case of Lagrangian intersection
Floer homology these
are $r$ pairs of strips
$
W:=(u_1,\hat{v}_1;\dots;
u_r,\hat{v}_r) :
{\mathbb{R}}\times [0,1]\to
(T^*M\times\overline{T^*M})^{\times r}
$
satisfying
\begin{equation}\label{eq:U-V}
{\partial}_s W+J_r(W)
{\partial}_\tau W
=0,\qquad
W(s,0)\in\Delta_r,\qquad
W(s,1)\in
\Gamma_r(\varphi^h).
\end{equation}
Here we suggest to use the
almost complex structure
\begin{equation}\label{eq:J_r}
J_r=J_{r,\tau}:=\bigoplus_1^r J_g\oplus
-(\varphi_{-\frac{\tau}{r}}
^h)^* J_g
\end{equation}
in order to make the key
idea in~\cite{V96} work:
achieve matching boundary
conditions by redefining the
$\hat{v}_j$'s
in the form
$v_j(s,\tau):=
\varphi_{-\frac{\tau}{r}}^h\circ
\hat{v}_j(s,\tau)$.
With this definition~(\ref{eq:U-V})
is equivalent to
\begin{equation}\label{eq:u-v}
\begin{aligned}
{\partial}_s u_j+J_g(u_j){\partial}_\tau u_j
&=0,&
v_j(s,0)
&=u_j(s,0),
\\
{\partial}_s v_j-J_g(v_j){\partial}_\tau v_j
&=\frac{1}{r}\nabla
h(v_j),\qquad&
u_j(s,1)
&=v_{j+1}(s,1),
\end{aligned}
\end{equation}
for $j=1,\dots,r$.
Here and throughout we
identify $r+1$ and $1$.
Note that the minus sign in the second
PDE is fine, since we need to
reverse time in the $v_j$'s
when fitting them together
with the $u_j$'s to obtain
a perturbed $J$-holomorphic map
$\tilde{w}:{\mathbb{R}}\times S^1\to T^*M$
(see Figure~\ref{fig:fig-wjuj}a).
\begin{figure}[ht]
\centering
\epsfig{figure=fig-wjuj.eps}
\caption{a) Cylinder of
$J$-holomorphic strips.
b) Time dependence of $\nabla F$.}
\label{fig:fig-wjuj}
\end{figure}
More precisely, define
\begin{equation}\label{eq:w-tilde}
\tilde{w}(s,t)
:=
\begin{cases}
v_j(2rs,1-2rt+2j-2)
&,\, j\in\{1,\dots,r\},\,
t\in [\frac{2j-2}{2r},
\frac{2j-1}{2r}],
\\
u_j(2rs,2rt-2j+1)
&,\, j\in\{1,\dots,r\},\,
t\in [\frac{2j-1}{2r},
\frac{2j}{2r}],
\end{cases}
\end{equation}
and consider the perturbation
associated to the Hamiltonian
(see Figure~\ref{fig:fig-wjuj}b)
\begin{equation*}
F_t(q,p)
:=
\begin{cases}
2h(\abs{p})
&,\, j\in\{1,\dots,r\},\,
t\in [\frac{2j-2}{2r},
\frac{2j-1}{2r}],
\\
0
&,\,\text{else}.
\end{cases}
\end{equation*}
The argument
in~\cite{V96} concludes as follows:
the time-1-maps associated to
$F$ and $h$ coincide, hence
${\mathcal{P}}(F)\simeq {\mathcal{P}}(h)$,
and $\tilde{w}$ solves
\begin{equation*
{\partial}_s \tilde{w}
+J_g(\tilde{w}){\partial}_t \tilde{w}
-\nabla F_t(\tilde{w})
=0
\end{equation*}
iff the pairs
$(u_j,v_j)$ solve~(\ref{eq:u-v}).
Hence both
chain complexes are equal and
$$
\HF_*(\Delta_r,
\Gamma_r(\varphi^h),
(T^*M\times
\overline{T^*M})^{\times r},J_r)
\simeq
\HF_*(T^*M,F,J_g).
$$
According to~\cite{V96}
continuation\footnote{After writing
this paper Viterbo informed us
that here continuation
does not refer to Floer continuation
but to the following:
assume the 1-periodic orbits do not depend
on the parameter $\lambda\in[0,1]$
of a homotopy $f_\lambda$
between $F$ and $h$ and
consider two 1-periodic orbits $z^\pm$
of index difference one. Set
$X=\{(\lambda,w)\mid{\partial}_s w
+J_g(w){\partial}_t w
=\nabla f_\lambda(w),\;
\lim_{s\to \pm \infty} w(s,t)
=z^\pm(t)\}$.
Let ${\mathbb{R}}$ act on $X$ by $s$-shift of $w$.
Then the projection
$\pi:X/{\mathbb{R}}\to[0,1]$,
$(\lambda,[w])\mapsto\lambda$,
is Fredholm of index $0$.
Hence the algebraic numbers of
$\pi^{-1}(0)$ and $\pi^{-1}(1)$ are equal,
provided we have
compactness control.
But these numbers represent
the boundary matrix elements between
$z^-$ and $z^+$ in
the Floer complexes associated
to $F$ and $h$, respectively.
In the general case, as long as
the 1-periodic orbits do not bifurcate,
the same argument applies:
the boundary conditions now
depend on $\lambda$, but
remain nondegenerate of constant
index for all $\lambda$.
} shows
$
\HF_*(T^*M,F,J_g)
\simeq
\HF_*(T^*M,h,J_g)
$.
\medskip
The crucial point
in defining the Floer homology
associated to $F$
and constructing a Floer
continuation homomorphism is
to prove apriori $C^0$ estimates
for the corresponding solutions.
Both problems are
nonstandard in the sense that
$F$ as well as
the homotopy of Hamiltonians
are \emph{asymptotically
nonconstant}.
Whereas a convexity
argument can probably
be adapted to solve
the first problem, we don't
see how to get the
$C^0$ estimates needed
for Floer continuation.
Hence we propose a proof along
different lines:
in Remark~\ref{rmk:nonsmooth}
we switch on the Hamiltonian
perturbation smoothly --
simply to remain
in the familiar
setting of \emph{smooth}
Hamiltonians.
Then, in Remark~\ref{re:continuation},
we avoid asymptotically
nonconstant Floer continuation
altogether by introducing
an intermediate step. All
$C^0$ estimates we need will follow
from~\cite[Proposition~2.3]{JOA4}.
For convenience we
recall this result below.
Roughly speaking, it asserts
that a Floer cylinder
whose ends are located
\emph{inside} $D T^*M$
cannot leave this set,
whenever outside the Hamiltonian is
radial with second
derivative bounded below or above.
\begin{proposition}[\cite{JOA4}]
\label{pr:subsolution}
Given $R,c\ge0$,
let $f\inC^{\infty}({\mathbb{R}}\times S^1\times [R,\infty))$
satisfy ${\partial}_s f_{s,t}'\ge0$ and
$f_{s,t}''\ge -c$ (or $f_{s,t}''\le c$)
for all $s$ and $t$.
Here $f_{s,t}(r):=f(s,t,r)$
and $f_{s,t}':=\frac{d}{dr} f_{s,t}$.
Assume further that
$H\in C^\infty({\mathbb{R}}\times S^1\times TM)$
satisfies
$H(s,t,x,y)=f_{s,t}(\Abs{y})$
whenever $\Abs{y}\ge R$,
that the pair
$(u,v)\in C^\infty({\mathbb{R}}\times S^1,TM)$
satisfies
\begin{equation*}\label{eq:l2}
\begin{pmatrix}
{\partial}_su-\Nabla{t}v \\
\Nabla{s}v+{\partial}_tu
\end{pmatrix}
-\nabla H(s,t,u,v) =0,
\end{equation*}
and that there exists $T>0$ such that
$\Abs{v(s,\cdot)}\le R$
whenever $\Abs{s}\ge T$.
Then $\Abs{v}\le R$
on ${\mathbb{R}}\times S^1$.
\end{proposition}
\begin{remark}[Smooth Hamiltonian]
\label{rmk:nonsmooth}\rm
We smoothly switch off the Hamiltonian
perturbation near the boundary
of the $\hat{v}_j$-strips:
fix a nondecreasing smooth map
$\alpha:[0,1]\to{\mathbb{R}}$ with
$\alpha\equiv0$ near 0 and
$\alpha\equiv\frac{1}{r}$ near 1
(see Figure~\ref{fig:fig-alph}).
\begin{figure}[ht]
\centering
\epsfig{figure=fig-alph.eps}
\caption{The function $\alpha$
and its derivative.}
\label{fig:fig-alph}
\end{figure}
\noindent
Define
$$
v_j(s,\tau)
:=\varphi^h
_{-\alpha(\tau)}\circ
\hat{v}_j(s,\tau),\qquad
J_r=J_{r,\tau}:=\bigoplus_1^r J_g\oplus
-(\varphi^h
_{-\alpha(\tau)})^* J_g.
$$
Then the Lagrangian boundary value
problem~(\ref{eq:U-V})
is equivalent to
\begin{equation}\label{eq:u-v-2}
\begin{aligned}
{\partial}_s u_j+J_g(u_j){\partial}_\tau u_j
&=0,&
v_j(s,0)
&=u_j(s,0),
\\
{\partial}_s v_j-J_g(v_j){\partial}_\tau v_j
&=\dot\alpha(\tau) \nabla
h(v_j),\qquad&
u_j(s,1)
&=v_{j+1}(s,1),
\end{aligned}
\end{equation}
for $j=1,\dots,r$.
Define a $1/r$-periodic
function
(see Figure~\ref{fig:fig-fnew})
by
\begin{equation*}
\beta(t)
:=
\begin{cases}
2r\dot\alpha(1-2rt+2j-2)
&,\, j\in\{1,\dots,r\},\,
t\in [\frac{2j-2}{2r},
\frac{2j-1}{2r}],
\\
0
&,\,\text{else}.
\end{cases}
\end{equation*}
\begin{figure}[ht]
\centering
\epsfig{figure=fig-beta.eps}
\caption{Smooth time dependence
of Hamiltonian
perturbation.}
\label{fig:fig-fnew}
\end{figure}
\noindent
Consider
the Hamiltonian $(t,q,p)\mapsto
\beta(t) h(\abs{p})$, and
let $\tilde{w}:{\mathbb{R}}\times S^1\to T^*M$
be given by~(\ref{eq:w-tilde}).
Then~(\ref{eq:u-v-2}) is equivalent to
$$
{\partial}_s \tilde{w}+J_g(\tilde{w}){\partial}_t \tilde{w}
-\beta(t) \nabla h(\tilde{w})
=0.
$$
With these definitions
the Hamiltonian perturbations
on the strips $(u_j,v_j)$
fit together smoothly. Indeed
$\nabla (\beta h)=\beta\nabla h$
depends smoothly on $t$
(see Figure~\ref{fig:fig-fnew}).
\\
The fact that
$\int_0^{1/r}\beta(t)\: dt=1/r$ and
the identity $X_{\beta h}=\beta X_h$
together imply that the time-$1/r$-maps
associated to $\beta h$
and $h$, respectively, are equal.
This proves
$\Fix\:\varphi^h=
\Fix\:\varphi^{\beta h}$ and
${\mathcal{P}}(h)\simeq{\mathcal{P}}(\beta h)$.
The latter correspondence is given by
mapping $z\in{\mathcal{P}}(h)$ to
$\tilde{z}(\cdot):=z(\int_0^\cdot \beta(t)\:dt)$.
Hence both chain complexes are equal
and therefore
$$
\HF_*(\Delta_r,
\Gamma_r(\varphi^h),
(T^*M\times
\overline{T^*M})^{\times r},
J_r;{\mathbb{Z}}_2)
\simeq
\HF_*(T^*M,\beta h,J_g;{\mathbb{Z}}_2).
$$
Again we remark that
it is an open problem to
prove equality of the gradings.
Since the Hamiltonian $\beta h$
is not well behaved at
infinity, definition of the
right hand side
requires an additional
argument to obtain a
uniform $C^0$-bound for
Floer trajectories:
given our knowledge that all elements
of ${\mathcal{P}}(\beta h)$ take values in
$DT^*M$, the $C^0$ estimate
for radial time-dependent Hamiltonians
Proposition~\ref{pr:subsolution}
shows that \emph{all} Floer cylinders
connecting elements of ${\mathcal{P}}(\beta h)$
also take values in
the bounded set $DT^*M$.
\end{remark}
\begin{remark}[Floer continuation]
\label{re:continuation}\rm
To conclude the proof of Step~1
we need to show that the Floer
homologies of $\beta h$ and $h$,
respectively, are isomorphic.
Choosing a different function
$\beta$, if necessary,
we may assume without loss of
generality that
$\beta_m:={\rm norm}{\beta}_\infty=3$.
(Any real number strictly larger
than $2$ can be realized
as such a maximum.)
Pick a sufficiently large real
$R_0>1$ and a smooth nondecreasing
cutoff function $\rho$
which equals 0 on $(-\infty,1]$
and 1 on $[R_0,\infty)$
such that the radial Hamiltonian
$$
h^{\rho,\beta(t)}(r)
:=\Bigl( \rho(r)+\beta(t)
\bigl(1-\rho(r)\bigr)\Bigr)
h(r),\qquad r=\abs{p},
$$
(see Figure~\ref{fig:fig-hrho})
is nondecreasing for every $t\in[0,1]$.
To check that such $R_0$ and $\rho$
exist is left as an exercise.
(Hint: start with
$$
R_0:=-11\frac{c_\lambda}{\lambda}+12,
\qquad
\tilde{\rho}(r)
:=\begin{cases}
0
&, r\le 1 \\
-\frac{(r-R_0)^2}{(1-R_0)^2}+1
&, r\in [1,R_0] \\
1
&, r\ge R_0
\end{cases}
$$
and check that indeed
$(h^{\tilde{\rho},\beta(t)})^\prime\ge 0$
for every $t\in[0,1]$.
Then smooth out
$\tilde{\rho}$ near $r=1$ and $r=R_0$
using cutoff functions
whose derivatives are
supported in $[1,1+\mu]$
and $[R_0-\mu,R_0]$, respectively,
for appropriate $\mu>0$.)
In the $t$-variable
$h^{\rho,\beta(t)}$ oscillates
between $h^{\rho,\beta_m}$
and $\rho h$
(see Figure~\ref{fig:fig-hrho}).
For $r\le 1$ the Hamiltonian
$h^{\rho,\beta}$
coincides with $\beta h$
and for $r\ge R_0$
with $h(r)=-c_\lambda+\lambda r$.
\begin{figure}[ht]
\centering
\epsfig{figure=fig-hrho.eps}
\caption{The family of Hamiltonians
$h^{\rho,\beta(t)}$
and a cutoff function $\rho$.}
\label{fig:fig-hrho}
\end{figure}
Since $h$ and $h^{\rho,\beta}$ are both
linear of slope $\lambda$
outside $D_{R_0}T^*M$,
any homotopy $f_s$
given by a convex combination
of the two Hamiltonians satisfies
$f_s^\prime\equiv \lambda$
outside $D_{R_0}T^*M$.
Hence the two conditions
$f_s^{\prime\prime}\equiv 0$
and ${\partial}_s f_s^\prime\equiv 0$
in Proposition~\ref{pr:subsolution}
are satisfied for $R=R_0$
and the homotopy
gives rise to a Floer continuation
homomorphism. It is an isomorphism
by the reverse homotopy argument.
Hence
$$
\HF_*(T^*M,h,J_g;{\mathbb{Z}}_2)
\simeq
\HF_*(T^*M,h^{\rho,\beta},J_g;{\mathbb{Z}}_2).
$$
By the end of Remark~\ref{rmk:nonsmooth}
all elements of ${\mathcal{P}}(\beta h)$
and all connecting Floer
trajectories associated to $\beta h$
take values in $DT^*M$.
We claim that all elements
of ${\mathcal{P}}(h^{\rho,\beta})$
take values
in $DT^*M$, too.
Then by Proposition~\ref{pr:subsolution}
the same will be true
for all connecting
Floer cylinders associated to
$h^{\rho,\beta}$.
But on $S^1\times DT^*M$
both Hamiltonians coincide,
hence both chain complexes are equal
and
$$
\HF_*(T^*M,h^{\rho,\beta},J_g;{\mathbb{Z}}_2)
\simeq
\HF_*(T^*M,\beta h,J_g;{\mathbb{Z}}_2).
$$
To prove the claim
consider the Hamiltonian vector field
$X_{\abs{p}^2/2}$ which
generates the geodesic flow on $T^*M$.
Since the Hamiltonian $h^{\rho,\beta}$
is radial, it holds
$$
X_{h^{\rho,\beta}}(q,p)
=r^{-1} (h^{\rho,\beta})^\prime (r)
X_{r^2/2}(q,p),\qquad r=\abs{p}.
$$
This shows that the projection
to the zero section $M$
of a $X_{h^{\rho,\beta}}$-trajectory
is a reparametrized geodesic.
In particular, if
$(x,y)\in{\mathcal{P}}(h^{\rho,\beta})$,
then $x:S^1\to M$ is a closed geodesic
(not necessarily parametrized
with respect to arc length)
and $\abs{y(t)}$ is
independent of $t$. Assume
by contradiction
$r:=\abs{y(t)}\ge 1$, then the
length of the closed geodesic $x$
is given by
\begin{equation*}
\begin{split}
\ell(x)
&=\int_0^1\Abs{\dot x(t)} dt
=\int_0^1\Abs{
\frac{(h^{\rho,\beta})^\prime(r)}{r}
g(x(t))^{-1} y(t)} dt \\
&=\int_0^1\Abs{
(h^{\rho,\beta})^\prime(r)} dt
=\int_0^1
\rho^\prime(1-\beta)h
+\bigl(\rho+\beta(1-\rho)\bigr)
\lambda \,dt \\
&=\rho^\prime h\int_0^1(1-\beta)\, dt
+\rho\lambda\int_0^1(1-\beta)\, dt
+\lambda \int_0^1\beta\, dt
=\lambda.
\end{split}
\end{equation*}
Here we used $(h^{\rho,\beta})^\prime\ge 0$,
the fact that $h^\prime(r)=\lambda$
whenever $r\ge 1$, and $\int_0^1 \beta=1$.
But $\lambda$ was chosen in the
complement of the length spectrum of $M$.
\end{remark}
\begin{remark}[Closed aspherical case]\rm
Replacing the cotangent bundle
by a closed \emph{symplectically
aspherical} manifold $(N,\omega)$,
i.e. $\omega$ and the first
Chern class vanish over $\pi_2(N)$,
the argument of Step~1 in the
case $r=1$ provides a new
proof of the known fact
$$
HF_*(N,H;{\mathbb{Z}})
\simeq
HF_*(\Delta,\graph\:\varphi^H;
N\times \overline{N};{\mathbb{Z}}).
$$
Note that
Remark~\ref{re:continuation}
can be replaced by
taking right away a homotopy
between $\beta H$ and $H$
and applying the standard
reverse homotopy continuation argument.
It is interesting to compare with
the proof in~\cite{BPS03}.
\end{remark}
\section{Example: The euclidean torus}
\label{sec:torus}
We compute $L^2$ Morse homology
of the loop space
of the euclidean torus ${\mathbb{T}}^n={\mathbb{R}}^n/2\pi{\mathbb{Z}}^n$
and Floer homology of its cotangent bundle.
Since the differential equations decouple,
the calculation essentially reduces to
the case $n=1$.
\subsubsection*{\boldmath$L^2$ Morse homology
for $\Ll_\alpha{\mathbb{T}}^1$}
\label{subsubsec:HM(T^1)}
Given $\alpha\in{\mathbb{Z}}$,
we think of $x\in\Ll_\alpha{\mathbb{T}}^1$ as a smooth
map $x:{\mathbb{R}}\to{\mathbb{R}}$ satisfying
$x(t+1)=x(t)+2\pi\alpha$.
Hence $\alpha$
is the winding number of $x$.
We fix $\alpha$ and
analyze the problem
for each component of the
free loop space separately.
The equation for the free pendulum
is ${\partial}_t{\partial}_t x=0$
and its solutions are given by
$\{ x(t)=2\pi\alpha t+x_0\mid
x_0\in{\mathbb{T}}^1\}$.
Upon introducing a
time-dependent potential
on $S^1\times{\mathbb{T}}^1$ of the form
$
V_\alpha(t,q)
:=-\cos \left(q-2\pi\alpha t\right)
$,
the circle of solutions splits into two
nondegenerate critical points.
Equation~(\ref{eq:crit})
for the critical points of
the functional
${\mathcal{S}}_{V_\alpha}$ defined on
$\Ll_\alpha {\mathbb{T}}^1$ becomes
\begin{equation}\label{eq:crit-1}
{\partial}_t{\partial}_t x
=-\sin\left(x-2\pi\alpha t\right).
\end{equation}
Observe that the case $\alpha=0$ describes
the mathematical pendulum with gravity.
Obvious 1-periodic solutions
of~(\ref{eq:crit-1}) are
the two equilibrium states
\emph{pendulum up} and
\emph{pendulum down}.
They are described by
the constant functions $x^{(1)}=\pi$
and $x^{(0)}=0$, respectively.
For general $\alpha$
equation~(\ref{eq:crit-1})
still describes
the same mathematical pendulum,
but now the observer rotates
with constant angular
speed $2\pi\alpha$.
The two equilibrium states
of the pendulum are then
seen by the observer
as rotations and described by
\begin{equation}\label{eq:equilibrium}
x^{(1)}=2\pi\alpha t+\pi,\qquad
x^{(0)}=2\pi\alpha t.
\end{equation}
Of course, the pendulum
admits plenty of nonstationary
periodic solutions, but their
periods are strictly bigger
than one. Hence ${\mathcal{S}}_{V_\alpha}$
has only the two critical points
$x^{(1)}$ and $x^{(0)}$.
Their actions are
$2\pi^2\alpha^2+1$ and
$2\pi^2\alpha^2-1$, respectively.
To compute the Morse index of $x^{(1)}$
we linearize equation~(\ref{eq:crit-1})
at $x^{(1)}$ and solve the
resulting eigenvalue problem
$-{\partial}_t{\partial}_t \xi-\xi=\lambda\xi$
for $\xi:{\mathbb{R}}\to{\mathbb{R}}$ with
$\xi(t+1)=\xi(t)$.
The solution to this ODE is
given by $\xi(t)=\sin \sqrt{\lambda+1}t$
and the periodicity condition implies
$\lambda\in\{4\pi^2k^2-1\mid
k=0,1,2,\dots\}$.
This shows $\IND_{V_\alpha}(x^{(1)})=1$.
Linearizing at $x^{(0)}$ we arrive at
$-{\partial}_t{\partial}_t\xi+\xi=\lambda\xi$
with solution
$\xi(t)=\sin \sqrt{\lambda-1}t$
and $\lambda\in\{4\pi^2k^2+1\mid
k=0,1,2,\dots\}$. Hence
$\IND_{V_\alpha}(x^{(0)})=0$.
Therefore the only
nontrivial chain groups
are given by
\begin{equation}\label{eq:chain-1}
C_0({V_\alpha})
={\mathbb{Z}} x^{(0)},\qquad
C_1({V_\alpha})
={\mathbb{Z}} x^{(1)}.
\end{equation}
There must be precisely two connecting
trajectories up to $s$-shift,
because the unstable manifold
of $x^{(1)}$ is 1-dimensional.
For $\sigma\in C^\infty({\mathbb{R}},{\mathbb{R}})$
consider the \emph{Ansatz}
$u(s,t):=2\pi\alpha t+\sigma(s)$.
Then the heat equation~(\ref{eq:heat})
is equivalent to
\begin{equation}\label{eq:sigma}
\sigma^\prime(s)=-\sin \sigma(s).
\end{equation}
The initial conditions
$\sigma(0)=0$ and $\sigma(0)=\pi$
produce the constant trajectories
$u(s,t)=x^{(0)}(t)$
and $u(s,t)=x^{(1)}(t)$, respectively.
The two 1-parameter families
of connecting trajectories
are obtained by choosing
$\sigma(0)$ in $(0,\pi)$ or in
$(-\pi,0)$.
Choosing an orientation of the unstable
manifold of $x^{(1)}$ induces an
orientation of each of the
two 1-parameter families.
For one family this orientation
coincides with the flow direction
and for the other one it does not,
so one characteristic sign
is positive and one is negative.
Hence
${\partial}^M x^{(1)}=x^{(0)}-x^{(0)}=0$.
This proves that
$L^2$ Morse homology of
$(\Ll_\alpha {\mathbb{T}}^1,{\mathcal{S}}_{V_\alpha})$
is in fact given by its chain
groups~(\ref{eq:chain-1}).
\subsubsection*{Floer homology
of \boldmath$T^*{\mathbb{T}}^1$}
\label{subsubsec:HF(T^*T^1)}
Let $V_\alpha$ be as above
and define the Hamiltonian
$H_{V_\alpha}(t,q,p):=\frac12 p^2
+V_\alpha(t,q)$ for
$(t,q,p)\in S^1\times{\mathbb{T}}^1\times{\mathbb{R}}$.
Hamilton's equations and their
solutions are given by
$$
\begin{pmatrix}\dot x(t)\\
\dot y(t)\end{pmatrix}
=\begin{pmatrix} y(t)\\
-\sin\left( x(t)-2\pi\alpha t\right)
\end{pmatrix},\;\;
z^{(1)}(t)
=\begin{pmatrix} 2\pi\alpha t+\pi\\
2\pi\alpha\end{pmatrix},\;\;
z^{(0)}(t)
=\begin{pmatrix} 2\pi\alpha t\\
2\pi\alpha\end{pmatrix}.
$$
To calculate the two connecting
trajectories let $u$ be as above
and set $v(s,t):=2\pi\alpha$.
Floer's equations~(\ref{eq:heat})
are then equivalent to~(\ref{eq:sigma})
and the argument continues as above
showing that
$\HF_*(T^*{\mathbb{T}}^1,H_{V_\alpha},J_1;\alpha)$
equals the chain
groups~(\ref{eq:chain-1}).
One might wonder what happens in the case of
the ${\varepsilon}$-Floer equations~(\ref{eq:floer-eps}).
With our Ansatz for $u$ and $v$ these are
again equivalent to~(\ref{eq:sigma}),
in particular ${\varepsilon}$ disappears.
This means that the map
${\mathcal{T}}^{\varepsilon}:\Mm^0\to\Mm^{\varepsilon}$
is essentially the identity.
It maps $u$ to $(u,{\partial}_tu)$
for any ${\varepsilon}>0$.
The case ${\varepsilon}=1$ and our
knowledge in the Morse case
show that up to $s$-shift
there are precisely
two Floer trajectories
\subsubsection*{Higher dimensional case}
\label{subsubsec:general}
Fix $n\ge1$ and
$\alpha=(\alpha_1,\dots,\alpha_n)\in{\mathbb{Z}}^n$.
On the euclidean torus
${\mathbb{T}}^n={\mathbb{R}}^n/{\mathbb{Z}}^n$ consider
the potential
$V_\alpha(t,q)$
given by the sum of $n$
one-dimensional potentials.
The critical points of
${\mathcal{S}}_{V_\alpha}$ on
$\Ll_\alpha{\mathbb{T}}^n$
are smooth functions
$x:{\mathbb{R}}\to{\mathbb{R}}^n$ with
$x(t+1)=x(t)+2\pi\alpha$
satisfying the ODE
$$
{\partial}_t{\partial}_t x(t)
=-\nabla V_\alpha(t,x(t))
=\left(
-\sin\left(x_1(t)-2\pi\alpha_1 t\right)
,\dots,
-\sin\left(x_n(t)-2\pi\alpha_n t\right)
\right).
$$
Its solutions
are of the form
$(x^{(\nu_1)},\dots,
x^{(\nu_n)})$ where the components
$x^{(\nu_k)}$ are given by one of the
two equilibrium states
in~(\ref{eq:equilibrium}).
Hence the assignment
$(x^{(\nu_1)},\dots,
x^{(\nu_n)})\mapsto
(\nu_1,\dots,\nu_n)=:\nu$
identifies the set of critical points with
the direct sum $({\mathbb{Z}}_2)^n$
of $n$ copies of ${\mathbb{Z}}_2$,
and the number of
critical points is $2^n$.
Moreover, the Morse index of
$x^{(\nu)}:=(x^{(\nu_1)},\dots,x^{(\nu_n)})$
equals $\Abs{\nu}$ and
\begin{equation}\label{eq:chain-n}
C_k(V_\alpha)
=\bigoplus_{\nu\in({\mathbb{Z}}_2)^n,\,
\Abs{\nu}=k}\;
x^{(\nu)}{\mathbb{Z}}
\simeq {\mathbb{Z}}^{\binom{n}{k}}.
\end{equation}
Arguing as above one observes that
trajectories connecting critical
points of index difference one
come in pairs of opposite
characteristic signs
showing that all
boundary operators are zero.
(For example, in the case $n=3$
the critical point $(1,0,1)$
admits two connecting trajectories
to $(0,0,1)$ and two to $(1,0,0)$).
It follows that
$L^2$ Morse homology and
Floer homology both
coincide with the chain groups~(\ref{eq:chain-n})
and, by Theorem~\ref{thm:main},
so does $\Ho_*(\Ll_\alpha {\mathbb{T}}^n)$.
\section{Application:
Noncontractible periodic orbits}
\label{sec:noncontractible}
The search for noncontractible
1-periodic orbits is a fairly recent
branch of symplectic geometry
pioneered by Gatien-Lalonde~\cite{GL00}
and Biran-Polterovich-Salamon~\cite{BPS03}.
(See also~\cite{L03} for some
generalizations of~\cite{GL00}).
These authors obtain existence results
under certain geometric assumptions,
for instance flatness of the metric.
More precisely, Theorem~\ref{thm:exist-orbit}
below is proved in~\cite{BPS03}
for the euclidean $n$-torus
and in the case of negative curvature
using the known simple structure
of the set of periodic geodesics.
In contrast our extension
to the case of \emph{arbitrary}
Riemannian metric
is based on Theorem~\ref{thm:main}.
Let $M$ be a closed connected
Riemannian manifold. Given a
homotopy class $\alpha$ of free
loops in $M$, define
the \emph{marked length
spectrum} $\Lambda_\alpha$
to be the set of lengths of all periodic
geodesics representing $\alpha$.
This set is closed and nowhere dense in ${\mathbb{R}}$.
Hence $\ell_\alpha:=\inf\Lambda_\alpha$
belongs to $\Lambda_\alpha$.
Let the open unit disc bundle
$DT^*M\subset T^*M$ be equipped
with the canonical
symplectic form $\omega_0$.
\begin{theorem}[\cite{JOA4}]
\label{thm:exist-orbit}
Let $\alpha$ be a homotopy class
of free loops in $M$.
Then every compactly supported
Hamiltonian
$H\inC^{\infty}([0,1]\times DT^*M)$
satisfying
$$
\sup_{[0,1]\times M} H
=:-c\le-\ell_\alpha
$$
admits a 1-periodic orbit $z=(x,y)$
with $[x]=\alpha$ and
action $\Aa_H(z)\ge c$.
\end{theorem}
The idea of proof is to sandwich
the Hamiltonian $H$ between two
Hamiltonians $f$ and $h$ (see
Figure~\ref{fig:fig-idea}) whose
action filtered
Floer homologies are computable and nonzero.
Then prove that the
\emph{monotonicity homomorphism}
associated to $f$ and $h$
is nonzero. This is a homomorphism
from Floer homology of a given Hamiltonian
to Floer homology of any pointwise larger
Hamiltonian. Its crucial properties are,
firstly, it respects the action window
and, secondly, it factors through Floer
homology of any third intermediate Hamiltonian.
In our case it factors
through $\HF_*^{(a,\infty)}(H;\alpha)$,
which therefore
is nonzero. Roughly speaking,
if $c$ is a regular
value of $\Aa_H|_{\Ll_\alpha T^*M}$,
set $a:=c$ and we are done.
The key tool to calculate
the Floer homologies of $f$ and $h$
is a refinement of the
Floer continuation principle.
Namely, two Hamiltonians connected
by an \emph{action regular homotopy}
(meaning that the boundary of the action
window consists of regular values
throughout the homotopy)
have the same Floer homology.
Action regularity is most easily
checked for radial Hamiltonians.
\begin{remark}[Radial Hamiltonians]
\label{rmk:radial}\rm
Let $h:{\mathbb{R}}\to{\mathbb{R}}$
be a smooth symmetric function.
A Hamiltonian of the form
$H(t,q,p)=h(\abs{p})$
is called \emph{radial}.
Whenever the slope of $h$ at
a point $r_\ell$
is equal to an element
$\ell\in\Lambda_\alpha$,
then there is a 1-periodic orbit $z_\ell$
of the Hamiltonian flow
on the sphere bundle of radius
$r_\ell$ and all 1-periodic
orbits arise that way.
Moreover, the symplectic
action of $z_\ell$ equals
minus the intercept of the tangent
to the graph at the point $r_\ell$.
\end{remark}
Hence we choose the Hamiltonians
$f$ and $h$ radial.
Then in each case we construct an
action regular homotopy towards
a \emph{convex} radial Hamiltonian,
since for these
Floer homology is computed
in~\cite{JOA4} on the basis
of Theorem~\ref{thm:main}.
\begin{figure}[ht]
\parbox[b]{3cm}{
\epsfig{figure=fig-idea.eps,width=\linewidth}
}
\hspace{-.2cm}
\parbox[b]{9.1cm}{
\begin{equation*}
\xymatrix{
&
\underrightarrow{SH}^{(a,\infty);c;\alpha}_*
\ar[r]^\simeq_{a\in(0,c]}
&
H_*(\Ll_\alpha M)
\\
HF_*^{(a,\infty)}(H;\alpha)
\ar[ur]^{\iota_H}
\ar @{} [r] |{\qquad\circlearrowleft}
&
\ar @{} [r] |{\circlearrowleft}
&
\\
&
\underleftarrow{SH}^{(a,\infty);\alpha}_*
\ar[ul]^{\pi_H}
\ar[uu]_T
\ar[r]^\simeq_{a\in{\mathbb{R}}^+\setminus
\Lambda_\alpha}
&
H_*(\Ll^{\frac{a^2}{2}}_\alpha M)
\ar[uu]_{\overset{\scriptstyle
[\iota]\not=0\quad\;\;}
{\text{if $a\ge\ell_\alpha$}}}
}
\end{equation*}
}
\caption{The Hamiltonian sandwich.}
\label{fig:fig-idea}
\end{figure}
Of course, for different $H$
we may have to take new choices for
$f$ and $h$. A convenient book
keeping tool
to deal with this problem
is to take the inverse limit
over all $f$ and the direct
limit over all $h$ subject to
the restriction
$\sup_{[0,1]\times M} h\le-c$.
These limits are called symplectic and
relative symplectic homology
and were introduced in~\cite{FH94,CFH95}
and~\cite{BPS03}, respectively.
The monotone homomorphisms
descend to a natural homomorphism $T$.
The main part of~\cite{JOA4} is devoted
to establish and prove commutativity
of the rectangular
part of the diagram in
Figure~\ref{fig:fig-idea}.
Here the homomorphism $[\iota]$
induced by inclusion
does not vanish
whenever $a\ge\ell_\alpha$.
The monotone homomorphisms
descend to a natural homomorphism
$T$. It follows that $T$
is nonzero whenever
$a\in[\ell_\alpha,c]\setminus\Lambda_\alpha$.
In the case that $c$
is a regular value of
$\Aa_H|_{\Ll_\alpha T^*M}$
and $-c<-\ell_\alpha$
we can choose $a=c$
and are done.
Otherwise choose a sequence
of Hamiltonians $H_\nu$
converging to $H$ in $C^\infty$ and
such that the corresponding $c_\nu$
satisfy both requirements above.
From the resulting
sequence of periodic orbits
extract a subsequence
whose limit is the periodic orbit
claimed by Theorem~\ref{thm:exist-orbit}.
Consider the radial Hamiltonian
whose graph in ${\mathbb{R}}^2$ consists
of a straight line from
$(0,-\ell_\alpha)$ to $(1,0)$
and which is zero elsewhere.
Approximate it by a smooth
function all of whose slopes are
strictly less then $\ell_\alpha$
and which therefore does not
admit any 1-periodic
orbit representing $\alpha$.
This example shows that the condition
in Theorem~\ref{thm:exist-orbit}
is sharp. It follows that the relative
capacity defined in~\cite{BPS03}
and associated to
$(DT^*M,M,\alpha)$
equals $\ell_\alpha$.
As a byproduct we obtain
in~\cite{JOA4}
a multiplicity version of
the Weinstein conjecture
for compact hypersurfaces
$Q\subset T^*M$ of contact type
which enclose
the zero section $M$.
More precisely, for every nontrivial
$\alpha$ we obtain existence
of a closed characteristic
on $Q$ whose projection to $M$
represents $\alpha$.
In the nonsimply connected case
this refines the result of Hofer and
Viterbo~\cite{HV88}
in the case $\alpha=0$.
Viterbo informed us that
their techniques should also provide
multiplicities.
|
3,212,635,537,422 | arxiv | \section{Introduction}
Neutrinos play a decisive role in core-collapse supernova explosions
since they carry most of the gravitational binding energy released.
The transport of neutrinos through the hot and dense stellar
environment is believed to ultimately be responsible for a
successful explosion, although the details are not fully understood
yet. The present paper addresses the role of thermal effects in the
inelastic neutrino-nucleus scattering in the iron core during infall
and shortly after bounce.
At the end of 1980th it was pointed out by W.~C.~Haxton that
inelastic neutrino-nucleus scattering (INNS) mediated by the
neutral-current can be of importance comparable with the other
processes of neutrino down-scattering \cite{Haxton88}. The INNS
contributes to the neutrino opacities and thermalization during the
collapse phase, the revival of the stalled shock wave in the delayed
explosion mechanism, and to explosive nucleosynthesis. The
estimates by Haxton were based on nuclei in their respective ground
states, i.e. for a ``cold'' nuclei. Subsequently, it was realized
that the INNS occurs in hot stellar environment ($T \geq 0.8$~MeV)
and, due to the thermal population of nuclear excited states,
sizeable changes of the INNS cross section are to be expected. The
effect was firstly analyzed in \cite{AstrPhysJ91} and then in
\cite{PLB02} on the basis of large-scale shell-model (LSSM)
calculations. In Refs.~\cite{PLB02,NPA05}, it was found that the
INNS cross section noticeably increases at $T \neq 0$ and for
neutrino energies $E_{\nu} \lesssim 10~\text{MeV}$, especially for
neutrino scattering off even-even nuclides.
However, in the subsequent core-collapse supernova simulations
\cite{PRL08} including several dozens of nuclides, it was
demonstrated that the inclusion of the INNS process does not have
a large effect on the collapse dynamics and the shock wave
propagation. But it significantly modifies the spectrum of
neutrinos generated in the $\nu_e$ burst.
Here, we apply an alternative approach for treating the thermal
effects for INNS cross sections. In essence, our approach is based
on the thermal quasiparticle random phase approximation (TQRPA). We
apply it in the context of Thermo-Field-Dynamics (TFD), which
enables a transparent treatment of thermal excitation and
de-excitation processes and offers the possibility for systematic
improvements. This approach has recently been used in studies of the
electron capture on hot iron and germanium nuclei under stellar
conditions \cite{PRC81}.
\section{FORMALISM}
\subsection{Fundamentals of the Thermo-Field-Dynamics}
Thermo-Field-Dynamics \cite{TFD1,TFD2,Ojima81} is a real-time
formalism for treating thermal effects in quantum field theory and
non-relativistic many-body theories. The standard TFD formalism
treats a many-body
system in thermal equilibrium with a heat bath and a particle
reservoir in the grand canonical ensemble. The thermal average of a
given operator $A$ is calculated as the expectation value in a
specially constructed, temperature-dependent state $|0(T)\rangle$
which is termed the thermal vacuum. This expectation value is equal
to the usual grand canonical average of $A$. In this sense, the
thermal vacuum describes the hot system in the thermal equilibrium.
To construct the state $|0(T)\rangle$, a formal doubling of the
system degrees of freedom is introduced. In TFD, a tilde conjugate
operator~$\widetilde A$ -- acting in the independent Hilbert space
-- is associated with $A$, in accordance with properly formulated
tilde conjugation rules~\cite{TFD1,TFD2,Ojima81}. For a system
governed by the Hamiltonian~$H$ at $T=0$, the whole Hilbert space
at $T\neq 0$ is spanned by the direct product of the eigenstates
of~$H$ (${H|n\rangle=E_n|n\rangle}$) and those of the tilde
Hamiltonian~$\widetilde H$ having the same eigenvalues (${\widetilde
H|\widetilde n\rangle=E_n|\widetilde n\rangle}$). The important
point is that, in the doubled Hilbert space, the time-translation
operator is not the initial Hamiltonian~$H$, but instead the thermal
Hamiltonian~${\mathcal H}=H-\widetilde H$. This implies that the
excitations of the thermal system are obtained by the
diagonalization of~${\cal H}$.
The thermal vacuum is the zero-energy eigenstate of the thermal
Hamiltonian $\mathcal H$ and satisfies the thermal state
condition~\cite{TFD1,TFD2,Ojima81}
\begin{equation}\label{TSC}
A|0(T)\rangle = \sigma\,{\rm e}^{{\mathcal H}/2T} {\widetilde
A}^\dag|0(T)\rangle,
\end{equation}
where $\sigma=1$ for bosonic~$A$ and $\sigma=i$ for fermionic $A$.
As it follows from the definition of $\mathcal H$ each of its
eigenstates with positive energy has the counterpart -- the
tilde-conjugate eigenstate -- with negative but the
same absolute energy value. This allows to treat excitation-
and de-excitation processes at finite temperatures.
Obviously, in most practical cases one cannot diagonalize
$\mathcal H$ exactly. Usually, one resorts to certain approximations
such as Hartree-Fock-Bogoliubov mean field theory (HFB) and the
Random-Phase Approximation (RPA) (see e.g.~\cite{Hat89}). In what
follows the TFD studies for neutrino induced charge-neutral
excitations in hot nuclei are based in part on the results
of~\cite{DzhVdo09,VdoDzh10} (see also~\cite{PRC81}).
\subsection{Charge-neutral excitations in hot nuclei}
In what follows we employ the Hamiltonian of the Quasiparticle-Phonon Model (QPM)
$H_{\rm QPM}$ \cite{sol92} which consists of proton and neutron mean
fields $H_{\rm sp}$, the BCS pairing interactions $H_{\rm pair}$ and
isoscalar and isovector separable particle-hole interactions. Since
the inelastic neutrino-nucleus scattering involves nuclear $J^\pi$
excitations of both natural ($\pi=(-1)^J$) and unnatural
($\pi=(-1)^{J+1}$) parities both the separable multipole $H^{\rm
ph}_{\rm M}$ and spin-multipole $H^{\rm ph}_{\rm SM}$ interactions
are included in the particle-hole channel
\begin{equation}\label{QPM}
H_{\rm QPM} = H_{\rm sp} + H_{\rm pair}+ H^{\rm ph}_{\rm M}+H^{\rm
ph}_{\rm SM}.
\end{equation}
The four terms of $H_{\rm QPM}$ read
\begin{align*}
H_{\rm sp} & = \sum_{\tau=p,n}{\sum_{jm}}^{\tau}(E_{j}-\lambda_\tau)
a^\dag_{jm}a^{\phantom{\dag}}_{jm}~, \\
H_{\rm pair}& =-\frac14\sum_{\tau=p,n} G_{\tau}{\sum_{\substack{jm \\
j'm'}}}^{\tau}
a^\dag_{jm}a^\dag_{\overline{\jmath m}}
a^{\phantom{\dag}}_{\overline{\jmath'm'}}a^{\phantom{\dag}}_{j'm'},
\\
H^{\rm ph}_{\rm M}&=-\frac12\sum_{\lambda\mu}\sum_{\tau\rho=\pm1}
(\kappa_0^{(\lambda)}+\rho\kappa_1^{(\lambda)})M^+_{\lambda\mu}(\tau)M^{\phantom{+}}_{\lambda\mu}(\rho\tau)~.
\\
H^{\rm ph}_{\rm SM} &=
-\frac12\sum_{L\lambda\mu}\sum_{\tau\rho=\pm1}(\kappa^{(L\lambda)}_0+\rho\kappa^{(L\lambda)}_1)
S^\dag_{L\lambda\mu}(\tau)
S^{\phantom{\dag}}_{L\lambda\mu}(\rho\tau),
\end{align*}
Here, we use standard notations of the QPM. Namely, $a^\dag_{jm}$ and
$a^{\phantom{\dag}}_{jm}$ are the creation and annihilation
operators of particle with quantum numbers $jm\equiv n,l,j,m$ and
energy $E_j$; $\overline{jm}$ stands for the time reversed
single-particle states; the index $\tau$ is isotopic one and
changing the sign of $\tau$ means changing $n \leftrightarrow p$\,;
the parameter $G_\tau$ is the constant of pairing interaction;
$\lambda_\tau$ is the chemical potential; the parameters
$\kappa_{0}^{(a)}$ ($\kappa_{1}^{(a)}$) denote the strength
parameters of the isoscalar (isovector) multipole ($a\equiv\lambda$
is a multipole index) and spin-multipole ($a\equiv L\lambda$ is a
spin-multipole index) forces. The multipole $M^+_{\lambda\mu}(\tau)$
and spin-multipole $S^+_{L\lambda\mu}(\tau)$ single-particle
operators read as
\begin{align}\label{mult}
M^+_{\lambda\mu}(\tau)&={\sum_{\genfrac{}{}{0pt}{1}{j_1m_1}{j_2m_2}}}^{\tau}
\langle j_1m_1|i^\lambda R_\lambda(r)Y_{\lambda\mu}|j_2m_2\rangle
a^\dag_{j_1m_1}a^{\phantom{\dag}}_{j_2m_2}~,
\notag \\
S^\dag_{L\lambda\mu}(\tau) &= {\sum_{\genfrac{}{}{0pt}{1}{j_1m_1}{j_2m_2}}}^{\tau}
\langle j_1m_1| i^L R_L(r) [Y_{L}\vec\sigma]^\lambda_\mu|j_2m_2\rangle
a^\dag_{j_1m_1}a^{\phantom{\dag}}_{j_2m_2}\, ,
\end{align}
where
\begin{equation*}
\bigl[Y_L\,\sigma\bigr]^\lambda_\mu=\sum_{M,\,m}\langle LM\,1m|\lambda\mu\rangle
Y_{LM}(\theta,\phi)\sigma_m~,
\end{equation*}
and the notation ${\sum}^\tau$ implies a summation over neutron
($\tau=n$) or proton ($\tau=p$) single-particle states only. The
excitations of natural parity are generated by the multipole and
spin-multipole $L=\lambda$ interactions, while the spin-multipole
interactions with $L=\lambda\pm1$ are responsible for the states of
unnatural parity.
To determine the thermal behavior of a nucleus governed by the
Hamiltonian (\ref{QPM}) we should diagonalize the thermal
Hamiltonian $\mathcal{H}_{\rm QPM} = H_{\rm QPM} -
\widetilde{H}_{\rm QPM}$ and find the corresponding thermal vacuum
state. This will be done in two steps.
In a first step, the sum of single-particle and pairing
terms $\mathcal{H}_{\rm BCS}=\mathcal{H}_{\rm sp} +\mathcal{H}_{\rm pair}$
is diagonalized. To this end two subsequent unitary transformations
are made. The first is the usual Bogoliubov $u, v$ transformation
from the original particle operators
$a^\dag_{jm},~a^{\phantom{\dag}}_{jm}$ to the quasiparticle ones
$\alpha^\dag_{jm},~\alpha^{\phantom{\dag}}_{jm}$. The same
transformation is applied to the tilde operators $\widetilde
a^\dag_{jm},\ \widetilde a^{\phantom\dag}_{jm}$, thus producing the
tilde quasiparticle operators $\widetilde\alpha^\dag_{jm},\
\widetilde\alpha^{\phantom\dag}_{jm}$. The second, unitary thermal
Bogoliubov transformation mixes the original and tilde degrees of
freedom
\begin{align}\label{TBt}
\beta^\dag_{jm}&=x_j\alpha^\dag_{jm}\!-\!i y_j\widetilde\alpha_{jm}\\
\widetilde\beta^\dag_{jm}&=x_j\widetilde\alpha^\dag_{jm}\!+\!i
y_j\alpha_{jm}~~~(x^2_j+y^2=1). \nonumber
\end{align}
The operators $\beta^\dag_{jm}, \beta_{jm}, \widetilde
\beta^\dag_{jm}$, and $\widetilde\beta_{jm}$ are called thermal
quasiparticle operators.
The coefficients $u_j,\ v_j,\ x_j,\ y_j$ are found by
diagonalizing~${\cal H}_{\rm BCS}$ and demanding that the vacuum of
thermal quasiparticles is the thermal vacuum in the BCS
approximation, i.e., it obeys the thermal state
condition~\eqref{TSC}. As a result one obtains the following
equations for $u_j,\ v_j$ and $x_j,\ y_j$:
\begin{eqnarray}
v_j & = & \frac{1}{\sqrt
2}\left(1-\frac{E_j-\lambda_{\tau}}{\varepsilon_j}\right)^{1/2},\
u_j=(1-v_j^2)^{1/2}, \label{u&v} \\
y_j & = &
\left[1+\exp\left(\frac{\varepsilon_j}{T}\right)\right]^{-1/2},\
x_j=\bigl(1-y^2_j\bigr)^{1/2}, \label{x&y}
\end{eqnarray}
where $\varepsilon_j=\sqrt{(E_j-\lambda_{\tau})^2+\Delta^2_{\tau}}$.
The coefficients $y^2_j$ determine the average number of thermally
excited Bogoliubov quasiparticles in the BCS thermal vacuum
\begin{equation}
\langle 0(T);{\rm qp}|
\alpha^\dag_{jm}\alpha^{\phantom{\dag}}_{jm}
|0(T);{\rm qp}\rangle=y^2_{j}
\end{equation}
and, thus, coincide with the thermal occupation factors of the
Fermi-Dirac statistics.
The pairing gap $\Delta_{\tau}$ and the chemical potential
$\lambda_\tau$ are the solutions to the finite-temperature BCS
equations
\begin{align}\label{BCS}
\Delta_\tau(T)&=\frac{G_\tau}{2}{\sum_j}^\tau(2j+1)(1-2y^2_j)u_jv_j,\nonumber\\
N_\tau&={\sum_j}^\tau(2j+1)(v^2_jx^2_j+u^2_jy^2_j),
\end{align}
where $N_\tau$ is the number of neutrons or protons in a nucleus.
At this stage, the thermal BCS Hamiltonian ${\mathcal H}_{\rm BCS}$
is diagonal
\begin{equation*}
{\mathcal H}_{\rm BCS}
\simeq\sum_\tau{\sum_{jm}}^\tau\varepsilon_j(T)
(\beta^\dag_{jm}\beta^{\phantom{\dag}}_{jm}-\widetilde\beta^\dag_{jm}\widetilde\beta^{\phantom{\dag}}_{jm}),
\end{equation*}
and corresponds to a system of non-interacting thermal
quasiparticles. The vacuum for thermal quasiparticles ${|0(T);{\rm
qp}\rangle}$ is the thermal vacuum in the BCS approximation. The
states $\beta^\dag_{jm}|0(T);{\rm qp}\rangle$ have positive
excitation energies whereas the corresponding tilde-states
$\widetilde\beta^\dag_{jm}|0(T);{\rm qp}\rangle$ have negative
energies. Since the thermal vacuum contains a certain number of
Bogoliubov quasiparticles, excited states can be built on
$|0(T);{\rm qp}\rangle$ by either adding or removing a Bogoliubov
quasiparticle. The first process corresponds to the creation of a
non-tilde thermal quasiparticle with positive energy, whereas the
second process creates a tilde quasiparticle with negative energy.
At the second step of the approximate diagonalization of ${\mathcal
H}_{\rm QPM}$, long-range correlations due to the particle-hole
interaction are taken into account within the thermal QRPA (TQRPA).
Within the TFD formalism the terms ${\cal H}^{\rm ph}_{\rm M}$ and
${\cal H}^{\rm ph}_{\rm SM}$ are written in terms of the thermal
quasiparticle operators determined above. Then, ${\cal H}_{\rm QPM}$
is approximately diagonalized within a basis of thermal phonon
operators
\begin{multline}\label{phonon}
Q^\dag_{\lambda \mu i}=\frac12\sum_\tau{\sum_{j_1j_2}}^\tau
\Bigl\{\psi^{\lambda i}_{j_1j_2}[\beta^\dag_{j_1}\beta^\dag_{j_2}]^\lambda_\mu +
\widetilde\psi^{\lambda i}_{j_1j_2}[\widetilde\beta^\dag_{\overline{\jmath_1}}
\widetilde\beta^\dag_{\overline{\jmath_2}}]^\lambda_\mu +
2i\,\eta^{\lambda i}_{j_1j_2}[\beta^\dag_{j_1}
\widetilde\beta^\dag_{\overline{\jmath_2}}]^\lambda_\mu\\
+
\phi^{\lambda i}_{j_1j_2}[\beta_{\overline{\jmath_1}}\beta_{\overline{\jmath_2}}]^\lambda_{\mu} +
\widetilde\phi^{\lambda i}_{j_1j_2}[\widetilde\beta_{j_1}
\widetilde\beta_{j_2}]^\lambda_{\mu} +
2i\,\xi^{\lambda i}_{j_1j_2}[\beta_{\overline{\jmath_1}}
\widetilde\beta_{j_2}]^\lambda_{\mu}\Bigr\},
\end{multline}
where $[~]^\lambda_\mu$ denotes the coupling of single-particle
angular momenta $j_1, j_2$ to a total angular momentum $\lambda$.
Now the thermal equilibrium state is treated as the vacuum
$|0(T);{\rm ph}\rangle$ for the thermal phonon annihilation
operators.
The thermal phonon operators are considered as bosonic ones which
imposes certain constraint on the phonon amplitudes. To find the
amplitudes and energies of the thermal phonons, the variational
principle is used, i.e., we find the minimum of the average value
of thermal Hamiltonian with respect to the one-phonon
states~${Q^\dag_{\lambda\mu i}|0(T);{\rm ph}\rangle}$ or
${\widetilde Q^\dag_{\overline{\lambda\mu i}}|0(T);{\rm ph}\rangle}$
under the aforementioned constraint.
After variation one obtains a system of linear equations for the
amplitudes $\psi^{\lambda i}_{j_1j_2},\ \widetilde\psi^{\lambda
i}_{j_1j_2},\ \eta^{\lambda i}_{j_1j_2}$, etc. as well as for the energies
(details can be found in ref. \cite{DzhVdo09}).
These constitute the equations for the thermal quasiparticle
random phase approximation. In contrast to the zero temperature
case, the negative solutions of the secular equation
have a physical meaning. They correspond to the tilde thermal
one-phonon states and arise from
$\widetilde\beta^\dag\widetilde\beta^\dag$ terms in the thermal
phonon operator. As it was noted above, creation of a tilde thermal
quasiparticle corresponds to the annihilation of a thermally excited
Bogoliubov quasiparticle. Consequently, excitations of
negative-energy thermal phonons correspond to transitions from
thermally excited nuclear states.
After diagonalization in terms of thermal phonon operators the TQRPA
part of the ${\cal H}_{\rm QPM}$ takes the form
\begin{equation}
{\cal H}_{\rm TRPA}=\sum_{\lambda\mu i}\omega_{\lambda i}
(Q^\dag_{\lambda\mu i}Q^{\phantom{\dag}}_{\lambda\mu i}
-\widetilde Q^\dag_{\lambda\mu i}\widetilde Q^{\phantom{\dag}}_{\lambda\mu i}).
\end{equation}
To fix properly the thermal vacuum state $|0(T);{\rm ph}\rangle$
corresponding to TRPA we once again turn to the thermal state
condition \eqref{TSC} and derive the final expressions for the
amplitudes of the thermal phonon operator \eqref{phonon}.
Once the structure of thermal phonons is determined, one can determine
the transition probabilities from the thermal vacuum to thermal
one-phonon states. They are given by the squared reduced matrix
elements of the corresponding transition operator
$\mathcal{T}_{\lambda\mu}$
\begin{align}\label{trans_ampl}
\Phi_{\lambda i}&=\bigl|\langle Q_{\lambda
i}\|\mathcal{T}_{\lambda}\|0(T);\mathrm{ph}\rangle\bigr|^2,
\notag\\
\widetilde \Phi_{\lambda i}&=\bigl|\langle\widetilde Q_{\lambda
i}\|\mathcal{T}_{\lambda}\|0(T);\mathrm{ph}\rangle\bigr|^2.
\end{align}
Thus, the probability to excite the hot nucleus is given by
$\Phi_{\lambda i}$, while $\widetilde\Phi_{\lambda i}$ is the
probability to de-excite it.
\subsection{Cross section of inelastic neutrino-nucleus scattering}
Considering neutrino-nucleus inelastic scattering in stellar
environments we assume that a nucleus is in thermal equilibrium
with a heat bath and particle reservoir
or, in TFD terms, in the thermal (phonon) vacuum state. An
inelastic collision of a hot nucleus with neutrinos leads to
transitions from the thermal vacuum to thermal one-phonon states.
In the derivation of the relevant cross section at finite
temperature we follow the formalism by Walecka-Donnelly \cite{Wal75,Don79},
which describes in a unified way electromagnetic and weak
semileptonic processes by taking advantage of the
multipole decomposition of the relevant hadronic current density
operator. In the case of neutral-current neutrino-nucleus
scattering, the differential cross section for a transition from an
initial nuclear state ($i$) to a final state ($f$) can be written as
a sum over all allowed multipolarities $J^\pi$
\begin{equation}\label{dif_cr_sect}
\frac{ d\sigma_{i\to f}}{d\Omega} = \frac{2G^2}{\pi}\frac {(E_\nu-\omega_{if})^2 \cos^2\frac{\Theta}{2} }{2J_i+1}
\Bigl\{\sum_{J=0}^\infty \sigma^J_{\rm CL} + \sum_{J=1}^\infty \sigma^J_{\rm T}
\Bigr\},
\end{equation}
where
\begin{equation}\label{CL}
\sigma^J_{\rm CL} = \big|\langle J_f\| \hat M_J + \frac{\omega_{if}}{q} \hat L_J\|J_i\rangle \big|^2
\end{equation}
and
\begin{multline}\label{T}
\sigma^J_{\rm T}=\Bigl(-\frac{q^2_\mu}{2q^2} + \tan^2\frac{\Theta}{2} \Bigr)
\Bigl[ |\langle J_f\| \hat J^{\rm mag}_J\|J_i\rangle|^2 + |\langle J_f\| \hat J^{\rm el}_J\|J_i\rangle|^2 \Bigr]
\\
-\tan\frac{\Theta}{2}\sqrt{-\frac{q^2_\mu}{2q^2} + \tan^2\frac{\Theta}{2}}
\Bigl[ 2 \mathrm{Re} \langle J_f\| \hat J^{\rm mag}_J\|J_i\rangle\langle J_f\| \hat J^{\rm el}_J\|J_i\rangle^* \Bigr].
\end{multline}
Here $G$ is the electroweak coupling constant, $\Theta$ is the
scattering angle, $E_\nu$ is the incoming neutrino energy,
$\omega_{if}$ is the transition energy from the initial nuclear
state ($i$) to the final state ($f$), and $q_\mu=(\omega_{if}, \vec
q)$ $\Big(q=|\vec
q|=\sqrt{\omega_{if}^2+4E_\nu(E_\nu-\omega_{if})\sin^2\frac{\Theta}{2}}~\Big)$
is the four-momentum transfer. The operators $\hat M_J$, $\hat L_J$,
$\hat J^{\rm el}_J$, and $\hat J^{\rm mag}_J$ are the multipole
operators for the charge, longitudinal, and the transverse electric
and magnetic parts of the four-current, respectively. Following
\cite{Wal75} they can be written in terms of one-body operators in
the nuclear many-body Hilbert space.
The cross section involves the reduced matrix elements of these
operators between the initial and final nuclear states. Within the
present approach, the initial nuclear state is the thermal phonon
vacuum (TV) and the final states are the thermal one-phonon states.
Therefore, at $T\neq 0$ all the reduced matrix elements in
Eqs.~(\ref{CL},\ref{T}) are calculated in accordance with
Eqs.~\eqref{trans_ampl}. The total cross section is obtained from
the differential cross sections by summing over all possible
one-phonon states of different multipolarity and by numerical
integration over scattering angles
\begin{equation}
\sigma(E_\nu)=2\pi\sum_{f\in\{\lambda i\}} \int_1^{-1} \frac{ d\sigma_{\mathrm{TV}\to f}}{d\Omega}d \cos\Theta.
\end{equation}
Up to moderate energies ($E_{\nu}\sim 15-20$~MeV), the inelastic
neutrino-nucleus scattering is dominated by the neutral-channel
Gamow-Teller transitions $J^\pi=1^+$. Moreover, in the $q \to 0$ limit,
the full operator exciting $1^+$ states is reduced to the following
Gamow-Teller operator:
\begin{equation}\label{GT0}
\mathrm{GT}_0= \Big(\frac{g_{A}}{g_{V}}\Bigr)\vec\sigma t_0
\end{equation}
where $(g_{A}/g_{V})=-1.2599$~\cite{Towner95} is the ratio of the
axial and vector weak coupling constants, $\vec \sigma$ is the spin
operator and $t_0$ is the zero-component of the isospin operator in
spherical coordinates.
To circumvent computational limitations in the LSSM calculations
\cite{PLB02,NPA05,PRL08} the total INNS cross section
$\sigma(E_\nu)$ was split into two parts -- a down-scattering part
$\sigma_{d}(E_\nu)$ and the up-scattering part $\sigma_{u}(E_\nu)$.
The term $\sigma_d(E_\nu)$ includes transitions where the scattered
neutrino loses energy whereas the term $\sigma_{u}(E_\nu)$ includes
those transitions where the neutrino gains energy from a hot
nucleus. Assuming the validity of the Brink hypothesis for the
GT$_0$ resonance, the down-scattering term was transformed to a sum
over only those final excited nuclear states which are coupled by a
direct GT$_0$ transition with the nuclear ground state. As a result,
$\sigma_{d}(E_\nu)$ appeared to be independent of $T$.
In our case, the part $\sigma_d(E_\nu)$ corresponds to transitions
from $|0(T);\mathrm{ph}\rangle$ to $|Q_{\lambda i}\rangle$ states
with positive energies whereas the $\sigma_{u}(E_\nu)$ term is the
sum of transitions $|0(T);\mathrm{ph}\rangle \to
|\widetilde{Q_{\lambda i}}\rangle$ where the tilde-states have negative
energies. In the latter transitions a neutrino gains energy due
to nuclear de-excitation.
Thus within the present approach the GT$_0$ ($J^\pi=1^+$)
contribution to the cross section reads
\begin{equation}\label{INNS-TFD}
\sigma(E_\nu)=\sigma_d(E_\nu)+\sigma_{u}(E_\nu) =
\frac{G^2}{\pi}\sum_{i}(E_\nu -\omega_{J i})^2\Phi_{J
i}+\frac{G^2}{\pi}\sum_{i}(E_\nu +\omega_{J i})^2\widetilde\Phi_{J
i},
\end{equation}
The probabilities $\Phi_{J i}$ and $\widetilde\Phi_{J i}$ are given
in \eqref{trans_ampl} with $\mathcal{T} = \mathrm{GT_0}$. Since
$\omega_{J i}$, $\Phi_{J i}$ and $\widetilde\Phi_{J i}$ are
functions of $T$, both terms $\sigma_d$ and $\sigma_u$ depend on
temperature.
Whereas the GT$_0$ component determines the neutrino-nucleus cross
section at low $E_\nu$, higher multipole contributions become
increasingly important at higher neutrino energies. Moreover, at
higher neutrino energies Eq.~\eqref{GT0} for GT$_0$ is not valid and
the $1^+$ transition operator will depend on transfer momentum $q$.
According to Refs.~\cite{Kolbe1999, Hektor2000} the $q$-dependence
reduces the cross section.
\section{Calculations for the hot nucleus $^{54}$Fe}
Numerical calculations have been performed for $^{54}$Fe.
The single-particle wave functions and energies were calculated in a
spherically symmetric Woods-Saxon potential. The constants of the
pairing interaction were determined to reproduce experimental
pairing energies in the BCS approximation. All parameters are
the same as in our previous calculations \cite{PRC81,BulRAS08} for
electron capture rates on the same nucleus at $T\neq 0$.
The radial dependence of the residual multipole and spin-multipole
forces is chosen in the form $R_\lambda(r)=\partial U(r)/\partial r$
where $U(r)$ is the central part of the single-particle Woods-Saxon
potential. Thus, $R_\lambda(r)$ as well as the parameters
$\kappa^{(\lambda)}_{0,1}$ and $\kappa^{(L\lambda)}_{0,1}$ do not
depend on $\lambda$. The isovector parameters
$\kappa^{(\lambda)}_{1}$ and $\kappa^{(L\lambda)}_{1}$ are fitted to
the experimental position of the E1~\cite{Nor78} and
M1~\cite{Richter85} resonances in $^{54}$Fe. According to the
estimates in Refs.~\cite{DTKPV86,PVV87}, the isoscalar
spin-multipole interaction is very weak in comparison with the
isovector one. Following~\cite{PVV87}, we take
$\kappa^{(L\lambda)}_{0}/\kappa^{(L\lambda)}_{1}=0.1$.
First, we have performed TQRPA calculations of the GT$_0$ strength
distribution in $^{54}$Fe. As in the LSSM
calculations~\cite{NPA05}, the GT$_0$ operator~\eqref{GT0} have been
scaled by a quenching factor~$0.74$. In Fig.~\ref{figure1}, we
display the GT$_0$ strength distributions for the ground state
($T=0$) of $^{54}$Fe and at three stellar temperature values,
occurring at different collapse stages: $T=0.86$~MeV corresponds to
the condition in the core of a presupernova model for a
$15\text{M}_{\odot}$ star; $T=1.29$~MeV and $T=1.72$~MeV relate
approximately to the neutrino trapping and neutrino thermalization
stages, respectively. All results are plotted as a function of the
energy transfer to $^{54}$Fe. For charge-neutral reactions this
energy is equal to a thermal phonon energy $\omega_{J i}$.
At $T=0$, the transition strength is concentrated mostly in
one-phonon $1^+$ state forming the GT$_0$ resonance near $\omega
\approx 10$~MeV. The main contribution to the phonon structure comes
from the proton and neutron single-particle transitions $1f_{7/2}
\to 1f_{5/2}$. With temperature increase the fraction of low-energy
transitions in the GT$_0$ strength distribution increases. The physical
reason is the weakening and subsequent collapse of pairing
correlations (at $T \approx 0.8$~MeV) and appearance of low-energy
particle-particle and hole-hole transitions due to thermal smearing
of neutron and proton Fermi surfaces. Moreover, at finite
temperature the ``negative energy'' transitions to tilde one-phonon
states appear. As a result, the GT$_0$ energy centroid is shifted
down by 1.1~MeV at $T=1.72$~MeV. This indicates a violation of the
Brink hypothesis within the present approach.
The contribution of $1^+$ transitions to the INNS cross section is
shown in Fig.~\ref{figure2}(a) for different temperatures. The
calculations have been performed with the exact $q$-dependent $1^+$
multipole transition operator~\cite{Wal75}. As in the LSSM
calculations \cite{PLB02}, the cross section $\sigma(E_\nu)$ at
$T=0$ is equal to zero when $E_\nu$ is less than the energy of the
lowest $1^+$ state in $^{54}$Fe. Within the QRPA, the lowest $1^+$
state in $^{54}$Fe has an excitation energy of $\omega(1^+)\approx
7.5$~MeV (see Fig.~\ref{figure1}). The GT$_0$ transitions at $T\ne
0$ do not show such a gap due to thermally unblocked low- and
negative-energy transitions. As a consequence, there is no a
threshold energy for neutrinos at finite temperatures and the INNS
cross section appears to be quite sensitive to $T$ at neutrino
energies $E_\nu<10$~MeV. As it follows from the present calculations
as well as from the LSSM study~\cite{PLB02}, thermal effects can
increase the low energy cross section by up to two orders of
magnitude when the temperature rises from 0.86~MeV to 1.72~MeV.
Finite temperature effects are unimportant for $E_\nu>15$~MeV where
excitation of the GT$_0$ resonance becomes possible and dominates
the cross section. These features were pointed in \cite{PLB02} as
well.
To check the influence of finite momentum transfer on the INNS cross
section we also have performed calculations with the GT$_0$
transition operator~\eqref{GT0}. A comparison of $1^+$ and GT$_0$
cross sections is shown in Fig.~\ref{figure2}(b) for $T=0.86$~MeV.
The $q$-dependence becomes important at $E_\nu>30$~MeV. At
$E_\nu=35$~MeV the INNS cross section calculated with the
$q$-dependent $1^+$ operator is by 20\% less than that calculated
with the GT$_0$ operator \eqref{GT0}. At $E_\nu=50$~MeV the
difference is by about factor of 2. The effect does not change with
temperature.
The contribution of first-forbidden transitions $0^-,~1^-$, and
$2^-$ to the INNS cross section were also calculated within the TQRPA,
taking into account the $q$-dependence as given in~\cite{Wal75}. The
results are presented in Fig.~\ref{figure3}. As it can be seen, a
temperature increase enhances the cross sections at low
and moderate $E_\nu$. The main reason is thermally unblocked
low-energy first-forbidden transitions. According to our
calculations $2^-$ transitions dominate the total contribution of
first-forbidden transitions to the cross section at low neutrino
energies, while at higher energies the total contribution is
mainly determined by the $1^-$ transitions.
In Fig.~\ref{figure4}, the INNS cross sections at different
temperatures are shown as a sum of $1^+, 0^-,1^-$, and $2^-$
contributions (we omit the contribution of the $0^+$ multipole
because it is negligible). At low $E_\nu$ the cross sections are
almost completely dominated by the GT$_0$ transitions. The part of
the cross sections arising from the first-forbidden transitions
becomes increasingly important at larger $E_\nu$. We find that for
$E_\nu=30$~MeV up to 20\% of the cross section is due to
first-forbidden transitions. For $E_\nu=40$~MeV allowed and
forbidden transitions contribute about equally, while at
$E_\nu=50$~MeV the contribution of first-forbidden transitions is
nearly twice as large as that of $1^+$ transitions.
In the LSSM calculations, the temperature-related enhancement of
$\sigma(E_\nu)$ was only due to the neutrino up-scattering. In our
approach both the up-scattering and down-scattering parts of
$\sigma(E_\nu)$ are temperature dependent. To analyze the relative
importance of these two types of scattering processes we
display them separately as the functions of $E_\nu$ for different
values of $T$ in Fig.~\ref{figure5}.
A weak $T$-dependence of $\sigma_d$ is seen at low neutrino
energies $E_\nu < 12$~MeV. At higher energies $\sigma_d$ practically
does not depend on $T$. As the function of $E_\nu$ the
down-scattering cross section sharply increases at low neutrino
energies and then grows more slowly. Instead, $\sigma_u$
is quite sensitive to temperature but its dependence on $E_\nu$ is
obviously smoother than that of $\sigma_d$ (at least at $E_\nu <
15$~MeV). The absolute values of $\sigma_d$ and $\sigma_u$ are of
the same order of magnitude only at quite low neutrino energies
$E_\nu \lesssim 4-10$~MeV.
Thus the conclusion is that the $T$-dependence of the INNS cross
section at low neutrino energies is mainly due to up-scattering
process whereas at neutrino energies $E_\nu > 15$~MeV when the
thermal effects are much less important the INNS cross section is
determined by the neutrino down-scattering.
The above conclusions agree well with the results of the LSSM
studies for even-even nuclei \cite{PLB02,NPA05}. Furthermore, our
results for $\sigma_d$ confirm the applicability of approximations
based on the Brink hypothesis, which has been used in calculations
of $\sigma_d$ in the LSSM.
\section{Conclusions}
We have performed studies of the temperature dependence of the cross
section for inelastic neutrino-nucleus scattering off the hot nucleus
$^{54}$Fe. Thermal effects were treated within the thermal
quasiparticle random phase approximation in the context of the TFD
formalism. These studies are relevant for supernova simulations.
In contrast to the large-scale shell-model studies
\cite{PLB02,NPA05} we do not assume the Brink hypothesis when
treating the down-scattering component of the cross section
$\sigma(E_\nu)$. Moreover, we take into account thermal effects
not only for the allowed $1^+$ transitions but also for the
first-forbidden transitions $0^-,~1^-$, and $2^-$. For all
multipole contributions we have performed the calculations with momentum
dependent multipole operators.
Despite these differences between the two approaches, our
calculations have revealed the same thermal effects as were found
in~\cite{PLB02,NPA05}: A temperature increase leads to a
considerable enhance of the INNS cross section for neutrino energies
lower than the energy of the GT$_0$ resonance. This enhancement is
mainly due to neutrino up-scattering at finite temperature.
The calculated cross sections for $^{54}$Fe are very close
to those given in \cite{NPA05}. Thus, the results of our study show
that the present approach provides a valuable tool for the
evaluation of the inelastic neutrino-nucleus cross sections under
stellar conditions. The approach can be easily adopted to calculate
the INNS cross sections as a function of scattering angle.
\section*{Acknowledgments}
The fruitful discussions with K.\,Langanke and G.\,Mart\'inez-Pinedo
are gratefully acknowledged. This work is supported in part by the
Heisenberg-Landau Program and the DFG grant (SFB 634).
|
3,212,635,537,423 | arxiv | \section{A$_\infty$-algebras}\label{2}
We first review the definition of A$_\infty$-algebras that are used
in the discussion of this paper.
\begin{defn}\label{2.1}
A \textbf{coalgebra} $(C,\Delta)$ over a ring $R$ consists of an
$R$-module $C$ and a comultiplication $\Delta:C\to C\otimes C$ of
degree $0$ satisfying coassociativity:
\[
\begin{diagram}
\node{C}\arrow{e,t}{\Delta} \arrow{s,l}{\Delta}
\node{C\otimes C}\arrow{s,r}{\Delta\otimes id}\\
\node{C\otimes C}\arrow{e,b}{id\otimes \Delta} \node{C\otimes
C\otimes C}
\end{diagram}
\]
Then a \textbf{coderivation} on $C$ is a map $f:C\to C$ such that
\[
\begin{diagram}
\node{C}\arrow{e,t}{\Delta} \arrow{s,l}{f}
\node{C\otimes C}\arrow{s,r}{f\otimes id+id\otimes f}\\
\node{C}\arrow{e,b}{\Delta} \node{C\otimes C}
\end{diagram}
\]
\end{defn}
\begin{defn}\label{2.2}
Let $V=\bigoplus_{j\in \mathbb{Z}} V_{j}$ be a graded module over a
given ground ring $R$. The \textbf{tensor-coalgebra} of $V$ over the
ring $R$ is given by
$$ TV:=\bigoplus_{i\geq 0} V^{\otimes i}, $$
$$ \Delta:TV\to TV\otimes TV, \quad
\Delta(v_{1},...,v_{n}):=\sum_{i=0}^{n} (v_{1},...,v_{i})
\otimes(v_{i+1},...,v_{n}).$$
Let $A=\bigoplus_{j\in \mathbb{Z}} A_{j}$ be a graded module over the
given ground ring $R$. Define its \textbf{suspension} $sA$ to be the
graded module $sA= \bigoplus_{j\in \mathbb{Z}} (sA)_{j}$ with
$(sA)_{j}:= A_{j-1}$. The suspension map $s:A\to sA$, $s:a\mapsto
sa:=a$ is an isomorphism of degree +1.
Now the \textbf{bar complex} of $A$ is given by $BA:=T(sA)$.
An \textbf{A$_\infty$-algebra} on $A$ is given by a coderivation $D$
on $BA$ of degree $-1$ such that $D^{2}=0$.
\end{defn}
The tensor-coalgebra has the property to lift every module map
$f:TV\to V$ to a coalgebra-map $F:TV \to TV$:
\[
\begin{diagram}
\node{} \node{TV}\arrow{s,r}{projection}\\
\node{TV}\arrow{e,b}{f}\arrow{ne,t}{F} \node{V}
\end{diagram}
\]
A similar property for coderivations on $TV$ will lead to give an
alternative description of A$_\infty$-algebras.
\begin{lem}\label{2.3}
\begin{itemize}
\item [(a)]
Let $\varrho:V^{\otimes n}\to V$, with $n\geq 0$, be a map of degree $|\varrho|$,
which can be viewed as $\varrho:TV\to V$ by letting its only
non-zero component being given by the original $\varrho$ on
$V^{\otimes n}$. Then $\varrho$ lifts uniquely to a coderivation
$\tilde{\varrho} :TV \to TV$ with
\[
\begin{diagram}
\node{} \node{TV}\arrow{s,r}{projection} \\
\node{TV}\arrow{ne,t}{\tilde{\varrho}} \arrow{e,b}{\varrho}
\node{V}
\end{diagram}
\]
by taking
$$ \tilde{\varrho}(v_{1},...,v_{k}):=0, \quad \text{for } k<n, $$
\begin{multline*}
\quad\quad\quad \tilde{\varrho}(v_{1},...,v_{k}):=\sum_{i=0}^{k-n}
(-1)^{|\varrho|\cdot(|v_{1}|+...+|v_{i}|)}(v_{1},...,\varrho
(v_{i+1},...,v_{i+n}),...,v_{k}), \\
\text{for } k\geq n.
\end{multline*}
Thus, $\tilde{\varrho}\mid_{V^{\otimes k}}:V^{\otimes k} \to
V^{\otimes k-n+1} $.
\item [(b)] There is a one-to-one correspondence between
coderivations $\sigma:TV\to TV$ and systems of maps
$\{\varrho_{i}:V^{\otimes i}\to V\}_{i\geq 0}$, given by
$\sigma=\sum_{i\geq 0} \tilde{\varrho_{i}}$.
\end{itemize}
\end{lem}
\begin{proof}
\begin{itemize}
\item [(a)]
Denote by $\tilde{\varrho}^{j}$ the component of $\tilde{\varrho}$
mapping $TV\to V^{\otimes j}$. Then
$\tilde{\varrho}^{1},...,\tilde{\varrho}^ {m-1}$ uniquely determine
the component $\tilde{\varrho}^{m}$, using the coderivation property
of $\tilde{\varrho}$.
\begin{eqnarray*}
\quad\quad
\Delta(\tilde{\varrho}(v_{1},...,v_{k}))&=&(\tilde{\varrho}\otimes
id+id\otimes \tilde{\varrho})(\Delta(v_{1},...,v_{k})) \\
&=& \sum_{i=0}^{k}\tilde{\varrho}(v_{1},...,v_{i})\otimes
(v_{i+1},...,v_{k}) \\
& & \,\,\, +(-1)^{|\tilde{\varrho}|\cdot(|v_{1}|+...+|v_{i}|)}
(v_{1},...,v_{i})\otimes \tilde{\varrho} (v_{i+1},...,v_{k}).
\end{eqnarray*}
Projecting both sides to $V^{\otimes i} \otimes
V^{\otimes j} \subset TV\otimes TV$, with $i+j=m$, yields
\begin{multline*}
\quad\quad\quad \Delta(\tilde{\varrho}^{m}(v_{1},...,v_{k}))|_{V^{\otimes i}\otimes V^{\otimes j}}=
\tilde{\varrho}^{i}(v_{1},...,v_{k-j})\otimes (v_{k-j+1},...,v_{k}) \\
+(-1)^{|\tilde{\varrho}|\cdot(|v_{1}|+...+|v_{i}|)}
(v_{1},...,v_{i})\otimes \tilde{\varrho}^{j} (v_{i+1},...,v_{k}).
\end{multline*}
For $m=i=1$ and $j=0$, this shows that $\tilde{\varrho}^0=0$. $\tilde{\varrho}^1=\varrho$ by the condition of the Lemma, and for $m\geq 2$, choosing $i=m-1,\,\,\, j=1$ uniquely determines $\tilde{\varrho}^m$ by lower components. Thus, an induction shows, that $\tilde{\varrho}^{m}$ is only nonzero on
$V^{\otimes k}$ for $k=m+n-1$, where $\tilde{\varrho}^{m}
(v_{1},...,v_{m+n-1})$ is given by $ \sum_{i=0}^{m-1}
(-1)^{|\varrho|\cdot(|v_{1}|+...+|v_{i}|)} (v_{1},..., \varrho
(v_{i+1},...,v_{i+n}),...,v_{m+n-1})$.
\item [(b)]
The map
$$ \quad\quad\quad
\alpha:\{\{\varrho_{i}:V^{\otimes i}\rightarrow V\}_{i\geq 0}\} \to
Coder(TV), \quad \{\varrho_{i}: V^{\otimes i}\rightarrow V\}_{i\geq
0} \mapsto \sum_{i\geq 0} \tilde{\varrho_{i}} $$ is well defined.
Its inverse $\beta$ is given by
$\beta:\sigma\mapsto\{pr_{V}\circ\sigma|_{V^{\otimes i }}\}_{i\geq
0}$, because the explicit lifting property of (a) shows that
$\beta\circ\alpha=id$, and the uniqueness part of (a) shows that
$\alpha\circ\beta=id$.
\end{itemize}
\end{proof}
Application to Definition \ref{2.2} gives the following
\begin{prop}\label{2.4}
Let $(A,D)$ be an A$_\infty$-algebra, and let $D$ be given by a
system of maps $\{D_{i}:sA^{\otimes i} \to sA\}_{i\geq 1}$, with
$D_{0}=0$. Let $m_{i}:A^{\otimes i} \to A$ be given by $D_{i}=s\circ
m_{i}\circ (s^{-1})^{\otimes i}$. Then the condition $D^{2}=0$ is
equivalent to the following system of equations:
\begin{eqnarray*}
m_{1}(m_{1}(a_{1}))&=&0,\\
m_{1}(m_{2}(a_{1},a_{2}))-m_{2}(m_{1}(a_{1}),a_{2})-(-1)^{|a_{1}|}
m_{2}(a_{1},m_{1}(a_{2}))&=&0,\\
m_{1}(m_{3}(a_{1},a_{2},a_{3}))-m_{2}(m_{2}(a_{1},a_{2}),a_{3})+
m_{2}(a_{1},m_{2}(a_{2},a_{3}))& &\\
+m_{3}(m_{1}(a_{1}),a_{2},a_{3})+(-1)^{|a_{1}|}m_{3}
(a_{1},m_{1}(a_{2}),a_{3})& &\\
+(-1)^{|a_{1}|+|a_{2}|}m_{3}(a_{1},a_{2},m_{1}(a_{3}))&=&0,\\
...\\
\sum_{i=1}^{k} \sum_{j=0}^{k-i+1} (-1)^{\varepsilon} \cdot
m_{k-i+1} (a_{1},...,m_{i} (a_{j},...,a_{j+i-1}),...,a_{k})&=&0,\\
where \,\,\,\varepsilon=i\cdot \sum_{l=1}^{j-1}|a_{l}|+ (j-1)
\cdot(i+1)+k-i\\
...
\end{eqnarray*}
\end{prop}
\begin{proof}
This follows from Lemma \ref{2.3} after a careful check of the
involved signs.
\end{proof}
\begin{expl}\label{2.5}
Any differential graded algebra $(A,\partial,\mu)$ gives an
A$_\infty$-algebra-structure on $A$ by taking $m_{1}:=\partial$,
$m_{2}:=\mu$ and $m_{k}:=0$ for $k\geq 3$. The equations from
Proposition \ref{2.4} are the defining conditions of a differential
graded algebra:
\begin{eqnarray*}
\partial^{2}(a)&=&0,\\
\partial (a\cdot b)&=& \partial(a)\cdot b+ (-1)^{|a|}a\cdot \partial(b), \\
(a\cdot b)\cdot c &=& a\cdot (b\cdot c). \\
\end{eqnarray*}
There are no higher equations.
\end{expl}
\begin{defn}\label{2.6}
Let $(A,D)$ be an A$_\infty$-algebra. The
\textbf{Hochschild-cochain-complex of $A$} is defined to be the
space $C^{*}(A):=CoDer(BA,BA)$ of coderivations on $BA$ with the
differential $\delta:C^{*}(A)\to C^{*}(A)$ given by
$\delta(f):=[D,f]=D\circ f-(-1)^{|f|}f\circ D$. We have
$\delta^{2}=0$, because with $D$ of degree $-1$ and $D^{2}=0$, it
follows that $\delta^{2}(f)=[D,D\circ f-(-1)^{|f|}f\circ D]= D\circ
D\circ f -(-1)^{|f|} D\circ f\circ D -(-1)^{|f|+1} D\circ f \circ D
+ (-1)^{|f|+|f|+1} f\circ D \circ D=0$.
\end{defn}
\section{A$_\infty$-bimodules}\label{3}
Let $(A,D)$ be an A$_\infty$-algebra. We now define the concept of
an A$_\infty$-bimodule over $A$, which was also considered in
\cite{GJ} and \cite{M}. This should be a generalization of two
facts. First, it is possible to define the
Hochschild-cochain-complex for any algebra with values in a
bimodule, which we would also like to do in the A$_\infty$ case.
Second, any algebra is a bimodule over itself by left- and
right-multiplication, which should also hold in the A$_\infty$ case.
The following space and map are important ingredients.
\begin{defn}\label{3.1}
For modules $V$ and $W$ over $R$, we define
$$T^{W}V:=R\oplus\bigoplus_{k\geq 0, l\geq 0} V^{\otimes k} \otimes W
\otimes V^{\otimes l}.$$
Furthermore, let
\begin{eqnarray*}
\Delta^{W}:T^{W}V&\to & (TV\otimes T^{W}V)\oplus (T^{W}V\otimes TV),\\
\Delta^{W}(v_{1},...,v_{k},w,v_{k+1},...,v_{k+l})&:=&
\sum_{i=0}^{k}(v_{1},...,v_{i}) \otimes(v_{i+1},...,w,...,v_{n})\\
& &+\sum_{i=k}^{k+l} (v_{1},...,w,...,v_{i})
\otimes(v_{i+1},...,v_{k+l}).
\end{eqnarray*}
Again for modules $A$ and $M$ let $B^{M}A$ be given by $T^{sM}sA$,
where $s$ is the suspension from Definition \ref{2.2}.
\end{defn}\label{3.2}
Observe that $T^{W}V$ is not a coalgebra, but rather a bi-comodule
over $TV$. We need the definition of a coderivation from $TV$ to
$T^{W}V$.
\begin{defn} A \textbf{coderivation} from $TV$ to $T^{W}V$ is a map
$f:TV\to T^{W}V$ so that the following diagram commutes:
\[
\begin{diagram}
\node{TV}\arrow{s,l}{f}\arrow{e,t}{\Delta}
\node{TV\otimes TV}\arrow{s,r}{id\otimes f+f\otimes id} \\
\node{T^{W}V}\arrow{e,b}{\Delta^{W}} \node{(TV\otimes
T^{W}V)\oplus (T^{W}V\otimes TV)}
\end{diagram}
\]
For modules $A$ and $M$ let $C^{*}(A,M):=CoDer(BA,B^{M}A)$ be the
space of coderivations in the above sense, called the
\textbf{Hochschild-cochain-complex of $A$ with values in $M$}.
\end{defn}
\begin{lem}\label{3.3}
\begin{itemize}
\item [(a)]
Let $\varrho:V^{\otimes n}\to W$ be a map of degree $|\varrho|$,
which can be viewed as a map $\varrho:TV\to W$ by letting its only
non-zero component being given by the original $\varrho$ on
$V^{\otimes n}$. Then $\varrho$ lifts uniquely to a coderivation
$\tilde{\varrho} :TV \to T^{W}V$ with
\[
\begin{diagram}
\node{} \node{T^{W}V}\arrow{s,r}{projection} \\
\node{TV}\arrow{ne,t}{\tilde{\varrho}} \arrow{e,b}{\varrho}
\node{W}
\end{diagram}
\]
by taking
$$ \tilde{\varrho}(v_{1},...,v_{k}):=0, \quad \text{for } k<n, $$
\begin{multline*}
\quad\quad\quad\tilde{\varrho}(v_{1},...,v_{k}):=\sum_{i=0}^{k-n}
(-1)^{|\varrho|\cdot(|v_{1}|+...+|v_{i}|)}
(v_{1},...,\varrho(v_{i+1},...,v_{i+n}),...,v_{k}),\\
\text{for } k\geq n.
\end{multline*}
Thus $\tilde{\varrho}\mid_{V^{\otimes k}}:V^{\otimes k} \to
\bigoplus_{i+j=k-n} V^{\otimes i} \otimes W \otimes V^{\otimes j}$.
\item [(b)] There is a one-to-one correspondence between
coderivations $\sigma:TV\to T^{W}V$ and systems of maps
$\{\varrho_{i}:V^{\otimes i}\to W\}_{i\geq 0}$, given by
$\sigma=\sum_{i\geq 0} \tilde{\varrho_{i}}$.
\end{itemize}
\end{lem}
\begin{proof}
\begin{itemize}
\item [(a)]
The proof is similar to the one of Lemma \ref{2.3} (a). Let
$\tilde{\varrho}^{j}$ be the component of $\tilde{\varrho}$ mapping
$TV\to\bigoplus_{r+s=j} V^{\otimes r} \otimes W\otimes V^{\otimes
s}$, and $\tilde{\varrho}^{-1}$ the component $TV\to R$. The equation
\begin{eqnarray*}
\quad\quad\quad
\Delta^{W}(\tilde{\varrho}(v_{1},...,v_{k}))&=&(\tilde{\varrho}\otimes
id+id\otimes \tilde{\varrho})(\Delta(v_{1},...,v_{k})) \\
&=& \sum_{i=0}^{k}\tilde{\varrho}(v_{1},...,v_{i})\otimes
(v_{i+1},...,v_{k}) \\
& & \,\,\, +(-1)^{|\tilde{\varrho}|\cdot(|v_{1}|+...+|v_{i}|)}
(v_{1},...,v_{i})\otimes \tilde{\varrho} (v_{i+1},...,v_{k})
\end{eqnarray*}
projected to $R\otimes TV$ shows that $\tilde{\varrho}^{-1}=0$. $\tilde{\varrho}^{0}=\varrho$ is uniquely determined by the statement of the Lemma, and projecting for fixed $i+j=m$, to the component
\begin{multline*}
\quad\quad\quad \bigoplus_{r+s=i} (V^{\otimes r}\otimes W\otimes
V^{\otimes s})\otimes V^{\otimes j}+V^{\otimes j}\otimes \bigoplus_{r+s=i} (V^{\otimes r}\otimes W\otimes V^{\otimes s})\\
\subset T^{W}V\otimes TV+TV\otimes T^{W}V,
\end{multline*}
shows that $ \Delta^{W}(\tilde{\varrho}^{m}(v_{1},...,v_{k}))|_{\bigoplus_{r+s=i} (V^{r}\otimes W\otimes V^{s})\otimes V^{j}+V^{j}\otimes \bigoplus_{r+s=i} (V^{r}\otimes W\otimes V^{s})}$ is given by
\begin{equation*}
\quad\quad \tilde{\varrho}^{i}(v_{1},...,v_{k-j})\otimes (v_{k-j+1},...,v_{k})+(-1)^{|\tilde{\varrho}|\cdot(|v_{1}|+...+|v_{j}|)} (v_{1},...,v_{j})\otimes \tilde{\varrho}^i (v_{j+1},...,v_{k}) .
\end{equation*}
For $m\geq 1$, choosing $i=m-1,\,\,\, j=1$ uniquely determines $\tilde{\varrho}^{m}$ by lower components. Thus, an induction shows, that $\tilde{\varrho}^{m}$ is only nonzero on
$V^{\otimes k}$ for $k=m+n-1$, where $ \tilde{\varrho}^{m}
(v_{1},...,v_{m+n-1})$ is given by $\sum_{i=0}^{m-1}
(-1)^{|\varrho|\cdot(|v_{1}|+...+|v_{i}|)}\cdot \\ (v_{1},...,\varrho
(v_{i+1},...,v_{i+n}),...,v_{m+n-1})$.
\item [(b)]
Then maps
\begin{eqnarray*}
& \alpha:\{\{\varrho_{i}:V^{\otimes i}\rightarrow W\}_{i\geq 0}\}
\to Coder(TV,T^{W}V), & \{\varrho_{i}: V^{\otimes i}\rightarrow
W\}_{i\geq 0} \mapsto
\sum_{i\geq 0} \tilde{\varrho_{i}} \\
& \beta:Coder(TV,T^{W}V) \to \{\{\varrho_{i}: V^{\otimes
i}\rightarrow W\}_{i\geq 0}\}, &
\sigma\mapsto\{pr_{W}\circ\sigma|_{V^{\otimes i }}\}_{i\geq 0}
\end{eqnarray*}
are inverse to each other by (a).
\end{itemize}
\end{proof}
We put a differential on $C^{*}(A,M)$, similar to the one from
section \ref{2}.
\begin{prop}\label{3.4}
Let $(A,D)$ be an A$_{\infty}$-algebra and $M$ be a graded
module. Let $D^{M}:B^{M}A\to B^{M}A$ be a map of degree $-1$. Then
the induced map $\delta^{M}:CoDer(BA,B^{M}A) \to CoDer(BA,B^{M}A)$,
given by $\delta^{M}(f):=D^{M}\circ f-(-1)^{|f|}f\circ D$, is
well-defined, (i.e. it maps coderivations to coderivations,) if and
only if the following diagram commutes:
\begin{equation}\label{eqn-3.1}
\begin{diagram}
\node{B^{M}A}\arrow{s,l}{D^{M}}\arrow{e,t}{\Delta^{M}}
\node{(BA\otimes B^{M}A)\oplus (B^{M}A\otimes BA)}\arrow{s,r}
{(id\otimes D^{M}+D\otimes id) \oplus (D^{M}\otimes id+id\otimes D)} \\
\node{B^{M}A}\arrow{e,b}{\Delta^{M}} \node{(BA\otimes
B^{M}A)\oplus (B^{M}A\otimes BA)}
\end{diagram}
\end{equation}
\end{prop}
\begin{proof}
Let $f:BA\to B^{M}A$ be a coderivation. Then, $\delta^{M}(f)$ is a
coderivation, if
\begin{multline*}
(id\otimes\delta^{M}(f)+\delta^{M}(f)\otimes id)\circ\Delta=
\Delta^{M}\circ\delta^{M}(f) \text{, i.e.}\\
(id\otimes(D^{M}\circ f)-(-1)^{|f|}id\otimes(f\circ D)+
(D^{M}\circ f)\otimes id-(-1)^{|f|}
(f\circ D)\otimes id)\circ\Delta \\= \Delta^{M}\circ D^{M}\circ f-(-1)^{|f|}
\Delta^{M}\circ f \circ D.
\end{multline*}
Using the coderivation property for $f$ and $D$, we get
\begin{eqnarray*}
\Delta^{M}\circ f \circ D &=&
(id\otimes f)\circ\Delta\circ D + (f\otimes id)\circ\Delta\circ D\\
&=& (id\otimes(f\circ D) +(-1)^{|f|} D\otimes f + f\otimes D +
(f\circ D)\otimes id) \circ \Delta,
\end{eqnarray*}
so that the requirement for $\delta^{M}(f)$ being a coderivation
reduces to
\begin{eqnarray*}
\Delta^{M}\circ D^{M}\circ f&=& (id\otimes(D^{M}\circ f)+
(D^{M}\circ f)\otimes id+ D\otimes f +(-1)^{|f|} f\otimes D)\circ \Delta\\
&=& (id\otimes D^{M}+ D\otimes id)\circ(id\otimes f)\circ \Delta\\
&& +
(D^{M}\otimes id+ id\otimes D)\circ(f\otimes id)\circ \Delta\\
&=& (id\otimes D^{M}+ D\otimes id)\circ\Delta^{M} \circ f +
(D^{M}\otimes id+ id\otimes D)\circ\Delta^{M} \circ f.
\end{eqnarray*}
Thus, we get the following condition for $D^{M}$,
$$ \Delta^{M}\circ D^{M}\circ f = (id\otimes D^{M}+ D\otimes id+
D^{M}\otimes id+ id\otimes D)\circ\Delta^{M} \circ f $$
for all coderivations $f:TA\to T^{M}A$. With Lemma \ref{3.3} this
condition reduces to $ \Delta^{M}\circ D^{M}= (id\otimes D^{M}+
D\otimes id+ D^{M}\otimes id+ id\otimes D)\circ\Delta^{M}$, which is
the claim.
\end{proof}
We can describe $D^{M}$ by a system of maps.
\begin{lem}\label{3.5}
\begin{itemize}
\item [(a)]
Let $V$ be a module, and let $\psi$ be a coderivation on $TV$
with associated system of maps $\{\psi_{i}:V^{\otimes i} \to
V\}_{i\geq 1}$ from Lemma \ref{2.3}. Then any map $\varrho:T^{W}V\to W$
given by $\varrho=\sum_{k\geq0, l\geq0} \varrho_{k,l}$, with
$\varrho_{k,l}: V^{\otimes k}\otimes W \otimes V^{\otimes l}
\to W$, lifts uniquely to a map
$\tilde{\varrho} :T^{W}V \to T^{W}V$
\[
\begin{diagram}
\node{} \node{T^{W}V}\arrow{s,r}{projection} \\
\node{T^{W}V}\arrow{e,b}{\varrho} \arrow{ne,t}{\tilde{\varrho}}
\node{W}
\end{diagram}
\]
which makes the following diagram commute
\begin{equation}\label{eqn-3.2}
\begin{diagram}
\node{T^{W}V}\arrow{s,l}{\tilde{\varrho}}\arrow{e,t}{\Delta^{W}}
\node{(TV\otimes T^{W}V)\oplus (T^{W}V\otimes TV)}\arrow{s,r}
{(id\otimes \tilde{\varrho}+\psi\otimes id) \oplus (\tilde{\varrho}
\otimes id+id\otimes \psi)} \\
\node{T^{W}V}\arrow{e,b}{\Delta^{W}} \node{(TV\otimes
T^{W}V)\oplus (T^{W}V\otimes TV)}
\end{diagram}
\end{equation}
This map is given
\begin{multline*}
\tilde{\varrho}(v_{1},...,v_{k},w,v_{k+1},...,v_{k+l}) \\
:= \sum_{i=1}^{k} \sum_{j=1}^{k-i+1}
(-1)^{|\psi_{i}|\sum_{r=1}^{j-1}|v_{r}|}
(v_{1},...,\psi_{i}(v_{j},...,v_{i+j-1}),...,w,...,v_{k+l}) \quad\quad\quad\quad\quad \\
+\sum_{i=0}^{k} \sum_{j=0}^{l}
(-1)^{|\varrho_{i,j}|\sum_{r=1}^{k-i}|v_{r}|}
(v_{1},...,\varrho_{i,j}(v_{k-i+1},...,w,...,v_{k+j}),...,v_{k+l}) \quad\quad\quad\quad \\
+\sum_{i=1}^{l} \sum_{j=1}^{l-i+1}
(-1)^{|\psi_{i}|(|w|+\sum_{r=1}^{k+j-1}|v_{r}|)}
(v_{1},...,w,...,\psi_{i}(v_{k+j},...,v_{k+i+j-1}),...,v_{k+l}).
\end{multline*}
(Notice that the condition of diagram \eqref{eqn-3.2} is not
linear.)
\item [(b)]
There is a one-to-one correspondence between maps $\sigma:T^{W}V\to
T^{W}V$ that make diagram \eqref{eqn-3.2} commute and maps
$\varrho=\sum\varrho_{k,l}$ from (a), given by
$\sigma=\tilde{\varrho}$.
\end{itemize}
\end{lem}
\begin{proof}
\begin{itemize}
\item [(a)]
As in the Lemmas \ref{2.3} and \ref{3.3}, we denote by $\tilde{\varrho}^{j}$, $j \geq 0$, the component of $\tilde{\varrho}$ mapping $T^{W}V\to \bigoplus _{k+l=j} V^{\otimes k}\otimes W\otimes V^{\otimes l}$ and by $\tilde{\varrho}^{-1}$ the component $T^W V\to R$. $\psi^{j}$, for $j\geq 1$, denotes the component of $\psi$ mapping
$TV\to V^{\otimes j}$. Then $\tilde {\varrho}^{m}$ is uniquely
determined by $\tilde{\varrho}^{0},...,\tilde{\varrho}^ {m-1}$.
\begin{multline*}
\quad\quad\quad \Delta^{W}(\tilde{\varrho}
(v_{1},...,v_{k},w,v_{k+1},...,v_{k+l})) \\
=(id\otimes \tilde{\varrho}+\psi\otimes id+ \tilde{\varrho}
\otimes id+id\otimes\psi) (\Delta^{W}
(v_{1},...,v_{k},w,v_{k+1},...,v_{k+l}))
\end{multline*}
\begin{eqnarray*}
&=& \sum_{i=0}^{k}(-1)^{|\tilde{\varrho}|\sum_{r=1}^{i}|v_{r}|}
(v_{1},...,v_{i}) \otimes \tilde{\varrho}(v_{i+1},...,w,...,v_{k+l})\\
& & +\sum_{i=0}^{k}\psi(v_{1},...,v_{i}) \otimes
(v_{i+1},...,w,...,v_{k+l})\\
& & +\sum_{i=k}^{k+l}\tilde{\varrho}(v_{1},...,w,...,v_{i})
\otimes (v_{i+1},...,v_{k+l})\\
& & +\sum_{i=k}^{k+l} (-1)^{|\psi|(|w|+\sum_{r=1}^{i}|v_{r}|)}
(v_{1},...,w,...,v_{i}) \otimes \psi(v_{i+1},...,v_{k+l}).\,\,
\end{eqnarray*}
Projecting both sides to $R\otimes TV$ shows that $\tilde{\varrho}^{-1}=0$, and projecting for fixed $i+j=m$, to the component
\begin{multline*}
\quad\quad\quad V^{\otimes j}\otimes \bigoplus_{r+s=i} (V^{\otimes r}\otimes W\otimes V^{\otimes s})+ \bigoplus_{r+s=i} (V^{\otimes r}\otimes W\otimes V^{\otimes s})\otimes V^{\otimes j} \\
\subset T^{W}V\otimes TV+TV\otimes T^{W}V,
\end{multline*}
shows that $ \Delta^{W}(\tilde{\varrho}^{m}(v_{1},...,w,...,v_{k}))|_{V^{\otimes j}\otimes \bigoplus_{r+s=i} (V^{\otimes r}\otimes W\otimes V^{\otimes s})+ \bigoplus_{r+s=i} (V^{\otimes r}\otimes W\otimes V^{\otimes s})\otimes V^{\otimes j} }$ is given by
\begin{multline*}
\pm (v_{1},...,v_{j})\otimes \tilde{\varrho}^i (v_{j+1},...,w,...,v_{k+l}) + \psi^{j}(v_1,...,v_{k+l-i})\otimes (v_{k+l-i+1},...,w,...,v_{k+l}) \\
+ \tilde{\varrho}^{i}(v_{1},...,w,...,v_{k+l-j})\otimes (v_{k+l-j+1},...,v_{k+l})\pm (v_1,...,w,...v_i)\otimes \psi^j (v_{i+1},...,v_{k+l}).
\end{multline*}
For $m\geq 1$, choosing $i=m-1,\,\,\, j=1$ uniquely determines $\tilde{\varrho}^{m}$ by lower components, and the $\psi^j$'s. Thus, an induction shows the claim of the Lemma.
\item [(b)]
Let $X:=\{\sigma:T^{W}V\to T^{W}V\,\,|\,\, \sigma$ makes diagram
\eqref{eqn-3.2} commute$\}$. Then
\begin{eqnarray*}
\alpha:\{\varrho:T^{W}V\to W\}\to X,
& & \varrho \mapsto \tilde{\varrho}, \\
\beta:X\to \{\varrho:T^{W}V\to W\}, & & \sigma \mapsto
pr_{W}\circ\sigma
\end{eqnarray*}
are inverse to each other by (a).
\end{itemize}
\end{proof}
\begin{defn}\label{3.6}
Let $(A,D)$ be an A$_{\infty}$-algebra. Then an
\textbf{A$_{\infty}$-bimodule} $(M,D^{M})$ consists of a graded
module $M$ together with a map $D^{M}:B^{M}A\to B^{M}A$ of degree
$-1$, which makes the diagram \eqref{eqn-3.1} of Proposition
\ref{3.4} commute, and satisfies $(D^{M})^{2} =0$.
By Proposition \ref{3.4}, we may put the differential $\delta^{M}:
CoDer(TA,T^{M}A) \to CoDer(TA,T^{M}A)$, $\delta(f) :=D^{M}\circ
f-(-1)^{|f|}f\circ D$ on the Hochschild-cochain-complex. It
satisfies $(\delta^{M})^{2}=0$, because with $(D^{M})^{2} =0$, we
get $(\delta^{M})^{2}(f)=D^{M}\circ D^{M}\circ
f-(-1)^{|f|}D^{M}\circ f\circ D -(-1)^{|f|+1} D^{M}\circ f \circ D +
(-1)^{|f|+|f|+1} f\circ D\circ D =0$.
The definition of an A$_{\infty}$-bimodule was already stated in
\cite{GJ} section 3 and also in \cite{M}.
\end{defn}
\begin{prop}\label{3.7}
Let $(A,D)$ be an A$_\infty$-algebra, and let $\{m_{i}:A^{\otimes i}
\to A\}_{i\geq 1}$ be the system of maps associated to $D$ by
Proposition \ref{2.4}, with $m_{0}=0$. Let $(M,D^{M})$ be an
A$_\infty$-bimodule over $A$, and let $\{D^{M}_{k,l} :sA^{\otimes
k}\otimes sM \otimes sA^{\otimes l} \to sM\}_{k\geq 0, l\geq 0}$ be
the system of maps associated to $D^{M}$ by Lemma \ref{3.5} (b). Let
$b_{k,l}:A^{\otimes k}\otimes M \otimes A^{\otimes l} \to M$ be the
induced maps by $D^{M}_{k,l}=s\circ b_{k,l}\circ (s^{-1})^{\otimes
k+l+1}$. Then the condition $(D^{M})^{2}=0$ is equivalent to the
following system of equations:
\begin{eqnarray*}
b_{0,0}(b_{0,0}(m))&=&0, \\
b_{0,0}(b_{0,1}(m,a_{1}))-b_{0,1}(b_{0,0}(m),a_{1})-(-1)^{|m|}
b_{0,1}(m,m_{1}(a_{1})) &=& 0,\\
b_{0,0}(b_{1,0}(a_{1},m))-b_{1,0}(m_{1}(a_{1}),m)-(-1)^{|a_{1}|}
b_{1,0}(a_{1},b_{0,0}(m)) &=& 0,\\
b_{0,0}(b_{1,1}(a_{1},m,a_{2}))-b_{0,1}(b_{1,0}(a_{1},m),a_{2})+
b_{1,0}(a_{1},b_{0,1}(m,a_{2}))& &\\
+b_{1,1}(m_{1}(a_{1}),m,a_{2})
+(-1)^{|a_{1}|} b_{1,1}(a_{1},b_{0,0}(m),a_{2})& &\\
+(-1)^{|a_{1}|+|m|}b_{1,1}(a_{1},m,m_{1}(a_{2})) &=& 0,\\
\end{eqnarray*}
$$...$$
\begin{eqnarray*}
\sum_{i=1}^{k} \sum_{j=1}^{k-i+1} \pm
b_{k-i+1,l}(a_{1},...,m_{i}(a_{j},...,a_{i+j-1}),...,m,...,a_{k+l}) & &\\
+\sum_{i=0}^{k} \sum_{j=0}^{l} \pm
b_{k-i,l-j}(a_{1},...,b_{i,j}(a_{k-i+1},...,m,...,a_{k+j}),...,a_{k+l}) & &\\
+\sum_{i=1}^{l} \sum_{j=1}^{l-i+1} \pm
b_{k,l-i+1}(a_{1},...,m,...,m_{i}(a_{k+j},...,a_{k+i+j-1}),...,a_{k+l})&=&0
\end{eqnarray*}
$$...$$
where the signs are analogous to the ones in Proposition \ref{2.4}.
\end{prop}
\begin{proof}
The result follows from Lemma \ref{3.5}, after rewriting
$D^{M}_{k,l}$ and $D_{j}$ by $b_{k,l}$ and $m_{j}$.
\end{proof}
\begin{expl}\label{3.8}
With this, Example \ref{2.5} may be extended in the following way.
Let $(A,\partial,\mu)$ be a differential graded algebra with the
A$_\infty$-algebra-structure $m_{1}:=\partial$, $m_{2}:=\mu$ and
$m_{k}:=0$ for $k\geq 3$. Let $(M,\partial',\lambda,\rho)$ be a
differential graded bimodule over $A$, where $\lambda:A\otimes M\to
M$ and $\rho :M\otimes A\to M$ denote the left- and right-action,
respectively. Then, $M$ is an A$_\infty$-bimodule over $A$ by taking
$b_{0,0}:=\partial'$, $b_{1,0}:=\lambda$, $b_{0,1}:=\rho$ and
$b_{k,l}:=0$ for $k+l>1$. The equations of Proposition \ref{3.7} are
the defining conditions for a differential bialgebra over $A$:
\begin{eqnarray*}
(\partial')^{2}(m)&=&0,\\
\partial'(m.a)&=&m.\partial(a)+(-1)^{|m|}\partial'(m).a,\\
\partial'(a.m)&=&\partial(a).m+(-1)^{|a|}a.\partial'(m),\\
(a.m).b&=&a.(m.b),\\
(m.a).b&=&m.(a\cdot b),\\
a.(b.m)&=&(a\cdot b).m.
\end{eqnarray*}
There are no higher equations.
\end{expl}
For later purposes it is convenient to have the following
\begin{lem}\label{3.9}
Given an A$_\infty$-algebra $(A,D)$ and an A$_\infty$-bimodule
$(M,D^{M})$, with system of maps $\{b_{k,l} :A^{\otimes k}\otimes M
\otimes A^{\otimes l} \to M\}_{k\geq 0, l\geq 0}$ from Proposition
\ref{3.7}, then the dual space $M^{*}:=Hom_{R}(M,R)$ has a canonical
A$_\infty$-bimodule-structure given by maps $\{b'_{k,l} :A^{\otimes
k}\otimes M^{*} \otimes A^{\otimes l} \to M^{*}\}_{k\geq 0, l\geq
0}$,
\begin{multline*}
(b'_{k,l}(a_{1},...,a_{k},m^{*},a_{k+1},...,a_{k+l}))(m):=(-1)^
{\varepsilon}\cdot m^{*}(b_{l,k}(a_{k+1},...,a_{k+l},m,a_{1},...,a_{k})),\\
\text{ where } \varepsilon:= (|a_{1}|+...+|a_{k}|)\cdot
(|m^{*}|+|a_{k+1}|+...+|a_{k+l}|+|m|)+|m^{*}|\cdot(k+l+1).
\end{multline*}
\end{lem}
\begin{proof}
To see, that $(D^{M^{*}})^{2}=0$, we can use the criterion from
Proposition \ref{3.7}. The top and the bottom term in the general
sum of Proposition \ref{3.7} convert to
\begin{multline*}
(b'_{k-i+1,l}(a_{1},...,m_{i}(a_{j},...,a_{i+j-1})
,...,m^{*},...,a_{k+l}))(m)\\
=\pm m^{*}(b_{l,k-i+1}(a_{k+1},...,a_{k+l},m,
a_{1},...,m_{i}(a_{j},...,a_{i+j-1}),...,a_{k}))\text{, and}
\end{multline*}
\begin{multline*}
(b'_{k,l-i+1}(a_{1},...,m^{*},...,m_{i}(a_{k+j},...,
a_{k+i+j-1}),...,a_{k+l}))(m)\\
=\pm m^{*}(b_{l-i+1,k}(a_{k+1},...,m_{i}(a_{k+j},...,
a_{k+i+j-1}),...,a_{k+l},m,a_{1},...,a_{k})).
\end{multline*}
These terms come from the A$_\infty$-bimodule-structure of $M$.
Similar arguments apply to the middle term:
\begin{multline*}
(b'_{k-i,l-j}(a_{1},...,b'_{i,j}(a_{k-i+1},...,
m^{*},...,a_{k+j}),...,a_{k+l}))(m)\\
= \pm (b'_{i,j}(a_{k-i+1},...,m^{*},...,a_{k+j}))(b_{l-j,k-i}
(a_{k+j+1},...,a_{k+l},m,a_{1},...,a_{k-i}))\quad\quad\quad\,\, \\
= \pm m^{*}(b_{j,i}(a_{k+1},...,a_{k+j},b_{l-j,k-i}
(a_{k+j+1},...,a_{k+l},m,a_{1},...,a_{k-i}),a_{k-i+1},...,a_{k})).
\end{multline*}
The sum from Proposition \ref{3.7} for the A$_\infty$-bimodule
$M^{*}$ contains exactly the terms of $m^{*}$ applied the the sum
for the A$_\infty$-bimodule $M$. A thorough check identifies the
signs.
\end{proof}
\section{Morphisms of A$_\infty$-bimodules}\label{4}
Let $(M,D^{M})$ and $(N,D^{N})$ be two A$_\infty$-bimodules over the
A$_\infty$-algebra $(A,D)$. We next define the notion of
A$_\infty$-bimodule-map between $(M,D^{M})$ and $(N,D^{N})$. Again a
motivation is to have an induced map of their
Hochschild-cochain-complexes.
\begin{prop}\label{4.1}
Let $V$, $W$ and $Z$ be modules, and let $F$ be a map $F:T^{W}V \to
T^{Z}V$. Then the induced map $F^{\sharp} :CoDer(TV,T^{W}V) \to
CoDer(TV,T^{Z}V)$, given by $F^{\sharp}(f):=F\circ f$, is
well-defined, (i.e. it maps coderivations to coderivations,) if and
only if the following diagram commutes:
\begin{equation}\label{eqn-4.1}
\begin{diagram}
\node{T^{W}V}\arrow{s,l}{F}\arrow{e,t}{\Delta^{W}}
\node{(TV\otimes T^{W}V)\oplus (T^{W}V\otimes TV)}\arrow{s,r}
{(id\otimes F) \oplus (F\otimes id)} \\
\node{T^{Z}V}\arrow{e,b}{\Delta^{Z}} \node{(TV\otimes
T^{Z}V)\oplus (T^{Z}V\otimes TV)}
\end{diagram}
\end{equation}
\end{prop}
\begin{proof}
If both $f:TV\to T^{W}V$ and $F\circ f:TV\to T^{Z}V$ are
coderivations, then the top diagram and the overall diagram below
commute.
\[
\begin{diagram}
\node{TV}\arrow{s,l}{f}\arrow{e,t}{\Delta}
\node{TV\otimes TV}\arrow{s,r} {(id\otimes f) + (f\otimes id)} \\
\node{T^{W}V}\arrow{s,l}{F}\arrow{e,t}{\Delta^{W}}
\node{(TV\otimes T^{W}V)\oplus (T^{W}V\otimes TV)}\arrow{s,r}
{(id\otimes F) \oplus (F\otimes id)} \\
\node{T^{Z}V}\arrow{e,b}{\Delta^{Z}} \node{(TV\otimes
T^{Z}V)\oplus (T^{Z}V\otimes TV)}
\end{diagram}
\]
Therefore, the lower diagram has to commute if applied to any
element in $Im(f)\subset T^{W}V$. By Lemma \ref{3.3} there are
enough coderivations to imply the claim.
\end{proof}
Again let us describe $F$ by a system of maps.
\begin{lem}\label{4.2}
\begin{itemize}
\item [(a)]
Let $V$, $W$ and $Z$ be modules, and let $\varrho:V^{\otimes
k}\otimes W\otimes V^{\otimes l}\to Z$ be a map, which may be viewed
as a map $\varrho: T^{W}V \to Z$ whose only nonzero component is the
original $\varrho$ on $V^{\otimes k}\otimes W\otimes V^{\otimes l}$.
Then $\varrho$ lifts uniquely to a map $\tilde{\varrho} :T^{W}V \to
T^{Z}V$
\[
\begin{diagram}
\node{} \node{T^{Z}V}\arrow{s,r}{projection} \\
\node{T^{W}V} \arrow{e,b}{\varrho} \arrow{ne,t}{\tilde{\varrho}}
\node{Z}
\end{diagram}
\]
which makes the diagram \eqref{eqn-4.1} in Proposition \ref{4.1}
commute. $\tilde{\varrho}$ is given by
$$ \tilde{\varrho}(v_{1},...,v_{r},w,v_{r+1},...,v_{r+s}):=0,
\quad \text{ for } r<k \text{ or } s<l, $$
\begin{multline*}
\quad\quad\quad \tilde{\varrho}(v_{1},...,v_{r},w,v_{r+1},...,v_{r+s})\\
:= (-1)^{|\varrho|\sum_{i=1}^{r-k}|v_{i}|}
(v_{1},...,\varrho(v_{r-k+1},...,w,...,v_{r+l}),...,v_{r+s}),\\
\text{for } r\geq k \text{ and } s\geq l.
\end{multline*}
Thus $\tilde{\varrho}\mid_{V^{\otimes r}\otimes W\otimes V^{\otimes
s}}:V^{\otimes r}\otimes W\otimes V^{\otimes s} \to V^{\otimes r-k}
\otimes Z\otimes V^{\otimes s-l}$.
\item [(b)]
There is a one-to-one correspondence between maps $\sigma:T^{W}V\to
T^{Z}V$ making diagram \eqref{eqn-4.1} commute and systems of maps
$\{\varrho_{k,l}:V^{\otimes k}\otimes W\otimes V^{\otimes l}\to
Z\}_{k\geq 0, l\geq 0}$, given by $\sigma=\sum_{k\geq 0, l\geq 0}
\widetilde{\varrho_{k,l}}$.
\end{itemize}
\end{lem}
\begin{proof}
\begin{itemize}
\item [(a)]
Denote by $\tilde{\varrho}^{j}$ the component of $\tilde{\varrho}$
mapping $T^{W}V\to \bigoplus _{r+s=j} V^{\otimes r}\otimes Z\otimes
V^{\otimes s}$, and $\tilde{\varrho}^{-1}$ the component $T^W V\to R$. Then, $\tilde{\varrho}^{0},...,\tilde{\varrho}^
{m-1}$ uniquely determine the component $\tilde{\varrho}^{m}$.
$$ \Delta^{Z}(\tilde{\varrho}(v_{1},...,v_{r},w,v_{r+1},...,
v_{r+s}))\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad $$
\begin{eqnarray*}
&=&(id\otimes \tilde{\varrho}+\tilde{\varrho} \otimes id)
(\Delta^{W} (v_{1},...,v_{r},w,v_{r+1},...,v_{r+s}))\\
&=& \sum_{i=0}^{r}(-1)^{|\tilde{\varrho}|\sum_{t=1}^{i}|v_{t}|}
(v_{1},...,v_{i}) \otimes \tilde{\varrho}(v_{i+1},...,w,...,v_{r+s})\\
& & +\sum_{i=r}^{r+s}\tilde{\varrho}(v_{1},...,w,...,v_{i})
\otimes (v_{i+1},...,v_{r+s}).\\
\end{eqnarray*}
Projecting both sides to $R\otimes TV$ shows that $\tilde{\varrho}^{-1}=0$, and projecting for fixed $i+j=m$, to the component
\begin{multline*}
\quad\quad\quad V^{\otimes j}\otimes \bigoplus_{r+s=i} (V^{\otimes r}\otimes Z\otimes V^{\otimes s})+\bigoplus_{r+s=i} (V^{\otimes r}\otimes Z\otimes V^{\otimes s})\otimes V^{\otimes j}\\
\subset T^{Z}V\otimes TV+TV\otimes T^{Z}V,
\end{multline*}
yields for $ \Delta^{Z}(\tilde{\varrho}^{m}(v_{1},...,w,...,v_{r+s}))|_{V^{\otimes j}\otimes \bigoplus_{r+s=i} (V^{\otimes r}\otimes Z\otimes V^{\otimes s})+\bigoplus_{r+s=i} (V^{\otimes r}\otimes Z\otimes V^{\otimes s})\otimes V^{\otimes j}}$ the expression
\begin{equation*}
\pm (v_{1},...,v_{j}) \otimes
\tilde{\varrho}^{i}(v_{j+1},...,w,...,v_{r+s})
+\tilde{\varrho}^{i}
(v_{1},...,w,...,v_{r+s-j}) \otimes (v_{r+s-j+1},...,v_{r+s}).
\end{equation*}
For $m\geq 1$, choosing $i=m-1,\,\,\, j=1$ uniquely determines $\tilde{\varrho}^{m}$ by lower components. Therefore, an induction shows that
$\tilde{\varrho}^{m}$ is only nonzero on $V^{\otimes r}\otimes
W\otimes V^{\otimes s}$ with $r-k+s-l=m$, where
$\tilde{\varrho}^{m}(v_{1},...,v_{r},w,v_{r+1},...,v_{r+s})$ is
given by $ (-1)^{|\varrho|\sum_{i=1}^{r-k}|v_{i}|}
(v_{1},...,\varrho(v_{r-k+1},...,w,...,v_{r+l}),...,v_{r+s})$.
\item [(b)]
Let $X:=\{\sigma:T^{W}V\to T^{Z}V\,\,|\,\, \sigma$ makes diagram
\eqref{eqn-4.1} commute$\}$. Then
\begin{eqnarray*}
& & \alpha:\{\{\varrho_{k,l}:V^{\otimes k}\otimes W\otimes
V^{\otimes l}\to Z\}_{k\geq 0, l\geq 0}\}
\to X, \\
& & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\{\varrho_{k,l}:V^{\otimes k}\otimes W\otimes V^{\otimes l}\to
Z\}_{k\geq 0, l\geq 0} \mapsto
\sum_{k\geq 0, l\geq 0} \widetilde{\varrho_{k,l}},\\
& & \beta:X \to \{\{\varrho_{k,l}:V^{\otimes k}\otimes
W\otimes V^{\otimes l}\to Z\}_{k\geq 0, l\geq 0}\},\\
& & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\sigma\mapsto\{pr_{Z}\circ\sigma|_{V^{\otimes k}\otimes W\otimes
V^{\otimes l}}\}_{k\geq 0, l\geq 0}
\end{eqnarray*}
are inverse to each other by (a).
\end{itemize}
\end{proof}
We may apply this to the Hochschild-complex.
\begin{defn}\label{4.3}
Let $(M,D^{M})$ and $(N,D^{N})$ be two A$_\infty$-bimodules over the
A$_\infty$-algebra $(A,D)$. Then a map $F: B^{M}A\to B^{N}A$ of
degree 0 is called an \textbf{A$_\infty$-bimodule-map}, if $F$ makes
the diagram
\[
\begin{diagram}
\node{B^{M}A}\arrow{s,l}{F}\arrow{e,t}{\Delta^{M}}
\node{(BA\otimes B^{M}A)\oplus (B^{M}A\otimes BA)}\arrow{s,r}
{(id\otimes F) \oplus (F\otimes id)} \\
\node{B^{N}A}\arrow{e,b}{\Delta^{N}} \node{(BA\otimes
B^{N}A)\oplus (B^{N}A\otimes BA)}
\end{diagram}
\]
commute, and in addition $F\circ D^{M}=D^{N}\circ F$.
By Proposition \ref{4.1}, every A$_\infty$-bimodule-map induces a
map $F^{\sharp}:f\mapsto F\circ f$ between the Hochschild-complexes,
which preserves the differentials, since $(F^{\sharp}\circ
\delta^{M})(f)=F^{\sharp}(D^{M}\circ f+(-1)^{|f|} f\circ D)= F\circ
D^{M}\circ f+(-1)^{|f|}F\circ f\circ D= D^{N} \circ F\circ
f+(-1)^{|f|} F\circ f\circ D=\delta^{N}(F\circ f)=(\delta^{N}\circ
F^{\sharp})(f)$.
\end{defn}
\begin{prop}\label{4.4}
Let $(A,D)$ be an A$_\infty$-algebra with system of maps
$\{m_{i}:A^{\otimes i} \to A\}_{i\geq 1}$ from Proposition \ref{2.4}
associated to $D$, where $m_{0}=0$. Let $(M,D^{M})$ and $(N,D^{N})$
be A$_\infty$-bimodules over $A$ with systems of maps $\{b_{k,l}
:A^{\otimes k}\otimes M \otimes A^{\otimes l} \to M\}_{k\geq 0,
l\geq 0}$ and $\{c_{k,l} :A^{\otimes k}\otimes N \otimes A^{\otimes
l} \to N\}_{k\geq 0, l\geq 0}$ from Proposition \ref{3.7} associated
to $D^{M}$ and $D^{N}$ respectively. Let $F:T^{M}A \to T^{N}A$ be an
A$_\infty$-bimodule-map between $M$ and $N$, and let $\{F_{k,l}
:sA^{\otimes k}\otimes sM \otimes sA^{\otimes l} \to sN\}_{k\geq 0,
l\geq 0}$ be a system of maps associated to $F$ by Lemma \ref{4.2}
(b). Rewrite the maps $F_{k,l}$ by $f_{k,l} :A^{\otimes k}\otimes M
\otimes A^{\otimes l} \to N$ by using the suspension map: $F_{k,l}=
s\circ f_{k,l}\circ (s^{-1})^{\otimes k+l+1}$. Then the condition
$F\circ D^{M}=D^{N}\circ F$ is equivalent to the following system of
equations:
$$ f_{0,0}(b_{0,0}(m))=c_{0,0} (f_{0,0}(m)), $$
\begin{multline*}
f_{0,0}(b_{0,1}(m,a))- f_{0,1}(b_{0,0}(m),a)- (-1)^{|m|}
f_{0,1}(m,m_{1}(a))\\=c_{0,0}(f_{0,1}(m,a))+ c_{0,1}(f_{0,0}(m),a),
\end{multline*}
\begin{multline*}
f_{0,0}(b_{1,0}(a,m))- f_{1,0}(m_{1}(a),m)- (-1)^{|a|}
f_{1,0}(a,b_{0,0}(m))\\=c_{0,0}(f_{1,0}(a,m))+
c_{1,0}(a,f_{0,0}(m)),
\end{multline*}
$$...$$
\begin{multline*}
\sum_{i=1}^{k} \sum_{j=1}^{k-i+1} (-1)^{\varepsilon}
f_{k-i+1,l}(a_{1},...,m_{i}(a_{j},...,a_{i+j-1}),...,m,...,a_{k+l+1}) \\
+\sum_{j=1}^{k} \sum_{i=k-j+2}^{k+l-j+2} (-1)^{\varepsilon}
f_{j,k+l-i-j+3} (a_{1},...,b_{k-j+1,i+j-k-2}(a_{j},...,m,...,a_{i+j-1}),...,a_{k+l+1}) \\
+\sum_{i=1}^{l} \sum_{j=k+2}^{k+l-i+2} (-1)^{\varepsilon}
f_{k,l-i+1}(a_{1},...,m,...,m_{i}(a_{j},...,a_{i+j-1}),...,a_{k+l+1}) \\
=\sum_{j=1}^{k+1} \sum_{i=k-j+2}^{k+l-j+2} (-1)^{\varepsilon'}
c_{j,k+l-i-j+3} (a_{1},...,f_{k-j+1,i+j-k-2}(a_{j},...,m,...,a_{i+j-1}),...,a_{k+l+1})
\end{multline*}
In order to simplify notation, it is assume that in $(a_{1},...
,a_{k+l+1})$ above, only the first $k$ and the last $l$ elements are
elements of $A$ and $a_{k+1}=m\in M$. Then the signs are given by
\begin{eqnarray*}
\varepsilon &=& i\cdot \sum_{r=1}^{j-1}|a_{r}|+ (j-1) \cdot(i+1)+
(k+l+1)-i, \\ \text{and } \quad \varepsilon' &=&(i+1)\cdot
(j+1+\sum_{r=1}^{j-1}|a_{r}|).
\end{eqnarray*}
$$...$$
\end{prop}
\begin{proof}
The formula follows immediately from the explicit lifting properties
in Lemma \ref{3.5} (a) and Lemma \ref{4.2} (a).
\end{proof}
\begin{expl}\label{4.5}
Examples \ref{2.5} and \ref{3.8} can be extended in the following
way. Let $(A,\partial, \mu)$ be a differential graded algebra with
the A$_\infty$-algebra-structure $m_{1}:=\partial$, $m_{2}:=\mu$ and
$m_{k}:=0$ for $k\geq 3$. Let $(M,\partial^{M},\lambda^{M},
\rho^{M})$ and $(N,\partial^{N},\lambda^{N},\rho^{N})$ be
differential graded bimodules over $A$, with the A$_\infty
$-bialgebra-structures given by $b_{0,0}:=\partial^{M}$,
$b_{1,0}:=\lambda^{M}$, $b_{0,1}:=\rho^{M}$ and $b_{k,l}:=0$ for
$k+l>1$, and $c_{0,0}:=\partial^{N}$, $c_{1,0}:=\lambda^{N}$,
$c_{0,1}:=\rho^{N}$ and $c_{k,l}:=0$ for $k+l>1$. Finally, let
$f:M\to N$ be a bialgebra map of degree 0. Then $f$ becomes a map of
A$_\infty $-bialgebras by taking $f_{0,0}:=f$ and $ f_{k,l}:=0$ for
$k+l>0$. The equations from Proposition \ref{4.4} are the defining
conditions of a differential bialgebra map from $M$ to $N$:
\begin{eqnarray*}
f\circ\partial^{M} (m) &=& \partial^{N}\circ f(m)\\
f(m.a) &=& f(m).a\\
f(a.m) &=& a.f(m)
\end{eqnarray*}
There are no higher equations.
\end{expl}
\section{$\infty$-inner-products on A$_\infty$-algebras}\label{5}
There are canonical A$_\infty$-bialgebra-structures on a given
A$_\infty$-algebra $A$ and on its dual space $A^*$. We will define
$\infty$-inner products as A$_\infty$-bialgebra-maps from $A$ to
$A^*$.
\begin{lem}\label{5.1}
Let $(A,D)$ be an A$_\infty$-algebra., and let $D$ be given by the
system of maps $\{m_{i}: A^{\otimes i} \to A\}_{i\geq 1}$ from
Proposition \ref{2.4}.
\begin{itemize}
\item [(a)]
There is a canonical A$_\infty$-bimodule-structure on $A$ given by
$b_{k,l}:A^{\otimes k}\otimes A \otimes A^{\otimes l}\to A,
b_{k,l}:=m_{k+l+1}$.
\item [(b)]
There is a canonical A$_\infty$-bimodule-structure on $A^{*}$ given
by $b_{k,l}:A^{\otimes k}\otimes A^{*} \otimes A^{\otimes l}\to
A^{*}$,
\begin{multline*}
\quad\quad\quad (b_{k,l}(a_{1},...,a_{k},a^{*},a_{k+1},...,a_{k+l}))(a) \\
:=\pm a^{*}(m_{k+l+1}(a_{k+1},...,a_{k+l},a,a_{1},...,a_{k})),
\end{multline*}
with the signs from Lemma \ref{3.9}.
\end{itemize}
\end{lem}
\begin{proof}
\begin{itemize}
\item [(a)]
The A$_\infty$-bialgebra extension described in Lemma \ref{3.5} (a)
becomes the extension by coderivation described in Lemma \ref{2.3}
(a). Equations of Proposition \ref{3.7} become the equations of
Proposition \ref{2.4} and the diagram \eqref{eqn-3.1} from
Proposition \ref{3.4} becomes the usual coderivation diagram for
$D$.
\item [(b)] This follows from (a) and Lemma \ref{3.9}.
\end{itemize}
\end{proof}
\begin{expl}\label{5.2}
For a differential algebra $(A,\partial, \mu)$, the above
A$_\infty$-bialgebra structure on $A$ is exactly the bialgebra
structure given by left- and right-multiplication, since
$b_{1,0}(a\otimes b)=m_{2}(a\otimes b)=a\cdot b$ and
$b_{0,1}(a\otimes b)=m_{2}(a\otimes b)=a\cdot b$, for $a,b\in A$.
Similarly the A$_\infty$-bialgebra structure on $A^{*}$ is given by
right- and left-multiplication in the arguments: $b_{1,0}(a\otimes
b^{*})(c)=b^{*}(m_{2}(c\otimes a))=b^{*}(c\cdot a)$ and
$b_{0,1}(a^{*}\otimes b)(c)=a^{*}(m_{2}(b\otimes c))=a^{*}(b\cdot
c)$, for $a,b,c\in A$, and $a^{*},b^{*}\in A^{*}$.
\end{expl}
\begin{defn}\label{5.3}
Let $(A,D)$ be an A$_\infty$-algebra. Then, we call any
A$_\infty$-bimodule-map $F$ from $A$ to $A^{*}$ an
\textbf{$\infty$-inner-product} on $A$.
\end{defn}
\begin{prop}\label{5.4}
Let $(A,D)$ be an A$_\infty$-algebra. Then, specifying an
$\infty$-inner product on $A$ is equivalent to specifying a system
of inner-products on $A$, $\{<.,.,...>_{k,l}:A^{\otimes k+l+2} \to
R\}_{k\geq 0, l\geq 0}$ which satisfy the following relations:
$$ \sum_{i=1}^{k+l+2}(-1)^{\sum_{j=1}^{i-1}|a_{j}|}
<a_{1},...,\partial(a_{i}),...,a_{k+l+2}>_{k,l}=\sum_{i,j,n}
\pm <a_{i},...,m_{j}(a_{n},...),...>_{r,s}, $$
where in the sum on the right side, there is exactly one
multiplication $m_{j}$ ($j\geq 2$) inside the inner-product
$<...>_{r,s}$ and this sum is taken over all i, j, n subject to the
following conditions:
\begin{itemize}
\item [(i)]
The cyclic order of the $(a_{1},...,a_{k+l+2})$ is preserved.
\item [(ii)]
$a_{k+l+2}$ is always in the last slot of $<...>_{r,s}$.
\item [(iii)]
$a_{k+l+2}$ could be inside some $m_{j}$. By (ii), this is only the
case, when the first argument in the inner product is $a_{i}\neq
a_{1}$, as for example in the expression $<a_{i},...,m_{j}
(a_{n},..., a_{k+l+2},a_{1},...,a_{i-1})>_{r,s}$ for $i> 1$.
\item [(iv)]
The special arguments $a_{k+1}$ and $a_{k+l+2}$ are never multiplied
by $m_{j}$ at the same time.
\item [(v)]
The numbers $r$ and $s$ are uniquely determined by the position of
the element $a_{k+1}$ in the inner-product $<...>_{r,s}$. More
precisely, $a_{k+1}$ is in $(r+1)$-th spot of $<...>_{r,s}$, and $s$
is determined by the inner product $<...>_{r,s}$ having $r+s+2$
arguments.
\end{itemize}
\end{prop}
A graphical representation of the above conditions is given in
Definition \ref{5.5} and Example \ref{exp-4.1} below.
\begin{proof}
We use the description from Proposition \ref{4.4} for
A$_\infty$-bimodule-maps. An A$_\infty$-bimodule-map from $A$ to
$A^{*}$ is given by maps $f_{k,l}:A^{\otimes k}\otimes A \otimes
A^{\otimes l}\to A^{*}$, for $k,\,\,\, l\geq 0$. These are
interpreted as maps $A^{\otimes k}\otimes A \otimes A^{\otimes
l}\otimes A\to R$, which we denoted by the inner-product-symbol
$<...>_{k,l}$:
$$ <a_{1},...,a_{k+l+1},a'>_{k,l}:=(-1)^{|a'|}
(f_{k,l}(a_{1},...,a_{k+l+1}))(a') $$
The A$_\infty$-bimodule-map condition from Proposition \ref{4.4}
becomes
\begin{multline}\label{eqn-cond}
\sum \pm f_{k,l}(...,m_{i}(...),...,a,...)+
\sum \pm f_{k,l}(...,b_{i,j}(...,a,...),...)\\
+ \sum \pm f_{k,l}(...,a,...,m_{i}(...),...)=
\sum \pm c_{i,j}(...,f_{k,l}(...,a,...),...).
\end{multline}
Here $a\in A$ is the $(k+1)$-th entry of an element in $A^{\otimes
k}\otimes A \otimes A^{\otimes l}$, so that it comes from the
\textit{A$_\infty$-bimodule} $A$, instead of the
\textit{A$_\infty$-algebra} $A$.
Now, by Lemma \ref{5.1} (a), $b_{i,j}=m_{i+j+1}$ is one of the
multiplications from the A$_\infty$-structure, and therefore the
left side of the equation is $f_{k,l}$ applied to all possible
multiplications $m_{i}$. As $f_{k,l}$ maps into $A^{*}$, we may
apply the left hand side of \eqref{eqn-cond} to an element $a'\in A$
and use the notation $<...>_{k,l}$:
\begin{equation}\label{eqn-5.1}
\sum \pm (f_{k,l}(...,m_{i}(...),...))(a') =
\sum \pm <...,m_{i}(...),...,a'>_{k,l}.
\end{equation}
Next, use Lemma \ref{5.1} (b) to rewrite the right hand side of
\eqref{eqn-cond}:
\begin{equation}\label{eqn-5.2}
\sum \pm (c_{i,j}(a_{1},...,f_{k,l}(...,a,...),...,a_{k+l+1}))(a')
\quad\quad\quad\quad\quad
\end{equation}
\begin{eqnarray*}
&=& \sum \pm (f_{k,l}(...,a,...))(m_{r}(...,a_{k+l+1},a',a_{1},...)) \\
&=& \sum \pm <...,a,...,m_{r}(...,a_{k+l+1},a',a_{1},...)>_{k,l}
\end{eqnarray*}
Equations \eqref{eqn-5.1} and \eqref{eqn-5.2} show that we take a
sum over all possibilities of applying one multiplication to the
arguments of the inner-product subject to the conditions (i)-(iv).
This is the statement of the Proposition, after isolating the
$\partial$-terms on the left. For condition (v), notice that the
extensions of $D$ and $D^{A}$ from Lemma \ref{2.3} (a) and Lemma
\ref{3.5} (a) record the special entry $a$ in the
A$_\infty$-bimodule $A$. Thus, the A$_\infty$-bimodule element $a$
determines the index $k$, and $l$ is determined by the number of
arguments of $<...>_{k,l}$.
An explicit check shows the correctness of the signs.
\end{proof}
There is a diagrammatic way of picturing Proposition \ref{5.4}.
\begin{defn}\label{5.5}
Let $(A,D)$ be an A$_\infty$-algebra with $\infty$-inner-product
$\{<.,.,...>_{k,l}:A^{\otimes k+l+2} \to R\}_{k\geq 0, l\geq 0}$. To
the inner-product $<...>_{k,l}$, we associate the symbol
\[
\begin{pspicture}(0,0)(4,4)
\psline(.5,2)(3.5,2)
\psline(2,2)(1.2,3)
\psline(2,2)(1.6,3)
\psline(2,2)(2,3)
\psline(2,2)(2.4,3)
\psline(2,2)(2.8,3)
\psline(2,2)(1.4,1)
\psline(2,2)(1.8,1)
\psline(2,2)(2.2,1)
\psline(2,2)(2.6,1)
\psdots[dotstyle=o,dotscale=2](2,2)
\rput[b](2.8,3.2){$1$} \rput[b](2.4,3.2){$2$}
\rput[b](1.8,3.2){$...$} \rput[b](1.2,3.2){$k$}
\rput[b](.5,2.2){$k+1$}
\rput[b](3.5,2.2){$k+l+2$}
\rput[tr](1.4,.8){$k+2$} \rput[t](2,.8){$...$} \rput[tl](2.6,.8){$k+l+1$}
\end{pspicture}
\]
More generally, to any inner-product which has (possibly iterated)
multiplications $m_{2}, m_{3}, m_{4}, ...$ (but without differential
$\partial=m_{1}$), such as
$$<a_{i},...,m_{j}(...),...,m_{p}(..., m_{q}(...) ,...),...>_{k,l},$$
we associate a diagram like above, by the following rules:
\begin{itemize}
\item[(i)]
To every multiplication $m_{j}$, associate a tree with $j$ inputs
and one output.
\[
\begin{pspicture}(0,0.5)(4,3.5)
\psline(2,2)(1.2,3)
\psline(2,2)(1.6,3)
\psline(2,1)(2,3)
\psline(2,2)(2.4,3)
\psline(2,2)(2.8,3)
\psdots[dotstyle=*,dotscale=2](2,2)
\rput[l](2.4,2){$m_{j}$}
\end{pspicture}
\]
The symbol for the multiplication may also occur in a rotated way.
\item[(ii)]
To the inner product $<...>_{r,s}$, associate the open circle:
\[
\begin{pspicture}(0,0)(4,4)
\psline(.5,2)(3.5,2)
\psline(2,2)(1.2,3)
\psline(2,2)(1.6,3)
\psline(2,2)(2,3)
\psline(2,2)(2.4,3)
\psline(2,2)(2.8,3)
\psline(2,2)(1.4,1)
\psline(2,2)(1.8,1)
\psline(2,2)(2.2,1)
\psline(2,2)(2.6,1)
\psdots[dotstyle=o,dotscale=2](2,2)
\rput[b](2.8,3.2){$1$} \rput[b](2.4,3.2){$2$}
\rput[b](1.8,3.2){$...$} \rput[b](1.2,3.2){$r$}
\rput[b](.5,2.2){$r+1$}
\rput[b](3.5,2.2){$r+s+2$}
\rput[tr](1.4,.8){$r+2$} \rput[t](2,.8){$...$} \rput[tl](2.6,.8){$r+s+1$}
\end{pspicture}
\]
There are $r$ elements attached at the top of the circle, and $s$
elements at the bottom of the circle, and the two (special) inputs
$(r+1)$ and $(r+s+2)$ are attached on the left and right. This gives
a total of $r+s+2$ inputs.
\item[(iii)]
The inputs $a_i$, for $i=1,\dots, r+s+2$, will be attached
counterclockwise, where the last element $a_{r+s+2}$ is in the far
right slot. For the multiplications $m_{j}$ of the graph, we use the
counterclockwise orientation of the plane to find the correct order
of the arguments $a_{i}$ in $m_{j}$.
\end{itemize}
We call these diagrams \textbf{inner-product-diagrams}.
\end{defn}
\begin{expl}\label{exp-4.1}
Let $a, b, c, d, e, f, g, h, i, j, k \in A$.
\begin{itemize}
\item
$<a,b,c,d>_{2,0}$, ($deg=2$):
\[
\begin{pspicture}(0,0)(4,3)
\psline(.5,1)(3.5,1)
\psline(2,1)(1.4,2)
\psline(2,1)(2.6,2)
\psdots[dotstyle=o,dotscale=2](2,1)
\rput[b](2.6,2.2){$a$} \rput[b](1.4,2.2){$b$}
\rput[b](0.5,1.2){$c$} \rput[b](3.5,1.2){$d$}
\end{pspicture}
\]
\item
$<a,b,c,d,e,f,g,h,i>_{3,4}$, ($deg=7$):
\[
\begin{pspicture}(0,0)(4,4)
\psline(.5,2)(3.5,2)
\psline(2,2)(1.6,3)
\psline(2,2)(2,3)
\psline(2,2)(2.4,3)
\psline(2,2)(1.4,1)
\psline(2,2)(1.8,1)
\psline(2,2)(2.2,1)
\psline(2,2)(2.6,1)
\psdots[dotstyle=o,dotscale=2](2,2)
\rput[b](2,3.2){$b$} \rput[b](2.4,3.2){$a$}
\rput[b](1.6,3.2){$c$}
\rput[b](.5,2.2){$d$} \rput[b](3.5,2.2){$i$}
\rput(1.4,.7){$e$} \rput(1.8,.7){$f$}
\rput(2.2,.7){$g$} \rput(2.6,.7){$h$}
\end{pspicture}
\]
\item
$<m_{2}(m_{2}(b,c),m_{2}(d,e)),m_{2}(f,a)>_{0,0}$, ($deg=0$):
\[
\begin{pspicture}(0,0)(4,4)
\psline(.5,2)(3.5,2)
\psline(3,2)(3.5,3)
\psline(1,2)(.5,3)
\psline(1.66,2)(.74,1)
\psline(1.2,1.5)(1.2,.8)
\psdots[dotstyle=*,dotscale=2](1.66,2)
\psdots[dotstyle=*,dotscale=2](1,2)
\psdots[dotstyle=*,dotscale=2](1.2,1.5)
\psdots[dotstyle=*,dotscale=2](3,2)
\psdots[dotstyle=o,dotscale=2](2.33,2)
\rput[b](.2,2){$c$} \rput[b](3.8,2){$f$}
\rput[b](3.5,3.2){$a$} \rput[b](0.5,3.2){$b$}
\rput[b](.5,.8){$d$} \rput[b](1.2,.5){$e$}
\end{pspicture}
\]
\item
$<a,b,m_{3}(c,d,m_{2}(e,f)),g,m_{2}(h,i))>_{1,2}$, ($deg=4$):
\[
\begin{pspicture}(0,-.8)(4,4)
\psline(.5,2)(3.5,2)
\psline(2,2)(2,3)
\psline(2,2)(.8,.8)
\psline(2,2)(2.6,1)
\psline(2.8,2)(3.4,1.4)
\psline(1.3,1.3)(.6,1.2)
\psline(1.3,1.3)(1.4,.6)
\psline(1.4,.6)(1.2,.2)
\psline(1.4,.6)(1.7,.2)
\psdots[dotstyle=o,dotscale=2](2,2)
\psdots[dotstyle=*,dotscale=2](2.8,2)
\psdots[dotstyle=*,dotscale=2](1.3,1.3)
\psdots[dotstyle=*,dotscale=2](1.4,.6)
\rput[b](2,3.2){$a$} \rput[b](.5,2.2){$b$}
\rput(2.6,.7){$g$} \rput[b](3.5,2.2){$i$}
\rput[b](3.6,1.2){$h$}
\rput[b](.4,1.1){$c$} \rput[b](.6,.4){$d$}
\rput[b](1.1,-.2){$e$} \rput[b](1.7,-.2){$f$}
\end{pspicture}
\]
\item
$<c,m_{2}(d,e),m_{2}(m_{2}(f,g),h),i,m_{4}(j,k,a,b)>_{2,1}$,
($deg=5$):
\[
\begin{pspicture}(0,0)(4,4)
\psline(.5,2)(3.5,2)
\psline(3,2)(3.5,3)
\psline(1,2)(.5,3)
\psline(1.66,2)(.74,1)
\psline(3,2)(3.6,2.5)
\psline(2.33,2)(2,1)
\psline(3,2)(3.6,1.2)
\psline(2.33,2)(2.6,3)
\psline(2.33,2)(2,2.5)
\psline(2,2.5)(2,3)
\psline(2,2.5)(1.5,3)
\psdots[dotstyle=*,dotscale=2](1.66,2)
\psdots[dotstyle=*,dotscale=2](1,2)
\psdots[dotstyle=*,dotscale=2](3,2)
\psdots[dotstyle=*,dotscale=2](2,2.5)
\psdots[dotstyle=o,dotscale=2](2.33,2)
\rput[b](.2,1.9){$g$} \rput[b](3.8,1.9){$k$}
\rput[b](3.5,3.2){$b$} \rput[b](0.5,3.2){$f$}
\rput[b](.5,.8){$h$} \rput[b](3.8,2.5){$a$}
\rput[b](2,.6){$i$} \rput[b](3.8,1){$j$}
\rput[b](2.6,3.2){$c$} \rput[b](2,3.2){$d$}
\rput[b](1.5,3.2){$e$}
\end{pspicture}
\]
\end{itemize}
\end{expl}
\begin{defn}\label{5.6}
We define a chain-complex associated to inner-product-diagrams.
We define the degree of the inner-product-diagram associated to
$<...>_{k,l}$ with multiplications $m_{i_{1}},...,m_{i_{n}}$ to be
$k+l+\sum_{j=1}^{n} (i_{j}-2)$. Examples are given in \ref{exp-4.1}.
For $n\geq 0$, let $C_{n}$ be the space generated by
inner-product-diagrams of degree $n$. Then let $C:=\bigoplus_{n\geq
0} C_{n}$.
As for the differential $d$ on $C$, we use the composition with the
operator $\tilde{\partial}:=\sum_{i} id\otimes ...\otimes id\otimes
\partial \otimes id\otimes ... \otimes id$, where $\partial=m_{1}$
is at the $i$-th spot:
\begin{multline*}
(d(<...,m(...,m(...),...),...>))(a_{1},...,a_{s})\\
:=(<...,m(...,m(...),...),...> ) (\sum_{i=1}^{s} (-1)^{\sum
_{j=1}^{i-1}|a_{j}|} (a_{1},...,\partial (a_{i}),...,a_{s}))
\end{multline*}
Some remarks and interpretations of this expression are in order.
First, consider the inner-product $<...>_{k,l}$ without any
multiplications. By Proposition \ref{5.4}, the differential applies
one multiplication into the inner-product-diagram in all possible
spots, such that the two lines on the far left and on the far right
are not being multiplied; compare Proposition \ref{5.4} (iv).
In the case, that multiplications are applied to the inner-product,
one can observe from Proposition \ref{2.3}, that $ \sum_{i}
m_{n}\circ(id\otimes ... \otimes \partial\otimes ... \otimes id) $
is given by the two terms
\begin{equation}\label{eqn-5.3}
\sum_{i} m_{n}\circ(id\otimes ... \otimes \partial\otimes ...
\otimes id)=\sum_{k=2}^{n-1} \sum_{i} m_{n+1-k} \circ(id\otimes ...
\otimes m_{k} \otimes ... \otimes id) + \partial \circ m_{n}.
\end{equation}
The sum over $i$ on both sides of the above equation applies $m_k$
to the $i$-th spot. The first term on the right hand side of
\eqref{eqn-5.3} transforms the multiplication $m_{n}$ into a sum of
all possible sompositions of $m_{n+1-k}$ and $m_{k}$:
\[
\begin{pspicture}(0,0.5)(4.5,3.5)
\psline(2,2)(1.2,3)
\psline(2,2)(1.6,3)
\psline(2,1)(2,3)
\psline(2,2)(2.4,3)
\psline(2,2)(2.8,3)
\psdots[dotstyle=*,dotscale=2](2,2)
\rput[l](2.4,2){$m_{n}$}
\rput(4,2){$\Longrightarrow$}
\end{pspicture}
\begin{pspicture}(0,0.5)(4,3.5)
\psline(2,1)(2,1.5)
\psline(2,1.5)(.5,3)
\psline(2,1.5)(3.2,3)
\psline(2,1.5)(3.6,3)
\psline(2,1.5)(1.6,2.3)
\psline(1.6,2.3)(1.3,3)
\psline(1.6,2.3)(1.5,3)
\psline(1.6,2.3)(1.7,3)
\psdots[dotstyle=*,dotscale=2](1.6,2.3)
\rput[l](1.9,2.3){$m_{k}$}
\psdots[dotstyle=*,dotscale=2](2,1.5)
\rput[l](2.4,1.5){$m_{n-k+1}$}
\end{pspicture}
\]
The last term \eqref{eqn-5.3} is used for an inductive argument of
the above. One gets a term $\partial (m_{n}(...))$ attached to the
inner-product or possibly another multiplication, that has arguments
with $\tilde{\partial}$ applied, so that the above discussion can be
continued inductively.
We conclude, that the differential applies exactly one
multiplication in all possible spots, without multiplying the given
far left and far right inputs. Examples are given in Example
\ref{5.7} below.
It is $d:C_{n}\to C_{n-1}$, and $d^{2}=0$.
\begin{proof}
According to the definition of the degrees above, a multiplication
$m_{n}$ with $n$ inputs contributes by $n-2$. Taking the
differential applies one more multiplication in all possible ways.
If we attach $m_{n}$ to the diagram, then it replaces $n$ arguments
with one argument in the higher level. Therefore,
\begin{eqnarray*}
\text{new degree} & = & (\text{old degree})-n+1+(n-2) \\
& = & (\text{old degree})-1.
\end{eqnarray*}
We can prove $d^{2}=0$ in two ways:
\begin{itemize}
\item
Algebraically:
The definition of $d$ on the inner-products is given by composition
with the operator $\tilde{\partial}=\sum_{i} id\otimes ...\otimes
id\otimes \partial \otimes id\otimes ... \otimes id$, where
$\partial$ is in the $i$-th spot. Thus $d^{2}$ is composition with
$$ \tilde{\partial}^{2}=\sum_{i,j} \pm id\otimes ... \otimes \partial
\otimes ...\otimes \partial \otimes ... \otimes id=0. $$ This
vanishes, since the sum has two terms, where $\partial$ occurs at
the $i$-th and the $j$-th spot. This is obtained, by either first
applying $\partial$ to the $i$-th and then to the $j$-th spot, or
vice versa. These two possibilities cancel as $\partial$ is of
degree $-1$ and the first $\partial$ either has to move over the
second $\partial$, by which an additional minus sign is introduced,
or not.
\item
Diagrammatically (without signs):
$d$ applies one new multiplication to the inner-product-diagram, so
that $d^{2}$ applies two new multiplications. For two
multiplications, we have the following two possibilities.
\begin{itemize}
\item[(i)]
In the first case, the multiplications are on different outputs.
\[
\begin{pspicture}(0,0)(8.5,3)
\psline(.5,.5)(.5,2.5)
\psline(.7,.5)(.7,2.5)
\psline(.9,.5)(.9,2.5)
\psline(1.1,.5)(1.1,2.5)
\psline(1.3,.5)(1.3,2.5)
\psline{->}(2,1.5)(3,1.5)
\psline(3.7,.5)(3.7,2.5)
\psline(3.9,.5)(3.9,2.5)
\psline(4.5,1.5)(4.3,2.5)
\psline(4.5,.5)(4.5,2.5)
\psline(4.5,1.5)(4.7,2.5)
\psdots[dotstyle=*,dotscale=2](4.5,1.5)
\psline{->}(5.4,1.5)(6.4,1.5)
\psline(7,2.5)(7.1,1.5)
\psline(7.2,2.5)(7.1,1.5)
\psline(7.1,1.5)(7.1,.5)
\psline(7.8,1.5)(7.6,2.5)
\psline(7.8,.5)(7.8,2.5)
\psline(7.8,1.5)(8,2.5)
\psdots[dotstyle=*,dotscale=2](7.8,1.5)
\psdots[dotstyle=*,dotscale=2](7.1,1.5)
\end{pspicture}
\]
\[
\begin{pspicture}(0,0)(8.5,3)
\psline(.5,.5)(.5,2.5)
\psline(.7,.5)(.7,2.5)
\psline(.9,.5)(.9,2.5)
\psline(1.1,.5)(1.1,2.5)
\psline(1.3,.5)(1.3,2.5)
\psline{->}(2,1.5)(3,1.5)
\psline(4.7,.5)(4.7,2.5)
\psline(4.5,.5)(4.5,2.5)
\psline(4.3,.5)(4.3,2.5)
\psline(3.7,2.5)(3.8,1.5)
\psline(3.9,2.5)(3.8,1.5)
\psline(3.8,1.5)(3.8,.5)
\psdots[dotstyle=*,dotscale=2](3.8,1.5)
\psline{->}(5.4,1.5)(6.4,1.5)
\psline(7,2.5)(7.1,1.5)
\psline(7.2,2.5)(7.1,1.5)
\psline(7.1,1.5)(7.1,.5)
\psline(7.8,1.5)(7.6,2.5)
\psline(7.8,.5)(7.8,2.5)
\psline(7.8,1.5)(8,2.5)
\psdots[dotstyle=*,dotscale=2](7.8,1.5)
\psdots[dotstyle=*,dotscale=2](7.1,1.5)
\end{pspicture}
\]
The above figure shows that the final terms are obtained in two
different ways, which in fact cancel each other.
\item[(ii)]
The other possibility is to have multiplications on the same output.
Again, these terms may be obtained in two ways that cancel each
other:
\[
\begin{pspicture}(0,0)(8.5,3)
\psline(.5,.5)(.5,2.5)
\psline(.7,.5)(.7,2.5)
\psline(.9,.5)(.9,2.5)
\psline(1.1,.5)(1.1,2.5)
\psline(1.3,.5)(1.3,2.5)
\psline{->}(2,1.5)(3,1.5)
\psline(3.7,.5)(3.7,2.5)
\psline(3.9,.5)(3.9,2.5)
\psline(4.5,1.5)(4.3,2.5)
\psline(4.5,.5)(4.5,2.5)
\psline(4.5,1.5)(4.7,2.5)
\psdots[dotstyle=*,dotscale=2](4.5,1.5)
\psline{->}(5.4,1.5)(6.4,1.5)
\psline(7,2.5)(7.3,1.2)
\psline(7.3,2.5)(7.3,.5)
\psline(7.3,1.2)(7.8,1.8)
\psline(7.8,1.8)(7.6,2.5)
\psline(7.8,1.8)(7.8,2.5)
\psline(7.8,1.8)(8,2.5)
\psdots[dotstyle=*,dotscale=2](7.8,1.8)
\psdots[dotstyle=*,dotscale=2](7.3,1.2)
\end{pspicture}
\]
\[
\begin{pspicture}(0,0)(8.5,3)
\psline(.5,.5)(.5,2.5)
\psline(.7,.5)(.7,2.5)
\psline(.9,.5)(.9,2.5)
\psline(1.1,.5)(1.1,2.5)
\psline(1.3,.5)(1.3,2.5)
\psline{->}(2,1.5)(3,1.5)
\psline(4.2,1.5)(3.8,2.5)
\psline(4.2,1.5)(4.0,2.5)
\psline(4.2,0.5)(4.2,2.5)
\psline(4.2,1.5)(4.4,2.5)
\psline(4.2,1.5)(4.6,2.5)
\psdots[dotstyle=*,dotscale=2](4.2,1.5)
\psline{->}(5.4,1.5)(6.4,1.5)
\psline(7,2.5)(7.3,1.2)
\psline(7.3,2.5)(7.3,.5)
\psline(7.3,1.2)(7.8,1.8)
\psline(7.8,1.8)(7.6,2.5)
\psline(7.8,1.8)(7.8,2.5)
\psline(7.8,1.8)(8,2.5)
\psdots[dotstyle=*,dotscale=2](7.8,1.8)
\psdots[dotstyle=*,dotscale=2](7.3,1.2)
\end{pspicture}
\]
\end{itemize}
\end{itemize}
\end{proof}
\end{defn}
\begin{expl}\label{5.7}
Let $a, b, c \in A$.
\begin{itemize}
\item
$k=0$, $l=0$: $d(<a,b>_{0,0})=0$
\[
\begin{pspicture}(0.6,0.4)(3.6,1.8)
\psline(1.3,1)(2.7,1)
\psdots[dotstyle=o,dotscale=2](2,1)
\rput[b](1.3,1.1){$a$} \rput[b](2.8,1.1){$b$}
\rput(.7,1.1){$\textrm{d}($} \rput(3.55,1.1){$)=0$}
\end{pspicture}
\]
\item
$k=1$, $l=0$: $d(<a,b,c>_{1,0})=<a\cdot b,c>_{0,0}\pm <b,c\cdot
a>_{0,0}$
\[
\begin{pspicture}(0,0)(10,2.5)
\psline(1.3,1)(2.7,1)
\psline(2,1)(2,1.8)
\psdots[dotstyle=o,dotscale=2](2,1)
\rput[b](1.3,1.1){$b$} \rput[b](2.7,1.1){$c$} \rput[b](2.2,1.6){$a$}
\rput(.7,1.1){$\textrm{d}($} \rput(3.4,1.1){$)=$}
\psline(4.3,1)(5.7,1)
\psline(4.8,1)(4.6,1.8)
\psdots[dotstyle=*,dotscale=2](4.8,1)
\psdots[dotstyle=o,dotscale=2](5.3,1)
\rput[b](4.3,1.1){$b$} \rput[b](5.7,1.1){$c$} \rput[b](4.8,1.8){$a$}
\rput(6.5,1.1){$\pm$}
\psline(7.3,1)(8.8,1)
\psline(8.3,1)(8.5,1.8)
\psdots[dotstyle=*,dotscale=2](8.3,1)
\psdots[dotstyle=o,dotscale=2](7.8,1)
\rput[b](7.3,1.1){$b$} \rput[b](8.8,1.1){$c$} \rput[b](8.3,1.8){$a$}
\end{pspicture}
\]
\item
$k=0$, $l=1$: $d(<a,b,c>_{0,1})=<a\cdot b,c>_{0,0}\pm <a,b\cdot
c>_{0,0}$
\[
\begin{pspicture}(0,-.2)(10,1.8)
\psline(1.3,1)(2.7,1)
\psline(2,1)(2,0.2)
\psdots[dotstyle=o,dotscale=2](2,1)
\rput[b](1.3,1.1){$a$} \rput[b](2.7,1.1){$c$} \rput[b](2.2,0.3){$b$}
\rput(.7,1.1){$\textrm{d}($} \rput(3.4,1.1){$)=$}
\psline(4.3,1)(5.7,1)
\psline(4.8,1)(4.6,0.2)
\psdots[dotstyle=*,dotscale=2](4.8,1)
\psdots[dotstyle=o,dotscale=2](5.3,1)
\rput[b](4.3,1.1){$a$} \rput[b](5.7,1.1){$c$} \rput[b](4.8,0.2){$b$}
\rput(6.5,1.1){$\pm$}
\psline(7.3,1)(8.8,1)
\psline(8.3,1)(8.5,0.2)
\psdots[dotstyle=*,dotscale=2](8.3,1)
\psdots[dotstyle=o,dotscale=2](7.8,1)
\rput[b](7.3,1.1){$a$} \rput[b](8.8,1.1){$c$} \rput[b](8.3,0.2){$b$}
\end{pspicture}
\]
\end{itemize}
In the following three figures, where $k+l=2$, the righthand side is
understood to be a sum over the five, or respectively six,
inner-product-diagrams. Then, as $d^{2}=0$, the terms may be
arranged according to their boundaries. We obtain the polyhedra
associated to the inner-products $<...>_{k,l}$.
\begin{itemize}
\item
$k=2$, $l=0$:
\[
\begin{pspicture}(0,0)(10,6)
\psline(1.3,2.9)(2.7,2.9) \psline(2,2.9)(2.3,3.4) \psline(2,2.9)(1.7,3.4)
\psdots[dotstyle=o,dotscale=2](2,2.9)
\rput(.7,3){$\textrm{d}($} \rput(3.4,3){$)=$}
\psline(5.2,3.2)(6.55,4.2) \psline(6.55,4.2)(7.9,3.2)
\psline(7.9,3.2)(7.3,1.8) \psline(7.3,1.8)(5.8,1.8) \psline(5.8,1.8)(5.2,3.2)
\psline(6,1.2)(7.1,1.2)
\psline(6.55,1.4)(6.75,1.6) \psline(6.55,1.4)(6.35,1.6)
\psdots[dotstyle=*,dotscale=1](6.55,1.4)
\psline(6.55,1.2)(6.55,1.4) \psdots[dotstyle=o,dotscale=1](6.55,1.2)
\psline(4,2.2)(5.1,2.2)
\psline(4.3,2.2)(4.5,2.6) \psdots[dotstyle=*,dotscale=1](4.3,2.2)
\psline(4.3,2.2)(4.1,2.6) \psdots[dotstyle=o,dotscale=1](4.8,2.2)
\psline(8,2.2)(9.1,2.2)
\psline(8.8,2.2)(8.6,2.6) \psdots[dotstyle=o,dotscale=1](8.3,2.2)
\psline(8.8,2.2)(9,2.6) \psdots[dotstyle=*,dotscale=1](8.8,2.2)
\psline(4.3,4)(5.4,4)
\psline(4.6,4)(4.4,4.4) \psdots[dotstyle=*,dotscale=1](4.6,4)
\psline(5.1,4)(5.1,4.4) \psdots[dotstyle=o,dotscale=1](5.1,4)
\psline(7.7,4)(8.8,4)
\psline(8,4)(8,4.4) \psdots[dotstyle=o,dotscale=1](8,4)
\psline(8.5,4)(8.7,4.4) \psdots[dotstyle=*,dotscale=1](8.5,4)
\end{pspicture}
\]
\item
$k=1$, $l=1$:
\[
\begin{pspicture}(0,0)(10,6)
\psline(1.3,2.9)(2.7,2.9) \psline(2,2.4)(2,3.4)
\psdots[dotstyle=o,dotscale=2](2,2.9)
\rput(.7,3){$\textrm{d}($} \rput(3.4,3){$)=$}
\psline(5,3)(5.8,4.2) \psline(5.8,4.2)(7.3,4.2) \psline(7.3,4.2)(8.1,3)
\psline(8.1,3)(7.3,1.8) \psline(7.3,1.8)(5.8,1.8) \psline(5.8,1.8)(5,3)
\psline(6,1.2)(7.1,1.2)
\psline(6.3,1.2)(6.1,.8) \psdots[dotstyle=*,dotscale=1](6.3,1.2)
\psline(6.8,1.2)(6.8,1.6) \psdots[dotstyle=o,dotscale=1](6.8,1.2)
\psline(6,4.8)(7.1,4.8)
\psline(6.3,4.8)(6.3,5.2) \psdots[dotstyle=o,dotscale=1](6.3,4.8)
\psline(6.8,4.8)(7,4.4) \psdots[dotstyle=*,dotscale=1](6.8,4.8)
\psline(4,2.2)(5.1,2.2)
\psline(4.3,2.2)(4.1,2.6) \psdots[dotstyle=*,dotscale=1](4.3,2.2)
\psline(4.3,2.2)(4.1,1.8) \psdots[dotstyle=o,dotscale=1](4.8,2.2)
\psline(8,2.2)(9.1,2.2)
\psline(8.3,2.2)(8.3,1.8) \psdots[dotstyle=o,dotscale=1](8.3,2.2)
\psline(8.8,2.2)(9,2.6) \psdots[dotstyle=*,dotscale=1](8.8,2.2)
\psline(4,3.8)(5.1,3.8)
\psline(4.3,3.8)(4.1,4.2) \psdots[dotstyle=*,dotscale=1](4.3,3.8)
\psline(4.8,3.8)(4.8,3.4) \psdots[dotstyle=o,dotscale=1](4.8,3.8)
\psline(8,3.8)(9.1,3.8)
\psline(8.8,3.8)(9,4.2) \psdots[dotstyle=o,dotscale=1](8.3,3.8)
\psline(8.8,3.8)(9,3.4) \psdots[dotstyle=*,dotscale=1](8.8,3.8)
\end{pspicture}
\]
\item
$k=0$, $l=2$:
\[
\begin{pspicture}(0,0)(10,6)
\psline(1.3,2.9)(2.7,2.9) \psline(2,2.9)(2.3,2.4) \psline(2,2.9)(1.7,2.4)
\psdots[dotstyle=o,dotscale=2](2,2.9)
\rput(.7,3){$\textrm{d}($} \rput(3.4,3){$)=$}
\psline(5.2,3.2)(6.55,4.2) \psline(6.55,4.2)(7.9,3.2)
\psline(7.9,3.2)(7.3,1.8) \psline(7.3,1.8)(5.8,1.8) \psline(5.8,1.8)(5.2,3.2)
\psline(6,1.2)(7.1,1.2)
\psline(6.55,1)(6.75,.8) \psline(6.55,1)(6.35,.8)
\psdots[dotstyle=*,dotscale=1](6.55,1)
\psline(6.55,1.2)(6.55,1) \psdots[dotstyle=o,dotscale=1](6.55,1.2)
\psline(4,2.2)(5.1,2.2)
\psline(4.3,2.2)(4.5,1.8) \psdots[dotstyle=*,dotscale=1](4.3,2.2)
\psline(4.3,2.2)(4.1,1.8) \psdots[dotstyle=o,dotscale=1](4.8,2.2)
\psline(8,2.2)(9.1,2.2)
\psline(8.8,2.2)(8.6,1.8) \psdots[dotstyle=o,dotscale=1](8.3,2.2)
\psline(8.8,2.2)(9,1.8) \psdots[dotstyle=*,dotscale=1](8.8,2.2)
\psline(4.3,4)(5.4,4)
\psline(4.6,4)(4.4,3.6) \psdots[dotstyle=*,dotscale=1](4.6,4)
\psline(5.1,4)(5.1,3.6) \psdots[dotstyle=o,dotscale=1](5.1,4)
\psline(7.7,4)(8.8,4)
\psline(8,4)(8,3.6) \psdots[dotstyle=o,dotscale=1](8,4)
\psline(8.5,4)(8.7,3.6) \psdots[dotstyle=*,dotscale=1](8.5,4)
\end{pspicture}
\]
\end{itemize}
Finally, we graph the polyhedra in the case $k+l=3$. In general, the
polyhedron associated to $<...>_{k,l}$ is isomorphic to the one from
$<...>_{l,k}$. Furthermore, the polyhedra for $<...>_{n,0}$ and
$<...>_{0,n}$ are the ones known as Stasheff's associahedra.
\begin{itemize}
\item
The polyhedron for $k=3, l=0$ and for $k=0, l=3$:
\[\begin{pspicture}(0,0.2)(4,4)
\psline(.4,3)(0,2.4) \psline(.4,1.6)(0,2.4)
\psline(.8,2.2)(.4,1.6) \psline(.8,2.2)(.4,3)
\psline(3.6,3)(4,2.4) \psline(3.6,1.6)(4,2.4)
\psline(3.2,2.2)(3.6,1.6) \psline(3.2,2.2)(3.6,3)
\psline(2,2.8)(1.6,2) \psline(2,2.8)(2.4,2)
\psline(2,1.2)(1.6,2) \psline(2,1.2)(2.4,2)
\psline(.4,3)(1.8,4) \psline(2,1.2)(2,0.2)
\psline(3.6,3)(1.8,4) \psline(3.6,1.6)(2,0.2)
\psline(2,2.8)(1.8,4) \psline(.4,1.6)(2,0.2)
\psline(.8,2.2)(1.6,2) \psline(3.2,2.2)(2.4,2)
\psline[linestyle=dashed](0,2.4)(4,2.4)
\end{pspicture}\]
\item
The polyhedron for $k=2, l=1$ and for $k=1, l=2$:
\[\begin{pspicture}(0,0)(3.4,4)
\psline(2.2,4)(.4,3.6) \psline(.4,3.6)(0,2.8)
\psline(0,2.8)(1.2,2.6) \psline(1.2,2.6)(2.6,3)
\psline(2.6,3)(3,3.6) \psline(3,3.6)(2.2,4)
\psline(3,3.6)(3.4,2.8) \psline(3.4,2.8)(3.3,1.6)
\psline(3.3,1.6)(2.6,3) \psline(1.2,2.6)(1.2,1.4)
\psline(2.4,.8)(1.2,1.4) \psline(2.4,.8)(3.3,1.6)
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\end{itemize}
\end{expl}
|
3,212,635,537,424 | arxiv |
\section{Approach}
\vspace{-0.1in}
\subsection{Problem statement}
Assume that we have a distribution $p(x)$ of data (e.g., images), with $x \in \mathcal{X}$ as the input. We are interested in the problem of robust representation learning, where we want to learn a representation $z$ of $x$ with the mapping $p_\theta(z|x)$, parameterized by $\theta$; so that $z$ is meaningful for downstream tasks (to be defined below), and that any classifier (based on $z$) of the downstream tasks should be robust against adversarial attacks. For computational reasons, we do not want to re-do the adversarial training, and desire that the robustness transfer directly to the downstream tasks. The representation $z$ can be probabilistic (e.g., $p_\theta(z|x)=\mathcal{N}(z;\mu_\theta(x),\sigma_\theta(x))$) or deterministic (i.e., $p_\theta(z|x)=\delta_{g_\theta(x)}(z)$ with a deterministic function $g_\theta$). In this paper, we will especially consider a probabilistic representation mapping. In practice, the representation can be learn by unsupervised learning methods (e.g., VAE) or self-supervised learning methods (with a pretext task).
\begin{remark}
A note on the choice of the representation distribution $p_\theta(z|x)$.
\label{gaussian}
\end{remark}
\vspace{-0.1in}
\quad As mentioned earlier, in this paper, we especially consider a probabilistic representation mapping. Specifically, we use a Gaussian distribution in all of our experiments, e.g., $p_\theta(z|x)=\mathcal{N}(z;\mu_\theta(x),\sigma_\theta(x))$. This is just a design choice that is simple and works well in practice, and our work is not limited by this choice of the distribution. We can use almost any other distribution (with a known parameterization trick to allow for backpropagation). Also, note that the Gaussian representation network is a generalized version of a deterministic network (it becomes a deterministic network when $\sigma_\theta(x) \rightarrow 0 \;\forall x$); therefore, this network choice is not at all restricted when compared to a typical deterministic network.
\vspace{0.15in}
Any downstream task $T$ is defined by a conditional distribution $p_T(y|x)$ where $y \in \mathcal{Y}$ is the label. The joint data distribution of this task is $p_T(x,y)=p(x)p_T(y|x)$. With the representation mapping $p_\theta(z|x)$ learned in advance, we want to learn a classifier $\hat{p}_T(y|z)$ (parameterized by $\omega_T$, which we will omit for notation simplicity) for the task $T$. This is often called an output head that classifies $y$ given $z$.\\
The predictive distribution of $y$ given $x$ for this task $T$ is:
\begin{align}
\mathbb{E}_{p_\theta(z|x)}[\hat{p}_T(y|z)]
\label{predictive}
\end{align}
(for a deterministic representation mapping, Eq.~\ref{predictive} simplifies into $\hat{p}_T(y|z=g_\theta(x))$)
\begin{remark}
On the inference complexity of a probabilistic representation.
\end{remark}
\vspace{-0.08in}
\quad Using a probabilistic representation, we need to sample multiple $z$ from $p_\theta(z|x)$ to estimate Eq.~\ref{predictive} with Monte Carlo sampling during test time. However, this is not a big issue for the representation learning framework, since we only need to run the representation network $p_\theta(z|x)$ (which is usually deep) once to get a distribution of $z$. After sampling multiple $z$ from that distribution, we only need to rerun the classifier $\hat{p}_T(y|z)$, which is usually a small network (e.g., often contains one or a few fully-connected layers). Furthermore, we can also run $\hat{p}_T(y|z)$ (a small network) in parallel for multiple $z$ to reduce inference time if necessary.
During training of the downstream task $T$, a single $z$ is sampled per input $x$ from the learned representation mapping $p_\theta(z|x)$; and the output head $\hat{p}_T(y|z)$ is trained via minimizing the following training objective:
\begin{equation}
\mathbb{E}_{p_T(x,y)}\left[\mathbb{E}_{p_\theta(z|x)}[-\log \hat{p}_T(y|z)]\right]
\label{train_obj}
\end{equation}
With common choices of the predictive distribution $\hat{p}_T(y|z)$, the quantity $-\log$\;$\hat{p}_T(y|z)$ is often non-negative. For example, with a categorical predictive distribution in a classification problem, this term is the cross-entropy loss; whereas with a Gaussian predictive distribution (with a fixed variance) in a regression problem, the above term becomes the squared error loss (with an additive constant).
\makebox[0.96\linewidth][s]{Note also that the objective in Eq.~\ref{train_obj} is an upper bound of the true loss for task $T$}\\$\mathbb{E}_{p_T(x,y)}\left[-\log \mathbb{E}_{p_\theta(z|x)}[\hat{p}_T(y|z)]\right]$ (due to Jensen's inequality), where $-\log \mathbb{E}_{p_\theta(z|x)}[\hat{p}_T(y|z)]$ is the loss of a datapoint $(x,y)$.
Now we formally define the adversarial robustness of the network on the downstream task. Denote $A(x)$ to be the set of adversarial examples of $x$. This set is different for different kinds of adversarial attack; for example, with an $l_\infty$ attack, $A(x)$ is the $l_\infty$-ball around $x$ with a predefined radius of $\epsilon$. The adversarial loss of the task $T$ is:
\begin{align}
\vspace{-0.1in}
\mathbb{E}_{p_T(x,y)}\left[\max_{x^{adv}\in A(x)} -\log \mathbb{E}_{p_\theta(z|x^{adv})}[\hat{p}_T(y|z)]\right]
\vspace{-0.1in}
\end{align}
Intuitively, this means that an attacker seeks to find an adversarial example $x^{adv}$ of each input $x$ that maximizes the loss w.r.t. its label $y$; and we, as the defender, want to minimize that loss. In the next subsections, we will discuss how we can minimize this adversarial loss, even in a task-agnostic manner during the representation learning phase.
\vspace{-0.1in}
\subsection{A bound on the adversarial loss}
We first propose a bound on the adversarial loss based on the downstream training objective and a robustness regularizer:
\begin{proposition} Assuming that $\forall x, \; p_\theta(z|x)$ has the same support set $\mathcal{Z}$ (e.g., $p_\theta(z|x)$ is Gaussian); and that $-\log \hat{p}_T(y|z) \leq M \;\forall z\in\mathcal{Z}, y\in\mathcal{Y}$ \footnote{In the classification problem, we can enforce this quite easily by augmenting the output softmax of the classifier $\hat{p}_T(y|z)$ so that each class probability is always at least $\exp{(-M)}$. For example, if we choose $M=3 \Rightarrow \exp{(-M)}\approx 0.05$, and if the output softmax is $(p_1,p_2,...,p_C)$, we can augment it into $(p_1\cdot K+0.05,p_2\cdot K+0.05,...,p_C\cdot K+0.05)$, where $K=1-0.05\cdot C$ and $C$ is the number of classes. This ensures the bound for the loss of a datapoint, while retaining the output prediction class.}, we have:
\vspace{-0.1in}
\begin{align}
\mathbb{E}_{p_T(x,y)}\left[\max_{x^{adv}\in A(x)} -\log \mathbb{E}_{p_\theta(z|x^{adv})}[\hat{p}_T(y|z)]\right] \leq\; \mathbb{E}_{p_T(x,y)}\left[\mathbb{E}_{p_\theta(z|x)}[-\log \hat{p}_T(y|z)]\right] \nonumber \\
+ \frac{M}{\sqrt{2}}\sqrt{\mathbb{E}_{p(x)}\left[\max_{x^{u\_adv}\in A(x)} \textup{KL}[p_\theta(z|x)|p_\theta(z|x^{u\_adv})]\right]}
\label{bound}
\end{align}
\end{proposition}
\begin{proof}
\vspace{-0.1in}
provided in the supplementary file.
\end{proof}
The first term $\mathbb{E}_{p_T(x,y)}\left[\mathbb{E}_{p_\theta(z|x)}[-\log \hat{p}_T(y|z)]\right]$ is the downstream training loss in Eq.~\ref{train_obj}, and will be minimized during the training of the output head $\hat{p}_T(y|z)$ for task $T$.
We call the second term $\mathbb{E}_{p(x)}\left[\max_{x^{u\_adv}\in A(x)} \text{KL}[p_\theta(z|x)|p_\theta(z|x^{u\_adv})]\right]$ a robustness regularizer. Since we do not want to perform adversarial training for the downstream tasks, we want to minimize this term during the representation learning phase. Since this term is label-free and task-independent, if we minimize it during the representation learning phase, it will transfer directly to the downstream task and help minimize the bound in Eq.~\ref{bound}. This will be discussed further in Subsection~\ref{application}. Recall that we use the Gaussian distribution of the per-image representation network, i.e., $p_\theta(z|x)=\mathcal{N}(z;\mu_\theta(x),\sigma_\theta(x))$, so this KL term can be computed analytically (and exactly). Also note that, as discussed in Remark~\ref{gaussian}, the Gaussian representation is a generalized version of a typical deterministic representation, so it is sufficiently expressive for typical Deep Learning problems.
\paragraph{\textbf{Comparison between our bound and TRADES~\cite{zhang2019theoretically}:}}
\begin{itemize}
\item The bound in TRADES only works for the case of binary classification, while our bound works for the general case of supervised learning (including multi-class classification and regression).
\item Our robustness regularizer is label-free and task-independent. Therefore, we can minimize it in the representation learning phase (with unsupervised or self-supervised tasks), and it will transfer directly to the downstream tasks. On the other hand, the robustness regularizer in TRADES is task-dependent, thus minimizing the term for a pretext task does not necessarily transfer to the downstream task. Furthermore, TRADES's robustness regularizer requires a predictive distribution of a task to compute, and this might not be applicable to many self-supervised learning methods where there is no prediction task (e.g., contrastive learning). These arguments are also true for almost all existing robustness methods.
\end{itemize}
\paragraph{\textbf{Trade-off between Clean Accuracy and Adversarial Robustness}} The trade-off between a model's performance on clean input and adversarial input has been well observed in practice \cite{zhang2019theoretically}; and this phenomenon can also be explained with our bound. Minimizing the first term in Eq.~\ref{bound} will help the model's performance on clean input, while minimizing the second term increase the model's robustness against adversarial input; and there is an inherent trade-off between them. Minimizing the second term $\mathbb{E}_{p(x)}\left[\max_{x^{u\_adv}\in A(x)} \text{KL}[p_\theta(z|x)|p_\theta(z|x^{u\_adv})]\right]$ too much will compress the representation, hurting its expressiveness and separability among classes. For example, consider the $l_2$ defense with radius $\epsilon$ ($A(x)$ will be the $l_2$-ball around $x$ with radius $\epsilon$). The above regularizer encourages the representation distribution of an input $x$ to be similar to that of its neighbours in the $l_2$-ball. Now, if there exist two inputs $x_1$ and $x_2$ from different classes such that $\epsilon < ||x_1-x_2||_2 < 2\epsilon$, then these two points do not belong to the other's adversarial set. Let $x'=(x_1+x_2)/2$, it follows that $||x_1-x'||_2 = ||x_2-x'||_2 = ||x_1-x_2||_2/2 < \epsilon$, meaning $x' \in A(x_1)$ and $x' \in A(x_2)$. Note that minimizing the regularizer term too much will encourage the representation distribution of both $x_1$ and $x_2$ to be similar to that of $x'$; and the classifier might fail to separate the two datapoints. Therefore, it might hurt the expressiveness and separability of the representation, especially for the inputs around the decision boundary.
\vspace{-0.1in}
\subsection{Applications and Use Cases}
\label{application}
\makebox[\linewidth][s]{In this subsection, we will discuss the use of our robustness regularizer} $\mathbb{E}_{p(x)}\left[\max_{x^{u\_adv}\in A(x)} \text{KL}[p_\theta(z|x)|p_\theta(z|x^{u\_adv})]\right]$. Although this term can be used directly in a supervised learning setting, a far more exiting application is to minimize it during the representation learning phase (of an unsupervised or self-supervised method). As mentioned earlier, since this term is task-independent, if we minimize it during the representation learning phase, it will transfer directly to downtream tasks, improving the model's adversarial robustness on these tasks (see Eq.~\ref{bound}). We name our regularizer \textbf{Urkle} (\textbf{U}nsupervised \textbf{R}obustness with \textbf{KL} divergenc\textbf{E}). In this subsection, we demonstrate some example scenarios to learn a meaningful (and robust) representation, namely with unsupervised learning (via VAE) and self-supervised learning (with any pretext task).
\vspace{-0.1in}
\subsubsection{With VAE}
\label{with_vae}
In VAE \cite{kingma2013auto}, we have an encoder $p_\theta(z|x)$ (which also acts as our representation mapping), a decoder $q_\phi(x|z)$, and a prior $p(z)$, the objective of VAE (negative ELBO) is:
\begin{align}
\mathbb{E}_{p(x)}\left[\mathbb{E}_{p_\theta(z|x)}[-\log q_\phi(x|z)]\right] + \mathbb{E}_{p(x)}\left[\text{KL}[p_\theta(z|x)|p(z)]\right]
\end{align}
Here we add our robustness regularizer to learn a robust encoder $p_\theta(z|x)$, leading to the below objective:
\begin{align}
\mathbb{E}_{p(x)}\left[\mathbb{E}_{p_\theta(z|x)}[-\log q_\phi(x|z)]\right] + \beta_{VAE}\mathbb{E}_{p(x)}\left[\text{KL}[p_\theta(z|x)|p(z)]\right] \nonumber\\
+ \beta_{robust} \mathbb{E}_{p(x)}\left[\max_{x^{u\_adv}\in A(x)} \text{KL}[p_\theta(z|x)|p_\theta(z|x^{u\_adv})]\right]
\label{obj_vae}
\end{align}
Note that we also add a coefficient $\beta_{VAE}$ for the VAE's regularizer
$\mathbb{E}_{p(x)}\left[\text{KL}[p_\theta(z|x)|p(z)]\right]$ (similar to $\beta$-VAE \cite{higgins2016beta}). We use a small value of $\beta_{VAE}$ in practice since we found that this term might hinder the expressiveness of the representation $p_\theta(z|x)$.
All three expectation terms in Eq~\ref{obj_vae} can be estimated with a minibatch of input $x$'s. For each input $x$, we find the unsupervised adversary $x^{u\_adv}$ by the inner maximization problem of $\max_{x^{u\_adv}\in A(x)} \text{KL}[p_\theta(z|x)|p_\theta(z|x^{u\_adv})]$ with, for example, the PGD algorithm. The objective in Eq~\ref{obj_vae} is minimized with respect to $\theta$ and $\phi$ (thus it will learn an encoder $p_\theta(z|x)$ such that the representation of an input is close to that of its adversaries).
\vspace{-0.1in}
\subsubsection{With a self-supervised learning task}
Let's assume we have a pretext task designed to learn a meaningful representation of $x$. Since self-supervised tasks and their loss functions are diverse, we will refer to the loss function as $L_{ssl}(p(x),p_\theta(z|x),\phi)$ in general, where $p_\theta(z|x)$ is a representation mapping used to solve the task (e.g., might be used for a pretext classification task, to solve a jigsaw puzzle, or to minimize the contrastive loss in the contrastive learning framework) and $\phi$ is any additional parameters (apart from $\theta$) used for this self-supervised task (e.g., parameters of the projector in SimCLR \cite{chen2020simple}, parameters of the output head of some pretext classification task).
Similarly, we can also add the robustness regularizer term here to to learn a robust representation mapping $p_\theta(z|x)$, leading to the following objective:
\begin{align}
L_{ssl}(p(x),p_\theta(z|x),\phi) + \beta_{robust} \mathbb{E}_{p(x)}\left[\max_{x^{u\_adv}\in A(x)} \text{KL}[p_\theta(z|x)|p_\theta(z|x^{u\_adv})]\right]
\end{align}
\paragraph{\textbf{Demonstration with SimCLR \cite{chen2020simple}:}} We re-emphasize that our proposed robustness regularizer can be applied to almost all unsupervised and self-supervised representation learning methods. However, since our main baseline \cite{kim2020adversarial} is built upon SimCLR, we also use SimCLR (with a slight adaptation of using a probabilistic representation network) in our experiments for a fair comparison (note that we conjecture using more recent and advanced self-supervised learning methods \cite{zbontar2021barlow,grill2020bootstrap} will likely improve further the performance of our model). We demonstrate the resulting method here in details.
First of all, SimCLR (and other contrastive learning methods) learns from two augmentations of each image. Let denote a minibatch of data as $\{(x_{1,1},x_{1,2})$, $(x_{2,1},x_{2,2}),...,(x_{b,1},x_{b,2})\}$, where $x_{i,1}$ and $x_{i,2}$ are two augmented version of a same original image $x_i$, and are called positive examples of each other. The goal of contrastive learning is to encourage the representation to be similar among the positive images. For each $x$ we sample a single $z$ from the network $p_\theta(z|x)$ to make the minibatch of representation $\{(z_{1,1},z_{1,2}),(z_{2,1},z_{2,2}),...,(z_{b,1},z_{b,2})\}$. Then the loss function of SimCLR is:
\begin{align}
\ell_{SimCLR} = \ell_{NT\_Xent}((z_{1,1},z_{1,2}),(z_{2,1},z_{2,2}),...,(z_{b,1},z_{b,2}))
\end{align}
where $\ell_{NT\_Xent}$ is the so-called ``normalized temperature-scaled cross entropy'' loss function that takes input as a batch of tuples $t_1,t_2,...,t_b$, with each $t_i$ is a tuple of representations from a set of positive images (in the case above each tuple would be of length 2). Specifically:
\begin{align}
&\ell_{NT\_Xent}(t_1,t_2,...,t_b) = \sum_{i=1}^b \sum_{z,z'\in t_i,z\neq z'}\frac{\exp(\mathrm{sim}(z,z')/\tau)}{\sum_{j\neq i}\sum_{z''\in t_j} \exp(\mathrm{sim}(z,z'')/\tau)}
\label{nt_xent}
\end{align}
where $\mathrm{sim}(z,z')$ is the cosine similarity between $z$ and $z'$ (we also often project the representation $z$ to a lower dimensional space before calculating the cosine similarity), and $\tau$ is the temperature (often set to $0.5$).
To implement our regularizer, we find the unsupervised adversaries\\ $\{(x^{u\_adv}_{1,1},x^{u\_adv}_{1,2}),...,(x^{u\_adv}_{b,1},x^{u\_adv}_{b,2})\}$ of $\{(x_{1,1},x_{1,2}),...,(x_{b,1},x_{b,2})\}$ that maximize:
\begin{align}
\frac{1}{2b}\sum_{i=1}^b \sum_{k=1}^2 \max_{x^{u\_adv}_{i,k} \in A(x_{i,k})} \text{KL}[p_\theta(z|x_{i,k}))|p_\theta(z|x^{u\_adv}_{i,k}))]
\end{align}
Following \cite{kim2020adversarial}, we also use PGD \cite{madry2017towards} to solve this inner maximization problem.
Let $\ell_{Urkle}$ be the value of the above maximum, i.e.:
\begin{align}
\ell_{Urkle} = \frac{1}{2b}\sum_{i=1}^b \sum_{k=1}^2 \text{KL}[p_\theta(z|x_{i,k}))|p_\theta(z|x^{u\_adv}_{i,k}))]
\end{align}
with $\{x^{u\_adv}_{i,k}\}$ found above.
Now we can add the regularizer $\ell_{Urkle}$ directly to the original loss function $\ell_{SimCLR}$. However, we note that $x^{u\_adv}_{i,1}$ and $x^{u\_adv}_{i,2}$ can also be treated as positive images of $x_{i,1}$ and $x_{i,2}$; therefore, we also include them when compute the $NT\_Xent$ loss. The final loss function of our model as:
\begin{align}
&\ell_{NT\_Xent}((z_{i,1},z_{i,2},z^{u\_adv}_{i,1},z^{u\_adv}_{i,2})_{i=1}^b) + \beta_{robust} \ell_{Urkle}
\end{align}
where $z^{u\_adv}_{i,k} \sim p_\theta(z|x^{u\_adv}_{i,k}) \;\forall i \in \overline{1,b}, k \in \{1,2\}$ , $\ell_{NT\_Xent}$ is calculated as in Eq~\ref{nt_xent} (in this case each tuple is of length 4), and $\beta_{robust}$ is a hyperparameter.
\vspace{-0.1in}
\paragraph{\textbf{PGD attack/defense:}} Since Projected Gradient Descent is used in both training and evaluation in our experiments, we briefly review it here to make the paper more self-contained. Let's assume that we need to find $x'$ within an $l_\infty$-norm (or other norms) ball of radius $\epsilon$ around $x$ that maximizes the function $f(x')$. The PGD algorithm is as follows:
\begin{enumerate}
\item Initialize $x^0$ to $x$ (possibly with a small added random noise).
\item Update $x^i=x^{i-1} + \alpha \texttt{sign}(\nabla_{x^{i-1}}f(x^{i-1}))$ with a step size $\alpha$, and clip $x^i$ to the $\epsilon$-ball that is being considered.
\item Repeat step 2 $k$ times, and set $x' = x^k$.
\end{enumerate}
Note that we can also use any other adversarial attack/defense methods for the inner maximization problem. We use PGD in this paper because it is considered ``gold-standard'' at the moment, and it is also used by our main baseline \cite{kim2020adversarial}.
\section{Conclusion}
To conclude, in this paper, we develop a task-agnostic robust representation learning method. The core idea behind our method is to minimize a task-independent robustness regularizer that enforces the representation of an image to be close to that of its adversarial examples. This is motivated by our theoretical result that, for a model using the learned representation for a downstream task, its adversarial loss is bounded by the loss on clean image plus the above task-independent regularizer. Our regularizer can be straightforwardly applied to almost any existing representation learning method (with only an adaptation to a probabilistic representation). To the best of our knowledge, our work is one of the first to study the problem of unsupervised robust representation learning in a principled way, and show that the robustness can be theoretically transferred to the downstream tasks. We demonstrate our proposed regularizer with several unsupervised/self-supervised methods (from VAE to SimCLR), and conduct extensive experiments on MNIST, CIFAR10, CIFAR100 and ImageNet to validate our method. Experimental results suggest that our proposed method (when used with SimCLR) achieves SOTA performance on the unsupervised robust representation learning task.
\section{Related Works}
\vspace{-0.1in}
\subsection{Adversarial Training}
It has been shown that neural networks, even with a high classification accuracy, are vulnerable to small bounded (but intentionally worst-case chosen) adversarial perturbations \cite{szegedy2013intriguing}. From this observation, many methods have been proposed to alter the typical training procedure of a neural network to improve its adversarial robustness (hence the name adversarial training). Two of the most often-used methods are AT \cite{madry2017towards} and TRADES \cite{zhang2019theoretically}, in which the algorithms try to find a worst-case perturbation of the input (called an adversarial example) with an inner maximization problem and minimize the loss (or a regularizer) with respect to that adversarial example. Studying the inner maximization problem is also an active and attractive research direction, with the aim to develop fast and/or accurate methods to find the adversarial examples. For example, PGD \cite{madry2017towards} uses multiple steps of projected gradient descent, which is accurate (and somewhat ``gold-standard'') but expensive. For this computational reason, many one-step algorithms have been proposed, including FGSM \cite{goodfellow2014explaining}, RS-FGSM \cite{wong2020fast} and GradAlign \cite{andriushchenko2020understanding}, with varying level of success. These adversarial defense/attack methods are relevant to our work since we also need to solve an inner maximization problem to find (unsupervised) adversarial examples for the robustness regularizer.
\vspace{-0.1in}
\subsection{Self-Supervised Learning}
As mentioned earlier, self-supervised learning has received great interest due to its ability to learn from unlabeled data, which reduces the need for expensive annotations of images. Self-supervised learning is based on a user-defined pretext task, which can be as simple as to predict the rotation angle of an image or more complex such as to solve a Jigsaw puzzle \cite{noroozi2016unsupervised}. Recently, a popular and successful self-supervised learning paradigm is to learn representations that are invariant under different
augmentations (also referred to as ‘distortions’) of an image \cite{chen2020simple,zbontar2021barlow,chen2021exploring}. The idea of this learning paradigm is to maximize the similarity between representations of two augmentations of an image, while avoiding network collapse (to a trivial and meaningless solution such as a constant function) by different objective functions. Since our proposed robustness regularizer is task-independent, it can be straightforwardly applied to most of these self-supervised learning methods.
\subsection{Robust Self-Supervised Learning}
Recently, the field of unsupervised robust representation learning has gained increasing interest, with the goal to leverage unlabeled data and learn a robust and meaningful representation for downstream tasks. One of the main baselines to our work is RoCL \cite{kim2020adversarial}, which applies adversarial training directly on the contrastive loss in SimCLR. However, it is not clear if a representation that is robust to the contrastive loss will be robust to the prediction loss of a downstream task. Some other methods \cite{jiang2020robust,hendrycks2019using} perform adversarial training on the self-supervised task to aid the adversarial training of the main task (not to replace), which are less related to our work. Similarly, \cite{alayrac2019labels} utilizes unlabeled data to help train a robust classifier in a semi-supervised manner. Meanwhile, \cite{awasthi2021adversarially} propose an algorithm to improve the robustness of PCA. However, PCA is not a common component of modern deep learning architectures/pipelines, thus its practicality might be limited.
\section{Introduction}
Deep learning has achieved state-of-the-art performance in many tasks such as image classification, object detection, and natural language processing. Instead of having to select handcrafted features and representation of the input as in classical machine learning, deep learning has the ability to automatically learn a meaningful representation with deep networks and gradient descent. However, the success of deep learning relies on the availability of a large amount of labeled data, which is expensive in practice. Self-supervised learning \cite{chen2020simple,grill2020bootstrap,chen2021exploring,zbontar2021barlow} has gained interest as a solution to the above problem due to its ability to learn from unlabeled data. However, although the representation learned via self-supervised learning is often meaningful for downstream tasks, the resulting prediction model usually lacks adversarial robustness. Currently, to the best of our knowledge, virtually all existing adversarial training methods \cite{madry2017towards,zhang2019theoretically} require labels and/or the prediction task. Furthermore, for computational reasons, we do not want to re-do the expensive adversarial training for the downstream tasks (this is also in the spirit of self-supervised learning); therefore, the (theoretical) transferability of a robust representation among different tasks is also crucial. However, this transferability aspect of robustness has not been well-studied. For example, \cite{kim2020adversarial} proposes a framework for adversarial contrastive learning, which enforces the robustness of the representation network by finding worst-case adversaries that maximize the contrastive loss, followed by minimizing that loss with respect to the network parameters. However, it is not clear how this robustness can be transferred to a downstream task.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.9\linewidth]{images/illustration.pdf}
\vspace{-0.1in}
\caption{\textbf{Illustration of our proposed regularizer}. This figure illustrates the (probabilistic) representation space, with each circle representing (the distribution of) the representation of an image. Non-filled circles depict the (probabilistic) representations of natural images and filled circles depict that of their adversarial examples. We use the KL divergence to pull the representations of an image and its adversarial images closer, which improves the bound of the adversarial loss on a downstream task.}
\label{illustration}
\end{figure*}
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.5\linewidth}
\centering
\includegraphics[width=\textwidth]{images/visualization_ori.pdf}
\end{subfigure
\hfill
\begin{subfigure}[b]{0.5\linewidth}
\centering
\includegraphics[width=\linewidth]{images/visualization_kl.pdf}
\end{subfigure}
\vspace{-0.2in}
\caption{\textbf{Visualization of the probablistic representation of our method}. Each color corresponds to a single image. For each image $x$, we sample 20 $z$'s from the probabilistic representation distribution $p_\theta(z|x)$ (hence the clusters of points). The left figure is for the original images and the right figure is for the unsupervised adversaries. Our method enforces the representation distribution of an image to be close to that of its adversaries, making the downstream models more robust.} \label{visualization}
\end{figure}
In this paper, we develop a task-independent robust representation learning method to tackle the above issue. Intuitively, if we could enforce the representation $z$ of an input $x$ to be close to that of its neighbors in the adversarial ball (for example, a $l_\infty$ ball around $x$), then it would make the downstream model more robust (Figure~\ref{illustration}). In particular, any adversarial example $x^{adv}$ of $x$ produced by an attack on the downstream task is within this adversarial ball, thus its representation would be close to that of $x$, making it harder for the task's decision boundary to separate them. However, there are some challenges when applying this idea naively. One might think that we can find a worst-case adversarial example $x^{u\_adv}$ (unsupervised adversary, as opposed to the typical supervised adversary $x^{adv}$) with the largest distance to $x$ in the representation space (based on some distance metrics), and minimize that distance with respect to the network's parameters. This, however, is problematic for a typical deterministic representation network, since it is not trivial how to define the distance on the representation space (for example, using the $l_2$ distance between the representations would not help since the model can ``cheat'' by making the norm of the representation smaller, which can be easily compensated by making the weights of the next fully connected layer bigger). Fortunately, in this paper, we observe that with a probabilistic representation network $p_\theta(z|x)$, we can define the ``closeness'' of the representations by the KL divergence, which allows for creating adversarial examples during unsupervised/self-supervised learning via an inner maximization of the KL term ($\text{KL}[p_\theta(z|x)|p_\theta(z|x^{u\_adv})]$). Visualization of the representation distribution of an image and its adversarial example of our method can be found in Figure~\ref{visualization}. We also show that this leads to a robustness regularizer for unsupervised learning that is \textbf{provably transferable} to downstream tasks (more details in the Approach section). Note that this idea can also be thought of as enforcing a low Lipschitzness of the representation network. However, as mentioned above, a typical norm-based (e.g., $l_1$ and $l_2$) Lipschitzness in the representation space is not meaningful (due to the aforementioned ``cheat''), whereas using KL divergence solves the problem and also leads to a theoretical guarantee on the downstream robustness.
In this work, we first propose an upper bound of the adversarial loss of a model for a certain prediction task based on its loss on the clean data and a robustness regularizer. The regularizer, which is based on the KL divergence as described above, only depends on the representation mapping and is independent of the task. Therefore, we can minimize this regularizer term during the representation learning phase (of an unsupervised or self-supervised method), so that the robustness will be transferred directly to a prediction model when the representation is used for a downstream task. Our method can be straightforwardly applied to any representation learning method, including unsupervised learning (e.g., VAE) or any self-supervised learning model.
Our contributions in this work are threefold:
\begin{itemize}
\item We derive an upper bound on the adversarial loss of a prediction model on a certain task, based on a task-independent robustness regularizer.
\item We propose to incorporate the above regularizer into existing representation learning frameworks to improve the adversarial robustness of the representation on downstream tasks.
\item We demonstrate the use of our proposed regularizer with existing representation learning frameworks, and show that the resulting models achieve SOTA results in the unsupervised robust representation learning task.
\end{itemize}
\section{Experiments}
\vspace{-0.1in}
We conduct extensive experiments to validate our method. In this section, we describe these experiments in details. For more information regarding the experimental settings and the baselines, please refer to our supplementary file.
\vspace{-0.25in}
\subsection{Datasets}
\vspace{-0.05in}
\paragraph{\textbf{MNIST \cite{lecun-mnisthandwrittendigit-2010}}} contains 70000 images of hand-written digits with the classification task of 10 digits.
\vspace{-0.07in}
\paragraph{\textbf{CIFAR10 \cite{krizhevsky2009learning}}} consists of 60000 images of size 32x32, and over ten classes: airplane, automobile, bird, cat, deer, dog, frog, horse, ship and truck.
\vspace{-0.07in}
\paragraph{\textbf{CIFAR100 \cite{krizhevsky2009learning}}} Similar to CIFAR10, CIFAR100 also consists of 60000 images with the size 32x32. The task is classification with 100 different classes.
\vspace{-0.07in}
\paragraph{\textbf{ImageNet \cite{deng2009imagenet}}} is a large scale real-world computer vision dataset, which consists of 1000 classes. To the best of our knowledge, this dataset has not been considered by previous adversarial self-supervised learning methods.
\vspace{-0.1in}
\subsection{Experimental Settings}
Within each experiment, we use the same network as the representation network for all models. Since our representation network is probabilistic, it only differs from the other deterministic networks in the last layer. In particular, for a representation of size $d_z$, the last layer's dimension of a deterministic representation network is $d_z$, while that of a probabilistic network is $2\cdot d_z$ ($d_z$ for $\mu$ and $d_z$ for $\sigma^2$). Also, although our method can be used for any adversarial attack (e.g., $l_1$ or $l_2$), we consider the $l_\infty$ adversaries in this experiments section. This is because $l_\infty$ is one of the most common adversarial attacks, and it is also used in our main baseline \cite{kim2020adversarial}.
\vspace{-0.05in}
\paragraph{\textbf{With VAE:}} We test the effectiveness of our robustness regularizer when used with VAE (as described in Section~\ref{with_vae}) with the MNIST dataset. The main baseline we consider in this experiment is AE (auto-encoder) with a TRADES-like regularizer. This is because the reconstruction in AE can be viewed as a prediction task, so we can use the regularizer in TRADES to force the reconstruction of an image to be similar to the reconstruction of its adversaries (more details of this baseline in the supplementary file). To make the comparison fair for AE (that has no built-in regularizer), we set $\beta_{VAE}=0$ in our experiment, although we note that slightly increasing this value leads to even better representation. Apart from this baseline, we also include supervised learning models (Standard Training, AT \cite{madry2017towards} and TRADES \cite{zhang2019theoretically}) for reference. For this experiment, we consider the $l_\infty$ perturbation with $\epsilon=0.1$. For this ``toy'' experiment, we use a simple convolutional neural network with four 3$\times$3 convolutional layers (followed by an average pooling layer) as the representation network.
\vspace{-0.05in}
\paragraph{\textbf{With Self-Supervised Learning (SimCLR):}} For the more challenging real-world datasets (CIFAR10, CIFAR100, ImageNet), learning generative features of images with VAE is difficult, so we use self-supervised learning methods (SimCLR) to validate our robustness regularizer. In this experiment, we consider RoCL \cite{kim2020adversarial} as our main baseline. We also include supervised learning models (Standard Training, AT \cite{madry2017towards} and TRADES \cite{zhang2019theoretically}) for reference.
We train our self-supervised model with 2000 epochs (except for ImageNet, where we train a total number of 200 epochs for computational reasons). Similar to \cite{kim2020adversarial}, we consider the $l_\infty$ attack and defense. Following standard in the adversarial robustness literature, we set the training perturbation radius as $\epsilon=8/255$ for CIFAR10/CIFAR100 and $\epsilon=2/255$ for ImageNet.
For CIFAR10, we use ResNet18 \cite{he2016deep} as the backbone network with a batchsize of 1024. As for the CIFAR100 dataset, we use ResNet50 as the backbone, and also with a batchsize of 1024. For ImageNet, we use a batchsize of 4096 with a ResNet50 network. For all experiments, we use a starting learning rate of $1.2$ and perform Cosine annealing on the learning rate over the course of training.
\vspace{-0.1in}
\subsection{Results}
\subsubsection{With VAE}
\begin{table*}[t!]
\centering
\captionsetup{justification=centering}
\caption{\textbf{MNIST} with $l_\infty$ adversaries. Training and testing $\epsilon$ are set to $0.1$ in this experiment. Our method (VAE+Urkle) outperforms the baseline AE+TRADES, while approaching the robustness similar to supervised adversarial training methods.}
\label{mnist}
\begin{tabular}{ccc}
\toprule
Models & Clean Acc & Adversarial Acc \\
\midrule
Standard Training & 99.3±0.1 & 1.0±0.3 \\
AT & 99.0±0.1 & \textbf{98.5±0.2} \\
TRADES & 99.1±0.1 & 98.3±0.1 \\
\midrule
AE + TRADES & 99.1±0.1 & 96.6±0.4 \\
VAE + Urkle (ours) & 99.1±0.1 & \textbf{98.0±0.1} \\
\bottomrule
\end{tabular}
\end{table*}
\begin{table*}[t!]
\centering
\captionsetup{justification=centering}
\caption{ \textbf{CIFAR10}: Results of supervised and self-supervised methods trained with $l_\infty$ adversaries and $\epsilon=8/255$ (when applicable). Our method (SimCLR+Urkle) significantly outperforms the baseline RoCL, especially with unseen (and stronger) attack $\epsilon=16/255$.}
\label{cifar10}
\centering
\small
\begin{tabular}{ccccccccccc}
\toprule
& \multicolumn{9}{c}{CIFAR10}\\
\cmidrule(r){2-10}
& \multicolumn{3}{c}{Fully Labeled Data} & \multicolumn{3}{c}{5000 Labeled Data} & \multicolumn{3}{c}{1000 Labeled Data}\\
\cmidrule(r){2-4} \cmidrule(r){5-7} \cmidrule(r){8-10}
Model & Clean & 8/255 & 16/255 & Clean & 8/255 & 16/255 & Clean & 8/255 & 16/255 \\
\midrule
Standard Training & 92.82 & 0.00 & 0.00 & 79.09 & 0.00 & 0.00 & 60.39 & 0.00 & 0.00 \\
AT & 81.63 & 44.50 & 14.47 & 64.97 & 24.52 & 6.69 & 50.03 & 15.26 & 3.91 \\
TRADES & 77.03 & \textbf{48.01} & \textbf{22.55} & 63.14 & \textbf{25.97} & \textbf{7.78} & 48.32 & \textbf{15.92} & \textbf{3.97} \\
\midrule
SimCLR & 91.25 & 0.63 & 0.15 & 84.31 & 0.84 & 0.12 & 82.15 & 0.55 & 0.11 \\
RoCL & 83.71 & 40.27 & 9.55 & 78.82 & 36.93 & 9.90 & 76.49 & 34.44 & 8.96 \\
SimCLR+Urkle (ours) & 82.31 & \textbf{42.56} & \textbf{14.29} & 77.47 & \textbf{38.76} & \textbf{12.94} & 74.82 & \textbf{37.56} & \textbf{12.22} \\
\bottomrule
\end{tabular}
\end{table*}
Table~\ref{mnist} shows the results of our model and the baselines. MNIST is a relatively easy dataset, so most methods perform reasonably well. Noticeably, our model (VAE+Urkle) outperforms AE+TRADES by 1.4\%, and approaches the performance of supervised adversarial training methods.
\vspace{-0.1in}
\subsubsection{With Self-Supervised Learning (SimCLR):} \;
\vspace{-0.1in}
\paragraph{\textbf{CIFAR10 and CIFAR100:}} As aforementioned, in the CIFAR10 and CIFAR100 experiments, we train all models (except for Standard Training and SimCLR) with $\epsilon = 8/255$. We evaluate these models against $l_\infty$ adversarial attack with strength $\epsilon=8/255$ and $\epsilon=16/255$. Table~\ref{cifar10} and Table~\ref{cifar100} show that our method (SimCLR+Urkle) clearly outperforms RoCL (with a slight trade-off of clean accuracy in some experiments), indicating the effectiveness of our robustness regularizer. Especially, our method is more robust against unseen attack strength ($\epsilon$=16/255).
As discussed earlier, there is a trade-off between the clean accuracy and adversarial robustness in our model (as well as other adversarial training methods such as TRADES), resulting in a slightly lower (around 1\%) clean accuracy of our model when compared to RoCL. However, our model outperforms RoCL significantly in terms of adversarial robustness (especially for $\epsilon=16/255$, which is an unseen attack strength). We find that this is reasonable and the improved robustness is well worth the trade-off. Note that a similar trend can be observed for TRADES and AT, where TRADES achieves lower clean accuracy but much better adversarial robustness when compared to AT.
\begin{table*}[t!]
\centering
\captionsetup{justification=centering}
\caption{\textbf{CIFAR100}: esults of supervised and self-supervised methods trained with $l_\infty$ adversaries and $\epsilon=8/255$ (when applicable). Our method (SimCLR+Urkle) significantly outperforms the baseline RoCL.}
\label{cifar100}
\centering
\small
\begin{tabular}{ccccccccccc}
\toprule
& \multicolumn{9}{c}{CIFAR100}\\
\cmidrule(r){2-10}
& \multicolumn{3}{c}{Fully Labeled Data} & \multicolumn{3}{c}{5000 Labeled Data} & \multicolumn{3}{c}{1000 Labeled Data}\\
\cmidrule(r){2-4} \cmidrule(r){5-7} \cmidrule(r){8-10}
Model & Clean & 8/255 & 16/255 & Clean & 8/255 & 16/255 & Clean & 8/255 & 16/255 \\
\midrule
Standard Training & 70.34 & 0.00 & 0.00 & 26.59 & 0.00 & 0.00 & 12.14 & 0.00 & 0.00 \\
AT & 52.87 & \textbf{19.46} & \textbf{6.80} & \textbf{21.05} & 5.30 & 1.52 & 12.46 & \textbf{3.26} & 0.95 \\
TRADES & 56.96 & 18.54 & 4.48 & 20.35 & \textbf{6.41} & \textbf{1.63} & 13.78 & 3.19 & \textbf{1.32} \\
\midrule
SimCLR & 58.79 & 0.47 & 0.00 & 53.23 & 0.46 & 0.12 & 44.16 & 0.57 & 0.26 \\
RoCL & 52.19 & 22.00 & 8.35 & 40.60 & 18.86 & 7.25 & 30.23 & 13.83 & 5.55 \\
SimCLR+Urkle (ours) & 53.81 & \textbf{24.82} & \textbf{10.63} & 42.66 & \textbf{22.34} & \textbf{11.06} & 32.12 & \textbf{14.40} & \textbf{6.71} \\
\bottomrule
\end{tabular}
\end{table*}
\begin{table}[t!]
\caption{\textbf{ImageNet} with $l_\infty$ adversaries and $\epsilon=2/255$.}
\label{imagenet}
\centering
\resizebox{0.7\columnwidth}{!}{
\begin{tabular}{c|cc}
\toprule
Models & Clean Acc & Adversarial Acc \\
\midrule
RoCL & 52.46 & 23.19 \\
SimCLR + Urkle (ours) & 51.19 & \textbf{25.69} \\
\bottomrule
\end{tabular}
}
\vspace{-0.1in}
\end{table}
\vspace{-0.1in}
\paragraph{\textbf{ImageNet:}} Preliminary result on the ImageNet dataset (Table~\ref{imagenet}) also indicates that our method outperforms RoCL on this large scale dataset.
\vspace{-0.1in}
\paragraph{\textbf{Experimental Results with limited numbers of labels:}} To take advantage of the unsupervised nature of our method, we also conduct the experiments when the number of labeled images is limited. To be re-emphasize, we train our SSL model and the unsupervised robustness regularizer \textbf{without any labels}, and the limited number of labels are only used for the training of the task-specific output head $\hat{p}(y|z)$ (without adversarial training). Table~\ref{cifar10} and Table~\ref{cifar100} report the results for CIFAR10 and CIFAR100 with 5000 and 1000 labels. It can be clearly seen that supervised methods fail to learn a robust model with such a few available labels. Among the unsupervised robust representation learning methods, our model also significantly outperforms the baseline RoCL in these scenarios.
\section{Proofs}
\subsection{Proposition 1}
We use a similar idea as the proof of Proposition 1 in \cite{nguyen2021kl}, which is a bound on the target loss in the domain adaptation problem. This is because our setting can somewhat be cast as a domain adaptation problem, where the source domain is the clean data distribution, and the target domain is the adversarial data distribution. The proof is as below:
\begin{proof} $ $\newline
Let $a(x) = \arg \max_{x^{adv}\in A(x)} -\log \mathbb{E}_{p_\theta(z|x^{adv})}[\hat{p}_T(y|z)]$ $\forall x \in \mathcal{X}$. We need to prove that:
\begin{align}
\mathbb{E}_{p_T(x,y)}\left[-\log \mathbb{E}_{p_\theta(z|a(x))}[\hat{p}_T(y|z)]\right]
\leq \mathbb{E}_{p_T(x,y)}\left[\mathbb{E}_{p_\theta(z|x)}[-\log \hat{p}_T(y|z)]\right] \nonumber \\
+ \frac{M}{\sqrt{2}}\sqrt{\mathbb{E}_{p(x)}\left[\max_{x^{u\_adv}\in A(x)} \text{KL}[p_\theta(z|x)|p_\theta(z|x^{u\_adv})]\right]}
\end{align}
Due to Jensen Inequality, we have:
\begin{align}
\mathbb{E}_{p_T(x,y)}\left[-\log \mathbb{E}_{p_\theta(z|a(x))}[\hat{p}_T(y|z)]\right]
\leq \mathbb{E}_{p_T(x,y)}\left[\mathbb{E}_{p_\theta(z|a(x))}[-\log \hat{p}_T(y|z)]\right]
\end{align}
Therefore, we only need to prove that:
\begin{align}
&\mathbb{E}_{p_T(x,y)}\left[\mathbb{E}_{p_\theta(z|a(x))}[-\log \hat{p}_T(y|z)]\right]
- \mathbb{E}_{p_T(x,y)}\left[\mathbb{E}_{p_\theta(z|x)}[-\log \hat{p}_T(y|z)]\right] \nonumber \\
\leq &\frac{M}{\sqrt{2}}\sqrt{\mathbb{E}_{p(x)}\left[\max_{x^{u\_adv}\in A(x)} \text{KL}[p_\theta(z|x)|p_\theta(z|x^{u\_adv})]\right]}
\end{align}
We have:
\begin{align}
&\mathbb{E}_{p_T(x,y)}\left[\mathbb{E}_{p_\theta(z|a(x))}[-\log \hat{p}_T(y|z)]\right]
- \mathbb{E}_{p_T(x,y)}\left[\mathbb{E}_{p_\theta(z|x)}[-\log \hat{p}_T(y|z)]\right] \\
= &\mathbb{E}_{p_T(x,y)}\left[\int_\mathcal{Z} p_\theta(z|a(x)) [-\log \hat{p}_T(y|z)] dz - \int_\mathcal{Z} p_\theta(z|x) [-\log \hat{p}_T(y|z)] dz \right]\\
= &\mathbb{E}_{p_T(x,y)}\left[\int_\mathcal{Z} -\log \hat{p}_T(y|z) \left[p_\theta(z|a(x))-p_\theta(z|x)\right] dz\right]
\end{align}
For all $x$, let $\mathcal{A}(x)=\{z \in \mathcal{Z}|p_\theta(z|a(x))-p_\theta(z|x) \geq 0\}$ and $\mathcal{B}(x)=\{z \in \mathcal{Z}|p_\theta(z|a(x))-p_\theta(z|x) < 0\}$, then:
\begin{align}
& \mathbb{E}_{p_T(x,y)}\left[\int_\mathcal{Z} -\log \hat{p}_T(y|z) \left[p_\theta(z|a(x))-p_\theta(z|x)\right] dz\right] \\
= &\mathbb{E}_{p_T(x,y)}\Bigg[\int_{\mathcal{A}(x)} -\log \hat{p}_T(y|z) \left[p_\theta(z|a(x))-p_\theta(z|x)\right] dz \nonumber\\
&\quad\quad\quad+ \int_{\mathcal{B}(x)} -\log \hat{p}_T(y|z) \left[p_\theta(z|a(x))-p_\theta(z|x)\right] dz\Bigg] \\
\leq &\mathbb{E}_{p_T(x,y)}\Bigg[\int_{\mathcal{A}(x)} -\log \hat{p}_T(y|z) \left[p_\theta(z|a(x))-p_\theta(z|x)\right] dz\Bigg] \\
& \quad\quad \text{(since $-\log \hat{p}_T(y|z)$ is a non-negative quantity)} \nonumber\\
\leq &\mathbb{E}_{p_T(x,y)}\left[\int_{\mathcal{A}(x)} M \left[p_\theta(z|a(x))-p_\theta(z|x)\right] dz\right] \\
= &M \mathbb{E}_{p_T(x,y)}\left[\int_{\mathcal{A}(x)} \left|p_\theta(z|a(x))-p_\theta(z|x)\right| dz\right]
\end{align}
We have:
\begin{align}
&\int_{\mathcal{Z}} \left[p_\theta(z|a(x))-p_\theta(z|x)\right] dz = 0 \\
\Rightarrow & \int_{\mathcal{A}(x)} \left[p_\theta(z|a(x))-p_\theta(z|x)\right] dz
+ \int_{\mathcal{B}(x)} \left[p_\theta(z|a(x))-p_\theta(z|x)\right] dz = 0 \\
\Rightarrow & \int_{\mathcal{A}(x)} \left|p_\theta(z|a(x))-p_\theta(z|x)\right| dz
= \int_{\mathcal{B}(x)} \left|p_\theta(z|a(x))-p_\theta(z|x)\right| dz \\
\Rightarrow & \int_{\mathcal{A}(x)} \left|p_\theta(z|a(x))-p_\theta(z|x)\right| dz
= \frac{1}{2}\int_{\mathcal{Z}} \left|p_\theta(z|a(x))-p_\theta(z|x)\right| dz
\end{align}
Due to the Pinsker's Inequality we have:
\begin{align}
&\frac{1}{2}\int_{\mathcal{Z}} \left|p_\theta(z|a(x))-p_\theta(z|x)\right| dz \nonumber\\
\leq &\frac{1}{2}\sqrt{2\int_\mathcal{Z}p_\theta(z|x)\log \frac{p_\theta(z|x)}{p_\theta(z|a(x))} dz} \\
= &\frac{1}{\sqrt{2}}\sqrt{\text{KL}[p_\theta(z|x)|p_\theta(z|a(x))]}
\end{align}
Therefore:
\begin{align}
& M \mathbb{E}_{p_T(x,y)}\left[\int_{\mathcal{A}(x)} \left|p_\theta(z|a(x))-p_\theta(z|x)\right| dz\right] \\
\leq &\frac{M}{\sqrt{2}} \mathbb{E}_{p_T(x,y)}\left[\sqrt{\text{KL}[p_\theta(z|x)|p_\theta(z|a(x))]}\right] \\
=& \frac{M}{\sqrt{2}} \mathbb{E}_{p(x)}\left[\sqrt{\text{KL}[p_\theta(z|x)|p_\theta(z|a(x))]}\right] \\
\leq& \frac{M}{\sqrt{2}} \sqrt{\mathbb{E}_{p(x)}\left[\text{KL}[p_\theta(z|x)|p_\theta(z|a(x))]\right]} \\
\leq &\frac{M}{\sqrt{2}} \sqrt{\mathbb{E}_{p(x)}\left[\max_{x^{u\_adv}\in A(x)}\text{KL}[p_\theta(z|x)|p_\theta(z|x^{u\_adv})]\right]}
\end{align}
We conclude our proof.
\end{proof}
\section{Experimental Results}
\subsection{Details on the baseline AE+TRADES}
Here we describe the baseline AE+TRADES, which we use in the VAE experiment, in more detail. In particular, an autoencoder (AE) consists of an encoder $g_\theta$ and a decoder $h_\phi$. The encoder $g$ transforms the input $x$ to a representation $z$ (often lower dimensional), i.e., $z=g_\theta(x)$; while the decoder $h$ tries to reconstruct the original input from the representation, i.e., $\hat{x}=h_\phi(z)$. Using the mean squared (l2) distance for the reconstruction, the objective of AE is:
\begin{align}
\mathbb{E}_{p(x)}[||x-h_\phi(g_\theta(x))||_2^2]
\end{align}
Since the reconstruction $h\circ g$ can be treated as a prediction task (predicting the original $x$), we can use a TRADES-like regularizer to make the model more robust, and thus the encoder is also more robust. Specially, we can enforce the reconstruction of an image to be similar to that of its adversaries. The final objective is:
\begin{align}
&\mathbb{E}_{p(x)}[||x-h_\phi(g_\theta(x))||_2^2] \nonumber\\
& + \beta \mathbb{E}_{p(x)}[\max_{x^{adv}\in A(x)}||h_\phi(g_\theta(x^{adv}))-h_\phi(g_\theta(x))||_2^2]
\end{align}
Note that we can only use this baseline with AE (not VAE) because the TRADES regularizer only works straightforwardly with a deterministic model / prediction.
\subsection{Experimental Settings}
\subsubsection{With VAE}
For the VAE MNIST experiment, the encoder (representation network) is a simple convolutional network with 4 \texttt{3x3} convolutional layers (with the last layer has 128 channels so that the representation has 128 dimension), followed by an average pooling layer.
With AE and VAE, the decoder consists of 4 ConvTranspose2d layers, mirroring the encoder.
The classifier (from a representation to the prediction label) is a composition of 3 fully connected layers (with batchnorm and ReLU activation in-between).
\subsubsection{With SimCLR}
For a ResNet18 backbone network, we set the representation dimension to 512, while that for a ResNet50 backbone is 2048. We set the initial learning rate to $1.2$ and do Cosine annealing to $0$. Other experiment details have been presented in the main paper. In addition, please also refer to our code for more details.
|
3,212,635,537,425 | arxiv | \section{Introduction}
\label{intro}
Graphs considered in the paper are finite, without loops or multiple edges. In a graph $G = (V , E)$, $V$ and $E$ ($V(G)$ and $E(G)$) denote the vertex set and the edge set of $G$, respectively. For undefined concepts and notation we refer the reader to \cite{BJD}.
For two vertices $u$ and $v$ in a graph $G$, a $uv$-geodesic is a shortest path between $u$ and $v$. A set $S$ of vertices of $G$ is convex if the vertices of every $uv$-geodesic is contained in $S$ for every $u, v \in S$. According to Duchet, convexity in graphs has been studied since the early seventies, when abstract convexity was studied in different contexts (\cite{D} is an outdated, but very nice, survey on the subject). Convexity in graphs has taken many different directions, and different related parameters have been defined and widely studied, e.g., the hull number \cite{ES}, the geodetic number \cite{HLT}, and the convexity number \cite{CWZ} of a graph. Recent papers on this subjects include \cite{ACGNSS,DPRS2,DPRS}, where the decision problem associated with these three parameters are shown to be $\mathcal{NP}$-complete, even when restricted to bipartite graphs, and in the case of the geodetic number, even when restricted to bipartite chordal graphs.
Chartrand, Fink and Zhang generalized the concept of convexity to oriented graphs, and defined the convexity number for an oriented graph; oriented analogues of the hull number and geodetic number are defined in \cite{CZ}. We focus on the convexity number of oriented graphs; although this generalization was introduced in 2002, and the proof given by Gimbel in \cite{G} on the $\mathcal{NP}$-completeness of determining the convexity number of an arbitrary graph is one of the shortest and neatest $\mathcal{NP}$-completeness proofs ever done, the problem of determining the convexity number of an oriented graph was not known to be $\mathcal{NP}$-complete until now. We prove that determining the convexity number of an oriented graph is $\mathcal{NP}$-complete even when restricted to bipartite graphs of girth $g$, with $g \ge 6$.
An \emph{oriented graph} is an orientation of some graph. In an oriented graph $D = (V , E)$, $V$ and $E$ ($V(D)$ and $E(D)$) denote the vertex set and the edge set of $D$, respectively. An \emph{oriented subgraph} $D' = (V', E')$ of an oriented graph $D = (V , E)$ is an oriented graph with $V' \subseteq V$ and $E' \subseteq E$. An oriented graph is \emph{connected} if its underlying graph is connected. A \emph{directed path} is a sequence $(v_1,v_2,...,v_k)$ of vertices of an oriented graph $D$ such that $v_1,v_2,...,v_k$ are distinct and $(v_i,v_{i+1}) \in E(D)$ for $i \in \{ 1, 2, . . . , k-1 \}$. An oriented graph is \emph{strongly connected} (or \emph{strong}) if for every pair of distinct vertices $u$ and $v$, there exists a directed path from $u$ to $v$. The \emph{girth} of an oriented graph is the length of a shortest directed cycle.
A $uv$-\emph{geodesic} in a digraph $D$ is a shortest $uv$-directed path and its length is $d_D (u, v)$. A nonempty subset, $S$, of the vertex set of a digraph, $D$, is called a \emph{convex set} of $D$ if, for every $u, v \in S$, every vertex lying on a $uv$- or $vu$-geodesic belongs to $S$. For a nonempty subset, $A$, of $V(D)$, the \emph{convex hull}, $[A]$, is the minimal convex set containing $A$. Thus $[S] = S$ if and only if $S$ is convex in $D$. The \emph{convexity number}, $\textnormal{con}(D)$, of a digraph $D$ is the maximum cardinality of a proper convex set of $D$. A \emph{maximum convex set} $S$, of a digraph $D$, is a convex set with cardinality $\textnormal{con}(D)$. Since every singleton vertex set is convex in a connected oriented graph $D$, $1\le \textnormal{con}(D) \le n-1$. The \emph{degree}, $d(v)$, of a vertex $v$ in an oriented graph is the sum of its in-degree and out-degree; this is, $d(v) = d^-(v) + d^+(v)$. A vertex, $v$, is an \emph{end-vertex} if $d(v) = 1$. A \emph{source} is a vertex having positive out-degree and in-degree $0$, while a \emph{sink} is a vertex having positive in-degree and out-degree $0$. For a vertex $v$ of $D$, the in-neighborhood of $v$, $N^-(v)$, is the set $\{ x \colon\ (v, x) \in E(D) \}$ and the out-neighborhood of $v$, $N^+ (v)$, is the set $\{ x \colon\ (x, v) \in E(D)\}$. A vertex $v$ of $D$ is a \emph{transitive vertex} if $d^+(v)>0$, $d^-(v) > 0$ and, for every $u \in N^+(v)$ and $w \in N^-(v)$, $(w, u) \in E(D)$.
For graphs $G$ and $H$, their {\em cartesian product}, $G \Box H$, is the graph with vertex set $V(G) \times V(H)$, and such that two vertices $(g_1,h_1)$ and $(g_2,h_2)$ are adjacent in $G \Box H$ if either $g_1=g_2$ and $h_1 h_2$ is an edge in $H$, or $h_1=h_2$ and $g_1 g_2$ is an edge in $G$. For a vertex $g$ of $G$, the subgraph of $G \Box H$ induced by the set $\{ (g,h) \colon\ h \in H \}$ is called an $H$-fiber and is denoted by $^gH$. Similarly, for $h \in H$, the $G$-fiber, $G^h$, is the subgraph induced by $\{ (g,h) \colon g \in G \}$. We will have occasion to use the fiber notation $G^h$ and $^gH$ to refer instead to the set of vertices in these subgraphs; the meaning will be clear from the context. It is clear that all $G$-fibers are isomorphic to $G$ and all $H$-fibers are isomorphic to $H$.
As mentioned, the concept of convexity number of an oriented graph was first introduced by Chartrand, Fink and Zhang in \cite{CFZ}, where they proved the following pair of theorems.
\begin{theorem}
Let $D$ be a connected oriented graph of order $n \ge 2$. Then $\textnormal{con} (D) = n-1$ if and only if $D$ contains a source, a sink or a transitive vertex.
\end{theorem}
\begin{theorem} \label{no2}
There is no connected graph of order at least $4$ with convexity number $2$.
\end{theorem}
Taking an interesting direction for the subject of convexity in oriented graphs, in \cite{TYF}, Tong, Yen and Farrugia introduced the concepts of convexity spectrum and strong convexity spectrum of a graph. For a nontrivial connected graph $G$, we define the \emph{convexity spectrum}, $S_C (G)$, of a graph $G$, as the set of convexity numbers of all orientations of $G$, and the \emph{strong convexity spectrum}, $S_{SC}(G)$, of a graph $G$ as the set of convexity numbers of all strongly connected orientations of $G$. If $G$ has no strongly connected orientation, then $S_{SC}(G)$ is empty. The \emph{lower orientable convexity number}, $\textnormal{con}^- (G)$, of $G$ is defined to be $\min S_C(G)$ and the \emph{upper orientable convexity number}, $\textnormal{con}^+(G)$, is defined to be $\max S_C(G)$. Hence, for every nontrivial connected graph $G$ of order $n$, $1\le \textnormal{con}^-(G) \le \textnormal{con}^+(G) \le n-1$.
Tong, Yen and Farrugia calculated the convexity and strong convexity spectra of complete graphs and also constructed, for every $a \in \mathbb{Z}^+$, a graph $G$ with convexity spectrum $\{ a, n-1 \}$. It is not very surprising that the strong convexity spectra of $K_n$ for $n \ge 7$ is a ``large'' set, $\{ 1, 3, 5,6, \dots, n-2 \}$, missing only $2, 4$ and $n-1$. Nonetheless, we find very surprising that, in one hand, Tong and Yen proved in \cite{TY} that $S_{SC} (K_{r,s}) = \{ 1 \}$ for every pair of integers $2 \le r, s$, and, in the other hand, we prove that the strong convexity spectrum of an $n \times m$ grid, for any pair of integers $n,m \ge 5$, only lacks the set of integers $\{ 2, 3, 5, n-1, n-2, n-3, n-4, n-5 \}$. So, an interesting question arises from the previous observation: What property in a graph determines a large strong convexity spectrum? We can discard regularity and high degrees; grids are not regular graphs, and have both small maximum and minimum degree. Although we did not find what is so special about grids in terms of convexity, we managed to calculate the strong convexity spectra of all grids.
The rest of this paper is ordered as follows.
Section \ref{SNP} is devoted to prove the $\mathcal{NP}$-completeness of the problem of determining the convexity number of a given oriented graph; the problem remains $\mathcal{NP}$-complete even when restricted to bipartite oriented graphs of arbitrarily large girth. In Section \ref{SGrids}, we prove some basic results on the convexity number of general oriented graphs, and also introduce a concept of main importance to this work: The whirlpool orientation of a grid. Using this concept, we prove that $1 \in S_{SC} (G)$ for any grid $G$. We finish the section with a result excluding some values from the strong convexity spectra of certain grids. Section \ref{SCSSG} is devoted to calculate the strong convexity spectra of $n \times 2$ and $n \times 3$ grids for every integer $n \ge 2$. In Section \ref{SMain}, the strong convexity spectra of $n \times m$ grids for every pair of integers $n, m \ge 4$ is calculated.
\section{$\mathcal{NP}$-completeness}
\label{SNP}
We define the problem \textsc{Oriented Convexity Number} as follows. Given an ordered pair $(D,k)$, consisting of an oriented graph $D$ and a positive integer $k$, determine whether $D$ has a convex set of size at least $k$.
This first section is devoted to prove the $\mathcal{NP}$-completeness of \textsc{Oriented Convexity Number}.
\begin{theorem}\label{redu}
\textsc{Oriented Convexity Number} restricted to bipartite oriented graphs of girth $6$ is $\mathcal{NP}$-complete.
\end{theorem}
\begin{proof}
Let $D$ be an oriented graph. Given a subset $C \subseteq V(D)$, it can be verified in polynomial time whether $C$ is a convex set. Hence, \textsc{Oriented Convexity Number} is in $\mathcal{NP}$.
In order to prove $\mathcal{NP}$-hardness (and hence $\mathcal{NP}$-completness), we reduce an instance $(G,k)$ of the well-known $\mathcal{NP}$-complete problem \textsc{Clique} to an instance $(D,k')$ of \textsc{Oriented Convexity Number} such that $\omega(G) \ge k$ if and only if $\textnormal{con}(D) \ge k'$, the encoding length of $(D,k')$ is polinomially bounded in terms of the encoding length of $(G,k)$, and $D$ is bipartite.
Let $(G,k)$ be an instance of \textsc{Clique}. Let us assume that $k \ge 3$ and $G$ is connected; we construct $D$ as follows. For every vertex $u$ of $G$ we create a directed hexagon, $H_u$, with two antipodal distinguished vertices $x_u$ and $y_u$. For every edge $uv \in E(G)$ we add the arcs $(x_u,y_v)$ and $(x_v,y_u)$ to $D$. We create four additional vertices $z_1, z_2, z_3, z_4$ with arcs $(z_i, z_{i+1})$ for $1 \le i \le 3$ and arcs $(x_u, z_1), (z_4, y_u)$ for every $u \in V(G)$.
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{Reduction.pdf}
\caption{The digraph $D$ of Theorem \ref{redu} when $G$ is $P_3 = (u,v,w)$.} \label{reduFig}
\end{center}
\end{figure}
Clearly, $|V(D)| = 6|V(G)| + 4$ and $|A(D)| = 8|V(G)| + 2|E(G)|+ 3$. It is direct to observe that the digraph $D$ is bipartite and strongly connected. In Figure \ref{reduFig} the classes of the bipartition are given by the vertices of the same color (black and white). We also claim the following statements to hold.
\begin{nclaim} \label{C1}
Let $u$ be a vertex in $G$ and let $C$ be a convex set of $D$ with $|C| \ge 2$. If $C \cap V(H_u) \ne \varnothing$, then $V(H_u) \subseteq C$.
\end{nclaim}
\begin{nclaim} \label{C2}
Let $C$ be a convex set of $D$ with $|C| \ge 2$. If $z_i \in C$ for $1 \le i \le 4$, then $C = V(D)$.
\end{nclaim}
\begin{nclaim}\label{C3}
Let $u, v \in V(G)$ be such that $d_G (u,v) \ge 2$. If $C$ is a convex set of $D$ such that $C \cap V(H_u) \ne \varnothing \ne C \cap V(H_v)$, then $C=V(D)$.
\end{nclaim}
\begin{nclaim} \label{C4}
If $S$ is a clique of $G$, then $C=\bigcup_{v \in S} V(H_v)$ is a convex set of $D$.
\end{nclaim}
It follows from Claims \ref{C1}-\ref{C4} that $C$ is a proper convex set of $D$ with $|C| \ge 2$, if and only if there exists a clique $S$ in $G$ such that $C=\bigcup_{v \in S} V(H_v)$. Considering a maximum clique of $G$, and a maximum proper convex set of $D$, we obtain $\omega(G) = 6 \textnormal{con}(D)$. Hence, if $k'=6k$, then $G$ contains a clique of size at least $k$ if and only if $D$ contains a convex set of size at least $k'$. Since the encoding length of $(D,k')$ is linearly bounded in terms of the encoding length of $(G,k)$, \textsc{Oriented Convexity Number} is $\mathcal{NP}$-complete.
\begin{proof}[of Claim \ref{C1}]
We will consider two cases. First, assume that $|C \cap V(H_u)| \ge 2$. Let $w_1$ and $w_2$ be vertices in $C \cap V(H_u)$. Clearly, $w_1 H_u w_2$ is a $w_1w_2$-geodesic in $D$ and $w_2 H_u w_1$ is a $w_2w_1$-geodesic in $D$. Hence, $V(H_u) \subseteq C$.
For the second case suppose that $w_1 \in V(H_u)$ and $w_2 \in C \setminus V(H_u)$. Therefore, every $w_1w_2$-directed path in $D$ uses the vertex $x_u$, and every $w_2w_1$-directed path in $D$ uses the vertex $y_u$. Thus, $|C \cap V(H_u)| \ge 2$ and we are back to the first case.
\end{proof}
\begin{proof}[of Claim \ref{C2}]
Suppose first that $z_j \in C$ with $i \ne j$. Assume without loss of generality that $i < j$. Every $z_j z_i$-directed path in $D$ uses the vertices $z_1$ and $z_4$, thus $z_1, z_4 \in C$. For every $u \in V(G)$, $(z_4, y_u) \cup (y_u H_u x_u) \cup (x_u, z_1)$ is a $z_4 z_1$-geodesic in $D$. We conclude from Claim \ref{C1} that $V(H_u) \subseteq C$ for every $u \in V(G)$, and it follows that $C = V(G)$.
If $z_i \in C$ for $1 \le i \le 4$ and $w \in C$ for some $w \in V(D) \setminus \{z_1, z_2, z_3, z_4\}$, then every $wz_i$-directed path in $D$ uses the vertex $z_1$ and every $z_iw$-directed path in $D$ uses the vertex $z_4$. Thus, $z_1, z_4 \in C$ and we are done.
\end{proof}
\begin{proof}[of Claim \ref{C3}]
It follows from Claim \ref{C1} that $V(H_u) \cup V(H_v) \subseteq C$. Since $d_G(u,v) \ge 2$, it is easy to observe that $(x_u, z_1, z_2, z_3, z_4, y_v)$ is an $x_u y_v$-geodesic in $D$. Hence, $z_1 \in C$ and Claim \ref{C2} guarantees $C=V(D)$.
\end{proof}
\begin{proof}[Proof of Claim \ref{C4}]
It is direct to verify that $V(H_u)$ is a convex set of $D$ for every $u \in V(G)$. Let $|S| \ge 2$ and $u,v \in S$. If $w_1 \in V(H_u)$ and $w_2 \in V(H_v)$, then every $w_1 w_2$-directed path in $D$ uses the vertices $x_u$ and $y_v$; moreover, it must contain an $x_uy_v$-directed path. Our claim follows from noting that $(x_u, y_v) \in A(D)$.
\end{proof}
\end{proof}
It is easy to observe that the directed $6$-cycles in the previous construction can be replaced by directed $2n$-cycles, and the directed path $(z_1, \dots, z_4)$ can be replaced by a directed path of length $n$, in order to get the result for an oriented bipartite graph of arbitrary large girth.
\section{Grids} \label{SGrids}
We begin this section with two straightforward lemmas regarding convex sets in strong oriented graphs. The first lemma is a direct observation, so the proof will be omitted.
\begin{lemma} \label{girthcon}
Let $D$ be a strongly connected oriented graph. If $C \subseteq V(D)$ is a convex set such that $|C| \ge 2$, then $C$ induces a strong subdigraph of $D$. Therefore, $\min (S_{SC} (G) \setminus \{ 1 \}) \ge g(G)$, where $g(G)$ stands for the girth of $G$.
\end{lemma}
\begin{lemma} \label{comcon}
Let $D$ be an oriented graph. If $C \subseteq V(D)$ is a maximal convex set, then $D - C$ is a connected subdigraph of $D$.
\end{lemma}
\begin{proof}
Otherwise, let $D_1, \dots, D_n$ be the connected components of $D-C$. Since $C$ is a convex set of $D$, $C \cup \bigcup_{i=1}^{n-1} V(D_i)$ is a convex set of $D$ properly containing $C$, a contradiction.
\end{proof}
The following observation is simple, but also very useful while searching for adequate orientations to realize specific convex numbers. The proof is straightforward and thus will be omitted.
\begin{observation} \label{ai-ao}
Let $G$ be a triangle-free graph and let $D$ be an orientation of $G$ with a convex set $C$. Let $x \in V \setminus C$ be adjacent to a vertex $y \in C$.
\begin{itemize}
\item If $(x,y) \in A(D)$, then $N(x) \cap C \subseteq N^+ (x)$.
\item If $(y,x) \in A(D)$, then $N(x) \cap C \subseteq N^- (x)$.
\end{itemize}
\end{observation}
Let $n$ be an integer. We denote by $P_n$ the path on $n$ vertices, and we will assume without loss of generality that $P_n = (1, \dots, n)$. The $(n \times m)$-\emph{grid} is the cartesian product $P_n \Box P_m$. Hence, if $G = P_n \Box P_m$, then $V(G) = \{ (i,j) \colon\ 1 \le i \le n, 1 \le i \le m \}$. Although it is not standard, and it can be impractical in a different context, for the sake of simplicity we will denote the ordered pair $(i,j)$ as $i_j$. Also, we will use the canonical embedding of the grid $G = P_n \Box P_m$ in the plane to define orientations of $G$, and directed paths in an orientation $D$ of $G$. To achieve this goal, we will use directions to ``move'' on the grid, denoted as a sequence of movements using the symbols $u, d, l, r$, which stand for up, down, left and right. As an example, consider a directed cycle denoted in the usual way, \emph{i.e.}, $(i_j, i_{j+1}, (i+1)_{j+1}, (i+1)_j, i_j)$; with our notation we have the sequence $(u,r,d,l)$, starting at vertex $i_j$. In the following paragraph, there is another example of the orientations that can be defined in this way.
Let $n, m \ge 2$ be integers and $G$ be the canonical plane embedding of the grid $P_n \Box P_m$. Let $H^\ast$ be a connected subgraph of the interior dual of $G$ and let $H$ be the subgraph of $G$ induced by the faces in $V(H^\ast)$ (hence, the interior dual of $H$ is $H^\ast$). We define a \emph{whirlpool} to be an oriented graph obtained from $H$ by the following orientation of its edges.
\begin{figure}
\begin{center}
\includegraphics{Figure8.pdf}
\caption{A whirlpool orientation of the subgraph induced by the vertices in the gray region.} \label{whirlpool}
\end{center}
\end{figure}
$$\textnormal{Orient } \left\{ \begin{array}{l} i_j i_{j+1} \textnormal{ as:} \left\{ \begin{array}{lc} (i_j,i_{j+1}) & \textnormal{if } i \stackrel{2}{\equiv} j \\ \\ (i_{j+1},i_j) & \textnormal{otherwise.} \end{array}\right. \\ \\ i_j (i+1)_j \textnormal{ as:} \left\{ \begin{array}{lc} ((i+1)_j,i_j) & \textnormal{if } i \stackrel{2}{\equiv} j \\ \\ (i_j,(i+1)_j) & \textnormal{otherwise.} \end{array} \right. \end{array} \right.$$ An example of a whirlpool is depicted in Figure \ref{whirlpool}, where the graph $H^\ast$ has the gray faces of the grid as its vertex set. In the rest of the figures, the gray squares will always correspond to directed cycles. As the following proposition shows, whirlpools have a very important property related to convexity.
\begin{proposition} \label{lcon1grids}
If $D$ is a whirlpool, then $\textnormal{con} (D) = 1$.
\end{proposition}
\begin{proof}
Let $D$ be a whirlpool. Hence, there exist a pair of integers $n, m \ge 2$ and a subgraph $H$ of the grid $P_n \Box P_m$ such that $D$ is obtained from $H$ by the aforementioned orientation of its edges. We affirm that every $4$-cycle in $H$ is an oriented $4$-cycle in $D$. If $C$ is a $4$-cycle in $H$, then $C = (i_j, i_{j+1},(i+1)_{j+1},(i+1)_j,i_j)$ for some $1 \le i \le n-1, 1 \le j \le m-1$. It is not difficult to observe that if $i \stackrel{2}{\equiv} j$, then $C$ is an oriented cycle in $D$. Else, $C^{-1}$ (the cycle $C$ in reverse order) is an oriented cycle in $D$. Hence, $D$ is strongly connected.
Since every edge of $H$ belongs to a $4$-cycle, it follows that every arc of $D$ belongs to a directed $4$-cycle. Recalling that $D$ is triangle free, it is clear that $u \to v$ implies $d(v,u) = 3$. Hence, if $u \to v$, then the vertex set of every $4$-cycle containing the arc $(u,v)$ is contained in the convex hull of $\{u,v\}$. But $D$ is strongly connected, so using the fact that the interior dual of $H$ is connected, it can be shown inductively that for every pair $u,v$ of vertices of $D$, the convex hull of $\{ u, v \}$ is $V(D)$. Therefore, $\textnormal{con} (D) = 1$.
\end{proof}
\begin{corollary} \label{con=1}
For every grid $G$, $1 \in S_{SG}(G)$.
\end{corollary}
Observe that if an oriented grid $D$ contains a whirlpool $W$ as a subdigraph, then every convex set containing at least two adjacent vertices of $W$ must contain $V(W)$. Also, if $W$ is a whirlpool, then the digraph $\overleftarrow{W}$ obtained by the reversal of every arc of $W$ has the same properties as $W$; we will call such a digraph an \emph{anti-whirlpool}.
For a given integer $k$ and an oriented graph $D$, it is easier to prove $k \in S_{SC} (D)$ than proving $k \notin S_{SC} (D)$. The following result excludes some values from the strong convexity spectra of grids. Although simple, the complete proof of the lemma is to long to be included here. The proof of the first item of the lemma is complete, as well as the cases $i=3$ and $i=4$ of the second item. The proofs for the cases $i=5$ and $i=6$ can be obtained with similar arguments.
\begin{lemma} \label{forbid}
Let $n, m \ge 2$ be integers. If $G = P_n \Box P_m$, then:
\begin{itemize}
\item $2,3,5, |V|-1 \notin S_{SC} (G)$.
\item For every $i \in \{ 3, 4, 5, 6 \}$, if $n,m \ge i$, then $|V|-(i-1) \notin S_{SC} (G)$.
\end{itemize}
\end{lemma}
\begin{proof}
By Theorem \ref{no2} we have $2 \notin S_{SC} (G)$. Observing that every connected subdigraph of $G$ with $3$ or $5$ vertices has at least one vertex of degree $1$, and thus does not admit a strong orientation, it follows from Lemma \ref{girthcon} that $3,5 \notin S_{SC} (G)$. Any strong orientation of $G$ has neither sinks nor sources. Also, $g(G)=4$ and hence the orientations of $G$ cannot have transitive vertices. It follows that $|V|-1 \notin S_{SC} (G)$.
Let $n,m$ be integers such that $n,m \ge 3$ and suppose that a strong orientation $D$ of $G$ has a convex set $C$ of cardinality $|V|-2$. Let $S=\{ x, y \}$ be the set $V \setminus C$ and assume without loss of generality that $(x,y) \in A(D)$. Since $D$ is strong, $d^-(x) \ge 1$ and $d^+(y) \ge 1$. It follows from Observation \ref{ai-ao} that $N^-(x) \cap C = N^-(x)$ and $N^+(y) \cap C = N^+(y)$. Let us denote by $x^u, x^d, x^l, x^r$ the vertices above, below, to the left and to the right of $x$, respectively, in G. Since $n,m \ge 3$, we can assume without loss of generality that either $(x^u, x, y, y^r)$ or $(x^l, x, y, y^u)$ is a directed path in $D$, and hence a geodesic. But this contradicts that $C$ is a convex set. Hence, a convex set of cardinality $|V|-2$ cannot exist.
Let $n,m$ be integers such that $n,m \ge 4$ and suppose that a strong orientation $D$ of $G$ has a convex set $C$ of cardinality $|V|-3$. Let $S=\{ x, y, z \}$ be the set $V \setminus C$. Let us assume without loss of generality that $(x,y) \in A(D)$ and $x^r = y$. We have two cases.
First consider $y^r = z$. Again, we have two cases. Our first subcase is $(z,y) \in A(D)$. As in the previous argument, either $(x^l, x, y, y^u)$ or $(z^r, z, y, y^u)$ is a geodesic in $D$, contradicting that $C$ is a convex set. Our second subcase is $(y,z) \in A(D)$. We will assume without loss of generality that $(y^u, y) \in A(D)$. Hence, either $(x^l, x, y, z, z^u)$ or $(y^u, y, z, z^r)$ is a geodesic in $D$, contradicting that $C$ is a convex set.
As a second case, consider $y^d = z$, with two subcases. Our first subcase is $(z,y) \in A(D)$. Either $(x^d, x, y, y^r)$ or $(z^l, z, y, y^u)$ is a geodesic in $D$, a contradiction. The second subcase is $(y,z) \in A(D)$. Hence, at least one of the following directed paths is present in $D$, and it is a geodesic: $(x^d, x, y, y^u), (x^d, x, y, y^r), (y^u, y, z, z^l), (y^r, y, z, z^l)$. But every case results in a contradiction. Hence, a convex set of cardnality $|V|-3$ cannot exist.
\end{proof}
\section{Convex spectra of small grids} \label{SCSSG}
The following pair of results deal with the convexity spectra of $n \times 2$ grids.
\begin{lemma} \label{2*n-part1}
Let $n \ge 2$ be an integer and let $G$ be the grid $P_n \Box P_2$. If $j$ is an integer such that $j \ne 1$ and $\left\lfloor \frac{n}{2} \right\rfloor \le j \le n-1$, then $2j \in S_{SC} (G)$.
\end{lemma}
\begin{proof}
Let $D_1$ be the orientation of $G_1 = P_j \Box P_2$ as a whirlpool. We will consider two cases.
First, suppose that $j \ge \frac{n}{2}$. Let $D_2$ be the orientation of $G_2 = G-G_1$ as a whirlpool, if $j$ is odd, or as anti-whirlpool if $j$ is even. In either case, the orientation of the edges $j_1 j_2$ and $(j+1)_1 (j+1)_2$ result in parallel arcs, \emph{i.e.}, we have either the arcs $(j_1,j_2)$ and $((j+1)_1, (j+1)_2)$ or the arcs $(j_2,j_1)$ and $((j+1)_2,(j+1)_1)$. We will assume that $j$ is odd, the remaining case can be dealt similarly. If $D$ is the digraph obtained by orienting the two remaining edges as $(j_1,(j+1)_1)$ and $((j+1)_2,j_2)$, then it is clear that $\partial^+(V(D_1)) = \{ (j_1,(j+1)_1) \}$ and $\partial^-(V(D_1)) = \{ ((j+1)_2,j_2) \}$. But $d_D(j_1,j_2)=1$, and hence, $V(D_1)$ is a convex set of cardinality $2j$. If $C$ is a convex set of $P_n \Box P_2$ such that $|C| > 2j$, then $C \cap V(D_1) \ne \varnothing \ne C \cap V(D_2)$. From the previous observations about $\partial^+ (V(D_1))$ and $\partial^- (V(D_1))$, we conclude that $j_1, j_2, (j+1)_1, (j+1)_2 \in C$. Since $D_1$ and $D_2$ are whirlpools, we obtain $C=V(P_n \Box P_2)$. Therefore, $V(D_1)$ is a maximum convex set of $D$.
If $j = \left\lfloor \frac{n}{2} \right\rfloor$, then we will assume that $n$ is odd, otherwise we are back in the previous case. Let $D_2$ be the orientation of $G_2 = G- G[V(D) \setminus V(G_1) \cup \{ (j+1)_1, (j+1)_2 \}]$ as a whirlpool, if $j$ is odd, or as anti-whirlpool if $j$ is even. Again, we will assume that $j$ is odd. Orient the remaining edges in the following way: $((j+1)_1, (j+1)_2), ((j+1)_2, j_2), ((j+1)_2, (j+2)_2), (j_1, (j+1)_1)$ and $ ((j+2)_1, (j+1)_1)$. An argument similar to the one used in the previous case shows that $V(G_1)$ and $V(G_2)$ are convex sets of $D$, and clearly $|V(G_1)| = |V(G_2)|$. If $C$ is a convex set of $P_n \Box P_2$ such that $|C| > 2j$, then $C \cap \{ (j+1)_1, (j+1)_2 \} \ne \varnothing$. Since $N^+((j+1)_1) = \{ (j+1)_2 \}$ and $N^-((j+1)_2) = \{ (j+1)_1 \}$, then $\{ (j+1)_1, (j+1)_2 \} \subseteq C$. But $((j+1)_2, j_2, (j-1)_2, (j-1)_1, j_1, (j+1)_1)$ and $((j+1)_2, (j+2)_2, (j+3)_2, (j+3)_1, (j+2)_1, (j+1)_1)$ are $(j+1)_2 (j+1)_1$-geodesics in $D$. Since $G_1$ and $G_2$ are whirlpools in $D$, we obtain $C = V(D)$. Hence, $V(G_1)$ is a maximum convex set of $D$.
\end{proof}
\begin{theorem}
If $n \ge 2$ is an integer and $G$ is the grid $P_n \Box P_2$, then $$S_{SC} (G) = \{ 1 \} \cup \{ 2j \colon\ \left\lfloor \tfrac{n}{2} \right\rfloor \le j \le n-1\} \setminus \{ 2 \}.$$
\end{theorem}
\begin{proof}
Let $D$ be a strong orientation of $G$ and $C$ a maximum convex set of $D$. It follows from Lemma \ref{comcon} and the fact that $D[C]$ is strong that, for some $2 \le j \le n-1$, either $V(C) = V(P_j \Box P_2)$, or $V(C) = V(D) \setminus V(P_j \Box P_2)$. Hence, there are not odd integers greater than $1$ in $S_{SC} (G)$.
Assume without loss of generality that $C = V(P_j \Box P_2)$ for some $2 \le j \le n-1$. Also assume without loss of generality that $(j_1, j_2) \in A(D)$. We will consider two cases.
Consider for the first case $((j+1)_2, (j+1)_1) \in A(D)$. Since $D$ is strong, either $((j+1)_1, j_1), (j_2, (j+1)_2) \in A(D)$ or $(j_1, (j+1)_1), ((j+1)_2, j_2) \in A(D)$. In the former case $(j_2, (j+1)_2, (j+1)_1, j_1)$ is a $j_2 j_1$-geodesic in $D$, a contradiction. In the latter case, it is direct to verify that $V(D) \setminus C$ is also a convex set; since $C$ is maximum, we get $j \ge \frac{n}{2}$.
For the second case consider $((j+1)_1, (j+1)_2) \in A(D)$. Again, since $D$ is strong, either $((j+1)_1, j_1), (j_2, (j+1)_2) \in A(D)$ or $(j_1, (j+1)_1), ((j+1)_2, j_2) \in A(D)$. In the former case it is easy to check that $V(D) \setminus C$ is a convex set of $D$, hence, $j \ge \frac{n}{2}$. In the latter case, if $((j+2)_2, (j+1)_2), ((j+1)_1, (j+2)_1) \in A(D)$, then $V(P_{j+1} \Box P_2)$ is a convex set of $D$, a contradiction. Hence $((j+1)_2, (j+2)_2), ((j+2)_1, (j+1)_1) \in A(D)$ and we have two further cases.
If $((j+2)_2, (j+2)_1) \in A(D)$, then $V(D) \setminus C$ is a convex set of $D$, thus $j \ge \frac{n}{2}$.
If $((j+2)_1, (j+2)_2) \in A(D)$, then $V(D) \setminus (C \cup \{ (j+1)_1, (j+1)_2 \})$ is a convex set of $D$, and hence $j \ge \left\lfloor \frac{n}{2} \right\rfloor$.
\end{proof}
Although a bit more complex, similar arguments can be used for the $n \times 3$ grids.
\begin{lemma} \label{3*n-c=3k}
Let $n \ge 2$ be an integer and let $G$ be the grid $P_n \Box P_3$. If $j$ is an integer such that $2 \le j \le n-1$, then $3j \in S_{SC} (G)$.
\end{lemma}
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{Figure10.pdf}
\caption{The orientations used in the proof of Lemma \ref{3*n-c=3k} with $n=7$ and $j \in \{ 4, 6 \}$.} \label{3*n-c=3kFig}
\end{center}
\end{figure}
\begin{proof}
We consider two cases. If $j=n-1$, then let $D_1$ be the orientation of $G_1 = P_{n-1} \Box P_3$ as a whirlpool. Suppose that $n$ is even, the remaining case is analogous. Orient the remaining edges as $((n-1)_3, n_3), ((n-1)_1, n_1), (n_2, (n-1)_2), (n_1, n_2), (n_3, n_2)$. It is straightforward to verify that this orientation $D$ of $G$ is strong and $V(D_1)$ is a maximum convex set of $D$.
For $2 \le j \le n-2$, let $D_1$ be the orientation of $G_1 = P_{j-2} \Box P_3$ as a whirlpool (note that $D_1$ is empty for $j=2$). Let $D_2$ be the orientation of $G_2 = G - (P_j \Box P_3)$ as a whirlpool. Orient $(j_3, (j-1)_3, (j-1)_2, (j-1)_1, j_1, j_2, j_3)$ as a directed cycle. Also, orient $\partial (G_1)$ as $((j-2)_3, (j-1)_3), ((j-1)_2, (j-2)_2), ((j-2)_1, (j-1)_1)$ if $j$ is even, and reverse each of these arcs if $j$ is odd. Finally, orient the remaining arcs as $(j_2, (j-1)_2), (j_3, (j+1)_3), (j_2, (j+1)_2), ((j+1)_1, j_1)$. Let $D$ be the resulting orientation of $G$, and let $D_3$ be the induced subdigraph $D[ V(P_j \Box P_3)]$ of $D$.
We will assume that $j$ is even, the remaining case can be dealt similarly. Observe that $\partial^+(D_3) = \partial^-(D_2) = \{ (j_3, (j+1)_3), (j_2, (j+1)_2) \}$, and $\partial^-(D_3) = \partial^+(D_2) = \{ ((j+1)_1, j_1) \}$. Since $d((j+1)_2, (j+1)_1)=3$, $d((j+1)_3, (j+1)_1)=4$, $d(j_3, j_1)=4$, and $d(j_2, j_1)=3$, it is clear that $V(D_3)$ is a convex set of $D$ with $|V(D_3)|=3j$.
Let $C$ be a convex set of $D$ such that $|C| > 3j$. Since every convex set in a strong digraph induces a strong subdigraph, we observe that $|C \cap V(D_2)| \ge 2$. But $D_2$ is a whirlpool, and hence $V(D_2) \subseteq C$. Note that $d((j+1)_1, (j+1)_3) = 4$ and also that $((j+1)_1, j_1, j_2, j_3, (j+1)_3)$ is a directed path in $D$. Thus, $j_1, j_2, j_3 \in C$. Recall that $d(j_3, j_1)=4$ and consider the directed path $(j_3, (j-1)_3, (j-1)_2, (j-1)_1, j_1)$ to conclude $(j-1)_1, (j-1)_2, (j-1)_3 \in C$. Since $j$ is even, $((j-1)_3, (j-1)_2, (j-2)_2, (j-2)_3, (j-1)_3)$ is a directed cycle in $D$. This implies $|C \cap V(D_1)| \ge 2$, but $D_1$ is a whirlpool, and hence $V(D_1) \subseteq C$ and $C=V(D)$. Therefore $\textnormal{con} (D) = 3j$.
\end{proof}
\begin{lemma} \label{3*n-c=3k+2}
Let $n \ge 2$ be an integer and let $G$ be the grid $P_n \Box P_3$. If $j$ is an integer such that $2 \le j \le n-2$, then $3j+2 \in S_{SC} (G)$.
\end{lemma}
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{Figure11.pdf}
\caption{The orientations used in the proof of Lemma \ref{3*n-c=3k+2} with $n=7$ and $j \in \{2,3,4,5\}$.} \label{3*n-c=3k+2Fig}
\end{center}
\end{figure}
\begin{proof}
Let $D_1$ be the orientation of $G_1 = (P_{j+1} \Box P_3) - (j+1)_3$ as a whirlpool. We will consider two cases.
For the first case consider $\left\lfloor \frac{n}{2} \right\rfloor \le j \le n-3$, and let $D_2$ be the orientation of $G_2 = G - (P_{j+1} \Box P_3)$ as a whirlpool, if $j$ is even, or as an anti-whirlpool, if $j$ is odd.
If $j$ is even, orient the remaining edges as $((j+1)_3, j_3), ((j+1)_3, (j+1)_2), ((j+2)_3, (j+1)_3), ((j+2)_2, (j+1)_2), ((j+1)_1, (j+2)_1)$ to obtain the orientation $D$ of $G$. It is direct to verify that $D$ is strong. Observing that $d((j+1)_1, (j+1)_2) = 1$ and $d((j+1)_1, j_3) = 3$ it is easy to verify that $V(D_1)$ is a convex set of $D$. If $C$ is a convex set of $D$ such that $|C| > 3j+2$, then $|C \cap (V(D) \setminus V(D_1))| \ne \varnothing$. But $\partial^+(D_1) = \{ ((j+1)_1, (j+2)_1) \}$, and hence $|C \cap V(D_2)| \ge 2$. Recalling that $D_2$ is a whirlpool and observing that $((j+2)_3, (j+1)_3, j_3)$ is a $(j+2)_3 j_3$-geodesic in $D$, we conclude that $C = V(D)$. Hence $\textnormal{con} (D) = 3j+2$.
If $j$ is odd, orient the remaining edges as $(j_3, (j+1)_3), ((j+1)_2, (j+1)_3), ((j+1)_3, (j+2)_3), ((j+1)_2, (j+2)_2), ((j+2)_1, (j+1)_1)$ to obtain the orientation $D$ of $G$. This orientation is, locally, the dual orientation of the case when $j$ is even, so analogous arguments show that $V(D_1)$ is a convex set and $\textnormal{con} (D) = 3j+2$.
As a second case, assume that $2 \le j < \left\lfloor \frac{n}{2} \right\rfloor $ or $j = n-2$. When $j$ is odd, orient $(j_3, (j+1)_3, (j+2)_3, (j+2)_2, (j+2)_1, (j+1)_1)$ as a directed path, and orient the arcs $((j+1)_2, (j+1)_3), ((j+1)_2, (j+2)_2)$. If $j \ne n-2$, let $D_2$ be the orientation of $G_2 = G \setminus (P_{j+2} \Box P_3)$ as an anti-whirlpool and orient the remaining edges of $G$ as $((j+2)_1, (j+3)_1), ((j+2)_2, (j+3)_2), ((j+3)_3, (j+2)_3)$ to obtain $D$. Clearly $D$ is strong. Also, it is direct to verify that $V(D_1)$ is a convex set of $D$ with $3j+2$ vertices.
Let $C$ be a convex set of $D$ such that $|C| > 3j+2$. If $|C \cap V(D_2)| \ge 2$, then $V(D_2) \subseteq C$. But $((j+3)_3, (j+2)_3, (j+2)_2, (j+2)_1, (j+3)_1)$ is a $(j+3)_3 (j+3)_1$-geodesic in $D$, and thus, $(j+2)_i \in C$ for $1 \le i \le 3$. Also, $((j+2)_1, (j+1)_1, j_1, j_2, (j+1)_2, (j+1)_3, (j+2)_3)$ is a $(j+2)_1 (j+2)_3$-geodesic in $D$. This implies $C=V(D)$, because $|C \cap V(D_1)| \ge 2$.
Otherwise, and because $j \ge 2$, $V(D_1) \subseteq C$ and $v \in C$ for some $v \in \{ (j+1)_3, (j+2)_1, (j+2)_2, (j+2)_3 \}$. In any case, $(j_3, (j+1)_3, (j+2)_3, (j+2)_2, (j+2)_1, (j+1)_1)$ is the union of a $j_3 v$-geodesic and a $v (j+1)_1$-geodesic in $D$ (and the case $j=n-2$ is finished) . Since $((j+2)_2, (j+3)_2, (j+3)_3, (j+2)_3)$ is a $(j+2)_2 (j+2)_3$-geodesic in $D$, we have $|C \cap V(D_2)| \ge 2$ and $C=V(D)$.
When $j$ is even, as in the previous case, we can orient the remaining edges of $G$ to obtain, locally, an orientation that is dual to the orientation when $j$ is odd. Hence, analogous arguments can be followed to prove that $V(D_1)$ is a maximum convex set of $D$.
Therefore, $\textnormal{con}(D)=3j+2$.
\end{proof}
\begin{lemma} \label{3*n-c=3k+1}
Let $n \ge 2$ be an integer and let $G$ be the grid $P_n \Box P_3$. If $j$ is an integer such that $3 \le j \le n-2$, then $3j+1 \in S_{SC} (G)$.
\end{lemma}
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{Figure12.pdf}
\caption{The orientation used in the proof of Lemma \ref{3*n-c=3k+1} for $n=7$ and $j \in \{3, 4, 5 \}$.} \label{3*n-c=3k+1Fig}
\end{center}
\end{figure}
\begin{proof}
We will assume that $j$ is odd, the remaining case can be dealt similarly. Orient $G'_1 = P_{j-1} \Box P_3$ as a whirlpool to obtain $D'_1$. We will consider two cases.
For the first case, suppose that $j = n-2$. Let $D_1$ be the digraph obtained from $G_1 = (P_{j+1} \Box P_3) - \{ j_3, (j+1)_3 \}$ by orienting $G'_1$ as $D'_1$, $((j-1)_1, j_1, (j+1)_1, (j+1)_2, j_2, (j-1)_2)$ as a directed path, and the remaining edge of $G[G_1]$ as $(j_2, j_1)$. Also, orient $((j+1)_1, (j+2)_1, (j+2)_2, (j+2)_3, (j+1)_3, j_3, (j-1)_3)$, and $((j+1)_3, (j+1)_2, (j+2)_2)$ as directed paths. Orient the remaining edge as $(j_3, j_2)$ to obtain the digraph $D$. It is immediate to verify that $D$ is a strong digraph.
Note that $\partial^+ (D_1) = \{ ((j+1)_1, (j+2)_1), ((j+1)_2, (j+2)_2) \}$ and $\partial^- (D_1) = \{ (j_3, (j-1)_3), (j_3, j_2), ((j+1)_3, (j+1)_2) \}$. Observing that $d((j+1)_2, j_2) = d((j+1)_1, (j+1)_2) = 1$, $d((j+1)_1, j_2)=2$, $d((j+1)_1, (j-1)_3)=4$, and $d((j+1)_2, (j-1)_3)=3$, it is easy to conclude that $V(D_1)$ is a convex set of $D$ with $3j+2$ vertices.
Let $C$ be a convex set of $D$ such that $|C| > 3j+1$. Since $j=n-2$, then $|C \cap V(D'_1)| \ge 2$ and hence $V(D'_1) \subseteq C$. Also, there is at least one vertex $v \in C \cap (V(D) \setminus V(D_1))$. Regardless of the choice of $v$, the directed path starting at $(j-1)_1$ and defined by the sequence $(r,r,r,u,u,l,l,l)$ results from the union of a $(j-1)_1 v$-geodesic and a $v (j-1)_3$-geodesic. Hence, $V(D) \subseteq C$ and therefore $\textnormal{con} (D) = 3j+1$
As a second case, assume that $j \le n-3$. Let $D_1$ and $D_2$ be the digraphs obtained by orienting both $G_1 = (P_{j+1} \Box P_3) - \{ j_3, (j+1)_3 \}$ and $G_2 = G - (V(G_1) \cup \{ j_3, (j+1)_3, (j+2)_1 \})$ as whirlpools. Orient $((j+1)_3, (j+1)_2, (j+2)_2, (j+2)_1, (j+1)_1)$ and $((j+2)_3, (j+1)_3, j_3,(j-1)_3)$ as directed paths. If $j = n-3$, orient $((j+3)_1, (j+2)_1)$, and orient the same edge as $(j+2)_1, (j+3)_1$ otherwise. Finally, orient the remaining edges as $(j_3, j_2)$ to obtain the digraph $D$. It is direct to verify that $D$ is strong.
Observe that $\partial^+ (D_1) = \{ ((j+1)_2, (j+2)_2) \}$ and $\partial^- (D_1) = \{ (j_3, j_2), (j_3, (j-1)_3), ((j+1)_3, (j+1)_2), ((j+2)_1, (j+1)_1) \}$. Noting that $d((j+1)_2, (j+1)_1)=1$, $d((j+1)_2, j_2)=3$ and $d((j+1)_2, (j-1)_3) = 5$, it is not hard to verify that $V(D_1)$ is a convex set of $D$.
Let $C$ be a convex set of $D$ such that $|C| > 3j+1$. If $j=n-3$, then $|V(D_1) \cap C| \ge 2$, and $V(D_1) \subseteq C$. If $j < n-3$ and $|C \cap V(D_2)| \ge 2$, then $V(D_2) \subseteq C$. But $((j+2)_2, (j+2)_1, (j+3)_1)$ is a $(j+2)_2 (j+3)_1$-geodesic in $D$, which implies $(j+2)_1 \in C$. The directed path with initial vertex $(j+2)_1$ and defined by the sequence $(l,l,u,r,r)$ is a $(j+2)_1 (j+2)_2$-geodesic in $D$. From here we observe that $|V(D_1) \cap C| \ge 2$ and thus $V(D_1) \subseteq C$. If $j < n-3$ and $|C \cap V(D_2)| \le 1$, then $|V(D_1) \cap C| \ge 2$, and $V(D_1) \subseteq C$. Hence, in every case $V(D_1) \subseteq C$. Since there is at least one vertex from $V(D) \setminus V(D_1)$ in $C$, necessarily $(j+2)_2 \in C$. The directed path starting at $(j+2)_2$ and defined by the sequence $(r,u,l,l,l,l)$ is a $(j+2)_2 (j-1)_3$-geodesic, and hence $V(D_2) \subseteq C$. But if $V(D_2) \cup V(D_1) \subseteq C$, it is easy to verify that $C = V(D)$. Hence, $\textnormal{con} (D) = 3j+1$.
\end{proof}
So far, we have every integer of the convexity spectrum of $P_n \Box P_3$, except for $4$. Our next theorem deals with the remaining case.
\begin{lemma} \label{n*3-c=4}
If $n \ge 3$ is an integer and $G$ is the grid $P_n \Box P_3$, then $4 \in S_{SC} (G)$
\end{lemma}
\begin{figure}
\begin{center}
\includegraphics{Figure13.pdf}
\caption{The orientation used in the proof of Lemma \ref{n*3-c=4} for $n=8$.} \label{n*3-c=4Fig}
\end{center}
\end{figure}
\begin{proof}
Consider the standard plane embedding of $G$ and color the interior faces gray and white with a checkerboard-like pattern, coloring the square on the bottom left corner with gray. We will define an orientation of the arcs of $G$ using this coloring, an example can be seen on Figure \ref{n*3-c=4Fig}.
There are two rows of squares. Enumerate the gray squares in each row from left to right. Orient the bottom left corner square as a whirlpool and, from here, orient all the gray squares in its row alternating whirlpool and anti-whirlpool orientations. Orient all the gray squares in the upper row following the same principle, but start orienting as an anti-whirlpool the first gray square. At this point, every arc dividing two interior faces of $G$ has received an orientation. Every remaining unoriented edge $e$ of $G$ divides a white square from the exterior face of $G$. Thus, the edge $e$ lies in exactly one square of $G$, and has one parallel arc $a$ in the square. It $e$ is not an edge of a corner square, orient it in the same direction as $a$. There are four edges belonging to the white corner squares that remain unoriented. Orient the remaining edges as $2$-paths in such way that there are not white oriented squares. Let $D$ be the digraph obtained by this orientation. Clearly, $D$ is strong and the vertices of each gray square conform a convex set.
Let $C$ be a convex set of $D$ such that $|C| > 4$. There must be two gray squares $S_1$ and $S_2$ such that $v \in V(S_1) \cap V(S_2)$ and $V(S_1) \cup V(S_2) \subseteq C$. Since $v$ is an interior vertex, it belongs to two white squares. Let $u$ be a vertex in the opposite corner in one of these white squares $S_3$. Assume without loss of generality that $v$ is the lower left corner of $S_1$, the upper right corner of $S_2$, and the lower right corner of $S_3$; the remaining cases can be dealt similarly.
If $u$ is the middle vertex of a $S_1 S_2$-path or a $S_2 S_1$-path of length two, then $u \in C$, and hence $V(S_3) \subseteq C$.
Otherwise, let $x$ and $y$ be the upper right and lower left corners of $S_3$, respectively. Hence, $d(x,y) = d(y,x)=4$ and either the sequence $(l,l,d,r)$ starting from $x$ determines an $xy$-directed path of length $4$, or the sequence $(l,u,r,r)$ determines a $yx$-directed path of length $4$. In either case, $V(S_3) \subseteq C$.
It can be verified inductively that $C = V(D)$, and hence, $\textnormal{con} (D) = 4$.
\end{proof}
\begin{theorem}
If $n \ge 3$ is an integer and $G$ is the grid $P_n \Box P_3$, then $$S_{SC} (G) = [1,3n-3] \setminus \{ 2, 3, 5, 7\}.$$
\end{theorem}
\begin{proof}
By virtue of Theorem \ref{no2} and Lemmas \ref{forbid}, \ref{3*n-c=3k}, \ref{3*n-c=3k+2}, \ref{3*n-c=3k+1} and \ref{n*3-c=4}, it remains to prove that $7 \notin S_{SC} (G)$. Let $D$ be a strong orientation of $G$ and $C$ a convex set of $D$. Lemmas \ref{girthcon} and \ref{comcon} imply that $C$ induces a strong subdigraph of $D$ and that $V(D) \setminus C$ induces a connected subdigraph of $D$, respectively. But every connected subgraph of $G$ with $7$ vertices has either a vertex of degree $1$, and thus does not admit a strong orientation; or does not have a connected complement. Hence, $7 \notin S_{SC} (G)$.
\end{proof}
\section{Convex spectra of general grids} \label{SMain}
The following lemma is the cornerstone of the vast majority of the arguments we will use in this section.
\begin{figure}
\begin{center}
\includegraphics{Figure1.pdf}
\caption{The digraph $H$: an orientation of $P_4 \Box P_4$ with convexity number $4$.} \label{4*4-c=4}
\end{center}
\end{figure}
\begin{lemma}
The oriented graph $H$ in Figure \ref{4*4-c=4} has convexity number $4$.
\end{lemma}
\begin{proof}
It is easy to check that the vertices on the boundary of each of the gray filled squares conform a convex set. We affirm that any convex set in $H$ has at most $4$ vertices. Let $C$ be a maximum convex set in $H$ and suppose that $|C| > 4$. Observe that the intersection of $C$ with the vertices of each gray filled square is either empty, or it has one vertex, or it has four vertices. Hence, since $|C| > 4$, the vertices of at least two squares are contained in $C$. If the vertices of two gray filled squares different from the central one are contained in $C$, then there are at least two vertices of the central square in $C$; thus, the vertices of the central square are contained in $C$. So, by symmetry, we need only to consider two cases.
The first case is when the square on the lower left corner and the central square are contained in $C$. Assume that the vertex in the lower left corner of $H$ is $(1,1)$. It is easy to check that $((2,3), (2,4), (1,4), (1,3), (1,2))$ and $((2,1), (3,1), (4,1), (4,2), (3,2))$ are $(2,3)(1,2)$- and $(2,1)(3,2)$-geodesics, respectively. From here, $(4,2), (3,3) \in C$, and $((4,2), (4,3), (3,3))$ is a $(4,2)(3,3)$-geodesic in $H$. Therefore, there are at least two vertices of each gray filled square in $C$ and we can conclude that $C = V(H)$, a contradiction.
In the second case, we have the upper left corner and the central square contained in $C$. Now, $((1,3), (1,2), (2,2))$ is a $(1,3)(2,2)$-geodesic in $H$. Hence, there are at least two vertices of the lower left corner square in $C$. So, the lower left corner square is contained in $C$ and we have the condition of the first case.
Since contradictions are obtained in both cases, we conclude that $|C| \le 4$. Hence, $\textnormal{con}(H) = 4$.
\end{proof}
As the reader would expect, the main part of the argument in the next lemma's proof is the construction of the orientation. The following lemmas will use similar orientations, so the descriptions will be very detailed in the first ones, and will loose detail as the lemmas progress.
\begin{lemma} \label{n*m-c=4}
Let $n,m \ge 4$ be integers. If $G= P_n \Box P_m$ is a grid, then $4 \in S_{SG} (G)$.
\end{lemma}
\begin{proof}
Consider the standard plane embedding of $G$ and color the interior faces gray and white with a checkerboard-like pattern, assigning gray to the square on the bottom left corner (like in Figure \ref{4*4-c=4}). We will define an orientation of the arcs of $G$ using this coloring.
Enumerate the rows of squares from bottom to top. Enumerate the gray squares in each row from left to right. Orient the bottom left corner square as a whirlpool and, from here, orient all the gray squares in the first column and first row alternating whirlpool and anti-whirlpool orientations. Now, the first square or every odd row is oriented, so we can orient all the gray squares in the odd rows alternating whirlpool and anti-whirlpool orientations. A similar idea can be used to orient all the gray squares in even rows, but start orienting as an anti-whirlpool the first gray square on the second row. At this point, every arc dividing two interior faces of $G$ has received an orientation. We will consider two cases.
First, suppose that $n$ and $m$ are even integers, hence every corner square of $G$ is gray, and every remaining unoriented edge $e$ of $G$ divides a white square from the exterior face of $G$. Thus, the edge $e$ lies in exactly one square of $G$, and has one parallel arc $a$ in the square. Orient $e$ in the same direction as $a$. All the edges of $G$ are now oriented; let $D$ be the resulting oriented graph. Figure \ref{4*4-c=4} is an example of this orientation. It is easy to verify that $D$ is strongly connected, and the vertices of every gray square conform a convex set of $D$. If $C$ is a convex set of $D$ such that $|C| > 4$, then $C$ intersects the vertices on at least two different gray squares $S_1$ and $S_2$ in odd columns and rows. Since $C$ induces a strong subdigraph of $D$, we may assume without loss of generality that $S_1$ and $S_2$ are gray squares in the same row and adjacent odd columns. If $S_1$ and $S_2$ are in row $i$ and columns $j$ and $j+2$, then the vertices of $S_1$ and $S_2$, together with the vertices of the squares $S_3$ and $S_4$ in row $i+2$ and columns $j$ and $j+2$, induce a subdigraph of $D$ isomorphic to the digraph $H$ of Figure \ref{4*4-c=4}. Therefore $\bigcup_{i=1}^4 V(S_i) \subseteq C$. We can repeat this argument using squares $S_2$ and $S_4$ and the squares in column $j+4$ and rows $i$ and $i+2$. Iterating this process we obtain $C = V(D)$. Hence, $\textnormal{con}(D)=4$.
For the second case, assume that $n$ or $m$ is an odd integer. By virtue of Lemma \ref{n*3-c=4}, we assume that $n, m \ge 4$. Observe that there are exactly two white corner squares in $G$. Except for the edges in the white corner squares, orient the remaining edges of $G$ as in the previous case. For each of the white corner squares we have the two cases depicted in Figure \ref{n*m-c=4Fig} (the squares we are interested in are the bottom right corners), and two isomorphic cases obtained by reversing all the arcs of the previous ones. We will consider two cases.
\begin{figure}
\begin{center}
\includegraphics{Figure2.pdf}
\caption{The two non-isomorphic cases for the white corners in the proof of Lemma \ref{n*m-c=4}.} \label{n*m-c=4Fig}
\end{center}
\end{figure}
If $n \ne 4 \ne m$, complete the orientation $D$ of $G$ as in Figure \ref{n*m-c=4Fig}. Again, it is direct to verify that $D$ is strong and the vertices of every gray square conform a convex set of $D$. Let $C$ be a convex set of $D$ such that $|C| > 4$. As in the previous case, it suffices to show that there are two consecutive gray squares in the same row or the same column that intersect $C$. Since $|C| > 4$, there $C$ intersects at least two different gray squares. Hence, the desired condition is held, unless the gray squares are precisely those adjacent to one of the white corners. We will assume without loss of generality that one white corner square is the one in the bottom right, as in Figure \ref{n*m-c=4Fig}. Let $S_1$ and $S_2$ be gray squares to the left and above the white corner, respectively. Let $v_i$ be the vertex in the upper left corner of $S_i$, $i \in \{ 1, 2 \}$.
Consider first the situation depicted by the digraph on the left in Figure \ref{n*m-c=4Fig}. It is clear that $d(v_1, v_2) = 4$, and also that, starting from $v_1$, the sequence $(u,u,r,d)$ determines a $v_1 v_2$-directed path of length $4$. Hence, $C$ intersects two consecutive gray squares in the same row.
For the situation depicted by the digraph on the right in Figure \ref{n*m-c=4Fig}, it is clear that $d(v_1, v_2)=6$. It is also clear that, starting from $v_1$, the sequence $(l,u,r,u,r,d)$ determines a $v_1 v_2$-directed path of length $6$. Since $C$ is convex, it intersects two consecutive gray squares in the same row.
As a final case, assume without loss of generality that $m=4$. Since $n$ is an odd integer, the two white corners are those on the right side of $G$. In the situation depicted by the digraph on the left in Figure \ref{n*m-c=4Fig}, use precisely that orientation and the same argument as in the previous case. In the remaining case, use the orientation of Figure \ref{n*m-c=4Fig} for the bottom right corner, and orient the upper right corner also as a directed path of length $2$ (assume that it goes up and left). This orientation is strong, the vertices of each gray square conform a convex set, and the same argument as the previous case shows that we can find two consecutive gray squares in the same column that intersect $C$, and hence $C = V(D)$.
\end{proof}
Our first modification to the previous orientation will be getting gray rectangles instead of $2 \times 2$ squares.
\begin{lemma} \label{n*m-c=ab}
Let $n,m \ge 4$ be integers and let $G= P_n \Box P_m$ be a grid. If $a,b \ge 2$ is a pair of integers such that $a \le n-1$ and $b \le m-1$, then $ab \in S_{SG} (G)$.
\end{lemma}
\begin{proof}
The idea of this proof is to generalize the orientation used in Lemma \ref{n*m-c=4}, but using grids of size $ab$ instead of squares in the odd-numbered rows and columns. An example of this orientation is depicted in Figure \ref{n*m-c=abFig}.
First, suppose that $a < n-1$ and $b < m-1$. Enumerate the rows and columns of squares of $G$ from down to up and from left to right, respectively. Color with gray and white the squares of $G$ in a checkerboard-like pattern, but considering rectangles of squares instead of single squares, in the following way. Color the squares in the first $a-1$ columns and $b-1$ rows, and the square in the $a$-th column and $b$-th row wih gray. Color the squares in the $a$-th column and the first $b-1$ rows, and the squares in the $b$-th column and the first $a-1$ rows in white. Color the rest of $G$ with a tiling of this coloring.
Enumerate the rows and columns of gray rectangles in the pattern from left to right and from down to up. Clearly, the gray rectangles in the even numbered rows (columns) are squares. We can assume that $ab > 4$, hence, the gray rectangles in the odd numbered rows (columns) are proper rectangles.
Orient the gray squares in the even rows as in the proof of Lemma \ref{n*m-c=4} (in particular, the first gray square of the second row is an anti-whirlpool). Orient the gray rectangles in the odd rows as whirlpools or anti-whirpools in such way that every gray square (possibly except the last gray square in every row or every column) in a even row, together with the four gray squares sharing a vertex with it, induce a digraph isomorphic to $H$ (Figure \ref{4*4-c=4}). Of course, to achieve this end, we also need to orient four additional arcs dividing either two white squares of $G$, or a white square and the exterior face of $G$. Now, every unoriented edge of $G$ divides two white squares, or a white square from the exterior face in an even row or column of gray squares. Orient every edge dividing two white squares in the same direction as the closest arc in a gray square in the same row or column. There are unoriented edges parallel to the arcs oriented in the previous step; orient those edges in the same direction as the arcs they are parallel to. The remaining unoriented edges form paths on the exterior face of $G$ joining pairs of gray rectangles. Orient those paths as directed paths in such way that, if any, the corner vertices of $G$ in a white square have in-degree and out-degree equal to one. Figure \ref{n*m-c=abFig} shows an example of this orientation.
\begin{figure}
\begin{center}
\includegraphics{Figure3.pdf}
\caption{The orientation described in the proof of Lemma \ref{n*m-c=ab} for $n=9$, $m=8$, $a=4$ and $b=3$.} \label{n*m-c=abFig}
\end{center}
\end{figure}
If $D$ is the digraph obtained from $G$ by means of the previously described orientation, then, by mimicking the arguments in the proof of Lemma \ref{n*m-c=4} we reach the desired conclusion.
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{Figure14.pdf}
\caption{The orientation used in the proof of Lemma \ref{n*m-c=ab} for $n=8$, $m=6$, $b=5$ and $a \in \{ 5,6 \}$. } \label{n*m-c=abFig2}
\end{center}
\end{figure}
If $a=n-1$ or $b=m-1$, we have to be careful with the white corners, but the simple modification shown in Figure \ref{n*m-c=abFig2} suffices to use the same argument as in the previous case.
\end{proof}
In the next lemma we use a simpler orientation, which is depicted in Figure \ref{n*m-c=nbFig}. This orientation resembles the one used in Lemma \ref{2*n-part1}.
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{Figure9.pdf}
\caption{The orientations used in the proof of Lemma \ref{n*m-c=nb} for $b=m-1$ (left) and $b<m-1$ (right).} \label{n*m-c=nbFig}
\end{center}
\end{figure}
\begin{lemma} \label{n*m-c=nb}
Let $n,m \ge 4$ be integers and let $G= P_n \Box P_m$ be a grid. If $b$ is an integer such that $\frac{m}{2} \le b \le m-1$, then $nb \in S_{SG} (G)$.
\end{lemma}
\begin{proof}
First, suppose that $b = m-1$. Orient $P_n \Box P_b$ as a whirlpool. Now, re-orient the corresponding arcs in the fiber $P_n^{m-1}$ to obtain a directed path in the same direction (left or right) as the arc between $1_{m-1}$ and $2_{m-1}$. Also, orient the fiber $P_n^m$ as a directed path in the same direction as the arc between $1_{m-1}$ and $2_{m-1}$. Assume without loss of generality that $P_n^m$ is oriented right. Finally orient down every edge in $\partial (P_n^m)$ except for $(1_{m-1}, 1_m)$, which is oriented up. If $D$ is the resulting oriented graph, then it is easy to verify that $V(P_n \Box P_b)$ is a convex set of $D$, and $\textnormal{con} (D) = nb$.
If $b < m-1$, then orient $G_1 = P_n \Box P_b$ as a whirlpool and $G_2 = G - G_1$ as a whirlpool or as an anti-whirlpool in such way that the fibers $P_n^b$ and $P_n^{b+1}$ are isomorphic. Orient the remaining edges arbitrarily, as long as there is one arc going up and one arc going down, to obtain $D$. We will show that $V(G_1)$ is a convex set of $D$.
Let $P$ be a $i_b j_b$-path in $D$ such that every intermediate vertex of $P$ belongs to $V(G_2)$. Then $P$ can be codified as $(u,x_2, \dots, x_{k-1}, d)$. Let $(y_2, \dots, y_{k-1})$ be the sequence such that $$y_i = \left\{ \begin{array}{ll} x_i & \textnormal{if } x_i \in \{ l, r \} \\ u & \textnormal{if } x_i = d \\ d & \textnormal{if } x_i = u. \end{array} \right.$$
From the way we oriented $G_1$ and $G_2$ we conclude that $P'=(y_2, \dots, y_{k-1})$ defines a $i_b j_b$-path in $D$, strictly shorter than $P$ and such that $V(P') \subseteq V(G_1)$. Hence, $V(G_1)$ is a convex set of $D$.
Recalling that $\frac{m}{2} \le b$, and using the fact that every convex set with more than $nb$ vertices has at least two vertices in $V(G_1)$ and at least two vertices in $V(G_2)$, we conclude that $\textnormal{con} (D) = nb$.
\end{proof}
The following three lemmas deal with the most complex cases.
\begin{lemma} \label{n*m-c=ab-k-l}
Let $n,m \ge 4$ be integers and let $G= P_n \Box P_m$ be a grid. If $a,b,k,l$ are integers such that $0 \le k \le a-2 \le n-3$, $0 \le l \le b-2 \le m-3$, and $a,b \ge 3$, then $ab-(k+l) \in S_{SG} (G)$.
\end{lemma}
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{Figure15.pdf}
\caption{The orientation used in the proof of Lemma \ref{n*m-c=ab-k-l} for $n=8$, $m=6$, $a=7$ $b=5$ and $(k,l) \in \{ (5,3), (3,2) \}$.} \label{n*m-c=ab-k-lFig}
\end{center}
\end{figure}
\begin{proof}
Consider a coloring of the squares of $G$ similar to the coloring used in the proof of Lemma \ref{n*m-c=ab}, with the following differences. In the lower left corner of the aforementioned orientation of $G$ we have a rectangle, $R_1$, of $(a-1)\times(b-1)$ gray squares. Consider the subgraph $G_1$ of $G$ obtained by deleting the first $(a-1)$ columns and the first $(b-1)$ rows of vertices of $G$. Color $G_1$ as in the proof of Lemma \ref{n*m-c=4}. Finally, complete the checkerboard-like pattern with gray rectangles of size $1 \times (a-1)$ squares in the first row, and with gray rectangles of size $(b-1) \times 1$ squares in the first column.
\begin{figure}
\begin{center}
\includegraphics{Figure4.pdf}
\caption{The orientation described in the proof of Lemma \ref{n*m-c=ab-k-l} for $n=12$, $m=10$, $a=4=b$, $k=2$ and $l=1$.} \label{n*m-c=ab-k-lFig}
\end{center}
\end{figure}
We want our largest convex set to be a segment of $R_1$, the gray rectangle in the lower left corner. Color white the squares in the first $k$ columns from the top row of $R_1$. Also color white the squares in the first $l$ rows of the rightmost column of $R_1$ to obtain the gray region $R$. Now, orient $G$ as in Lemma \ref{n*m-c=ab}, orienting $R$ as a whirlpool. Orient all the remaining edges to the right and down to obtain $D$. An example of this orientation is depicted in Figure \ref{n*m-c=ab-k-lFig}.
Now, observe that the top right corner of $R$ coincides with the top right corner of $R_1$. Hence, this orientation has the same properties as the orientation of Lemma \ref{n*m-c=ab}. We can assume without loss of generality that $k+l < \min \{ a, b\}$. Otherwise, assuming that $b \le a$, we have that $ab-(k+l)= a(b-1)-(k'+l')$ for some pair of integerers $k', l'$ such that $0 \le k' \le k$, $0 \le l' \le l$. Hence, $ab-(k+l) \ge 2a, 2b$. From here, it is easy to observe that the only proper convex sets of $D$ are the vertices in each of the gray regions of $D$. Since the largest one is $R$, which has $ab-(k+l)$ vertices, we conclude $S_{SC} (D) = ab-(k+l)$.
Again, we have to be careful when $a=n-1$ or $b=m-1$. The corresponding modifications to the previous orientation are depicted in Figure \ref{n*m-c=ab-k-lFig}.
\end{proof}
\begin{lemma} \label{5*4-c=nm-k}
Let $4 \le n,m \le 5$ be integers and let $G= P_n \Box P_m$ be a grid. If $k$ is an integer such that $6 \le k \le 2\max \{ n, m \}-1$, then $nm-k \in S_{SG} (G)$.
\end{lemma}
\begin{proof}
First, consider the case $n=5$ and $m=4$. If $k \in \{ 8,9 \}$, then the result follows from Lemma \ref{n*m-c=ab-k-l}. The orientations for $k \in \{ 6, 7 \}$ are shown in Figure \ref{5*4-c=nm-kFig}. It can be easily verified that the set of white vertices is a largest convex set for this orientation.
Now suppose that $n=4=m$. The case $k=7$ follows from Lemma \ref{n*m-c=ab-k-l}. The orientation for $k=6$ is depicted in Figure \ref{5*4-c=nm-kFig}. Again, it is not hard to verify that the sets of white vertices are largest convex sets for each of the orientations.
Finally, suppose that $n=5=m$. If $k=9$, then the result follows from Lemma \ref{n*m-c=ab-k-l}. The orientations for $k \in \{6, 7, 8\}$ are depicted in Figure \ref{5*4-c=nm-kFig}. As in the previous cases, the sets of white vertices are largest convex sets for the corresponding orientation.
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{Figure5.pdf}
\caption{The orientations used in the proof of Lemma \ref{5*4-c=nm-k}.} \label{5*4-c=nm-kFig}
\end{center}
\end{figure}
\end{proof}
\begin{lemma} \label{n*m-c=nm-k}
Let $n,m \ge 4$ be integers and let $G= P_n \Box P_m$ be a grid. If $k$ is an integer such that $6 \le k \le 2\max \{ n, m \}-1$, then $nm-k \in S_{SG} (G)$. If $m = 4$, then also $nm-5 \in S_{SC} (G)$.
\end{lemma}
\begin{proof}
If $n,m \le 5$, then the result follows from the previous lemma. Hence, we will suppose without loss of generality that $m \le n$ and $6 \le n$. Let us consider first that $m \ge 5$.
Observe the orientations of the $5 \times 4$ and $5 \times 5$ grids in Figure \ref{5*4-c=nm-kFig} for $k=6$. Clearly, the orientation of the latter can be obtained from the orientation of the former by adding an additional row of squares at the bottom of the grid, and orienting this new row as a whirlpool. Naturally, three arcs of the former orientation should change its direction, in this case, these are the arcs $(1_2, 1_3), (3_1,2_1)$ and $(5_1,4_1)$ of the former oriented graph. We have a similar situation with the orientation of the $4 \times 4$ and $5 \times 5$ grids for $k=6$ and $k=8$, respectively. Again, the latter can be obtained from the former by adding one row and one column of squares, orienting the new row as an anti-whirlpool and the new column following certain pattern. The idea of this proof is to observe that the three orientations of the $5 \times 5$ grids, shown in Figure \ref{5*4-c=nm-kFig}, can be naturally extended to obtain the desired orientations.
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{Figure6.pdf}
\caption{The orientations used in the proof of Lemma \ref{n*m-c=nm-k} for $n=7$ and $m=5$.} \label{n*m-c=nm-kFig}
\end{center}
\end{figure}
First, observe that, independently of the value of $n$, we can always add a new row of squares at the bottom of the grid. Notice that we are extending the largest convex set of the grid, but the complement of such set remains unchanged. Hence, if we consider the aforementioned orientations of the $5 \times 5$ grid, we can conclude that $6, 7, 8 \in S_{SC} (P_5 \Box P_m)$ for every integer $m \ge 5$.
To add a new column of squares we have two different behaviors. First, consider the orientations for $k=6$ and $k=7$. In this case, to add a new column, we need to change the direction of one arc, as in the $5 \times 4$ and $5 \times 5$ grids for the $k=6$ argument. But, to add further columns we will not need to change the direction of any arc, just add a column of squares oriented as a whirlpool. Following this procedure we will obtain an orientation of the grid $P_n \Box P_m$ for every pair of integers $n, m \ge 5$, for $k=6$ and $k=7$, respectively. In the other hand, when $k=8$, adding a new column will not preserve the complement of our largest convex set, the complement will grow larger. If we add a new column, following the pattern as in the example in Figure \ref{n*m-c=nm-kFig}, we will obtain an orientation of the grid $P_n \Box P_m$ for every pair of integers $n,m \ge 5$, for $k=2n-2$.
We can generalize the orientations used for $k=6$ and $k=7$, respectively, to obtain orientations of the grid $P_n \Box P_m$ for $k = 2n-4$ and $2n-3$. The idea is to extend the pattern horizontally, we will explain how it is done for the case $k=2n-4$, the remaining case is analogous. We want to preserve the $2_m n_{m-2}$ directed path defined by the sequence $(d,d,r,r, \dots, r)$, and the whirlpool (or anti-whirlpool) of $n-2$ squares in the upper right corner of the grid. The square of the upper left corner of the grid is oriented as a whirlpool and the rest of the grid is oriented as a whirlpool or anti-whirlpool, except for some alternating squares adjacent to the aforementioned directed path. The remaining unoriented edges are oriented as $(1_{m-2}, 1_{m-1})$ and, if it is still unoriented, $(n_{m-2}, n_{m-3})$. Following a similar idea, orientations for every grid $P_n \Box P_m$ for any pair of integers $n \ge m$ and any $k \in \{ 6, 7 \dots, 2n-2 \}$ can be obtained. We depict the orientations for $P_7 \Box P_5$ for $k \in \{ 6, 7, 8, 9, 10, 11, 12 \}$ in Figure \ref{n*m-c=nm-kFig}. Since all these orientations follow a similar pattern, it can be easily proved that each one has convexity number $nm-k$, as desired.
Finally, since we are assuming $n \ge m$, we have $2n-1 \ge n+m-1$, and the case $k \ge n+m-1$ is covered by Lemma \ref{n*m-c=ab-k-l}.
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{Figure7.pdf}
\caption{The orientations used in the proof of Lemma \ref{n*m-c=nm-k} for $m=4$.} \label{n*m-c=nm-kFig2}
\end{center}
\end{figure}
Now, consider the case $m=4$. If $k \equiv 0$ (mod 4), then Lemma \ref{n*m-c=nb} gives the desired orientation. Else, we give basic orientations that can be enlarged adding columns of whirlpools, either to the largest convex set, or to its complement. The orientation for $k=6$ is the orientation used for the $4 \times 4$ grid (Figure \ref{5*4-c=nm-kFig}). Clearly, additional rows of whirlpools can be added in the bottom of the grid. We give the orientations for $k \in \{ 5, 7, 9, 10, 11, 14\}$ in Figure \ref{n*m-c=nm-kFig2}. Again, the set of black vertices is the complement of the largest set of the oriented graph. Since it is easy to observe that the given orientations have the desired properties, and when we extend them the arguments remain the same, this concludes the proof ot the lemma.
\end{proof}
We present now the main theorem of this work.
\begin{theorem}
Let $4 \le m \le n$ be a pair of integers and let $G = P_n \Box P_m$ be a grid.
\begin{itemize}
\item If $m=4$, then $S_{SC} (G) = [1, nm-4] \setminus \{ 2, 3, 5 \}$.
\item If $m=5$, then $S_{SC} (G) = [1, nm-5] \setminus \{ 2, 3, 5 \}$.
\item If $m \ge 6$, then $S_{SC} (G) = [1, nm-6] \setminus \{ 2, 3, 5 \}$.
\end{itemize}
\end{theorem}
\begin{proof}
The excluded values are a consequence of Lemma \ref{forbid}. Corollary \ref{con=1} shows $1 \in S_{SC} (G)$. Let $r$ be an integer $4 \le r \le nm-i$, for $i \in \{ 4, 5, 6 \}$. The three cases are dealt similarly.
If $nm-2n+1 \le r$, then Lemma \ref{n*m-c=nm-k} implies that $r \in S_{SC} (G)$. Otherwise $r \le nm-2n \le (n-1)(m-1)$, and it follows from Lemmas \ref{n*m-c=4}, \ref{n*m-c=ab}, \ref{n*m-c=nb} and \ref{n*m-c=ab-k-l}, that $r \in S_{SC} (G)$.
\end{proof}
As a final remark, the reader might have noticed that the proofs are
lengthy and technical, which makes them hard to follow. It would be
a good problem to find a short proof for the main theorem of this article.
|
3,212,635,537,426 | arxiv | \section{introduction
Cavity quantum electrodynamics (QED) deals with
the interaction between a single atom and
a discretized photon mode confined in a resonator,
which is the simplest embodiment of quantum light-matter interaction.
The cavity QED systems have been realized in various physical platforms:
just to cite a few, single atoms coupled to an optical cavity,
a semiconductor quantum dot in a photonic-crystal cavity,
and a superconducting qubit coupled to a transmission-line resonator.
Interestingly, regardless of its physical platform,
a cavity QED system is characterized by several universal parameters,
such as $\om_a$ and $\om_c$ (atom and cavity frequencies),
$g$ (atom-photon coupling), $\kap$ (cavity decay rate), and
$\gam$ (atomic decay rate into environments).
In the history of cavity QED, extensive efforts have been made
to reach the strong-coupling regime ($g>\kap, \gam$),
where the vacuum Rabi oscillation and splitting become observable~\cite{sc1,sc2,sc3,sc4}.
In usual strong-coupling systems, the coupling is still by far smaller than
the resonance frequencies of the atom and cavity.
Recently,
attainments of the ultrastrong-coupling ($g \gtrsim \om_{a,c}/10$)
and deep-strong-coupling ($g \gtrsim \om_{a,c}$) regimes
have been reported~\cite{us1,us2,us3,us4,us5,deep1,deep2}.
In such ultrastrong-coupling systems,
the counter-rotating terms in the Hamiltonian,
which do not conserve the total number of excitations
and are usually negligible in the weakly coupled systems,
result in several intriguing physical phenomena,
such as the Bloch-Siegert shift~\cite{BS1,BS2},
virtual photons in the ground state~\cite{vp1,vp2,vp3,vp4,vp5},
and the number non-conserving optical processes such as
multiphoton vacuum Rabi oscillation~\cite{multi0,multi1,multi2}.
Waveguide QED deals with the interaction between a single atom and
a one-dimensional continuum of photon modes,
typically provided by a waveguide attached to the atom.
The parameters to characterize waveguide QED systems are
$\om_a$, $\gam_e$ (atomic decay rate into waveguide)
and $\gam_i$ (atomic decay rate into environments).
The strong-coupling regime in waveguide QED is defined by $\gam_e > \gam_i$,
namely, the condition that radiation from the atom is dominantly forwarded
to the waveguide~\cite{wQED1,wQED2,wQED3,wQED4,wQED5,wQED6}.
This is reflected in spectroscopy
as a strong suppression of transmission near the atomic resonance.
Following the definitions in cavity QED,
the ultra- and deep-strong waveguide QED
should be defined as $\gam_e \gtrsim \om_a/10$ and $\gam_e \gtrsim \om_a$, respectively.
The ultrastrong and deep-strong regimes of waveguide QED
have already been reached using a superconducting qubit~\cite{usQED1,usQED2}.
Theoretically, up to the usual strong-coupling regime,
perturbative treatment of dissipation
based on the rotating-wave and Born-Markov approximations
provides convenient and powerful theoretical tools,
such as the Lindblad master equation and the input-output formalism~\cite{th0a,th0b}.
However, this is not the case in highly dissipative regimes,
and rigorous numerical methods are actively developed~\cite{th1,th2,th3,th4}.
\begin{figure
\begin{center}
\includegraphics[width=70mm]{./Fig1.eps}
\end{center}
\caption{
Schematic of a cavity-waveguide system.
A cavity is coupled to a semi-infinite waveguide,
through which a monochromatic drive field is applied.
The $r<0$ ($r>0$) region in the waveguide
corresponds to the input (output) port.
}
\label{fig:setup}
\end{figure}
In this study, we investigate a linear waveguide QED setup,
namely, a harmonic oscillator coupled to a waveguide,
and investigate its optical response to a classical drive field
applied through this waveguide.
A merit of this system is that
the overall Hamiltonian is diagonalizable by the Fano's method~\cite{Fano1,Fano2,Fano3}
and rigorous optical response is accessible
even for highly dissipative situations.
We report an elliptic motion of the oscillator in the phase space,
which occurs, in principle, even in the usual waveguide QED setups
but becomes remarkable in the ultrastrong-coupling regime
due to the large Lamb shift.
However, in contrast with the intuition provided by the input-output theory,
such elliptic motion does not propagate into the waveguide.
We also obtain an analytic formula of the reflection/transmission coefficient,
which is asymmetric with respect to the renormalized cavity frequency.
We hope that the rigorous optical response presented here
would be useful for developing theoretical tools applicable
to highly dissipative cavity and waveguide QEDs.
\section{theoretical model}
\label{sec:model}
\subsection{Hamiltonian
In a setup considered in this study (Fig.~\ref{fig:setup}),
a cavity is coupled to a semi-infinite waveguide
and a monochromatic drive field is applied through this waveguide.
In the natural units of $\hbar=v=1$,
where $v$ is the photon velocity in the waveguide,
the Hamiltonian of the overall system is given by
\bea
\hH &=& \om_b \hb^{\dag} \hb + \int_0^{\infty}dk
\left[ k \hc_k^{\dag} \hc_k + \xi_k (\hb^{\dag}+\hb)(\hc_k^{\dag}+\hc_k) \right],
\label{eq:Ham}
\eea
where $\om_b$ is the {\it bare} cavity frequency, and
$\hb$ and $\hc_k$ are the annihilation operators of the cavity mode
and the waveguide mode with wave number $k$, respectively,
satisfying the bosonic commutation relations,
$[\hb, \hb^{\dag}]=1$ and $[\hc_k, \hc^{\dag}_{k'}]=\delta(k-k')$.
The cavity-waveguide coupling $\xi_k$ is a real function of $k$.
In this study, in order that the Fano diagonalization is applicable,
we assume the following conditions on $\xi_k$~\cite{Fano2}:
(i)~$\xi_k$ is nonzero for $k>0$,
(ii)~$\xi^2_k$ is an odd function of $k$, namely, $\xi^2_{-k}=-\xi^2_k$,
and (iii)~the coupling is weak enough to satisfy
\bea
\int_0^{\infty} dk \ \xi^2_k/k &<& \om_b/4.
\label{eq:xi_cond}
\eea
\subsection{Drude-form coupling
To be more concrete, we employ a Drude-form for the cavity-waveguide coupling,
\bea
\xi_k^2 &=& C\frac{k}{k^2+\om_x^2},
\label{eq:xik2}
\eea
where $C$ is a constant
and $\om_x$ is the cutoff frequency.
We assume $\om_x \gg \om_b$ so that
the coupling is Ohmic ($\propto k$) near the cavity resonance.
We set $\om_x=5~\om_b$ hereafter.
We denote the radiative decay rate of the cavity mode into the waveguide by $\kap$.
By naively applying the Fermi golden rule, we obtain $\kap=2\pi\xi_{\om_b}^2$.
Therefore, we set the constant $C$ as
\bea
C &=& \frac{\kap(\om_b^2+\om_x^2)}{2\pi\om_b}.
\label{eq:xik2C}
\eea
In this paper, we employ a dimensionless quantity $\kap/\om_b$
as a measure of the strength of the cavity-waveguide coupling.
\subsection{Renormalization of frequency and decay rate
\begin{figure
\begin{center}
\includegraphics[width=70mm]{./Fig2.eps}
\end{center}
\caption{
Dependences of the cavity decay rate ($\tkap/\kap$, solid) and
resonance frequency ($\tom_b/\om_b$, dashed)
on the cavity-waveguide coupling, $\kap/\om_b$.
Their ratio, $\tkap/\tom_b$, is plotted by a dotted line.
The ultrastrong coupling ($\tkap/\tom_b>0.1$)
is attained for $\kap/\om_b > 0.076$ and
the deep-strong coupling ($\tkap/\tom_b>1$)
is attained for $\kap/\om_b > 0.183$.
An alternative expression of the renormalized frequency,
Eq.~(\ref{eq:tomb2}), is also shown (thin solid).
}
\label{fig:tomb}
\end{figure}
Since the Fermi golden rule is in principle valid
only for a weak cavity-waveguide coupling,
$\kap$ may deviate from the actual decay rate $\tkap$,
particularly for a stronger coupling.
Furthermore, the resonance frequency $\om_b$ also acquires a Lamb shift
and takes a renormalized value $\tom_b$.
As we observe later in Sec.~\ref{ssec:3b},
$\tom_b$ and $\tkap$ are identified as
\bea
\tom_b &=& \mathrm{Re}(\lam_1),
\label{eq:tombdef}
\\
\tkap &=& 2 \ \mathrm{Im}(\lam_1),
\label{eq:tkapdef}
\eea
where $\lam_1$ is a complex cavity frequency,
which is a solution of the cubic equation~(\ref{eq:cubic})
in the first quadrant (Fig.~\ref{fig:lam123}).
From Eq.~(\ref{eq:xi_cond}), we have
$\kap/\om_b < \om_b\om_x/(\om_b^2+\om_x^2)$.
This inequality sets an upper bound for the coupling strength:
$\kap/\om_b < 0.192$ for $\om_x=5\om_b$.
However, as we discuss in Sec.~\ref{ssec:3b},
from the condition that the renormalized frequency $\tom_b$ is positive,
we have a more strict upper bound, $\kap/\om_b < 0.190$.
In Fig.~\ref{fig:tomb}, we plot the dependences of
$\tom_b$ and $\tkap$ on $\kap/\om_b$.
We observe that, beyond the perturbative regime of $\kap/\om_b \ll 1$,
the agreement between $\tkap$ and $\kap$ is fairly good even for stronger coupling.
In contrast, the renormalized cavity frequency decreases drastically
as the coupling becomes stronger. As a result,
not only the ultrastrong coupling regime ($\tkap/\tom_b > 0.1$)
but also the deep-strong coupling regime ($\tkap/\tom_b > 1$) is attainable
within this theoretical model.
\subsection{Initial state vector
In this study, we investigate the optical response of a cavity
driven by a monochromatic classical field
applied through the waveguide (Fig.~\ref{fig:setup}).
The positively rotating part of drive amplitude is given by
\bea
E(r,t) &=& E_d e^{ik_d(r-t)},
\label{eq:Ed}
\eea
where $E_d$ and $k_d$ are the complex amplitude and
wavenumber/frequency of the drive, respectively.
At the initial moment ($t=0$), we assume that
the whole system is in the ground state expect the drive field in the waveguide,
which is in a coherent state.
The initial state vector is then written as
\bea
|\psi_i\ra &=&
\exp\left(\sqrt{2\pi}E_{d}\hc_{k_{d}}^{\dag}
-\sqrt{2\pi}E^*_{d}\hc_{k_{d}}\right)|vac\ra,
\label{eq:psii}
\eea
where $|vac\ra$ is the overall ground state.
The real-space representation $\tc_r$ of the waveguide field operator
is defined as the Fourier transform of $\hc_k$,
\bea
\tc_r &=& \frac{1}{\sqrt{2\pi}} \int_0^{\infty} dk \ e^{ikr}\hc_k.
\label{eq:tcr}
\eea
We can check that
$\la \tc_r(0) \ra \equiv \la \psi_i|\tc_r(0)|\psi_i\ra = E(r,0)$.
Strictly speaking,
the real-space representation of the waveguide mode
depends on the boundary condition of the waveguide at $r=0$.
For example, for a closed boundary condition,
the waveguide mode function takes the form of
$f_k(r) =\sqrt{2/\pi} \sin(kr)=(ie^{-ikr}-ie^{ikr})/\sqrt{2\pi}$~\cite{norm}.
Therefore, we should add a phase factor $i$ ($-i$)
for the input (output) port in Eq.~(\ref{eq:tcr}),
which accounts for the sign flip upon reflection at a mirror.
However, we employ Eq.~(\ref{eq:tcr})
as the real-space representation of waveguide modes
for simplicity.
This introduces no problem except for definition of the relative phase
in the input and output ports.
\section{Diagonalization
\subsection{General formula
The Hamiltonian [Eq.~(\ref{eq:Ham})] is bilinear in bosonic operators
and can be diagonalized by the Fano's method.
When the cavity-waveguide coupling is weak enough to satisfy Eq.~(\ref{eq:xi_cond}),
we can rewrite the Hamiltonian as
\bea
\hH &=&
\int_0^{\infty} dk \ k \hd_k^{\dag} \hd_k,
\label{eq:Ham2}
\eea
where $\hd_k$ is an eigenmode annihilation operator
satisfying the bosonic commutation relation,
\bea
[\hd_k, \hd_{k'}^{\dagger}] &=& \delta(k-k').
\label{eq:norm}
\eea
$\hd_k$ is given by linear combination of the original bosonic operators as
\bea
\hd_k &=& \beta_1(k) \hb + \beta_2(k) \hb^{\dag} +
\int_0^{\infty} dq \left[ \gam_1(k,q) \hc_q + \gam_2(k,q) \hc_q^{\dag} \right],
\eea
where the coefficients are given by (see Appendix~\ref{app:deter} for derivation)
\bea
\beta_1(k) &=&
\frac{(k+\om_b)\xi_k}{k^2-\om_b^2 z(k)},
\label{eq:beta1}
\\
\beta_2(k) &=& \frac{(k-\om_b)\xi_k}{k^2-\om_b^2 z(k)},
\label{eq:beta2}
\\
\gam_1(k,q) &=& \delta(k-q) + \tgam_1(k,q),
\label{eq:gam1}
\\
\gam_2(k,q) &=& \frac{2\om_b \xi_k \xi_q}{(k+q)[k^2-\om_b^2 z(k)]},
\label{eq:gam2}
\eea
where
\bea
\tgam_1(k,q) &=& \frac{2\om_b \xi_k \xi_q}{(k-q-i0)[k^2-\om_b^2 z(k)]},
\label{eq:tgam1}
\eea
and $z(k)$ is a dimensionless quantity
representing the self-energy correction for the resonator frequency,
\bea
z(k) &=& 1 + \frac{2}{\om_b}\int_{-\infty}^{\infty}dq \frac{\xi_q^2}{k-q-i0}.
\label{eq:z(k)}
\eea
Inversely, the bare operators $\hb$ and $\hc_k$ are expressed
in terms of the eigenoperators by
\bea
\hb &=& \int_0^{\infty} dq [\beta_1^*(q) \hd_q - \beta_2(q) \hd_q^{\dag}],
\label{eq:hb}
\\
\hc_k &=& \int_0^{\infty} dq [\gam_1^*(q,k)\hd_q - \gam_2(q,k)\hd_q^{\dag}].
\label{eq:hck}
\eea
\subsection{Specific results for Drude-form coupling}
\label{ssec:3b}
\begin{figure}
\begin{center}
\includegraphics[width=70mm]{./Fig3.eps}
\end{center}
\caption{
$\lam_{1,2,3}$ on the complex plane. Arrows indicate the directions
as the cavity-waveguide coupling $\kap$ is increased.
}
\label{fig:lam123}
\end{figure}
When the cavity-waveguide coupling takes the Drude form
[Eqs.~(\ref{eq:xik2}) and (\ref{eq:xik2C})],
$z(k)$ and $k^2-\om_b^2 z(k)$ are rewritten as follows,
\bea
z(k) &=& 1+\frac{2\pi i C}{\om_b(k-i\om_x)},
\\
k^2-\om_b^2 z(k) &=& \frac{(k-\lam_1)(k-\lam_2)(k-\lam_3)}{k-i\om_x},
\label{eq:k2omb2}
\eea
where $\lam_{1,2,3}$ are the solutions of the following cubic equation for $k$,
\bea
k^3-i\om_x k^2 -\om_b^2 k + (i\om_x\om_b^2 - 2i\pi C \om_b) &=& 0.
\label{eq:cubic}
\eea
As shown in Fig.~\ref{fig:lam123},
$\lam_1$ ($\lam_2$) is on the first (seond) quadrant
and $\lam_3$ is on the positive imaginary axis.
The real and imaginary parts of $\lam_1$ correspond to
the Lamb-shifted resonance frequency $\tom_b$ and
half of the decay rate $\tkap/2$ [Eqs.~(\ref{eq:tombdef}) and (\ref{eq:tkapdef})].
For reference,
we present the perturbative solution of Eq.~(\ref{eq:cubic})
with respect to the cavity-waveguide coupling $\kap$.
The zeroth-order solutions are
$\lam_1^{(0)}=\om_b$, $\lam_2^{(0)}=-\om_b$, and $\lam_3^{(0)}=i\om_x$.
Up to the first order in $\kap$, the three solutions are given by
$\lam_1 \approx (\om_b-\kap\om_x/2\om_b) + i\kap/2$,
$\lam_2 \approx -(\om_b-\kap\om_x/2\om_b) + i\kap/2$,
and $\lam_3 \approx i\om_x - i\kap$.
For an extremely strong coupling,
$\lam_1$ and $\lam_2$ also become purely imaginary.
The condition that
the renormalized frequency $\tom_b$ remain positive,
in other words,
$\lam_1$ and $\lam_2$ are not purely imaginary,
is that $\kap < [\om_b^2\om_x-f(\mu_-)]/(\om_b^2+\om_x^2)$,
where $f(x)=x^3-\om_x x^2+\om_b^2 x$ and
$\mu_-$ is a smaller root of the $df/dx=0$,
namely, $\mu_- = (\om_x-\sqrt{\om_x^2-3\om_b^2})/3$.
For $\om_x = 5~\om_b$, this condition is $\kap/\om_b < 0.190$.
\section{optical response
\subsection{Cavity Amplitude}
\label{ssec:camp}
In this section, we investigate
time evolution of the whole system
from the initial state vector, Eq.~(\ref{eq:psii}).
We first observe the amplitude of the cavity mode,
$\la \hb(t) \ra \equiv \la \psi_i|\hb(t)|\psi_i\ra$.
Since $\hd_k$ is an eigenoperator of the Hamiltonian,
$\hb(t)$ is given, from Eq.~(\ref{eq:hb}), by
\bea
\hb(t) &=& \int_0^{\infty}dq \left[
e^{-iqt} \beta_1^*(q) \hd_q - e^{iqt} \beta_2(q) \hd^{\dag}_q
\right].
\label{eq:hbt}
\eea
Furthermore, $|\psi_i\ra$ is an eigenstate of $\hd_q$ and satisfies
\bea
\hd_q |\psi_i\ra &=&
\sqrt{2\pi} [E_d \gam_1(q,k_d) + E^*_d \gam_2(q,k_d)] |\psi_i\ra.
\label{eq:dqpsi}
\eea
From these results, $\la \hb(t) \ra$ is given by
\bea
\la \hb(t) \ra
&=&
\sqrt{2\pi}E_d \int_0^{\infty} dq \left[
e^{-iqt}\beta_1^*(q) \gam_1(q,k_d ) - e^{iqt}\beta_2(q) \gam_2^*(q,k_d )
\right]
\nonumber
\\
&+&
\sqrt{2\pi}E^*_d \int_0^{\infty} dq \left[
e^{-iqt}\beta_1^*(q) \gam_2(q,k_d ) - e^{iqt}\beta_2(q) \gam_1^*(q,k_d )
\right].
\eea
This is divided into stationary and transient components as
$\la \hb(t) \ra=\la \hb(t) \ra_s + \la \hb(t) \ra_t$.
The stationary component is given by
\bea
\la \hb(t) \ra_s &=&
\sqrt{2\pi} \beta_1^*(k_d) E_d e^{-ik_d t} -
\sqrt{2\pi} \beta_2(k_d) E_d ^* e^{ik_d t}.
\label{eq:bts}
\eea
The transient component is presented in Appendix~\ref{app:tra}.
Putting $E_d =|E_d |e^{i\theta_d}$, we have
\bea
\mathrm{Re}\la \hb(t) \ra_s &=&
\sqrt{8\pi}|E_d|\om_b\xi_{k_d}
\mathrm{Re}
\left(
\frac{e^{i(k_d t-\theta_d)}}{k_d^2-\om_b^2 z(k_d)}
\right),
\label{eq:Re}
\\
\mathrm{Im}\la \hb(t) \ra_s &=&
-\sqrt{8\pi}|E_d|k_d \xi_{k_d}
\mathrm{Im}
\left(
\frac{e^{i(k_d t-\theta_d)}}{k_d^2-\om_b^2 z(k_d)}
\right).
\label{eq:Im}
\eea
These equations indicate that
the motion of the cavity amplitude $\la \hb(t) \ra_s$
on the phase space is elliptical in general;
the ratio of the vertical (imaginary) radius
relative to the horizontal (real) radius
is $k_d/\om_b$, and thus depends on the drive frequency.
However, such elliptical motion is not remarkable
when the cavity-waveguide coupling $\kap$ is small.
For a small $\kappa$ case,
strong optical response is obtained within a narrow frequency region
around the renormalized cavity frequency $\tom_b$,
which is close to the bare frequency $\om_b$.
For example, when $\kap/\om_b=0.01$,
the renormalized frequency amounts to $\tom_b = 0.975~\om_b$ [Eq.~(\ref{eq:tomb2})].
Therefore, the motion is almost circular for small $\kap$,
as we observe in Figs.~\ref{fig:tra}~(a) and (c).
In contrast, for a large $\kappa$ case,
the motion on the phase space becomes highly elliptical,
as we observe in Figs.~\ref{fig:tra}~(b) and (d).
This is due to the large frequency renormalization (Lamb shift).
When $\kap/\om_b=0.15$, the renormalized frequency amounts to $\tom_b = 0.476~\om_b$.
\begin{figure}
\begin{center}
\includegraphics[width=150mm]{./Fig4.eps}
\end{center}
\caption{
Elliptical motion of the cavity amplitude.
(a)~Trajectories on the phase space for $\kap/\om_b=0.01$.
The drive frequency is set at
the renormalized resonance $\tom_b(=0.975~\om_b)$ (solid) and
the bare resonance $\om_b$ (dashed).
The photon rate of the drive field is set at $|E_d|^2=2.5~\kap$,
at which the mean intra-cavity photon number is estimated to be
$\la \hb^{\dag}\hb \ra = 4|E_d|^2/\kap =10$ on resonance,
following the input-output theory.
The uncertainty ellipse is also shown.
(b)~The same plot as (a) for $\kap/\om_b=0.15$.
The renormalized resonance is $\tom_b=0.476~\om_b$.
(c)~Dependence of the long (solid line) and short (dotted line) axial radii
on the drive frequency $k_d$ for $\kap/\om_b=0.01$.
(d)~The same plot as (c) for $\kap/\om_b=0.15$.
}
\label{fig:tra}
\end{figure}
\subsection{Quadrature Fluctuations
\begin{figure}
\begin{center}
\includegraphics[width=70mm]{./Fig5.eps}
\end{center}
\caption{Quadrature fluctuations:
$\Delta X$ (solid), $\Delta Y$ (dashed),
and $\sqrt{\Delta X \Delta Y}$ (thin dotted).
$\om_x/\om_b=5$.
}
\label{fig:DxDy}
\end{figure}
Here, we investigate the quadrature fluctuations of the cavity mode.
We define the $\hX$ and $\hY$ quadratures
by $\hX=(\hb+\hb^{\dagger})/2$ and $\hY=-i(\hb-\hb^{\dagger})/2$, respectively,
and their fluctuations by
$\Delta X=\sqrt{\la \hX^2 \ra - \la \hX \ra^2}$ and
$\Delta Y=\sqrt{\la \hY^2 \ra - \la \hY \ra^2}$, respectively,
where $\la \hO \ra = \la \psi_i|\hO|\psi_i\ra$.
From these definitions, we have
\bea
\Delta X
&=&
\frac{\sqrt{1 + 2\la \hb^{\dag}(t), \hb(t)\ra + 2\mathrm{Re}\la \hb(t), \hb(t) \ra}}{2},
\label{eq:DX2}
\\
\Delta Y
&=&
\frac{\sqrt{1 + 2\la \hb^{\dag}(t), \hb(t)\ra - 2\mathrm{Re}\la \hb(t), \hb(t) \ra}}{2},
\label{eq:DY2}
\eea
where $\la \hO, \hO' \ra \equiv \la \hO\hO' \ra-\la \hO \ra \la \hO' \ra$.
From Eqs.~(\ref{eq:hbt}) and (\ref{eq:dqpsi}),
we can confirm that
both $\la \hb^{\dag}(t), \hb(t) \ra$ and $\la \hb(t), \hb(t) \ra$
reduces to the following time-independent quantities,
\bea
\la \hb^{\dag}, \hb \ra
&=&
\int_0^{\infty}dq |\beta_2(q)|^2,
\label{eq:<b*,b>}
\\
\la \hb, \hb \ra &=&
-\int_0^{\infty}dq \beta_1^*(q) \beta_2(q),
\label{eq:<b,b>}
\eea
and that the quadrature fluctuations, $\Delta X$ and $\Delta Y$,
are identical to those of the vacuum fluctuations.
The integrals appearing in Eqs.~(\ref{eq:<b*,b>}) and (\ref{eq:<b,b>}) can be performed
analytically for the Drude-form coupling (Appendix~\ref{app:C}).
Figure~\ref{fig:DxDy} plots the dependences of $\Delta X$ and $\Delta Y$
on the cavity-waveguide coupling $\kap$.
We observe that there exists squeezing in $Y$ quadrature,
and the degree of squeezing increases for larger $\kap$.
The state is not a minimum uncertainty state,
since $\sqrt{\Delta X \Delta Y}>1/2$ as we observe in Fig.~\ref{fig:DxDy}.
\subsection{Amplitude of waveguide field
From Eqs.~(\ref{eq:hck}) and (\ref{eq:dqpsi}),
the amplitude of the waveguide field in the wavenumber representation is given by
\bea
\la \hc_k(t) \ra &=&
\sqrt{2\pi}E_d \int_0^{\infty} dq \left[
e^{-iqt}\gam_1^*(q,k) \gam_1(q,k_d ) - e^{iqt}\gam_2(q,k) \gam_2^*(q,k_d )
\right]
\nonumber
\\
&+&
\sqrt{2\pi}E^*_d \int_0^{\infty} dq \left[
e^{-iqt}\gam_1^*(q,k) \gam_2(q,k_d ) - e^{iqt}\gam_2(q,k) \gam_1^*(q,k_d )
\right].
\eea
Using Eqs.~(\ref{eq:gam1})--(\ref{eq:tgam1}), this quantity is rewritten as follows,
\bea
\la \hc_k(t) \ra &=&
\sqrt{2\pi}E_d \left[
e^{-ik_d t}\delta(k-k_d ) + e^{-ikt}\tgam_1(k,k_d ) + e^{-ik_d t}\tgam_1^*(k_d ,k)
\right]
\nonumber
\\
&-& i\sqrt{2/\pi} \om_b \xi_k \xi_{k_d } E_d
\int_{-\infty}^{\infty}dq \frac{e^{-iqt}}{(q-k+i0)(q-k_d -i0)}
\left(
\frac{1}{q^2-\om_b^2 z(q)}-\frac{1}{q^2-\om_b^2 z^*(q)}
\right)
\nonumber
\\
&+& \sqrt{2\pi}E_d ^* \left[
e^{-ikt}\gam_2(k,k_d ) - e^{ik_d t}\gam_2(k_d ,k)
\right]
\nonumber
\\
&+& i\sqrt{2/\pi} \om_b \xi_k \xi_{k_d } E_d ^*
\int_{-\infty}^{\infty}dq \frac{e^{iqt}}{(q+k-i0)(q-k_d +i0)}
\left(
\frac{1}{q^2-\om_b^2 z(q)}-\frac{1}{q^2-\om_b^2 z^*(q)}
\right).
\label{eq:<c_k(t)>}
\eea
The integral in the second line in the above equation
can be performed by employing the residue theorem.
The integrand has four poles in the lower complex plane of $q$
at $k-i0$, $\lam_1^*$, $\lam_2^*$, and $\lam_3^*$,
and the latter three poles yield transient components.
Therefore, the stationary component of the second line
comes from the pole at $k-i0$ and is given by
$-\sqrt{8\pi}\om_b\xi_k\xi_{k_d } \frac{E_d e^{-ikt}}{k-k_d -i0}
(\frac{1}{k^2-\om_b^2 z(k)}-\frac{1}{k^2-\om_b^2 z^*(k)})$.
Repeating the same arguments,
the stationary component of the fourth line of Eq.~(\ref{eq:<c_k(t)>}) is given by
$-\sqrt{8\pi}\om_b\xi_k\xi_{k_d } \frac{E_d ^*e^{-ikt}}{k+k_d }
(\frac{1}{k^2-\om_b^2 z(k)}-\frac{1}{k^2-\om_b^2 z^*(k)})$.
As a result, the stationary component of
the waveguide amplitude is written as
\bea
\la c_k(t) \ra &=&
\la c_k(t) \ra^{(1)} + \la c_k(t) \ra^{(2)} + \la c_k(t) \ra^{(3)},
\label{eq:<c_k(t)>v3}
\\
\la c_k(t) \ra^{(1)} &=&
\sqrt{2\pi}\delta(k-k_d ) E_d e^{-ik_d t},
\\
\la c_k(t) \ra^{(2)} &=&
\frac{\sqrt{8\pi}\om_b \xi_k \xi_{k_d } E_d }{k-k_d -i0}
\left(
\frac{e^{-ikt}}{k^2-\om_b^2 z^*(k)} -
\frac{e^{-ik_d t}}{k_d ^2-\om_b^2 z^*(k_d )}
\right),
\label{eq:<ckt>2}
\\
\la c_k(t) \ra^{(3)} &=&
\frac{\sqrt{8\pi}\om_b \xi_k \xi_{k_d } E_d ^*}{k+k_d }
\left(
\frac{e^{-ikt}}{k^2-\om_b^2 z^*(k)} -
\frac{e^{ik_d t}}{k_d ^2-\om_b^2 z(k_d )}
\right).
\eea
\begin{figure}
\begin{center}
\includegraphics[width=130mm]{./Fig6.eps}
\end{center}
\caption{
Normalized amplitude of the waveguide field, $\la \tc_r(t) \ra/E(r,t)$.
(a)~Real and (b)~imaginary parts.
Solid lines represent the rigorous numerical results,
and dotted lines represent the approximate one given by Eq.~(\ref{eq:<tcr>}).
The parameters are chosen as follows:
$\kap/\om_b=0.1$, $k_d/\om_b=0.6$, and $t=300/\om_b$.
}
\label{fig:crt2}
\end{figure}
We switch to the real-space representation, $\la \tc_r(t) \ra$,
using Eq.~(\ref{eq:tcr}).
$\la \tc_r(t) \ra^{(1)}$ is immediately given by
\bea
\la \tc_r(t) \ra^{(1)} &=& E_d e^{ik_d (r-t)}.
\eea
Obviously, this is nothing but the input drive field of Eq.~(\ref{eq:Ed}).
Regarding $\la \tc_r(t) \ra^{(2)}$,
the principal contribution comes from the pole at $k=k_d+i0$
in the right-hand-side of Eq.~(\ref{eq:<ckt>2}).
Therefore, we can employ the following approximation,
$\la c_k(t) \ra_s^{(2)} \approx
\sqrt{8\pi}\om_b \xi_{k_d }^2 E_d
[k_d ^2-\om_b^2 z^*(k_d )]^{-1} [k-k_d -i0]^{-1}
\left( e^{-ikt} - e^{-ik_d t} \right)$.
Then, we have
\bea
\la \tc_r(t) \ra^{(2)} & \approx &
-\frac{4\pi i \om_b \xi_{k_d }^2}{k_d ^2-\om_b^2 z^*(k_d )}
\theta(r)\theta(t-r) E_d e^{ik_d (r-t)},
\eea
where $\theta$ is the Heaviside step function.
This represents the radiation from the cavity
emitted into the positive $r$ region.
Finally, $\la \tc_r(t) \ra^{(3)}$ yields no propagating wave.
Combining these results,
we obtain the following analytic form of $\la \tc_r(t) \ra$:
\bea
\la \tc_r(t) \ra & \approx &
\left(
1-\frac{4\pi i \om_b \xi_{k_d }^2}{k_d ^2-\om_b^2 z^*(k_d )}
\theta(r)\theta(t-r)
\right) \times E_d e^{ik_d (r-t)}.
\label{eq:<tcr>}
\eea
The spatial shape of $\la \tc_r(t) \ra$ is plotted in Fig.~\ref{fig:crt2},
in which the rigorous shape [numerical Fourier transform of Eq.~(\ref{eq:<c_k(t)>v3})]
is plotted by solid lines
and the approximate form [Eq.~(\ref{eq:<tcr>})]
is plotted by dotted lines.
We observe good agreement between them,
except the deviations at the wavefront of the cavity radiation ($r \lesssim t$)
and at the cavity position ($r \sim 0$).
The former deviation originates in the transient cavity response,
which is not taken into account in Eq.~(\ref{eq:<tcr>}).
The transient response vanishes within a timescale of $\kap^{-1}$,
which agrees with our observation in Fig.~\ref{fig:crt2}.
On the other hand,
the latter deviation around the cavity position originates in the fact that
the cavity-waveguide interaction has a finite bandwidth in the wavenumber space
and therefore is not spatially local in the present theoretical model.
The bandwidth of the cavity-waveguide coupling
is of the order of $\om_b$ in the wavenumber space,
and is therefore of the order of $\om_b^{-1}$ in the real space.
This explains the deviation localized at the origin in Fig.~\ref{fig:crt2}.
A notable fact is that,
in contrast with the intracavity field amplitude [Eq.~(\ref{eq:bts})]
that is composed of both positively and negatively oscillating components,
the waveguide field amplitude in the output port [Eq.~(\ref{eq:<tcr>})]
is composed only of the positively oscillating one.
Therefore, the elliptic motion is specific to the intracavity amplitude.
\subsection{Refection coefficient
The refection coefficient is identified as
$R = \la \tc_r(t) \ra/E(r,t)$ at the output port ($r>0$).
From Eq.~(\ref{eq:<tcr>}), $R$ is identified as
\bea
R(k_d ) &=& 1-\frac{4\pi i \om_b \xi_{k_d }^2}{k_d ^2-\om_b^2 z^*(k_d )}.
\label{eq:Ref}
\eea
We can check that $|R|=1$ for any input frequency $k_d $.
This implies that input field is reflected completely coherently,
which is characteristic to linear optical response.
In Fig.~\ref{fig:ref}, we plot
the phase shift upon reflection, $\arg R$,
as a function of the drive frequency $k_d$,
varying the cavity-waveguide coupling.
As we increase the coupling,
we observe the broadening of the linewidth
and the redshift of the resonance frequency.
The spectrum takes a kink-shaped form around the renormalized frequency.
For a weak coupling, the spectrum is anti-symmetric
with respect to the renormalized frequency,
as is predicted by standard input-output theory.
However, for a stronger coupling, such symmetry is gradually lost.
We can determine the renormalized resonance frequency $\tom_b$
as the drive frequency achieving the $\pi$ phase shift, $R(\tom_b)=-1$.
From this condition, $\tom_b$ is analytically given by
\bea
\tom_b^2 &=&
\frac{\om_b^2-\om_x^2+\sqrt{(\om_b^2 + \om_x^2)(\om_b^2 + \om_x^2 -4\kap\om_x)}}{2}.
\label{eq:tomb2}
\eea
As we can confirm in Fig.~\ref{fig:tomb},
this is almost identical to the former
definition of $\tom_b$ by Eq.~(\ref{eq:tombdef}).
We observe in Fig.~\ref{fig:ref} that
the reflection coefficient becomes independent of the coupling strength
$\kap/\om_b$ at the bare cavity resonance, $k_d =\om_b$;
we can check that $R(\om_b)=(\om_x-i\om_b)/(\om_x+i\om_b)$.
\begin{figure}
\begin{center}
\includegraphics[width=70mm]{./Fig7.eps}
\end{center}
\caption{
Phase shift upon reflection as a function of the drive frequency.
The cavity-waveguide coupling strength, $\kap/\om_b$, is indicated.
}
\label{fig:ref}
\end{figure}
\subsection{Open waveguide
In the previous subsection,
we have determined the reflection coefficient $R$
when a semi-infinite waveguide is coupled to a cavity (Fig.~\ref{fig:setup}).
From this result, we can readily determine
the reflection and transmission coefficients $R'$ and $T'$,
when the cavity is coupled to an open waveguide [Fig.~\ref{fig:open}(a)].
The amplitude of waveguide field in this case is written as
\bea
E(r,t) &=& E_d e^{-i\om_d t} \times
\begin{cases}
e^{ik_dr} + R' e^{-ik_dr} & (r<0)
\\
T' e^{ik_dr} & (0<r)
\end{cases}.
\eea
We divide this field into even and odd components.
The even component interacts with the cavity
whereas the odd component does not.
The even component is defined by $E_s(r,t)=[E(r,t)+E(-r,t)]/2$
and is therefore given by
$E_s(r,t)=\frac{1}{2}E_d e^{-ik_d(r+t)} + \frac{R'+T'}{2}E_d e^{ik_d(r-t)}$ for $r>0$.
Since the first (second) term in the right-hand-side of this equation
represents the incoming (outgoing) field,
we have $R'+T'=R$.
Similarly,
the odd component is defined by $E_a(r,t)=[E(r,t)-E(-r,t)]/2$
and is therefore given by
$E_s(r,t)=-\frac{1}{2}E_d e^{-ik_d(r+t)} + \frac{T'-R'}{2}E_d e^{ik_d(r-t)}$ for $r>0$.
Since the incoming field simply transmits the cavity without interaction,
we have $T'-R'=1$. Therefore,
\bea
R' &=& (R-1)/2,
\\
T' &=& (R+1)/2.
\eea
We can readily confirm that $|R'|^2+|T'|^2=1$.
The transmissivity $|T'|^2$ is plotted in Fig.~\ref{fig:open}(b)
as a function of the drive frequency.
We observe that the symmetric transmission dip for a weak coupling case (solid line)
gradually becomes asymmetric
as the cavity-waveguide coupling increases (dashed and dotted lines).
\begin{figure}
\begin{center}
\includegraphics[width=140mm]{./Fig8.eps}
\end{center}
\caption{
(a)~Schematic of a cavity coupled to an open waveguide.
(b)~Transmissiveity $|T'|^2$ as a function of the drive frequency.
The cavity-waveguide coupling strength, $\kap/\om_b$, is indicated.
}
\label{fig:open}
\end{figure}
\section{summary}
In this study, we investigated optical response of
a linear waveguide QED system, namely,
an optical cavity coupled to a waveguide.
Our analysis is based on exact diagonalization of the overall Hamiltonian,
and is therefore rigorous
even in the ultrastrong and deep-strong coupling regimes of waveguide QED,
in which the perturbative treatments of dissipation
such as the Lindblad master equation are no longer valid.
We observed that the motion of the cavity amplitude
in the phase space is elliptical in general,
owing to the counter-rotating terms in the cavity-waveguide coupling.
Such elliptical motion becomes remarkable
in the ultrastrong coupling regime
due to the large Lamb shift of the cavity frequency
comparable to its bare frequency.
However, such an elliptical motion of the cavity amplitude
is not reflected in the output field,
contrary to the intuition by the input-output theory.
We obtained an analytic expression of the reflection/transmission coefficient,
which becomes asymmetric with respect to the resonance frequency
as the cavity-waveguide coupling is increased.
\section*{Acknowledgments}
The author acknowledges fruitful discussions with T. Shitara and I. Iakoupov.
This work is supported in part by
JST CREST (Grant No. JPMJCR1775), JST ERATO (Grant no. JPMJER1601),
MEXT Q-LEAP, and JSPS KAKENHI (Grant No. 19K03684).
|
3,212,635,537,427 | arxiv |
\section{INTRODUCTION}
Starting from the work by Giles, see \cite{Giles,Giles2,Giles3}, we developed a novel multilevel method called p-refined Multilevel Quasi Monte Carlo (p-MLQMC), see \cite{Blondeel2020}. Similar to classic Multilevel Monte Carlo, see \cite{Giles}, p-MLQMC uses a hierarchy of increasing resolution Finite Element meshes to achieve a computational speedup. Most of the samples are taken on coarse and computationally cheap meshes, while a decreasing number of samples are taken on finer and computationally expensive meshes. The major difference between classic Multilevel Monte Carlo and p-MLQMC resides in the refinement scheme used for constructing the mesh hierarchy. In classic Multilevel Monte Carlo (h-MLMC), an h-refinement scheme is used to build the mesh hierarchy, see, for example \cite{Cliffe}. The accuracy of the model is increased by increasing the number of elements in the Finite Element mesh. In p-MLQMC, a p-refinement scheme is used to construct the mesh hierarchy. The accuracy of the model is increased by increasing the polynomial order of the element's shape functions while retaining the same number of elements. This approach reduces the computational cost with respect to h-MLMC, as shown in \cite{Blondeel2020}. Furthermore, instead of using a random sampling rule, i.e, the Monte Carlo method, p-MLQMC uses a deterministic Quasi-Monte Carlo (QMC) sampling rule, yielding a further computational gain.
However, the p-MLQMC method presents the practitioner with a challenge. This challenge consists of adequately incorporating the uncertainty, modeled as a random field, into the Finite Element model. For classic Multilevel (Quasi)-Monte Carlo (h-ML(Q)MC) this is typically achieved by means of the midpoint method \cite{ChunChing}. The model uncertainty is represented as scalars resulting from the evaluation of the random field at centroids of the elements. These scalars are then assigned to the elements. With this method, the uncertainty is modeled as being constant inside each element. In h-refined multilevel methods, the midpoint method intrinsically links the spatial resolution of the mesh with the spatial resolution of the random field. An h-refinement of the mesh will result in a finer representation of the random field.
However, for the p-MLQMC method, the midpoint method cannot be used. This is because the refinement scheme used in p-MLQMC does not increase the number of elements. The p-MLQMC method makes use of the integration point method, see \cite{MATTHIES1997283}. Scalars resulting from the evaluation of the random field at certain spatial locations are taken into account during numerical integration of the element stiffness matrices. With this method, the uncertainty varies inside each element. In the present work, we investigate how to adequately select the spatial locations used for the evaluation of the random field. Specifically, we distinguish three different approaches how to select these random field evaluation points. The \emph{Non-Nested Approach} (NNA), the \emph{Global Nested Approach} (GNA), and the \emph{Local Nested Approach} (LNA). We investigate how these approaches affect the variance reduction in the p-MLQMC method, and how the total computational runtime increases over the levels. These approaches will be benchmarked on a model problem which consists of a slope stability problem, which assesses the stability of natural or man made slopes. The uncertainty is located in the soil's cohesion, and is represented as a two-dimensional lognormal random field.
The paper is structured as follows. First we give a theoretical background motivating our research, and give a concise overview of the building blocks of p-MLQMC. Second, we present the three approaches. Hereafter, we shortly discuss the underlying Finite Element solver, and introduce the model problem. Last, we present the results obtained with p-MLQMC for the three different approaches. Here we focus on the variance reduction over the levels and the effect on the total computational runtime.
\vspace{-0.5cm}
\section{Theoretical Background}
\label{sec:background}
Multilevel Monte Carlo methods rely on a hierarchy of meshes in order to achieve a speedup with respect to Monte Carlo. This speedup is achieved by taking a majority of samples on coarse and computationally cheap meshes, and by taking only a decreasing number of samples on finer and computationally expensive meshes. This hierarchy of meshes can be obtained by applying an h-refinement scheme or a p-refinement scheme to a coarse mesh model. We opt for a hierarchy based on p-refinement. The hierarchy applied to a discretized model of the slope stability problem is shown in Fig.\,\ref{fig:meshes_2_p}. Here, the Finite Element nodal points are represented as red dots. A more thorough discussion of the slope stability problem and the underlying Finite Element model is given in \S\ref{sec:FEM}.
\begin{figure}[H]
\hspace{0.08cm}
\begin{minipage}{0.3\textwidth}
\scalebox{0.8}{
\input{level0.tex}}
\end{minipage}
\begin{minipage}{0.3\textwidth}
\scalebox{0.8}{
\input{level1.tex}}
\end{minipage}
\begin{minipage}{0.3\textwidth}
\scalebox{0.8}{
\input{level2.tex}}
\end{minipage}
\\
\begin{minipage}{0.32\textwidth}
\scalebox{0.1}{
\includetikz{L1}{L1.tex}}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\scalebox{0.1}{
\includetikz{L2}{L2.tex}}
\end{minipage}
\begin{minipage}{0.33\textwidth}
\scalebox{0.1}{
\includetikz{L3}{L3.tex}}
\end{minipage}
\\
\begin{minipage}{0.3\textwidth}
\scalebox{0.8}{
\input{level3.tex}}
\end{minipage}
\begin{minipage}{0.3\textwidth}
\scalebox{0.8}{
\input{level4.tex}}
\end{minipage}
\begin{minipage}{0.3\textwidth}
\scalebox{0.8}{
\input{level5.tex}}
\end{minipage}
\\
\begin{minipage}{0.32\textwidth}
\scalebox{0.1}{
\includetikz{L4}{L4.tex}}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\scalebox{0.1}{
\includetikz{L5}{L5.tex}}
\end{minipage}
\begin{minipage}{0.33\textwidth}
\scalebox{0.1}{
\includetikz{L6}{L6.tex}}
\end{minipage}
\\
\vspace{-0.5cm}
\caption{p-refined hierarchy of meshes used for the slope stability problem.}
\label{fig:meshes_2_p}
\end{figure}
\vspace{-0.5cm}
In the Multilevel Monte Carlo setting, the meshes in the hierarchy are commonly referred to as `levels'. The coarsest mesh is referred to as level 0. Subsequent finer meshes are assigned the next cardinal number, e.g., level 1, level 2, $\ldots$
The number of samples to be taken on levels greater than 0 $\left(\ell>0\right)$, is proportional to the sample variance of the difference, $\V\left[\Delta P_\ell\right]$ with $\Delta P_\ell = P_\ell - P_{\ell-1}$ and $P$ a chosen quantity of interest (QoI). It is only for determining the number of samples on level 0 that the sample variance $\V\left[P_\ell\right]$ is used. In order to obtain a decreasing number of samples per increasing level, i.e., $N_0 > N_1 > \cdots > N_\text{L}$, it is necessary to have a variance reduction over the levels, i.e., $\V\left[\Delta P_1\right] > \V\left[\Delta P_2\right] > \cdots > \V\left[\Delta P_\text{L}\right]$, and an increasing cost `of one solve' per increasing level, i.e., $\mathcal{C}_0 < \mathcal{C}_1 < \cdots < \mathcal{C}_\text{L}$. This variance reduction is only obtained when a strong positive correlation is achieved between the results of two successive levels, i.e.,
\begin{equation}\label{eq:cov}
\begin{split}
\V\left[\Delta P_\ell\right] &= \V\left[P_\ell - P_{\ell-1}\right]\\
&= \V\left[P_\ell\right] + \V\left[P_{\ell-1}\right] -2\text{cov}\left(P_\ell,P_{\ell-1}\right),
\end{split}
\end{equation}
where $\text{cov}\left(P_\ell,P_{\ell-1}\right) = \rho_{\ell,\ell-1}\sqrt{\V\left[P_\ell\right]\V\left[P_{\ell-1}\right]}$ is the covariance between $P_\ell$ and $P_{\ell-1}$ with $\rho_{\ell,\ell-1}$ the correlation coefficient. The value of $\text{cov}\left(P_\ell,P_{\ell-1}\right)$ must be larger than 0 to have a large variance reduction, and hence an efficient multilevel method.
In our p-MLQMC algorithm applied to a slope stability problem, see \cite{Blondeel2020}, the model uncertainty representing the soil's cohesion is located in the elastic constitutive matrix $\mathbf{D}$. It is taken into account at the locations of the quadrature points when computing the integral in the element stiffness matrices $\mathbf{K^e}$, i.e.,
\begin{equation}\label{eq:K
\mathbf{K^e} = \int_{\Omega_e} \mathbf{B}^\text{T} \mathbf{D} \mathbf{B} d \Omega_e.
\end{equation}
This is calculated in practice as
\begin{equation}\label{eq:intpoint}
\mathbf{K^e} = \sum_{i=1}^{\card{\mathbf{q}}}\mathbf{B}_i^\text{T} \mathbf{D}_i\mathbf{B}_i \text{w}_i,
\end{equation}
where, the matrix $\mathbf{B}$ contains the derivatives of the shape functions, evaluated at the quadrature points $\mathbf{q}^i$, $\mathbf{B}_i = \mathbf{B}(\mathbf{q}^i)$, the matrix $\mathbf{D}_i = \mathbf{D}\left(\omega_i\right)$ contains the model uncertainty, and $\text{w}_i$ are the quadrature weights. The set of quadrature points $\mathbf{q}$ is expressed in a local coordinate system of the triangular reference element. The uncertainty in the matrix $\mathbf{D}\left(\omega_i\right)$ is represented by a scalar originating from the evaluation of the random field at a carefully chosen spatial location. This approach is commonly referred to as the \emph{integration point method} \cite{MATTHIES1997283}. Note that here the uncertainty is not constant in an element, i.e., $\mathbf{D}_1 \neq \mathbf{D}_2 \neq \cdots \neq \mathbf{D}_k$ in Eq.\eqref{eq:intpoint}.
The scalar used in the matrix $\mathbf{D}\left(\omega_i\right)$ originate from the evaluation of the random field at spatial location $\mathbf{x}$, in a global coordinate system of the mesh, by means of the Karhunen-Lo\`eve (KL) expansion with stochastic dimension $s$, i.e.,
\begin{equation}
Z(\mathbf{x},\omega)=\overline{Z}(\mathbf{x})+\sum_{n=1}^{s} \sqrt{\theta_n} \xi_n(\omega) b_n(\mathbf{x})\,,
\label{eq:KLExpansion}
\end{equation}
where $\overline{Z}(\mathbf{x})$ is the mean of the field and $\xi_n(\omega)$ denote i.i.d.\,standard normal random variables.
The symbols $\theta_n$ and $b_n(\mathbf{x})$ denote the eigenvalues and eigenfunctions respectively, which are the solutions of the eigenvalue problem $\int_D C(\mathbf{x},\mathbf{y})b_n({\mathbf{y}})\mathrm{d}\mathbf{y} = \theta_n b_n({\mathbf{x}})$ with a given covariance kernel $C(\mathbf{x},\mathbf{y})$. Note that in order to represent the uncertainty of the soil's cohesion in the considered slope stability problem, we do not use $Z(\mathbf{x},\omega)$ but $\text{exp}(Z(\mathbf{x},\omega))$, see \S\ref{sec:FEM}.
Our goal is to select evaluation points for Eq.\,\eqref{eq:KLExpansion}, grouped in sets $\left\lbrace\mathbf{x}_\ell\right\rbrace_{\ell=0}^\text{L}$, in order to ensure a good correlation between $P_\ell$ and $P_{\ell-1}$, i.e., such that the covariance, $\text{cov}\left(P_\ell,P_{\ell-1}\right)$, is as large as possible, see Eq.\,\eqref{eq:cov}. We distinguish three different approaches for selecting the evaluation points on the different levels, the \emph{Non-Nested Approach} (NNA), the \emph{Global Nested Approach} (GNA) and the \emph{Local Nested Approach} (LNA). All the approaches start from the given sets of quadrature points on the different levels $\left\lbrace\mathbf{q}_\ell\right\rbrace_{\ell=0}^\text{L}$. Note that the number of quadrature points per level increases, $\card{\mathbf{q}_0}<\card{\mathbf{q}_1}\cdots<\card{\mathbf{q}_\text{L}}$. Given the sets $\left\lbrace\mathbf{q}_\ell\right\rbrace_{\ell=0}^\text{L}$, we select evaluation points for the random field in a local coordinate system and group them in sets $\left\lbrace\mathbf{x}^\text{local}_\ell\right\rbrace_{\ell=0}^\text{L}$, with the condition that $ \card{\mathbf{x}_\ell^{\text{local}}} = \card{\mathbf{q}_\ell}$. The points in the sets $\left\lbrace\mathbf{x}^\text{local}_\ell\right\rbrace_{\ell=0}^\text{L}$ are then transformed to points in global coordinates, resulting in sets $\left\lbrace\mathbf{x}_\ell\right\rbrace_{\ell=0}^\text{L}$. The points belonging to $\left\lbrace\mathbf{x}_\ell\right\rbrace_{\ell=0}^\text{L}$ are then used in Eq.\,\eqref{algo:NN}. Note that with the integration point method, the spatial resolution of the field is proportional to the number of quadrature points. Increasing the number of quadrature points will result in a finer resolution of the random field.
Before elaborating further upon these approaches, we first introduce the estimator used in our p-MLQMC algorithm. The estimator is given by
\begin{equation}
Q^{\textrm{MLQMC}}_\text{L}:= \frac{1}{R_0}\sum_{r=1}^{R_0}\frac{1}{N_0}\sum_{n=1}^{N_0} P_0(\mathbf{u}_0^{(r,n)}) +
\sum_{\ell=1}^\text{L} \frac{1}{R_\ell}\sum_{r=1}^{R_\ell}\left \{ \frac{1}{N_\ell} \sum_{n=1}^{N_\ell} \left( P_\ell(\mathbf{u}_\ell^{(r,n)})-P_{\ell-1}(\mathbf{u}_\ell^{(r,n)})\right) \right \}.
\label{eq:MLQMC}
\end{equation}
It expresses the expected value of the quantity of interest on the finest level $\text{L}$ as the sample average of the quantity of interest on the coarsest level, plus a series of correction terms. In addition, MLQMC uses an average over a number of shifts $R_\ell$ on each level $\ell$ and a set of deterministic sample points given by
\begin{equation}
\mathbf{u}^{\left(r,n\right)} = {\Phi}^{-1}\left(\fracfun{\phi_2(n)\mathbf{z} + \Xi_r}\right) \quad \text{for} \quad n \in \mathbb{N},
\label{eq:qmc_pt}
\end{equation}
where ${\Phi}^{-1}$ is the inverse of the univariate standard normal cumulative distribution function, $\fracfun{x} = x - \floor{x}, x>0$, $\phi_2$ is the radical inverse function in base 2, $\mathbf{z}$ is an $s$-dimensional vector of positive integers, and $\Xi_r \in \left[0,1\right]^s$ is the random shift with $r=1,2,\ldots,R_\ell$, and $s$ the stochastic dimension. The representation of the points from Eq.\,\eqref{eq:qmc_pt} is known as a \emph{shifted rank-1 lattice rule}. The generating vector $\mathbf{z}$ was constructed with the component-by-component (CBC) algorithm with decreasing weights, $\gamma_j = 1/j^2$, see \cite{KuoGenVec}.
\section{Incorporating the uncertainty in the model}
\label{sec:approaches}
In this section, we will discuss the mechanics behind the three approaches. We will show how the evaluation points of the random field are selected on each level $\ell = \left\lbrace0,\ldots,\text{L}\right\rbrace$ for the different approaches. Each of the approaches selects the evaluation points differently. However, all approaches start from the given set of the quadrature points $\mathbf{q}_\ell$. The points $\mathbf{q}^i_\ell \in \mathbf{q}_\ell$ are represented by $\textcolor{blue}{{\triangle}}_\ell$, on a reference triangular finite element on level $\ell$. Given the sets of quadrature points, the evaluation points of the random field, represented by $\textcolor{red}{\CIRCLE}_\ell$, are selected on a reference triangular finite element on level $\ell$, and grouped in the set $\mathbf{x}_\ell^\text{local}$.
The quadrature points consist of a combination of points developed by Dunavant \cite{Dunavant} and Wandzurat \cite{WANDZURAT20031829}, see Table\,\ref{Tab:References}. The code used to generate the Wandzurat points can be found at \cite{JohnBurkardt}.
\subsection{Non-Nested Approach}
For the Non-Nested Approach, the quadrature points on each level are selected as the evaluation points of the random field, i.e., $\mathbf{x}^{\text{local}}_\ell = \mathbf{q}_\ell$ for $\ell=\{0,\ldots,\text{L}\}$. Because the sets of quadrature points are not nested over the levels, i.e., $\mathbf{q}_0 \not\subseteq \mathbf{q}_1 \not\subseteq \cdots \not\subseteq \mathbf{q}_\text{L}$, it follows that the sets of the evaluation points of the random field are not nested, i.e., $\mathbf{x}^{\text{local}}_0 \not\subseteq \mathbf{x}^{\text{local}}_1 \not\subseteq \cdots \not\subseteq \mathbf{x}^{\text{local}}_\text{L}$, and thus $\mathbf{x}_0 \not\subseteq \mathbf{x}_1 \not\subseteq \cdots \not\subseteq \mathbf{x}_\text{L}$. This approach is the most straightforward one, and is illustrated in Fig.\,\ref{fig:NonNested}. In Algorithm\,\ref{algo:NN}, we present the procedure which selects the evaluation points of the random field for each level in local coordinates and groups them in sets $\lbrace \mathbf{x}^\text{local}_\ell\rbrace_{\ell=0}^\text{L}$.
\begin{figure}[H]
\begin{adjustbox}{varwidth=1.2\textwidth,fbox}
\begin{minipage}{0.18\textwidth}
\scalebox{0.7}{
\input{level00.tex}}
\end{minipage}
\begin{minipage}{0.18\textwidth}
\scalebox{0.26}{
\includetikz{Triag_1_NN}{Triag_1_nn.tex}}
\end{minipage}
\end{adjustbox}
\begin{adjustbox}{varwidth=1.2\textwidth,fbox}
\begin{minipage}{0.18\textwidth}
\scalebox{0.7}{
\input{level11.tex}}
\end{minipage}
\begin{minipage}{.18\textwidth}
\scalebox{0.26}{
\includetikz{Triag_2_NN}{Triag_2_nn.tex}}
\end{minipage}
\end{adjustbox}
\begin{adjustbox}{varwidth=1.2\textwidth,fbox}
\begin{minipage}{0.18\textwidth}
\scalebox{0.7}{
\input{level22.tex}}
\end{minipage}
\begin{minipage}{.18\textwidth}
\scalebox{0.26}{
\includetikz{Triag_3_NN}{Triag_3_nn.tex}}
\end{minipage}
\end{adjustbox}
\begin{adjustbox}{varwidth=1.2\textwidth,fbox}
\begin{minipage}{0.18\textwidth}
\scalebox{0.7}{
\input{level33.tex}}
\end{minipage}
\begin{minipage}{.18\textwidth}
\scalebox{0.26}{
\includetikz{Triag_4_NN}{Triag_4_nn.tex}}
\end{minipage}
\end{adjustbox}
\begin{adjustbox}{varwidth=1.2\textwidth,fbox}
\begin{minipage}{0.18\textwidth}
\scalebox{0.7}{
\input{level44.tex}}
\end{minipage}
\begin{minipage}{.18\textwidth}
\scalebox{0.26}{
\includetikz{Triag_5_NN}{Triag_5_nn.tex}}
\end{minipage}
\end{adjustbox}
\vspace{0.5cm}
\begin{adjustbox}{varwidth=1.2\textwidth,fbox}
\begin{minipage}{0.2\textwidth}
\scalebox{0.7}{
\input{level55.tex}}
\end{minipage}
\begin{minipage}{0.2\textwidth}
\scalebox{0.26}{
\includetikz{Triag_6_NN}{Triag_6_nn.tex}}
\end{minipage}
\end{adjustbox}
\begin{adjustbox}{varwidth=1.2\textwidth,fbox}
\begin{minipage}{0.2\textwidth}
\scalebox{0.7}{
\input{level66.tex}}
\end{minipage}
\begin{minipage}{.2\textwidth}
\scalebox{0.26}{
\includetikz{Triag_7_NN}{Triag_7_nn.tex}}
\end{minipage}
\end{adjustbox}
\vspace{-0.8cm}
\caption{Locations of the quadrature points \textcolor{blue}{$\triangle$} and of the evaluation points of the random field \textcolor{red}{$\CIRCLE$} on a reference triangular element in NNA.}
\label{fig:NonNested}
\end{figure}
\begin{fullwidth}[width=\linewidth+3cm,leftmargin=-1.5cm,rightmargin=-1.5cm]
\scalebox{0.8}{
\begin{algorithm}[H]
\KwData{\\ Max level $\text{L}$, Set of quadrature points per level $\left\lbrace \mathbf{q}_\ell \right\rbrace_{\ell=0}^\text{L}$}
$\ell \gets L$\;
\While{$\ell \geq 0$}{
$\mathbf{x}^{\text{local}}_\ell \leftarrow \mathbf{q}_\ell$\;
$\ell \leftarrow \ell - 1$ \;
}
return $\left\lbrace \mathbf{x}^{\text{local}}_\ell \right\rbrace_{\ell=0}^\text{L}$
\vspace{0.4cm}
\caption{Generation of the evaluation points of the random field in NNA.}
\label{algo:NN}
\end{algorithm}
}
\end{fullwidth}
\subsection{Global Nested Approach}
\label{sec:gna}
\vspace{-0.3cm}
For the Global Nested Approach, we proceed in a different way. All the levels are correlated with each other. The sets of evaluation points of the random field are chosen such that they are nested over all the levels, i.e., $\mathbf{x}^{\text{local}}_0 \subseteq \mathbf{x}^{\text{local}}_1 \subseteq \cdots \subseteq \mathbf{x}^{\text{local}}_\text{L}$, and thus $\mathbf{x}_0 \subseteq \mathbf{x}_1 \subseteq \cdots \subseteq \mathbf{x}_\text{L}$. For GNA, the sets of evaluation points are not equal to the sets of quadrature points, except on the finest level, i.e., $\mathbf{x}^{\text{local}}_\ell \neq \mathbf{q}_\ell$ $\text{for}$ $\ell=\{0,\ldots,\text{L}-1\}$ and $\mathbf{x}^{\text{local}}_\text{L} = \mathbf{q}_\text{L}$. The approach for selecting the evaluation points of the random field is as follows. The quadrature points on the finest level $\text{L}$ are selected as the evaluation points of the random field, i.e., $\mathbf{x}^{\text{local}}_\text{L} = \mathbf{q}_\text{L}$. The points selected for $\mathbf{x}^{\text{local}}_\ell$ on levels $\ell=\{\text{L}-1, \ldots, 0 \} $, consist of a number of points $\card{\mathbf{q}_\ell}$, which are selected from the set $\mathbf{x}^{\text{local}}_{\ell+1}$, such that each selected point is the closest neighbor of a point of the set $\mathbf{q}_\ell$, i.e., $\mathbf{x}_\ell^\text{local} := \underset{\underset{\card{\mathbf{x}_\ell^\text{local}}=\card{\mathbf{q}_\ell} }{\mathbf{x}_\ell^\text{local} \subseteq \mathbf{x}_{\ell+1}^\text{local}}}{\mathrm{argmin}} \textbf{D}\left(\mathbf{x}_{\ell+1}^\text{local},\mathbf{q}_\ell\right)$, where $\textbf{D}\left(\mathbf{a},\mathbf{b}\right):= \underset{a\in \mathbf{a}}{\sum}\textbf{d}\left(a,\mathbf{b}\right)$ is the distance between two sets, and where $\textbf{d}\left(a,\textbf{b}\right):= \text{inf}\left\lbrace\text{d}\left(a,b\right)|b\in\mathbf{b}\right\rbrace$ is the minimal distance between a point and a set, with $\text{d}\left(a,b\right)$ the Euclidean distance between two points.
This is illustrated in Fig.\,\ref{fig:GlobalNested}. The procedure used to select the evaluation points for GNA is given in Algorithm\,\ref{algo:Global}.
\begin{figure}[H]
\vspace{-0.5cm}
\begin{adjustbox}{varwidth=1.0\textwidth,fbox,center}
\begin{minipage}{0.23\textwidth}
\scalebox{0.65}{
\input{level00.tex}}
\end{minipage}
\begin{minipage}{0.23\textwidth}
\scalebox{0.65}{
\input{level11.tex}}
\end{minipage}
\begin{minipage}{0.23\textwidth}
\scalebox{0.65}{
\input{level22.tex}}
\end{minipage}
\begin{minipage}{0.23\textwidth}
\scalebox{0.65}{
\input{level33.tex}}
\end{minipage}
\begin{minipage}{0.23\textwidth}
\scalebox{0.24}{
\includetikz{Triag_1}{Triag_1_Glob.tex}}
\end{minipage}
\begin{minipage}{.23\textwidth}
\scalebox{0.24}{
\includetikz{Triag_2}{Triag_2_Glob.tex}}
\end{minipage}
\begin{minipage}{.23\textwidth}
\scalebox{0.24}{
\includetikz{Triag_3}{Triag_3_Glob.tex}}
\end{minipage}
\begin{minipage}{.23\textwidth}
\scalebox{0.24}{
\includetikz{Triag_4}{Triag_4_Glob.tex}}
\end{minipage}
\centering
\begin{minipage}{0.25\textwidth}
\scalebox{0.65}{
\input{level44.tex}}
\end{minipage}
\begin{minipage}{0.25\textwidth}
\scalebox{0.65}{
\input{level55.tex}}
\end{minipage}
\begin{minipage}{0.25\textwidth}
\scalebox{0.65}{
\input{level66.tex}}
\end{minipage}
\begin{minipage}{.25\textwidth}
\scalebox{0.24}{
\includetikz{Triag_5}{Triag_5_Glob.tex}}
\end{minipage}
\begin{minipage}{0.25\textwidth}
\scalebox{0.24}{
\includetikz{Triag_6}{Triag_6_Glob.tex}}
\end{minipage}
\begin{minipage}{.25\textwidth}
\scalebox{0.24}{
\includetikz{Triag_7}{Triag_7_Glob.tex}}
\end{minipage}
\end{adjustbox}
\vspace{-0.6cm}
\caption{Locations of the quadrature points \textcolor{blue}{$\triangle$} and of the evaluation points of the random field \textcolor{red}{$\CIRCLE$} on a reference triangular element in GNA.}
\label{fig:GlobalNested}
\end{figure}
\vspace{-0.6cm}
\vspace{-0.4cm}
\begin{fullwidth}[width=\linewidth+3cm,leftmargin=-1.5cm,rightmargin=-1.5cm]
\scalebox{0.8}{
\begin{algorithm}[H]
\KwData{\\ Max level $\text{L}$, Set of quadrature points per level $\left\lbrace \mathbf{q}_\ell \right\rbrace_{\ell=0}^\text{L}$ }
$\mathbf{x}^{\text{local}}_\text{L} \leftarrow \mathbf{q}_\text{L}$\;
$\ell \gets \text{L}-1$\;
\While{$\ell \geq 0$}{
$i \gets 1 $ \;
$\mathbf{x}^{\text{local}}_\ell \gets \varnothing$ \;
\While{$i \leq \card{\mathbf{q}_\ell} $}{
Find the point $p \in \mathbf{x}^{\text{local}}_{\ell+1}$ ,which is not in $\mathbf{x}^{\text{local}}_\ell$, closest to $\mathbf{q}^i_\ell$ \;
$\mathbf{x}^{\text{local}}_\ell \leftarrow \mathbf{x}^{\text{local}}_\ell \cup \{p\}$; // Add it to the array \\
$i \leftarrow i + 1$ \;
}
$\ell \leftarrow \ell - 1$ \;
}
return $\left\lbrace \mathbf{x}^{\text{local}}_\ell \right\rbrace_{\ell=0}^\text{L}$
\vspace{0.4cm}
\caption{Generation of the evaluation points of the random field in GNA.}
\label{algo:Global}
\end{algorithm}}
\vspace{-1.8cm}
\end{fullwidth}
\vspace{-1.3cm}
\subsection{Local Nested Approach}
\vspace{-0.3cm}
The Local Nested Approach is as follows. Rather than correlating all the levels with each other, we now correlate them two-by-two. Each level $\ell=\{1,\ldots,\text{L}\}$ has two sets of evaluation points $\mathbf{x}_{\ell,\text{coarse}}$ and $\mathbf{x}_{\ell,\text{fine}}$, which are nested, i.e., $\mathbf{x}_{\ell,\text{coarse}} \subseteq \mathbf{x}_{\ell,\text{fine}}$. The points in these sets are used to generate a coarse and a fine representation of the random field on level $\ell$. NNA and GNA have only one set of points per level, and thus only one representation of the random field per level. The coarse representation of the random field essentially acts as a representation of the field on level $\ell-1$. This is because $\mathbf{q}_{\ell,\text{coarse}} = \mathbf{q}_{\ell-1,\text{fine}}$. The selection process is as follows. For each level $\ell =\left\lbrace0,\ldots,\text{L}\right\rbrace$, $\mathbf{x}^{\text{local}}_{\ell,\text{fine}} = \mathbf{q}_{\ell,\text{fine}}$. The points in $\mathbf{x}^{\text{local}}_{\ell,\text{coarse}}$ are selected according to the same methodology as in GNA, i.e., they are selected from the set $\mathbf{x}^{\text{local}}_{\ell,\text{fine}}$, such that each selected point is the closest neighbor to a point of the set $\mathbf{q}_{\ell,\text{coarse}}$. This is illustrated in Fig.\,\ref{fig:Local}. The main advantage of this approach is level exchangeability and extensibility. With exchangeability we mean that if one pair of correlated levels, say $\tau$ and $\tau-1$, exhibits a `sub-optimal' value of $\V\left[\Delta P_\tau\right]$ with respect to the variances $\V\left[\Delta P_\ell\right]$ on other levels, this pair can easily be exchanged against another newly computed pair with a different set of quadrature points. This is in contrast with GNA where the whole hierarchy needs to be recomputed. With level extensibility we mean that if for a user requested tolerance $\varepsilon$ and maximum level $\text{L}$, the tolerance is not reached, the hierarchy can easily be extended by supplying the extra needed level(s) and reusing the previously computed samples. In case of GNA, the whole hierarchy needs to be recomputed with extra level(s) and the previously computed samples cannot be reused. This level extensibility is the major advantage of LNA over GNA.
We will now discuss how the computational cost for each of the approaches can be determined. This is done regardless of the number of samples. The total computational cost is split in an offline part, i.e., the cost of computing the eigenvalues and eigenvectors of the random fields, and an online part, i.e., the cost of computing point evaluations of the random fields at the evaluation points. The offline cost is only accounted for at startup, while the online cost is accounted for when computing each sample.
For NNA, the offline cost is equal to $\sum_{\ell=0}^\text{L} C^{\text{eig}}_\ell$, with $C^{\text{eig}}_\ell$ the cost of computing the eigenvalues and eigenvectors of the random field on level $\ell$. For LNA, this cost is equal to $\sum_{\ell=0}^\text{L} C^{\text{eig}}_{\ell,\text{fine}}$. Only the eigenvalues and eigenvectors of the fine representation of the fields on each level need be computed. Note that $C^{\text{eig}}_\ell = C^{\text{eig}}_{\ell,\text{fine}}$. For GNA, the offline cost equals $\sum_{\ell=0}^\text{L} C^{\text{eig}}_\ell$. In case of GNA, the choice could be made to compute the eigenvectors and eigenfunctions only on the finest level L, resulting in the offline cost $C^{\text{eig}}_\text{L}$. This is only possible because of the property $\mathbf{x}_0 \subseteq \mathbf{x}_\ell \subseteq \cdots \subseteq \mathbf{x}_\text{L}$, see \S\,\ref{sec:gna}. In practice this will not be done, because of a drastic increase of the online cost, see further on. The online cost for NNA on levels $\ell > 0$ is equal to the cost of computing point evaluations of the random field on level $\ell$, $C^{\text{samp}}_\ell$, and on level $\ell-1$, $C^{\text{samp}}_{\ell-1}$. This yields a total cost per sample equal to $C^{\text{samp}}_\ell+C^{\text{samp}}_{\ell-1}$. For LNA, the online cost on level $\ell > 0$ is only equal to the cost of computing point evaluations of the fine representation of the field on level $\ell$, $C^{\text{samp}}_{\ell,\text{fine}}$. In order to represent the coarse field on level $\ell$, a restriction of the point evaluations of the fine random field on level $\ell$ are taken. This is because of the property $\mathbf{x}_{\ell,\text{coarse}} \subseteq \mathbf{x}_{\ell,\text{fine}}$, see Fig.\,\ref{fig:Local}. There is no cost associated with the restriction. Note that $C_\ell^\text{samp}=C_{\ell,\text{fine}}^\text{samp}$. For GNA, the online cost on level $\ell>0$ amounts to $C^{\text{samp}}_\ell$, if the eigenfunctions and eigenvectors have been computed for each level $\ell$. Then, the online cost is equal to the one of LNA. However, if the eigenfunctions and eigenvalues have only been computed on the finest level, the online cost for GNA equals $C^{\text{samp}}_\text{L}$ regardless of the level. Point evaluations of the random field are computed on level L and restricted to the desired level $\ell$. In practice this is not done because of the much higher cost this incurs, $C^{\text{samp}}_\text{L} \gg C^{\text{samp}}_\ell$.
\begin{figure}[H]
\vspace{-0.7cm}
\begin{minipage}{0.25\textwidth}
\scalebox{0.8}{
\input{level00.tex}}
\end{minipage}
\begin{adjustbox}{varwidth=0.90\textwidth,fbox}
\begin{minipage}{0.24\textwidth}
\scalebox{0.7}{
\input{level_fine.tex}}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\scalebox{0.26}{
\input{Triag_1_nn.tex}}
\end{minipage}
\end{adjustbox}
\vspace{0.1cm}
\begin{minipage}{0.45\textwidth}
\scalebox{0.8}{
\input{level11.tex}}
\end{minipage}
\vspace{0.1cm}
\begin{minipage}{0.3\textwidth}
\scalebox{0.8}{
\input{level22.tex}}
\end{minipage}
\begin{adjustbox}{varwidth=0.90\textwidth,fbox}
\begin{minipage}{0.24\textwidth}
\scalebox{0.7}{
\input{level_coarse.tex}}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\scalebox{0.7}{
\input{level_fine.tex}}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\scalebox{0.26}{
\input{Triag_21_Loc.tex}}
\end{minipage}
\begin{minipage}{.24\textwidth}
\scalebox{0.26}{
\input{Triag_22_Loc.tex}}
\end{minipage}
\end{adjustbox}
\begin{adjustbox}{varwidth=0.90\textwidth,fbox}
\begin{minipage}{0.24\textwidth}
\scalebox{0.7}{
\input{level_coarse.tex}}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\scalebox{0.7}{
\input{level_fine.tex}}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\scalebox{0.26}{
\input{Triag_32_Loc.tex}}
\end{minipage}
\begin{minipage}{.24\textwidth}
\scalebox{0.26}{
\input{Triag_33_Loc.tex}}
\end{minipage}
\end{adjustbox}
\vspace{0.1cm}
\begin{minipage}{0.45\textwidth}
\scalebox{0.8}{
\input{level33.tex}}
\end{minipage}
\vspace{0.1cm}
\begin{minipage}{0.3\textwidth}
\scalebox{0.8}{
\input{level44.tex}}
\end{minipage}
\begin{adjustbox}{varwidth=0.90\textwidth,fbox}
\begin{minipage}{0.24\textwidth}
\scalebox{0.7}{
\input{level_coarse.tex}}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\scalebox{0.7}{
\input{level_fine.tex}}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\scalebox{0.26}{
\input{Triag_43_Loc.tex}}
\end{minipage}
\begin{minipage}{.24\textwidth}
\scalebox{0.26}{
\input{Triag_44_Loc.tex}}
\end{minipage}
\end{adjustbox}
\begin{adjustbox}{varwidth=0.90\textwidth,fbox}
\begin{minipage}{0.24\textwidth}
\scalebox{0.7}{
\input{level_coarse.tex}}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\scalebox{0.7}{
\input{level_fine.tex}}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\scalebox{0.26}{
\input{Triag_54_Loc.tex}}
\end{minipage}
\begin{minipage}{.24\textwidth}
\scalebox{0.26}{
\input{Triag_55_Loc.tex}}
\end{minipage}
\end{adjustbox}
\vspace{0.1cm}
\begin{minipage}{0.45\textwidth}
\scalebox{0.8}{
\input{level55.tex}}
\end{minipage}
\vspace{0.1cm}
\begin{minipage}{0.3\textwidth}
\scalebox{0.8}{
\input{level66.tex}}
\end{minipage}
\begin{adjustbox}{varwidth=0.90\textwidth,fbox}
\begin{minipage}{0.24\textwidth}
\scalebox{0.7}{
\input{level_coarse.tex}}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\scalebox{0.7}{
\input{level_fine.tex}}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\scalebox{0.26}{
\input{Triag_65_Loc.tex}}
\end{minipage}
\begin{minipage}{.24\textwidth}
\scalebox{0.26}{
\input{Triag_66_Loc.tex}}
\end{minipage}
\end{adjustbox}
\begin{adjustbox}{varwidth=0.90\textwidth,fbox}
\begin{minipage}{0.24\textwidth}
\scalebox{0.7}{
\input{level_coarse.tex}}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\scalebox{0.7}{
\input{level_fine.tex}}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\scalebox{0.26}{
\input{Triag_76_Loc.tex}}
\end{minipage}
\begin{minipage}{.24\textwidth}
\scalebox{0.26}{
\input{Triag_77_Loc.tex}}
\end{minipage}
\end{adjustbox}
\vspace{-0.5cm}
\caption{Locations of the quadrature points \textcolor{blue}{$\triangle$} and of the evaluation points of the random field \textcolor{red}{$\CIRCLE$} on a reference triangular element in LNA.}
\label{fig:Local}
\end{figure}
\vspace{-0.3cm}
The procedure used to generate the point set for LNA is given in Algorithm\,\ref{algo:Local}. For LNA, each level has two representation of the random field, a coarse and a fine one, with sets $\mathbf{x}_{\ell,\text{coarse}}$ and $\mathbf{x}_{\ell,\text{fine}}$.
\begin{fullwidth}[width=\linewidth+3cm,leftmargin=-1.5cm,rightmargin=-1.5cm]
\vspace{2pt}
\scalebox{0.8}{
\begin{algorithm}[H]
\KwData{\\ Max level $\text{L}$, Set of quadrature points per level $\left\lbrace \mathbf{q}_{\ell,\text{coarse}} \right\rbrace_{\ell=0}^\text{L}$ and $\left\lbrace \mathbf{q}_{\ell,\text{fine}} \right\rbrace_{\ell=0}^\text{L}$}
$\ell \gets \text{L}$ \;
\While{$\mathcal{\ell}>0$}{
$\mathbf{x}^{\text{local}}_{\ell,\text{fine}} \gets \mathbf{q}_{\ell,\text{fine}}$\;
$\mathbf{x}^{\text{local}}_{\ell,\text{coarse}} \gets \varnothing$ \;
$i \gets 1 $ \;
$k \gets \card{\mathbf{q}_{\ell,\text{coarse}}}$\;
\While{$i \leq k$}{
Find the point $p \in \mathbf{x}^{\text{local}}_{\ell,\text{fine}}$ ,which is not in $\mathbf{x}^{\text{local}}_{\ell,\text{coarse}}$, closest to $\mathbf{q}^i_{\ell,\text{coarse}}$ \;
$\mathbf{x}^{\text{local}}_{\ell,\text{coarse}} \leftarrow \mathbf{x}^{\text{local}}_{\ell,\text{coarse}} \cup \{p\}$; // Add it to the array \\
$i \leftarrow i + 1$ \;
}
$\ell \gets \ell-1$\;
}
$\mathbf{x}^{\text{local}}_{0,\text{fine}} \leftarrow \mathbf{q}_{0,\text{fine}}$\;
return $\left\lbrace \mathbf{x}^{\text{local}}_{0,\text{fine}}, \mathbf{x}^{\text{local}}_{1,\text{fine}}, \mathbf{x}^{\text{local}}_{1,\text{coarse}}, \mathbf{x}^{\text{local}}_{2,\text{fine}}, \mathbf{x}^{\text{local}}_{2,\text{coarse}}, \ldots \right\rbrace$
\vspace{0.4cm}
\caption{Generation of the evaluation points of the random field in LNA.}
\label{algo:Local}
\end{algorithm}}
\vspace{-0.9cm}
\end{fullwidth}
\section{Model Problem}
\label{sec:FEM}
The model problem we consider for benchmarking the three approaches consists of a slope stability problem where the soil's cohesion has a spatially varying uncertainty \cite{Whenham}. In a slope stability problem the safety of the slope can be assessed by evaluating the vertical displacement of the top of the slope when sustaining its own weight. We consider the displacement in the plastic domain, which is governed by the Drucker--Prager yield criterion. A small amount of isotropic linear hardening is taken into account for numerical stability reasons. Because of the nonlinear stress-strain relation arising in the plastic domain, a Newton--Raphson iterative solver is used. In order to compute the displacement in a slope stability problem, an incremental load approach is used, i.e., the total load resulting from the slope's weight is added in steps starting with a force of $0\,\mathrm{N}$ until the downward force resulting from the slope's weight is reached. This approach results in the following system of equations for the displacement,
\begin{equation}\label{Displacement_eq_plast}
\mathbf{K} \Delta\mathbf{u} = \mathbf{r},
\end{equation}
where $\Delta\mathbf{u}$ stands for the displacement increment, $\mathbf{K}$ the global stiffness matrix resulting from the assembly of element stiffness matrices $\mathbf{K^e}$, see Eq.\,\eqref{eq:K}. The vector $\mathbf{r}$ is the residual,
\begin{equation}
\mathbf{r}=\mathbf{f}+\Delta\mathbf{f}-\mathbf{k},
\end{equation}
where $\mathbf{f}$ stands for the sum of the external force increments applied in the previous steps, $\Delta\mathbf{f}$ for the applied load increment of the current step and $\mathbf{k}$ for the internal force resulting from the stresses. For a more thorough explanation on the methods used to solve the slope stability problem we refer to \cite[Chapter 2 $\S$4 and Chapter 7 $\S$3 and $\S$4]{Borst}.
The mesh hierarchy shown in Fig.\,\ref{fig:meshes_2_p} is generated by using a combination of the open source mesh generator GMSH \cite{GMSH} and \textsc{Matlab} \cite{MATLAB:2017}. Table\,\ref{Tab:References} lists the number of elements (Nel), degrees of freedom (DOF), element order per level (Order), the number of quadrature points per element (Nquad), and the reference for the quadrature points (Ref) per level for p-MLQMC. The number of quadrature points is chosen as to increase the spatial resolution of the field per increasing level, and to ensure numerical stability of the computations of the displacement in the plastic domain. In this paper we consider two-dimensional uniform, Lagrange triangular elements.
\begin{table}[H]
\caption{Number of elements, degrees of freedom, element order and number of quadrature points for the model problem.}
\label{Tab:References}
\centering
\scalebox{1.0}{
\begin{tabular}{cccccc}
\toprule
\multicolumn{6}{c}{p-MLQMC} \\
\cmidrule(rl{4pt}){1-6}
{Level} & {Nel} & {DOF} & {Order} & {Nquad} & Ref.\\
0 & 33 & 160 &2 &16 &\cite{Dunavant} \\
1 & 33 & 338 &3 &19 &\cite{Dunavant} \\
2 & 33 & 582 &4 &28 &\cite{Dunavant}\\
3 & 33 & 892 &5 &37 &\cite{Dunavant}\\
4 & 33 & 1268 &6 &61 &\cite{Dunavant}\\
5 & 33 & 1710 &7 &73 &\cite{Dunavant}\\
6 & 33 & 2218 &8 &126 &\cite{WANDZURAT20031829}\\
\bottomrule
\end{tabular}}
\end{table}
We consider the vertical displacement in meters of the upper left node of the model as a quantity of interest (QoI). This is depicted in Fig.\,\ref{fig:QoI2} by the arrow.
\begin{figure}[H]
\sidecaption
\scalebox{0.1}{
\input{p_0_S_node.tex}}
\caption{The vertical displacement of the upper left node as the QoI, indicated with an arrow.}
\label{fig:QoI2}
\end{figure}
The uncertainty of the soil's cohesion is represented by means of a lognormal random field. This field is obtained by applying the exponential to the field obtained in Eq.\,\eqref{eq:KLExpansion}, $Z_{\text{lognormal}}(\mathbf{x},\omega)=\exp(Z(\mathbf{x},\omega))$.
For the covariance Kernel of the random field, we use the Mat\'ern covariance kernel,
\begin{equation}\label{eq:covariance_kernel}
C(\mathbf{x},\mathbf{y}):= \sigma^2 \frac{1}{2^{\nu-1} \Gamma\left(\nu\right)}\left(\sqrt{2\nu}\dfrac{\norm{\mathbf{x}-\mathbf{y}}_2}{\lambda}\right)^\nu K_\nu \left(\sqrt{2\nu}\dfrac{\norm{\mathbf{x}-\mathbf{y}}_2}{\lambda}\right) \,,
\end{equation}
with $\nu = 2.0$ the smoothness parameter, $K_\nu$ the modified Bessel function of the second kind, $\sigma^2 = 1$ the variance and $\lambda = 0.3$ the correlation length. The characteristics of the lognormal distribution used to represent the uncertainty of the soil's cohesion are as follows: a mean of $8.02\,\mathrm{kPa}$ and a standard deviation of $400\,\mathrm{Pa}$. The spatial dimensions of the slope are: a length of $20\,\mathrm{m}$, a height of $14\,\mathrm{m}$ and a slope angle of $\ang{30}$. The material characteristics are: a Young's modulus of $30\,\mathrm{MPa}$, a Poisson ratio of $0.25$, a density of $1330\,\mathrm{kg/m^3}$ and a friction angle of $\ang{20}$. Plane strain is considered for this problem. The number of stochastic dimensions considered for the generation of the Gaussian random field is $s \!= 400$, see Eq.\,\eqref{eq:KLExpansion}. With a value $s \!= 400$, 99$\%$ of the variability of the random field is accounted for.
The stochastic part of our simulations was performed with the Julia packages \textbf{MultilevelEstimators.jl}, see \cite{PieterJanGit1}, and \textbf{GaussianRandomFields.jl}, see \cite{PieterJanGit2}. The Finite Element code used is an in-house \textsc{Matlab} code developed by the Structural Mechanics Section of the KU Leuven. All the results have been computed on a workstation equipped with 2 physical cores, Intel Xeon E5-2680 v3 CPU's, each with 12 logical cores, clocked at 2.50 GHz, and a total of 128 GB RAM.
\vspace{-0.6cm}
\section{Numerical Results}
\vspace{-0.4cm}
In this section we present our numerical results obtained with the p-MLQMC method.
\vspace{-1.2cm}
\subsection{Displacement}
\vspace{-0.2cm}
In Fig.\,\ref{fig:Disp}, we show the displacement of the mesh and the nodes for a single sample of the random field on different levels. For better visualization, the displacement of the mesh and nodes in the figure have been exaggerated by a factor 20. The value of the QoI is listed beneath each figure depicting the displacement.
\vspace{-0.6cm}
\begin{figure}[H]
\hspace{0.08cm}
\begin{minipage}{0.3\textwidth}
\scalebox{0.75}{
\input{level0.tex}}
\end{minipage}
\begin{minipage}{0.3\textwidth}
\scalebox{0.75}{
\input{level1.tex}}
\end{minipage}
\begin{minipage}{0.3\textwidth}
\scalebox{0.7}{
\input{level2.tex}}
\end{minipage}
\\
\begin{minipage}{0.32\textwidth}
\scalebox{0.1}{
\input{Displacement/M0_S0}}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\scalebox{0.1}{
\input{Displacement/M1_S1}}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\scalebox{0.1}{
\input{Displacement/M2_S2}}
\end{minipage}
\\
\begin{minipage}{0.3\textwidth}
\scalebox{0.75}{
\input{level3.tex}}
\end{minipage}
\begin{minipage}{0.3\textwidth}
\scalebox{0.75}{
\input{level4.tex}}
\end{minipage}
\begin{minipage}{0.3\textwidth}
\scalebox{0.75}{
\input{level5.tex}}
\end{minipage}
\\
\begin{minipage}{0.32\textwidth}
\scalebox{0.1}{
\input{Displacement/M3_S3}}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\scalebox{0.1}{
\input{Displacement/M4_S4}}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\scalebox{0.1}{
\input{Displacement/M5_S5}}
\end{minipage}
\caption{Displacement of the mesh and QoI for different samples of the random field.}
\label{fig:Disp}
\end{figure}
\subsection{Variance and Expected value over the Levels}
\label{sec:var}
In Fig.\,\ref{fig:variance} we show the sample variance over the levels $\V\left[P_\ell\right]$, the sample variance of the difference over the levels $ \V\left[\Delta P_\ell\right]$, the expected value over the levels $\EE\left[P_\ell\right]$ and the expected value of the difference over the levels $ \EE\left[\Delta P_\ell\right]$.
\vspace{-0.5cm}
\begin{figure}[H]
\hspace{-0.7cm}
\begin{minipage}{.6\textwidth}
\scalebox{0.7}{
\input{Rates/V_case}}
\end{minipage}
\begin{minipage}{.4\textwidth}
\scalebox{0.7}{
\input{Rates/E_case}}
\end{minipage}
\vspace{-0.3cm}
\begin{minipage}{0.4\textwidth}
\hspace{-1.3cm}
\scalebox{0.9}{
\input{Rates/General_Legend}}
\end{minipage}
\vspace{-3.9cm}
\caption{Variance and Expected Value over the levels.}
\label{fig:variance}
\end{figure}
\vspace{-0.6cm}
As expected with multilevel methods, we observe that $\EE\left[P_\ell\right]$ remains constant over the levels, while $ \EE\left[\Delta P_\ell\right]$ decreases with increasing level. This is the case for all approaches we introduced in \S\ref{sec:approaches}.
As explained in \S\ref{sec:background}, multilevel methods are based on a variance reduction by means of a hierarchical refinement of Finite Element meshes. In practice this means that the sample variance $\V\left[P_\ell\right]$ remains constant across the levels, while the sample variance of the difference over the levels $ \V\left[\Delta P_\ell\right]$ decreases for increasing level.
This is indeed what we observe for GNA and LNA. For NNA we observe that $ \V\left[\Delta P_\ell\right]$ does not decrease.
From Fig.\,\ref{fig:variance}, we can conclude that the choice of using nested over non-nested spatial locations over the levels as evaluation points for the random field greatly improves the behavior of $ \V\left[\Delta P_\ell\right]$. This influence stems from a `bad' correlation between the results of two successive levels in the NNA case, see Eq.\,\eqref{eq:cov}. We will show in the next section that the number of samples per level required by NNA will be larger than GNA or LNA. This will impact the total runtime.
\vspace{-.8cm}
\subsection{Number of Samples}
In Fig.\,\ref{fig:samples}, we show the number of samples for the three approaches for thirteen different tolerances on the RMSE. These numbers do not include the number of shifts, which value is taken to be $R_\ell=10$ $\text{for}$ $\ell=\left\lbrace 0,\ldots,\text{L}\right\rbrace$.
\begin{figure}[H]
\hspace{-3.cm}
\begin{minipage}{0.3\textwidth}
\scalebox{.68}{
\input{samples/samples_1}
}
\end{minipage}
\hspace{2.2cm}
\begin{minipage}{0.3\textwidth}
\scalebox{.68}{
\input{samples/samples_2}
}
\end{minipage}
\hspace{1.5cm}
\begin{minipage}{0.3\textwidth}
\scalebox{.68}{
\input{samples/samples_3}
}
\end{minipage}
\begin{minipage}[b]{1.\textwidth}
\hspace{-2.8cm}
\scalebox{1.}{
\input{samples/General_Legend_case2}}
\end{minipage}
\vspace{-5.2cm}
\caption{Number of samples for the three implementations for a given tolerance $\varepsilon$.}
\label{fig:samples}
\end{figure}
We observe that for a given tolerance $\varepsilon$, the number of samples for NNA for levels greater than 0 is higher than for GNA and LNA. This is due to the slow decrease of $ \V\left[\Delta P_\ell\right]$, see Fig.\,\ref{fig:variance}.
\vspace{-0.9cm}
\subsection{Runtimes}
\vspace{-0.5cm}
We show the absolute and relative runtime as a function of the user requested tolerance $\varepsilon$ on the RMSE for the different implementations in Fig.\,\ref{fig:runtime_case}.
\vspace{-.3cm}
\begin{figure}[H]
\hspace{-2.8cm}
\begin{minipage}{0.45\textwidth}
\scalebox{.84}{
\input{Runtime/run_time_case_MEAN}
}
\end{minipage}
\hspace{2.4cm}
\begin{minipage}{0.45\textwidth}
\scalebox{.84}{
\input{Runtime/run_time_case_Normalized_MEAN}
}
\end{minipage}
\begin{minipage}[b]{0.6\textwidth}
\scalebox{1.}{
\hspace{-0.5cm}
\input{Runtime/General_Legend_case2}}
\end{minipage}
\vspace{-5.3cm}
\caption{Runtimes in function of requested user tolerance.}
\label{fig:runtime_case}
\end{figure}
The results for the absolute runtime are expressed in seconds. For the relative runtime, we have normalized the computational cost of all three approaches such that the results for LNA for each tolerance have unity cost. For both the results of the absolute and the relative runtime, we show the average, computed over three independent simulation runs, together with the minimum and maximum bounds.
We observe that LNA achieves a speedup up to a factor 5 with respect to NNA and a factor 1.5 with respect to GNA. GNA also outperforms NNA in terms of lower computational cost. This better performance of GNA and LNA is due to a lower number of samples per level, resulting from a better correlation between the successive levels. We can thus state that the Local and Global Nested Approaches achieve a lower computational cost with respect to the Non-Nested Approach.
\section{CONCLUSIONS}
In this work, we investigated how the spatial locations used for the evaluation of the random field by means of a Karhunen-Lo\`eve expansion impact the performance of the p-MLQMC method. We distinguished three different approaches, the \emph{Non-Nested Approach}, the \emph{Global Nested Approach} and the \emph{Local Nested Approach}. We demonstrated that the choice of the evaluation points of the random field impacts the variance reduction over the levels $\V\left[\Delta P_\ell \right]$. We showed that the Global and Local Nested approaches exhibit a much better decrease of $\V\left[\Delta P_\ell \right]$ due to a better correlation between the levels than the Non-Nested Approach. This leads to a lower number of samples for the Nested Approaches and thus a lower total runtime for a given tolerance. Furthermore we have shown that the Local Nested Approach has the additional properties of level exchangeability and extensibility with respect to the Global Nested Approach. By correlating the levels two-by-two, in the Local Nested Approach, one pair of levels can easily be exchanged for another computed pair, if needed. In addition, the hierarchy can also easily be extended by adding a newly computed pair. When exchanging a pair of levels or extending the hierarchy, the previously computed samples can be reused. The Global Nested Approach does not have these properties. There, the whole mesh hierarchy needs to be recomputed with the extra added and/or exchanged level(s). The previously computed samples cannot be reused. In addition to these properties, the Local Nested Approach also has a smaller runtime for a given tolerance on the RMSE than the Global Nested Approach. Based on the results in this work, we conclude that for selecting the random field evaluation points, an approach where the points are nested across the mesh hierarchy provides superior results compared to a non-nested approach. Of the nested approaches we consider in this paper, the Local Nested Approach has the smallest computational runtime, outperforming the Non-Nested Approach by a factor 5.
\section*{Acknowledgments}
The authors gratefully acknowledge the support from the Research Council of KU Leuven through project C16/17/008 ``Efficient methods for large-scale PDE-constrained optimization in the presence of uncertainty and complex technological constraints". The computational resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation - Flanders (FWO) and the Flemish Government – department EWI.
|
3,212,635,537,428 | arxiv | \section{Introduction}
\subsection{Background}
Affine Lie algebras distinguish themselves among Kac-Moody Lie algebras, largely due to 2 distinct constructions: a Serre presentation (as definition) and a loop algebra realization. Such a dual nature of affine Lie algebras has led to numerous applications to string theory, modular forms, algebraic combinatorics, and so on.
Remarkably, these constructions repeat themselves at the quantum level for Drinfeld-Jimbo quantum groups: besides the Serre presentation (as definition) \cite{Dr87, Jim85}, an affine quantum group admits a current realization, also known as Drinfeld's new presentation \cite{Dr88}.
An explicit isomorphism between these 2 presentations with proof was supplied by Beck \cite{Be94} in the untwisted affine type. Damiani \cite{Da12, Da15} consolidated part of Beck's arguments and established interconnections among different relations in the current realization.
Drinfeld's current presentation of affine quantum groups has played a fundamental role in numerous subsequent algebraic, geometric, and categorical developments. It has been instrumental in the active area of (finite-dimensional) representation theory of affine quantum groups ${\mathbf U}$, in algebraic or combinatorial approach, by numerous authors including Chari, Kashiwara, and their collaborators and others; see the survey paper \cite{CH10} for partial references; there have been connections with cluster algebras and monoidal categorification \cite{HL15}. A powerful geometric approach to representation theory of ${\mathbf U}$ was developed by Ginzburg, Vasserot, and Nakajima (see \cite{V98, Nak00}).
Moreover, the Drinfeld presentation arises categorically in Hall algebras of coherent sheaves over (weighted) projective lines as initiated by Kapranov; see \cite{Ka97, BKa01, Sch04, DJX12}.
According to the $\imath$program as outlined in \cite{BW18}, various algebraic, geometric, and categorical constructions of quantum groups should be generalizable to $\imath$quantum groups arising from quantum symmetric pairs. As an $\imath$-analogue of Drinfeld double quantum groups $\widetilde{\mathbf U}$, the universal $\imath$quantum groups $\widetilde{{\mathbf U}}^\imath$ was introduced by the authors in \cite{LW19a} (also see \cite{LW20}) as they arise naturally from the $\imath$Hall algebra constructions of $\imath$quivers; moreover, there is a braid group action on $\widetilde{{\mathbf U}}^\imath$ which is realized by reflection functors in $\imath$Hall algebras \cite{LW19b}. A central reduction of $\widetilde{{\mathbf U}}^\imath$ recovers the $\imath$quantum groups ${\mathbf U}^\imath ={\mathbf U}^\imath_{\boldsymbol{\varsigma}}$ with parameters ${\boldsymbol{\varsigma}} \in \mathbb Q(v)^{\times,\mathbb{I}}$ introduced earlier by G.~Letzter and generalized by Kolb \cite{Let99, Let02, Ko14}.
We view $\imath$quantum groups as a vast generalization of quantum groups, as quantum groups can be regarded as $\imath$quantum groups of diagonal type. The $\imath$quantum groups of finite type are classified by Satake diagrams (or the real forms of complex simple Lie algebras) of roughly 2 dozens types, and there is an even richer family of affine $\imath$quantum groups.
The split and more generally quasi-split $\imath$quantum groups make sense in Kac-Moody generality and form a distinguished class of $\imath$quantum groups. For some recent progress in representation theory of $\imath$quantum groups, see \cite{Wat19}.
\subsection{Goal}
The goal of this paper is to initiate a current realization of affine $\imath$quantum groups. We shall restrict ourselves to the split affine ADE type in this paper, where most of the new features are already present; the Drinfeld type presentation for affine $\imath$quantum group of rank one (known as $q$-Onsager algebra) is already new.
We expect that the current realizations of affine $\imath$quantum groups will open up algebraic, categorical, and geometric developments as did Drinfeld's presentation for the affine quantum group ${\mathbf U}={\mathbf U}(\widehat{\mathfrak{g}})$.
Throughout the paper, we shall work with the universal $\imath$quantum group $\widetilde{{\mathbf U}}^\imath$, which contains central generators $\mathbb{K}_i$, for $i\in \mathbb{I}$. The corresponding results for ${\mathbf U}^\imath$ (with arbitrary parameters) can be obtained from $\widetilde{{\mathbf U}}^\imath$ readily by a central reduction which specializes $\mathbb{K}_i$ to scalars.
\subsection{Rank one}
The split affine $\imath$quantum group ${\mathbf U}^\imath ={\mathbf U}^\imath_{\boldsymbol{\varsigma}} (\widehat{\mathfrak{sl}}_2)$, with parameters ${\boldsymbol{\varsigma}} =\{\varsigma_0, \varsigma_1\} \in (\mathbb Q(v)^\times)^2$, is known as $q$-Onsager algebra in the literature.
There has been some attempts with mixed success by Baseilhac and collaborators toward a Dinfeld type presentation for ${\mathbf U}^\imath$, cf. \cite{BS10, BK20} and references therein.
Making an ansatz with the constructions for affine quantum group ${\mathbf U}(\widehat{\mathfrak{sl}}_2)$ by Damiani \cite{Da93}, Baseilhac and Kolb \cite{BK20} defined the $q$-root vectors in the $q$-Onsager algebra ${\mathbf U}^\imath$ (where both parameters were set to be equal: $\varsigma_0=\varsigma_1$, for technical reasons), and established a PBW basis for ${\mathbf U}^\imath$. Along the way, an affine braid group action on ${\mathbf U}^\imath$ (when $\varsigma_0=\varsigma_1$) is given. Various relations among the $q$-root vectors were computed, but clearly they do not resemble the relations in Drinfeld's current presentation for ${\mathbf U}(\widehat{\mathfrak{sl}}_2)$.
In this paper, we first upgrade the main results of \cite{BK20} for ${\mathbf U}^\imath$ to $\widetilde{{\mathbf U}}^\imath$, such as the constructions of $v$-root vectors (denoted by $B_{1,k}, \acute{\Theta}_m$, for $k\in \mathbb Z, m\in \mathbb Z_{\ge 1}$) and their relations, with the help of a braid group action coming from the $\imath$Hall algebra realization of $\widetilde{{\mathbf U}}^\imath$. By a central reduction this in turn allows us to obtain the $v$-root vectors and PBW basis for ${\mathbf U}^\imath$ with 2 arbitrary parameters $\{\varsigma_0, \varsigma_1\}$, somewhat improving \cite{BK20}.
As a point of departure, we introduce new $v$-imaginary root vectors $\Theta_m$ and especially $H_m$, for $m\ge 1$, and work with generating functions. This allows us to greatly simplify some of the BK relations to be Drinfeld type relations. The normalization from $\acute{\Theta}_m$ to $\Theta_m$ in rank 1 is inspired by the realization of $q$-Onsager algebra via $\imath$Hall algebra of the projective line in a forthcoming work \cite{LRW20}. We also obtain new Drinfeld type relations among real $v$-root vectors. In this way, we formulate a Drinfeld type presentation for $\widetilde{{\mathbf U}}^\imath$ as an algebra isomorphism ${}^{\text{Dr}}\tUi \cong \widetilde{{\mathbf U}}^\imath$; see Definition~\ref{def:DrOnsa} and Theorem~\ref{thm:Dr1}.
Via generating functions in a variable $z$,
\[
{ \boldsymbol{\Theta}} (z)= 1+(v-v^{-1}) \sum_{m\ge 1} \Theta_{m} z^m, \qquad
{\mathbf B }(z)=\sum_{r\in \mathbb Z} B_{1,r} z^r, \qquad
\boldsymbol{\Delta}(z) = \sum_{k\in\mathbb Z} C^k z^k,
\]
where $C=\mathbb{K}_\delta$, the Drinfeld presentation for the $q$-Onsager algebra $\widetilde{{\mathbf U}}^\imath$ is reformulated as:
\begin{align}
{ \boldsymbol{\Theta}} (z) \boldsymbol{\Theta}(w)
&= { \boldsymbol{\Theta}} (w) \boldsymbol{\Theta}(z),
\label{gf1} \\
{ \boldsymbol{\Theta}} (z) {\mathbf B }(w)
& = \frac{(1 -v^{-2}zw^{-1}) (1 -v^{2} zw C)}{(1 -v^{2}zw^{-1})(1 -v^{-2}zw C)}
{\mathbf B }(w) { \boldsymbol{\Theta}} (z),
\\
(v^2z-w) \bB(z) \bB(w) & +(v^{2}w-z) \bB(w) \bB(z)
\label{gf3} \\
& =\frac{v^{-2} \mathbb{K}_1}{v-v^{-1}} \boldsymbol{\Delta}(zw) \big( (v^2z-w) \boldsymbol{\Theta}(w) +(v^2w-z) \boldsymbol{\Theta}(z) \big).
\notag
\end{align}
\subsection{Higher rank}
Now let $\widetilde{{\mathbf U}}^\imath =\widetilde{{\mathbf U}}^\imath_{{\boldsymbol{\varsigma}}} (\widehat{\mathfrak{g}})$ be the universal affine $\imath$quantum group of split ADE type, where $\mathfrak{g}$ is the Lie algebra of type ADE with root datum $\mathbb{I}_0$. (The corresponding $\imath$quntum groups ${\mathbf U}^\imath$ were introduced in \cite{BB10} and referred to as generalized $q$-Onsager algebras; cf. \cite{Ko14}.) By definition, $\widetilde{{\mathbf U}}^\imath$ is generated by $B_i, \mathbb{K}_i, \mathbb{K}_i^{-1}$, for $i\in \mathbb{I} =\mathbb{I}_0 \cup \{0\}$, where $\mathbb{K}_i$ are central and the $B_i$ satisfy some inhomogenous Serre relations; see \eqref{eq:S2}--\eqref{eq:S3}. The algebra $\widetilde{{\mathbf U}}^\imath$ can be realized via an $\imath$Hall algebra construction associated to $\imath$quivers \cite{LW19b, LW20}; moreover, there are automorphisms $\texttt{\rm T}_i$ ($i\in \mathbb{I}$) of $\widetilde{{\mathbf U}}^\imath$ which are realized as reflection functors on $\imath$Hall algebras, which gives rise to an affine braid group action on $\widetilde{{\mathbf U}}^\imath$.
It is worth pointing out that such a braid group action is not a restriction of the braid group action of Lusztig on quantum groups \cite{Lus90, Lus94}.
In a way similar to \cite{Be94}, with the help of the braid group action and the rank one construction above, we construct real $v$-root vectors $B_{i,k}$ and imaginary $v$-root vectors $\acute{\Theta}_{i,m}$ (or $\Theta_{i,m}$, $H_{i,m}$), for $i\in \mathbb{I}_0, k\in \mathbb Z, m\in \mathbb Z_{\ge 1}$.
Then the formulation of relations among these elements are used to define a new Drinfeld type algebra ${}^{\text{Dr}}\tUi$; see Definition~\ref{def:iDR}. Our main result (see Theorem~\ref{thm:ADE}) asserts that ${}^{\text{Dr}}\tUi \cong \widetilde{{\mathbf U}}^\imath$, providing a new presentation for the affine $\imath$quantum group $\widetilde{{\mathbf U}}^\imath$.
The defining relations~\eqref{iDR1}--\eqref{iDR5} for ${}^{\text{Dr}}\tUi$ turn out to be strikingly neat and similar to Drinfeld's current relations for ${\mathbf U}(\widehat{\mathfrak{g}})$. Actually, one can regard half the Drinfeld's realization of ${\mathbf U}(\widehat{\mathfrak{g}})$ as the associated graded algebra with respect to a filtration of ${}^{\text{Dr}}\tUi$ over its central subalgebra. In other words, the relations \eqref{iDR2}, \eqref{iDR3b} and \eqref{iDR5} for ${}^{\text{Dr}}\tUi$ look like Drinfeld's relations for ${\mathbf U}(\widehat{\mathfrak{g}})$ plus lower terms (involving powers of $C=\mathbb{K}_\delta$). Yet another view of the relations \eqref{iDR2}, \eqref{iDR3b} and \eqref{iDR5} for ${}^{\text{Dr}}\tUi$ is that they exhibit a hybrid phenomenon mixing relations in the current negative half with relations between current positive and negative generators for ${\mathbf U}(\widehat{\mathfrak{g}})$.
One new relation which was not present in the rank one case is the Serre type relation \eqref{iDR5}. As the Serre relations among $B_i =B_{i,0}$ \eqref{eq:S2} are inhomogeneous, it is understandable that the general Serre type relations \eqref{iDR5} among $B_{i,k}$ for ${}^{\text{Dr}}\tUi$ are much more challenging to formulate than its counterpart for ${\mathbf U}(\widehat{\mathfrak{g}})$, where the RHS of \eqref{iDR5} is simply set to 0 as in a standard Serre relation. What is perhaps surprising to us is that such a relation can be formulated concretely after all.
Damiani \cite{Da12} made a careful analysis of the relations in Drinfeld's current realization of ${\mathbf U}(\widehat{\mathfrak{g}})$, and showed that they can be derived from a few distinguished relations. To that end, the triangular decomposition of ${\mathbf U}(\widehat{\mathfrak{g}})$ was very helpful.
In contrast, $\widetilde{{\mathbf U}}^\imath$ does not admit a triangular decomposition. Because of this, the verifications of the new relations for $\widetilde{{\mathbf U}}^\imath$, in particular \eqref{iDR3b}--\eqref{iDR5}, require a very different strategy from \cite{Be94, Da12}, though our overall plan is somewhat similar by showing that all the new relations for $\widetilde{{\mathbf U}}^\imath$ can be derived from a few simpler ones.
By verifying all the new relations in $\widetilde{{\mathbf U}}^\imath$ we obtain a homomorphism $\Phi: {}^{\text{Dr}}\tUi \rightarrow \widetilde{{\mathbf U}}^\imath$, and it remains to show that $\Phi$ is an isomorphism. Our argument of the surjectivity of $\Phi$ is adapted from the proof of Damiani \cite[Theorem~12.11]{Da12}.
The injectivity of $\Phi$ follows by applying some filtered algebra argument to reduce to the corresponding injectivity in the affine quantum group setting.
We obtain several natural variants of the current presentation of $\widetilde{{\mathbf U}}^\imath$, including one in the generating function form similar to \eqref{gf1}--\eqref{gf3} for $q$-Onsager algebra; see Theorem~\ref{thm:ADEgf}.
A surprising bonus of working with $\widetilde{{\mathbf U}}^\imath$ (instead of ${\mathbf U}^\imath$) is that the canonical central element $C$ in affine quantum group naturally appear as $\mathbb{K}_\delta =\prod_{i\in \mathbb{I}} \mathbb{K}_i^{a_i}$ associated to the basic imaginary root $\delta =\sum_{i\in \mathbb{I}} a_i \alpha_i$. This is especially clear in a symmetrized variant of the current presentation of $\widetilde{{\mathbf U}}^\imath$ (see Definition~\ref{def:i-DR-ref} and Proposition~\ref{prop:symm}).
This phenomenon is even more remarkable, as the classical ($v\mapsto 1$) limit of $\widetilde{{\mathbf U}}^\imath$ or ${\mathbf U}^\imath$ does not contain the canonical central element of the affine Lie algebra $\widehat \mathfrak{g}$; see \S\ref{subsec:classical}.
\subsection{Applications}
While the $\imath$Hall algebras are not used in this paper in any explicit manner, they have played a fundamental role in guiding our work. Since we have realized the universal $\imath$quantum group in its Serre presentation via $\imath$Hall algebra of $\imath$quivers \cite{LW19a, LW20}, it is natural to expect that $\imath$Hall algebras of coherent sheaves over (weighted) projective lines should provide a realization of $\widetilde{{\mathbf U}}^\imath$ in a new {\em current} presentation (keeping in mind the classic works \cite{Ka97, BKa01, Sch04, DJX12} relating (weighted) projective lines to the current realization of affine quantum groups). Preliminary computations at earlier stages of \cite{LRW20, LR20} on $\imath$Hall algebras of (weighted) projective lines have been very helpful in pinning down some new relations for $\widetilde{{\mathbf U}}^\imath$. The current presentation of $\widetilde{{\mathbf U}}^\imath$ in this paper will play an essential role in the $\imath$Hall algebra realization therein.
It is our hope that this work can be of interest to people with diverse algebraic, categorical, geometric backgrounds. With this in mind, we have tried to make the presentation in this paper to be self-contained and come up with proofs independent of $\imath$Hall algebras. (As a result, a multiple of proofs for various relations in ${\mathbf U}^\imath$ are known to us.)
In sequels to this paper, we shall further generalize this work to obtain Drinfeld type presentations for affine $\imath$quantum groups beyond split ADE type. The new current presentation should be very helpful in developing an algebraic approach toward the representation theory of affine $\imath$quantum groups (cf., e.g., \cite{CH10}), to which we hope to return elsewhere.
In yet another direction, this work makes it possible to develop the geometric realization of affine $\imath$quantum groups via equivariant K-theory \cite{SuW20}, building on the works of Y.~Li and collaborators \cite{BKLW, Li19} and generalizing earlier works of Vasserot and Nakajima \cite{V98, Nak00}.
\subsection{Organization}
In Section~\ref{sec:Onsager}, we define the $v$-root vectors and give a Drinfeld type presentation for the $q$-Onsager algebra $\widetilde{{\mathbf U}}^\imath$.
In Section~\ref{sec:main}, we define the $v$-root vectors for $\widetilde{{\mathbf U}}^\imath$ of split affine ADE type, and formulate a Drinfeld type presentation for $\widetilde{{\mathbf U}}^\imath$ in the form of isomorphism $\Phi: {}^{\text{Dr}}\tUi \rightarrow \widetilde{{\mathbf U}}^\imath$. We show $\Phi$ is a bijection under the assumption that $\Phi$ is an algebra homomorphism, while postponing the verification of the new relations in $\widetilde{{\mathbf U}}^\imath$ to Section~\ref{sec:relation1}.
We verify the new relations in $\widetilde{{\mathbf U}}^\imath$ one-by-one in Section~\ref{sec:relation1}, with the most challenging ones being \eqref{iDR5} and \eqref{iDR2} for $c_{ij}=-1$.
In Section~\ref{sec:variants}, we offer some variants of the new presentation for $\widetilde{{\mathbf U}}^\imath$, one in generating function format (Theorem~\ref{thm:ADEgf}), and in a symmetrized form which resembles Drinfeld's realization for ${\mathbf U}$ better (Definition~\ref{def:i-DR-ref} and Proposition~\ref{prop:symm}), and yet another one in terms of different $v$-imaginary root vectors (Theorem~\ref{thm:ADE1}). We also formulate a Drinfeld type presentation for ${\mathbf U}^\imath_{\boldsymbol{\varsigma}}$ (Theorem~\ref{thm:ADE2}).
\vspace{2mm}
\noindent {\bf Acknowledgement.}
We thank Shiquan Ruan for his collaboration on related Hall algebra projects, and thank Weinan Zhang who inspired us to simplify much our earlier proofs in \S\ref{subsec:iDR2=>iDR5}--\ref{subsec:iDR5=>iDR2}. ML thanks University of Virginia for hospitality and support. WW is partially supported by the NSF grant DMS-1702254 and DMS-2001351.
\section{The $q$-Onsager algebra and its Drinfeld type presentation}
\label{sec:Onsager}
In this section, we derive Drinfeld type new relations among the generators of the universal $q$-Onsager algebra, and recast them in the form of generating functions. This is built on a reformulation and enhancement of the results in \cite{BK20} for $q$-Onsager algebra.
We introduce new imaginary root vectors in the universal $q$-Onsager algebra (with motivation coming from the $\imath$Hall algebra of the projective line), and establish a Drinfeld type presentation.
\subsection{Root vectors}
For $n\in \mathbb Z, r\in \mathbb N$, denote by
\[
[n] =\frac{v^n -v^{-n}}{v-v^{-1}},\qquad
\qbinom{n}{r} =\frac{[n][n-1]\ldots [n-r+1]}{[r]!}.
\]
For $A, B$ in a $\mathbb Q(v)$-algebra, we shall denote $[A,B]_{v^a} =AB -v^aBA$, and $[A,B] =AB - BA$.
\begin{definition}
\label{def:Onsager}
The {\em (universal) $q$-Onsager algebra $\widetilde{{\mathbf U}}^\imath =\widetilde{{\mathbf U}}^\imath(\widehat{\mathfrak{sl}}_2)$} is the $\mathbb Q(v)$-algebra generated by $B_0,B_1$, $\mathbb{K}_0,\mathbb{K}_1$, subject to the following relations: $\mathbb{K}_0,\mathbb{K}_1$ are central, and
\begin{align}
\sum_{r=0}^3 (-1)^r \qbinom{3}{r} B_i^{3-r} B_j B_i^{r}&= -v^{-1} [2]^2 (B_iB_j-B_jB_i) \mathbb{K}_i,
\quad \text{ for } i\neq j.
\label{relation:s3}
\end{align}
(This algebra is also known as the {\rm universal $\imath$quantum group of split type $A_1^{(1)}$}.)
\end{definition}
\begin{remark} \label{rem:Ui}
The generator $\mathbb{K}_i$ here is related to $\Bbbk_i$ used in \cite{LW19a} by $\mathbb{K}_i=-v^2 \Bbbk_i$; see Remark~\ref{rem:Kk}. The $q$-Onsager algebra ${\mathbf U}^\imath_{\boldsymbol{\varsigma}}$, for ${\boldsymbol{\varsigma}} =(\varsigma_0, \varsigma_1) \in (\mathbb Q(v)^\times)^2$, is obtained from $\widetilde{{\mathbf U}}^\imath$ by a central reduction ${\mathbf U}^\imath_{\boldsymbol{\varsigma}} = \widetilde{{\mathbf U}}^\imath /( \mathbb{K}_i +v^2 \varsigma_i | i=0,1)$, where $(-)$ denotes an ideal. The 1-parameter specialization ${\mathbf U}^\imath_{\boldsymbol{\varsigma}}$ by taking $\varsigma_0=\varsigma_1=-c$ recovers the $q$-Onsager algebra $\mathcal B_c$ studied in \cite{BK20}.
\end{remark}
Let $\{\alpha_0, \alpha_1\}$ be the simple roots of the affine Lie algebra $\widehat{\mathfrak{sl}}_2$, and $\delta =\alpha_0 +\alpha_1$ is the basic imaginary root. The root system for $\widehat{\mathfrak{sl}}_2$ is
${\mathcal R} =\{\pm (\alpha_1 + k \delta), m\delta | k,m \in \mathbb Z, m\neq 0\}.$
For $\mu, \nu \in \mathbb Z \alpha_0 \oplus \mathbb Z \alpha_1$ and $i=0,1$, set
\begin{align}
\mathbb{K}_{\alpha_i} =\mathbb{K}_i, \quad
\mathbb{K}_{\alpha_i}^{-1} =\mathbb{K}_i^{-1}, \quad
\mathbb{K}_\delta =\mathbb{K}_0 \mathbb{K}_1,
\quad \mathbb{K}_{\mu +\nu} =\mathbb{K}_{\mu} \mathbb{K}_{\nu}.
\end{align}
Let $\dag$ be the involution of the $\mathbb Q(v)$-algebra $\widetilde{{\mathbf U}}^\imath$ such that
\begin{align}
\dag:B_0\leftrightarrow B_1, \qquad \mathbb{K}_0\leftrightarrow \mathbb{K}_1.
\end{align}
We have the following two automorphisms $\texttt{\rm T}} %{\mathbf T_0,\texttt{\rm T}} %{\mathbf T_1$, which has an interpretation in $\imath$Hall algebras, see \cite{LW19b} and a forthcoming sequel in the Kac-Moody setting (This shows some conceptual advantage of $\widetilde{{\mathbf U}}^\imath$ over ${\mathbf U}^\imath_{\boldsymbol{\varsigma}}$ \cite{BK20}, where the parameters $\varsigma_i$ are set to be equal):
\begin{align}
\texttt{\rm T}} %{\mathbf T_1 (\mathbb{K}_1) &=\mathbb{K}_1^{-1},\qquad \texttt{\rm T}} %{\mathbf T_1(\mathbb{K}_0)= \mathbb{K}_{\delta} \mathbb{K}_1,
\\
\texttt{\rm T}} %{\mathbf T_1(B_1)&= \mathbb{K}_1^{-1} B_1,
\\
\texttt{\rm T}} %{\mathbf T_1(B_0)&= [2]^{-1} \big(B_0B_1^{2} -v[2] B_1 B_0B_1 +v^2 B_1^{2} B_0 \big) + B_0\mathbb{K}_1,
\label{T1B0}
\\
\texttt{\rm T}} %{\mathbf T_1^{-1}(B_0)&= [2]^{-1} \big( B_1^{2}B_0-v[2] B_1B_0B_1 +v^2 B_0B_1^{2} \big) +B_0\mathbb{K}_1.
\label{T1B0-2}
\end{align}
The action of $\texttt{\rm T}} %{\mathbf T_0$ is obtained from the above formulas by switching indices $0,1$, that is,
\begin{align}
\texttt{\rm T}} %{\mathbf T_0=\dag \circ \texttt{\rm T}} %{\mathbf T_1 \circ \dag.
\end{align}
For $n\in\mathbb Z$, following \cite{BK20}, we define the real $v$-root vectors
\begin{align}
B_{1,n} &=(\dag \texttt{\rm T}} %{\mathbf T_1)^{-n}(B_1).
\label{eq:B1n}
\end{align}
Slightly modifying \cite{BK20}, we further define, for $m\ge 1$,
\begin{align}
\acute{\Theta}_{m} &= -B_{1,m-1} B_0+v^{2} B_0B_{1,m-1} + (v^{2}-1)\sum_{p=0}^{m-2} B_{1,p} B_{1,m-p-2} \mathbb{K}_0.
\label{eq:dB1}
\end{align}
Note that $B_{1,0}=B_1$ by definition. (Our $- v^{-2} \acute{\Theta}_{m}$ corresponds to $ B_{m\delta}$ in \cite[(3.11)]{BK20}.)
In particular, we have
\acute{\Theta}_1 = -B_1B_0+v^{2} B_0B_1.
$
We also set
\begin{align}
\acute{\Theta}_{m} :=\begin{cases}
\frac{1}{v-v^{-1}} & \text{ if }m=0,
\\
0& \text{ if }m<0.
\end{cases}
\end{align}
From \eqref{eq:B1n}, we have
\begin{align*}
B_{1,-1}=(\dag \texttt{\rm T}} %{\mathbf T_1)(B_1)=B_0 \mathbb{K}_0^{-1},
\qquad
B_0=B_{1,-1}\mathbb{K}_0.
\end{align*}
So \eqref{eq:dB1} can be rewritten as
\begin{align}
\label{eq:reformTheta}
\acute{\Theta}_{m} &= \big(-B_{1,m-1} B_{1,-1}+v^{2} B_{1,-1}B_{1,m-1} + (v^{2}-1)\sum_{p=0}^{m-2} B_{1,p} B_{1,m-p-2} \big)\mathbb{K}_0
\\\notag
&=-\sum_{p=0}^{m-1} [B_{1,p},B_{1,m-2-p}]_{v^2} \mathbb{K}_0.
\end{align}
We note that \cite[Corollary~ 5.12]{BK20} in our setting of $\widetilde{{\mathbf U}}^\imath$ reads
\begin{align} \label{eq:Tm}
\dag \texttt{\rm T}} %{\mathbf T_1 (\acute{\Theta}_{m})=\acute{\Theta}_{m}.
\end{align}
Applying $(\dag\texttt{\rm T}} %{\mathbf T_1)^{-1}$ to \eqref{eq:reformTheta},
we have
\acute{\Theta}_{m} = -\sum_{p=1}^{m} [B_{1,p},B_{1,m-p}]_{v^2} \mathbb{K}_1^{-1}.
\subsection{Relations \`a la Baseilhac-Kolb}
The following relations in $\widetilde{{\mathbf U}}^\imath$ are the counterparts of the main relations for $\mathcal B_c$ established in \cite{BK20}; see Remark~\ref{rem:Ui}. They are obtained by literally repeating the arguments {\em loc. cit.}, and hence we shall skip the proofs altogether in this subsection. Our formulation in turn strengthens somewhat the results in \cite{BK20}, as the central reduction in Remark~\ref{rem:Ui} provides relations and then a presentation for ${\mathbf U}^\imath$ with {\em arbitrary} 2 parameters $\varsigma_i$ $(i=0, 1)$.
\begin{proposition}
[\text{\cite[Corollary 5.11]{BK20}}]
\label{prop:ThTh1}
We have $[\acute{\Theta}_{n},\acute{\Theta}_{m}]=0$ holds in $\widetilde{{\mathbf U}}^\imath$, for $n,m\ge 1$.
\end{proposition}
For $m\in\mathbb N$, define
\begin{align}
\label{eq:coeff}
a_p^m:=\left\{ \begin{array}{ll} v^{2(p-1)}(1+v^{2}), & \text{ if }p=1,2,\dots,\lfloor \frac{m-1}{2}\rfloor,
\\
v^{m-2}, & \text{ if }2 | m \text{ and } p=\frac{m}{2}. \end{array} \right.
\end{align}
\begin{proposition}
[\text{\cite[Proposition 5.5, Corollary 5.13]{BK20}}]
The following relation holds in $\widetilde{{\mathbf U}}^\imath$, for $m\in\mathbb N$ and $r\in\mathbb Z$:
\begin{align}
\label{eq:BK581}
[ B_{1,r+m+1},&B_{1,r}]_{v^{2}}= -\acute{\Theta}_{m+1} \mathbb{K}_{r\delta+\alpha_1}-(v^{2}-1) \sum_{p=1}^{\lfloor \frac{m}{2}\rfloor} v^{2(p-1)} \acute{\Theta}_{m-2p+1}\mathbb{K}_{(p+r)\delta+\alpha_1}
\\
&\qquad \qquad +(v^{2}-1) \sum_{p=1}^{\lfloor \frac{m+1}{2}\rfloor} a_p^{m+1} B_{1,r+p}B_{1,m+r-p+1}. \notag
\end{align}
\end{proposition}
\begin{proposition}
[\text{\cite[Proposition 5.8, Corollary 5.13]{BK20}}]
The following relation holds in $\widetilde{{\mathbf U}}^\imath$, for $m\geq 1$ and $r\in\mathbb Z$:
\begin{align}
\label{eq:im-real2}
&[\acute{\Theta}_{m}, B_{1,r}]
\\
&= [2] \Big(
v^{2(m-1)} B_{1,r+m}
- (v^2 -v^{-2}) \sum_{h=1}^{m-1} v^{2(m-2h)} B_{1,r+m-2h} \mathbb{K}_{h\delta}
- v^{2(1-m)} B_{1,r-m} \mathbb{K}_{m \delta}
\Big)
\notag
\\
& + (v^2 -v^{-2}) \times
\notag
\\
& \sum_{a=1}^{m-1}
\Big(
v^{2(a-1)} B_{1,r+a}
- (v^2 -v^{-2}) \sum_{h=1}^{a-1} v^{2(a-2h)} B_{1,r+a-2h} \mathbb{K}_{h\delta}
- v^{2(1-a)} B_{1,r-a} \mathbb{K}_{a \delta}
\Big)
\acute{\Theta}_{m-a}.
\notag
\end{align}
\end{proposition}
\subsection{Drinfeld type relations in rank 1}
We shall introduce a new imagnary $v$-root vectors $\acute{H}_m$ and
formulate several Drinfeld type relations in $\widetilde{{\mathbf U}}^\imath$ among the $v$-root vectors.
\subsubsection{} We start with the relations among real $v$-root vectors $B_{1,k}$.
\begin{proposition} \label{prop:iDr}
The following relation holds in $\widetilde{{\mathbf U}}^\imath$, for $r, s \in \mathbb Z$:
\begin{align}
\label{rel:iDr}
[B_{1,r}, B_{1,s+1}]&_{v^{-2}} -v^{-2}[B_{1,r+1}, B_{1,s}]_{v^{2}} = v^{-2}\acute{\Theta}_{r-s+1}\mathbb{K}_{s\delta+\alpha_1} -v^{-2}\acute{\Theta}_{r-s-1}\mathbb{K}_{(s+1)\delta+\alpha_1} \\
&+v^{-2}\acute{\Theta}_{s-r+1}\mathbb{K}_{r\delta+\alpha_1} -v^{-2}\acute{\Theta}_{s-r-1}\mathbb{K}_{(r+1)\delta+\alpha_1}.\notag
\end{align}
\end{proposition}
\begin{proof}
Note that $-v^{-2}[B_{1,r+1}, B_{1,s}]_{v^{2}} = [B_{1,s}, B_{1,r+1}]_{v^{-2}}$, and hence \eqref{rel:iDr} is invariant under $r\leftrightarrow s$. So we can assume $m=s-r\geq0$.
By combining 2 relations in \eqref{eq:BK581} for $[B_{1,r+m+1},B_{1,r}]_{v^2}$ and $[B_{1,r+m},B_{1,r+1}]_{v^2}$ for $m\ge 2$, respectively, we obtain
\begin{align}
\label{rel:iDr1}
&[B_{1,r+m+1},B_{1,r}]_{v^2} - v^2 [B_{1,r+m},B_{1,r+1}]_{v^2}
\\
=&-\acute{\Theta}_{m+1} \mathbb{K}_{r\delta+\alpha_1} + \acute{\Theta}_{m-1}\mathbb{K}_{(r+1)\delta+\alpha_1} +(v^4-1) B_{1,r+1} B_{1,m+r}.\notag
\end{align}
One then rewrites \eqref{rel:iDr1} equivalently as \eqref{rel:iDr}, for $m\ge 2$.
Recalling $\acute{\Theta}_{0}=\frac{1}{v-v^{-1}}$, one observes that \eqref{rel:iDr} for $m=s-r=1$ is equivalent to the relation
$
[B_{1,r}, B_{1,r+2}]_{v^{-2}} = v^{-2}\acute{\Theta}_{2} \mathbb{K}_{r\delta+\alpha_1} + (v^{-2} -1) B_{1,r+1}^2,$
and then equivalent to the following relation in \eqref{eq:BK581}:
\[
[B_{1,r+2},B_{1,r}]_{v^{2}} = -\acute{\Theta}_{2} \mathbb{K}_{r\delta+\alpha_1} + (v^2 -1) B_{1,r+1}^2.
\]
The relation \eqref{rel:iDr} for $m=s-r=0$ is equivalent to $[B_{1,r}, B_{1,r+1}]_{v^{-2}} = v^{-2}\acute{\Theta}_{1}\mathbb{K}_{r\delta+\alpha_1}$, another relation in \eqref{eq:BK581}.
The proof of \eqref{rel:iDr} is completed.
\end{proof}
\begin{remark}
The relation \eqref{rel:iDr} was derived from \eqref{eq:BK581} above; they are actually equivalent, as the converse can be shown by induction on $m$.
\end{remark}
\subsubsection{}
Define elements $\acute{H}_m$ in $\widetilde{{\mathbf U}}^\imath$, for $m\ge 1$, by the following equation:
\begin{align}
\label{eq:exp}
1+ \sum_{m\geq 1} (v-v^{-1})\acute{\Theta}_{m} z^m = \exp\Big( (v-v^{-1}) \sum_{m\ge 1} \acute{H}_m z^m \Big).
\end{align}
Introduce the following generating functions in a variable $z$:
\[
\acute{\mathbf H}(z) =\sum_{m\ge 1} \acute{H}_m z^m,
\quad
{\mathbf B }(z) =\sum_{r\in \mathbb Z} B_{1,r} z^r,
\quad
\acute \boldsymbol{\Theta} (z) = 1+ (v-v^{-1})\sum_{m\ge 1} \acute{\Theta}_{m} z^m.
\]
Then we have
\begin{align} \label{Theta:z}
\acute \boldsymbol{\Theta} (z) = \exp\big( (v-v^{-1}) \acute{\mathbf H} (z) \big).
\end{align}
The relations between imaginary and real $v$-root vectors can now be formulated as follows; here the imaginary $v$-root vectors refer to $\acute{H}_m$.
\begin{proposition}
\label{prop:equiv4}
The following identities hold, for $m \ge 1, l\in \mathbb Z$:
\begin{align}
\label{eq:hB}
[\acute{H}_m, B_{1,l}] &=\frac{[2m]}{m} B_{1,l+m}-\frac{[2m]}{m} B_{1,l-m}\mathbb{K}_{m\delta},
\\
\label{eq:eHBe}
\acute \boldsymbol{\Theta} (z) {\mathbf B }(w)
& = \frac{(1 -v^{-2}zw^{-1}) (1 -v^{2} zw \mathbb{K}_\delta)}{(1 -v^{2}zw^{-1})(1 -v^{-2}zw \mathbb{K}_\delta)}
{\mathbf B }(w) \acute \boldsymbol{\Theta} (z),
\\
\label{eq:hB1}
[\acute{\Theta}_{m},B_{1,l}]+[\acute{\Theta}_{m-2},B_{1,l}]\mathbb{K}_\delta
& =v^{2}[\acute{\Theta}_{m-1},B_{1,l+1}]_{v^{-4}}+v^{-2}[\acute{\Theta}_{m-1},B_{1,l-1}]_{v^{4}}\mathbb{K}_\delta.
\end{align}
Indeed, the identities \eqref{eq:im-real2}, \eqref{eq:hB}, \eqref{eq:eHBe} and \eqref{eq:hB1} are all equivalent.
\end{proposition}
The proof of Proposition~\ref{prop:equiv4} will be given in \S\ref{subsec:proof} below.
\subsubsection{}
One also has the following variant \eqref{eq:im-real} of \eqref{eq:im-real2}, which we will not use. We skip a similar proof.
\begin{align}
\label{eq:im-real}
&[\acute{\Theta}_{m}, B_{1,l}]
\\
& = [2] \Big(
v^{2(1-m)} B_{1,l+m}
+ (v^2 -v^{-2}) \sum_{h=1}^{m-1} v^{2(2h-m)} B_{1,l+m-2h} \mathbb{K}_{h\delta}
- v^{2(m-1)} B_{1,l-m} \mathbb{K}_{m \delta}
\Big)
\notag
\\
& + (v^2 -v^{-2}) \times
\notag
\\
& \sum_{a=1}^{m-1} \acute{\Theta}_{m-a}
\Big(
v^{2(1-a)} B_{1,l+a}
+ (v^2 -v^{-2}) \sum_{h=1}^{a-1} v^{2(2h-a)} B_{1,l+a-2h} \mathbb{K}_{h\delta}
- v^{2(a-1)} B_{1,l-a} \mathbb{K}_{a \delta}
\Big). \notag
\end{align}
\iffalse
\subsubsection{Proof of equivalence of \eqref{eq:eHBe} and \eqref{eq:im-real}}
\label{subsec:hB-im-real}
\red{SKIP}
The identity \eqref{eq:eHBe} can be reformulated as
\begin{align}
\label{eq:eHBe2}
{\mathbf B }(w) e^{(v-v^{-1}) \acute{\mathbf H}(z)}
= \left(
\frac{1 -v^{2}zw^{-1}}{1 -v^{-2}zw^{-1}} \cdot \frac{1 -v^{-2}zw \mathbb{K}_\delta}{1 -v^{2} zw \mathbb{K}_\delta}
\right)
e^{(v-v^{-1}) \acute{\mathbf H}(z)} {\mathbf B }(w).
\end{align}
We have an identity
\begin{align*}
&\frac{1 -v^{2}zw^{-1}}{1 -v^{-2} zw^{-1}} \cdot \frac{1 -v^{-2}zw \mathbb{K}_\delta}{1 -v^{2} zw \mathbb{K}_\delta}
\\
&= 1 - \sum_{l \ge 1} z^l (v^2 -v^{-2})
\left( v^{2(1-l)} w^{-l}
+ (v^2 -v^{-2}) \sum_{h=1}^{l-1} v^{2(2h-l)} w^{2h-l} \mathbb{K}_{h\delta}
-v^{2(l-1)} w^l \mathbb{K}_{l \delta}
\right).
\end{align*}
It follows by this identity and \eqref{eq:exp} that \eqref{eq:eHBe2} is equivalent to
\begin{align*}
& {\mathbf B }(w) \Big(1+ \sum_{m \geq 1} (v-v^{-1})\acute{\Theta}_{m} z^m \Big)
\\
&= \Big(1+ \sum_{n \geq 1} (v-v^{-1})\acute{\Theta}_{n} z^n \Big) {\mathbf B }(w)
\\
& \quad - \sum_{l \ge 1} z^l (v^2 -v^{-2})
\left( v^{2(1-l)} w^{-l}
+ (v^2 -v^{-2}) \sum_{h=1}^{l-1} v^{2(2h-l)} w^{2h-l} \mathbb{K}_{h\delta}
-v^{2(l-1)} w^l \mathbb{K}_{l \delta}
\right) \times
\\
&\qquad\qquad
\Big(1+ \sum_{n \geq 1} (v-v^{-1})\acute{\Theta}_{n \delta} z^n \Big) {\mathbf B }(w),
\end{align*}
Equating the coefficients of $(-z^m w^r)$ on both sides, for $m\ge 1$, we obtain
\begin{align*}
&(v-v^{-1}) [\acute{\Theta}_{m}, B_{1,r} ]
\\
& = (v^2 -v^{-2}) \times
\\
& \Big(
v^{2(1-m)} B_{1,r+m}
+ (v^2 -v^{-2}) \sum_{h=1}^{m-1} v^{2(2h-m)} B_{1,r+m-2h} \mathbb{K}_{h\delta}
- v^{2(m-1)} B_{1,r-m} \mathbb{K}_{m \delta}
\Big)
\\
& + (v -v^{-1}) (v^2 -v^{-2}) \times
\\
& \sum_{l=1}^{m-1} \acute{\Theta}_{m-l}
\Big(
v^{2(1-l)} B_{1,r+l}
+ (v^2 -v^{-2}) \sum_{h=1}^{l-1} v^{2(2h-l)} B_{1,r+l-2h} \mathbb{K}_{h\delta}
- v^{2(l-1)} B_{1,r-l} \mathbb{K}_{l \delta}
\Big).
\end{align*}
\fi
\subsection{Proof of Proposition~\ref{prop:equiv4} }
\label{subsec:proof}
We shall establish the equivalences among identities \eqref{eq:im-real2}, \eqref{eq:hB}, \eqref{eq:eHBe} and \eqref{eq:hB1}.
\subsubsection{Proof of equivalences of \eqref{eq:im-real2}, \eqref{eq:hB} and \eqref{eq:eHBe} }
The identity \eqref{eq:hB} can be equivalently reformulated via generating functions as
\begin{align*}
(v-v^{-1}) [\acute{\mathbf H}(z), {\mathbf B }(w)]
& =\sum_{k\ge 1,m\in\mathbb Z} B_{1,m+k} w^{m+k} \left( \frac{(v^{2} z w^{-1})^k}{k} - \frac{(v^{-2} z w^{-1})^k}{k}
\right)
\\
&\quad -\sum_{k\ge 1,m\in\mathbb Z} B_{1,m-k} w^{m-k} \left( \frac{ (v^{2} zw\mathbb{K}_\delta )^k}{k} - \frac{ (v^{-2} zw\mathbb{K}_\delta )^k}{k} \right)
\\
&= \ln \left(
\frac{1-v^{-2}zw^{-1}}{1-v^{2}zw^{-1}} \cdot \frac{1-v^{2} zw \mathbb{K}_\delta}{1-v^{-2}zw \mathbb{K}_\delta}
\right)
{\mathbf B }(w).
\end{align*}
Via integration this is then equivalent to
\begin{align} \label{eq:eHBe1}
e^{(v-v^{-1}) \acute{\mathbf H}(z)} {\mathbf B }(w) e^{ - (v-v^{-1}) \acute{\mathbf H}(z)}
& = {\mathbf B }(w)
\frac{(1 -v^{-2}zw^{-1}) (1 -v^{2} zw \mathbb{K}_\delta)}{(1 -v^{2}zw^{-1})(1 -v^{-2}zw \mathbb{K}_\delta)},
\end{align}
which can be reformulated as \eqref{eq:eHBe}.
We have the following identities:
\begin{align*}
\frac{1 -v^{-2}zw^{-1}}{1 -v^{2}zw^{-1}}
&= 1 +(v^2 -v^{-2}) \sum_{h\ge 1} v^{2(h-1)} z^h w^{-h},
\\
\frac{1 -v^{2} zw \mathbb{K}_\delta}{1 -v^{-2}zw \mathbb{K}_\delta}
&= 1 - (v^2 -v^{-2}) \sum_{h\ge 1} v^{2(1-h)} z^h w^{h} \mathbb{K}_{h\delta},
\end{align*}
and hence
\begin{align}
&\frac{(1 -v^{-2}zw^{-1}) (1 -v^{2} zw \mathbb{K}_\delta)}{(1 -v^{2}zw^{-1})(1 -v^{-2}zw \mathbb{K}_\delta)}
\label{eq:zwzw} \\
&= 1 + \sum_{a \ge 1} z^a (v^2 -v^{-2})
\Big(
v^{2(a-1)} w^{-a}
- (v^2 -v^{-2}) \sum_{h=1}^{a-1} v^{2(a-2h)} w^{2h-a} \mathbb{K}_{h\delta} -v^{2(1-a)} w^a \mathbb{K}_{a \delta}
\Big).
\notag
\end{align}
It follows by \eqref{eq:zwzw} that the identity \eqref{eq:eHBe} is equivalent to the following identity:
\begin{align*}
& \Big(1+ \sum_{m \geq 1} (v-v^{-1})\acute{\Theta}_{m } z^m \Big) {\mathbf B }(w)
=
{\mathbf B }(w) \Big(1+ \sum_{m \geq 1} (v-v^{-1})\acute{\Theta}_{m} z^m \Big) +
\\
& \quad \sum_{a \ge 1} z^a (v^2 -v^{-2})
\left(
v^{2(a-1)} w^{-a}
- (v^2 -v^{-2}) \sum_{h=1}^{a-1} v^{2(a-2h)} w^{2h-a} \mathbb{K}_{h\delta}
-v^{2(1-a)} w^a \mathbb{K}_{a \delta}
\right) \times
\\
&\qquad\qquad
{\mathbf B }(w) \Big(1+ \sum_{n \geq 1} (v-v^{-1})\acute{\Theta}_{n} z^n \Big).
\end{align*}
Equating the coefficients of $z^m w^l$ on both sides, for $m\ge 1$, we obtain
\begin{align*}
& (v-v^{-1}) [\acute{\Theta}_{m}, B_{1,l} ]
\\
& = (v^2 -v^{-2}) \times
\\
& \Big(
v^{2(m-1)} B_{1,l+m}
- (v^2 -v^{-2}) \sum_{h=1}^{m-1} v^{2(m-2h)} B_{1,l+m-2h} \mathbb{K}_{h\delta}
- v^{2(1-m)} B_{1,l-m} \mathbb{K}_{m \delta}
\Big)
\\
& + (v -v^{-1}) (v^2 -v^{-2}) \times
\\
& \sum_{a=1}^{m-1}
\Big(
v^{2(a-1)} B_{1,l+a}
- (v^2 -v^{-2}) \sum_{h=1}^{a-1} v^{2(a-2h)} B_{1,l+a-2h} \mathbb{K}_{h\delta}
- v^{2(1-a)} B_{1,l-a} \mathbb{K}_{a \delta}
\Big)
\acute{\Theta}_{m-a},
\end{align*}
which is equivalent to the identity \eqref{eq:im-real2}.
\subsubsection{Equivalence of \eqref{eq:eHBe} and \eqref{eq:hB1} }
The identity \eqref{eq:eHBe} can be rephrased as
\begin{align*}
(1-v^2zw^{-1})(1-v^{-2}zw\mathbb{K}_\delta) \acute{ \boldsymbol{\Theta}}(z) {\mathbf B }(w)
=(1-v^{-2}zw^{-1})(1-v^2zw\mathbb{K}_\delta){\mathbf B }(w) \acute{ \boldsymbol{\Theta}}(z).
\end{align*}
This can be rewritten as
\begin{align} \label{eq:TBBT}
&(1-v^2zw^{-1})(1-v^{-2}zw\mathbb{K}_\delta)\big(1+\sum_{m\geq1} (v-v^{-1})\acute{\Theta}_{m}z^m \big) \sum_{r\in\mathbb Z}B_{1,r}w^r
\\
=&(1-v^{-2}zw^{-1})(1-v^2zw\mathbb{K}_\delta) \sum_{r\in\mathbb Z}B_{1,r}w^r\big(1+\sum_{m\geq1} (v-v^{-1})\acute{\Theta}_{m}z^m \big).
\notag
\end{align}
Equating the coefficients of $z^m w^l$ on both sides of \eqref{eq:TBBT}, for $m\ge 1$, we obtain the following.
If $m\geq3$, then
\begin{align*}
\acute{\Theta}_m & B_{1,l}-v^2\acute{\Theta}_{m-1}B_{1,l+1} -v^{-2}\acute{\Theta}_{m-1}B_{1,l-1}\mathbb{K}_\delta+\acute{\Theta}_{m-2}B_{1,l}\mathbb{K}_\delta
\\
=&B_{1,l}\acute{\Theta}_m-v^{-2}B_{1,l+1}\acute{\Theta}_{m-1}-v^2B_{1,l-1}\acute{\Theta}_{m-1}\mathbb{K}_\delta+B_{1,l}\acute{\Theta}_{m-2}\mathbb{K}_\delta,
\end{align*}
which can be transformed into \eqref{eq:hB1}.
If $m=2$, then
\begin{align*}
&\acute{\Theta}_2B_{1,l}-v^2\acute{\Theta}_1 B_{1,l+1} -v^{-2}\acute{\Theta}_1 B_{1,l-1}\mathbb{K}_\delta
=B_{1,l}\acute{\Theta}_2-v^{-2}B_{1,l+1}\acute{\Theta}_1-v^{2}B_{1,l-1}\acute{\Theta}_1\mathbb{K}_\delta.
\end{align*}
If $m=1$, then
\begin{align*}
(v-v^{-1}) & \acute{\Theta}_1B_{1,l}-v^2B_{1,l+1} -v^{-2} B_{1,l-1}\mathbb{K}_\delta
\\
=&(v-v^{-1})B_{1,l}\acute{\Theta}_1-v^{-2}B_{1,l+1}-v^{2}B_{1,l-1}\mathbb{K}_\delta.
\end{align*}
The above identities in both cases for $m=1,2$ coincide with \eqref{eq:hB1}. Hence, the equivalence between \eqref{eq:eHBe} and \eqref{eq:hB1} in all cases are established.
\subsection{New imaginary root vectors}
With motivation from and application to $\imath$Hall algebra of the projective line \cite{LRW20} in mind, we shall introduce a somewhat normalized versions of the elements $\acute{\Theta}_{m}$ and $\acute{H}_m$, denoted by $\Theta_m$ and $H_m$, respectively.
For $m \geq 1$, define
\begin{align}
\label{eq:dBB}
\Theta_{m}=\acute{\Theta}_{m} - \sum\limits_{a=1}^{\lfloor\frac{m-1}{2}\rfloor}(v^2-1) v^{-2a} \acute{\Theta}_{m-2a}\mathbb{K}_{a\delta} -\delta_{m,ev}v^{1-m} \mathbb{K}_{\frac{m}{2}\delta},
\end{align}
where
\begin{align*}
\delta_{m,ev}= \begin{cases}
1, & \text{ for $m$ even}, \\
0, & \text{ for $m$ odd}.
\end{cases}
\end{align*}
Note that $\Theta_{1}=\acute{\Theta}_{1}$.
We also set $\Theta_{0}=\frac{1}{v-v^{-1}}$, and $\Theta_{m}=0$ for $m<0$.
Let
\begin{align} \label{Theta:z2}
\boldsymbol{\Theta} (z) = 1+(v-v^{-1}) \sum_{m\ge 1} \Theta_{m} z^m.
\end{align}
We define the new {\em imaginary $v$-root vectors} $H_m$ by letting
\begin{align} \label{eq:HTh}
1+ \sum_{m\geq 1} (v-v^{-1})\Theta_{m} u^m = \exp\Big( (v-v^{-1}) \sum_{m\ge 1} H_m u^m \Big).
\end{align}
\begin{lemma}
The identity \eqref{eq:dBB} can be reformulated as a generating function identity:
\begin{align}
\label{eq:Theta2}
\boldsymbol{\Theta}(z) = \frac{1- \mathbb{K}_\delta z^2}{1-v^{-2}\mathbb{K}_\delta z^2} \acute{ \boldsymbol{\Theta}} (z).
\end{align}
\end{lemma}
\begin{proof}
By \eqref{eq:dBB}, we have (with a change of variables $k=m-2a$ in the double summand below)
\begin{align*}
\boldsymbol{\Theta} (z) &=
1 + (v-v^{-1}) \sum_{m\ge 1} \acute{\Theta}_{m} z^m
- (v-v^{-1}) \sum_{m\ge 1} \sum_{m/2> a\ge 1} (v^2 -1) v^{-2a} \acute{\Theta}_{m-2a} \mathbb{K}_{a \delta} z^m
\\
&\quad - (v-v^{-1}) \sum_{n\ge 1} v^{1-2n} \mathbb{K}_{n \delta} z^{2n}
\\
&= \acute{ \boldsymbol{\Theta}}(z) - (v^2-1) \Big(\sum_{a\ge 1} v^{-2a} \mathbb{K}_{a \delta} z^{2a} \Big) \cdot \Big( \sum_{k\ge 1} (v -v^{-1}) \acute{\Theta}_{k} z^{k}\Big)
- \frac{(v-v^{-1}) v^{-1}\mathbb{K}_\delta z^2}{1-v^{-2}\mathbb{K}_\delta z^2}
\\
&= \acute{ \boldsymbol{\Theta}}(z) - (v^2-1) \frac{v^{-2}\mathbb{K}_\delta z^2}{1-v^{-2}\mathbb{K}_\delta z^2} \big(\acute{ \boldsymbol{\Theta}}(z) -1 \big)
- \frac{(v-v^{-1}) v^{-1}\mathbb{K}_\delta z^2}{1-v^{-2}\mathbb{K}_\delta z^2}
\\
&= \frac{1 -\mathbb{K}_\delta z^2}{1-v^{-2}\mathbb{K}_\delta z^2} \acute{ \boldsymbol{\Theta}}(z).
\end{align*}
The lemma is proved.
\end{proof}
\begin{lemma}
We have, for $m\in\mathbb Z$,
\begin{align}
\label{eqn:real1}
\Theta_{m+1}-v^{-2}\Theta_{m-1} \mathbb{K}_{\delta}=\acute{\Theta}_{m+1}-\acute{\Theta}_{m-1} \mathbb{K}_{\delta}.
\end{align}
\end{lemma}
\begin{proof}
By \eqref{eq:Theta2}, we have $(1-v^{-2}\mathbb{K}_\delta z^2) \boldsymbol{\Theta}(z) = (1- \mathbb{K}_\delta z^2) \acute{ \boldsymbol{\Theta}} (z).$ The lemma follows by comparing the coefficients of $z^{m+1}$ on both sides of this identity.
\end{proof}
\begin{proposition}
\label{prop:HH1}
We have $[H_m, H_n] =0 = [\Theta_m, \Theta_n]$, for $m,n\ge 1$.
\end{proposition}
\begin{proof}
Follows from Proposition~\ref{prop:ThTh1} by using \eqref{eq:Theta2} and noting $\mathbb{K}_\delta$ is central.
\end{proof}
The following is a counterpart via $H_m$ and $\Theta_m$ of the identities \eqref{eq:hB} and \eqref{eq:hB1}, and they look formally the same.
\begin{proposition}
\label{prop:HB1}
The following identity holds in $\widetilde{{\mathbf U}}^\imath$, for $m\ge 1$ and $r\in\mathbb Z$:
\begin{align}
[H_m, B_{1,l}] &=\frac{[2m]}{m} B_{1,l+m}-\frac{[2m]}{m} B_{1,l-m}\mathbb{K}_{m\delta},
\label{eq:hB3}
\\
[\Theta_{m},B_{1,r}]+[\Theta_{m-2},B_{1,r}]\mathbb{K}_\delta& =v^{2}[\Theta_{m-1},B_{1,r+1}]_{v^{-4}}+v^{-2}[\Theta_{m-1},B_{1,r-1}]_{v^{4}}\mathbb{K}_\delta.
\label{eq:hB2}
\end{align}
\end{proposition}
\begin{proof}
By \eqref{eq:eHBe}, \eqref{eq:Theta2} and noting $\mathbb{K}_\delta$ is central, we have
\begin{align*}
{ \boldsymbol{\Theta}} (z) {\mathbf B }(w)
& =
\frac{(1 -v^{-2}zw^{-1}) (1 -v^{2} zw \mathbb{K}_\delta)}{(1 -v^{2}zw^{-1})(1 -v^{-2}zw \mathbb{K}_\delta)}
{\mathbf B }(w) { \boldsymbol{\Theta}} (z),
\end{align*}
which takes the same form as \eqref{eq:eHBe}. Now \eqref{eq:hB2} (which takes the same form as \eqref{eq:hB1}) follows by exactly the same argument for the equivalence between \eqref{eq:eHBe} and \eqref{eq:hB1} in \S\ref{subsec:proof}. Then \eqref{eq:hB3} follows from \eqref{eq:hB2} exactly as in the proof of Proposition~\ref{prop:equiv4}.
\end{proof}
\begin{proposition}
\label{prop:BB1}
We have, for $r,s\in \mathbb Z$,
\begin{align}
\label{rel:iDr1b}
[B_{1,r},& B_{1,s+1}]_{v^{-2}} -v^{-2} [B_{1,r+1}, B_{1,s}]_{v^{2}} \\\notag
&= v^{-2}\Theta_{(s-r+1)\delta} \mathbb{K}_{r\delta +\alpha_1}-v^{-\red{4}}\Theta_{(s-r-1)\delta} \mathbb{K}_{(r+1)\delta+\alpha_1}
\\
&\quad +v^{-2}\Theta_{(r-s+1)\delta} \mathbb{K}_{s\delta +\alpha_1}-v^{-\red{4}}\Theta_{(r-s-1)\delta} \mathbb{K}_{(s+1)\delta+\alpha_1}.\notag
\end{align}
\end{proposition}
\begin{proof}
Follows from \eqref{rel:iDr} by using \eqref{eqn:real1}.
\end{proof}
\subsection{Grading and filtration}
We shall relabel the $v$-root vectors by a set of roots $\{\alpha_1 + k \delta \mid k\in \mathbb Z \} \cup \{m \delta \mid m \ge 1\}$:
\begin{equation} \label{eq:weight}
B_{\alpha_1+k\delta} =B_{1,k},
\quad B_{m\delta} =H_m, \quad \text{ for } k\in \mathbb Z, m\ge 1.
\end{equation}
Denote by $\widetilde{{\mathbf U}}^\imath_0$ the subalgebra of $\widetilde{{\mathbf U}}^\imath$ generated by $\Theta_m$, for $m\ge 1$. By Definition~\ref{def:Onsager}, the algebra $\widetilde{{\mathbf U}}^\imath$ is $\mathbb Z \mathbb{I} (\equiv \mathbb Z\alpha_0\oplus \mathbb Z \alpha_1)$-graded by weights, with
\begin{align} \label{eq:grade1}
\text{wt} (B_i) =\alpha_i, \quad \text{wt} (\mathbb{K}_i) =2\alpha_i, \quad \text{ for } i \in \{0,1\}.
\end{align}
It follows that $\text{wt} (B_{1,k}) =\alpha_1+k\delta, \text{wt} (H_m) =m\delta$, whence the notation \eqref{eq:weight}.
For any fixed positive integer $\texttt{h}$ (with a standard choice being $\texttt h=1$), the algebra $\widetilde{{\mathbf U}}^\imath$ is endowed with a filtered algebra structure $|\cdot |_{\texttt h}$ by setting
\begin{align}
\label{eq:hfilt}
|B_1|_{\texttt h} =1, \quad
|B_0|_{\texttt h} =\texttt{h}, \quad
|\mathbb{K}_i|_{\texttt h} =0, \quad
\quad \text{ for } i\in \{0,1\}.
\end{align}
Then there is an algebra isomorphism relating the associated graded $\text{gr} \widetilde{{\mathbf U}}^\imath$ to half a quantum affine $\mathfrak{sl}_2$, ${\mathbf U}^- =\langle F_0, F_1 \rangle$ in the setting of \cite{LW20}, which goes back to \cite{Let02, Ko14}:
\begin{align} \label{eq:filter1}
\text{gr} \widetilde{{\mathbf U}}^\imath \cong {\mathbf U}^- \otimes \mathbb Q(v)[\mathbb{K}_1^{\pm 1},\mathbb{K}_\delta^{\pm 1}],
\qquad \overline{B_i} \mapsto F_i \; (i=0,1).
\end{align}
\begin{proposition} [cf. \cite{BK20}]
\label{prop:PBW1}
The following holds in $\widetilde{{\mathbf U}}^\imath:$
\begin{enumerate}
\item
The subalgebra $\widetilde{{\mathbf U}}^\imath_0$ is a polynomial algebra in $\Theta_m$, for $m\ge 1$; it is also a polynomial algebra in $H_m$, for $m\ge 1$.
\item
Fix any total order $<$ on the roots $\{\alpha_1 + k \delta \mid k\in \mathbb Z \} \cup \{m \delta \mid m \ge 1\}$. Then $\widetilde{{\mathbf U}}^\imath$ admits a basis
\[
\big \{\mathbb{K}_1^{r} \mathbb{K}_\delta^{s} B_{\gamma_1}^{a_1} B_{\gamma_2}^{a_2} \ldots B_{\gamma_N}^{a_N} \mid
r, s\in\mathbb Z, a_1, a_2, \ldots, a_N \in \mathbb N, N\in \mathbb N, \gamma_1<\gamma_2 <\ldots < \gamma_N \big \}.
\]
\end{enumerate}
\end{proposition}
\begin{proof}
Part (2) follows as a variant of \cite[Theorem 4.5]{BK20} on a PBW basis for ${\mathbf U}^\imath$, which is proved by using the filtration \eqref{eq:hfilt}--\eqref{eq:filter1} and comparing with the $v$-root vectors in ${\mathbf U}$ given in \cite{Da93}. (A mild difference is that $\widetilde{{\mathbf U}}^\imath$ admits the central elements $\mathbb{K}_\beta$, for $\beta \in {\mathcal R}$, and $B_{m\delta}$ for ${\mathbf U}^\imath$ used {\em loc. cit.} is understood as a version of $\acute\Theta_m$.)
Clearly, the algebraic independence among $\{\acute\Theta_m\mid m\ge 1\}$ implies the algebraic independence of
$\{\Theta_m\mid m\ge 1\}$ as well as of $\{H_m\mid m\ge 1\}$.
Part (1) follows.
\end{proof}
\subsection{A Drinfeld type presentation in rank 1}
\label{subsec:Dr1}
\begin{definition}
\label{def:DrOnsa}
Let ${}^{\text{Dr}}\tUi ={}^{\text{Dr}}\tUi (\widehat{\mathfrak{sl}}_2)$ be the $\mathbb Q(v)$-algebra generated by $\mathbb{K}_1^{\pm1}$, $C^{\pm1}$, $H_{m}$ and $B_{1,r}$, where $m\geq1$, $r\in\mathbb Z$, subject to the following defining relations, for $m,n\geq1$ and $r,s\in \mathbb Z$:
\begin{align}
\mathbb{K}_1\mathbb{K}_1^{-1} =1, \quad C C^{-1} &=1, \quad \mathbb{K}_1, C \text{ are central},
\label{iDRo0}
\\
[H_m,H_n] &=0, \label{iDRo1}
\\
[H_m, B_{1,r}] &=\frac{[2m]}{m} B_{1,r+m}-\frac{[2m]}{m} B_{1,r-m}C^m,
\label{iDRo2}
\\
\label{iDRo3}
[B_{1,r}, B_{1,s+1}]_{v^{-2}} -v^{-2} [B_{1,r+1}, B_{1,s}]_{v^{2}}
&= v^{-2}\Theta_{s-r+1} C^r \mathbb{K}_1-v^{-4}\Theta_{s-r-1} C^{r+1} \mathbb{K}_1 \\ \notag
&\quad +v^{-2}\Theta_{r-s+1} C^s \mathbb{K}_1-v^{-4}\Theta_{r-s-1} C^{s+1} \mathbb{K}_1, \notag
\end{align}
where
$1+ \sum_{m\geq 1} (v-v^{-1})\Theta_{m} z^m = \exp\big( (v-v^{-1}) \sum_{m\ge 1} H_m z^m \big).$
\end{definition}
\begin{theorem}
\label{thm:Dr1}
There is a $\mathbb Q(v)$-algebra isomorphism ${\Phi}: {}^{\text{Dr}}\tUi \rightarrow\widetilde{{\mathbf U}}^\imath$, which sends
\begin{align*}
B_{1,r}\mapsto B_{1,r}, \quad \Theta_m \mapsto \Theta_m,
\quad
\mathbb{K}_1\mapsto \mathbb{K}_1, \quad C\mapsto \mathbb{K}_\delta,
\quad \text{ for } m\ge 1, r\in \mathbb Z.
\end{align*}
The inverse ${\Phi}^{-1} : \widetilde{{\mathbf U}}^\imath \rightarrow {}^{\text{Dr}}\tUi$ sends
$\mathbb{K}_1\mapsto \mathbb{K}_1,
\mathbb{K}_0\mapsto C \mathbb{K}_1^{-1},
B_1\mapsto B_{1,0},
B_0\mapsto B_{1,-1}C \mathbb{K}_1^{-1}.$
\end{theorem}
\begin{proof}
The relations \eqref{iDRo1}, \eqref{iDRo2} and \eqref{iDRo3} in ${}^{\text{Dr}}\widetilde{\mathbf U}$ hold for the images of the generators of ${}^{\text{Dr}}\tUi$ under $\Phi$, thanks to Propositions~\ref{prop:HH1}, \ref{prop:HB1}, and \ref{prop:BB1}, respectively. (The relation \eqref{iDRo0} under $\Phi$ holds trivially.) Thus $\Phi$ is a homomorphism.
By the defining relations of ${}^{\text{Dr}}\tUi$, one shows that ${}^{\text{Dr}}\tUi$ has a spanning set
\[
\{\mathbb{K}_1^{r} C^{s} B_{\gamma_1}^{a_1} B_{\gamma_2}^{a_2} \ldots B_{\gamma_N}^{a_N} \mid
r, s\in\mathbb Z, a_1, a_2, \ldots, a_N \in \mathbb N, N\in \mathbb N, \gamma_1<\gamma_2 <\ldots < \gamma_N \}.
\]
Since this set is mapped to a basis of $\widetilde{{\mathbf U}}^\imath$ according to Proposition~\ref{prop:PBW1}, it must be a basis for ${}^{\text{Dr}}\tUi$ as well. Therefore, $\Phi$ is an isomorphism.
\end{proof}
\begin{remark}
There is another Drinfeld type presentation of the $\mathbb Q(v)$-algebra $\widetilde{{\mathbf U}}^\imath$ with the same set of generators as in Theorem~\ref{thm:Dr1}, subject to the relations \eqref{iDRo0}--\eqref{iDRo2} and \eqref{iDRo3b} below (in place of \eqref{iDRo3}):
\begin{align}
\label{iDRo3b}
[B_{1,r}, B_{1,s+1}]_{v^{-2}} -v^{-2} [B_{1,r+1}, B_{1,s}]_{v^{2}}
&= v^{-2}\Theta_{s-r+1} C^r \mathbb{K}_1-v^{-2}\Theta_{s-r-1} C^{r+1} \mathbb{K}_1 \\ \notag
&\quad +v^{-2}\Theta_{r-s+1} C^s \mathbb{K}_1-v^{-2}\Theta_{r-s-1} C^{s+1} \mathbb{K}_1. \notag
\end{align}
One should (secretly) regard the $\Theta_m$ in this presentation as $\acute{\Theta}_m$ defined earlier, and then the relation \eqref{iDRo3b} is simply \eqref{rel:iDr}. In this way, the equivalence between \eqref{rel:iDr} and \eqref{rel:iDr1b} (or \eqref{iDRo3}) has been established earlier. Hence the equivalence between this presentation and the presentation ${}^{\text{Dr}}\tUi$ in Definition~\ref{def:DrOnsa} follows.
\end{remark}
\section{A Drinfeld type presentation of affine $\imath$quantum groups }
\label{sec:main}
In this section, we study in depth the $\imath$quantum groups $\widetilde{{\mathbf U}}^\imath$ of split affine ADE type. We introduce a new set of generators for $\widetilde{{\mathbf U}}^\imath$ and use them to formulate a Drinfeld type presentation for $\widetilde{{\mathbf U}}^\imath$, generalizing the one for $q$-Onsager algebra in \S\ref{subsec:Dr1}.
\subsection{Affine Weyl and braid groups}
Let $(c_{ij})_{i,j\in \mathbb{I}_0}$ be the Cartan matrix of the simple Lie algebra $\mathfrak{g}$ of type ADE. Let ${\mathcal R}_0$ be the set of roots for $\mathfrak{g}$, and fix a set ${\mathcal R}^+_0$ of positive roots with simple roots $\alpha_i$ $(i\in \mathbb{I}_0)$. Denote by $\theta$ the highest root of $\mathfrak{g}$.
Let $\widehat{\mathfrak{g}}$ be the (untwisted) affine Lie algebra with affine Cartan matrix denoted by $(c_{ij})_{i,j\in\mathbb{I}}$, where $\mathbb{I}=\{0\} \cup \mathbb{I}_0$ with the affine node $0$. Let $\alpha_i$ $(i\in \mathbb{I})$ be the simple roots of $\widehat{\mathfrak{g}}$, and $\alpha_0=\delta -\theta$, where $\delta$ denotes the basic imaginary root. The root system ${\mathcal R}$ for $\widehat{\mathfrak{g}}$ and its positive system ${\mathcal R}^+$ are defined to be
\begin{align}
{\mathcal R} &=\{\pm (\beta + k \delta) \mid \beta \in {\mathcal R}_0^+, k \in \mathbb Z\} \cup \{m \delta \mid m \in \mathbb Z\backslash \{0\} \},
\label{eq:roots} \\
{\mathcal R}^+ &= \{k \delta +\beta \mid \beta \in {\mathcal R}_0^+, k \ge 0\}
\cup \{k \delta -\beta \mid \beta \in {\mathcal R}_0^+, k > 0\}
\cup \{m \delta \mid m \ge 1\}.
\label{eq:roots+}
\end{align}
For $\gamma =\sum_{i\in \mathbb{I}} n_i \alpha_i \in \mathbb N \mathbb{I}$, the height $\text{ht} (\gamma)$ is defined as $\text{ht} (\gamma) =\sum_{i\in \mathbb{I}} n_i$.
Let $P$ and $Q$ denote the weight and root lattices of $\mathfrak{g}$, respectively. Let $\omega_i \in P$ $(i\in \mathbb{I}_0)$ be the fundamental weights of $\mathfrak{g}$. Note $\alpha_i =\sum_{j\in \mathbb{I}_0} c_{ij}\omega_j$. We define a bilinear pairing $\langle \cdot, \cdot \rangle : P\times Q \rightarrow \mathbb Z$ such that $\langle \omega_i, \alpha_j \rangle =\delta_{i,j}$, for $i,j \in \mathbb{I}_0$, and thus $\langle \alpha_i, \alpha_j \rangle = c_{ij}$.
The Weyl group $W_0$ of $\mathfrak{g}$ is generated by the simple reflection $s_i$, for $i \in \mathbb{I}_0$. It acts on $P$ by
$s_i(x)=x-\langle x, \alpha_i \rangle\alpha_i$ for $x\in P$. The extended affine Weyl group $\widetilde{W}$ is the semi-direct product $W_0 \ltimes P$, which contains the affine Weyl group $W:=W_0 \ltimes Q =\langle s_i \mid i \in \mathbb{I} \rangle$ as a subgroup; we denote
\[
t_\omega =(1, \omega) \in \widetilde W, \quad \text{ for } \omega \in P.
\]
We identify $P/Q$ with a finite group $\Omega$ of Dynkin diagram automorphism, and so $\widetilde{W} =\Omega . W$. There is a length function $\ell(\cdot)$ on $\widetilde{W}$ such that $\ell(s_i)=1$, for $i\in \mathbb{I}$, and each element in $\Omega$ has length 0.
For $i\in \mathbb{I}_0$, as in \cite{Be94}, we have
\begin{equation} \label{eq:tomega}
\ell(t_{\omega_i}) =\ell(\omega_i')+1, \qquad
\text{ where } \omega_i':= t_{\omega_i} s_i \in W.
\end{equation}
Let ${\mathbf U} ={\mathbf U}(\widehat{\mathfrak{g}})$ denote the Drinfeld-Jimbo affine quantum group, a $\mathbb Q(v)$-algebra generated by $E_i, F_i, K_i^{\pm 1}$, for $i\in \mathbb{I}$.
Following Lusztig \cite{Lus90, Lus94}, we have the braid group action of $T_w$ on $\widetilde{\mathbf U}$, for $w \in \widetilde W$; for example $T_i$, for $i\in \mathbb{I}$, acts on ${\mathbf U}$ by, for $i\neq j \in \mathbb{I}, \mu \in \mathbb Z\mathbb{I}$ (our $K_i$ corresponds to $\widetilde{K}_i$ in \cite{Lus94}):
\begin{equation}\label{eq:braidgroup}
\begin{split}
&T_i (E_i) = -F_i {K}_{i} , \quad T_i (F_i) = - {K}_{i}^{-1}E_i, \quad T_i (K_\mu) = K_{s_i(\mu)}, \\
&T_i (E_j) = \sum_{r+s = - c_{ij}} (-1)^r v^{-r}_{i} E^{(s)}_i E_j E^{(r)}_i, \\
&T_i (F_j) = \sum_{r+s = - c_{ij}} (-1)^r v^{r}_{i} F^{(r)}_i F_j F^{(s)}_i.
\end{split}
\end{equation}
The formulas above are written in terms of $v_i$ (dependent on the length of $\alpha_i$) for the convenience of future references. In the current ADE setting, we always have $v_i =v$. Similar remarks apply elsewhere, e.g., $[n]_i =[n]_{v_i}$.
The following was known to Bernstein and Lusztig.
\begin{lemma} [\text{\cite[\S2.7]{Lus89}}]
\label{lem:Braid-Lus}
Let $x\in P$, $i, j \in \mathbb{I}_0$.
\begin{enumerate}
\item[(a)]
If $s_ix=xs_i$, then $T_iT_x=T_xT_i$.
\item[(b)]
If $s_i xs_i=t_{\alpha_i}^{-1}x=\prod_{k\in \mathbb{I}_0} t_{\omega_k}^{a_k}$, then we have
$T_i^{-1}T_xT_i=\prod_k T_{\omega_k}^{a_k}$; in particular, we have $T_i^{-1}T_{\omega_i}T_i^{-1}=T_{\omega_i}^{-1}\prod_{k\neq i}T_{\omega_k}^{-c_{ik}}$.
\item[(c)]
$T_{\omega_i} T_{\omega_j} = T_{\omega_j} T_{\omega_i}.$
\end{enumerate}
\end{lemma}
We record some useful formulas:
\begin{align}
T_i \big(T_{\omega_i'}(F_i) \big)
&=T_{\omega_i'}(F_i) F_i^{(2)} -v_i F_i T_{\omega_i'}(F_i) F_i +v_i^2 F_i^{(2)} T_{\omega_i'}(F_i),
\label{eq:TiX}
\\
T_i^{-1} \big(T_{\omega_i'}(F_i) \big)
&= F_i^{(2)} T_{\omega_i'}(F_i) -v_i F_i T_{\omega_i'}(F_i) F_i +v_i^2 T_{\omega_i'}(F_i) F_i^{(2)},
\label{eq:TiX2}
\\
T_i \big([T_{\omega_i'}(F_i), F_i]_{v_i^{-2}} \big) &=[F_i, T_{\omega_i'}(F_i)]_{v_i^{-2}}.
\label{T-imagine}
\end{align}
The formula \eqref{eq:TiX} follows by applying the Chevalley involution to its $E$-version, which can be found in \cite{Be94}. The formula \eqref{eq:TiX2} can be derived (and is equivalent to) \eqref{eq:TiX}. Formula \eqref{T-imagine} can also be derived directly from \eqref{eq:TiX}.
\subsection{Drinfeld presentation of an affine quantum group}
Let $C=(c_{ij})_{i,j\in\mathbb{I}}$ be the Cartan matrix of untwisted affine type. Let $^{\text{Dr}}{\mathbf U}$ be the $\mathbb Q(v)$-algebra generated by $x_{i k}^{\pm}$, $h_{i l}$, $K_i^{\pm 1}$, $C^{\pm \frac12}$, for $i\in\mathbb{I}$, $k\in\mathbb Z$, and $l\in\mathbb Z\backslash\{0\}$, subject to the following relations: $C^{\frac12}, C^{- \frac12}$ are central such that
\begin{align*}
[K_i,K_j] & = [K_i,h_{j l}] =0, \quad K_i K_i^{-1} =C^{\frac12} C^{- \frac12} =1,
\\
[h_{ik},h_{jl}] &= \delta_{k, -l} \frac{[k c_{ij}]_i}{k} \frac{C^k -C^{-k}}{v_j -v_j^{-1}},
\\
K_ix_{jk}^{\pm} K_i^{-1} &=v^{\pm c_{ij}} x_{jk}^{\pm},
\\
[h_{i k},x_{j l}^{\pm}] &=\pm\frac{[kc_{ij}]_i}{k} C^{\mp \frac{|k|}2} x_{j,k+l}^{\pm},
\\
[x_{i k}^+,x_{j l}^-] &=\delta_{ij} {(C^{\frac{k-l}2} K_i\psi_{i,k+l} - C^{\frac{l-k}2} K_i^{-1} \varphi_{i,k+l})},
\\
x_{i,k+1}^{\pm} x_{j,l}^{\pm}-v^{\pm c_{ij}} x_{j,l}^{\pm} x_{i,k+1}^{\pm} &=v^{\pm c_{ij}} x_{i,k}^{\pm} x_{j,l+1}^{\pm}- x_{j,l+1}^{\pm} x_{i,k}^{\pm},
\\
\operatorname{Sym}\nolimits_{k_1,\dots,k_r}\sum_{t=0}^{r} (-1)^t \qbinom{r}{t}_i
& x_{i,k_1}^{\pm}\cdots
x_{i,k_t}^{\pm} x_{j,l}^{\pm} x_{i,k_t+1}^{\pm} \cdots x_{i,k_n}^{\pm} =0, \text{ for } r= 1-c_{ij}\; (i\neq j),
\end{align*}
where
$\operatorname{Sym}\nolimits_{k_1,\dots,k_r}$ denotes the symmetrization with respect to the indices $k_1,\dots,k_r$, $\psi_{i,k}$ and $\varphi_{i,k}$ are defined by the following functional equations:
\begin{align*}
1+ \sum_{m\geq 1} (v_i-v_i^{-1})\psi_{i,m}u^m &= \exp\Big((v_i -v_i^{-1}) \sum_{m\ge 1} h_{i,m}u^m\Big),
\\
1+ \sum_{m\geq1 } (v_i-v_i^{-1}) \varphi_{i, -m}u^{-m} &= \exp \Big((v_i -v_i^{-1}) \sum_{m\ge 1} h_{i,-m}u^{-m}\Big).
\end{align*}
(We omit a degree operator $D$ in the version of ${}^{\text{Dr}}{\mathbf U}$ above.)
It was stated by Drinfeld \cite{Dr88} (and proved by Beck \cite{Be94} and Damiani \cite{Da12, Da15})
that there exists a $\mathbb Q(v)$-algebra isomorphism
\begin{align} \label{eq:phi1}
\phi: {}^{\text{Dr}}{\mathbf U} \longrightarrow {\mathbf U}.
\end{align}
We omit the explicit formulas for the images under $\phi$ of generators of ${}^{\text{Dr}}{\mathbf U}$; see \cite{Be94, BCP99}.
\subsection{Affine $\imath$quantum groups of split ADE type}
Recall the Cartan matrix $(c_{ij})_{i,j\in \mathbb{I}}$ of affine ADE type, for $\mathbb{I} = \mathbb{I}_0 \cup \{0\}$ with the affine node $0$.
The notion of (quasi-split) universal $\imath$quantum groups $\widetilde{{\mathbf U}}^\imath$ was formulated in \cite{LW19a}.
The {\em universal affine $\imath$quantum group of split ADE type} is the $\mathbb Q(v)$-algebra $\widetilde{{\mathbf U}}^\imath =\widetilde{{\mathbf U}}^\imath(\widehat{\mathfrak{g}})$ with generators $B_i$, $\mathbb{K}_i^{\pm 1}$ $(i\in \mathbb{I})$, subject to the following relations, for $i, j\in \mathbb{I}$:
\begin{align}
\mathbb{K}_i\mathbb{K}_i^{-1} =\mathbb{K}_i^{-1}\mathbb{K}_i=1, & \quad \mathbb{K}_i \text{ is central},
\\
B_iB_j -B_j B_i&=0, \qquad\qquad\qquad\qquad\qquad \text{ if } c_{ij}=0,
\label{eq:S1} \\
B_i^2 B_j -[2] B_i B_j B_i +B_j B_i^2 &= - v^{-1} B_j \mathbb{K}_i, \qquad\qquad\qquad \text{ if }c_{ij}=-1,
\label{eq:S2} \\
\sum_{r=0}^3 (-1)^r \qbinom{3}{r} B_i^{3-r} B_j B_i^{r} &= -v^{-1} [2]^2 (B_iB_j-B_jB_i) \mathbb{K}_i, \text{ if }c_{ij}=-2.
\label{eq:S3}
\end{align}
The universal split $\imath$quantum group of rank 1 is also known as the $q$-Onsager algebra, and this is the only case where the relation \eqref{eq:S3} is needed; see Definition~\ref{def:Onsager}.
For $\mu = \mu' +\mu'' \in \mathbb Z \mathbb{I} := \oplus_{i\in \mathbb{I}} \mathbb Z \alpha_i$, define $\mathbb{K}_\mu$ such that
\begin{align}
\mathbb{K}_{\alpha_i} =\mathbb{K}_i, \quad
\mathbb{K}_{-\alpha_i} =\mathbb{K}_i^{-1}, \quad
\mathbb{K}_{\mu} =\mathbb{K}_{\mu'} \mathbb{K}_{\mu''},
\quad \mathbb{K}_\delta =\mathbb{K}_0 \mathbb{K}_\theta.
\end{align}
The algebra $\widetilde{{\mathbf U}}^\imath$ is endowed with a filtered algebra structure
\begin{align} \label{eq:filt1}
\widetilde{{\mathbf U}}^{\imath,0} \subset \widetilde{{\mathbf U}}^{\imath,1} \subset \cdots \subset \widetilde{{\mathbf U}}^{\imath,m} \subset \cdots
\end{align}
by setting
\begin{align} \label{eq:filt}
\widetilde{{\mathbf U}}^{\imath,m} =\mathbb Q(v)\text{-span} \{ B_{i_1} B_{i_2} \ldots B_{i_r} \mathbb{K}_\mu \mid \mu \in \mathbb N\mathbb{I}, i_1, \ldots, i_r \in \mathbb{I}, r\le m \}.
\end{align}
Note that
\begin{align} \label{eq:UiCartan}
\widetilde{{\mathbf U}}^{\imath,0} =\bigoplus_{\mu \in \mathbb N\mathbb{I}} \mathbb Q(v) \mathbb{K}_\mu,
\end{align}
is the $\mathbb Q(v)$-subalgebra generated by $\mathbb{K}_i$ for $i\in \mathbb{I}$.
Then, according to a basic result of Letzter and Kolb on quantum symmetric pairs adapted to our setting of $\widetilde{{\mathbf U}}^\imath$ (cf. \cite{Let02, Ko14}), the associated graded $\text{gr} \widetilde{{\mathbf U}}^\imath$ with respect to \eqref{eq:filt1}--\eqref{eq:filt} can be identified with
\begin{align} \label{eq:filter}
\text{gr} \widetilde{{\mathbf U}}^\imath \cong {\mathbf U}^- \otimes \mathbb Q(v)[\mathbb{K}_i^\pm | i\in \mathbb{I}],
\qquad \overline{B_i}\mapsto F_i, \quad
\overline{\mathbb{K}}_i \mapsto \mathbb{K}_i \; (i\in \mathbb{I}).
\end{align}
\begin{remark}
\label{rem:Kk}
The generator $\mathbb{K}_i$ here, which corresponds to a (generalized) simple module in the $\imath$Hall algebra, is related to the generator $\tilde{k}_i$ used in \cite{LW19a, LW20} (which is natural from the viewpoint of Drinfeld doubles) by a rescaling:
$\mathbb{K}_i = -v^2\tilde{k}_i$.
The precise relation between the algebra $\widetilde{{\mathbf U}}^\imath$ and the $\imath$quantum group ${\mathbf U}^\imath$ arising from quantum symmetric pairs \cite{Ko14} is explained {\em loc. cit.}; also see \S\ref{subsec:parameter} below.
\end{remark}
\begin{remark}
The $\mathbb Q(v)$-algebra $\widetilde{{\mathbf U}}^\imath$ is $\mathbb Z \mathbb{I}$-graded by letting
\begin{align}
\label{eq:deg}
\text{wt} (B_i) =\alpha_i, \quad \text{wt} (\mathbb{K}_i) =2\alpha_i, \quad \text{ for } i \in \mathbb{I}.
\end{align}
A variant of $\widetilde{{\mathbf U}}^\imath$, in which $\mathbb{K}_i$ is not assumed to be invertible, is $\mathbb N\mathbb{I}$-graded by \eqref{eq:deg}. \end{remark}
\begin{lemma} [\text{also cf. \cite{KP11, BK20}}]
\label{lem:Ti}
For $i\in I$, there exists an automorphism $\texttt{\rm T}} %{\mathbf T_i$ of the $\mathbb Q(v)$-algebra $\widetilde{{\mathbf U}}^\imath$ such that
$\texttt{\rm T}} %{\mathbf T_i(\mathbb{K}_\mu) =\mathbb{K}_{s_i\mu}$, and
\[
\texttt{\rm T}} %{\mathbf T_i(B_j)= \begin{cases}
\mathbb{K}_i^{-1} B_i, &\text{ if }j=i,\\
B_j, &\text{ if } c_{ij}=0, \\
B_jB_i-vB_iB_j, & \text{ if }c_{ij}=-1, \\
{[}2]^{-1} \big(B_jB_i^{2} -v[2] B_i B_jB_i +v^2 B_i^{2} B_j \big) + B_j\mathbb{K}_i, & \text{ if }c_{ij}=-2,
\end{cases}
\]
for $\mu\in \mathbb Z\mathbb{I}$ and $j\in \mathbb{I}$.
Moreover, $\texttt{\rm T}} %{\mathbf T_i$ $(i\in \mathbb{I})$ satisfy the braid group relations, i.e., $\texttt{\rm T}} %{\mathbf T_i \texttt{\rm T}} %{\mathbf T_j =\texttt{\rm T}} %{\mathbf T_j \texttt{\rm T}} %{\mathbf T_i$ if $c_{ij}=0$, and $\texttt{\rm T}} %{\mathbf T_i \texttt{\rm T}} %{\mathbf T_j \texttt{\rm T}} %{\mathbf T_i =\texttt{\rm T}} %{\mathbf T_j \texttt{\rm T}} %{\mathbf T_i \texttt{\rm T}} %{\mathbf T_j$ if $c_{ij}=-1$.
\end{lemma}
\begin{proof}
The formulas for $\texttt{\rm T}} %{\mathbf T_i$ when $c_{ij}=0, -1$ were obtained in \cite{LW19b} via reflection functors in an $\imath$Hall algebra approach and the braid relations are verified therein; the formulas for $\texttt{\rm T}} %{\mathbf T_i$ when $c_{ij}= -2$ can be similarly obtained (as a special case of $\imath$quantum groups of Kac-Moody type in a forthcoming sequel to \cite{LW19b}); also see \cite{KP11} and \cite{BK20} for closely related versions on ${\mathbf U}^\imath$ via computer packages. It follows from the reflection functor construction that these formulas define an automorphism $\texttt{\rm T}} %{\mathbf T_i$ of $\widetilde{{\mathbf U}}^\imath$.
\end{proof}
We also have $\texttt{\rm T}} %{\mathbf T_i^{-1} (\mathbb{K}_\mu) =\mathbb{K}_{s_i\mu}$, and
\[
\texttt{\rm T}} %{\mathbf T_i^{-1} (B_j)=
\begin{cases}
\mathbb{K}_i^{-1} B_i, &\text{ if }j=i,\\
B_j, &\text{ if } c_{ij}=0, \\
B_iB_j -vB_jB_i, & \text{ if }c_{ij}=-1, \\
{[}2]^{-1} \big( B_i^{2}B_j-v[2] B_iB_jB_i+v^2 B_jB_i^{2} \big) +B_j\mathbb{K}_i, & \text{ if }c_{ij}=-2.
\end{cases}
\]
Just as the quantum group setting \cite{Lus94}, $\texttt{\rm T}} %{\mathbf T_i^{-1}$ is related to $\texttt{\rm T}} %{\mathbf T_i$ by
\begin{align}
\label{antiT}
\texttt{\rm T}} %{\mathbf T_i^{-1} = \dag \circ \texttt{\rm T}} %{\mathbf T_i \circ \dag,
\end{align}
where $\dag$ denotes the anti-involution of the $\mathbb Q(v)$-algebra $\widetilde{{\mathbf U}}^\imath$, which fixes the generators $B_i, \mathbb{K}_i$.
For $w\in \widetilde{W}$ with a reduced expression $w =\sigma s_{i_1} \ldots s_{i_r}$ and $\sigma \in \Omega$, we define $\texttt{\rm T}} %{\mathbf T_w = \sigma \texttt{\rm T}} %{\mathbf T_{i_1} \ldots \texttt{\rm T}} %{\mathbf T_{i_r}$, where $\sigma$ acts on $\widetilde{{\mathbf U}}^\imath$ by permuting the indices of generators, $\sigma(B_i) =B_{\sigma i}, \sigma(\mathbb{K}_i) =\mathbb{K}_{\sigma i}$, for all $i\in \mathbb{I}$. Hence, $\texttt{\rm T}} %{\mathbf T_w$ is well defined by Lemma~\ref{lem:Ti}. In particular, the standard results such as Lemma~\ref{lem:Braid-Lus} on braid groups associated to $\widetilde W$ apply to $\texttt{\rm T}} %{\mathbf T_w$.
\begin{lemma}
\label{lem:fixB}
We have $\texttt{\rm T}} %{\mathbf T_w (B_i) = B_{w i}$, for $i\in \mathbb{I}$ and $w \in W$ such that $wi \in \mathbb{I}$.
\end{lemma}
\begin{proof}
This statement is well known for quantum groups \cite{Lus94}, and the proof here is adapted from the proof in \cite[Lemma~8.20]{J95} in the usual quantum group setting.
We first prove the lemma in the rank 2 setting. Assume that $w \in \langle s_i, s_j \rangle$, for some $j\in \mathbb{I}$. The case when $s_i s_j =s_j s_i$ is trivial as we must have $wi=i$. Otherwise, $c_{ij}=-1$, $i\in \mathbb{I}$ such that $wi \in \mathbb{I}$ only happen when $w=s_is_j$, and in this case, a direct computation shows $wi=j$ and $\texttt{\rm T}} %{\mathbf T_i\texttt{\rm T}} %{\mathbf T_j (B_i) =B_j$.
In general, we prove by induction on $\ell(w)$.
We shall assume $\ell(w)>0$ and let $j \in \mathbb{I}$ such that $w \alpha_j<0$ (clearly $j \neq i$). Then, as in the proof of \cite[Lemma~8.20]{J95}, we have a decomposition $w=w'w''$ such that $w'' \in \langle s_i, s_j \rangle$, $\ell(w)=\ell(w') +\ell(w'')$, $w\alpha_i >0,$ and $w\alpha_j<0$. Thus, $w''\alpha_i>0, w''\alpha_j<0$. Following the proof {\em loc. cit.}, $wi\in \mathbb{I}$ implies $w''i\in \mathbb{I}$.
The inductive assumption applied to $w'$ (and $w'' i \in \mathbb{I}$) yields $\texttt{\rm T}} %{\mathbf T_{w'} (B_{w'' i}) =B_{w'w''i} =B_{wi}$. The rank~2 result in a previous paragraph shows $B_{w'' i} =\texttt{\rm T}} %{\mathbf T_{w''} B_i$. Therefore, we obtain
$\texttt{\rm T}} %{\mathbf T_w (B_i) = \texttt{\rm T}} %{\mathbf T_{w'} \texttt{\rm T}} %{\mathbf T_{w''} (B_i)
= \texttt{\rm T}} %{\mathbf T_{w'} (B_{w''i}) = B_{w i}.$
\end{proof}
For $i \in \mathbb{I}_0$, let $\widetilde{{\mathbf U}}^\imath_{[i]}$ be the subalgebra of $\widetilde{{\mathbf U}}^\imath$ generated by
$B_i, \texttt{\rm T}} %{\mathbf T_{\omega'_i}(B_i), \mathbb{K}_i, \mathbb{K}_{\delta-\alpha_i}.
$
\begin{lemma} \label{lem:TiFix}
Let $i, j\in \mathbb{I}_0$ be such that $i\neq j$. Then $\texttt{\rm T}} %{\mathbf T_{\omega_i}(x)=x$, for all $x\in\widetilde{{\mathbf U}}^\imath_{[j]}$.
\end{lemma}
\begin{proof}
Clearly, $\texttt{\rm T}} %{\mathbf T_{\omega_i}(\mathbb{K}_j)=\mathbb{K}_j$ and $\texttt{\rm T}} %{\mathbf T_{\omega_i}(\mathbb{K}_{\delta-\alpha_j})=\mathbb{K}_{\delta-\alpha_j}$.
By Lemma \ref{lem:fixB}, $\texttt{\rm T}} %{\mathbf T_{\omega_i}(B_j)=B_j$.
By Lemma \ref{lem:Braid-Lus}(b) and noting $\texttt{\rm T}} %{\mathbf T_j^{-1}(B_i) =\texttt{\rm T}} %{\mathbf T_i(B_j)$, we have
\begin{align}
\texttt{\rm T}} %{\mathbf T_{\omega_i}^{-1} \texttt{\rm T}} %{\mathbf T_{\omega_j}(B_j)=&\texttt{\rm T}} %{\mathbf T_i^{-1}\texttt{\rm T}} %{\mathbf T_{\omega_i}\texttt{\rm T}} %{\mathbf T_i^{-1}(B_j)
\label{eq:TTB1} \\
=&\texttt{\rm T}} %{\mathbf T_i^{-1} \texttt{\rm T}} %{\mathbf T_{\omega_j} \texttt{\rm T}} %{\mathbf T_j^{-1}(B_i)
=\texttt{\rm T}} %{\mathbf T_{\omega_j}\texttt{\rm T}} %{\mathbf T_i^{-1}\texttt{\rm T}} %{\mathbf T_j^{-1}(B_i)
=\texttt{\rm T}} %{\mathbf T_{\omega_j}(B_j).
\notag
\end{align}
On the other hand, we have
\begin{align} \label{eq:TTB2}
\texttt{\rm T}} %{\mathbf T_{\omega_j}(B_j)=\texttt{\rm T}} %{\mathbf T_{\omega_j'}\texttt{\rm T}} %{\mathbf T_j(B_j)=\texttt{\rm T}} %{\mathbf T_{\omega_j'}(B_j\mathbb{K}_j^{-1})= \texttt{\rm T}} %{\mathbf T_{\omega_j'}(B_j)\mathbb{K}_{\delta-\alpha_j}.
\end{align}
If follows by \eqref{eq:TTB1}--\eqref{eq:TTB2} that $\texttt{\rm T}} %{\mathbf T_{\omega_i}^{-1} \texttt{\rm T}} %{\mathbf T_{\omega_j'}(B_j)=\texttt{\rm T}} %{\mathbf T_{\omega_j'}(B_j).$
As $\texttt{\rm T}} %{\mathbf T_{\omega_i}$ fixes all generators of $\widetilde{{\mathbf U}}^\imath_{[j]}$, the lemma follows.
\end{proof}
\begin{lemma}
\label{lem:T-T}
We have
$\texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i) =\texttt{\rm T}} %{\mathbf T_{\omega_i'}^{-1}(B_i),$ for $i\in \mathbb{I}_0$.
\end{lemma}
\begin{proof}
Note that Lemma~\ref{lem:Braid-Lus} remains valid when the $T$'s therein are replaced by $\texttt{\rm T}} %{\mathbf T$'s. Then we have $\texttt{\rm T}} %{\mathbf T_{\omega_i'}=\texttt{\rm T}} %{\mathbf T_{\omega_i'}^{-1}\prod_{j\neq i} \texttt{\rm T}} %{\mathbf T_{\omega_j}^{-c_{ij}}$, thanks to $\texttt{\rm T}} %{\mathbf T_{\omega_i} =\texttt{\rm T}} %{\mathbf T_{\omega_i'} \texttt{\rm T}} %{\mathbf T_i.$ Therefore, it follows by Lemma~\ref{lem:TiFix} that
$\texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i) =\texttt{\rm T}} %{\mathbf T_{\omega_i'}^{-1}\prod_{j\neq i} \texttt{\rm T}} %{\mathbf T_{\omega_j}^{-c_{ij}} (B_i) =\texttt{\rm T}} %{\mathbf T_{\omega_i'}^{-1}(B_i).$
\end{proof}
Here are some additional useful formulas:
\begin{align}
\texttt{\rm T}} %{\mathbf T_i \big( \texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i) \big)
&= \texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i) B_i^{(2)} -v_i B_i \texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i) B_i +v_i^2 B_i^{(2)} \texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i) + \texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i) \mathbb{K}_i,
\label{eq:TTiX}
\\
\texttt{\rm T}} %{\mathbf T_i^{-1} \big( \texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i) \big)
&= B_i^{(2)} \texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i) -v_i B_i \texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i) B_i +v_i^2 \texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i) B_i^{(2)} + \texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i) \mathbb{K}_i,
\label{eq:TTiX2}
\\
\texttt{\rm T}} %{\mathbf T_i \big([\texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i), B_i]_{v_i^{-2}} \big) &=[B_i, \texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i)]_{v_i^{-2}}.
\label{TT-imagine}
\end{align}
In the rank 1 case, formulas \eqref{eq:TTiX}--\eqref{eq:TTiX2} reduce to \eqref{T1B0}--\eqref{T1B0-2} while \eqref{TT-imagine} reduces to \eqref{eq:Tm}. The formulas \eqref{eq:TTiX}--\eqref{TT-imagine} are obtained by an Ansatz with the corresponding formulas \eqref{eq:TiX}--\eqref{T-imagine} in the setting of affine quantum groups. Formulas \eqref{eq:TTiX} and \eqref{eq:TTiX2} are equivalent to each other in the presence of \eqref{TT-imagine} or in light of \eqref{antiT}; Equation~\eqref{TT-imagine} is compatible with \eqref{antiT} as well. These formulas shall have interpretations in the $\imath$Hall algebras of $\imath$quivers and $\imath$-weighted projective lines. We skip the details.
\begin{lemma}
\label{lem:iSerre-rank1}
For $i \in \mathbb{I}_0$, we have
\begin{align}
\sum_{a=0}^3 (-1)^a \qbinom{3}{a}_i B_i^{3-a} \texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i) B_i^{a}
&= -v_i^{-1} [2]_i^2 [B_i, \texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i)] \mathbb{K}_i,
\label{Se01}\\
\sum_{a=0}^3 (-1)^a \qbinom{3}{a}_i (\texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i))^{3-a} B_i (\texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i))^{a}
&= -v_i^{-1} [2]_i^2 [\texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i), B_i] \mathbb{K}_\delta \mathbb{K}_i^{-1}.
\label{Se01-2}
\end{align}
\end{lemma}
\begin{proof}
Recall $\mathbb{K}_i$ is central in $\widetilde{{\mathbf U}}^\imath$. Using \eqref{eq:TTiX2}--\eqref{TT-imagine}, we compute
\begin{align*}
[2]_i^{-1} \sum_{a=0}^3 (-1)^a \qbinom{3}{a}_i B_i^{3-a} \texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i) B_i^{a}
& = [2]_i^{-1} \Big[ B_i, \big[ B_i, [B_i, \texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i)]_{v_i^2} \big]_1 \Big]_{v_i^{-2}}
\\
&= \big[ B_i, \texttt{\rm T}} %{\mathbf T_i^{-1} (\texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i)) - \texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i) \mathbb{K}_i \big]_{v_i^{-2}}
\\
&= \big[ B_i, \texttt{\rm T}} %{\mathbf T_i^{-1} (\texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i)) \big]_{v_i^{-2}} - \big[ B_i, \texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i) \mathbb{K}_i \big]_{v_i^{-2}}
\\
&= \mathbb{K}_i \texttt{\rm T}} %{\mathbf T_i^{-1} \big( [ B_i, \texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i)]_{v_i^{-2}} \big) - \big[ B_i, \texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i) \big]_{v_i^{-2}} \mathbb{K}_i
\\
&= \mathbb{K}_i [\texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i), B_i]_{v_i^{-2}} - \big[ B_i, \texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i) \big]_{v_i^{-2}} \mathbb{K}_i
\\
&= -v_i^{-1} [2]_i [B_i, \texttt{\rm T}} %{\mathbf T_{\omega_i'}(B_i)] \mathbb{K}_i.
\end{align*}
The first formula \eqref{Se01} follows.
Applying $\texttt{\rm T}} %{\mathbf T_{\omega_i'}$ to both sides of the first formula \eqref{Se01}, we obtain the second formula \eqref{Se01-2} by using $\texttt{\rm T}} %{\mathbf T_{\omega_i'}(\mathbb{K}_i) =\mathbb{K}_\delta \mathbb{K}_i^{-1}$ and Lemma~\ref{lem:T-T}.
\end{proof}
Let us denote the $q$-Onsager algebra in Definition~\ref{def:Onsager} by $\widetilde{{\mathbf U}}^\imath(\widehat{\mathfrak{sl}}_2)$ in the proposition below, in order to distinguish from $\widetilde{{\mathbf U}}^\imath$ of higher rank.
\begin{proposition}
\label{prop:rank1iso}
Let $i\in \mathbb{I}_0$. There is a $\mathbb Q(v)$-algebra isomorphism $\aleph_i: \widetilde{{\mathbf U}}^\imath(\widehat{\mathfrak{sl}}_2) \rightarrow \widetilde{{\mathbf U}}^\imath_{[i]}$, which sends $B_1 \mapsto B_i, B_0 \mapsto \texttt{\rm T}} %{\mathbf T_{\omega_i'} (B_i), \mathbb{K}_1 \mapsto \mathbb{K}_i, \mathbb{K}_0 \mapsto \mathbb{K}_\delta \mathbb{K}_i^{-1}$.
\end{proposition}
\begin{proof}
It follows by Definition~\ref{def:Onsager} and Lemma~\ref{lem:iSerre-rank1} that $\aleph_i$ is a surjective homomorphism.
It remains to show that $\aleph_i$ is injective. Let $h$ be the Coxeter number of $\mathfrak{g}$. Recall from \eqref{eq:hfilt}--\eqref{eq:filter1} a filtered algebra structure on $\widetilde{{\mathbf U}}^\imath(\widehat{\mathfrak{sl}}_2)$ given by $|\cdot |_{h-1}$ with $|B_1 |_{h-1} =1, |B_0 |_{h} =h-1, |\mathbb{K}_1 |_{h} = |\mathbb{K}_\delta |_{h} =0$. As $\text{wt}( \texttt{\rm T}} %{\mathbf T_{\omega_i'} (B_i) ) =\delta-\alpha_i$, we have $|\texttt{\rm T}} %{\mathbf T_{\omega_i'} (B_i)| =h-1$, cf. \eqref{eq:filter}. Hence $\aleph_i$ is a homomorphism of filtered algebras, and it induces a homomorphism of the associated graded:
\[
\aleph_i^{\text{gr}}: \text{gr} \widetilde{{\mathbf U}}^\imath(\widehat{\mathfrak{sl}}_2) \longrightarrow \text{gr} \widetilde{{\mathbf U}}^\imath_{[i]},
\]
which can be identified with
\[
\aleph_i^{-}: {\mathbf U}^-(\widehat{\mathfrak{sl}}_2) \otimes \mathbb Q(v)[\mathbb{K}_1^\pm, \mathbb{K}_\delta^\pm]
\longrightarrow {\mathbf U}^- \otimes \mathbb Q(v)[\mathbb{K}_i^\pm, \mathbb{K}_\delta^\pm],
\]
sending $F_1 \mapsto F_i, F_0 \mapsto T_{\omega_i'} (F_i)$ and $\mathbb{K}_1 \mapsto \mathbb{K}_i, \mathbb{K}_\delta \mapsto \mathbb{K}_\delta$.
The homomorphism $\aleph_i^{-}|_{{\mathbf U}^-(\widehat{\mathfrak{sl}}_2)}: {\mathbf U}^-(\widehat{\mathfrak{sl}}_2) \rightarrow {\mathbf U}^-$ is well known to be injective; cf. \cite{Be94}. It follows that $\aleph_i^-$ and $\aleph_i^{\text{gr}}$ are injective, hence so is $\aleph_i$.
\end{proof}
As in \cite{Be94}, we have (cf. \eqref{eq:TTiX})
\begin{align}
\label{eq:T1Ti}
{\texttt{\rm T}} %{\mathbf T_i}_{|\widetilde{{\mathbf U}}^\imath_{[i]}} = \aleph_i \circ \texttt{\rm T}} %{\mathbf T_1 \circ \aleph_i^{-1},
\qquad
{\texttt{\rm T}} %{\mathbf T_{\omega_i}}_{|\widetilde{{\mathbf U}}^\imath_{[i]}} = \aleph_i \circ \texttt{\rm T}} %{\mathbf T_{\omega_1} \circ \aleph_i^{-1}, \quad \text{ for } i\in \mathbb{I}_0.
\end{align}
Define a sign function
\[
o(\cdot): \mathbb{I}_0 \longrightarrow \{\pm 1\}
\]
such that $o(i) o(j)=-1$ whenever $c_{ij} <0$ (there are clearly exactly 2 such functions).
We define the following elements in $\widetilde{{\mathbf U}}^\imath$, for $i\in \mathbb{I}_0$, $k\in \mathbb Z$ and $m\ge 1$ (compare with the rank 1 formulas \eqref{eq:B1n}--\eqref{eq:dB1} and \eqref{eq:dBB}, where $\texttt{\rm T}} %{\mathbf T_{\omega_1} =\dag \texttt{\rm T}} %{\mathbf T_1$):
\begin{align}
B_{i,k} &= o(i)^k \texttt{\rm T}} %{\mathbf T_{\omega_i}^{-k} (B_i),
\label{Bik} \\
\acute{\Theta}_{i,m} &= o(i)^m \Big(-B_{i,m-1} \texttt{\rm T}} %{\mathbf T_{\omega_i'} (B_i) +v^{2} \texttt{\rm T}} %{\mathbf T_{\omega_i'} (B_i) B_{i,m-1}
\label{Thim1} \\
& \qquad\qquad\qquad\qquad + (v^{2}-1)\sum_{p=0}^{m-2} B_{i,p} B_{i,m-p-2} \mathbb{K}_{i}^{-1}\mathbb{K}_{\delta} \Big),
\notag \\
\Theta_{i,m} &=\acute{\Theta}_{i,m} - \sum\limits_{a=1}^{\lfloor\frac{m-1}{2}\rfloor}(v^2-1) v^{-2a} \acute{\Theta}_{i,m-2a}\mathbb{K}_{a\delta} -\delta_{m,ev}v^{1-m} \mathbb{K}_{\frac{m}{2}\delta}.
\label{Thim}
\end{align}
In particular, $B_{i,0}=B_i$.
\subsection{A Drinfeld type presentation}
\label{subsec:Dr2}
Let $k_1, k_2, l\in \mathbb Z$ and $i,j \in \mathbb{I}_0$. We introduce shorthand notations:
\begin{align}
\begin{split}
S(k_1,k_2|l;i,j)
&= B_{i,k_1} B_{i,k_2} B_{j,l} -[2] B_{i,k_1} B_{j,l} B_{i,k_2} + B_{j,l} B_{i,k_1} B_{i,k_2},
\\
\mathbb{S}(k_1,k_2|l;i,j)
&= S(k_1,k_2|l;i,j) + \{k_1 \leftrightarrow k_2 \}.
\label{eq:Skk}
\end{split}
\end{align}
Here and below, $\{k_1 \leftrightarrow k_2 \}$ stands for repeating the previous summand with $k_1, k_2$ switched, so the sums over $k_1, k_2$ are symmetrized.
We also denote
\begin{align}
\begin{split}
R(k_1,k_2|l; i,j)
&= \mathbb{K}_i C^{k_1}
\Big(-\sum_{p\geq0} v^{2p} [2] [\Theta _{i,k_2-k_1-2p-1},{B}_{j,l-1}]_{v^{-2}}C^{p+1}
\label{eq:Rkk} \\
&\qquad\qquad -\sum_{p\geq 1} v^{2p-1} [2] [{B}_{j,l},\Theta _{i,k_2-k_1-2p}]_{v^{-2}} C^{p}
- [{B}_{j,l}, \Theta _{i,k_2-k_1}]_{v^{-2}} \Big),
\\
\mathbb{R}(k_1,k_2|l; i,j) &= R(k_1,k_2|l;i,j) + \{k_1 \leftrightarrow k_2\}.
\end{split}
\end{align}
Sometimes, it is convenient to rewrite part of the summands in \eqref{eq:Rkk} as
\begin{align*}
&-\sum_{p\geq 1} v^{2p-1} [2] [{B}_{j,l},\Theta _{i,k_2-k_1-2p}]_{v^{-2}} C^{p}
- [{B}_{j,l}, \Theta _{i,k_2-k_1}]_{v^{-2}}\\
=&
-\sum_{p\geq \red{0}} v^{2p-1} [2] [{B}_{j,l},\Theta _{i,k_2-k_1-2p}]_{v^{-2}} C^{p}
+v^{-2}[{B}_{j,l}, \Theta _{i,k_2-k_1}]_{v^{-2}}.
\end{align*}
We shall often omit $i,j$ and write $S(k_1,k_2|l)= S(k_1,k_2|l;i,j)$, $\mathbb{S}(k_1,k_2|l)= \mathbb{S}(k_1,k_2|l;i,j)$, and similarly for $R$ and $\mathbb{R}$, whenever $i,j$ are clear from the context.
\begin{definition}
\label{def:iDR}
Let ${}^{\text{Dr}}\tUi$ be the $\mathbb Q(v)$-algebra generated by $\mathbb{K}_{i}^{\pm1}$, $C^{\pm1}$, $H_{i,m}$ and ${B}_{i,l}$, where $i\in \mathbb{I}_0$, $m \in \mathbb Z_{\geq1}$, $l\in\mathbb Z$, subject to the following relations, for $m,n \in \mathbb Z_{\geq1}$ and $k,l\in \mathbb Z$:
\begin{align}
& \mathbb{K}_i, C \text{ are central, }\quad
[H_{i,m},H_{j,n}]=0, \quad \mathbb{K}_i\mathbb{K}_i^{-1}=1, \;\; C C^{-1}=1,
\label{iDR1}
\\%2
&[H_{i,m},{B}_{j,l}]=\frac{[mc_{ij}]}{m} {B}_{j,l+m}-\frac{[mc_{ij}]}{m} {B}_{j,l-m}C^m,
\label{iDR2}
\\%3
&[{B}_{i,k} ,{B}_{j,l}]=0, \text{ if }c_{ij}=0, \label{iDR4}
\\%4
&[{B}_{i,k}, {B}_{j,l+1}]_{v^{-c_{ij}}} -v^{-c_{ij}} [{B}_{i,k+1}, {B}_{j,l}]_{v^{c_{ij}}}=0, \text{ if }i\neq j,
\label{iDR3a}
\\
&[{B}_{i,k}, {B}_{i,l+1}]_{v^{-2}} -v^{-2} [{B}_{i,k+1}, {B}_{i,l}]_{v^{2}}
=v^{-2}\Theta_{i,l-k+1} C^k \mathbb{K}_i-v^{-4}\Theta_{i,l-k-1} C^{k+1} \mathbb{K}_i
\label{iDR3b} \\
&\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad\quad
+v^{-2}\Theta_{i,k-l+1} C^l \mathbb{K}_i-v^{-4}\Theta_{i,k-l-1} C^{l+1} \mathbb{K}_i, \notag
\\
\label{iDR5}
& \mathbb{S}(k_1,k_2|l; i,j) = \mathbb{R}(k_1,k_2|l; i,j), \text{ if }c_{ij}=-1.
\end{align}
Here we set
\begin{align} \label{Hm0}
{\Theta}_{i,0} =(v-v^{-1})^{-1}, \qquad {\Theta}_{i,m} =0, \; \text{ for }m<0,
\end{align}
and $H_{i,m}$ are related to $\Theta_{i,m}$ by the following equation:
\begin{align}
\label{exp h}
1+ \sum_{m\geq 1} (v-v^{-1})\Theta_{i,m} u^m = \exp\Big( (v-v^{-1}) \sum_{m\geq 1} H_{i,m} u^m \Big).
\end{align}
\end{definition}
Let us mention that in spite of its appearance the RHS of \eqref{iDR3b} typically has two nonzero terms, thanks to the convention \eqref{Hm0}.
The $\mathbb Q(v)$-algebra ${}^{\text{Dr}}\tUi$ clearly exhibits certain symmetries as follows.
\begin{lemma} \label{lem:aut}
For each $i\in \mathbb{I}_0$, we have an automorphism $\t_i$ of the algebra ${}^{\text{Dr}}\tUi$ given by
\[
\t_i (B_{j,r}) =B_{j,r -\delta_{i,j}}, \quad
\t_i (H_{j,m}) =H_{j,m}, \quad
\t_i (\mathbb{K}_j) = \mathbb{K}_j C^{-\delta_{ij}}, \quad
\t_i(C) =C,
\]
(and hence $\t_i (\Theta_{j,m}) = \Theta_{j,m}$), for all $j\in \mathbb{I}_0, r\in \mathbb Z, m\ge 1$. Moreover, $\t_i \t_k =\t_k\t_i$ for all $i,k \in \mathbb{I}_0$.
\end{lemma}
\begin{proof}
Follows by inspection on the defining relations for ${}^{\text{Dr}}\tUi$ in Definition~\ref{def:iDR}.
\end{proof}
Define the generating functions, for $i\in \mathbb{I}_0$:
\begin{align}
\label{eq:Genfun}
\begin{cases}
&\bB_{i}(z) = \sum\limits_{k\in\mathbb Z} {B}_{i,k}z^{k}, \\
& \boldsymbol{\Theta}_{i}(z) = 1+ \sum\limits_{m>0} (v-v^{-1})\Theta_{i,m}z^{m},
\\
& \boldsymbol{\Delta}(z) = \sum\limits_{k\in\mathbb Z} C^k z^k.
\end{cases}
\end{align}
The following is a generalization of Proposition~\ref{prop:equiv4}, which follows by a similar formal manipulation of generating functions as in \S\ref{subsec:proof}; we skip the detail.
\begin{proposition}
\label{prop:equivij}
The identity \eqref{iDR2} is equivalent to either of the following identities, for $m \ge 1, l\in \mathbb Z$ and $i,j \in \mathbb{I}_0$:
\begin{align}
\boldsymbol{\Theta}_i (z) {\mathbf B }_j(w)
& = \frac{(1 -v^{-c_{ij}}zw^{-1}) (1 -v^{c_{ij}} zw C)}{(1 -v^{c_{ij}}zw^{-1})(1 -v^{-c_{ij}}zw C)}
{\mathbf B }_j(w) \boldsymbol{\Theta}_i (z),
\\
[\Theta_{i,m},B_{j,l}]+[\Theta_{i,m-2},B_{j,l}]C &=v^{c_{ij}}[\Theta_{i,m-1},B_{j,l+1}]_{v^{-2c_{ij}}}+v^{-c_{ij}} [\Theta_{i,m-1},B_{j,l-1}]_{v^{2c_{ij}}}C.
\end{align}
\end{proposition}
Recall the elements $B_{i,k}, \Theta_{i,m}$ in $\widetilde{{\mathbf U}}^\imath$ defined in \eqref{Bik} and \eqref{Thim}, while elements in the same notation are generators for the algebra ${}^{\text{Dr}}\tUi$. Below is the main result of this paper.
\begin{theorem}
\label{thm:ADE}
There is a $\mathbb Q(v)$-algebra isomorphism ${\Phi}: {}^{\text{Dr}}\tUi \rightarrow\widetilde{{\mathbf U}}^\imath$, which sends
\begin{align} \label{eq:map}
{B}_{i,k}\mapsto B_{i,k}, \quad H_{i,m}\mapsto H_{i,m}, \quad \Theta_{i,m}\mapsto \Theta_{i,m}, \quad
\mathbb{K}_i\mapsto \mathbb{K}_i,
\quad C\mapsto \mathbb{K}_\delta,
\end{align}
for $i\in \mathbb{I}_0, k\in \mathbb Z$, and $m \ge 1$.
\end{theorem}
The proof of Theorem~\ref{thm:ADE} will be given in \S\ref{subsec:inj} and Section~\ref{sec:relation1} below.
Note that the algebra ${}^{\text{Dr}}\tUi$ is $\mathbb Z \mathbb{I}$-graded by letting
\begin{align}
\label{eq:wtgrading}
\text{wt}( B_{i,k}) =\alpha_i +k\delta, \quad
\text{wt} (\Theta_{i,m}) = m\delta, \quad
\text{wt} (\mathbb{K}_i) =2\alpha_i, \quad
\text{wt} (C) =2\delta,
\end{align}
for $i \in \mathbb{I}_0, k\in \mathbb Z, m\ge 1.$ (It follows that $\text{wt} (\mathbb{K}_0) =2\alpha_0$.) Moreover, $\Phi$ preserves the $\mathbb Z \mathbb{I}$-gradings.
\subsection{Proof of the main isomorphism}
\label{subsec:inj}
The details of the proof of Proposition~\ref{thm:iDB hom} below will occupy Section~\ref{sec:relation1}.
\begin{proposition}
\label{thm:iDB hom}
There is a homomorphism ${\Phi}: {}^{\text{Dr}}\tUi \rightarrow\widetilde{{\mathbf U}}^\imath$ as prescribed in \eqref{eq:map}.
\end{proposition}
\begin{proof}
By Propositions~\ref{prop:iDR4}, \ref{prop:iDR3a}, \ref{prop:iDR2}, \ref{prop:iDR31}, \ref{prop:iDR1} and
\ref{prop:iDR25} in a later Section~\ref{sec:relation1}, all the defining relations \eqref{iDR1}--\eqref{iDR5} for the generators in ${}^{\text{Dr}}\tUi$ are satisfied by their images under $\Phi$ in $\widetilde{{\mathbf U}}^\imath$.
We shall specify precisely what relations are used when deriving a given relation, to make sure our proofs of these propositions do not run circularly.
Hence ${\Phi}: {}^{\text{Dr}}\tUi \rightarrow\widetilde{{\mathbf U}}^\imath$ is a homomorphism.
\end{proof}
Assuming Proposition~\ref{thm:iDB hom} (whose proof is much more difficult), we can now complete the proof of Theorem \ref{thm:ADE}.
\begin{proof}[Proof of Theorem \ref{thm:ADE}]
We first show that $\Phi: {}^{\text{Dr}}\tUi \rightarrow\widetilde{{\mathbf U}}^\imath$ is surjective. All generators for $\widetilde{{\mathbf U}}^\imath$ except $B_0$ are clearly in the image of $\Phi$, and so it remains to show that $B_0\in \text{Im} (\Phi)$. We shall adapt and modify the arguments in the proof of \cite[Theorem~12.11]{Da12}.
The automorphisms $\t_i \in \operatorname{Aut}\nolimits ({}^{\text{Dr}}\tUi)$ (see Lemma~\ref{lem:aut}) and $\texttt{\rm T}_{\omega_i} \in \operatorname{Aut}\nolimits (\widetilde{{\mathbf U}}^\imath)$, for $i\in \mathbb{I}_0$, satisfy
\[
\Phi \circ \t_i =\texttt{\rm T}_{\omega_i} \circ \Phi, \qquad \text{ for } i\in \mathbb{I}_0.
\]
It follows that $\text{Im}(\Phi)$ is $\texttt{\rm T}_{\omega_i}$-stable, for each $i \in \mathbb{I}_0$.
Recall $\theta$ is the highest root in ${\mathcal R}^+_0$. We can choose and fix ${\bf i} \in \mathbb{I}_0$ such that
\begin{align} \label{theta}
\alpha_0 + \alpha_{\bf i} \in {\mathcal R}^+, \quad
\theta_{\bf i} =\theta -\alpha_{\bf i} \in {\mathcal R}^+_0, \quad
\text{ and } s_{\theta_{\bf i}}(\alpha_{\bf i})=\theta.
\end{align}
Write $t_{\omega_{\bf i}} =\sigma_{\bf i} s_{i_1} \ldots s_{i_N}$ with $\sigma_{\bf i} \in \Omega$. Then $t_{\omega_{\bf i}}(\theta) =\theta-\delta\in -{\mathcal R}^+$, and so there exists $p$ such that $s_{i_N} \ldots s_{i_{p+1}} (\alpha_{i_p}) =\theta$. Hence, $y:=\texttt{\rm T}_{i_N}^{-1} \ldots \texttt{\rm T}_{i_{p+1}}^{-1} (B_{i_p}) \in \text{Im} (\Phi)$; note $\text{wt} (y) =\theta$.
Consider $\texttt{\rm T}_{\omega_{\bf i}} (y) = \texttt{\rm T}_{\sigma_{\bf i}} \texttt{\rm T}_{i_1} \ldots \texttt{\rm T}_{i_{p-1}} (B_{i_p}) \in \text{Im} (\Phi)$, since $\text{Im}(\Phi)$ is $\texttt{\rm T}_{\omega_{\bf i}}$-stable. Note that $\sigma_{\bf i} s_{i_1} = s_0 \sigma_{\bf i}$ (see \cite[Lemma 3.1]{Be94}), and so $\texttt{\rm T}_{s_0\omega_{\bf i}} =\texttt{\rm T}_0^{-1} \texttt{\rm T}_{\omega_{\bf i}}$.
Since $\text{wt} (\texttt{\rm T}_0^{-1} \texttt{\rm T}_{\omega_{\bf i}} (y) )=\alpha_0$ (i.e., $s_0\omega_{\bf i} s_{i_N} \ldots s_{i_{p+1}} (\alpha_{i_p}) =\alpha_0$), it follows by Lemma~\ref{lem:fixB} that $\texttt{\rm T}_0^{-1} \texttt{\rm T}_{\omega_{\bf i}} (y) =B_0$, and hence $\texttt{\rm T}_{\omega_{\bf i}} (y) =\texttt{\rm T}_0(B_0) =\mathbb{K}_{0} B_0$. Therefore, we have $\mathbb{K}_{0} B_0\in \text{Im} (\Phi)$ and thus $B_0\in \text{Im} (\Phi)$.
\vspace{2mm}
It remains to show that the map $\Phi: {}^{\text{Dr}}\tUi \rightarrow\widetilde{{\mathbf U}}^\imath$ is injective. Let us explain the simple underlying idea of the arguments for injectivity: we shall first show that $\Phi$ on the associated graded level when restricted to ``a positive half" of $\widetilde{{\mathbf U}}^\imath$ (which correspond to ``a quarter" in the affine quantum group ${\mathbf U}$, for which the roots share the same sign in Kac-Moody sense and in the Drinfeld current sense) is injective; this is summarized by the commutative diagram \eqref{diag:1} below. Then we show the injectivity on ``the positive half" implies the injectivity of $\Phi: {}^{\text{Dr}}\tUi \rightarrow\widetilde{{\mathbf U}}^\imath$ fully via the translation automorphisms $\t_i$.
Denote by $\widetilde{{\mathbf U}}^\imath_>$ (respectively, ${}^{\text{Dr}}\tUi_>$) the subalgebra of $\widetilde{{\mathbf U}}^\imath$ (respectively, ${}^{\text{Dr}}\tUi$) generated by $B_{i,m}$, $H_{i,m}, \mathbb{K}_i$, for $m\ge 1$, and $i\in \mathbb{I}_0$. Then ${\Phi}: {}^{\text{Dr}}\tUi \longrightarrow\widetilde{{\mathbf U}}^\imath$ restricts to a surjective homomorphism ${\Phi}: {}^{\text{Dr}}\tUi_> \longrightarrow\widetilde{{\mathbf U}}^\imath_>$.
Define a filtration on ${}^{\text{Dr}}\tUi_>$ by
\begin{align} \label{eq:filt1D}
({}^{\text{Dr}}\tUi_>)^0 \subset ({}^{\text{Dr}}\tUi_>)^1 \subset \cdots \subset ({}^{\text{Dr}}\tUi_>)^m \subset \cdots
\end{align}
by setting
\begin{align} \label{eq:filtD}
({}^{\text{Dr}}\tUi_>)^m &=\mathbb Q(v)\text{-span} \big\{x=B_{i_1,m_1} B_{i_2,m_2} \ldots B_{i_r,m_r} \Theta_{j_1,n_1} \Theta_{j_2,n_2} \ldots \Theta_{j_s,n_s} \mathbb{K}_\mu
\\
&\quad
\mid \mu \in \mathbb N\mathbb{I}, i_1, \ldots, i_r, j_1, \ldots j_s, \in \mathbb{I}_0, m_1,\ldots, m_r, n_1, \ldots, n_s \ge 1, \text{ht}^+(x)\leq m \big\}.
\notag
\end{align}
Here we have denoted
\begin{align} \label{eq:ht}
\text{ht}^+(x) :=\sum_{a=1}^r \text{ht}(m_a\delta +\alpha_{i_a}) +\sum_{b=1}^s n_b \text{ht}(\delta),
\end{align}
where $\text{ht}(\beta)$ denotes the height of a positive root $\beta$; compare with the $\mathbb N\mathbb{I}$-grading on $\widetilde{{\mathbf U}}^\imath$ by \eqref{eq:wtgrading}. Recalling $\widetilde{{\mathbf U}}^{\imath,0}$ from \eqref{eq:UiCartan}, we have
\[
({}^{\text{Dr}}\tUi_>)^0 = \widetilde{{\mathbf U}}^{\imath,0} =\mathbb Q(v) [ \mathbb{K}_i^{\pm 1} \mid i\in \mathbb{I} ].
\]
The filtration \eqref{eq:filt1D}--\eqref{eq:filtD} on ${}^{\text{Dr}}\tUi_>$ defined via a height function is compatible with the filtration \eqref{eq:filt1}--\eqref{eq:filt} on $\widetilde{{\mathbf U}}^\imath$ under $\Phi$, and thus the surjective homomorphism ${\Phi}: {}^{\text{Dr}}\tUi_> \longrightarrow\widetilde{{\mathbf U}}^\imath_>$ induces a surjective homomorphism
\begin{align} \label{eq:gradeP}
{}^{\text{gr}} {\Phi}_>: {\text{gr}}\tUiD_> \longrightarrow{\text{gr}}\tUi_>.
\end{align}
Recall from \eqref{eq:phi1} an isomorphism $\phi: {}^{\text{Dr}}{\mathbf U} \rightarrow {\mathbf U}$ for the affine quantum group ${\mathbf U}$. Denote by ${}^{\text{Dr}}{\mathbf U}^-_<$ the $\mathbb Q(v)$-subalgebra of ${\mathbf U}^-$ generated by $x^-_{i,-k}$, for $i\in \mathbb{I}_0, k>0$, and denote by ${\mathbf U}^-_< =\phi({}^{\text{Dr}}{\mathbf U}^-_<)$. Then $\phi$ restricts to an isomorphism
\begin{align} \label{eq:phi}
\phi: {}^{\text{Dr}}{\mathbf U}^-_< \stackrel{\cong}{\longrightarrow} {\mathbf U}^-_<.
\end{align}
On the other hand, consider the associated graded ${\text{gr}}\tUi$ with respect to the filtration on $\widetilde{{\mathbf U}}^\imath$ given by \eqref{eq:filt1}--\eqref{eq:filt}. Recall the algebra isomorphism in \eqref{eq:filter}
\begin{align*}
\mathbb G: {\mathbf U}^- \otimes \widetilde{{\mathbf U}}^{\imath,0} \longrightarrow {\text{gr}}\tUi,
\qquad
F_i \mapsto \overline{B}_i, \quad
\mathbb{K}_i \mapsto \overline{\mathbb{K}}_i,
\end{align*}
where ${\mathbf U}^-$ denotes half a Drinfeld-Jimbo quantum group generated by $F_i$, for $i\in \mathbb{I}$.
The homomorphism $\mathbb G$ above restricts to an isomorphism
\begin{align} \label{eq:grade1}
\mathbb G: {\mathbf U}^-_< \otimes \widetilde{{\mathbf U}}^{\imath,0} \stackrel{\cong}{\longrightarrow} {\text{gr}}\tUi_>.
\end{align}
Finally, by definition \eqref{eq:filtD}--\eqref{eq:ht} of the filtration on ${}^{\text{Dr}}\tUi_>$, we have a surjective homomorphism
\begin{align} \label{eq:Xi}
\Xi: {}^{\text{Dr}}{\mathbf U}^-_< \otimes \widetilde{{\mathbf U}}^{\imath,0}
\longrightarrow
{\text{gr}}\tUiD_>,
\end{align}
which sends $x_{i,-k}^- \mapsto \overline{B}_{i,k}$, for $k>0$ (note the opposite sign in indices).
Collecting \eqref{eq:gradeP}, \eqref{eq:grade1}, \eqref{eq:Xi} and \eqref{eq:phi}, we obtain the following commutative diagram
\begin{align} \label{diag:1}
\xymatrix{
{}^{\text{Dr}}{\mathbf U}^-_< \otimes \widetilde{{\mathbf U}}^{\imath,0}
\ar[rr]^{\Xi}
\ar[d]^{\phi,\cong}
&& {\text{gr}}\tUiD_>
\ar[d]^{{}^{\text{gr}}\Phi_>}
\\
{\mathbf U}^-_< \otimes \widetilde{{\mathbf U}}^{\imath,0}
\ar[rr]^{\mathbb G, \cong}
&& {\text{gr}}\tUi_> }
\end{align}
Since the homomorphisms $\Xi$ and ${}^{\text{gr}}\Phi_>$ are surjective while $\phi$ and $\mathbb G$ are isomorphisms, we conclude that ${}^{\text{gr}}\Phi_>: {\text{gr}}\tUiD_> \longrightarrow{\text{gr}}\tUi_>$ is injective (and an isomorphism).
Now we show that ${}^{\text{gr}}\Phi: {\text{gr}}\tUiD \longrightarrow{\text{gr}}\tUi$ is injective (and hence an isomorphism); a similar argument was used in \cite[Proposition~5.2]{Da15}. Recall $\rho =\sum_{i\in \mathbb{I}_0} \omega_i$ is half the sum of positive roots in $\Phi$, and thus $\texttt{\rm T}_\rho =\prod_{i\in \mathbb{I}_0} \texttt{\rm T}_{\omega_i}$; the automorphism $\texttt{\rm T}_\rho$ on $\widetilde{{\mathbf U}}^\imath$ induces an automorphism (with the same notation) on ${\text{gr}}\tUi$.
Assume that a finite linear combination $X =\sum (*) B_{i_1,r_1} B_{i_2,r_2} \ldots B_{i_t,r_t} \Theta_{j_1,n_1} \Theta_{j_2,n_2} \ldots \Theta_{j_s,n_s} \mathbb{K}_\mu$ (with $r_a\in \mathbb Z, n_b \ge 1$, for various $a, b$) lies in the kernel of ${}^{\text{gr}}\Phi$. Recall the automorphisms $\t_i$ of ${}^{\text{Dr}}\tUi$ from Lemma~\ref{lem:aut}. Applying an automorphism $\prod_{i\in \mathbb{I}_0} \t_{i}^{-N}$ to $X$ produces an element $X_N =\prod_{i\in \mathbb{I}_0} \t_{i}^{-N}(X)$, which is obtained from $X$ with all indices $r_a$ of $B$'s in each summand of $X$ shifted to $r_a+N$. Pick and fix an $N$ large enough so that all relevant $r_a+N$ are positive, that is, $X_N \in {\text{gr}}\tUiD_>$. Thanks to the following commutative diagram
\begin{align*}
\xymatrix{
{\text{gr}}\tUiD_>
\ar[rr]^{\prod_{i\in \mathbb{I}_0} \t_i^{N}}
\ar[d]^{{}^{\text{gr}}\Phi_>}
&& {\text{gr}}\tUiD
\ar[d]^{{}^{\text{gr}}\Phi}
\\
{\text{gr}}\tUi_>
\ar[rr]^{{\texttt{\rm T}}_{\rho}^N}
&& {\text{gr}}\tUi }
\end{align*}
We have $\texttt{\rm T}_\rho^N \circ {}^{\text{gr}}\Phi_> (X_N) = {}^{\text{gr}}\Phi \circ (\prod_{i\in \mathbb{I}_0} \t_i^{N}) (X_N) = {}^{\text{gr}}\Phi (X) =0$, and hence ${}^{\text{gr}}\Phi_> (X_N) =0$. Since ${}^{\text{gr}}\Phi_>: {\text{gr}}\tUiD_> \longrightarrow{\text{gr}}\tUi_>$ is injective, we must have $X_N=0$ and hence $X=0$.
This proves the injectivity of ${}^{\text{gr}}\Phi$. It follows that $\Phi: {}^{\text{Dr}}\tUi \longrightarrow\widetilde{{\mathbf U}}^\imath$ is injective. This completes the proof of Theorem \ref{thm:ADE}.
\end{proof}
\begin{remark}
It follows by Lemma~\ref{lem:fixB} that $B_0=\texttt{\rm T}_{\theta_{\bf i}} \texttt{\rm T}_{\omega_{\bf i}'}(B_{\bf i})= o({\bf i}) \mathbb{K}_0\texttt{\rm T}_{\theta_{\bf i}} (B_{{\bf i},-1})$, where $\bf i \in \mathbb{I}_0$ is chosen as in \eqref{theta}.
The inverse ${\Phi}^{-1}: \widetilde{{\mathbf U}}^\imath \rightarrow {}^{\text{Dr}}\tUi$ to the isomorphism $\Phi$ in \eqref{eq:map} is given by
\begin{align*}
\mathbb{K}_j \mapsto \mathbb{K}_j, &\quad \mathbb{K}_0\mapsto C \mathbb{K}_\theta^{-1},
\quad
B_{j}\mapsto {B}_{j,0}, \quad B_0\mapsto o({\bf i}) \texttt{\rm T}} %{\mathbf T_{\theta_{\bf i}} ({B}_{{\bf i},-1})C \mathbb{K}_\theta^{-1},
\quad \text{for } j\in \mathbb{I}_0.
\end{align*}
Another formula for $\Phi^{-1}(B_0)$ can be read off from the proof of Theorem \ref{thm:ADE}.
\end{remark}
\begin{remark}
One can construct all real $v$-root vectors in $\widetilde{{\mathbf U}}^\imath$ with the help of braid group action. Together with the imaginary $v$-root vectors which have been constructed, one can write down a natural PBW basis for $\widetilde{{\mathbf U}}^\imath$, following \cite{BCP99}.
\end{remark}
\begin{remark}
Definition~\ref{def:iDR} for ${}^{\text{Dr}}\tUi$ formally makes sense for a generalized Cartan matrix (GCM) $(c_{ij})_{i,j\in \mathbb{I}}$ of a simply-laced Kac-Moody algebra $\mathfrak{g}$, and hence can be regarded as a definition of {\em $\imath$quantum loop Kac-Moody algebras} for $\mathfrak{g}$. A most interesting subclass of these new algebras will be {\em $\imath$quantum toroidal algebras} when $\mathfrak{g}$ is of affine type. Once we have extended Definition~\ref{def:iDR} to a wide class of affine $\imath$quantum groups, similar relaxing of conditions on GCMs will allow us to define more general {\em $\imath$quantum loop Kac-Moody algebras}.
\end{remark}
\begin{remark}
There is a nonstandard comultiplication $\Delta^{\text{n-std}}$ (due to Drinfeld) on the affine quantum group ${}^{\text{Dr}} {\mathbf U}$ (or its Drinfeld double $\widetilde{\mathbf U}$) via its Drinfeld presentation. It will be interesting to ask if there is a natural coideal subalgebra of ${}^{\text{Dr}} {\mathbf U}$ with respect to $\Delta^{\text{n-std}}$ (which, if it exists, could be different from ${}^{\text{Dr}}\tUi$).
\end{remark}
\subsection{Classical limit}
\label{subsec:classical}
Denote the Chevalley generators of the semisimple (or even Kac-Moody) Lie algebra $\mathfrak{g}$ over $\mathbb Q$ by $e_i, f_i, h_i$, for $i\in \mathbb{I}_0$.
Denote $L\mathfrak{g} =\mathfrak{g} \otimes \mathbb Q[t,t^{-1}]$, and the affine Lie algebra (as a vector space) $\widehat{\mathfrak{g}} = L\mathfrak{g} \oplus \mathbb Q c$; set $x_k =x\otimes t^k$, for $x\in \mathfrak{g}, k\in \mathbb Z$. Denote by $\omega$ the Chevalley involution on $\mathfrak{g}$ such that $\omega(c)= -c, \omega(e_i)= -f_i, \omega(f_i) = -e_i, \omega(h_i)= -h_i$, for all $i$. (More generally, one can take $\omega =\omega_a$, for any fixed nonzero scalar $a$, such that $\omega_a(c)= -c, \omega_a(e_i)= -af_i, \omega_a(f_i) = -a^{-1} e_i, \omega_a(h_i)= -h_i$.) Then $\omega$ induces an involution $\widehat{\omega}$ on $\widehat{\mathfrak{g}}$ such that $\widehat{\omega} (x_k) =\omega(x)_{-k}$, for all $x\in \mathfrak{g}, k\in \mathbb Z$. The algebra $\widetilde{{\mathbf U}}^\imath$ of split affine type ADE specializes at $v=1$ to the enveloping algebra of the ${\widehat{\omega}}$-fixed point subalgebra $(L\mathfrak{g})^{\widehat{\omega}}$.
Let us examine in detail $(L\mathfrak{g})^{\widehat{\omega}}$ in the case when $\mathfrak{g} =\mathfrak{sl}_2$ with standard basis $\{e,h,f\}$. Set $b_r :=f_r + e_{-r}, t_r :=h_r - h_{-r}$, for $r\in \mathbb Z$. Note $t_{-r} = -t_r$ and $t_0=0$. Then $\{b_r, t_m\mid r\in \mathbb Z, m\ge 1\}$ forms a basis for $(L\mathfrak{sl}_2)^{\widehat{\omega}}$. They satisfy the relations, for $r, s \in \mathbb Z, m\ge 1$,
\begin{align}
[b_r, b_s] &= t_{s-r},
\label{eq:bb1}
\\
[t_m, t_n] &=0,
\label{eq:tt} \\
[t_m, b_r] &=-2 b_{m+r} + 2b_{-m+r},
\label{eq:tb} \\
[b_r, b_{s+1}] - [b_{r+1}, b_s] &= t_{s-r+1} -t_{s-r-1}.
\label{eq:bb2}
\end{align}
Clearly, \eqref{eq:bb1}--\eqref{eq:tb} are defining relations for $(L\mathfrak{sl}_2)^{\widehat{\omega}}$.
One checks that \eqref{eq:bb1} and \eqref{eq:bb2} are equivalent. Hence \eqref{eq:tt}--\eqref{eq:bb2} are also defining relations for $(L\mathfrak{sl}_2)^{\widehat{\omega}}$, providing a non-standard presentation for the Onsager algebra, which is compatible with our Drinfeld type presentation up to suitable changes of indices: $b_r \leftrightarrow B_{1,-r}$ and $t_r \leftrightarrow H_{r}$; this index issue originates from the identification of the associated graded of the $q$-Onsager algebra with the positive (instead of the negative) half of quantum loop $\mathfrak{sl}_2$ in \cite{BK20} (which is followed in our Section~\ref{sec:Onsager}).
A Drinfeld type presentation of $(L\mathfrak{g})^{\widehat{\omega}}$ in ADE case (generalizing \eqref{eq:tt}--\eqref{eq:bb2} for $\mathfrak{g}=\mathfrak{sl}_2$) can be similarly written down, compatible with Definition~\ref{def:iDR}.
\section{Verification of Drinfeld type new relations}
\label{sec:relation1}
In this section, we prove that ${\Phi}: {}^{\text{Dr}}\tUi \rightarrow\widetilde{{\mathbf U}}^\imath$ defined by \eqref{eq:map} is a homomorphism. We shall first establish the relations \eqref{iDR1}, \eqref{iDR4}--\eqref{iDR3b}, and an easier case of \eqref{iDR2} in $\widetilde{{\mathbf U}}^\imath$. We then establish the more challenging relations \eqref{iDR5} and \eqref{iDR2} (when $c_{ij}=-1$) in $\widetilde{{\mathbf U}}^\imath$.
\subsection{Relation \eqref{iDR4}}
\begin{proposition}
\label{prop:iDR4}
Assume $c_{ij}=0$, for $i,j \in \mathbb{I}_0$. Then
$[B_{i,k},B_{j,l}]=0,$ for all $k,l\in\mathbb Z$.
\end{proposition}
\begin{proof}
The identity for $k=l=0$, i.e., $[B_{i},B_{j}]=0,$ is exactly the defining relation \eqref{eq:S1} for $\widetilde{{\mathbf U}}^\imath$.
The identity for general $k,l$ follows by applying $\texttt{\rm T}} %{\mathbf T_{\omega_i}^{-k} \texttt{\rm T}} %{\mathbf T_{\omega_j}^{-l}$ to the above identity and using Lemma~\ref{lem:Braid-Lus}(c) and Lemma~\ref{lem:TiFix}.
\end{proof}
\subsection{Relation \eqref{iDR3a}}
\begin{lemma} [\text{cf. \cite[Lemma 3.3]{Be94}}]
\label{lem:TXij}
For $i\neq j\in\mathbb{I}_0$ such that $c_{ij}=-1$, denote
\begin{align*}
X_{ij}:=\texttt{\rm T}} %{\mathbf T_j^{-1}B_i= B_jB_i-vB_iB_j.
\end{align*}
Then we have
\texttt{\rm T}} %{\mathbf T_{\omega_i}(X_{ji})=\texttt{\rm T}} %{\mathbf T_{\omega_j}(X_{ij}).
\end{lemma}
\begin{proof}
This follows by a direct computation using Lemma~\ref{lem:Braid-Lus} and Lemma~\ref{lem:TiFix}:
\begin{align*}
\texttt{\rm T}} %{\mathbf T_{\omega_j}(X_{ij})= &\texttt{\rm T}} %{\mathbf T_{\omega_j} \texttt{\rm T}} %{\mathbf T_j^{-1}(B_i)= \texttt{\rm T}} %{\mathbf T_j\texttt{\rm T}} %{\mathbf T_{\omega_j}^{-1}\texttt{\rm T}} %{\mathbf T_{\omega_i}(B_i)
=\texttt{\rm T}} %{\mathbf T_{\omega_i}\texttt{\rm T}} %{\mathbf T_j(B_i)=\texttt{\rm T}} %{\mathbf T_{\omega_i}(X_{ji}).
\end{align*}
\end{proof}
Now we are ready to establish the relation \eqref{iDR3a}.
\begin{proposition}
\label{prop:iDR3a}
We have $[{B}_{i,k}, {B}_{j,l+1}]_{v^{-c_{ij}}} -v^{-c_{ij}} [{B}_{i,k+1}, {B}_{j,l}]_{v^{c_{ij}}}=0$, for $i\neq j \in \mathbb{I}_0$ and $k, l \in \mathbb Z.$
\end{proposition}
\begin{proof}
If $c_{ij}=0$, then the identity in the proposition follows directly by \eqref{iDR4}.
Assume $c_{ij}=-1$.
Note that
v[B_{i,k+1},B_{j}]_{v^{-1}}= - o(i)^{k+1} \texttt{\rm T}} %{\mathbf T_{\omega_i}^{-(k+1)}(X_{ij}).
$
Hence we have
\begin{align*}
[B_{i,k},B_{j,1}]_{v}&= B_{i,k}B_{j,1}-vB_{j,1}B_{i,k}
\\
& = o(i)^k o(j) \texttt{\rm T}} %{\mathbf T_{\omega_i}^{-k} \texttt{\rm T}} %{\mathbf T_{\omega_j}^{-1}(X_{ji})
= - o(i)^{k+1} \texttt{\rm T}} %{\mathbf T_{\omega_i}^{-k} \texttt{\rm T}} %{\mathbf T_{\omega_i}^{-1} (X_{ij})
\\
&= - o(i)^{k+1} \texttt{\rm T}} %{\mathbf T_{\omega_i}^{-(k+1)}(X_{ij})=v[B_{i,k+1},B_{j}]_{v^{-1}}.
\end{align*}
So we have obtained an identity
$[B_{i,k},B_{j,1}]_{v}-v[B_{i,k+1},B_{j}]_{v^{-1}}=0.$
The identity in the proposition follows by applying $\texttt{\rm T}} %{\mathbf T_{\omega_j}^{-l}$ to this identity.
\end{proof}
\subsection{Relation \eqref{iDR2} for $c_{ij}=0$}
We shall identify $C=\mathbb{K}_\delta$ below.
\begin{lemma}
\label{lem:comb}
For $j\in\mathbb{I}_0$, we have
\begin{align*}
\Theta_{j,n}=
\begin{cases}
v^{-2} C \Theta_{j,n-2}+v^2\mathbb{K}_j^{-1} \big( [B_{j,0},B_{j,n}]_{v^{-2}}+[B_{j,n-1},B_{j,1}]_{v^{-2}} \big), & \text{ if } n\ge 3,
\\
-v^{-1} C +v^2\mathbb{K}_j^{-1} \big( [B_{j,0},B_{j,2}]_{v^{-2}}+[B_{j,1},B_{j,1}]_{v^{-2}} \big), & \text{ if } n=2,
\\
v^2 \mathbb{K}_{j}^{-1} [{B}_{j,0}, {B}_{j,1}]_{v^{-2}}, & \text{ if } n= 1.
\end{cases}
\end{align*}
In particular, for any $n\ge 1$, the element $\Theta_{j,n}$ is a $\mathbb Q(v)[C^{\pm 1},\mathbb{K}_j^{\pm 1}]$-linear combination of $1$ and $[{B}_{j,k}, {B}_{j,l+1}]_{v^{-2}} +[{B}_{j,l}, {B}_{j,k+1}]_{v^{-2}}$, for $l, k \in \mathbb Z$.
\end{lemma}
\begin{proof}
The recursion formulas in the lemma are reformulations of \eqref{iDR3b} with $k=0$ and $l=n-1$. The second statement follows by an induction on $n$ using the recursion formulas. (A precise linear combination can be written down, but will not be needed.)
\end{proof}
\begin{proposition}
\label{prop:iDR2}
Assume $c_{ij}=0$, for $i,j\in \mathbb{I}_0$. Then, for $m\geq 1$ and $r \in\mathbb Z$, we have
\begin{align*}
[\Theta_{i,m},B_{j,r}] &=0 =[H_{i,m},B_{j,r}].
\end{align*}
\end{proposition}
\begin{proof}
We shall prove the first equality only (as $H_{i,n}$ can be expressed in terms of $\Theta_{i,m}$ for various $m$).
By Lemma~\ref{lem:comb} (with index $j$ replaced by $i$), it suffices to check that $[B_{i,k}, B_{i,l}]_{v^{-2}}$ commutes with $B_{j,r}$ for all $k,l,r$. But this clearly follows by the commutativity between $B_{i,k}$ and $B_{j,r}$ \eqref{iDR4}.
\end{proof}
\subsection{Relations \eqref{iDR3b} and \eqref{iDR1}--\eqref{iDR2} for $i=j$}
\label{subsec:rank1}
\begin{proposition}
\label{prop:iDR31}
Relation \eqref{iDR3b} and relations \eqref{iDR1}--\eqref{iDR2} for $i=j \in \mathbb{I}_0$ hold in $\widetilde{{\mathbf U}}^\imath$.
\end{proposition}
\begin{proof}
These rank one relations (for a fixed $i\in \mathbb{I}_0$) follow by transporting the corresponding relations in $q$-Onsager algebra (see Theorem~\ref{thm:Dr1}) to $\widetilde{{\mathbf U}}^\imath$ using Proposition~\ref{prop:rank1iso}. Note that for \eqref{iDR3b}, the overall sign $o(i)^{k+l+1}$ originated from \eqref{Bik}--\eqref{Thim} cancels out.
\end{proof}
\subsection{Relation \eqref{iDR1} for $i\neq j$}
We shall derive the identity $[H_{i,m},H_{j,n}]=0$ in \eqref{iDR1}, for $i\neq j \in \mathbb{I}_0$, from the relations \eqref{iDR3b} (proved above) and \eqref{iDR2}. The proof of \eqref{iDR2} in the following subsections will not use the relation \eqref{iDR1}.
\begin{lemma}
\label{lem:comm}
For $i,j \in \mathbb{I}_0$, $l, k \in \mathbb Z$ and $m\ge 1$, we have
\[
\Big[ H_{i,m}, [{B}_{j,k}, {B}_{j,l+1}]_{v^{-2}} +[{B}_{j,l}, {B}_{j,k+1}]_{v^{-2}} \Big] =0.
\]
\end{lemma}
\begin{proof}
Using \eqref{iDR2} we have
\begin{align*}
\Big[ H_{i,m}, & [{B}_{j,k}, {B}_{j,l+1}]_{v^{-2}} +[{B}_{j,l}, {B}_{j,k+1}]_{v^{-2}} \Big]
\\\notag
= & [H_{i,m}, {B}_{j,k}] {B}_{j,l+1} +v^{-2} {B}_{j,l+1} [{B}_{j,k}, H_{i,m}]
\\\notag
& + [H_{i,m}, {B}_{j,l}] {B}_{j,k+1} +v^{-2} {B}_{j,k+1} [{B}_{j,l}, H_{i,m}]
\\\notag
& +{B}_{j,l} [H_{i,m}, {B}_{j,k+1}] +v^{-2} [{B}_{j,k+1}, H_{i,m}] {B}_{j,l}
\\\notag
& + {B}_{j,k} [H_{i,m}, {B}_{j,l+1}] +v^{-2} [{B}_{j,l+1}, H_{i,m}] {B}_{j,k}\\\notag
= &\frac{[mc_{ij}]}{m}\Big( ({B}_{j, k+m} -{B}_{j,k-m} C^m) {B}_{j, l+1} -v^{-2} {B}_{j,l+1} ({B}_{j,k+m} -{B}_{j,k-m} C^m)
\\\notag
& + ({B}_{j, l+m} -{B}_{j,l-m} C^m) {B}_{j, k+1} -v^{-2} {B}_{j,k+1} ({B}_{j,l+m} -{B}_{j,l-m} C^m)
\\\notag
& + {B}_{j, l}({B}_{j, k+m+1} -{B}_{j,k-m+1} C^m) -v^{-2} ({B}_{j,k+m+1} -{B}_{j,k-m+1} C^m) {B}_{j, l}
\\\notag
& + {B}_{j, k}({B}_{j, l+m+1} -{B}_{j,l-m+1} C^m) -v^{-2} ({B}_{j,l+m+1} -{B}_{j,l-m+1} C^m) {B}_{j, k}\Big).
\end{align*}
The above equality can be further rewritten as
\begin{align*}
&\Big[ H_{i,m}, [{B}_{j,k}, {B}_{j,l+1}]_{v^{-2}} +[{B}_{j,l}, {B}_{j,k+1}]_{v^{-2}} \Big]= \frac{[mc_{ij}]}{m}\times
\\
&\Big([{B}_{j,k+m}, {B}_{j,l+1}]_{v^{-2}} - [{B}_{j,k-m}, {B}_{j,l+1}]_{v^{-2}} C^m
+ [{B}_{j,l+m}, {B}_{j,k+1}]_{v^{-2}} - [{B}_{j,l-m}, {B}_{j,k+1}]_{v^{-2}} C^m
\\
&+ [{B}_{j,l}, {B}_{j,k+m+1}]_{v^{-2}} - [{B}_{j,l}, {B}_{j,k-m+1}]_{v^{-2}} C^m
+ [{B}_{j,k}, {B}_{j,l+m+1}]_{v^{-2}} - [{B}_{j,k}, {B}_{j,l-m+1}]_{v^{-2}} C^m\Big).
\end{align*}
There are 4 terms in each of the above 2 lines, and we add up column by column using \eqref{iDR3b}. Note that the sum of column 1 cancels with the sum of column 4 since both are equal to (modulo an opposite sign)
\begin{align*}
\begin{cases}
v^{-2} \Theta_{j, l-k-m+1} C^{k+m} \mathbb{K}_j -v^{-2} \Theta_{j, l-k-m-1} C^{k+m+1} \mathbb{K}_j, & \text{ if } l>k+m,
\\
v^{-2} \Theta_{j, k+m-l+1} C^l \mathbb{K}_j -v^{-2} \Theta_{j, k+m-l-1} C^{l+1} \mathbb{K}_j, & \text{ if } l<k+m,
\\
2 v^{-2} \Theta_{j, 1} C^l \mathbb{K}_j, & \text{ if } l =k+m.
\end{cases}
\end{align*}
Similarly, the sum of column 2 cancels with the sum of column 3 since both are equal to (modulo an opposite sign)
\begin{align*}
\begin{cases}
v^{-2} \Theta_{j, k-l-m+1} C^{l+m} \mathbb{K}_j -v^{-2} \Theta_{j, k-l-m-1} C^{l+m+1} \mathbb{K}_j, & \text{ if } k>l+m,
\\
v^{-2} \Theta_{j, l+m-k+1} C^k \mathbb{K}_j -v^{-2} \Theta_{j, l+m-k-1} C^{k+1} \mathbb{K}_j , & \text{ if } k<l+m,
\\
2 v^{-2} \Theta_{j, 1} C^k \mathbb{K}_j, & \text{ if } k =l+m.
\end{cases}
\end{align*}
This proves the lemma.
\end{proof}
\begin{proposition}
\label{prop:iDR1}
Relation \eqref{iDR1} for $i\neq j \in \mathbb{I}_0$ in $\widetilde{{\mathbf U}}^\imath$ follows from the relations \eqref{iDR2} and \eqref{iDR3b} in $\widetilde{{\mathbf U}}^\imath$.
\end{proposition}
\begin{proof}
It follows by Lemma~\ref{lem:comb} and Lemma~\ref{lem:comm} that $[ H_{i,m}, \Theta_{j,a}] =0$, for all $m, a\ge 1.$
Since $H_{j,n}$ for any $n \ge 1$ is a linear combination of monomials in $\Theta_{j,a}$, for various $a\ge 1$ by \eqref{exp h}, we conclude that $[ H_{i,m}, H_{j,n}] =0$, whence \eqref{iDR1}.
\end{proof}
\subsection{Two more relations rephrased}
It remains to establish relations \eqref{iDR5} and \eqref{iDR2} for $c_{ij}=-1$ in $\widetilde{{\mathbf U}}^\imath$, with the help of the finite type Serre relation \eqref{eq:S2} and \eqref{iDR3a}--\eqref{iDR3b}.
By Proposition~\ref{prop:equivij}, the relation \eqref{iDR2} (for $c_{ij}=-1$) is equivalent to the following relation
\begin{align}
\label{iDR2-reform}
[ & \Theta_{i,k},B_{j,r}]+[\Theta_{i,k-2},B_{j,r}]\mathbb{K}_\delta
\\
&= v^{-1}[\Theta_{i,k-1},B_{j,r+1}]_{v^2}+v[\Theta_{i,k-1},B_{j,r-1}]_{v^{-2}}\mathbb{K}_\delta, \quad \text{ for } k \ge 0 \text{ and } r \in \mathbb Z.
\notag
\end{align}
Clearly \eqref{iDR2-reform} also holds for $k< 0$.
The relation \eqref{iDR5} in $\widetilde{{\mathbf U}}^\imath$ (where we can assume $k_2\ge k_1$ without loss of generality) reads:
\begin{align}
\label{iDR5-reform}
\mathbb{S}(k_1,k_2|l) =\mathbb{R}(k_1,k_2|l), \quad \text{ for } k= k_2 -k_1 \ge 0 \text{ and } l \in \mathbb Z.
\end{align}
We shall prove \eqref{iDR2-reform} and \eqref{iDR5-reform} simultaneously and inductively on $k$ in \S\ref{subsec:iDR2=>iDR5}--\ref{subsec:iDR5=>iDR2} below. To that end, we shall refer to \eqref{iDR2-reform} and \eqref{iDR5-reform} as \eqref{iDR2-reform}$_k$ and \eqref{iDR5-reform}$_k$, respectively. We then denote by \eqref{iDR2-reform}$_{\le k}$ (respectively, \eqref{iDR2-reform}$_{< k}$) the identities \eqref{iDR2-reform}$_\ell$ for all $\ell \le k$ (respectively, for all $\ell <k$); and similarly for \eqref{iDR5-reform}$_{\le k}$.
\subsection{Implication from $\eqref{iDR2-reform}_{< k}$ to $\eqref{iDR5-reform}_{\leq k}$ }
\label{subsec:iDR2=>iDR5}
We shall fix $i,j \in \mathbb{I}_0$ such that $c_{ij}=-1$ throughout this subsection.
We identify $C=\mathbb{K}_\delta$ below.
\begin{lemma} \label{lem:SSSa}
For $k_1, k_2, l \in \mathbb Z$, we have
\begin{align*}
& \mathbb{S}(k_1,k_2+1 |l) + \mathbb{S}(k_1+1,k_2|l) -[2] \mathbb{S}(k_1+1,k_2+1 |l-1)
\\
&= \Big( -[B_{jl}, \Theta_{i, k_2-k_1+1}]_{v^{-2}} C^{k_1} +v^{-2} [B_{jl}, \Theta_{i, k_2-k_1-1}]_{v^{-2}} C^{k_1+1} \Big) \mathbb{K}_i + \{k_1 \leftrightarrow k_2\}.
\end{align*}
\end{lemma}
(The proof of the lemma uses only the relations \eqref{iDR3a}--\eqref{iDR3b}.)
\begin{proof}
We rewrite \eqref{eq:Skk} as
\begin{align*}
\mathbb{S}(k_1,k_2|l)
&= B_{i,k_2} [B_{i,k_1}, B_{j,l}]_{v^{-1}} - v [B_{i,k_1}, B_{j,l}]_{v^{-1}} B_{i,k_2}
+\{k_1 \leftrightarrow k_2\}.
\end{align*}
This together with \eqref{iDR3a} implies that
\begin{align}
&\mathbb{S}(k_1+1,k_2+1|l-1)
\label{eq:Skk1a} \\
&= (B_{i,k_2+1} [B_{i,k_1+1}, B_{j,l-1}]_{v^{-1}} - v [B_{i,k_1+1}, B_{j,l-1}]_{v^{-1}} B_{i,k_2+1})
+\{k_1 \leftrightarrow k_2\}
\notag \\
&= (v^{-1} B_{i,k_2+1} [B_{i,k_1}, B_{j,l}]_{v} - [B_{i,k_1}, B_{j,l}]_{v} B_{i,k_2+1})
+\{k_1 \leftrightarrow k_2\}. \notag
\end{align}
Using \eqref{eq:Skk1a}, we compute
\begin{align}
\label{eq:Skk2a}
& \mathbb{S}(k_1,k_2+1 |l) + \mathbb{S}(k_1+1,k_2|l) -[2] \mathbb{S}(k_1+1,k_2+1|l-1)
\\
&= \Big( \mathbb{S}(k_1,k_2+1 |l)
-[2] \big(v^{-1} B_{i,k_2+1} [B_{i,k_1}, B_{j,l}]_{v} - [B_{i,k_1}, B_{j,l}]_{v} B_{i,k_2+1} \big) \Big)
+\{k_1 \leftrightarrow k_2\}
\notag \\
&=
\Big( \big( B_{i,k_1} [B_{i,k_2+1}, B_{j,l}]_{v} - v^{-1} [B_{i,k_2+1}, B_{j,l}]_{v} B_{i,k_1}
+B_{i,k_2+1} [B_{i,k_1}, B_{j,l}]_{v} - v^{-1} [B_{i,k_1}, B_{j,l}]_{v} B_{i,k_2+1} \big)
\notag \\
& \quad -[2] \big(v^{-1} B_{i,k_2+1} [B_{i,k_1}, B_{j,l}]_{v} - [B_{i,k_1}, B_{j,l}]_{v} B_{i,k_2+1} \big) \Big)
+ \{k_1 \leftrightarrow k_2\}
\notag \\
&= \big(B_{jl} [B_{i, k_2+1}, B_{i, k_1}]_{v^2} -v^{-2} [B_{i, k_2+1}, B_{i, k_1}]_{v^2} B_{jl} \big)
+ \{k_1 \leftrightarrow k_2\},
\notag
\end{align}
where the last identity is obtained by first combining the third and fifth terms (and respectively, the fourth and sixth terms) and then further adding the first and second terms.
The relation \eqref{iDR3b} can be rewritten as
\begin{align}
\label{eq:BBBB}
[{B}_{i,l+1}, {B}_{i,k}]_{v^{2}} + [{B}_{i,k+1}, {B}_{i,l}]_{v^{2}}
&= -(\Theta_{i,l-k+1} C^k -v^{-2}\Theta_{i,l-k-1} C^{k+1}) \mathbb{K}_i
+ \{k \leftrightarrow l \}.
\end{align}
Using \eqref{eq:BBBB}, we rewrite the RHS of the identity \eqref{eq:Skk2a} as
\begin{align*}
& \big(B_{jl} [B_{i, k_2+1}, B_{i, k_1}]_{v^2} -v^{-2} [B_{i, k_2+1}, B_{i, k_1}]_{v^2} B_{jl} \big)
+ \{k_1 \leftrightarrow k_2\}
\\
&=B_{jl} \big( [B_{i, k_2+1}, B_{i, k_1}]_{v^2} + [B_{i, k_1+1}, B_{i, k_2}]_{v^2} \big)
-v^{-2} \big( [B_{i, k_2+1}, B_{i, k_1}]_{v^2} + [B_{i, k_1+1}, B_{i, k_2}]_{v^2} \big) B_{jl}
\\
&= -B_{jl} \big( \Theta_{i, k_2-k_1+1} C^{k_1} -v^{-2}\Theta_{i,k_2-k_1-1} C^{k_1+1} \big) \mathbb{K}_i
\\
&\quad + v^{-2} \big(\Theta_{i, k_2-k_1+1} C^{k_1} -v^{-2}\Theta_{i,k_2-k_1-1} C^{k_1+1} \big) B_{jl} \mathbb{K}_i
+ \{k_1 \leftrightarrow k_2\}
\\
&= \Big( -[B_{jl}, \Theta_{i, k_2-k_1+1}]_{v^{-2}} C^{k_1} +v^{-2} [B_{jl}, \Theta_{i, k_2-k_1-1}]_{v^{-2}} C^{k_1+1} \Big) \mathbb{K}_i + \{k_1 \leftrightarrow k_2\}.
\end{align*}
The lemma is proved.
\end{proof}
Denote, for $n\in \mathbb Z$,
\begin{align}
\label{X}
X_{n |l}=&\sum_{p\ge 0} v^{2p+1}[B_{j,l+1},\Theta_{i,n-2p-1}]_{v^{-2}} C^{p+1}
+\sum_{p\ge 1} v^{2p-1} [2] [\Theta_{i,n-2p},B_{j,l}] C^{p+1}
\\
&\quad -\sum_{p\ge 0} v^{2p+1}[\Theta_{i,n-2p-1},B_{j,l-1}]_{v^{-2}} C^{p+2}
+ [\Theta_{i,n},B_{j,l}] C.
\notag
\end{align}
Clearly, $X_{n |l}=0$, for $n\leq 0$.
\begin{lemma}
\label{lem:vanish1}
If \eqref{iDR2-reform}$_{\le k}$ holds, then $X_{n |l-1}=0$, for all $n \leq k$ and $l\in \mathbb Z$. (The converse is also true.)
\end{lemma}
\begin{proof}
Recalling \eqref{X}, we compute
\begin{align*}
& X_{n |l-1} - v^2 C X_{n-2 |l-1}
\\
&= v [B_{j,l},\Theta_{i,n-1}]_{v^{-2}} + v[2][\Theta_{i,n-2},B_{j,l-1}]C - v[\Theta_{i,n-1},B_{j,l-2}]_{v^{-2}}C
\\
&\quad + [\Theta_{i,n},B_{j,l-1}] -v^2[\Theta_{i,n-2},B_{j,l-1}]C
\\
&= -v^{-1}[\Theta_{i,n-1},B_{j,l}]_{v^{2}}+ [\Theta_{i,n-2},B_{j,l-1}]C + [\Theta_{i,n},B_{j,l-1}] -v [\Theta_{i,n-1},B_{j,l-2}]_{v^{-2}}C.
\end{align*}
For $0\le n\le k$, the RHS is $0$, which is equivalent to the assumption that \eqref{iDR2-reform}$_{n}$ holds.
The lemma follows by an induction on $n$ and noting that $X_{-1 |l-1} = X_{0 |l-1}=0$.
\end{proof}
Recall $R(k_1,k_2 |l)$ from \eqref{eq:Rkk}. Let us establish an $R$-counterpart of Lemma~\ref{lem:SSSa}.
\begin{lemma} \label{lem:RRRa}
For $k_1, k_2, l \in \mathbb Z$ with $k_2\ge k_1$, we have
\begin{align*}
& R(k_1,k_2+1 |l) + R(k_1+1,k_2|l) -[2] R(k_1+1,k_2+1|l-1)
\\
&= - [2]^2 X_{k_2-k_1 |l-1} C^{k_1} \mathbb{K}_i
+ \big( -[B_{j,l}, \Theta_{i, k_2-k_1+1}]_{v^{-2}} C^{k_1} +v^{-2} [B_{j,l}, \Theta_{i, k_2-k_1-1}]_{v^{-2}} C^{k_1+1} \big) \mathbb{K}_i.
\end{align*}
\end{lemma}
\begin{proof}
This proof is based on only formal algebraic manipulations, and does not use any nontrivial relations in $\widetilde{{\mathbf U}}^\imath$.
Following the format of \eqref{eq:Rkk} for $R$, we write
\begin{align*}
R(k_1,k_2+1 |l) &=\big( R_1' +R_2' - [{B}_{j,l}, \Theta _{i,k_2-k_1+1}]_{v^{-2}} \big) C^{k_1} \mathbb{K}_i,
\\
R(k_1+1,k_2|l) &= \big( R_1+R_2 - [{B}_{j,l}, \Theta _{i,k_2-k_1-1}]_{v^{-2}} C \big) C^{k_1} \mathbb{K}_i,
\\
R(k_1+1,k_2+1 |l-1) &= \big( R_3 +R_4 - [{B}_{j,l-1}, \Theta _{i,k_2-k_1}]_{v^{-2}} C \big) C^{k_1} \mathbb{K}_i.
\end{align*}
where $R_1'$ and $R_2'$ denote the first and second summands of $R(k_1,k_2+1 |l)$ as in \eqref{eq:Rkk}; the notations in the other two identities are understood similarly.
By a direct computation, we have
\begin{align*}
R_2' + R_2 &= -\sum_{p\geq0} v^{2p} [2]^2 [B_{j,l}, \Theta _{i,k_2-k_1-2p-1}]_{v^{-2}} C^{p+1}
+v^{-1}[2] [B_{j,l}, \Theta _{i,k_2-k_1-1}]_{v^{-2}}C.
\end{align*}
By first summing up $R_1' + R_1$, we also obtain by a direct computation that
\begin{align*}
R_1' + R_1 -[2] R_4
&= -\sum_{p\geq 1} v^{2p-2} [2]^3 [\Theta _{i,k_2-k_1-2p},{B}_{j,l-1}] C^{p+1}
- [2] [\Theta _{i,k_2-k_1}, B_{j,l-1}]_{v^{-2}} C.
\end{align*}
Note the main summands in $(R_2' + R_2)$, $(R_1' + R_1 -[2] R_4)$ and $-[2] R_3$ above are precisely $- v^{-1}[2]^2$ times the three main summands in $X_{k_2-k_1 |l-1}$ defined in \eqref{X}. Hence we have
\begin{align*}
& R(k_1,k_2+1 |l) + R(k_1+1,k_2|l) -[2] R(k_1+1,k_2+1 |l-1)
\\
&= (R_2' + R_2) + (R_1' + R_1 -[2] R_4) -[2] R_3
\notag \\
&\quad - [B_{j,l}, \Theta _{i,k_2-k_1+1}]_{v^{-2}} - [B_{j,l}, \Theta _{i,k_2-k_1-1}]_{v^{-2}}C
+ [2] [B_{j,l-1}, \Theta_{i,k_2-k_1}]_{v^{-2}} C
\notag \\
%
& =(- v^{-1} [2]^2 X_{k_2-k_1 |l-1} + v^{-1} [2]^2 [\Theta_{i,k_2-k_1}, B_{j,l-1}] C )
\notag \\
&\quad +v^{-1}[2] [B_{j,l}, \Theta _{i,k_2-k_1-1}]_{v^{-2}}C
- [2] [\Theta _{i,k_2-k_1}, B_{j,l-1}]_{v^{-2}} C
\notag \\
&\quad - [B_{j,l}, \Theta _{i,k_2-k_1+1}]_{v^{-2}} - [B_{j,l}, \Theta _{i,k_2-k_1-1}]_{v^{-2}}C
+[2] [B_{j,l-1}, \Theta _{i,k_2-k_1}]_{v^{-2}} C
\notag \\
&= - v^{-1} [2]^2 X_{k_2-k_1 |l-1}
- [B_{j,l}, \Theta _{i,k_2-k_1+1}]_{v^{-2}} +v^{-2} [B_{j,l}, \Theta _{i,k_2-k_1-1}]_{v^{-2}}C.
\notag
\end{align*}
The lemma is proved.
\end{proof}
Now we can show that \eqref{iDR2-reform}$_{< k}$ (together with \eqref{eq:S2} and \eqref{iDR3a}--\eqref{iDR3b}) implies \eqref{iDR5-reform}$_{\le k}$, for $k\ge 1$.
\begin{proposition}
\label{prop:2to5}
Let $k\ge 0$. If \eqref{iDR2-reform}$_{\le k}$ holds, then \eqref{iDR5-reform}$_{\leq k+1}$ holds.
\end{proposition}
\begin{proof}
Assume \eqref{iDR2-reform}$_{\le k}$ holds. Then by Lemma~\ref{lem:vanish1}, $X_{k_2-k_1 |l-1}=0$. Hence by comparing Lemma~\ref{lem:SSSa} and Lemma~\ref{lem:RRRa} we obtain, for $k=k_2-k_1\ge 0$,
\begin{align}
&\mathbb{S}(k_1,k_2+1 |l) + \mathbb{S}(k_1+1,k_2|l) -[2] \mathbb{S}(k_1+1,k_2+1 |l-1)
\label{eq:SR} \\
=& \mathbb{R}(k_1,k_2+1 |l) + \mathbb{R}(k_1+1,k_2|l) -[2] \mathbb{R}(k_1+1,k_2+1|l-1).
\notag
\end{align}
We induct on $k$. Consider the base case when $k=0$. We have $\mathbb{S}(r,r |l-1)= \mathbb{R}(r,r |l-1)$, for all $r, l$, by applying $\texttt{\rm T}} %{\mathbf T_i^{-r} \texttt{\rm T}} %{\mathbf T_j^{1-l}$ to the finite type Serre relation \eqref{eq:S2}. Hence \eqref{iDR5-reform}$_{\le 1}$ holds by \eqref{eq:SR} for $k_1=k_2$ and noting $\mathbb{S}(k_1,k_1+1 |l) =\mathbb{S}(k_1+1,k_1 |l)$ and $\mathbb{R}(k_1,k_1+1 |l-1)
=\mathbb{R}(k_1+1,k_1|l-1)$ by the symmetrization definition of $\mathbb{S}$ and $\mathbb{R}$.
By the inductive assumption, \eqref{iDR5-reform}$_{\le k}$ holds; in particular, we have, for $k=k_2-k_1 >0$,
\[ \mathbb{S}(k_1+1,k_2|l) = \mathbb{R}(k_1+1,k_2 |l),\qquad \mathbb{S}(k_1+1,k_2+1 |l-1) =\mathbb{R}(k_1+1,k_2+1|l-1).
\]
We conclude from this and \eqref{eq:SR} that $\mathbb{S}(k_1,k_2+1 |l) = \mathbb{R}(k_1,k_2+1 |l)$, whence \eqref{iDR5-reform}$_{k+1}$.
\end{proof}
\subsection{Implication from $\eqref{iDR5-reform}_{\le k}$ to $\eqref{iDR2-reform}_{\leq k}$ }
\label{subsec:iDR5=>iDR2}
We shall fix $i,j \in \mathbb{I}_0$ such that $c_{ij}=-1$ throughout this subsection.
The strategy here is similar to the strategy used in the previous subsection on the opposite implication. We start with a variant of Lemma~\ref{lem:SSSa}.
\begin{lemma} \label{lem:SSS}
For $k_1, k_2, l \in \mathbb Z$, we have
\begin{align*}
& \mathbb{S}(k_1,k_2+1 |l) + \mathbb{S}(k_1+1,k_2|l) -[2] \mathbb{S}(k_1,k_2|l+1)
\\
&= \Big( -[\Theta_{i, k_2-k_1+1}, B_{jl}]_{v^{-2}} C^{k_1} +v^{-2} [\Theta_{i, k_2-k_1-1}, B_{jl}]_{v^{-2}} C^{k_1+1} \Big) \mathbb{K}_i + \{k_1 \leftrightarrow k_2\}.
\end{align*}
\end{lemma}
(The proof of the lemma uses only the relations \eqref{iDR3a}--\eqref{iDR3b}.)
\begin{proof}
We rewrite \eqref{eq:Skk} as
\begin{align*}
\mathbb{S}(k_1,k_2|l)
&= B_{i,k_1} [B_{i,k_2}, B_{j,l}]_{v} - v^{-1} [B_{i,k_2}, B_{j,l}]_{v} B_{i,k_1}
+\{k_1 \leftrightarrow k_2\}.
\end{align*}
This together with \eqref{iDR3a} implies that
\begin{align}
\mathbb{S}(k_1,k_2|l+1)
&= (B_{i,k_1} [B_{i,k_2}, B_{j,l+1}]_{v} - v^{-1} [B_{i,k_2}, B_{j,l+1}]_{v} B_{i,k_1})
+\{k_1 \leftrightarrow k_2\}
\label{eq:Skk1} \\
&= (v B_{i,k_1} [B_{i,k_2+1}, B_{j,l}]_{v^{-1}} - [B_{i,k_2+1}, B_{j,l}]_{v^{-1}} B_{i,k_1})
+\{k_1 \leftrightarrow k_2\}. \notag
\end{align}
Using \eqref{eq:Skk1}, we compute
\begin{align}
\label{eq:Skk2}
& \mathbb{S}(k_1,k_2+1 |l) + \mathbb{S}(k_1+1,k_2|l) -[2] \mathbb{S}(k_1,k_2|l+1)
\\
&= \Big( \mathbb{S}(k_1,k_2+1 |l)
-[2] \big(v B_{i,k_1} [B_{i,k_2+1}, B_{j,l}]_{v^{-1}} - [B_{i,k_2+1}, B_{j,l}]_{v^{-1}} B_{i,k_1} \big) \Big)
+\{k_1 \leftrightarrow k_2\}
\notag \\
&=
\Big( \big( B_{i,k_1} [B_{i,k_2+1}, B_{j,l}]_{v} - v^{-1} [B_{i,k_2+1}, B_{j,l}]_{v} B_{i,k_1}
+B_{i,k_2+1} [B_{i,k_1}, B_{j,l}]_{v} - v^{-1} [B_{i,k_1}, B_{j,l}]_{v} B_{i,k_2+1} \big)
\notag \\
& \quad -[2] \big( v B_{i,k_1} [B_{i,k_2+1}, B_{j,l}]_{v^{-1}} - [B_{i,k_2+1}, B_{j,l}]_{v^{-1}} B_{i,k_1} \big) \Big)
+ \{k_1 \leftrightarrow k_2\}
\notag \\
&= \big( [B_{i, k_2+1}, B_{i, k_1}]_{v^2} B_{jl} -v^{-2} B_{jl} [B_{i, k_2+1}, B_{i, k_1}]_{v^2} \big)
+ \{k_1 \leftrightarrow k_2\},
\notag
\end{align}
where the last identity is obtained by first adding the first and fifth terms (and respectively, the second and sixth terms) and then further simplifying when adding with the third and fourth terms.
Using \eqref{eq:BBBB}, we rewrite the RHS of the identity \eqref{eq:Skk2} as
\begin{align*}
& \big( [B_{i, k_2+1}, B_{i, k_1}]_{v^2} B_{jl} -v^{-2} B_{jl} [B_{i, k_2+1}, B_{i, k_1}]_{v^2} \big)
+ \{k_1 \leftrightarrow k_2\}
\\
&=\big( [B_{i, k_2+1}, B_{i, k_1}]_{v^2} + [B_{i, k_1+1}, B_{i, k_2}]_{v^2} \big) B_{jl}
-v^{-2} B_{jl} \big( [B_{i, k_2+1}, B_{i, k_1}]_{v^2} +[B_{i, k_1+1}, B_{i, k_2}]_{v^2} \big)
\\
&= -\big( \Theta_{i, k_2-k_1+1} C^{k_1} -v^{-2}\Theta_{i,k_2-k_1-1} C^{k_1+1} \big) \mathbb{K}_i B_{jl}
\\
&\quad + v^{-2} B_{jl} \big(\Theta_{i, k_2-k_1+1} C^{k_1} -v^{-2}\Theta_{i,k_2-k_1-1} C^{k_1+1} \big) \mathbb{K}_i
+ \{k_1 \leftrightarrow k_2\}
\\
&= \Big( -[\Theta_{i, k_2-k_1+1}, B_{jl}]_{v^{-2}} C^{k_1} +v^{-2} [\Theta_{i, k_2-k_1-1}, B_{jl}]_{v^{-2}} C^{k_1+1} \Big) \mathbb{K}_i + \{k_1 \leftrightarrow k_2\}.
\end{align*}
The lemma is proved.
\end{proof}
Denote
\begin{align} \label{Y}
Y_{k+1} &=[{B}_{j,l+1}, \Theta _{i,k}]_{v^{-2}} - [\Theta _{i,k},{B}_{j,l-1}]_{v^{-2}}C
+v^{-1} [\Theta_{i,k+1},B_{j,l}] + v^{-1} [\Theta_{i,k-1},B_{j,l}] C.
\end{align}
(The identity \eqref{iDR2-reform} can be stated equivalently as $Y_{k+1} =0$.)
Here is a variant of Lemma~\ref{lem:RRRa}.
\begin{lemma} \label{lem:TTT}
For $k_1, k_2, l \in \mathbb Z$ with $k_2\ge k_1$, we have
\begin{align*}
& R(k_1,k_2+1 |l) + R(k_1+1,k_2|l) -[2] R(k_1,k_2|l+1)
\\
&= [2]^2 X_{k_2-k_1-1 |l} C^{k_1}\mathbb{K}_i + [2] Y_{k_2-k_1+1} C^{k_1} \mathbb{K}_i
\\
&\quad + \big( -[\Theta_{i, k_2-k_1+1}, B_{j,l}]_{v^{-2}} C^{k_1} +v^{-2} [\Theta_{i, k_2-k_1-1}, B_{j,l}]_{v^{-2}} C^{k_1+1} \big) \mathbb{K}_i.
\end{align*}
\end{lemma}
\begin{proof}
This proof is based on only formal algebraic manipulations, and does not use any nontrivial relations in $\widetilde{{\mathbf U}}^\imath$.
Following the format of \eqref{eq:Rkk} for $R$, we write
\begin{align*}
R(k_1,k_2+1 |l) &=\big( R_1' +R_2' - [{B}_{j,l}, \Theta _{i,k_2-k_1+1}]_{v^{-2}} \big) C^{k_1} \mathbb{K}_i,
\\
R(k_1+1,k_2|l) &= \big( R_1+R_2 - [{B}_{j,l}, \Theta _{i,k_2-k_1-1}]_{v^{-2}} C \big) C^{k_1} \mathbb{K}_i,
\\
R(k_1,k_2|l+1) &= \big( R_5 +R_6 - [{B}_{j,l+1}, \Theta _{i,k_2-k_1}]_{v^{-2}} \big) C^{k_1} \mathbb{K}_i.
\end{align*}
where $R_1'$ and $R_2'$ denote the first and second summands of $R(k_1,k_2+1 |l)$ as in \eqref{eq:Rkk}; the notations in the other two identities are understood similarly.
By a direct computation, we have
\begin{align*}
R_1' + R_1 =-\sum_{p\geq0} v^{2p+1} [2]^2 [\Theta _{i,k_2-k_1-2p-2},{B}_{j,l-1}]_{v^{-2}}C^{p+2}
- [2] [\Theta _{i,k_2-k_1},{B}_{j,l-1}]_{v^{-2}}C.
\end{align*}
By first summing up $R_2' + R_2$, we also obtain by a direct computation that
\begin{align*}
R_2' + R_2 -[2] R_5
&= -\sum_{p\geq 1} v^{2p-1} [2]^3 [\Theta _{i,k_2-k_1-2p-1},{B}_{j,l}] C^{p}
\\
&\quad - v[2] [B_{j,l}, \Theta _{i,k_2-k_1-1}]_{v^{-2}}C
+[2]^2 [\Theta _{i,k_2-k_1-1},{B}_{j,l}]_{v^{-2}}C
\end{align*}
Note the main summands in $(R_1' + R_1)$, $(R_2' + R_2 -[2] R_5)$ and $-[2] R_6$ above are precisely $[2]^2$ times the three main summands (in reversed order) in $X_{k_2-k_1}$ defined in \eqref{X}. Hence we have
\begin{align}
& R(k_1,k_2+1 |l) + R(k_1+1,k_2|l) -[2] R(k_1,k_2|l+1)
\label{eq:TTT2} \\
&= (R_1' + R_1) + (R_2' + R_2 -[2] R_5) -[2] R_6
\notag \\
&\quad - [{B}_{j,l}, \Theta _{i,k_2-k_1+1}]_{v^{-2}}
- [{B}_{j,l}, \Theta _{i,k_2-k_1-1}]_{v^{-2}} C
+[2] [{B}_{j,l+1}, \Theta _{i,k_2-k_1}]_{v^{-2}}
\notag \\
&= [2]^2 (X_{k_2-k_1-1 |l} - [\Theta_{i,k_2-k_1-1},B_{j,l}] C)
- [2] [\Theta _{i,k_2-k_1},{B}_{j,l-1}]_{v^{-2}}C
\notag \\
&\quad - v[2] [B_{j,l}, \Theta _{i,k_2-k_1-1}]_{v^{-2}}C
+[2]^2 [\Theta _{i,k_2-k_1-1},{B}_{j,l}]_{v^{-2}}C
\notag \\
&\quad - [{B}_{j,l}, \Theta _{i,k_2-k_1+1}]_{v^{-2}}
- [{B}_{j,l}, \Theta _{i,k_2-k_1-1}]_{v^{-2}} C
+[2] [{B}_{j,l+1}, \Theta _{i,k_2-k_1}]_{v^{-2}}.
\notag
\end{align}
On the RHS of \eqref{eq:TTT2} above, there is exactly one term involving $B _{i,l+1}$ and one term involving $B_{i,l-1}$, with opposite coefficients, just as those appearing in $Y_{k_2-k_1+1}$ \eqref{Y}. This allows us to rewrite the RHS of \eqref{eq:TTT2} in terms of $X_{k_2-k_1-1 |l}, Y_{k_2-k_1+1}$ and the terms involving $B_{j,l}$ only. The terms involving $B_{j,l}$ can then be simplified by some direct computation (without using any relations) to the formula stated in the lemma.
\end{proof}
Now we can show that \eqref{iDR5-reform}$_{\le k}$ (together with \eqref{eq:S2} and \eqref{iDR3a}--\eqref{iDR3b}) implies \eqref{iDR2-reform}$_{\le k}$.
\begin{proposition}
\label{prop:5to2}
Let $k\ge 0$. If \eqref{iDR5-reform}$_{\le k}$ holds, then \eqref{iDR2-reform}$_{\le k}$ holds.
\end{proposition}
\begin{proof}
We prove by induction on $k$. The identity \eqref{iDR2-reform}$_{k}$ with $k=0$ is trivial.
The case \eqref{iDR2-reform}$_{k+1}$ (i.e., $Y_{k+1}=0$) follows by comparing the identities in Lemma~\ref{lem:SSS}
and Lemma~\ref{lem:TTT} (where $k=k_2 -k_1 \ge 0$), with help from Lemma~\ref{lem:vanish1} and
the inductive assumption \eqref{iDR5-reform}$_{\le k+1}$. (See the proof of Proposition~\ref{prop:2to5} for more details of the same type of arguments.)
\end{proof}
\begin{proposition}
\label{prop:iDR25}
The relations \eqref{iDR2} and \eqref{iDR5} for $c_{ij}=-1$ hold in $\widetilde{{\mathbf U}}^\imath$.
\end{proposition}
\begin{proof}
These relations follow by the equivalent identities \eqref{iDR2-reform}--\eqref{iDR5-reform}, which have been established inductively and simultaneously in Proposition~\ref{prop:2to5} and Proposition~ \ref{prop:5to2}.
\end{proof}
\section{Variants of Drinfeld type presentations}
\label{sec:variants}
In this section, we formulate several variants of the Drinfeld type presentation for $\widetilde{{\mathbf U}}^\imath$, one in generating function formalism, one in a more symmetrized form, and another via different imaginary root vectors. We also deduce a Drinfeld type presentation for the $\imath$quantum group ${\mathbf U}^\imath_{\boldsymbol{\varsigma}}$ with parameters ${\boldsymbol{\varsigma}} =(\varsigma_i)_{i\in \mathbb{I}}$.
\subsection{Presentation via generating functions}
Recall the generating functions $\bB_{i}(z), \boldsymbol{\Theta}_i(z)$ and $ \boldsymbol{\Delta}(z)$ from \eqref{eq:Genfun}.
\begin{theorem}
\label{thm:ADEgf}
${}^{\text{Dr}}\tUi$ is generated by $\mathbb{K}_{i}^{\pm1}$, $C^{\pm1}$, $\Theta_{i,k}$ and ${B}_{i,l}$ $(i\in \mathbb{I}_0$, $k\geq1$, $l\in\mathbb Z)$, subject to the following relations, for $i, j \in \mathbb{I}_0$:
\begin{align}
&\mathbb{K}_i \text{ are central, }\quad \boldsymbol{\Theta}_i(z) \boldsymbol{\Theta}_j(w) = \boldsymbol{\Theta}_j(w) \boldsymbol{\Theta}_i(z),
\label{iDRG1}
\\
& \boldsymbol{\Theta}_i (z) {\mathbf B }_j(w)
= \frac{(1 -v^{-c_{ij}}zw^{-1}) (1 -v^{c_{ij}} zw C)}{(1 -v^{c_{ij}}zw^{-1})(1 -v^{-c_{ij}}zw C)}
{\mathbf B }_j(w) \boldsymbol{\Theta}_i (z),
\label{iDRG2}
\\
&\bB_i(w)\bB_j(z)=\bB_j(z)\bB_i(w), \qquad\qquad\qquad\text{ if }c_{ij}=0, \label{iDRG4}
\\
&(v^{c_{ij}}z -w) \bB_i(z) \bB_j(w) +(v^{c_{ij}}w-z) \bB_j(w) \bB_i(z)=0, \qquad \text{ if }i\neq j, \label{iDRG3a}
\\
&(v^2z-w) \bB_i(z) \bB_i(w) +(v^{2}w-z) \bB_i(w) \bB_i(z) \label{iDRG3b}
\\\notag
&\quad =\frac{v^{-2}}{v-v^{-1}} \mathbb{K}_{i} \boldsymbol{\Delta}(zw) \big( (v^2z-w) \boldsymbol{\Theta}_i(w) +(v^2w-z) \boldsymbol{\Theta}_i(z) \big),
\\
&\bB_i(w_1)\bB_i(w_2)\bB_{j}(z) -[2]\bB_i(w_1)\bB_{j}(z)\bB_i(w_2)+\bB_{j}(z)\bB_i(w_1)\bB_i(w_2)+ \{w_1\leftrightarrow w_2\}
\label{iDRG5} \\
=& -\mathbb{K}_{i} \frac{ \boldsymbol{\Delta}(w_1w_2)}{v-v^{-1}} \Big( [ \boldsymbol{\Theta}_i(w_2),{\mathbf B }_j(z)]_{v^{-2}} \frac{[2] z w_1^{-1} }{1 -v^{2}w_2w_1^{-1}}
+ [{\mathbf B }_j(z), \boldsymbol{\Theta}_i(w_2)]_{v^{-2}} \frac{1 +w_2w_1^{-1}}{1 -v^{2}w_2w_1^{-1}} \Big)
\notag \\
& \qquad + \{w_1\leftrightarrow w_2\},
\qquad\qquad
\text{ if }c_{ij}=-1.
\notag
\end{align}
\end{theorem}
\begin{proof}
We simply rewrite the relations \eqref{iDR1}--\eqref{iDR5} in Theorem \ref{def:iDR} by using the generating functions \eqref{eq:Genfun}.
The relation \eqref{iDRG1} is clear. As formulated in Proposition~\ref{prop:equivij}, the relation \eqref{iDRG2} is equivalent to \eqref{iDR2}.
The relation \eqref{iDRG3b} is obtained from \eqref{iDR3b} by multiplying both side of the relation \eqref{iDR3b} by $v^2z^{r+1} w^{s+1}$ and summing over $r,s\in\mathbb Z$.
Similarly, the relations \eqref{iDRG4} and \eqref{iDRG3a} are equivalent to the relations \eqref{iDR4} and \eqref{iDR3a}, respectively.
Finally, the relation \eqref{iDRG5} is obtained by multiplying both side of the relation \eqref{iDR5} by $w_1^{k_1}w_2^{k_2}z^{r}$ and summing over $k_1,k_2,r\in\mathbb Z$.
\end{proof}
\subsection{A symmetrized presentation}
We add two central generators $C^{\pm\frac{1}{2}}$ such that $C^{\frac{1}{2}} C^{-\frac{1}{2}}=1=C^{-\frac{1}{2}}C^{\frac{1}{2}}$ and $(C^{\frac{1}{2}})^2=C$.
\begin{definition}
\label{def:i-DR-ref}
Let ${}^{\text{DR}}\widetilde{{\mathbf U}}^\imath$ be the $\mathbb Q(v)$-algebra generated by $\mathbb{K}_{i}^{\pm1}$, $C^{\pm\frac{1}{2}}$, ${\widehat{H}}_{i,m}$ and ${B}_{i,l}$, where $i\in \mathbb{I}_0$, $m\geq1$, $l\in\mathbb Z$, subject to the following relations, for $m,n\ge 1$ and $k,l\in \mathbb Z$:
\begin{align}
&\mathbb{K}_i, C^{\pm\frac{1}{2}} \text{ are central, } \quad [{\widehat{H}}_{i,m},{\widehat{H}}_{j,n}]=0, \label{iDRC1}
\\
&[{\widehat{H}}_{i,m},{B}_{jl}]=\frac{[mc_{ij}]}{m} {B}_{j,l+m}C^{-\frac{m}{2}}-\frac{[mc_{ij}]}{m} {B}_{j,l-m}C^{\frac{m}{2}}, \label{iDRC2}
\\
\label{iDRC3a}
&[{B}_{i,k},{B}_{j,l+1}]_{v^{-c_{ij}}}-v^{-c_{ij}}[{B}_{i,k+1},{B}_{j,l}]_{v^{c_{ij}}}=0, \text{ if }i\neq j,
\\
&[{B}_{i,k} ,{B}_{j,l}]=0, \quad \text{ if }c_{ij}=0, \label{iDRC4}
\\
&[{B}_{i,k},{B}_{i,l+1}]_{v^{-2}}-v^{-2}[{B}_{i,k+1},{B}_{i,l}]_{v^{2}} \label{iDRC3}
\\\notag
&= \mathbb{K}_i C^{\frac{k+l+1}{2}}
\Big( v^{-2}\widehat{\Theta}_{i,l-k+1} -v^{-4} \widehat{\Theta}_{i,l-k-1} +v^{-2} \widehat{\Theta}_{i,k-l+1} -v^{-4} \widehat{\Theta}_{i,k-l-1}\Big),
\\
& B_{i,k_1} B_{i,k_2} B_{j,l} -[2] B_{i,k_1} B_{j,l} B_{i,k_2} + B_{j,l} B_{i,k_1} B_{i,k_2} +\{k_1 \leftrightarrow k_2\}
\label{iDRC5} \\
&= \mathbb{K}_i C^{ \frac{k_1+k_2}{2}}
\Big(-\sum_{p\geq0}v^{2p}[2] [\widehat{\Theta}_{i,k_2-k_1-2p-1},{B}_{j,l-1}]_{v^{-2}} C^{\frac{1}{2}}\notag
\\\notag
&-\sum_{p\geq0}v^{2p-1}[2][{B}_{j,l},\widehat{\Theta}_{i,k_2-k_1-2p}]_{v^{-2}}
+v^2[{B}_{j,l}, \widehat{\Theta}_{i,k_2-k_1}]_{v^{-2}} \Big) +\{k_1 \leftrightarrow k_2\},
\text{ if }c_{ij}=-1.
\notag
\end{align}
Here
$\widehat{\Theta}_{i,m}$, for $i \in \mathbb{I}_0$ and $m\ge 1$, are defined by the following equation:
\begin{align}
\label{exp hC}
1+ \sum_{m\geq 1} &(v-v^{-1})\widehat{\Theta}_{i,m} u^m = \exp\big( (v-v^{-1}) \sum_{m=1}^\infty {\widehat{H}}_{i,m} u^m \big);
\end{align}
we also set $\widehat{\Theta}_{i,0}=\frac{1}{v-v^{-1}}$, $\widehat{\Theta}_{i,m}=0$ for $m<0$.
\end{definition}
We enlarge ${}^{\text{Dr}}\tUi$ to a $\mathbb Q(v)$-algebra ${}^{\text{Dr}}\tUi [C^{\pm\frac12}] := {}^{\text{Dr}}\widetilde{{\mathbf U}}^\imath \otimes_{\mathbb Q(v)[C^{\pm 1}]} \mathbb Q(v)[C^{\pm \frac12}]$; recall it contains the generators $H_m$.
For $i \in \mathbb{I}_0$ and $m\ge 1$, we identify
\begin{align} \label{eq:HH2}
{\widehat{H}}_{i,m}=H_{i,m} C^{-\frac{m}{2}},\qquad \widehat{\Theta}_{i,m}=\Theta_{i,m} C^{-\frac{m}{2}}.
\end{align}
Then \eqref{exp hC} holds.
Now such an identification \eqref{eq:HH2} leads to an isomorphism ${}^{\text{DR}}\widetilde{{\mathbf U}}^\imath \cong {}^{\text{Dr}}\widetilde{{\mathbf U}}^\imath [C^{\pm\frac12}]$ in the proposition below, which also identifies elements in the same notation. We skip the detail as the verification is straightforward.
\begin{proposition}
\label{prop:symm}
We have a natural $\mathbb Q(v)$-algebra isomorphism
${}^{\text{DR}}\widetilde{{\mathbf U}}^\imath \cong {}^{\text{Dr}}\widetilde{{\mathbf U}}^\imath [C^{\pm\frac12}]$.
\end{proposition}
\subsection{Presentation via different root vectors}
Recall starting from the rank 1 case treated in Section~\ref{sec:Onsager}, we have preferred the imaginary root vectors $\Theta_m$ over $\acute{\Theta}_m$, because of a consideration from Hall algebra \cite{LRW20}. Choosing $\acute{\Theta}_m$ will lead to the following presentation for $\widetilde{{\mathbf U}}^\imath$.
\begin{theorem}
\label{thm:ADE1}
The algebra $\widetilde{{\mathbf U}}^\imath$ admits a presentation with the same set of generators of ${}^{\text{Dr}}\widetilde{{\mathbf U}}^\imath$ as in Definition~\ref{def:iDR}, subject to the relations~\eqref{iDR1}--\eqref{iDR3a} for ${}^{\text{Dr}}\widetilde{{\mathbf U}}^\imath$ and the following 2 relations \eqref{DrBB}--\eqref{DrSerre} (in place of \eqref{iDR3b}--\eqref{iDR5} for ${}^{\text{Dr}}\widetilde{{\mathbf U}}^\imath$):
\begin{align}
&[{B}_{i,r}, {B}_{i,s+1}]_{v^{-2}} -v^{-2} [{B}_{i,r}, {B}_{i,s}]_{v^{2}}
\label{DrBB}
=v^{-2} {\Theta}_{i,s-r+1} C^r \mathbb{K}_i-v^{-2} {\Theta}_{i,s-r-1} C^{r+1} \mathbb{K}_i \\
\notag
&+v^{-2} {\Theta}_{i,r-s+1} C^s \mathbb{K}_i-v^{-2} {\Theta}_{i,r-s-1} C^{s+1} \mathbb{K}_i,
\\
& B_{i,k_1} B_{i,k_2} B_{j,l} -[2] B_{i,k_1} B_{j,l} B_{i,k_2} + B_{j,l} B_{i,k_1} B_{i,k_2} +\{k_1 \leftrightarrow k_2\}
\label{DrSerre}
\\
&=
- \textstyle C^{k_1} \mathbb{K}_i \Big(\sum_{p\geq0} (v^{-2p-1} +v^{2p+1}) [ {\Theta}_{i,k_2-k_1-2p-1},{B}_{j,l-1}]_{v^{-2}}C^{p+1} \notag
\\\notag
& \textstyle
\quad +\sum_{p\geq \red{1}} (v^{-2p} +v^{2p} ) [{B}_{j,l}, {\Theta} _{i,k_2-k_1-2p}]_{v^{-2}} C^{p}
+[{B}_{j,l}, {\Theta} _{i,k_2-k_1}]_{v^{-2}} \Big)
\notag \\
&\quad +\{k_1 \leftrightarrow k_2\},
\qquad \text{ if }c_{ij}=-1.
\notag
\end{align}
\end{theorem}
\begin{proof}
In this proof, we shall denote the generators ${\Theta}_{i,k}$ in the proposition by $\acute{\Theta}_{i,k}$, in order to distinguish the ${\Theta}_{i,k}$ used for ${}^{\text{Dr}}\tUi$ in Definition~\ref{def:iDR}.
By Theorem~\ref{thm:ADE}, it suffices to show the algebra with presentation given by the proposition is isomorphic to ${}^{\text{Dr}}\tUi$. The isomorphism is given by match generators in the same notation and in addition imposing the relation between ${\Theta}_{i,k}$ and $\acute{\Theta}_{i,k}$ as follows (cf. \eqref{eq:Theta2}):
\begin{align*}
\boldsymbol{\Theta}_i(z) = \frac{1- Cz^2}{1-v^{-2}C z^2} \acute{ \boldsymbol{\Theta}}_i(z).
\end{align*}
Then the equivalence between the relations \eqref{iDR3b} and \eqref{DrBB} follows from the equivalence between \eqref{rel:iDr1} and \eqref{rel:iDr1b}. Finally, the equivalence between the relations \eqref{iDR5} and \eqref{DrSerre} follows by a direct computation, which we omit here.
\end{proof}
\subsection{Presentation of affine $\imath$quantum groups with parameters}
\label{subsec:parameter}
Fix ${\boldsymbol{\varsigma}} =(\varsigma_i)_{i\in \mathbb{I}}$, where $\varsigma_i \in \mathbb Q(v)^\times$, for each $i$. A quantum symmetric pair $({\mathbf U}, {\mathbf U}^\imath_{\boldsymbol{\varsigma}})$ was first formulated by G. Letzter in finite type and then generalized by Kolb \cite{Ko14}.
The $\imath$quantum group ${\mathbf U}^\imath_{\boldsymbol{\varsigma}}$ is the $\mathbb Q(v)$-algebra with generators $B_i (i\in \mathbb{I})$, and it can be obtained from $\widetilde{{\mathbf U}}^\imath$ by a central reduction \cite{LW19a} (recall $\tilde{k}_i$ was used {\em loc. cit.} and it is related to $\mathbb{K}_i$ in this paper by $\mathbb{K}_i =-v^2\tilde{k}_i$):
\begin{align} \label{Ui}
{\mathbf U}^\imath_{\boldsymbol{\varsigma}} =\widetilde{{\mathbf U}}^\imath / (\mathbb{K}_i +v^2 \varsigma_i \mid i \in \mathbb{I}).
\end{align}
For $\delta=\sum_{i\in \mathbb{I}}a_i\alpha_i$, we define
\[
\delta_{\boldsymbol{\varsigma}}=\prod_{i\in \mathbb{I}} (-v^2\varsigma_i)^{a_i} \in \mathbb Q(v).
\]
From \eqref{Ui} we obtain a natural surjective homomorphism of $\mathbb Q(v)$-algebra
\[
\pi: \widetilde{{\mathbf U}}^\imath \longrightarrow {\mathbf U}^\imath_{\boldsymbol{\varsigma}}, \qquad B_i \mapsto B_i \; (i\in \mathbb{I}).
\]
By abuse of notations, we shall keep using the same notations for the images under $\pi$ of various elements such as $B_{i,k}, \Theta_{i,m}, H_{i,m}$, for $i\in \mathbb{I}_0, k\in \mathbb Z, m\ge 1$. The algebra ${\mathbf U}^\imath_{\boldsymbol{\varsigma}}$ has a Serre-type presentation with generators $B_i$, for $i\in \mathbb{I}$, and defining relations \eqref{eq:S1}--\eqref{eq:S3}, where $\mathbb{K}_i$ is replaced by $-v^2 \varsigma_i$, cf. \cite{BB10, Ko14}. Below we present a Drinfeld type presentation for ${\mathbf U}^\imath_{\boldsymbol{\varsigma}}$.
\begin{theorem} \label{thm:ADE2}
Let ${\boldsymbol{\varsigma}}=(\varsigma_i)_{i\in \mathbb{I}}\in \mathbb Q(v)^{\times, \mathbb{I}}$. Then the $\mathbb Q(v)$-algebra ${\mathbf U}^\imath_{\boldsymbol{\varsigma}}$ has a presentation with generators
$H_{i,m}$ and ${B}_{i,l}$, where $i\in \mathbb{I}_0$, $m\geq1$, $l\in\mathbb Z$ and the following relations, for $r,s \in \mathbb Z$, $m,n \ge 1$, $i,j\in \mathbb{I}_0$:
\begin{align}
&[H_{i,m},H_{j,n}]=0,
\\
&[H_{i,m},{B}_{j,l}]=\frac{[mc_{ij}]}{m} {B}_{j,l+m}-\frac{[mc_{ij}]}{m} \delta_{{\boldsymbol{\varsigma}}}^m {B}_{j,l-m},
\\
&[{B}_{i,k} ,{B}_{j,l}]=0, \text{ if }c_{ij}=0,
\\
&[{B}_{i,k}, {B}_{j,l+1}]_{v^{-c_{ij}}} -v^{-c_{ij}} [{B}_{i,k}, {B}_{j,l}]_{v^{c_{ij}}}=0, \text{ if }i\neq j,
\\
&[{B}_{i,k}, {B}_{i,l+1}]_{v^{-2}} -v^{-2} [{B}_{i,k}, {B}_{i,l}]_{v^{2}}
=- \delta_{{\boldsymbol{\varsigma}}}^k\varsigma_i \Theta_{i,l-k+1} +v^{-2} \delta_{{\boldsymbol{\varsigma}}}^{k+1} \varsigma_i \Theta_{i,l-k-1} \\\notag
&\qquad\qquad\quad\qquad\qquad\qquad\qquad\qquad
- \delta_{{\boldsymbol{\varsigma}}}^l \varsigma_i \Theta_{i,k-l+1} +v^{-2} \delta_{{\boldsymbol{\varsigma}}}^{l+1} \varsigma_i\Theta_{i,k-l-1},
\notag
\\
& B_{i,k_1} B_{i,k_2} B_{j,l} -[2] B_{i,k_1} B_{j,l} B_{i,k_2} + B_{j,l} B_{i,k_1} B_{i,k_2} +\{k_1 \leftrightarrow k_2\}
\\
=& v^2 \varsigma_i \delta_{{\boldsymbol{\varsigma}}}^{k_1}
\left(\sum_{p\geq0} v^{2p}[2] \delta_{{\boldsymbol{\varsigma}}}^{p+1} [\Theta _{i,k_2-k_1-2p-1},{B}_{j,l-1}]_{v^{-2}} \right. \notag
\\
& \left. +\sum_{p\geq 1} v^{2p-1}[2] \delta_{{\boldsymbol{\varsigma}}}^{p} [{B}_{j,l},\Theta _{i,k_2-k_1-2p}]_{v^{-2}}
+ [{B}_{j,l}, \Theta _{i,k_2-k_1}]_{v^{-2}} \right)
+\{k_1 \leftrightarrow k_2\},
\text{ if }c_{ij}=-1.
\notag
\end{align}
\end{theorem}
The theorem above, for which we continue to adopt the convention \eqref{Hm0}--\eqref{exp h}, follows directly from Theorem~\ref{thm:ADE} and \eqref{Ui}.
|
3,212,635,537,429 | arxiv | \section{Introduction}
The probability of magnetic field emergence is not uniform at all solar longitudes. Numerous works have been devoted to identify and follow the most active longitudinal belts but the different datasets, time intervals, approaches and preassumptions resulted in a broad variety of spatial patterns and temporal behaviours. In the history of this research field two larger groups can be separated. One of them disregards the spatially resolved active region data and focuses on the temporal variation of the activity by assuming that the active zones belong to a frame having different rotation rate from that of the Carrington frame. This approach is restricted by the preassumption that this different rotation rate is constant (Bai, 1987; Balthasar, 2007; Bogart, 1982; Jetsu et al., 1997; Olemskoy and Kitchatinov, 2007). The other group uses the position data of active regions but also with restricting preassumptions allowing e.g. the impact of differential rotation which necessarily implies a cyclic behaviour (Berdyugina and Usoskin, 2003; Usoskin et al., 2005; Zhang et al., 2011).
Our work tries to combine these two approaches. At first we want to identify the active belts and its motion with respect to the Carrington frame without assuming that this motion is invariable or cycle dependent. In the next phase the study of the temporal variation can focus on the moving active longitudinal zone.
Our previous paper (Gyenge et al., 2012, hereinafter Paper I) presented a possible method to localise and follow the longitudinal zone of enhanced activity. The diagrams of that paper show a characteristic migration pattern in the time-longitude diagram. In contrast to the well recognizable Sp\"orer pattern in the time-latitude diagram, the longitudinal migration of enhanced activity does not show connection with the cycle profiles. A parabola has been fitted to the migration path and along this curve the width of the active zone was about 30 degrees, the flip-flop phenomenon was well identifiable. The migration path comprised the decreasing phase of cycle 21, the entire cycle 22 and the beginning of cycle 23 so it is highly improbable that the phenomenon of active longitude is connected with the solar cycle or the differential rotation. That work was restricted to the time interval of the Debrecen Photoheliographic Data (DPD): 1979-2011. The aim of the recent work is to extend the study to earlier cycles and to scrutinize closer the dynamics of flux emergence within the activity belt.
\section{Time-Longitude Analysis of Sunspot Distributions}
The work is based on the two detailed sunspot catalogue, the DPD (Gy\H ori et al., 2011) and the GPR (Greenwich Photoheliographic Results, Royal Observatory Greenwich). The procedure was similar to that of Paper I. The $360^{\circ}$ longitudinal circumference of the Sun was divided into $10^{\circ}$ bins and the normalized weight of activity has been computed in each bin and each Carrington rotation by the formula:
\begin{eqnarray}
W_{i} = \frac{A_{i}}{ \sum_{j=1}^{36} A_{j} }
\end{eqnarray}
\begin{figure}[h!]
\begin{center}
\epsfig{file=22NorthLONG.eps,width=6.4cm,angle=-90}
\epsfig{file=22NorthDIS.eps,width=4.8cm,angle=-90}
\end{center}
\caption{Upper panel: migration of the active longitudinal zone with respect to the Carrington frame in the northern hemisphere between Carrington rotations 1740\,--\,1935 (between years 1984\,--\,1996). On the vertical axis the solar circumference is plotted three times. The lowest part of the panel contains the simultaneous cycle profiles plotted by using the International Sunspot Number (SIDC-team). Lower panel: longitudinal distribution of activity as measured from the parabola.}
\end{figure}
The $W_{i}$ quantity represents the fraction of the total activity emerging at a certain longitudinal bin so if its highest values are tracked through the rotations it may reveal the migration of the most active longitudinal zone with respect to the Carrington frame. The first step in the identification of the path was a search by visual pattern recognition in the time\,--\,longitude diagram, this was a forward and backward shift of the active zone between the Carrington rotations 1740\,--\,1935 (between years 1984\,--\,1996) in the northern hemisphere. This subjective choice has been checked with more objective procedures: fitting of a parabola on the points of the highest activity, the determination of the width of active zone along the parabola path, the detection of a flip-flop phenomenon. In the present work this procedure has been repeated with a somewhat different approach: in each Carrington rotation all longitudinal bins were disregarded in which the fraction of entire activity in the given rotation was smaller than 0.28 which is twice as high as the mean standard deviation of the averaged $W_{i}$ in the rotations. The remaining bins show up the migration path of the active zone, the parabola fitted on them is practically the same as in Paper I:
\begin{eqnarray}
l=-0.081(r-1837)^{2}+720
\end{eqnarray}
Where {\it l} is the longitude, and {\it r} is the Carrington rotation number.
\begin{figure}[h!]
\begin{center}
\epsfig{file=22NorthGRP.eps,width=8cm}
\epsfig{file=22NorthACF.eps,width=8cm}
\epsfig{file=22NorthACFmind.eps,width=8cm}
\end{center}
\caption{Upper panel: temporal variation of monthly total sunspot area in the migrating active zone. Middle panel: the autocorrelogram of this data series. Lower panel: the autocorrelogram of the entire activity in the same time interval.}
\end{figure}
Figure 1 shows the path of the active zone, the selected points of the time\,--\,longitude diagram and the fitted parabola. The solar longitudinal circumference is plotted three times on the vertical axis in order to follow the migration. It is remarkable that the forward motion in the Carrington frame starts in the decreasing phase of cycle 21, it returns at the time of maximum of cycle 22 and it ends during the rising phase of cycle 23. The lower panel shows the longitudinal distribution of the activity in a moving reference frame in which the position of $60^{\circ}$ mark moves along the parabola of the migration path. The distributions in each Carrington rotation were averaged for the total length of the path. The activity distribution has a smaller secondary maximum at the opposite longitude of the main maximum.
\begin{figure}[h!]
\begin{center}
\epsfig{file=14SouthLONG.eps,width=6.4cm,angle=-90}
\epsfig{file=14SouthDIS.eps,width=4.8cm,angle=-90}
\end{center}
\caption{The same diagrams as in Figure 1 for the migration path between Carrington rotations 420\,--\,620 (years 1885\,--\,1900) }
\end{figure}
The temporal behaviour of the active zone was investigated by using autocorrelation analysis. Figure 2 shows the temporal variation and the autocorrelogram of monthly total sunspot area in the longitudinal zone of $\pm 15^{\circ}$ width on both sides of the parabola curve. The highest peak of the curve is at the rotation 18 which corresponds nearly to 1.3 years. To check whether this period belongs really to the active zone the autocorrelation of the entire activity has also been computed, see the lower panel of Figure 1, it does not contain this peak or some other signatures of any periodicity.
\begin{figure}[h!]
\begin{center}
\epsfig{file=14SouthACF.eps,width=8cm}
\epsfig{file=14SouthACFmind.eps,width=8cm}
\end{center}
\caption{The same diagrams as in the two lower panels of Figure 2 for the path shown in Figure 3. }
\end{figure}
The above presented investigations have been carried out by using the data of DPD. In order to extend the investigations the time-longitude diagrams of the GPR-period have been studied and another similar migration path has been selected in the southern hemisphere at the time of cycle 14 between Carrington rotations 420\,--\,620 (years 1885\,--\,1900). The procedure of curve fitting was the same as in the case of the path presented above. The obtained curve and the longitudinal activity distribution around it are plotted in Figure 3.
There are remarkable similarities between these diagrams and those of Figure 1, primarily the parabola shape of the migration, the return of the migration at the time of cycle 14 maximum, and the main and secondary maxima at opposite positions in the longitudinal distribution of the activity obtained along the path. The most important difference is that the lengths of the forward-backward migrations are not the same, at around the rotation No. 600 a new migration path starts in forward direction, not followed here.
The most intriguing property of the northern migration path between 1984\,--\,1996 is that the activity within its narrow belt exhibits a variation with a period of about 1.3 years and this period cannot be detected in the entire activity of the same time interval. The same variation has also been studied in the migration path coinciding with cycles 13-14, the results are shown in Figure 4. In the autocorrelogram of Figure 4 the significant peak also exists at rotation 18 (1.33 years) but two smaller peaks are also present at rotations 22 and 24 (1.62 and 1.77 years). The check of the curve, the autocorrelogram of the entire activity is similar to that of the studied interval at cycles 21\,--\,23, it has no significant peaks at all.
\section{Discussion}
The obtained results of the above case studies may have connections to several other findings. The shapes of the migration paths of the active longitudinal zones are fairly similar to those variations which have been found by Juckett (2006) with a quite different method. The variations of the flux emergences within the active zones exhibit a period of 1.3 years, the active zone during cycle 14 has also two further periods of smaller peaks. These periods are absent in the entire activity. This may imply that the magnetic fluxes of these active zones emerge from the bottom of the convective zone where the 1.3 year radial torsional oscillation has been detected (Howe et al, 2000). This interpretation is supported by the theoretical considerations of Bigazzi and Ruzmaikin (2004) that the active longitude can only be pertinent at the bottom of the convection zone because at higher layers the differential rotation would disarrange it. If the conjectured connection to the depth of the tachocline zone really exists it could support the "shallow layer" model of the active longitudes proposed by Dikpati and Gilman (2005).
The present work studied a northern and a southern migration path at an interval of about a century from each other and several similarities have been found between them. A large statistical study is in preparation for the detailed search for further identifiable migrating active longitudes in the entire time interval covered by the DPD and GPR.
\section*{Acknowledgements}
The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2012-2015) under grant agreement No. 284461.
\section*{References}
\begin{itemize}
\small
\itemsep -2pt
\itemindent -20pt
\item[] Bai, T., 1987, {\it \apj}, 314, 795-807.
\item[] Balthasar, H., 2007, {\it \aap} 471, 281-287.
\item[] Berdyugina, S. V. \& Usoskin, I. G., 2003, {\it \aap} 405, 1121-1128.
\item[] Bigazzi, A. \& Ruzmaikin, A., 2004, {\it \apj}, 604, 944-959.
\item[] Bogart, R. S., 1982, {\it \solphys}, 76, 155-165.
\item[] Dikpati, M. \& Gilman, P., 2005, {\it \apj}, 635, L193-L196.
\item[] Gyenge, N., Baranyi, T. \& Ludm\'any, A., 2012, {\it CEAB}, 36, No.1., 9-16. Paper I.
\item[] Gy\H ori, L., Baranyi, T., \& Ludm\'any, A. 2011, {\it IAUS} 273, 403. \\ see: http://fenyi.solarobs.unideb.hu/DPD/index.html
\item[] Howe, R., Christensen-Dalsgaard, J., Hill, F., et al., 2000, {\it Science}, 287, 2456-2460.
\item[] Jetsu, L., Pohjolainen, S., Pelt, J., \& Tuominen, I., 1997, {\it \aap}, 318, 293-307.
\item[] Juckett, D. A., 2006, {\it \solphys}, 245, 37-53.
\item[] Olemskoy, S. V. \& Kitchatinov, L. L., 2007, {\it Geomagn.Aeron.}, 49, 866-870.
\item[] Royal Observatory Greenwich, Greenwich Photoheliographic Results, 1874-1976, in 103 volumes, see: http://solarscience.msfc.nasa.gov/greenwch.shtml
\item[] SIDC-team, World Data Center for the Sunspot Index, Royal Observatory of Belgium, Monthly Report on the International Sunspot Number, online catalogue of the sunspot index: http://www.sidc.be/sunspot-data/
\item[] Usoskin, I. G., Berdyugina, S. V. \& Poutanen, J., 2005, {\it \aap}, 441, 347-352.
\item[] Zhang, L.Y., Mursula, K., Usoskin, I. G. \& Wang, H.N., 2011, {\it \aap}, 529, 23.
\end{itemize}
\end{document}
|
3,212,635,537,430 | arxiv | \section{INTRODUCTION}
\label{sec:intro}
\textbf{SIMULATOR} stands for \textbf{S}peckle \textbf{I}mager via \textbf{MU}lti \textbf{L}ayer \textbf{A}tmospheric \textbf{T}urbulence \textbf{O}bject \textbf{R}econstructor. It is a speckle imager that can mimic the exact characteristics of various observation sites as requested. This instrument has been made with the sole purpose of applying post-processing techniques like speckle correlation-based imaging\cite{beavers1989speckle,horch2004speckle} in image reconstruction under diverse conditions, including telescope cites turbulence strength, wind profiles, sky background variations etc. However, this instrument can be used as a testbed against post-processing techniques or algorithms like lucky imaging\cite{staley2014lucky,law2006lucky}, phase diversity method\cite{gonsalves2018phase} etc. The idea of this instrument was inspired by the MAPS (Multi-Atmospheric Phase screens and Stars) instrument, developed under the European Space Organisation (ESO) Very Large Telescope (VLT) team, solely to test the Multi-Conjugate Adaptive Optics (MCAO) project\cite{kolb2004maps}. With this instrument in hand, we can overcome the time request for a telescope which is extremely hard to get these days. We can even strengthen our methods/algorithms under investigation against various instrument errors. This project aims to mimic the three-dimensional evolving turbulence effect, thus stimulating turbulence behaviour of the atmosphere, covering all layers of the atmosphere up to 83 km.
The early-stage conceptual design of SIMULATOR can be found in fig~\ref{fig:conceptual_design}. Upon user request, the source plane can provide adjustable fed lines from where the light originates. They can act as natural guide stars (NGS) depending on the field of view (FoV). A set of lenses combined to collimate the entire FoV. This collimated beam then passes through three turbulence phase screens before reaching the pupil plane. After exiting from the pupil stop, a combination of lenses re-images the incoming beam to the focus or sensor plane. With the help of mirrors, a slow converging beam can be brought to the camera plane. Here, the EMCCD camera is a sensor for fast frame capturing, rotating, and translating stages to simulate atmospheric turbulence effects by changing the atmosphere's $C_n^{2}$ profile.
\begin{figure}[h!]
\centerline{\includegraphics[angle=0,width=0.99\textwidth]{conceptual_design.png}}
\caption{Early stage conceptual design of SIMULATOR\label{fig:conceptual_design} }
\end{figure}
\section{Characterization of SIMULATOR}
\label{sec:characterization}
\subsection{Turbulence phase screen}
\label{sec:SIMULATOR_phase_screen}
The turbulence in the atmosphere can be approximated by engraving the phase map profile on physical material disc named as turbulent phase screen, shown in fig.~\ref{fig:phase_screen}. The optical path difference gets generated for beams passing through different points across the field of observation. To design this model, we have adopted a three turbulence layer model. It means the three layers of the atmosphere can characterise an entire 3D turbulence volume. In this model, the turbulence strength profile parameter $C_n^2$ is considered constant across the plane for different height profiles\cite{wyngaard1971behavior}. Working with a single layer lowers this model's ability to mimic speckle features, including anisoplanatism and small-scale fluctuations effect at the pupil plane. These two effects can only be emulated if separate turbulence layers exist at both ends, ground and high altitude\cite{bos2012technique}. \\
\subsection{Scaling factors }
\label{sec:SIMULATOR_scalings}
\subsubsection{Site features}
\label{chap4:sec:SIMULATOR_scaling_telescope}
To incorporate an infinite distant long real imaging system within a finite-size optical table space of $0.8\times0.5$ m (based on initial calculations), we need to regulate all the parameters and scale them to smaller equivalent numbers. That's includes bringing telescope size $D$ to scaled-down version $d$, global fried parameter $r_0^{global}$ to $r_0^{scaled}$, turbulence wind velocity from $V(H)$ at height $H$ above telescope to scaled velocity $v(h)$ at scaled height $h$ above the pupil plane.\\
\subsubsection{Science features}
\label{chap4:sec:SIMULATOR_scaling_science}
To emulate the features of a large telescope observatory within the lab, we need to do matching of the characteristics parameters in a small space within the optical bench. A speckle image can be characterised by simple parameters ratio number $D/r_0$, where $D$ is the size of the telescope and $r_0^{global}$ is the global fried parameter of any particular site. This ratio estimates the number of coherence patches through the telescope imprinted within a single speckle image. This number automatically becomes significant for a large observatory. For example, the SOAR Observatory in Chile corresponds to 27 at $0.5\mu m$ wavelength. Thus, any site characteristic features can be emulated through this instrument, given a provision is in place to play around with the size of the simulated telescope number and fried parameter.
This instrument aims to have the largest possible $D/r_0$ value. Thus, after giving thorough research, we decided to mimic telescope of size $D = 25 $ m in diameter with fried parameter of $r_0^{global} = 14.4 $ cm (average number for Paranal observatory site\cite{kolb2004maps}). This results in coherence patches count to $\approx 174$. Our research includes the maximum size of the turbulent phase screen that can be manufactured. Thus limiting maximum height up to which atmosphere can be covered within the imaging system, i.e maximum footprint size(shown in fig. \ref{fig:phase_screen}).
\subsubsection{Atmospheric height profile}
\label{chap4:sec:SIMULATOR_scaling_turbulence}
Our atmospheric turbulence ranges up to 100 km (roughly ) as measured from the telescope pupil plane. Thus, the ground layer location (PS00) plays a vital role in incorporating maximum turbulent height within the minor separation. Ideally speaking ground layer of the atmosphere should be situated next to the pupil plane. Physically this is impossible to achieve. For reference, the ground layer can be called to be located at 250 m above the telescope plane. The minimum separation feasible between PS00 and pupil stop (PS) is 0.5 mm. This constraint comes from the mechanical regulation of the model. The separation between the PS00 exit face and the exit face of PS must be 0.5 mm. A more appropriate solution has been adopted in the mechanical design and covered in section~\ref{sec:opto_final_design}. A simple derivation tells height of the atmosphere falls as square of the height above the telescope. Height $H$ and scaled height $h$ above the telescope pupil goes as follows\\
\begin{equation}
\frac{h}{H} = \Big(\frac{d}{D}\Big)^2
\end{equation}
\begin{figure}[p]
\centering
\begin{subfigure}{0.5\linewidth}
\centering
\includegraphics[width=\linewidth]{location.png}
\caption{\label{fig:scaling_location}}
\end{subfigure}
\hfill
\begin{subfigure}{0.5\linewidth}
\centering
\includegraphics[width=\linewidth]{footprint.png}
\caption{\label{fig:scaling_footprint} }
\end{subfigure}
\\[\baselineskip]
\begin{subfigure}[H]{0.5\linewidth}
\centering
\includegraphics[width=\linewidth]{wind.png}
\caption{\label{fig:scaling_wind} }
\end{subfigure}
\centering
\caption{(a) Phase screen location for $d_{min} = 0.5$ mm , $D = 25 $ m. (b) Phase screen location based on FoV = $0.3^{\circ}$ , maximum footprint size at PS02 buffer = 52.5 mm. (c) Differential wind profile across PS02B versus pupil diameter. }
\end{figure}
Thus, for fixed aperture of size $D$ = 25 m, and minimum step size of h = 0.5 mm (corresponding H = 0.25 km), gives scaled pupil stop $d$ = 35.355 mm. Fig~\ref{fig:scaling_location}, plots pupil stop size versus phase screen distances from pupil stop. For min PS00 = 0.5 mm , and corresponding d = 35.355 mm, gives PS01 = 22 mm, PS02 = 100 mm and PS02B (buffer space for linear translation of PS02) = 166 mm. Here PS01, PS02 are mid and high altitude turbulence phase screen and more details are covered in section~\ref{sec:SIMULATOR_phasescreen}.
The three turbulent phase screens used for this experiment are of sizes PS00 = PS01 = 100 mm (85 mm clear aperture) and PS02 = 125 mm, each 1 mm thick are shown in fig~\ref{fig:phase_screen}. These phase screens are procured from Silios technologies, located in France. Thus maximum footprint allowed over each phase screens are PS00 = PS01$<$ 42.5 mm and PS02 $<$52.5 mm. Fig.~\ref{fig:scaling_footprint} shows plots for footprint size versus various pupil stop sizes. Thus, for PS00 = 35.38 mm ($<$ 45 mm), PS01 = 36.71 mm ($<$ 45 mm), PS02 = 41.53 mm ($<$ 52.5 mm)and PS02B = 45.59 mm ($<$ 52.5 mm). Calculation for footprint size $y$ goes as follows
\begin{equation}
\label{eq:footprint}
y = d+2h tan(\theta)
\end{equation}
where, FoV = 2$\theta$ scaled as
\begin{equation}
\label{eq:fov}
\frac{\phi}{\theta} = \frac{d}{D}
\end{equation}
where, $\phi$ is the scaled version of half of FoV. Thus, for 2$\theta = 0.3^{'}$ (Table.~\ref{tab:instrument_table}), $\phi = 1.76^{\circ}$ .
\begin{figure}
\centering
\begin{subfigure}[b]{0.5\linewidth}
\centering
\includegraphics[width=\linewidth]{phase_screen.png}
\caption{\label{fig:phase_screen}}
\end{subfigure}
\begin{subfigure}[b]{0.38\linewidth}
\centering
\includegraphics[width=\linewidth]{rin_rout.png}
\caption{\label{fig:footprint}}
\end{subfigure}
\caption{(a) Turbulence phase screens PS00, PS01 and PS02 physical features. (b) Footprint impression on phase screen, where $r_{in}$ and $r_{out}$ are inner and outer radius. For our case, $r_{in}$ = 0}
\label{fig:roc_curve}
\end{figure}
The wind gradient profile is the next important factor to incorporate within this instrument. The speckle imaging technique works within the frozen flow hypothesis, which means the speckle should be fully developed during an exposure period of a single speckle. Statistically, the atmosphere effects remain more or less the same except for the wind effect. For instance, when we model atmosphere, we assume a constant wind profile at a given height $h$ above the ground, just like the $C_n^2$ profile. It is impossible to incorporate this into our system because extended phase screens (ranging from 100 mm to 125 mm in OD) will experience a differential wind profile across the area of interest (or footprint). However, this can be minimised by bringing phase screens closer to each other, thus helping to downsize footprint size. This is not fruitful given our desired goals due to the limited coverage of turbulence layers of the atmosphere. We aim to keep differential speed less than 50\% across the footprint region. Data also shows for VLT 8 m telescope aperture average number is close to 50\%\cite{kolb2004maps}. For three-phase screens, we have picked wind velocity PS00 = 7 $\pm$ 3 $ms^{-1}$, PS01 = 30 $\pm$ 15 $ms^{-1}$ and PS02 = 60$\pm$ 30 $ms^{-1}$\cite{kolb2004maps}. The scaling factor for wind velocities goes linearly with pupil size as,
\begin{equation}
\label{eq:wind_scaling}
\frac{V}{v}= \frac{D}{d}
\end{equation}
Thus, velocity by which phase screen needs to be constantly rotated are PS00 = $9.9\pm4.2 mm/s (7\pm 3 m/s)
$, PS01 = $42.4\pm21.2 mm/s (30\pm 15 m/s)$
and PS02 = $84.9\pm42.4 mm/s (60\pm 30 m/s)$. The following equations have been employed to find the gradient change across the PS02 screen.
\begin{equation}
\label{eq:wind_1}
\frac{V_{in}}{V_{out}}= \frac{r_{in}}{r_{out}}
\end{equation}
\begin{equation}
\label{eq:wind_2}
\frac{V_{in}}{V_{out}} = \frac{r_{out}-y}{r_{out}}
\end{equation}
where footprint $y$ is used from eq.~\ref{eq:footprint} and also shown in fig~\ref{fig:footprint}. Fig~\ref{fig:scaling_wind} plots differential velocity plot ($V_{in}/V_{out}$) versus pupil stop diameter. For pupil stop size of 35.355 mm, differential wind across footprint PS02B is within the tolerance level of the atmosphere ($<$ 50\%). Results are shown only for the case of PS02B, which can encapsulate maximum error because it is located farthest from the pupil stop.
\subsection{Fried parameter to PS00, PS01 and PS02}
\label{sec:SIMULATOR_fried}
As per paranal site characteristics, the global fried parameter $r_0^{G}$ is 14.4 cm at 0.5$\mu m$ in median seeing condition \cite{kolb2004maps}. Thus, a scaled version of the fried number goes as
\begin{equation}
\label{eq:fried_scaling}
\frac{r_0^{G}}{r_{0}^{g}} = \frac{D}{d}
\end{equation}
Thus producing scaled global $r_{0}^{g}$ value of 0.203 mm. Next, we aim to parametrise each phase screen fried number for given global $r_{0}^{g}$. Table~\ref{tab:phase_screen_fried_parameter} has been used to find fried parameters for each independent phase screens. The $i^{th}$ layer fried parameter in terms of $C_n^{2}$ profile is given as\cite{roggemann1996imaging}
\begin{equation}
\label{eq:ith_layer}
r_{0_i} = [0.423\kappa^2C_{n_i}^2\Delta h_i]^{-3/5}
\end{equation}
where, $\Delta h_i$ marks distance between $i^{th}$ and $(i-1)^{th}$ layer. We can write the desired optical field parameters isoplanatic angle $\theta_{0}$ and log amplitude variance $\sigma
_{\chi}^2$, in terms of the phase-screen $r_{0_i}$ values as\cite{roggemann1996imaging}
\begin{equation}
\label{eq:ith_layer_fried}
r_0 = \Bigg[\sum_{i=0}^{i=n}r_{0_i}^{-5/3}(h/\Delta h)^{5/3}\Bigg]^{-3/5}
\end{equation}
\begin{equation}
\label{eq:ith_layer_iso}
\theta_0 = \Bigg[6.8794L^{5/3}\sum_{i=0}^{i=n}r_{0_i}^{-5/3}(1-h_i/\Delta h)^{5/3}\Delta h_i\Bigg]^{-3/5}
\end{equation}
\begin{equation}
\label{eq:ith_layer_sigma}
\sigma_{\chi}^2 = 1.331\kappa^{-5/6}\Delta h^{5/6}\sum_{i=0}^{i=n}r_{0_i}^{-5/3}(h_i/\Delta h)^{5/6}(1-h_i/\Delta h)^{5/6}\Delta h_i
\end{equation}
Next step is to find the individual fried parameters $r_0^{PS00}$, $r_0^{PS01}$ and $r_0^{PS02}$. Above equations eq.~\ref{eq:ith_layer_fried} and eq.~\ref{eq:ith_layer_sigma} can be rewritten in terms of matrix notation, and can be solved using fmincon function within MATLAB on eq.~\ref{eq:fmincon} by imposing constraint on log amplitude variance cannot exceed more than 20\% by each screen\cite{schmidt2010numerical} and remaining fixed parameters from table.~\ref{tab:instrument_table}.
\begin{equation}
\label{eq:fmincon}
\resizebox{0.95\hsize}{!}{$
\left[\begin{array}{c}
\left(\hat{r}_{0}\right)^{-5 / 3} \\
\frac{\hat{\sigma}_{x}^{2}}{1.331 \kappa^{-5 / 6} L^{5 / 6}}
\end{array}\right]=\left[\begin{array}{cccc}
\left(h_{1} / L\right)^{5 / 3} & \left(h_{2} / L\right)^{5 / 3} & \ldots & \left(h_{N} / L\right)^{5 / 3} \\
\left(h_{1} / L\right)^{5 / 6}\left(1-\frac{h_{1}}{L}\right)^{5 / 6} & \left(h_{2} / L\right)^{5 / 6}\left(1-\frac{h_{2}}{L}\right)^{5 / 6} & \ldots & \left(h_{N} / L\right)^{5 / 6}\left(1-\frac{h_{N}}{L}\right)^{5 / 6}
\end{array}\right] \times\left[\begin{array}{c}
r_{01}^{-5 / 3} \\
r_{0_{2}}^{-5 / 3} \\
\ldots r_{0_{N}}^{-5 / 3}
\end{array}\right]$}
\end{equation}
$N$ represents a number of phase screens (3 for our case), $\kappa$ is the wavenumber. The output from \textit{fmincon} are $r_0^{PS00}$ = 0.308 mm, $r_0^{PS01}$ = 0.3039 mm and $r_0^{PS02}$ = 0.3366 mm. The final output can be cross-checked using eq.~\ref{eq:ith_layer_iso} and verified against theoretical isoplanatic results. Our calculation shows maximum of 0.04\% error remains in isoplanatic measurements.
\section{PRELIMINARY DESIGN}
\label{sec:concept}
Our design comprises three stages. The first stage demands three sets of phase screens engraved with turbulence properties, as per fig.~\ref{fig:phase_maps}. The second stage covers a set of optics required to produce a collimated beam from natural guide star/point sources located at the source plane for the desired FoV. The third stage requires a set of optics for focusing distorted wavefront, slow converging beam towards camera plane for diffraction-limited results with minimal seidal coefficients (in fig.~\ref{fig:seidel}).
Table~\ref{tab:instrument_table} contains a list of parameters that have been incorporated within the instrument during its design. We must test our speckle imaging technique against a broad wavelength range of 150 nm in the optical regime. The requisite space of 20 mm at the source plane corresponds to the $0.3^{'}$ of FoV (or $3.534^\circ$ scaled FoV ). Thus, the input f-number of 4.95 is chosen for that reason. Since this system has been designed to have a diffraction-limited result for a telescope size of 25 m, it thus requires a slow converging beam for output. Output F-number of 51 corresponds to a back focal length of 1800 mm, and this has been achieved using multiple fold mirrors for compactness. The following subsections will explain the zemax output design for various subsystems within the instrument. \\
\subsection{Fixing phase screens}
\label{sec:SIMULATOR_phasescreen}
Fig~\ref{fig:phase_screen_layout} contains first stage information, which contains detailed information of three-phase screen location (shown in table.~\ref{fig:phase_screen_layout} \& \ref{tab:phase_screen_fried_parameter}). Each phase screen is placed off-centred w.r.t. principle axis of the beam because of the symmetric nature of turbulence profile engraved to it, as shown in fig~\ref{fig:phase_screen} and. We want the beam to be passed only through one-half of the turbulent phase. Corning C7980 is the material the manufacturer uses for making, and each phase screen is only 1 mm thick. PS00 and PS01 are 100 mm in diameter and PS02 is 125 mm in diameter. Thus, these numbers are initially put within the zemax at the very early design stage and kept fixed. Although for design purposes, phase screens PS01 and PS02 have been kept fixed to 22 mm and 100 mm from pupil stop. A provision in mechanical design is made for the movement of the phase screen by $\pm$ 30 mm and $\pm$ 15 mm. This instrument must bring turbulence profile variability for testing purposes (details covered in table \ref{tab:phase_screen_variability}).
The memory cell \cite{roddier1981atmospheric,roddier1982isoplanatic} or fried parameter is another critical parameter that gets affected by variability in phase screen location. Memory cell size is about 1.5 times larger than the fried parameter. For configuration where PS01 and PS02 have located at 22 km and 83 km, respectively, memory cell size turns out to be $0.0702^\circ$ ( entire FoV: $3.534^\circ$ ). Thus, leaving a total number of memory cells across one axis around 50 or 2500 cells within the $10\times 10$ mm region of the object plane. Each cell turns out to be 400 $\mu$m in size at the source plane. Atleast a couple of source is required to be fitted within 400 $\mu$m for testing.
\subsection{Collimating optics}
\label{sec:SIMULATOR_s2p}
After fixing up phase screens at desired locations, the next task is to focus light from $10\times 10$ mm within the source plane that must be collimated before entering the phase screen model. With the help of zemax, we have used a set of four custom-made collimating lenses (CL) for this particular task CL1, CL2, CL3 and CL4. For design optimisation, two important parameters had taken under consideration. Footprint shift per unit memory cell (or fried parameter) across three phase screens and pupil shift due to chromatic effects. Ideally, wavefront or light exiting from CL4 (collimated) originated from one ideal point source should overlap for the entire wavelength band. But, due to different refractive materials, each light ray bends or interacts with lenses differently (governed by Cauchy's equation\cite{roddier1999adaptive}). This leads to a shift in wavefront w.r.t. to the central propagation wavelength. This shift in wavefront w.r.t. to central wavelength is called footprint shift. Zemax helps in eliminating this shift as much as close to ideal figures. But due to the restrained number of lens catalogues, achieving an ideal figure is almost impossible. Thus a tolerance in final results is tested against this shift, and a compromise has been made. Our scientific goal requires that incoming light passes through the same coherence patch of sky or, in other words, speckles must be fully developed during exposure. As shown in table~\ref{tab:table_fixed_2}, we manage to bring footprint shift down to 5.7\% $r_0$ for PS02 i.e less than 19 $\mu$m, 0.02\% $r_0$ for PS01 i.e less than 0.06 $\mu$m and 0.1\% $r_0$ for PS00 i.e less than 0.34 $\mu$m. Fig~\ref{fig:pupil_footprint} shows footprint diagram covering entire FoV of $10\times 10$ mm and entire broadband range of 0.486 to 0.656 $\mu$m.
\begin{figure}[h!]
\centerline{\includegraphics[width=0.6\textwidth]{phase_maps.png}}
\caption{Phase screen maps of PS00, PS01 and PS02 respectively\label{fig:phase_maps} }
\end{figure}
\begin{figure}[h!]
\centerline{\includegraphics[angle=0,width=0.9\textwidth]{phase_screen_table.png}}
\caption{Technical description of all three phase screens PS00, PS01 and PS02.\label{fig:phase_screen_layout} }
\end{figure}
\begin{figure}[h!]
\centerline{\includegraphics[angle=0,width=0.8\textwidth]{phase_screen_layout.png}}
\caption{Proposed phase screen layout for PS00, PS01 and PS02. All distances are in mm\label{fig:phase_screen_layout} }
\end{figure}
\subsection{Re-imaging optics}
\label{sec:SIMULATOR_p2c}
Our next scientific target is a diffraction-limited result at the focal or camera plane. Our sensor area is only $8.2\times 8.2$ mm ( EMCCD camera iXon Ultra 897, covered in tab ~\ref{tab:EMCCD_1} ). We finally decided to focus on smaller FoV $1\times 1$ mm (corresponding plate scale of 1.9 arcmin/mm, output F-number 51 and back focal length of 1805 mm). With the help of zemax, the best combination of spherical lenses is selected by playing around with merit function parameters. Our major focusing parameters include WFNO (working f-number), EFLX - Effective focal length ( back focal length), and optimisation parameter covers RMS spot radius (centroid) at the focal plane. Finally, we ended up with four imaging lenses IL1, IL2, IL3 and IL4. The separation from the IL4 lens's last surface to the camera plane accounts for about 1333 mm. We have used five-fold mirrors (M1-M5) to make the system compact. However, depending on the optomechanical constraint (e.g. space constraint within an optical bench ), this design can be readjusted without adjusting other lenses combination (shown in fig.~\ref{fig:OM_mirror}).
\section{Final design}
\label{sec:final_design}
Fig.~\ref{fig:final_design} shows the final zemax design of instrument SIMULATOR. Fig.~\ref{fig:final_design_errors} covers final residual errors in footprint shift, RMS spot radius, net seidel coefficients and geometric encircled energy within FoV 2mm. RMS spot radius for the outer most field is 15.5 $\mu m$, which is well within airy disc radius 29.79 $\mu m$ at 0.5 $\mu m$. Maximum aberration scale is 0.1 mm at 0.486 $\mu m$. 80\% of geometric encircled energy can be covered within 3.75$\times$3.75 pixels.
\begin{figure}[h!]
\centerline{\includegraphics[angle=0,width=0.9\textwidth]{final_design.png}}
\caption{Final zemax design for SIMULATOR\label{fig:final_design} }
\end{figure}
\begin{figure*}
\centering
\begin{subfigure}[t]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{pupil_footprint.png}
\caption{\label{fig:pupil_footprint} Footprint diagram at pupil plane for entire field of view $10\times 10$ mm, covering entire wavelength range 0.486 to 0.656 $\mu$m}
\end{subfigure}%
\hfill
\begin{subfigure}[t]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{spot_diagram.png}
\caption{\label{fig:spot_diagram} Spot diagram for 2 mm FoV, covering bandwidth range of 0.4861 to 0.6563 $\mu$m }
\end{subfigure}
\bigskip
\begin{subfigure}[t]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{seidel_diagram.png}
\caption{\label{fig:seidel} Net seidel coefficients at camera plane, each square dimensions are 0.01$\times$0.01 mm}
\end{subfigure}%
\hfill
\begin{subfigure}[t]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{encicled_energy.png}
\caption{\label{fig:encircled} Radius of centroid in $\mu$m vs fraction of geometric encircled energy for 2 mm FoV}
\end{subfigure}
\caption{Final design residual errors in terms of footprint shift, rms spot radius, net seidel coefficients and geometric encircled energy, within FoV 2mm }
\label{fig:final_design_errors}
\end{figure*}
\begin{table}[h]
\begin{subtable}[h]{0.45\textwidth}
\centering
\resizebox{1\textwidth}{!}{%
\begin{tabular}{l | l | l}
\textbf{Item} & \textbf{Value} & \textbf{Comments} \\
\hline \hline
{ $r_{0}^{g}$} & { 0.204 mm} & { scaled fried parameter} \\ \hline
{ $\bar{\lambda}$} & { 0.5$\mu$ m} & { central wavelength} \\ \hline
{D} & {25 m} & { aperture size} \\ \hline
{ d} & { 35.355 mm} & { pupil stop size} \\ \hline
{height scaling} & {2 $\mu$m} & { $\Big(\frac{d}{D}\Big)^2$} \\ \hline
{h\_PS00} & {0.5 mm} & { H\_PS00 = 0.25 km actual height} \\ \hline
{ h\_PS01} & { 22 mm} & { H\_PS01 = 11 km, actual height} \\ \hline
{ h\_PS02} & { 100 mm} & { H\_PS02 = 50 km, actual height} \\ \hline
{ L} & { 166 mm} & { \begin{tabular}[c]{@{}c@{}}Total length of atmosphere, \\ corresponds to 83 km.\end{tabular}}
\end{tabular}}
\caption{}
\label{tab:phase_screen_fried_parameter}
\end{subtable}
\hfill
\begin{subtable}[h]{0.45\textwidth}
\centering
\resizebox{1\textwidth}{!}{%
\begin{tabular}{l | l | l}
\textbf{Item} & \textbf{Value} & \textbf{Comments} \\
\hline \hline
Wavelength range & 0.486 - 0.656$ \mu$m & Optical range \\ \hline
FOV: & 0.3 arcmin ($3.534^\circ$) & Scaled FoV in brackets \\ \hline
\begin{tabular}[c]{@{}c@{}}F\#\\ \\ Input/Output\end{tabular} & \begin{tabular}[c]{@{}c@{}}Input : 4.95\\ \\ Output : 51\end{tabular} & \\ \hline
Plate scale & 1.9 arcmin/mm & $\lambda\times$ F\#(output) \\ \hline
Phase Screen Thickness & 1 mm & manufacturing constraints \\ \hline
No. of screens & 3 & three layer turbulent model \\ \hline
Screen positions & 0.25/11/50 km & Flexible upto 83 km \\ \hline
Pupil Stop Size & 35.355 mm & Fixed, as discussed earlier \\ \hline
\end{tabular}}
\caption{}
\label{tab:instrument_table}
\end{subtable}
\caption{a) Parameters assigned to evaluate fried parameters for each phase screen, b) Important characterisations for instrument designing }
\end{table}
\begin{table}[]
\centering
\resizebox{1\textwidth}{!}{%
\begin{tabular}{|c|c|c|}
\hline
\textbf{Parameter} & \textbf{Value} & \textbf{Comment} \\ \hline
\begin{tabular}[c]{@{}c@{}}2nd screen range 16 - 44 mm (8-22 km);\\ {[}$r_0^{g}$, $D/r_0^{g}${]}\end{tabular} & {[} 0.201- 0.217 mm, 176 - 163 mm (8 - 22 km){]} & moving only PS01 screen \\ \hline
\begin{tabular}[c]{@{}c@{}}3nd screen range 44 - 166 mm (22 - 83 km);\\ {[} $r_0^{g}$, $D/r_0^{g}$ {]}\end{tabular} & {[} 0.186 - 0.216 mm, 190 - 163 mm (22 - 83 km){]} & moving only PS02 screen \\ \hline
$D/r_0^{g}$ over allowed range & 152 - 193 & \begin{tabular}[c]{@{}c@{}}193: 8/22 km (PS01/PS02)\\ 152: 22/83 km (PS01/PS02)\end{tabular} \\ \hline
Simulated telescope size & 21.8 - 27.79 & \\ \hline
\end{tabular}%
}
\caption{Variability in the simulated telescope size with changing phase screen positions}
\label{tab:phase_screen_variability}
\end{table}
\begin{table}[h]
\begin{subtable}[h]{0.35\textwidth}
\centering
\resizebox{1\textwidth}{!}{%
\begin{tabular}{l | l }
\textbf{Parameter} &\textbf{Value} \\
\hline \hline
\begin{tabular}[c]{@{}l@{}}Chromatic effect :\\ Pupil Shift (total)\end{tabular} & \begin{tabular}[c]{@{}l@{}}\textless 58$\mu$m (28\% $r_0$) (across 20 cm FoV)\\ \textless 23 $\mu$m (11\% $r_0$) (across 4 cm FoV)\end{tabular} \\ \hline
\begin{tabular}[c]{@{}l@{}}Footprint Shift per cell:\\ In footprint planes\end{tabular} & \begin{tabular}[c]{@{}l@{}}PS02 : \textless 19 $\mu$m ( 5.7\% $r_0$ )\\ PS01 : \textless 0.06 $\mu$m ( 0.02\% $r_0$ )\\ PS00 : \textless 0.34 $\mu$m ( 0.1\% $r_0$ )\end{tabular}
\end{tabular}}
\caption{}
\label{tab:table_fixed_2}
\end{subtable}
\hfill
\begin{subtable}[h]{0.65\textwidth}
\centering
\resizebox{1\textwidth}{!}{%
\begin{tabular}{l | l | l}
\textbf{Parameter} & \textbf{Formula} & \textbf{Value} \\
\hline \hline
Memory cell size on EMCCD plane & $1/\text{plate scale} \times \text{cell size} (0.0707^{\circ})$ & 2.32 mm \\ \hline
Total central width & $\lambda \times \text{output f-number( = 51)}$, $\lambda$ = 0.487 $\mu$ m & 30.3 $\mu$m \\ \hline
Total FoV & plate scale $\times$ No of pixel $\times$ pixel size & 15.56 arcmin \\ \hline
\begin{tabular}[c]{@{}l@{}}Total number of cells across along one axis (within CCD \\ area)\end{tabular} & FoV/(size of each cell) & 3.67 \\ \hline
Geometric encircled energy & 60 $\mu$m @487 nm & 16 $\mu$m ; 3.75$\times$3.75 pixels \\ \hline
\end{tabular}}
\caption{}
\label{tab:EMCCD_1}
\end{subtable}
\caption{a) Important parameters encapsulating best collimated beam, b) Technical details of memory cells falling on the sensor }
\end{table}
\subsection{Tolerance analysis}
\label{sec:SIMULATOR_tolerance}
\begin{figure}[h!]
\centerline{\includegraphics[width=0.33\textwidth]{montecarlo.png}}
\caption{\label{fig:monte_carlo} Monte carlo outcome from 2000 iterations using normal distribution. Diffraction limit spot size at 0.633 $\mu$m is 39.38 $\mu$m }
\end{figure}
Fig.~\ref{fig:monte_carlo} shows the outcome of Monte Carlo with iterations over 2000, using a normal distribution function. For an air disk radius of 40 $\mu$m at 0.633 $\mu$m, 90\% of the iterations are within an airy disk radius. This result corresponds to a confidence level of 1.5 sigmas (or corresponds to 87\% ). That means over 90\% of the time, the RMS radius stays within airy disc size.
\section{Optomechanical design}
\label{sec:opto_final_design}
Opto-mechanical design for SIMULATOR has been led by two core team members of IUCAA: Abhay and Bhushan. Abhay has expertise in SolidWorks, and Bhushan has PCB layout and design expertise. This section reviews the preliminary design and final opto-mechanical design of SIMULATOR.
Fig.~\ref{fig:preliminary_phasescreen} shows a preliminary design for all three turbulence phase screens adjusted on a base plate. All three bearings have been procured from SKF with sizes and thicknesses relevant to each turbulence phase screen. All three stands for phase screens have a rotating mechanism at the bottom of each plate. Additionally, two linear translation stages are set at the top of the base plate for PS01 and PS02. Relevant motors with enough high torque to balance out the heavyweight of the entire structure have been procured from robokits.
Fig.~\ref{fig:OM_design} shows the final optomechanical design of SIMULATOR. Starting from the source plane sitting on a stand, a ray of light first passes through a set of fixed collimation lenses (as one body system: Fixed CL1-3) and to the compensator CL4. After passing through three turbulent rotating phase screens, it will produce a collimated beam of pupil size 35.355 mm. The pupil plane has been placed within PS00 (not visible in this picture). The final pupil location has been adjusted (a little further away from PS00 due to space constraints), and thus accordingly, size has been adapted for the pupil plane. Adjustable lens IL1 (independent from the rest of the lenses) and three fixed lens systems (IL2-4) are aligned to produce a slow converging beam. Last, five-fold mirrors M1-5 have been deployed to make the system compact and finally to the EMCCD. EMCCD holder can withstand small translation in the y-axis and z-axis for readjustment.
\begin{figure}
\centering
\begin{subfigure}[b]{0.48\linewidth}
\centering
\includegraphics[width=\linewidth]{preliminary_1.png}
\caption{\label{fig:preliminary_phasescreen}}
\end{subfigure}
\begin{subfigure}[b]{0.48\linewidth}
\centering
\includegraphics[width=\linewidth]{optomechanicaldesign_1.png}
\caption{\label{fig:OM_mirror}}
\end{subfigure}
\caption{(a) Preliminary design of all three phase screens adjusted on base plate, along with rotation and translation stages, (b) Schematic layout for proposed mirror design for optimal space coverage}
\label{fig:roc_curve}
\end{figure}
\begin{figure}[h!]
\centerline{\includegraphics[angle=0,width=0.7\textwidth]{optomechanicaldesign_5.png}}
\caption{\label{fig:OM_design} Optomechanical design of SIMULATOR }
\end{figure}
\section{Assembly, integration and testing (AIT)}
\label{sec:AIT}
Fig.~\ref{fig:stage_testing} shows the setup for in-house made translation stages for three turbulent phase screens fitted over a flat stage. Motors can be read through an in-house hardware element called white dwarf (made by Bhushan and tested by Rani, instrumentation lab, IUCAA). This element will read user commands from the PC, act as an inter-mediator and pass signals in step voltages to the motor with the best accuracy. Hardware and software integration of the motors are already in place.
\begin{figure}[h!]
\centerline{\includegraphics[width=0.6\textwidth]{stage_testing.png}}
\caption{\label{fig:stage_testing} Integration and testing of translation stages (both hardware and software) of three turbulent phase screen.}
\end{figure}
\subsection{Electronics design}
\label{sec:electronics_final_design}
Fig.~\ref{fig:OM_source} shows a final 3d printed design for source plane box. The box's front end is equipped with a 5-micron hole (as an example), and just behind that (inside the box), a target has been placed of 2000 lpi Ronchi ruling, which corresponds to 7 microns roughly. The back end of the box is equipped with a PCB board
controlled by a USB connection to a computer. GUI has been designed through the MATLAB platform to control LEDs. Fig.~\ref{fig:OM_PCB} shows the circuit diagram layout for all 5 LED's covering wavelength bands from 520 - 640 nm. They have been aligned symmetrically with the central LED $L_0$ to affect the target ruling uniformly. For the experiment purpose, one LED will be used at a time. GUI for LED control has been produced and will be integrated with the rest of the instrument control (shown in fig.~\ref{fig:GUI}).
\begin{figure}
\centering
\begin{subfigure}[b]{0.48\linewidth}
\centering
\includegraphics[width=\linewidth]{optomechanicaldesign_4.png}
\caption{\label{fig:OM_source}a) 3D printed design of source plane box, b) Target: ronchi ruling 2000 lpi and c) Dimensions of source plane box}
\end{subfigure}
\begin{subfigure}[b]{0.48\linewidth}
\centering
\includegraphics[width=\linewidth]{optomechanicaldesign_6.png}
\caption{\label{fig:OM_PCB} a) Circuit diagram, b) Layout, b.1) LED layout and c) PCB design of LED's control board}
\end{subfigure}
\caption{Electronics circuit boards setup for SIMULATOR}
\end{figure}
\subsection{Graphic user interface}
\label{sec:GUI}
Fig.~\ref{fig:GUI} shows a user interfacing window for complete automation of SIMULATOR. The current version can control commands for LEDs, background luminosity and site parameters. However, work is still in progress, and the final design will also incorporate commands for controlling EMCCD.
\begin{figure}[h!]
\centerline{\includegraphics[width=0.5\textwidth]{GUI.png}}
\caption{\label{fig:GUI} Graphic user interface (GUI) window for complete automation of SIMULATOR }
\end{figure}
\acknowledgments
|
3,212,635,537,431 | arxiv | \section{Introduction}
The use of group theory to study bound and scattering states has found wide
application in physics\cite{intro}. A common starting point is to identify a
spectrum generating algebra or SGA for the problem of interest. This is
possible when the Hamiltonian can be expressed in terms of generators of
some algebra $G$. Representation theory then can be used to identify exactly
solvable limits of the theory, which can then be translated to explicit
forms of the Hamiltonian\cite{prldk}. Connections between bound state or
scattering state problems and representations of compact and non-compact
groups are well understood. The remaining category, having to do with band
structure and periodic potentials, was suggested initially in
\cite{gursey}, but remained an open problem until recently. In \cite{prllk}
it was shown that dynamical symmetry techniques and representation
theory can be used to solve band structure problems. In
particular, for the Scarf Hamiltonian,
band structure arises when one uses the complementary
series of the projective representations of $su(1,1)$ (and $so(2,2)$ for the
extended Scarf potential). Further, these dynamical symmetries allowed
simple evaluation of the transfer matrix and dispersion relations. In the
present study, we consider the band structure problem associated with the
Lam\'e equation, which is not a dynamical symmetry situation, but more
generally that of a SGA. We discuss several realizations of the Lam\'e
equation and its relation to $su(2)$ and $su(1,1)$, including a discussion
of dynamical symmetry limits, generators, dispersion relation and transfer
matrix.
While the Lam\'e equation at first might seem to be an obscure differential
equation, it does find surprisingly wide application in physics. For
instance, it has been shown that if one uses the periodic potentials in
super--symmetric quantum mechanics, the super partners of the Lam\'{e}
potentials are distinctly different from the original except for $n=1$ which
is not self-isospectral. This provides new solvable periodic potentials\cite
{a}. The Lam\'{e} equation also appears in the topics ranging from solitons
to exactly-solvable models. This includes associations with solutions of the
periodic KdV equation\cite{novikov}, BPS monopoles\cite{sutcliffe},
sphaleron solutions of the (1+1)-dimensional abelian Higgs model\cite
{brihaye}, sine-Gordon solitons\cite{Liang}, as well as relations to
Calogero-Moser systems\cite{enolskii}. The analysis of this equation has
many group theoretical aspects as well. For instance, the Lam\'{e} equation
can be written in terms of the composition of two first-order matrix
operators, where the coefficients of which satisfy so(3) Nahm's equation\cite
{b}. It also arises naturally in the context of $su(2)$ when one tries to
separate Laplace's equation in certain coordinate systems\cite{miller,patera}
as well as in the general classification of Lie algebraic potentials\cite
{karman} and in quasi-exactly solvable $sl(2)$ models\cite{turbiner}. The
discussion of band structure in the context of group theory is more recent
however\cite{gursey}, although the system was not solved and the transfer
matrix was not determined.
The Jacobian form of the Lam\'e equation is
\begin{equation}
-\frac{d^{2}}{dx^{2}}+\kappa ^{2}\ell (\ell +1){\rm sn}^{2}x={\cal E}.
\end{equation}
We will view this as a Schr\"odinger equation with mass $M=1/2$ when $x
$ is real valued. This equation was first studied group
theoretically in \cite{patera}. The elliptic functions sn$\alpha =$sn$(\alpha |\kappa )$,
cn$\alpha ={\rm cn(}\alpha |\kappa )$, and ${\rm dn}\alpha ={\rm dn(}\alpha
|\kappa )$ are doubly-periodic functions in the complex plane, of modulus $
\kappa $, where $0\leq \kappa \leq 1$. We will omit the modulus except in
functions where it differs from $\kappa $. The complementary modulus is
defined as $\kappa ^{\prime }=(1-\kappa ^{2})^{1/2}$. The periods of the
Jacobi elliptic functions are related to the complete elliptic integrals $
K=(\pi /2)F(1/2,1/2,1;\kappa ^{2})$ and $K^{\prime }=(\pi
/2)F(1/2,1/2,1;\kappa ^{\prime 2})$, where $F$ is the hypergeometric
function, by:
\begin{equation}
\begin{array}[t]{ll}
{\rm sn}\,(\alpha +\tau )={\rm sn}\,\alpha \qquad & \tau =2iK^{\prime
},\quad 4K+4iK^{\prime },\quad 4K \\
{\rm cn}\,(\alpha +\tau )={\rm cn}\,\alpha & \tau =4iK^{\prime },\quad
2K+2iK^{\prime },\quad 4K \\
{\rm dn}\,(\alpha +\tau )={\rm dn}\,\alpha & \tau =4iK^{\prime },\quad
4K+4iK^{\prime },\quad 2K
\end{array}
\end{equation}
In the limit $\kappa =1$, the real period related to $K$ becomes infinite,
and the functions are no longer periodic on the real axis. \ Along the real
axis, the potential ${\rm sn}^{2}x$ is periodic and bounded, while along the
imaginary axis, it has periodic singularities of the type $1/x^{2}.$
Examples are shown in Fig. 1 for $\kappa ^{2}=1/2$ for potentials $V(x)=
{\rm sn}^{2}(ax+b)$, with several choices of $a$ and $b$.
The Lam\'e equation has also been related to the band structure problem
associated with the P\"oschl-Teller potential, $V_{pt}(x)\sim 1/\cosh ^{2}x$
\cite{sutherland,prllk}, by using the expansions for the elliptic
functions in terms of trigonometric functions found in the exercises of
Whittaker and Watson\cite{whitwat}. Specifically, one can start with the 1-d
band structure problem
\begin{equation}
H=-\frac{d^{2}}{dx^{2}}+\sum_{k=-\infty }^{\infty }\frac{g}{\cosh
^{2}((x-ka)/x_{0})}.
\end{equation}
If we write the coupling constant as
\begin{equation}
g=-\frac{\ell (\ell +1)}{x_{0}^{2}}
\end{equation}
we can make the coordinate transformation
\begin{equation}
x=i\left( \frac{\pi x_{0}}{2K}\right) z-\left( \frac{a+i\pi x_{0}}{2}\right)
\end{equation}
which transforms the Hamiltonian to the Lam\'e equation. In this article, we
first discuss the relation of $su(2)$ Hamiltonians to the Lam\'e equation,
followed by a discussion of band edges. Next we consider $su(1,1)$ and the
relation to scattering states in the dynamical symmetry limits.
Finally we derive the transfer matrix and dispersion relation for the Lam\'e
equation followed by the relation to the group theoretical SGA\ Hamiltonians.
\section{$su(2)$ Realizations of the Lam\'e Equation}
\subsection{Sphero-conal Coordinates for $su(2)$}
\bigskip
In order to realize the Lam\'e equation from $su(2)$\ or $su(1,1)$, we will
use coordinate systems defined in terms of Jacobi elliptic functions. These
are often refered to as sphero-conal\cite{Arscott} or conical\cite{miller}
coordinates. One can find variations that have been considered in the past,
and not all provide a good description for all values of the modulus $\kappa
.$ Consider first the mapping:
\begin{equation}
x=\kappa \,{\rm sn}\,\alpha \,{\rm sn}\,\beta ,\,y=i\frac{\kappa }{\kappa
^{\prime }}\,{\rm cn}\,\alpha \,{\rm cn}\,\beta ,\,z=\frac{1}{\kappa
^{\prime }}\,{\rm dn}\,\alpha \,{\rm dn}\,\beta ,
\end{equation}
The ranges of the angles $\alpha ,\beta $ are defined in terms of the
elliptic integrals $K$ and $K^{\prime }$, so that they depend on the value
of $\kappa $. Specifically $-2K<\alpha <2K$ and $\,K\leq \beta
<K+2iK^{^{\prime }}$ where $\kappa ^{\prime 2}=1-\kappa ^{2}$, and $
x^{2}+y^{2}+z^{2}=1.$ The contours of constant $\alpha $ and $\beta $ cover
the sphere as illustrated in Fig. 2(a). For $\kappa =1/2$, four typical
contours (for $\alpha =\pm K/2$ and $\beta =K+iK^{\prime }/2,K+3iK^{\prime
}/2$)\ are shown in\ Fig. 2(b). This parametrization can be used to
construct the generators of $su(2)$. A direct computation of the generators
yields the realization (I) given in Table 1. These
generators satisfy the commutation relations in the form $\left[ L_{a},L_{b}
\right] =\epsilon _{abc}L_{c}$ with Casimir invariant
\begin{eqnarray}
C_{2} &=&L_{x}^{2}+L_{y}^{2}+L_{z}^{2} \nonumber \\
&=&\frac{1}{\kappa ^{2}({\rm sn}^{2}\,\alpha -{\rm sn}\,^{2}\beta )}\left(
\frac{\partial ^{2}}{\partial \alpha ^{2}}-\frac{\partial ^{2}}{\partial
\beta ^{2}}\right) .
\end{eqnarray}
The $su(2)$ algebras we discuss here have Casimir invariants which have
expectation value $-\ell (\ell +1)$. We now consider which types of
quadratic Hamiltonians of the form
\begin{equation}
H=\sum_{j}\eta _{j}L_{j}^{2}
\end{equation}
can be constructed which result in Schr\"odinger equations in $\alpha $ and $
\beta .$ Specifically, when the coordinate representation of $H$ contains
differential opertors only of the type $\partial ^{2}/\partial \alpha ^{2}$
and $\partial ^{2}/\partial \beta ^{2}$, one can use $H$ together with $C_{2}
$ to perform a separation of variables resulting in two independent Lam\'e
equations. We find the three forms in the top right of Table 1, which are
simply related through the Casimir operator. To demonstrate the separation,
consider $H_{3}$,
\begin{eqnarray}
H_{3} &=&L_{z}^{2}+\kappa ^{2}L_{y}^{2} \nonumber \\
&=&\frac{1}{({\rm sn}^{2}\,\beta -{\rm sn}\,^{2}\alpha )}\left( {\rm sn}
^{2}\,\beta \frac{\partial ^{2}}{\partial \alpha ^{2}}-{\rm sn}^{2}\,\alpha
\frac{\partial ^{2}}{\partial \beta ^{2}}\right) , \label{ham1}
\end{eqnarray}
and require it to be a constant of the motion with eigenvalue $H_{3}={\cal E}
$. Using basis states denoted by the direct product, $\Psi (\alpha ,\beta
)=\psi (\alpha )\phi (\beta )$, we arrive at two identical Lam\'{e}
equations with identical eigenvalue:
\begin{eqnarray}
\ \left[ -\frac{d^{2}}{d\alpha ^{2}}+\kappa ^{2}\ell (\ell +1)\,{\rm sn}
^{2}\alpha \right] \psi (\alpha ) &=&{\cal E}\psi (\alpha ), \\
\ \left[ -\frac{d^{2}}{d\beta ^{2}}+\kappa ^{2}\ell (\ell +1)\,{\rm sn}
^{2}\beta \right] \phi (\beta ) &=&{\cal E}\phi (\beta ). \nonumber
\end{eqnarray}
Thus solution of the eigenvalue problem for $H_{3}=L_{z}^{2}+\kappa
^{2}L_{y}^{2}$ yields the single valued solutions to these equations, which
will correspond to the band edges. One should keep in mind that while $
\alpha $ is defined on the real axis, $\beta $ is complex. Also, while the
Hamiltonians $H_{k}$ are well defined in their algebraic form, the
generators and coordinates have singularities in both $\kappa =0$ and $
\kappa =1$ limits. Hence we consider a different realization below.
To reduce these Hamiltonians to more conventional Schr\"odinger
equations in $\alpha$ and $\beta$, we would
like the coordinates to be defined along the real axis. To do so, we make a
shift (i) $\beta \rightarrow \beta +K+iK^{^{\prime }}$ followed by (ii) $
\beta \rightarrow -i\beta $. This results in the transformations:
\begin{eqnarray}
{\rm sn}(\beta |\kappa ) &\rightarrow &\frac{1}{\kappa }{\rm dn}(\beta
|\kappa ^{\prime }), \nonumber \\
{\rm cn}(\beta |\kappa ) &\rightarrow &\frac{\kappa ^{\prime }}{i\kappa }
{\rm cn}(\beta |\kappa ^{\prime }), \\
{\rm dn}(\beta |\kappa ) &\rightarrow &\kappa ^{\prime }{\rm sn}(\beta
|\kappa ^{\prime }). \nonumber
\end{eqnarray}
The new coordinate system is no longer singular, and is parameterized as
\begin{equation}
x={\rm sn}\,\alpha \,{\rm dn}(\beta |\kappa ^{\prime }),\,y={\rm cn}\,\alpha
\,{\rm cn}(\beta |\kappa ^{\prime }),\,z={\rm dn}\,\alpha \,{\rm sn}(\beta
|\kappa ^{\prime }),
\end{equation}
where $-2K<\alpha <2K,\,-K^{^{\prime }}\leq \beta <K^{^{\prime }}$ are now
both real valued. Contours of constant $\alpha $ and $\beta $ are shown in
Fig. 2(c) for $\alpha =\pm K/2$ and $\beta =\pm K^{\prime }/2$. If we
construct the generators of $su(2)$ in this basis, we obtain the realization
(II)\ given in Table 1. The Casimir invariant now has the form
\[
C_{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}=\frac{-1}{\kappa ^{2}{\rm cn}^{2}\alpha
+\kappa ^{\prime 2}{\rm cn}^{2}(\beta |\kappa ^{\prime })}\left( \frac{
\partial ^{2}}{\partial \alpha ^{2}}+{\rm \ }\frac{\partial ^{2}}{\partial
\beta ^{2}}\right)
\]
The quadratic Hamiltonians which give rise to Schr\"odinger equations when
combined with $C_{2}$ are given in the lower right of Table 1. Consider
again $H_{3}=L_{z}^{2}+\kappa ^{2}L_{y}^{2}$. If we set $H_{3}={\cal E}$ and
$C_{2}=-\ell (\ell +1)$, we obtain
\begin{eqnarray}
\left[ -\frac{d^{2}}{d\alpha ^{2}}+\kappa ^{2}\ell (\ell +1){\rm sn}
^{2}\alpha \right] \psi (\alpha ) &=&{\cal E}\psi (\alpha ), \\
\ \left[ -\frac{d^{2}}{d\beta ^{2}}+\kappa ^{\prime 2}\ell (\ell +1){\rm sn}
^{2}(\beta |\kappa ^{\prime })\right] \phi (\beta ) &=&(\ell (\ell +1)-{\cal
E})\phi (\beta ).
\end{eqnarray}
In this case the coordinates are both real valued. For integer $\ell $, and
both potentials are non-negative, \ we see that $0\leq {\cal E}\leq \ell
(\ell +1).$ Further the ground state of one Hamiltonian corresponds the the
highest energy state of the other, and vice-versa. (Similar equations
are obtained if we use $H_{2}$ or $H_{1}.$) Before we discuss some of the
general properties of these equations, we consider first the dynamical
symmetry limits which are when $\kappa =0$ or $1$.
\subsection{$\kappa =0$ Limit}
In the limit $\kappa =0$, and $H_{3}=L_{z}^{2}=$ $m^{2}$, so that ${\cal E}
=m^{2}$, and the coordinate Hamiltonians are:
\begin{eqnarray}
\ -\frac{d^{2}}{d\alpha ^{2}}\psi (\alpha ) &=&{\cal E}\psi (\alpha ), \\
\ \left[ -\frac{d^{2}}{d\beta ^{2}}-\frac{\ell (\ell +1)}{\cosh ^{2}\beta }
\right] \phi (\beta ) &=&-{\cal E}\phi (\beta ).
\end{eqnarray}
The generators in this limit have the simpler form:
\begin{eqnarray}
I_{\pm } &=&\ e^{\pm i\alpha }\left[ \mp \cosh \beta \ \frac{\partial }{
\partial \beta }+i\sinh \beta \frac{\partial }{\partial \alpha }\right]
\nonumber \\
I_{3} &=&\ -i\frac{\partial }{\partial \alpha }
\end{eqnarray}
The first equation is free motion, while the second corresponds to bound
states of the P\"oschl-Teller potential. For the discrete representations of $
su(2)$, one obtains the bound state spectrum $\ -{\cal E}=-m^{2}.$ The
wavefunctions are of the form:
\begin{equation}
\Psi (\alpha ,\beta ;\kappa =0)\sim P_{\ell }^{m}(\tanh \beta )\exp (\pm
im\alpha ),\qquad m=-\ell ,...,\ell ;\quad \ell =0,1,2,...
\end{equation}
with $-\pi <\alpha <\pi ,\,-\infty <\beta <\infty .$
\subsection{$\kappa =1$ Limit}
In the limit $\kappa =1$, $H_{3}=L^{2}-L_{x}^{2},$ the equations reduce to
\begin{eqnarray}
\ \left[ -\frac{d^{2}}{d\alpha ^{2}}-\frac{\ell (\ell +1)}{\cosh ^{2}\alpha }
\right] \ \psi (\alpha ) &=&[{\cal E}-\ell (\ell +1)]\psi (\alpha )={\cal E}
^{\prime }\psi (\alpha ) \\
-\frac{d^{2}}{d\beta ^{2}}\phi (\beta ) &=&[\ell (\ell +1)-{\cal E}]\phi
(\beta )=-{\cal E}^{\prime }\phi (\beta )
\end{eqnarray}
The generators are the same as in the $\kappa =0$ limit, with $\alpha $ and $
\beta $ interchanged. The second equation is now free motion, while the
first corresponds to bound states of the P\"oschl-Teller potential with a
shifted eigenvalue. For the discrete representations of $su(2)$, one obtains
the bound state spectrum ${\cal E}^{\prime }=-m^{2}.$ The wavefunctions are
of the form:
\begin{equation}
\Psi (\alpha ,\beta ;\kappa =1)\sim P_{\ell }^{m}(\tanh \alpha )\exp (\pm
im\beta ),\qquad m=-\ell ,...,\ell ;\quad \ell =0,1,2,...
\end{equation}
with $-\infty <\alpha <\infty ,\,-\pi <\beta <\pi .$ \
\subsection{ Band Edges and $su(2)$}
For general values of $\kappa $, the Lam\'e Hamiltonians are periodic (as seen
in Fig. 1), and will have band structure. As the discrete representations of
$su(2)$ correspond to single-valued wavefunctions, the discrete
representations can at most describe the band edges. Since we have an
algebraic realization of the Hamiltonians whose spectrum corresponds to the
Lam\'e equations, we can obtain the eigenvalues of the Hamiltonian $H_{k}$
(and hence the band edges) by a direct diagonalization in the spherical
harmonic basis $\left| \ell m\right\rangle $, which can then be related to
the results for the elliptic basis through the coordinate transformations.
The resulting functions are
doubly periodic solutions of Lam\'e's equation known as Lam\'e
polynomials\cite{Arscott}. These polynomials are of the form
sn$^a x$cn$^b x$dn$^c x F_p($sn$^2 x )$, where $a,b,c=0,1$ and
$a+b+c+2p=\ell$. Here $F_p(z)$ is a polynomial in $z$ of order
$p$.
A discussion of these functions can be found in \cite{bateman,whitwat,Arscott}.
For $\ell =1$, there are three eigenstates of $H_{3}$ given by
\begin{eqnarray}
\Psi _{1}(\alpha ,\beta ;\kappa ) &=&\left| 10\right\rangle \sim {\rm sn}
\,\alpha {\rm dn}(\beta |\kappa ^{\prime }) \nonumber \\
\Psi _{2}(\alpha ,\beta ;\kappa ) &=&\frac{1}{\sqrt{2}}\left[ \left|
11\right\rangle +\left| 1-1\right\rangle \right] \sim {\rm \ cn}\,\alpha
{\rm cn}(\beta |\kappa ^{\prime }) \\
\Psi _{3}(\alpha ,\beta ;\kappa ) &=&\frac{1}{\sqrt{2}}\left[ \left|
11\right\rangle -\left| 1-1\right\rangle \right] \sim {\rm \ dn}\,\alpha
{\rm sn}(\beta |\kappa ^{\prime }) \nonumber
\end{eqnarray}
with eigenvalues $E_{1}=1+\kappa ^{2}$, $E_{2}=1$, $E_{3}=\kappa ^{2}$. In
the dynamical symmetry limits, these three states reduce to those discussed
above.
For $\ell =2$, we have
\begin{eqnarray}
\Psi _{1}(\alpha ,\beta ;\kappa ) &=&\frac{1}{\sqrt{2}}\left[ \left|
22\right\rangle -\left| 2-2\right\rangle \right] \sim {\rm cn}\,\alpha {\rm
dn}\,\alpha {\rm sn}(\beta |\kappa ^{\prime }){\rm cn}(\beta |\kappa
^{\prime }) \nonumber \\
\Psi _{2}(\alpha ,\beta ;\kappa ) &=&\frac{1}{\sqrt{2}}\left[ \left|
21\right\rangle +\left| 2-1\right\rangle \right] \sim {\rm sn}\,\alpha {\rm
cn}\,\alpha {\rm cn}(\beta |\kappa ^{\prime }){\rm dn}(\beta |\kappa
^{\prime }) \nonumber \\
\Psi _{3}(\alpha ,\beta ;\kappa ) &=&\frac{1}{\sqrt{2}}\left[ \left|
21\right\rangle -\left| 2-1\right\rangle \right] \sim {\rm sn}\,\alpha {\rm
dn}\,\alpha {\rm sn}(\beta |\kappa ^{\prime }){\rm dn}(\beta |\kappa
^{\prime }) \\
\Psi _{4}(\alpha ,\beta ;\kappa ) &\sim &\left| 22\right\rangle +\left|
2-2\right\rangle +\sqrt{\frac{2}{3}}\frac{2f_{+}(\kappa )-1-\kappa ^{2}}{
1-\kappa ^{2}}\left| 20\right\rangle \nonumber \\
&\sim &(1-f_{+}(\kappa ){\rm sn}\,^{2}\alpha )(1-f_{-}(\kappa ^{\prime })
{\rm sn}\,^{2}(\beta |\kappa ^{\prime })) \nonumber \\
\Psi _{5}(\alpha ,\beta ;\kappa ) &\sim &\left| 22\right\rangle +\left|
2-2\right\rangle +\sqrt{\frac{2}{3}}\frac{2f_{-}(\kappa )-1-\kappa ^{2}}{
1-\kappa ^{2}}\left| 20\right\rangle \nonumber \\
&\sim &(1-f_{-}(\kappa ){\rm sn}\,^{2}\alpha )(1-f_{+}(\kappa ^{\prime })
{\rm sn}\,^{2}(\beta |\kappa ^{\prime })) \nonumber
\end{eqnarray}
with eigenvalues
\begin{eqnarray}
E_{1} &=&1+\kappa ^{2},\qquad E_{2}=4+\kappa ^{2},\qquad E_{3}=1+4\kappa
^{2},\qquad \\
E_{4} &=&2f_{+}(\kappa ),\qquad E_{5}=\ 2f_{-}(\kappa ),\qquad
\end{eqnarray}
where $f_{\pm }(\kappa )=\left( 1+\kappa ^{2}\pm \sqrt{1-\kappa
^{2}\kappa ^{\prime 2}}\right) $.
In general there will
be a relation between the eigenstates $\left| \ell
m\right\rangle $ of the $2\ell+1$ band edges and
products of Lam\'e polynomials. The band edges for $\ell =1,2$ are shown in
Fig. 3 as a function of $\kappa ^{2}$. The bands are indicated by the shaded
regions. The dashed line indicates the height of the potential. One can see
that the bands merge at $\kappa =0$, and pass to the bound and scattering
states of the P\"oschl-Teller potential for $\kappa =1$.
\section{su(1,1) Realizations of the Lam\'e Equation}
\subsection{Elliptic Parametrization of su(1,1)}
In order to discuss the eigenstates in the band, and develop the dispersion
relation, we require more general representations. In the spirit of recent
work on the Scarf potential where it was shown that one can use a $su(1,1)$
dynamical symmetry to analytically solve for the dispersion relation\cite
{prllk}, states and transfer matrix, we consider transforming our generators
to $su(1,1)$. Consider the transformation of our previous coordinates
associated with $x\rightarrow -ix,\,y\rightarrow -iy:$
\begin{equation}
x=-i\kappa \,{\rm sn}\,\alpha \,{\rm sn}\,\beta ,\,y=\frac{\kappa }{\kappa
^{\prime }}\,{\rm cn}\,\alpha \,{\rm cn}\,\beta ,\,z=\frac{1}{\kappa
^{\prime }}\,{\rm dn}\,\alpha \,{\rm dn}\,\beta ,
\end{equation}
where $0\leq \alpha <4K,\,0\leq \beta <iK^{^{\prime }}$\cite{patera}.
This now parametrizes the hyperbolic surface $z^{2}-x^{2}-y^{2}=1$
illustrated in Fig. 4(a) for $\kappa ^{2}=1/2$. Contours are shown for
selected values of $\alpha $ and $\beta $ in Fig. 4(b). The generators of $
su(1,1)$ algebra are given in Table 2 as realization (I), and satisfy the
commutation relations $\left[ L_{x},L_{y}\right] =-L_{z}$, $\left[
L_{y},L_{z}\right] =L_{x}$, and $\left[ L_{z},L_{x}\right]
=L_{y}.$ The
Casimir invariant now has the form
\begin{equation}
C_{2}=L_{z}^{2}-L_{x}^{2}-L_{y}^{2}=\frac{-1}{\kappa ^{2}({\rm sn}
^{2}\,\alpha -{\rm sn}^{2}\,\beta )}\left( \frac{\partial ^{2}}{\partial
\alpha ^{2}}-\frac{\partial ^{2}}{\partial \beta ^{2}}\right)
\end{equation}
As in the $su(2)$ case, there are three forms of bilinear Hamiltonians which
lead to Schr\"odinger equations in $\alpha $ and $\beta $. These are given in
the top right of Table 2. If we choose \ $H_{3}=L_{z}^{2}-\kappa
^{2}L_{x}^{2}={\cal E}$, together with the Casimir invariant $C_{2}=-\ell
(\ell +1)$, we obtain the decoupled Lam\'e equations:
\begin{eqnarray}
\ \left[ -\frac{d^{2}}{d\alpha ^{2}}+\kappa ^{2}\ell (\ell +1){\rm sn}
^{2}\alpha \right] \psi (\alpha ) &=&-{\cal E}\psi (\alpha ), \label{sua} \\
\ \left[ -\frac{d^{2}}{d\beta ^{2}}+\kappa ^{2}\ell (\ell +1){\rm sn}
^{2}\beta \right] \phi (\beta ) &=&-{\cal E}\phi (\beta ). \label{sub}
\end{eqnarray}
Again similar results are obtained if we use $H_{1}$ or $H_{2}$, the only
difference arising in the definition of the eigenvalue. Note that the range
of $\beta $ is along the imaginary axis. This realization is
problematic in the $\kappa =0$ and $\kappa =1$ limits since the
generators become singular. Consequently we consider a slightly different realization of
$su(1,1)$ below.
\subsection{Another su(1,1) Realization}
While we do not have a coordinate system which avoids the singularities at $
\kappa =1$, it is possible to at least allow a study of the scattering
states when $\kappa =0.$ To do this we make the transformations (i) $\beta
\rightarrow \beta +K+iK^{^{\prime }}$ followed by (ii) $\beta \rightarrow
-i\beta $. As a result:
\begin{eqnarray}
{\rm sn}(\beta |\kappa ) &\rightarrow &\frac{1}{\kappa }{\rm dn}(\beta
|\kappa ^{\prime }), \nonumber \\
{\rm cn}(\beta |\kappa ) &\rightarrow &\frac{\kappa ^{\prime }}{i\kappa }
{\rm cn}(\beta |\kappa ^{\prime }), \nonumber \\
{\rm dn}(\beta |\kappa ) &\rightarrow &\kappa ^{\prime }{\rm sn}(\beta
|\kappa ^{\prime }).
\end{eqnarray}
This results in coordinates\cite{patera}
\begin{equation}
x=-i\,{\rm sn}\,\alpha \,{\rm dn}(\beta |\kappa ^{\prime }),\,y=-i\,{\rm cn}
\,\alpha \,{\rm cn}(\beta |\kappa ^{\prime }),\,z={\rm dn}\,\alpha \,{\rm sn}
(\beta |\kappa ^{\prime }),
\end{equation}
where $0\leq \alpha <4K,\,-iK\leq \beta <-iK+K^{^{\prime }}$, also
satisfying $z^{2}-x^{2}-y^{2}=1$. Typical contours for this parametrization
are shown in Fig. 4(c). The generators are given in Table 2 as realization
(II) and satisfy $\left[ L_{x},L_{y}\right] =-L_{z}$, $\left[ L_{y},L_{z}
\right] =L_{x}$, and $\left[ L_{z},L_{x}\right] =L_{y}$ with Casimir
invariant
\begin{equation}
C=L_{z}^{2}-L_{x}^{2}-L_{y}^{2}=\ \frac{1}{\kappa ^{2}{\rm cn}^{2}\alpha
+\kappa ^{\prime 2}{\rm cn}^{2}(\beta |\kappa ^{\prime })}\left( \frac{
\partial ^{2}}{\partial \alpha ^{2}}+\frac{\partial ^{2}}{\partial \beta ^{2}
}\right)
\end{equation}
\qquad The Hamiltonians which result in seperable Hamiltonians are given in
the bottom right of Table 2. Using $H_{1}=L_{x}^{2}+\kappa ^{\prime
2}L_{y}^{2}={\cal E}\ $and $C_{2}=-\ell (\ell +1)$ leads to the two Lam\'{e}
equations:
\begin{eqnarray}
\ \left[ -\frac{d^{2}}{d\alpha ^{2}}+\kappa ^{2}\ell (\ell +1){\rm sn}
^{2}\alpha \right] \psi (\alpha ) &=&{\cal E}\psi (\alpha ), \nonumber \\
\ \left[ -\frac{d^{2}}{d\beta ^{2}}+\kappa ^{\prime 2}\ell (\ell +1){\rm sn}
^{2}(\beta |\kappa ^{\prime })\right] \phi (\beta ) &=&(\ell (\ell +1)-{\cal
E})\phi (\beta ).
\end{eqnarray}
It should be kept in mind that $\beta $ is still complex valued. Never the
less this realization allows an analysis of the $\kappa =0$ dynamical
symmetry.
\subsection{$\kappa =0$ Limit}
\bigskip
This realization is not singular in the $\kappa =0$ limit (although it is in
the $\kappa =1$ case). Taking $\kappa =0$, and shifting $\beta $ to be on
the real axis by $\beta =\theta -i\pi /2$, we find
\begin{eqnarray}
L_{\pm } &=&\pm e^{\pm i\alpha }\left( \mp \sinh \theta \frac{\partial }{
\partial \theta }+i\cosh \theta \frac{\partial }{\partial \alpha }\right)
\nonumber \\
L_{z} &=&-\ \frac{\partial }{\partial \alpha }.
\end{eqnarray}
To recover the usual $su(1,1)$ commutation relations $[I_{z},I_{\pm }]=\pm
I_{\pm }$, $[I_{+},I_{-}]=-2I_{z}$, we then identify $I_{\pm }=\pm L_{\pm }$
and $I_{z}=iL_{z}$. It is convient to perform the
transformations: $\theta \rightarrow \theta -i\pi /2$, $\tanh \theta
\rightarrow \cos \theta $, followed by a similarity transformation $f(\theta
)=\sqrt{\sin \theta }$, and $\theta \rightarrow i\theta $. Then we have the
form:
\begin{eqnarray}
\ -\frac{d^{2}}{d\alpha ^{2}}\ \psi (\alpha ) &=&m^{2}\psi (\alpha ),
\nonumber \\
\ \left[ -\frac{d^{2}}{d\theta ^{2}}+\frac{m^{2}-1/4}{\sinh ^{2}\theta }
\right] \phi (\theta ) &=&-(\ell +\frac{1}{2})^{2}\phi (\theta ).
\end{eqnarray}
The principal series $\ell =-1/2+i\rho $ of the projective representations
of $su(1,1)$ now describe these scattering states, and the eigenfunctions
are of the form:
\begin{equation}
\Psi _{\ell }^{m}\sim \sqrt{i\sinh \theta }P_{\ell }^{m}(\cosh \theta
)e^{\pm im\alpha }\quad \qquad m\in \Re ,\ \ell =-1/2+i\rho ,\ \rho >0.
\end{equation}
\section{Band Structure of the Lam\'{e} Hamiltonian}
We now focus on the properties of the Lam\'e equation in the form of Eq. (1).
As we are interested in band structure, \ the eigenstates must satisfy
Bloch's theorem. For a potential which is periodic with period $a$, $
V(x+a)=V(x)$, the wavefunctions must be of the form
\begin{equation}
\Psi _{k}(x)=u_{k}(x)\exp [-ikx],
\end{equation}
where $k$ is the wavenumber and $u_{k}(x)$ has the periodicity of the
lattice: $u_{k}(x+a)=u_{k}(x)$. As the eigenstates $\Psi _{k}(x)$ are not
periodic, the doubly periodic solutions of the
Lam\'e equation do not play a role for energies in the band. Rather, we look
to the more general class of solutions expressed in terms of Jacobi theta
functions\cite{whitwat}. Starting with the Hamiltonian
\begin{equation}
H\psi =\left[ -\frac{d^{2}}{dx^{2}}+\kappa ^{2}\ell (\ell +1){\rm sn}
^{2}(x|\kappa )\right] \psi (x)={\cal E}\psi (x), \label{Lameham}
\end{equation}
the solutions for positive integer $\ell $ are given parametrically by
\begin{equation}
\psi (x)=\prod_{n=1}^{\ell }\left[ \frac{{\cal H}(x+\alpha _{n})}{\theta (x)}
e^{-xZ(\alpha _{n})}\right]
\end{equation}
where ${\cal H}$ and $\theta$ are theta functions, and
$\alpha _{1},\alpha _{2},...\alpha _{\ell }$ are constants
determined by the constraints:
\begin{eqnarray}
{\cal E} &=&\sum_{n=1}^{\ell }\frac{1}{{\rm sn}^{2}\alpha _{n}}-\left[ \sum_{n=1}^{\ell }
{\rm cn}\alpha _{n}{\rm dn}\alpha _{n}/{\rm sn}\alpha _{n}\right] ^{2}
\label{eq:const} \\
0 &=&\sum_{p=1}^{\ell }\frac{{\rm sn}\alpha _{p}{\rm cn}\alpha _{p}{\rm dn}
\alpha _{p}+{\rm sn}\alpha _{n}{\rm cn}\alpha _{n}{\rm dn}\alpha _{n}}{{\rm
sn}^{2}\alpha _{p}-{\rm sn}^{2}\alpha _{n}}\quad (p\neq n)
\end{eqnarray}
If this solution is not doubly periodic, a second solution is
\begin{equation}
\psi ^{^{\prime }}(x)=\prod_{n=1}^{\ell }\left[ \frac{{\cal H}(x-\alpha _{n})}{
\theta (x)}e^{xZ(\alpha _{n})}\right].
\end{equation}
We can then identify the dispersion relation by putting these
wavefunctions in Bloch form and by using the periodicity of the
theta functions to extract $u_k(x)$. We find
\begin{equation}
k({\cal E})=-i\sum_{n=1}^{\ell }Z(\alpha _{n}|\kappa ^{2})+\frac{\ell \pi }{2K}.
\label{eq:disp}
\end{equation}
We will start with the case of $\ell =1$, where simple anlytic
results can be obtained. We then derive the transfer matrix for
the general case of integer $\ell .$
\subsection{$ \ell =1$ Results}
Since there is only one parameter $\alpha ,$ the constraint equation is
simply
\begin{equation}
{\rm dn}^{2}\alpha ={\cal E}-\kappa ^{2}.
\end{equation}
The condition that the dispersion relation is real,
Re$Z(\alpha|\kappa^2)=0$, results in two energy bands given by
\begin{equation}
\kappa ^{2}\leq {\cal E}\leq 1,\quad 1+\kappa ^{2}\leq {\cal E}.
\end{equation}
These are shown in Fig. 3 (top). In the lower band, $\alpha $ has the form $
\alpha =K+i\eta $, where $\eta $ ranges from $K^{\prime }$ at ${\cal E}
=\kappa ^{2}$, to $0$ at ${\cal E}=1.$ In the upper band, $\alpha =i\eta $,
where $\eta $ ranges from $0$ at ${\cal E}=1+\kappa ^{2}$, to $K^{\prime }$
as ${\cal E}\rightarrow \infty $. This path traced out by the
parameter $\alpha$ as a function energy ${\cal E}$ is shown schematically in Fig. 5. The upper
and lower sides correspond to band gaps while the right and left
edges are the energy bands.
Using the specific forms of $\alpha $ for each band, the dispersion relation
becomes
\begin{equation}
k({\cal E})=\left\{
\begin{array}{ll}
-Z(\eta |\kappa ^{\prime 2})+\frac{\pi }{2K}(1-\frac{\eta }{K^{\prime }})+
\sqrt{\frac{({\cal E}-\kappa ^{2})(1-{\cal E)}}{1+\kappa ^{2}-{\cal E}}} &
\kappa ^{2}\leq {\cal E}\leq 1, \\
-Z(\eta |\kappa ^{\prime 2})+\frac{\pi }{2K}(1-\frac{\eta }{K^{\prime }})+
\sqrt{\frac{({\cal E}-\kappa ^{2}-1)({\cal E}-\kappa ^{2})}{{\cal E}-1}} &
1+\kappa ^{2}\leq {\cal E}
\end{array}
\right.
\end{equation}
This is plotted in Fig. 6 (a) for the case of $\ell=1$. The
momentum $k$ is plotted up to the edge of the Brillouin zone,
which is $k=\pi/2K$. In the figure we use $\kappa^2=1/2$, so
that the band edges are ${\cal E}=1/2$, 1 and 3/2. (The
analogous behavior for $\ell=2$ is indicated in Fig. 6(b).)
The solution of the Lam\'e equation in Bloch form is now
\begin{equation}
\psi _{k}(x)=\left\{
\begin{array}{ll}
\left[ \frac{H_{1}(x+i\eta )}{\Theta (x)}\exp (i\pi x/2K)\right] \exp (-ikx)
& \kappa ^{2}\leq {\cal E}\leq 1, \\
\left[ \frac{H(x+i\eta )}{\Theta (x)}\exp (i\pi x/2K)\right] \exp (-ikx) &
1+\kappa ^{2}\leq {\cal E}
\end{array}
\right. .
\end{equation}
The component of the wavefunction in square brackets can be checked to be
periodic with the periodicity of the direct lattice: $x\rightarrow x+2K.$
The dispersion relation displays the desired limits. One can see that $k(
{\cal E})\rightarrow 0$ as ${\cal E}\rightarrow \kappa ^{2}$, and $k({\cal E}
)\rightarrow \pi /2K$ as ${\cal E}\rightarrow 1.$ Further, as the modulus of
the elliptic function vanishes, $\kappa ^{2}\rightarrow 0$, the Hamiltonian
becomes that of a free system, and we find $k({\cal E})\rightarrow \sqrt{
{\cal E}}$ as desired. In the P\"oschl-Teller limit, $\kappa ^{2}\rightarrow 1$
, the band vanishes, and we find $k({\cal E})\rightarrow 0$. Similar
results hold for the upper band as well.
From the dispersion relation we can compute the group velocity and effective
mass. To do so, we use the relation between the zeta function and the
elliptic integrals of the first and second kind
\begin{equation}
Z(\alpha )=E(\alpha )-\frac{E(\kappa ^{2})}{K}\alpha
\end{equation}
where $E(\kappa ^{2})=\frac{\pi }{2}F(-1/2,1/2;1;\kappa ^{2})$ and $K=\frac{
\pi }{2}F(1/2,1/2;1;\kappa ^{2})$ are the complete elliptic integrals and $
E(\alpha )$ is incomplete. Then, the group velocity is given by:
\begin{equation}
\frac{1}{\nu }=\frac{dk({\cal E})}{d{\cal E}}
\end{equation}
or
\begin{equation}
\nu ({\cal E})=\frac{\ 2\sqrt{(1-{\cal E})({\cal E}-\kappa ^{2})(1+\kappa
^{2}-{\cal E})}}{\kappa ^{2}+\frac{E(\kappa ^{2})}{K}-{\cal E}}.
\end{equation}
We plot $\nu ^{2}$ as a function of energy in Fig. 7 for several values of $
\kappa .$ As $\kappa$ approaches zero, the energy gaps vanish,
and the group velocity approaches the free particle limit $E=M\nu^2/2=\nu^2/4$
(dot-dashed line). As $\kappa$ approaches unity, the lower band
vanishes becoming a bound state, and the group velocity is only
non-vanishing for the continuum states of the P\"oschl-Teller
potential with ${\cal E}\ge 2$.
The effective mass $M^{\ast }$ is determined by
\begin{eqnarray}
\frac{1}{M^{\ast }} &=&\frac{d\nu }{dk} \nonumber \\
&=&-2\frac{({\cal E}-\kappa ^{2})(1+\kappa ^{2}-{\cal E})+({\cal E}
-1)(1+\kappa ^{2}-{\cal E})-({\cal E}-1)({\cal E}-\kappa ^{2})}{({\cal E}
-\kappa ^{2}-\frac{E(\kappa ^{2})}{K})^{2}} \nonumber \\
&&+4\frac{({\cal E}-1)({\cal E}-\kappa ^{2})(1+\kappa ^{2}-{\cal E})}{({\cal
E}-\kappa ^{2}-\frac{E(\kappa ^{2})}{K})^{3}}
\end{eqnarray}
We plot $1/M^*$ in Fig. 8 for selected values of $\kappa .$ One can see that
as $\kappa ^{2}\rightarrow 0$, the gaps vanish and $M^{\ast }\rightarrow M=1/2$ as expected for
the free particle.
\subsection{Transfer Matrix for the Lam\'e Hamiltonian}
The general form of the transfer matrix is computed using the
definitions in the Appendix. Using the wavefunctions and Eq. (A1), we have
\begin{equation}
r=-\sum_{n=1}^{\ell }[Z(\alpha _{n})-\frac{{\rm sn}\,\alpha _{n}\,{\rm dn}
\,\alpha _{n}}{{\rm cn}\,\alpha _{n}}]-i\frac{\ell \pi }{2k}.
\end{equation}
Consequently, we see that
\begin{equation}
r+ik({\cal E})=\sum_{n=1}^{\ell }\frac{{\rm sn}\,\alpha _{n}\,{\rm dn}
\,\alpha _{n}}{{\rm cn}\,\alpha _{n}}.
\end{equation}
The transfer matrix then has the form
\begin{equation}
T\,=\,\left(
\begin{array}{cc}
\cos 2k({\cal E})K & i\left( \sum_{n=1}^{\ell }\frac{{\rm sn}\,\alpha _{n}\,
{\rm dn}\,\alpha _{n}}{{\rm cn}\,\alpha _{n}}\right) ^{-1}\sin 2k({\cal E})K
\\
i(\sum_{n=1}^{\ell }\frac{{\rm sn}\,\alpha _{n}\,{\rm dn}\,\alpha _{n}}{{\rm
cn}\,\alpha _{n}})\sin 2k({\cal E})K & \cos 2k({\cal E})K
\end{array}
\right) .
\end{equation}
(Note that this is in the form of Eq. (A.3) rather than (A.5).)
While this expression requires knowledge of the parameters $\alpha _{n}$,
one can obtain various limits of this for the free particle and
P\"oschl-Teller potentials.
\subsection{The $\kappa =0$ Free Particle Limit}
We start first with the $\ell =1$ case. Taking $\kappa =0$ in our transfer
matrix, we obtain for the upper and lower bands
\begin{equation}
T\,=\,\left(
\begin{array}{cc}
\cosh 2k({\cal E})K & i\frac{{\rm cn}\,\alpha }{{\rm sn}\,\alpha \,{\rm dn}
\,\alpha }\ \sinh 2k({\cal E})K \\
i\frac{{\rm sn}\,\alpha \,{\rm dn}\,\alpha }{{\rm cn}\,\alpha }\sinh 2k(
{\cal E})K & \cosh 2k({\cal E})K\
\end{array}
\right) .
\end{equation}
$\ $According to Eq. (\ref{eq:const}),
\begin{equation}
\frac{{\rm sn}\,\alpha \,{\rm dn}\,\alpha }{{\rm cn}\,\alpha }=\left[ \frac{
(1+\kappa ^{2}-{\cal E})({\cal E}-\kappa ^{2})}{({\cal E}-1)}\right] ^{1/2}.
\label{eq:ratio}
\end{equation}
Taking $\kappa =0$, we have $r\rightarrow 0$, so that
\begin{equation}
k({\cal E})=-i\frac{{\rm sn}\,\alpha \,{\rm dn}\,\alpha }{{\rm cn}\,\alpha }=
\sqrt{{\cal E}}.
\end{equation}
The transfer matrix becomes that for a free particle, given by:
\begin{equation}
T\,=\,\left(
\begin{array}{cc}
\cos \pi \sqrt{{\cal E}} & {\cal E}^{-1/2}\sin \pi \sqrt{{\cal E}} \\
-{\cal E}^{1/2}\sin \pi \sqrt{{\cal E}}\ & \ \cos \pi \sqrt{{\cal E}}
\end{array}
\right) . \label{eq:freetrans}
\end{equation}
For the general case of integer $\ell $, we start with
the constraint equations (\ref{eq:const}) and take the limit $\kappa
\rightarrow 0.$ We first must show that
\begin{equation}
r+ik({\cal E})=\sum_{n=1}^{\ell }\tan \alpha _{n}\rightarrow i\sqrt{{\cal E}}
. \label{eq:sumtan}
\end{equation}
Then, from the definition of $r$ in the appendix, it must vanish for free
motion, so that $r\rightarrow 0$ implies $k({\cal E})\rightarrow \sqrt{{\cal
E}}$. Then we recover the free particle transfer matrix. While we can show
that the sum in Eq. (\ref{eq:sumtan}) tends to $i\sqrt{{\cal E}}$ for small $
\ell $ on a case by case basis, we do not yet have a general proof. However,
since the transfer matrix must be that of a free particle in this limit, it
is clear that Eq. (\ref{eq:sumtan}) must hold, and we can use this instead
to provide an additional relation among the parameters $\alpha _{n}$.
\subsection{The $\kappa =1$ P\"oschl-Teller Limit}
\bigskip
In the limit $\kappa =1$, the Hamiltonian $H$ becomes the P\"oschl-Teller
Hamiltonian, and our transfer matrix should reduce to that case. We start
first with $\ell =1$ and examine the upper band.(The lower band becomes
degenerate at $\kappa =1$). In this limit, $K\rightarrow \infty $ and $
K^{\prime }\rightarrow \pi /2$. For the upper band,
\begin{equation}
k({\cal E})\rightarrow \sqrt{{\cal E}-2}+\frac{\pi +2i\alpha }{2K},
\end{equation}
From Eq. (\ref{eq:ratio}), we have in the $\kappa =1$ limit:
\begin{equation}
\frac{{\rm sn}\,\alpha \,{\rm dn}\,\alpha }{{\rm cn}\,\alpha }=\tanh \alpha
=i\sqrt{{\cal E}-2}. \label{leq:ratioa}
\end{equation}
The asymptotic form of the transfer matrix (as $K\rightarrow \infty $)
becomes:
\begin{equation}
T\,=\,\left(
\begin{array}{cc}
\cos (2K\sqrt{{\cal E}-2}+2i\alpha ) & ({\cal E}-2)^{-1/2}\sin (2K\sqrt{
{\cal E}-2}+2i\alpha ) \\
-\ ({\cal E}-2)^{1/2}\sin (2K\sqrt{{\cal E}-2}+2i\alpha ) & \cos (2K\sqrt{
{\cal E}-2}+2i\alpha )\
\end{array}
\right) .
\end{equation}
We must now change this form of the transfer matrix, defined for periodic
potentials $T$, to the form used in the P\"oschl-Teller case which is not
periodic, ${\cal T}\,$(see Appendix). Using $k({\cal E})=\sqrt{{\cal E}-2}$
, we find:
\begin{equation}
{\cal T}\,_{\kappa =1}=\,\left(
\begin{array}{cc}
\frac{\Gamma (ik)\Gamma (1+ik)}{\Gamma (2+ik)\Gamma (ik-1)} & \ 0 \\
0 & \ \frac{\Gamma (-ik)\Gamma (1-ik)}{\Gamma (2-ik)\Gamma (-ik-1)}
\end{array}
\right) .
\end{equation}
For the case of arbitrary $\ell $, we have the more general relations:
\begin{equation}
\sum_{n=1}^{\ell }\frac{\,{\rm sn}\alpha _{n}{\rm cn}\alpha _{n}}{{\rm dn}
\alpha _{n}}\rightarrow \sum_{n=1}^{\ell }\tanh \alpha _{n}=i\sqrt{{\cal E}
-\ell (\ell +1)}.
\end{equation}
The dispersion relation becomes
\begin{equation}
k({\cal E})\rightarrow -i\sum_{n=1}^{\ell }(\tanh \alpha _{n}-\frac{\alpha
_{n}}{K})+\ell \frac{\pi }{2K}=\sqrt{{\cal E}-\ell (\ell +1)}+\ell \frac{\pi
}{2K}+i\frac{1}{K}\sum_{n=1}^{\ell }\alpha _{n}.
\end{equation}
To compute the form of the transfer matrix for the P\"oschl-Teller potential,
we use the notation
\begin{equation}
{\cal T}\,=\,\left(
\begin{array}{cc}
F & G\ \\
G^{\ast } & F^{\ast }
\end{array}
\right) .
\end{equation}
where, from the appendix, we have:
\begin{equation}
F=(-)^{\ell }\exp \left[ -2\sum_{n=1}^{\ell }\alpha _{n}\right]
=\prod_{n=1}^{\ell }\frac{\tanh \alpha _{n}-1}{\tanh \alpha _{n}+1}.
\end{equation}
This can be further simplified using the limiting form of the constraint
equations (\ref{eq:const}):
\begin{eqnarray}
{\cal E} &=&-\left[ \sum_{n=1}^{\ell }\frac{1-\tanh ^{2}\alpha _{n}}{\tanh
\alpha _{n}}\right] ^{2}+\sum_{n=1}^{\ell }\coth ^{2}\alpha _{n}\ ,
\nonumber \\
0 &=&\sum_{p=1(p\neq n)}^{\ell }\frac{(1-\tanh ^{2}\alpha _{n})\tanh \alpha
_{n}+(1-\tanh ^{2}\alpha _{p})\tanh \alpha _{p}}{\tanh ^{2}\alpha _{p}-\tanh
^{2}\alpha _{n}}\ .
\end{eqnarray}
For small $\ell $ ($\ell =1,2$) we can solve these equations to show that
\begin{equation}
F=\frac{\Gamma (ik)\Gamma (1+ik)}{\Gamma (1+ik+\ell )\Gamma (ik-\ell )}
,\qquad G=0.
\end{equation}
This reduces to the correct form of the transfer matrix. For arbitrary $\ell
$, we have not been able to solve these equations, but the result must still
hold, since it is governed by the form of the Hamiltonian. Again, we can use
these relations to provide additional relations between the parameters $
\alpha _{n}$.
\section{Conclusions}
We have presented a group theoretical analysis of the Lam\'e equation, which
is an example of a SGA band structure problem for $su(2)$ and $su(1,1)$. We
have computed the dispersion relation and transfer matrix, and discussed the
limiting dynamical symmetry limits of these results which correspond to the
P\"oschl-Teller and free particle Hamiltonians. Because the general
Hamiltonian is not of the dynamical symmetry the spectrum cannot be obtained
in closed form. Never the less, a diagonalization of Hamiltonians which are
bilinear in the angular momentum generators will provide the general
solution. There are still many open questions associtated with the group
theoretical treatment of the Lam\'e equation. It would be nice to develop a $
su(1,1)$ parametrization which is non-singular for all values of $\kappa .$
In addition, the
Scarf and Mathieu equation limits would be interesting to realize more expliticly in
the transfer matrix and dispersion relations. Finally, the diagonalization
of the algebraic Hamiltonians in the continuum $su(1,1)$ bases would be
interesting to study. It is clear that the result of the diagonalization
must yield the same transcendental equations for the parameters $\alpha _{n}$
, but their origin would be different.
\vspace{2cm}
\section*{\bf Appendix: Form of the transfer matrix for periodic potentials}
\setcounter{equation}{1}
\renewcommand{\theequation}{A.\arabic{equation}}
For symmetric periodic potentials with period $\tau $, we can derive a
general formular of the transfer matrix. Suppose for a specific energy E, we
have two bloch solutions $u_{+}(x)e^{ikx}$, $u_{-}(x)e^{-ikx}$, where $x$ is
the coordinate, $k=k({\cal E})$ is the dispersion relation. Since the
potential is symmetric, we can define\cite{James}
\begin{equation}
r=\frac{u_{+}^{^{\prime }}(\tau /2)}{u_{+}(\tau /2)}=-\frac{u_{-}^{^{\prime
}}(\tau /2)}{u_{-}(\tau /2)}.
\end{equation}
Then the transfer matrix is
\begin{equation}
T\,=\,\left(
\begin{array}{cc}
\cos k\tau & \frac{i}{ik+r}\sin k\tau \\
i(ik+r)\sin k\tau & \cos k\tau
\end{array}
\right) .
\end{equation}
The forms of transfer matrices used for periodic and non-periodic potentials
is different. In the discussion of the Lam\'e equation, the $\kappa
\rightarrow 1$ limit takes a periodic potential to a non-periodic one, so
that we require the transformations that take us from one standard form to
the other. If we express the transfer matrix $T$ for the periodic potential
as
\begin{equation}
T=\left(
\begin{array}{cc}
{\rm Re}(F\exp (ik\tau )+G) & \frac{1}{k}{\rm Im}(F\exp (ik\tau )+G) \\
k{\rm Im}(F\exp (ik\tau )-G)\ & {\rm Re}(F\exp (ik\tau )-G)\
\end{array}
\right) ,
\end{equation}
then that of the non-periodic limit will have the form
\begin{equation}
{\cal T}=\left(
\begin{array}{cc}
F & G \\
G^{\ast }\ & F^{\ast }
\end{array}
\right) .
\end{equation}
In this notation, $k=k({\cal E})$ is the dispersion relation, and $\tau $ is
the period of the periodic potential which tends to $\infty .$
\newpage
|
3,212,635,537,432 | arxiv | \section*{Acknowledgments}
We gratefully acknowledge, for support and valuable conversations, Johannes Gehrke, Li Deng, Qi Lu, Yi-Min Wang, Harry Shum, Eric Horvitz, Susan Dumais, Xiaodong He, Asl{\i} \c{C}eliky{\i}l\-maz, Chris Meek, Hamid Palangi, Qiuyuan Huang, Nebojsa Jojic, Imanol Schlag, Kezhen Chen, Shuai Tang, Laurel Brehm, Najoung Kim, Matthias Lalisse, Paul Soulos, Eric Rosen, Caitlin Smith, Coleman Haley, G\'{e}raldine Legendre, Jason Eisner, Ben Van Durme, Alan Yuille, Hynek Hermansky,
Tal Linzen, Robert Frank, J\"{u}rgen Schmidhuber, Ken Forbus, Gary Marcus, Yoshua Bengio, Steven Pinker, Jay McClelland, Alan Prince,
Ewan Dunbar, Dapeng Wu, Randy O'Reilly, Fran\c{c}ois Charton, Guillaume Lample, Peter beim Graben, Daniel Crevier,
and all our collaborators on the papers reviewed here.
The work reported here was supported in part by NSF (GRFP 1746891, BCS-1344269, DGE-0549379) and by Microsoft Research.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation or Microsoft.
\end{document}
|
3,212,635,537,433 | arxiv | \section{Introduction}
The nested error regression (NER) model with the normality assumption for both the random effects or model error terms and the unit-level error terms has played a key role in analyzing unit-level data in small area estimation. Many popular small area estimation methods have been developed under this model. In the frequentist approach, Battese et al. (1988), Prasad and Rao (1990), and Datta and Lahiri (2000), for example, derived empirical best linear unbiased predictors (EBLUPs) of small area means. These authors used various estimation methods for the variance components and derived approximately accurate estimators of mean squared error (MSEs) of the EBLUPs. On the other hand, Datta and Ghosh (1991) followed the hierarchical Bayesian (HB) approach to derive posterior means as HB predictors and variances of the small area means. While the underlying normality assumptions for all the random quantities are appropriate for regular data, they fail to adequately accommodate outliers. Consequently, these frequentist/Bayesian methods are highly influenced by major outliers in the data, or break down if the outliers grossly violate distributional assumptions.
\bigskip
\noindent
Sinha and Rao (2009) investigated the robustness, or lack thereof, of the EBLUPs from the usual normal NER model in the presence of ``representative outliers''. According to Chambers (1986), a representative outlier is a ``sample element with a value that has been correctly recorded and cannot be regarded as unique. In particular, there is no reason to assume that there are no more similar outliers in the nonsampled part of the population.'' Sinha and Rao (2009) showed via simulations for the NER model that while the EBLUPs are efficient under normality, they are very sensitive to outliers that deviate from the assumed model.
\bigskip
\noindent
To address the non-robustness issue of EBLUPs, Sinha and Rao (2009) used the $\psi$-function, Huber's Proposal 2 influence function in M-estimation, to downweight the contribution of outliers in the BLUPs and the estimators of the model parameters, both regression coefficients and variance components. Using M-estimation for robust maximum likelihood, estimators of model parameters, and robust predictors of random effects, Sinha and Rao (2009) for mixed linear models proposed a robust EBLUP (REBLUP) of mixed effects, which they used to estimate small area means for the NER model. By using a parametric bootstrap procedure they have also developed estimators of the MSEs of the REBLUPs. We refer to Sinha and Rao (2009) for details of this method. Their simulations show that when the normality assumptions hold, the proposed REBLUPs perform similar to the EBLUPs in terms of empirical bias and empirical MSE. But, in presence of outliers in the unit-level errors, while both EBLUPs and REBLUPs remain approximately unbiased, the empirical MSEs of the EBLUPs are significantly larger than those of the REBLUPs.
\bigskip
\noindent
Datta and Ghosh (1991) proposed a noninformative HB model to predict finite population small area means. In this article we follow the approach to finite population sampling which was also followed by Datta and Ghosh (1991). Our suggested model includes the treatment of the NER model by Datta and Ghosh (1991) as a special case. Our model facilitates accommodating outliers in the population and in the sample values. We replace the normality of the unit-level error terms by a two-component mixture of normal distributions, each component centered at zero. As in Datta and Ghosh (1991), we assume normality of the small area effects.
\bigskip
\noindent
Simulation results of Sinha and Rao (2009) indicated that there was not enough improvement in performance of the REBLUP procedures over the EBLUPs when they considered outliers in both the unit-level error and the model error terms. To keep both analytical and computational challenges for our noninformative HB analysis manageable, we use a realistic framework and we restrict ourselves to the normality assumption for the random effects. Moreover, the assumption of zero means for the unit-level error terms is similar to the assumption made by Sinha and Rao (2009). While allowing the component of the unit-level error terms with the bigger variance to also have non-zero means to accommodate outliers might appear attractive, we note later that it is not possible to conduct a noninformative Bayesian analysis with an improper prior on the new parameter.
\bigskip
\noindent
We focus only {on} unit-level model robust small area estimation in this article. There is a substantial literature on small area estimation based on area-level data using the Fay-Herriot model (see Fay and Herriot, 1979; Prasad and Rao, 1990). The paper by Sinha and Rao (2009) also discussed robust small area estimation for an area-level model. In another paper, Lahiri and Rao (1995) discussed EBLUP and estimation of MSE under a non-normality assumption for the random effects. An early robust Bayesian approach for area-level models is due to Datta and Lahiri (1995), where they used a scale mixture of normal distributions for the random effects. It is worth mentioning that the $t$-distributions are special cases of the scale mixture of normal distributions. While Datta and Lahiri (1995) assumed long-tailed distributions for the random effects, Bell and Huang (2006) used the HB method based on the $t$ distribution, either only for the sampling errors or only for the model errors.
\bigskip
\noindent
The scale mixture of normal distributions requires specification of the mixing distribution, or in the specific case for $t$ distributions, it requires the degrees of freedom. In an attempt to avoid this specification, in a recent article Chakraborty et al. (2016) proposed a simple alternative via a two-component mixture of normal distributions in terms of the variance components for the model errors.
\section{Unit-Level HB Models for Small Area Estimation } \label{sec2}
The model-based approach to finite population sampling is very useful for modeling unit-level data in small area estimation. The NER model of Battese et al. (1988) is a popular model for unit-level data. Suppose a finite population is partitioned into $m$ small areas, with the $i$th area having $N_i$ units. The NER model relates $Y_{ij}$, the value of a response variable $Y$ for the $j$th unit in the $i$th small area, with $x_{ij} =(x_{ij1},\cdots, x_{ijp})^T$, the value of a $p$-component covariate vector associated with that unit, through a mixed linear model given by
\begin{equation}
Y_{ij}= x_{ij}^T\beta +v_i +e_{ij}, ~ j=1,\cdots, N_i, ~i=1,\cdots, m,
\label{NER-BHF}
\end{equation}
where all the random variables $v_i$'s and $e_{ij}$'s are assumed independent. Distributions of these variables are specified by assuming that random effects $v_i\stackrel{iid}\sim N(0,\sigma_v^2)$ and unit-level errors $e_{ij}\stackrel{iid}\sim N(0,\sigma_e^2)$. Here $\beta =(\beta_1,\cdots, \beta_p)^T$ is the regression coefficient vector. We want to predict the $i$th small area finite population mean ${\bar Y}_i= N_i^{-1}\sum_{j=1}^{N_i} Y_{ij}$, $i=1,\cdots,m$. We assume that the population level model (\ref{NER-BHF}) holds for any sample from the population.
\bigskip
\noindent
Battese et al. (1988) and Prasad and Rao (1990), among others, considered noninformative sampling, where a simple random sample of size $n_i$ is selected from the $i$th small area. For notational simplicity we denote the sample by $Y_{ij}, j=1,\cdots, n_i, i=1,\cdots,m$. To develop predictors of the small area means ${\bar Y}_i, i=1,\cdots,m$, these authors first derived, for known model parameters, the conditional distribution of the {\it unsampled} values, $Y_{ij}, j=n_i +1,\cdots, N_{i}, i=1,\cdots,m$, given the sampled values $Y_{ij}, j=1,\cdots, n_{i}, i=1,\cdots,m$. Under squared error loss, the best predictor of ${\bar Y}_i$ is its mean with respect to this conditional distribution, also known as the predictive distribution. In the frequentist approach, Battese et al. (1988) and Prasad and Rao (1990) obtained the EBLUP of $\bar Y_i$ by replacing in the conditional mean the unknown model parameters $(\beta^T,\sigma_e^2,\sigma_v^2)^T$ by their estimators using $Y_{ij}, j=1,\cdots, n_i, i=1,\cdots, m$. In the Bayesian approach, on the other hand, Datta and Ghosh (1991) developed HB predictors of ${\bar Y}_i$ by integrating out these parameters in the conditional mean of $\bar Y_i$ with respect to their posterior density, which is derived based on a prior distribution on the parameters and the distribution of the sample $Y_{ij}, j=1,\cdots, n_i, i=1,\cdots, m$, derived under the model (\ref{NER-BHF}).
\bigskip
\noindent
While the frequentist approach for the NER model under the distributional assumptions in (\ref{NER-BHF}) continues with accurate approximation and estimation of the MSEs of the EBLUPs, the Bayesian approach typically proceeds under some noninformative priors, and computes numerically, usually by the MCMC method, the exact posterior means and {posterior variances of the area means} $\bar Y_i$'s. Among various noninformative priors for $\beta, \sigma_e^2, \sigma_v^2$, a popular choice is
\begin{equation}
\pi_P(\beta,\sigma_e^2,\sigma_v^2) = \frac 1{\sigma_e^2},
\label{Pop-prior-BHF}
\end{equation}
(see, for example, Datta and Ghosh, 1991).
\bigskip
\noindent
The standard NER model in (\ref{NER-BHF}) is unable to explain outlier behavior of unit-level error terms. To avoid the breakdown of EBLUPs and their MSEs in the presence of outliers, Sinha and Rao (2009) modified all estimating equations for the model parameters and random effects terms by robustifying various ``standardized residuals'' that appear in the estimating equations by using Huber's $\psi$-function, which truncates large absolute values to a certain threshold. They did not replace the working NER model in (\ref{NER-BHF}) to accommodate outliers, but they accounted for their potential impacts on the EBLUPs and estimated MSEs by downweighting large standardized residuals that appear in various estimating equations through Huber's $\psi$-function. Their approach, in the terminology of Chambers et al. (2014), may be termed {\it robust projective}, where they estimated the working model in a robust fashion and used that to project sample non-outlier behavior to the unsampled part of the model.
\bigskip
\noindent
To investigate the effectiveness of their proposal, Sinha and Rao (2009) conducted simulations based on various long-tailed distributions for the random effects and/or the unit-level error terms. In one of their simulation scenarios which is reasonably simple but useful, they used a two-component mixture of normal distributions for the unit-level error terms, with both components centered at zero but with unequal variances, and the component with the larger variance appearing with a small probability. This modifies the regular setup of the NER model with the possibility of outliers arising as a small fraction of contamination caused by the error corresponding to the larger variance component. {Simulation results in Table 2 of Sinha and Rao (2009) report that outliers in the random effect have little impact on the EBLUP. Hence we could focus on the unit-level error only}. In this article, we incorporate this mixture distribution to modify the model in (\ref{NER-BHF}) to develop new Bayesian methods that would be robust to outliers. Our proposed population level HB model is given by
\bigskip
\noindent
Normal Mixture (NM) HB Model:
\begin{itemize}
\item[(I)] Conditional on $\beta =(\beta_1,\cdots,\beta_p)^T, v_1,\cdots, v_m, z_{ij}, j=1,\cdots, N_i, i=1,\cdots, m, p_e, \sigma_1^2, \sigma_2^2$ and $\sigma_v^2$,
$$
Y_{ij} \stackrel{ind} \sim z_{ij} N(x_{ij}^T\beta+v_i , \sigma_1^2) + (1-z_{ij}) N(x_{ij}^T\beta+v_i , \sigma_2^2), ~j=1,\cdots, N_i, i=1,\cdots, m.
$$
\item[(II)] The indicator variables $z_{ij}$'s are iid with $P(z_{ij}=1|p_e)=p_e, ~j=1,\cdots, N_i, i=1,\cdots, m,$ and are independent of $\beta =(\beta_1,\cdots,\beta_p)^T, v_1,\cdots, v_m, \sigma_1^2, \sigma_2^2$ and $\sigma_v^2$.
\item[(III)] Conditional on $\beta, z=(z_{11},\cdots, z_{1N_1},\cdots, z_{m1},\cdots, z_{mN_m})^T, p_e, \sigma_1^2, \sigma_2^2$ and $\sigma_v^2$, random small area effects $v_i\stackrel{iid}\sim N(0,\sigma_v^2)$ for $i=1,\cdots, m$.
\end{itemize}
For simplicity, we assume the contamination probability $p_e$ to remain the same for all units in all small areas. Gershunskaya (2010) proposed this mixture model for empirical Bayes point estimation of small area means. We assume independent simple random samples of size $n_1, \cdots, n_m$ from the $m$ small areas. The Simple Random Sampling results in a noninformative sample and the joint distribution of responses of the sampled units can be obtained from the NM HB model above by replacing $N_i$ by $n_i$. This marginal distribution in combination with the prior distribution provided below will yield the posterior distribution of the $v_i$'s, and of all the parameters in the model. For the informative sampling developments in small area estimation we refer to Pfeffermann and Sverchkov (2007) and Verret et al. (2015).
\bigskip
\noindent
Two components of the normal mixture distribution differ only by their variances. We will assume the variance component $\sigma_2^2$ is larger than $\sigma_1^2$ and is intended to explain any outliers in a data set. However, if a data set does not include any outliers, the two component variances $\sigma_1^2, \sigma_2^2$ may only minimally differ. In such situation, the likelihood based on the sample will include limited information to distinguish between these variance parameters, and consequently, the likelihood will also have little information about the mixing proportion $p_e$. We notice this behavior in our application to a subset of the corn data in Section \ref{sec:data}.
\bigskip
\noindent
In this article, we carry out an objective Bayesian analysis by assigning a noninformative prior to the model parameters. In particular, we propose a noninformative prior
\begin{equation}
\pi(\beta,\sigma_1^2,\sigma_2^2,\sigma_v^2,p_e) = \frac{I(0<\sigma_1^2<\sigma_2^2 <\infty)}{{(\sigma_2^2)}^2} ,
\label{new-prior}
\end{equation}
where we have assigned an improper prior on $\beta, \sigma_v^2, \sigma_1^2, \sigma_2^2$ and a proper uniform prior on the mixing proportion $p_e$. However, subjective priors could also be assigned when such subjective information is available. Notably, it is possible to use some other proper prior on $p_e$ that may elicit the extent of contamination to the basic model to reflect prevalence of outliers. While many such subjective priors can be reasonably modeled by a beta distribution, we use a {\it uniform distribution} from this class to reflect noninformativeness or little information about this parameter. We also use the traditional uniform priors on $\beta$ and $\sigma_v^2$. In the Supplementary materials, we explore the propriety of the posterior distribution corresponding to the improper priors in (\ref{new-prior}).
\bigskip
\noindent
{The improper prior distribution on the two variances for the mixture distribution has been carefully chosen so that the prior will yield conditionally proper distributions for each parameter given the other. The proper conditional densities given $\sigma_2^2$ (or $\sigma_1^2$) respectively are
\begin{equation*}
\pi(\sigma_1^2|\sigma_2^2) = \dfrac{1}{\sigma_2^2}I(0<\sigma_1^2<\sigma_2^2), ~~~
\pi(\sigma_2^2|\sigma_1^2) = \dfrac{\sigma_1^2}{(\sigma_2^2)^2}I(\sigma_1^2<\sigma_2^2 <\infty).
\end{equation*}
This conditional propriety is {\it necessary} for parameters appearing in the mixture distribution in order to ensure under suitable conditions the propriety of the posterior density resulting from the HB model. Alternatively, if we used, $\pi(\sigma_1^2,\sigma_2^2) \propto (\sigma^2_1)^{-1}(\sigma^2_2)^{-1}$, the posterior distribution would be improper for situations when there are no observations from the outlying distribution. Prior ($\ref{new-prior}$) can accommodate these situations. The specific prior distribution that we propose above is such that the resulting marginal densities for $\sigma_1^2$ and $\sigma_2^2$ respectively, are $\pi_{\sigma_1^2}(\sigma_1^2) =(\sigma_1^2)^{-1}$ and $\pi_{\sigma_2^2}(\sigma_2^2) =(\sigma_2^2)^{-1}$. These two densities are of the same form as that of $\sigma_e^2$ in the regular model in (\ref{Pop-prior-BHF}) introduced earlier. Indeed by setting $p_e=0 $ or $1$ in our analysis, we can reproduce the HB analysis of the regular model given by (\ref{NER-BHF}) and (\ref{Pop-prior-BHF}).}
\bigskip
\noindent
We use the NM HB Model under noninformative sampling and the noninformative priors given by (\ref{new-prior}) to derive the posterior predictive distribution of $\bar{Y}_i, i=1,\cdots, m$. The NM HB model and noninformative sampling that we propose here facilitate building model for {\it representative outliers} (Chambers, 1986). According to Chambers, a representative outlier is a value of a sampled unit which is not regarded as unique in the population, and one can expect existence of similar values in the non-sampled part of the population which will influence the value of the finite population means $\bar Y_i$'s or the other parameters involved in the superpopulation model.
\bigskip
\noindent
Following the practice of Battese et al. (1988) and Prasad and Rao (1990), we approximated the predictand $\bar Y_i$ by $\theta_i =\bar X_i^T\beta +v_i$ to draw inference on the finite population small area means. Here $\bar X_i = N_i^{-1}\sum_{j=1}^{N_i} x_{ij}$ is assumed known. This approximation works well for small sampling fractions $n_i/N_i$ and large $N_i$'s. It has been noted by these authors, and by Sinha and Rao (2009), that even for the case of outliers in the sample the difference between the inference results for $\bar Y_i$ and $\theta_i$ is negligible. Our own simulations for our model also confirm that {observation}. Once MCMC samples from the posterior distribution of $\beta, v_i$'s and $\sigma_v^2, \sigma_1^2, \sigma_2^2, p_e$ have been generated, using the NM HB Model the MCMC samples of $Y_{ij}, j=n_i+1,\cdots, N_i, i=1,\cdots, m$ from their posterior predictive distributions can be easily generated. Finally, using the relation $\bar Y_i =N_i^{-1}[\sum_{j=1}^{n_i} y_{ij} + \sum_{j=n_i +1}^{N_i} Y_{ij}]$, (posterior predictive) MCMC samples for $\bar Y_i$'s can be easily generated for inference on these quantities. In our own data anaysis, where the sampling fractions are negligible, we do inference for the approximated predictands $\theta_i$'s.
\bigskip
\noindent
Chambers and Tzavidis (2006) took a new frequentist approach to small area estimation that is different from the mixed model prediction used in EBLUP. Instead of using a mixed model for the response, they suggeted a method based on quantile regression. We briefly review their M-quantile small area estimation method in Section \ref{MQ-rev}. They also proposed an estimator of MSE of their point estimators.
\bigskip
\noindent
Our Bayesian proposal has two advantages over the REBLUP of Sinha and Rao (2009). First, instead of a working model for the non-outliers, we use an explicit mixture model to specify the joint distribution of responses of all the units in the population, and not only the non-outliers part of the population. It enables us to use all the sampled observations to predict the entire non-sampled part, consisting of outliers and non-outliers, of the population. Our method is robust predictive and the noninformative HB predictors are less susceptible to bias. Second, the main thrust of the EBLUP approach in small area estimation is to develop accurate approximations and estimation of MSEs of EBLUPs (cf. Prasad and Rao, 1990). Datta and Lahiri (2000) and Datta et al. (2005) termed this approximation as second-order accurate approximation, which neglects terms lower order than $m^{-1}$ in the approximation. Second-order accurate approximation results for REBLUPs have not been obtained by Sinha and Rao (2009). Also, their bootstrap proposal to estimation of the MSE under the working model has not been shown to be second-order accurate. Our HB proposal does not rely on any asymptotic approximations. Analysis of the corn data set and simulation study show less uncertainty {(and better stability of this measure)} of our method compared to the M-quantile method.
\section{M-quantile Small Area Estimation}\label{MQ-rev}
Small area estimation is dominated by linear mixed effects models where the conditional mean of $Y_{ij}$, the response of the $j$th unit in the $i$th small area, is expressed as $E(Y_{ij}|x_{ij}, v_i)=x_{ij}^T\beta + z_{ij}^Tv_i,$ where $x_{ij}$ and $z_{ij}$ are suitable known covariates, $v_i$ is a random effects vector and $\beta$ is a common regression coefficient vector. This assumption is the building block for EBLUPs of small area means, based on suitable additional assumptions for this conditional distribution and the distribution of the random effects. Also with suitable prior distribution on the model parameters, HB methodology for prediction of small area means is developed.
\bigskip
\noindent
As an alternative to linear regression which models $E(Y|x)$, the mean of the conditional distribution of $Y$ given covariates $x$, quantile regression has been developed by modeling suitable quantiles of the conditional distribution of $Y$ given $x$. In particular in quantile linear regression, for $0<q<1$, the $q$th quantile $Q_q(Y|x)$ of this distribution is modeled as $Q_q(Y|x)=x^T\beta_q$, where $\beta_q$ is a suitable parameter modeling the linear quantile function. For a given quantile regression function, the quantile coefficient $q_i\in (0,1)$ of an observation $y_i$ satisfies $Q_{q_i}(Y|x_i)=y_i$. In particular, for a linear quantile function, for given $y_i, x_i$, the $q_i$ satisfies $x_i^T\beta_{q_i}=y_i$.
\bigskip
\noindent
While in the linear regression setup the regression coefficient $\beta$ is estimated from a set of data $\{y_i,x_i:i=1,\cdots, n\}$ by minimizing the sum of squared errors $\sum_{i=1}^n (y_i-x_i^T\beta)^2$ with respect to $\beta$, the quantile regression coefficient $\beta_q$ for a fixed $q\in (0,1)$ is obtained by minimizing the loss function $\sum_{i=1}^n |y_i-x_i^Tb| \{(1-q)I(y_i-x_i^T b \le 0) + q I(y_i-x_i^T b >0) \}$ with respect to $b$. Here $I(\cdot)$ is a usual indicator function.
\bigskip
\noindent
Following the idea of M-estimation in robust linear regression, Breckling and Chambers (1988) generalized quantile regression by minimizing an objective function $\sum_{i=1}^n d(|y_i-x_i^Tb|)\{(1-q)I(y_i-x_i^T b \le 0) + q I(y_i-x_i^T b >0) \}$ with respect to $b$ for some given loss function $d(\cdot)$. [Linear regression is a special case for $q=.5$ and $d(u)=u^2$.] Estimator of $\beta_q$ is obtained by solving the equation
$$
\sum_{i=1}^n \psi_q(r_{iq}) x_i =0,
$$
where $r_{iq} = y_i - x_i^T\beta_q$, $ \psi_q(r_{iq}) = \psi(s^{-1} r_{iq})\{(1-q)I(r_{iq} \le 0) +
q I(r_{iq}>0) \}$, the function $\psi(\cdot)$, known as the influence function in M-estimation, is determined by $d(\cdot)$ (actually, $\psi(u)$ is related to the derivative of $d(u)$, assuming it is differentiable). The quantity $s$ is a suitable scale factor determined from the data (cf. Chambers and Tzavidis, 2006). In M-quantile regression, these authors suggested using $\psi(\cdot)$ as the Huber Proposal 2 influence function $\psi(u) = u I(|u| \le c) + c \mbox{ sign}(u) I(|u| > c)$, where $c$ is a given positive number bounded away from $0$.
\bigskip
\noindent
To apply the M-quantile method in small area estimation for a set of data $\{y_{ij}, x_{ij},j=1,\cdots, n_i, i=1,\cdots, m\}$, Fabrizi et al. (2012) followed Chambers and Tzavidis (2006) and suggested determining a set of $\hat{\beta}_q$ in a fine grid for $q\in (0,1)$ by solving
$$
\sum_{i=1}^m\sum_{j=1}^{n_i} \psi_q(r_{ijq})x_{ij} =0,
$$
where $r_{ijq} = y_{ij} -x_{ij}^T\hat{\beta}_q$. Fabrizi et al. (2012) defined M-quantile estimator of $\bar Y_i $
by
\begin{equation}
\hat{\bar Y}_{i,MQ} = \frac 1{N_i} [\sum_{j=1}^{n_i}y_{ij} + \sum_{j=n_i +1}^{N_i} x_{ij}^T\hat{\beta}_{\bar{q}_i}
+ (N_i-n_i)(\bar y_i - \bar x_i^T\hat{\beta}_{\bar{q}_i}) ],
\label{MQest}
\end{equation}
where $(\bar y_i, \bar x_i)$ is the sample mean of $\{(y_{ij}, x_{ij}), j=1,\cdots, n_i\}$.
Here $\bar q_i = \frac 1{n_i}\sum_{j=1}^{n_i} q_{ij}$ is the average estimated quantile coefficient of the $i$th small area, where $q_{ij}$ is obtained by solving $x_{ij}^T\hat{\beta}_{q} =y_{ij}$, based on the set $\{\hat{\beta}_q\}$ described above (if necessary, interpolation for $q$ is made to solve $x_{ij}^T\hat{\beta}_{q} =y_{ij}$ accurately). Here we suppress the dependence of $\hat{\beta}_q$ and $q_{ij}$ on the influence function $\psi(\cdot)$. For details on M-quantile small area estimators and associated estimators of MSE based on a pseudo-linearization method, we refer to Tzavidis and Chambers (2005) and Chambers et al. (2014).
\section{Robust Empirical Best Linear Unbiased Prediction}\label{sec:REBLUP}
Empirical best linear unbiased predictors (EBLUPs) of small area means, developed under normality assumptions for the random effects and the unit-level errors, play a very useful role in production of reliable model-based estimation methods. While the EBLUPs are efficient under the normality assumptions, they may be highly influenced by outliers in the data. Sinha and Rao (2009) investigated the robustness of the classical EBLUPs to the departure from normality assumptions and proposed a new class of predictors which are resistant to outliers. Their proposed robust modification of EBLUPs of small area means, which they termed robust EBLUP (REBLUP), downweight any influential observations in the data in estimating the model parameters and the random effects.
\bigskip
\noindent
Sinha and Rao (2009) considered a general linear mixed effects model with a block-diagonal variance-covariance matrix. Their model, which is sufficiently general to include the popular Fay-Herriot model and the nested error regression model as special cases, is given by
\begin{equation}
y_i= X_i\beta + Z_i v_i +e_i, i=1,\cdots, m,
\label{lme}
\end{equation}
for specified design matrices $X_i, Z_i$, random effects vector $v_i$ and unit-level error vector $e_i$ associated with the data $y_i$ from the $i$th small area. They assumed normality and independence of the random vectors $v_1,\cdots, v_m, e_1,\cdots, e_m$, where $v_i\sim N(0, G_i(\delta))$ and $e_i\sim N(0, R_i(\delta))$. Here $\delta$ includes the variance parameters associated with the model (\ref{lme}).
\bigskip
\noindent
To develop a robust predictor of a mixed effect $\mu_i= h_i^T\beta + k_i^Tv_i$, Sinha and Rao (2009) started with the well-known mixed model equations given by
\begin{equation}
\sum_{i=1}^m X_i^TR_i^{-1}(y_i-X_i\beta - Z_i v_i) =0, ~ Z_i^TR_i^{-1}(y_i-X_i\beta - Z_i v_i)- G_i^{-1}v_i =0,~i=1,\cdots, m,
\label{mixedmodeq}
\end{equation}
which are derived as estimating equations by differentiating the joint density of $y_1,\cdots, y_m,$ and $ v_1,\cdots, v_m$ with respect to $\beta$, and $v_1,\cdots, v_m$ to obtain``maximum likelihood'' estimators of $\beta, v_1,\cdots, v_m$ for known $\delta$. The unique solution $\tilde\beta(\delta),\tilde{v}_1(\delta),\cdots, \tilde{v}_m(\delta)$ to these equations leads to the BLUP $h_i^T\tilde\beta + k_i^T\tilde{v}_i$ of $\mu_i$. To estimate the variance parameters $\delta$, Sinha and Rao (2009) maximized the profile likelihood of $\delta$, which is the value of the likelihood of $\beta$ and $\delta$ based on the joint distribution of the data $y_1,\cdots, y_m$ at $\beta=\tilde{\beta}(\delta)$.
\bigskip
\noindent
To mitigate the impact of outliers on the estimators of the variance parameters, the regression coefficients and the random effects, Sinha and Rao (2009) extended the work of Fellner (1986) to robustify all the ``estimating equations'' by using Huber's $\psi$-function in M-estimation. Based on the robustified estimating equations, Sinha and Rao (2009) obtained the robust estimators of $\beta, \delta$ and $v_i, i=1,\cdots, m$, denoted respectively by $\hat{\beta}_M,\hat{\delta}_M$ and $\hat{v}_{iM}, i=1,\cdots, m$. These estimators lead to the REBLUP of $\mu_i$ given by $h_i^T\hat{\beta}_M + k_i^T\hat{v}_{iM}$. For details of the REBLUP and the associated parametric bootstrap estimators of the MSE of the REBLUPs of $\mu_i$, we refer the readers to the paper by Sinha and Rao (2009).
\section{Data Analysis}\label{sec:data}
We illustrate our method by analyzing the crop areas data {reported} by Battese et al. (1988) who considered EBLUP prediction of county crop areas for 12 counties in Iowa. Based on U.S. farm survey data in conjunction with LANDSAT satellite data they developed predictors of county means of hectares of corn and soybeans. Battese et al. (1988) were the first to put forward the nested error regression model for the prediction of the county crop areas. Datta and Ghosh (1991) later used the HB prediction approach on this data to illustrate Bayesian treatment of the nested error regression model. In the USDA farm survey data on 37 sampled segments from these 12 counties, Battese et al. (1988) determined in their reported data that the second observation for corn in Hardin county was an outlier so that this outlier would not unduly affect the model-based estimates of the small area means, Battese et al. (1988) initially recommended, and Datta and Ghosh (1991) subsequently followed, to remove this suspected outlier observation from their analyses. Discarding this observation results in a better fit for the nested error regression model. However, removing any data which may not be a non-representative outlier from analysis will result in loss of valuable information about a part of the non-sampled units of the population which may contain outliers.
\begin{table}[ht]\caption{Various point estimates and standard errors of county hectares of corn }\label{tab1:BHFanalysis}
\tiny
\begin{center}
\begin{tabular}{|l|crr|rr|rr|rr|crr|rr|rr|rr|}
\hline
SA & \multicolumn{9}{|c|}{Full Data} & \multicolumn{9}{|c|}{Reduced Data} \\
& {$n_i$} & \multicolumn{2}{c}{DG HB} &\multicolumn{2}{c}{NM HB} & \multicolumn{2}{c}{SR} & \multicolumn{2}{c|}{MQ}& {$n_i$} & \multicolumn{2}{c}{DG HB} &\multicolumn{2}{c}{NM HB} & \multicolumn{2}{c}{SR} & \multicolumn{2}{c|}{MQ}\\
& & Mean & SD & Mean & SD & Mean & SD & Mean & SD & & Mean & SD & Mean & SD & Mean & SD & Mean & SD \\
\hline
1 & 1 & 123.8 & 11.7 & 123.4 & 9.8 & 123.7 & 9.9 & 130.0 & 5.7 & 1 & 122.0 & 11.6 & 121.7 & 9.7 & 122.2 & 9.9 & 128.0 & 3.7 \\
2 & 1 & 124.9 & 11.4 & 126.6 & 10.3 & 125.3 & 9.7 & 134.2 & 8.4 & 1 & 126.4 & 10.9 & 127.2 & 9.7 & 126.5 & 9.5 & 133.4 & 6.0 \\
3 & 1 & 110.0 & 12.3 & 108.0 & 11.3 & 110.3 & 9.4 & 86.0 & 18.3 & 1 & 107.6 & 12.4 & 105.6 & 10.1 & 106.7 & 9.5 & 94.6 & 14.4 \\
4 & 2 & 114.2 & 10.7 & 112.3 & 10.2 & 114.1 & 8.8 & 114.4 & 3.4 & 2 & 108.9 & 10.5 & 108.2 & 8.7 & 111.0 & 8.3 & 113.3 & 3.7 \\
5 & 3 & 140.3 & 10.8 & 142.1 & 8.1 & 140.8 & 7.8 & 144.2 & 11.3 & 3 & 143.6 & 9.7 & 144.1 & 7.0 & 143.3 & 7.1 & 144.2 & 9.3 \\
6 & 3 & 110.0 & 9.6 & 111.4 & 7.6 & 110.8 & 7.6 & 108.6 & 3.9 & 3 & 112.3 & 9.7 & 112.5 & 6.5 & 112.3 & 7.1 & 114.5 & 5.4 \\
7 & 3 & 116.0 & 9.7 & 114.3 & 7.6 & 115.2 & 7.3 & 116.3 & 4.2 & 3 & 113.4 & 9.1 & 112.5 & 6.8 & 112.9 & 7.1 & 115.4 & 3.8 \\
8 & 3 & 123.2 & 9.5 & 122.7 & 7.9 & 122.7 & 7.5 & 122.5 & 3.9 & 3 & 121.9 & 8.8 & 121.9 & 6.6 & 121.9 & 7.1 & 122.7 & 4.0 \\
9 & 4 & 112.6 & 9.9 & 113.9 & 6.9 & 113.5 & 6.5 & 115.3 & 5.8 & 4 & 115.5 & 9.2 & 115.7 & 5.7 & 115.3 & 6.4 & 115.7 & 4.6 \\
10 & 5 & 124.4 & 8.9 & 123.5 & 6.1 & 124.1 & 6.3 & 121.6 & 4.7 & 5 & 124.8 & 8.4 & 124.4 & 5.4 & 124.5 & 5.3 & 123.1 & 4.0 \\
11 & 5 & 111.3 & 8.9 & 108.2 & 6.8 & 109.5 & 6.2 & 106.9 & 10.6 & 5 & 107.7 & 8.5 & 106.3 & 5.7 & 106.8 & 5.4 & 105.5 & 7.0 \\
12 & 6 & 130.7 & 8.3 & 135.3 & 7.5 & 136.9 & 6.0 & 135.8 & 4.3 & 5 & 142.6 & 9.0 & 143.5 & 5.9 & 143.1 & 5.8 & 140.6 & 4.9 \\
\hline
\end{tabular}
\end{center}
\end{table}
\bigskip
\noindent
We reanalyze the full data set for corn using our proposed HB method as well as the other methods we reviewed above.
In Table~\ref{tab1:BHFanalysis} we report various point estimates and standard error estimates. We compare our proposed robust HB prediction method with the standard HB method of Datta and Ghosh (1991), and two robust frequentist methods, the REBLUP method of Sinha and Rao (2009) and the MQ method of Chambers and Tzavidis (2006).
We list in the table various estimates of county hectares of corn, along with their estimated standard errors or posterior standard deviations. Our analysis of the full data set including the potential outlier from the last small area shows that for the first 11 small areas there is a close agreement among the three sets of point estimates by Datta and Ghosh (1991), Sinha and Rao (2009) and the proposed normal mixture HB method. The Datta and Ghosh method, which was not developed to handle outliers, yields a point estimate for the 12th small area that is much different from the point estimates from Sinha-Rao or the proposed NM HB method. The latter two robust estimates are very similar in terms of point estimates for all the small areas. But when we compare these two sets of robust estimates with those from another robust method, namely, the MQ estimates, we find that the MQ estimates for the first three small areas are widely different from those for the other two methods. These numbers possibly indicate a potential bias of the MQ estimates.
\bigskip
\noindent
To compare performance of all these methods in the absence of any potential outliers, we reanalyzed the corn data by removing the suspected outlier (our robust HB analysis confirmed the outlier status of this observation, cf. Figure \ref{BHF-Reduced} below). When we compare the MQ estimates with the four other sets of estimates, the DG HB, the SR, the NM, which are reported in Table \ref{tab1:BHFanalysis}, and the EB estimates from Table 3 of Fabrizi et al. (2012), we notice a great divide between the MQ estimates and the other estimates. Out of the twelve small areas, the estimates for areas 1, 2, 3, 5, and 6 from the MQ method differ substantially from the estimates from the other four methods. On the other hand, the close agreement among the last four sets of estimates also shows in general the usefulness of the robust predictors, the proposed HB predictors and the Sinha-Rao robust EBLUP predictors.
\bigskip
\noindent
To examine the influence of the outlier on the estimates we compare changes in the estimates from both the full and reduced data. We find that the largest change occurs, not surprisingly, for the DG HB method for the small area suspected of the outlier. Such a large change occurred since the DG method cannot suitably downweight an outlier, consequently, it treated the outlier value of 88.59 in the same manner as it treated any other non-outlier observation. As a result, the predictor substantially underestimated the true mean $\bar Y_i$ for Hardin county. The next largest difference occurred for the MQ method for small area 3 which is not known to include any outlier. Such a large change is contrary to behavior of a robust method.
\begin{figure}[h]
\begin{center}
\begin{tabular}{cc}
\includegraphics[scale=.35]{BHF_Res_Outlier_Post_Prob.eps} & \includegraphics[scale=.35]{BHF_Res_Outlier_Post_Prob_reducedData.eps}
\end{tabular}
\caption{Posterior probabilities of observations being outliers in {\it full} and {\it reduced} data }\label{BHF-Reduced}
\end{center}
\end{figure}
\bigskip
\noindent
The changes in point estimates for the robust HB and the REBLUP methods are moderate for the areas not known to include any outliers, and the changes seem proportionate for the small area suspected of an outlier. The corresponding changes in the estimates from the MQ method for some of the areas not including any outlier seem disproportionately large, and the change in the estimate for the area suspected of an outlier is not as large. This behavior to some extent indicates a lack of robustness of the MQ method to outliers.
\bigskip
\noindent
An inspection of the posterior standard deviations of the two Bayesian methods reveals some interesting points.
First, the posterior SDs of the small area means for the proposed mixture model appear to be substantially smaller than the posterior SDs associated with the Datta-Ghosh HB estimators. Smaller posterior SDs suggest the posterior distribution of the small area means under the mixture model are more concentrated than those under the Datta-Ghosh model. This has been confirmed by simulation study, reported in the next section.
\bigskip
\noindent
Next, when we compare the posterior SDs of small area means for our proposed method based on the full data and the reduced data, all posterior SDs increase for the full data (which likely contain an outlier). In the presence of outliers, the unit-level variance is expected to be large. Even though the posterior SDs of the small area means do not depend entirely only on the unit-level error variance, they are expected to increase with this variance. This monotonic increase appears reasonable due to the suspected outlier. While this intuitive property holds for our proposed method, it does not hold for the standard Datta-Ghosh method.
\bigskip
\noindent
For further demonstration of the effectiveness of our proposed robust HB method, we computed model parameter estimates for both the reduced and the full data sets. These estimates are displayed in Table \ref{tabMPE:BHFanalysis}. The HB estimate of the larger variance component (976, based on mean) of the mixture is much larger than the estimate of the smaller component (182) for the full data, indicating a necessity of the mixture model. On the other hand, for the reduced data the estimates of variances for the two mixing components, 231 and 121, respectively are very similar and can be argued identical within errors in estimation, indicating limited need of the mixture distribution. A comparison of the estimates of $p_e$ for the two cases also reveals the appropriateness of the mixture model for the full data. It also shows the redundancy of including $p_e$ in the modeling of the reduced data as explained below.
\begin{table}[h]\caption{Parameter estimates for various models with and without the suspected outlier }\label{tabMPE:BHFanalysis}
\begin{center}
\scriptsize
\begin{tabular}{c|rr|rr|rr|rr|rr}
\hline
Estimates & \multicolumn{2}{c}{Datta-Ghosh HB} & \multicolumn{2}{c}{Datta-Ghosh HB} & \multicolumn{2}{c}{Proposed Mixture HB} & \multicolumn{2}{c}{Proposed Mixture HB} & \multicolumn{2}{c}{Sinha-Rao}\\
Estimates & \multicolumn{2}{c}{MEAN } & \multicolumn{2}{c}{MEDIAN } & \multicolumn{2}{c}{MEAN } & \multicolumn{2}{c}{MEDIAN } & \multicolumn{2}{c}{Sinha-Rao}\\
& Full & Reduced & Full & Reduced & Full & Reduced & Full & Reduced & Full & Reduced \\
& Data & Data & Data & Data & Data & Data & Data & Data & Data & Data \\
\hline
$\hat{\beta}_0$ & $ 17.29$ & $ 50.35$ & $ 16.17$ & $ 50.92$ & $ 30.89$ & $ 49.98$ & $ 31.46$ & $ 50.78$ & $ 29.14$ & $ 48.20$ \\
$\hat{\beta}_1$ & $ 0.37$ & $ 0.33$ & $ 0.37$ & $ 0.33$ & $ 0.35$ & $ 0.33$ & $ 0.35$ & $ 0.33$ & $ 0.36$ & $ 0.34$ \\
$\hat{\beta}_2$ & $ -0.03$ & $ -0.13$ & $ -0.03$ & $ -0.13$ & $ -0.07$ & $ -0.13$ & $ -0.07$ & $ -0.13$ & $ -0.07$ & $ -0.13$ \\
$\hat{p}_e$ & $ -$ & $ -$ & $ -$ & $ -$ & $ 0.62$ & $ 0.50$ & $ 0.68$ & $ 0.49$ & $ -$ & $ -$ \\
$\hat{\sigma}_v^2$ & $175.68$ & $231.87$ & $127.68$ & $186.07$ & $205.01$ & $238.42$ & $160.22$ & $203.55$ & $102.74$ & $155.15$ \\
$\hat{\sigma}_1^2$ & $ -$ & $ -$ & $ -$ & $ -$ & $182.01$ & $121.40$ & $170.64$ & $119.49$ & $ -$ & $ -$ \\
$\hat{\sigma}_2^2$ & $370.00$ & $216.00$ & $341.00$ & $192.00$ & $976.00$ & $231.00$ & $483.00$ & $188.00$ & $225.60$ & $161.50$ \\
\hline
\end{tabular}
\end{center}
\end{table}
\bigskip
\noindent
The posterior density in a reasonable noninformative Bayesian analysis is usually dominated by the likelihood of the parameters generated by the data. In case the data do not provide much information about some parameters to the likelihood, posterior densities of such parameters will be dominated by their prior information. Consequently, the posterior distribution for some of them may be very similar to the prior distribution. An overparameterized likelihood usually carries little information for some parameters responsible for overparameterization. In particular, if our mixture model is overparameterized in the sense that variances of mixture components are similar, then the integrated likelihood may be flat on the mixing proportion. We observe this scenario in our data analysis when we removed the suspected outlier observation from analysis based on our model. Since our mixture model is meant to accommodate outliers based on unequal variances for the mixing components, in the absence of any outliers the mixture of two normal distributions may not be required. In particular, we noticed earlier that with the suspected outlier removed the estimates of the two variance components $\sigma_1^2$ and $\sigma_2^2$ are very similar. Also, the posterior histogram of the mixing proportion $p_e$, not presented here, resembles a uniform distrubution, the prior distribution assigned in our Bayesian analysis. In fact, the posterior mean of this parameter for the reduced data is the same as the prior mean $0.5$. This essentially says that the likelihood is devoid of any information about $p_e$ to update the prior distribution.
\bigskip
\noindent
One advantage of our mixture model is that it explicitly models any representative outlier through the latent indicator variable $z_{ij}$. By computing the posterior probability of $z_{ij}=0$ we can compute the posterior probability that an observed $y_{ij}$ is an outlier. While the REBLUP method does not give a similar measure for an observation, one can determine the outlier status by computing the standardized residual associated with an observation. To show the effectiveness of our method, in Figure \ref{BHF-Reduced}, we plotted the posterior probabilities of an individual observation being an outlier against the observation's standardized residual. In the left panel, we showed the plot of these posterior probabilities for the full data, and in the right panel we included the same by removing the suspected outlier. These two figures are in sharp contrast; the left panel clearly showed that there is a high probability (0.86) that the second observation in Hardin county is an outlier. The associated large negative standardized residual of this observation also confirmed that, and from this plot an approximate monotonicity of these posterior probabilities with respect to the absolute values of the standardized residuals may also be discerned. However, the right panel shows that for the reduced data excluding the suspected outlier, the standardized residuals for the remaining observations are between $-3$ and $3$, with the associated posterior probabilities of being outlier observations are all between 0.44 and 0.64. None of these probabilities is particularly larger than prior probability 0.5 to indicate outlier status of that corresponding observation. This little change of the outlier prior probabilities in the posterior distribution for the reduced data essentially confirms that a discrete scale mixture of normal distributions is not supported by the data, or in other words, the scale mixture model is not required to explain the data, which is the same as that there are possibly no outliers in the data set.
\section{A Simulation Study}\label{sec:simul}
In our extensive simulation study, we followed the simulation setup used by Sinha and Rao (2009). Corresponding to the model in (\ref{NER-BHF}), we use a single auxiliary variable $x$, which we generated independently from a normal distribution with mean $1$ and variance $1$. In our simulations we use $m=40$. We generated 40 sets of 200 ($=N_i$) values of $x$ to create the finite population of covariates for the 40 small areas. Based on these simulated values we computed $\bar X_i = \frac 1{N_i}\sum_{j=1}^{N_i} x_{ij}$. Throughout our simulations we keep the generated $x$ values fixed. We used these generated $x_{ij}$ values and generated $v_i, i=1,\cdots, m$ independently from $N(0,\sigma_v^2)$ with $\sigma_v^2=1$. We generated $e_{ij}, j=1,\cdots, N_i, i=1,\cdots,m$ as iid from one of three possible distributions: (i) the case of no outliers where $e_{ij}$ are generated from $N(0,1)$ distribution; (ii) a mixture of normal distributions, with 10\% outliers from a $N(0,5^2)$ distribution and the remaining 90\% from the $N(0,1)$ distribution; and (iii) $e_{ij}$'s are iid from a $t$-distribution with 4 degrees of freedom. We also took $\beta_0 =1$ and $\beta_1 =1$ as in Sinha and Rao (2009), and generated $m$ small area finite populations based on the generated $x_{ij}$'s, $v_i$'s and $e_{ij}$'s by computing $Y_{ij} =\beta_0 +\beta_1 x_{ij} +v_i +e_{ij}$ based on the NER model in (\ref{NER-BHF}). Our goal is prediction of finite population small area means $\bar Y_i =\frac 1{N_i}\sum_{j=1}^{N_i} Y_{ij}, i=1,\cdots, m$. After examining no significant difference between $\bar Y_i$ and $\beta_0+\beta_1\bar X_i +v_i=\theta_i$ (say) in the simulated populations, as in Sinha and Rao (2009), we also consider prediction of $\theta_i$.
\bigskip
\noindent
From each simulated small area finite population we selected a simple random sample of size $n_i=4$ for each small area. Based on the selected samples we derived the HB predictors of Datta and Ghosh (1991) (referred to as DG), the REBLUPs of Sinha and Rao (2009) (referred to as SR), the MQ predictors of Chambers et al. (2014) (referred to as CCST-MQ, based on their equation (38)) and our proposed robust HB predictors (referred to as NM). In addition to the point predictors we also obtained the posterior variances of both the HB predictors and the estimates of the MSE of the REBLUPs based on the bootstrap method proposed by Sinha and Rao (2009), and the estimates of MSE of the MQ predictors, obtained by using pseudo-linearization in equation (39) of Chambers et al. (2014).
\bigskip
\noindent
For each simulation setup, we have simulated $S=100$ populations. For the $s$th created population, $s=1,\cdots, S$, we computed the values of $\theta_i^{(s)}$, which will be treated as the true values. We denote the $s$th simulation sample by $d^{(s)}$, and based on this data we calculate the REBLUP predictors $\hat{\theta}_{i,SR}^{(s)}$ and their estimated MSE, $mse(\hat{\theta}_{i,SR}^{(s)})$ using the procedure proposed by Sinha and Rao (2009).
To assess the accuracy of the point predictors we computed the empirical bias $eB_{i,SR}=\frac 1S \sum_{s=1}^S (\hat{\theta}_{i,SR}^{(s)} - \theta_i^{(s)})$ and empirical MSE $eM_{i,SR}=\frac 1S \sum_{s=1}^S (\hat{\theta}_{i,SR}^{(s)} - \theta_i^{(s)})^2$. Treating $eM_{i,SR}$ as the ``true'' measure of variability of $\hat{\theta}_{i,SR}$, we also evaluate the accuracy of the MSE estimator $mse(\hat{\theta}_{i,SR})$, suggested by Sinha and Rao (2009). Accuracy of the MSE estimator is evaluated by the relative difference between the empirical MSE and the average (over simulations) estimated MSE, given by $RE_{mse-SR,i} = \{ (1/S)\sum_{s=1}^S mse(\hat{\theta}_{i,SR}^{(s)}) - eM_{i,SR} \}/ eM_{i,SR}$. Similarly, we obtained the predictors $\hat{\theta}_{i,CCST}^{(s)}$, estimated MSEs $mse(\hat{\theta}_{i,CCST}^{(s)})$ of Chambers et al. (2014), empirical biases and empirical MSEs of point estimators and relative biases of the estimated MSEs. Using the point estimates and MSE estimates we created approximate 90\% prediction intervals $I_{i,SR,90}^{(s)} = [\hat{\theta}_{i,SR}^{(s)} -1.645\sqrt{mse(\hat{\theta}_{i,SR}^{(s)})},\hat{\theta}_{i,SR}^{(s)} +1.645\sqrt{mse(\hat{\theta}_{i,SR}^{(s)})}] $ and 95\% prediction intervals $I_{i,SR,95}^{(s)} = [\hat{\theta}_{i,SR}^{(s)} -1.96\sqrt{mse(\hat{\theta}_{i,SR}^{(s)})},\hat{\theta}_{i,SR}^{(s)} +1.96\sqrt{mse(\hat{\theta}_{i,SR}^{(s)})}] $. We also obtained similar intervals for the MQ method of Chambers et al. (2014).
We evaluated empirical biases, empirical MSEs, relative biases of estimated MSEs, and empirical coverage probabilities of prediction intervals for all four methods. These quantities for all 40 small areas are plotted in Figures \ref{tout:BiasVar3}, \ref{tout:EMSE4} and \ref{tout:CI2}.
\bigskip
\noindent
We plotted the empirical biases on the left panel and the empirical MSEs on the right panel of Figure \ref{tout:BiasVar3}. These estimators do not show any systematic bias. In terms of {\it eM}, the REBLUP and the proposed NM HB predictor appear to be most accurate and perform similarly (in fact, based on all evaluation criteria considered here, the proposed NM HB and the REBLUP methods have equivalent performance). In terms of {\it eM}, the MQ predictor has maximum variability and the standard DG HB predictor is in third place. In the case of no outliers, while the other three predictors have the same {\it eM}, the MQ predictor is slightly more variable. Moreover, we examined how closely the posterior variances of the Bayesian predictors and the MSE estimators of the frequentist robust predictors track their respective {\it eM} of prediction (see Figure \ref{tout:EMSE4}). The posterior variance of the proposed NM HB predictor and the estimated MSE of REBLUP appear to track the {\it eM} the best without any evidence of bias. The posterior variance of the standard HB predictor appears to overestimate the {\it eM} and the estimated MSE of the MQ predictor appears to underestimate. An undesirable consequence of this negative bias of the MSE estimator of the MQ method is that the related prediction intervals often fail to cover the true small area means (see the plots in Figure \ref{tout:CI2}).
\bigskip
\noindent
Our sampling-based Bayesian approach allowed us to create credible intervals for the small area means at the nominal levels of 0.90 and 0.95 based on sample quantiles of the Gibbs samples of the $\theta_i$'s. For the Sinha-Rao and the Chambers et al. methods we used their respective estimated root MSE of the REBLUPs or MQ-predictors to create symmetric approximate 90\% and 95\% prediction intervals of the small area means.
\bigskip
\noindent
To assess the coverage rate of these prediction intervals we computed empirical coverage probabilities $eC_{i,SR,90}=\frac 1S \sum_{s=1}^S I[\theta_i^{(s)} \in I_{i,SR,90}^{(s)}]$ and $eC_{i,SR,95}=\frac 1S \sum_{s=1}^S I[\theta_i^{(s)} \in I_{i,SR,95}^{(s)}]$, where $I[x\in A]$ is the usual indicator function that is one for $x \in A$ and 0 otherwise.
\bigskip
\noindent
Based on the same setup and same set of simulated data we also evaluated the two HB procedures. In the Bayesian approach, the point predictor, the posterior variance and the credible intervals for $\theta_i^{(s)}$ in the $s$th simulation were computed based on the MCMC samples of $\theta_i^{(s)}$ from its posterior distribution, generated by Gibbs sampling. The posterior mean and posterior variance are computed by the sample mean and the sample variance of the MCMC samples. An equi-tailed $100(1-2\alpha)\%$ credible interval for $\theta_i^{(s)}$ is created, where the lower limit is the $100\alpha$th sample percentile and the upper limit is the $100(1-\alpha)$th sample percentile of the MCMC samples of $\theta_i^{(s)}$ from the $s$th simulation.
\bigskip
\noindent
Suppose in the $s$th simulation $\hat{\theta}_{i,DG}^{(s)}$ denotes the Datta-Ghosh HB predictor of $\theta_i$ and $V_{i,DG}^{(s)}$ denotes the posterior variance. The empirical bias of the Datta-Ghosh predictor of $\theta_i$ is defined by $eB_{i,DG}=\frac 1S \sum_{s=1}^S (\hat{\theta}_{i,DG}^{(s)} - \theta_i^{(s)})$ and empirical MSE by $eM_{i,DG}=\frac 1S \sum_{s=1}^S (\hat{\theta}_{i,DG}^{(s)} - \theta_i^{(s)})^2$. To investigate the extent $V_{i,DG}^{(s)}$ may be interpreted as an estimated mse of the predictor $\hat{\theta}_{i,DG}$, we compute the relative difference between the empirical MSE and the average (over simulations) posterior variance, given by $RE_{V-DG,i} = \{ (1/S)\sum_{s=1}^S V_{i,DG}^{(s)} - eM_{i,DG} \}/ eM_{i,DG}$. These quantities for all 40 small areas are plotted in Figure \ref{tout:EMSE4}.
\bigskip
\noindent
Based on the MCMC samples of $\theta_i$'s for the $s$th simulated data set, let $I_{i,DG,90}^{(s)}$ be the 90\% credible interval for $\theta_i$. To evaluate the frequentist coverage probability of the credible interval for $\theta_i$ we computed empirical coverage probabilities $eC_{i,DG,90}=\frac 1S \sum_{s=1}^S I[\theta_i^{(s)} \in I_{i,DG,90}^{(s)}]$.
Corresponding to a credible interval $I_{i,DG,90}^{(s)}$, we use $L_{i,DG,90}^{(s)}$ to denote its length, and computed empirical average length of a 90\% credible interval for $\theta_i$ based on Datta-Ghosh approach by ${\bar L}_{i,DG,90} = \frac 1S \sum_{s=1}^S L_{i,DG,90}^{(s)}$. Similarly, we computed $eC_{i,DG,95}$ and ${\bar L}_{i,DG,95}$ for the 95\% credible intervals for $\theta_i$.
\bigskip
\noindent
Finally, as we did for the Datta-Ghosh HB predictor, we computed similar quantities for our new robust HB predictor. Specifically, suppose $\hat{\theta}_{i,NM}^{(s)}$ is the newly proposed NM HB predictor of $\theta_i^{(s)}$ and $V_{i,NM}^{(s)}$ is the posterior variance. For the new predictor we define the empirical bias by $eB_{i,NM}=\frac 1S \sum_{s=1}^S (\hat{\theta}_{i,NM}^{(s)} - \theta_i^{(s)})$ and empirical MSE by $eM_{i,NM}=\frac 1S \sum_{s=1}^S (\hat{\theta}_{i,NM}^{(s)} - \theta_i^{(s)})^2$. Again, to investigate the extent $V_{i,NM}^{(s)}$ may be viewed as an estimated MSE of the predictor $\hat{\theta}_{i,NM}$, we computed the relative difference between the emprical MSE and the average (over simulations) posterior variance, given by $RE_{V-NM,i} = \{ (1/S)\sum_{s=1}^S V_{i,NM}^{(s)} - eM_{i,NM} \}/ eM_{i,NM}$. These quantities for all 40 small areas are plotted in Figure \ref{tout:EMSE4}. Based on the MCMC samples of $\theta_i$'s for the $s$th simulated data set, let $I_{i,NM,90}^{(s)}$ be the 90\% credible interval for $\theta_i$. To evaluate the frequentist coverage probability of the credible interval for $\theta_i$ we computed empirical coverage probabilities $eC_{i,NM,90}=\frac 1S \sum_{s=1}^S I[\theta_i^{(s)} \in I_{i,NM,90}^{(s)}]$.
Corresponding to a credible interval $I_{i,NM,90}^{(s)}$, we use $L_{i,NM,90}^{(s)}$ to denote its length, and computed empirical average length of a 90\% credible interval for $\theta_i$ based on new approach by ${\bar L}_{i,NM,90} = \frac 1S \sum_{s=1}^S L_{i,NM,90}^{(s)}$. Similarly, we computed $eC_{i,NM,95}$ and ${\bar L}_{i,NM,95}$ for the 95\% credible intervals for $\theta_i$.
\bigskip
\noindent
We plotted the empirical coverage probabilities for the four methods that we considered in this article. The plot reveals significant undercoverage of the approximate prediction intervals created by using the estimated prediction MSE proposed by Chambers et al. (2014). This undercoverage is not surprising since their estimated MSE mostly underestimates the true MSE (measured by the {\it eM}) (see Figure \ref{tout:EMSE4}). Coverage probabilities of the Sinha-Rao prediction intervals and the two Bayesian credible intervals are remarkably accurate. This lends dual interpretation of our proposed credible intervals, Bayesian by construction, and frequentist by simulation validation. This property is highly desirable to practitioners, who often do not care about a paradigm or a philosophy. In the same plot, we also plotted the ratio of the average lengths of the DG credible intervals to the newly proposed robust HB credible intervals. These plots show the superiority of the proposed method, yielding intervals which meet coverage accurately with average lengths about 25-30\% shorter compared to the DG method for normal mixture model with 10\% contamination. Again these two intervals meet the coverage accurately when the unit-level errors are generated from normal (no outliers) or a moderately heavy-tail distribution ($t_4$). In these cases, the reduction in length of the intervals is less, which is about 10\%. This shorter prediction intervals from the new method even for normal distribution for the unit-level error is interesting; it shows that the proposed method does not lose any efficiency in comparison with the Datta-Ghosh method even when the normality of the unit-level errors holds.
\bigskip
\noindent
The comparison of NM HB prediction intervals and the Sinha-Rao prediction intervals yields a mixed picture. In the mixture setup, the NM HB prediction intervals attained coverage probability more accurately than the Sinha-Rao intervals, which undercover by 1\%, and on an average the Bayesian prediction intervals are about 2\% shorter than the frequentist intervals. When the data are simulated from a $t_4$ distribution, the coverage probabilities of the Sinha-Rao prediction intervals are about 1\% below the target, but these intervals are about 3\% shorter than the NM HB prediction intervals, which attained the nominal coverage. Finally, when the population does not include any outlier, these two methods perform the same, both attained the nominal coverage and yield the same average length.
\section{Conclusion}
The NER model by Battese et al. (1988) plays an important role in small area estimation for unit-level data. While Battese et al. (1988), Prasad and Rao (1990) and Datta and Lahiri (2000) investigated EBLUPs of small area means, Datta and Ghosh (1991) proposed an HB approach for this model. Sinha and Rao (2009) investigated robustness of the MSE estimates of EBLUPs in Prasad and Rao (1990) for outliers in the response. They showed in presence of outliers robustness of their REBLUPs and lack of robustness of the EBLUPs.
\bigskip
\noindent
In this article we showed that non-robustness also persists for the HB predictors by Datta and Ghosh (1991). To deal with this undesirable issue we proposed an alternative to the HB predictors by using a mixture of normal distributions for the unit-level error part of the NER model. An illustrative application and simulation study show the superiority of our proposed method over the existing HB, EBLUP and M-quantile solutions. Indeed simulation results show the superiority of our method over the Datta and Ghosh (1991) HB predictors and the M-quantile small area estimators of Chambers et al. (2014). Performance of our proposed NM HB method is found to be as good as the frequentist solution of Sinha and Rao (2009). Our proposed Bayesian intervals also achieve the corresponding frequentist coverage. Thus, unlike the frequentist solutions, our proposed HB solution enjoys dual interpretation, Bayesian by construction, and frequentist via simulation, a feature attractive to practitioners. Moreover, suggested credible intervals are shorter in length in comparison with the other nominal prediction intervals. In fact, the application and simulations show the proposed NM HB method is the best among the four methods in presence of outliers. Our proposed method is as good as the HB method of Datta and Ghosh (1991), even in absence of outliers. Thus there will be no loss in using the proposed HB method for all data sets. It is not clear to us that why M-quantile performs poorly. However, we note that in our simulations, all the errors are centered at zero. Alternatively, one can explore the performance of these methods when the outlier parts of the respective error components are generated from a distribution which is not centered at zero. This remains a topic of future research.
\section{Acknowledgment}
Authors are thankful to Drs. Bill Bell and Jerry Maples for their insightful comments.
\section*{Supporting Information}
Property of the posterior distribution corresponding to the proposed
model has been discussed in the supplementary material.
\clearpage
\section*{References}
\def\beginref{\begingroup
\clubpenalty=10000
\widowpenalty=10000
\normalbaselines\parindent 0pt
\parskip.0\baselineskip
\everypar{\hangindent1em}}
\def\par\endgroup{\par\endgroup}
\beginref
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Breckling, J. and Chambers, R. (1988), M-quantiles, {\it Biometrika}, {\bf 75}, $761-771$
Chakraborty, A., Datta, G. S. and Mandal, A. (2016), A two-component normal mixture alternative to the Fay-Herriot model, {\em Statistics in Transition new series and Survey Methodology Joint Issue: Small Area Estimation 2014}, {\bf 17}, 67--90.
Chambers, R. L. (1986), Outlier robust finite population estimation, {\em Journal of the American Statistical Association}, {\bf 81}, 1063--1069.
Chambers, R., Chandra, H., Salvati, N. and Tzavidis, N. (2014), Outlier robust small area estimation. {\em Journal of the Royal Statistical Society Series B}, {\bf 76}, 47–-69.
Chambers, R.L. and Tzavidis, N. (2006), M-quantile models for small area estimation, {\it Biometrika}, {\bf 93}, $255-268.$.
Datta, G. and Ghosh, M. (1991), Bayesian prediction in linear models: Applications to small area estimation, {\em Annals of Statistics}, {\bf 19}, 1748--1770.
Datta, G. S. and Lahiri, P. (1995), Robust hierarchical Bayesian estimation of small area characteristics in presence of covariates and outliers, {\em Journal of Multivariate Analysis}, {\bf 54}, 310--328.
Datta, G. S. and Lahiri, P. (2000), A unified measure of uncertainty of estimated best linear unbiased predictors in small area estimation problems, {\em Statistica Sinica}, {\bf 10}, 613--627.
Datta, G. S., Rao, J. N. K. and Smith, D. D. (2005), On measuring the variability of small area estimators under a basic area level model, {\em Biometrika}, {\bf 92}, 183--196.
Fay, R. E. and Herriot, R. A. (1979), Estimates of income for small places:
an application of James-Stein procedures to census data, {\em Journal of the American Statistical Association}, {\bf 74}, 269--277.
Fellner, W. H. (1986), Robust estimation of variance components, {\em Technometrics}, {\bf 28}, 51--60.
Fabrizi, E., Salvati, N. and Pratesi (2012), M. Constrained small area estimators based on M-quantile methods. {\em Journal of Official Statistics}, {\bf 28}, 89–-106.
Gershunskaya, J. (2010). Robust Small Area Estimation Using a Mixture Model. {\it Proceedings of the Section on Survey Research Methods Section}, Washington, DC: American Statistical Association.
Hobert, J. and Casella, G. (1996), Effect of improper priors on Gibbs sampling in hierarchical linear mixed models, {\em Journal of the American Statistical Association}, {\bf 91}, 1461--1473.
Lahiri, P. and Rao, J.N.K. (1995), Robust estimation of mean square error of small area estimators, {\em Journal of the American Statistical Association}, {\bf 90}, 758--766.
Pfeffermann, D. and Sverchkov, M. (2007), Small area estimation under informative probability sampling of areas and within the selected areas, {\em Journal of the American Statistical Association}, {\bf 102}, 1427--1439.
Prasad, N. G. N. and Rao, J. N. K. (1990), On the estimation of mean square error of small area predictors, {\em Journal of the American Statistical Association}, {\bf 85}, 163--171.
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{Tzavidis, N. and Chambers, R. (2005), Bias adjusted estimation for small areas with M-quantile models,
{\em Statistics in Transition} {\bf 7}, 707--713.}
Verret, F., Rao, J. N. K. and Hidiroglou, M. (2015), Model-based small area estimation under informative sampling, {\em Survey Methodology}, {\bf 41}, 333--347.
\par\endgroup
\clearpage
\begin{figure}[h]
\begin{center}
\begin{tabular}{ccc}
\raisebox{3cm}{10\%\hspace{.75cm}} & \includegraphics[scale=.375]{Bias_10_Out.eps} & \includegraphics[scale=.375]{EMSE_10_Out.eps} \\
\raisebox{3cm}{$t_{(4)}$\hspace{.75cm}} & \includegraphics[scale=.375]{Bias_t_Out.eps} & \includegraphics[scale=.375]{EMSE_t_Out.eps} \\
\raisebox{3cm}{No outlier} & \includegraphics[scale=.375]{Bias_No_Out.eps} & \includegraphics[scale=.375]{EMSE_No_Out.eps}
\end{tabular}
\caption{Plot of empirical biases and empirical MSEs of $\hat{\theta}$s}\label{tout:BiasVar3}
\end{center}
\end{figure}
\clearpage
\begin{figure}[h]
\begin{center}
\begin{tabular}{ccc}
\raisebox{3cm}{10\%\hspace{.75cm}} & \includegraphics[scale=.375] {Var_10_Out.eps}& \includegraphics[scale=.375]{Rel_bias_10_Out.eps} \\
\raisebox{3cm}{$t_{(4)}$\hspace{.75cm}} & \includegraphics[scale=.375] {Var_t_Out.eps} & \includegraphics[scale=.375] {Rel_bias_t_Out.eps} \\
\raisebox{3cm}{No outlier} & \includegraphics[scale=.375]{Var_No_Out.eps} & \includegraphics[scale=.375]{Rel_bias_No_Out.eps}
\end{tabular}
\caption{Plot of posterior variances and MSE estimates and their empirical relative biases }\label{tout:EMSE4}
\end{center}
\end{figure}
\clearpage
\begin{figure}[h]
\vspace{-1.75cm}
\begin{center}
\begin{tabular}{ccc}
\raisebox{3cm}{10\%\hspace{.75cm}} & \includegraphics[scale=.375]{CI90_10_Out.eps} & \includegraphics[scale=.375]{CI95_10_Out.eps} \\
\raisebox{3cm}{$t_{(4)}$\hspace{.75cm}} & \includegraphics[scale=.375]{CI90_t_Out.eps} & \includegraphics[scale=.375]{CI95_t_Out.eps} \\
\raisebox{3cm}{No outlier} & \includegraphics[scale=.375]{CI90_No_Out.eps} & \includegraphics[scale=.375]{CI95_No_Out.eps} \\
\end{tabular}
\caption{Plot of lengths and coverages of credible and prediction intervals}\label{tout:CI2}
\vspace{-1.075cm}
\includegraphics[scale=.35,trim = 0 500 0 100]{CI_Legend1.eps} \includegraphics[scale=.35,trim = 0 500 0 010]{CI_Legend2.eps} \includegraphics[scale=.35,trim = 0 500 0 010]{CI_Legend3.eps}
\end{center}
\end{figure}
\end{document}
|
3,212,635,537,434 | arxiv | \section{Introduction}
Vehicular networks is a typical application of 5G ultra-reliable and low-latency communication (URLLC). It is also an important mean to realize automatic driving in intelligent transportation systems (ITS) \cite{1}. Automatic driving technology is helpful to avoid traffic accidents and reduce traffic congestion \cite{2,3}. In order to realize automatic driving technology, the accuracy of positioning and controlling for automatic driving vehicles (ADVs) should be up to the centimeter level. Therefore, a series of sensor information should be provided for the ADVs in order to support the centimeter accuracy \cite{4}.\par
A rapid lane detection algorithm was proposed based on machine vision, which is accurate and robust under different conditions, such as lane line missing and obstacle appearance in the track \cite{5}. However, many vision-based research challenges have not yet been solved, such as the low-definition image clarity and poor visibility in rainy, hazy weather and night conditions \cite{6}. Since vision-based automatic driving probably has safety risks in the situations mentioned above, high-definition (HD) map emerges to support automatic driving. The road information contained by HD map has enough precision to help ADVs identify the road signs with a centimeter accuracy. On the other hand, HD map also contains real-time traffic information, such as the state information of running cars, pedestrians and cyclists, which will be helpful to avoid accidents in critical situations with fast response times \cite{7}. Moreover, HD map-based vehicle localization and predictive cruise control have been studied in \cite{8,9}. However, how to transmit the HD map by wireless network is still an open problem to our best knowledge. Since HD map contains more road and traffic related information than traditional maps, the data volume is huge for the network to delivery. Therefore, it is necessary to design some transmission schemes to support high transmission rate with low network cost and latency. \par
\parskip=0pt
Vehicle-to-infrastructure (V2I) makes vehicles connect to the neworks via roadside units (RSUs) and supports high-speed short-range communications \cite{10}. Therefore, a transmission scheme based on V2I communication has been proposed through jointly optimizing the traffic flow rate and the power consumption of the network \cite{11}. On the other hand, in addition to transmitting HD map by RSU, two cars driving in the opposite direction contain the HD map information needed by each other. Therefore, vehicle-to-vehicle (V2V) communication can also be employed for HD map transmission. Besides, V2V communications have shorter communication distance than V2I, which can reduce the path loss and transmission delay \cite{12,13}.
\par
Motivated by this, this paper studies the HD map transmission for automatic driving. Considering both the power efficiency and communication efficiency, a collaborative V2X transmission scheme is proposed in order to achieve high-speed HD map transmission with low power cost. The proposed scheme combines both V2I and V2V communications and adaptively allocates the power between the RSU and vehicle. On the other hand, a more realistic estimation expression of the transmission rate is adopted in order to reflect the influence of the decoding error probability. The simulation results indicate that the proposed transmission scheme can reduce power consumption while guaranteeing the transmission rate requirement of HD map.\par
The remainder of this paper is organized as follows. Section \uppercase\expandafter{\romannumeral2} describes the system model and formulates the HD map transmission problem with power consumption minimization. A cooperative transmission scheme is proposed based on mobile vehicle rule in Section \uppercase\expandafter{\romannumeral3}. The simulation results and analysis are given in Section \uppercase\expandafter{\romannumeral4}. Section \uppercase\expandafter{\romannumeral5} concludes this paper.
\par
\begin{figure*}[!t]
\centering
\subfigure[V2I transmission only]{
\label{Fig.sub.1}
\includegraphics[width=0.31\textwidth]{SystemModel1.eps}}
\subfigure[Cooperative V2X transmission]{
\label{Fig.sub.2}
\includegraphics[width=0.31\textwidth]{SystemModel2.eps}}
\subfigure[V2V transmission only]{
\label{Fig.sub.3}
\includegraphics[width=0.31\textwidth]{SystemModel3.eps}}
\vspace{-6pt}
\caption{Illustration of cooperative transmission of high definition map.}
\vspace{-1.3em}
\label{Fig.main}
\end{figure*}
\section{System Model and Problem Formulation}
\subsection{System Model}
As shown in Fig. 1, two vehicles driving in the opposite directions on an urban road, where the red one is considered as the targeted vehicle of the HD map requirement, the blue one has stored the required HD map information in the forward direction of the red vehicle.
The different frequency bands with bandwidths $B_{\rm V}$ and $B_{\rm R}$ are employed by V2V and V2I in order to avoid interference. Then, different parts of HD map information can be simultaneously transmitted by the blue vehicle and the RSU for the targeted vehicle. Moreover, it is assumed that the HD map data volume per meter $q$ bits/m follows truncated Gaussian distribution, i.e., $q \sim \mathbb{N}(\mu,{{\sigma} ^2},0,+ \infty )$, where $\mu$ and ${\sigma}$ are the mean and variance, and the data volume is positive. When the vehicle speed is $v$ m/s, the transmission rate of the networks at least is $vq$ bits/s.\par
We assume that ${d_{{\rm{V}}}}$ denotes the distance between the two vehicles and ${d_{{\rm{VM}}}}$ indicates the maximum V2V communication range. As shown in Fig. 1, three cases of transmission schemes are then discussed based on the position relationship of the two vehicles. In Fig. 1(a), only RSU transmits the HD map information to the targeted vehicle when ${d_{{\rm{V}}}}>{d_{{\rm{VM}}}}$. In Fig. 1(b), when ${d_{{\rm{V}}}}<{d_{{\rm{VM}}}}$, both RSU and vehicle transmit the HD map information to the targeted vehicle. In Fig. 1(c), ${d_{{\rm{V}}}}$ becomes so small that the HD map transmission rate can be satisfied only by the V2V communication. Next, the channel models, transmission model and transmission rate are given in details.
\subsubsection{Channel Model}
Denote $\phi_0 {\it{d}_{\rm{V}}^{ - \alpha }}$ and $\phi_0 {\it{d}_{\rm{R}}^{ - \alpha }}$ as the gains of downlink large-scale channels for V2V and V2I, respectively, where $\phi_0$ is a channel constant related to the antenna gain and carrier frequency, $\alpha$ is the path-loss exponent, $d_{\rm R}$ represents the distance between the red vehicle and RSU. On the other hand, denote ${h}_{\rm V}$ and ${h}_{\rm R}$ as the fast fading coefficients of the V2V and V2I channels, respectively, and ${h}_{\rm V}$, ${h}_{\rm R} \sim \mathbb{CN}(0,1)$. Then, the V2V and V2I channels can be expressed as ${g_{\rm V}} \rm{ = } \phi_0 {\it{d}_{\rm V}^{ - \alpha }}{\it h}_{\rm V}$ and ${g_{\rm R}} \rm{ = } \phi_0 {\it{d}_{\rm{R}}^{ - \alpha }} \it{h}_{\rm R}$, respectively.
\subsubsection{Transmission Model}
Assuming that the transmit power values of the blue vehicle and RSU are $p_{\rm V}$ and $p_{\rm R}$, and $s_{\rm V}$ with $\left| s_{\rm V} \right| \rm{ = } 1 $ and $s_{\rm R}$ with $\left| s_{\rm R} \right| \rm{ = } 1 $ represent the transmitted symbols from the blue vehicle and RSU, respectively. Then the transmission signals of the blue vehicle and RSU can be written respectively by
\vspace{-0.5em}
\begin{align}
\vspace{-0.3em}
&x_{\rm V} { = } \sqrt{p_{\rm{V}}}s_{\rm V},\\
&x_{\rm R} { = } \sqrt{p_{\rm{R}}}s_{\rm R},
\end{align}
and the received signals of the red vehicle are respectively given by
\vspace{-0.5em}
\begin{align}
\vspace{-0.5em}
\notag y_{\rm V} &= {{g}_{\rm V} \it {x_{\rm V} }+n_{\rm V}}\\
&={{{\phi_0 {\it{d}_{\rm V}^{ - \alpha }}{{h}_{\rm V}}\sqrt {p_{\rm{V}}} {s_{\rm V}}} + n_{\rm V}}},\\
\notag y_{\rm R} &= {{g}_{\rm R} \it {x_{\rm R} }+n_{\rm R}}\\
&={{{\phi_0 {\it{d}_{\rm R}^{ - \alpha }}{{h}_{\rm R}}\sqrt {p_{{\rm R}}} {s_{\rm R}}} + n_{\rm R}}},
\end{align}
where the additive white Gaussian noises $n_{\rm V}\sim \mathbb{CN}(0,{B_{\rm V}N_0}), n_{\rm R} \sim \mathbb{CN}(0,{B_{\rm R}N_0})$. Therefore, the signal to noise ratios (SNRs) of the red vehicle can be expressed respectively as
\begin{align}
&\rho_{\rm V}= {\frac{{{\phi _0}d_{\rm{V}}^{ - \alpha }{h_{\rm{V}}}{p_{\rm{V}}}}}{{{\phi _1}{N_0}{B_{\rm{V}}}}}},\\
&\rho_{\rm R}= {\frac{{{\phi _0}d_{\rm{R}}^{ - \alpha }{h_{\rm{R}}}{p_{\rm{R}}}}}{{{\phi _1}{N_0}{B_{\rm{R}}}}}},
\end{align}
where $\phi _1$ is a SNR loss coefficient due to non-ideal channel state information at the transmitter.
\par
\subsubsection{Transmission rate}
Considering the decoding error probability, the transmission rates of vehicle and RSU can be expressed respectively as
\begin{align}
{r_{\rm{V}}}\! =\! \frac{{{B_{\rm{V}}}}}{{\ln 2}}\!\left[\! {\ln \!\left(\! {1\!+\!\frac{{{\phi _0}d_{\rm{V}}^{\! -\! \alpha }{h_{\rm{V}}}{p_{\rm{V}}}}}{{{\phi _1}{N_0}{B_{\rm{V}}}}}} \!\right) \!\!-\!\! \sqrt {\frac{1}{{\tau {B_{\rm{V}}}}}} Q_{\rm{G}}^{ \!- \!1}\!\left(\! {\varepsilon _{\rm{V}}^{{\rm{C,V}}}}\! \right)}\! \right]\!\!,
\end{align}
\begin{align}
{r_{\rm{R}}}\! = \!\frac{{{B_{\rm{R}}}}}{{\ln 2}}\!\left[\! {\ln \!\left(\! {1 \!+\! \frac{{{\phi _0}d_{\rm{R}}^{\! -\! \alpha }{h_{\rm{R}}}{p_{\rm{R}}}}}{{{\phi _1}{N_0}{B_{\rm{R}}}}}}\! \right)\! \!-\!\! \sqrt {\frac{1}{{\tau {B_{\rm{R}}}}}} Q_{\rm{G}}^{ \!- \!1}\!\left(\! {\varepsilon _{\rm{R}}^{{\rm{C,R}}}} \!\right)}\! \right]\!\!,
\end{align}
where $\tau$ is the duration of transmission, $ {\varepsilon _{\rm{V}}^{{\rm{C,V}}}}$ and $ {\varepsilon _{\rm{R}}^{{\rm{C,R}}}} $ are decoding error probabilities in downlink of V2V channel and V2I channel, respectively, and $Q_{\rm{G}}^{ - 1}\left( x \right)$ denotes the inverse of the Gaussian Q-function \cite{14,15}.
\par
\subsection{Problem Formulation}
Based on the proposed collaborative V2X transmission scheme, the objective of this paper is to minimize the total power consumption of both the RSU and vehicles as well as ensure the transmission rate requirement of the HD map. Therefore, the optimization problem of this paper can be formulated as \par
\vspace{-1em}
\begin{align}
&\mathop {\min }\limits_{{p_{\rm{R}}},{p_{\rm{V}}}} \left\{ {{p_{\rm{R}}} + {p_{\rm{V}}}} \right\}\\
{\rm{s}}{\rm{.t}}&{\rm{. 0}} \le {p_{\rm{R}}} \le {p_{{\rm{RM}}}},\\
&{\rm{ 0}} \le {p_{\rm{V}}} \le {p_{{\rm{VM}}}},\\
&{\rm{ P}}\left\{ {{r_{\rm{R}}} + {r_{\rm{V}}} < qv} \right\} \le \delta ,\\
&{\rm{ 0}} \le {d_{\rm{V}}} \le {d_{{\rm{VM}}}},
\end{align}
where (10) and (11) are the transmit power constraints of vehicle and RSU, ${p_{{\rm{RM}}}}$ and ${p_{{\rm{VM}}}}$ denote the maximum transmit power values of RSU and vehicle, respectively, (12) describes the outage probability requirement of HD map transmission and $\delta$ is the maximum violation probability.
\par
\section{Cooperative power allocation for HD Map Transmission}
In order to solve the formulated problem described in (9)-(13), we need to transform the transmission rate outage constraint (12) into a constraint of $p_{\rm{R}}$ or $p_{\rm{V}}$ by considering the distribution of the HD map volume $q$. Since the expression of tansmission rate in (7) or (8) contains a constant term, (12) has different expressions in different situations, i.e.
\vspace{-0.5em}
\begin{align}
&{{\rm{P}}\left\{ {{r_{\rm{R}}} < qv} \right\} \le \delta ,}\\
&{{\rm{ P}}\left\{ {{r_{\rm{V}}} < qv} \right\} \le \delta ,{\rm{ }}}\\
&{{\rm{ P}}\left\{ {{r_{\rm{R}}} + {r_{\rm{v}}} < qv} \right\} \le \delta .}
\end{align}
We will solve the problem in (9)-(13) under the conditions (14)-(16), respectively, i.e., V2I transmission only, V2V transmission only, and cooperative V2X transmission. Then, we can obtain three suboptimal power allocation schemes. Finally, the optimal power allocation can be obtained by comprehensively considering the sum transmit power values of both RSU and vehicle.
\subsection{Suboptimal Power Allocation}
\newtheoremstyle{mythm}{}{}{\normalfont}{}{ \bfseries \it}{\normalfont:}{.5em}{}
\theoremstyle{mythm}
\newtheorem{theorem}{\quad \it Proposition}
\newtheorem{Commen}{\quad \it Comment }
\renewenvironment{proof}{{\it Proof \normalfont:}}{ \hfill $\blacksquare$ }
\subsubsection{V2I transmission only}
We have the following results.
\begin{theorem}
When the HD map is transmitted by V2I only, i.e., $p_{\rm{V}}=0$, the power allocation of the vehicle and the RSU is given by
\begin{align}
{\psi _1} = \left( {{p_{\rm{V}}},{p_{\rm{R}}}} \right),
\end{align}
where
\begin{align}
\notag {p_{\rm{R}}} =& \!\left\{\! {\frac{{{\phi _1}{N_0}{B_{\rm{R}}}}}{{{\phi _0}d_{\rm{R}}^{ \!-\! \alpha }{h_{\rm{R}}}}}\!\!\left[\! {\exp \!\!\left(\! {\frac{{v\!\!\left(\! {\sigma {\Phi ^{ \!-\!1}}\!\left(\! {1 \!-\! \delta \!\left[\! {1\!-\! \Phi \!\left(\! { \!-\!{\textstyle{\mu \over \sigma }}} \!\right)\!} \right]\!} \right)\! \!+\! \mu } \!\right)\!\ln 2}}{{{B_{\rm{R}}}}}} \right.} \right.} \right.\\
&\left. {\left. {\left. { \!+\! \sqrt {\frac{1}{{\tau {B_{\rm{R}}}}}} Q_{\rm{G}}^{ \!-\! 1}\!\left(\! {\varepsilon _{\rm{R}}^{{\rm{C,R}}}} \!\right)}\! \right)\! \!-\! 1} \!\right]\!} \right\}_0^{{p_{{\rm{RM}}}}}\!\!\!\!,
\end{align}
where $\Phi \left( x \right)$ is the cumulative distribution function of standard normal distribution, and $\{x\}_a^b=\min \{b, \max \{a,x\}\}$.
\end{theorem}
\begin{proof}
To obtain the power allocation under the conditions (14),
we need to transform the transmission rate constraint (12). The probability density function of $q$ can be given by
\begin{align}
\!\!f\left( q \right) &\!=\! \frac{{{\textstyle{1 \over {{\sigma }}}}{f_N}\left( {{\textstyle{{q \!-\! \mu } \over {{\sigma }}}}} \right)}}{{1 \!-\! \Phi \left( { \!-\! {\textstyle{\mu \over {{\sigma }}}}} \right)}}\!\!=\!\!\frac{{\exp \left( { \!-\! \frac{1}{2}{{\left( {{\textstyle{{q\!-\!\mu } \over {{\sigma }}}}} \right)}^2}} \right)}}{{\sqrt {2\pi } {\sigma}\left[ {1 \!-\! \Phi \left( {\!-\!{\textstyle{\mu \over {{\sigma}}}}} \right)} \right]}},
\end{align}
where ${f_N}\left( x \right)$ is the probability density function of standard normal distribution. Then, the left side of inequality (12) can be transformed into
\begin{align}
\notag &{\rm{P}}\!\left\{\!\! {\frac{{{B_{\rm{R}}}}}{{\ln 2}}\!\!\left[\! {\ln \!\!\left(\!\! {1 \!+\! \frac{{{\phi _0}d_{\rm{R}}^{ \!-\! \alpha }{h_{\rm{R}}}{P_{\rm{R}}}}}{{{\phi _1}{N_0}{B_{\rm{R}}}}}}\! \!\right)\! \!\!-\!\! \sqrt {\frac{1}{{\tau {B_{\rm{R}}}}}} Q_{\rm{G}}^{ \!-\! 1}\!\left(\! {\varepsilon _{\rm{R}}^{{\rm{C,R}}}} \!\right)}\! \right]\!\! <\! qv} \!\right\}\\
\!=&{\rm{P}}\!\left\{\!\! {q \!>\!\! \frac{{{B_{\rm{R}}}}}{{v\ln 2}}\!\!\left[\! {\ln \!\!\left(\!\! {1 \!+\! \frac{{{\phi _0}d_{\rm{R}}^{ \!-\! \alpha }{h_{\rm{R}}}{P_{\rm{R}}}}}{{{\phi _1}{N_0}{B_{\rm{R}}}}}} \!\!\right)\! \!-\!\! \sqrt {\frac{1}{{\tau {B_{\rm{R}}}}}} Q_{\rm{G}}^{ \!-\! 1}\!\left(\! {\varepsilon _{\rm{R}}^{{\rm{C,R}}}}\! \right)\!} \!\right]\!\!} \right\}\!\!\!.
\end{align}\par
In order to simplify the expression, we define the constant
\begin{align}
\kappa {\rm{ = }} \frac{{{B_{\rm{R}}}}}{{v\ln 2}}\!\!\left[\! {\ln \!\!\left(\!\! {1 \!+\! \frac{{{\phi _0}d_{\rm{R}}^{ \!-\! \alpha }{h_{\rm{R}}}{P_{\rm{R}}}}}{{{\phi _1}{N_0}{B_{\rm{R}}}}}} \!\!\right)\! \!-\!\! \sqrt {\frac{1}{{\tau {B_{\rm{R}}}}}} Q_{\rm{G}}^{ \!-\! 1}\!\left(\! {\varepsilon _{\rm{R}}^{{\rm{C,R}}}}\! \right)\!} \!\right]\!\!.
\end{align}
Then, based on (19) and (21), the outage probability in (20) can be expressed as
\begin{align}
\notag \int_\kappa ^{\! +\! \infty }\!\! {f\!\!\left( q \right)\!{\rm d}q} =& \int_\kappa ^{ \!+\! \infty } \!\!{\frac{{\exp \left( { \!-\! \frac{1}{2}{{\!\left( \!{{\textstyle{{q\! -\! \mu } \over {{\sigma }}}}}\! \right)\!}^2}} \right)}}{{\sqrt {2\pi } {\sigma }\left[ {1 \!-\! \Phi \!\left(\! { - {\textstyle{\mu \over {{\sigma }}}}}\! \right)\!} \right]}}{\rm d}q} \\
=& \frac{{1 \!-\! \Phi \!\left(\! {{\textstyle{{\kappa \!-\! \mu } \over {{\sigma}}}}} \!\right)\!}}{{1 \!-\! \Phi \!\left(\! { \!-\! {\textstyle{\mu \over {{\sigma }}}}} \!\right)\!}},
\end{align}
and therefore (12) can be rewritten as
\begin{align}
\frac{{1 \!-\! \Phi \!\left(\! {{\textstyle{{\kappa \!-\! \mu } \over {{\sigma }}}}} \!\right)\!}}{{1 \!-\! \Phi \!\left(\! { \!-\! {\textstyle{\mu \over {{\sigma }}}}} \!\right)\!}} \le \delta .
\end{align}
Further, taking into account equation (21), constraint (12) can be transformed into
\begin{align}
\notag {p_{\rm{R}}} \!\ge\! \!\frac{{{\phi _1}{N_0}{B_{\rm{R}}}}}{{{\phi _0}d_{\rm{R}}^{ - \alpha }{h_{\rm{R}}}}}\!\!&\left[ \!{\exp\! \left(\!\! {\frac{{v\!\left(\! {\sigma {\Phi ^{ \!-\! 1}}\!\!\left(\! {1 \!-\! \delta\! \left[\! {1\! - \!\Phi \!\left(\! {\! - \!{\textstyle{\mu \over \sigma }}}\! \right)} \!\right]\!} \right) \!+\! \mu } \!\right)\!\ln 2}}{{{B_{\rm{R}}}}}} \right.} \right.\\
&\left. {\left. {\! +\! \sqrt {\frac{1}{{\tau {B_{\rm{R}}}}}} Q_{\rm{G}}^{ \!-\! 1}\!\left(\! {\varepsilon _{\rm{R}}^{{\rm{C,R}}}}\! \right)} \!\right)\!\! - \!1} \right].\end{align}
\par
Considering constraints (10) and (11), the power allocation under V2I transmission only can be given by (17).
\end{proof}
\subsubsection{V2V transmission only}
We have the following results.
\begin{theorem}
When the HD map is transmitted by V2V only, i.e., $p_{\rm R}=0$, the power allocation of the vehicle and the RSU is given by
\begin{align}
{\psi _2} = \left( {{p_{\rm{V}}},{p_{\rm{R}}}} \right),
\end{align}
where
\begin{align}
\notag {p_{\rm{V}}} =& \!\left\{\! {\frac{{{\phi _1}{N_0}{B_{\rm{V}}}}}{{{\phi _0}d_{\rm{V}}^{ \!-\! \alpha }{h_{\rm{V}}}}}\!\!\left[\! {\exp \!\!\left(\! {\frac{{v\!\!\left(\! {\sigma {\Phi ^{ \!-\!1}}\!\left(\! {1 \!-\! \delta \!\left[\! {1\!-\! \Phi \!\left(\! { \!-\!{\textstyle{\mu \over \sigma }}} \!\right)\!} \right]\!} \right)\! \!+\! \mu } \!\right)\!\ln 2}}{{{B_{\rm{V}}}}}} \right.} \right.} \right.\\
&\left. {\left. {\left. { \!+\! \sqrt {\frac{1}{{\tau {B_{\rm{V}}}}}} Q_{\rm{G}}^{ \!-\! 1}\!\left(\! {\varepsilon _{\rm{V}}^{{\rm{C,V}}}} \!\right)}\! \right)\! \!-\! 1} \!\right]\!} \right\}_0^{{p_{{\rm{VM}}}}}\!\!\!\!.
\end{align}
\end{theorem}
\begin{proof}
The proof is similar to that of {\it Proposition} 1 and therefore is ignored.
\end{proof}
\subsubsection{Cooperative V2X transmission}We have the following results.
\begin{theorem}
When the HD map is cooperatively transmitted by V2V and V2I, the power allocation of vehicle and RSU is given by
\begin{align}
{\psi _3}= \left( {{{p}_{\rm{V}}},{p_{\rm{R}}}} \right),
\end{align}
where
\begin{align}
&{p_{\rm{V}}}=\!{{\frac{{{\phi _1}{N_0}{B_{\rm{V}}}}}{{{\phi _0}d_{\rm{V}}^{ \!-\! \alpha }{h_{\rm{V}}}}}\!\!\!\left[ \!\!{{{\left(\! {\frac{{d_{\rm{R}}^{ -\! \alpha }{h_{\rm{R}}}}}{{d_{\rm{V}}^{ -\! \alpha }{h_{\rm{V}}}{{\rm e}^{{\chi}}}}}} \!\right)}^{ \!-\! \frac{{{B_{\rm{R}}}}}{{{B_{\rm{V}}} \!+\! {B_{\rm{V}}}}}}} \!\!\!\!\!\!-\! 1}\! \right]_0^{{p_{{\rm{VM}}}}}} }\! \!\!\!\!,\\
&{p_{\rm{R}}}=\!\! {{\frac{{{\phi _1}{N_0}{B_{\rm{R}}}}}{{{\phi _0}d_{\rm{R}}^{ -\! \alpha }{h_{\rm{R}}}}}\!\!\left[\! {{e^{{\chi}}}{{\left(\! {\frac{{d_{\rm{R}}^{ -\! \alpha }{h_{\rm{R}}}}}{{d_{\rm{V}}^{ \!-\! \alpha }{h_{\rm{V}}}{{\rm e}^{{\chi }}}}}} \!\right)}^{\frac{{{B_{\rm{V}}}}}{{{B_{\rm{V}}} \!+\! {B_{\rm{R}}}}}}} \!\!\!\!\!\!-\! 1} \!\right]_0^{{p_{{\rm{RM}}}}}}\!\!}\!\!.
\end{align}
\end{theorem}
\begin{proof}
To obtain the power allocation of cooperative HD map transmission, we use the method in {\it Proposition} 1 to transform (12) into
\begin{align}
{p_{\rm{R}}} \!\ge\! g\!\left(\! {{p_{\rm{V}}}} \!\right),
\end{align}
where
\begin{align}
\!\!\!\!g\!\left(\! {{p_{\rm{V}}}} \!\right) \!=\! \frac{{{\phi _1}{N_0}{B_{\rm{R}}}}}{{{\phi _0}d_{\rm{R}}^{\! -\! \alpha }{h_{\rm{R}}}}}\!\!\left[\!\! {{{\rm e}^{{\chi}}}{{\!\left(\! {1 \!+\! \frac{{{\phi _0}d_{\rm{V}}^{ \!-\! \alpha }{h_{\rm{V}}}{p_{\rm{V}}}}}{{{\phi _1}{N_0}{B_{\rm{v}}}}}} \!\right)\!}^{\! -\! \frac{{{B_{\rm{V}}}}}{{{B_{\rm{R}}}}}}} \!\!\!-\! 1} \!\right]\!\!,
\end{align}
with
\begin{align}
\!\!\!\!&\chi \!=\! \frac{{ \{\sigma {\Phi ^{ \!-\! 1}}\left[ {1 \!-\! \delta \left( {1 \!-\! \Phi \left( { \!-\! {\textstyle{\mu \over \sigma }}} \right)} \right)} \right] \!+\! \mu {\rm{\!-\! }}{\textstyle{\vartheta \over v}}\}v\ln 2}}{{{B_{\rm{r}}}}},\\
\!\!\!\!&\vartheta \!=\! \! -\! \frac{{{B_{\rm{r}}}}}{{\ln 2}}\!\sqrt {\frac{1}{{\tau {B_{\rm{r}}}}}} Q_{\rm{G}}^{ \!-\! 1}\left(\! {\varepsilon _{\rm{r}}^{{\rm{c,r}}}}\! \right) \!\!-\!\! \frac{{{B_{\rm{v}}}}}{{\ln 2}}\!\sqrt {\frac{1}{{\tau {B_{\rm{v}}}}}} Q_{\rm{G}}^{\! -\! 1}\left(\! {\varepsilon _{\rm{v}}^{{\rm{c,v}}}} \!\right)\!\!.
\end{align}
\par
Without consideration of (13), the partial Lagrange function of problem in (9) is given by
\begin{align}
\!\!\notag {\mathcal L}\!\left(\! {{p_{\rm{R}}},{p_{\rm{V}}},\lambda }\! \right)\!&\!=\!{p_{\rm{R}}} \!+\! {p_{\rm{V}}} \!-\! {\lambda _1}{p_{\rm{R}}} \!-\! {\lambda _2}{p_{\rm{V}}} \!+\! {\lambda _3}\!\left(\! {{p_{\rm{R}}} \!-\! {p_{{\rm{RM}}}}} \!\right)\\
&\!+\!{\lambda _4}\left( {{p_{\rm{V}}} \!-\! {p_{{\rm{VM}}}}} \right) \!-\! {\lambda _5}\left[ {{p_{\rm{R}}} \!-\! g\left( {{p_{\rm{V}}}} \right)} \!\right]\!\!,
\end{align}
where ${\bm \lambda} {\rm{ = }}\left\{ {{\lambda _i} \ge 0,i = 1,..., 5} \right\}$ is the Lagrange multiplier vector. We can prove that the first order derivative of $g\left( {{p_{\rm{V}}}} \right)$ with respect to $p_{\rm{V}}$ is less than zero and the second-order derivative of $g\left( {{p_{\rm{V}}}} \right)$ with respect to $p_{\rm{V}}$ is greater than zero, therefore the considered optimization problem is a convex problem. Then, the optimization problem can be solved based on Karush-Kuhn-Tucker (KKT) conditions. With ${{ p}_{\rm{R}}} \ne 0$and ${{ p}_{\rm{V}}} \ne 0$, the KKT conditions are given by
\begin{align}
&0 < {{ p}_{\rm{R}}} \le {p_{{\rm{RM}}}},\\
&0 < {{ p}_{\rm{V}}} \le {p_{{\rm{VM}}}},\\
&{{ p}_{\rm{R}}} \!-\! g\left( {{p_{\rm{V}}}} \right)\!\ge\! 0,\\
&{{ \lambda }_i} \ge 0,i = 1,...,5,\\
&{{ \lambda }_1}{\rm{ = }}{{ \lambda }_2}{\rm{ = }}0,\\
&{{ \lambda }_3}\left( {{{ p}_{\rm{R}}} - {p_{{\rm{RM}}}}} \right){\rm{ = }}0,\\
&{{ \lambda }_4}\left( {{{ p}_{\rm{V}}} - {p_{{\rm{VM}}}}} \right){\rm{ = }}0,\\
&{{ \lambda }_5}\left({ p}_{\rm{R}}- g\left({p_{\rm{V}}} \right)\right)=0,\\
&\nabla {\mathcal L}=0.
\end{align}
Then, substituting (34) into (43) results in
\begin{align}
&{\rm{1}} \!+\! {{ \lambda } _3} \!-\! {{ \lambda } _5}{\rm{ = }}0,\\
&{\rm{1}} \!+\! {{ \lambda } _4} \!-\! {{ \lambda } _5}\frac{{d_{\rm{V}}^{\!-\! \alpha }{h_{\rm{V}}}}}{{d_{\rm{R}}^{ \!-\! \alpha }{h_{\rm{R}}}}}{e^{{{\rm{A}}_3}}}{\!\left(\! {1 \!+\! \frac{{{\phi _0}d_{\rm{V}}^{ \!-\! \alpha }{h_{\rm{V}}}{{ P}_{\rm{V}}}}}{{{\phi _1}{N_0}{B_{\rm{V}}}}}} \!\right)^{ \!\!\!-\! \frac{{{B_{\rm{V}}}}}{{{B_{\rm{R}}}}} \!-\! 1}} \!\!\!\!\!\!\!=\! 0.
\end{align}
Finally, according to KKT conditions (35)-(45), we can obtain the power allocation given in (27).
\end{proof}
\subsection{Optimal Power Allocation}
By comprehensively considering the three cases in Sec. III-A, the optimal power allocation can be expressed by
\begin{align}
{\psi ^*}\left( {p_{\rm{V}}^*,p_{\rm{R}}^*} \right){\rm{ = argmin}}&\left\{ {{p_{{\rm{V}\it{i}}}} + {p_{{\rm{R}\it{i}}}},\left( {{p_{{\rm{V}\it{i}}}},{p_{{\rm{R}\it{i}}}}} \right)} \right.\\
&\left. { \in \left\{ {{\psi _1},{\psi _2},{\psi _3}} \right\}} \right\}.
\end{align}
\begin{table}[!b]
\centering
\scriptsize
\renewcommand{\arraystretch}{1.4}
\vspace{-1.5em}
\caption{Simulation Parameters}
\vspace{0pt}
\label{parameters}
\begin{tabular}{|p{4.7cm}<{\centering}|p{2.1cm}<{\centering}|}\hline
\footnotesize \textbf {Parameter} & \footnotesize \textbf {Value}\\\hline
Channel constant ($\phi_0$) &$10^{-3}$ \\\hline
path-loss exponent ($\alpha$) &3\\\hline
SNR loss coefficient ($\phi_1$) &1.5\\\hline
Duration of transmission ($\tau$) &$10^{-3}$ s\\\hline
Noise power spectrum density (${N_0}$) &-174 dBm/Hz\\\hline
RSU and vehicle bandwidth ($B_{\rm{R}}$, $B_{\rm{V}}$) & 1 MHz, 0.5 MHz\\\hline
Maximum transmit power of RSU and vehicle ($p_{\rm{RM}}$,$p_{\rm{VM}}$) &40 dBm, 36 dBm \\\hline
Communication range of V2V ($d_{\rm{VM}}$) &150 m\\\hline
Decoding error probabilities ($ {\varepsilon _{\rm{V}}^{{\rm{C,V}}}}\!={\varepsilon _{\rm{R}}^{{\rm{C,R}}}} $) &$10^{-4}$\\\hline
Mean and variance of HD map data volume ($\mu$, $\sigma$) &0.8 kbits/m, $100$\\\hline
Maximum violation probability ($\delta$) &$10^{-4}$ \\\hline
\end{tabular}
\end{table}
\section{Simulation Results and Analysis}
In this section, numerical results and analysis are provided to show the performance of the proposed cooperative V2X transmission scheme.
\subsection{Parameter Setup}
The simulation parameters are listed in Table \uppercase\expandafter{\romannumeral1}. The road length covered by the RSU is 432 m, and the distance between the RSU and the road is 250 m, the lane width is 3.5 m in Fig. 1. In order to simplify the simulation, we assume that the two vehicles have the same speed.
\par
\subsection{Results and Analysis}
Fig. 2 shows the average transmit power of RSU versus different locations of the targeted vehicle. With the horizontal change of the targeted vehicle, both the distances $d_{\rm R} $, $d_{\rm V} $ first decrease and then gradually increase. This change rule gives rise to the same trend of total transmit power. When the location of the targeted vehicle between 144 m - 288 m, we have $d_{\rm V} < d_{\rm VM}$; therefore V2V communication exists and the average transmit power obviously decreases. It indicates that the proposed collaborative transmission can significantly reduce the power consumption.
Fig. 3 and Fig. 4 illustrate the average transmission rate and average power allocation versus vehicle speed, respectively. The transmission rate requirement is related to both vehicle speed and HD map data volume per meter. Due to the power limitation of RSU, the V2I transmission only can not satisfy the transmission rate requirement when the vehicle speed exceeds 22 m/s. On the other hand, limited by the communication range of V2V, V2V transmission only can not meet the transmission rate requirement of the HD map. However, under the proposed cooperative V2X transmission, the transmission rate requirement can be satisfied when the vehicle speed is under 30 m/s, which approaches the maximum speed limitation of vehicle. Moreover, Fig. 4 shows that the average transmit power under the cooperative V2X transmission is significantly reduced than that under V2I transmission only. Therefore, the proposed cooperative transmission can achieve the HD map transmission with low power consumption.
\par
\begin{figure}[!t]
\centering
\includegraphics[width=0.44\textwidth]{power.eps}
\caption{Average transmit power v.s. the location of the targeted vehicle.}
\vspace{-8pt}
\end{figure}
\section{CONCLUSION}
This paper studied the power efficient transmission of HD map, which is significant for automatic driving. In order to reduce power consumption while guaranteeing the transmission rate requirement, a cooperative V2V/V2I transmission was proposed for HD map transmission. To realize the cooperative transmission scheme, the power allocation at both RSU and vehicle are given through solving three sub optimization problems. Finally, the simulation results indicated that the proposed scheme can significantly reduce the total power consumption compared to the V2I transmission scheme while meeting the transmission rate requirement of HD map.\par
\begin{figure}[!t]
\centering
\includegraphics[width=0.44\textwidth]{speedrate.eps}
\caption{
Average transmission rate v.s. vehicle speed.}
\vspace{-12pt}
\end{figure}
\section*{Acknowledgement}
This work was supported by the China Natural Science Funding under Grant 61731004.
\begin{figure}[!t]
\centering
\includegraphics[width=0.44\textwidth]{speedpower.eps}
\caption{
Average transmit power v.s. vehicle speed.}
\vspace{-10pt}
\end{figure}
|
3,212,635,537,435 | arxiv | \section{Introduction}
Consider a connected 1-graph $G$ whose arcs are denoted by 1, 2, ..., $m$ and let some quantities $b_{i}, c_{i}$ be such that
\[
-\infty \le b_{i} \le c_{i} \le +\infty
\]
with the conditions:
1) $b_{i} = 0 \qquad (i = 1, 2, ..., m)$;
2) $c_{i} \ge 0 \qquad$ for all $i$, and $c_{i} = +\infty$;
3) Arc $i$ = 1 is the arc $(b, a)$ which connects a point $b$ named the {\it output} with a point $a$ named the {\it input}, these two points verifying:
\[
\omega^-(a) = (1, 0, 0, ..., 0),
\]
\[
\omega^+(a) = (1, 0, 0, ..., 0);
\]
4) $G$ is an antisymmetric 1-graph.
The arc 1 = ($b, a$), that will not be drawn, is named the {\it return arc} and is just introduced to maintain the Kirchoff law at the vertices $a$ and $b$.\\
\begin{dfn}
A graph $G$, with a capacity $c_{i}$ associated to any arc $i$, and which satisfies all these conditions, is called a {\it transportation network} (see \cite{For})\footnote{Historically, before Ford and Fulkerson, it seems that interest for combinatorial optimization may be found in an article of A. N. Tolsto\u{\i} from 1930, in which the transportation problem is studied, as well as an, until recently secret, RAND report of T. E. Harris and F. S. Ross from 1955, that Ford and Fulkerson mention as motivation to study the maximum flow problem. These papers have in common that they both apply their methods to the Soviet railway network. As Schrijver recalled, the transportation problem was formulated by (\cite{Hit}, and a cycle criterion for optimality was considered by \cite{Kan1}, \cite{Kan2}, \cite{Koo1}, \cite{Koo2}, \cite{Rob1}, \cite{Rob2} \cite{Gal1}, \cite{Gal2}, \cite{Lur}, \cite{Ful} and \cite{Kle}. On all that, see \cite{Sch}.}
and it will be denoted by:
\[
N = (X, U, c(u)).
\]
\end{dfn}
In the following, as we will not pay attention to capacities, the previous network will be reduced to a connected 1-graph $G$.
\section{Flows and tensions in networks}
\begin{dfn}
A flow in a connected graph $G$ is usually defined as a vector $\phi = (\phi_{1}, \phi_{2}, ... , \phi_{m}) \in \mathbb{Z}^m$ such that:
(1) $\phi_{i} \in \mathbb{Z}$ for $i =1, 2, ... , m$. (The integer $\phi_{i}$ is called an {\it arc flow} and may be regarded as the number of vehicules (signals, etc.) travelling through arc $i$ along its direction if $\phi_{i} \ge 0$ or against its direction if $\phi_{i} < 0$.)
(2) For each vertex $x$, the sum of the arc flows entering $x$ equals the sum of the arc flows leaving $x$ (Kirchoff law), i.e.,
\[
\sum_{i \in \omega^-(x)} \phi_{i} = \sum_{j \in \omega^+(x)} \phi_{j} \qquad (x \in X).
\]
\end{dfn}
According to Berge (see \cite{Ber}, 85), it is possible to develop an algebraic study of flows in such a graph.
Firstable, as $\mathbb{Z}^m$ is a module on $\mathbb{Z}$ (not a vector space, because $\mathbb{Z}$ is not a field), the set $\Phi$ of all flows in the graph $G$ constitutes a submodule of $\mathbb{Z}^m$, i.e we have:
\[
\phi^1, \phi^2 \in \Phi \Rightarrow \phi^1 + \phi^2 \in \Phi,
\]
\[
s \in \mathbb{Z}, \phi \in \Phi \Rightarrow s\phi \in \Phi.
\]
Berge proves the following theorem:\\
\begin{thm}
Let $G = (X, U)$ a connected graph; $H = (X, V)$ an arbitrary tree of $G$; 1, 2, ..., $k$, the arcs of $U-V$; $\mu^1, \mu^2, ..., \mu^k$ the cycles associated with $H$. A flow $\phi$ is uniquely defined by its values $\phi_{1}, \phi_{2}, ..., \phi_{k} \in U-V$ by:
\[
\phi = \phi_{1}\vec{\mu}^1 + \phi_{2}\vec{\mu}^2+ ... + \phi_{k}\vec{\mu}^k,
\]
where the $\phi_{i}$ are scalars and the $\vec{\mu}^i$ are vectors associated with independent elementary cycles.
\end{thm}
This means that a flow is uniquely defined by its components on a cotree of $G$.
Let now come to tensions.\\
\begin{dfn}
A tension (or potential difference) in a connected graph $G$ is defined to be a vector $\theta = (\theta_{1}, \theta_{2}, ... , \theta_{m}) \in \mathbb{Z}^m$ such that, for each elementary cycle $\mu$,
\[
\sum_{i \in \mu^+} \theta_{i} = \sum_{i \in \mu^-} \theta_{i}.
\]
\end{dfn}
For every arc $i$, we have: $\theta_{i}= t$ (terminal end of arc $i$) - $t$ (initial end of arc $i$).
Let $\Theta$ denote the set of all tensions. Note that $\Theta$ is also a submodule of $\mathbb{Z}^m$, i.e.,
\[
\theta^1, \theta^2 \in \Theta \Rightarrow \theta^1 + \theta^2 \in \Theta,
\]
\[
s \in \mathbb{Z}, \theta \in \Theta \Rightarrow s\theta \in \Theta.
\]
Here again, Berge proves the following theorem:\\
\begin{thm}
Let $G = (X, U)$ a connected graph; $H = (X, V)$ an arbitrary tree of $G$; 1, 2, ..., $k$, the arcs of this tree; $\vec{\omega}^1, \vec{\omega}^2, ..., \vec{\omega}^\ell$ the cocycles associated with $H$. A tension $\theta$ is uniquely defined by its values $\theta_{1}, \theta_{2}, ..., \theta_{\ell}$ on the arcs of the tree by:
\[
\theta = \theta_{1}\vec{\omega}^1 + \theta_{2}\vec{\omega}^2+ ... + \theta_{\ell}\vec{\omega}^\ell,
\]
where the $\theta_{i}$ are scalars and the $\vec{\omega}^i$ are vectors associated with independent elementary cocycles.
\end{thm}
This means that a tension is uniquely defined by its components on a tree of $G$.
We can easily see that $\Theta$ and $\Phi$ are two orthogonal submodules of $\mathbb{Z}^m$, which means that, for every elementary cycle $\mu$, we have:
\[
\langle \phi, \theta \rangle =\sum_{i=1}^m \phi_{i}\theta_{i} = 0.
\]
\section{Algebraic lattices}
We propose to extend the previous model. Let us consider now the set of all possible values of tensions or flows in some network $N$. We will prove that this set can be associated to a metanetwork $G_{k}(\Gamma)$ which satisfies good properties. Recall first the following definition: \\
\begin{dfn}
A lattice $\Gamma$, in an $\mathbb{R}$-vector space $V$ of finite dimension, is a subgroup of $V$ verifying one of the following equivalent conditions enumerated by Serre (see \cite{Ser}, 133):
1) $\Gamma$ is discrete and $V/ \Gamma$ is compact;
2) $\Gamma$ is discrete and generates the $\mathbb{R}$-vector space $V$;
3) There exists an $\mathbb{R}$-basis $\{e_{1}, ... e_{n} \}$ of $V$, which is a $\mathbb{Z}$-basis of $\Gamma$ and $\Gamma = \mathbb{Z}e_{1} \oplus ... \oplus \mathbb{Z}e_{n}$.
\end{dfn}
Now, let us choose values of flows (or tensions) in an $\mathbb{R}$-vector space $V= \mathbb{R}^n$.\\
\begin{thm}
The set of all possible flow (resp. tension) values of the network $N$ is a lattice in $\mathbb{R}^n$.
\end{thm}
\begin{proof}
Let $\epsilon = (\epsilon_{1}, \epsilon_{2}, ... , \epsilon_{n})$, a flow (resp. a tension) in some arc(s) of $G$. By definition, $\epsilon$ belongs to $\mathbb{R}^n$, viewed as a vector space on $\mathbb{R}$. Moreover, according to the definition of flows (Def. 2.1) and of tensions (Def. 2.2), the set $\Gamma$, of all flow (resp. tension) values in the graph $G$, is the subgroup of all linear combinations with integer coefficients of the basis vectors of $\mathbb{R}^n$ (cycles, resp. cocyles). So it is such that:
\[
\Gamma = \mathbb{Z}\epsilon_{1} \oplus ... \oplus \mathbb{Z}\epsilon_{n},
\]
for any basis of $\mathbb{R}^n$. In other words, it forms a lattice in $\mathbb{R}^n$.
\end{proof}
\section{The lattices of $\mathbb{C}$}
Assume now that the flow (resp. tension) values of $G$ are in $\mathbb{C}$, and consider only two-valued flows (resp. tensions).
Let us call $\mathcal{R}$ the set of lattices of $\mathbb{C}$, considered as an $\mathbb{R}$-vector space, and let us now choose a pair of flow (resp. tension) values ($\alpha_{1}, \alpha_2) \in \mathbb{C}^*$ so that Im($\alpha_{1}/ \alpha_2) > 0$. $M$ will be the set of these pairs.
To such a pair ($\alpha_{1}, \alpha_2)$, we associate the lattice:
\[
\Gamma(\alpha_{1}, \alpha_{2}) = \mathbb{Z}\alpha_{1} \oplus \mathbb{Z}\alpha_{2}.
\]
with basis $\{\alpha_{1}, \alpha_2\}$.
Thus we get a map $M \rightarrow \mathcal{R}$, which is clearly surjective.
Now let:
\[
g =
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\in \textnormal{SL}(2, \mathbb{Z})
\]
the special linear group of square matrices $2 \times 2$ with relative coefficients, and let $(\alpha_{1}, \alpha_2) \in M.$
One proves (see \cite{Ser}, 134) the following theorem:\\
\begin{thm}
For two elements of $M$ to define the same lattice, it is necessary and sufficient that they are congruent modulo \textnormal{SL(}$2, \mathbb{Z}$\textnormal{)}.
\end{thm}
\begin{proof}(Serre)
The condition is sufficient. Let us put:
\[
\alpha'_{1} = a \alpha_{1} + b \alpha_{2} \ \textnormal{and} \ \alpha'_{2} = c \alpha_{1} + d \alpha_{2}.
\]
Il is clear that $\{\alpha'_{1}, \alpha'_{2}\}$ is a basis of $\Gamma(\alpha_{1}, \alpha_{2})$. Moreover, if the set $z =
\alpha_{1}/\alpha_{2}$ and $z' = \{\alpha'_{1}/ \alpha'_{2}\}$, we have:
\[
z' = \frac{az + b}{cz+d} = gz.
\]
This shows that Im$(z')>0$, hence that $(\alpha'_{1}, \alpha'_{2})$ belongs to $M$.
Conversely, if $(\alpha_{1}, \alpha_{2})$ and $(\alpha'_{1}, \alpha'_{2})$ are two elements of $M$ which define the same lattice, there exists an integer matrix
\[
g =
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\]
of determinant $\pm 1$ which transforms the first basis into the second. If det($g$) was $<0$, the sign of Im$(\alpha'_{1}/ \alpha'_{2})$ would be the opposite of Im$(\alpha_{1}/ \alpha_{2})$ as one sees by an immediate computation. The two signs being the same, we have necessarily det($g$) = 1, which proves the theorem.
\end{proof}
Hence we can identify the set $\mathcal{R}$ of all the lattices of $\mathbb{C}$ (which are, for us, sets of flow (or tension) values associated to connected 1-graphs (or networks) with the quotient of $M$ by the action of SL($2, \mathbb{Z}$).
\section{Modular functions}
Let now $F$ be a function on $\mathcal{R}$, with complex values, and let $k \inÊ\mathbb{Z}$. We say (with Serre) that $F$ is of weight $2k$ if:
\begin{equation}
F(\lambda\Gamma) = \lambda^{-2k}F(\Gamma),
\end{equation}
for all lattices $\Gamma$ and all $\lambda \in \mathbb{C}^*$.
Let $F$ be such a function. If $(\alpha_{1}, \alpha_{2}) \in M$, we denote by $F(\alpha_{1}, \alpha_{2})$ the value of $F$ on the lattice $\Gamma(\alpha_{1}, \alpha_{2})$. The formula (1) translates to:
\begin{equation}
F(\lambda\alpha_{1}, \lambda\alpha_{2}) = \lambda^{-2k}F(\alpha_{1}, \alpha_{2}).
\end{equation}
Writing that $F$ is invariant by SL(2, $\mathbb{Z})$, we can see that it satisfies the identity:
\begin{equation}
F(z) = (cz + d)^{-2k}F(\frac{az+b}{cz+d}),
\end{equation}
for all:
\[
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\in \textnormal{SL(}2,\mathbb{Z}).
\]
Conversely, if $F$ verifies (3), $F$ is a function on $\mathcal{R}$ which is of weight $2k$. We can thus identify {\it modular functions of weight 2k} with some {\it lattice functions of weight 2k}.
Then we know that some lattice functions, that are modular functions, can be identified with Eisenstein series, which are themselves convergent. Serre (1973) proves the following lemma: \\
\begin{lem}
Let $\Gamma$ be a lattice in $\mathbb{C}$. The series:
\[
\sum'_{\gamma \in \Gamma} 1/|\gamma|^{\sigma}
\]
is convergent for $\sigma >2$.
\end{lem}
(The symbol $\sum'$ signifies that the summation runs over the nonzero elements of $\Gamma$.)
Now let $k$ be an integer >1. If $\Gamma$ is a lattice of $\mathbb{C}$, put:
\[
G_{k}(\Gamma) = \sum_{\gamma \in \Gamma}' 1/\gamma^{2k}.
\]
This series converges absolutely thanks to the preceding lemma. It proves the existence of a lattice function on the set $\mathcal{R}$ of lattices of $\mathbb{C}$.
In other words, all the lattices of $\mathbb{C}$, which represent sets of flow (or tension) values in connected 1-graphs (or networks) are themselves connected by this lattice function.
Let now $k$ be an integer >1. Like all the Eisentein series of the type $G_{k}(z)$:
1) $G_{k}(\Gamma)$, which is a modular form of weight $2k$, is holomorphic everywhere (including at the infinite);
2) $G_{k}(\infty) = 2\zeta(2k)$;
3) $G_{k}$ has a limit for Im$(z) \rightarrow \infty, \ z$ being the value for which $\Gamma$ vanishes at one and only one point.
\section{Siegel space}
We can still extend the previous construction.
Let $N_{1}, N_{2}, ..., N_{m}$ be some finite connected 1-graphs and consider, for each of them, their associated matrices of flow (or tension) values. Let $Z_{1}, Z_{2}, ..., Z_{m}$ be such matrices with complex coefficients.
Let $L$ be the set of all $n \times n$ complex symmetric matrices and $C_{n}$ the set of matrices $Z$ of $L$ such that
the hermitian matrix $I - Z\bar{Z}$ is strictly positive.
Let now $S_{n}$ (the Siegel space) be the set of matrices $Z$ of $L$ whose imaginary part Im$\ Z = (1/ 2i) (Z - \bar{Z})$ is strictly positive. It is well known that the so-called "Cayley transformations" apply $C_{n}$ to $S_{n}$ and vice versa (see \cite{Deh}, 437-438).
Hence, the real symplectic group Sp$(2n, \mathbb{R}$) plays the same role, with respect to the Siegel space $S_n$, than the group Sp(2, $\mathbb{R})$ = SL(2, $\mathbb{R}$) with respect to the upper half-plane of the complex plane. When the group SL(2, $\mathbb{R}$) operates in $\mathbb{C}$ by the Poincar\'{e} Fuchsian transformations, the group Sp$(2n, \mathbb{R}$) now operates in the Siegel space $S_{n}$ by the transformations:
\begin{equation}
g' =
\begin{pmatrix}
A & B \\
C & D,
\end{pmatrix}
\in \textnormal{Sp}(2n, \mathbb{R}).
\end{equation}
So we have:
\[
g'Z = (AZ + B) (CZ + D)^{-1}.
\]
Now let us call $\mathcal{R}'$ the set of all the matrix lattices of $C_{n}$, and let $M'$ be the set of pairs ($A_{1}, A_{2}) \in C_{n}$, such that Im$(A_{1}, A_{2}^{-}) >0$, which supposes that $A_{2}$ is inversible.
To such a pair ($A_{1}, A_2)$, we associate now the lattice:
\[
\Gamma'(A_{1}, A_{2}) = \mathbb{Z}A_{1} \oplus \mathbb{Z}A_{2}.
\]
with basis $\{A_{1}, A_2\}$. Thus, we get a map $M' \rightarrow R'$, which is clearly surjective.
One gets the following theorem:\\
\begin{thm}
So that two elements of $M'$ define the same lattice, it is necessary and sufficient that they are congruent modulo \textnormal{Sp(}$2n,\mathbb{R}$\textnormal{)}.
\end{thm}
\begin{proof}
The condition is sufficient. Let $A_{1}, A_{2} \in M'$. Then, put :
\[
A'_{1} = a A_{1} + b A_{2} \ \textnormal{and} \ A'_{2} = c A_{1} + d A_{2}.
\]
Il is clear that $\{A'_{1}, A'_{2}\}$ is a basis of $\Gamma(A_{1}, A_{2})$. Moreover, if $Z = A_{1}A_{2}^{-}$ and $Z' = A'_{1} A'^{-}_{2}$,
\[
Z' = (AZ + B)(CZ+D)^- = g'Z.
\]
This shows that Im$(Z')>0$, hence that $(A'_{1}, A'_{2})$ belongs to $M'$.
Conversely, if $(A_{1}, A_{2})$ and $(A'_{1}, A'_{2})$ are two elements of $M'$ which define the same lattice, there exists an integer matrix
\[
g' =
\begin{pmatrix}
A & B \\
C & D
\end{pmatrix}
\]
of determinant >0 which transforms the first basis into the second. If det($g'$) was <0, the sign of Im$(A'_{1} A'^{-}_{2})$ would be the opposite of Im$(A_{1} A^{-}_{2})$ as one sees by an immediate computation. The two signs being the same, we have necessarily det($g'$) >0, which proves the theorem.
Thus, we can identify the set $M'$ of all the lattice matrices of $C_{n}$ with the quotient of $S_{n}$ by the action of Sp(2$n,\mathbb{R}$).
\end{proof}
For the same reasons, we can also define, as previously, a lattice function of weight $2k$.
Let $F'$ be such a function. If $(A_{1}, A_{2}) \in M'$, we denote by $F'(A_{1}, A_{2})$ the value of $F'$ on the lattice $\Gamma'(A_{1}, A_{2})$. The formula (2) translates to:
\begin{equation}
F'(\lambda A_{1}, \lambda A_{2}) = \lambda^{-2k}F'(A_{1}, A_{2}).
\end{equation}
Writing now that $F'$ is invariant by Sp($2n, \mathbb{R})$, we can see that this function satisfies the identity:
\begin{equation}
F'(Z) = (XZ+D)^{-2k}f(\frac{AZ+B}{CZ+D}),
\end{equation}
for all:
\[
\begin{pmatrix}
A & B \\
C & D
\end{pmatrix}
\in \textnormal{Sp(}2n,\mathbb{R}).
\]
As previously, this function can be identified with an Eisenstein series $G'_{k}(\Gamma')$ on the set $\mathcal{R}'$ of the matrix lattices of $C_{n}$, which is absolutely convergent.
In other words, all the $n \times n$ matrix lattices of $C_{n}$, which represent sets of subsets of flows (or tensions) in connected 1-graphs (or networks), are linked by this lattice function.
If we associate networks with subsets of flow (or tension) values, this proves the existence of a "network of networks".
\section{The tree of minimal length}
Let $G_{k}(\Gamma)$ be the graph associated with the set of all subsets of flows and $A_{0}$, the minimal tree of $G_{k}(\Gamma)$. If $U$ is the set of arcs of $G_{k}(\Gamma), \ U - A_{0} = A'_{0}$ is the maximal cotree of $G_{k}(\Gamma)$\footnote{On trees and co-trees, see (\cite{Gon}, 103-128.}. Now, it is easy to see that :
(1) The smallest arc of all cocycles (tensions) is in $A_{0}$;
(2) The greatest arc of all cycles (flows) is in $U - A_{0} = A'_{0}$.
We finally obtain a set of arcs without a maximal cycle and we can always find an optimal flow in the graph because any flow does not circulate in all the arcs of the whole graph but only in those of a tree whose capacities, which do not admit a higher bound, are infinite.
Let us now precise the form of the tree of minimal length $A_{0}$. Let $n$ be the number of vertices of $A_{0}, \ A$ the set of its arcs, $a$ an arc of $A, \ d$ the distance between two vertices $s$ and $s'$. We have:
(1) Card($A) = n(n-1)/2$;
(2) $a = \{s,s'\} : d(a) = d(s,s')$.
Now let $P$ be a polygon, i.e. a set of edges which is a subset of $A_{0}$. The support of $P$ will be the union of the edges of $A_{0}$, that is, the set of vertices of $G_{k}(\Gamma)$ which are ends of at least one edge of $P$. One can speak of polygon $P$ on $G_{k}(\Gamma)$ (resp. in $G_{k}(\Gamma)$) according to whether the support of $P$ is $G_{k}(\Gamma)$ or a subset of $G_{k}(\Gamma)$ distinct from itself.
$A_{0}$, which is the set of all possible edges on $G_{k}(\Gamma)$, is a complete polygon of $G_{k}(\Gamma)$.
A graph being the conjunction of a polygon and its support, a chain $C$ will be a polygon in $G_{k}(\Gamma)$ whose vertices that form its support can be ordered in a sequence ($s_{0}, s_{1}, ... s_{p}$). We have:
(1) For every $i \in \ ]P[, \{s_{i-1}, s_{i}\} \in C$;
(2) For every $i, j \in [P], i \neq j \Rightarrow s_{i} \neq s_{j}.$
A cycle is a chain where condition (2) holds for all the points of its support except $s_{0}$ and $s_{p}$ which are merged (the ends of $C$).
To exhibit $G_{k}(\Gamma)$, we need the following complementary considerations:
A) A tree is a connected polygon that does not contain a loop.
B) The length of a polygon is the sum of the lengths of all its edges.
C) The width of a polygon is the length of its longest edge.
Suppose that the polygon reduced to the edge $(s, s')$ represents the chain of $A$ with minimum width joining $s$ to $s'$, then $\{s, s'\}$ is an element of the tree of minimal length $T$ on $G_{k}(\Gamma)$ and there exists at least one such edge on $A$, the edge of minimal length.
Conversely, if $\{r, s\}$ is an element of the tree $T$ on $G_{k}(\Gamma)$, then $\{r, s\}$ is the chain of $A$ having the smallest width and joining $r$ to $s$.
In this context, $G_{k}(\Gamma)$ can be identified with a classification of classifications. This would amount to doing a factor analysis on all parts of the representative tree. Such a classification would correspond to all the axes of a factor analysis, with an original calculation on the first axis.
\section{Construction of $G_{k}(\Gamma)$}
In order to construct $G_{k}(\Gamma)$, we must first look at the lattice $\Gamma = \mathbb{Z}\alpha_{1} \oplus \mathbb{Z}\alpha_{2}$, which makes possible to distinguish a lattice function and a modular function. It must be assumed that the minimal bases of this lattice suppose a matroid $M$. If $B$ is the set of these bases, then $C$, the set of cycles of $M$ (resp. $D$ the set of cocycles of $M$) is the set of subsets which are not included in any basis (resp. which have a non-empty intersection with any basis) and minimal for inclusion with this property.
Let now $B \in \mathcal{B},\ b \in B,\ c \in X - B$. Let $D (b, B)$ be the unique cocycle satisfying $B \cap D(b, B) = \{b\}$ and $C(c, B)$, the unique cycle satisfying $C(c, B) - B = \{c\}$. We then have:
\[
B \in C (c, B) \iff c \in D (b, B) \iff B - {b} \cup {c} \in \mathcal{B}.
\]
$C$ and $D$ are the sets of cycles and minimal cocycles for the inclusion of the graph.
The minimum tree of a graph is the set of minimum edges of a cocycle, its complement being the set of the maximum edges of a cycle.
If we consider the lattice $(F, \cup, \cap$), a sublattice of $M$, the algebraic properties of $F$ (distributivity) are stronger than those of $M$ (semi-modularity). It is thus possible to construct $G_{k}(\Gamma)$, the super-lattice, by defining it as the set of distributive sub-lattices of any geometric lattice, that is to say, a sub-lattice of the semi-geometric lattice associated with $M$.
\section{Possible applications}
Let's finish with some more epistemological considerations: after all, mathematical physics and philosophy are not so far apart (see \cite{Par4}).
The space associated with this "network of networks", that is, the zeros and poles of the modular function of all networks, has been studied in hard proof theorems, because one does not define a structure of complex analytic variety on the single compactified network. (A natural way of proceeding would be to define a compactified isomorphism on the Riemann sphere $S = \mathbb{C} \cup \{\infty\}.)$
Whatever the difficulties of study, it is proved that this network function exists, and we have thus proved also that the set of all sets of possible flows exists as a modular function of all networks in the algebraic sense of the term.
Let us now consider some possible applications of the previous formalism.
1. Since the old work of [Von Neuman 1946], quantum mechanics represents all the physical states of the universe by a vector space of infinite dimension called "Hilbert space". However, the separability property and the convergence condition make it possible to reduce to closed subspaces. In this case, the complex vectors form a finite dimensional subspace and their mathematics is identical to that of flows or tensions on a graph, except that their coefficients can take on complex values. This situation makes it possible, as we have seen, to apply known theorems of arithmetic to them.
2. Because of the flow-tension duality, the network function defines as well the set of all the sets of possible tensions, and hence it specifies the shortest path in the total set of all possible paths, as well as the most rational scheduling of tasks in the set of all possible actions. Here we have a theorem of the existence of an optimal behavior, whatever the field we consider.
3. Moreover, the problem of the shortest path in a graph is related to the question of the tree of minimum length, which itself formalizes the notion of classification. A "network of networks" with a maximum voltage would thus make it possible both: to confirm the existence of a tree of minimum length of the network of all networks, and hence, of a classification of classifications (see \cite{Par3}).
4. In general, the variable "weights" can receive different meanings (reliability, economy, etc.) on a tree, other than the length of the arcs. So the network of networks $G_{k}(\Gamma)$ can still make it possible to calculate the maximum reliability path, or the most economical route, etc., in the set of all possible paths.
5. I will say a final word about the aim of this construction : though the world may be multiple and chaotic, circulations and actions can be ordered in relation to the same structure, which is expressed - in the linear case - through the form of this remarkable holomorphic function which has been here constructed. Doing that, we tried in fact to formalize the intuition of a "network of networks", as it is expressed in the conclusion of our book on networks (see \cite{Par1}, 265-286). This is also the achievement of what we called elsewhere a "rationalit\'{e} r\'{e}ticulaire" ({\it reticular rationality})(see \cite{Par2}).
|
3,212,635,537,436 | arxiv | \section{Introduction}
Quantum states with small (zero) transition dipole moment with respect to the ground state, referred to as dark states, are notoriously difficult to investigate by linear spectroscopic techniques. The most widespread approach is detection via their low emission yields \cite{Cook1985,Yip1998}. Alternative methods are fluorescence blinking, transient absorption, and comparisons with modified samples suppressing the presumed dark state \cite{Ferretti2016,Krecik2015,Berkeland2002}. The fact that whole classes of electronic transitions are dipole forbidden lends fundamental relevance to their study. Dark states are often very stable and keep phase relation with other states, leading to small decoherence rates. Additionally, dark states are essential for understanding energy and charge transport phenomena in fields ranging from quantum optics to solid state physics.
Examples are singlet - triplet spin structures, appearing from interactions of two spin 1/2 particles; triplet states are dark for dipole transitions from the singlet ground state. \santi{Here we focus on the class of states which are dark due to spin selection rules (henceforth spin dark states), in contrast to dark states which are dark owing to parity selection rules. Spin dark states such as the triplets investigated here have been demonstrated to be relevant for charge separation in artificial light harvesting, where they might hold the key to enhanced photocurrent generation \cite{Rao2013a,Chang2015a}}. Furthermore, they are responsible for efficient transport within quantum wells and two-dimensional materials, or for optical control in solid-state and semiconductor systems \cite{Snoke2002,Ye2014,Yale2013}. The control of spin dark states, in combination with advancing me\-thods of spintronics, holds the potential to all-optical control of such quantum systems. This line of research offers intriguing possibilities in quantum technological applications, such as quantum computing or sensing.
One possibility to overcome the dipole-forbidden character of dark states is to employ field-matter interactions of higher order, i.e. multipoles. However, this approach is hampered by the high field strengths necessary, making material damage threshold a limiting factor.
Instead of utilizing higher electric multipoles of dark states, we focus on their magnetic dipole moments. For triplet states, which are the subject of our research, the origin of magnetic dipole moments are their characteristic unpaired spins. Conventional techniques for the study of triplet states such as electron paramagnetic resonance (EPR) and multidimensional variants thereof are highly sensitive and insightful in terms of electronic structure determination and reactivity \cite{Chechik}. With respect to time-resolution, EPR-experiments are typically limited to the microsecond regime due to restrictions in microwave pulse technology. In the femtosecond regime, time resolved Two-Dimensional Electronic Spectroscopy (2DES) has emerged as the most comprehensive technique for the study of dynamics, due to its versatility and by its main characteristic of resolving a non-linear signal in excitation and emission frequencies \cite{Jonas2003}. Dark states occurring during relaxation can be detected here by their (potential) transitions to higher lying states, i.e. as excited state absorption. It has to be noted however that such ESA-features are often broad, featureless, convoluted, and therefore hard to analyze \cite{Polivka2004,Read2009,Perlik2015}. \santi{For some materials, singlet fission leads to allowed formation of triplet states which can be detected via 2DES \cite{Bakulin2016}.} In this article, we will propose a modification of 2DES allo\-wing to observe spin-dependent dark states. We argue that, apart from the linear spectroscopic signal and ESA-signatures, spin properties of quantum states can be exploited to make dark states responsive to direct spectroscopic probing.
It was recently demonstrated that a static magnetic field can manipulate charge transfer states, owing to the non-trivial spin and small Coulomb binding properties \cite{Oviedo2017}. Moreover, these
states can be inferred through the behavior of the resultant photocurrent when affected by magnetic fields of varying intensity \cite{Oviedo2018}. In particular, it was shown that a magnetic field
can be used to induce dipole moment redistribution between a bright singlet charge transfer state and a dark triplet charge transfer state, making the latter bright. This result suggests that the same principle could be applied to any
such dark state that is spin-connected to a bright state. Recently, static magnetic fields as high as 25 Tesla have been employed to study energy transport and charge transfer in biomolecules \cite{Scholes2018}. Despite their immense field strength, such static fields are not capable of modifying inter-system crossing (i.e. singlet-triplet transitions) in a time-scale relevant for 2DES, as spin precession depends on the coupling to the magnetic field, which is controlled by the Bohr magneton, and limits the precession speed to nanoseconds at most for attainable fields. The solution we propose in this article is employing a terahertz magnetic pulse, which we demonstrate is able to modify the dynamics of spin precession within femtoseconds. At the same time, it achieves a redistribution of dipole moment that has the potential to turn dark states bright at a time-scale relevant for the 2DES experiment. \santi{THz electric transients have already been used in combination with 2DES for the study of Raman spectroscopy, rotational and vibrational spectroscopy of molecules, or phonon excitations \cite{Woerner2013,Nelson2015,Finneran2016,Lu2019}.} By combining such magnetic pulses with a conventional 2DES scheme (see Fig. \ref{Fig1}), we demonstrate that it is possible to study the properties of dark states exhaustively, due to the spectral resolution offered by 2DES.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{./2Dfoldedboxcarwithtimelineandmagneticfield-eps-converted-to.pdf}
\caption{Schematic representation of the proposed magnetic field enhanced two-dimensional spectroscopic experiment. In (a) the three laser pulse sequence is shown. The first pulse excites a bright state in the system, which evolves freely for a variable time $t_1$ (coherence time), during which the magnetic pulse (which acts all through the laser pulses sequence) induces coherent transitions from the excited bright state to a dark state. The second pulse stabilizes the population in the dark state while the third produces a rephasing signal. (b) displays the fully non-collinear phase matching geometry of the laser pulses which in our design are immersed in a magnetic pulse (violet filled curve).}
\label{Fig1}
\end{figure}
\section{Model}
To illustrate the mechanism of the magnetic field enhanced 2DES, we use a minimal model containing an electronic ground state $|g,S\rangle$, a spin one singlet electron-hole (e-h) pair $\ket{CT,S} \propto \frac{1}{\sqrt{2}} (\ket{\uparrow}_e \ket{\downarrow}_h -\ket{\downarrow}_e \ket{\uparrow}_h)$ which carries all dipole moment in the system, and triplet states $\ket{CT,T_0} \propto \frac{1}{\sqrt{2}} (\ket{\uparrow}_e \ket{\downarrow}_h +\ket{\downarrow}_e \ket{\uparrow}_h)$, $\ket{CT,T_+} \propto \ket{\uparrow}_e \ket{\uparrow}_h $, $\ket{CT,T_-} \propto \ket{\downarrow}_e \ket{\downarrow}_h $ which are dipole forbidden; i.e. for this model $\hat{\mu} = \mu \ket{CT,S}\bra{g,S} + h.c.$.
We choose the e-h pair to have low Coulomb binding, \santi{a feature of charge transfer or polaron states in, for example, organic polymer systems \cite{Veldman2009}.} Our choice is based on the importance that these states have in a wide variety of systems ranging from quasi 2D quantum wells to natural and artificial light conducting and harvesting systems \cite{Rao2013a,Gelinas2014a,Chang2015a,Novoderezhkin2007,Ye2014,Snoke2002}. Low Coulomb binding bright states are expected to have low dipole moments. Moreover, owing to the loose nature of the e-h Coulomb binding in charge transfer states, they will be less sensitive to strong magnetic fields. States with just one electron are easier to manipulate with a magnetic field; composite states with stronger binding will typically have a higher dipole moment and will provide a clearer optical signal.
The electronic level diagram we employ in our model is depicted in Fig. \ref{Fig2}. At zero magnetic fields the triplet states are degenerate
\begin{equation}
H_0= E_S\ket{CT,S}\bra{CT,S}+ E_T\sum_{i=0,+,-} \ket{CT,T_i}\bra{ CT,T_i}.
\end{equation}
This singlet – triplet structure will be subjected to inter-system crossing (ISC) by the external magnetic pulse, whose dynamics are explained in detail in section \ref{sect3}.
The singlet and triplet charge transfer states are energetically separated ($E_S\neq E_T$) for ultrashort timescales of 2DES experiments, where the decoherence times impose a limit of few picoseconds \cite{Lienau2016}. Natural inter-system crossing can occur as a result of thermal fluctuations of the fine structure, spin-orbit coupling, hyperfine interaction, or wavefunction spatial overlap. These effects appear on much longer (hundred picosecond to microsecond) timescales
\cite{Kohler2009,Cohen2009,Zhang2012}, or break the degeneracy of the triplet states by a fraction of meV \cite{Cohen2009}. Throughout this paper, only hyperfine interaction will be considered.
In addition to possible coherent singlet-triplet transitions, spin-pair states are subject to the action of an environment composed of electronic and vibrational degrees of freedom causing decoherence and dephasing of the quantum states. This dynamics will be accounted for by Lindblad formalism as described in section \ref{Section4}.
\begin{figure}[h!]
\centering
\includegraphics[width=\columnwidth]{./Fig2model-eps-converted-to.pdf}
\caption{Model system employed to demonstrate how a magnetic pulse turns dark states bright in a 2DES experimental scheme. In the model, a singlet charge transfer state is excited through laser light. The three components of the corresponding triplet charge transfer state (degenerated in absence of a magnetic field) are completely dark with respect to the electronic ground state as spin transitions are prohibited upon excitation. An appropriately tuned external magnetic pulse, $B(t)$, is able to induce spin flips on the individual components of an e-h pair by modifying the precession frequency of the spins. Consequently, a magnetic pulse is used to redistribute dipole moment from the singlet state to the spin triplet states. In that way, an initial population in the singlet state can populate first the spin zero triplet state and, subsequently the spin $\pm$ one states, according to Eqs. (\ref{eq2}).}
\label{Fig2}
\end{figure}
\section{Terahertz inter-system crossing}\label{sect3}
Nowadays, controlling spins with an external magnetic field is technically achievable \cite{Alegre2007,Fuchs2009,London2014}. Terahertz magnetic field pulses have become only recently available \cite{Ferguson2002,Tonouchi2007}, offering the possibility of accessing ultrafast magnetization dynamics on the femtosecond time-scale \cite{Zhao2008,Yamaguchi2010,Nakajima2010,Kampfrath2010,Yamaguchi2013,Kim2014}. Moreover, contrary to what happens with optical radiation, which interacts with valence electrons and interferes with the 2DES target states, terahertz photons have energies around the meV, which will not produce optical transitions \cite{Kampfrath2013}. \santi{The coupling of the electric field associated with the magnetic pulse to the dipoles of the system is expected to be small ($\sim$ meV), consequently it will only induce vibrations on the sample that add-up to the dephasing noise. Furthermore, the small pulse energy, in the order of femtojoules, means that thermal effects are negligible \cite{Yamaguchi2013}. Experiments show that upon interaction with a terahertz pulse, the relatively large electric field associated does not produce appreciable short-term effects, although in the long run, damage is possible \cite{Kampfrath2010,Vicario2013}. This means that for several 2DES with terahertz magnetic pulse cycles the sample is assured to survive. In general, no significant change of the interactions or deformation of the wavefunctions is to be expected even for the fields involved \cite{Scholes2018}. In fact, comparable or stronger electric fields have already been used in combination with 2DES \cite{Loukianov2017}, and applied to organic samples, with positive results. }
To produce spin flips --as it is needed for inter-system crossing-- the coupling induced by the magnetic field has to be strong enough to ensure that the precession of the spins is faster than the
relaxation time of the states. However, the typical coupling of an electron (hole) to a magnetic field is mediated by the Bohr magneton and the Land{\'e} factor. The small value of these two factors
in typical e-h pairs has to be compensated with a magnetic field of at least kiloteslas in order to produce the sub-picosecond inter-system crossing necessary for having an effect on the 2D spectra
\cite{Oviedo2017}. Such magnetic fields are beyond any experimental instrument available nowadays, and are likely to disrupt the sample. To circumvent the limitation, we propose employing
terahertz magnetic pulses, whose interaction with the spin components is strong enough to produce the necessary spin flips \cite{Zhao2008,Nakajima2012,Vicario2013,Kim2014}. As we demonstrate in this
article, the key factor which allows to reduce the magnitude of the fields employed, is not the interaction strength, but rather the frequency with which the magnetic pulse oscillates.
The interaction of a spin with an external oscillating magnetic field with components in the z and x directions ($\vec{B} = B_z \vec{z} + B_x \vec{x}$) is governed by the interaction Hamiltonian for spin pair dynamics \cite{Malla2017}
\begin{equation}
H = B_e S_e^z + B_h S_h^z + B (S_e^x + S_h^x),
\label{ham1}
\end{equation}
where each component $B_{e,h}$ references the total magnetic field acting on either the electron or the hole, thus including the hyperfine interaction component felt by each of the spins, which by convention is taken in the z direction. $B_{e,h} (B)$ in Eq. (\ref{ham1}) already includes the coupling of the magnetic field with the spin in the form $g_{i}\mu_B/2$, with g being the Land{\'e} factor and $\mu_B$ the Bohr magneton constant. The Land{\'e} factor is specific for either the electron or the hole. Thus $B_{e} = -\mu_B g_e(B_z + a_eI_e)$, and $B_{h} = -\mu_B g_h(B_z + a_hI_h)$, where I is the hyperfine magnetic field.
The hyperfine component is created by the interaction of the magnetic moment with the atomic nuclei environment surrounding it. At the relevant 2DES time-scale, i.e. femtoseconds, the motion of such nuclei is slow and can be considered static. This fact, together with the large amount of nuclei contributing to the hyperfine field, means that the total hyperfine component can be approximated by a Gaussian distribution \cite{Schulten1978}, where the width reflects the typical strength of the hyperfine component for each material. In the case of most organic materials this width is approximately 1 meV \cite{Flatte2012}. We calculate the hyperfine contribution to the ISC by averaging over 10$^3$ random instances drawn from such Gaussian distribution.
The hyperfine component alone is not enough to produce meaningful inter-system crossing. However, in separating the effect on the electron and the hole, it allows for the external magnetic field to have an amplifying effect which results in greater inter-system crossing \cite{Flatte2016}.
The unitary singlet-triplet dynamics is described by the Schr\"odinger equation for the time evolution of a wavefunction in the singlet-triplet basis spanned by the total spin of the charge transfer state. For reasons of simplicity we omit here possible non-unitary effects, such as dephasing, which will be accounted for in the 2DES simulations. In particular, the dynamics is described by amplitudes S(t), T$_0$(t), T$_+$(t) and T$_-$(t), respectively, i.e.
\begin{equation}
e^{iE_T t/\hbar} \ket{\psi (t)}= S(t)\ket{CT,S} + \sum_{i=+,-,0} T_i(t) \ket{CT,T_i}.
\end{equation}
This base is a logical choice as excitation by incoming pulses in 2DES occurs directly in the singlet state (we assume the impulsive limit in which the laser pulses is much faster than any other timescale in the dynamics). The time evolution of the amplitudes is governed by the equations of motion
\begin{align}
\begin{split}
{\hbar}\frac{dS(t)}{dt} &= - i\delta_0(t) T_0(t) - i(E_S-E_T)S(t), \\
{\hbar}\frac{dT_0(t)}{dt} &= -i\delta_0(t) S(t) - i\sqrt{2}B(t)(T_+(t) + T_-(t)), \\
{\hbar}\frac{dT_+(t)}{dt} &= -i {\bar{B}}(t)T_+(t) - i\sqrt{2}B(t)T_0(t), \\
{\hbar}\frac{dT_-(t)}{dt} &= i{\bar{B}}(t)T_-(t) - i\sqrt{2}B(t)T_0(t).
\label{eq2}
\end{split}
\end{align}
\begin{figure}
\includegraphics[width=\columnwidth]{./Fig3-eps-converted-to.pdf}
\caption{Numerical solution of Eqs. (\ref{eq2}) for a set of singlet-triplet charge transfer states, subject to the effect of a magnetic pulse of the form in Eq. (\ref{eq3}). In the left column, (a) displays a resonant pulse $B \approx \omega$ with $B_0$ = 1 Tesla, which induces coherent Rabi oscillations on an initially populated singlet charge transfer state (b), transferring population to the initially dark triplet states (c). After the pulse ends, no population remains in the triplet state. In the right column, the magnetic pulse is off-resonant ($\omega \approx 0.1B$) with $B_0$ = 1 Tesla, but $\omega = 100$ GHz, as shown in (d). Such a pulse produces coherent population transfer from an initially populated singlet state (e) to initially dark triplet states in (f). In this case, the oscillations not only have a much bigger amplitude than in (a), but also they have more frequencies intermixed. Moreover, triplet states remain partially populated at the end of the pulse. Triplet states $\pm$ 1 have identical population time evolution.}
\label{Figmagnetic}
\end{figure}
In Eqs. (\ref{eq2}), $B(t)$ represents the magnetic field, while $\delta_0(t)$ and ${\bar{B}}(t)$ are, respectively, the difference and sum of the interaction of the magnetic field with each of the spins, i.e.
$\delta_0=\frac{B_e(t) - B_h(t)}{2}$,${\bar{B}}$ =$\frac{B_e(t) + B_h(t)}{2}$. From these equations two conclusions emerge immediately: first, why it is required that the Land{\'e} factor has to be different for
the electron and the hole, for otherwise $\delta$ = 0 and the dynamics only mix the triplet states. And second: the magnetic field must exhibit z- and x- polarization components. \santi{The reason is that circular polarization allows spin mixing between the singlet and the triplet spin zero components --both polarized along the z-direction-- and also from the triplet spin zero to the triplet $\pm$ one components, which will be Zeeman split resulting in distinct peaks in a 2D map. Here, we consider the possible phase between z- and x- pulse directions to be $\pi/2$, as it will not be important for spin dark states spectroscopy. Nonetheless the effect of finite phase might be worth exploring in a future work, since it can be used to (for example) delay the onset of transfer to certain components of the triplet states, with relevance not only in spectroscopy but also in spintronics.} The previous point relates to the aim of spin state mixing not only between the singlet and triplet spin zero states --both polarized along the z-direction-- but also among the three triplet spin states, which will be Zeeman split resulting in distinct peaks in a 2D map. \santi{Importantly, it is not necessary for the magnetic pulse to be resonant with the energy difference among singlet and triplet states, though if the condition is met, transfer would happen at lower intensity pulses.}
The magnetic pulse
\begin{align}
\begin{split}
B_x&= B_0 \sin(\omega t-\frac{\pi}{2}) \exp\left[\frac{(t-t_0)^2}{2\sigma_t^2}\right] \\
B_z&= B_0 \sin(\omega t) \exp\left[\frac{(t-t_0)^2}{2\sigma_t^2}\right]
\label{eq3}
\end{split}
\end{align}
has an oscillating component and a Gaussian envelope. The behavior of each spin of the spin-pair in an oscillating magnetic field under strong drive, described by Eqs. (\ref{eq2}) and displayed in Figs. \ref{Figmagnetic}, goes as $\cos(\frac{B_0}{\omega}\cos\omega t)$ for an envelope that is sufficiently wide such that its effect can be neglected for the purposes of dynamics of the spin pair. The ratio $\frac{B_0}{\omega}$ controls the amplitude of the oscillations, while the frequency depends both on the ratio $\frac{B_0}{\omega}$ and on $\omega$. Thus, we speak of two regimes. The first is called resonant and occurs when $B_0 \approx \omega$, while the second is denominated off-resonant \cite{Malla2017}. Notice that here resonance does not have anything to do with energy levels.
Figs. \ref{Figmagnetic}(a) and (d), show singlet triplet dynamics for two example magnetic pulses. In both cases, the amplitude of the
field $B_0$ is chosen to be 1 Tesla while its frequency $\omega$ is in an order of magnitude larger in Fig. \ref{Figmagnetic}(a) as compared to Fig. \ref{Figmagnetic}(d).
In both left and right panels on Fig. \ref{Figmagnetic} the initial state witnesses the spin singlet charge transfer state (Figs. \ref{Figmagnetic}(b) and (e)) completely
populated while the three triplet states have no population at all. Initially, as the pulse interacts with the spin pair, the population is partially transferred to the spin zero triplet charge transfer state and afterwards into the spin $\pm$ 1 triplet charge transfer states, as represented in Figs. \ref{Figmagnetic}(c) and (f) (only the spin +1 case is shown as the spin -1 behaves in exactly the same way).
From the analysis in Figs. \ref{Figmagnetic} we observe that a resonant pulse produces coherent Rabi oscillations that vanish as the pulse ends. In this case, very clean oscillations in the population of the charge transfer states can be observed, but there will be no final steady state population in the dark triplet charge transfer state (see \ref{Figmagnetic}(c)). However, working out of resonance means that on the one hand, several frequency components show up in the population oscillations and on the other hand that the steady state after the pulse has ended shows population in all four spin charge transfer states, which will decay on a slow timescale. Consequently, effective population transfer to the triplet states occurs with pulses as the one represented in Fig. \ref{Figmagnetic}(f). We conclude that for the purposes of this article where clean oscillations are ideal, resonant pulses are much better suited, while for applications where the interest lays in transferring population and controlling the spin states, off-resonant pulses would be required.
\section{2DES with a magnetic pulse}
\label{Section4}
The development of the theory of nonlinear spectroscopy \cite{Mukamelbook}, together with the progress of ultrafast laser technologies and multidimensional spectroscopic methods \cite{Hamm1998}
represented a decisive step forward in the study of dynamics in condensed-matter quantum systems. The result was a deeper understanding of processes such as light absorption, energy transport, and
quantum dynamics in open systems. 2DES represents a recent highlight in this development
\cite{Jonas2003,Fleming2005,Zigmantas2006,Engel2007,Schlaucohen2011,Plenio2013,Lim2015,Lienau2016,Prior2017,Zigmantas2018}.
2DES is a powerful technique to study nuclear and electronic correlations between different transitions or initial and final states. It utilizes three ultrashort, spectrally broad laser pulses
separated by controlled time delays (see Fig. \ref{Fig1}(a)) together with a local oscillator. The Fourier transform of the system response with respect to the coherence time $t_1$ (time between the
first and second pulses) and with respect to the rephasing time $t_3$ (time between the third pulse and the local oscillator) yields a 2D spectrum in the frequency domain which correlates absorption and
emission frequencies at each population time $t_2$ (time between the second and third pulses). To increase the number of coherent superpositions between quantum states, broad-band excitation lasers are
used. Each specific feature in the 2D spectrum then corresponds with one superposition between quantum states and provides real-time information about the both population and coherence
dynamics in the system.
The simulation of the 2DES-signals requires calculating the third-order non-linear response function $S^{(3)}(t_3,t_2,t_1)$ of the material which relates the driving fields of pulses coming at
intervals $t_1$, $t_2$ to the induced nonlinear polarization at delay $t_3$ after last pulse. $S^{(3)}(t_3,t_2,t_1)$ is only defined for positive times and reads
\begin{equation}
\begin{split}
S(t_3,t_2,t_1) &=\left(\frac{i}{{\hbar}}\right)^3 {\rm Tr} \hat{\mu} \mathcal{U}(t_1+t_2+t_3,t_2+t_1)\mu^{(-)}\mathcal{U}(t_1+t_2,t_1) \\& \mu^{(-)} \mathcal{U}(t_1,0)\mu^{(-)} \rho(0)
\label{eq7}
\end{split}
\end{equation}
where $\mu^{(-)} \ldots = [\hat{\mu}, \ldots ] $ is superoperator notation for commutator with dipole $\mu$. And where the $\mathcal{U}(t_a,t_b)$ is evolution superoperator which bring the density
matrix from time $t_b$ to time $t_a$ between the pulses, $\rho(t_b)= \mathcal{U}(t_a,t_b)\rho(t_b)$, in other words it is the Green function solution of the equation of motion governing the evolution of the density matrix
\begin{equation}
\frac{d\rho}{dt} =-\frac{i}{{\hbar}}\left[\mathcal{H}(t),\rho\right] + L (\rho),
\label{eqdensity}
\end{equation}
between two times. Here $L$ describes pure dephasing processes
\begin{equation}
L = \sum_{\alpha=S,T} \gamma_{i} \left[ \sigma_{\alpha} \rho(t) \sigma_{\alpha}^\dagger - \frac{1}{2} \left\{ \sigma_{\alpha}^{\dagger}\sigma_{\alpha},\rho(t) \right\}\right],
\label{Eq1}
\end{equation}
Where the fluctuations of singlet state $\sigma_{S}= \ket{CT,S}\bra{CT,S}$ are associated with dephasing rate $\gamma_{S}$ and (less intense) fluctuations of triplet states
$\sigma_{T}= \sum_{i \in \{0,+,-\}} \ket{CT,T_i}\bra{CT,T_i}$ with dephasing rate $\gamma_{T}$. This asymmetry originates from the magnetic noise affecting each of the states and to which triplet states can be more resilient. Note however that, since the key point regarding dephasing is that coherence survives long enough for singlet states generation, the main results of this article are unchanged by considering both dephasing rates as equal. The dephasing among triplet states is considered negligible.
The 2D signals are usually displayed in a mixed time-frequency domain
\begin{equation}
S(\Omega_3, t_2, \Omega_1)=\int_0^{\infty} dt_1 \int_0^{\infty} dt_3 S(t_3,t_2,t_1)e^{i\Omega_3t_3} e^{\pm i\Omega_1t_1}
\end{equation}
where the + (-) sign is applied for rephasing (nonrephasing) contributions to signal \cite{Schlaucohen2011,Plenio2013,Lim2015}. The double time dependence $t_a, t_b$ of evolution operator also challenges the proper definition of absorption spectra. We remain with $t_b=0$ , i.e. we define $I(\Omega)= \int_0^{\infty} dt e^{i\Omega t}\int_0^{\infty} {\rm Tr} \hat{\mu} \mathcal{U}(t,0)\mu^{(-)} \rho(0)$ when the absorption spectrum is simply obtained from 2D by integrating over $\Omega_3$, i.e. by using projection-slice theorem
$\int_0^{\infty} d\Omega_3 S(\Omega_3, t_2=0, \Omega_1)=S(t_3=0, t_2=0, \Omega_1)\propto I(\Omega_1)$.
For the coherent evolution $\mathcal{H}(t)$ we propose a scheme that includes a magnetic pulse that will be acting on the sample through the duration of the laser pulses sequence (see
Fig.~\ref{Fig1}); consequently, the Hamiltonian depends explicitly on time (and evolution operator $\mathcal{U}$ indeed depends on both initial and final time). Concretely, the Hamiltonian for the system interacting with the magnetic pulse reads
\begin{equation}
\begin{split}
\mathcal{H}(t) &= E_S\ket{S}\bra{S} + E_{T}\ket{T_0}\bra{T_0} + (E_{T} + Z(t))\ket{T_+}\bra{T_+} \\ & + (E_{T} - Z(t) )\ket{T_-}\bra{T_-}+ \mathcal{H_I^B(t)},
\end{split}
\end{equation}
where $Z(t)={\bar{B}}$ describes the (time-dependent) Zeeman splitting while $\mathcal{H_I^B}(t)$ describes part of the interaction which induces coherent transport among energy levels, and which reads
\begin{equation}
\mathcal{H_I^B}(t) = \delta_0 \ket{S}\bra{T_0} + B \ket{T_0}\left( \bra{T_+} + \bra{T_-} \right) + c.c..
\end{equation}
Eq. \ref{eqdensity} describes the evolution of the density matrix interacting with impulsive laser pulses \cite{Plenio2013}. Note that while the laser pulses are treated perturbatively, the magnetic pulse interaction with the system is treated exactly. The evolution is averaged over the phase of magnetic fields (as these are not phase synchronized with laser fields).
\section{Spin dark states detection}
We simulate the dynamics of a singlet-triplet system interacting with a resonant magnetic pulse in different spectroscopic configurations. \santi{We choose Landé factor values that mimic organic compounds used for solar cells fabrication, with $g_e = g_h \approx$ 2, and $\Delta g = 10^{-3}$ when the difference is relevant for dynamics}. The magnetic pulse is chosen to have amplitude $B_0$ = 1 T,
oscillation frequency $\omega$ = 1 THz, and a Gaussian envelope that guarantees the pulse has approximately constant amplitude through the duration of the laser sequence simulation, namely $>$ 400 fs.
Initially, dipole moment is associated only with the singlet state, meaning that the triplet state is completely dark in all the simulations. Consequently, in the absence of a magnetic field, the only
contribution comes from the singlet charge transfer state absorption. We note that the model presented here is not substantially altered by marginal singlet-triplet coupling or non-zero dipole moment
of the dark state.
2DES is a costly technique and its use has to be well-motivated when simpler spectroscopy techniques might suffice. In Fig. \ref{Figure2D2}(a), we plot the absorption spectrum in the absence and in
the presence of a magnetic pulse. These results demonstrate that in absence of B-induced interaction, there is only one absorption peak corresponding with the singlet state. Yet in the presence of
such an interaction, a complicated spectral signature emerges (see green line in Fig. \ref{Figure2D2}(a)). \santi{Though a four peak structure emerges, as expected from the singlet-triplet peak splitting due to the magnetic field}, any further analysis is hampered by the convoluted character of the signal, leading to complicated lineshapes. \santi{Note that the slight displacement of the central (main) peak is due to the interaction of the singlet state with the magnetic pulse.}
\begin{figure}
\includegraphics[width=10cm]{./Fig4new-eps-converted-to.pdf}
\caption{Numerical simulation of absorption spectra (a) and transient absorption spectra (b) in the model with a singlet and triplet charge transfer states with energies in 885.6 (1.4) and 892 (1.39)
nm (eV) respectively, in the absence (grey) and presence (green) of a magnetic pulse of frequency 1 THz and amplitude 1 Tesla, with an envelope that guarantees the pulse duration for the whole
experiment. Both plots are calculated at 400 fs population time. Notice that the peaks have been scaled to appear with similar height. In the absence of magnetic interaction only an absorption peak for the singlet charge transfer state shows in the
spectrum while in the presence of interaction a complicated peaks structure develops.}
\label{Figure2D2}
\end{figure}
Moving on to time-resolved methods, transient absorption spectroscopy (pump-probe) yields information about the energy of the quantum states and the transition and relaxation rates among them. In Fig.
\ref{Figure2D2}(b), we plot the pump-probe spectrum at a pump-probe delay of 400 fs. In the absence of a magnetic field, the lineshape of the singlet absorption peak is a well-defined Lorentzian. Similar to the case of the linear
spectrum discussed above, the spectral features are convoluted in a non-trivial manner in the presence of a magnetic field. \santi{The presence of the same four peaks as in absorption spectra Fig. \ref{Figure2D2} but with different relative intensity indicates energy transfer and, therefore, electronic coupling.} Since the only source of peak splitting is the magnetic pulse, resolution
depends on the interaction strength of the magnetic pulse with the electron and the hole, which is in principle unknown. The information about the interaction is encoded in the dipole moment redistribution and the coherence transfer among states, as a consequence of the magnetic pulse, and conspicuous either as oscillations in the population peaks or as distinctive in 2DES.
In Fig. \ref{Figure2D1}, we plot the real part of the spectrum resulting from the sum of all rephasing and non-rephasing components of ground state bleaching (GSB) and stimulated emission (SE). The
polarization of the excitation pulses was set to all-parallel. Such a configuration yields the strongest overall 2D-signal, and also contains coherence dynamics from intramolecular
states \cite{Mancal2012},
which is readily accounted for in simulations. Fig. \ref{Figure2D1}(a) displays a characteristic star-shaped 2D-peak corresponding with the singlet state which in absence of a magnetic pulse is the only
bright state of the system. Interaction of the states with a magnetic pulse through the duration of the 2DES sequence produces a spectrum richly populated with distinctive features in Fig.
\ref{Figure2D1}(b).
Focusing the analysis on the central part of the spectrum, we observe several peaks along the diagonal (see also Figs. \ref{Figure2D2}(a) and(b)) that correspond to absorption peaks from
different states composing the system. We are witnessing the triplet dark states mixed with the singlet state by the magnetic pulse. In addition, the numerous crosspeaks correspond to the different
coherences that gather the information of the corresponding redistribution of dipole moment among states and the coherence transfer between singlet and triplet ground state to excited state coherences.
\begin{figure}
\includegraphics[width=\columnwidth]{./2DwithandwithoutB-eps-converted-to.pdf}
\caption{Numerical simulation of a 2D electronic spectrum according to the model with a singlet and triplet charge transfer states with energies in 885.6 (1.4) and 892 (1.39) nm (eV) respectively. In (a), in the
absence of a magnetic pulse the absorption peak corresponding to the singlet charge transfer state. In (b), the action of a magnetic pulse of frequency 1 THz and amplitude 1 Tesla, with an envelope
that guarantees the pulse duration for the whole experiment, permits dipole moment redistribution from the singlet to the various triplet charge transfer states, as well as ground-excited states coherences transfer, as evidenced by several diagonal peaks and non-diagonal structures which demonstrate the presence of coherent transport among different quantum states. The latter are Zeeman split, with a gap of 1.1 (1) nm (meV). The coupling strength of the magnetic pulse to the charge transfer states is 112 (100) nm (meV) while the lifetimes are 80 fs for the singlet charge transfer state and 200 fs for the triplet components. Both plots are at population time 400 fs.}
\label{Figure2D1}
\end{figure}
We have demonstrated that 2DES can reveal the presence and dynamics of dark states if supplemented with a magnetic pulse. As a next step, we reconstruct the properties that characterize these dark states. In Eq.
(\ref{eq6}), we present the original Hamiltonian which needs to be reconstructed. The diagonal terms are the energies of the states prior to the interaction with the magnetic pulse with an extra term
we name Z accounting for the Zeeman splitting. The non-diagonal terms are the different interactions according to Eqs. (\ref{eq2}). Conversely, the diagonal peaks in the 2D spectrum in Fig.
\ref{Figure2D1}(b) provide information about the exciton energies, namely, about the eigenvalues of Eq. (\ref{eq6}). Therefore, to reconstruct the original energies we need the interaction terms of
Eq. (\ref{eq6}) and then revert the diagonalization procedure.
\begin{equation}
\mathcal{H} = \left(
\begin{array}{cccc}
S & 0 & A & 0 \\
0 & T_{-} - Z & B & 0 \\
A & B & T_{0} & B \\
0 & 0 & B & T_{+} + Z
\end{array}
\right).
\label{eq6}
\end{equation}
The diagonal peaks in Fig. \ref{Figure2D1}(b) are located at 866 (1.432), 882 (1.406), 895 (1.385), and 920 (1.348) nm (eV). The information about the interaction terms is encoded in the
non-diagonal peaks of Fig. \ref{Figure2D1}(b). The location of these peaks tells us about between which two states the coherence transport is happening. The frequency of oscillation of these peaks in
population time encodes the information about the interaction strength between energy levels. Hence, after multi-exponential fitting to get rid of dephasing, the Fourier transform in population time
of the non-diagonal peaks
provides us with the interaction terms. These give us the following numbers: 11 (0.009) nm (eV) for A and 16 (0.013) for B. With them we obtain that the interaction of the magnetic
pulse with the electron and the hole is 155 (0.124) nm (eV), which gives us a Zeeman splitting of 8 (0.006) nm (meV) and a reconstructed states at 880 (1.41), 894 (1.387), 894.6 (1.386), and 895
(1.385) nm (eV). Hence we are able to reconstruct the original states
with an error smaller than 2\%.
Note that the information required to perform the full reconstruction of the system’s Hamiltonian is attainable neither from linear spectroscopy nor from transient absorption spectroscopy, \santi{which can only reveal the presence of additional states with energy transfer among them}. 2DES on the other hand, fully resolves the peak structure and is necessary to determine the properties of the sates.
\section{Conclusions}
Summarizing, we propose and test numerically a modified version of 2DES in which a terahertz magnetic pulse is employed to create coherent population transfer from a bright spin singlet electron-hole
pair to the components of the corresponding triplet state. The magnetic pulse is able to modify the spins precession in a time-scale relevant for 2DES. The effect can be understood as a dipole moment
redistribution with coherence transfer that allows transitions from the ground state to the triplet states, which then appear as distinctive peaks in the 2D-spectrum. State reconstruction from the position of the peaks and
the oscillation frequencies of the coherences allows one to infer the properties of the original states, a feature that simpler spectroscopic techniques do not allow either for lack of information or
lack of resolution.
The magnetic pulse employed to demonstrate the feasibility of the proposal is realistic by today standards. The parameters describing the charge transfer states are as well within the typical range.
Tuning the magnetic pulse so as to produce the maximum effect requires some work, as it is the balance between the amplitude of the pulse and the frequency that dictates the transfer rate among states
and the frequency at which the coherences will oscillate. Nonetheless, the split of peaks and the unveiling of spin dark states is observed for almost any sensible magnetic pulse; it is therefore easy to
obtain a first estimation that permits fine-tuning the experiment. The strength with which the magnetic pulse couples to the spin states depends on the nature of the later, and has to be elucidated
experimentally. However, knowing the value of this coupling is necessary in order to reconstruct the in principle unknown energies of the states analyzed. Hence the need for 2DES experiments is
warranted.
Notice that different kinds of dark states with different spin configurations will have different equations of motion than Eq. (\ref{eq2}), and the constraints imposed by the Land{\'e} factor will not
necessarily apply. However, the background physics remains unchanged, and any spin dark state can be manipulated so, being it singlet or triplet. Therefore, the conclusions obtained in this article are valid for a wide class of
dark states.
\section{Acknowledgement}
S.O.C. and J.P. are grateful for financial support from MCIU
(SPAIN), including FEDER funds: FIS2015-69512-R and PGC2018-097328-B-100 together with Fundaci{\'o}n S{\'e}neca (Murcia, Spain) Project No. 19882/GERM/15. S.O.C.is supported by
the Fundación Ramón Areces postdoctoral fellowship (XXXI edition of grants for Postgraduate Studies in Life and Matter
Sciences in Foreign Universities and Research Centers 2019/2020). F.\v{S}. acknowledges support by Czech Science Foundation
(Grant No. 17-22160S). F.\v{S}. and J.H. acknowledge the mobility project “Exciton-exciton annihilation probed by non-linear spectroscopy ” (MSMT Grant No. 8J19DE009, DAAD-Projekt 57444962). J.H. acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy EXC-2089.
|
3,212,635,537,437 | arxiv | \section{Introduction}
Planetary atmospheric compositions offer valuable clues to the planet formation and evolution process, \rev{especially for giant planets with primordial atmospheres}.
Over the past decade a number of studies have suggested that atmospheric elemental ratios, such as the carbon-to-oxygen ratio (C/O), can diagnose the orbital distance where a planet initially forms \citep[e.g.,][]{Oberg+11,Madhusudhan+14,Madhusudhan+17,Ali-dib+14,Helling+14,Thiabaud+15,Piso+15,Piso+16,Oberg&Bergin16,Cridland+16,Cridland+17,Cridland+19,Espinoza+17,Eistrup+16,Eistrup+18,Eistrup+22,Booth+17,Booth&Ilee19,Oberg&Wordsworth19,Ohno&Ueda21,Turrini+22,Schneider&Bitsch21,Molliere+22,Pacetti+22,Bitsch+22,Notsu+22,Eistrup22}.
Many previous studies focused on the atmospheric C/O ratio, as it has significant impacts on atmospheric chemistry and likely leaves observable fingerprints \citep[e.g.,][]{Madhusudhan+12,Moses+13,Moses+13b,Molliere+15,Drummond+19,Notsu+20,Dash+22}.
Beyond the C/O ratio, several recent studies also have also discussed the potential importance of other elements, such as nitrogen \citep{Piso+16,Cridland+20,Ohno&Ueda21,Notsu+22}, sulfur \citep{Turrini+22,Pacetti+22}, and refractory metals \citep{Lothringer+21,Schneider&Bitsch21b,Hands&Helled22,Chachan+22}.
Nitrogen is the third most abundant volatile element in solar composition and may provide important constrains on the planetary formation environments.
Nitrogen has particular advantages to probe the formation locations.
\rev{
\citet{Piso+16} first pointed out that the N/O ratio of disk gas is always higher than stellar N/O by a factor of $\ge$2 and monotonically increases with radial distance, which provides additional clues to constrain planetary formation location from a planet's atmospheric N/O ratio.
\citet{Cridland+20} studied the atmospheric compositions of warm Jupiters using a population synthesis model and suggested that combining C/O and N/O helps to probe the formation history, such as whether the planet acquired its atmosphere outside of the refractory carbon erosion front.
\citet{Ohno&Ueda21} also stressed that the atmospheric N/O is expected to be sensitive to formation location if disk solids, such as pebbles and planetesimals, determine the atmospheric composition.
This is because the solid N/O ratio has an order-of-magnitude variation as a function of a radial distance (see also \citealt{Notsu+22} for the discussion based on a disk chemistry model).
\citet{Turrini+22} and \citet{Pacetti+22} suggested that C/N, N/O, and S/N ratios help to constrain the formation and migration pathways of giant planets.
}
\rev{
To clarify the usefulness of the N/O ratio for example, Figure \ref{fig:NtoO} shows the nitrogen-to-oxygen ratio (N/O) of solids and gas in a protoplanetary disk computed by the phase equilibrium model of \citet{Ohno&Ueda21} for the protosolar disk model of \citet{Oberg&Wordsworth19}.
The gas-phase N/O monotonically increases with orbital distance, as the O-bearing molecules (e.g., H$_2$O, CO$_2$) are gradually removed from gas phase via condensation while most of N remains as a highly volatile N$_2$ gas within N$_2$ snowline.
The solid-phase N/O shows an order-of-magnitude orbital variation because of large abundance difference between NH$_3$ and N$_2$ \citep[e.g.,][]{Oberg&Bergin21}.
The latter indicates the strong dependence of atmospheric N/O to formation location if solid (e.g., planetesimal) accretion predominantly determines the atmospheric composition.
}
It is also worth noting that several recent studies have discussed formation scenarios for Jupiter in our solar system.
Motivated by the nitrogen abundance being comparable to other heavy elements in the Jovian atmosphere \citep[for recent review, see][]{Guillot+22,Atreya+22}, \citet{Oberg&Wordsworth19} and \citet{Bosman+19} proposed that Jupiter might have initially formed outward of the N$_2$ snowline beyond $30~{\rm AU}$ where solid elemental ratios coincide with solar values \citep[see also][]{Owen+99}.
\citet{Ohno&Ueda21} suggested that the Jovian atmospheric composition could also be explained if Jupiter formed at a locally cold disk region caused by the shadow cast by a disk substructure, such as a dust pileup at H$_2$O snowline, which is not nearly at far away from the central star.
\begin{figure}[t]
\centering
\includegraphics[clip, width=\hsize]{Figure1.pdf}
\caption{The nitrogen-to-oxygen ratio as a function of orbital distance in a protoplanetary disk model. The black dashed and solid lines show the N/O ratio of disk gas, and solids, respectively. The solar ratio is shown in dotted blue. We compute this profile using the phase equilibrium model of \citet{Ohno&Ueda21} assuming the disk model of \citet{Oberg&Wordsworth19}.
}
\label{fig:NtoO}
\end{figure}
In substellar atmospheres, in the absence of ionizing flux, N$_2$ and NH$_3$, are the main nitrogen reservoirs \citep{Lodders&Fegley02}.
HCN can also be abundant if photochemical processes are at work \citep{Moses+13}.
NH$_3$ and HCN would be likely detectable by near future infrared observations by JWST and Ariel \citep{MacDonald&Madhusudhan17}, while N$_2$ is in general not observable due to \rev{the negligibly low visible and infrared opacity for the temperature regime of exoplanets.}
\citet{Hobbs+19} used a photochemical kinetic model to show that the abundances of C- and O-bearing species, such as H$_2$O and CO, are insensitive to N/H ratio in hot Jupiters like HD 209458b.
\citet{Ramirez+20} also investigated the impact of N/H ratio on TiO abundances in ultra-hot Jupiters and found that the TiO abundance is nearly independent of N/H.
Since C- and O-bearing species abundances are insensitive to bulk nitrogen abundance \rev{in sub-stellar atmospheres}, it appears that the only route to diagnose \rev{the bulk nitrogen abundance of a giant planet is from NH$_3$ and/or HCN.
\rev{However}, constraining the bulk nitrogen abundance from NH$_3$ and HCN is a complex task.
The NH$_3$ and HCN abundances in the observable atmosphere readily deviates from thermochemical equilibrium abundances because of disequilibrium effects, such as vertical mixing and photochemistry \citep[e.g.,][]{Moses+11,Line+11,Venot+13}.
For warm planets of $T_{\rm eq}\la1000~{\rm K}$, \citet{Fortney+20} investigated disequilibrium NH$_3$ abundances on Saturn-like planets with various $T_{\rm eq}$ and found that NH$_3$ abundance depends on a number of factors, such as planetary mass, age, and metallicity.
They also suggested that N$_2$ will actually dominate over NH$_3$ over a very wide range of temperature and ages, making the observable NH$_3$ abundance only a lower limit of bulk nitrogen abundance.
\citet{Hu21} also investigated photochemistry on temperate/cold H$_2$-rich planets and found that NH$_3$ tends to be depleted due to photodissociation, especially on planets around G/K stars.
In this study, we expand the work of \citet{Fortney+20} with a particular focus on nitrogen chemistry.
\rev{Here in Paper I we systematically investigate the thermal structure of planetary deep atmosphere, which significantly affects the disequilibrium abundance of NH$_3$, as demonstrated by \citet{Fortney+20}.}
\rev{While \citet{Fortney+20} investigated the effects of planetary deep atmospheres using numerical models, this study advances the field by establishing a semi-analytical model that explicitly links planetary gravity, intrinsic temperature, metallicity, bulk nitrogen abundance, and disequilibrium NH$_3$ abundance. The model is readily applicable to arbitrarily planets and will be useful to interpret the retrieved NH$_3$ abundance in future observations.}
The organization of this paper is as follows.
In Section \ref{sec:overview}, we introduce a basic background of nitrogen equilibrium and disequilibrium chemistry.
In Section \ref{sec:deep_adiabat}, we investigate atmospheric pressure-temperature (\emph{P--T}) profiles for a wide range of planetary parameters. We derive a semi-analytical fit to understand why giant planets typically have a universal deep adiabat, irrespective of incident flux, which has a major impact on NH$_3$ abundances from disequilibrium chemistry from vertical mixing.
In Section \ref{sec:N_map}, we identify the relation between NH$_3$ and bulk nitrogen abundances as a function of planetary parameters from semi-analytical arguments.
In Section \ref{sec:discussion}, we \rev{describe caveats of this study.}
In Section \ref{sec:summary}, we summarize our findings.
\rev{In the paper II of this series \citep{Ohno&Fortney22b}, we verify our semi-analytical predictions using a photochemical kinetics model and discuss the observational implications for atmospheric nitrogen species on transmission and emission spectra.
\section{Nitrogen Chemistry: The Importance of The Deep Atmosphere Structure}\label{sec:overview}
One of the important factors in controlling the observable NH$_3$ abundance is vertical vertical mixing within an atmosphere.
Atmospheric compositions follow thermochemical equilibrium in the deep hot atmospheres, while the abundances at lower pressure, where it is colder, tend to be out of equilibrium, vertically constant, and reflect the equilibrium compositions of the deep atmosphere \rev{(though see Section \ref{sec:caveat} for a caveat on this picture)}.
This phenomena if often called ``quenching'' \citep[e.g.,][]{Fegley&Prinn85,Fegley&Lodders94,Zahnle+14,Tsai+18} and was originally identified for CO/CH$_4$ in the Jovian atmosphere, where detected CO abundances are many orders of magnitude higher than thermochemical equilibrium calculations \citep[][]{Prinn&Barshay77}.
This quenching is caused by the slow thermochemical conversion, compared to relatively fast vertical mixing \citep[e.g.,][]{Moses+11}.
Several recent studies have attempted to constrain the strength of vertical mixing in brown dwarf and giant planet atmospheres from quenched molecular abundances \citep{Miles+20,Kawashima&Min21,Mukherjee+22b}.
Since the upper atmospheric composition is related to the composition of deep hot atmosphere, it is necessary to understand the planetary deep atmosphere and interior to relate the observable NH$_3$ abundance with bulk nitrogen abundance \citep{Fortney+20}.
To this end, we first introduce the nitrogen chemistry in deep atmospheres where thermochemical equilibrium is expected.
\subsection{Thermochemical equilibrium and vertical quenching of NH$_3$}\label{sec:nitrogen_chem}
\begin{figure*}[t]
\centering
\includegraphics[clip, width=\hsize]{Figure2.pdf}
\caption{(Left) \emph{P--T} profiles of solar composition atmospheres computed in radiative-convective equilibrium (see Section \ref{sec:RCE_result} for details). Different colored lines show the \emph{P--T} profiles for different equilibrium temperatures. We assume a surface gravity of $g=10~{\rm m~s^{-2}}$ and planetary intrinsic temperature of $T_{\rm int}=100~{\rm K}$. The gray dashed lines show the abundance ratio contours of ${\rm NH_3/N_2}=0.01$, $0.1$, $1$, $10$, $100$ from top to bottom, computed by Equation \eqref{eq:contour_N2_NH3_1}. \rev{(Right) Vertical distribution of NH$_3$ volume mixing ratio for the \emph{P--T} profiles in the left panel. The solid and dash-dot lines show the distributions for eddy diffusion coefficients of $10^{8}$ and ${10}^{10}~{\rm cm^2~s^{-1}}$, respectively. Note that the distributions of different eddy diffusion coefficients are almost superimposed on each other at $T_{\rm eq}<1000~{\rm K}$. We have turned off photochemistry for the sake of simplicity.}}
\label{fig:PT_example}
\end{figure*}
The quenching behavior of NH$_3$ has an interesting characteristic: the quenched NH$_3$ abundance is insensitive to the strength of vertical mixing \citep{Saumon+06,Zahnle+14,Fortney+20}.
As discussed in \citet{Zahnle+14}, this is caused by the abundance ratio contours of $\rm NH_3/N_2$ being nearly parallel to the adiabatic profiles of substellar atmospheres (see also Figure \ref{fig:PT_example}).
The vertically quenched abundance is determined, to a good approximation, by the equilibrium abundance at certain depth where thermochemical interconversion timescale becomes equal to the vertical mixing timescale.
However, since the deep adiabat is nearly along the contour of constant $\rm NH_3/N_2$ ratio, the quenched NH$_3$ abundance is nearly the same wherever the quenching takes place.
This characteristic has an advantage in interpreting the quenched NH$_3$: one does not need to worry too much about the uncertainty of vertical mixing strength, parameterized by $K_{\rm zz}$.
\rev{To further clarify the quenching behavior of NH$_3$, the right panel of Figure \ref{fig:PT_example} shows the vertical distribution of NH$_3$ in a solar composition atmosphere computed by the chemical kinetics code VULCAN \citep{Tsai+17,Tsai+21} for various planetary equilibrium temperature and eddy diffusion coefficients. While the NH$_3$ distribution in the upper atmosphere depends on the eddy diffusion coefficient for hot ($T_{\rm eq}\ga1000~{\rm K}$) planets where the quenching occurs at shallower radiative parts of the atmosphere, the abundances are nearly independent of the eddy diffusion at warm ($T_{\rm eq}<1000~{\rm K}$) planets where the quenching occurs at deep adiabatic atmospheres.}
Since the quenched abundance is determined by the composition of a deep atmosphere point where thermochemical equilibrium is valid, it is worth first examining the equilibrium abundance of nitrogen species.
The law of mass action provides the relation of N$_2$ and NH$_3$ that should be satisfied in thermochemical equilibrium, given by
\begin{equation}\label{eq:nitrogen_eq}
\frac{P_{NH_3}^2}{P_{N_2}P_{H_2}^3}=K_{\rm N_2 \rightleftharpoons NH_3}=Ae^{B/T},
\end{equation}
where $P_{\rm H_2}$, $P_{\rm N_2}$, and $P_{\rm NH_3}$ are the partial pressure of H$_2$, N$_2$, and NH$_3$, respectively, $K_{\rm N_2 \rightleftharpoons NH_3}$ is the equilibrium constant of N$_2$-NH$_3$ interconversion (N$_2$+3H$_{\rm 2}$ $\rightleftharpoons$2NH$_3$), $A=5.90\times{10}^{-13}~{\rm bar^{-2}}$, and $B=13207~{\rm K}$ \citep{Zahnle+14}.
Assuming that N$_2$ and NH$_3$ accommodate most of the nitrogen, we can approximate the nitrogen conservation as $f_{\rm N}\approx f_{\rm NH_3}+2f_{\rm N_2}$, where $f_{\rm N}$, $f_{\rm N_2}$, and $f_{\rm NH_3}$ are volume mixing ratios of total nitrogen, N$_2$, and NH$_3$.
Then, one can obtain the pressure-temperature relation of $\rm NH_3/N_2=\xi$ contour as \citep{Zahnle+14}
\begin{equation}\label{eq:contour_N2_NH3_1}
f_{\rm N}\left( \frac{\xi^2}{2+\xi }\right) =P^2 f_{\rm H_2}^3Ae^{B/T},
\end{equation}
where $f_{\rm H2}={\rm H_2/(H_2+He)}=0.859$ and $f_{\rm N}=1.16\times{10}^{-4}$ in solar elemental abundances of \citet{Asplund+21}.
We note that $f_{\rm N}$ is not identical to the N/H ratio, as it is given by
\begin{equation}\label{eq:N/H}
f_{\rm N}={\rm \frac{N}{H_2+He } }={\rm \frac{2N/H}{ 1+2He/H } }=2f_{\rm H_2}{\rm N/H},
\end{equation}
where $\rm N/H=6.76\times{10}^{-5}$ is the value for solar composition \citep{Asplund+21}.
Equation \eqref{eq:contour_N2_NH3_1} is inconvenient from an observational perspective, as both $f_{\rm N}$ and $\xi$ are unknown.
Instead, eliminating $f_{\rm N_2}$ in Equation \eqref{eq:nitrogen_eq} using $f_{\rm N}\approx f_{\rm NH_3}+2f_{\rm N_2}$, we obtain
\begin{equation}\label{eq:f_N1}
{\rm N/H}=\frac{f_{\rm NH_3}}{2f_{\rm H_2}}\left[ 1 + \frac{2f_{\rm NH_3}e^{-B/T}}{Af_{\rm H_2}^3P^2} \right],
\end{equation}
where we use Equation \eqref{eq:N/H}.
Under chemical equilibrium, Equation \eqref{eq:f_N1} can straightforwardly constrain the bulk nitrogen abundance from the NH$_3$ abundance.
In addition, as introduced above, the equilibrium NH$_3$ abundance is approximately constant along the deep adiabatic profiles.
Thus, it is expected that the quenched NH$_3$ abundance is mostly determined by the deep adiabatic profile alone.
\section{Constraining Thermal Structures of Deep Atmospheres}\label{sec:deep_adiabat}
\subsection{Radiative Zero Solution of Irradiated Exoplanets}
The preceding argument highlights the importance of identifying the thermal structures of deep atmospheres (below the photosphere) to relate the quenched NH$_3$ abundance with bulk nitrogen abundance.
Here, we point out an interesting trend of deep atmospheres: many planets with different equilibrium temperatures\footnote{We refer the equilibrium temperature to the temperature for zero Bond albedo with full heat redistribution unless otherwise indicated.} ($T_{\rm eq}=250$--$1200~{\rm K}$) have nearly the same deep adiabatic profile as seen in Figure \ref{fig:PT_example}.
\citet{Fortney+07} first reported such a universal deep adiabat in their radiative-convective models.
Motivated by simplified calculations with dual-band radiative transfer \citep{Guillot10}, \citet{Fortney+20} speculated that the universal deep adiabat may emerge owing to the steep change of visible-to-infrared opacity ratio caused by the loss of gas-phase alkali metals, although the actual cause still remains unclear.
\rev{The universality of the deep adiabat has a crucial impact on the disequilibrium abundance of NH$_3$: the quenched NH$_3$ abundance is nearly independent of the equilibrium temperature for temperate to warm exoplanets, as seen in the right panel of Figure \ref{fig:PT_example}.}
Here, we elaborate why many planets have nearly the same deep adiabatic profile for a wide range of equilibrium temperatures.
The common thermal structure independent of upper boundary conditions is reminiscent of the ``radiative zero solution'' discussed in the context of stellar and protoplanetary envelope structures \citep[e.g.,][]{Hayashi+62,Mizuno80,Stevenson+82,Kippenhahn+94}.
In purely radiative atmospheres without convection, the atmospheric temperature structure follows
\begin{eqnarray}\label{eq:nabla_rad2}
\left( \frac{d\ln T}{d\ln P}\right)_{\rm rad}&=&\frac{3\kappa L_{\rm int}}{64\pi \sigma GM}\frac{P}{T^4}=\frac{3\kappa_{\rm 0} P^{1+\alpha}T^{\rm \beta}}{16 g}\frac{T_{\rm int}^4}{T^4},
\end{eqnarray}
where $L_{\rm int}=4\pi R^2 \sigma T_{\rm int}^4$ is the planetary intrinsic luminosity, $T_{\rm int}$ is the planetary intrinsic temperature, $\sigma$ is the Stefan-Boltzmann constant, $g=GM/R^2$ is the planetary gravity, and $\kappa=\kappa_{\rm 0}P^{\alpha}T^{\beta}$ is the atmospheric Rosseland-mean opacity.
Assuming constant gravity, $\alpha$, and $\beta$, Equation \eqref{eq:nabla_rad2} yields an analytical solution of
\begin{equation}\label{eq:rad_zero}
T=T_{\rm 0}+\left[ \frac{3\kappa_{\rm 0}T_{\rm int}^4(4-\beta)}{16g(1+\alpha)}\right]^{1/(4-\beta)} (P^{(1+\alpha)/(4-\beta)}-P_{\rm 0}^{(1+\alpha)/(4-\beta)}),
\end{equation}
where $P_{\rm 0}$ and $T_{\rm 0}$ are the pressure and temperature of the upper boundary.
For $(4-\beta)/(1+\alpha)>0$ \footnote{For $(4-\beta)/(1+\alpha)<0$, the temperature structure converges to the following isothermal profile in the limit of $P\gg P_{\rm 0}$:
\begin{equation}
\nonumber
T\approx T_{\rm 0}-\left[ \frac{3\kappa_{\rm 0}T_{\rm int}^4P_{\rm 0}^{1+\alpha}(4-\beta)}{16g(1+\alpha)}\right]^{1/(4-\beta)}.
\end{equation}
In this case, the upper boundary conditions controls the temperature structure of the deep atmosphere.
}, since $P\gg P_{\rm 0}$ and $T\gg T_{\rm 0}$ in the limit of a deep atmosphere, the temperature structure asymptotically approaches the same temperature structure relation with $P_{\rm 0}=0$ and $T_{\rm 0}=0$ in Equation \eqref{eq:rad_zero} regardless of the upper boundary condition, which is called the radiative-zero solution \citep[e.g.,][]{Hayashi+62,Mizuno80,Stevenson+82,Kippenhahn+94}.
\begin{figure*}[t]
\centering
\includegraphics[clip, width=0.49\hsize]{Figure3_1.pdf}
\includegraphics[clip, width=0.49\hsize]{Figure3_2.pdf}
\includegraphics[clip, width=0.49\hsize]{Figure3_3.pdf}
\includegraphics[clip, width=0.49\hsize]{Figure3_4.pdf}
\caption{(Upper left and right panels) Pressure and temperature dependence of the Rosseland mean opacity of a solar composition gas, based on the \citet{Freedman+14} analytic opacity fit.
(Lower left) The ratio of the RCB pressure to the threshold RCB pressure estimated by Equation \eqref{eq:metric_adiabat}. The \emph{P--T} profile converges to the radiative zero solution before it meets the RCB for $P_{\rm rcb}/P_{\rm thr}>1$ (redder colors), leading to the deep adiabatic profile that is insensitive to stellar insolation. The black contour denotes $P_{\rm rcb}/P_{\rm thr}=1$.
(Lower right) The pressure dependence of the radiative zero solution, $(1+\alpha)/(4-\beta)$. It is expected that the \emph{P--T} profiles tend to converge to the same radiative zero solution of $T\propto P^{(1+\alpha)/(4-\beta)}$ in the \emph{P--T} space with the same value of $(1+\alpha)/(4-\beta)$. We have filled the space of $(1+\alpha)/(4-\beta)<0$ in white for clarity. \rev{In computing the bottom left panel, we set $\phi=1$ when $\phi$ exceeds unity, as the \emph{P--T} profile always converges to the radiative-zero solution at $\phi\ge1$.}
}
\label{fig:dependence}
\end{figure*}
\begin{figure*}[t]
\centering
\includegraphics[clip, width=0.4\hsize]{Figure4_1.pdf}
\includegraphics[clip, width=0.4\hsize]{Figure4_2.pdf}
\includegraphics[clip, width=0.33\hsize]{Figure4_3.pdf}
\includegraphics[clip, width=0.33\hsize]{Figure4_4.pdf}
\includegraphics[clip, width=0.33\hsize]{Figure4_5.pdf}
\includegraphics[clip,width=0.33\hsize]{Figure4_6.pdf}
\includegraphics[clip,width=0.33\hsize]{Figure4_7.pdf}
\includegraphics[clip,width=0.33\hsize]{Figure4_8.pdf}
\caption
\emph{P--T} profiles for a variety of planetary surface gravities, intrinsic temperatures, atmospheric metallicities, and C/O ratios. All models are 1D radiative-convective equilibrium. The thicker lines denote convective regions, and thin lines show the radiative regions. The gray dashed line plots the RCB pressure estimated from Equation \eqref{eq:P_RCB1} with the Rosseland mean opacity of \citet{Freedman+14}, which reasonably explains the deep innermost RCB found in numerical results. The black dotted line shows our semi-analytical fit of the deep adiabatic \emph{P--T} profile (Equation \ref{eq:Pad_fit}), which the \emph{P--T} profiles converge on, for $T_{\rm eq}\sim250$--$1200~{\rm K}$.}
\label{fig:PT_summary}
\end{figure*}
The radiative zero solution does not necessarily apply for atmospheric structures, as convection sets in to force the temperature gradient to the adiabatic temperature gradient $\nabla_{\rm ad}$.
In the convective region, from the definition of the adiabatic gradient $(d\ln{T}/d\ln{P})=\nabla_{\rm ad}$, the temperature structure follows
\begin{equation}
T = T_{\rm rcb}\left( \frac{P}{P_{\rm rcb}}\right)^{\nabla_{\rm ad}},
\end{equation}
where $P_{\rm rcb}$ and $T_{\rm rcb}$ are pressure and temperature of radiative-convective boundary (RCB), and we assume a constant adiabatic gradient for the sake of simplicity.
Inserting Equation \eqref{eq:rad_zero} into this equation with $P=P_{\rm rcb}\gg P_{\rm 0}$, the deep adiabatic temperature can be expressed by
\begin{equation}\label{eq:T_ad}
T = \left[ T_{\rm 0}+ \left(\frac{3\kappa_{\rm 0}T_{\rm int}^4P_{\rm rcb}^{1+\alpha}(4-\beta)}{16g(1+\alpha)} \right)^{1/(4-\beta)}\right]\left( \frac{P}{P_{\rm rcb}}\right)^{\nabla_{\rm ad}},
\end{equation}
This equation could strongly depend on the upper boundary condition if the first term in the prefactor (i.e., $T_{\rm 0}$) dominates over the second term.
In other words, the deep adiabatic profile does depend on the upper boundary condition if the atmospheric \emph{P--T} profile meets the RCB before it converges to the radiative-zero solution.
Based on the preceding argument, we suggest that planets have the common deep adiabatic profile regardless of stellar insolation if the \emph{P--T} profile \rev{converges to the radiative zero solution above the RCB pressure level.
Equating Equation \eqref{eq:nabla_rad2} and $\nabla_{\rm ad}$, the RCB pressure is given by
\begin{equation}\label{eq:P_RCB1}
P_{\rm rcb}=\left(\frac{16gT_{\rm rcb}^{4-\beta}}{3 \kappa_{\rm 0}T_{\rm int}^4} \nabla_{\rm ad}\right)^{1/(1+\alpha)}.
\end{equation}
\rev{Inserting Equation \eqref{eq:P_RCB1} into \eqref{eq:rad_zero}, we obtain the relation between the RCB temperature and upper boundary temperature $T_{\rm 0}$ as
\begin{equation}\label{eq:T0Trcb}
\frac{T_{\rm 0}}{T_{\rm rcb}}\approx 1-\phi^{1/(4-\beta)},
\end{equation}
where we have approximated $P_{\rm 0}=0$, as $P_{\rm rcb}\gg P_{\rm 0}$. We have introduced a dimensionless parameter defined as
\begin{equation}\label{eq:phi}
\phi \equiv \frac{4-\beta}{1+\alpha}\nabla_{\rm ad}.
\end{equation}
The $\phi$ parameter is equivalent to the ratio of adiabatic temperature gradient to the radiative temperature gradient in the limit of deep atmospheres.
Equation \eqref{eq:T0Trcb} is invalid at $\phi>1$ because one cannot define an RCB in the atmosphere with $\phi>1$, where convection does not occur (see Appendix \ref{sec:appendix}).
}
Meanwhile, solving the equality of the first and second terms in the prefactor of Equation \eqref{eq:T_ad} with respect to $P_{\rm rcb}$, we can evaluate a threshold RCB pressure above which the thermal structure converges to the radiative zero solution before it meets the RCB, as
\begin{equation}\label{eq:RCB_cri}
P_{\rm thr}=\left[ \frac{16gT_{\rm 0}^{4-\beta}(1+\alpha)}{3\kappa_{\rm 0}T_{\rm int}^4(4-\beta)} \right]^{1/(1+\alpha)}.
\end{equation}
Taking the ratio of Equation \eqref{eq:P_RCB1} to \eqref{eq:RCB_cri} \rev{with \eqref{eq:T0Trcb},
we achieve the diagnostic metric, given by
\rev{
\begin{eqnarray}\label{eq:metric_adiabat}
\nonumber
\frac{P_{\rm rcb}}{P_{\rm thr}}&=&\left( \frac{4-\beta}{1+\alpha}\nabla_{\rm ad}\right)^{1/(1+\alpha)}\left(\frac{T_{\rm rcb}}{T_{\rm 0}}\right)^{(4-\beta)/(1+\alpha)}\\
&=&\left[ \phi^{1/(\beta-4)}-1\right]^{(\beta-4)/(\alpha+1)}
\end{eqnarray}}
If $P_{\rm rcb}/P_{\rm thr}>1$, the \emph{P--T} profile converges to the radiative zero solution before it meets the RCB, resulting in a deep adiabatic profile being independent of upper boundary condition, i.e., level of stellar insolation.
Interestingly, Equation \eqref{eq:metric_adiabat} indicates that whether or not stellar insolation affects the deep profile only depends on the adiabatic gradient and atmospheric opacity law.
Next we investigate Equation \eqref{eq:metric_adiabat} using the opacity of solar composition gas, from \citet{Freedman+14}.
We numerically compute $\alpha$ and $\beta$ from the analytical fit of the Rosseland-mean opacity obtained by \citet{Freedman+14} as a function of pressure and temperature, as shown in the upper two panels of Figure \ref{fig:dependence}.
While previous studies adopted single values of $\alpha$ and $\beta$ \citep[e.g.,][]{Rogers&Seager10,Owen&Wu17,Ginzburg+18}, these values differ at different pressure and temperature conditions.
We also note that since the complex \citet{Freedman+14} analytic fitting formula changes at 800 K, which is necessary since the opacities change with $T$, this leads to plots of $\beta$ that somewhat exaggerate the sharpness of this change to the opacities.
The adiabatic gradient is taken from the equation of state (EOS) for H/He mixtures \citep{Chabrier+19}, which has updated the widely used SCvH EOS \citep{Saumon+95}.
The lower left panel of Figure \ref{fig:dependence} shows $P_{\rm rcb}/P_{\rm thr}$ as a function of pressure and temperature.
We find that the RCB pressure is \rev{much higher} than the threshold pressure in the temperature range of $\sim800$--$1400~{\rm K}$.
\rev{Actually, the $\phi$ parameter takes a value of $\phi\ga1$ in that temperature range, which prohibits the transition from radiative to convective atmospheres.}
This indicates that the \emph{P--T} profiles tend to converge to the radiative zero solution before reaching the RCB in that temperature range.
In Figure \ref{fig:PT_example}, \emph{P--T} profiles indeed converge to the same deep adiabatic profile when the temperature at the second (deeper) nearly isothermal region at $P\sim 10$--$100~{\rm bar}$ falls into $\sim800$--$1500~{\rm K}$, consistent with the phase space of \rev{$P_{\rm rcb}/P_{\rm thr}\gg 1$} in Figure \ref{fig:dependence}.
\citet{Fortney+07} and \citet{Fortney+20} also obtained the \emph{P--T} profiles converging to same deep adiabatic line when the same condition applies.
We note that the deep adiabatic profile might diverge even if the \emph{P--T} profile converges to the radiative zero solution.
This is because the temperature structure obeys $T\propto P^{(1+\alpha)/(4-\beta)}$ in the radiative zero solution (Equation \ref{eq:rad_zero}), where different power-law index of $(1+\alpha)/(4-\beta)$ yields different temperature structure lines.
As shown in the bottom right panel of Figure \ref{fig:dependence}, the index is roughly constant, $(1+\alpha)/(4-\beta)\sim 0.25$, in the temperature range of $\sim800$--$1300~{\rm K}$.
Thus, it might be expected that the \emph{P--T} profiles tend to converge to the radiative-zero solution of $T\propto P^{0.25}$ in this temperature range where Equation \eqref{eq:metric_adiabat} predicts $P_{\rm rcb}\ga P_{\rm thr}$.
\subsection{Numerical Exploration of P--T profiles}\label{sec:RCE_result}
To further study the thermal structure of deep atmospheres, we explore the PT profiles at wide range of planetary properties using EGP, a version of the 1D radiative-convective equilibrium model of \citet{McKay+89} and \citet{Marley&McKay99} \footnote{\rev{A Python version of the adopted model has now been made publicly available \citep{Mukherjee+22a}.}}.
The model solves for radiative-convective equilibrium in a plane-parallel atmosphere based on the algorithm of \citet{Toon+89} with thermochemical equilibrium accounting for rain-out effects \citep{Fegley&Lodders94,Lodders&Fegley02,Visscher+06,Visscher&Lodders10}.
The model implements non-gray atmospheric opacity with the correlated-k approximation, where we adopt correlated k-coefficients datasets calculated for 1060 pressure-temperature grid points (\citealt{Lupu+21}, see references therein for the details of opacity sources)
We note that our calculations omit TiO/VO opacity, except for the C/O$=1.1$ models.
In convective layers, the model switches to use the adiabatic temperature gradient extracted from the equation of state for H/He mixture with $Y=0.292$ in \citet{Chabrier+19}.
The model has been extensively applied for solar system objects \citep{McKay+89,Marley&McKay99,Fortney+11}, exoplanets \citep{Fortney+05,Fortney+07,Fortney+08,Fortney+20,Morley+13,Morley+15,Morley+17,Thorngren+19,Gao+20,Mayorga+21}, and brown dwarfs \citep{Marley+96,Marley+21,Saumon&Marley08,Morley+12,Morley+14,Robinson&Marley14,Tang+21,Karalidi+21,Mukherjee+22b}.
We refer readers to \citet{Marley&Robinson15} and \citet{Marley+21} for further details of the radiative-convective equilibrium model.
Figure \ref{fig:PT_summary} shows \emph{P--T} profiles for various values of the planetary equilibrium temperature, surface gravity, intrinsic temperature, atmospheric metallicity, and C/O ratio.
As found in previous studies, cooler planets ($T_{\rm eq}\la1000~{\rm K}$) tend to have steeper temperature gradients, which yields hotter middle atmospheres ($P\sim0.1$--$1~{\rm bar}$) as compared to the equilibrium temperature.
The cooler atmospheres ($T_{\rm eq}\la250$--$600~{\rm K}$) also develop detached convective layers at around $P\sim 0.1$--$10~{\rm bar}$. A convective zone at these pressures is found in non-irradiated models at these $T_{\rm eff}$ values, and is the ``natural'' outcome for these atmospheres, given the high thermal infrared opacities \citep{Marley&Robinson15}. Such convective zones are not possible in the more highly irradiated objects given the high temperatures at low pressure, which forces a shallower-than-adiabatic temperature gradient throughout much of the atmosphere.
The \emph{P--T} profiles converge to the same deep adiabatic profile in the equilibrium temperature range of $T_{\rm eq}\sim 250$--$1200~{\rm K}$ for a given set of planetary intrinsic temperature, gravity, and atmospheric compositions.
For solar metallicity with $g=1$, $10$, and $100~{\rm m^2~s^{-1}}$ and $T_{\rm int}=50$, $100$, and $200~{\rm K}$ (top five panels), planets with the equilibrium temperature of $T_{\rm eq}=250$--$1200~{\rm K}$ have nearly the same \emph{P--T} profiles in deep convective layers.
These atmospheric models have temperatures in the middle atmosphere ($P\sim10$--$100~{\rm bar}$) between $\sim 1000$--$1500~{\rm K}$, which is consistent with the criterion argued in the previous section.
\emph{P--T} profiles also tend to converge to the same adiabatic profile for high metallicity models of $\rm [Fe/H]=+1$ and $+2$, as well as a higher C/O model of $\rm C/O=1.1$.
At very high metallicities, the deep adiabatic profiles starts to deviate from the same deep profile at a lower equilibrium temperature, for instance at $T_{\rm eq}=1000~{\rm K}$ for $\rm [Fe/H]=2$.
This could be attributed to the middle atmosphere temperature being relatively hotter than that for low-metallicity models, which acts to cause the RCB before the temperature structure converges to the radiative zero solution.
\subsection{Semi-analytical Model of the Deep Adiabat}
We now derive a semi-analytical fit to the universal thermal structure of deep atmospheres for $T_{\rm eq}\sim250$--$1200~{\rm K}$.
Since the deep adiabatic profile would be scaled by the RCB, we infer the parameter dependence of $P\propto (\kappa_{\rm 0}g)^{1/(1+\alpha)}T_{\rm int}^{-4/(1+\alpha)}T^{(4-\beta)/(1+\alpha)}$ from Equation \eqref{eq:P_RCB1}.
According to Figure \ref{fig:dependence}, the opacity law approximately follows $\alpha\sim0.5$ and $(4-\beta)/(1+\alpha)\sim4$ in the temperature range of interest.
In addition, the reference opacity $\kappa_{\rm 0}$ depends on the metallicity.
We assume the metallicity dependence of $\kappa_{\rm 0} \propto {10}^{c{\rm [Fe/H]}}$, where $c$ is a fitting constant \footnote{\rev{This dependence is motivated by the analytic Rosseland mean opacity of \citet{Freedman+14}, who also assumed the opacity proportional to $10^{c{\rm[Fe/H]}}$. \citet{Freedman+14} considered different $c$ coefficients between high and low pressure limits and also considered a temperature dependence of $c$ at the high pressure limit. Here, We have assumed a constant value of $c$ for the sake of simplicity.}}.
Inserting these values and determining the reference pressure to match numerical results, we achieve the following analytical form of the deep adiabatic \emph{P--T} profile which \emph{P--T} profiles converge at $T_{\rm eq}\sim250$--$1200~{\rm K}$, as
\begin{equation}\label{eq:Pad_fit}
P\approx 70\times{10}^{-0.4{\rm [Fe/H]}}~{\rm bar}~\left( \frac{T}{1000~{\rm K}}\right)^{4}\left( \frac{g}{10~{\rm m~s^{-2}}}\right)^{2/3}\left( \frac{T_{\rm int}}{100~{\rm K}}\right)^{-8/3},
\end{equation}
or equivalently
\begin{equation}\label{eq:Pad_fit2}
T\approx 1090\times{10}^{0.1{\rm [Fe/H]}}~{\rm K}~\left( \frac{P}{100~{\rm bar}}\right)^{1/4}\left( \frac{g}{10~{\rm m~s^{-2}}}\right)^{-1/6}\left( \frac{T_{\rm int}}{100~{\rm K}}\right)^{2/3},
\end{equation}
where we have inserted $c\approx0.6$ to fit the metallicity dependence of the numerical results.
Equation \eqref{eq:Pad_fit} indicates that the deep adiabat becomes hotter at higher intrinsic temperature, metallicity, and lower surface gravity.
\rev{The black dotted lines in Figure \ref{fig:PT_summary} show the analytic deep adiabatic \emph{P--T} profile of Equation \eqref{eq:Pad_fit}.} As seen in the Figure, Equation \eqref{eq:Pad_fit} explains the common deep adiabatic profile for $T_{\rm eq}\sim250$--$1200~{\rm K}$ very well, including its dependence on surface gravity, intrinsic temperature, and atmospheric metallicity.
Thus, for cool to warm exoplanets with $T_{\rm eq}\sim250$--$1200~{\rm K}$, one can utilize Equation \eqref{eq:Pad_fit} to estimate the thermal structure of the deep atmosphere, such as for estimating the quenched abundance of disequilibrium chemical species.
\section{Exploring the Relationship Between NH$_3$ and Bulk Nitrogen Abundances}\label{sec:N_map}
In the previous section we worked to derive a semi-analytic theory of the deep atmosphere temperature structure as a step towards a semi-analytic understanding of an atmospheres NH$_3$ abundance. We continue along this path here. In this section we explore the relationship between observable NH$_3$ and the bulk nitrogen abundance based on semi-analytical arguments
\begin{figure*}[t]
\centering
\includegraphics[clip, width=0.47\hsize]{Figure5_1.pdf}
\includegraphics[clip, width=0.47\hsize]{Figure5_2.pdf}
\includegraphics[clip, width=0.47\hsize]{Figure5_3.pdf}
\includegraphics[clip, width=0.47\hsize]{Figure5_4.pdf}
\caption{The quenched NH$_3$ abundance (top panels, Equation \ref{eq:NH3_analytic}) and the ratio of the bulk nitrogen to the quenched NH$_3$ abundance ratio $f_{\rm N}/f_{\rm NH3}$ (bottom panels, Equation \ref{eq:N_diagnostic}) as a function of a planetary mass and age, \rev{applicable to planets that have the universal deep adiabat ($T_{\rm eq}\sim250$--$1200$ K)}.
The black line denote the abundance contours of $f_{\rm NH_3}={10}^{-5}$ and ${10}^{-6}$ for the top panels and the contours of $f_{\rm N}/f_{\rm NH3}=3$, $10$, and $30$ for the bottom panels. \rev{The yellow contours in the upper panels also denote the contours of NH$_3$ abundances corresponding to the 90\% and 50\% of bulk nitrogen budget.} The left and right columns show the results for solar metallicity and $10\times$ solar metallicity atmospheres, respectively, where we have assumed that the N/H ratio is scaled by the metallicity.
}
\label{fig:NH3_map}
\end{figure*}
\subsection{Semi-analytical predictions}\label{seq:N_analytic}
We first estimate how the vertically quenched NH$_3$ abundance relates to the bulk nitrogen abundance based on the semi-analytical argument established in previous sections.
Since exoplanets with $T_{\rm eq}\sim250$--$1200~{\rm K}$ have nearly the same deep adiabatic profile (Section \ref{sec:deep_adiabat}) and $\rm NH_3/N_2$ ratio is nearly constant along the deep adiabat (Section \ref{sec:nitrogen_chem}), the quenched NH$_3$ abundance would be nearly independent of the equilibrium temperature, as \rev{previously} demonstrated in \citet{Fortney+20}.
For cool to warm exoplanets with $T_{\rm eq}\sim250$--$1200~{\rm K}$, solving Equation \eqref{eq:f_N1} with respect to $f_{\rm NH3}$, we predict the quenched NH$_3$ abundance of
\begin{equation}\label{eq:NH3_analytic}
\frac{f_{\rm NH3}}{f_{\rm N}}=\frac{\sqrt{1+8\mathcal{K}^{-1}}-1}{4}\mathcal{K},
\end{equation}
where
\begin{eqnarray}\label{eq:K}
\mathcal{K}&=& P^2 f_{\rm H2}^3f_{\rm N}^{-1} Ae^{B/T}\\
\nonumber
&\approx& 3.46\times{10}^{-0.8{\rm [Fe/H]}}\left( \frac{f_{\rm N}}{10^{-4}}\right)^{-1}\left( \frac{g}{10~{\rm m~s^{-2}}}\right)^{4/3}\left( \frac{T_{\rm int}}{100~{\rm K}}\right)^{-16/3},
\end{eqnarray}
where we have inserted the semi-analytic deep adiabat (Equation \ref{eq:Pad_fit}) with \rev{$T=2000~{\rm K}$, where the temperature is chosen arbitrary as the equilibrium NH$_3$ abundance is approximately constant along the deep adiabat.}.
Qualitatively speaking, low intrinsic temperature, low atmospheric metallicity, and high surface gravity lead to colder deep atmospheres, which corresponds to large $\mathcal{K}$.
Thus, Equation \eqref{eq:NH3_analytic} yields NH$_3$ rich deep atmospheres of $f_{\rm NH_{\rm 3}}=f_{\rm N}$ in the limit of high $\mathcal{K}$ and vice versa for low $\mathcal{K}$.
Meanwhile, substitution of Equation \eqref{eq:Pad_fit} into \eqref{eq:f_N1} with $T_{\rm }=2000~{\rm K}$ yields
\begin{eqnarray}\label{eq:N_diagnostic}
\frac{f_{\rm N}}{f_{\rm NH_3}}&\approx&1 + 0.58\times{10}^{0.8{\rm [Fe/H]}}\left( \frac{f_{\rm NH_3}}{10^{-4}}\right)\left( \frac{g}{10~{\rm m~s^{-2}}}\right)^{-4/3}\left( \frac{T_{\rm int}}{100~{\rm K}}\right)^{16/3}.
\end{eqnarray}
Equation \eqref{eq:N_diagnostic} enables us to constrain the bulk nitrogen abundance for a given quenched NH$_3$ abundance, atmospheric metallicity, surface gravity, and intrinsic temperature.
The former three values could be constrained by observations, while the intrinsic temperature could be constrained either by thermal evolution models \citep[e.g.,][]{Guillot+96,Burrows+97,Guillot&Showman02,Baraffe+03,Fortney+07,Mordasini+12,Valencia+13,Lopez&Fortney14,Kurosaki+14,Kurokawa&Nakamoto14,Vazan+15,Thorngren+16,Chen&Rogers16,Kubyshkina+20} and/or emission spectroscopy in the limit of very high $T_{\rm int}$ \citep{Morley+17}.
Focusing on cool to warm exoplanets, we here predict the quenched NH$_3$ abundance and its fraction to the bulk nitrogen, $f_{\rm NH_3}/f_{\rm N}$, over a wide range of planetary mass and age.
We combine Equations \eqref{eq:NH3_analytic} and \eqref{eq:N_diagnostic} with the thermal evolution tracks of \citet{Fortney+07} to predict $g$ and $T_{\rm int}$ for given planetary masses and ages, where we adopted the evolution track for the core mass of $10~M_{\rm \oplus}$ and semi-major axis of $0.1~{\rm AU}$ \footnote{Grids of the evolution tracks are available at \url{https://www.ucolick.org/~jfortney/models.htm}}.
Figure \ref{fig:NH3_map} shows the predicted NH$_3$ abundance and its fraction of the bulk nitrogen abundance.
The quenched NH$_3$ abundance is in general higher at lower mass and older planets, as these planets have cooler interiors and deep atmospheres that allow an NH$_3$-rich deep atmosphere.
In many cases, the quenched NH$_3$ abundance exceeds what is a potentially detectable mixing ratio \footnote{\rev{We note that the actual detectable abundance would depend on a number of other factors, such as wavelength range, spectral resolution, chemical species of interest, and abundances of other chemical species.}} of $\ga {10}^{-6}$ (see \citealt{Fortney+20} for the discussion on the threshold), except for super-Jupiter mass planets at very young ages of $\la 0.01~{\rm Myr}$.
In terms of the NH$_3$ fraction to the total nitrogen, for solar metallicity atmospheres (left column of Figure \ref{fig:NH3_map}), the quenched NH$_3$ abundance is almost identical to the bulk nitrogen abundance if the planet has a sub-Jupiter mass ($\la 1M_{\rm j}$) and old ages ($\ga 1~{\rm Gyr}$).
For more massive and younger planets, the quenched NH$_3$ abundance starts to deviate from the bulk nitrogen abundance.
For example, the NH$_3$ abundance is approximately an order of magnitude lower than the bulk nitrogen abundance in Jupiter-mass planets at $0.1~{\rm Gyr}$.
Thus, for massive and young planets, the observed NH$_3$ abundance would only constrains the lower limit of the bulk nitrogen abundance.
The discrepancy between the NH$_3$ and bulk nitrogen abundance becomes even larger if the planet has a higher metallicity atmosphere.
The right column of Figure \ref{fig:NH3_map} shows the quenched NH$_3$ abundance and $f_{\rm N}/f_{\rm NH3}$ for 10$\times$ solar metallicity atmospheres.
Interestingly, the expected quenched NH$_3$ abundance is almost comparable to that expected for solar metallicity atmospheres, as N$_2$ is favored for both higher N/H and hotter deep atmospheres due to higher metallicities.
This can also be understood as follows.
Assuming a high metallicity atmosphere with $\mathcal{K}\ll8$ and $f_{\rm N}\propto {10}^{\rm [Fe/H]}$, Equation \eqref{eq:NH3_analytic} approximately yields the NH$_3$ abundance of
\begin{equation}
f_{\rm NH_{\rm 3}}\approx f_{\rm N}\sqrt{\frac{\mathcal{K}}{2}}\propto 10^{0.1{\rm [Fe/H]}}.
\end{equation}
Thus, the NH$_3$ abundance is insensitive to [Fe/H] for high metallicity atmospheres.
The weak metallicity dependence of $f_{\rm NH3}$ leads to the \emph{fraction} of NH$_3$ to the bulk nitrogen, i.e., $f_{\rm NH3}/f_{\rm N}$, being lower in the higher metallicity atmospheres at a given planetary mass and age.
\rev{In other words, the fraction of observable nitrogen (i.e., NH$_3$) decreases with an increased atmospheric metallicity because of the conversion of NH$_3$ to N$_2$.}
Importantly then, it is necessary to assess the overall atmospheric metallicity from other spectral features for correctly inferring the bulk nitrogen abundance from NH$_3$.
\section{Discussion}\label{sec:discussion}
\subsection{\rev{Issues of the Strong Dependence on Metallicity}}\label{sec:metallicity_issue}
\rev{
One of our main findings is the strong metallicity dependence of the ratio of the quenched NH$_3$ abundance to the bulk nitrogen abundance.
Atmospheric metallicity could be constrained by the presence of chemical species sensitive to the metallicity, such as CO$_2$ \citep[e.g.,][]{ERS+22,ERS+22_G395,ERS+22_PRISM} and SO$_2$ \citep[][]{Polman+22,ERS+22_SO2}.
Broad wavelength coverage and the unprecedented precision of JWST may also help to better constrain the metallicity.
Thus, we anticipate that observers can use a planet's observationally constrained metallicity for converting the retrieved NH$_3$ abundance to bulk nitrogen abundance through Equation \eqref{eq:N_diagnostic}.
However, the inference would be further complicated if a planetry atmosphere has strongly non-solar elemental ratios (e.g., C/O$\gg$1), as it may cause a distinct deep adiabatic profile from our semi-analytic \emph{P--T} profile.
}
\rev{
It is difficult to predict whether the quenched NH$_3$ abundance is comparable to the bulk nitrogen abundance before observations.
There a few potential ways to coarsely aid such predictions, however.
Interior structure models could set an upper limit on atmospheric metallicity assuming that the metals in the planetary interior is fully mixed to the atmosphere \citep{Thorngren&Fortney19}.
The estimated upper limit may be used to predict the largest discrepancy between the quenched NH$_3$ and bulk nitrogen abundance.
One might eventually also be able to utilize the relation between planetary mass and atmospheric metallicity suggested from Solar System giant planets \citep[e.g.,][]{Kreidberg+14b,Wakeford+17,Welbanks&Madhusudhan19}.
However, exoplanets have not shown a clear mass-metallicity relation as of yet \citep{Wakeford&Dalba20,Guillot+22,Edwards+22}. We need better knowledge about the population-level metallicity trend of exoplanetary atmospheres to make a reliable prediction.
}
\subsection{\rev{Caveats}}\label{sec:caveat}
\rev{We have assumed that the NH$_3$ abundance is vertically constant above the quench level. While several studies assumed the same approximation to model the transport-driven disequilibrium chemistry \citep[e.g.,][]{Morley+17,Fortney+20,Mukherjee+22b}, the assumption is not always valid. For example, \citet{Moses+11} showed that NH$_3$ abundance gradually decreases with decreasing pressure above the quench level in hot Jupiter HD189733b.
\citet{Moses+21} also obtained similar NH$_3$ profiles in their pseudo-2D photochemical simulations for many planetary equlibrium temperature.
These vertically nonuniform profile could occur when the eddy diffusion timescale is not sufficiently short as compared to chemical interconversion timescale.
We anticipate that the vertically constant abundance would be reasonable for warm to cool exoplanets where the chemical timescale quickly increases with altitude \citep[see][]{Tsai+18}. However, one should always be encouraged to verify the assumption using kinetic chemical model for a specific planet of interest.
}
\rev{
We have assumed nearly constant equilibrium abundance of NH$_3$ along the deep adiabat.
While the assumption is reasonably valid at hydrogen-dominated substellar atmospheres, the assumption would be no longer valid if an atmosphere has different a primary composition with different adiabatic index.
}
\rev{
We have only considered the transport-induced disequilibrium chemistry, while other physical processes can also affect the NH$_3$ vertical profile. \citet{Molaverdikhani+19} provides an in depth discussion about how the photochemistry and molecular diffusion could cause the discrepancy from vertically constant profile. \citet{Hu21} showed that NH$_3$ tends to be depleted by photodissociation in temperate to cold exoplanets. We investigate the effect of photochemistry and how the observable NH$_3$ abundance relates with bulk nitrogen abundance in our Paper II.
}
\subsection{Relevance for Cold and Directly Imaged Planets}
\rev{Planets that lack strong radiative forcing from their parent star would in some ways be simpler to interpret, from an observational perspective. First, lacking external forcing, their interiors would cool off somewhat faster, into the NH$_3$ dominated chemical T-P phase space. While their deep atmospheres would not share the radiative-zero solution, they do have the real added benefit that their intrinsic temperature (the object's effective temperature in this case) and the T-P conditions of their deep atmosphere adiabat can be directly constrained by thermal infrared observations. Moreover, those isolated objects can avoid NH$_3$ depletion by photodissociation, which limits the observability of NH$_3$ for irradiated planets \citep{Hu21}. Nitrogen disequilibrium in the atmospheres of such isolated objects was recently modeled in \citet{Karalidi+21,Mukherjee+22b}.
}
\rev{
For the very coldest planets whether irradiated or not, one needs to further consider relevant condensation physics.
In cold exoplanets where NH$_3$ clouds form, NH$_3$ must be depleted above the NH$_3$ cloud base, like solar system giant planets.
The formation of H$_2$O clouds may also affect NH$_3$ abundances.
Recent microwave observations of Jupiter by JUNO revealed that NH$_3$ abundance is still partly depleted even below the NH$_3$ cloud base \citep{Bolton+17,Li+17}.
\citet{Guillot+20,Guillot+20b} suggested that such NH$_3$ depletion could be explained by the formation of NH${_3}\cdot$H$_2$O condensate.
NH$_3$ can also be depleted because of the dissolution into liquid H$_2$O clouds \citep{Hu19}.
Thus, one needs to be cautious in interpreting the NH$_3$ abundances on very cool planets where NH$_3$ and/or H$_2$O clouds potentially form.
}
\section{Summary}\label{sec:summary}
In this study, we have investigated how observable NH$_3$ abundances relate to bulk nitrogen abundances of exoplanetary atmospheres.
We first identified that irradiated substellar atmospheres follow nearly the same deep adiabatic profile over a wide range of equilibrium temperatures ($T_{\rm eq}\sim250$--$1200~{\rm K}$).
We have derived a semi-analytical model of such a universal deep adiabat (Equation \ref{eq:Pad_fit}) that readily explains the radiative-convective equilibrium model.
Then, we established a semi-analytical model that relates vertically quenched NH$_3$ abundances with the bulk nitrogen abundance of the atmosphere (Equation \ref{eq:N_diagnostic}).
\rev{Based on the semi-analytical model, we predict the relation between the quenched NH$_3$ and bulk nitrogen abundances as a function of planetary mass and age. We verify our semi-analytical model using a photochemical kinetic model in Paper II.}
Our key findings are summarized as follows:
\begin{enumerate}
\item Irradiated giant planet atmospheres have nearly the same deep adiabatic profile for the equilibrium temperature of $T_{\rm eq}\sim250$--$1200~{\rm K}$ for a given set of planetary gravity and intrinsic temperature.
This is caused by the fact that their atmospheric \emph{P--T} profiles tend to converge to the radiative-zero solution that is independent of the upper boundary conditions before they meet the radiative convective boundary (Section \ref{sec:deep_adiabat}).
Based on the series of radiative-convective equilibrium calculations, we have derived a semi-analytical model of such universal deep adiabats applicable to planets with $T_{\rm eq}\sim250$--$1200~{\rm K}$ (Section \ref{sec:RCE_result}, Equations \ref{eq:Pad_fit} or \ref{eq:Pad_fit2}).
\item We have established a semi-analytical model that relates the vertically quenched NH$_3$ abundance with the bulk nitrogen abundance (Equations \ref{eq:NH3_analytic} and \ref{eq:N_diagnostic}).
\rev{Our model is applicable to warm irradiated giant exoplanets. We are able to readily assess discrepancies between the quenched NH$_3$ and bulk nitrogen abundances. This aids when attempting to infer the bulk nitrogen abundance from an observed NH$_3$ abundance.}
\item \rev{At solar composition in a giant planet atmosphere,} the vertically quenched NH$_3$ abundance nearly coincides with the bulk nitrogen abundance \emph{only} when a planet has a sub-Jupiter mass ($\la1~{\rm M_{\rm J}}$) and old age ($\ga 1~{\rm Gyr}$).
For planets with super-Jupiter mass and/or age younger than $1~{\rm Gyr}$, in contrast, the quenched NH$_3$ abundance is considerably lower than the bulk nitrogen abundance, as the deep atmosphere is so hot that N$_2$ dominates over NH$_3$.(Section \ref{sec:N_map} and Figure \ref{fig:NH3_map}).
\item \rev{As the atmospheric metallicity increases, while the predicted quenched NH$_3$ mixing ratio remains constant at a given mass and age, the ratio of NH$_3$ to the bulk atmospheric nitrogen abundance decreases significantly. The issue of NH$_3$ only containing a fraction of the bulk nitrogen abundance then occurs across all giant planet phase space, at sub-Jupiter planet masses and old ages. This ``missing nitrogen'' problem can be corrected with an assessment of the deep atmospheric T-P profile (likely from structure or evolution models) and an overall assessment of atmospheric metallicity from other species, such a C- and O-bearing molecules, or alkali metals.}
\end{enumerate}
\section*{Acknowledgements}
We are grateful to anonymous reviewer for their insightful comments that greatly improved the quality of this paper.
We thank Masahiro Ikoma for helpful comments on the thermal structures of deep atmospheres.
We also thank Neel Patel, Xinting Yu, Ben Lew, Eliza Kempton, Yuichi Ito, Yui Kawashima, Shota Notsu, Tatsuya Yoshida, and Akifumi Nakayama for fruitful discussions.
This work benefited from the 2022 Exoplanet Summer Program in the Other Worlds Laboratory (OWL) at the University of California, Santa Cruz, a program funded by the Heising-Simons Foundation.
Most of numerical computations were carried out on PC cluster at Center for Computational Astrophysics, National Astronomical Observatory of Japan.
K.O. was supported by JSPS Overseas Research Fellowship. J.J.F. is supported by
an award from the Simons Foundation.
\rev{ |
3,212,635,537,438 | arxiv | \section{Introduction}
A thorough understanding of crack formation in brittle materials is of great interest in both experimental sciences and theoretical studies. Starting with the seminal contribution by Francfort and Marigo \cite{Francfort-Marigo:1998}, where the displacements and crack paths are determined from an energy minimization principle, various variational models in the framework of free discontinuity problems have appeared in the literature over the past years. These so-called Griffith functionals comprising elastic and surface contributions generalize the original Griffith theory (see \cite{Griffith:1921}) which is based on the fundamental idea that the formation of fracture may be regarded as the competition of elastic bulk and surface energies.
For the sake of a simplified mathematical description the investigation of fracture models in the realm of linearized elasticity is widely adopted (see e.g. \cite{Ambrosio-Coscia-Dal Maso:1997, Bellettini-Coscia-DalMaso:98, Bourdin-Francfort-Marigo:2008, Chambolle:2003, Chambolle:2004, Iurlano:13}) and has led to a lot of realistic applications in engineering as well as to efficient numerical approximation schemes (we refer to \cite{Bourdin:07, Bourdin-Francfort-Marigo:2000, Ortner:13, Focardi-Iurlano:13, Negri:2003, Negri:06, SchmidtFraternaliOrtiz:2009} making no claim to be exhaustive). On the contrary, their nonlinear counterparts are usually significantly more difficult to treat since in the regime of finite elasticity the energy density of the elastic contributions is genuinely geometrically nonlinear due to frame indifference rendering the problem highly non-convex. Consequently, in contrast to linear models already the fundamental question if minimizing configurations for given boundary data exist at all is a major difficulty. Even more challenging tasks in this context are the determination of the material behavior under expansion or compression, in particular the derivation of specific cleavage laws.
Consequently, for a deeper understanding of nonlinear models the identification of an effective linearized theory is desirable as in this way one may rigorously show that in the small displacement regime the neglection of effects arising from the non-linearities is a good approximation of the problem. Moreover, such a derivation is also interesting in the context of discrete systems. Previous investigations which were motivated by the analysis of cleavage laws for brittle crystals (see \cite{FriedrichSchmidt:2011, FriedrichSchmidt:2014.1} or the seminal paper \cite{Braides-Lew-Ortiz:06}) have shown that the most interesting regime for the elastic strains is given by $\sqrt{\varepsilon}$, where $\varepsilon$ denotes the typical interatomic distance. Consequently, a passage from discrete-to-continuum systems naturally involves a simultaneous linearization process.
In elasticity theory the nonlinear-to-linear limit is by now well understood in various different settings via $\Gamma$-convergence (cf. \cite{Braides-Solci-Vitali:07, DalMasoNegriPercivale:02, Schmidt:08, Schmidt:2009}), where the passage is performed in terms of suitably rescaled displacement fields measuring the distance of the deformation from a rigid motion and being the fundamental quantity on which the linearized elastic energy depends. In fracture mechanics, however, the relation between the deformation of a material and corresponding displacements is more complicated since the body may be disconnected by the jump set into various components. In fact, it turns out that, without passing to rescaled configurations, in the small strain limit nonlinear Griffith energies converge to a limiting functional which is finite for piecewise rigid motions and measures the \textit{segmentation energy} which is necessary to disconnect the body.
Obviously a major drawback of this simple limiting model appears to be the fact that it does not capture the elastic deformations which are typically present in the nonlinear models. Consequently, in order to arrive at a limiting model showing coexistence of elastic and surface contributions it is indispensable to pass to rescaled configurations similarly as in \cite{DalMasoNegriPercivale:02}. The goal of this article is to identify such an effective linearized Griffith energy as the $\Gamma$-limit of nonlinear and frame indifferent models in the small strain regime. To the best of our knowledge such a result has not yet been derived in the general setting of free discontinuity problems introduced by Ambrosio and De Giorgi \cite{DeGiorgi-Ambrosio:1988}.
The farthest reaching result in this direction seems to be a recent contribution by Negri and Toader \cite{NegriToader:2013} where a nonlinear-to-linear analysis is performed in the context of quasistatic evolution for a restricted class of admissible cracks. In particular, in their model the different components of the jump set are supposed to have a least positive distance rendering the problem considerably easier from an analytical point of view. In particular, the specimen cannot be separated into different parts effectively leading to a simple relation between the deformation and the rescaled displacement field. On the other hand, in \cite{FriedrichSchmidt:2014.2} we have performed a simultaneous discrete-to-continuum and nonlinear-to-linear analysis for general crack geometries, but under the simplifying assumption that all deformations lie close to the identity mapping.
In the present context we establish a limiting linearized Griffith functional in a planar setting without any a priori assumptions on the deformation and the crack geometry. We identify an effective model which appears to be more general than the energies which are widely investigated in the literature. Whereas in elasticity theory, in the approaches \cite{FriedrichSchmidt:2014.2, NegriToader:2013} mentioned before and in most linear fracture models there is a simple relation between the deformation of the material and the associated infinitesimal displacement field, in our framework the deformation is related to a triple consisting of a partition of the domain, a corresponding piecewise rigid motion being constant on each connected component of the cracked body and a displacement field which is defined separately on each piece of the specimen.
On each component of the partition the energy is of Griffith-type in the realm of linearized elasticity. In addition, the functional contains the segmentation energy which is necessary to disconnected the parts of the body. In particular, the latter contribution is a specific feature of our general model where we do not restrict the analysis to a linearization around a fixed rigid motion.
Let us briefly note that although all arguments used in the proofs of this article are valid in any space dimension, we have to restrict our analysis to two dimensions as one of the ingredients of our analysis, an $SBD$-rigidity result (see \cite{Friedrich-Schmidt:15}), has only been derived in a planar setting for isotropic surface energies. However, we believe that the estimate in \cite{Friedrich-Schmidt:15} may be generalized in the future and then the generalizations for the results in the work at hand immediately follow.
As applications of our result \BBB we investigate problems with external forces and also \EEE present a cleavage law in a continuum setting with isotropic surface energies. As discussed before, the identification of critical loads and the investigation of crack paths is a challenging problem particularly for nonlinear models. The arguments in \cite{FriedrichSchmidt:2011, FriedrichSchmidt:2014.1, Mora:2010}, where boundary value problems of uniaxial extension for brittle materials were investigated, fundamentally relied on the application of certain slicing techniques and due to the lack of convexity were not adapted to treat the case of compression. Our general $\Gamma$-limit result can now be applied to solve also boundary value problems of uniaxial compression which is as the uniaxial tension test a natural and interesting problem. Hereby we may complete the picture about the derivation of cleavage laws in \cite{FriedrichSchmidt:2011, FriedrichSchmidt:2014.1}.
One essential point in our investigation is the establishing of a compactness result providing limiting configurations which consist of piecewise rigid motions and corresponding displacement fields. Similarly as in the derivation of linearized systems for elastic materials (see e.g. \cite{DalMasoNegriPercivale:02}), where the main ingredient is a quantitative geometric rigidity estimate by Friesecke, James and M\"uller \cite{FrieseckeJamesMueller:02}, the starting point of our analysis is a quantitative $SBD$-rigidity result (see \cite{Friedrich-Schmidt:15}) in the framework of special functions of bounded deformation (see \cite{Ambrosio-Coscia-Dal Maso:1997, Bellettini-Coscia-DalMaso:98}), which is tailor-made for general Griffith models with coexistence of both both energy forms.
As there is no uniform bound on the functions, it turns out that the limiting displacements are generically not summable and we naturally end up in the space of $GSBD$ functions (for the definition and basic properties we refer to \cite{DalMaso:13}). We believe that our results are interesting also outside of this specific context as they allow to solve more general variational problems in fracture mechanics. Typically, for compactness results in function spaces as $SBV$ (see \cite{Ambrosio-Fusco-Pallara:2000} for the definition and basic properties) and $SBD$ one needs $L^\infty$ or $L^1$ bounds on the functions (see \cite{Ambrosio:90, Bellettini-Coscia-DalMaso:98, DalMaso:13}). However, in many applications, in particular for atomistic systems and for models dealing with rescaled deformations, such estimates cannot be inferred from energy bounds. Nevertheless, we are able to treat problems without any a priori bound by passing from the deformations to displacement fields whose distance from rigid motions can be controlled.
The other essential point in our analysis is the investigation of the limiting configurations. In particular, we study the properties of the partition which disconnects the body into various parts. It turns out that an even finer segmentation may occur if on a connected component of the partition the jump set of the corresponding displacement field further separates the body. Here it becomes apparent that we treat a real multiscale model as the jump heights at the boundaries associated to the coarse partition are of order $\gg \sqrt{\varepsilon}$ ($\sqrt{\varepsilon}$ denotes the regime of the typical elastic strain), whereas the jump heights of the finer partition are of order $\sqrt{\varepsilon}$. Moreover, it is evident that the choice of the limiting partition is not unique. However, we propose a selection principle and show existence and uniqueness of a \emph{coaresest partition}.
The paper is organized as follows. In Section \ref{rig-sec: main} we state the main compactness and $\Gamma$-convergence results and discuss properties of the limiting linearized Griffith functional. Moreover, we present our application to cleavage laws for uniaxially extended or compressed brittle materials.
Section \ref{rig-sec: pre} is devoted to some preliminaries. We first give the definition of special functions of bounded variation and deformation and discuss basic properties. Afterwards, we recall the notion of Caccioppoli partitions which will be fundamental in our analysis to analyze the properties of limiting configurations. Moreover, we recall geometric rigidity results for elastic and brittle materials, in particular the $SBD$-rigidity result proved in \cite{Friedrich-Schmidt:15}.
In Section \ref{rig-sec: sub, comp1} we then establish the main compactness result for a sequence of deformations $(y_\varepsilon)_\varepsilon$, where $\varepsilon$ stands for the order of the elastic energy. First, the convergence of the partitions and the corresponding rigid motions is based on compactness theorems for Caccioppoli partitions and piecewise constant functions (see \cite{Ambrosio-Fusco-Pallara:2000} or Section \ref{rig-sec: sub, cacciop} below).
Although the $SBD$-rigidity estimate is a fundamental ingredient in our analysis giving $L^2$ bounds for rescaled displacement fields, we still have to face major difficulties since the rigidity estimate provides a family of displacement fields $(u^\rho_\varepsilon)^\rho_\varepsilon$ with an additional parameter $\rho$ representing a `modification error' between $y_\varepsilon$ and $u^\rho_\varepsilon$. Consequently, the goal will be to choose an appropriate diagonal sequence.
An additional challenge is the fact that the bounds in the $SBD$-rigidity estimate depend on $\rho$ and blow up for $\rho \to 0$. For the symmetric part of the gradient this problem can be bypassed by a Taylor expansion taking the nonlinear elastic energy $\varepsilon$ and a higher order term into account, which shows that the constant may be chosen independently of $\rho$. For the function itself, however, the problem is more subtle since a uniform bound cannot be inferred by energies bounds. In particular, generically the limiting configurations are not in $L^2$, but only finite almost everywhere. The strategy to establish the latter assertion is to show that for fixed $\varepsilon$ the functions $(u^\rho_{\varepsilon})_\rho$ essentially coincide in a certain sense on the bulk part of the domain. Afterwards, by a careful analysis we can derive that such a property is preserved in the limit $\varepsilon \to 0$, whereby we can establish a kind of equi-integrability of the configurations.
In Section \ref{rig-sec: sub, comp2} we concern ourselves with the limiting configurations consisting of a partition, a corresponding piecewise rigid motion and a displacement field. Recalling that genuinely the limits provided by the compactness result are highly non-unique we introduce the notion of a \textit{coarsest partition}. Roughly speaking, the definition states that the jump heights at the boundaries associated to this partition are of order $\gg \sqrt{\varepsilon}$ leading to a meaningful mathematical description of the observation that the size of the crack opening is a multiscale phenomenon in our model.
The fundamental point is the proof of existence and uniqueness of the coarsest partition. Uniqueness follows from the fact that under the assumption that there are two different coarsest partitions one always can find an even coarser partition. Existence is a more challenging problem. We first give an alternative characterization and identify coarsest partitions as the maximal elements of the partial order on the set of admissible
partitions which is induced by subordination. We then show that each chain of the partial order has an upper bound repeating some arguments of the main compactness result. Consequently, the claim is inferred by an application of Zorn's lemma. Finally, having found the coarsest partition we can then show that the corresponding admissible displacement field is uniquely determined up to piecewise infinitesimal rigid motions.
In Section \ref{rig-sec: sub, gamma1} we derive the main $\Gamma$-limit, where the elastic part can be treated as in \cite
{FrieseckeJamesMueller:02, Schmidt:2009} and for the surface energy we separate the effects arising from the segmentation energy and the crack energy inside the components by employing a structure theorem for Caccioppoli partitions (see Theorem \ref{th: local structure} below). \BBB At this point we also establish a result including external loads. \EEE
Finally, in Section \ref{rig-sec: sub, gamma2} we prove a cleavage law and extend the results obtained in \cite{FriedrichSchmidt:2011, FriedrichSchmidt:2014.1, Mora:2010} to the case of uniaxial compression, where we essentially follow the proof in \cite{FriedrichSchmidt:2014.2, Mora:2010}, in particular using a piecewise rigidity result in $SBD$ (see \cite{Chambolle-Giacomini-Ponsiglione:2007}) and a structure theorem on the boundary of sets of finite perimeter (see \cite{Federer:1969}). It turns out that in the linearized limit the behavior for compression and extension is virtually identical. We briefly note that to avoid unphysical effects such as self-penetrability further modeling assumptions would be necessary.
\section{The model and main results}\label{rig-sec: main}
\subsection{The nonlinear model}
Let $\Omega \subset \Bbb R^2$ open, bounded with Lipschitz boundary. Recall the properties of the space $SBV(\Omega,\Bbb R^2)$, frequently abbreviated as $SBV(\Omega)$ hereafter, in Section \ref{rig-sec: sub, sbv}. Fix a (large) constant $M>0$ and define
\begin{align}\label{rig-eq: SBVfirstdef}
SBV_M(\Omega) = \Big\{ y \in SBV(\Omega,\Bbb R^2): \Vert y\Vert_{\infty} + \Vert \nabla y\Vert_{\infty} \le M, \ {\cal H}^1(J_y) < + \infty \Big\}.
\end{align}
Let $W:\Bbb R^{2 \times 2} \to [0,\infty)$ be a frame-indifferent stored energy density with $W(F) = 0$ iff $F \in SO(2)$. Assume that $W$ is continuous, $C^3$ in a neighborhood of $SO(2)$ and scales quadratically at $SO(2)$ in the direction perpendicular to infinitesimal rotations. In other words, there is a positive constant $c$ such that
\begin{align}\label{eq:W}
W(F) \ge c\operatorname{dist}^2(F,SO(2)) \ \ \ \BBB \text{for all} \ \ \ F \in \Bbb R^{2 \times 2} \ \ \ \text{with} \ \ \ |F|\le M. \EEE
\end{align}For $\varepsilon >0$ define the Griffith-energy $E_\varepsilon : SBV_M(\Omega) \to [0,\infty)$ by
\begin{align}\label{rig-eq: Griffith en}
E_\varepsilon(y) = \frac{1}{\varepsilon}\int_\Omega W(\nabla y(x)) \,dx + {\cal H}^1(J_y).
\end{align}
We briefly note that we can also treat inhomogeneous materials where the energy density has the form $W: \Omega \times \Bbb R^{2 \times 2} \to [0,\infty)$. Moreover, it suffices to assume $W \in C^{2,\alpha}$, where $C^{2,\alpha}$ is the H\"older space with exponent $\alpha >0$.
The main goal of the present work is the identification of an effective linearized Griffith energy in the small strain limit which is related to the nonlinear energies $E_\varepsilon$ through $\Gamma$-convergence. \BBB In this context, we also discuss minimization problems associated to $E_\varepsilon$ for given body forces or boundary data. \EEE Moreover, we will investigate the limiting model which appears to be more general than many other Griffith functionals in the realm of linearized elasticity (cf. e.g. \cite{Bourdin-Francfort-Marigo:2008, Chambolle:2003, Chambolle:2004, Iurlano:13, SchmidtFraternaliOrtiz:2009}) as the limiting configuration not only consists of a displacement field, but also of a coarse partition of the domain and associated rigid motions. In particular, it will turn out that there are various scales for the size of the crack opening occurring in the system.
\BBB
\begin{rem}\label{rem:M}
{\normalfont The threshold $M$ in \eqref{rig-eq: SBVfirstdef} may be chosen arbitrarily large, but is fixed. Confining $y$ in this way effectively models a large box containing
the deformed specimen. The restriction on $\nabla y$ is necessary for technical reasons as it allows us to
apply a quantitative piecewise rigidity estimate, see Theorem \ref{rig-th: rigidity}.
Let us mention that (almost) minimizers of the nonlinear energy $\int_{\Omega\setminus \overline{J_y}}W(\nabla y)$ (for given boundary data) are possibly not Lipschitz continuous as particularly at nonsmooth points of the boundary $\partial (\Omega\setminus \overline{J_y})$ (e.g. at crack tips) the deformation gradient is expected to form singularities. Consequently, the constraint $\Vert \nabla y\Vert_{\infty} \le M$ is a real restriction on the class of admissible configurations from a mathematical point of view.
On the other hand, for materials undergoing brittle fracture there is typically a critical strain (and stress), beyond which failure occurs, and therefore the uniform bound on the absolute continuous part of the gradient has a reasonable mechanical interpretation. Moreover, the energy of certain atomistic systems can be related to \eqref{rig-eq: Griffith en} when deformations are identified with piecewise affine interpolations on cells of microscopic size (see e.g. \cite{Braides-Gelli:2002-2, FriedrichSchmidt:2014.2}). (Note that in discrete systems the parameter $\varepsilon$ represents not only the order of the elastic energy, but also the typical interatomic distance.) In this context, the bound $\Vert \nabla y\Vert_{\infty} \le M$ is naturally satisfied. In fact, on cells exceeding such a threshold, also called \emph{ultimate strain} (see \cite{Braides-DalMaso-Garroni:1999}), a discontinuous interpolation with bounded deformation gradient is introduced and their contribution to the energy then enters through the surface part of the energy functional.
Finally, let us mention that particularly from a computational point of view it is interesting to combine a continuum model as \eqref{rig-eq: Griffith en} with an atomistic approach using the quasicontinuum method introduced in \cite{Tadmor}. Here the underlying idea is to split the domain into a bulk part with a coarse, continuum description, and into certain \emph{critical regions} characterized by fast variations of the deformation gradient (such as regions near a dislocation core or a crack tip) where the problem is treated as a fully atomistic system at scale $\varepsilon$ (see \cite{Miller}).
}
\end{rem}
\EEE
\subsection{The segmentation problem}\label{rig-sec: sub, seg}
As a first natural approach to the problem we concern ourselves with the question if the functionals $E_\varepsilon$ can be related to a limiting functional for $\varepsilon \to 0$ in terms of the deformations. We observe that for configurations with uniformly bounded energy $E_\varepsilon(y_\varepsilon)$ the absolute continuous part of the gradient satisfies $\nabla y_\varepsilon \approx SO(2)$ as the stored energy density is frame-indifferent and minimized on $SO(2)$. Assuming that $y_\varepsilon \to y$ in $L^1$, one can show that $\nabla y \in SO(2)$ a.e. applying lower semicontinuity results for $SBV$ functions (see \cite{Kristensen:1999}) and the fact that the quasiconvex envelope of $W$ is minimized exactly on $SO(2)$ (see \cite{Zhang:2004}).
A piecewise rigidity result by Chambolle, Giacomini and Ponsiglione (see Theorem \ref{rig-cor: cgp} below) generalizing the classical Liouville result for smooth functions now states that an $SBV$ function $y$ satisfying the constraint $\nabla y \in SO(2)$ a.e. is a collection of an at most countable family of rigid deformations, i.e. the body may be divided into different components each of which subject to a different rigid motion.
Consequently, the limit of the sequence $E_\varepsilon$ (in the sense of $\Gamma$-convergence) is given by the functional which is finite for piecewise rigid motions and measures the \textit{segmentation energy} which is necessary to disconnect the body. The exact statement is formulated in Corollary \ref{rig-cor: gamma} as a direct consequence of our main $\Gamma$-convergence result in Theorem \ref{rig-th: gammaconv}.
\BBB Apparently this simple limiting model does not account for \EEE the elastic deformations which are typically present in the nonlinear models. Consequently, to obtain a better understanding of the problem it is desirable to pass to rescaled configurations and to derive a limiting linearized energy as it was performed in \cite{DalMasoNegriPercivale:02} in the framework of nonlinear elasticity theory. The main ingredient in that analysis is a quantitative rigidity result due to Friesecke, James and M\"uller (see Theorem \ref{rig-th: geo rig}). The starting point for our analysis will be a corresponding quantitative result in the $SBD$ setting (see \cite{Friedrich-Schmidt:15} or Theorem \ref{rig-th: rigidity}) adapted for Griffith functionals of the form \eqref{rig-eq: Griffith en} where both elastic bulk and surface contributions are present.
\subsection{Compactness \BBB and limiting configurations\EEE}\label{rig-sec: sub, main com}
We now present our main compactness result for rescaled displacement fields. As a preparation, recall the notion and basic properties of a \textit{Caccioppoli partition} in Section \ref{rig-sec: sub, cacciop}. For a given (ordered) Caccioppoli partition ${\cal P} = (P_j)_j$ of $\Omega$ let
\begin{align}\label{rig-eq: defA}
{\cal R}({\cal P}) = \Big\{ T: \Omega \to \Bbb R^2: \ T(x) = \sum\nolimits_j \chi_{P_j}(x) (R_j \, x + b_j), \ R_j \in SO(2), \ b_j \in \Bbb R^2 \Big\}
\end{align}
be the set of corresponding piecewise rigid motions. Likewise we define the set of piecewise infinitesimal rigid motions, denoted by ${\cal A}({\cal P})$, replacing $R_j \in SO(2)$ by $A_j \in \Bbb R^{2 \times 2}_{\rm skew} = \lbrace A \in \Bbb R^{2 \times 2}: A=-A^T\rbrace$. Moreover, we define the triples
\begin{align}\label{eq:triples}
{\cal D} &:= \big\{ (u,{\cal P}, T): \ u \in SBV(\Omega), \ {\cal P} \text{ C.-partition of } \Omega,\ T \in {\cal R}({\cal P}) \big\}, \\
{\cal D}_\infty &:= \big\{ (u,{\cal P}, T): \, {\cal P} \text{ C.-partition of } \Omega,\, T \in {\cal R}({\cal P}), \, \nabla T^T u \in GSBD^2(\Omega) \big\}.\notag
\end{align}
Here $\nabla T$ denotes the absolutely continuous part of $DT$. The space $GSBD^2(\Omega)$ generalizes the definition of the space $SBD(\Omega)$ based on certain slicing properties, see Section \ref{rig-sec: sub, sbv}. Define $e(G) = \frac{G^T + G}{2}$ for all $G \in \Bbb R^{2 \times 2}$ and denote by $\partial^*$ the \emph{essential boundary} (see below \eqref{eq: essential boundary}). \BBB Let $A \triangle B$ be the symmetric difference of two sets $A,B \subset \Bbb R^2$. \EEE We now formulate the main compactness theorem.
\begin{theorem}\label{rig-th: comp1}
Let $\Omega \subset \Bbb R^2$ open, bounded with Lipschitz boundary. Let $M>0$ and $\varepsilon_k \to 0$ as $k \to \infty$. If $E_{\varepsilon_k}(y_{k}) \le C$ for a sequence $(y_k)_k \subset SBV_M(\Omega)$, then there exists a subsequence (not relabeled) such that the following holds: \\
There are triples $(u_k, \BBB {\cal P}_k, \EEE T_k) \in {\cal D}$, where ${\cal P}_k = (P^k_j)_j$, \BBB and $c>0$ with \EEE
\begin{align}\label{rig-eq: comp2}
\begin{split}
(i)& \ \ u_k(x) - \varepsilon_k^{-1/2} (y_k(x) -T_k(x)) \to 0 \ \text{for a.e. } x \in \Omega \ \ \text{ for $k \to \infty$},\\
(ii)& \ \ \Vert\nabla u_k\Vert_{L^\infty(\Omega)} \le c\varepsilon_k^{-1/8} \ \ \text{ for $k \in \Bbb N$}
\end{split}
\end{align}
such that we find a limiting triple $(u, {\cal P}, T) \in {\cal D}_\infty$ with
\begin{align}\label{rig-eq: comp1}
\begin{split}
(i)& \ \ \BBB |P^k_j \triangle P_j| \to 0 \EEE \ \ \ \text{ for all} \ j \in \Bbb N,\\
(ii)& \ \ T_k \to T \text{ in } L^2(\Omega, \Bbb R^2), \ \ \ \nabla T_k \to \nabla T \text{ in } L^2(\Omega, \Bbb R^{2 \times 2})
\end{split}
\end{align}
for $k \to \infty$ and
\begin{align}\label{rig-eq: comp1-2}
\begin{split}
(i)&\ \ u_k \to u \ \ \text{ a.e. in } \ \Omega,\\
(ii)& \ \ e ( \nabla T^T_k \nabla u_k ) \rightharpoonup e (\nabla T^T \nabla u) \ \ \text{ weakly in} \ L^2(\Omega,\Bbb R^{2\times 2}_{\rm sym})
\end{split}
\end{align}
for $k \to \infty$. Moreover, \BBB for the elastic and crack \EEE energy we obtain
\begin{align}\label{rig-eq: comp3}
(i)&\ \ \frac{1}{\varepsilon_k}\int_\Omega W(\nabla y_k) + o(1) \ge \frac{1}{\varepsilon_k}\int_\Omega W(\mathbf{Id} + \sqrt{\varepsilon_k} \nabla T^T_k \nabla u_k ) \ \ \ \text{as} \ k\to\infty,\notag\\
(ii)&\ \ \liminf_{k \to \infty} {\cal H}^1(J_{y_k}) \ge {\cal H}^1\Big(\bigcup\nolimits_j \partial^* P_j \cap \Omega\Big) + {\cal H}^1\Big(J_u \setminus \bigcup\nolimits_j \partial^* P_j\Big).
\end{align}
\end{theorem}
\BBB
Recall that the central object in linearized elasticity is the symmetric part of the gradient, which comes from the fact that (1) deformations are linearized around the identity and (2) the orthogonal space to $SO(d)$ at the identity is given by the symmetric matrices (see e.g. \cite{DalMasoNegriPercivale:02}). In the present context, where we possibly linearize around different rigid motions, the symmetrized gradient is accordingly replaced by $e(\nabla T^T u)$ in both the limiting description and the convergence (see \eqref{eq:triples} and \eqref{rig-eq: comp1-2}(ii), respectively).
In \eqref{rig-eq: comp1} and \eqref{rig-eq: comp1-2} the convergence for the partitions, rigid motions and displacement fields is given, respectively. Moreover, \eqref{rig-eq: comp2} and \eqref{rig-eq: comp3} represent \emph{compatibility conditions} for the triple $(u_k,{\cal P}_k,T_k)$: In general, $u_k$ is a modification of the rescaled displacement $\varepsilon_k^{-1/2} (y_k -T_k)$, but asymptotically both configurations coincide (see \eqref{rig-eq: comp2}(i)). Moreover, the modifications can be constructed such that $\nabla u_k$ is suitably controllable. (The exponent $-\frac{1}{8}$ is chosen for definiteness only and could be replaced by any small negative exponent, cf. \eqref{rig-eq: main properties2}(iv) and the paragraph below Theorem \ref{rig-th: rigidity}.) Finally, the elastic and crack energy associated to the triples are controlled by the corresponding energies of $y_k$ up to small errors vanishing in the limit (see \eqref{rig-eq: comp3}).
\begin{definition}\label{def:conv}
{\normalfont
We say a sequence $(y_k)_k \subset SBV_M(\Omega)$ is \emph{asymptotically represented} by a limiting triple $(u,{\cal P},T) \in {\cal D}_\infty$, and write $y_k \to (u,{\cal P},T)$, if there is a sequence of triples $(u_k,{\cal P}_k,T_k) \in {\cal D}$ such that \eqref{rig-eq: comp2}-\eqref{rig-eq: comp3} are satisfied.
}
\end{definition}
Although we use the notation $\to$ and call $(u,{\cal P},T)$ a limiting triple, it is clear that Definition \ref{def:conv} cannot be understood as a convergence in the usual sense. In particular, in the small strain limit a tripling of the variables occurs, which is a specific feature of our limiting model. Additionally, the triples $(u,{\cal P},T)$ given by the main compactness theorem for a sequence $(y_k)_k$ are not determined uniquely, but crucially depend on the choice of the sequences $({\cal P}_k)_k$ and $(T_k)_k$. To illustrate the latter phenomenon, \EEE we consider the following example.
\begin{example}\label{ex}
{\normalfont
Consider $\Omega= (0,3) \times (0,1)$, $\Omega_1 = (0,1) \times (0,1)$, $\Omega_2 = (1,2) \times (0,1)$, \BBB $\Omega_3 = (2,3) \times (0,1)$ \EEE and
$$y_k = \mathbf{id} \chi_{\Omega_1} + (\mathbf{id} + \alpha\sqrt{{\varepsilon_k}})\chi_{\Omega_2} + \BBB (\mathbf{id} + {\varepsilon_k}^{1/4})\chi_{\Omega_3} \EEE$$for $\alpha \in \Bbb R^2$. Then for $b \in \Bbb R^2$ possible alternatives are e.g.
\BBB
\begin{align*}
(1)& \ \ P^1_1 = \Omega_1, \ P^1_2 = \Omega_2, \ P^1_3 = \Omega_3 \ \ \text{with} \ \ T^1_k = y_k \ \text{on} \ \Omega,\\
(2)& \ \ P^2_1 = \Omega_1 \cup \Omega_2, \ P^2_2 = \Omega_3 \ \ \text{with} \ \ T^2_k = \mathbf{id} \chi_{\Omega_1\cup \Omega_2} + (\mathbf{id} + {\varepsilon_k}^{1/4} -b \sqrt{\varepsilon_k})\chi_{\Omega_3}.
\end{align*}
\EEE Letting $u^i_{k} = {\varepsilon_k}^{-\frac{1}{2}}\big(y_{k} - T_k^i\big)$ for $i=1,2$ we obtain in the limit $\varepsilon_k \to 0$ \BBB the unique rigid motion $T=\mathbf{id}$ \EEE and the different configurations \BBB
\begin{align*}
&(1) \ \ u^1 = 0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ P^1_1 = \Omega_1, \ P^1_2 = \Omega_2, \ P^1_3 = \Omega_3,\\
&(2) \ \ u^2 = 0 \cdot \chi_{\Omega_1} + \alpha\chi_{\Omega_2} + b\chi_{\Omega_3}, \ \ \ \ \ \ \ P^2_1 = \Omega_1 \cup \Omega_2, \ P^2_2 = \Omega_3.
\end{align*}\EEE}
\end{example}
\vspace{-0.8cm}
We now introduce a special subclass of partitions in which uniqueness will be guaranteed. The above example already shows that different partitions are not equivalent in the sense that they may contain a different `amount of information'. Note that on the various elements of the partition the configuration $u$ is defined separately and the different pieces of the domain are not `aware of each other'. In particular, \BBB the difference of the traces of $u$ on $\partial^* P_i \cap \partial^* P_j$, $i\neq j$, \EEE does not have any physically reasonable interpretation. On the contrary, in example \BBB (2) \EEE where we did not split up \BBB $\Omega_1 \cup \Omega_2$, \EEE we gain the jump height on \BBB $\partial \Omega_1 \cap \partial \Omega_2$ \EEE as an additional information. The observation that coarser partitions provide more information about the behavior at the jump set motivates the definition of the \textit{coarsest partition}.
\begin{definition}\label{rig-def: ad,coar}
{\normalfont
Let $(y_k)_k$ be a given (sub-)sequence as in Theorem \ref{rig-th: comp1}.
\begin{itemize}
\item[(i)] We say a partition ${\cal P}$ of $\Omega$ is \textit{admissible} for $(y_k)_k$, and write ${\cal P} \in {\cal Z}_P((y_k)_k)$, if there exist $u,T$ such that $(u, {\cal P}, T) \in {\cal D}_\infty$ and \BBB $y_k \to (u,{\cal P},T)$. \EEE
\item[(ii)] We say a piecewise rigid motion $T$ is \textit{admissible} for \BBB $(y_k)_k$, \EEE and write $T \in {\cal Z}_T((y_k)_k)$, if there exist $u, \BBB {\cal P} \EEE $ such that $(u, {\cal P}, T) \in {\cal D}_\infty$ and \BBB $y_k \to (u,{\cal P},T)$\EEE .
\item[(iii)] We say a configuration $u$ is \textit{admissible} for $(y_k)_k$ and ${\cal P}$, and write $u \in {\cal Z}_u((y_k)_k,{\cal P})$, if there exists $T$ such that $(u, {\cal P}, T) \in {\cal D}_\infty$ and \BBB $y_k \to (u,{\cal P},T)$.\EEE
\item[(iv)] We say a partition ${\cal P}$ of $\Omega$ is a \textit{coarsest partition} for $(y_k)_k$ if the following holds: The partition is admissible, i.e. ${\cal P} \in {\cal Z}_P((y_k)_k)$. Moreover, for all admissible $u \in {\cal Z}_u((y_k)_k, {\cal P})$ and \BBB all corresponding triples $(u_k,{\cal P}_k,T_k) \in {\cal D}$ satisfying \eqref{rig-eq: comp2}-\eqref{rig-eq: comp3} \EEE the mappings $T_k = \sum_j(R_j^k \cdot + b_j^k)\chi_{P^k_j}$ fulfill
\begin{align}\label{rig-eq: toinfty}
\frac{|R^k_{i} - R^k_{j}| + |b^k_{i} - b^k_{j}|}{\sqrt{\varepsilon_k}} \to \infty
\end{align}
for all $i,j \in \Bbb N$, $i \neq j$ and $k \to \infty$.
\end{itemize}
}
\end{definition}
In Lemma \ref{rig-lemma: comp2.1} below we find an equivalent characterization of coarsest partitions being the maximal elements of the partial order on the sets of admissible partitions which is induced by subordination. Loosely speaking, the above definition particularly implies that given a coarsest partition a region of the domain is partitioned into different sets $(P_j)_j$ if and only if the \BBB (scaled) jump height $\varepsilon_k^{-1/2}[y_k]$ \EEE on $(\partial^* P_j)_j$ tends to infinity \BBB (cf. Example \ref{ex}). \EEE
Recall the definition of the piecewise infinitesimal rigid motions ${\cal A}({\cal P})$ below \eqref{rig-eq: defA}. We now obtain a unique characterization of the limiting configuration up to piecewise infinitesimal rigid motions.
\begin{theorem}\label{rig-th: comp2}
Let $\varepsilon_k \to 0$ be given. \BBB Let $(y_k)_k \subset SBV_M(\Omega)$ be a sequence \EEE for which the assertion of Theorem \ref{rig-th: comp1} holds. Then we have the following:
\begin{itemize}
\item[(i)] There is a unique $T \in {\cal Z}_T((\BBB y_{k})_k \EEE )$.
\item[(ii)] There is a unique coarsest partition $\bar{\cal P}$ of $\Omega$.
\item[(iii)] Given some $u \in {\cal Z}_u((y_{k})_k, \bar{\cal P})$ all admissible limiting configurations are of the form $u + \nabla T{\cal A}(\bar{\cal P})$, i.e. the limiting configuration is determined uniquely up to piecewise infinitesimal rigid motions.
\end{itemize}
\end{theorem}
\BBB Going back to Example \ref{ex}, we observe that $T = \mathbf{id}$ is uniquely given and that the partition in (2) is the coarsest partition. The non-uniqueness in (iii) is a consequence of the fact that the nonlinear energy is invariant under rigid motions (see also (2) in Example \ref{ex} for $b \in \Bbb R^2$). \EEE
\subsection{The limiting linearized model and $\Gamma$-convergence}\label{rig-sec: sub, main gamma}
We now introduce the limiting linearized model, discuss its properties and show that it can be identified as the $\Gamma$-limit of the nonlinear energies $E_\varepsilon$. Let $Q = D^2W(\mathbf{Id})$ be the Hessian of the stored energy density $W$ at the identity. Define $E : {\cal D}_\infty \to [0,\infty)$ by
\begin{align}\label{rig-eq: Griffith en-lim}
E(u,{\cal P},T) = \int_\Omega \frac{1}{2} Q(e(\nabla T^T \nabla u)) + {\cal H}^1\Big(J_u \setminus \bigcup\nolimits_j\partial^* P_j\Big) + {\cal H}^1\Big(\bigcup\nolimits_j \partial^* P_j \cap \Omega\Big),
\end{align}
where as before ${\cal P} = (P_j)_j$. Recall that a triple of the limiting model consists of a partition, a corresponding piecewise rigid motion and a displacement field. \BBB We emphasize that in contrast to the nonlinear model (see \eqref{rig-eq: SBVfirstdef} and Remark \ref{rem:M}) there are no restrictive bounds on the functions $u$ and their derivatives\EEE .
The surface energy of $E$ has two parts. Similarly as discussed in Section \ref{rig-sec: sub, seg}, on the right we have the \textit{segmentation energy} which is necessary to disconnect the components of the body. Moreover, on the left we have the \textit{inner crack energy} associated to the discontinuity set of the displacement field in each part of the material \BBB (see also Remark \ref{rem:new}(ii) below). \EEE Whereas the first two terms of the functional typically appear in the study of linearized Griffith energies, the segmentation energy is a characteristic feature of our general model where the analysis is not restricted to a linearization around a fixed rigid motion.
We now present our main $\Gamma$-convergence result. Recall \BBB Definition \ref{def:conv}. \EEE
\begin{theorem}\label{rig-th: gammaconv}
Let $\Omega \subset \Bbb R^2$ open, bounded with Lipschitz boundary. Let $M> 0$ and $\varepsilon_k \to 0$. Then \BBB $E_{\varepsilon_k}$ converges to $E$ in the sense of $\Gamma$-convergence, \EEE i.e.
\begin{itemize}
\item[(i)] $\Gamma-\liminf$ inequality: For all $(u,{\cal P},T) \in {\cal D}_\infty$ and for all sequences $(y_k)_k \subset SBV_M(\Omega)$ \BBB with $y_k \to (u,{\cal P},T)$ \EEE we have
$$\liminf_{k \to \infty} E_{\varepsilon_k}(y_k) \ge E(u,{\cal P},T).$$
\item[(ii)] Existence of recovery sequences: For every $(u,{\cal P},T) \in {\cal D}_\infty$ with $u \in L^2(\Omega)$ we find a sequence $(y_k)_k \subset SBV_M(\Omega)$ \BBB such that $y_k \to (u,{\cal P},T)$ and \EEE
$$\lim_{k \to \infty} E_{\varepsilon_k}(y_k) = E(u,{\cal P},T).$$
\end{itemize}
\end{theorem}
\vspace{-0.6cm}
\begin{rem}\label{rem:new}
{\normalfont
(i) \BBB The limiting model could equivalently be formulated with $v = \nabla T^T u$ in place of the displacement field $u$. (Accordingly, replace $u_k$ by $v_k = \nabla T_k^T u_k$ in Theorem \ref{rig-th: comp1}). This alternative notation simplifies the description of the elastic energy in \eqref{rig-eq: Griffith en-lim}, but does not account for the fact that the linearization was possibly performed around different rigid motions.
(ii) Using the local structure of Caccioppoli partitions (see Theorem \ref{th: local structure} and recall \eqref{eq: essential boundary}) the limiting energy can equivalently be written as
$$\sum\nolimits_j \Big( \int_{P_j} \frac{1}{2} Q(e(R_j^T \nabla u)) + \mathcal{H}^1(J_u \cap (P_j)^1) + \frac{1}{2}\mathcal{H}^1(\partial^* P_j \cap \Omega) \Big).$$ \EEE
(iii) For configurations $(u,\bar{\cal P},T)$ defined in terms of the coarsest partition $\bar{\cal P}$ there is an additional interpretation for the crack opening of the sequence of deformations $y_\varepsilon$: (1) The jumps on $\bigcup_j\partial^* P_j$ are associated to jump heights $\gg \sqrt{\varepsilon}$ and (2) the jump heights corresponding to the inner crack energy are of the order $\sqrt{\varepsilon}$, \BBB which illustrates the multiscale nature of the model. \EEE In fact, (1) follows from \eqref{rig-eq: toinfty} and (2) is a consequence of \eqref{rig-eq: comp2}(i).
(iv) On a component $P_j$ of $\bar{\cal P}$ the body may still be disconnected by the jump set \BBB $(P_j)^1 \cap J_u$ \EEE forming a finer partition of the specimen. However, in contrast to the boundary of $\bar{\cal P}$ the jump heights have a meaningful physical interpretation.
\BBB (v) In general, the partition induced by the \emph{macroscopic jumps} (represented by $J_T$) is coarser than $\bar{\cal P}$, i.e. $\mathcal{H}^1(\bigcup_j\partial^* P_j \setminus (\partial \Omega \cup J_T))>0$, cf. Example \ref{ex}. \EEE
}
\end{rem}
As a direct consequence of Theorem \ref{rig-th: gammaconv} we get that the $\Gamma$-limit \BBB of the same functionals $E_{\varepsilon_k}$ with respect to the much weaker notion of $L^1$-convergence of the unrescaled deformations $y_k$ is given by the segmentation energy. \EEE
\begin{corollary}\label{rig-cor: gamma}
Let $\Omega \subset \Bbb R^2$ open, bounded with Lipschitz boundary. Let $M> 0$ and $\varepsilon_k \to 0$. Then $E_{\varepsilon_k}$ $\Gamma$-converge to $E_{\rm seg}$ with respect to the $L^1(\Omega)$-convergence, where
$$E_{\rm seg}(y) = \begin{cases} {\cal H}^1\big( \BBB J_T \EEE \big)& y = T \in {\cal R}({\cal P}) \ \text{ for a Caccioppoli partition } {\cal P}, \\ + \infty & \text{else.}\end{cases} $$
\end{corollary}
\BBB Note that the segmentation energy in Corollary \ref{rig-cor: gamma} differs from the one in \eqref{rig-eq: Griffith en-lim}, see Remark \ref{rem:new}(v).
\BBB
\subsection{Application: External loads}\label{rig-sec: sub, cleavage2}
For the investigation of minimization problems associated to $E_\varepsilon$ it is interesting to take external loads into account. In the context of brittle materials, however, the incorporation of body forces is a delicate problem. Indeed, assumptions on the class of admissible loads have to ensure that no part of the body is broken apart and sent to infinity, which clearly excludes the case of a constant body force (see \cite[Remark 3.1]{DalMaso-Francfort-Toader:2005}). To avoid the occurrence of such phenomena, it is natural to assume a uniform $L^\infty$ bound on the admissible functions, see e.g. \cite{DFT2, DalMaso-Lazzaroni:2010}. Unfortunately, this is not expedient in our setting since the bound $\Vert y \Vert_\infty \le M$ is futile after passage to rescaled configurations in Theorem \ref{rig-th: comp1}.
Let us mention that in \cite{DalMaso-Francfort-Toader:2005} body forces were indispensable to ensure reasonable compactness properties for sequences and to guarantee existence of minimizers. In our setting, however, due to the subtraction of suitable rigid motions on a partition of the domain (cf. \eqref{rig-eq: comp2}(i)) we obtain a compactness result without the necessity of additional loading terms.
We fix a sequence $\varepsilon_k \to 0$ and consider the following prototype problem $F_{\varepsilon_k} : SBV_M(\Omega) \to [0,\infty)$ with
\begin{align}\label{ennew}
F_{\varepsilon_k}(y) = E_{\varepsilon_k}(y) + \frac{\lambda}{\varepsilon_k} \Vert y - f_k \Vert^2_{L^2(\Omega)},
\end{align}
where $\lambda >0$ and $(f_k)_k \subset SBV_M(\Omega)$ a sequence with $\sup_k E_{\varepsilon_k}(f_k) < \infty$. An expansion yields the constant $ \lambda\varepsilon_k^{-1} \int_\Omega|f_k|^2$, the \emph{external load} $-2\lambda\varepsilon_k^{-1} \int_\Omega f_k \cdot y$ and the term $\lambda \varepsilon_k^{-1} \int_\Omega |y|^2$. The latter can be interpreted as an \emph{ artificial confining potential}, which prevents parts of the body from being sent to infinity.
We assume that there is a triple $(g,{\cal P}_g,T_g) \in \mathcal{D}_\infty$ such that $f_k \to (g,{\cal P}_g,T_g)$ in the sense of Definition \ref{def:conv} and that for the associated triples $(g_k,{\cal P}_k^g,T_k^g) \in {\cal D}$ satisfying \eqref{rig-eq: comp2}-\eqref{rig-eq: comp3} we have $\varepsilon_k^{-1/2}(f_k - T_k^g) \to g$ in $L^2(\Omega)$ and ${\cal P}_k^g = {\cal P}_g$ for all $k \in \Bbb N$. (Note that up to a subsequence the convergence in the sense of Definition \ref{def:conv} is already guaranteed by Theorem \ref{rig-th: comp1}.) Moreover, we suppose that ${\cal P}_g$ is the coarsest partition given by Theorem \ref{rig-th: comp2}(ii) and write ${\cal P}_g = (P^g_j)_j$.
By ${\cal C}_g \subset {\cal D}_\infty$ we denote the set of triples $(u,{\cal P},T) \in {\cal D}_\infty$ with $T=T_g$ and the property that ${\cal P}_g$ is coarser than ${\cal P}$, i.e. for each $P_j$ there exists $P^g_i$ with $|P_j \setminus P_i^g|= 0$.
\begin{lemma}\label{lemmanew}
Let $(y_k)_k \subset SBV_M(\Omega)$ be a sequence with $F_{\varepsilon_k}(y_{k}) \le C$ and $y_k \to (u,{\cal P}, T) \in {\cal D}_\infty$ in the sense of Definition \ref{def:conv}. Then $(u,{\cal P},T) \in {\cal C}_g$.
\end{lemma}
Recalling \eqref{rig-eq: Griffith en-lim} we introduce the limiting energy $F_g : {\cal D}_\infty \to [0,\infty]$ by
\begin{align*}
F_g(u,{\cal P},T) = \begin{cases} E(u,{\cal P},T) + \min_{v \in u + \nabla T \mathcal{A}(\mathcal{P})} \lambda\Vert v- g \Vert^2_{L^2(\Omega)} & \text{if } (u,{\cal P},T) \in {\cal C}_g,\\ + \infty &\text{else}, \end{cases}
\end{align*}
where $\mathcal{A}(\mathcal{P})$ as defined below \eqref{rig-eq: defA}. Similarly as the functional in \eqref{rig-eq: Griffith en-lim}, $F_g$ is invariant under infinitesimal rigid motions on the components of the partition ${\cal P}$. However, the additional term on the right induces a symmetry breaking and there is exactly one distinguished configuration $u^*$ in the class $u + \nabla T \mathcal{A}(\mathcal{P})$ (cf. (iii) in Theorem \ref{rig-th: comp2}) which satisfies $\min_{v \in u + \nabla T \mathcal{A}(\mathcal{P})}\Vert v- g \Vert^2_{L^2(\Omega)} = $ $ \Vert u^*- g \Vert^2_{L^2(\Omega)}$. We close this section with a corresponding $\Gamma$-convergence result.
\begin{theorem}\label{rig-th: gammaconv2}
Let $\Omega \subset \Bbb R^2$ open, bounded with Lipschitz boundary. Let $M> 0$, $\varepsilon_k \to 0$ and $(f_k)_k, g$ as above. Then $F_{\varepsilon_k}$ converges to $F_g$ in the sense of $\Gamma$-convergence. (Replace $E_{\varepsilon_k}$ by $F_{\varepsilon_k}$ and $E$ by $F_g$ in (i),(ii) of Theorem \ref{rig-th: gammaconv}.) Moreover, we have
$$\lim_{k \to \infty} \ \ \inf_{y \in SBV_M(\Omega)} F_{\varepsilon_k}(y) = \min_{(u,{\cal P},T) \in {\cal D}_\infty} F_g(u,{\cal P},T)$$ and (almost) minimzers of $F_{\varepsilon_k}$ converge (up to subsequences) to minimizers of $F_g$ in the sense of Definition \ref{def:conv}.
\end{theorem}
\EEE
\subsection{\BBB Application: Cleavage laws \EEE }\label{rig-sec: sub, cleavage}
In fracture mechanics it is a major challenge to identify critical loads at which
a body fails and to determine the geometry of crack paths that occur in the
fractured regime. As \BBB another \EEE application of the above results we now finally derive such a cleavage law. We consider a special boundary value problem of uniaxial compression/extension. Let $\Omega = (0,l) \times (0,1)$, $\Omega' = (-\eta, l + \eta) \times (0,1)$ for $l>0$, $\eta >0$ and for $a_\varepsilon \in \Bbb R$ define
$${\cal A}(a_\varepsilon) := \lbrace y \in SBV_M(\Omega'): y_1 (x) = (1+ a_\varepsilon)x_1 \text{ for } x_1 \le 0 \ \text{or} \ x_1 \ge l \rbrace.$$
As usual in the theory of $SBV$ functions the boundary values have to be imposed in small neighborhoods of the boundary. In what follows, the elastic part of the energy \eqref{rig-eq: Griffith en} still only depends on $y|_\Omega$, whereas the surface energy is given by ${\cal H}^1(J_y)$ with $J_y \subset \Omega'$. In particular, jumps on $\lbrace0,l\rbrace \times (0,1)$ contribute to the energy $E_\varepsilon(y)$. (Also compare a similar discussion before \cite[Theorem 2.2]{FriedrichSchmidt:2014.2}.) The present problem in the framework of continuum fracture mechanics with isotropic surface energies is a slightly simplified model of the problem considered in \cite{FriedrichSchmidt:2011, FriedrichSchmidt:2014.2}.
As a preparation, define $\alpha$ such that $\inf \lbrace Q(F): \mathbf{e}_1^T F \mathbf{e}_1 = 1\rbrace = \alpha$ and observe $\inf \lbrace Q(F): \mathbf{e}_1^T F \mathbf{e}_1 = a\rbrace = \alpha a^2$ for all $a \in \Bbb R$. Moreover, let $F^a \in \Bbb R^{2 \times 2}_{\rm sym}$ be the unique matrix such that $\mathbf{e}_1^T F^a \mathbf{e}_1 = a$ and $Q(F^a) = \inf \lbrace Q(F): \mathbf{e}_1^T F \mathbf{e}_1 = a\rbrace = \alpha a^2$.
We recall that the proof of the cleavage laws in \cite{FriedrichSchmidt:2011, FriedrichSchmidt:2014.1, Mora:2010} fundamentally relied on the application of certain slicing techniques which were not suitable to treat the case of compression. Having general compactness and $\Gamma$-convergence results we can now complete the picture about cleavage laws by extending the results to the case of uniaxial compression.
\begin{theorem}\label{rig-th: cleavage-cont}
Suppose $a_\varepsilon/\sqrt{\varepsilon} \to a \in [-\infty,\infty]$. The limiting minimal energy is given by
\begin{align}\label{rig-eq: cleavage en}
\lim_{\varepsilon \to 0} \inf \lbrace E_\varepsilon(y): y \in {\cal A}(a_\varepsilon)\rbrace = \min\Big\{ \frac{1}{2} \alpha l a^2,1\Big\}.
\end{align}
Let $a_{\rm crit}:= \sqrt{\frac{2\alpha}{l}}$. For every sequence $(y_\varepsilon)_\varepsilon$ of almost minimizers, up to passing to subsequences, we get $\varepsilon^{-1/2}(y_\varepsilon(x) - x) \to u(x)$ for a.e. $x \in \Omega$, where
\begin{itemize}
\item[(i)] if $|a| < a_{\rm crit}$, $u(x) = (0,s) + F^a x$ for $s\in \Bbb R$,
\item[(ii)] if $|a| > a_{\rm crit}$, $u(x) = \begin{cases} (0,s) & x_1 < p, \\ (l a,t) & x_1 > p, \end{cases}$ for $s,t \in \Bbb R$, $p\in (0,l)$.
\end{itemize}
\end{theorem}
\BBB Let us emphasize that the cleavage law is derived for a special geometry of $\Omega$ by solving a static, global minimization problem similarly as in \cite{Braides-Lew-Ortiz:06, FriedrichSchmidt:2014.1, Mora:2010}. An accurate prediction of crack propagation under tensile loading is beyond the scope of the present contribution. \EEE
\section{Preliminaries}\label{rig-sec: pre}
In this section we collect the definitions as well as basic properties of $SBV$ and $SBD$ functions and state the rigidity estimates which are necessary for the derivation of our main compactness result.
\subsection{(G)SBV and (G)SBD functions}\label{rig-sec: sub, sbv}
Let $\Omega \subset \Bbb R^d$ open, bounded with Lipschitz boundary. Recall that the space $SBV(\Omega, \Bbb R^d)$, abbreviated as $SBV(\Omega)$ hereafter, of \emph{special functions of bounded variation} consists of functions $y \in L^1(\Omega, \Bbb R^d)$ whose distributional derivative $Dy$ is a finite Radon measure, which splits into an absolutely continuous part with density $\nabla y$ with respect to Lebesgue measure and a singular part $D^s y$ whose Cantor part vanishes and thus is of the form
$$ D^s y = [y] \otimes \nu_y {\cal H}^{d-1} \lfloor J_y. $$
Here ${\cal H}^{d-1}$ denotes the $(d-1)$-dimensional Hausdorff measure, $J_y$ (the `crack path') is an ${\cal H}^{d-1}$-rectifiable set in $\Omega$, $\nu_y$ is a normal of $J_y$ and $[y] = y^+ - y^-$ (the `crack opening') with $y^{\pm}$ being the one-sided limits of $y$ at $J_y$. If in addition $\nabla y \in L^2(\Omega)$ and ${\cal H}^{d-1}(J_y) < \infty$, we write $y \in SBV^2(\Omega)$. See \cite{Ambrosio-Fusco-Pallara:2000} for the basic properties of this function space.
Likewise, we say that a function $y \in L^1(\Omega, \Bbb R^d)$ is a \emph{special function of bounded deformation} if the symmetrized distributional derivative $Ey := \frac{(Dy)^T + Dy}{2}$ is a finite \BBB $\Bbb R^{d \times d}_{\rm sym}$-valued \EEE Radon measure with vanishing Cantor part. It can be decomposed as
\begin{align}\label{rig-eq: symmeas}
Ey = e(\nabla y) {\cal L}^d + E^s y = e(\nabla y) {\cal L}^d + [y] \odot \nu_y {\cal H}^{d-1}|_{J_y},
\end{align}
where $e(\nabla y)$ is the absolutely continuous part of $Ey$ with respect to the Lebesgue measure ${\cal L}^d$, $[y]$, $\nu_y$, $J_y$ as before and $a \odot b = \frac{1}{2}(a \otimes b + b \otimes a)$. For basic properties of this function space we refer to \cite
{Ambrosio-Coscia-Dal Maso:1997, Bellettini-Coscia-DalMaso:98}.
To treat variational problems as considered in Section \ref{rig-sec: main} (see in particular \eqref{rig-eq: Griffith en}) the spaces $SBV(\Omega)$ and $SBD(\Omega)$ are not adequate due to the lacking $L^\infty$-bound being essential in the compactness theorems. To overcome this difficulty the space of $GSBV(\Omega)$ was introduced consisting of all ${\cal L}^d$-measurable functions $y: \Omega \to \Bbb R^d$ such that for every $\phi \in C^1(\Bbb R^d)$ with the support of $\nabla \phi$ compact, the composition $\phi \circ y $ belongs to $SBV_{\rm loc}(\Omega)$ (see \cite{DeGiorgi-Ambrosio:1988}). In this new setting one may obtain a more general compactness result (see \cite[Theorem 4.36]{Ambrosio-Fusco-Pallara:2000}). Unfortunately, this approach cannot be pursued in the framework of $SBD$ functions as for a function $y \in SBD(\Omega)$ \BBB the composition \EEE $\phi \circ y$ typically does not lie in $SBD(\Omega)$. In \cite{DalMaso:13}, Dal Maso suggested another approach which is based on certain properties of one-dimensional slices.
First we have to introduce some notation. For every $\xi \in \Bbb R^d \setminus \lbrace 0 \rbrace$, for every $s \in \Bbb R^d$ and for every $B \subset \Omega$ we let
\begin{align}\label{eq: slicing 1}
B^{\xi,s} = \lbrace t \in \Bbb R: s + t\xi \in B\rbrace.
\end{align}
Furthermore, define the hyperplane $\Pi^\xi = \lbrace x \in \Bbb R^d: x \cdot \xi = 0\rbrace$. Moreover, for every function $y: B \to \Bbb R^d$ we introduce the function $y^{\xi,s} : B^{\xi,s} \to \Bbb R^d$ by
\begin{align}\label{eq: slicing 2}
y^{\xi,s}(t) = y(s + t\xi)
\end{align}
and $\hat{y}^{\xi,s} : B^{\xi,s} \to \Bbb R$ by $\hat{y}^{\xi,s}(t) = y(s + t\xi) \cdot \xi$. If $\hat{y}^{\xi,s} \in SBV(B^{\xi,s},\Bbb R)$ and $J_{\hat{y}^{\xi,s}}$ denotes \BBB the \EEE \textit{approximate jump set}, we define
$$J^1_{\hat{y}^{\xi,s}} := \lbrace t \in J_{\hat{y}^{\xi,s}}: |[\hat{y}^{\xi,s}]( t)| \ge 1 \rbrace.$$
The space $GSBD(\Omega,\Bbb R^d)$ of \emph{generalized functions of bounded deformation} is the space of all ${\cal L}^d$-measurable functions $y: \Omega \to \Bbb R^d$ with the following property: There exists a nonnegative bounded Radon measure $\lambda$ on $\Omega$ such that for all $\xi \in S^{d-1}:=\lbrace x \in \Bbb R^d: |x|=1 \rbrace$ we have that for ${\cal H}^{d-1}$-a.e. $s \in \Pi^\xi$ the function $\hat{y}^{\xi,s} = y^{\xi,s} \cdot \xi$ belongs to $SBV_{\rm loc}(\Omega^{\xi,s})$ and
$$\int_{\Pi^\xi} \Big( |D\hat{y}^{\xi,s}|(B^{\xi,s} \setminus J^1_{\hat{y}^{\xi,s}}) + {\cal H}^0(B^{\xi,s} \cap J^1_{\hat{y}^{\xi,s}})\Big)\, d{\cal H}^{d-1}(s) \le \lambda(B)$$
for all Borel sets $B \subset \Omega$.
We refer to \cite{DalMaso:13} for basic properties of this space. In particular, for later reference we now recall fundamental slicing, compactness and approximation results. We first briefly state the main slicing properties of $GSBD$ functions (see \cite[Section 8,9]{DalMaso:13}.) Recall definitions \eqref{eq: slicing 1} and \eqref{eq: slicing 2} and let $J_y^\xi = \lbrace x \in J_y: [y](x) \cdot \xi \neq 0 \rbrace$.
\begin{theorem}\label{clea-th: slic}
Let $y \in GSBD(\Omega)$. For all $\xi \in S^{d-1}$ and ${\cal H}^{d-1}$-a.e.\ $s$ in $\Pi^\xi = \lbrace x: x\cdot \xi = 0\rbrace$ we have $J_{\hat{y}^{\xi,s}} = (J^\xi_y)^{\xi,s}$ and
\begin{align*}
\int_{\Pi^\xi} \# J_{\hat{y}^{\xi,s}} \, d{\cal H}^{d-1}(s) = \int_{J^\xi_y} |\nu_y \cdot \xi| \, d{\cal H}^{d-1}.
\end{align*}
Moreover, the approximate symmetrized gradient $e(\nabla y)$ exists in the sense of \cite[(9.1)]{DalMaso:13}, satisfies $e(\nabla y) \in L^1(\Omega, \Bbb R_{\rm sym}^{d \times d})$ and for all $\xi \in S^{d-1}$ and ${\cal H}^{d-1}$-a.e.\ $s$ in $\Pi^\xi$ we have
\begin{align*}
\xi^T e(\nabla y(s+ t\xi))\xi = (\hat{y}^{\xi,s})'(t) \ \text{ for a.e. } t \in {\Omega}^{\xi,s}.
\end{align*}
\end{theorem}
If in addition $e(\nabla y) \in L^2(\Omega)$ and ${\cal H}^{d-1}(J_y) < \infty$, we write $y \in GSBD^2(\Omega)$. Similar properties for $SBV$ functions may be found in \cite[Section 3.11]{Ambrosio-Fusco-Pallara:2000}. We now state a general compactness result in $GSBD$ proved in \cite[Theorem 11.3]{DalMaso:13} which we slightly adapt for our purposes.
\begin{theorem}\label{rig-th: GSBD comp}
Let $(y_k)_k$ be a sequence in \BBB $GSBD^2(\Omega)$. \EEE Suppose that there exist a constant $M>0$ and an increasing continuous functions $\psi:[0,\infty) \to [0,\infty)$ with $\lim_{ \BBB t \EEE \to \infty} \psi( \BBB t \EEE ) = + \infty$ such that
$$\int_{\Omega} \psi(|y_k|) + \int_{\Omega} |e(\nabla y_k)|^2 + {\cal H}^{d-1}(J_{y_k}) \le M $$
for every $k \in \Bbb N$. Then there exist a subsequence, still denoted by $(y_k)_k$, and a function $y \in GSBD^2(\Omega
)$ such that
\begin{align}\label{rig-eq: convergence sense}
\begin{split}
& y_k \to y \ \ \ \text{pointwise a.e. in} \ \ \ \Omega,\\
&e (\nabla y_k) \rightharpoonup e (\nabla y) \ \ \text{ weakly in} \ L^2(\Omega,\Bbb R^{d\times d}_{\rm sym}),\\
& \liminf_{k \to \infty} {\cal H}^{d-1}(J_{y_k}) \ge {\cal H}^{d-1}(J_y).
\end{split}
\end{align}
\end{theorem}
The lower semicontinuity result for the jump set can be generalized considering one-dimensional slices. Define $\theta_\sigma: [0,\infty) \to [0,1]$ by $\theta_\sigma(t) = \min \lbrace \frac{t}{\sigma},1 \rbrace$ for $\sigma > 0$ and additionally $\theta_0 \equiv 1$. Let
\begin{align}\label{rig-eq: lemma2**}
\hat{\mu}^{\sigma,\xi}_y(B) := \int_{\Pi^\xi} \int_{B^{\xi,s} \cap J_{\hat{y}^{\xi,s}}} \theta_\sigma(|[\hat{y}^{\xi,s}](t) | )\, d{\cal H}^0(t) \, d{\cal H}^{d-1}(s)
\end{align}
for all Borel sets $B \subset \Omega$.
\begin{lemma}\label{rig-lemma: lemma}
Let $(y_k)_k$ be a sequence in \BBB $GSBD^2(\Omega)$ \EEE converging to a function $y \in \BBB GSBD^2(\Omega) \EEE $ in the sense of \eqref{rig-eq: convergence sense}. Then
\begin{align*}
\hat{\mu}^{\sigma,\xi}_y(U) \le \liminf_{k \to \infty} \hat{\mu}^{\sigma,\xi}_{y_k}(U)
\end{align*}
for all $\sigma \ge 0$, every $\xi \in S^{d-1}$ and for all open sets $U \subset \Omega$.
\end{lemma}
\par\noindent{\em Proof. } As $y_k \to y$ in the sense of \eqref{rig-eq: convergence sense}, we may assume that $ (\hat{y}_k)^{\xi,s} \to \hat{y}^{\xi,s}$ in $GSBV(U^{\xi,s})$ for ${\cal H}^{d-1}$-a.e. $s \in U^\xi := \lbrace s \in \Pi^\xi: U^{\xi,s} \neq \emptyset\rbrace$. This is one of the essential steps in the proof of Theorem \ref{rig-th: GSBD comp} (cf. \cite[Theorem 11.3]{DalMaso:13} or \cite[Theorem 1.1]{Bellettini-Coscia-DalMaso:98} for an elaborated proof in the $SBD$-setting). The desired claim now follows from the corresponding lower semicontinuity result for $GSBV$ functions (see e.g. \cite[Theorem 4.36]{Ambrosio-Fusco-Pallara:2000}) and Fatou's lemma. \nopagebreak\hspace*{\fill}$\Box$\smallskip
We briefly note that using the area formula (see e.g. \cite[Theorem 2.71]{Ambrosio-Fusco-Pallara:2000})) and fine properties of $GSBD$ functions (see \cite{DalMaso:13}), $\hat{\mu}^{\sigma,\xi}_y(B)$ can be written equivalently as
\begin{align}\label{rig-eq: lemma2}
\hat{\mu}^{\sigma,\xi}_y(B) = \int_{\BBB J^\xi_y \EEE \cap B} \theta_\sigma(|[y] \cdot \xi|) |\nu_y \cdot \xi| \, d{\cal H}^{d-1}
\end{align}
for all \BBB $\sigma \ge 0$, \EEE for all $\xi \in S^{d-1}$ and all Borel sets $B \subset \Omega$ (see also \cite[Remark 9.3]{DalMaso:13}). Finally, we recall a density result in $GSBD$ (see \cite{Iurlano:13}).
\begin{theorem}\label{rig-th: cortesani2}
Let $y \in GSBD^2(\Omega) \cap L^2(\Omega)$. Then there exists a sequence $y_k \in SBV^2(\Omega)$ such that each $J_{y_k}$ is contained in the union of a finite number of closed connected pieces of $C^{1}$-hypersurfaces, each $y_k$ belongs to
$W^{1,\infty}(\Omega \setminus \overline{J_{y_k}},\Bbb R^d)$ and the following properties hold:
\begin{align*}
(i) & \ \ \Vert y_k - y \Vert_{L^2(\Omega)} \to 0,\\
(ii) & \ \ \Vert e(\nabla y_k) - e(\nabla y) \Vert_{L^2(\Omega)} \to 0,\\
(iii) & \ \ {\cal H}^{d-1}(J_{y_k} \BBB \triangle \EEE J_y) \to 0.
\end{align*}
\end{theorem}
\subsection{Caccioppoli partitions}\label{rig-sec: sub, cacciop}
Let $\Omega \subset \Bbb R^d$ open and $E \subset \Omega$ measurable. For \BBB $t \in [0,1]$ we define the points of density $t$ by
\begin{align}\label{eq: essential boundary}
E^t = \left\{ x \in \Bbb R^d: \lim\nolimits_{\varrho \downarrow 0} \frac{|E \cap B_\varrho(x)|}{|B_\varrho(x)|} = t\right\}
\end{align}
(see \cite[Definition 3.60]{Ambrosio-Fusco-Pallara:2000}). By $\partial^* E = \Bbb R^d \setminus (E^0 \cup E^1)$ we denote the \emph{essential boundary} of $E$ and $\mathcal{H}^1(\partial^*E \cap \Omega)$ denotes the \emph{perimeter} of $E$ in $\Omega$ (cf. \cite[(3.62)]{Ambrosio-Fusco-Pallara:2000})\EEE .
We say a partition ${\cal P} = (P_j)_{j\in\Bbb N}$ of $\Omega$ is a \textit{Caccioppoli partition} of $\Omega$ if $\sum_j \mathcal{H}^1(\partial^*P_j) < + \infty$. We say a partition is \textit{ordered} if $|P_i| \ge |P_j|$ for $i \le j$. In the whole paper we will always tacitly assume that partitions are ordered. Given a rectifiable set $S$ we say that a Caccioppoli partition is \textit{subordinated} to $S$ if (up to an ${\cal H}^{d-1}$-negligible set) the essential boundary $\partial^* P_j$ of $P_j$ is contained in $S$ for every $j \in \Bbb N$.
The local structure of Caccioppoli partitions can be characterized as follows (see \cite[Theorem 4.17]{Ambrosio-Fusco-Pallara:2000}).
\begin{theorem}\label{th: local structure}
Let $(P_j)_j$ be a Caccioppoli partition of $\Omega$. Then
$$\bigcup\nolimits_j (P_j)^1 \cup \bigcup\nolimits_{i \neq j} \partial^* P_i \cap \partial^* P_j$$
contains ${\cal H}^{d-1}$-almost all of $\Omega$, \BBB where $(P_j)^1$ as defined in \eqref{eq: essential boundary}. \EEE
\end{theorem}
Essentially, the theorem states that ${\cal H}^{d-1}$-a.e. point of $\Omega$ either belongs to exactly one element of the partition or to the intersection of exactly two sets $\partial^* P_i$, $\partial^* P_j$. We now state a compactness result for ordered Caccioppoli partitions (see \cite[Theorem 4.19, Remark 4.20]{Ambrosio-Fusco-Pallara:2000}).
\begin{theorem}\label{th: comp cacciop}
Let $\Omega \subset \Bbb R^d$ open, bounded with Lipschitz boundary. Let ${\cal P}_i = (P_{j,i})_j$, $i \in \Bbb N$, be a sequence of ordered Caccioppoli partitions of $\Omega$ fulfilling $\sup_i \sum_j \mathcal{H}^{d-1}(\partial^* P_{j,i}) < \infty$. Then there exists a Caccioppoli partition ${\cal P} = (P_j)_j$ and a not relabeled subsequence such that \BBB $|P_{j,i} \triangle P_j| \to 0$ \EEE for all $j \in \Bbb N$ as $i \to \infty$.
\end{theorem}
We will also use the fact that $|P_{j,i} \triangle P_j| \to 0$ for all $j \in \Bbb N$ is equivalent to $\sum_j|P_{j,i} \triangle P_j| \to 0$. Caccioppoli partitions are naturally associated to piecewise constant functions. We say $y: \Omega \to \BBB \Bbb R^m \EEE $ is \emph{piecewiese constant in $\Omega$} if there exists a Caccioppoli partition $(P_j)_j$ of $\Omega$ and a sequence $(t_j)_j \subset \BBB \Bbb R^m \EEE $ such that $y = \sum_j t_j \chi_{P_j}$. We close this section with a compactness result for piecewise constant functions (see \cite[Theorem 4.25]{Ambrosio-Fusco-Pallara:2000}).
\begin{theorem}\label{th: piecewise const}
Let $\Omega \subset \Bbb R^d$ open, bounded with Lipschitz boundary. Let $(y_i)_i \subset SBV(\Omega, \BBB \Bbb R^m \EEE )$ be a sequence of piecewise constant functions such that $\sup_i (\Vert y_i \Vert_\infty + {\cal H}^{d-1}(J_{y_i})) < \infty$. Then there exists a not relabeled subsequence converging in measure to a piecewise constant function $y$.
\end{theorem}
\subsection{Rigidity estimates}\label{sec: rig}
In this section we first recall a geometric rigidity result obtained in the framework of nonlinear elasticity and a piecewise rigidity estimate for brittle materials for the sake of completeness. Afterwards we introduce a quantitative result in $SBD$ adapted for Griffith energies of the form \eqref{rig-eq: Griffith en} which will be the starting point for our analysis.
We begin with the quantitative geometric rigidity result by Friesecke, James, M\"uller \cite{FrieseckeJamesMueller:02} generalizing the classical Liouville theorem.
\begin{theorem}\label{rig-th: geo rig}
Let $\Omega \subset \Bbb R^d$ a (connected) Lipschitz domain and $1 < p < \infty$. Then there exists a constant $C = C(\Omega,p)$ such that for any $y \in W^{1,p}(\Omega,\Bbb R^d)$ there is a rotation $R \in SO(d)$ such that
\begin{align*}
\left\|\nabla y - R\right\|_{L^p(\Omega)} \leq C \left\|\operatorname{dist}(\nabla y, SO(d))\right\|_{L^p(\Omega)}.
\end{align*}
\end{theorem}
In the theory of fracture mechanics the problem is more involved as global rigidity can fail if the crack disconnects the body. Chambolle, Giacomini and Ponsiglione \cite{Chambolle-Giacomini-Ponsiglione:2007} have proved the following qualitative result for brittle materials which do not store elastic energy (i.e. $\nabla y \in SO(d)$ a.e. in $\Omega$).
\begin{theorem}\label{rig-cor: cgp}
Let $y \in SBV(\Omega)$ such that ${\cal H}^{d-1}(J_y) < +\infty$ and $\nabla y \in SO(d)$ a.e. in $\Omega$. Then $y$ is a collection of an at most countable family of rigid deformations, i.e., there exists a Caccioppoli partition ${\cal P} = (P_j)_j$ subordinated to $J_y$ such that
$$y(x) = \sum\nolimits_j (R_j \, x + b_j) \chi_{P_j}(x),$$
where $R_j \in SO(d)$ and $b_j \in \Bbb R^d$.
\end{theorem}
Loosely speaking, the result states that the only way that rigidity may fail is that the body is divided into at most countably many parts each of which subject to a different rigid motion. We briefly note that there is an analogous result in the geometrically linear setting (see \cite[Theorem A.1]{Chambolle-Giacomini-Ponsiglione:2007}): A function $u \in SBD(\Omega)$ with ${\cal H}^{d-1}(J_u) < +\infty$ and $e(\nabla u) = 0$ a.e. in $\Omega$ has the form $u(x) = \sum\nolimits_j (A_j \, x + b_j) \chi_{P_j}(x)$ for $A_j \in \Bbb R^{d \times d}_{\rm skew}$ and $b_j \in \Bbb R^d$.
We now introduce a quantitative $SBD$-rigidity result which may be seen as a suitable combination of the above estimates and is tailor-made for general Griffith functionals of the form \eqref{rig-eq: Griffith en} where both energy forms are coexistent (see \cite[Theorem 2.1, Remark 2.2]{Friedrich-Schmidt:15}). Let $\Omega_\rho = \lbrace x\in\Omega: \operatorname{dist}(x, \partial \Omega) > \BBB \rho \EEE \rbrace$ for $\rho>0$. Recall \eqref{rig-eq: SBVfirstdef}, \eqref{rig-eq: Griffith en} and introduce \BBB an auxiliary \EEE energy functional by
\begin{align}\label{rig-eq: Griffith en2}
E_\varepsilon^\rho(y,U) = \frac{1}{\varepsilon}\int_U W(\nabla y(x)) \,dx + \int_{J_y \cap U} f_\varepsilon^\rho(|[y](x )|)\,d{\cal H}^1(x).
\end{align}
for $\rho > 0$, $\varepsilon > 0$ and $U \subset \Omega$, where $f_\varepsilon^\rho(\BBB t \EEE ) := \min\lbrace\frac{\BBB t \EEE}{\sqrt{\varepsilon}\rho} ,1 \rbrace$. Recall the definition $e(G) = \frac{G + G^T}{2}$ for all $G \in \Bbb R^{2 \times 2}$.
\begin{theorem}\label{rig-th: rigidity}
Let $\Omega \subset \Bbb R^2$ open, bounded with Lipschitz boundary. Let $M>0$ and $0 < \eta, \rho < 1$. Then there are a universal constant $c>0$, constants $\bar{C}=\bar{C}(\Omega,M,\eta)>0$, \BBB $\hat{C}=\hat{C}(\Omega,M,\eta,\rho)>0$, \EEE and \BBB $\varepsilon_0 = \varepsilon_0(M,\eta,\rho)>0$ \EEE such that the following holds for all \BBB $0 < \varepsilon \le \varepsilon_0$: \EEE \\
For each $y \in SBV_{M}(\Omega)$ with ${\cal H}^{1}(J_y) \le M$ and $\int_\Omega \operatorname{dist}^2(\nabla y,SO(2) ) \le M\varepsilon$ there is an open set $\Omega_y \subset \Omega$ and a modification $\hat{y} \in SBV_{cM}(\Omega)$ satisfying
\begin{align}\label{rig-eq: energy le}
\begin{split}
\BBB (i) \EEE & \ \ \Vert \hat{y} - y \Vert^2_{L^2(\Omega_y)} + \Vert \nabla \hat{y} - \nabla y \Vert^2_{L^2(\Omega_y)}\le \bar{C}\varepsilon\rho, \ \ \ \ |\Omega\setminus\Omega_y| \le \bar{C}\rho,\\
(ii) & \ \ E_\varepsilon^\rho(\hat{y},\Omega_\rho) \le E_\varepsilon(y) + \bar{C}\rho
\end{split}
\end{align}
with the following properties: We find a Caccioppoli partition ${\cal P} = (P_j)_j$ of $\Omega_\rho$ with $\sum_j \mathcal{H}^1(\partial^* P_j\cap \Omega_{\rho}) \le \bar{C}$ and for each $P_j$ a corresponding rigid motion $R_j \cdot +b_j$, $R_j \in SO(2)$ and $b_j \in \Bbb R^2$, such that the function $u: \Omega \to \Bbb R^2$ defined by
$$
u(x) := \begin{cases} \hat{y}(x) - (R_j\,x +b_j) & \ \ \text{ for } x \in P_j \\
0 & \ \ \text{ for } x \in \Omega \setminus \Omega_\rho \end{cases}
$$
satisfies the estimates
\begin{align}\label{rig-eq: main properties2}
\begin{split}
(i) & \ \, {\cal H}^{1}(J_u) \le \bar{C}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (ii) \, \ \Vert u\Vert^2_{L^2(\Omega_\rho)} \le \hat{C}\varepsilon, \\
(iii) & \ \, \sum\nolimits_j \Vert e(R^T_j \nabla u)\Vert^2_{L^2(P_j)} \le \hat{C}\varepsilon, \ \ \ \ \ \ \ \, (iv) \ \, \Vert \nabla u\Vert^2_{L^2(\Omega_\rho)} \le \hat{C}\varepsilon^{1-\eta}.
\end{split}
\end{align}
\end{theorem}
We remark that we get a sufficiently strong bound only for the symmetric part of the gradient (see \eqref{rig-eq: energy le}(iii)) which is not surprising due to the fact that there is no direct analogue of Korn's inequality in $SBV$. However, there is at least a weaker bound on the full absolutely continuous part of the gradient $\nabla u$ (see \eqref{rig-eq: energy le}(iv)) which will essentially be needed to estimate the elastic part of the energy in the passage to the linearized theory (see \eqref{rig-eq: ff} and \eqref{rig-eq: ff2} below). \BBB In particular, it will allow us to obtain \eqref{rig-eq: comp2}(ii) after modification of $u$ on a set of small measure. \EEE
Furthermore, let as briefly note that the uniform bound on the gradient (see \eqref{rig-eq: SBVfirstdef}) in the setting of the nonlinear model is only needed for the application of the rigidity estimate. The condition essentially ensures that the elastic energy cannot concentrate on scales being much smaller than $\varepsilon$. In particular, this is a natural assumption in the investigation of discrete systems, where $\varepsilon$ may be interpreted as the typical interatomic distance.
\begin{rem}\label{rem: rig}
{\normalfont
(i) Estimate \eqref{rig-eq: energy le}(ii) can be refined. Indeed, we obtain
\begin{align}\label{rig-eq: part + crack}
\begin{split}
(i) & \ \ \BBB \frac{1}{\varepsilon}\int_{\Omega_\rho}W(\nabla \hat{y}(x)) \,dx \le \frac{1}{\varepsilon}\int_{\Omega}W(\nabla y(x)) \, dx + \bar{C}\rho, \EEE \\
(ii) & \ \ \sum\nolimits_j \frac{1}{2} \mathcal{H}^1(\partial^* P_j \cap \Omega_\rho) + \int_{J_{\hat{y}} \setminus \bigcup_j \partial^* P_j} f_\varepsilon^\rho(|[\hat{y}]|) \,d{\cal H}^1 \le {\cal H}^1(J_y) + \bar{C}\rho.
\end{split}
\end{align}
\BBB We remark that it is indispensable to allow for a small modification of the deformation in Theorem \ref{rig-th: rigidity} in order to guarantee the sharp energy estimate \eqref{rig-eq: part + crack}(ii). \EEE
(ii) To derive \eqref{rig-eq: main properties2}\BBB (iii) \EEE one essentially shows
$$
\Vert \nabla u\Vert^4_{L^4(\Omega_\rho)} = \sum\nolimits_j\Vert \nabla \hat{y}- R_j\Vert^4_{L^4(P_j)} \le \hat{C}\varepsilon.
$$
The claim then follows from \BBB \eqref{eq:W}, \eqref{rig-eq: part + crack}(i), \EEE $\Vert\operatorname{dist}( \BBB \nabla y, \EEE SO(2))\Vert^2_{L^2(\Omega)} \le M\varepsilon$ and the linearization formula \BBB (see \cite[(3.20)]{FrieseckeJamesMueller:02}) \EEE
\begin{align}\label{rig-eq: linearization}
|e(R^T G -\mathbf{Id})| = \operatorname{dist}(G,SO(2)) + \BBB O\EEE(|G- R|^2)
\end{align}
for $G \in \Bbb R^{2 \times 2}$ and $R \in SO(2)$, where $\mathbf{Id}$ denotes the identity matrix.
}
\end{rem}
\section{Compactness of rescaled configurations}\label{rig-sec: sub, comp1}
This section is devoted to the proof of the main compactness result given in Theorem \ref{rig-th: comp1}. Moreover, we also show that Theorem \ref{rig-th: comp1} provides an alternative proof of the piecewise rigidity result stated in Theorem \ref{rig-cor: cgp}.
\subsection{Preparations}
For the compactness theorem in $GSBD$ (see Theorem \ref{rig-th: GSBD comp}) it is necessary that the integral for some integrand $\psi$ with $\lim_{\BBB t \EEE \to \infty} \psi( \BBB t \EEE ) = \infty$ is uniformly bounded. We first give a simple criterion for the existence of such a function which is, loosely speaking, based on the condition that the functions coincide in a certain sense on the bulk part of the domain.
\begin{lemma}\label{rig-lemma: concave function}
For every increasing sequence $(b_i)_i \subset (0,\infty)$ with $b_i \to \infty$ there is an increasing concave function $\psi:[0,\infty) \to [0,\infty)$ with $\lim_{t \to \infty} \psi(t) = \infty$ and $\psi(b_i) \le 2^{i}$ for all $i \in \Bbb N$.
\end{lemma}
\par\noindent{\em Proof. } Let $f:[0,\infty) \to [0,\infty)$ be the function with $f(0) = 0$, $f(b_i) = 2^{i}$ which is affine on each segment $[b_i,b_{i+1}]$. Clearly, $f$ is increasing and satisfies $f(t) \to \infty$ for $t \to \infty$, but is possibly not concave. We now construct $\psi$ and first let $\psi = f$ on $[0,b_1]$. Assume $\psi$ has been defined on $[0,b_i]$ and that \BBB $\psi$ is increasing, concave, satisfies $\psi \le f$ and \EEE $\psi(b_i) = f(b_i) = 2^{i}$. If $f'(b_i-)\ge f'(b_i+)$, we set $\psi = f$ on $[b_{i},b_{i+1}]$. Here, $f'(t\pm)$ denote the one-sided limits of the derivative at point $t$. \BBB This implies that $\psi$ is concave on $[0,b_{i+1}]$ since $\psi'(b_i -) \ge f'(b_i-)$. \EEE
Otherwise, we let $\psi(t) = f(b_i) + f'(b_i-)(t-b_i)$ for $t \in [b_i,\bar{t}]$, where $\bar{t}$ is the smallest value larger than $b_i$ such that $f(\bar{t}) = f(b_i) + f'(b_i-)(\bar{t}-b_i)$. If $\bar{t}$ does not exist, we are done. If $\bar{t}$ exists, we assume $\bar{t} \in (b_{j-1},b_j]$ and define $\psi=f$ on $[\bar{t}, b_j]$. \BBB Note that if $\bar{t} \in (b_{j-1},b_j)$, we have $\psi'(\bar{t}-) \ge \psi'(\bar{t}+)$ since $f$ is affine on $[b_{j-1},b_j]$ and $\psi < f$ on $(b_i,\bar{t})$. Thus, $\psi$ is concave on $[0,b_j]$.
Repeating the construction \EEE we end up with an increasing concave function $\psi$ with $\psi \le f$ and $\psi(t) \to \infty$ for $t \to \infty$. \nopagebreak\hspace*{\fill}$\Box$\smallskip
\begin{lemma}\label{rig-lemma: concave function2}
Let $\Omega \subset \Bbb R^2$ and let $(y^l)_l \subset L^1(\Omega)$ be a sequence satisfying $|\Omega \setminus \bigcup_{n\in \Bbb N} \bigcap_{l \ge n} \lbrace |y^n - y^l| \le 1 \rbrace|=0$. Then there is a not relabeled subsequence such that
$$\int_{\Omega}\psi(|y^l|)\le C$$
for a constant $C>0$ independent of $l$, where $\psi$ is an increasing continuous function with $\lim_{t \to \infty} \psi(t) = \infty$.
\end{lemma}
\par\noindent{\em Proof. } Define $C_l := \max_{1 \le i \le l} \Vert y^i \Vert_{L^1(\Omega)}$ for all $l \in \Bbb N$. Let $A_n = \bigcap_{l \ge n} \lbrace |y^n - y^l|\le 1 \rbrace$ and set $B_1 = A_1$ as well as $B_n = A_n \setminus \bigcup^{n-1}_{m=1} B_{m}$ for all $n \in \Bbb N$. The sets $(B_n)_n$ are pairwise disjoint with $\sum_n |B_n| = |\Omega|$. We choose $0=n_1 < n_2 < \ldots$ such that $\sum_{1\le n \le n_{i}} \frac{|B_n|}{|\Omega|} \ge 1 - 4^{-i}$. We let $B^i = \bigcup^{n_{i+1}}_{n=n_i+1} B_n$ and observe $|B^i|\le 4^{-i}|\Omega| $.
We pass to the subsequence $(n_i)_i \subset \Bbb N$ and choose $ E^i \supset B^i$ such that $| E^i| = 4^{-i} |\Omega|$. Let $b_i = \frac{C_{n_{i+1}}}{| E^i|} + 2= 4^{i}\frac{C_{n_{i+1}}}{|\Omega|}+ 2$ for $i\in \Bbb N$ and note that $(b_i)_i$ is increasing with $b_i \to \infty$. By Lemma \ref{rig-lemma: concave function} we get an increasing concave function $\psi:[0,\infty) \to [0,\infty)$ with $\lim_{t \to \infty} \psi(t) = \infty$ and $\psi(b_i) \le 2^{i}$ for all $i \in \Bbb N$. Clearly, $\psi$ is also continuous.
For $\hat{B}^{ i} := \Omega \setminus \bigcup^{n_i}_{n=1} B_n$ we have $|\hat{B}^{ i}| \le 4^{-i}|\Omega|$ and choose $\hat{E}^{ i} \supset \hat{B}^i$ with $|\hat{E}^{ i}| = 4^{-i} |\Omega|$. We then obtain $\frac{C_{n_i}}{|\hat{E}^{ i}|}= 4^{i}\frac{C_{n_{i}}}{|\Omega|} \le b_{i}$. Now let $l = n_i$. Using Jensen's inequality, the definition of the sets $B^i$, $\Vert y^l \Vert_{L^1(\Omega)} \le C_l$ and the monotonicity of $\psi$ we compute
\begin{align}\label{rig-eq: psi est}
\begin{split}
\int_\Omega \psi(|y^l|) &=\sum\nolimits_{1 \le j \le i-1} \int_{B^j} \psi(|y^l|) + \int_{\hat{B}^{ i}} \psi(|y^l|) \\
& \BBB \le \EEE \sum\nolimits_{1 \le j \le i-1} \int_{B^j} \psi(|y^{n_{j+1}}|+ 2) + \int_{\hat{B}^{i}} \psi(|y^l|) \\
& \le \sum\nolimits_{1 \le j \le i-1} |E^j| \psi\Big(\Xint-_{E^j} |y^{n_{j+1}}|+2\Big) + |\hat{E}^{ i}| \psi\Big(\Xint-_{\hat{E}^{ i}} |y^l|\Big) \\
& \le \sum\nolimits_{1 \le j \le i-1} 4^{-j}|\Omega| 2^{j} + 4^{-i}|\Omega| 2^{i} \le |\Omega|\sum\nolimits_{j \in \Bbb N} 2^{-j}.
\end{split}
\end{align}
As the estimate is independent of $l \in (n_i)_i$, this yields $\int_\Omega \psi(|y^l|) \le C$ uniformly in $l$, as desired. \nopagebreak\hspace*{\fill}$\Box$\smallskip
\subsection{Proof of Theorem \ref{rig-th: comp1}}
Now we are in a position to give the proof of the main compactness result. In the first part we show that \eqref{rig-eq: comp2}-\eqref{rig-eq: comp1-2} hold.
\noindent {\em Proof of Theorem \ref{rig-th: comp1}, part 1.} Let $(\varepsilon_k)_k$ be a sequence \BBB with $\varepsilon_k \to 0$. \EEE Let $y_k \in SBV_M(\Omega)$ with $E_{\varepsilon_k}(y_k) \le C$ be given. \BBB Possibly passing to a larger $M$, \EEE we get $\Vert \operatorname{dist} (\nabla y_k,SO(2)) \Vert^2_{L^2(\Omega)} \le M\varepsilon_k$ \BBB by \eqref{eq:W} and ${\cal H}^1(J_{y_k}) \le M$ for all $k \in \Bbb N$. \BBB In the following generic constants only depending on $\Omega$ and $M$ will be denoted by $C$. \EEE
\smallskip
\BBB\emph{Step I:} \EEE Choose $\rho_0 >0$ small and let $\rho_l = 2^{- 3l} \rho_0$ for all $l\in \Bbb N$. \BBB We apply Theorem \ref{rig-th: rigidity} for $\rho = \rho_l$ and $\eta=\frac{1}{5}$ (the choice of $\eta$ is related to the exponent $-\frac{1}{8}$ in \eqref{rig-eq: comp2}(ii)). Denote by $c$, $\bar{C}=\bar{C}(\Omega,M,\eta)$, $\hat{C}_l=\hat{C}_l(\Omega,M,\eta,\rho_l)$ the constants in Theorem \ref{rig-th: rigidity}. For each $l \in \Bbb N$ there exists $\kappa_l = \kappa_l(M,\eta,\rho_l)$ such that for $k \ge \kappa_l$ \EEE we find modifications $y^l_k \in SBV_{cM}(\Omega, \Bbb R^2)$ with $E_{\varepsilon_k}^{\rho_l} (y^l_k, \Omega_{\rho_l}) \le E_{\varepsilon_k}(y_k) + \bar{C}\rho_l$ and
\begin{align}\label{rig-eq: approx en}
\Vert y^l_k - y_k \Vert^2_{L^2(\Omega^l_k)} + \Vert \nabla y^l_k - \nabla y_k \Vert^2_{L^2(\Omega^l_k)} \le \bar{C}\varepsilon_k\rho_l,
\end{align}
where $\Omega^l_k := \Omega_{y^l_k}$ with $|\Omega \setminus \Omega^l_k| \le \bar{C}\rho_l$. We further get Caccioppoli partitions $(P^{k,l}_j)_j$ of $\Omega_{\rho_l}$ with $\sum_j \mathcal{H}^1(\partial^* P^{k,l}_j \cap \Omega_{\rho_l}) \le \bar{C}$ and corresponding piecewise rigid motions $T_k^l:= \sum_j(R^{k,l}_j \cdot + b^{k,l}_j) \chi_{P^{k,l}_j} \BBB + \mathbf{id}\chi_{\Omega \setminus \Omega_{\rho_l}} \EEE $ such that the functions $v^l_k : \Omega \to \Bbb R^2$ defined by
\begin{align}\label{rig-eq: comp13}
v^l_k (x) = \begin{cases}
\frac{ 1}{\sqrt{\varepsilon_k}} (R^{k,l}_j)^T \big( y^l_k(x) - (R^{k,l}_j\,x + b^{k,l}_j)\big) & \text{ for } x \in P^{k,l}_j, \ j \in \Bbb N, \\
0 & \text{ else,}
\end{cases}
\end{align}
satisfy by \eqref{rig-eq: main properties2}
\begin{align}\label{rig-eq: comp11}
{\cal H}^1(J_{v^l_k}) \le \bar{C}, \ \ \ \Vert v^l_k \Vert_{L^2(\Omega)} + \Vert e (\nabla v^l_k) \Vert_{L^2(\Omega)} \le \hat{C}_l, \ \ \ \Vert \nabla v^l_k \Vert^2_{L^2(\Omega)} \le \hat{C}_l\varepsilon_k^{-\BBB 1/5 \EEE }.
\end{align}
\BBB We recall $\Vert y^l_k \Vert_\infty \le cM$ for all $k \ge \kappa_l$. Thus, possibly passing to other (not relabeled) constants $b^{k,l}_j$ in \eqref{rig-eq: comp13}, we can assume that $|b^{k,l}_j| \le CM$ for $C=C(\Omega,c)$ and that \eqref{rig-eq: comp11} still holds. \EEE Each partition may be extended to $\Omega$ by adding the element $\Omega \setminus \Omega_{\rho_l}$. \BBB As for $\rho_0$ small enough we get ${\cal H}^1(\partial \Omega_{\rho_l}) \le C{\cal H}^1(\partial \Omega)$ (see \cite[Theorem 4.1]{Doktor}) for all $l \in \Bbb N$, \EEE there is $C=C(\bar{C},\Omega)$ such that
\begin{align}\label{eq: new last}
\sum\nolimits_j \mathcal{H}^1(\partial^* P^{k,l}_j) \le C.
\end{align}
\BBB\emph{Step II:} \EEE Using a diagonal argument we get a (not relabeled) subsequence of $(\varepsilon_k)_k$ such that by Theorem \ref{rig-th: GSBD comp} for every $l \in \Bbb N$ we find $v^l \in GSBD^2(\Omega)$ with
\begin{align}\label{rig-eq: T conv3}
v^l_k \to v^l \text{ a.e. in } \Omega \text{ \ \ \ and \ \ \ } e(\nabla v^l_k) \rightharpoonup e(\nabla v^l) \text{ weakly in } L^2(\Omega, \Bbb R^{2 \times 2}_{\rm sym})
\end{align}
for $k \to \infty$. By \BBB Theorem \ref{th: comp cacciop}, Theorem \ref{th: piecewise const}, \eqref{eq: new last} and the fact that $|b^{k,l}_j| \le CM$ \EEE we obtain an (ordered) partition $(P^l_j)_j$ of $\Omega$ with $\sum_j \mathcal{H}^1(\partial^* P^l_j) \le C$ and a piecewise rigid motion $T^l := \sum_j(R^l_j \cdot + b^l_j)\chi_{P^l_j}$ such that for all $l \in \Bbb N$ we get (again up to a subsequence) \BBB $|P^{k,l}_j\triangle P^l_j| \to 0$, $R^{k,l}_j \to R^l_j $, and $b^{k,l}_j \to b^l_j $ \EEE for all $j \in \Bbb N$ as $k \to \infty$. This also implies
\begin{align}\label{rig-eq: T conv2}
\sum\nolimits_j |P^{k,l}_j \triangle P^{l}_j| + \Vert T_k^l - T^l \Vert_{L^2(\Omega)} + \Vert \nabla T_k^l - \nabla T^l \Vert_{L^2(\Omega)}\to 0
\end{align}
for $k \to \infty$. We now show that
\begin{align}\label{rig-eq: comp12}
\Vert v^l\Vert_{L^1(\Omega)} \le C\Vert v^l \Vert_{L^2(\Omega)} \le C \hat{C}_l, \ \ \ {\cal H}^1(J_{v^l}) \le \bar{C}, \ \ \ \Vert e(\nabla v^l)\Vert^2_{L^2(\Omega)} \le C.
\end{align}
The first two claims follow directly from \eqref{rig-eq: comp11} and \eqref{rig-eq: convergence sense}. To see the third estimate, we let $\BBB \phi^l_k (x) \EEE:= \chi_{[0,\varepsilon_k^{-1/8}]} (|\nabla v^l_k (x)|)$ \BBB (cf. \eqref{rig-eq: comp2}(ii)). \EEE Moreover, we obtain by an elementary computation (cf. \eqref{rig-eq: linearization}) $\operatorname{dist}^2(G,SO(2)) = | e(R^T G - \mathbf{Id})|^2 + \omega_{\rm dist }(R^TG-\mathbf{Id})$ for $G \in \Bbb R^{2 \times 2}$, $R \in SO(2)$ with $\sup\lbrace |G|^{-3}\omega_{\rm dist}(G): |G| \le 1\rbrace \le C$. We compute by \eqref{eq:W} and \eqref{rig-eq: comp13}
\begin{align}\label{rig-eq: ff}
\begin{split}
C & \ge E^{\rho_l}_{\varepsilon_k} (y^l_{k},\Omega_{\rho_l}) \geq \frac{C}{\varepsilon_k} \int_{\Omega_{\rho_l}} \operatorname{dist}^2(\nabla y^l_k,SO(2) )\\ & \ge \frac{C}{\varepsilon_k} \sum\nolimits_j\int_{P^{k,l}_j \BBB \cap \Omega_{\rho_l} \EEE } \phi^l_k \Big(|{e}( (R^{k,l}_j)^T\nabla y^l_k - \mathbf{Id})|^2 + \omega_{\rm dist}((R^{k,l}_j)^T\nabla y^l_k - \mathbf{Id}) \Big) \\ & = C \int_{\Omega} \phi^l_k \Big(|e(\nabla v^l_k)|^2 + \frac{1}{\varepsilon_k}\omega_{\rm \operatorname{dist}}(\sqrt{\varepsilon_k} \nabla v^l_k ) \Big).
\end{split}
\end{align}
The second term of the integral can be estimated by
\begin{align}\label{rig-eq: ff2}
\int_{\Omega} \phi^l_k \frac{1}{\varepsilon_k}\omega_{\rm dist}(\sqrt{\varepsilon_k} \nabla v^l_k ) = \int_{\Omega}\phi^l_k \sqrt{\varepsilon_k} |\nabla v^l_k |^3 \frac{\omega_{\rm dist}(\sqrt{\varepsilon_k} \nabla v^l_k) }{|\sqrt{\varepsilon_k} \nabla v^l_k |^3} \le C\varepsilon_k^{\frac{1}{8}} \to 0.
\end{align}
As $e(\nabla v^l_k) \rightharpoonup e(\nabla v^l)$ weakly in $L^2(\Omega)$ and $\phi^l_{k} \rightarrow 1$ boundedly in measure on $\Omega$ by \eqref{rig-eq: comp11}, it follows $\phi^l_{k} e(\nabla v^l_k) \rightharpoonup e(\nabla v^l)$ weakly in $L^2(\Omega)$. By lower semicontinuity we obtain $\Vert e(\nabla v^l)\Vert^2_{L^2(\Omega)} \le C$ for a constant particularly independent of $\rho_l$ which concludes \eqref{rig-eq: comp12}.
\smallskip
\BBB\emph{Step III:} \EEE We now want to pass to the limit $l \to \infty$. Similarly as in the argumentation leading to \eqref{rig-eq: T conv2}, by the compactness result for piecewise constant functions (see Theorem \ref{th: piecewise const}) we find a partition $(P_j)_j$ of $\Omega$ and a piecewise rigid motion $T := \sum_j(R_j \cdot +b_j)\chi_{P_j}$ such that for a suitable (not relabeled) subsequence
\begin{align}\label{rig-eq: T conv1}
\sum\nolimits_j |P^{l}_j \triangle P_j| + \Vert T^l - T \Vert_{L^2(\Omega)} + \Vert \nabla T^l - \nabla T \Vert_{L^2(\Omega)}\to 0
\end{align}
for $l \to \infty$. Recalling \eqref{rig-eq: T conv2} and using a diagonal argument we can choose a (not relabeled) subsequence of $(\rho_l)_l$ and afterwards of $(\varepsilon_k)_k$ such that for all $l$ we have
\begin{align}\label{rig-eq: subsubseq}
\sum\nolimits_j |P^l_j \triangle P_j| \le 2^{-l}, \ \ \ \sum\nolimits_j |P^{k,l}_j \triangle P^{l}_j| \le 2^{-l} \ \text{ for all } \ k \ge l.
\end{align}
We see that the compactness result in $GSBD$ cannot be applied directly on the sequence $(v^l)_l$ as the $L^2$ bound in \eqref{rig-eq: comp12} depends on $\rho_l$. We now show that by choosing the rigid motions on the elements of the partitions appropriately (see \eqref{rig-eq: comp13}) we can construct the sequence $(v^l)_l$ such that we obtain
\begin{align}\label{rig-eq: Asetprep}
\Big|\Omega \setminus \bigcup\nolimits_{n \in \Bbb N} \bigcap\nolimits_{ m \ge n}\lbrace |v^n - v^{m}| \le 1\rbrace\Big|=0
\end{align}
and thus Lemma \ref{rig-lemma: concave function2} is applicable.
\smallskip
\BBB\emph{Step IV:} \EEE We fix $k \in \Bbb N$ and describe an iterative procedure to redefine $R^{k,l}_j, b^{k,l}_j$ for all $l$ with \BBB $k \ge \kappa_l$ \EEE and $j \in \Bbb N$. Let $\tilde{v}^1_k= {v}^1_k$ as defined in \eqref{rig-eq: comp13} and assume \BBB $\tilde{v}^l_k$ with corresponding \EEE $\tilde{R}^{k,l}_j, \tilde{b}^{k,l}_j$ have been chosen (which possibly differ from $R^{k,l}_j, b^{k,l}_j$) such that \eqref{rig-eq: comp11} still holds possibly passing to a larger constant \BBB $\tilde{C}_l= \tilde{C}_l(\Omega,M,\eta,l)$. Let $I^{k,l}_1 = \lbrace j: |P^{k,l+1}_{j} \cap P^{k,l}_{j}| \ge 4\bar{C}\rho_l\rbrace$ and $I^{k,l}_2 = \Bbb N \setminus I^{k,l}_1$. \EEE Define
\begin{align}\label{new3}
\begin{split}
&\tilde{R}^{k,l+1}_{j} = \tilde{R}^{k,l}_{j}, \ \ \ \ \ \ \tilde{b}^{k,l+1}_{j} = \tilde{b}^{k,l}_{j} \ \ \ \ \ \, \text{ for } j \in I^{k,l}_1,\\
&\tilde{R}^{k,l+1}_{j} = R^{k,l+1}_{j}, \ \ \ \tilde{b}^{k,l+1}_{j} = b^{k,l+1}_{j} \ \ \ \text{ for } j \in I^{k,l}_2.
\end{split}
\end{align}
\BBB Consider $j \in I^{k,l}_1$ and define $R_j' = {R}^{k,l+1}_{j} - \tilde{R}^{k,l}_{j}$, $b_j' = {b}^{k,l+1}_{j} - \tilde{b}^{k,l}_{j}$, $P'_j = P^{k,l+1}_{j} \cap P^{k,l}_{j} \cap \Omega^l_k \cap \Omega_k^{l+1}$ for shorthand. By the triangle inequality, \eqref{rig-eq: approx en} and \eqref{rig-eq: comp13} we get
\begin{align}\label{eq:NEW}
\begin{split}
\Vert R_j' \cdot + b_j' \Vert_{L^2(P'_j)} &\le \sqrt{\varepsilon_k}(\Vert \tilde{v}^l_k \Vert_{L^2(\Omega)} + \Vert {v}^{l+1}_k \Vert_{L^2(\Omega)}) + \Vert y^l_k - y^{l+1}_k \Vert_{L^2(\Omega^l_k \cap \Omega_k^{l+1})} \\
& \le \sqrt{\varepsilon_k} (\tilde{C}_l + \hat{C}_{l+1} + C\sqrt{\rho_l}+ C\sqrt{\rho_{l+1}}) \le C'_l \sqrt{\varepsilon_k}
\end{split}
\end{align}
for a constant $C'_l = C'_l(\Omega,M,\eta, l)$, where in the penultimate step we used that \eqref{rig-eq: comp11} holds for $\tilde{v}^l_k$ and ${v}^{l+1}_k$. Herefrom we now derive $|R_j'| \le C'_l \sqrt{\varepsilon_k}$. Indeed, if $R_j' \neq 0$, then $R_j'$ is invertible and a short computation yields
\begin{align}\label{eq:NEW2}
\tfrac{1}{\sqrt{2}}|R_j'|\Vert \cdot - z\Vert_{L^2(P'_j \setminus B_\lambda(z))} \le \Vert R_j' \cdot + b_j' \Vert_{L^2(P'_j)} \le C'_l \sqrt{\varepsilon_k},
\end{align}
where $z:= - (R_j')^{-1}b_j'$ and $B_\lambda(z)$ denotes the ball with center $z$ and radius $\lambda = (\pi^{-1}\bar{C}\rho_l)^{1/2}$. Then by definition of $I^{k,l}_1$ and $|\Omega \setminus \Omega_k^l| \le \bar{C}\rho_l$ we find $|P_j' \setminus B_\lambda(z)| \ge |P^{k,l+1}_{j} \cap P^{k,l}_{j}| - |\Omega \setminus \Omega_k^l| - |\Omega \setminus \Omega_k^{l+1}| - |B_\lambda(z)| \ge \bar{C}\rho_l$, which together with \eqref{eq:NEW2} implies the claim for $C'_l$ sufficiently large.
Recalling \eqref{eq:NEW} we then also find $|b_j'| \le C'_l \sqrt{\varepsilon_k}$ and summing over all components we derive
$$\sum_{j \in I^{k,l}_1} \Big( \Vert R_j' \Vert^2_{L^2(P_j^{k,l+1})} + \Vert R_j' \Vert^4_{L^4(P_j^{k,l+1})} + \Vert R_j' \, \cdot + b_j'\Vert^2_{L^2(P_j^{k,l+1})} \Big) \le \# I^{k,l}_1 C'_l \varepsilon_k \le \frac{|\Omega|C'_l}{4\bar{C}\rho_l} \varepsilon_k,$$
where in the last step we used the definition of $I^{k,l}_1$. Define $\tilde{v}_k^{l+1}$ as in \eqref{rig-eq: comp13} with $\tilde{R}^{k,l+1}_j, \tilde{b}^{k,l+1}_j$ instead of ${R}^{k,l+1}_j, {b}^{k,l+1}_j$. The previous estimate together with the fact that \eqref{rig-eq: comp11} holds for ${v}^{l+1}_k$ now shows \eqref{rig-eq: comp11} for $\tilde{v}_k^{l+1}$. Indeed, the estimates for $\Vert \tilde{v}^{l+1}_k \Vert_{L^2(\Omega)}$, $\Vert \nabla \tilde{v}^{l+1}_k \Vert_{L^2(\Omega)}$ follow directly and for $\Vert e (\nabla \tilde{v}^{l+1}_k) \Vert_{L^2(\Omega)}$ we argue as in Remark \ref{rem: rig}(ii). \EEE
\BBB Note that as $(\hat{C}_l)_l$ also $(\tilde{C}_l)_l$ converges to infinity. For simplicity the modified functions and rigid motions will still be denoted by $v^l_k$, $R_j^{k,l}$ and $b_j^{k,l}$ in the following.
By a diagonal argument we can choose a further (not relabeled) subsequence of $(\varepsilon_k)_k$ such that the modifications $v^l_k$ exist for all $l \in \Bbb N$ and $k \ge l$\EEE.
\smallskip
\BBB\emph{Step V:} \EEE We define $ \BBB A^n_{k,l} \EEE = \bigcap_{n \le m \le l} \lbrace |v^m_k - v^n_k| \le \frac{1}{2} \rbrace$ for all $n \in \Bbb N$ and $n \le l \le k$. If we show
\begin{align}\label{rig-eq: Akl2}
|\Omega \setminus A^n_{k,l}| \le C2^{-n},
\end{align}
then \eqref{rig-eq: Asetprep} follows. Indeed, for given $l\ge n$ we can choose $K=K(l)\ge l$ so large that $|\lbrace |v^m_K - v^m| > \frac{1}{4} \rbrace| \le 2^{-m}$ for all $n \le m \le l$ since $v^m_k \to v^m$ in measure for $k \to \infty$. This implies
$$\big|\Omega \setminus \bigcap\nolimits_{n \le m \le l} \lbrace |v^m -v^n|\le 1 \rbrace\big|\le |\Omega \setminus A^n_{K,l}| + \sum\nolimits_{n \le m \le l} |\lbrace |v^m_K - v^m| > \text{\scriptsize $\frac{1}{4}$} \rbrace| \le C2^{-n}.$$
Passing to the limit $l \to \infty$ we find $|\Omega \setminus \bigcap\nolimits_{ m \ge n} \lbrace |v^{m} -v^n| \le 1 \rbrace| \le C2^{-n}$ and taking the union over all $n \in \Bbb N$ we derive \eqref{rig-eq: Asetprep}.
To show \eqref{rig-eq: Akl2} we proceed in two steps. Employing the redefinition of the piecewise rigid motions we first show that the set where $T^{m}_k, n \le m \le l$, differ is small. Afterwards, we use \eqref{rig-eq: approx en} to find that the set where $y^{m}_k, n \le m \le l$, differ is small. We define $ \BBB B^n_{k,l} \EEE = \bigcap_{n \le m \le l} \lbrace T^m_k = T^n_k \rbrace$ for $k \ge l \ge n$ and prove that
\begin{align}\label{rig-eq: Bkl}
|\Omega \setminus B^n_{k,l}| \le C2^{-n}
\end{align}
for all $k \ge l \ge n$. To this end, consider $\lbrace T^m_k = T^{m+1}_k \rbrace$ for $n \le m \le l-1$ and first note that by \eqref{rig-eq: subsubseq} we have $\sum_j |P_j^{k, m+1} \triangle P_j^{k,m}| \le 3 \cdot 2^{-m}$. Define $J_1 \subset \Bbb N$ such that \BBB $|P_j^{k,m+1}| \le 8\bar{C}\rho_m$ \EEE for all $j \in J_1$ and let $J_2 \subset \Bbb N \setminus J_1$ such that $|P^{k, m+1}_j \cap P^{k, m}_j| > \frac{1}{2} |P_j^{k, m+1}|$ for all $j \in J_2$. Observe that $|P_j^{k, m+1}| \le 2 |P_j^{k, m+1} \setminus P_j^{k,m}|$ for $j \in J_3 := \Bbb N \setminus (J_1 \cup J_2)$. \BBB Using the isoperimetric inequality and the the fact that $(\rho_m)_m \subset (2^{- 3m}\rho_0)_m$ we find by \eqref{eq: new last}
\begin{align*}
\sum_{j \in J_1} |P_j^{k,m+1}| \le (8\bar{C}\rho_m)^{\frac{1}{2}} \sum_{j \in J_1} |P_j^{k,m+1}|^{\frac{1}{2}} \le C2^{-m}\sum\nolimits_j \mathcal{H}^1(\partial^* P_j^{k,m+1})\le C2^{-m}.
\end{align*}
\EEE Due to the above construction of the rigid motions \BBB (see \eqref{new3}) \EEE we obtain $\lbrace T^m_k = T^{m+1}_k \rbrace \supset \bigcup_{j \in J_2} (P^{k,m+1}_j \cap P^{k,m}_j)$ and therefore
\begin{align*}
|\Omega \setminus \lbrace T^m_k = T^{m+1}_k \rbrace| & \le \sum\nolimits_{j \in J_2} |P_j^{k, m+1} \setminus P_j^{k, m}| + \sum\nolimits_{j \in J_1 \cup J_3} |P_j^{k, m+1}| \\
&\le \sum\nolimits_{j \in J_2} |P_j^{k, m+1} \setminus P_j^{k, m}| + \sum\nolimits_{j \in J_3} 2|P_j^{k, m+1} \setminus P_j^{k, m}| \BBB + C2^{-m} \EEE \\
& \le \BBB 2\sum\nolimits_j |P_j^{k, m+1} \triangle P_j^{k,m}| \EEE + C2^{-m} \le C2^{-m}.
\end{align*}
Summing over $n \le m \le l-1$ we establish \eqref{rig-eq: Bkl}. Now recalling \eqref{rig-eq: approx en}, \eqref{rig-eq: Bkl}, $|\Omega \setminus \Omega_k^l| \le \bar{C}\rho_{l}$ and the fact that $(\rho_l)_l \subset (2^{- 3l}\rho_0)_l$ we find
$$|\Omega \setminus A^n_{k,l}| \le |\Omega \setminus B^n_{k,l}| + \sum\nolimits_{n \le m \le l-1} |\lbrace |y^{m+1}_k - y^m_k| > 2^{-m -1} \sqrt{\varepsilon_k} \rbrace| \le C2^{-n}$$
for all $k \ge l \ge n$, as desired.
\smallskip
\BBB\emph{Step VI:} \EEE By \eqref{rig-eq: comp12} and \eqref{rig-eq: Asetprep} we can apply Lemma \ref{rig-lemma: concave function2} on the sequence $(v^l)_l$. We employ Theorem \ref{rig-th: GSBD comp} and obtain a function $v \in GSBD(\Omega)$ and a further not relabeled subsequence with $v^l \to v$ a.e in $\Omega$ and $e(\nabla v^l) \rightharpoonup e(\nabla v)$ weakly in $L^2(\Omega, \Bbb R^{2 \times 2}_{\rm sym})$.
We now select a suitable diagonal sequence such that \BBB \eqref{rig-eq: comp2}-\eqref{rig-eq: comp1-2} hold. \EEE First, we may suppose that after an infinitesimal modification we have $v_k^l \in W^{2,\infty}(\Omega \setminus \overline{J_{v^l_k}})$ (see \cite{Cortesani-Toader:1999}). Consequently, by the coarea formula \cite[ Theorem 3.40]{Ambrosio-Fusco-Pallara:2000} we get $\mathcal{H}^1(\partial^*\lbrace |\nabla v^l_k| \le \lambda\varepsilon_k^{-1/8} \rbrace) < \infty$ for all $\lambda \in (\frac{1}{2},1)\setminus H_k^l$, where $H_k^l$ is an $\mathcal{L}^1$-negligible set. Choosing $\lambda \in (\frac{1}{2},1)\setminus \bigcup_{k,l \in \Bbb N}H_k^l$ and defining $\hat{\phi}^l_k = \chi_{[0,\lambda\varepsilon_k^{-1/8}]} (|\nabla v^l_k (x)|)$, the functions $\hat{v}_k^l := \hat{\phi}_k^l v^l_k$ lie in $SBV(\Omega)$ by \cite[ Theorem 3.84]{Ambrosio-Fusco-Pallara:2000}. Recalling the definition of $\phi^l_k$ before \eqref{rig-eq: ff}, we observe that by \eqref{rig-eq: ff}, \eqref{rig-eq: ff2} the functions fulfill $\Vert e(\nabla\hat{v}_k^l)\Vert_{L^2(\Omega)} \le C$
and $\Vert \nabla \hat{v}_k^l \Vert_\infty \le \varepsilon_k^{-1/8}$ for a constant independent of $k,l \in \Bbb N$. Moreover, by \eqref{rig-eq: comp11} we get $\hat{\phi}^l_k \to 1$ in measure on $\Omega$ as $k \to \infty$.
\EEE As weak convergence in $L^2$ is metrizable on bounded sets and convergence in measure is metrizable (take $(f,g) \mapsto \int_\Omega \min\lbrace |f-g|,1\rbrace$), we can apply a diagonal sequence argument and find a not relabeled subsequence $(y_n)_{n}$ and a corresponding diagonal sequence $(w_n)_{n \in \Bbb N} \subset (\hat{v}_k^l)_{k,l} $ with corresponding partitions $(P^n_j)_j$ and piecewise rigid motions $(T_n)_n$ such that by \eqref{rig-eq: T conv3}, \eqref{rig-eq: T conv2} and \eqref{rig-eq: T conv1}
\begin{align}\label{new4}
& w_n \to v \ \text{ in measure on} \ \Omega, \ \ \ e(\nabla w_n) \rightharpoonup e(\nabla v) \text{ weakly in } L^2(\Omega), \\
& T_n \to T \text{ in } L^2(\Omega), \ \ \ \nabla T_n \to \nabla T \text{ in } L^2(\Omega), \ \ \ \BBB |P^n_j \triangle P_j| \to 0 \EEE \ \ \text{for all } j \in \Bbb N \notag
\end{align}
for $n \to \infty$. Up to a further subsequence we can assume $w_n \to v$ a.e. and $\nabla T_n \to \nabla T$ a.e. in $\Omega$. Finally, define $u_n = \nabla T_n w_n$ for all $n \in \Bbb N$ and let $u= \nabla T v$. Observe that \eqref{rig-eq: comp2}(ii), \eqref{rig-eq: comp1}, and \eqref{rig-eq: comp1-2} hold. Moreover, as $ \hat{\phi}_k^l \to 1$ in measure on $\Omega$ and $|\Omega \setminus \Omega_k^l| \to 0$ for $k,l \to \infty$, we also get \eqref{rig-eq: comp2}(i) recalling \eqref{rig-eq: approx en}, \eqref{rig-eq: comp13} and possibly passing to a further subsequence. \nopagebreak\hspace*{\fill}$\Box$\smallskip
To complete the proof of Theorem \ref{rig-th: comp1}, it remains to show \eqref{rig-eq: comp3}.
\noindent {\em Proof of Theorem \ref{rig-th: comp1}, part 2.} \BBB To see \eqref{rig-eq: comp3}(i), it suffices to recall that each $\nabla T_n^T u_n$ coincides with some $ \hat{\phi}^l_k v^l_k$ and thus $\mathbf{Id} + \sqrt{\varepsilon_n}\nabla T_n^T \nabla u_n = (\nabla T^l_k)^T\nabla y^l_k$ a.e. on $\lbrace \hat{\phi}^l_k =1 \rbrace \cap \Omega_{\rho_l}$ by \eqref{rig-eq: comp13}. The assertion then follows from \eqref{rig-eq: part + crack}(i) and the frame indifference of $W$. We now show \eqref{rig-eq: comp3}(ii). \EEE To this end, the estimate is first carried out in terms of the \BBB auxiliary \EEE functionals (see \eqref{rig-eq: Griffith en2}). Afterwards, we conclude by passing to the limit $\rho \to 0$.
\BBB Let $v$ as given in \eqref{new4} and recall $u = \nabla T v$. \EEE The sets $J_v^{\BBB d \EEE } := \lbrace x \in J_v: [v](x) = d \rbrace$ for $d \in B_1(0)\setminus \lbrace 0 \rbrace$ are pairwise disjoint with ${\cal H}^1$-$\sigma$ finite union, i.e. ${\cal H}^1(J_v^d) = 0$ up to at most countable values of $d$. Consequently, we can choose a sequence $(b_j)_j$ with $0 < |b_j| \le 1$ such that $b_i \neq b_j$ and ${\cal H}^1(J_v^{b_i - b_j}) = 0$ for $i \neq j$. Replacing $v$ by $\tilde{v} = v + \sum_j b_j \chi_{P_j}$, we thus obtain ${\cal H}^1 ( (\bigcup_j \partial^* P_j \BBB \cap \Omega ) \EEE \setminus J_{\tilde{v}}) = 0$. We first show that for \eqref{rig-eq: comp3}(ii) it suffices to prove
\begin{align}\label{rig-eq: compa1}
\liminf\nolimits_{k \to \infty} {\cal H}^1(J_{y_k} ) \ge {\cal H}^1(J_{\tilde{v}}).
\end{align}
Indeed, we get ${\cal H}^1(J_{\tilde{v}}) = {\cal H}^1(J_{u} \setminus \partial P) + {\cal H}^1(\partial P \cap \Omega)$, \BBB where for shorthand $\partial P = \bigcup_j \partial^* P_j$: \BBB We have ${\cal H}^1(J_{\tilde{v}}) = {\cal H}^1(J_{\tilde{v}} \cup (\partial P \cap \Omega)) = {\cal H}^1(\partial P \cap \Omega) + {\cal H}^1(J_{\tilde{v}} \setminus \partial P)$. Then it suffices to note ${\cal H}^1(J_{\tilde{v}} \setminus \partial P) = {\cal H}^1(J_{u} \setminus \partial P)$. \EEE
We now show \eqref{rig-eq: compa1} in two steps first passing to the limit $k \to \infty$ and then letting $l \to \infty$. We replace $v^l_k$ \BBB (see \eqref{rig-eq: comp13}) \EEE by $\tilde{v}^l_k = v^l_k + \sum_j b_j \chi_{P^{ k,l}_j}$ and $v^l$ by $\tilde{v}^l = v^l + \sum_j b_j \chi_{P^{l}_j}$ noting that $\tilde{v}^l_k \to \tilde{v}^l$ for $k \to \infty$ \BBB (cf. \eqref{rig-eq: T conv3}) \EEE and $\tilde{v}^l \to \tilde{v}$ for $l \to \infty$ in the sense of \eqref{rig-eq: convergence sense}. In the following we write $J^l_k = J_{\tilde{v}^l_k} \cap \Omega_{\rho_l}$ and $\partial P^{k,l} := \bigcup_j \partial^* P^{k,l}_j$ for shorthand, \BBB where $\Omega_{\rho_l}$ was defined before \eqref{rig-eq: Griffith en2}. \EEE We obtain by \eqref{rig-eq: comp13}, \eqref{rig-eq: part + crack}(ii) \BBB and Theorem \ref{th: local structure} \EEE
\begin{align}\label{rig-eq: comp1.2}
{\cal H}^1(J_{y_k}) + C\rho_l & \ge \BBB \int_{J_{y^l_k} \setminus \partial P^{k,l}} f^{\rho_l}_{\varepsilon_k}(|[y^l_k]|) \, d{\cal H}^1 + \mathcal{H}^1( \partial P^{k,l} \cap \Omega_{\rho_l}) \EEE \\
& \ge \int_{J^l_k\setminus \partial P^{k,l}} \theta_{\rho_l}(|[\tilde{v}^l_k]|) \, d{\cal H}^1 + \mathcal{H}^1( \partial P^{k,l} \cap \Omega_{\rho_l}) \ge \int_{J^l_k} \theta_{\rho_l}(|[\tilde{v}^l_k]|) \, d{\cal H}^1,\notag
\end{align}
where $\theta_{\sigma}(\BBB t \EEE ) := \min\lbrace \frac{t}{\sigma},1\rbrace$ for $\sigma>0$. We cannot directly apply lower semicontinuity results for $GSBD$ functions due to the involved function $\theta_{\rho_l}$. We therefore pass to the limit $k \to \infty$ on one-dimensional sections.
Recall the measure $\hat{\mu}^{\sigma,\xi}_{\tilde{v}^l}$ defined in \eqref{rig-eq: lemma2**} for $\sigma \ge 0$. By Lemma \ref{rig-lemma: lemma} we have
$$\hat{\mu}^{\sigma,\xi}_{\tilde{v}^l}(U) \le \liminf_{k \to \infty} \hat{\mu}^{\sigma,\xi}_{\tilde{v}^l_k}(U) $$
for all $\sigma\ge 0$, $\xi \in S^1$ and for every open set $U \subset \Omega$. Let $\kappa_1 = \int_{S^1} |\nu \cdot \xi| \,d{\cal H}^1(\xi)$ for some $\nu \in S^1$ which clearly does not depend on the particular choice of $\nu$. Using Fatou's lemma and \eqref{rig-eq: lemma2} we compute
\begin{align*}
\liminf_{k\to \infty} {\cal H}^1(J_{y_k}) &+ C\rho_l \ge \liminf_{k \to \infty} \int_{J^l_k} \theta_\sigma(|[\tilde{v}^l_k]|) \, d{\cal H}^1 \\
& \ge \kappa_1^{-1} \int_{S^1} \liminf_{k \to \infty} \int_{J^l_k} \theta_\sigma(|[\tilde{v}^l_k](x)|) |\nu_{\tilde{v}^l_k}(x) \cdot \xi| \, d{\cal H}^1(x) \, d{\cal H}^1(\xi) \\
& \ge \kappa_1^{-1} \int_{S^1} \liminf_{k \to \infty} \hat{\mu}^{\sigma,\xi}_{\tilde{v}^l_k}(\Omega_{\rho_l}) \, d{\cal H}^1(\xi) \ge \kappa_1^{-1} \int_{S^1} \hat{\mu}^{\sigma,\xi}_{\tilde{v}^l}(\Omega_{\rho_l}) \, d{\cal H}^1(\xi).
\end{align*}
We pass to the limit $l \to \infty$ (i.e. $\rho_l \to 0$) and obtain \BBB by Lemma \ref{rig-lemma: lemma} and \EEE the dominated convergence theorem
$$\liminf_{k\to \infty} {\cal H}^1(J_{y_k}) \ge \kappa_1^{-1} \int_{S^1} \hat{\mu}^{\sigma,\xi}_{\tilde{v}}(\Omega) \, d{\cal H}^1(\xi).$$
Recall that $\theta_\sigma \to 1$ pointwise for $\sigma \to 0$. Now letting $\sigma \to 0$ we obtain by the dominated convergence theorem and \eqref{rig-eq: lemma2}
\begin{align*}
\liminf_{k \to \infty} {\cal H}^1(J_{y_k}) & \ge \kappa_1^{-1} \int_{S^1} \hat{\mu}^{0,\xi}_{\tilde{v}}(\Omega) \, d{\cal H}^1(\xi) \\
& = \kappa_1^{-1} \int_{S^1} \int_{J^\xi_{\tilde{v}}} |\nu_{\tilde{v}}(x) \cdot \xi| \, d{\cal H}^1(x)\, d{\cal H}^1(\xi) = {\cal H}^1( J_{\tilde{v}} ).
\end{align*}
This gives \eqref{rig-eq: compa1} and completes the proof. \nopagebreak\hspace*{\fill}$\Box$\smallskip
\begin{rem}\label{rem: NNNN}
{\normalfont
Using \eqref{rig-eq: comp2}(ii), \eqref{rig-eq: comp3}(i) and arguing as in \eqref{rig-eq: ff}, \eqref{rig-eq: ff2}, we observe that all sequences $(u_k, \mathcal{P}_k, T_k)$ in Definition \ref{def:conv} satisfy $\Vert e(\nabla T_k^T \nabla u_k) \Vert_{L^2(\Omega)} \le C$ for $C>0$ only depending on $\sup_k E_{\varepsilon_k}(y_k)$, $\Omega$, and the constant in \eqref{eq:W}.}
\end{rem}
At the end of this section we briefly note that our compactness result provides an alternative proof of the piecewise rigidity result given in Theorem \ref{rig-cor: cgp} (at least in a planar setting).
\noindent {\em Proof of Theorem \ref{rig-cor: cgp} for $d=2$.} Let $y \in SBV(\Omega)$ with ${\cal H}^1(J_y) < \infty$ as well as $\int_\Omega \operatorname{dist}^2(\nabla y,SO(2)) = 0$ be given. \BBB First, assume $y \in L^\infty(\Omega)$. \EEE Define an arbitrary infinitesimal sequence $(\varepsilon_k)_k$ and the sequence $y_k = y$ for all $k \in \Bbb N$. Applying Theorem \ref{rig-th: comp1} we obtain piecewise rigid motions $T,T_k$ such that $T_k \to T$, $\nabla T_k \to \nabla T$ in $L^2(\Omega)$ by \eqref{rig-eq: comp1} up to passing to a subsequence. Moreover, $y_k - T_k \to 0$ a.e. in $\Omega$ for $k \to \infty$ by \eqref{rig-eq: comp2}(i). This implies $y = T$ is a piecewise rigid motion. \BBB If $y \notin L^\infty(\Omega)$, using the $BV$ coarea formula we can approximate $y$ by a sequence $y \chi_{\Phi_k} + \mathbf{id}\chi_{\Omega \setminus \Phi_k} \in SBV(\Omega) \cap L^\infty(\Omega)$ with $\sup_{k} {\cal H}^1(\partial^* \Phi_k )<\infty$, $|\Phi_k| \to 0$ for $k \to \infty$ and conclude by Theorem \ref{th: piecewise const}. \EEE \nopagebreak\hspace*{\fill}$\Box$\smallskip
\section{Admissible \BBB and \EEE coarsest partitions and limiting configurations}\label{rig-sec: sub, comp2}
In this section we will prove Theorem \ref{rig-th: comp2}. Let $(y_k)_k$ be a (sub-)sequence as considered in Theorem \ref{rig-th: comp1}. Recall Definition \ref{rig-def: ad,coar}. For notational convenience we will drop the dependence of $(y_k)_k$ in the sets ${\cal Z}_P, {\cal Z}_u, {\cal Z}_T$. We introduce a partial order on the admissible partitions ${\cal Z}_P$: Given two partitions ${\cal P}^1 :=(P^1_j)_j, {\cal P}^2:=(P^2_j)_j$ in ${\cal Z}_P$ we say ${\cal P}^2 \ge {\cal P}^1$ if \BBB
\begin{align}\label{new10}
\text{for all \ $P^1_{j_1}$ \ there exists \ $P^2_{j_2}$ \ such that \ $|P^1_{j_1} \setminus P^2_{j_2}|=0$.}
\end{align}
Note that Theorem \ref{th: local structure} implies $\bigcup_j \partial^* P^1_j \supset \bigcup_j \partial^* P^2_j$ \EEE up to an ${\cal H}^1$-negligible set. We observe that if ${\cal P}^1 \ge {\cal P}^2$ and ${\cal P}^2 \ge {\cal P}^1$, abbreviated by ${\cal P}^1 = {\cal P}^2$ hereafter, then the Caccioppoli partitions coincide: After a possible reordering of the sets we find $|P^1_j \triangle P^2_j| = 0$ for all $j \in \Bbb N$.
We begin with the observation that the piecewise rigid motion is uniquely determined in the limit.
\begin{lemma}\label{rig-lemma: comp2.5}
Let $(y_k)_k$ be a (sub-)sequence as considered in Theorem \ref{rig-th: comp1}. Then there is a unique $T \in {\cal Z}_T$.
\end{lemma}
\par\noindent{\em Proof. } Assume there are $T,\hat{T} \in {\cal Z}_T$. Let $(u,{\cal P},T), (\hat{u},\hat{\cal P},\hat{T}) \in {\cal D}_\infty$
according to Definition \ref{rig-def: ad,coar}(ii) and \BBB let $(u_k,{\cal P}_k,T_k), (\hat{u}_k,\hat{\cal P}_k,\hat{T}_k) \in {\cal D}$ for $k \in \Bbb N$ be triples given by Definition \ref{def:conv}. \EEE As $u_k - \hat{u}_k - (\varepsilon_k^{-1/2} (T_k - \hat{T}_k)) \to 0$ a.e. by \eqref{rig-eq: comp2}(i) and $u_k - \hat{u}_k$ converges pointwise a.e. (and the limits lie in $\Bbb R$ a.e.) by \eqref{rig-eq: comp1-2}(i), we get $T_k - \hat{T}_k \to 0$ pointwise almost everywhere. \BBB By \eqref{rig-eq: comp1}(ii) \EEE this implies $T = \hat{T}$. \nopagebreak\hspace*{\fill}$\Box$\smallskip
From now on $T$ will always stand for the rigid motion given by Lemma \ref{rig-lemma: comp2.5}.
\subsection{Equivalent characterization of the coarsest partition}
We state a lemma giving an equivalent characterization of the coarsest partition (recall Definition \ref{rig-def: ad,coar}(iv)).
\begin{lemma}\label{rig-lemma: comp2.1}
Let $(y_k)_k$ be a (sub-)sequence as considered in Theorem \ref{rig-th: comp1}. Then ${\cal P} \in {\cal Z}_P$ is coarsest if and only if it is a maximal element in the partial order $({\cal Z}_P, \ge)$, i.e. $\hat{\cal P} \ge {\cal P} $ implies $\hat{\cal P} = {\cal P} $.
\end{lemma}
\par\noindent{\em Proof. } (1) Assume ${\cal P} = (P_j)_j$ was not coarsest. According to Definition \ref{rig-def: ad,coar}(iv) let $u$ and $(u_k, {\cal P}_k, T_k)\in {\cal D}$ be given such that $(u,{\cal P},T) \in {\cal D}_\infty$ and \eqref{rig-eq: comp2}-\eqref{rig-eq: comp3} hold. Without restriction, possibly passing to a subsequence \BBB and reordering the partition, \EEE we assume that $ \varepsilon_k^{-1/2} \big(|R^k_1 - R^k_2| + |b_1^k - b_2^k| \big) \le C$ for all $k \in \Bbb N$ (cf. \eqref{rig-eq: toinfty}). By \eqref{rig-eq: linearization} we obtain $A^k \in \Bbb R^{2 \times 2}_{\rm skew}$ for $k \in \Bbb N$ with $|A^k| \le C$ such that $R_2^k - R^k_1 = R_1^k ((R_1^k)^T R^k_2 - \mathbf{Id}) = R_1^k (\sqrt{\varepsilon_k} A^k + O(\varepsilon_k))$. Passing to a (not relabeled) subsequence we then obtain for all $x \in \Omega$
\begin{align}\label{rig-eq: Tdef}
\begin{split}
S(x) & := \lim_{k \to \infty} \frac{1}{\sqrt{\varepsilon_k}} \big((R^k_2 - R^k_1)\,x + b_2^k - b_1^k \big) \\
& = \lim_{k \to \infty} \frac{1}{\sqrt{\varepsilon_k}} \big( \sqrt{\varepsilon_k} R^k_1 A^k\,x + b_2^k - b_1^k \big) + O(\sqrt{\varepsilon_k}) = R A\,x +b
\end{split}
\end{align}
for some $A \in \Bbb R^{2 \times 2}_{\rm skew}$, $b \in \Bbb R^2$ and $R = \lim_{k \to \infty} R^k_1$. We now introduce $\hat{ \cal P}_k$, $\hat{ \cal P}$, $\hat{T}_k$, $\hat{u}_k, \hat{u}$ as follows. Let $\hat{P}^k_1 = P^k_1 \cup P^k_2$, $\hat{P}^k_2 = \emptyset$, $\hat{P}^k_j = P^k_j$ for $j \ge 3$ and likewise for the limiting partition $\hat{\cal P}$. Let $\hat{T}_k(x) = R^k_1 \, x + b^k_1$ for $x \in \hat{P}^k_1$ and $\hat{T}_k(x) = T_k(x)$ for $x \in \Omega \setminus \hat{P}^k_1$. Furthermore, we let
$$\hat{u}_k = u_k + \frac{1}{\sqrt{\varepsilon_k}} \big((R^k_2 - R^k_1)\cdot + b_2^k - b_1^k \big) \chi_{P^k_2}$$
and $\hat{u} = u + (RA\cdot+b) \chi_{P_2}$ (see \eqref{rig-eq: Tdef}). \BBB We now show that $(\hat{u}_k, \hat{\cal P}_k, \hat{T}_k)$ converges to $(\hat{u},\hat{\cal P},T)$ in the sense of \eqref{rig-eq: comp2}-\eqref{rig-eq: comp3}. First, \EEE \eqref{rig-eq: comp2}(i) clearly holds since $\hat{T}_k - T_k = \big((R^k_1 - R^k_2)\cdot+ b_1^k - b_2^k \big)\chi_{P_2^k}$. Moreover, we derive that \eqref{rig-eq: comp1} holds as $|R^k_1 - R^k_2| + |b_1^k - b_2^k| \to 0$ for $k \to \infty$. Since ${\cal H}^1(J_u \setminus \bigcup_j \partial^* P_j) + {\cal H}^1(\bigcup_j \partial^* P_j \cap \Omega) \ge {\cal H}^1(J_{\hat{u}} \setminus \bigcup_j \partial^* \hat{P}_j) + {\cal H}^1(\bigcup_j \partial^* \hat{P}_j \cap \Omega)$, also \eqref{rig-eq: comp3}(ii) is satisfied.
\BBB As $|R^k_2 - R^k_1| \le C\sqrt{\varepsilon_k}$, we note that $\Vert \nabla \hat{u}_k\Vert_{L^\infty(\Omega)} \le c\varepsilon_k^{-1/8} $ for $c>0$ large enough and thus \eqref{rig-eq: comp2}(ii) holds. \EEE It remains to verify \eqref{rig-eq: comp1-2} and \eqref{rig-eq: comp3}(i). First, \eqref{rig-eq: comp1-2}(i) follows from \eqref{rig-eq: Tdef} and the definition of $\hat{u}$. We use $R^k_2 = R_1^k + \sqrt{\varepsilon_k} R^k_1 A^k + O(\varepsilon_k)$, $|A^k|\le C$ and $\Vert \nabla u_k \Vert_{L^\infty(\Omega)}\le c\varepsilon_k^{-1/8}$ to find a.e. on $P_2^k$
\begin{align}\label{rig-eq: bar-nonbar}
\nabla \hat{T}_k^T \nabla \hat{u}_k = (R_1^k)^T \nabla u_k + A^k + O(\sqrt{\varepsilon_k}) = (R_2^k)^T \nabla u_k + A^k + O(\varepsilon^{3/8}_k).
\end{align}
Now we get
\begin{align*}
\chi_{\hat{P}^k_1} e(\nabla \hat{T}_k^T \nabla \hat{u}_k) & = \sum\nolimits_{j=1,2}\chi_{P^k_j} e((R^k_j)^T \nabla u_k) + \chi_{P^k_2} e( A^k) + \BBB O(\varepsilon^{3/8}_k) \EEE \\
& \rightharpoonup \sum\nolimits_{j=1,2} \chi_{P_j} e(R_j^T \nabla u) = \chi_{\hat{P}_1} e(R^T\nabla \hat{u}) \BBB = \chi_{\hat{P}_1} e(\nabla T^T\nabla \hat{u}) \EEE
\end{align*}
weakly in $L^2(\Omega,\Bbb R^{2 \times 2}_{\rm sym}).$ \BBB This gives \eqref{rig-eq: comp1-2}(ii). By the assumptions on $W$, a Taylor expansion yields \EEE $W(G) = \frac{1}{2} Q( e(G-\mathbf{Id})) + \omega_{\rm \BBB W \EEE}(G-\mathbf{Id})$ for $G \in \Bbb R^{2 \times 2}$, where $\sup\lbrace |F|^{-3}\omega_W(F): |F| \le 1\rbrace \le C$ and $Q = D^2 W(\mathbf{Id})$. Thus, we obtain by \eqref{rig-eq: bar-nonbar} and \BBB $\Vert \nabla u_k \Vert_{L^\infty(\Omega)}\le c\varepsilon_k^{-1/8}$ \EEE
\begin{align*}
\frac{1}{\varepsilon_k} \int_{P_2^k} \hspace{-0.1cm} W(\mathbf{Id} + \sqrt{\varepsilon_k} \nabla \hat{T}_k^T \nabla \hat{u}_k) & = \int_{P_2^k} \Big(\frac{1}{2}Q(e(\nabla \hat{T}_k^T \nabla \hat{u}_k)) + \frac{1}{\varepsilon_k}\omega_W(\sqrt{\varepsilon_k}\nabla\hat{T}_k^T \nabla \hat{u}_k) \Big) \\
& = \int_{P_2^k} \hspace{-0.1cm} \Big(\frac{1}{2}Q(e(\nabla T_k^T \nabla u_k)) + \frac{\omega_W(\sqrt{\varepsilon_k} \nabla \hat{u}_k)}{\varepsilon_k}\Big) + O(\varepsilon_k^{\frac{\BBB 1 \EEE}{4}})
\end{align*}
and likewise
\begin{align*}
\frac{1}{\varepsilon_k} \int_{P_2^k} W (\mathbf{Id} + \sqrt{\varepsilon_k} \nabla T_k^T \nabla u_k) = \int_{P_2^k} \Big(\frac{1}{2}Q(e(\nabla T_k^T \nabla u_k)) + \frac{1}{\varepsilon_k}\omega_W(\sqrt{\varepsilon_k} \nabla u_k)\Big).
\end{align*}
In both estimates the second terms converge to $0$ using $\Vert \nabla u_k \Vert_\infty + \Vert \nabla \hat{u}_k \Vert_\infty \le c\varepsilon_k^{-1/8} $ and arguing as in \eqref{rig-eq: ff2}. Consequently, we get
\begin{align}\label{rig-eq: bar-nonbar2}
\frac{1}{\varepsilon_k} \int_{P_2^k} W(\mathbf{Id} + \sqrt{\varepsilon_k} \nabla \hat{T}_k^T \nabla \hat{u}_k) = \frac{1}{\varepsilon_k} \int_{P_2^k} W(\mathbf{Id} + \sqrt{\varepsilon_k} \nabla T_k^T \nabla u_k) + o(1)
\end{align}
for $\varepsilon_k \to 0$, i.e. \eqref{rig-eq: comp3}(i) holds. Therefore, $\hat{\cal P}$ is an admissible partition and thus ${\cal P}$ is not maximal.
\smallskip
(2) Conversely, assume that ${\cal P} = (P_j)_j$ was not maximal, i.e. we find $\hat{\cal P} = (\hat{P}_j)_j$ with $\hat{\cal P} \ge {\cal P}$, $\hat{\cal P} \neq {\cal P}$. \BBB Upon reordering \EEE we may assume that $ P_1 \cap \hat{P}_1$ and $P_2 \cap \hat{P}_1$ have positive ${\cal L}^2$-measure. According to Definition \ref{rig-def: ad,coar}(i) let $u,\hat{u}$ and $(u_k, {\cal P}_k,T_k), (\hat{u}_k, \hat{\cal P}_k, \hat{T}_k) \in {\cal D}$ be given such that $(u,{\cal P},T), (\hat{u},\hat{\cal P},T) \in {\cal D}_\infty$ and \eqref{rig-eq: comp2}-\eqref{rig-eq: comp3} hold. As by \eqref{rig-eq: comp1-2}(i) $u_k$ and $\hat{u}_k$ convergence pointwise a.e., by \eqref{rig-eq: comp2} also $ \varepsilon_k^{-1/2}(T_k - \hat{T}_k)$ converges pointwise a.e. (and the limits lie in $\Bbb R$ a.e.). But this implies $ \varepsilon_k^{-1/2} \big(|R^k_j - \hat{R}^k_1| + |b_j^k - \hat{b}_1^k| \big) \le C$ for $j=1,2$ and $k \in \Bbb N$. \BBB Then $ \varepsilon_k^{-1/2}\big(|R^k_1 - {R}^k_2| + |b_1^k - {b}_2^k| \big) \le C$ by the triangle inequality. Thus, \EEE \eqref{rig-eq: toinfty} is violated and ${\cal P}$ is not a coarsest partition. \nopagebreak\hspace*{\fill}$\Box$\smallskip
The alternative characterization now directly implies that there is at most one coarsest partition.
\begin{lemma}\label{rig-lemma: comp2.2}
Let $(y_k)_k$ be a (sub-)sequence as considered in Theorem \ref{rig-th: comp1}. Then there is at most one maximal element in $({\cal Z}_P, \ge)$.
\end{lemma}
\par\noindent{\em Proof. } Assume there are two maximal elements ${\cal P}^1 =(P^1_j)_j,{\cal P}^2 = (P^2_j)_j \in {\cal Z}_P$ with ${\cal P}^1 \neq {\cal P}^2$. As before, without restriction we may assume that $P^1_1 \cap P^2_1$ and $P^1_2 \cap P^2_1$ have positive ${\cal L}^2$-measure. We proceed as in the proof of Lemma \ref{rig-lemma: comp2.1}(2) to see that ${\cal P}^1$ is not coarsest and thus not a maximal element in $({\cal Z}_P, \ge)$. \nopagebreak\hspace*{\fill}$\Box$\smallskip
\subsection{Admissible configurations}
We now analyze the admissible configurations if the partitions are given. Recall the definition of the set of piecewise infinitesimal rigid motions ${\cal A}({\cal P})$ below \eqref{rig-eq: defA}.
\begin{lemma}\label{rig-lemma: comp2.4}
Let $(y_k)_k$ be a (sub-)sequence as considered in Theorem \ref{rig-th: comp1} and \BBB let $T \in {\cal Z}_T$ be the unique mapping given by Lemma \ref{rig-lemma: comp2.5}. \EEE Let ${\cal P},\hat{\cal P} \in {\cal Z}_P$ such that $\hat{\cal P} \ge {\cal P}$ and $\hat{u} \in {\cal Z}_u(\hat{\cal P})$. Then ${\cal Z}_u({\cal P}) = \hat{u} + \nabla T {\cal A}({\cal P})$.
\end{lemma}
\par\noindent{\em Proof. } (1) To see ${\cal Z}_u({\cal P}) \subset \hat{u} + \nabla T {\cal A}({\cal P})$, we have to show that $u - \hat{u} \in \nabla T {\cal A}({\cal P})$ for all $u \in {\cal Z}_u({\cal P})$. To this end, consider $P_{j} \in {\cal P}$, $\hat{P}_{i} \in \hat{\cal P}$ such that $|P_{j} \setminus \hat{P}_{i}| = 0$. Let $(u_k,{\cal P}_k,T_k), (\hat{u}_k,\hat{\cal P}_k,\hat{T}_k) \in {\cal D}$ be given according to \BBB Definition \ref{def:conv}. \EEE As $u_k - \hat{u}_k$ and thus $ \varepsilon_k^{-1/2}(T_k - \hat{T}_k)$ converge pointwise a.e. \BBB by \eqref{rig-eq: comp2}(i), \eqref{rig-eq: comp1-2}(i), \EEE we find $|R^k_{j} - \hat{R}^k_{i}| + |b^k_{j} - \hat{b}^k_{i}| \le C\sqrt{\varepsilon_k}$. Repeating the argument in \eqref{rig-eq: Tdef} we find some $A \in \Bbb R^{2 \times 2}_{\rm skew}$, $b \in \Bbb R^2$ such that for a.e. $x \in P_j$
$$u(x)- \hat{u}(x) = \lim_{k \to \infty} u_k(x)- \hat{u}_k(x) = \lim_{k \to \infty} \varepsilon_k^{-1/2} (\hat{T}_k(x)- {T}_k(x)) = \nabla T(x) (A \,x + b).$$
(2) Conversely, to see ${\cal Z}_u({\cal P}) \supset \hat{u} + \nabla T{\cal A}({\cal P})$ we first consider the special case ${\cal P} = \hat{\cal P} = (P_j)_j$. Let $\bar{u} \in {\cal Z}_u( { \cal P})$ and $\bar{A} = \sum_{j} (A_j\cdot + d_j) \chi_{P_j} \BBB \in {\cal A}({\cal P}) \EEE $ be given. We have to show that $u := \bar{u}+ \nabla T \bar{A} \in {\cal Z}_u( {\cal P})$.
According to Definition \ref{rig-def: ad,coar}(iii) let $(\bar{u}_k, {\cal P}_k, \bar{T}_k) \in {\cal D}$ be given such that \eqref{rig-eq: comp2}-\eqref{rig-eq: comp3} hold \BBB with the limiting triple $(\bar{u},{\cal P},T)$. \EEE Assume that $\bar{T}_k$ has the form $\bar{T}_k = \sum_j( \bar{R}^k_j \cdot + \bar{b}^k_j) \chi_{P^k_j}$. Now choose $R^k_j$ such that $| R^k_j - \bar{R}^k_j (\mathbf{Id} - \sqrt{\varepsilon_k} A_{j})| = \operatorname{dist}(\bar{R}^k_j (\mathbf{Id} - \sqrt{\varepsilon_k} A_{j}), SO(2))$ and let $b^k_j = \bar{b}^k_j -\sqrt{\varepsilon_k} \bar{R}^k_j d_j$. \BBB By \eqref{rig-eq: linearization} we have
\begin{align}\label{new5}
R^k_j = \bar{R}^k_j - \sqrt{\varepsilon_k} \bar{R}^k_j A_{j}- \omega_{j,k} \text{ with $|\omega_{j,k}| \le C\varepsilon_k|A_j|^2$ for all $j \in \Bbb N$.}
\end{align}
Let $I_k = \lbrace j \in \Bbb N: |A_j| + |d_j| \le \varepsilon_k^{-1/8} \rbrace$ and $V_k = \bigcup_{j \in \Bbb N \setminus I_k} P_j^k$. Note that $|V_k| \to 0$ for $k \to \infty$ and $|\varepsilon_k^{-1/2}\omega_{j,k}| \le C\varepsilon^{1/4}_k$ for $j \in I_k$. \EEE Define
$$ \BBB T_k = \sum\nolimits_{j \in I_k} (R^k_j \cdot + b^k_j ) \chi_{{P}^k_j} + \bar{T}_k\chi_{V_k}, \EEE \ \ \ \ u_k = \bar{u}_k + \frac{1}{\sqrt{\varepsilon_k}}(\bar{T}_k - T_k). $$\BBB We now show that $(u_k, {\cal P}_k,T_k)$ converges to $(u,{\cal P},T)$ in the sense of \eqref{rig-eq: comp2}-\eqref{rig-eq: comp3}. First, \EEE \eqref{rig-eq: comp2}(i) clearly holds. Moreover, ${\cal H}^1(J_u \setminus \bigcup_j \partial^* P_j) = {\cal H}^1(J_{\bar{u}} \setminus \bigcup_j \partial^* P_j)$ and thus \eqref{rig-eq: comp3}(ii) is satisfied. \BBB By \eqref{new5} \EEE we find
$$ T_k = \bar{T}_k - \sum\nolimits_{\BBB j\in I_k \EEE } \big(\sqrt{\varepsilon_k} \bar{R}^k_j A_j \cdot + \omega_{j,k} \cdot + \sqrt{\varepsilon_k}\bar{R}_j^k d_{j}\big) \chi_{{P}^k_j} \to T$$
in measure for $k \to \infty$. Then it is not hard to see that $T_k \to T$ and $\nabla T_k \to \nabla T$ in $L^2(\Omega)$ which gives \eqref{rig-eq: comp1}. Likewise, we obtain
\begin{align*}
u_k - \bar{u}_k &= \frac{1}{\sqrt{\varepsilon_k}}\big(\bar{T}_k - T_k\big) = \sum\nolimits_{j \in I_k} \Big( \bar{R}^k_j (A_j\cdot + d_j) + \frac{1}{\sqrt{\varepsilon_k}}\omega_{j,k}\cdot\Big)\chi_{{P}^k_j} \\ & \to \nabla T \sum\nolimits_j (A_j \cdot + d_j)\chi_{P_j} = \nabla T \bar{A}
\end{align*}
pointwise a.e. which implies $u_k \to \bar{u} + \nabla T \bar{A}$ and shows \eqref{rig-eq: comp1-2}(i). \BBB By the definition of $V_k$ we have \EEE
$$\Vert \nabla u_k - \nabla \bar{u}_k\Vert_{L^\infty(\BBB \Omega \EEE)} \le \Vert \sum\nolimits_j \chi_{P^k_j}\big( \BBB \bar{R}_j^k A_j \EEE + \varepsilon_k^{-1/2}\omega_{j,k}\big)\Vert_{L^\infty(\Omega \setminus V_k)}\le C\varepsilon_k^{-1/8}.$$Therefore, \BBB $\Vert \nabla u_k \Vert_\infty \le c\varepsilon_k^{-1/8}$ for $c$ large enough, which shows \eqref{rig-eq: comp2}(ii). \EEE Arguing as in \eqref{rig-eq: bar-nonbar} we get by $\Vert \nabla \bar{u}_k \Vert_{L^\infty(\Omega)} \le c\varepsilon_k^{-1/8}$ and \BBB \eqref{new5}\EEE
\begin{align*}
(R_j^k)^T \nabla u_k(x) &= (R_j^k)^T \nabla \bar{u}_k(x) + \BBB (R_j^k)^T\bar{R}_j^k \EEE A_j + (R_j^k)^T \varepsilon_k^{-1/2}w_{j,k}\\
& =(\bar{R}_j^k)^T \nabla \bar{u}_k(x) + A_j + O(\varepsilon_k^{1/4})
\end{align*}
for a.e. $x \in P_j^k$, \BBB $j \in I_k$. \EEE Thus, \eqref{rig-eq: comp1-2}(ii) follows from the fact that \eqref{rig-eq: comp1-2}(ii) holds for the sequence $\bar{u}_k$ and
\begin{align*}
\sum\nolimits_{\BBB j\in I_k} \int\nolimits_{{P}^k_j} |e\big(( R^k_j)^T\nabla u_k \big)& - e\big(( \bar{R}^k_j)^T\nabla \bar{u}_k \big)|^2 \le C\varepsilon_k^{1/2} \to 0.
\end{align*}
Finally, the above estimates together with a similar argumentation as in \eqref{rig-eq: bar-nonbar2} yield \eqref{rig-eq: comp3}(i).
In the general case we have to show $u := \hat{u}+ \nabla T \bar{A} \in {\cal Z}_u( {\cal P})$ for given $\hat{u} \in {\cal Z}_u( {\cal \hat{P}})$, $\hat{\cal P} \ge {\cal P}$, and $\bar{A} \in {\cal A}({\cal P})$. As ${\cal P} \in {\cal Z}_P$, we find some $\bar{u} \in {\cal Z}_u({\cal P})$ which by (1) satisfies $\bar{u} - \hat{u} = \nabla T \hat{A}$ for some $\hat{A} \in {\cal A}({\cal P})$. Thus, we get $u = \bar{u} + \nabla T(\bar{A} - \hat{A})$ and by the special case in (2) we know that $u \in {\cal Z}_u({\cal P})$, as desired. \nopagebreak\hspace*{\fill}$\Box$\smallskip
\subsection{Existence of coarsest partitions}
To guarantee existence of coarsest partitions we show that each totally ordered subset has upper bounds such that afterwards we may apply Zorn's lemma.
\begin{lemma}\label{rig-lemma: comp2.3}
Let $(y_k)_k$ be a (sub-)sequence as considered in Theorem \ref{rig-th: comp1}. Let $I$ be an arbitrary index set and let $\lbrace {\cal P}_i = (P_{i,j})_j: i \in I\rbrace \subset{\cal Z}_P$ be a totally ordered subset, i.e. for each $i_1,i_2 \in I$ we have ${\cal P}_{i_1} \le {\cal P}_{i_2}$ or ${\cal P}_{i_2} \le {\cal P}_{i_1}$. Then there is a partition ${\cal P} \in {\cal Z}_P$ with ${\cal P}_i \le {\cal P}$ for all $i \in I$.
\end{lemma}
\par\noindent{\em Proof. } \BBB \emph{Step I:} \EEE To prove the existence of an upper bound we first show that it suffices to consider a suitable countable subset of $\lbrace{\cal P}_i: i \in I \rbrace$. For notational convenience we write $i_1 \le i_2$ for $i_1, i_2 \in I$ if ${\cal P}_{i_1} \le {\cal P}_{i_2}$. Choose an arbitrary $i_0 \in I$ and note that it suffices to find an upper bound for all $i \ge i_0$. \BBB For each $k \in \Bbb N$ we introduce partitions ${\cal P}^k_i = (P^k_{i,j})_{j \ge 0}$ consisting of the components $P^k_{i,j} = P_{i,j} \setminus \bigcup\nolimits_{l\ge k}P_{i_0,l}$ for $j \in \Bbb N$ and $P^k_{i,0} = \bigcup\nolimits_{l\ge k}P_{i_0,l}$. (Note that the partitions ${\cal P}^k_i$ are possibly not ordered.) By \eqref{new10} we get that ${\cal P}^k_{i_1} \le {\cal P}^k_{i_2}$ if $i_0 \le i_1\le i_2$. Typically, ${\cal P}^k_i$ are not elements of $\lbrace {\cal P}_i: i \in I \rbrace$, but satisfy for $i \ge i_0$
\begin{align}\label{new11}
\begin{split}
&\text{$|P_{i,j} \triangle P^k_{i,j}| \le \big|\bigcup\nolimits_{l\ge k}P_{i_0,l} \big| \le \omega(k)$ for all $j \ge 0$}
\end{split}
\end{align}
with $\omega(k) \to 0$ for $k \to \infty$, where we set $P_{i,0} = \emptyset$. For all $k \in \Bbb N$ we observe that $\lbrace {\cal P}^k_i: i \ge i_0\rbrace$ contains only a finite number of different elements and therefore contains a maximal element ${\cal P}^k = (P^k_j)_j$. Now we can choose $i_0 \le i_1 \le i_2 \le \ldots$ such that ${\cal P}^k = {\cal P}^k_{i_k}$ for $k\in\Bbb N$. It now suffices to construct an upper bound ${\cal P} = (P_j)_j \in {\cal Z}_P$ with ${\cal P} \ge {\cal P}_{i_k}$ for all $k \in \Bbb N$. Indeed, we then obtain ${\cal P} \ge {\cal P}_{i}$ for all $i_0 \le i$ as follows:
For each $P_{i,j}$ and each $k \in \Bbb N$ we find $P_{j_k}$ with $|P_{i,j} \setminus P_{j_k}| \le 2\omega(k)$. In fact, using repetitively \eqref{new10} and \eqref{new11} we get $j',j_k$ such that $|P_{i,j} \setminus P_{i,j}^k| \le \omega(k)$, $|P_{i,j}^k \setminus P^k_{i_k,j'}|=0$, $|P^k_{i_k,j'} \setminus P_{i_k,j'}| \le \omega(k)$, $|P_{i_k,j'} \setminus P_{j_k}|=0$ and thus $|P_{i,j} \setminus P_{j_k}| \le 2\omega(k)$. As $\omega(k) \to 0$ for $k \to \infty$ and ${\cal P}$ contains only a finite number of components with $\mathcal{L}^2$-measure larger than $\frac{1}{2}|P_{i,j}|$, we indeed find $P_{j_*}$ with $|P_{i,j} \setminus P_{j_*}| = 0$, as desired. \EEE
\smallskip
\BBB \emph{Step II:} \EEE Now consider the totally ordered sequence of partitions $({\cal P}_{i_k})_k$. For notational convenience we will denote the sequence by $({\cal P}_{i})_{i \in \Bbb N}$ in the following. By the compactness theorem for Caccioppoli partitions (see Theorem \ref{th: comp cacciop}) we get an (ordered) Caccioppoli partition ${\cal P}=(P_j)_j$ such that \BBB $|P_{i,j} \triangle P_j| \to 0$ \EEE for $i \to \infty$ for all $j \in \Bbb N$. Thus, for all $j \in \Bbb N$ there exists $I_j \in \Bbb N$ such that $|P_{i_1,j} \setminus P_{i_2,j}| \le \frac{1}{2}|P_{i_1,j}|$ for all $I_j \le i_1 \le i_2$. As $({\cal P}_{i})_{i \in \Bbb N}$ is totally ordered, \eqref{new10} then gives $|P_{i_1,j} \setminus P_{i_2,j}|=0$ for all $I_j \le i_1 \le i_2$ and this monotonicity yields $|P_{i_1,j} \setminus P_j|=0$ for $i_1 \ge I_j$. Eventually, fixing $P_{i,j}$ for $i,j \in \Bbb N$, by the above arguments there exists $j' \in \Bbb N$ such that $|P_{i,j} \setminus P_{i',j'}|=0$ for all $i'$ large enough and thus $|P_{i,j} \setminus P_{j'}|=0$.
This implies ${\cal P} \ge {\cal P}_i$ for all $i \in \Bbb N$ and therefore it suffices to show that ${\cal P} \in {\cal Z}_P$. To this end, we will construct partitions ${\cal P}^n$, rigid motions $T_n \in {\cal R}({\cal P}^n)$ and a limiting function $u$ by a diagonal sequence argument.
For all $i \in \Bbb N$, according to Definition \ref{rig-def: ad,coar}(i), we find $(u^k_i, {\cal P}^k_i, T^k_i) \in {\cal D}$, an admissible limiting configurations $u_i \in {\cal Z}_u( {\cal P}_i)$ and \BBB $T \in {\cal Z}_T$ as in Lemma \ref{rig-lemma: comp2.5} \EEE such that \eqref{rig-eq: comp2}-\eqref{rig-eq: comp3} hold as $k \to \infty$. The strategy is to select $u_i$ in a suitable way such that we find a limiting configuration $u \in GSBD(\Omega)$ with
\begin{align}\label{rig-eq: vi to v}
\begin{split}
&u_i \to u \ \text{ a.e. in $\Omega$, } \\ & e(\nabla T^T \nabla u_i) \rightharpoonup e(\nabla T^T \nabla u) \ \ \ \text{weakly in} \ L^2(\Omega,\Bbb R^{2\times 2}_{\rm sym}),\\ & \liminf\nolimits_{i \to \infty} {\cal H}^1(J_{u_i}) \ge {\cal H}^1(J_u).
\end{split}
\end{align}
\BBB We defer the selection of the sequence $(u_i)_i$ to Step III below. \EEE Then we can choose a diagonal sequence $(\bar{u}_n) := (u^{k(n)}_n)_n$ converging to the triple $(u,{\cal P},T)$ in the sense of \eqref{rig-eq: comp2}-\eqref{rig-eq: comp3}. Indeed, $k(n)$ can be selected such that letting $\bar{\cal P}^n = (\bar{P}^n_j)_j = {\cal P}^{k(n)}_n$ and $\bar{T}_n = T^{k(n)}_n \in {\cal R}(\bar{\cal P}^n)$, we find \BBB $|\bar{P}^n_j \triangle P_j| \to 0$ \EEE for all $j \in \Bbb N$ (even $\sum_j|\bar{P}^n_j \triangle P_j| \to 0$, cf. below Theorem \ref{th: comp cacciop}) and $\bar{T}_n \to T$, $\nabla \bar{T}_n \to \nabla T$ in $L^2(\Omega)$. This gives \eqref{rig-eq: comp1}. Moreover, as measure convergence is metrizable, this can be done in a way that $\bar{u}_n\to u$ in measure and $\bar{u}_n - \varepsilon_n^{-1/2}(y_n - \bar{T}_n) \to 0$ in measure. Then, possibly passing to a further subsequence, we can assume that the convergence also holds a.e. in $\Omega$ and thus \eqref{rig-eq: comp2}(i), \eqref{rig-eq: comp1-2}(i) are satisfied.
Likewise, \eqref{rig-eq: comp1-2}(ii) can be achieved by \eqref{rig-eq: vi to v} and the fact the weak convergence is metrizable as by Remark \ref{rem: NNNN} we get $\Vert e( (\nabla T^{k}_i)^T \nabla u^k_i) \Vert_{L^2(\Omega)} \le C$ for a constant independent of $k,i$. Moreover, \eqref{rig-eq: comp2}(ii) and \eqref{rig-eq: comp3}(i) directly follow from the corresponding estimates for the functions $u^k_i$. Finally, to see \eqref{rig-eq: comp3}(ii) it suffices to prove
\begin{align}\label{new15}
\liminf\nolimits_{i \to \infty} \big({\cal H}^1(J_{u_i} \setminus \partial P_i) + {\cal H}^1(\partial P_i\cap \Omega)\big) \ge {\cal H}^1(J_{u} \setminus \partial P) + {\cal H}^1(\partial P\cap \Omega),
\end{align}
where for shorthand $\partial P_i = \bigcup_j \partial^* P_{i,j}$ and $\partial P = \bigcup_j \partial^* P_{j}$. This can be derived arguing as in \eqref{rig-eq: compa1}: We may consider an infinitesimal perturbation of the form $\tilde{u}_i = u_i + \sum_j b_j \chi_{P_{i,j}}$, $\tilde{u} = u + \sum_j b_j \chi_{P_{j}}$ with $b_j$ small such that ${\cal H}^1(\partial P_i \setminus J_{\tilde{u}_i}) ={\cal H}^1(\partial P \setminus J_{\tilde{u}})= 0$ and the convergence in \eqref{rig-eq: vi to v} still holds after replacing $u_i$, $u$ by $\tilde{u}_i$, $\tilde{u}$, respectively. Then the claim follows from \eqref{rig-eq: vi to v}. Consequently, ${\cal P} \in {\cal Z}_P$ due to Definition \ref{rig-def: ad,coar}(i).
\smallskip
\BBB \emph{Step III:} \EEE It remains to show \eqref{rig-eq: vi to v}. Clearly, we have $\Vert e(\nabla T^T \nabla u_i)\Vert^2_{L^2(\Omega)} \le C$ and ${\cal H}^1(J_{u_i}) \le C$ for a constant independent of $i \in \Bbb N$. This follows by a lower semicontinuity argument using \eqref{rig-eq: comp3}(ii) and $\Vert e((\nabla T^{k}_i)^T \nabla u^k_i) \Vert_{L^2(\Omega)} \le C$ by Remark \ref{rem: NNNN}. Thus, in order to apply Theorem \ref{rig-th: GSBD comp}, we have to \BBB select $u_i \in {\cal Z}_u({\cal P}_i)$ suitably and to \EEE find an increasing continuous function $\psi: [0,\infty) \to [0,\infty)$ with $\lim_{\BBB t \EEE \to \infty} \psi(\BBB t \EEE ) = + \infty$ such that $\int_\Omega \psi(|u_i|) \le C$ \BBB independently of $i \in \Bbb N$. \EEE
We proceed similarly as in the proof of Theorem \ref{rig-th: comp1} and define $u_i$ iteratively. Choose $u_1 \in {\cal Z}_u({\cal P}_1)$ arbitrarily. Given $u_i$ we define $u_{i+1}$ as follows. Consider some $P_{i+1,j}$ and \BBB recalling \eqref{new10} \EEE choose $l_{1,j} <l_{2,j} < \ldots$ such that $P_{i+1,j} = \bigcup^\infty_{k=1} P_{i,l_{k,j}}$ up to an ${\cal L}^2$- negligible set (observe that the union may also be finite). Choose some $\tilde{u}_{i+1} \in {\cal Z}_u({\cal P}_{i+1})$. By Lemma \ref{rig-lemma: comp2.4} for ${\cal P} = {\cal P}_i, \hat{\cal P} = {\cal P}_{i+1}$ we get $(u_i- \tilde{u}_{i+1})\chi_{P_{i+1,j}} = \sum^\infty_{k=1} \BBB \nabla T \EEE(A_{l_{k,j}}\cdot + b_{l_{k,j}})\chi_{P_{i,l_{k,j}}}$ for $A_{l_{k,j}} \in \Bbb R^{2 \times 2}_{\rm skew}$, $b_{l_{k,j}} \in \Bbb R^2$. Now define
$$u_{i+1}(x) = \tilde{u}_{i+1}(x) + \BBB \nabla T(x) \EEE (A_{l_{1,j}}\, x + b_{l_{1,j}})$$for $x \in P_{i+1,j}$ and observe that $u_i = u_{i+1}$ on $P_{i,l_{1,j}}$. Proceeding in this way on all $P_{i+1,j}$ we find some $\tilde{A}^{i+1} \in {\cal A}({\cal P}_{i+1})$ such that $u_{i+1}:= \tilde{u}_{i+1} + \nabla T \tilde{A}^{i+1} \in {\cal Z}_u({\cal P}_{i+1})$ applying Lemma \ref{rig-lemma: comp2.4} for ${\cal P} = \hat{\cal P} = {\cal P}_{i+1}$. Moreover, there is a corresponding $A^i \in {\cal A}({\cal P}_{i})$ such that $u_{i+1} = u_i + \nabla T A^i$ with $A^i =0$ on $\bigcup_j P_{i,l_{1,j}}$.
We now show that $\sum_{i \in \Bbb N} |A^i(x)| < + \infty$ \BBB for a.e. $x\in\Omega$. \EEE To see this, we recall that \BBB $|P_{i,j} \triangle P_{j}| \to 0$ \EEE for all $j \in \Bbb N$. Consequently, as due to the total order of the partitions the sets $P_{i,j}$ are increasing for fixed $j \in \Bbb N$, the construction of the functions $(u_i)_i$ implies $A^i = 0$ on $P_{i,j}$ for $i$ so large that $ |P_{i,j}| > \frac{1}{2} |P_j|$. Thus, for a.e. $x \in P_{j}$ the sum $\sum_{i \ge 1} |A^i(x)|$ is a finite sum and therefore finite. As $j \in \Bbb N$ \BBB was arbitrary, \EEE we obtain $\sum_{i \in \Bbb N} |A^i| < + \infty$ almost everywhere.
Therefore, the function $v:= |u_1| + \sum_{l\in \Bbb N} |A^l|$ is finite a.e. in $\Omega$ and we apply Lemma \ref{rig-lemma: concave function2} on the sequence $v_k = v\chi_{\lbrace v \le k\rbrace}$ to find an increasing continuous function ${\psi}: [0,\infty) \to [0,\infty)$ with $\lim_{t \to \infty}{\psi}(t) = \infty$ such that by Fatou's lemma $\Vert {\psi}(v) \Vert_{L^1(\Omega)} \le \liminf_{k \to \infty} \Vert {\psi}(v_k) \Vert_{L^1(\Omega)} \le C < \infty$. Using the definition $u_{i+1} = u_i + \nabla T A^i$ and the monotonicity of ${\psi}$ we find $\Vert {\psi}(|u_i|) \Vert_{L^1(\Omega)} \le \Vert {\psi}(|u_1| + \sum_{l \in \Bbb N} |A^l|) \Vert_{L^1(\Omega)} \BBB \le C \EEE < \infty$ for all $i\in\Bbb N$, as desired. \nopagebreak\hspace*{\fill}$\Box$\smallskip
After these preparatory lemmas we are in a position to prove Theorem \ref{rig-th: comp2}.
\noindent {\em Proof of Theorem \ref{rig-th: comp2}.} First, (i) follows from Lemma \ref{rig-lemma: comp2.5}. The uniqueness of the coarsest partition is a consequence of Lemma \ref{rig-lemma: comp2.2} and Lemma \ref{rig-lemma: comp2.1}. We obtain existence by Zorn's lemma: As $({\cal Z}_P, \ge)$ is a partial order and every chain has an upper bound by Lemma \ref{rig-lemma: comp2.3}, there exists a maximal element $\bar{ \cal P} \in {\cal Z}_P$. Lemma \ref{rig-lemma: comp2.1} shows that $\bar{ \cal P}$ is a coarsest partition which gives (ii). Finally, assertion (iii), namely ${\cal Z}_u(\bar{\cal P}) = v + \nabla T {\cal A}(\bar{\cal P})$ for some $v \in {\cal Z}_u(\bar{\cal P})$, follows from Lemma \ref{rig-lemma: comp2.4} for the choice ${\cal P} = \hat{\cal P} = \bar{\cal P}$. \nopagebreak\hspace*{\fill}$\Box$\smallskip
\section{The effective linearized Griffith model}\label{rig-sec: gamma}
In this final section we identify the effective linearized Griffith functional via $\Gamma$-convergence and derive a cleavage law for the limiting model.
\subsection{Derivation of linearized models via $\Gamma$-convergence}\label{rig-sec: sub, gamma1}
We now give the proof of Theorem \ref{rig-th: gammaconv}.
\noindent {\em Proof of Theorem \ref{rig-th: gammaconv}.} (i) Thanks to the preparations in the last section the lower bound is almost immediate. Let $(u,{\cal P},T) \in {\cal D}_\infty$ be given as well as a sequence $(y_k)_k \subset SBV_M(\Omega)$ with \BBB $y_k \to (u,{\cal P},T)$, i.e. by Definition \ref{def:conv} the are triples \EEE $(u_k,{\cal P}_k,T_k) \in {\cal D}$ such that \eqref{rig-eq: comp2}-\eqref{rig-eq: comp3} hold. Due to \eqref{rig-eq: comp3}(ii) it suffices to show
$$\liminf_{k \to \infty} \frac{1}{\varepsilon_k}\int_\Omega W(\nabla y_k) \ge \int_\Omega \frac{1}{2} Q(e(\nabla T^T \nabla u) ).$$
We proceed as in \eqref{rig-eq: ff}: Recall that $W(G) = \frac{1}{2}Q( e(G-\mathbf{Id})) + \omega_{\BBB W \EEE } (G-\mathbf{Id})$ with $\sup\lbrace |F|^{-3}\omega_W(F): |F| \le 1\rbrace \le C$ by the assumptions on $W$, where $Q = D^2 W(\mathbf{Id})$. We compute by \eqref{rig-eq: comp3}(i)
\begin{align*}
\frac{1}{\varepsilon_k} \int_{\Omega} W(\nabla y_k ) & \ge \frac{1}{\varepsilon_k} \int_{\Omega} W(\mathbf{Id} + \sqrt{\varepsilon_k}\nabla T_k^T \nabla u_k ) + o(1)\\
& = \int_{\Omega} \frac{1}{2}\Big( Q(e(\nabla T_k^T \nabla u_k)) + \frac{1}{\varepsilon_k}\omega_W(\sqrt{\varepsilon_k} \nabla T_k^T\nabla u_k) \Big) + o(1)
\end{align*}
as $k \to \infty$. The second term converges to $0$ arguing as in \eqref{rig-eq: ff2} and using $\Vert \nabla u_k\Vert_\infty \le c\varepsilon_k^{-1/8}$ (see \eqref {rig-eq: comp2}(ii)). As $e( \nabla T_k^T\nabla u_k) \rightharpoonup e(\nabla T^T \nabla u)$ weakly in $L^2(\Omega,\Bbb R^{2\times 2}_{\rm sym})$ by \eqref{rig-eq: comp1-2}(ii) and $Q$ is convex, we conclude
$$\liminf_{k \to \infty} \frac{1}{\varepsilon_k} \int_{\Omega} W(\nabla y_k)\ge \int_{\Omega} \frac{1}{2} Q(e(\nabla T^T \nabla u)).$$
(ii) By a general density result in the theory of $\Gamma$-convergence together with Theorem \ref{rig-th: cortesani2} and the fact that the limiting functional $E(u,{\cal P}, T)$ is continuous in $u$ with respect to the convergence given in Theorem \ref{rig-th: cortesani2}, it suffices to provide recovery sequences for functions $u$ with $u \in W^{1,\infty}(\Omega \setminus \overline{J_u})$, where $J_u$ is contained in the union of a finite number of closed connected pieces of $C^1$- curves. Moreover, as in the proof of Theorem \ref{rig-th: comp1} \BBB (see paragraph before \eqref{rig-eq: compa1}) \EEE we may assume that ${\cal H}^1( (\bigcup_j \partial^* P_j \cap \Omega) \setminus J_{u} ) = 0$ up to an infinitesimal small perturbation of $u$ (a similar argument was used below \BBB \eqref{new15}). \EEE
Let $(u,{\cal P},T) \in {\cal D}_\infty$ and $\varepsilon_k \to 0$ be given. Define $y_k(x) = T(x) + \sqrt{\varepsilon_k} u(x)$ for all $x \in \Omega$. It is not hard to see that $(y_k)_k \subset SBV_M(\Omega)$ for $\varepsilon_k$ small enough \BBB (and $M$ not too small). \EEE Moreover, define ${\cal P}_k = {\cal P}$, $T_k= T$ and $u_k = \varepsilon_k^{-1/2} \big( y_k - T_k\big) \equiv u$ for all $k \in \Bbb N$. Then \eqref{rig-eq: comp2}(i) and \eqref{rig-eq: comp1}-\eqref{rig-eq: comp3} hold trivially. To see \eqref{rig-eq: comp2}(ii), it suffices to note that $\Vert \nabla u_k \Vert_\infty = \Vert \nabla u \Vert_\infty \le C \le C\varepsilon_k^{-1/8}$. Consequently, $y_k \to (u, {\cal P},T)$ in the sense of Definition \ref{def:conv}. \EEE
We finally confirm $\lim_{k \to \infty} E_{\varepsilon_k}(y_k) = E(u,{\cal P},T)$. As clearly $\lim_{k \to \infty}{\cal H}^1(\BBB J_{y_k} \EEE ) = {\cal H}^1(\bigcup_j\partial^* P_j\cap \Omega) + {\cal H}^1(J_u \setminus \bigcup_j\partial^* P_j)$, it suffices to show $\lim_{k \to \infty} \frac{1}{\varepsilon_k}\int_\Omega W(\nabla y_k) = \int_\Omega \frac{1}{2} Q(e(\nabla T^T \nabla u))$. Using again that $W(G) = \frac{1}{2}Q( e(G-\mathbf{Id})) + \omega_W(G-\mathbf{Id})$ \BBB and the frame indifference of $W$ \EEE we compute
\begin{align}\label{new22}
\begin{split}
\frac{1}{\varepsilon_k} \int_{\Omega} W(\nabla y_k ) & =\frac{1}{\varepsilon_k} \int_{\Omega} W( \BBB \mathbf{Id} + \sqrt{\varepsilon_k} \nabla T_k^T \nabla u_k \EEE )\\
& = \int_{\Omega} \Big( \frac{1}{2} Q(e(\nabla T_k^T \nabla u_k)) + \frac{1}{\varepsilon_k}\omega_W(\sqrt{\varepsilon_k} \nabla T_k^T\nabla u_k) \Big) \\ &= \int_{\Omega} \frac{1}{2} Q(e(\nabla T^T \nabla u)) + O(\sqrt{\varepsilon_k})\to \int_{\Omega} \frac{1}{2} Q(e(\nabla T^T \nabla u)).
\end{split}
\end{align}
This concludes the proof. \nopagebreak\hspace*{\fill}$\Box$\smallskip
The proof of Corollary \ref{rig-cor: gamma} is now straightforward.
\noindent {\em Proof of Corollary \ref{rig-cor: gamma}.} To see the liminf-inequality, assume $y_{\varepsilon_k} \to y$ in $L^1(\Omega)$ for $k \to \infty$ and without restriction that $E_{\varepsilon_k}(y_{\varepsilon_k}) \le C$. \BBB By Theorem \ref{rig-th: comp1} we find a limiting triple $(u,{\cal P},T) \in {\cal D}_\infty$ such that $y_{\varepsilon_k} \to (u,{\cal P},T)$ in the sense of Definition \ref{def:conv}. \EEE By \eqref{rig-eq: comp2}(i), \eqref{rig-eq: comp1}(ii) we obtain $y = T$. \BBB As $T \in {\cal R}({\cal P})$, we get $\mathcal{H}^1(J_T) \le \mathcal{H}^1(\bigcup_j \partial^* P_j \cap \Omega)$, where ${\cal P} = (P_j)_j$. \EEE Thus, Theorem \ref{rig-th: gammaconv}(i) yields $\liminf_{k \to \infty} E_{\varepsilon_k}(y_k) \BBB \ge E(u,{\cal P},T) \ge {\cal H}^1(J_T) \EEE = E_{\rm seg}(y)$. A recovery sequence is obviously given by $y_k = y$ for all $k \in \Bbb N$. \nopagebreak\hspace*{\fill}$\Box$\smallskip
We close this section with the proof of Lemma \ref{lemmanew} and Theorem \ref{rig-th: gammaconv2}.
\BBB
\noindent {\em Proof of Lemma \ref{lemmanew}.} Consider triples $(u_k,{\cal P}_k,T_k)$ and $(g_k,{\cal P}^g_k,T^g_k)$ such that the triples converge to $(u, {\cal P},T)$ and $(g,{\cal P}_g,T_g)$, respectively, in the sense of \eqref{rig-eq: comp2}-\eqref{rig-eq: comp3}. Since $\varepsilon_k^{-1/2}(y_k - f_k)$ is bounded in $L^2(\Omega)$ by \eqref{ennew}, $\varepsilon_k^{-1/2}(T_k - T_k^g) + (u_k - g_k) - \varepsilon_k^{-1/2}(y_k - f_k)$ converges a.e. by \eqref{rig-eq: comp2}(i), and $u_k - g_k$ converges a.e. by \eqref{rig-eq: comp1-2}(i), we get $\varepsilon_k^{-1/2}(T_k - T_k^g)$ converges (up to a subsequence) in measure on $\Omega$. This implies $T = T_g$ by \eqref{rig-eq: comp1}(ii).
Moreover, suppose ${\cal P}_g \ge {\cal P}$ was wrong (recall \eqref{new10}). We may then assume after reordering that $P_1^g \cap {P}_1$ and $P_2^g \cap {P}_1$ have positive ${\cal L}^2$-measure. Since $\varepsilon_k^{-1/2}(T_k - T_k^g)$ converges in measure on $P_1$, we can argue exactly as in the proof of Lemma \ref{rig-lemma: comp2.1}(2) to see that the partition ${\cal P}_g$ is not coarsest, which contradicts the assumption. \nopagebreak\hspace*{\fill}$\Box$\smallskip
\noindent {\em Proof of Theorem \ref{rig-th: gammaconv2}.} First consider the lower bound. Let $(u,{\cal P},T) \in {\cal D}_\infty$ be given as well as a sequence $(y_k)_k \subset SBV_M(\Omega)$ with $y_k \to (u,{\cal P},T)$. If $(u,{\cal P},T) \notin {\cal C}_g$, Lemma \ref{lemmanew} implies $\liminf_{k \to \infty} F_{\varepsilon_k}(y_k) = \infty$. If $(u,{\cal P},T) \in {\cal C}_g$, we recall Theorem \ref{rig-th: gammaconv}(i) and see that it suffices to show
\begin{align}\label{new21}
\liminf_{k \to \infty}\varepsilon_k^{-1} \Vert y_k - f_k \Vert^2_{L^2(\Omega)} \ge \min_{v \in u + \nabla T \mathcal{A}(\mathcal{P})} \Vert v- g \Vert^2_{L^2(\Omega)}.
\end{align}
Consider the triples $(u_k,{\cal P}_k,T_k)$ and $(g_k,{\cal P}^g_k,T^g_k)$ as in the previous proof. Recall $\varepsilon_k^{-1/2}(y_k - f_k ) - \varepsilon_k^{-1/2}(T_k - T_k^g) - (u_k - g_k) \to 0$ a.e. by \eqref{rig-eq: comp2}(i) and $u_k - g_k \to u - g$ a.e. by \eqref{rig-eq: comp1-2}(i). Now it is enough to show that for each $P_j$ there are $A_j \in \Bbb R^{2 \times 2}_{\rm skew}$
and $d_j \in \Bbb R^2$ such that $\varepsilon_k^{-1/2}(T_k - T_k^g) \to \nabla T(A_j\cdot+d_j)$ a.e. on $P_j$. Then \eqref{new21} follows from Fatou's lemma. We recall from the proof of Lemma \ref{lemmanew} that $\varepsilon_k^{-1/2}(T_k - T_k^g)$ converges in measure on $P_j$ and that $P_j \subset P_i^g$ for some $P_i^g$. In particular, this implies $|R^k_j - R_i^{g,k}| + |b^k_j - b_i^{g,k}| \le C\sqrt{\varepsilon_k}$, where $R_i^{g,k}\cdot + b_i^{g,k}$ denotes the rigid motion associated to $T_k^g$. Repeating the argument in \eqref{rig-eq: Tdef} we obtain the desired convergence.
For the construction of recovery sequences we mainly follow Theorem \ref{rig-th: gammaconv}(ii) and only indicate the necessary adaptions. Let $(u,{\cal P},T) \in {\cal C}_g$ be given with $u$ having the specific regularity assumed in Theorem \ref{rig-th: gammaconv}(ii), particularly $\nabla u \in L^\infty(\Omega)$. Let $A \in {\cal A}({\cal P})$ such that $v:= u + \nabla T A$ realizes the minimum in \eqref{new21}.
As $u,g \in L^2(\Omega)$ and thus $\nabla T A \in L^2(\Omega)$, we can choose a sequence $(A^k)_k \subset {\cal A}({\cal P}_k^g) $ such that $\nabla T_k^g A^k \to \nabla T_g A = \nabla T A$ in $L^2(\Omega)$ and $\sqrt{\varepsilon_k}|\nabla A^k|^2 \to 0$ in $L^2(\Omega)$. Select $T_k = \sum_j (R^{k}_j\cdot+b^{k}_j)\chi_{P^{g,k}_j} \in {\cal R}({\cal P}_k^g)$ such that (cf. before \eqref{new5} for a similar construction)
$$\operatorname{dist}(\nabla (T_k^g + \sqrt{\varepsilon_k}\nabla T_k^g A^k) ,SO(2)) = |\nabla (T_k^g + \sqrt{\varepsilon_k}\nabla T_k^g A^k) - \nabla T_k| \text{ on } \Omega$$
and on each component the translations of $T_k$ and $T_k^g + \sqrt{\varepsilon_k}\nabla T_k^g A^k$ coincide, i.e.
$$b^{k}_j = T_k(x) - \nabla T_k(x)\,x = (T_k^g + \sqrt{\varepsilon_k}\nabla T_k^g A^k)(x) - \nabla (T_k^g + \sqrt{\varepsilon_k}\nabla T_k^g A^k)(x) \, x$$
for all $x \in P^{g,k}_j$ and all $j \in \Bbb N$. Note that using \eqref{rig-eq: linearization} a short calculation implies
\begin{align}\label{new25}
| T_k^g + \sqrt{\varepsilon_k}\nabla T_k^g A^k - T_k| + |\nabla T_k^g + \sqrt{\varepsilon_k}\nabla T_k^g \nabla A^k - \nabla T_k| \le C\varepsilon_k|\nabla A^k|^2
\end{align}
pointwise a.e. in $\Omega$. We define $y_k = T_k + \sqrt{\varepsilon_k} u$, $u_k = u$ and ${\cal P}_k = {\cal P}$ for all $k \in \Bbb N$. We note that $T_k \in {\cal R}({\cal P}_k)$ since $T_k \in {\cal R}({\cal P}^g_k)$, ${\cal P}_k \le {\cal P}_g$ by $(u,{\cal P},T) \in {\cal C}_g$ and $ {\cal P}_k^g = {\cal P}^g$ by assumption. Moreover, $\nabla T_k^T \nabla y_k = \mathbf{Id} + \sqrt{\varepsilon_k}\nabla T_k^T \nabla u$ and by \eqref{new25}
\begin{align}\label{new23}
\nabla T_k^T \nabla u = (\nabla T_k^g)^T \nabla u + O(\Vert \nabla u\Vert_\infty \,\varepsilon_k|\nabla A^k|^2) + O(\Vert \nabla u\Vert_\infty \sqrt{\varepsilon_k}|\nabla A^k|).
\end{align}
Using $\sqrt{\varepsilon_k}|\nabla A^k|^2 \to 0$ in $L^2(\Omega)$ and $T_k^g \to T, \nabla T_k^g \to \nabla T$ in $L^2(\Omega)$, one can check that $(y_k)_k \subset SBV_M(\Omega)$ and that \eqref{rig-eq: comp2}-\eqref{rig-eq: comp3} hold, i.e. $y_k \to (u,{\cal P},T)$ (see also proof of Theorem \ref{rig-th: gammaconv}). Likewise, \eqref{new23} together with the calculation in \eqref{new22} shows that $\lim_{k\to \infty} E_{\varepsilon_k}(y_k) = E(u,{\cal P},T)$. Finally, recalling $\varepsilon_k^{-1/2}(f_k - T_k^g) \to g$ in $L^2(\Omega)$ (see before Lemma \ref{lemmanew}), $\nabla T_k^g A^k \to \nabla T A$ in $L^2(\Omega)$ and \eqref{new25} we conclude
\begin{align*}
\varepsilon_k^{-1/2}\Vert y_k - f_k \Vert_{L^2(\Omega)} &= \Vert \varepsilon_k^{-1/2} (T_k - T_k^g) + u - \varepsilon_k^{-1/2}(f_k - T_k^g)\Vert_{L^2(\Omega)}\\& =
\Vert \nabla T_k^g A^k + u - \varepsilon_k^{-1/2}(f_k - T_k^g) \Vert_{L^2(\Omega)} + O(\Vert \varepsilon_k^{1/2}|\nabla A^k|^2 \Vert_{L^2(\Omega)})\\&\to \Vert \nabla T A + u - g\Vert_{L^2(\Omega)} =\Vert v - g\Vert_{L^2(\Omega)}.
\end{align*}
Finally, the convergence result for minimum problems and minimizers follows from a general result in the theory of $\Gamma$-convergence (see \cite[Chapter 7]{DalMaso:93}). \nopagebreak\hspace*{\fill}$\Box$\smallskip
\EEE
\subsection{An application to cleavage laws}\label{rig-sec: sub, gamma2}
We are finally in a position to prove the cleavage law in Theorem \ref{rig-th: cleavage-cont}. Analogous results for the case of expansive boundary values have been obtained in \cite{Mora:2010} and \cite{FriedrichSchmidt:2014.2}. We thus do not repeat all the steps of these proofs but rather concentrate on the additional arguments necessary in our general setting (see \eqref{rig-eq: Griffith en-lim}) in which we particularly can extend the aforementioned results to the case of compression.
\smallskip
\noindent {\em Proof of Theorem \ref{rig-th: cleavage-cont}.} Let $(y_{\varepsilon_k})_{k}$ be a sequence of almost minimizers. Passing to a suitable subsequence, by Theorem \ref{rig-th: comp1} we obtain a triple $(u_k, {\cal P}_k, T_k) \in {\cal D}$ and a limiting triple $(u, {\cal P}, T) \in {\cal D}_\infty$ such that \eqref{rig-eq: comp2}-\eqref{rig-eq: comp3} hold and
$$ E(u, {\cal P}, T) \le \liminf\nolimits_{\varepsilon \to 0} \inf \lbrace E_\varepsilon(y): y \in {\cal A}(a_\varepsilon)\rbrace$$
by Theorem \ref{rig-th: gammaconv}(i). \BBB Write $T_k = \sum_j (R_j^k\cdot + b_j^k)\chi_{P_j^k}$ and ${\cal P} =(P_j)_j$. \EEE Due to the boundary conditions \BBB and \eqref{rig-eq: comp2}(i),\eqref{rig-eq: comp1-2}(i), \EEE on each component $P_j \in {\cal P}$ we find $A_j \in \Bbb R^{2 \times 2}_{\rm skew}$ and $b_j \in \Bbb R^2$ such that
\begin{align}\label{rig-eq: bc-cleav2}
\begin{split}
u_1(x) &= \lim\nolimits_{k \to \infty} \varepsilon_k^{-1/2}( \mathbf{e}_1 \cdot (\mathbf{Id} - R^k_j)\, x - \mathbf{e}_1 \cdot b^k_j+ a_{\varepsilon_k} x_1 ) \\ &= \mathbf{e}_1 \cdot A_j\, x +\mathbf{e}_1 \cdot b_j + a x_1
\end{split}
\end{align}
for a.e. $x \in \Omega'$ with $x_1 <0$ or $x_1 > l$ and $x \in P_j$. In particular, this implies
\begin{align}\label{rig-eq: bc-cleav}
u_1(x_1,x_2) - u_1(\hat{x}_1,x_2) = |x_1 - \hat{x}_1| a
\end{align}
for a.e. $x \in \Omega'$ with $\hat{x}_1 < 0$, $x_1 > l$ and $(x_1,x_2), (\hat{x}_1,x_2) \in P_j$.
We first derive the limiting minimal energy and postpone the characterization of the sequence of almost minimizers to the end of the proof. The argument in \eqref{rig-eq: bc-cleav2} shows that $\nabla T = \mathbf{Id}$ on $P_j$ if $|P_j \cap \lbrace x: x_1 < 0 \text{ or } x_1 > l \rbrace|>0$. It is not restrictive to assume $\nabla T^T \nabla u = \nabla u$ a.e. Indeed, we may replace $u$ by \BBB $0$ \EEE in a component $P_j$ which does not intersect the boundaries without \BBB increasing \EEE the energy. By \eqref{rig-eq: Griffith en-lim}, a slicing argument in $GSBD$ (see Theorem \ref{clea-th: slic}) and the fact that $\inf \lbrace Q(F): \mathbf{e}_1^T F \mathbf{e}_1 = a\rbrace = \alpha a^2$ (see Section \ref{rig-sec: sub, cleavage}) we obtain
\begin{align}\label{eq:E-uS}
\begin{split}
E(u, {\cal P}, T) &\ge \int_{\BBB \Omega \EEE } \frac{1}{2} Q(e(\nabla u)) + \int_{J_u} | \BBB \nu_u \EEE \cdot \mathbf{e}_1| d{\cal H}^1 + {\cal E}(u) \\
&\ge \int_0^1 \Big( \int_0^l \frac{\alpha}{2} (\mathbf{e}_1^T \nabla u(x) \mathbf{e}_1)^2 \, dx_1 + S^{x_2}(u) \Big) \, dx_2 + {\cal E}(u),
\end{split}
\end{align}
where $S^{x_2}$ denotes the number of jumps of $u_1$ on a slice $(-\eta,l+\eta) \times \lbrace x_2 \rbrace$ and ${\cal E}(u) = \int_{J_u} (1 -|\nu_u \cdot \mathbf{e}_1|) d{\cal H}^1$. If $S^{x_2} \ge 1$, the inner integral is bounded from below by $1$. By the structure theorem for Caccioppoli partitions (see Theorem \ref{th: local structure}) we find that $((-\eta ,0) \cup (l, l +\eta)) \times \lbrace x_2 \rbrace \subset P_j$ for some $j \in \Bbb N$ for ${\cal H}^1$-a.e. $x_2$ with $S^{x_2} = 0$. Consequently, if $\#S^{x_2} =0$, by applying Jensen's inequality we derive that the term is bounded from below by $\frac{1}{2} \alpha l a^2$ due to the boundary conditions \eqref{rig-eq: bc-cleav}. This implies $E(u) \ge \min \lbrace \frac{1}{2} \alpha l a^2,1 \rbrace$.
Otherwise, it is not hard to see that the configurations $y^{\rm el}_{\varepsilon_k}(x) = x + F^{a_{\varepsilon_k}} \, x$ for $x \in \Omega'$ satisfy $E_{\varepsilon_k}(y^{\rm el}_{\varepsilon_k}) \to \frac{1}{2}\alpha l a^2$ for $k\to \infty$. Likewise, we get $E_{\varepsilon_k}(y^{\rm cr}_{\varepsilon_k}) = 1$ for all $k \in \Bbb N$, where $y^{\rm cr}_{\varepsilon_k}(x) = x \chi_{x_1 < \frac{1}{2}} + (x + (l a_{\varepsilon_k},0)) \chi_{x_1 > \frac{1}{2}}$ for $x \in \Omega$ and $y^{\rm cr}_{\varepsilon_k} = (x_1(1+a_{\varepsilon_k}) , x_2 )$ for $x \in \Omega' \setminus \Omega$. This completes \eqref{rig-eq: cleavage en}.
\smallskip
It remains to characterize the sequences of almost minimizers. \BBB Let $(y_{\varepsilon_k})_{k}$ be a sequence of almost minimizers
and $(u,{\cal P},T) \in {\cal D}_\infty$ a limiting triple as considered before \eqref{rig-eq: bc-cleav2}. Again we may suppose $\nabla T^T \nabla u = \nabla u$ a.e. \EEE We let first $|a| < a_{\rm crit}$ and follow the arguments in the proof of \cite[Theorem 2.4]{FriedrichSchmidt:2014.2}. Since $E(u,{\cal P},T) = \frac{1}{2}\alpha la^2$, we infer from \eqref{eq:E-uS} that $u$
has no jump on a.e.\ slice $(-\eta,l+\eta) \times \left\{x_2\right\}$ and satisfies $\mathbf{e}_1^{T} \nabla u \, \mathbf{e}_1 = a$ a.e.\ by the imposed boundary values and the strict convexity of the mapping $t \mapsto t^2$ on \BBB $\Bbb R$. \EEE Thus, if $J_{u} \neq
\emptyset$, a crack normal must satisfy $\nu_{u} = \pm \mathbf{e}_2$ ${\cal H}^1$-a.e.
Taking additionally ${\cal E}(u)$ into account we find $J_{u} = \emptyset$ up to an ${\cal H}^1$ negligible set, i.e., $u \in H^1(\Omega')$. By the strict convexity of $Q$ on symmetric
matrices and the boundary values \eqref{rig-eq: bc-cleav2} we see that the derivative has the form
$$ \nabla u(x) = \begin{footnotesize} F^a + A \end{footnotesize} \text{ for a.e.\ $x \in \Omega$} $$
for a suitable $A \in \Bbb R^{2 \times 2}_{\rm skew}$. Since $\Omega$ is connected, we conclude
$$u(x) = F^a \,x + A\,x + b$$
for $x \in \Omega$ and some $b \in \Bbb R^2$. In particular, this implies ${\cal P}$ consists only of $P_1 = \Omega'$ and thus by \eqref{rig-eq: bc-cleav2} we get $A = \lim_{k \to \infty} \varepsilon_k^{-1/2}(\mathbf{Id} - R^k_1)$ and $\mathbf{e}_1 \cdot b = - \lim_{k \to \infty} \varepsilon^{-1/2}_{k}\mathbf{e}_1 \cdot b^k_1$. Let $ s= \lim_{k \to \infty} \mathbf{e}_2 \cdot (\varepsilon_k^{-1/2} b^k_1 +b)$, which exists by \eqref{rig-eq: comp2}(i), \eqref{rig-eq: comp1-2}(i). We now conclude by \eqref{rig-eq: comp2}(i), \BBB \eqref{rig-eq: comp1-2}(i) \EEE for a.e. $x \in \Omega$
\begin{align}\label{rig-eq: lincleav}
\begin{split}
\BBB \bar{u}(x):= \EEE \lim_{k\to \infty} \varepsilon_k^{-1/2}(y_{\varepsilon_k}(x) - x) & = u(x) + \lim_{k\to \infty} \varepsilon_k^{-1/2} \big((R^k_1 - \mathbf{Id})\,x + b^k_1 \big) \\
& = u(x) - A\,x - b + (0,s)= (0,s) + F^a \, x,
\end{split}
\end{align}
\BBB i.e. $\bar{u}$ fulfills Theorem \ref{rig-th: cleavage-cont}(i). \EEE If $|a| > a_{\rm crit}$, we again consider the lower bound \eqref{eq:E-uS} and now obtain that on
a.e.\ slice $(0,l) \times \left\{x_2\right\}$ a minimizer $u$ has precisely one jump
and that $\mathbf{e}_1^{T} \nabla u \, \mathbf{e}_1 = 0$ a.e. By the strict convexity of $Q$ on symmetric matrices we then derive that
$\nabla u$ is antisymmetric a.e. As a consequence, the linearized piecewise rigidity estimate for $SBD$ functions (see \cite[Theorem A.1]{Chambolle-Giacomini-Ponsiglione:2007} or the remark below Theorem \ref{rig-cor: cgp}) yields that there is a Caccioppoli partition $(E_i)_i$ of $\Omega$ such that
$$ u(x) = \sum\nolimits_{i} (A_i x + b_i) \chi_{E_i}(x) \quad \text{and} \quad
J_u = \bigcup\nolimits_{i} \partial^* E_i \cap \Omega, $$
where $A_i\in \Bbb R^{2 \times 2}_{\rm skew}$ and $b_i \in \Bbb R^2$. (Note that indeed the linearized rigidity estimate can also be applied in the $GSBD$-setting as it relies on a slicing argument and an approximation which is also available in the generalized framework, see \cite[Section 3.3]{Iurlano:13}. The only difference is that the approximation does not converge in $L^1$ but only pointwise a.e. which does not affect the argument.)
As ${\cal E}(u) = 0$, we also note that $\nu_u = \pm \mathbf{e}_1$ a.e.\ on $J_u$. Following the arguments in \cite{Mora:2010}, in particular using regularity results for boundary curves of sets of finite perimeter and exhausting the sets $\partial^* E_i$ with Jordan curves, we find that
$$ J_u = \bigcup\nolimits_{i} \partial^* E_i \cap \Omega \subset (p, 0) + \Bbb R \mathbf{e}_1 $$
for some $p \in (0,l)$. We thus obtain that $(E_i)_i$ consists of only two sets and $u$ has the form
$$u(x) = \begin{cases} A_1 \, x + b_1 & \text{for } x_1 < p, \\ A_2 \, x + b_2 & \text{for } x_1 > p, \end{cases} $$
for $A_i \in \Bbb R^{2 \times 2}_{\rm skew}$and $b_i \in \Bbb R^2$, $i=1,2$. Now repeating the calculation in \eqref{rig-eq: lincleav} for the sets $P_1 = \lbrace x \in \Omega': x_1 < p\rbrace$ and $P_2 = \Omega' \setminus P_1$ we find $s, t \in \Bbb R$ such that for $x \in \Omega$ a.e.
\begin{align*}
\BBB \bar{u}(x):= \EEE \lim_{k\to \infty} \varepsilon_k^{-\frac{1}{2}}(y_{\varepsilon_k}(x) - x) & = u(x) - (A_1\,x + b_1)\chi_{x_1 < p}(x) - (A_2\,x + b_2)\chi_{x_1> p}(x) \\ & \ \ \ + (0,s)\chi_{x_1 < p}(x) + ((l a,t))\chi_{x_1 > p}(x).
\end{align*}
\BBB Then $\bar{u}$ satisfies Theorem \ref{rig-th: cleavage-cont}(ii). \EEE This concludes the proof. \nopagebreak\hspace*{\fill}$\Box$\smallskip
\textbf{Acknowledgements} I am very grateful to Bernd Schmidt for many stimulating discussions and valuable comments from which the results of this paper and their exposition have benefited a lot. \BBB Moreover, I am gratefully indebted to the referee for her/his careful reading of the manuscript and many helpful suggestions. \EEE
\typeout{References}
|
3,212,635,537,439 | arxiv | \section{Introduction}
We consider the derivative nonlinear Schr\"odinger equation (DNLS)
\begin{equation}\label{DNLS}
{i\mkern1mu} \partial_t q + \partial^2_x q + {i\mkern1mu} \partial_x(|q|^2q) =0,
\end{equation}
where \(q\colon \ensuremath{\mathbb{R}}\times \ensuremath{\mathbb{R}}\rightarrow \ensuremath{\mathbb{C}}\). \eqref{DNLS} is $L^2$-critical since the dilation
\begin{align}\label{scaling}
q(t,x)\mapsto q_{\lambda}(t,x)=\lambda^{1/2}q(\lambda^2t, \lambda x)
\end{align}
leaves both \eqref{DNLS} and the $L^2$ norm invariant. The derivative nonlinear Schr\"odinger equation appears in plasma physics \cite{MOMT-PHY, M-PHY,
SuSu-book}, and references therein.
Local well-posedness result for \eqref{DNLS} in the energy space was
worked out by N. Hayashi and T. Ozawa \cite{HaOz-94, Oz-96}. They
combined the fixed point argument with the $L^4_tW^{1, \infty}_x$
estimate to construct local-in-time solution with arbitrary data in
energy space. For other results, we can refer to \cite{Ha-93,
HaOz-92}. Since \eqref{DNLS} is energy subcritical case, the maximal time interval of existence only depends on $H^1$ norm of
initial data. Later, local well-posedness result for \eqref{DNLS} in $H^s, s\geq 1/2$ is due to
H. Takaoka \cite{Ta-99} by Bourgain's Fourier restriction method.
The sharpness is shown in \cite{Ta-01} in the sense that nonlinear
evolution $u(0)\mapsto u(t)$ fails to be $C^3$ or even uniformly
$C^0$ in this topology, even when $t$ is arbitrarily close to zero
and $H^s$ norm of the data is small (see also Biagioni-Linares
\cite{BiLi-01-Illposed-DNLS-BO}).
Global well-posedness is shown for \eqref{DNLS}
in the energy space in \cite{Oz-96}, under the smallness condition
\begin{align}\label{Cond:smalldata}
\|u_0\|^2_{L^2} < 2 \pi,
\end{align}
the argument is based on the sharp Gagliardo-Nirenberg inequality
and the energy method (conservation of mass and energy). This result is
improved by H. Takaoka \cite{Ta-01} by Bourgain's
restriction method, who proved global
well-posedness in $H^s$ for $s>32/33$ under the condition
\eqref{Cond:smalldata}. In \cite{CKSTT-01, CKSTT-02}, I-team make use of almost conservation law \cite{Tao:book:Nonlinear Dispersive Equations} to
show global well-posedness in $H^s, s>1/2$ under
\eqref{Cond:smalldata}. Miao, Wu and Xu \cite{MiaoWX-2011} combine almost conservation law and the refined resonant
decomposition technique to obtain the
global well-posedness in $H^{1/2}$ under
\eqref{Cond:smalldata}. Later, Wu use the generalized Gagliardo-Nirenberg inequality to improve the global well-posedness of \eqref{DNLS} in the energy space under the condition
\begin{align}\label{Cond:small:4pi}
\|u_0\|^2_{L^2} < 4 \pi
\end{align}
in \cite{Wu-DNLS}, where $4\pi$ is the mass of the solitary waves with critical parameters of \eqref{DNLS}. Miao, Tang and Xu use the structure analysis and classical variational argument to show the existence of solitary waves with two parameters and improve the global result of \eqref{DNLS} in the energy space in \cite{MTX:DNLS:Exist}, and further use perturbation argument, modulation analysis and Lyapunov stability to show the orbital stability of weak interaction multi-soliton solution with subcritical parameters in the energy space in \cite{MTX:DNLS:stab}.
We can also refer to \cite{ColOhta:DNLS:stab, GNW:gDNLS:instab, LeW:gDNLS:stab, MTX:gDNLS:instab, TX:gDNLS:stab} for the stability analysis of the solitary waves of the (generalized) derivative nonlinear Schr\"odinger equation in the energy space and to \cite{GW:DNLS:GWP} for lower regularity result of \eqref{DNLS} by almost conservation law in \cite{Tao:book:Nonlinear Dispersive Equations}.
Since \eqref{DNLS} is an integrable system in \cite{AbCl:book, KaupN:DNLS}, there are lots of global well-posedness of \eqref{DNLS} with mass restriction in the weighted Sobolev spapce based on the inverse scattering method, please refer to \cite{JLPS:DNLS:APDE, JLPS:DNLS:QJPAM, JLPS:DNLS:CPDE, PelinSS:DNLS:DPDE, PelinSS:DNLS:IMRN}and reference therein.
The conjecture about \eqref{DNLS} is the following.
\begin{conj}
Let \(s>0\). \eqref{DNLS} is globally well-posed for all initial data in \(H^s(\ensuremath{\mathbb{R}})\) in the sense that the solution map \(\Phi\) extends uniquely from Schwartz space to a jointly continuous map \(\Phi\colon \ensuremath{\mathbb{R}}\times H^s(\ensuremath{\mathbb{R}})\rightarrow H^s(\ensuremath{\mathbb{R}})\).
\end{conj}
According to the above well known result, we need loosen the continuous dependence of the solution on initial data to consider the solution of \eqref{DNLS} in $H^s(\ensuremath{\mathbb{R}})$ with $s\in (0,1/2)$. The basic question is that how to control the uniform estimates of the solution of \eqref{DNLS}.
Motivated by Killip-Visan-Zhang's argument in \cite{KVZ:KdV:GAFA}, Klaus and Schippa combine the integrability of \eqref{DNLS} with the series expansion of the perturbation determinant \cite{Rybkin:KdV:Cons Law, Simon:Trace} to obtain macroscopic conservation law for the Schwartz solution of \eqref{DNLS} with small mass in \cite{KlausS:DNLS}, In this paper, we will show the corresponding microscopic form and obtain one-parameter family of microscopic conservation laws for the $A(\kappa)$ flows (see \eqref{A-def}) and the DNLS flow by Harrop-Griffiths-Killip-Visan's argument in \cite{HKV:NLS, KV:KdV:AnnMath}. Compared with macroscopic form, microscopic conservation law with coercivity helps to show the local smoothing effect for \eqref{DNLS} in $H^s(\ensuremath{\mathbb{R}})$ and can be further applied into global wellposedness analysis. We can refer to \cite{HKV:NLS,KV:KdV:AnnMath, Tal:BO} and reference therein.
We now recall some Hamiltonian mechanics background. \eqref{DNLS} is Hamiltonian equation with respect to the following Poisson structure:
\begin{equation}\label{PoissonBracket}
\{F,G\} :
= \int \tfrac{\delta F}{\delta r}\partial_x \bigl(\tfrac{\delta G}{\delta q}\bigr) + \tfrac{\delta F}{\delta q}\partial_x \bigl(\tfrac{\delta G}{\delta r}\bigr) \,{\rm{d}}x,
\end{equation}
where the operators $\tfrac{\delta }{\delta q}$ and $\tfrac{\delta }{\delta r}$ denote the functional Fr\'echet derivatives. Any Hamiltonian $H(q, r, t)$ generates a flow via the equation
\begin{align}\label{HFlow}
\partial_t \begin{bmatrix} q \\ r \end{bmatrix}
= \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \partial_x
\begin{bmatrix} \frac{\delta H}{\delta q} \\ \frac{\delta H}{\delta r} \end{bmatrix}.
\end{align}
Correspondingly, \eqref{DNLS} is the Hamiltonian flow associated to
\begin{equation}\label{HDNLS}
H_{\mr{DNLS}} := \int _{\ensuremath{\mathbb{R}}}-{i\mkern1mu} qr' + \frac12 q^2r^2\,{\rm{d}}x,
\end{equation}
where $r=-\bar{q}$. Two other important Hamiltonian quantities for \eqref{DNLS} are
\begin{equation}\label{Mass Energy}
M:= \int qr\,{\rm{d}}x, \quad E_{\mr{DNLS}} :=\int q' r' - \frac32 {i\mkern1mu} q^2 rr' + \frac12 q^3 r^3 \,{\rm{d}}x.
\end{equation}
Conservations of $M$, $H_{\mr{DNLS}}$ and $E_{\mr{DNLS}}$ is due to gauge, space translation and time translation invariance of \eqref{DNLS}, the commutativity of $H_{\mr{DNLS}}$ and $E_{\mr{DNLS}}$ is based on the fact that they are completely integrable, which means the existence of an infinite family of commuting flows for \eqref{DNLS}. We can refer to \cite{AbCl:book} \cite{KlausS:DNLS} for more details. Based on the recent breakthrough in KdV, mKdV and NLS by B. Harrop-Griffiths, R. Killip and M. Visan in \cite{HKV:NLS, KV:KdV:AnnMath}, the commuting flow approximation will be the robust method for showing global well-posedness theory of \eqref{DNLS} in the lower regularity space. We can also refer to \cite{KTataru:NLS:DMJ, NRTataru:DS:InventM} and reference thereein.
Let us write the Lax operator related to \eqref{DNLS} and its unperturbed one
\begin{equation}\label{Intro KN L}
L(\varkappa) := \begin{bmatrix}\ensuremath{\partial}+{i\mkern1mu}\varkappa^2 & -\varkappa q\\ -\varkappa r&\ensuremath{\partial}-{i\mkern1mu}\varkappa^2\end{bmatrix} \text{~~and~~} L_0(\varkappa) := \begin{bmatrix}\ensuremath{\partial}+{i\mkern1mu}\varkappa^2 & 0\\0& \ensuremath{\partial}-{i\mkern1mu}\varkappa^2 \end{bmatrix}.
\end{equation}
By simple calculations, we know that
\begin{equation*}
R_0(\kappa) := L_0(\kappa)^{-1} = \begin{bmatrix}(\ensuremath{\partial}+{i\mkern1mu}\kappa^2)^{-1} & 0\\0&(\ensuremath{\partial}-{i\mkern1mu}\kappa^2)^{-1}\end{bmatrix}
\end{equation*}
admits the integral kernel
\begin{equation*}\label{G_0}
G_0(x,y;\kappa) = e^{-{i\mkern1mu}\kappa^2|x - y|}\begin{bmatrix}\ensuremath{\mathbbm 1}_{\{y<x\}}&0\\0&-\ensuremath{\mathbbm 1}_{\{x<y\}}\end{bmatrix} \quad\text{for ${i\mkern1mu}\kappa^2 >0$}.
\end{equation*}
For ${i\mkern1mu}\kappa^2<0$, we may use ${G}_0(x,y;\kappa)=-G_0(y,x;-\bar{\kappa})$. The resolvent operator $R(\kappa):=L(\kappa)^{-1}$ for Schwartz function $q$ with small mass also has the integral kernel $G(x,y;\kappa)$ (See Proposition \ref{P:R})
\begin{equation*}
\begin{bmatrix}
G_{11}(x,y,\kappa) & G_{12}(x,y,\kappa) \\
G_{21}(x,y,\kappa) & G_{22}(x,y,\kappa)
\end{bmatrix}.
\end{equation*}
Let us define three key functionals as follows.
\begin{align*}
\gamma(x;\kappa) &:=\sgn({i\mkern1mu}\kappa^2) \bigl[ G_{11}(x,x;\kappa) - G_{22}(x,x;\kappa) \bigr] - 1, \\
g_{12}(x;\kappa) &:= \sgn({i\mkern1mu}\kappa^2) G_{12}(x,x;\kappa), \\
g_{21}(x;\kappa) &:= \sgn({i\mkern1mu}\kappa^2) G_{21}(x,x;\kappa),
\end{align*}
With these preparations, the main result in this paper is
\begin{thrm}
Let $s\in (0,1/2)$, ${i\mkern1mu}\varkappa^2\in\ensuremath{\mathbb{R}}\setminus(-1,1)$, $q(0)\in H^s (\ensuremath{\mathbb{R}})\cap \ensuremath{\mathscr{S}}$ with small mass. Suppose that $q$ is a solution to \eqref{DNLS}, then we have
$$\partial_t \rho(\varkappa) + \partial_x j_{\mr{DNLS}}(\varkappa) =0, $$ where
the density $ \rho$ and the flux $ j_{\mr{DNLS}}$ are defined as follows \begin{align*}
\rho(\varkappa): =\, & -\varkappa\frac{qg_{21}(\varkappa)+rg_{12}(\varkappa)}{2+\gamma(\varkappa)},\\
j_{\mr{DNLS}}(\varkappa):
= \, &
i \varkappa \frac{q'g_{21}(\varkappa) - r' g_{12}(\varkappa)}{2+\gamma(\varkappa)}
- i\varkappa^2 q r
- 2\varkappa^2 \rho(\varkappa)
- qr\rho(\varkappa).
\end{align*}
\end{thrm}
\begin{rem} We give some remarks as follows.
\begin{enumerate}
\item In this paper, we consider microscopic conservation law for the solution of \eqref{DNLS}. That is the reason why we consider the Schwartz solution. In addition, mass threshold is another interesting problem. Please refer to \cite{HaOz-94, MTX:DNLS:Exist}\cite{Wu-DNLS} and reference therein. Recently, it is very interesting that Bahouri and Perelman obtain global well-posedness for \eqref{DNLS} in $H^{1/2}(\ensuremath{\mathbb{R}})$ without mass restriction by combining the profile decomposition techniques with the integrability structure in \cite{BahouriPerelman:DNLS:GWP}.
\item The above microscopic conservation law corresponds to macroscopic conservation law of \eqref{DNLS} in \cite{KlausS:DNLS}. In fact, we have
\begin{align*}A(\varkappa) = & \sgn({i\mkern1mu}\varkappa^2)\sum_{j=1}^\infty\frac{(-1)^{j-1}}j \tr\left\{\left(\sqrt{R_0}\left(L - L_0\right)\sqrt{R_0}\right)^j\right\} \\
= & \int_{\ensuremath{\mathbb{R}}} \rho(\varkappa) \,{\rm{d}}x.
\end{align*}
Compared to macroscopic conservation law, microscopic conservation law with coercivity can be used to show the local smoothing estimate for the solution of \eqref{DNLS} with small mass in $H^s\cap \ensuremath{\mathscr{S}}$, $s\in (0,1/2)$ and others. Please refer to \cite{KV:KdV:AnnMath, KVZ:KdV:GAFA}.
\item The leading order term in $\rho$ has the following form
$$\left|\Re \int_{\ensuremath{\mathbb{R}}}\rho^{[2]}(\varkappa) dx \right| \approx |\varkappa|^2 \|q\|^2_{H^{-1/2}_{\varkappa}},$$
which captures $L^2$ norm of the part of $q$ living at frequencies $|\xi|\lesssim |\kappa|^2$ (See also \eqref{LogarithmicBound} and \eqref{A-def'}). Combining the above conservation law and the similar argument on Besov norm estimate in \cite[Section ~$3$]{KVZ:KdV:GAFA, KlausS:DNLS} for any Hamiltonian flow preserving $A(\varkappa)$ for all $|\varkappa|\geq 1$, we can obtain a uniform bound of $H^s(\ensuremath{\mathbb{R}})$ norm of the Schwartz solution of \eqref{DNLS} with small mass
\begin{equation}\label{APBound}
\|q(t)\|_{H^s}\lesssim \|q(0)\|_{H^s}, \text{~~for~~} s\in (0,1/2).
\end{equation}
\end{enumerate}
\end{rem}
Lastly, the paper is organized as follows. In Section \ref{S:2}, we recall some notations and preliminary estimates. In Section \ref{S:3}, we show the existence and some properties of the Green's function related to the Lax operator $L(\kappa)$. In Section \ref{S:4}, we introduce the invairant quantity $A(\kappa)$ from the logarithmic perturbation determinant and show its microscopic conservation laws for the $A(\kappa)$ flow and the DNLS flow.
\subsection*{Acknowledgements}
X. Tang was supported by NSFC (No. 12001284), and G. Xu was supported by NSFC (No. 11671046, and No. 11831004) and by National Key Research and Development Program of China (No. 2020YFA0712900). The authors would like to thank Professor Monica Visan for her valuable comments and suggestions.
\section{Some notation and preliminary estimates}\label{S:2}
In this paper, we take $ r = -\bar{q}$ and choose
$
s\in (0, \tfrac12), $
and all implicit constants can depend on $s$. We denote $\ensuremath{\mathscr{S}}$ the Schwartz function, and introduce the notation
\begin{equation}\label{Bdelta}
B_\delta := \left\{q\in H^s:\|q\|_{H^s}\leq \delta\right\}.
\end{equation}
which can be ensured by scaling argument under the assumption that the mass $\|q\|_{L^2}$ is small enough.
We use the inner product on $L^2(\ensuremath{\mathbb{R}})$ as follows
$$
\langle f, g\rangle = \int \overline{f(x)} g(x)\,dx,
$$
which also gives the dual product between $H^s(\ensuremath{\mathbb{R}})$ and $H^{-s}(\ensuremath{\mathbb{R}})$. In addtion, If $F:\ensuremath{\mathscr{S}}\to\ensuremath{\mathbb{C}}$ is $C^1$, we have
\begin{equation}\label{FunctDeriv}
\tfrac{d\ }{d\theta}\Big|_{\theta=0} F(q+\theta f) = \bigl\langle \bar f, \tfrac{\delta F}{\delta q}\bigr\rangle - \bigl\langle f, \tfrac{\delta F}{\delta r}\bigr\rangle.
\end{equation}
The Fourier transform is defined by
\begin{align*}
\hat f(\xi) = \tfrac{1}{\sqrt{2\pi}} \int_\ensuremath{\mathbb{R}} e^{-i\xi x} f(x)\,dx, \qtq{whence} \widehat{fg}(\xi) = \tfrac{1}{\sqrt{2\pi}} [\hat f * \hat g] (\xi).
\end{align*}
\subsection{Sobolev spaces}
For complex $\kappa$ with $|\kappa| \geq 1$ and \(\sigma\in \ensuremath{\mathbb{R}}\) we define the norm
\[
\|q\|_{H^{\sigma}_\kappa}^2 := \int_{\ensuremath{\mathbb{R}}} \left(4|\kappa|^4 + \xi^2\right)^\sigma |\hat q(\xi)|^2\,d\xi
\]
and write \(H^\sigma := H^{\sigma}_1\).
For $0<s<\frac12$, simple calculation yields the Sobolev inequality
\begin{align}\label{Linfty bdd}
\|f\|_{L^\infty} \lesssim \|\hat{f}\|_{L^1}\leq \|f\|_{H^{s+\frac{1}2 }_\kappa} \bigl\|(|\xi|^2+4|\kappa|^4)^{-\frac{2s+1}4}\bigr\|_{L^2}\lesssim |\kappa|^{-2s}\|f\|_{H^{s+\frac{1}2 } _\kappa}.
\end{align}
Consequently, we have the following algebra property of $H^{s+\frac{1}2 }_\kappa$ space:
\begin{equation}\label{E:algebra}
\| f g \|_{H^{s+\frac{1}2 }_\kappa} \lesssim |\kappa|^{-2s} \| f \|_{H^{s+\frac{1}2}_\kappa} \| g \|_{H^{s+\frac{1}2}_\kappa}.
\end{equation}
By duality and the fractional product rule in \cite{ChristW:FracRule}, Sobolev embedding, and \eqref{Linfty bdd}, we obtain
\begin{align}\label{multiplier bdd on ss}
\|qf\|_{H^{s-1/2}}\lesssim
& |\kappa|^{-2s} \|q\|_{H^{s-1/2}}\|f\|_{H^{s+\frac12}_\kappa}.
\end{align}
\subsection{Operator estimates and Trace}
For \(0<\sigma<1\) and \(i \kappa^2 \in \ensuremath{\mathbb{R}}\), \(|\kappa|\geq 1\) we define the operator \((i \kappa^2 \mp \ensuremath{\partial})^{-\sigma}\) using the Fourier multiplier \((i \kappa^2 \mp i\xi)^{-\sigma}\) where, for \(\arg z\in (-\pi,\pi]\), we define
\begin{equation}\label{z to the sigma}
z^{-\sigma} = |z|^{-\sigma}e^{-i\sigma \arg z}.
\end{equation}
Therefore for all ${i\mkern1mu} \kappa^2 \in \ensuremath{\mathbb{R}}$ with \(|\kappa|\geq 1\) we have
\[
\left((i\kappa^2 \mp \ensuremath{\partial})^{-\sigma}\right)^* = (i \kappa^2 \pm \ensuremath{\partial})^{-\sigma},
\]
and
\[
\left\|(i\kappa^2 \mp \ensuremath{\partial})^{-\sigma}\right\|_{\mr{op}}\leq |\kappa|^{-2\sigma}.
\]
We denote $\mf I_p$ the Schatten class of compact operators on $L^2(\ensuremath{\mathbb{R}})$ whose singular
numbers are $l^p$ summable. $\mf I_p$ is complete and is an embedded subalgebra of bounded operators on $L^2(\ensuremath{\mathbb{R}})$. Moreover, we have
\begin{equation}\label{Ip: embedding}
\mf I_p \subset \mf I_q, \text{~~for~~} p\leq q.
\end{equation}
Let us recall some facts about the class $\mf I_p$ that we will use repeatedly in Section \ref{S:3}: An operator A on $L^2(\ensuremath{\mathbb{R}})$ is Hilbert–Schmidt class ($\mf I_2$) if and only if it admits an integral kernel $a(x, y) \in L^2(\ensuremath{\mathbb{R}} \times \ensuremath{\mathbb{R}})$, and
$$\|A\|^2_{\mr{op}}\leq \|A\|_{\mf I_2}^2=\iint_{\ensuremath{\mathbb{R}}\times \ensuremath{\mathbb{R}}} |a(x,y)|^2 \, \,{\rm{d}}x \,{\rm{d}}y. $$
The product of two Hilbert–Schmidt operators is trace class,; Moreover, we have
\begin{align*}
\tr(AB):= & \iint_{\ensuremath{\mathbb{R}}\times \ensuremath{\mathbb{R}}} a(x,y)b(y,x) \,{\rm{d}}y \,{\rm{d}}x = \tr(BA), \\ |\tr(AB)|& \leq \|AB\|_{\mf I_1} \leq \|A\|_{\mf I_2}\|B\|_{\mf I_2}
\end{align*}
The class $\mf I_p$ forms a two-sided ideal in the algebra of
bounded operators on $L^2$; indeed, for any bounded operators $B, C $ on $L^2(\ensuremath{\mathbb{R}})$, we have
$$\|BAC\|_{ \mf I_p} \leq \|B\|_{\mr{op}} \|A\|_{\mf I_p} \|C\|_{\mr{op}}.$$
We can refer to \cite{GohGoldK:book, Simon:Trace} for more details.
The following estimates are the elementary estimates in this paper.
\begin{lem}\label{L:BasicBounds}
Let $s\in (0, 1/2)$, ${i\mkern1mu} \kappa^2 \in \ensuremath{\mathbb{R}}$, we have
\begin{align}
\| (\ensuremath{\partial} + {i\mkern1mu} \kappa^2 )^{-\frac12}\kappa q(\ensuremath{\partial}- {i\mkern1mu} \kappa^2 )^{ - \frac12}\|_{\mf I_2}
& \lesssim \|q\|_{L^2}, \label{LogarithmicBound}\\
\| (\ensuremath{\partial} + {i\mkern1mu} \kappa^2 )^{s-\frac12}\kappa q(\ensuremath{\partial}- {i\mkern1mu} \kappa^2 )^{ - \frac12}\|_{\mf I_2} &\lesssim |\kappa| \|q\|_{H^{s-\frac12}_\kappa}, \label{BasicBound}
\end{align}
and
\begin{align}
\|(\ensuremath{\partial}+ i \kappa^2 )^{-s-\frac12} f (\ensuremath{\partial}- i \kappa^2 )^{-s-\frac12} \|_{\mr{op}} &\lesssim |\kappa|^{-2s} \|f\|_{H^{-s-\frac12 }_\kappa}. \label{BasicOpBound}
\end{align}
\end{lem}
\begin{proof} The estimate \eqref{LogarithmicBound} can be proved by \cite[Lemma~4.1]{KVZ:KdV:GAFA}. For \eqref{BasicBound}, it suffices to consider the case \({i\mkern1mu} \kappa^2 = 1\) by scaling argument. By Plancherel's Theorem, we have
\begin{align*}
\|(1 - \ensuremath{\partial})^{s-\frac12} q(1 + \ensuremath{\partial})^{-\frac12}\|_{\mf I_2}^2 &= \tr\left\{(1 - \ensuremath{\partial}^2)^{-\alpha}q(1 - \ensuremath{\partial}^2)^{-\beta}\bar q\right\}\\
&= \tfrac{1}{2\pi} \iint_{\ensuremath{\mathbb{R}}^2} \frac{|\hat q(\xi-\eta)|^2\,}{\left(1 + \xi ^2\right)^{\frac12-s}\left(1 + \eta^2\right)^{\frac12}}d\eta\, d\xi,
\end{align*}
Note that
\begin{equation*}
\int_{\ensuremath{\mathbb{R}}} \frac1{\left(1 + (\xi + \eta)^2\right)^{\frac12 -s}\left(1 + \eta^2\right)^{\frac12}}\,d\eta \lesssim (4 + \xi^2)^{s-\frac12},
\end{equation*}
we can obtain the estimate \eqref{BasicBound}.
Lastly, we can obtain \eqref{BasicOpBound} by duality and \eqref{E:algebra} as follows
\begin{equation*}
\biggl| \int_{\ensuremath{\mathbb{R}}} f g h \, dx \biggr| \lesssim \| f \|_{H^{-s-\frac12}_\kappa} \| gh \|_{H^{s+\frac12}_\kappa} \lesssim |\kappa|^{- 2s}\| f \|_{H^{-s-\frac12}_\kappa} \| g \|_{H^{s+\frac12}_\kappa} \| h \|_{H^{s+\frac12}_\kappa}.
\end{equation*}
This completes the proof.
\end{proof}
\section{The diagonal Green's functions}\label{S:3}
In this Section, motivated by the ideas in \cite{HKV:NLS, KV:KdV:AnnMath}, we introduce three key quantities $g_{12}$, $g_{21}$, and $\gamma$ from the diagonal Green's function related to the Lax operator $L(\kappa)$ for \eqref{DNLS}, and establish some elementary estimates about them. Recall that
\begin{equation}\label{Lax L'}
L(\kappa) = L_0(\kappa) + \begin{bmatrix}0 & -\kappa q\\-\kappa r&0 \end{bmatrix} \text{~~where~~} L_0(\kappa) := \begin{bmatrix}\ensuremath{\partial}+i\kappa^2 & 0\\0&\ensuremath{\partial}-i\kappa^2\end{bmatrix}.
\end{equation}
Since we only consider the case ${i\mkern1mu}\kappa^2\in\ensuremath{\mathbb{R}}$ with $|\kappa|\geq 1$, we have
\begin{align}\label{L conjugation}
L(\kappa)^* = \begin{bmatrix} -\ensuremath{\partial}-{i\mkern1mu} \bar{k}^2 & -\bar{\kappa} \bar{r} \\-\bar{\kappa}\bar{q} & -\ensuremath{\partial}+i\bar{\kappa}^2 \end{bmatrix}
= - \begin{bmatrix}1 & 0 \\
0 & -1 \end{bmatrix}
L(-\bar{\kappa}) \begin{bmatrix} 1 & 0 \\
0 & -1 \end{bmatrix}.
\end{align}
We now construct the Green's function associated to $L_0(\kappa)$ and $L(\kappa)$, respectively. By the Fourier transformation, the resolvent operator
\begin{equation*}
R_0(\kappa) := L_0(\kappa)^{-1} = \begin{bmatrix}(\ensuremath{\partial}+{i\mkern1mu}\kappa^2)^{-1} & 0\\0&(\ensuremath{\partial}-{i\mkern1mu}\kappa^2)^{-1}\end{bmatrix}
\end{equation*}
admits the integral kernel
\begin{equation}\label{G_0}
G_0(x,y;\kappa) = e^{-{i\mkern1mu}\kappa^2|x - y|}\begin{bmatrix}\ensuremath{\mathbbm 1}_{\{y<x\}}&0\\0&-\ensuremath{\mathbbm 1}_{\{x<y\}}\end{bmatrix} \quad\text{for ${i\mkern1mu}\kappa^2 >0$}.
\end{equation}
For ${i\mkern1mu}\kappa^2<0$, we may use ${G}_0(x,y;\kappa)=-G_0(y,x;-\bar{\kappa})$ by \eqref{L conjugation}.
By the perturbation theory and the resolvent identity, the resolvent operator $R(\kappa):=L(\kappa)^{-1}$ can be formally expressed as
\begin{align}
R = & R_0 + \sum_{\ell = 1}^\infty (-1)^{\ell}\sqrt{R_0}\left(\sqrt{R_0}(L - L_0)\sqrt{R_0}\right)^\ell\sqrt{R_0},
\label{Resolvent}
\end{align}
where
\begin{align}
\sqrt{R_0}(L - L_0)\sqrt{R_0} = & -\begin{bmatrix} 0 &\Lambda \\ \Gamma & 0\end{bmatrix}, \label{LambdaGammaJob} \\
\Lambda := (\ensuremath{\partial}+{i\mkern1mu}\kappa^2)^{-\tfrac12} \kappa q(\ensuremath{\partial}-{i\mkern1mu}\kappa^2)^{-\tfrac12}
\text{~and~} &
\Gamma := (\ensuremath{\partial}-{i\mkern1mu}\kappa^2)^{-\tfrac12} \kappa r( \ensuremath{\partial}+{i\mkern1mu}\kappa^2)^{-\tfrac12}, \label{D:LambdaGamma}
\end{align}
and fractional powers of $R_0$ are defined via \eqref{z to the sigma}. By \eqref{LogarithmicBound}, we have
\begin{equation}\label{Lambda}
\|\Lambda\|_{\mf I_2} = \|\Gamma\|_{\mf I_2} \lesssim \|q\|_{L^2} \lesssim \delta.
\end{equation}
Now we have the convergence of series \eqref{Resolvent} as part of the following result.
\begin{prop}[Existence of the Green's function]\label{P:R}
There exists \(\delta>0\) such that $L(\kappa)$ is invertible as an operator on $L^2(\ensuremath{\mathbb{R}})$, for all \(q\in B_\delta\) and all ${i\mkern1mu} \kappa^2 \in \ensuremath{\mathbb{R}}$ with \(|\kappa|\geq 1\). The resolvent operator $R(\kappa):=L(\kappa)^{-1}$ admits an integral kernel $G(x,y;\kappa)$ satisfying
\begin{equation}\label{G conjugation}
\begin{bmatrix}
G_{11}(x,y,\kappa) & G_{12}(x,y,\kappa) \\
G_{21}(x,y,\kappa) & G_{22}(x,y,\kappa) \\
\end{bmatrix}
=
-
\begin{bmatrix}
\bar{G}_{11}(y,x,-\bar{\kappa}) & \bar{G}_{21}(y,x,-\bar{\kappa}) \\
\bar{G}_{12}(y,x,-\bar{\kappa}) & \bar{G}_{22}(y,x,-\bar{\kappa}) \\
\end{bmatrix}
\end{equation}
such that the mapping
\begin{equation}\label{tensor eqn}
H^s_\kappa(\ensuremath{\mathbb{R}})\ni q \mapsto G - G_0 \in H^{ s+ \frac12}_\kappa\otimes H^{ s + \frac12 }_\kappa
\end{equation}
is continuous. Moreover, $G-G_0$ is continuous as a function of $(x,y)\in\ensuremath{\mathbb{R}}\times \ensuremath{\mathbb{R}}$. Lastly, we have
\begin{align}
\ensuremath{\partial}_xG(x,y;\kappa) &= \begin{bmatrix}-{i\mkern1mu}\kappa^2&\kappa q(x)\\ \kappa r(x)& {i\mkern1mu}\kappa^2\end{bmatrix}G(x,y;\kappa) + \begin{bmatrix}\delta(x-y)&0\\0&\delta(x-y)\end{bmatrix},\label{xID}\\
\ensuremath{\partial}_yG(x,y;\kappa) &= G(x,y;\kappa)\begin{bmatrix}{i\mkern1mu}\kappa^2&- \kappa q(y)\\ -\kappa r(y)&-{i\mkern1mu}\kappa^2\end{bmatrix} - \begin{bmatrix}\delta(x-y)&0\\0&\delta(x-y)\end{bmatrix}\label{yID}
\end{align}
in the sense of distributions.
\end{prop}
\begin{proof} The proof is similar as those in \cite[ Proposition $3.1$]{HKV:NLS}. We sketch the proof for completeness. From \eqref{Lambda}, we have\footnote{Since the convergence of the tails of the series \eqref{Resolvent} is key to the existence of the Green's function, we can pay some regularity $s>s_0>0$ with $s_0\in (0, 1/2)$, use the $\mf I_p$ estimate instead of the $\mf I_2$ estimate to remove small mass assumption. Here we pay attention to the regularity problem of the solution for \eqref{DNLS} in this paper. }
$$
\bigl\| \sqrt{R_0}(L - L_0)\sqrt{R_0} \,\bigr\|_{\mf I_2}\leq \sqrt2 \|\Lambda\|_{\mf I_2} \lesssim \|q\|_{L^2} \lesssim \delta
$$
uniformly for $|\kappa|\geq 1$. Thus, for $\delta>0$ sufficiently small, the series \eqref{Resolvent} converges in operator norm uniformly for $|\kappa|\geq 1$. It is easy to verify that the sum acts as an inverse to $L(\kappa)$.
From \eqref{Lambda}, we also have $R-R_0 \in \mf I_2$. In particular, the operator $R-R_0$ admits an integral kernel $G-G_0$ in $L^2(\ensuremath{\mathbb{R}}\times \ensuremath{\mathbb{R}})$. Moreover, by \eqref{BasicBound}, we know that $R-R_0$ converges in the sense of Hilbert--Schmidt operators from $H^{ -s -\frac12 }_\kappa$ to $H^{s+\frac 12 }_\kappa$, which implies \eqref{tensor eqn}.
By \eqref{tensor eqn} , we obtain that the kernel function $G-G_0$ is continuous in $(x,y)$ since $s+ \frac 12 >\frac12$.
For regular $q$, the identities \eqref{xID} and \eqref{yID} precisely express the fact that $G$ is an integral kernel for $R(\kappa)$. They also hold for irregular $q$ by \eqref{tensor eqn}.
\end{proof}
Let us define $\gamma$, $g_{12}$ and $g_{21}$ as follows:
\begin{align}
\gamma(x;\kappa) &:=\sgn({i\mkern1mu}\kappa^2) \bigl[ G_{11}(x,x;\kappa) - G_{22}(x,x;\kappa) \bigr] - 1, \label{def gama}\\
g_{12}(x;\kappa) &:= \sgn({i\mkern1mu}\kappa^2) G_{12}(x,x;\kappa), \label{def g12}\\
g_{21}(x;\kappa) &:= \sgn({i\mkern1mu}\kappa^2) G_{21}(x,x;\kappa), \label{def g21}
\end{align}
where $G_{ij}(x,y,\kappa)$, $1\leq i, j\leq 2$, are the entries of the integral kernel $G(x,y,\kappa)$. If $q\in \Bd\cap \Schwartz$, we may use \eqref{xID} and \eqref{yID} to obtain
\begin{align}
\gamma' &= 2\kappa\left(qg_{21} - rg_{12}\right),\label{rho-ID}\\
g_{12}' &= -2i\kappa^2 g_{12} -\kappa q[\gamma + 1],\label{g12-ID}\\
g_{21}' &= 2i\kappa^2 g_{21} +\kappa r[\gamma + 1].\label{g21-ID}
\end{align}
By \eqref{G conjugation}, we have
\begin{align}
\gamma(\kappa) &= \bar \gamma(-\bar{\kappa}) \qtq{and} g_{12}(\kappa) = - \bar g_{21}(-\bar{\kappa}).\label{grho-symmetries}
\end{align}
Moreover, by \eqref{rho-ID}, \eqref{g12-ID}, and \eqref{g21-ID} , we have the following identity
\begin{align}
& \frac{\kappa^2-\varkappa^2}{\kappa^2}[ g_{12}'(\kappa)g_{21}(\varkappa)+g_{21}'(\kappa)g_{12}(\varkappa) ]
\notag \\
&=
[ g_{12}(\kappa)g_{21}(\varkappa)+g_{21}(\kappa)g_{12}(\varkappa) ]'
+
\frac{\varkappa}{2\kappa}[ (\gamma(\kappa)+1)(\gamma(\varkappa)+1) ]',\label{micro As commute}
\end{align}
which is closely connected to the commutativity of the $A(\kappa)$'s flows under the Poisson bracket \eqref{PoissonBracket} .
From the series representation \eqref{Resolvent} of $R(\kappa)$ in $q$ and $r$, we can deduce the corresponding series representations of $g_{12}$, $g_{21}$, and $\gamma$ in $q$ and $r$. We use the square brackets notation as follows
\begin{align}\label{g12-Series terms}
g_{12}\sbrack{2m+1}(\kappa) &:= \sgn({i\mkern1mu}\kappa^2)\Big\< \delta_x,\ (\ensuremath{\partial}+i\kappa^2)^{-\frac12}\Lambda \left(\Gamma\Lambda\right)^m(\ensuremath{\partial}-i\kappa^2)^{-\frac12}\delta_x\Big\>, \\
\label{g21-Series terms}
g_{21}\sbrack{2m+1}(\kappa) &:= \sgn({i\mkern1mu}\kappa^2)\Big\< \delta_x,\ (\ensuremath{\partial}-i\kappa^2)^{-\frac12}\Gamma \left(\Lambda\Gamma\right)^m(\ensuremath{\partial}+i\kappa^2)^{-\frac12}\delta_x\Big\>,
\end{align}
with $g_{12}\sbrack{2m}(\kappa)=g_{21}\sbrack{2m}(\kappa):= 0$, and similarly, $\gamma\sbrack{2m+1}(\kappa):=0$ and
\begin{align}
\gamma\sbrack{2m}(\kappa) :=& \sgn({i\mkern1mu}\kappa^2)\Big\< \delta_x,\ (\ensuremath{\partial}+i\kappa^2)^{-\frac12}(\Lambda\Gamma)^m(\ensuremath{\partial}+i\kappa^2)^{-\frac12}\delta_x\Big\> \notag \\
&
- \sgn({i\mkern1mu}\kappa^2)\Big\< \delta_x,\ (\ensuremath{\partial}-i\kappa^2)^{-\frac12}(\Gamma\Lambda)^m (\ensuremath{\partial}-i\kappa^2)^{-\frac12}\delta_x\Big\>. \label{rho-Series terms}
\end{align}
then by \eqref{Resolvent}, we have
\begin{equation}\label{g12-gamma-Series}
g_{12}(\kappa) = \sum_{\ell=1}^\infty g_{12}\sbrack{\ell}(\kappa),\quad g_{21}(\kappa) = \sum_{\ell=1}^\infty g_{21}\sbrack{\ell}(\kappa), \text{~and~} \gamma(\kappa) = \sum_{\ell=2}^\infty \gamma\sbrack{\ell}(\kappa) .
\end{equation}
We also write the tails of these series as
\begin{align*}
g_{12}\sbrack{\geq m}(\kappa) := & \sum_{\ell=m}^\infty g_{12}\sbrack{\ell}(\kappa), \quad g_{21}\sbrack{\geq m}(\kappa) := \sum_{\ell=m}^\infty g_{21}\sbrack{\ell}(\kappa), \\
& \gamma \sbrack{\geq m}(\kappa) := \sum_{\ell=m}^\infty \gamma \sbrack{\ell}(\kappa).
\end{align*}
By \eqref{g12-ID} \eqref{g21-ID} and \eqref{QuadraticID}, we obtain the identities
$$
g_{12} = - (2i \kappa^2+ \ensuremath{\partial})^{-1} [\kappa q+\kappa \gamma q],\quad g_{21} = - (2i \kappa^2-\ensuremath{\partial})^{-1} [\kappa r+\kappa \gamma r],
$$
and
$$
\gamma = -2g_{12} g_{21} -\tfrac12 \gamma^2,
$$
from which we have the explicit expressions for the leading order terms
\begin{gather}
g_{12}\sbrack{1}(\kappa) = - \tfrac {\kappa q}{2i \kappa^2 + \ensuremath{\partial}},\qquad g_{12}\sbrack{3} (\kappa)= \tfrac2{2i\kappa^2 + \ensuremath{\partial}}\big(\kappa q\cdot\tfrac {\kappa r}{2i\kappa^2 - \ensuremath{\partial}}\cdot \tfrac {\kappa q}{2i \kappa^2 + \ensuremath{\partial}}\big), \label{g12 1 3}\\
g_{21}\sbrack{1}(\kappa) = - \tfrac {\kappa r}{2i \kappa^2 - \ensuremath{\partial}},\qquad\hphantom{-} g_{21}\sbrack{3} (\kappa)= \tfrac{2}{2i\kappa^2 - \ensuremath{\partial}}\big(\kappa r\cdot \tfrac {\kappa q}{2i\kappa^2 + \ensuremath{\partial}}\cdot\tfrac {\kappa r}{2i\kappa^2 - \ensuremath{\partial}}\big),\label{g21 1 3}
\end{gather}
and
\begin{align}
\gamma\sbrack{2}(\kappa) &= - 2\,\tfrac {\kappa q}{2i\kappa^2 + \ensuremath{\partial}}\cdot\tfrac{\kappa r} {2i\kappa^2 - \ensuremath{\partial}}, \label{gamma 2}\\
\gamma\sbrack{4}(\kappa) &= \tfrac {\kappa q}{2i\kappa^2 + \ensuremath{\partial}}\cdot \tfrac 4{2i\kappa^2- \ensuremath{\partial}}\big(\kappa r\cdot\tfrac {\kappa q}{2i\kappa^2 + \ensuremath{\partial}}\cdot \tfrac {\kappa r}{2i\kappa^2 - \ensuremath{\partial}}\big) \notag \\
&\quad
+ \tfrac4{2i\kappa^2 + \ensuremath{\partial}}\big(\kappa q\cdot\tfrac {\kappa r}{2i \kappa^2 - \ensuremath{\partial}}\cdot \tfrac {\kappa q}{2i\kappa^2 + \ensuremath{\partial}}\big)\cdot \tfrac{\kappa r} {2i\kappa^2 - \ensuremath{\partial}} \notag\\
&\quad - 2\,\tfrac {\kappa q}{2i\kappa^2 + \ensuremath{\partial}}\cdot\tfrac {\kappa r}{2i\kappa^2 - \ensuremath{\partial}}\cdot \tfrac {\kappa q}{2i\kappa^2 + \ensuremath{\partial}}\cdot\tfrac {\kappa r}{2i\kappa^2 - \ensuremath{\partial}}. \label{gamma 4}
\end{align}
We are now ready to obtain some basic estimates on $g_{12}$, $g_{21}$.
\begin{prop}[Properties of \(g_{12}\) and \(g_{21}\)]\label{prop:g}
There exists \(\delta>0\) such that for all ${i\mkern1mu} \kappa^2\in \ensuremath{\mathbb{R}}$ with \(|\kappa|\geq 1\) the maps \(q\mapsto g_{12}(\kappa)\) and \(q\mapsto g_{21}(\kappa)\) are (real analytic) diffeomorphisms of \(B_\delta\) into \(H^{s+\frac{1}2}\) satisfying the estimates
\begin{equation}\label{g12-Hs}
\|g_{12}(\kappa)\|_{H^{s+\frac{1}2}_\kappa} + \|g_{21}(\kappa)\|_{H^{s+\frac{1}2}_\kappa} \lesssim |\kappa|\, \|q\|_{H^{s-\frac12 }_\kappa}.
\end{equation}
Further, the remainders satisfy the estimate
\begin{equation}\label{g12-LO}
\|g_{12}\sbrack{\geq 3}(\kappa)\|_{H^{s+\frac{1}2}_\kappa} + \|g_{21}\sbrack{\geq 3}(\kappa)\|_{H^{s+\frac{1}2}_\kappa} \lesssim |\kappa|\|q\|_{H^{s-\frac{1}2}_\kappa}\|q\|^2_{L^2},
\end{equation}
uniformly in $\kappa$. Finally, if \(q\) is Schwartz then so are \(g_{12}(\kappa)\) and \(g_{21}(\kappa)\).
\end{prop}
\begin{proof}
It suffices to consider the case \({i\mkern1mu} \kappa^2\geq 1\) as the case \({i\mkern1mu} \kappa^2\leq -1\) is similar, and by \eqref{grho-symmetries}, it suffices to consider $g_{12}(\kappa)$. Recalling \eqref{g12 1 3}, we obtain
\begin{equation}\label{linear isomorphism}
\|g_{12}\sbrack{1}(\kappa)\|_{H^{s+\frac{1}2}_\kappa} = |\kappa|\, \|q\|_{H^{s-\frac 12 }_\kappa}.
\end{equation}
To bound the remainder terms in the series\footnote{If we use $\mf I_p$ with $p>2$ instead of $\mf I_2$ once again, we may pay some regularity on $s>s_0>0$ to remove the small mass assumption.}, we employ duality and Lemma~\ref{L:BasicBounds}:
\begin{align}\label{trilinear est}
\bigl|\<f, g_{12}\sbrack{\geq 3}(\kappa)\>\bigr|
&\leq \|(\ensuremath{\partial}-i\kappa^2 )^{-s-\frac{1}2}\bar f(\ensuremath{\partial} + i\kappa^2 )^{-s-\frac{1}2}\|_{\mr{op}} \notag\\
&\qquad\qquad \times\sum_{\ell=1}^\infty\|(\ensuremath{\partial}+i\kappa^2 )^{s-\frac12}kq(\ensuremath{\partial}-i\kappa^2 )^{-\frac12}\|_{\mf I_2}^{2\ell+1}|\kappa|^{-2s(2\ell-1)} \notag\\
&\lesssim |\kappa|^{-2s} \|f\|_{H^{-s-\frac{1}2}_\kappa}\sum_{\ell=1}^\infty \left(|\kappa| \|q\|_{H_\kappa^{s-\frac{1}2}} \right)^{2\ell+1} |\kappa|^{-2s(2\ell-1)} \notag\\
&\lesssim\|f\|_{H^{-s-\frac{1}2}_\kappa} |\kappa|\, \|q\|_{H_\kappa^{s-\frac{1}2}}\sum_{\ell=1}^\infty \left(|\kappa|^{1-2s} \|q\|_{H_\kappa^{s-\frac{1}2}} \right)^{2\ell} \notag\\
&\lesssim\|f\|_{H^{-s-\frac{1}2}_\kappa} |\kappa|\, \|q\|_{H_\kappa^{s-\frac{1}2}} \|q\|^{2}_{L^2}
\end{align}
provided $\delta>0$ is sufficiently small. This proves \eqref{g12-LO} and completes the proof of \eqref{g12-Hs}.
In addtion, we have
\begin{equation*}
\tfrac{\delta g_{12}}{\delta q}(\kappa)\bigr|_{q=0} = - \frac{\kappa}{ 2i\kappa^2 + \ensuremath{\partial}} \qtq{and} \tfrac{\delta g_{12}}{\delta r}(\kappa)\bigr|_{q=0} = 0
\end{equation*}
which is an isomorphism, as noted already in \eqref{linear isomorphism}. Furthermore, for any $f\in \ensuremath{\mathscr{S}}$, we have
\begin{align*}
\left.\frac d{d\epsilon}\right|_{\epsilon = 0} G(x,z;q + \epsilon f) &= -\int G(x,y;q)\begin{bmatrix}0&f(y)\\ - \bar f(y)&0\end{bmatrix}G(y,z;q)\,dy.
\end{align*}
By similar analysis as those to prove \eqref{g12-LO}, we have
\begin{align*}
& \bigl\| \tfrac{\delta g_{12}}{\delta r}(\kappa) \bigr\|_{H^{s-\frac 12}_\kappa\rightarrow H^{s+\frac{1}2}_\kappa} + \bigl\| \tfrac{\delta g_{12}}{\delta q}(\kappa)+ \frac{\kappa}{ 2i\kappa^2 + \ensuremath{\partial}} \bigr\|_{H^{s-\frac 12}_\kappa\rightarrow H^{s+\frac{1}2}_\kappa}
\lesssim \|q\|^2_{L^2}
\lesssim \delta^2,
\end{align*}
and so the inverse function theorem implies the diffeomorphism property for sufficiently small $\delta$ .
The regulairty result can be easily obtained in a similar argument as those in \cite[Proposition $2.2$]{KV:KdV:AnnMath} and \cite[Proposition~3.2]{HKV:NLS}.
\end{proof}
We also have some estimates on $\gamma$ as follows.
\begin{prop}[Properties of \(\gamma\)]\label{prop:rho}
There exists \(\delta>0\) such that for all ${i\mkern1mu} \kappa^2\in \ensuremath{\mathbb{R}}$ with \(|\kappa|\geq 1\) the map \(q\mapsto \gamma(\kappa)\) is bounded from \(B_\delta\) to \(L^1\cap H^{s+\frac{1}2}\) and we have the estimates
\begin{align}
\|\gamma(\kappa)\|_{H^{s+\frac{1}2}_\kappa} &\lesssim |\kappa|^{2-2s}\|q\|_{H^{s-\frac{1}2}_\kappa}^2,\label{rho-Hs}\\
\|\gamma(\kappa)\|_{L^\infty} &\lesssim |\kappa|^{2-4s}\|q\|_{H^{s-\frac 12}_\kappa}^2,\label{rho-Linfty}\\
\|\gamma(\kappa)\|_{L^1} &\lesssim |\kappa |^{2} \|q\|_{H^{-1}_\kappa}^2 + |\kappa|^{-2(4s-1)}\|q\|_{H^{s-\frac 12}_\kappa}^4,\label{rho-L1}\\
\|\gamma\sbrack{\geq 4}(\kappa)\|_{L^1}&\lesssim |\kappa|^{-2(4s-1) }\|q\|_{H^{s-\frac 12}_\kappa}^4,\label{rho-LO}
\end{align}
uniformly in \(\kappa\).
Further, we have the quadratic identity
\begin{equation}
\label{QuadraticID}
\gamma + \frac12 \gamma^2 = -2 g_{12}g_{21},
\end{equation}
and if \(q\) is Schwartz, then so is \(\gamma(\kappa)\).
\end{prop}
\begin{proof}
Once again it suffices to consider the case \({i\mkern1mu} \kappa^2\geq1\). Using \eqref{gamma 2} and \eqref{E:algebra}, we obtain
\begin{align*}
\|\gamma\sbrack{2}\|_{H_\kappa^{s+\frac12}}\lesssim \|\tfrac {\kappa q}{2i\kappa^2 + \ensuremath{\partial}}\cdot\tfrac{\kappa r} {2i\kappa^2 - \ensuremath{\partial}} \|_{H_\kappa^{s+\frac12}}\lesssim |\kappa|^{2-2s}\| q\|_{H^{s-\frac12}_\kappa}^2.
\end{align*}
To handle $\gamma\sbrack{\geq 4}$ we use the series representation \eqref{g12-gamma-Series} and the same dual argument used to prove \eqref{g12-LO}. The estimate \eqref{rho-Linfty} then follows from \eqref{rho-Hs} via~\eqref{Linfty bdd}.
Choosing $\varkappa=\kappa$ in \eqref{micro As commute}, we obtain
$$
\partial_x \bigl\{ 2 g_{12}(x;\kappa)g_{21}(x;\kappa) + \tfrac12\gamma(x;\kappa)^2 + \gamma(x;\kappa) \bigr\} = 0.
$$
By \eqref{g12-Hs} and \eqref{rho-Hs}, we know that the term in the braces vanishes as $|x|\to\infty$. Thus the identity \eqref{QuadraticID} follows by integration.
By using this quadratic identity, we may write
\begin{equation}\label{E:vr gr 4}
\gamma\sbrack{\geq 4} = - \tfrac12 \gamma^2 - 2g_{12}\sbrack{\geq 3}\cdot g_{21} - 2g_{12}\sbrack{1}\cdot g_{21}\sbrack{\geq 3}.
\end{equation}
By Proposition~\ref{prop:g} and \eqref{rho-Hs}, we have
\begin{align*}
\|g_{12}\sbrack{\geq 3}\|_{L^2}+\|g_{21}\sbrack{\geq 3}\|_{L^2}&\lesssim |\kappa|^{-2(s+\frac12)}\bigl[\|g_{12}\sbrack{\geq 3}\|_{H^{s+\frac12}_\kappa}+ \|g_{21}\sbrack{\geq 3}\|_{H^{s+\frac12}_\kappa}\bigr] \lesssim |\kappa|^{-6s+2}\|q\|_{H^{s-\frac12 }_\kappa}^3,\\
\|g_{12}\|_{L^2}+\|g_{21}\|_{L^2}&\lesssim |\kappa|^{-2(s+\frac12)} \bigl( \|g_{12}\|_{H_\kappa^{s+\frac12}}+\|g_{21}\|_{H_\kappa^{s+\frac12}}\bigr)\lesssim |\kappa|^{-2s}\|q\|_{H^{s-\frac12}_\kappa},
\end{align*} and
\begin{align*}
\|\gamma\|_{L^2}&\lesssim |\kappa|^{-2(s+\frac12)}\|\gamma\|_{H^{s+\frac12}_\kappa}\lesssim|\kappa|^{-4s+1}\|q\|_{H^{s-\frac12}_\kappa}^2,
\end{align*}
which together with H\"older's inequality imply that
\begin{align*}
\|\gamma\sbrack{\geq 4}\|_{L^1} &\lesssim \|\gamma\|_{L^2}^2 + \|g_{12}\sbrack{\geq 3}\|_{L^2}\|g_{21}\|_{L^2} + \|g_{12}\sbrack{1}\|_{L^2}\|g_{21}\sbrack{\geq 3}\|_{L^2}\lesssim |\kappa|^{-2(4s-1)}\|q\|_{H^{s-\frac12}_\kappa}^4,
\end{align*}
thus we obtain \eqref{rho-LO}. The estimate \eqref{rho-L1} then follows from applying the Cauchy-Schwarz inequality to \eqref{gamma 2}.
The regulairty result can also be deduced by a similar argument as those in \cite[Proposition $2.2$]{KV:KdV:AnnMath} and \cite[Proposition~3.3]{HKV:NLS}.
\end{proof}
Due to the structure of microscopic conservation law in \eqref{micro A}, the combination function \(\frac{g_{12}(\kappa)}{2 + \gamma(\kappa)}\) will be also used later. We now give the analogue estimates. Firstly, we denote:
\begin{equation*}
\tfrac{g_{12}}{2 + \gamma} = \big(\tfrac{g_{12}}{2 + \gamma}\big)\sbrack{1} + \big(\tfrac{g_{12}}{2 + \gamma}\big)\sbrack{ 3} + \big(\tfrac{g_{12}}{2 + \gamma}\big)\sbrack{\geq 5},
\end{equation*}
where the leading order terms are given by
\begin{equation}\label{more sbrack}
\big(\tfrac{g_{12}}{2 + \gamma}\big)\sbrack{1} = \tfrac12 g_{12}\sbrack{1} \qtq{and} \big(\tfrac{g_{12}}{2 + \gamma}\big)\sbrack{3} = \tfrac12 g_{12}\sbrack{3} - \tfrac14g_{12}\sbrack{1}\gamma\sbrack{2},
\end{equation}
and the remainder term is given by
\begin{align}
\big(\tfrac{g_{12}}{2 + \gamma}\big)\sbrack{\geq 3} &= \tfrac12 g_{12}\sbrack{\geq 3} - \tfrac{g_{12} \gamma}{2(2 + \gamma)}. \label{more sbrack'}
\end{align}
We can now show the following estimates about $\tfrac{g_{12}(\kappa)}{2 + \gamma(\kappa)}$.
\begin{cor}\label{C:ET}
Let $s\in (0,1/2)$ and $q\in B_\delta$. there exists \(\delta>0\) such that for all ${i\mkern1mu} \kappa^2\in \ensuremath{\mathbb{R}}$ with $|\kappa|\geq 1$, we have the estimates
\begin{align}
|\kappa|^2\bigl\|\tfrac{g_{12}(\kappa)}{2 + \gamma(\kappa)}\bigr\|_{H^{s-1/2}} + \bigl\| \tfrac{g_{12}(\kappa)}{2 + \gamma(\kappa)}\bigr\|_{H^{s+1/2}}
&\lesssim |\kappa| \|q\|_{H^{s-1/2}},\label{ET Sob} \\
|\kappa|^2\bigl\|\big(\tfrac{g_{12}(\kappa)}{2 + \gamma(\kappa)}\big)\sbrack{\geq 3}\bigr\|_{H^{s-1/2}} + \bigl\| \big(\tfrac{g_{12}(\kappa)}{2 + \gamma(\kappa)}\big)\sbrack{\geq 3}\bigr\|_{ H^{s+1/2}}
&\lesssim |\kappa| \|q\|_{H^{s-1/2}}\|q\|_{L^2}^2. \label{ET1 Sob}
\end{align}
\end{cor}
\begin{proof}
From \eqref{more sbrack} and \eqref{g12 1 3}, we see that
$$
|\kappa|^2\bigl\|\big(\tfrac{g_{12}}{2 + \gamma}\big)\sbrack{1}\bigr\|_{H^{s-1/2}} + \bigl\| \big(\tfrac{g_{12}}{2 + \gamma}\big)\sbrack{1}\bigr\|_{ H^{s+1/2}}
\approx \bigl\| (2i\kappa^2+\ensuremath{\partial}) \big(\tfrac{g_{12}(\varkappa)}{2 + \gamma(\varkappa)}\big)\sbrack{1}\bigr\|_{ H^s}
\approx|\kappa| \|q\|_{H^{s-1/2}}.
$$
Thus it suffice for \eqref{ET Sob} to show \eqref{ET1 Sob}. Moreover, by \eqref{g12-ID}, we have
$$
(2i\kappa^2+\ensuremath{\partial}) \big(\tfrac{g_{12}}{2 + \gamma}\big)\sbrack{\geq 3} = - \kappa \tfrac{\gamma}{2(2+\gamma)} q + \tfrac{g_{12}}{(2 + \gamma)^2} \gamma',
$$
and therefore, we have
\begin{align*}
\text{LHS of \eqref{ET1 Sob}} &\lesssim |\kappa| \bigl\| \tfrac{\gamma}{2+\gamma} q \bigr\|_{H^{s-1/2}} + \bigl\| \tfrac{g_{12}}{(2 + \gamma)^2} \gamma' \bigr\|_{H^{s-1/2}} \\
&\lesssim |\kappa|^{1-2s} \|q\|_{H^{s-1/2}} \bigl\| \tfrac{\gamma}{2+\gamma} \bigr\|_{ H^{s+1/2}_\kappa}
+ |\kappa|^{-2s} \|\gamma'\|_{H^{s-1/2}} \bigl\| \tfrac{g_{12}}{(2 + \gamma)^2} \bigr\|_{ H^{s+1/2}_\kappa}\\
&\lesssim |\kappa|^{1-2s} \|q\|_{H^{s-1/2}} \bigl\|\gamma \bigr\|_{ H^{s+1/2}_\kappa} \\
&\lesssim |\kappa| \|q\|_{H^{s-1/2}} \|q\|^2_{L^2},
\end{align*}
where the second step we use \eqref{multiplier bdd on ss} and \eqref{rho-Hs}, and the third step we expand as series and employ the algebra property \eqref{E:algebra}, together with \eqref{g12-Hs} and \eqref{rho-Hs}. This yields \eqref{ET Sob} for sufficiently small $\delta$.
\end{proof}
\section{Conservation laws and dynamics}\label{S:4}
In this section, we will firstly introduce the invariant quantity $A(\kappa)$ from the logrithmic perturbation determinant, which is related to the integrability and spectral invariance of \eqref{DNLS}, then we show the dynamics and microscopic conservation laws of the $A(\kappa)$'s flow and the DNLS (i.e. $H_{\mr{DNLS}}$) flow, respectively.
\subsection{Conservation Laws}
Inspired by \cite{HKV:NLS, KV:KdV:AnnMath,KVZ:KdV:GAFA, Rybkin:KdV:Cons Law}, we formally define the logarithmic perturbation determinant $\sgn({i\mkern1mu}\kappa^2)\log\det(L_0^{-1}L)$ as follows:
\begin{equation}\label{A-def}
A(\kappa; q,r) := \sgn({i\mkern1mu}\kappa^2)\sum_{j=1}^\infty\frac{(-1)^{j-1}}j \tr\left\{\left(\sqrt{R_0}\left(L - L_0\right)\sqrt{R_0}\right)^j\right\}.
\end{equation}
By \eqref{LambdaGammaJob}, simple calculations deduce that for $q,r\in \ensuremath{\mathscr{S}}$, we have
\begin{equation}\label{A-def'}
A(\kappa; q,r) = - \sgn({i\mkern1mu}\kappa^2)\sum_{m=1}^\infty \tfrac{1}{m} \tr\left\{(\Lambda\Gamma)^m \right\}.
\end{equation}
In the following, we will use \eqref{A-def'} as the definition of $A(\kappa; q,r)$.
For the sake of simplicity, we write
\begin{equation}\label{Am to A}
A(\kappa; q,r) = \sum\limits_{m=1}^{\infty} A_m(\kappa; q,r), \; A_m(\kappa; q,r) :=-\sgn({i\mkern1mu}\kappa^2)\tfrac{1}{m} \tr\left\{(\Lambda\Gamma)^m \right\}.
\end{equation}
Firstly, we have
\begin{lem}[Properties of $A$]\label{L:A}
There exists $\delta>0$ such that for all $q\in B_\delta \cap \ensuremath{\mathscr{S}}$ and $i \kappa^2\in \ensuremath{\mathbb{R}}$ with $|\kappa|\geq 1$, the series \eqref{A-def'} converges absolutely. Moreover, we have
\begin{gather}
\tfrac{\delta\,}{\delta q} A(\kappa) =-\kappa g_{21}(\kappa),
\quad \tfrac{\delta\,}{\delta r}A(\kappa) = -\kappa g_{12}(\kappa), \label{gij from A}\\
\; \gamma'(\kappa) = 2\left(-q\tfrac{\delta\,}{\delta q}A(\kappa) + r\tfrac{\delta\,}{\delta r}A(\kappa) \right).
\label{rho from A}
\end{gather}
\end{lem}
\begin{proof}
By \eqref{Lambda}, we know that $ A(\kappa; q,r)$ converges absolutely for all $q\in B_\delta\cap \ensuremath{\mathscr{S}}$.
By simple calculations, we have
\begin{equation}\label{rho from Am}
\tfrac{\delta\,\,}{\delta q}A_m =-\kappa g_{21}\sbrack{2m-1}
\qtq{and}
\tfrac{\delta\,\,}{\delta r}A_m =-\kappa g_{21}\sbrack{2m-1},
\end{equation}
which together with \eqref{rho-ID}, \eqref{g12-gamma-Series} implies \eqref{rho from A}.
\end{proof}
Next, we show that the mass, the Hamiltonian and the energy defined by \eqref{HDNLS} and \eqref{Mass Energy} respectively, arise as the coefficients in the asymptotic expansion of $A(\kappa)$ as $|\kappa| \to\infty$. More precisely, we have.
\begin{lem}[Asymptotic expansion of $A(\kappa)$]\label{L:A asymptotics} For $q\in B_\delta\cap \ensuremath{\mathscr{S}}$, we have, as $|\kappa|\to\infty$,
\begin{equation}\label{A asymptotics}
A(\kappa)=(-{i\mkern1mu})\tfrac{M}{2} - \tfrac{(-{i\mkern1mu})^2}{2i\kappa^2}\frac{H_{\mr{DNLS}}}{2} +\frac{(-i)^3}{(2i\kappa^2)^2} \frac{E_{\mr{DNLS}}}{2}+ O(|\kappa|^{-6}).
\end{equation}
\end{lem}
\begin{proof}
By \eqref{g12-ID} and \eqref{g21-ID}, and \eqref{rho from A}, we have
\begin{equation*}
\begin{split}
2 {i\mkern1mu} \kappa ^2 \tfrac{\delta A}{\delta q}
= & \;
\partial \tfrac{\delta A}{\delta q} + 2 \kappa^2 r \cdot \partial^{-1}\left(-q \tfrac{\delta A}{\delta q} + r \tfrac{\delta A}{\delta r}\right) + \kappa^2 r ,
\\
2 {i\mkern1mu} \kappa ^2 \tfrac{\delta A}{\delta r}
= & -
\partial \tfrac{\delta A}{\delta r} + 2 \kappa^2 q \cdot \partial^{-1}\left(-q \tfrac{\delta A}{\delta q} + r \tfrac{\delta A}{\delta r}\right) + \kappa^2 q.
\end{split}
\end{equation*}
By the asymptotic analysis, we have
\begin{equation}\label{biHam2}
\begin{split}
\tfrac{\delta A}{\delta q} &= -\kappa g_{21} \\
& =-{i\mkern1mu}
\frac{1}{2}r+\frac{1}{2{i\mkern1mu}\kappa^2}\bigl(-\frac{ {i\mkern1mu}}{2} r'+\frac{1}{2} qr^2 \bigr)+ \frac{1}{(2{i\mkern1mu} \kappa^2)^2} \left(-\frac{{i\mkern1mu}}{2}r'' + \frac{3}{2} qr r' + \frac{3}{4} {i\mkern1mu} q^2r^3\right) + O(|\kappa|^{-6}),
\\
\tfrac{\delta A}{\delta r} &= -\kappa g_{12} \\
& =-{i\mkern1mu}
\frac{1}{2}q+\frac{1}{2{i\mkern1mu}\kappa^2}\bigl(\frac{ {i\mkern1mu}}{2} q'+\frac{1}{2} q^2r \bigr)+ \frac{1}{(2{i\mkern1mu} \kappa^2)^2} \left(-\frac{{i\mkern1mu}}{2}q'' - \frac{3}{2} qq'r + \frac{3}{4} {i\mkern1mu} q^3r^2\right) + O(|\kappa|^{-6}),
\end{split}
\end{equation}
which together with the fact that
\begin{equation}\label{recovA}
A(q,r) = \int_0^1 \partial_\theta A(\theta q,\theta r)\,{\rm{d}}\theta
\end{equation}
imply \eqref{A asymptotics}.
\end{proof}
\begin{rem}
By computing $\gamma= 2 \partial^{-1}\left( - q\tfrac{\delta A}{\delta q} + r\tfrac{\delta A}{\delta r}\right)$,
we can obtain asymptotic expansion for $\gamma$,
\begin{align}
\label{vr asymptotics}
\gamma =
\frac{1}{2\kappa^2}qr-\frac{1}{4\kappa^4}\bigl( {i\mkern1mu} q'r-{i\mkern1mu} qr'+\frac{3}{2}q^2r^2 \bigr)+O(|\kappa|^{-6}).
\end{align}
\end{rem}
\begin{lem}[Density function of $A(\kappa)$]\label{L:micro A}
For all $q\in B_\delta\cap\ensuremath{\mathscr{S}}$ and $i \kappa^2\in \ensuremath{\mathbb{R}}$ with $|\kappa|\geq 1$, we have
\begin{gather}
A(\kappa) = - \bar A(-\bar{\kappa}), \label{A-Symmetries} \\
\tfrac{\partial\, }{\partial \kappa}A(\kappa) =
\int_{\ensuremath{\mathbb{R}}}\Bigl[ 2i\kappa \cdot \gamma(\kappa)- \bigl(qg_{21}(\kappa)+rg_{12}(\kappa)\bigr)\Bigr]\,{\rm{d}}x ,
\label{rho-to-A} \\
A (\kappa) = \int_{\ensuremath{\mathbb{R}}} \rho(\kappa)\,{\rm{d}}x, \qtq{where} \rho(\kappa)=-\kappa\frac{qg_{21}(\kappa)+rg_{12}(\kappa)}{2+\gamma(\kappa)}. \label{micro A}
\end{gather}
\end{lem}
\begin{proof}
Firstly, we show that \eqref{rho-to-A}. In fact, simple calculations imply that
\begin{equation}\label{rho to Am}
\frac{\partial\ }{\partial \kappa}A_m
=
\int_{\ensuremath{\mathbb{R}}}\Bigl[ 2{i\mkern1mu}\kappa\gamma\sbrack{2m} - \bigl(q g_{21}\sbrack{2m-1}+ r g_{12}\sbrack{2m-1} \bigr)\Bigr]\,{\rm{d}}x.
\end{equation}
By summation with respect to $m$, we can obtain \eqref{rho-to-A}.
We now estimate \eqref{micro A}.
On one hand,
by differentiating \eqref{g12-ID}, \eqref{g21-ID}, and \eqref{QuadraticID} with respect to $\kappa$,
we obtain that,
\begin{align*}
\partial_x\Bigl(g_{12}\tfrac{\partial g_{21}}{\partial\kappa} - \tfrac{\partial g_{12}}{\partial\kappa} g_{21}\Bigr)
=
&
\kappa(qg_{21}+rg_{12})\tfrac{\partial\ }{\partial\kappa}(\gamma+1)
-\kappa(\gamma+1)\tfrac{\partial\ }{\partial\kappa}(qg_{21}+rg_{12})
\\
&
-2{i\mkern1mu}\kappa\gamma(\gamma+2)
+(\gamma+1)(qg_{21}+rg_{12}).
\end{align*}
Using \eqref{rho-ID}, we have
\begin{align*}
- \bigl( g_{12}\tfrac{\partial g_{21}}{\partial\kappa} - \tfrac{\partial g_{12}}{\partial\kappa} g_{21}\bigr) \gamma' = -\gamma(2+\gamma)\tfrac{\partial\ }{\partial\kappa}\bigl(qg_{21}-rg_{12}\bigr)+ \bigl(qg_{21}-rg_{12}\bigr)(1+\gamma)\tfrac{\partial\gamma}{\partial\kappa}.
\end{align*}
Combining the above two identities, we get
\begin{align*}
\partial_x\frac{g_{12}\tfrac{\partial g_{21}}{\partial\kappa} - \tfrac{\partial g_{12}}{\partial\kappa} g_{21}}{2+\gamma} &= \bigl(qg_{21}+rg_{12}-2{i\mkern1mu}\kappa\gamma\bigr) - \frac{\partial\ }{\partial\kappa}\bigl(\kappa\frac{qg_{21}+rg_{12}}{2+\gamma}\bigr) ,
\end{align*}
which can be integrated in $x$ to yield
\begin{align}\label{iden dAdk}
\frac{\partial\ }{\partial\kappa} \int\kappa \frac{qg_{21}+rg_{12}}{2+\gamma}\,{\rm{d}}x = \int qg_{21}+rg_{12}-2{i\mkern1mu}\kappa\gamma\,{\rm{d}}x = -\frac{\partial A}{\partial\kappa} .
\end{align}
On the other hand, by \eqref{A asymptotics}, we have
\begin{equation}\label{A expand}
A(\kappa)=-i\tfrac{1}{2} M+O( |\kappa|^{-2} ), \qtq{as} \kappa\to\infty.
\end{equation} and by \eqref{biHam2} and \eqref{vr asymptotics}, we obtain
\begin{equation}\label{vr expand}
\rho(\kappa)=-{i\mkern1mu}\frac{1}{2}qr + O(|\kappa|^{-2}), \qtq{as} \kappa\to\infty.
\end{equation}
Combining \eqref{iden dAdk}, \eqref{A expand} and \eqref{vr expand}, we can obtain \eqref{micro A}.
Lastly, \eqref{A-Symmetries} is obvious from \eqref{grho-symmetries} and \eqref{micro A}.
\end{proof}
Next, we show the commutation of $A(\kappa)$'s under the Poisson bracket \eqref{PoissonBracket}, which implies that $A(\varkappa)$ is an invariant quantity under the $A(\kappa)$'s flows.
\begin{lem}[Poisson brackets]\label{lem:PoissonBrackets}
There exists $\delta>0$ such that for any $\kappa$ and $\varkappa$ with ${i\mkern1mu}\kappa^2, \, {i\mkern1mu} \varkappa^2 \in \ensuremath{\mathbb{R}}\setminus(-1,1)$ with $\kappa^2\neq\varkappa^2$ and any $q\in \Bd\cap \Schwartz$ we have
\begin{equation}\label{APoissonBracket}
\{A(\kappa),A(\varkappa)\} = 0.
\end{equation}
\end{lem}
\begin{proof}
By \eqref{rho from A} and \eqref{PoissonBracket}, we have
\begin{align*}
\{A(\kappa),A(\varkappa)\} &=\kappa\varkappa \int g_{12}(\kappa)g_{21}'(\varkappa) + g_{21}(\kappa)g_{12}'(\varkappa)\,{\rm{d}}x,
\end{align*}
which together with \eqref{micro As commute} implies that \eqref{APoissonBracket} holds. .
\end{proof}
Next we will exhibit the dynamics of $g_{12}$, $g_{21}$ and $\gamma$ defined by \eqref{def gama}, \eqref{def g12} and \eqref{def g21} along the $A(\kappa)$ flow and DNLS flow respectively.
\subsection{Dynamics I: the $A(\kappa)$ flow.}
Firstly, we have
\begin{lem}[Dynamics of $A(\kappa)$ flow]\label{L:A flow}
Let $i\kappa^2\in\ensuremath{\mathbb{R}}$ with $|\kappa|>1$. Under the $A(\kappa)$ flow, we have
\begin{align}\label{qr under A}
\frac{\,{\rm{d}}\ }{\,{\rm{d}} t}q = - \kappa g_{12}'(\kappa) \qtq{and} \frac{\,{\rm{d}}\ }{\,{\rm{d}} t}r =- \kappa g_{21}'(\kappa).
\end{align}
\end{lem}
\begin{proof}
It is obvious from \eqref{rho from A} and \eqref{HFlow}.
\end{proof}
\begin{lem}[Lax pair for the $A(\kappa)$ flow]\label{P:Lax A}
Let ${i\mkern1mu}\kappa^2, {i\mkern1mu}\varkappa^2\in\ensuremath{\mathbb{R}}\setminus(-1,1)$ with $\kappa^2\neq\varkappa^2$, and $L(\varkappa)$ be defined by \eqref{Intro KN L}. Under the $A(\kappa)$ flow, we have
\begin{align}\label{Lax rep A flow}
\frac{\,{\rm{d}}\ }{\,{\rm{d}} t} L(\varkappa) = [ P_{A(\kappa)}, L(\varkappa)],
\end{align}
where
\begin{equation}\label{A B}
P_{A(\kappa)}=
\begin{bmatrix}-\frac 12 \Xi (\gamma(\kappa) + 1) & - \Theta g_{12}(\kappa)\\-\Theta g_{21}(\kappa)&\frac 12 \Xi (\gamma(\kappa) + 1)\end{bmatrix} \text{~~with~~}
\Theta=\tfrac{\varkappa \kappa^3}{\kappa^2-\varkappa^2}, ~~ \Xi=\tfrac{\varkappa^2\kappa^2}{\kappa^2-\varkappa^2}.
\end{equation}
\end{lem}
\begin{proof}
Firstly, by \eqref{rho-ID} and the fact that
\begin{align*} \gamma'(\kappa)
= & (\partial+{i\mkern1mu}\varkappa^2)\bigl( \gamma(\kappa)+1 \bigr)-\bigl( \gamma(\kappa)+1 \bigr)(\partial+{i\mkern1mu}\varkappa^2) \\
= & (\partial-{i\mkern1mu}\varkappa^2)\bigl( \gamma(\kappa)+1 \bigr)-\bigl( \gamma(\kappa)+1 \bigr)(\partial-{i\mkern1mu}\varkappa^2),
\end{align*}
we have
\begin{align}
\label{L11}
(\partial+{i\mkern1mu}\varkappa^2)\bigl( \gamma(\kappa)+1 \bigr)-\bigl( \gamma(\kappa)+1 \bigr)(\partial+{i\mkern1mu}\varkappa^2)-2\kappa\bigl(qg_{21}(\kappa) - rg_{12}(\kappa)\bigr)=0.
\\
\label{L22}
(\partial-{i\mkern1mu}\varkappa^2)\bigl( \gamma(\kappa)+1 \bigr)-\bigl( \gamma(\kappa)+1 \bigr)(\partial-{i\mkern1mu}\varkappa^2)-2\kappa\bigl(qg_{21}(\kappa) - rg_{12}(\kappa)\bigr)=0.
\end{align}
Next, by \eqref{g12-ID} and the fact that
\begin{equation*}
g_{12}'(\kappa)+2{i\mkern1mu}\varkappa^2g_{12}(\kappa)
=
(\partial+{i\mkern1mu}\varkappa^2)g_{12}(\kappa) -g_{12}(\kappa)(\partial-{i\mkern1mu}\varkappa^2),
\end{equation*}
we get
\begin{align}\label{L12}
(\kappa^2- \varkappa^2 )g_{12}'(\kappa)
=
\kappa^2\bigl[ (\partial+{i\mkern1mu}\varkappa^2)g_{12}(\kappa) -g_{12}(\kappa) (\partial-{i\mkern1mu}\varkappa^2)\bigr]+\kappa\varkappa^2 q\bigl(\gamma(\kappa)+1\bigr).
\end{align}
Finally, by \eqref{g21-ID} and the fact that
\begin{equation*}
g_{21}'(\kappa)-2{i\mkern1mu}\varkappa^2g_{21}(\kappa)
=
(\partial-{i\mkern1mu}\varkappa^2)g_{21}(\kappa) -g_{21}(\kappa)(\partial+{i\mkern1mu}\varkappa^2),
\end{equation*}
we obtain
\begin{align}\label{L21}
(\kappa^2 - \varkappa^2 )g_{21}'(\kappa)
=
\kappa^2\bigl[ (\partial-{i\mkern1mu}\varkappa^2)g_{21}(\kappa) - g_{21}(\kappa)(\partial+{i\mkern1mu}\varkappa^2) \bigr]-\kappa\varkappa^2 r\bigl(\gamma(\kappa)+1\bigr).
\end{align}
Combining \eqref{Intro KN L}, \eqref{qr under A} with
\eqref{L11}, \eqref{L22}, \eqref{L12}, \eqref{L21}, we can obtain \eqref{Lax rep A flow}.
\end{proof}
\begin{prop}[Microscopic conservation law for the $A(\kappa)$ flow]\label{Thm:A flow}
Let ${i\mkern1mu}\varkappa^2, {i\mkern1mu} \kappa^2\in\ensuremath{\mathbb{R}}\setminus(-1,1)$ with $\kappa^2 \neq\varkappa^2$. Under the $A(\kappa)$ flow, we have
\begin{align}
\frac{\,{\rm{d}}\ }{\,{\rm{d}} t} g_{12}(\varkappa) &=-\Xi g_{12}(\varkappa) \left[\gamma(\kappa)+1\right] + \Theta g_{12}(\kappa)\left[\gamma(\varkappa)+1\right] , \label{Ady g12}\\
\frac{\,{\rm{d}}\ }{\,{\rm{d}} t} g_{21}(\varkappa) &= \Xi g_{21}(\varkappa) \left[\gamma(\kappa)+1\right] - \Theta g_{21}(\kappa)\left[\gamma(\varkappa)+1\right] , \label{Ady g21}\\
\frac{\,{\rm{d}}\ }{\,{\rm{d}} t} \gamma(\varkappa) &=-
2 \Theta \left[ g_{12}(\kappa) g_{21}(\varkappa) - g_{21}(\kappa)g_{12}(\varkappa) \right],\label{Ady gam}
\end{align}
and the following microscopic conservation laws
\begin{align}
\label{Ady gam claw}
\partial_t\bigl\{ 2i\varkappa\cdot \gamma(\varkappa) -\left(q g_{21}(\varkappa)+r g_{12}(\varkappa)\right)\bigr\}
+ & \partial_x j_{\gamma}(\varkappa,\kappa)=0,\\
\label{A flow micr cons law}
\partial_t \rho(\varkappa) + \partial_x j_{A(\kappa)}(\varkappa,\kappa) =0, &
\end{align}
where the flux functions $ j_{\gamma}$ and $j_{A(\kappa)}$ are determined by
\begin{align}\label{A:rho dot}
j_{\gamma}(\varkappa,\kappa):
= & -
\tfrac{ \kappa^3(\kappa^2+\varkappa^2) }{ ( \kappa^2-\varkappa^2 )^2 }
\bigl[ g_{12}(\kappa)g_{21}(\varkappa)+g_{21}(\kappa)g_{12}(\varkappa)\bigr] \notag \\&
-
\tfrac{\kappa^4\varkappa}{ ( \kappa^2-\varkappa^2 )^2 }
\bigl[( \gamma(\kappa)+1 )( \gamma(\varkappa)+1 ) \bigr],
\\ \label{j sub A}
j_{A(\kappa)}(\varkappa,\kappa):= & -\Theta \tfrac{ g_{12}(\kappa)g_{21}(\varkappa)+g_{12}(\varkappa)g_{21}(\kappa) }{2 + \gamma(\varkappa) }
-\Theta \frac{\varkappa}{2\kappa}
\gamma(\kappa)
.
\end{align}
\end{prop}
\begin{rem}Due to the coercivity of the quadratic term of the flux, the conservation law \eqref{A flow micr cons law} is more useful than \eqref{Ady gam claw}.
\end{rem}
\begin{proof}
Firstly, we show the dynamics \eqref{Ady g12}, \eqref{Ady g21} and \eqref{Ady gam}. By \eqref{HFlow}, Proposition \ref{P:R} and Lemma \ref{P:Lax A}, we have
\begin{align*}
\frac{\,{\rm{d}}\ }{\,{\rm{d}} t} G(x,z;\varkappa)
&=
- \int G(x,y;\varkappa)\frac{\,{\rm{d}}\ }{\,{\rm{d}} t} L(\varkappa)G(y,z;\varkappa)\,{\rm{d}}{y} \\
&= \int G(x,y;\varkappa)
\begin{bmatrix}
0 & -\kappa^2g_{12}'(\kappa) \\
-\kappa^2g_{21}'(\kappa) & 0
\end{bmatrix}G(y,z;\varkappa)\,{\rm{d}}{y}
\\
&=
P_{A(\kappa)}(x;\kappa,\varkappa)G(x,z;\varkappa)-G(x,z;\varkappa)P_{A(\kappa)}(z;\kappa,\varkappa).
\end{align*}
By choosing $x=z$ and \eqref{Lax rep A flow}, we have
\begin{align*}
\frac{\,{\rm{d}}\ }{\,{\rm{d}} t} G(x,x;\varkappa)
= &
- \Xi
\begin{bmatrix}
0
&g_{12}(\varkappa)\bigl( \gamma(\kappa)+1 \bigr)
\\
-g_{21}(\varkappa)\bigl( \gamma(\kappa)+1 \bigr)
& 0
\end{bmatrix}
\\ &
-
\Theta
\begin{bmatrix}
g_{12}(\kappa)g_{21}(\varkappa)-g_{21}(\kappa)g_{12}(\varkappa)
& -g_{12}(\kappa)\bigl( \gamma(\varkappa)+1 \bigr)
\\
g_{21}(\kappa)\bigl( \gamma(\varkappa)+1 \bigr)
& -g_{12}(\kappa)g_{21}(\varkappa)+g_{21}(\kappa)g_{12}(\varkappa)
\end{bmatrix},
\end{align*}
which implies \eqref{Ady g12}, \eqref{Ady g21} and \eqref{Ady gam}.
Next, we prove \eqref{Ady gam claw}. A direct computation implies that
\begin{align}
\notag &
\partial_t\bigl\{2i\varkappa\gamma(\varkappa)- q g_{21}(\varkappa)-r g_{12}(\varkappa)\bigr\}
\\
=&
\label{A:rho dot 1}
- g_{21}(\varkappa)\frac{\,{\rm{d}}\ }{\,{\rm{d}} t}q
- g_{12}(\varkappa)\frac{\,{\rm{d}}\ }{\,{\rm{d}} t}r
+ 2{i\mkern1mu}\varkappa\frac{\,{\rm{d}}\ }{\,{\rm{d}} t}\gamma(\varkappa)
- q \frac{\,{\rm{d}}\ }{\,{\rm{d}} t}g_{21}(\varkappa)
- r \frac{\,{\rm{d}}\ }{\,{\rm{d}} t}g_{12}(\varkappa).
\end{align}
On the one hand, by Lemma \ref{L:A flow}, we have
\begin{equation}\label{A:rho dot 11}
- g_{21}(\varkappa)\frac{\,{\rm{d}}\ }{\,{\rm{d}} t}q
- g_{12}(\varkappa)\frac{\,{\rm{d}}\ }{\,{\rm{d}} t}r
=
\kappa\left[ g_{21}(\varkappa)g_{12}'(\kappa) + g_{12}(\varkappa)g_{21}'(\kappa) \right].
\end{equation}
On the other hand, by \eqref{Ady g12}, \eqref{Ady g21}, and \eqref{Ady gam}, we obtain that
\begin{align}
\notag
& 2{i\mkern1mu}\varkappa\frac{\,{\rm{d}}\ }{\,{\rm{d}} t}\gamma(\varkappa)
- q \frac{\,{\rm{d}}\ }{\,{\rm{d}} t}g_{21}(\varkappa)
- r \frac{\,{\rm{d}}\ }{\,{\rm{d}} t}g_{12}(\varkappa)
\\
=
\notag
&
-4 i\varkappa\Theta g_{12}(\kappa)g_{21}(\varkappa)
-\Xi g_{21}(\varkappa) q\left[ \gamma(\kappa)+1 \right]
+\Theta g_{21}(\kappa) q\left[ \gamma(\varkappa)+1 \right]
\\
\notag
&+4 i\varkappa\Theta g_{12}(\kappa)g_{21}(\varkappa)
+\Xi g_{12}(\varkappa) r\left[ \gamma(\kappa)+1 \right]
-\Theta g_{12}(\kappa) r\left[ \gamma(\varkappa)+1 \right]
\\
=
&
\frac{\varkappa\Theta}{\kappa^2}\left[ g_{21}(\varkappa)g_{12}'(\kappa) + g_{12}(\varkappa)g_{21}'(\kappa) \right]
-\frac{\Theta}{\varkappa}\left[ g_{21}(\kappa)g_{12}'(\varkappa) + g_{12}(\kappa)g_{21}'(\varkappa) \right].
\label{A:rho dot 21}
\end{align}
Combining \eqref{A:rho dot 1}, \eqref{A:rho dot 11}, \eqref{A:rho dot 21} with \eqref{micro As commute}, we can obtain
\eqref{A:rho dot}.
Finally, let us show that \eqref{j sub A} holds. A direct calculation implies that
\begin{align}
& \; \bigl( \gamma(\varkappa)+2 \bigr)^2\partial_t \rho(\varkappa)
\notag \\
=
&
\label{j sub A 1}
-\varkappa\bigl[
g_{21}(\varkappa)\frac{\,{\rm{d}}\ }{\,{\rm{d}} t}q+ g_{12}(\varkappa)\frac{\,{\rm{d}}\ }{\,{\rm{d}} t}r
\bigr]\bigl( \gamma(\varkappa)+2 \bigr)
\\
\label{j sub A 2}
&
-
\varkappa\bigl[
q \frac{\,{\rm{d}}\ }{\,{\rm{d}} t}g_{21}(\varkappa)+r \frac{\,{\rm{d}}\ }{\,{\rm{d}} t}g_{12}(\varkappa)
\bigr]\bigl( \gamma(\varkappa)+2 \bigr)
\\
\label{j sub A 3}
&
+\varkappa\bigl[
q g_{21}(\varkappa)+r g_{12}(\varkappa)
\bigr]\partial_t \gamma(\varkappa).
\end{align}
By Lemma \ref{L:A flow} and \eqref{Ady g12}, \eqref{Ady g21}, we have
\begin{align}\label{j sub A 12}
\notag
& \eqref{j sub A 1}+\eqref{j sub A 2}
\\
\notag
= &
\varkappa\kappa\left[ g_{21}(\varkappa)g_{12}'(\kappa)+g_{12}(\varkappa)g_{21}'(\kappa) \right]\bigl( \gamma(\varkappa)+2 \bigr)
\\
\notag
&+\frac{\varkappa\Theta}{2\kappa}\gamma'(\kappa)\bigl( \gamma(\varkappa)+1 \bigr)\bigl( \gamma(\varkappa)+2 \bigr)
-\frac{\Xi}{2}\gamma'(\varkappa)\bigl( \gamma(\kappa)+1 \bigr)\bigl( \gamma(\varkappa)+2 \bigr)
\\
\notag
= &
\varkappa\kappa\left[ g_{21}(\varkappa)g_{12}'(\kappa)+g_{12}(\varkappa)g_{21}'(\kappa) \right]\bigl( \gamma(\varkappa)+2 \bigr)
\\
&-\frac{\Xi}{2}\left[\bigl(\gamma(\kappa)+1\bigr)\bigl( \gamma(\varkappa)+1 \bigr)\right]'\bigl( \gamma(\varkappa)+2 \bigr)
+\Xi\gamma'(\kappa)\bigl( \gamma(\varkappa)+1 \bigr)\bigl( \gamma(\varkappa)+2 \bigr).
\end{align}
By \eqref{Ady gam}, we have
\begin{align}\label{j sub A 3 1}
\notag
\eqref{j sub A 3}
= &
-2\varkappa\Theta
\left[qg_{21}(\varkappa)+rg_{12}(\varkappa)\right]
\left[ g_{12}(\kappa)g_{21}(\varkappa)-g_{21}(\kappa)g_{12}(\varkappa) \right]
\\
\notag
=&
-2\varkappa\Theta
\left[qg_{21}(\varkappa)-rg_{12}(\varkappa)\right]
\left[ g_{12}(\kappa)g_{21}(\varkappa)+g_{21}(\kappa)g_{12}(\varkappa) \right]
\\
\notag
&
+4\Theta\varkappa g_{12}(\varkappa)g_{21}(\varkappa)\left[qg_{21}(\kappa)-rg_{12}(\kappa)\right]
\\
=&
-\Theta
\gamma'(\varkappa)
\left[ g_{12}(\kappa)g_{21}(\varkappa)+g_{21}(\kappa)g_{12}(\varkappa) \right]
- \frac{\varkappa\Theta}{2\kappa}\gamma'(\kappa) \gamma(\varkappa)\left[ \gamma(\varkappa)+2 \right].
\end{align}
Combining \eqref{j sub A 12}, \eqref{j sub A 3 1} and \eqref{micro As commute}, we can deduce
\eqref{j sub A} .
\end{proof}
\subsection{Dynamics II: the DNLS flow:.} Now we turn to the $H_{\mr{DNLS}}$ flow. We firstly recall the Lax representation for \eqref{DNLS} as follows.
\begin{lem}[Lax Pair for the DNLS flow, \cite{AbCl:book,KaupN:DNLS}]\label{P:Lax H} Let ${i\mkern1mu}\varkappa^2\in\ensuremath{\mathbb{R}}\setminus(-1,1)$, $L(\varkappa)$ be defined by \eqref{Intro KN L}. Under the DNLS flow, we have
\begin{align}\label{Lax repres}
\frac{\,{\rm{d}}\ }{\,{\rm{d}} t} L(\varkappa) = [ P_{H_{\mr{DNLS}}}, L(\varkappa)],
\end{align}
where $
P_{H_{\mr{DNLS}}}=
\begin{bmatrix} -2i \varkappa^4-i\varkappa^2 q r & 2\varkappa^3q + i\varkappa q' + \varkappa q^2 r\\
2\varkappa^3 r - i \varkappa r' + \varkappa qr^2 &2i \varkappa^4 + i \varkappa^2 q r \end{bmatrix}$.
\end{lem}
Following the analogue argument as those in Proposition \ref{Thm:A flow}, we can obtain main result in this paper.
\begin{thrm}[Microscopic conservation law for the DNLS flow]\label{Thm:H flow}
Let ${i\mkern1mu}\varkappa^2\in\ensuremath{\mathbb{R}}\setminus(-1,1)$. Under the DNLS flow, we have
\begin{align}\label{qr under DNLS}
\frac{\,{\rm{d}}\ }{\,{\rm{d}} t}q = i q''+(q^2r)' \qtq{and} \frac{\,{\rm{d}}\ }{\,{\rm{d}} t}r =- i r''+(qr^2)'.
\end{align}
and
\begin{equation}
\begin{aligned}
\frac{\,{\rm{d}}\ }{\,{\rm{d}} t} g_{12}(\varkappa) &=-2\left(2{i\mkern1mu} \varkappa^4 + i\varkappa^2 q r \right) g_{12}(\varkappa) - \left(2\varkappa^3 q + i\varkappa q' + \varkappa q^2 r\right) \left(\gamma(\varkappa) + 1\right) , \\
\frac{\,{\rm{d}}\ }{\,{\rm{d}} t} g_{21}(\varkappa) &= 2\left(2{i\mkern1mu} \varkappa^4 + i\varkappa^2 q r \right) g_{21}(\varkappa) + \left(2\varkappa^3 r - i\varkappa r' + \varkappa q r^2\right) \left(\gamma(\varkappa) + 1\right) , \\
\frac{\,{\rm{d}}\ }{\,{\rm{d}} t} \gamma(\varkappa) &= 2\varkappa^2\gamma'(\varkappa) + 2i\varkappa(q'g_{21}(\varkappa)+r'g_{12}(\varkappa)) + qr\gamma'(\varkappa).
\end{aligned}
\end{equation}
Moreover, we have the following microscopic conservation law
\begin{equation}\label{MCL: H Flow}
\partial_t \rho(\varkappa) + \partial_x j_{\mr{DNLS}}(\varkappa)=0,
\end{equation}
where the density $\rho$ and the flux $j_{\mr{DNLS}}$ are defined by \eqref{micro A} and
\begin{align}\label{j sub H}
j_{\mr{DNLS}}(\varkappa):
= \, &
i \varkappa \frac{q'g_{21}(\varkappa) - r' g_{12}(\varkappa)}{2+\gamma(\varkappa)}
- i\varkappa^2 q r
- 2\varkappa^2 \rho(\varkappa)
- qr\rho(\varkappa).
\end{align}
\end{thrm}
\begin{proof}
By \eqref{HFlow}, Proposition \ref{P:R} and Lemma \ref{P:Lax H}, it is easy to obtain the dynamics of $q$, $r$ and $g_{12}$, $g_{21}$ and $\gamma$ under the $H_{\mr{DNLS}}$ flow. Combining the dynamics of $q$, $r$, $g_{12}$, $g_{21}$, $\gamma$ and \eqref{micro As commute}, we can complete the proof of the microscopic conservation law \eqref{MCL: H Flow}.
\end{proof}
\bibliographystyle{plain}
|
3,212,635,537,440 | arxiv | \section{Introduction}
The neutrino oscillations between different flavour states were measured in a series of
experiments with atmospheric neutrinos~\cite{Fukuda:1998mi}, solar
neutrinos~\cite{Cleveland:1998nv}, and neutrinos produced in nuclear
reactors~\cite{Eguchi:2002dm} and accelerators~\cite{Ahn:2002up}. As a result of the global
combined analysis including all dominant and subdominant oscillation effects, the
difference of the squared neutrino masses and the mixing angles in the lepton mixing
matrix, $U_{ _{PMNS} }$, were determined at $1 \sigma$ ($3 \sigma$) confidence
level~\cite{GonzalezGarcia:2007ib}:
\begin{equation}\label{MCGCdatos}
\begin{array}{l}
\Delta m_{ 21 }^{ 2 } = 7.67^{ + 0.22 }_{ - 0.21 } \left( _{ -0.61 }^{ +0.67 } \right)
\times 10^{ -5 }~\textrm{eV}^{2}, \\ \\
\begin{array}{l}
\Delta m_{ 31 }^{ 2 } = \left \{
\begin{array}{l}
-2.37 \pm 0.15 \left( _{-0.46}^{+0.43} \right) \times 10^{ -3 }~\textrm{eV}^{2}, \\
\quad ( m_{ \nu _{ 2 } } > m_{ \nu _{ 1 } } > m_{ \nu _{ 3 } } ). \\ \\
+2.46 \pm 0.15 \left( _{-0.42}^{+0.47} \right) \times 10^{ -3 }~\textrm{eV}^{2}, \\
\quad ( m_{ \nu _{ 3 } } > m_{ \nu _{ 2 } } > m_{ \nu _{ 1 } } ).
\end{array} \right.
\end{array}
\end{array}
\end{equation}
{\small \begin{equation}
\begin{array}{ll}
\theta_{12}^{ l } = 34.5^{o} \pm 1.4 \left( _{-4.0 }^{+4.8 } \right), &
\theta_{23}^{ l } = 42.3^{o + 5.1 }_{ \;\; -3.3 }
\left( _{ \; \; -7.7 }^{ +11.3 } \right), \\ \\
\theta_{13}^{ l } = 0.0_{ \; \;-0.0 }^{ o +7.9 }
\left( _{-0.0}^{+12.9} \right).
\end{array}
\end{equation} }
Thus, values of the magnitudes of all nine elements of the lepton mixing matrix,
$U_{ _{ PMNS } }$, at $90\%$ CL, are:
{\small \begin{equation}\label{GG:UPMNS}
U_{ _{ PMNS } } = \left( \begin{array}{ccc}
0.80 \rightarrow 0.84 & 0.53 \rightarrow 0.60 & 0.00 \rightarrow 0.17 \\
0.29 \rightarrow 0.52 & 0.51 \rightarrow 0.69 & 0.61 \rightarrow 0.76 \\
0.26 \rightarrow 0.50 & 0.46 \rightarrow 0.66 & 0.64 \rightarrow 0.79
\end{array} \right).
\end{equation} }
The CHOOZ experiment determined an upper bound for the $\theta_{13}^{ l }$ mixing
angle~\cite{Apollonio:1999ae}. The latest analyses give the following best
values~\cite{PhysRevLett.103.061804,GonzalezGarcia:2010er}:
\begin{equation}
\theta_{13}^{ l } = -0.07_{-0.11 }^{+0.18}
\end{equation}
and (at 1$\sigma$(3$\sigma$))
\begin{equation}
\begin{array}{l}
\theta_{13}^{ l } = 5.6_{-2.7 }^{+3.0}\left(\leq 12.5\right)^{o},\;
\theta_{13}^{ l } = 5.1_{-3.3 }^{+3.0}\left(\leq 12.0\right)^{o},
\end{array}
\end{equation}
see also~\cite{Maltoni:2008ka}.
On the other hand, in the last years extensive research has been done in the precise
determination of the values of the $V_{ _{ CKM } }$ quark mixing matrix elements. The most
precise fit results for the values of the magnitudes of all nine $CKM$ elements
are~\cite{Amsler:2008zzb}:
{\small \begin{equation}
\begin{array}{l}
V_{ _{ CKM } } = \\
\left( \begin{array}{lll}
0.97419 \pm .00022 & 0.2257 \pm .0010 & 0.00359 \pm .00016 \\
0.2256 \pm .0010 & 0.97334 \pm .00023 & 0.0415_{ -.0011 }^{+.0010} \\
0.00874_{ -.00037 }^{ +.00026 } & 0.0407 \pm .0010 &
0.999133_{ -.000043 }^{ +.000044 }
\end{array} \right)
\end{array}
\end{equation} }
and the Jarlskog invariant is
\begin{equation}
J^{q} = \left( 3.05_{ - 0.20 }^{ +0.19 } \right) \times 10^{ -5 }.
\end{equation}
We also have the three angles of the unitarity triangle with the following reported best
values~\cite{Amsler:2008zzb}:
\begin{equation}
\alpha = \left( 88_{ -5 }^{ +6 } \right)^{o}, \;
\beta = \left( 21.46 \pm 0.71 \right)^{o}, \;
\gamma = \left( 77_{ -32 }^{ +30 } \right)^{o}.
\end{equation}
Each of the elements of the $V_{ _{ CKM } }$ matrix can be extracted from a large number of
decays and, for the purpose of our analysis, will be considered as independent. Hence,
current knowledge of the mixing angles for the quark sector can be summarized at 1$\sigma$
as~\cite{Amsler:2008zzb}:
\begin{equation}\label{PDGdatosang}
\begin{array}{l}
\sin \theta_{12}^{q} = 0.2257 \pm 0.0010, \;
\sin \theta_{23}^{q} = 0.0415_{ -0.0011 }^{ +0.0010 }, \\ \\
\sin \theta_{13}^{q} = 0.00359 \pm 0.00016.
\end{array}
\end{equation}
The solar mixing angle $\theta^{l}_{12}$ and the correponding mixing angle in the quark sector,
the Cabibbo angle~$\theta^{q}_{12}$, satisfy an interesting and intriguing numerical relation
(at $90~\% $ confidence level)~\cite{Smirnov:2004ju},
\begin{equation}\label{t12}
\theta^{l}_{12} + \theta^{q}_{12} \approx 45^{o} + 2.5^{o} \pm 1.5^{o},
\end{equation}
see also~\cite{Smirnov:2009dk}. The equation (\ref{t12}) relates the 1-2 mixing angles in
the quark and lepton sectors, it is commonly known as Quark-Lepton Complementarity relation
(QLC) and, if not accidental, it could imply a quark-lepton symmetry. A second QLC relation
between the atmospheric and 2-3 mixing angles, is also
satisfied~\cite{GonzalezCanales:2009zz},
\begin{equation}
\theta_{23}^{ l } + \theta_{ 23 }^{ q } = \left( 44.67^{+5.1}_{ - 3.3 } \right)^{o} .
\end{equation}
However, this is not as interesting as (\ref{t12}) because $\theta_{23}^{q}$ is only about
$2^{o}$, and the corresponding QLC relation would be satisfied, within the errors, even
if the angle $\theta_{23}^{q}$ had been zero, as long as $\theta_{23}^{ l }$ is close to the
maximal value $\pi/4$. A third possible QLC relation is not realized at all, or at least not
realized in the same way, since it is less than $10^{o}$~\cite{GonzalezCanales:2009zz}.
\begin{equation}\label{t13}
\theta_{13}^{ l } + \theta_{13}^{q} < 8.1^{o}.
\end{equation}
Equations (\ref{t12})-(\ref{t13}) are known as the extended quark lepton Complementarity, for a
review see~\cite{Minakata:2005rf}. The extended QLC relations could imply a quark-lepton
symmetry~\cite{Minakata:2005rf} or a quark lepton unification~\cite{Frampton:2004ud}. \\
A systematic numerical exploration of all CP conserving textures of the neutrino mass matrix
compatible the QLC relations and the experimental information on neutrino mixings is given
in~\cite{Plentinger:2006nb}.
The neutrino oscillations do not provide information about either the absolute mass scale or
if neutrinos are Dirac or Majorana particles~\cite{Camilleri:2008zz}. Thus, one of the most
fundamental problems of the neutrinos physics is the question of the nature of massive
neutrinos. A direct way to reveal the nature of massive neutrinos is to investigate
processes in which the total lepton number is not conserved~\cite{PhysRevD.76.116008}. The
matrix elements for these processes are proportional to the effective Majorana neutrino
masses, which are defined as
\begin{equation}\label{masa_eff.1}
\langle m_{ll} \rangle \equiv \sum_{j=1}^{3} m_{ \nu_{j} } U_{lj}^{2} ,\qquad
l = e, \mu, \tau,
\end{equation}
where $m_{ \nu_{j} }$ are the neutrino Majorana masses and $U_{lj}$ are the elements of the
lepton mixing matrix.
In this work, we will focus our attention on understanding the nature of the QLC relation,
and finding possible values for the effective Majorana neutrino masses. Thus, we made a
unified treatment of quarks and leptons, where we assumed that the charged lepton and quark
mass matrices have the same generic form with four texture zeroes from a universal $S_{3}$
flavor symmetry and its sequential explicit breaking.
\section{Universal mass matrix with a four zeroes texture}
In particle physics, the imposition of a flavour symmetry has been successful in reducing
the number of parameters of the Standard Model. Recent flavour symmetry models are reviewed
in~\cite{Ishimori:2010au}; see also the references therein. In particular, a permutational
$S_{3}$ flavor symmetry and its sequential explicit breaking allows us to take the same
generic form for the mass matrices of all Dirac fermions, conventionally called the
generalized Fritzsch ansatz with four texture zeroes~\cite{PhysRevD.61.113002, Fritzsch:1999ee}:
\begin{equation}\label{T_fritzsch}
{\bf M_{ i } = }\left( \begin{array}{ccc}
0 & A_{ i } & 0 \\
A^{*}_{ i } & B_{ i } & C_{ i } \\
0 & C_{ i } & D_{ i } \end{array}\right), \quad i=u,d,l,\nu_{ _D }.
\end{equation}
where $B_{ i }$, $C_{ i }$ and $D_{ i }$ are real, while
$ A_{ i } = \left| A_{ i } \right|e^{ i \phi_{i} }$ with
$ \phi_{i} = \arg \left \{ A_{ i } \right \} $.
In the most general case, all entries in the Hermitian mass matrix $M_{ i }$ are complex and
nonvanishing. However, without loss of generality, by means of a common unitary transformation
of the Dirac fields $\Psi_{ u, \nu_{ _D } }$ and $\Psi_{ d, l }$, it is always possible to
change to a new flavour basis where the off-diagonal elements
$\left( M_{ i } \right)_{13} = \left( M_{ i } \right)_{31}$
vanish~\cite{Fritzsch:1999ee}. The vanishing of the diagonal elements
$ \left( M_{ u, \nu_{ _D } }\right)_{11}$ and $\left( M_{ d, l } \right)_{11}$,
constrains the physics and allows for the predictions of the Cabibbo angle as funtion of
the $u$ and $d$-type quark masses in the quark sector and the solar angle as funtion of the
charged leptons and Majorana neutrinos masses in the leptonic sector in good agreement with
the experimental values.
Then, in the quark sector, $M_{u}$ and $M_{d}$ totally have four texture zeroes and, in the
leptonic sector, $M_{ \nu_{ _D } }$ and $M_{e}$, totally have four texture zeroes (here a
pair of off-diagonal texture zeroes are counted as one zero, due to the Hermiticity of
$M_{ i }$)~\cite{Fritzsch:1999ee}. Hence, following a common convention we will refer to
$M_{ i }$ as a generalized Fritzsch ansatz with four texture zeros.
Some reasons to propose the validity of a generalized Fritzsch ansatz with four texture
zeros as a universal form for the mass matrix of all Dirac fermions in the theory are the
following:
\begin{enumerate}
\item The idea of $S_{3}$ flavor symmetry and its explicit breaking has been succesfully
realized as a mass matrix with four texture zeroes in the quark sector to interpret the
strong mass hierarchy of up and down type quarks~\cite{Fritzsch:1977za}.
\item The quark mixing angles and the CP violating phase, appearing in the $V_{ _{CKM} }$
mixing matrix, were computed as explicit, exact functions of the four quark mass ratios
{\small $(m_{u}/m_{t},m_{c}/m_{t},m_{d}/m_{b},m_{s}/m_{b})$,} one symmetry breaking
parameter defined as $Z^{1/2} \equiv \frac{ C_{i} }{ B_{ i } }$ and one CP violating phase
$\phi_{ _{u-d} }= \phi_{u} - \phi_{d}$. Asuming that $Z_{u} = Z_{d} = Z$, a $\chi^{2}$
fit of the theoretical expresion for $V_{ _{CKM } }^{ th }$ to the experimentally
determined $V_{ _{CKM } }^{ exp }$ gave $Z^{1/2} = \left(\frac{81}{32}\right)^{1/2}$ and
$\phi_{ _{u-d} }= 90^{o}$, in good agreement with the experimetal
data~\cite{PhysRevD.61.113002}. This agreement with improved as the precision of the
experimental data has improved and, now, it is very good~\cite{Amsler:2008zzb}.
\item Since the mass spectrum of the charged leptons exhibits a hierarchy similar to the
quark's one, it would be natural to consider the same $S_{3}$ symmetry and its explicit
breaking to justify the use of the same generic form with four texture zeroes for the
charged lepton mass matrix.
\item As for the Dirac neutrinos, we have no direct information about the absolute values
or the relative values of the neutrino masses, but the mass matrix with four texture
zeroes can be obtained from an $SO(10)$ neutrino model which describes these the data on
neutrino masses and mixings well~\cite{Buchmuller:2001dc}. Furthermore, from
supersymmetry arguments, it would be sensible to assume that the Dirac neutrinos have a
mass hierarchy similar to that of the u-quarks and it would be natural to take for the
Dirac neutrino mass matrix also a matrix with four texture zeroes.
\end{enumerate}
The Hermitian mass matrix (\ref{T_fritzsch}) may be written in terms of a real symmetric
matrix $\bar{M}_{ i }$ and a diagonal matrix of phases
$P_{ i }\equiv \textrm{diag}\left[ 1, e^{ i\phi_{ i } }, e^{ i\phi_{ i } } \right]$ as
follows:
\begin{equation}\label{Polar:FT}
M_{ i } = P^{\dagger}_{ i } \bar{M}_{ i } P_{ i }\; .
\end{equation}
The real symetric matrix $\bar{M}_{ i }$ may be brought to diagonal form by means of an
orthogonal transformation,
\begin{equation}\label{Defi:Oreal}
\bar{M}_{ i } = {\bf O }_{ i } \textrm{diag} \left \{ m_{i1} , m_{i2} , m_{i3} \right \}
{\bf O }^{ T }_{ i } ,
\end{equation}
where the $m_{ i }$'s are the eigenvalues of $M_{ i }$ and ${\bf O }_{ i }$ is a real
orthogonal matrix. Now computing the invariants of the real symetric matrix
$\bar{M}_{ i }$, $\textrm{tr}\left \{ \bar{M}_{ i } \right \}$,
$\textrm{tr}\left \{ \bar{M}_{ i }^{2} \right \}$ and
$\textrm{det}\left \{ \bar{M}_{ i } \right \}$, we may express the parameters $A_{ i }$,
$B_{ i }$, $C_{ i }$ and $D_{ i }$ occuring in (\ref{T_fritzsch}) in terms of the mass
eigenvalues. In this way, we get that the $\bar{M}_{ i }$ matrix $(i=u,d,l,\nu_{ _D })$,
reparametrized in terms of its eigenvalues and the parameter
$D_{ i } \equiv 1 - \delta_{ i } $ is
\begin{equation}\label{T_fritzsch:ev}
\bar{ M }_{ i } = \left( \begin{array}{ccc}
0 & \sqrt{ \frac{ \widetilde{ m }_{ i1 }
\widetilde{ m }_{ i2 } }{ 1 - \delta_{ i } } } & 0 \\
\sqrt{ \frac{ \widetilde{ m }_{ i1 } \widetilde{ m }_{ i2 } }{ 1 - \delta_{ i } } } &
\widetilde{ m }_{ i1 } -
\widetilde{ m }_{ i2 } + \delta_{ i } &
\sqrt{ \frac{\delta_{ i } }{ ( 1 - \delta_{ i } ) } f_{ i1 } f_{ i2 } } \\
0 & \sqrt{ \frac{ \delta_{ i } }{ ( 1 - \delta_{ i } ) } f_{ i1 } f_{ i2 } } &
1 - \delta_{ i }
\end{array}\right),
\end{equation}
where $\widetilde{m}_{i1} = \frac{ m_{i1} }{ m_{i3} }$,
$\widetilde{m}_{i2} = \frac{ | m_{i2} | }{ m_{i3} }$,
\begin{equation}\label{fs}
f_{i1}=1-\widetilde{m}_{i1}-\delta_{i}, \quad f_{i2} =1+ \widetilde{m}_{i2} - \delta_{i}.
\end{equation}
The small parameters $\delta_{i}$ are also functions of the mass ratios and the flavor
symmetry breaking parameter $Z^{1/2}_{i}$~\cite{PhysRevD.61.113002}. The flavor
symmetry breaking parameter $Z^{1/2}_{i}$, which measures the mixing of singlet and
doublet irreducible representations of $S_{3}$, is defined as the ratio
\begin{equation}\label{def:Z}
Z^{1/2}_{i} = \frac{ \left( M_{ i } \right)_{23} }{ \left( M_{ i } \right)_{22} }.
\end{equation}
It is related with the parameters $\delta_{i}$ by the following cubic
equation~\cite{PhysRevD.61.113002}:
\begin{equation}\label{ecu:cubica}
\begin{array}{l}
\delta_{i}^{ 3 } - \frac{ 1 }{ Z_{i} + 1 }
\left(2 + \widetilde{m}_{i2} - \widetilde{m}_{i1} +
\left( 1 + 2 \left( \widetilde{m}_{i2} - \widetilde{m}_{i1} \right) \right)
Z_{i} \right) \delta_{i}^{2} + \\ \\
+ \frac{ 1 }{ Z_{i} + 1 }
\left( Z_{i} \left( \widetilde{m}_{i2} - \widetilde{m}_{i1} \right)
\left( 2 + \widetilde{m}_{i2} - \widetilde{m}_{i1} \right)
+ \right. \\ \\ \left.
+ \left( 1 + \widetilde{m}_{i2} \right) \left( 1 - \widetilde{m}_{i1} \right)
\right)\delta_{i}
+ \frac{ Z_{ i } \left( \widetilde{m}_{i2} - \widetilde{m}_{i1} \right)^{2} }{
Z_{ i } + 1 } = 0.
\end{array}
\end{equation}
Thus, the small parameter $\delta_{i}$ is obtained as the solution of the cubic equation
(\ref{ecu:cubica}), which vanishes when $Z_{i}$ vanishes. The last term in the left-hand
side of (\ref{ecu:cubica}) is equal to the product of the three roots of
(\ref{ecu:cubica}). Therefore, the root that vanishes when $Z_{i}$ vanishes may be written
as
\begin{equation}
\delta_{i} = \frac{ Z_{ i } }{ Z_{ i } + 1 }
\frac{ \left( \widetilde{m}_{i2} - \widetilde{m}_{i1} \right)^{2} }{
W _{i}\left( Z \right) }
\end{equation}
where $W _{i}\left( Z \right)$ is the product of the two roots of (\ref{ecu:cubica}) which
do not vanish when $Z_{i}$ vanishes. The explicit form of $W _{i}\left( Z \right)$
is~\cite{PhysRevD.61.113002}:
\begin{equation}
\begin{array}{l}
W _{i}\left( Z \right) = \left[ p^{3}_{i} + 2 q^{2}_{i} + 2q \sqrt{ p^{3}_{i} +
q^{2}_{i} } \right]^{ \frac{ 1 }{ 3 } } - | p_{i} | + \\
+ \left[ p^{3}_{i} + 2 q^{2}_{i} - 2q_{i} \sqrt{ p^{3}_{i} + q^{2}_{i} }
\right]^{ \frac{ 1 }{ 3 } } + \\
+ \frac{1}{9} \left( Z_{i} \left( 2 \left( \widetilde{m}_{i2} -
\widetilde{m}_{i1}\right) + 1 \right) + \left( \widetilde{m}_{i2} -
\widetilde{m}_{i1}\right) + 2 \right)^{2} \\
-\frac{1}{3} \left( \left[ q_{i} + \sqrt{ p^{3}_{i} + q^{2}_{i} }
\right]^{ \frac{ 1 }{ 3 } } +
\left[ q_{i}- \sqrt{ p^{3}_{i} + q^{2}_{i} } \right]^{ \frac{ 1 }{ 3 } } \right)
\times \\ \times
\left( Z_{i} \left( 2 \left( \widetilde{m}_{i2} -
\widetilde{m}_{i1}\right) + 1 \right) + \left( \widetilde{m}_{i2} -
\widetilde{m}_{i1}\right) + 2 \right)
\end{array}
\end{equation}
with
\begin{equation}
\begin{array}{l}
p_{i} = -\frac{1}{3} \frac{ Z_{ i } }{ Z_{ i } + 1 } \left( Z_{i}
\left( 2 \left( \widetilde{m}_{i2} -
\widetilde{m}_{i1}\right) + 1 \right) + \widetilde{m}_{i2} - \right. \\ \left.
\widetilde{m}_{i1} + 2 \right)^{2} + \frac{ 1 }{ Z_{ i } + 1 }
\left[ Z_{i} \left( \widetilde{m}_{i2} - \widetilde{m}_{i1} \right)
\left( \widetilde{m}_{i2} - \widetilde{m}_{i1} +
\right. \right. \\ \left. \left. + 2 \right)
\left(1 + \widetilde{m}_{i2} \right) \left( 1 - \widetilde{m}_{i1} \right) \right],
\end{array}
\end{equation}
\begin{equation}
\begin{array}{l}
q_{i} = -\frac{1}{27} \frac{ 1 }{ \left( Z_{ i } + 1 \right)^{ 3 } } \left( Z_{i} \left( 2
\left( \widetilde{m}_{i2} - \widetilde{m}_{i1}\right) + 1 \right) + \widetilde{m}_{i2} -
\right. \\ \left.
\widetilde{m}_{i1} + 2 \right)^{3} + \frac{1}{6} \frac{ 1 }{ \left( Z_{ i } + 1
\right)^{2} } \left[ Z_{i} \left( \widetilde{m}_{i2} - \widetilde{m}_{i1} \right)
\left( \widetilde{m}_{i2} -
\right. \right. \\ \left. \left. - \widetilde{m}_{i1} + 2 \right)
\left(1 + \widetilde{m}_{i2} \right) \left( 1 - \widetilde{m}_{i1} \right) \right]
\left( Z_{i} \left( 2 \left( \widetilde{m}_{i2} -
\right.\right. \right. \\ \left.\left.\left.
- \widetilde{m}_{i1} \right)+ 1 \right)
+ \widetilde{m}_{i2} - \widetilde{m}_{i1} + 2 \right).
\end{array}
\end{equation}
Also, the values allowed for the parameters $\delta_{i}$ are in the following range
$ 0 < \delta_{ i } < 1 - \widetilde{m}_{ i1 }$. \\
Now, the entries in the real orthogonal matrix ${\bf O}$, eq.~(\ref{Defi:Oreal}),
may also be expressed in terms of the eigenvalues of the mass matrix (\ref{T_fritzsch}) as
{\small \begin{equation}\label{M_ortogonal}
{\bf O_{i}=}\left(\begin{array}{ccc}
\left[ \frac{ \widetilde{m}_{i2} f_{i1} }{ {\cal D}_{ i1 } } \right]^{ \frac{1}{2} } &
-\left[ \frac{ \widetilde{m}_{i1} f_{i2} }{ {\cal D}_{ i2 } } \right]^{ \frac{1}{2} } &
\left[ \frac{ \widetilde{m}_{i1} \widetilde{m}_{i2} \delta_{i} }{ {\cal D}_{i3} }
\right]^{ \frac{1}{2} } \\
\left[ \frac{ \widetilde{m}_{i1} ( 1 - \delta_{i} ) f_{i1} }{ {\cal D}_{i1} }
\right]^{ \frac{1}{2} } &
\left[ \frac{ \widetilde{m}_{i2} ( 1 - \delta_{i} ) f_{i2} }{ {\cal D}_{i2} }
\right]^{ \frac{1}{2} } &
\left[ \frac{ ( 1 - \delta_{i} ) \delta_{i} }{ {\cal D}_{i3} } \right]^{ \frac{1}{2} } \\
-\left[ \frac{ \widetilde{m}_{i1} f_{i2} \delta_{i} }{ {\cal D}_{i1} }
\right]^{ \frac{1}{2} } &
-\left[ \frac{ \widetilde{m}_{i2} f_{i1} \delta_{i} }{ {\cal D}_{i2} }
\right]^{ \frac{1}{2} } &
\left[ \frac{ f_{i1} f_{i2} }{ {\cal D}_{i3} } \right]^{ \frac{1}{2} }
\end{array}\right) ,
\end{equation} }
where,
\begin{equation}\label{Ds}
\begin{array}{l}
{\cal D}_{i1} = ( 1 - \delta_{i} )( \widetilde{m}_{i1} + \widetilde{m}_{i2} )
( 1 - \widetilde{m}_{i1} ), \\\\
{\cal D}_{i2} = ( 1 - \delta_{i} )( \widetilde{m}_{i1} + \widetilde{m}_{i2} )
( 1 + \widetilde{m}_{i2} ), \\\\
{\cal D}_{i3} = ( 1 - \delta_{i} )( 1 - \widetilde{m}_{i1} )( 1 + \widetilde{m}_{i2} ).
\end{array}
\end{equation}
\section{SEESAW MECHANISM AND PHASES OF THE LEFT-HANDED NEUTRINO MASS MATRIX}
The left-handed Majorana neutrinos naturally acquire their small masses through an
effective type I seesaw mechanism of the form
\begin{equation}\label{subibajadef}
M_{ \nu_{L} } = M_{ \nu_{D} } M_{ \nu_R }^{-1} M_{ \nu_{D} }^{T},
\end{equation}
where $M_{ \nu_{D} }$ and $ M_{ \nu_{R} }$ denote the Dirac and right handed Majorana
neutrino mass matrices, respectively. The symmetry of the mass matrix of the left-handed
Majorana neutrinos, $M_{\nu_{L}} =M_{\nu_{L}}^{ T }$, and the seesaw mechanism of type I,
eq. (\ref{subibajadef}), fix the form of the right handed Majorana neutrinos mass matrix,
$M_{ \nu_{R} }$, which has to be nonsingular and symmetric.
Further restrictions on $M_{ \nu_{R} }$, follow from requiring that $M_{ \nu_{L} }$ also
has a texture with four zeroes, as will be shown below. With this purpose in mind, the
seesaw mechanism, eq.~(\ref{subibajadef}), may be written in a more explicit form as:
\begin{equation}\label{defseesaw1}
M_{ \nu_{L} } = \frac{ 1 }{ \det \left( M_{ \nu_R } \right) }
M_{ \nu_{D} } \textrm{adj} \left( M_{ \nu_R } \right) M_{ \nu_{D} }^{T},
\end{equation}
where $\det ( M_{ \nu_R } ) $ and $\textrm{adj} \left( M_{ \nu_R } \right)$ are the
determinant and adjugate matrix of $M_{ \nu_R }$, respectively. \\
Now, if we consider the more general form of a complex symmetric matrix of $3 \times 3$
\begin{equation}\label{Matriz:MR}
M_{ \nu_R } = \left( \begin{array}{ccc}
g_{ \nu_{ _R } } & a_{ \nu_{ _R } } & e_{ \nu_{ _R } } \\
a_{ \nu_{ _R } } & b_{ \nu_{ _R } } & c_{ \nu_{ _R } } \\
e_{ \nu_{ _R } } & c_{ \nu_{ _R } } & d_{ \nu_{ _R } }
\end{array}\right)
\end{equation}
to represent the right handed Majorana neutrinos mass matrix, we may write
eq.~(\ref{defseesaw1}) in a more explicit form if we express
$\det ( M_{ \nu_R } ) $ and $\textrm{adj} \left( M_{ \nu_R } \right)$ in terms of the
cofactors of the elements of the matrix $M_{ \nu_R }$. Then,
\begin{equation}
\det ( M_{ \nu_{ _R } } ) = g_{ \nu_{ _R } } X_{11} -
a_{ \nu_{ _R } } X_{ 12 } + e_{ \nu_{ _R } } X_{ 13 }
\end{equation}
and
\begin{equation}\label{subibajagenraltipoI}
M_{ \nu_{L} } = \frac{ 1 }{ \det \left( M_{ \nu_R } \right) }
\left( \begin{array}{ccc}
G_{ \nu_{ _L } } & A_{ \nu_{ _L } } & E_{ \nu_{ _L } } \\
A_{ \nu_{ _L } } & B_{ \nu_{ _L } } & C_{ \nu_{ _L } } \\
E_{ \nu_{ _L } } & C_{ \nu_{ _L } } & D_{ \nu_{ _L } }
\end{array}\right),
\end{equation}
where
{\small \begin{equation}\label{subibajagenraltipoI-2}
\begin{array}{l}
G_{ \nu_{ _L } } = X_{22} A_{ \nu_{ _D } }^{2} ,\\ \\
A_{ \nu_{ _L } } = -X_{12} | A_{ \nu_{ _D } }|^{2} + X_{ 22 } A_{ \nu_{ _D } }
B_{ \nu_{ _D } } - X_{23} A_{ \nu_{ _D } } C_{ \nu_{ _D } } ,\\\\
B_{ \nu_{ _L } } = X_{11} A_{ \nu_{ _D } }^{*2} + X_{ 22 } B_{ \nu_{ _D } }^{2}
+ X_{ 33 } C_{ \nu_{ _D } }^{ 2 }
\\ \quad
-2 X_{ 12 } A_{ \nu_{ _D } }^{*} B_{ \nu_{ _D } }
+ 2 X_{ 13 } A_{ \nu_{ _D } }^{*} C_{ \nu_{ _D } }
- 2 X_{23} B_{ \nu_{ _D } } C_{ \nu_{ _D } }, \\ \\
E_{ \nu_{ _L } } = X_{ 22 } A_{ \nu_{ _D } } C_{ \nu_{ _D } } -
X_{ 23 } A_{ \nu_{ _D } } D_{ \nu_{ _D } }, \\ \\
C_{ \nu_{ _L } } = X_{ 13 } A_{ \nu_{ _D } }^{*} D_{ \nu_{ _D } }
- X_{ 12 } A_{ \nu_{ _D } }^{*} C_{ \nu_{ _D } }
+ X_{ 22 } B_{ \nu_{ _D } } C_{ \nu_{ _D } } \\ \qquad
- X_{ 23 } \left( B_{ \nu_{ _D } } D_{ \nu_{ _D } } + C_{ \nu_{ _D } }^{ 2 } \right)
+ X_{ 33 } C_{ \nu_{ _D } } D_{ \nu_{ _D } }, \\ \\
D_{ \nu_{ _L } } = X_{ 22 } C_{ \nu_{ _D } }^{ 2 }
- 2 X_{ 23 } C_{ \nu_{ _D } } D_{ \nu_{ _D } } + X_{ 33 } D_{ \nu_{ _D } }^{ 2 }.
\end{array}
\end{equation} }
In these expresions, the $X_{ nm }$ ($m,n= 1,2,3$) are the cofactors of the correponding
elements of the $\textrm{adj} \left( M_{ \nu_R } \right)$
matrix~\footnote{ The cofactors of the elements of $M_{ \nu_R }$ matrix, are defined as
$X_{ nm } = ( -1 )^{ n + m } \det \left( H_{nm} \right)$, where $H_{nm}$ is obtained by
deleting the $n$ row and the $m$ column of $M_{ \nu_R }$ matrix.}.
From eqs.~(\ref{subibajagenraltipoI})~and~(\ref{subibajagenraltipoI-2}), when conditions
$X_{22} = X_{23}= 0$ are satisfied, the mass matrix of the left-handed Majorana neutrinos will
have the same universal form with four texture zeroes as the Dirac mass matrices. These
conditions are equivalent to
\begin{equation}\label{ConFourZeros}
\begin{array}{l}
g_{ \nu_{ _R } } d_{ \nu_{ _R } } = e_{ \nu_{ _R } }^{2}, \qquad
g_{ \nu_{ _R } } c_{ \nu_{ _R } } = a_{ \nu_{ _R } } e_{ \nu_{ _R } },
\end{array}
\end{equation}
Thus, we obtain the relation
\begin{equation}
\begin{array}{c}
\frac{ a_{ \nu_{ _R } } }{ c_{ \nu_{ _R } } } =
\frac{ e_{ \nu_{ _R } } }{ d_{ \nu_{ _R } } }.
\end{array}
\end{equation}
For non vanishing $\det ( M_{ \nu_R } )$, these conditions~(\ref{ConFourZeros}) are satisfied,
if
\begin{equation}\label{condseesawinv}
g_{ \nu_{ _R } }=0 \quad \textrm{and} \quad e_{ \nu_{ _R } }=0.
\end{equation}
If we extend the meaning of a mass matrix with four texture zeroes, defined
in~(\ref{T_fritzsch}), to include the symmetric mass matrix of the right-handed Majorana
neutrinos, $M_{ \nu_{ _R } }$~\cite{Xing:2003zd}, which is non-Hermitian, we could say
that the matrix with four zeroes texture is invariant under the action of the seesaw
mechanism of type I~\cite{Xing:2003zd,Fritzsch:1999ee, GonzalezCanales:2009zz}. \\
It may also be noticed that, if we set $b_{ \nu_{ _R } } = 0$ or/and
$c_{ \nu_{ _R } } = 0$, the resulting expression for $M_{ \nu_{ _L } }$ still has four
texture zeroes. Therefore, $M_{ \nu_{ _L } }$ may also have a four texture zeroes
when $M_{ \nu_{ _R } }$ has four, three or two texture zeroes (the two last cases
are called Fritzsch textures).
Let us further assume that the phases in the entries of the $M_{ \nu_{ R} }$ may be
factorized out as
\begin{equation}
M_{ \nu_{ _R } } = R \bar{M}_{ \nu_{ _R } }R,
\end{equation}
where
\begin{equation}
\bar{M}_{ \nu_{ _R } } = \left( \begin{array}{ccc}
0 & a_{ \nu_{ _R } } & 0 \\
a_{ \nu_{ _R } } & | b_{ \nu_{ _R } } | &| c_{ \nu_{ _R } } | \\
0 & | c_{ \nu_{ _R } } | & d_{ \nu_{ _R } }
\end{array}\right),
\end{equation}
and $R \equiv \textrm{diag}\left[ e^{ - i\phi_{ c }}, e^{ i\phi_{ c } }, 1 \right]$ with
$\phi_{ c } \equiv \arg \left \{ c_{ \nu_{ _R } } \right \}$. \\
Then, the type I seesaw mechanism takes the form:
\begin{equation}\label{seesaw9}
M_{ \nu_{ _L } } = P_{ _D }^{\dagger} \bar{M}_{\nu_{ _D } } P_{ _D }
R^{ \dagger } \bar{M}_{ \nu_{ _R } } ^{-1} R^{ \dagger }
P_{ _D } \bar{M}_{\nu_{ _D } } P_{ _D }^{ \dagger },
\end{equation}
and the mass matrix of the left-handed neutrinos has the following form with four texture
zeroes~\footnote{The seesaw invariance of the four zeroes mass matrix of the Majorana
neutrino is also derived in~\cite{Xing:2003zd}. However, this authors ignored the phases in
the elements of mass matrices in their discussion.}:
\begin{equation}\label{seesaw:F}
M_{ \nu_{ _L } } = \left( \begin{array}{ccc}
0 & a_{ \nu_{ _L } } & 0 \\
a_{ \nu_{ _L } } & b_{ \nu_{ _L } } & c_{ \nu_{ _L } } \\
0 & c_{ \nu_{ _L } } & d_{ \nu_{ _L } }
\end{array}\right),
\end{equation}
where
{\small \begin{equation}\label{seesaw:F:elem}
\begin{array}{l}
a_{ \nu_{ _L } } = \frac{ | a_{ \nu_{ _D } } |^{2} }{ a_{ \nu_{ _R } } } ,\\ \\
b_{ \nu_{ _L } } = \frac{ c_{ \nu_{ _D } }^{ 2 } }{ d_{ \nu_{ _R } } } + \frac{ | c_{
\nu_{ _R } }|^{2} - | b_{ \nu_{ _R } }| d_{ \nu_{ _R } } }{ d_{ \nu_{ _R } } }
\frac{ | a_{ \nu_{ _D } } |^{ 2 } }{ a_{ \nu_{ _R } }^{2} } e^{ i 2\left( \phi_{ c } -
\phi_{ \nu_{ D } } \right)} \\
\qquad + 2 \frac{ | a_{ \nu_{ _D } }| }{ |a_{ \nu_{ _R } } | } \left( b_{ \nu_{ _D } }
e^{- i \phi_{ \nu_{ D } } }
- \frac{c_{ \nu_{ _D } } |c_{ \nu_{ _R } }| }{ d_{ \nu_{ _R } } } e^{ i \left( \phi_{ c
} - \phi_{ \nu_{ D } } \right) } \right) , \\ \\
c_{ \nu_{ _L } } = \frac{ c_{ \nu_{ _D } } d_{ \nu_{ _D } } }{ d _{ \nu_{ _R } } }+
\\ \quad +
\frac{ | a_{ \nu_{ _D } } | }{ | a_{ \nu_{ _R } } |}
\left ( c_{ \nu_{ _D } }e^{- i \phi_{ \nu_{ D } } } - \frac{ |c_{ \nu_{ _R } }| d_{ \nu_{
_D } } }{ d_{ \nu_{ _R }
} } e^{ i \left( \phi_{ c } - \phi_{ \nu_{ D } } \right) } \right) , \\ \\
d_{ \nu_{ _L } } = \frac{ d_{ \nu_{ _D } }^{2} }{ d_{ \nu_{ _R } } }.
\end{array}
\end{equation} }
The elements $a_{ \nu_{ _L } }$ and $d_{ \nu_{ _L } }$ are real , while
$b_{ \nu_{ _L } }$ and $c_{ \nu_{ _L } }$ are complex.
Notice that the phase factors appearing in eqs. (\ref{seesaw9}) and (\ref{seesaw:F:elem})
are fully determined by the seesaw mechanism and our choice of a generalized Fritzsch ansatz
with four texture zeroes for the mass matrices of all Dirac fermions and the complex
symetric, but non-Hermitian, mass matrix of the right handed Majorana neutrinos.
Now, to diagonalize the left-handed Majorana neutrino mass matrix
$M_{ \nu_{ _L } }$ by means of a unitary matrix, we need to construct the hermitian
matrices $M_{ \nu_{ _L } }M_{ \nu_{ _L } }^{ \dagger }$ and
$M_{ \nu_{ _L } }^{ \dagger }M_{ \nu_{ _L } }$, which can be diagonalized with unitary
matrices through of the following transformations:
\begin{equation}\label{bilineal:RL}
\begin{array}{l}
U_{ _R }^{ \dagger } M_{ \nu_{ _L } }^{ \dagger } M_{ \nu_{ _L } } U_{ _R }
= \textrm{diag} \left[ \left| m_{ \nu_{1} }^{ s } \right|^{ 2 },
\left| m_{ \nu_{2} }^{ s }\right|^{ 2 }, \left|
m_{ \nu_{3} }^{ s } \right|^{ 2 }\right], \\ \\
U_{ _L }^{ \dagger } M_{ \nu_{ _L } } M_{ \nu_{ _L } }^{ \dagger } U_{ _L } =
\textrm{diag} \left[ \left|
m_{ \nu_{1} }^{ s } \right|^{ 2 }, \left| m_{ \nu_{2} }^{ s }\right|^{ 2 }, \left| m_{
\nu_{3} }^{ s } \right|^{ 2 }\right],
\end{array}
\end{equation}
where the $ m_{ \nu_{j} }^{ s }$ $(j=1,2,3)$ are the singular values of the
$M_{ \nu_{ _L } }$ matrix. Thus, with the help of the symmetry of the matrix
(\ref{seesaw:F}) and the transformations (\ref{bilineal:RL}), the left-handed
Majorana neutrino mass matrix, $M_{ \nu_{ _L } }$, is diagonalized by a unitary matrix
\begin{equation}
U_{ \nu }^{ \dagger } M_{ \nu_{ _L } } U_{ \nu }^{ * } =
\textrm{diag}\left[ \left| m_{ \nu_{1} }^{ s } \right|,
\left| m_{ \nu_{2} }^{ s } \right|, \left| m_{ \nu_{3} }^{ s }\right| \right],
\end{equation}
where $U_{ \nu }\equiv U_{ _L } {\cal K }$ and
${\cal K } \equiv \textrm{diag}\left[ e^{i\eta_{1}/2}, e^{i\eta_{2}/2}, e^{i\eta_{3}/2}
\right]$ is the diagonal matrix of the Majorana phases. \\
From the previous analysis, the matrix $M_{ \nu_{ _L } }$ has two non-ignorable phases which
are
\begin{equation}\label{fases:ml}
\phi_{1} \equiv \arg \left\{ b_{ \nu_{ _L } } \right \} \quad \textrm{and} \quad
\phi_{2} \equiv \arg \left\{ c_{ \nu_{ _L } } \right \} .
\end{equation}
However, to discribe the phenomenology of neutrinos masses and mixing, only one phase in
$ M_{ \nu_{ _L } }$ is required. Therefore, without loss of generality, we may chose
$\phi_{1}=2\phi_{2}= 2\varphi$ and the following relationship is fulfilled\footnote{The
general case, when $\phi_{1} \neq 2 \phi_{2}$ is slightly more complicated. This case will
be trated in detail in a following paper.}:
\begin{equation}
\tan \phi_{1} = \frac{ 2 \Im m \; c_{ \nu_{ _L } } \Re e \; c_{ \nu_{ _L } } }{
\left( \Re e \; c_{ \nu_{ _L } } \right)^{ 2 } - \left( \Im m\; c_{ \nu_{ _L } }
\right)^{ 2 } } .
\end{equation}
In this case, the analysis simplifies since the phases in $M_{ \nu_{ _L } }$ may be
factorized out as
\begin{equation}
M_{ \nu_{ _L } } = Q \bar{M}_{ \nu_{ _L } } Q,
\end{equation}
where $Q$ is a diagonal matrix of phases
{\small $Q \equiv \textrm{diag}\left[ e^{ -i \varphi }, e^{ i \varphi }, 1 \right] $ }
and $\bar{M}_{ \nu_{ _L } }$ is a real symetric matrix. Then, the matrix $M_{ \nu_{ _L } }$,
can be diagonalized by a unitary matrix through the transformation
\begin{equation}
U_{ \nu }^{ \dagger } M_{ \nu_{ _L } } U_{ \nu }^{ * } = \textrm{diag}\left[ m_{ \nu_{1} },
m_{ \nu_{2} }, m_{ \nu_{3} } \right];
\end{equation}
where $m_{ \nu_{j} }$ ($j=1,2,3$) are the eigenvalues of the matrix $M_{ \nu_{ _L } }$, and
the unitary matrix is
$U_{ \nu } \equiv Q { \bf O_{\nu} } {\cal K }$ where $ {\bf O_{\nu} }$ is the orthogonal
real matrix (\ref{M_ortogonal}), that diagonalizes the real symetric matrix
$\bar{M}_{ \nu_{ _L } }$.
It is also important to mention that when the Hermitian matrix with four texture zeroes
defined in eq. (\ref{T_fritzsch}), is taken as a universal mass matrix for all
Dirac fermions and right handed Majorana neutrinos~\cite{GonzalezCanales:2009zz}, the phases
of all entries in the right handed Majorana neutrino mass matrix are fixed at the numerical
value of $\phi_{ \nu_{ R } } = n \pi$. Thus, the right handed Majorana neutrinos mass matrix
is real and symmetric and has the form with four texture zeroes shown in (\ref{T_fritzsch}).
In the more general case in which the Dirac fermions and right handed neutrino mass matrices
are represented by Hermitian matrices, that can be written in polar form as
$A=P^{\dagger}\bar{A}P$, where $P$ is a diagonal matrix of phases and $\bar{A}$ is a real
symmetric matrix, the symmetry of the left-handed Majorana neutrino mass matrix also fixes
all phases in the mass matrix of the right handed neutrinos at the numerical value
$\phi_{ \nu_{ R } } = n \pi$. Hence, the only undetermined phases in the mass matrix of the
left-handed Majorana neutrinos $M_{ \nu_{ _L } }$ are the phases $\phi_{ \nu_{ D } }$,
coming from the mass matrix of the Dirac neutrinos.
\section{Mixing Matrices}
The quark and lepton flavor mixing matrices, $U_{ _{PMNS} }$ and $V_{ _{CKM } }$, arise
from the mismatch between diagonalization of the mass matrices of $u$ and $d$ type
quarks~\cite{Amsler:2008zzb} and the diagonalization of the mass matrices of
charged leptons and left-handed neutrinos~\cite{Hochmuth:2007wq} respectively,
\begin{equation}\label{M_unitarias}
U_{ _{PMNS} } = U_{l}^{\dagger}U_{\nu}, \quad V_{ _{CKM} } = U_{u}U_{d}^{\dagger}.
\end{equation}
Therefore, in order to obtain the unitary matrices appearing in~(\ref{M_unitarias}) and get
predictions for the flavor mixing angles and CP violating phases, we should specify the
mass matrices. \\
In the quark sector, the unitarity of $V_{ _{CKM} }$ leads to the relations
$\sum_{i} V_{ij}V_{ik}^{*} = \delta_{jk}$ and $\sum_{j} V_{ij}V_{kj}^{*} = \delta_{ik}$. The
vanishing combinations can be represented as triangles in a complex plane. The area of all
triangles is equal to half of the Jarlskog invariant, $J_{ q }$~\cite{Jarlskog:1985cw},
which is a rephasing invariant measure of CP violation. The term unitarity triangle is
usually reserved for the tringle obtained from the relation
$V_{ud}V_{ub}^{*} + V_{cd}V_{cb}^{*} + V_{td}V_{tb}^{*} = 0$. In this case de Jarlskog
invariant is
\begin{equation}\label{quarks:JCP}
J_{ q } = \Im m \left[ V_{us}V_{cs}^{*}V_{ub}^{*}V_{cb} \right],
\end{equation}
and the inner angles of the unitarity triangle are
\begin{equation}\label{quarks:InerAng}
\begin{array}{l}
\alpha \equiv \arg\left( -\frac{ V_{td}V_{tb}^{*} }{ V_{ud}V_{ub}^{*} }\right), \quad
\beta \equiv \arg\left( -\frac{ V_{cd}V_{cb}^{*} }{ V_{td}V_{tb}^{*} }\right), \\ \\
\gamma \equiv \arg\left( -\frac{ V_{ud}V_{ub}^{*} }{ V_{cd}V_{cb}^{*} }\right).
\end{array}
\end{equation}
For the lepton sector, when the left-handed neutrinos are Majorana particles,
the mixing matrix is defined as~\cite{Mohapatra:2006gs}
$U_{ _{PMNS} } = U_{l}^{\dagger}U_{ _L } K$ where
$K\equiv \textrm{diag}\left[1, e^{i\beta_{1}}, e^{i\beta_{2}} \right]$ is the diagonal
matrix of the Majorana CP violating phases. Also in the case of three neutrino mixing there
are three CP violation rephasing invariants~\cite{Hochmuth:2007wq}, associated with the
three CP violating phases present in the $U_{ _{PMNS} }$ matrix. The rephasing invariant
related to the Dirac phase, analogous to the Jarlskog invariant in the quark sector,
is given by:
\begin{equation}\label{InvJl}
J_{ l } \equiv \Im m \left[ U_{e1}^{ * } U_{ \mu 3 }^{ * } U_{ e3 } U_{ \mu 1 }\right] .
\end{equation}
The rephasing invariant $J_{ l }$ controls the magnitude of CP violation effects in neutrino
oscillations and is a directly observable quantity. The other two rephasing invariants
associated with the two Majorana phases in the $U_{ _{PMNS} }$ matrix, can be chosen as:
\begin{equation}\label{InvS1S2}
S_{1} \equiv \Im m \left[ U_{e1}U_{ e3 }^{ * }\right], \quad S_{2} \equiv \Im m \left[
U_{e2}U_{ e3 }^{ * }\right].
\end{equation}
These rephasing invariants are not uniquely defined, but the ones shown in the
eqs.~(\ref{InvJl})~and~(\ref{InvS1S2}) are relevant for the definition of the effective Majorana
neutrino mass, $m_{ee}$, in the neutrinoless double beta decay.
\subsection{Mixing Matrices as Functions of the Fermion Masses}
The unitary matrices $U_{u,d}$ occurring in the definition of $V_{ _{CKM} }$,
eq.~(\ref{M_unitarias}), may be written in polar form as
$U_{ u,d } = {\bf O}_{ u,d }^{T} P_{ u,d }$.
In this expresion, $P_{ u,d }$ is the diagonal matrix of phases appearing in the four texutre
zeroes mass matrix~(\ref{Polar:FT}). Then, from~(\ref{M_unitarias}), the quark mixing matrix
takes the form
\begin{equation}\label{M_unitaria2}
V_{_{CKM} }^{ ^{th} } = {\bf O_{u} }^{T} P^{(u-d)} {\bf O}_{d},
\end{equation}
where $P^{ (u-d) } = \textrm{diag}\left[1, e^{i\phi}, e^{i\phi} \right]$ with $\phi = \phi_{u}
- \phi_{d}$, and $ {\bf O }_{ u,d }$, are the real orthogonal matrices~(\ref{M_ortogonal}) that
diagonalize the real symmetric mass matrices $\bar{M}_{i}$.
A similar analysis shows that $U_{ _{PMNS} }$ may also be written as
$U_{ _{PMNS} } = U_{l}^{\dagger}U_{\nu}$, with
$U_{ \nu, l } = P_{ \nu, l } {\bf O_{ \nu, l } }$, this matrix takes the form
\begin{equation}\label{M_unitaria3}
U_{ _{PMNS } }^{ ^{th} } = { \bf O}_{l}^{T}P^{ ( \nu - l ) } {\bf O}_{\nu} K,
\end{equation}
where $P^{ ( \nu -l )} = \textrm{diag}\left[1, e^{ i \Phi_{1} }, e^{ i \Phi_{2} } \right]$
is the diagonal matrix of the Dirac phases, with $\Phi_{1} = 2\varphi - \phi_{ l }$ and
$\Phi_{2} = \varphi - \phi_{l}$. The real orthogonal matrices ${ \bf O}_{ \nu, l } $ are
defined in eq.~(\ref{M_ortogonal}).
Substitution of the expressions (\ref{fs})-(\ref{Ds}) in the unitary matices
(\ref{M_unitaria2}) and (\ref{M_unitaria3}) allows us to express the mixing
matrices~$V_{_{CKM}}^{ ^{th} }$~and~$U_{_{PMNS}}^{ ^{th} }$ as explicit functions of the masses
of quarks and leptons. For the elements of the $V_{_{CKM}}^{ ^{th} }$ mixing matrix, we obtained
the same theoretical expressions given by Mondrag\'on and
Rodr\'{\i}guez-Jauregui~\cite{PhysRevD.61.113002}:
\begin{equation}
V_{_{CKM}}^{ ^{th} } =
\left( \begin{array}{ccc}
V_{ud}^{ ^{th} } & V_{us}^{ ^{th} } & V_{ub}^{ ^{th} } \\
V_{cd}^{ ^{th} } & V_{cs}^{ ^{th} } & V_{cb}^{ ^{th} } \\
V_{td}^{ ^{th} } & V_{ts}^{ ^{th} } & V_{tb}^{ ^{th} }
\end{array} \right),
\end{equation}
where
\begin{widetext}
\begin{equation}\label{elem:ckm}
\begin{split}
\begin{array}{l}
V_{ ud }^{ ^{th} } =
\sqrt{ \frac{ \widetilde{m}_{c} \widetilde{m}_{s} f_{ u1 } f_{ d1} }{
{\cal D}_{ u 1 } {\cal D}_{ d1 } } }
+ \sqrt{ \frac{ \widetilde{m}_{u} \widetilde{m}_{d} }{
{\cal D}_{ u 1 } {\cal D}_{ d1 } } } \left( \sqrt{ \left( 1 - \delta_{ u } \right)
\left( 1 - \delta_{d} \right) f_{ u1 } f_{ d1 } } + \sqrt{ \delta_{u} \delta_{d} f_{ u2 }
f_{ d2 } } \right) e^{ i \phi }, \\
V_{us}^{ ^{th} } =
- \sqrt{ \frac{ \widetilde{m}_{c} \widetilde{m}_{d} f_{ u1 } f_{ d2 } }{
{\cal D}_{ u1 } {\cal D}_{ d2 } } } + \sqrt{ \frac{ \widetilde{m}_{u} \widetilde{m}_{s} }{
{\cal D}_{ u1 } {\cal D}_{ d2 } } } \left( \sqrt{ \left( 1 - \delta_{u} \right) \left( 1 -
\delta_{d} \right) f_{ u1 } f_{ d2 }} + \sqrt{ \delta_{u} \delta_{d} f_{ u2 } f_{ d1 } }
\right) e^{ i \phi }, \\
V_{ub}^{ ^{th} } =
\sqrt{ \frac{ \widetilde{m}_{c} \widetilde{m}_{d} \widetilde{m}_{s} \delta_{d} f_{ u1 } }{
{\cal D}_{ u1 } {\cal D}_{ d3 } } } + \sqrt{ \frac{ \widetilde{m}_{u} }{
{\cal D}_{ u1 } {\cal D}_{ d3 } } } \left( \sqrt{ \left( 1 - \delta_{u} \right) \left( 1 -
\delta_{d} \right) \delta_{d} f_{ u1 } } - \sqrt{ \delta_{u} f_{ u2 } f_{ d1 } f_{ d2 } }
\right) e^{ i \phi },\\
V_{cd}^{ ^{th} } =
- \sqrt{ \frac{ \widetilde{m}_{u} \widetilde{m}_{s} f_{ u2 } f_{ d1} }{
{\cal D}_{ u2 } {\cal D}_{ d1 } } } + \sqrt{ \frac{ \widetilde{m}_{c} \widetilde{m}_{d} }{
{\cal D}_{ u2 } {\cal D}_{ d1 } } } \left( \sqrt{ \left( 1 - \delta_{u} \right) \left( 1 -
\delta_{d} \right) f_{ u2 } f_{ d1 } } + \sqrt{ \delta_{u} \delta_{d} f_{ u1 } f_{ d2 } }
\right) e^{ i \phi },\\
V_{cs}^{ ^{th} } =
\sqrt{ \frac{ \widetilde{m}_{u} \widetilde{m}_{d} f_{ u2 } f_{ d2} }{
{\cal D}_{ u2 } {\cal D}_{ d2 } } } + \sqrt{ \frac{ \widetilde{m}_{c} \widetilde{m}_{s} }{
{\cal D}_{ u2 } {\cal D}_{ d2 } } } \left( \sqrt{ \left( 1 - \delta_{u} \right) \left( 1 -
\delta_{d} \right) f_{ u2 } f_{ d2 } } + \sqrt{ \delta_{u} \delta_{d} f_{ u1 } f_{ d1 } }
\right) e^{ i \phi }, \\
V_{cb}^{ ^{th} } =
- \sqrt{ \frac{ \widetilde{m}_{u} \widetilde{m}_{d} \widetilde{m}_{s} \delta_{d} f_{ u2 }
}{ {\cal D}_{ u2 } {\cal D}_{ d3 } } } + \sqrt{ \frac{ \widetilde{m}_{c} }{
{\cal D}_{ u2 } {\cal D}_{ d3 } } } \left( \sqrt{ \left( 1 - \delta_{u} \right) \left( 1
- \delta_{d} \right) \delta_{d} f_{ u2 } } - \sqrt{ \delta_{u} f_{ u1 } f_{ d1 } f_{ d2 }
} \right) e^{ i \phi } , \\
V_{td}^{ ^{th} } =
\sqrt{ \frac{ \widetilde{m}_{u} \widetilde{m}_{c} \widetilde{m}_{s} \delta_{u} f_{ d1 } }{
{\cal D}_{ u3 } {\cal D}_{ d1 } } } + \sqrt{ \frac{ \widetilde{m}_{d} }{
{\cal D}_{ u3 } {\cal D}_{ d1 } } } \left( \sqrt{ \delta_{u} \left( 1 - \delta_{u} \right)
\left( 1 - \delta_{d} \right) f_{ d1 } } - \sqrt{ \delta_{d} f_{ u1 } f_{ u2 } f_{ d2 } }
\right) e^{ i \phi }, \\
\end{array}
\end{split}
\end{equation}
\begin{displaymath}
\begin{split}
\begin{array}{l}
V_{ts}^{ ^{th} } =
- \sqrt{ \frac{ \widetilde{m}_{u} \widetilde{m}_{c} \widetilde{m}_{d} \delta_{u} f_{ d2} }{
{\cal D}_{ u3 } {\cal D}_{ d2 } } } + \sqrt{ \frac{ \widetilde{m}_{s} }{
{\cal D}_{ u3 } {\cal D}_{ d2 } } } \left( \sqrt{ \delta_{u} \left( 1 - \delta_{u} \right)
\left( 1 - \delta_{d} \right) f_{ d2 } } - \sqrt{ \delta_{d} f_{ u1 } f_{ u2 } f_{ d1 } }
\right) e^{ i \phi },\\
V_{tb}^{ ^{th} } =
\sqrt{ \frac{ \widetilde{m}_{u} \widetilde{m}_{c} \widetilde{m}_{d} \widetilde{m}_{s}
\delta_{u} \delta_{d} }{ {\cal D}_{ u3 } {\cal D}_{ d3 } } } + \left( \sqrt{
\frac{ f_{ u1 } f_{ u2 } f_{ d1 } f_{ d2 } }{ {\cal D}_{ u3 } {\cal D}_{ d3 } } }
+ \sqrt{ \frac{ \delta_{u} \delta_{d} \left( 1 - \delta_{u} \right) \left( 1 - \delta_{d}
\right) }{ {\cal D}_{ u3 } D_{ d3 } } } \right) e^{ i \phi }.
\end{array}
\end{split}
\end{displaymath}
\end{widetext}
Here, the $m$'s, $f$'s and ${\cal D}$'s are defined in (\ref{fs}) and
(\ref{Ds}), respectively. And takes the form
\begin{equation}\label{MsFsDs:quarks}
\begin{array}{l}
\widetilde{m}_{u(d)} = \frac{ m_{u(d)} }{ m_{t(b)} },\\
\widetilde{m}_{c(s)} = \frac{ m_{c(s)} }{ m_{t(b)} },\\
f_{ u(d)1 } = \left( 1 - \widetilde{m}_{u(d)} - \delta_{u(d)} \right), \\
f_{ u(d)2 } = \left( 1 + \widetilde{m}_{c(s)} - \delta_{u(d)} \right), \\
{\cal D}_{u(d)1} = ( 1 - \delta_{u(d)} )( \widetilde{m}_{u(d)} + \widetilde{m}_{c(s)} )
( 1 - \widetilde{m}_{u(d)} ), \\
{\cal D}_{u(d)2} = ( 1 - \delta_{u(d)} )( \widetilde{m}_{u(d)} + \widetilde{m}_{c(s)} )
( 1 + \widetilde{m}_{u(d)} ), \\
{\cal D}_{u(d)3} = ( 1 - \delta_{u(d)} )( 1 - \widetilde{m}_{u(d)} )( 1 +
\widetilde{m}_{c(s)} ).
\end{array}
\end{equation}
Now, with the help of the equations (\ref{M_ortogonal}) and (\ref{M_unitaria3}), we obtain
the theoretical expresion of the elements of the lepton mixing matrix,
$U_{_{PMNS}}^{ ^{th} }$. This expresions have the following form:
\begin{equation}
U_{_{ PMNS } }^{ ^{th} } =
\left( \begin{array}{ccc}
U_{ e 1 }^{ ^{th} } & U_{ e 2 }^{ ^{th} } e^{ i \beta_{ 1 } } & U_{ e 3 }^{ ^{th} }
e^{ i \beta_{ 2 } } \\
U_{ \mu 1 }^{ ^{th} } & U_{ \mu 2 }^{ ^{th} } e^{ i \beta_{ 1 } } & U_{ \mu 3 }^{ ^{th} }
e^{ i \beta_{ 2 } } \\
U_{ \tau 1 }^{ ^{th} } & U_{ \tau 2 }^{ ^{th} } e^{ i \beta_{ 1 } } & U_{ \tau 3 }^{ ^{th} }
e^{ i \beta_{ 2 } }
\end{array} \right)
\end{equation}
where
\begin{widetext}
\begin{equation}\label{elem:pmns}
\begin{split}
\begin{array}{l}
U_{e1}^{ ^{th} } =
\sqrt{ \frac{\widetilde{m}_{\mu} \widetilde{m}_{\nu_{2}} f_{ l1 } f_{ \nu1 } }{
{\cal D}_{ l1 } {\cal D}_{ \nu1 } } } + \sqrt{ \frac{ \widetilde{m}_{e}
\widetilde{m}_{\nu_{1}} }{ {\cal D}_{ l1 } {\cal D}_{ \nu 1 } } }
\left( \sqrt{ ( 1 - \delta_{l} )( 1 - \delta_{\nu} ) f_{ l1 } f_{ \nu1 } }
e^{ i \Phi_{1} } + \sqrt{ \delta_{l} \delta_{\nu} f_{ l2 } f_{ \nu2 } } e^{ i \Phi_{2} }
\right), \\
U_{e2}^{ ^{th} } =
- \sqrt{ \frac{\widetilde{m}_{\mu} \widetilde{m}_{\nu_{1}} f_{ l1 } f_{ \nu2 } }{
{\cal D}_{ l1 } {\cal D}_{ \nu2 } } } + \sqrt{ \frac{ \widetilde{m}_{e}
\widetilde{m}_{\nu_{2}} }{ {\cal D}_{ l1 } {\cal D}_{ \nu2 } } } \left( \sqrt{ ( 1 -
\delta_{l} )( 1 - \delta_{\nu} ) f_{ l1 } f_{ \nu2 } } e^{ i \Phi_{ 1 } } +
\sqrt{ \delta_{l} \delta_{\nu} f_{ l2 } f_{ \nu1 } } e^{ i \Phi_{2} } \right) , \\
U_{e3}^{ ^{th} } =
\sqrt{ \frac{ \widetilde{m}_{\mu} \widetilde{m}_{\nu_{1}} \widetilde{m}_{\nu_{2}}
\delta_{\nu} f_{ l1 } }{ {\cal D}_{ l1 } {\cal D}_{ \nu3} } } + \sqrt{ \frac{
\widetilde{m}_{e} }{ {\cal D}_{ l1 } {\cal D}_{ \nu3 } } } \left( \sqrt{
\delta_{\nu} ( 1 - \delta_{l} ) ( 1 - \delta_{\nu} ) f_{ l1 } } e^{ i \Phi_{1} }
- \sqrt{ \delta_{e} f_{ l2 } f_{ \nu1 } f_{ \nu2 } } e^{ i \Phi_{2} }\right) , \\
U_{ \mu1 }^{ ^{th} } =
-\sqrt{ \frac{ \widetilde{m}_{e} \widetilde{m}_{\nu_{2}} f_{ l2 } f_{ \nu1 } }{
{\cal D}_{ l2 } {\cal D}_{ \nu1 } } } + \sqrt{ \frac{ \widetilde{m}_{\mu}
\widetilde{m}_{\nu_{1}} }{ {\cal D}_{ l2 } {\cal D}_{ \nu1 } } } \left( \sqrt{ ( 1 -
\delta_{l} )( 1 - \delta_{\nu} ) f_{ l2 } f_{ \nu1 } } e^{ i \Phi_{1} } + \sqrt{
\delta_{l} \delta_{\nu} f_{ l1 } f_{ \nu2 } } e^{ i \Phi_{2} } \right) ,\\
U_{ \mu2 }^{ ^{th} } =
\sqrt{ \frac{ \widetilde{m}_{e} \widetilde{m}_{\nu_{1} } f_{ l2 } f_{ \nu2 } }{
{\cal D}_{ l2 } {\cal D}_{\nu 2} } } + \sqrt{ \frac{ \widetilde{m}_{\mu}
\widetilde{m}_{\nu_{2}} }{ {\cal D}_{ l2 } {\cal D}_{ \nu2 } } } \left( \sqrt{ ( 1 -
\delta_{l} ) ( 1 - \delta_{\nu} ) f_{ l2 } f_{ \nu2 } } e^{ i \Phi_{1} } + \sqrt{
\delta_{l} \delta_{\nu} f_{ l1 } f_{ \nu1 } } e^{ i \Phi_{2} } \right ) , \\
U_{ \mu3 }^{ ^{th} } =
-\sqrt{ \frac{\widetilde{m}_{e} \widetilde{m}_{\nu_{1}} \widetilde{m}_{\nu_{2}}
\delta_{\nu} f_{ l2 } }{ {\cal D}_{ l2 } {\cal D}_{ \nu3 } } } + \sqrt{
\frac{ \widetilde{m}_{\mu} }{ {\cal D}_{ l2 } {\cal D}_{ \nu3 } } } \left( \sqrt{
\delta_{\nu} ( 1 - \delta_{l} ) ( 1 - \delta_{\nu} ) f_{ l2 } } e^{ i \Phi_{1} }
- \sqrt{ \delta_{l} f_{ l1 } f_{ \nu1 } f_{ \nu2 } } e^{ i \Phi_{2} } \right ), \\
U_{ \tau1 }^{ ^{th} } =
\sqrt{ \frac{ \widetilde{m}_{e} \widetilde{m}_{\mu} \widetilde{m}_{\nu_{2}} \delta_{l}
f_{ \nu1 } }{ {\cal D}_{ l3 } {\cal D}_{ \nu1 } } } + \sqrt{ \frac{ \widetilde{m}_{\nu_{1}}
}{ {\cal D}_{ l3 } {\cal D}_{ \nu1 } } } \left( \sqrt{ \delta_{l} ( 1 - \delta_{l} )( 1 -
\delta_{\nu} ) f_{ \nu1 } } e^{ i \Phi_{1} } - \sqrt{ \delta_{ \nu } f_{ l1 } f_{ l2 }
f_{ \nu2 } } e^{ i \Phi_{2} } \right), \\
U_{ \tau2 }^{ ^{th} } =
- \sqrt{ \frac{ \widetilde{m}_{e} \widetilde{m}_{\mu} \widetilde{m}_{\nu_{1}} \delta_{l}
f_{ \nu2 } }{ {\cal D}_{ l3 } {\cal D}_{\nu 2} } } + \sqrt{ \frac{ \widetilde{m}_{\nu_{2}}
}{ {\cal D}_{ l3 } {\cal D}_{ \nu2 } } } \left( \sqrt{ \delta_{l} ( 1 - \delta_{l} )( 1 -
\delta_{\nu} ) f_{ \nu2 } } e^{ i \Phi_{1} } - \sqrt{ \delta_{\nu} f_{ l1 } f_{ l2 }
f_{ \nu1 } } e^{ i \Phi_{2} } \right ), \\
U_{ \tau3 }^{ ^{th} } =
\sqrt{ \frac{ \widetilde{m}_{e} \widetilde{m}_{\mu} \widetilde{m}_{\nu_{1}}
\widetilde{m}_{\nu_{2}} \delta_{l} \delta_{\nu} }{ {\cal D}_{ l3 } {\cal D}_{ \nu3 } } } +
\sqrt{ \frac{ \delta_{l} \delta_{\nu}( 1 - \delta_{l} ) ( 1 - \delta_{\nu} ) }{
{\cal D}_{ l3 } {\cal D}_{ \nu3 } } } e^{ i \Phi_{1} } + \sqrt{ \frac{ f_{l1 } f_{ l2 }
f_{ \nu1 } f_{ \nu2 } }{ {\cal D}_{ l3 } {\cal D}_{ \nu3 } } } e^{ i \Phi_{2} } ,
\end{array}
\end{split}
\end{equation}
\end{widetext}
in these expresions the $\widetilde{m}$'s, $f$'s and ${\cal D}$'s are defined in (\ref{fs}) and
(\ref{Ds}), respectively. And takes the form
{\small \begin{equation}\label{MsFsDs:leptones}
\begin{array}{l}
\widetilde{m}_{\nu_{1}(e)} = \frac{ m_{\nu_{1}(e)} }{ m_{\nu_{3}(\tau)} },\\
\widetilde{m}_{\nu_{2}(\mu)} = \frac{ m_{\nu_{2}(\mu)} }{ m_{\nu_{3}(\tau)} },\\
f_{ \nu(l)1 } = \left( 1 - \widetilde{m}_{\nu_{1}(e)} - \delta_{\nu(l)} \right), \\
f_{ \nu(l)2 } = \left( 1 + \widetilde{m}_{\nu_{2}(\mu)} - \delta_{\nu(l)} \right), \\
{\cal D}_{\nu(l)1} = ( 1 - \delta_{\nu(l)} )( \widetilde{m}_{\nu_{1}(e)} +
\widetilde{m}_{\nu_{2}(\mu)} ) ( 1 - \widetilde{m}_{\nu_{1}(e)} ), \\
{\cal D}_{ \nu(l)2} = ( 1 - \delta_{\nu(l)} )( \widetilde{m}_{\nu_{1}(e)} +
\widetilde{m}_{\nu_{2}(\mu)} ) ( 1 + \widetilde{m}_{\nu_{2}(\mu)} ), \\
{\cal D}_{\nu(l)3} = ( 1 - \delta_{\nu(l)} )( 1 - \widetilde{m}_{\nu_{1}(e)} )( 1 +
\widetilde{m}_{\nu_{2}(\mu)} ).
\end{array}
\end{equation} }
\subsection{The $\chi^{2}$ fit for the Quark Mixing Matrix}
We made a $\chi^{2}$ fit of the exact theoretical expressions for the modulii of the entries
of the quark mixing matrix $| ( V_{ _{ CKM } }^{ ^{th} } )_{ij} |$ and the inner angles of the
unitarity triangle $\alpha^{ ^{th} }$, $\beta^{ ^{th} }$ and $\gamma^{ ^{th} }$ to the
experimental values given by Amsler~\cite{Amsler:2008zzb}. In this fit, we computed the modulii
of the entries of the quark mixing matrix and the inner angles of the unitarity triangle from
the theoretical expresion (\ref{elem:ckm}) with the following numerical values of the quark mass
ratios~\cite{Amsler:2008zzb}:
\begin{equation}\label{quark-rat}
\begin{array}{l}
\widetilde{m}_{u} = 2.5469 \times 10^{ -5}, \quad
\widetilde{m}_{c} = 3.9918 \times 10^{ -3}, \\
\widetilde{m}_{d} = 1.5261 \times 10^{ -3}, \quad
\widetilde{m}_{s} = 3.2319 \times 10^{ -2}.
\end{array}
\end{equation}
The numerical values of the mass ratios were left fixed at the values given in
eq.~(\ref{quark-rat}) and the parameters $\delta_{u}$ and $\delta_{d}$ were left as
free parameters to be varied. Hence, in the $\chi^{2}$ fit we have six degrees of freedom
($d.o.f.$), namely, the nine observable modulii of the entries in the $V_{ _{CKM} }$
matrix less the three free parameters to be varied. Once the best values of the parameters
$\delta_{u}$, $\delta_{d}$ and $\phi$ were determined , we computed the three inner angles
of the unitary triangle from eq.~(\ref{quarks:InerAng}) and the Jarlskog invariant from
eq.~(\ref{quarks:JCP}).
The resulting best values of the parameters $\delta_{u}$ and $\delta_{d}$ are
\begin{equation}\label{X2:Qds}
\delta_{u} = 3.829 \times 10^{-3}, \quad \delta_{d} = 4.08 \times 10^{ -4 }
\end{equation}
and the Dirac CP violating phase is $\phi = 90^{o}$. The best values for the moduli of the entries of the $CKM$ mixing matrix are given in the
following expresion
\begin{equation}
\left| V_{ _{CKM} }^{ ^{th} } \right| = \left(\begin{array}{ccc}
0.97421 & 0.22560 & 0.003369 \\
0.22545 & 0.97335 & 0.041736 \\
0.008754 & 0.04094 & 0.99912
\end{array} \right)
\end{equation}
and inner angles of the unitary triangle
\begin{equation}
\alpha^{ ^{th} } = 91.24^{o}, \quad \beta^{ ^{th} } = 20.41^{o}, \quad
\gamma^{ ^{th} } = 68.33^{o}.
\end{equation}
The Jarlskog invariant takes the value
\begin{equation}
J_{q}^{ ^{th} } = 2.9 \times 10^{-5}.
\end{equation}
All these results are in good agreement with the experimental values. The minimun value of
$\chi^{2}$ obtained in this fit is 4.6 and the resulting value of $\chi^{2}$ for degree of
freedom is {\small $\frac{\chi^{2}_{min} }{ d.o.f. }=0.77$}.
\subsection{ The $\chi^{2}$ fit for the Lepton Mixing Matrix}
In the case of the lepton mixing matrix, we made a $\chi^{2}$ fit of the theoretical
expressions for the modulii of the entries of the lepton mixing matrix
$| ( U_{ _{ PMNS } }^{ ^{th} } )_{ij} |$ given in eq.~(\ref{elem:pmns}) to the values
extracted from experiment as given by Gonzalez-Garcia~\cite{GonzalezGarcia:2007ib} and
quoted in eq.~(\ref{GG:UPMNS}).The computation was made using the following values for the
charged lepton masses~\cite{Amsler:2008zzb}:
\begin{equation}\label{massChL}
\begin{array}{l}
m_{e} = 0.5109~\textrm{MeV}, \;\; m_{\mu}= 105.685~\textrm{MeV}, \;\; \\
m_{\tau}=1776.99~\textrm{MeV}.
\end{array}
\end{equation}
We took for the masses of the left-handed Majorana neutrinos a normal hierarchy. This allows
us to write the left-handed Majorana neutrinos mass ratios in terms of the neutrino
squared mass differences and the neutrino mass $m_{ \nu_{3} }$ in the following form:
\begin{equation}
\begin{array}{l}
\widetilde{m}_{ \nu_{1} } = \sqrt{ 1 - \frac{ \left( \Delta m_{ 32 }^{ 2 } +
\Delta m_{ 21 }^{ 2 } \right) }{
m_{ \nu_{3} }^{ 2 } } }, \;
\widetilde{m}_{ \nu_{2} } =
\sqrt{ 1 - \frac{ \Delta m_{ 32 }^{ 2 } }{ m_{ \nu_{3} }^{ 2 } } }.
\end{array}
\end{equation}
The neutrino squared mass differences were obtained from the experimental data on neutrino
oscillations given in Gonzalez-Garcia~\cite{GonzalezGarcia:2007ib} and we left the mass
$m_{ \nu_{3} }$ as a free parameter of the $\chi^{2}$ fit. Also, the parameters
$\delta_{e}$, $\delta_{\nu}$, $\Phi_{1}$ and $\Phi_{2} $ were left as frees parameters to
be varied. Hence, in this $\chi^{2}$ fit we have four degrees of freedom. \\
From the best values obtained for $m_{ \nu_{3} }$ and the experimental values of the
$\Delta m_{ 32 }^{ 2 }$ and $\Delta m_{ 21 }^{ 2 }$, we obtained the following best
values for the neutrino masses
\begin{equation}\label{X2:Mnus}
\begin{array}{l}
m_{\nu_{1}} = 2.7 \times 10^{-3}\textrm{eV}, \quad m_{\nu_{2}} = 9.1 \times
10^{-3}\textrm{eV}, \\ m_{\nu_{3}} = 4.7 \times 10^{-2}\textrm{eV}.
\end{array}
\end{equation}
The resulting best values of the parameters $\delta_{e}$ and $\delta_{\nu}$ are
\begin{equation}\label{X2:ds}
\delta_{l} = 0.06, \qquad \delta_{ \nu } = 0.522 ,
\end{equation}
and the best values of the Dirac CP violating phases are
$\Phi_{1} = \pi \quad \textrm{and} \quad \Phi_{2} = 3\pi/2$.
The best values for the modulii of the entries of the $PMNS$ mixing matrix are given in the
following expresion
\begin{equation}
\left| U_{ _{PMNS} }^{ ^{th} } \right| =
\left(\begin{array}{ccc}
0.820421 & 0.568408 & 0.061817 \\
0.385027 & 0.613436 & 0.689529 \\
0.422689 & 0.548277 & 0.721615
\end{array} \right).
\end{equation}
The value of the rephasing invariant related to the Dirac phase is
\begin{equation}
J_{ l }^{ ^{th} } = 8.8 \times 10^{ -3}.
\end{equation}
In the absence of experimental information about the Majorana phases
$\beta_{1 }$ and $\beta_{ 2 }$, the two rephasing invariants $S_{1}$ and $S_{2}$,
eq.~(\ref{InvS1S2}), associated with the two Majorana phases in the $U_{ _{PMNS} }$ matrix,
could not be determined from experimental values. Therefore, in order to make a numerical
estimate of Majorana phases, we maximized the rephasing invariants $S_{1}$ and $S_{2}$, thus
obtaining a numerical value for the Majorana phases $\beta_{1 }$ and $\beta_{ 2 }$. Then,
the maximum values of the rephasing invariants,
eq(\ref{InvS1S2}), are:
\begin{equation}\label{valor:S1S2}
S_{1}^{ max } = -4.9 \times 10^{ -2 }, \quad S_{2}^{ max } = 3.4 \times 10^{ -2 },
\end{equation}
with $\beta_{1 }= -1.4^{ o }$ and $\beta_{ 2 } = 77^{o}$. In this numerical analysis, the
minimum value of the $\chi^{2}$, corresponding to the best fit, is $\chi^{2}=0.288$ and
the resulting value of $\chi^{2}$ for degree of freedom is
{\small $\frac{\chi^{2}_{min} }{ d.o.f. }=0.075$}. All numerical results of the fit are in
very good agreement with the values of the moduli of the entries in the matrix
$U_{ _{PMNS} }$ as given in Gonzalez-Garcia~\cite{GonzalezGarcia:2007ib}.
\section{The Mixing Angles}
In the standard PDG parametrization, the entries in the quark and lepton mixing matrices are
parametrized in terms of the mixing angles and phases. Thus, the mixing angles are related
to the observable moduli of quark (lepton) $ V_{ _{ CKM } } ( U_{ _{ PMNS } } )$ through the
relations:
\begin{equation}\label{angulosMezclas}
\begin{array}{l}
\sin^{2}{\theta_{12}^{q ( l ) } } = \frac{ \left| V_{us} \left( U_{ e2 } \right)
\right|^{2} }{ 1 - \left| V_{ub} \left( U_{ e3 } \right) \right|^{2} }, \\\\
\sin^{2} \theta_{23}^{ q ( l ) } = \frac{ \left| V_{cb} \left( U_{ \mu 3 } \right)
\right|^{2} }{ 1 - \left| V_{ub} \left( U_{ e3 } \right)\right|^{2} }, \\\\
\sin^{2} \theta_{13}^{ q ( l ) } = \left| V_{ub} \left( U_{ e3 } \right) \right|^{2}.
\end{array}
\end{equation}
Then, theoretical expression for the quark mixing angles as functions of the quark mass
ratios are readily obtained when the theoretical expressions for the modulii of
the entries in the $CKM$ mixing matrix, given in eqs.~(\ref{elem:ckm}) and~(\ref{Ds}), are
substituted for $\left|V_{ij}\right|$ in the right hand side of eqs.~(\ref{angulosMezclas}). In
this way,and keeping only the leading order terms, we get :
\begin{equation}
\sin^{2}{\theta_{12}^{q ^{th} } } \approx
\frac{ \frac{ \widetilde{m}_{d} }{ \widetilde{m}_{s} } + \frac{ \widetilde{m}_{u} }{
\widetilde{m}_{c} } - 2 \sqrt{ \frac{ \widetilde{m}_{u} }{ \widetilde{m}_{c} }
\frac{ \widetilde{m}_{d} }{ \widetilde{m}_{s} } } \cos{ \phi } }{ \left( 1 +
\frac{ \widetilde{m}_{u} }{ \widetilde{m}_{c} } \right) \left( 1 + \frac{ \widetilde{m}_{d}
}{ \widetilde{m}_{s} } \right) },
\end{equation}
\begin{equation}
\sin^{2} \theta_{23}^{ q ^{th} } \approx
\frac{ \left( \sqrt{ \delta_{ u } } - \sqrt{ \delta_{ d } } \right)^{2} }{ \left( 1 +
\frac{ \widetilde{m}_{u} }{ \widetilde{m}_{c} } \right) },
\end{equation}
\begin{equation}
\sin^{2} \theta_{13}^{ q ^{th} } \approx
\frac{ \frac{ \widetilde{m}_{ u } }{ \widetilde{m}_{ c } } \left( \sqrt{ \delta_{ u }
} - \sqrt{ \delta_{ d } } \right)^{2} }{ \left( 1 + \frac{ \widetilde{m}_{u}
}{ \widetilde{m}_{c}} \right) }.
\end{equation}
Now, the numerical values of the quark mixing angles may be computed from
eq.(\ref{elem:ckm}) and the numerical values of the parameters $\delta_{ u }$ and
$\delta_{ d} $,eq. (\ref{X2:Qds}), and the CP violating phase $\phi=90^{o}$ obtained from
$\chi^{2}$ fit of $\left| V_{ _{CKM} }^{ th } \right| $ to the experimentally determined
values $\left| V_{ _{CKM} }^{ exp } \right| $. In this way we obtain
\begin{equation}
\theta_{12}^{q^{th}} = 13^{o},\quad \theta_{23}^{q^{th}} = 2.38^{o}, \quad
\theta_{13}^{q^{th}} = 0.19^{o},
\end{equation}
in very good agreement with the latest analysis of the experimental
data~\cite{Mateu:2005wi}, see (\ref{PDGdatosang}). \\
The numerical values of the leptonic mixing angles are computed in a similar fashion.
The theoretical expressions for the lepton mixing angles as funtion of the charged lepton
and neutrino mass ratios are obtained from eqs (\ref{angulosMezclas}) when the theoretical
expressions for the modulii of the entries in the $PMNS$ mixing matrix, given in
eqs.~(\ref{elem:pmns}) and (\ref{Ds}), are substituted for $\left| U_{ ij } \right| $ in the
right hand side of eqs.(\ref{angulosMezclas}). If we keep only the leading orders terms, we
obtain:
\begin{equation}\label{S12L}
\begin{array}{l}
\sin^{2}{\theta_{12}^{ l^{th} }} \approx
\frac{ 1 + \widetilde{m}_{\nu_{2}} - \delta_{ \nu } }{ \left( 1 + \widetilde{m}_{\nu_{2}}
\right) \left( 1 - \delta_{ \nu } \right) \left( 1 + \frac{\widetilde{m}_{\nu_{1}} }{
\widetilde{m}_{\nu_{2}} } \right) \left( 1 + \frac{ \widetilde{m}_{e} }{
\widetilde{m}_{\mu} } \right)} \left \{ \frac{ \widetilde{m}_{\nu_{1}} }{
\widetilde{m}_{\nu_{2}} } +
\right. \\ \qquad \left.
+ \frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu} } \left( 1 - \delta_{ \nu } \right)
+ 2 \sqrt{\frac{ \widetilde{m}_{\nu_{1}} }{ \widetilde{m}_{\nu_{2}} }
\frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu} } \left( 1 - \delta_{ \nu } \right) }
\cos{ \Phi_{ _1 } } \right \},
\end{array}
\end{equation}
\begin{equation}\label{S23L}
\sin^{ 2 } \theta_{23}^{ l^{th} } \approx
\frac{ \delta_{ \nu } + \delta_{ e } f_{ \nu2 } - \sqrt{ \delta_{ \nu } \delta_{ e }
f_{ \nu2 } } \cos \left( \Phi_{ _1 } - \Phi_{ _2 } \right) }{ \left( 1 +
\frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu} } \right) \left( 1 +
\widetilde{m}_{\nu_{2} } \right) },
\end{equation}
\begin{equation}\label{S13L}
\begin{array}{l}
\sin^{ 2 } \theta_{13}^{ l^{th} } \approx
\frac{ \delta_{ \nu } }{ \left( 1 + \frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu} }
\right) \left( 1 + \widetilde{m}_{\nu_{2} } \right) } \left \{ \frac{ \widetilde{m}_{e} }{
\widetilde{m}_{\mu} } + \frac{ \widetilde{m}_{\nu_{1} } \widetilde{m}_{\nu_{2}} }{
\left( 1 - \delta_{ \nu } \right)} -
\right. \\ \left.
\qquad \qquad -2 \sqrt{ \frac{ \widetilde{m}_{e} }{
\widetilde{m}_{\mu} } \frac{ \widetilde{m}_{\nu_{1}} \widetilde{m}_{ \nu_{2} } }{
\left( 1 - \delta_{ \nu } \right) } } \cos \Phi_{ _1 } \right \}.
\end{array}
\end{equation}
From eqs.~(\ref{MsFsDs:leptones}) we have that
$f_{ \nu2 } = 1 + \widetilde{m}_{\nu_{2}} - \delta_{ \nu }$.
The expressions quoted above are written in terms of the ratios of the lepton masses. When
the well known values of the charged lepton masses, the values of the neutrino masses,
eq.~(\ref{X2:Mnus}), the values of the delta parameters eq.~(\ref{X2:ds}) and the values
of the Dirac CP violating phases obtained from $\chi^2$ fit in the lepton sector, are
inserted in eqs.~(\ref{S12L})-(\ref{S13L}), we obtain the following numerical values for
the mixing angles
\begin{equation}
\theta_{12}^{l^{th}} = 34.7^{o}, \quad \theta_{23}^{l^{th}} = 43.6^{o}, \quad
\theta_{13}^{l^{th}} = 3.5^{o},
\end{equation}
which are in very good agreement with the latest experimental
data~\cite{GonzalezGarcia:2007ib, GonzalezGarcia:2010er}.
\section{Quark-Lepton Complementarity}
The relations between mixing angles and the moduli of the entries of the mixing matrices given
in eqs.~(\ref{angulosMezclas}) allow us to write the following identities:
\begin{equation}\label{QLC-M12}
\tan{\left( \theta_{12}^{q} + \theta_{12}^{l } \right)} = 1 + \Delta_{12},
\end{equation}
where
\begin{equation}\label{QLC-M12:Delta}
\begin{array}{l}
\Delta_{12} = \frac{ \left| V_{ us } \right| \left( \left| U_{ e1 } \right| +
\left| U_{ e2 } \right| \right) - \left| V_{ ud } \right| \left( \left| U_{ e1 } \right| -
\left| U_{ e2 } \right| \right) }{ \left| U_{ e1 } \right|
\left| V_{ ud } \right|- \left| U_{ e2 } \right| \left| V_{ us }\right|}.
\end{array}
\end{equation}
and
\begin{equation}\label{QLC-M23}
\tan{ \left( \theta_{23}^{ q } + \theta_{23}^{ l }\right)} = 1 + \Delta_{23},
\end{equation}
where
\begin{equation}
\begin{array}{l}
\Delta_{23} =
\frac{ \left| V_{ cb } \right| \left( \left| U_{ \tau3 } \right| +
\left| U_{ \mu3 } \right| \right) - \left| V_{ tb } \right| \left( \left| U_{ \tau3 }
\right| - \left| U_{ \mu3 } \right| \right) }{ \left| U_{ \tau3 } \right|
\left| V_{ tb } \right|- \left| U_{ \mu3 } \right| \left| V_{ cb }\right|}.
\end{array}
\end{equation}
and
\begin{equation}\label{QLC-M13}
\begin{array}{l}
\tan{ \left( \theta_{13}^{ q } + \theta_{13}^{ l }\right)} =
\frac{\left| V_{ ub } \right|\sqrt{ 1 - \left| U_{ e3 } \right|^{2}} +
\left| U_{ e3 } \right| \sqrt{ 1 - \left| V_{ ub } \right|^{2}} }{
\sqrt{ 1 - \left| V_{ ub } \right|^{2}} \sqrt{ 1 - \left| U_{ e3 }
\right|^{2}} - \left| U_{ e3 } \right| \left| V_{ ub } \right|}
\end{array}
\end{equation}
We notice that numerical values of $\Delta_{12}$ and $\Delta_{23}$ obtained from the
experimentally determined $\left| V_{_{CKM}} \right|$ and $\left|U_{_{ PMNS } } \right|$ are
much smaller than one,
\begin{displaymath}
\Delta_{12} \ll 1 \quad \textrm{and} \quad \Delta_{23} \ll 1 ,
\end{displaymath}
for this reason, the identities~(\ref{QLC-M12})-(\ref{QLC-M13}) are sometimes called
Quark Lepton Complementarity relations (QLC).
The substitution of expresions (\ref{elem:ckm}) and (\ref{elem:pmns}) for the
modulii of the elements of the mixing matrices $V_{_{CKM}}^{ th }$ and $U_{_{ PMNS } }^{ th }$,
allows us express the small terms $\Delta_{12}$ and $\Delta_{23}$ as funtions of the mass
ratios of quarks and leptons. Then, the eqs.~(\ref{QLC-M12})-(\ref{QLC-M13}) take the following
form:
\begin{equation}\label{QLCTAN12}
\tan{\left( \theta_{12}^{q^{th}} + \theta_{12}^{l^{th} } \right)} =
1 + \Delta_{12}^{^{th}} \left(
\frac{ \widetilde{m}_{u} }{ \widetilde{m}_{c} },
\frac{ \widetilde{m}_{d} }{ \widetilde{m}_{s} },
\frac{\widetilde{m}_{\nu_1} }{ \widetilde{m}_{\nu_2}},
\frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu} }
\right),
\end{equation}
where
\begin{widetext}
\begin{equation}\label{DELTA}
\begin{split}
\begin{array}{l}
\Delta_{12}^{ ^{th} } \approx
\frac{
\sqrt{ \frac{ \widetilde{m}_{d} }{ \widetilde{m}_{s} } +
\frac{ \widetilde{m}_{u} }{ \widetilde{m}_{c} } }
\left[ \sqrt{ \frac{ \widetilde{m}_{\nu_1} }{ \widetilde{m}_{\nu_2} } f_{ \nu2 } }
\left( 1 + \sqrt{ \frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu} }
\frac{ \widetilde{m}_{\nu_2} }{ \widetilde{m}_{\nu_1} } \left( 1 - \delta_{ \nu } \right)
} \right) + \sqrt{ \left( 1 + \widetilde{m}_{\nu_2} \right)
\left( 1 - \delta_{ \nu } \right) } \right]
- \left[ \sqrt{ \left( 1 + \widetilde{m}_{\nu_2} \right) f_{ \nu1 } } -
\sqrt{ \frac{ \widetilde{m}_{\nu_1} }{ \widetilde{m}_{\nu_2} } f_{ \nu2 } }
\left( 1 + \sqrt{ \frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu} }
\frac{ \widetilde{m}_{\nu_2} }{ \widetilde{m}_{\nu_1} } \left( 1 - \delta_{ \nu } \right)
} \right) \right]
}{ \sqrt{ \left( 1 + \widetilde{m}_{\nu_2} \right)
\left( 1 - \delta_{ \nu } \right) } -
\sqrt{ \frac{ \widetilde{m}_{d} }{ \widetilde{m}_{s} } +
\frac{ \widetilde{m}_{u} }{ \widetilde{m}_{c} } }
\left( 1 + \sqrt{ \frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu} }
\frac{ \widetilde{m}_{\nu_2} }{ \widetilde{m}_{\nu_1} } \left( 1 - \delta_{ \nu } \right)
} \right) }
\end{array}
\end{split}
\end{equation}
\end{widetext}
Here, rather than writing a lenghty but not very illuminating exact expresion, we give
an approximate expression for $\Delta_{12}^{ ^{th} }$, whose numerical value differs from the
exact expresion in $12\%$.
In the derivation of eq.~(\ref{DELTA}) from (\ref{QLC-M12:Delta}) we used the following
approxinations
\begin{equation}
\frac{ \left| V_{us}^{ ^{th} } \right| }{ \left| V_{ud}^{ ^{th} } \right| } \approx
\sqrt{ \frac{ \widetilde{m}_{d} }{ \widetilde{m}_{s} }
+ \frac{ \widetilde{m}_{u} }{ \widetilde{m}_{c} } } \approx 0.23152,
\end{equation}
which differs from the exact value in less than $1\%$, and
\begin{equation}
\begin{array}{l}
\frac{ \left| U_{ e2 }^{ ^{th} }\right| }{ \left| U_{ e1 }^{ ^{th} }\right|} \approx
\sqrt{ \frac{ \frac{\widetilde{m}_{\nu_1} }{ \widetilde{m}_{\nu_2} } }{
1 + \widetilde{m}_{\nu_2} } }
\sqrt{ \frac{ 1 + \widetilde{m}_{\nu_2} - \delta_{ \nu } }{ 1 - \widetilde{m}_{\nu_1} -
\delta_{ \nu } } } \left\{ 1 +
\right. \\ \left.
\qquad \quad + \sqrt{ \frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu}}
\frac{\widetilde{m}_{\nu_2} }{ \widetilde{m}_{\nu_1} } \left( 1 - \delta_{ \nu } \right) }
\right\}
\approx 0.688,
\end{array}
\end{equation}
which differs from the exact value in less than $1\%$. \\
The identity~(\ref{QLCTAN12}) that defines
{\small $\Delta_{12}^{^{th}} \left( \frac{ \widetilde{m}_{u} }{ \widetilde{m}_{c} },
\frac{ \widetilde{m}_{d} }{ \widetilde{m}_{s} }, \frac{\widetilde{m}_{\nu_1} }{
\widetilde{m}_{\nu_2}}, \frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu} }
\right)$} is frequently written in terms of the angle
$ \varepsilon^{ ^{th} }_{_{12}}$ that measures the desviation of
$\left( \theta_{12}^{q^{th}} + \theta_{12}^{l^{th} } \right)$ from
$\frac{\pi}{4}$. Then, eq.~(\ref{QLCTAN12}) may also be written as
\begin{equation}
\tan{\left( \theta_{12}^{q^{th}} + \theta_{12}^{l^{th} } \right)} =
\tan{\left( \frac{\pi}{4} + \varepsilon^{ ^{th} }_{_{12}} \right)} = 1 + \Delta_{12}^{^{th}}.
\end{equation}
From this expression, we get
\begin{equation}\label{corre:epsilon12}
\varepsilon^{ ^{th} }_{_{12}} = \arctan \left\{ \frac{\Delta_{12}^{^{th}} }{
2 + \Delta_{12}^{^{th}} } \right\}, \quad \left| \varepsilon^{ ^{th} }_{_{12}} \right|
< \frac{\pi}{2}
\end{equation}
which given $ \varepsilon^{ ^{th} }_{_{12}}$ as funtion of the mass ratios of quarks
and leptons. \\
Similarly,
\begin{equation}
\tan{ \left( \theta_{23}^{ q ^{th} } + \theta_{23}^{ l^{th} }\right)} = 1 +
\Delta_{23}^{ ^{th} }\left(
\frac{ \widetilde{m}_{u} }{ \widetilde{m}_{c} },
\frac{ \widetilde{m}_{d} }{ \widetilde{m}_{s} },
\frac{\widetilde{m}_{\nu_1} }{ \widetilde{m}_{\nu_2}},
\frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu} }
\right),
\end{equation}
where
\begin{widetext}
\begin{equation}
\begin{split}
\begin{array}{l}
\Delta_{23}^{ ^{th} } \approx
\frac{ \left( \left[ \left( 1 + \frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu} } \right)
\left( 1 + \widetilde{m}_{ \nu_2 } \right) - \delta_{ \nu } - \delta_{ e } f_{ \nu 2 }
\right]^{\frac{1}{2}} + \sqrt{ \delta_{ \nu } + \delta_{ e } f_{ \nu 2 } } \right)
\left( \sqrt{ 1 + \frac{ \widetilde{m}_{u} }{ \widetilde{m}_{c} } - \left( \sqrt{
\delta_{ u} } - \sqrt{ \delta_{ d } } \right)^{2} } + \left( \sqrt{ \delta_{ u} } - \sqrt{
\delta_{ d } } \right) \right) }{\left[ \left( 1 + \frac{ \widetilde{m}_{e} }{
\widetilde{m}_{\mu} } \right)
\left( 1 + \widetilde{m}_{ \nu_2 } \right) - \delta_{ \nu } - \delta_{ e } f_{ \nu 2 }
\right]^{\frac{1}{2}} \sqrt{ 1 + \frac{ \widetilde{m}_{u} }{ \widetilde{m}_{c} } - \left(
\sqrt{ \delta_{ u} } - \sqrt{ \delta_{ d } } \right)^{2} }- \left( \sqrt{ \delta_{ u} } -
\sqrt{ \delta_{ d } } \right) \sqrt{ \delta_{ \nu } +
\delta_{ e } f_{ \nu 2 } } }
\end{array}
\end{split}
\end{equation}
Also,
\begin{equation}\label{QLC-T13}
\begin{split}
\begin{array}{l}
\tan{ \left( \theta_{13}^{ q^{th} } + \theta_{13}^{ l^{th} }\right)} \approx
\frac{
\sqrt{ \frac{\widetilde{m}_{u} }{ \widetilde{m}_{c}} }
\left( \sqrt{ \delta_{ u} } - \sqrt{ \delta_{ d } } \right)
\left[ \left( 1 + \frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu} } \right) \left( 1 +
\widetilde{m}_{\nu_2} \right) - \delta_{\nu} \left( \sqrt{ \frac{ \widetilde{m}_{ \nu_1 }
\widetilde{m}_{ \nu_2 } }{ \left( 1 - \delta_{ \nu } \right) } } -
\sqrt{ \frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu} } } \right)^{ 2 }
\right]^{ \frac{1}{2}} +
}{
\sqrt{ 1 + \frac{ \widetilde{m}_{u} }{\widetilde{m}_{c}} - \frac{ \widetilde{m}_{u}
}{ \widetilde{m}_{c}} \left( \sqrt{ \delta_{ u} } -\sqrt{ \delta_{ d } } \right)^{2} }
\left[ \left( 1 + \frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu} } \right) \left( 1 +
\widetilde{m}_{ \nu_2 } \right) - \delta_{\nu} \left( \sqrt{
\frac{ \widetilde{m}_{ \nu_1 } \widetilde{m}_{ \nu_2 } }{ \left( 1 - \delta_{ \nu }
\right) } } - \sqrt{ \frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu} } } \right)^{ 2 }
\right]^{\frac{1}{2}} - } \\
\frac{
+ \sqrt{ \delta_{ \nu } } \left( \sqrt{ \frac{ \widetilde{m}_{ \nu_1 }
\widetilde{m}_{ \nu_2 } }{ \left( 1 - \delta_{\nu} \right) } } - \sqrt{
\frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu} } } \right) \sqrt{ 1 +
\frac{ \widetilde{m}_{u} }{ \widetilde{m}_{c} } - \frac{ \widetilde{m}_{u} }{
\widetilde{m}_{c} }\left( \sqrt{ \delta_{ u} } - \sqrt{ \delta_{ d } } \right)^{2} } }{ -
\sqrt{ \frac{\widetilde{m}_{u}}{\widetilde{m}_{c}} } \left( \sqrt{ \delta_{ u} } -
\sqrt{ \delta_{ d } } \right) \sqrt{ \delta_{ \nu } } \left( \sqrt{
\frac{ \widetilde{m}_{ \nu_1 } \widetilde{m}_{ \nu_2 } }{
\left( 1 - \delta_{ \nu } \right) } } - \sqrt{ \frac{ \widetilde{m}_{e} }{
\widetilde{m}_{\mu} } } \right) }.
\end{array}
\end{split}
\end{equation}
\end{widetext}
After substitution of the numerical values of the mass ratios of quarks and leptons in
eqs.~(\ref{DELTA})-(\ref{QLC-T13}), we obtain,
\begin{equation}
\begin{array}{l}
\Delta_{12}^{ ^{th} }= 0.1, \quad \Delta_{23}^{ ^{th} } = 3.23 \times 10^{-2}, \\\\
\tan{ \left( \theta_{23}^{ q ^{th} } + \theta_{23}^{ l^{th} }\right)} =
6.53 \times 10^{-2}.
\end{array}
\end{equation}
Hence,
\begin{equation}
\quad \theta_{12}^{q^{th}} + \theta_{12}^{l^{th}} = 45^{o} + 2.7^{o}.
\end{equation}
\begin{equation}
\theta_{23}^{ q^{th} } + \theta_{23}^{ l ^{th} } = 45^{o} + 1^{o},
\end{equation}
\begin{equation}
\theta_{13}^{ q^{th} } + \theta_{13}^{ l^{th}} = 3.7^{o} .
\end{equation}
The equations (\ref{QLCTAN12}) and (\ref {DELTA}) are obtained from an exact analytical
expression for $ \tan{\left( \theta_{12}^{q^{th}} + \theta_{12}^{l ^{th}} \right)} $
as a funtion of the absolute values of the entries in the mixing matrices
$V_{ _{CKM} }^{^{th}}$ and
$U_{ _{PMNS} }^{^{th}}$, eqs (\ref{QLC-M12}) and (\ref{QLC-M12:Delta}).
In eqs.~(\ref{elem:ckm}) and (\ref{elem:pmns}), the elements of the mixing matrices
$V_{ _{CKM} }^{^{th}}$ and $U_{ _{PMNS} }^{^{th}}$ are given as exact, explicit analytical
funtions of the quark and lepton mass ratios. Let us stress that these expressions are exact and
valid for any possible values of the quark and lepton mass ratios. From~(\ref{DELTA}), it
becomes evident that the small numerical value of $\Delta_{12}^{ ^{th} }$ is due to the partial
cancellation of two large terms of almost the same magnitude but opposite sign appearing in the
numerator of the expresion in the right hand side of the eq.~(\ref{DELTA}), namely,
{\small \begin{equation}
\begin{array}{l}
\sqrt{ \frac{ \widetilde{m}_{d} }{ \widetilde{m}_{s} } +
\frac{ \widetilde{m}_{u} }{ \widetilde{m}_{c} } }
\left[ \sqrt{ \frac{ \widetilde{m}_{\nu_1} }{ \widetilde{m}_{\nu_2} } f_{ \nu2 } }
\left( 1 + \sqrt{ \frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu} }
\frac{ \widetilde{m}_{\nu_2} }{ \widetilde{m}_{\nu_1} } \left( 1 - \delta_{ \nu } \right)
} \right) +
\right. \\ \left.
\qquad + \sqrt{ \left( 1 + \widetilde{m}_{\nu_2} \right)
\left( 1 - \delta_{ \nu } \right) } \right] =0.287,
\end{array}
\end{equation} }
and
{\small \begin{equation}
\begin{array}{l}
\sqrt{ \left( 1 + \widetilde{m}_{\nu_2} \right) f_{ \nu1 } } -
\sqrt{ \frac{ \widetilde{m}_{\nu_1} }{ \widetilde{m}_{\nu_2} } f_{ \nu2 } }
\left( \; 1 +
\right. \\ \left. \qquad
+ \sqrt{ \frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu} }
\frac{ \widetilde{m}_{\nu_2} }{ \widetilde{m}_{\nu_1} } \left( 1 - \delta_{ \nu } \right)
} \; \right) = 0.22.
\end{array}
\end{equation} }
The approximate numerical equality of these two expressions has its origin in the combined
effect of the strong hierarchy of charged leptons and $u$ and $d$-type quarks which yields
small and very small mass ratios, and the seesaw mechanism type~I which gives very small
neutrino masses but relatively large neutrino mass ratios. \\
We may conclude that the so called Quark-Lepton Complementarity as expresed in~(\ref{QLCTAN12})
and~(\ref{DELTA}) is more than a numerical coincidence, it is the result of the combined
effect of two factors:
\begin{enumerate}
\item The strong mass hierarchy of the Dirac fermions which produces small and very small
mass ratios of $u$ and $d$-type quarks and charged leptons. The quark mass hierarchy is
then reflected in a similar hierarchy of small and very small quark mixing angles.
\item The normal seesaw mechanism type~I which gives very small masses to the left-handed
Majorana neutrinos with relatively large values of the neutrino mass ratio
$m_{\nu_1}/m_{\nu_{2}}$ and allows for large $\theta_{12}^{l}$ and $\theta_{23}^{l}$ mixing
angles (see eqs.~(\ref{S12L})-(\ref{S13L})) .
\end{enumerate}
The two factors just mentioned contribute to numerator of $\Delta_{12}^{q^{th}}$ with two terms
of almost equal magnitud but opposite sign. Hence, the small numerical value of
$\Delta_{12}^{q^{th}}$ ocurring by partial cancellation of this two terms.
\section{The effective Majorana masses}
The square of the magnitudes of the effective Majorana neutrino masses,
eq.(\ref{masa_eff.1}), are
\begin{equation}\label{masa_eff.19}
\begin{array}{l}
\left| \langle m_{ll} \rangle \right|^{2} = \sum_{j=1}^{3} m_{ \nu_{j} }^{ 2 }
\left| U_{ lj } \right|^{ 4 } + 2 \sum_{j<k}^{3} m_{ \nu_{j} } m_{ \nu_{k} } \times \\
\; \;
\times \left| U_{ lj } \right|^{ 2 } \left| U_{ lk } \right|^{ 2 } \cos 2\left( w_{lj} -
w_{lk} \right),
\end{array}
\end{equation}
where $ w_{lj} = \arg \left \{ U_{ lj } \right \}$; this term includes phases of
both types, Dirac and Majorana.
The theoretical expression for the squared magnitud of the effective Majorana neutrino mass
of electron neutrino, written in terms of the ratios of the lepton masses, is:
\begin{equation}\label{eff-ele}
\begin{array}{l}
\left| \langle m_{ee} \rangle \right|^{2} \approx
\frac{ 1 }{ \left( 1 + \frac{ \widetilde{m}_{ e } }{ \widetilde{m}_{ \mu }} \right)^{ 2 }
\left( 1 + \frac{\widetilde{m}_{\nu_1}}{\widetilde{m}_{\nu_2}} \right)^{ 2 } } \left
\{ m_{\nu_1}^{ 2 } \left( 1 - \right. \right. \\ \left. \left.
-4 \sqrt{ \frac{ \widetilde{m}_{ e } }{ \widetilde{m}_{ \mu }} \frac{
\widetilde{m}_{\nu_1} }{ \widetilde{m}_{\nu_2} } \left( 1 - \delta_{ \nu } \right) }
\right) + \frac{ m_{\nu_2}^{ 2 } f_{\nu2}^2 }{ \left( 1 + \widetilde{m}_{\nu_2}
\right)^{2} \left( 1 - \delta_{ \nu } \right)^{2} }
\frac{ \widetilde{m}_{\nu_1} }{ \widetilde{m}_{\nu_2} } \left( \frac{
\widetilde{m}_{\nu_1} }{ \widetilde{m}_{\nu_2} }
\right. \right. \\ \left. \left.
+ 4 \sqrt{ \frac{ \widetilde{m}_{ e } }{ \widetilde{m}_{ \mu } } \frac{
\widetilde{m}_{\nu_1} }{ \widetilde{m}_{\nu_2} }
\left( 1 - \delta_{ \nu } \right) } + 6 \frac{ \widetilde{m}_{ e } }{
\widetilde{m}_{ \mu }} \left( 1 - \delta_{ \nu } \right) \right)
\right. \\ \left.
+ 2 \frac{ m_{\nu_1} m_{\nu_3} \delta_{ \nu} }{ \left( 1 + \widetilde{m}_{\nu_2} \right) }
\left( 1 + \frac{ \widetilde{m}_{\nu_1} }{ \widetilde{m}_{\nu_2} } \right)
\left( \sqrt{ \frac{ \widetilde{m}_{\nu_1} \widetilde{m}_{\nu_2}}{ \left( 1
- \delta_{ \nu } \right) } } - \sqrt{ \frac{ \widetilde{m}_{ e } }{
\widetilde{m}_{ \mu }} } \right)^{ 2 } \times \right. \\ \left. \times
\cos 2( w_{e1} - w_{e3} ) +2 \frac{ m_{\nu_1} m_{\nu_2} f_{\nu2} }{ \left( 1 +
\widetilde{m}_{\nu_2} \right) \left( 1 - \delta_{ \nu } \right) } \left(
\frac{\widetilde{m}_{\nu_1}}{\widetilde{m}_{\nu_2}}
\right. \right. \\ \left.\left.
+ 2 \left( 1 - \frac{\widetilde{m}_{\nu_1}}{\widetilde{m}_{\nu_2}} \right)
\sqrt{ \frac{\widetilde{m}_{\nu_1}}{\widetilde{m}_{\nu_2}}
\frac{ \widetilde{m}_{ e } }{ \widetilde{m}_{ \mu }} \left( 1 - \delta_{ \nu } \right) }
\right) \cos 2( w_{e1} - w_{e2} )
\right. \\ \left.
+ 2 \frac{ m_{\nu_2} m_{\nu_3} f_{\nu2} \delta_{ \nu } }{ \left( 1 +
\widetilde{m}_{\nu_2} \right)^{2} \left( 1 - \delta_{ \nu } \right)^{2} }
\left( 1 + \frac{ \widetilde{m}_{\nu_1} }{ \widetilde{m}_{\nu_2} } \right)
\left( 2 \widetilde{m}_{\nu_1} \widetilde{m}_{\nu_2}
\right. \right. \\ \left.\left.
+ \sqrt{ \frac{ \widetilde{m}_{ e } }{ \widetilde{m}_{ \mu }}
\frac{\widetilde{m}_{\nu_1}}{\widetilde{m}_{\nu_2}} \left( 1 - \delta_{ \nu } \right) }
\right) \cos 2( w_{e2} - w_{e3} )
\right\}
\end{array}
\end{equation}
where $w_{ e2 } \approx \beta_{1} $ and
\begin{equation}
w_{ e1 } = \arctan \left\{ - \frac{ \sqrt{ \frac{\widetilde{m}_{\nu_1}}
{\widetilde{m}_{\nu_2}} \frac{\widetilde{m}_{e}
}{\widetilde{m}_{\mu}} \delta_{e} \delta_{ \nu } f_{\nu2} } }{ \sqrt{ \left( 1
- \delta_{ \nu } \right) } + \sqrt{ \frac{\widetilde{m}_{\nu_1} }{
\widetilde{m}_{\nu_2} } \frac{\widetilde{m}_{e}}{\widetilde{m}_{\mu}} } \left( 1
- \delta_{ \nu } \right) } \right \},
\end{equation}
\begin{equation}
\begin{array}{l}
w_{ e3 } \approx \arctan
\left\{ \frac{ \sqrt{ \frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu} } \delta_{ e }
f_{ \nu 2} \left( 1 - \delta_{ \nu } \right) } + }{ - \sqrt{ \frac{ \widetilde{m}_{e}
}{ \widetilde{m}_{\mu} } \delta_{ e } f_{ \nu 2} \left( 1 - \delta_{ \nu } \right) }
\tan \beta_{ 2 } + }
\right. \\ \left. \qquad \quad
\frac{+\sqrt{ \delta_{ \nu } } \left( \sqrt{ \widetilde{m}_{\nu_1} \widetilde{m}_{\nu_2} }
- \sqrt{ \frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu} } \left( 1 - \delta_{ \nu }
\right) } \right) \tan \beta_{2} }{ + \sqrt{ \delta_{ \nu } } \left( \sqrt{
\widetilde{m}_{\nu_1} \widetilde{m}_{\nu_2} } - \sqrt{ \frac{\widetilde{m}_{e} }{
\widetilde{m}_{\mu} } \left( 1 - \delta_{ \nu } \right) } \right) } \right \}.
\end{array}
\end{equation}
In a similar way, the theoretical expression for the squared magnitud of the effective
Majorana neutrino mass of the muon neutrino is:
\begin{equation}\label{eff-mu}
\begin{array}{l}
\left| \langle m_{\mu \mu} \rangle \right|^{2} \approx \frac{1}{ \left( 1 +
\frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu} } \right)^{2} \left( 1 +
\frac{ \widetilde{m}_{\nu_1} }{ \widetilde{m}_{\nu_2} } \right)^{2} \left( 1
+ \widetilde{m}_{\nu_2} \right) } \left \{ \frac{ m_{\nu_3}^{ 2 } }{ \left( 1 +
\widetilde{m}_{\nu_2} \right) } \right. \\ \left.
\left( 1 + \frac{\widetilde{m}_{\nu_1}}{\widetilde{m}_{\nu_2}} \right)^{2}
\left( \delta_{ \nu } + 2 \delta_{ e } f_{\nu2} \right) + \frac{ m_{\nu_2}^{ 2 } }{
\left( 1 + \widetilde{m}_{\nu_2} \right) \left( 1 - \delta_{ \nu } \right) }
\left( 1 - \delta_{ \nu }
\right. \right. \\ \left. \left.
-4 \sqrt{ \frac{\widetilde{m}_{e}}{\widetilde{m}_{\mu}} \frac{ \widetilde{m}_{\nu_1}
}{ \widetilde{m}_{\nu_2}} \left( 1 - \delta_{ \nu } \right) } + 6
\frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu}} \frac{\widetilde{m}_{\nu_1} }{
\widetilde{m}_{\nu_2}} \right) +2 m_{\nu_1}m_{\nu_2} f_{\nu2}
\right. \\ \left.
\left( \frac{\widetilde{m}_{\nu_1}}{\widetilde{m}_{\nu_2}} \left( 1 - \delta_{ \nu }
\right) +2 \sqrt{ \frac{\widetilde{m}_{\nu_1}}{\widetilde{m}_{\nu_2}}
\frac{\widetilde{m}_{e}}{\widetilde{m}_{\mu}}
\left( 1 - \delta_{ \nu } \right) } \left( 1 - \frac{ \widetilde{m}_{\nu_1} }{
\widetilde{m}_{\nu_2}} \right) \right)
\right. \\ \left.\cos 2(w_{\mu 1} -w_{ \mu 2} )
+ 2 m_{\nu_1}m_{\nu_3} \left( 1 + \frac{\widetilde{m}_{\nu_1}}{\widetilde{m}_{\nu_2}}
\right) \left( 2 \delta_{ \nu }
\times \right. \right. \\ \left.\left. \times \sqrt{ \frac{\widetilde{m}_{\nu_1} }{
\widetilde{m}_{\nu_2}} \frac{\widetilde{m}_{e}}{\widetilde{m}_{\mu}} \left( 1 -
\delta_{ \nu } \right) } + \frac{\widetilde{m}_{\nu_1}}
{\widetilde{m}_{\nu_2}} \left( 1 - \delta_{ \nu } \right) \left( \delta_{ \nu } +
\delta_{ e } f_{\nu2} \right)\right)
\right. \\ \left.
\cos 2(w_{\mu 1} -w_{ \mu 3} ) +2 \frac{ m_{\nu_2}m_{\nu_3} f_{\nu2} }{ \left( 1 +
\widetilde{m}_{\nu_2} \right) \left( 1 - \delta_{ \nu } \right) } \left( 1 +
\frac{\widetilde{m}_{\nu_1}}{\widetilde{m}_{\nu_2}} \right)
\right. \\ \left.
\left( \left( 1 - \delta_{ \nu } \right) \left( \delta_{ \nu } + \delta_{ e } f_{\nu2}
\right) - 2 \delta_{ \nu } \sqrt{ \frac{\widetilde{m}_{\nu_1}}{\widetilde{m}_{\nu_2}}
\frac{ \widetilde{m}_{e} }{ \widetilde{m}_{\mu} } \left( 1 - \delta_{ \nu } \right) }
\right)
\right. \\ \left. \cos 2(w_{\mu 2} -w_{ \mu 3} ) \right \}
\end{array}
\end{equation}
where
\begin{equation}
w_{ \mu 1 } \approx \arctan \left \{
\frac{ \sqrt{ \frac{\widetilde{m}_{\nu_1}}{\widetilde{m}_{\nu_2} } \delta_{ e }
\delta_{ \nu } f_{\nu2} } }{ \sqrt{ \frac{\widetilde{m}_{e}}{\widetilde{m}_{\mu}}
\left( 1 - \delta_{ \nu } \right) } +
\sqrt{ \frac{\widetilde{m}_{\nu_1}}{\widetilde{m}_{\nu_2} } } \left( 1 - \delta_{ \nu }
\right) } \right \} ,
\end{equation}
and
\begin{equation}
w_{ \mu 2} \approx \arctan \left \{ \frac{ \sqrt{ f_{\nu2} } \tan \beta_{1} + \sqrt{
\delta_{e} \delta_{ \nu } } }{ \sqrt{ f_{\nu2} } - \sqrt{ \delta_{e} \delta_{ \nu } }
\tan \beta_{1} } \right \},
\end{equation}
\begin{equation}
w_{ \mu 3 } \approx
\arctan \left \{ \frac{ \tan \beta_{2} - \sqrt{ f_{ \nu2 } } }{ 1 +
\sqrt{ f_{ \nu2 } } \tan \beta_{2} } \right \}.
\end{equation}
From these expresions and the numerical values of the neutrinos masses given in
eq.~(\ref{X2:Mnus}), we obtain the following expresions for effective Majorana masses with
the phases as a free parameters:
{\small \begin{equation}
\begin{array}{l}
\left| \langle m_{ee} \rangle \right|^{2} \approx
\left \{ 9.41 + 8.29 \cos ( 1^{o} - 2\beta_{1} ) + 4.3 \cos ( 1^{o} - 2w_{e3} )
\right. \\ \left.
+ 4.31 \cos 2( \beta_{1} - w_{e3} )
\right \} \times 10^{-6}~\textrm{eV}^2
\end{array}
\end{equation} }
where
\begin{equation}
w_{e3} = \arctan \left \{ \frac{0.15 \tan \beta_{2} - 0.013 }{
0.15 + 0.013 \tan \beta_{2} } \right \}.
\end{equation}
Similarly,
\begin{equation}
\begin{array}{l}
\left| \langle m_{\mu \mu} \rangle \right|^{2} \approx
\left \{ 4.8 + 0.17 \cos 2( 44^{o} - w_{\mu 2} )
\right. \\ \left.
+ 1.8 \cos 2( w_{\mu 2} - w_{\mu 3} )
\right \} \times 10^{-4}~\textrm{eV}^2
\end{array}
\end{equation}
where
\begin{equation}
w_{ \mu 2} \approx \arctan \left \{
\frac{ 0.65 \tan \beta_{1} + 0.13 }{ 0.65 - 0.13 \tan \beta_{1} } \right \},
\end{equation}
\begin{equation}
w_{ \mu 3 } \approx
\arctan \left \{ \frac{ \tan \beta_{2} - 0.13 }{ 1 +
0.13 \tan \beta_{2} } \right \}.
\end{equation}
In order to make a numerical estimate of the effective Majorana neutrinos masses
$\left| \langle m_{ee} \rangle \right|$ and $\left| \langle m_{\mu \mu} \rangle \right|$,
we used the following values for the Majorana phases $\beta_{1 }=-1.4^{o}$ and
$\beta_{ 2 }=77^{o}$ obtained by maximizing the rephasing invariants $S_{1}$ and $S_{2}$,
eq.~(\ref{valor:S1S2}). Then, the numerical value of the effective Majorana neutrino
masses are:
\begin{equation}
\left| \langle m_{ee} \rangle \right| \approx 4.6 \times 10^{ -3 }~\textrm{eV} , \quad
\left| \langle m_{\mu \mu} \rangle \right| \approx 2.1 \times 10^{ -2 }~\textrm{eV} .
\end{equation}
These numerical values are consistent with the very small experimentally determined upper
bounds for the reactor neutrino mixing angle
$\theta_{13}^{l}$~\cite{PhysRevLett.101.141801}.
\section{Conclusions}
In this communication, we outlined a unified treatment of masses and mixings of quarks and
leptons in which the left-handed Majorana neutrinos acquire their masses via the type I
seesaw mechanism, and the mass matrices of all Dirac fermions have a similar form with
four texture zeroes and a normal hierarchy. Then, the mass matrix of the left-handed Majorana
neutrinos also has a texture with four zeros. In this scheme, we derived exact, explicit
expressions for the Cabibbo ($\theta_{12}^{q}$) and solar $(\theta_{12}^{l})$ mixing
angles as functions of the quark and lepton masses, respectively. The so called Quark-Lepton
Complementarity relation takes the form,
\begin{equation}
\theta_{12}^{q^{th}} + \theta_{12}^{l^{th}} = 45^{o} + \varepsilon^{ ^{th} }_{_{12}}.
\end{equation}
The correction term, $\varepsilon^{ ^{th} }_{_{12}}$, is an explicit function of the ratios of
quark and lepton masses, given in eq.~(\ref{corre:epsilon12}), which reproduces the
experimentally determined value,
\begin{equation}
\varepsilon_{12}^{^{exp}} \approx 2.7^{o},
\end{equation}
when the numerical values of the quark and lepton masses are substituted
in~(\ref{corre:epsilon12}).
Three essential ingredients are needed to explain the correlations implicit in the small
numerical value of $\varepsilon^{ ^{th} }_{_{12}}$:
\begin{enumerate}
\item The strong hierarchy in the mass spectra of the quarks and charged leptons, realized
in our scheme through the explicit breaking of the $S_{3}$ flavor symmetry in the
mass matrices with four texture zeroes, explains the resulting small or very small
quark mixing angles, the very small charged lepton mass ratios explain the very small
value of $\theta_{13}^{l}$.
\item The normal seesaw mechanism that gives very small masses to the left-handed Majorana
neutrinos with relatively large values of the neutrino mass ratio $m_{\nu_1}/m_{\nu_{2}}$
and allows for large $\theta_{12}^{l}$ and $\theta_{23}^{l}$ mixing angles.
\item The assumption of a normal hierarchy for the masses of the Majorana neutrinos.
\end{enumerate}
\acknowledgments{ We thank Dr. Myriam Mondrag\'on for many inspiring discussions on this
exciting problem. This work was partially supported by CONACyT Mexico under Contract No.
51554-F and 82291, and DGAPA-UNAM Contract No. PAPIIT IN112709.}
|
3,212,635,537,441 | arxiv | \section{Introduction}
The problem of quickly detecting a change in sequentially acquired data dates back to the pioneering works \cite{shewhart1931economic} and \cite{page1954continuous} and is motivated by a wide range of engineering and scientific applications. Examples of
such applications can be found in industrial process quality control~(\cite{bissell1969cusum,hawkins2003changepoint,joe2003statistical}), target detection and identification (\cite{ru2009detection,blackman2004multiple}), integrity monitoring of navigation systems (\cite{nikiforov1993application,bakhache1999reliable}),
target tracking (\cite{tartakovsky2003sequential}), network intrusion detection (\cite{tartakovsky2006detection,tartakovsky2006novel}), bioterrorism (\cite{rolka2007issues,fienberg2005statistical}), and genomics (\cite{siegmund2013change}). In most of these applications, there are many possible
types of change and it is useful, if not critical, to not only detect the change quickly but also to correctly identify its type upon stopping.
The problem of simultaneously detecting a change in the distribution of sequentially collected data and identifying the correct post-change distribution among a finite set of alternatives is known as \textit{sequential change diagnosis}. The literature on this problem can be broadly classified into two categories. In the first one (see, e.g., \cite{dayanik2008bayesian,dayanik2013asymptotically, ma2021two,tartakovsky2021asymptotic}), a Bayesian formulation is adopted that assumes a prior distribution on the change-point, and sometimes on the type of change. In the second (see, e.g., \cite{nikiforov1995generalized, tartakovsky2008multidecision, lai2000sequential, nikiforov2000simple, pergamenchtchikov2022minimax, oskiper2002online}), no such prior information is assumed. Both setups are considered in some works, such as \cite{lai2000sequential, pergamenchtchikov2022minimax}.
A Bayesian formulation of the sequential change \textit{detection} problem was first proposed in \cite{shiryaev1963optimum} (see also \cite{shiryaev2007optimal}), where the Bayes rule was derived under the assumptions of a Geometric prior for the change-point, iid observations before and after the change conditionally on the change-point, and completely specified distributions. A Bayesian formulation of the sequential change \textit{diagnosis} problem was first proposed in \cite{malladi1999generalized} and \cite{dayanik2008bayesian}. In the latter work, given a Geometric prior distribution for the change-point, the Bayes rule is shown to raise an alarm the first time the multi-dimensional posterior probability process enters a union of disjoint regions, each of which corresponds to a posited post-change distribution. These regions need to be computed off-line, via dynamic programming, and the necessary calculations can be very demanding.
To address this issue, a computationally feasible procedure was proposed in \cite{dayanik2013asymptotically} and was shown to achieve the Bayes risk asymptotically
as the probabilities of false alarm and false identification go to 0. A two-stage approach was considered in \cite{ma2021two}, where a sequential change detection algorithm is applied and, after it is declared that the change has occurred, a sequential hypothesis test is then performed to determine the post-change distribution. Non-iid models for the pre-change and post-change regimes, as well as a non-geometric prior for the change-point, were allowed in \cite{tartakovsky2021asymptotic} in a special case of the \textit{multichannel} problem. The latter refers to a case of the sequential diagnosis problem wherein a number of data channels are observed in parallel and a change occurs in the distribution of an unknown subset of the channels. In \cite{tartakovsky2021asymptotic}, the channels are assumed to be independent of each other and it is a priori known that the change occurs in only one of them.
In the absence of a priori information regarding the change-point, the sequential change diagnosis problem turns out to be significantly more complex than the pure sequential change detection problem, even in the case of iid observations before and after the change. Indeed, a recursive algorithm for the sequential change detection problem, Page's Cumulative Sum (CuSum), was proposed early on in \cite{page1954continuous} and, since then, has been grounded on strong theoretical support. In particular, assuming completely specified pre-change and post-change distributions, it was shown in \cite{moustakides1986optimal} to minimize \textit{Lorden's criterion} (\cite{lorden1971procedures}), i.e., the worst-case conditional expected detection delay with respect to both the change-point and the history of the observations up
to the change-point, subject to a user-specified bound on the false alarm rate. Modifications of the CuSum algorithm have been introduced to allow for parametric uncertainty and/or temporal dependence in the pre-change and post-change models, e.g., \cite{mei2006sequential, siegmund1995using, lai1998information, lai2010sequential, lai1999efficient}.
The first sequential change diagnosis scheme that does not utilize prior information regarding the change-point (and in general) was
proposed in \cite{nikiforov1995generalized}, and was a generalization of the CuSum algorithm, termed the ``Generalized CuSum.'' However, unlike the CuSum, this algorithm does not admit a recursive computational structure, even in the iid setting, and the number of operations it requires per time instance grows with the number of collected observations. Since the change can take a very long time to occur, this procedure needs to be modified to be applicable in practice. One such modification, suggested in \cite{lai2000sequential}, uses at any given time only the most recent data via a sliding window of deterministic size, and is referred to as the ``window-limited Generalized CuSum.'' A different modification of the Generalized CuSum was proposed in \cite{oskiper2002online} (see also \cite{tartakovsky2008multidecision}), where the order of maximizing over possible change-points and minimizing over post-change scenarios is reversed. This procedure was termed the ``Matrix CuSum,'' as it requires the parallel computation of a matrix of CuSum statistics.
Independently of their very different computational requirements, the Generalized CuSum, its window-limited modification, and the Matrix CuSum all control Lorden's worst-case criterion for the detection delay and have been shown theoretically to control false identification metrics \textit{only when the change occurs from the beginning of monitoring}. This, however, may be the \textit{best} possible scenario for these algorithms with respect to the identification task. In fact, the Generalized CuSum and the Matrix CuSum have both been reported to have, in certain cases, large probabilities of false isolation when the change-point is large \cite[Section 4.3.1]{nikiforov2016sequential}. Despite this, mainly due to its practically convenient, recursive structure, the Matrix CuSum continues to inspire formulations of the sequential change diagnosis problem (see, e.g., \cite{huang2021asymptotic}) in which only this
``easy'' case is considered for the false isolation metric.
Intuitively, the identification ability of the Generalized CuSum and the Matrix CuSum is compromised
by the impact of data from before the change in estimating the post-change distribution. This impact is reduced when applying the window-limited Generalized CuSum in \cite{lai2000sequential}. However, the size of the deterministic window is a tuning parameter, which is not determined by the user-specified error constraints and whose choice is characterized by a fundamental trade-off. Indeed, the window should be large enough for the algorithm to be able to detect the change, especially when the actual post-change alternative is ``close'' to the pre-change distribution, but not too large to compromise its identification ability. An asymptotic lower bound is obtained for this window in \cite{lai2000sequential}, but to the best of our knowledge it is not clear how to precisely select this window, in general, in order to simultaneously guarantee the desired isolation control.
A recursive algorithm that does not require any tuning parameters was proposed and analyzed in \cite{nikiforov2000simple}. This algorithm requires the computation of the CuSum statistics that compare the post-change distributions against the pre-change, and was termed the ``Vector CuSum,'' as its implementation requires the recursive calculation of a vector of CuSum statistics. It was argued that it can control the conditional probability of a wrong identification, given that there was no false alarm, uniformly with respect to the change-point. However, its detection delay analysis was conducted using Pollak's criterion (\cite{pollak1985optimal}), not Lorden's. That is, the Vector CuSum was shown to control the worst-case conditional expected detection delay with respect to the change-point, but not also the data before the change. In \cite{lai2000sequential} it was shown that a window-limited algorithm modification of the Vector CuSum
can control the probability of false identification within a fixed window from the change-point, uniformly with respect to the change-point. However, in addition to requiring the specification of a tuning parameter, this algorithm was not shown to control a worst-case criterion for the detection delay, but only weighted averages of the expected detection delay.
Finally, more recently, an algorithm was proposed in \cite{pergamenchtchikov2022minimax} that controls the conditional probability of a false identification, given that there was no false alarm, in the special case of the multichannel problem where the channels are independent and the change can occur in only one of them. However, the conditional probability of false identification is only controlled when the change occurs within a fixed window of the outset. Similarly to the above window-limited algorithms, this procedure requires the specification of certain tuning parameters, in addition to the thresholds that guarantee the prescribed error constraints. Similarly to the Vector CuSum, it was not shown to control's Lorden's criterion for the delay, but a Pollak-type delay criterion. \\
The first goal of this paper is to introduce and analyze a novel sequential change diagnosis algorithm that (i) can provide the desired guarantees for false isolation control, (ii) does not require the specification of tuning parameters, (iii) has a practically convenient, recursive structure, and (iv) controls Lorden's criterion for the detection delay. Specifically, the proposed scheme is a modification of the Matrix CuSum, in which each of its statistics is reset at some \textit{random, data-dependent} time that corresponds to an estimate of the change-point and, for this reason, we refer to it as the ``Adaptive Matrix CuSum.''
These adaptive resets suppress the inclusion of pre-change data when determining the post-change regime, and, in this way, they lead to the desired false identification error control without sacrificing the detection ability of the algorithm.
The second goal of the present work is to establish a general theory for a family of sequential change diagnosis procedures, which encompasses the Vector CuSum, the Matrix CuSum, and
its proposed modification. In particular, we establish a uniform exponential bound for the tail probabilities under the pre-change distribution of the statistics of the Vector CuSum and of the proposed scheme. This implies a novel lower bound on their expected time to false alarm, and suggests that pre-change data do not influence the decisions of these two schemes about the post-change distribution. However, we argue that, based on the techniques used in the present paper as well as in \cite{nikiforov2000simple}, the false identification control of the Vector CuSum can indeed be established for a certain family of change-points, but not necessarily for all possible change-points, as reported in \cite{nikiforov2000simple}.
A final goal of the present paper is to develop a comprehensive, computationally efficient framework for the design and fair comparison of sequential change diagnosis schemes. Based on this framework, we conduct a number of simulation studies in which we compare various sequential change diagnosis schemes under two different setups of the multichannel problem.
The remainder of the paper is organized as follows: In Section \ref{sec:state} we formulate mathematically the problem of sequential change diagnosis that we consider in this work.
In Section~\ref{sec:CUSUM}
we review the CuSum statistic and the min-CuSum procedure.
In Section~\ref{sec:family} we introduce a family of sequential change diagnosis procedures and the proposed novel procedure. In Section~\ref{sec:suppressing} we provide a description of how, for some algorithms, pre-change data can systematically lead to an erroneous decision regarding the post-change distribution. In
Section~\ref{sec:analysis} we analyze the performance of schemes in the family which we introduce, with an emphasis on the proposed procedure. In Section~\ref{sec:design} we introduce the proposed framework for the design and comparison of sequential change diagnosis procedures in this family, which is followed by simulation studies in said framework in Section~\ref{sec:simulation}. Lastly, concluding remarks are provided in Section~\ref{sec:conclusion}. Appendixes~A-F are dedicated to the proofs of all main results in this work, as well as to the statement and proof of supporting lemmas, and Appendix G contains additional figures.
We end this section by introducing some notation that we use throughout the paper. Thus, $\mathbb{N}$ is the set of positive integers, $\mathbb{N}_0$ the set of non-negative integers, and $\mathbb{R}$ the set of real numbers. For $n \in \mathbb{N}$, we set ${[n] \equiv \lbrace 1,
\ldots, n \rbrace}$. For $x \in \mathbb{R}$, $x^+$ is its positive part and $x^-$ its negative part, i.e., $ x^+ \equiv \max\{x,0\}$ and $ x^- \equiv \max\{-x,0\}$. We denote by $ x \wedge y $ the minimum of $ x, y $. If $(x_n), (y_n)$ are two families of positive numbers, where $n \in \mathbb{N}$, then $x_n \sim y_n$ and $ x_n \gg y_n$ stand for $ x_n/y_n \to 1 $ and $ x_n - y_n \to \infty $, respectively, as $ n \to \infty. $
We denote by $\mathcal{N}(\mu, 1)$ the density of a Gaussian random variable with mean $\mu$ and standard deviation $1$.
The indicator function is written as $ \mathds{1}(\cdot)$.
\section{Problem Statement} \label{sec:state}
Let $X\equiv \{X_n, n \in \mathbb{N}\}$ be a sequence of independent \mbox{$\mathbb{S}$-valued} random elements, where $\mathbb{S}$ is an arbitrary Polish space. We assume that each term of $X$ has a density with respect to a $\sigma$-finite measure $\lambda$, which is $f$ up to and including some deterministic time $\nu \in \mathbb{N}_0 $ and is $g$ after $\nu$. The pre-change density, $f$, is completely specified, but there is a finite number, $K$, of posited alternatives for the post-change density, i.e., ${g \in \{g_1, \ldots, g_K\} }$, and the \textit{change-point},~$\nu$, is completely unknown. The terms of the sequence $X$ are observed sequentially, and the goal is to quickly detect the change and also identify the correct post-change distribution upon detection. Thus, we need to specify an $\mathbb{N}$-valued random time, $T$, at which we declare that the change has occurred, and a $[K]$-valued random variable, $D$, that represents the decision regarding the post-change density at the time of stopping. That is, for any $n \in \mathbb{N}$ and $i \in[K]$, the alarm is raised and $g_i$ is declared as the correct post-change density after having taken $n$ observations on the event $\{T=n, D=i \}$. We refer to such a pair $(T,D)$ as a \textit{sequential change diagnosis procedure} if
$T$ is an $\{\mathcal{F}_n, n \in \mathbb{N} \}$-stopping time and $D$ is an $\mathcal{F}_T$-measurable, $[K]$-valued random variable, or in other words if
$$\{T=n \}, \{T=n, D=i \} \in \mathcal{F}_n \quad \text{ for all } \; n \in \mathbb{N}, \, i \in [K],$$
where $\mathcal{F}_n$ is the $\sigma$-algebra generated by the first $n$ observations, $X_1, \ldots, X_n$, and $ \mathcal{F}_0 $ is the trivial $\sigma$-algebra. We denote by $ \mathcal{C}$ the family of all sequential change diagnosis procedures.
We denote by~$ \mathbb{P}_{\infty}$ the distribution of $X$, and by~$\mathbb{E}_{\infty}$ the corresponding expectation, when the change never occurs, i.e., when
$X$ is a sequence of independent random elements with common density $f$. We denote by $ \mathbb{P}_{\nu,i} $ the distribution of $X$, and by $ \mathbb{E}_{\nu,i} $ the corresponding expectation, when the change occurs at time $ \nu $ and the post-change density is $g_i$. For simplicity, when the change occurs from the very beginning, we suppress the dependence on the change-point and set $ \mathbb{P}_i \equiv\mathbb{P}_{0,i}$ and $\mathbb{E}_i \equiv \mathbb{E}_{0,i}$. Moreover, without loss of generality, we restrict ourselves to stopping times that are not almost surely bounded under any ${\mathbb{P}}_{\nu,i}$, where $\nu \in \mathbb{N}_0$, $i \in [K]$.
To measure the ability of a sequential change diagnosis procedure $(T,D) \in \mathcal{C}$ to avoid false alarms we use the average number of observations until stopping under ${\mathbb{P}}_\infty$, i.e., ${\mathbb{E}}_\infty[T]$. We denote by $\mathcal{C}(\alpha)$ the subfamily of diagnosis procedures whose expected time until stopping under ${\mathbb{P}}_\infty$ is at least $1/ \alpha $, i.e.,
\begin{equation}
\mathcal{C}(\alpha) \equiv \{(T,D) \in \mathcal{C}: {\mathbb{E}}_{\infty}[T] \geq 1/\alpha\},
\end{equation}
where $\alpha \in (0,1)$ is a user-specified value that represents tolerance to false alarms.
To measure the ability of $(T,D) \in \mathcal{C}$ to isolate the post-change density $g_i$ when the change occurs at time $\nu$, we use the conditional probability of an incorrect identification given that there was no false alarm, i.e.,
$ {\mathbb{P}}_{\nu, i}(D \neq i | T > \nu)$. Thus, we denote by $\mathcal{C}(\alpha, \beta, N)$ the subfamily of change diagnosis procedures in $\mathcal{C}(\alpha)$ that control the conditional probability of false isolation below $ \beta$ when the change-point belongs to a set $N \subseteq \mathbb{N}_0$, i.e.,
\begin{equation}
\mathcal{C}(\alpha, \beta, N) \equiv \big\lbrace (T,D) \in \mathcal{C}(\alpha) :\max_{i \in [K]} \sup_{\nu \in N} {\mathbb{P}}_{\nu, i}(D \neq i \, | \, T > \nu)\leq \beta \big\rbrace,
\end{equation}
where $\beta \in (0,1)$ is a user-specified value that represents tolerance to false isolations.
Finally, to measure the ability of $(T,D) \in \mathcal{C}$ to quickly detect the change when the post-change density is $g_i$ for some $i \in [K]$, we adopt Lorden's criterion (\cite{lorden1971procedures}) by employing the worst-case conditional expected detection delay with respect to both the change-point and the data up to the change:
\begin{equation}\label{eqn:lorden_delay_def}
\mathcal{J}_i[T]\equiv \sup_{\nu \in \mathbb{N}_0} \esssup {\mathbb{E}}_{\nu,i}[ T - \nu \, | \, \mathcal{F}_{\nu}, T>\nu].
\end{equation}
The problem we consider in this work is to find a change diagnosis scheme that can be designed to belong to $\mathcal{C}(\alpha, \beta, N)$ for arbitrary $\alpha, \beta \in (0,1)$, with $N$ being as large as possible, and achieve
\begin{equation} \label{infimum}
\inf_{ (T,D) \in \mathcal{C}(\alpha, \beta, N)} \mathcal{J}_i[T],
\end{equation}
to a first-order asymptotic approximation as $ \alpha$ and $ \beta $ go to 0, simultaneously for every possible post-change alternative, i.e., for every $i \in [K]$.
\subsection{Assumptions} \label{sec: notation}
Our standing assumption throughout the paper is that the Kullback-Leibler divergences,
\begin{align} \label{KL}
\begin{split}
I_{i} &\equiv {\mathsf{Div}}(g_i \, || \,f) \equiv \int \log(g_i/f) \, g_i \, d \lambda,\\
I_{ij} &\equiv {\mathsf{Div}}(g_i \, || \, g_j) \equiv \int \log(g_i/g_j) \, g_i \, d \lambda,
\end{split}
\end{align}
are positive and finite for every $i, j \in [K]$ such that $i \neq j$. For every $n \in \mathbb{N}$ and $i,j \in [K]$ with $j \neq i$ we set
\begin{equation} \label{LLR}
\ell_{i}(n) \equiv \log \frac{ g_i (X_n)}{ f(X_n) }, \quad \quad
\ell_{ij}(n) \equiv \log \frac{ g_i (X_n)}{ g_j(X_n)},
\end{equation}
so that $I_i= {\mathbb{E}}_i[ \ell_i(n)]$ and $I_{ij}= {\mathbb{E}}_i[ \ell_{ij}(n)],$ and we denote by $\psi_{ij}$ the cumulant generating function of~$\ell_{ij}(1)$ under ${\mathbb{P}}_\infty$, i.e.,
\begin{equation}
\psi_{ij}(\theta) \equiv \log \left( {\mathbb{E}}_{\infty} \left[ \exp [ \theta \, \ell_{ij}(1) ] \right] \right), \quad \theta \in \mathbb{R}.
\end{equation}
For the main results of this work we need to assume that
\begin{equation}\label{assum:A1_prime}
\psi_{ij} \; \; \text{is finite around 0 for every} \; i,j \in [K], \, i \neq j,
\end{equation}
but we state this assumption explicitly when we make it.
\subsection{Example: the multichannel problem}\label{subsec:multichannel}
We illustrate the above sequential change diagnosis problem in the special case of the
multichannel problem, where $d$ independent channels are simultaneously monitored and there is a change in the marginal distributions of an unknown subset of them at some unknown time, $\nu$. Specifically, suppose that channel~$i$ generates a sequence of independent $\mathbb{S}_i$-valued random elements, where $\mathbb{S}_i$ is some Polish space, and let $X_{i,n}$ denote the observation from channel $i$ at time $n$. If the change does not occur in channel~$i$, then $X_{i,n}$ has density~$p_i$ with respect to a $\sigma$-finite measure $\lambda_i$ on $\mathbb{S}_i$, whereas if the change does occur in that channel at time $\nu$, then the density of $X_{i,n}$ is~$p_i$ for $n \leq \nu$ and $q_i$ for $n > \nu$.
In this context we have
\begin{align} \label{multichannel_X}
X_n &= (X_{1,n}, X_{2,n}, \ldots, X_{d,n}) \in \mathbb{S} \equiv \mathbb{S}_1 \otimes \cdots \otimes \mathbb{S}_d,
\end{align}
and the pre-change density is
\begin{equation}
f(x_1, \ldots , x_d) = \prod_{i=1}^d p_i(x_{i}), \quad \quad (x_1, \ldots, x_d) \in \mathbb{S}.
\end{equation}
If the change can occur in only one channel, as it is often assumed in the literature, the number of posited post-change distributions, $K$, is equal to the number of channels, $d$, and the post-change densities are
\begin{equation} \label{multichannel_single}
g_i (x_1, \ldots, x_d) = q_i(x_i) \prod_{j \in [d]: \, j \neq i} p_j(x_j), \quad \quad
(x_1, \ldots, x_d) \in \mathbb{S}, \quad i \in [d].
\end{equation}
However, it is conceptually and practically relevant to allow for the possibility of the change occurring in multiple channels simultaneously (\cite{mei2010efficient, xie2013sequential, fellouris2016second}). As we will see in Section \ref{sec:simulation} and elsewhere, this more general setup turns out to be much more challenging for existing change diagnosis algorithms even in the case of two channels ($d=2$), in which case the total number of post-change alternatives is only $K=3$, specifically,
\begin{align}\label{multichannel_simultaneous}
g_1(x_1, x_2) \equiv q_1 \left( x_1 \right) \cdot p_2 \left( x_{2} \right), \quad g_2(x_1, x_2) \equiv p_1 \left( x_{1} \right) \cdot q_2 \left( x_{2} \right), \quad
g_3(x_1, x_2) \equiv q_1(x_1) \cdot q_2(x_2).
\end{align}
\section{The min-CuSum Algorithm}\label{sec:CUSUM}
In this section we review the sequential change detection algorithm that provides the basis for the change diagnosis schemes that we consider in this work. Page's CuSum algorithm (\cite{page1954continuous}) for detecting a change from $f$ to $g_i$, for some fixed $i \in [K]$, raises an alarm as soon as the statistic
\begin{align} \label{cusum_statistic}
Y_{i} (n) &\equiv \max_{ 0 \leq t \leq n} \sum_{u=t+1}^n \ell_{i}(u),
\quad n \in \mathbb{N},
\end{align}
exceeds a threshold $b_i>0$, i.e., at
\begin{align} \label{cusum}
\begin{split}
\sigma_i (b_i) &\equiv \inf \lbrace n \geq 1 : Y_i(n) \geq b_i \rbrace,
\end{split}
\end{align}
where we adopt the convention $ \sum_{n+1}^{n} = 0. $
An important property of this stopping rule concerning its implementation in practice is that its statistic can be computed via the following recursion:
\begin{align}\label{cusum_recursion}
Y_{i} (n) &= \left(Y_{i}(n-1) + \ell_{i}(n) \right)^+, \quad n \in \mathbb{N},
\end{align}
with $Y_{i}(0) = 0$.
Of course, this algorithm is directly applicable in our setup only when $K=1$. When $K>1$, a standard approach for detecting the change is to run in parallel the $K$ CuSum statistics, $Y_1, \ldots, Y_K$, and stop as soon as one of them exceeds its corresponding threshold, i.e., at $ \min_{i \in [K]} \sigma_i(b_i).$
In what follows, we set $ b_1 = \ldots = b_K = b$ and refer to the stopping time
\begin{equation}
\sigma(b) \equiv \min_{i \in [K]} \sigma_i(b),
\end{equation}
as the ``min-CuSum'' stopping time. It is known \cite[Chapter 9.2]{tartakovsky2014sequential} that, for any $b>0$,
\begin{align} \label{CUSUM_ARL}
{\mathbb{E}}_{\infty}[\sigma(b)] &\geq e^{b}/K.
\end{align}
As a result, $\sigma(b_\alpha) \in \mathcal{C} (\alpha)$ for any $\alpha \in (0,1)$, where
\begin{equation}\label{eqn:b_alpha}
b_\alpha \equiv |\log \alpha| + \log K.
\end{equation}
It is also well known (see, e.g., \cite{lorden1971procedures}) that $\sigma (b_\alpha)$ minimizes $\mathcal{J}_i$ in $\mathcal{C} (\alpha)$, for every $i \in [K]$, to a first-order asymptotic approximation as $\alpha \to 0$, i.e, for every $i \in [K]$, as $\alpha \to 0$,
\begin{align} \label{CUSUM_AO}
\mathcal{J}_{i} \left[ \sigma(b_\alpha) \right] \sim
\frac{|\log \alpha|}{I_{i}} \sim \inf_{(T, D) \in \mathcal{C}(\alpha)} \mathcal{J}_i[T].
\end{align}
The stopping time $\sigma(b)$ is associated with a natural identification rule, $ \widehat{\sigma}(b)$, which is to select a post-change alternative that corresponds to the largest CuSum statistic at the time of stopping, i.e.,
\begin{equation}\label{CUSUM_isolation_rule}
\widehat{\sigma}(b) \in \argmax_{i \in [K]} Y_i(\sigma(b)) ,
\end{equation}
solving ties, if any, in some arbitrary way. The min-CuSum procedure, i.e., the pair $(\sigma(b), \widehat{\sigma}(b))$, is often used in applications as an ad-hoc solution to the sequential change diagnosis problem (see, e.g., \cite{chen2015quickest}), whereas
there is some theoretical justification for its use (\cite{warnercusum, han2007detection}), which we return to in Section \ref{sec:conclusion}.
\section{Diagnosis Procedures}\label{sec:family}
In this section we introduce a novel sequential change diagnosis scheme in the context of a more general family that encompasses many existing procedures in the literature.
\subsection{A family of diagnosis procedures}
For each $i \in [K]$ and $n \in \mathbb{N}$, let $W_{i}(n)$ be an \mbox{$\mathcal{F}_n$-measurable} statistic, large values of which should provide evidence that the correct post-change hypothesis is $g_i$. Given such statistics, for every $i \in [K]$ we denote by $\tau_i(b,h)$ the first time the CuSum statistic $Y_i$ is equal to or exceeds a threshold $b>0$, and, at the same time, $W_i$ is equal to or exceeds a distinct threshold $h>0$, i.e.,
\begin{align} \label{general_stopping_time}
\tau_i(b,h) &\equiv \inf \left\lbrace n \in \mathbb{N} : Y_i(n) \geq b \quad \& \quad W_i(n) \geq h \right\rbrace.
\end{align}
Then, a natural diagnosis procedure is
\begin{align} \label{family}
\tau(b,h) &\equiv \min_{i \in [K]} \tau_i(b,h), \quad \widehat{\tau}(b,h) \in \argmin_{i \in [K]} \tau_i(b,h),
\end{align}
where ties in the decision rule are settled in some arbitrary way.
With an appropriate selection of $W_1, \ldots, W_K$,
we can recover many of the sequential change diagnosis algorithms that have been proposed in the literature. For example, when $W_{i}(n)$ is of the form
\begin{align} \label{window_limited}
\max_{ M_i(n) \leq t \leq n } \, \min \left\lbrace \sum_{u=t+1}^n \ell_{i}(u), \min_{ j \in [K]: j \neq i}
\, \sum_{u=t+1}^n \ell_{ij}(u) \right\rbrace ,
\end{align}
we recover the Generalized CuSum in \cite{nikiforov1995generalized} when ${M_i(n)=0}$, and the window-limited Generalized CuSum in \cite{lai2000sequential} when ${M_i(n)=n-m}$ for some fixed $m \in \mathbb{N}$.
Other change diagnosis schemes in the literature can be recovered by setting $W_i(n)$ equal to
\begin{equation}\label{family4}
\min_{ j \in [K]: j \neq i} W_{ij}(n),
\end{equation}
where each $W_{ij}(n)$ is an $\mathcal{F}_n$-measurable statistic, large values of which should provide evidence that $g_i$ is a more plausible post-change alternative than $g_j$. For example, when \begin{equation}
W_{ij}(n) = Y_i(n) - Y_j(n),
\end{equation} where $Y_i$ is the CuSum statistic defined in \eqref{cusum_statistic}, we recover the Vector CuSum in \cite{nikiforov2000simple}, whereas when $W_{ij}$ is selected as the CuSum statistic for detecting a change from $g_j$ to $g_i$, i.e.,
\begin{align}\label{matrix_cusum}
Y_{ij} (n) &\equiv \max_{0 \leq t \leq n }\sum_{u=t+1}^n \ell_{ij}(u),
\end{align}
or equivalently
\begin{align}
\begin{split}
Y_{ij} (n) &= \left( Y_{ij}(n-1) + \ell_{ij}(n) \right)^+, \quad n \in \mathbb{N},\\
Y_{ij}(0) &= 0,
\end{split}
\end{align}
we recover the Matrix CuSum in \cite{oskiper2002online}.
\subsection{The Adaptive Matrix CuSum}
In this work, we propose a novel modification of the Matrix CuSum that is obtained by resetting each $Y_{ij}$ whenever $Y_i$ becomes $0$. Specifically, for each $i,j \in [K]$ with $ j \neq i$, we propose selecting $W_{ij}$ in \eqref{family4} as
\begin{align}\label{proposed_procedure}
Y'_{ij} (n) &\equiv \max_{R_i(n) \leq t \leq n} \sum_{u=t+1}^n \ell_{ij}(u),
\end{align}
where $R_i(n)$ is, at time $n$, the most recent time that $Y_i$ was at $0$, i.e.,
\begin{align} \label{R}
R_i(n) &\equiv \max \left\lbrace 0 \leq t \leq n : Y_{i}(t) = 0 \right\rbrace.
\end{align}
This scheme admits the same recursive structure as the Matrix CuSum. Indeed, for every $n \in \mathbb{N}$ and $i, j \in [K]$ with $i \neq j$ we have
\begin{align} \label{proposed_recursion}
Y^{\prime}_{ij}(n) &= \left(
Y^{\prime}_{ij}(n-1) + \ell_{ij}(n) \right)^+ \cdot \mathds{1} \left( \{ Y_i(n) > 0\} \right) ,
\end{align}
where $ Y^{\prime}_{ij}(0) = 0$. The main reason for proposing it is that it suppresses an excessive use of pre-change data for deciding the post-change distribution that,
as we will see later, in some cases characterizes the Matrix CuSum and leads to very large conditional probabilities of false identification. Indeed, for each $ i \in [K]$, $ R_i(n)$ is an estimate, based on the data up to time $n \in \mathbb{N}$, of the time at which the density changes from $f$ to $g_i$, which goes back to \cite{hinkley1970} and has been used in the literature of sequential change detection for different purposes (see, e.g., \cite{yang2017quickest}). Therefore, discarding the data up to and including time $R_i(n)$ in the evaluation of $W_{ij}(n)$, for each $ j \neq i$ and $n \in \mathbb{N}$, allows the estimate of the post-change density to be based mostly on data from after the change, no matter when it occurs. In Section \ref{sec:analysis} we conduct a theoretical analysis that supports the above claims, but, first, in Section \ref{sec:suppressing}, we provide an intuitive explanation of why the Matrix CuSum, as well as the Generalized CuSum, can systematically fail to correctly isolate the post-change distribution in certain setups.\\
\noindent \textbf{\underline{Remark:}} An alternative sequential change diagnosis scheme can be obtained by setting $M_i(n) = R_i(n)$ in \eqref{window_limited}.
However, the calculation of $ W_i(n)$ in this case would require a number of operations of the order of $n - R_i(n) $. Specifically, the computational cost and memory requirement of the procedure would be dictated by a random variable with a long tail, which may not be desirable in practice. On the other hand, applying the window $ R_i(n)$ to the Matrix CuSum statistics leads to a recursive structure \eqref{proposed_recursion}, which is another motivation for our proposal of this scheme. \\
\section{The Effect of Pre-Change Data}\label{sec:suppressing}
In this section we explain intuitively how the Matrix CuSum and Generalized CuSum can systematically fail, in certain setups, to correctly isolate the post-change regime. To this end, we focus on the case that the true post-change density is $g_K$ and there is another density, say $g_1$, such that $g_1$ and $g_K$ are closer together than $g_K$ and $f$, in the sense that
\begin{equation}
{\mathsf{Div}} (g_K \, || \, g_1) < {\mathsf{Div}}(g_K \, || \, f)
\end{equation}
or, equivalently,
\begin{align}
{\mathbb{E}}_K[\ell_{1}(n)] &>0 \quad \forall \; n \in \mathbb{N}. \label{cond2}
\end{align}
This is the case, for example, when there is a change in the mean of a sequence of Gaussian random variables from $0$ to some positive number, i.e.
\begin{equation} \label{Gaussian}
f = \mathcal{N}(0,1) \quad \text{and} \quad g_i = \mathcal{N}(\theta_i, 1), \quad i \in [K], \quad \quad
\text{where} \quad 0 <\theta_1 < \ldots < \theta_K,
\end{equation}
or in the multichannel problem of Section \ref{subsec:multichannel} when more than one channel may be affected by the change. Consider, for simplicity, the case with $ d= 2$ channels, where $ K = 3$ and $ g_1, g_2, g_3$ are given by \eqref{multichannel_simultaneous}, i.e., $ g_i$ is the density when the change affects only channel $i$, where $ i \in \lbrace 1, 2 \rbrace$, and $ g_3 $ is the density when the change affects both channels.
Then, \eqref{cond2} holds, since
\begin{align}
\begin{split}
{\mathsf{Div}}(g_3 \, || \, f ) &= {\mathsf{Div}}(q_2 \, || \, p_2) + {\mathsf{Div}}(q_1 \, || \, p_1) \\
&> {\mathsf{Div}}(q_2 \, || \, p_2) = {\mathsf{Div}}(g_3 \, || \, g_1 ).
\end{split}
\end{align}
By the standard properties of the CuSum statistics and condition \eqref{cond2} it follows that $Y_K$ will tend to be close to 0 before the change and will increase after the change, and that this will also be the case for $Y_1$, although its growth after the change will be smaller than that of $Y_K$. As a result, if also $W_1$ is large, especially if it is larger than $W_K$, for a certain time period after the change, it becomes likely to incorrectly identify $g_1$ as the post-change density. This is the case, for example, if, in addition to \eqref{cond2}, $g_1$ is the unique density that is the closest to $f$ before the change in the sense that
\begin{equation}
{\mathsf{Div}}(f \, || \, g_1) < {\mathsf{Div}}(f \, || \, g_j) \text{ for all } j \in \{ 2, \ldots, K\},
\end{equation}
or equivalently
\begin{align}
{\mathbb{E}}_\infty[\ell_{1j}(n)] &> 0 \text{ for all } j \in \{ 2, \ldots, K\}, \quad n \in \mathbb{N}.\label{cond1}
\end{align}
The latter condition is satisfied for example in the Gaussian mean shift problem \eqref{Gaussian}.
To explain how condition \eqref{cond1} implies a large value of $W_1$, much larger than that of $W_K$, for a certain period of time after the change, for simplicity we focus on the simplest case of $ K = 2$ post-change alternatives and we illustrate our explanation
(Fig. \ref{fig:matrix_cusum_vs_proposed} \& \ref{fig:WLGC_paths}) in the context of Gaussian mean shift problem. Thus, in the following discussion, the true post-change density is $g_2$.
\subsection{Matrix CuSum and the proposed adaptive modification} We start with the Matrix CuSum, i.e., when
\begin{equation}
W_1= W_{12}=Y_{12} \quad \text{and} \quad W_2= W_{21}=Y_{21}.
\end{equation}
Condition \eqref{cond1} then implies that, before the change, $ W_2$ will tend to be close to 0 and $W_{1}$ will behave like a random walk with positive drift. As a result, the expected value of $W_1$ at the time of the change will be proportional to the change-point. Therefore, even though $W_1$ will start decreasing after the change, there will be some period after the change that it will be large and, in fact, much larger than $ W_2 $, as the latter will start growing only after the change occurs. Most importantly, the length of this period is increasing in the change-point. The longer it takes for the change to happen, the larger the expected value of $W_1$ at the time of the change, the longer it take for its values after the change to become small. Thus, it is not possible to control this problematic behavior unless an upper bound is imposed on the change-point.
On the contrary, using the proposed adaptive window, i.e., setting
\begin{equation}
W_1= W_{12}=Y'_{12} \quad \text{and} \quad W_2= W_{21}=Y'_{21},
\end{equation}
effectively removes pre-change data from the decision-making process and prevents the erroneous growth of $W_1$ before the change, without affecting the correct behavior of $W_2$.
These points are illustrated in Fig. \ref{fig:matrix_cusum_vs_proposed}, where we see that
$ Y'_{12}$, unlike $Y_{12}$, does not grows before the change, whereas $Y'_{21} $ behaves very similarly to $Y_{21}$.
\\
\subsection{ Generalized CuSum and window-limited modifications} We continue with the Generalized CuSum, i.e., when
\begin{align} \label{gen_cum_stat}
\begin{split}
W_1(n) &= \max_{0 \leq k \leq n} \left\lbrace \sum_{u = k+1}^n \ell_1(u) \wedge \sum_{u = k+1}^n \ell_{12}(u) \right\rbrace, \\
W_2(n) &= \max_{0 \leq k \leq n} \left\lbrace \sum_{u = k+1}^n \ell_2(u) \wedge \sum_{u = k+1}^n \ell_{21}(u) \right\rbrace.
\end{split}
\end{align}
Then, standard properties of likelihood ratios and condition \eqref{cond1} imply that $W_2 $ does not grow before the change and starts increasing only after the change. On the other hand, while $W_1$ does not grow before the change, as in the case of the Matrix CuSum, it does grow for a certain period of time \textit{after} the change occurs. To see this, observe that for ${ k \geq \nu }$, the sum $\sum_{u = k+1}^n \ell_{12}(u)$ is generally small/negative, which is conducive to correct identification, while $ \sum_{u = k+1}^n \ell_1 (u )$ is generally positive. However, both of them can be large for $ k < \nu $, i.e., when pre-change data are used, as it is illustrated in
Fig. \ref{fig:partial_sums}, where these two sums are plotted as functions of $k$ when $n=75$ and $\nu=50$.
In this Figure we see that the first sum is maximized when $ k \approx \nu$, as expected, and the second is negative when $ k \geq \nu $, as desired. However, both of them are positive and quite large for $ k$ around $35$, which results in a rather large value for $W_1$ at time $n$.
To sum up, when applying the Generalized CuSum, $W_1$ does not grow before the change, but it does grow
for a certain time-period \textit{after the change} during which it is likely to be much larger than $ W_2 $, as the latter grows only from post-change data. Most importantly, the length of this problematic time-period increases with $\nu$, i.e., a later change-point results in an erroneous post-change growth of $W_1$ for a longer period of time, and therefore $W_1 $ becomes more likely to be larger than $W_2$. This point is illustrated in the right-hand side of Fig. \ref{fig:WLGC_paths}.
This phenomenon is mitigated when using the window-limited Generalized CuSum, i.e., when restricting the maxima in \eqref{gen_cum_stat} to $n-m\leq k \leq n$, where $m$ is some deterministic constant. Then, the post-change growth of $W_1$ can last for at most $m$ observations, after which pre-change data are removed from the calculation of the statistic. This is illustrated in the left-hand plot in Fig. \ref{fig:WLGC_paths} when $\nu=50$ and $m=15$. On the other hand, as it shown in \cite{lai2000sequential}, $m$ should be selected large enough for the detection of the change to occur within the window, i.e., within $m$ observations after the change occurs. Therefore, the selection of $m$ is characterized by a trade-off, which is not clear how to resolve in a satisfactory way. As we discussed in the final remark of the previous section, a solution to this problem would be to replace the fixed window $m$ by the adaptive window in \eqref{R}, i.e., to restrict the first maximum in \eqref{gen_cum_stat} to $R_1(n)\leq k \leq n$ and the second to $R_2(n)\leq k \leq n$. \\
\noindent \textbf{\underline{Remark:}} Conditions \eqref{cond2} and \eqref{cond1} provide only a particular class of examples in which the Matrix CuSum and the Generalized CuSum can have difficulties in identifying the post-change. For example, condition \eqref{cond1} is not satisfied with strict inequality in the case of the multichannel problem with simultaneous faults. Nevertheless, a similar reasoning can be applied to argue that when the change does not occur from the outset and it affects more than one channels, then these two schemes are likely to make a wrong identification. In fact, this is shown for the Matrix CuSum, and to a lesser extent the Generalized CuSum, in our simulation studies in Section \ref{sec:simulation}.
\begin{figure}[tb]
\begin{center}
\includegraphics[scale=1]{matrix_cusum_statistics_vs_proposed_paths_with_cusums.eps}
\caption{Sample paths of the Matrix CuSum and Adaptive Matrix CuSum statistics are plotted in the Gaussian mean shift problem where $ \nu = 50.$}
\label{fig:matrix_cusum_vs_proposed}
\end{center}
\end{figure}
\begin{figure}[tb]
\begin{center}
\includegraphics[scale=1]{WLGC_paths.eps}
\caption{Sample paths of the Generalized CuSum statistics and the window-limited modification are plotted in the Gaussian mean shift problem where $ \nu = 50.$}
\label{fig:WLGC_paths}
\end{center}
\end{figure}
\begin{figure}[tb]
\begin{center}
\includegraphics[scale=1]{partial_sums.eps}
\caption{The partial sums $ \sum_{u = k+1}^n \ell_1 (u ), \sum_{u = k+1}^n \ell_{12}(u)$ are plotted as a function of the lower limit $k$ when $n = 75$ and $ \nu = 50$, in the Gaussian mean shift problem.}
\label{fig:partial_sums}
\end{center}
\end{figure}
\section{Performance analysis}\label{sec:analysis}
In this section we establish some general results regarding the performance of sequential change diagnosis procedures of the form \eqref{general_stopping_time}-\eqref{family}, where also each statistic $W_i$ is of the form \eqref{family4}. We focus, in particular, on the proposed scheme, for which we establish (i)~an upper bound on its Lorden delay,
(ii)~a novel non-asymptotic lower bound for its expected time to false alarm, (iii)~an upper bound for its conditional probability of false isolation for a certain family of change-points, and (iv)~an asymptotic optimality property as $\alpha$ goes to 0 sufficiently faster than $\beta$.
\subsection{Delay analysis}
We start our delay analysis with
an upper bound for the expected delay of $\tau(b,h)$ when $\nu=0$, i.e., when the change has occurred once the monitoring begins. For the following lemma, we introduce $I_i^*$ as the minimum of the Kullback-Leibler information numbers in ${ \{I_{ij}: j \in [K], j \neq i\} }$, i.e.,
\begin{equation}
I_i^* \equiv \min_{j \in [K]: j \neq i} I_{ij}.
\end{equation}
\begin{lemma} \label{lem:new_lemma_for delay}
Fix $i \in [K]$ and suppose that, for every $j \in [K]$ such that $j \neq i$,
\begin{align} \label{summable2}
\sum_{n=1}^\infty {\mathbb{P}}_i \left( W_{ij}(n) \leq \rho n \right) &< \infty \quad \text{for all} \; \rho < I_{ij}.
\end{align}
Then, for all $\delta > 0$, there is a constant $C_\delta>0$ such that
\begin{align} \label{delay under 0}
{\mathbb{E}}_i \left[ \tau_i(b,h) \right] \leq \max \lbrace b / I_{i}, h / I^*_i \rbrace (1+\delta) +C_\delta
\end{align}
for all $b,h \geq 0.$
\end{lemma}
\begin{IEEEproof} Appendix C.\\
\end{IEEEproof}
Condition \eqref{summable2} is satisfied when $W_{ij}$ is either $Y_{ij}$ or $Y_{i}-Y_j$, which means that the upper bound \eqref{delay under 0} applies to both the Vector CuSum and the Matrix CuSum. Since, by definition, $Y'_{ij} \leq Y_{ij}$, it is not obvious that \eqref{summable2} holds for the proposed scheme, i.e., when $W_{ij}=Y'_{ij}$. In the following lemma we show that this is indeed the case under mild moment conditions.
\begin{lemma}\label{lem:det_delay_bound}
Fix $i, j \in [K], j \neq i$. If there is a $ p > 1$ such that
\begin{equation}
{\mathbb{E}}_i[|\ell_{ij}(1)|^{p}] < \infty,
\end{equation}
and an $\epsilon>0$ so that
\begin{equation}
{\mathbb{E}}_i \left[ \left( \ell_i^-(1) \right)^{2 + p/(p-1) + \epsilon} \right] < \infty,
\end{equation}
then \eqref{summable2} holds when $W_{ij}=Y'_{ij}$.
\end{lemma}
\begin{IEEEproof}
Appendix C.\\
\end{IEEEproof}
In the case of the Matrix CuSum, i.e., when $W_{ij}=Y_{ij}$, it is well known that the Lorden delay of $\tau(b,h)$ agrees with the expected delay of $\tau(b,h)$ when $\nu=0$, i.e., for all $ i \in [K]$ and $b,h>0$,\begin{equation}
\mathcal{J}_i[\tau(b,h)]={\mathbb{E}}_i[\tau(b,h)].
\end{equation}
In the following lemma we show that this is also the case for the proposed scheme, i.e., when $W_{ij}=Y'_{ij}$, despite the presence of the adaptive window. We emphasize that this is \textit{not} the case for the Vector CuSum, whose worst-case delay analysis was conducted in \cite{nikiforov2000simple,nikiforov2003lower} using Pollak's criterion, not Lorden's.
\begin{lemma}\label{lem:worst_case}
If $W_{ij}=Y'_{ij}$ for every $i,j \in [K], j \neq i$, then
\begin{align} \label{worst case at 0}
\mathcal{J}_i[\tau(b,h)] = {\mathbb{E}}_i[\tau(b,h)] \quad \text{ for all } i \in [K], \; b, h\geq 0.
\end{align}
\end{lemma}
\begin{IEEEproof}
Fix $ i \in [K]$ and $b, h\geq 0$. By the definition of $\tau(b,h)$ and recursions \eqref{cusum_recursion} and \eqref{proposed_recursion} it follows that, for any $\nu \in \mathbb{N}_0$, the conditional expectation
\begin{equation}\label{eqn:conditional_expectation}
{\mathbb{E}}_{\nu, i} \left[\tau(b,h) - \nu \; | \; \mathcal{F}_{\nu }, \tau(b,h) > \nu \right]
\end{equation}
depends on $\mathcal{F}_\nu$ only through
\begin{equation}
\mathcal{Y}(\nu) \equiv \{ Y_k(\nu), Y_{kj}'(\nu), \; k, j \in [K], j \neq k \}.
\end{equation}
Thus, for every $\nu \in \mathbb{N}_0$,
\begin{align}
\begin{split}
& {\mathbb{E}}_{\nu, i} \left[ \tau(b,h) - \nu \; | \; \mathcal{F}_{\nu}, \tau(b,h) > \nu \right] \\
&= {\mathbb{E}}_{\nu, i} \left[\tau(b,h) - \nu \;|\; \mathcal{Y}(\nu), \tau(b,h) > \nu \right].
\end{split}
\end{align}
From recursion \eqref{proposed_recursion} it also follows that the latter conditional expectation is decreasing, thus, it is maximized when all components of $\mathcal{Y}(\nu)$ are as small as possible, i.e., $0$. When this happens, this conditional expectation becomes equal to ${\mathbb{E}}_i[\tau(b,h)]$, since $ X_{\nu + 1}, X_{\nu + 2}, \ldots$ are independent of $\mathcal{F}_\nu$ and have the same distribution under~$ {\mathbb{P}}_{\nu,i}$ as $X_1, X_2, \ldots$ under~${\mathbb{P}}_i$. \\
\end{IEEEproof}
Combining the above results, we establish an upper bound for the Lorden delay of the proposed scheme.
\begin{theorem}\label{thm:delay_bound}
If $W_{ij}=Y'_{ij}$ for every $i,j \in [K], j \neq i$ and the conditions of Lemma \ref{lem:det_delay_bound} hold, then for every $\delta>0$ there is a constant $C_\delta>0$ so that, for every $b>0$ and $h>0$,
\begin{align} \label{Lorden delay}
\mathcal{J}_i \left[ \tau(b,h) \right] \leq \max \lbrace b / I_{i}, h / I^*_i \rbrace (1+\delta) +C_\delta.
\end{align}
\end{theorem}
\begin{IEEEproof}
The proof follows directly by combining the lemmas of this subsection.\\
\end{IEEEproof}
\subsection{False alarm control}
Independently of the choice of the statistics $W_{i}$, $i \in [K]$, by the definition of $\tau(b,h)$ it is clear that, for any $b, h\geq 0$, it holds that $\tau(b,h) \geq \sigma (b)$ and, in view of \eqref{CUSUM_ARL},
\begin{align*}
{\mathbb{E}}_\infty \left[ \tau (b,h)\right] &\geq e^{b}/K.
\end{align*}
Thus, it is always possible to guarantee that the false alarm rate of $\tau (b,h)$ does not exceed an arbitrary $\alpha \in (0,1)$ by simply selecting $b$ to be larger than or equal to $b_\alpha$, which was defined in \eqref{eqn:b_alpha}. Indeed, this is the approach typically taken in the literature, at least for the purpose of analysis. However, as we show in the next lemma, if the $W_{ij}$s are stochastically small under ${\mathbb{P}}_\infty$, then we also have a lower bound for the expected time to false alarm that is independent of $b$ and exponential in $h$.
\begin{lemma}\label{lem:false_alarm_bound}
If, for every $ i, j \in [K]$ such that $j \neq i$, there are $Q_{ij}, q_{ij} > 0$ so that, for all $ n \in \mathbb{N}$,
\begin{equation} \label{tail}
{\mathbb{P}}_{\infty} \left( W_{ij}(n) \geq x \right) \leq Q_{ij}\, e^{-q_{ij} x}, \quad \forall x \geq 0,
\end{equation}
then, for any $b>0$ and $h>0$,
\begin{align}
{\mathbb{E}}_\infty \left[ \tau(b,h) \right] &\geq \frac{1}{2} \left( \sum_{i=1}^K \min_{j \neq i} \left\{ Q_{ij} e^{-q_{ij} h} \right\} \right)^{-1}.
\end{align}
\end{lemma}
\begin{IEEEproof}
Appendix D.\\
\end{IEEEproof}
Condition \eqref{tail} is satisfied for all $n \in \mathbb{N}$ in the case of the Vector CuSum, i.e., when $W_{ij}=Y_i-Y_j$. Indeed, for any such $i,j$, $n \in \mathbb{N}$ and $x \geq 0$,
\begin{equation}
{\mathbb{P}}_{\infty}(Y_i(n) - Y_j(n) \geq x) \leq {\mathbb{P}}_{\infty}(Y_i(n) \geq x) \leq e^{-x},
\end{equation}
where the second inequality is a well-known property of the CuSum statistic (restated in Appendix A). On the other hand, condition \eqref{tail} is not, in general, satisfied in the case of the Matrix CuSum, i.e., when $W_{ij}=Y_{ij}$. Indeed, when ${\mathbb{E}}_\infty[\ell_{ij}(n)] >0$, $Y_{ij}$ essentially behaves as a random walk with positive drift before the change. In the next theorem we show that condition \eqref{tail} is always satisfied for the proposed scheme as long as condition \eqref{assum:A1_prime} holds.
\begin{theorem}\label{thm:uniform_exponential_bound}
If condition \eqref{assum:A1_prime} holds, then for every $i, j \in [K]$ such that $j \neq i$ there are $Q_{ij}, q_{ij} >0$, such that \eqref{tail} holds for all $ n \in \mathbb{N}_0$ when $W_{ij}=Y'_{ij}$.
\end{theorem}
\begin{IEEEproof}
Appendix E.\\
\end{IEEEproof}
In summary, the results of this section reveal that both the proposed diagnosis scheme and the Vector CuSum can control the false alarm rate below any user-specified value $\alpha$ with a suitable selection of threshold $h$ alone, independently of the value of $b$.
\subsection{False isolation control}
To establish our main result regarding false isolation control, we need $W_{ij}$ to be pathwise smaller than or equal to $Y_{ij}$, i.e.,
\begin{equation} \label{pathwise bound}
W_{ij}(n) \leq Y_{ij}(n) \quad \text{for all} \quad
n \in \mathbb{N}
\end{equation}
for all $i, j \in [K], j \neq i$. This holds, trivially, for the Matrix CuSum and the proposed scheme, whereas it also holds for the Vector CuSum.
In fact, in our analysis, we use the same technique as in the proof for the false isolation control of the Vector CuSum in \cite[Theorem 2]{nikiforov2000simple}, but, as we will see, our conclusions are somewhat different.
To state our results in the appropriate generality, we need to introduce,
for each $i, j \in [K], j \neq i$, the fictitious statistic that utilizes the CuSum statistic $Y_{ij}$, instead of $W_{ij}$, after $\nu$, i.e.,
\begin{align} \label{W_nu}
W_{ij}^{\nu}(n) \equiv
\begin{cases}
W_{ij}(n), & \quad n \leq \nu, \\
\left(W^{\nu}_{ij}(n - 1) + \ell_{ij}(n) \right)^+, & \quad n > \nu,
\end{cases}
\end{align}
and the stopping time
\begin{align}\label{tau_nu}
\tau^\nu_{ij}(h) &\equiv \inf \lbrace n > \nu : W^\nu_{ij}(n) \geq h \rbrace.
\end{align}
Specifically, the statistic $ W_{ij}^{\nu}(n)$ is equal to $ W_{ij}(n)$ before the change occurs, and behaves like a CuSum statistic with initialization $ W_{ij}(\nu)$ after the change-point, $\nu$.
\begin{theorem}
\label{th:family_misspecification_bound_theorem}
Fix $b >0 $, $h \geq 1$, $\nu \in \mathbb{N}_0$,
$i, j \in [K]$ with $i \neq j$. Let $ C_{ij} > 0$, $ c_{ij} \in (0,1)$. If $W_{ij}$ satisfies \eqref{pathwise bound} and
\begin{equation} \label{condition on the change-point}
{\mathbb{P}}_{\infty}(W_{ij}(\nu) \geq x \, | \, \tau(b,h) > \nu) \leq C_{ij}e^{-c_{ij}x} \qquad \text{for all } \; x \in (0, h],
\end{equation}
and
\begin{align}
{\mathbb{P}}_{\nu, j} \left( \tau_{ij}^{\nu}(h) > \tau_j(b,h) \, |\, \tau(b,h) > \nu \right) > 0, \label{condition technical}
\end{align}
then there exists a function
$$ \phi_{ij}: (0, \infty) \to (0, \infty)$$
such that $\phi_{ij}(h) \to 0$ as $h \to \infty$, and
\begin{align} \label{show}
\begin{split}
{\mathbb{P}}_{\nu, j} (\widehat{\tau}(b,h) & =i \,| \, \tau(b,h) > \nu ) \\
& \leq e^{-h} \, {\mathbb{E}}_{\nu, j}[\tau_j(b,h) - \nu \,|\, \tau(b,h) > \nu] + \frac{C_{ij}}{1 - c_{ij}}e^{-c_{ij}h} \left( 1 + \phi_{ij}(h) \right).
\end{split}
\end{align}
\end{theorem}
\begin{IEEEproof}
Appendix F. \\
\end{IEEEproof}
\noindent \underline{\textbf{Remark}}: If condition \eqref{condition on the change-point} holds with $c_{ij} \geq 1 $, then a tighter upper bound can be derived with minor adjustments to the proof.\\
To explore the implications of Theorem \ref{th:family_misspecification_bound_theorem} we need to introduce some additional notation. Thus, for any family of real numbers
\begin{equation}
\boldsymbol{C} \equiv \left\lbrace c_{ij} \in (0,1), C_{ij}>0: \, i, j \in [K], i \neq j \right\rbrace,
\end{equation}
we introduce the set of numbers in $\mathbb{N}_0$ for which \eqref{condition on the change-point} holds for all $ b > 0, h \geq 1$:
\begin{align}
N_{\boldsymbol{C}} \equiv \big\lbrace n \in \mathbb{N}_0 : {\mathbb{P}}_{\infty}(W_{ij}(n) \geq x \, | \, \tau(b,h) > n) \leq C_{ij}e^{-c_{ij}x} \quad \forall \, x \in (0, h], \, b > 0, \, h \geq 1 \big\rbrace .
\end{align}
\begin{lemma}
If $W_{ij}(0)= 0$ for every $i \neq j$, then $0\in N_{\boldsymbol{C}}$ for any $\boldsymbol{C}$. If $m \in \mathbb{N}$ and condition \eqref{tail} holds for every $n \in [m]$, then there is a $ \boldsymbol{C}$ so that $[m] \subseteq N_{\boldsymbol{C}}$.
\end{lemma}
\begin{IEEEproof}
The first claim is obvious. For the second, observe that, for every $n \in \mathbb{N}$,
\begin{equation}
{\mathbb{P}}_{\infty}(W_{ij}(n) \geq x \, | \, \tau(b,h) > n) \leq \frac{{\mathbb{P}}_{\infty}(W_{ij}(n) \geq x )}{ {\mathbb{P}}_{\infty}( \tau(b,h) > n)}.
\end{equation}
By standard properties of the CuSum statistics,
\begin{equation}
{\mathbb{P}}_{\infty}( \tau(b,h) > n) \geq {\mathbb{P}}_{\infty}( \sigma(b) > n) > 0.
\end{equation}
Therefore, if condition \eqref{tail} holds for every $n \in [m]$, then there is a $ \boldsymbol{C}$, which depends on $m$, such that $[m] \subseteq N_{\boldsymbol{C}}$. \\
\end{IEEEproof}
\noindent \underline{\textbf{Remark}}: As we showed earlier,
\eqref{tail} holds \textit{uniformly in $n$} for the proposed procedure and the Vector CuSum. In view of this, we conjecture that, for these two schemes, there is a $ \boldsymbol{C} $ such that $ N_{\boldsymbol{C}} = \mathbb{N}_0$. However, this cannot be shown with the previous argument, since
\begin{equation}
\inf_{n \in \mathbb{N}_0} {\mathbb{P}}_{\infty}( \sigma(b) > n) =
0.
\end{equation}
We next specialize Theorem \ref{th:family_misspecification_bound_theorem} to the proposed scheme.
\begin{corollary}\label{coro: h_choice}
Let $W_{ij}=Y'_{ij}$ for all $i, j \in [K]$ such that $ j \neq i$. Suppose that condition \eqref{assum:A1_prime} holds, and the conditions of Lemma \ref{lem:det_delay_bound} hold for all $i, j \in [K]$ such that $ j \neq i$. Then, for any $\alpha, \beta \in (0,1)$ and any $ \boldsymbol{C}$, we can select $h$ large enough so that
\begin{equation}
(\tau (b_\alpha, h), \widehat{\tau}(b_{\alpha}, h)) \in \mathcal{C}(\alpha, \beta, N_{\boldsymbol{C}}).
\end{equation}
\end{corollary}
\begin{IEEEproof}
Appendix F. \\
\end{IEEEproof}
\noindent \underline{\textbf{Remark}}: A similar corollary of Theorem \ref{th:family_misspecification_bound_theorem} can be stated for the Vector CuSum, i.e., when ${W_{ij}=Y_i-Y_j}$ for every $i \neq j$. Indeed, the conditional expected delay in the upper bound of \eqref{show} can be upper bounded using \cite[Theorem 1]{nikiforov2000simple}. However, it is not clear whether condition \eqref{condition technical} is satisfied for all $ \nu \in \mathbb{N}_0$, at least for $h$ large enough, in the case of the Vector CuSum. Even if it is, the corresponding corollary would not guarantee false isolation control uniformly in $ \nu \in \mathbb{N}_0 $, as it is implied by \cite[Theorem 2]{nikiforov2000simple}. The reason is that in the proof presented in \cite{nikiforov2000simple} the conditioning in \eqref{condition on the change-point} is ignored, but it is not clear to us whether this is indeed possible.
\subsection{An asymptotic optimality property}
We close this section by showing that the proposed procedure achieves the infimum in \eqref{infimum} to a first-order asymptotic approximation as $\alpha$ goes to $ 0 $ sufficiently faster than $\beta$, for a given $ \boldsymbol{C}$.
\begin{theorem}\label{th:optimality}
Let $W_{ij}=Y'_{ij}$ for all $i, j \in [K]$ such that $ j \neq i$ and suppose that \eqref{assum:A1_prime}, the conditions of Lemma \ref{lem:det_delay_bound} hold.
Then, for any $ \boldsymbol{C}$, there is family of thresholds $(h_\beta)$ such that, for all $ i \in [K]$,
\begin{align}
& \sup_{\nu \in N_{\boldsymbol{C}}} {\mathbb{P}}_{\nu, i}(\widehat{\tau}(b_{\alpha}, h_\beta) \neq i| \tau(b_{\alpha}, h_\beta) > \nu) \lesssim \beta, \quad \text{and}\\
& \mathcal{J}_i[\tau(b_{\alpha}, h_{ \beta})] \sim \inf_{(T, D) \in \mathcal{C}(\alpha, \beta, N_{\boldsymbol{C}})} \mathcal{J}_i[T],
\end{align}
as $ \alpha, \beta \to 0 $ such that \begin{equation}
\log | \log \beta | \ll \log | \log \alpha | \ll| \log \beta |.
\end{equation}
\end{theorem}
\begin{IEEEproof}
Appendix F. \\
\end{IEEEproof}
\section{Design } \label{sec:design}
The formulation of Section \ref{sec:state} suggests
that thresholds $b$ and $h$ of a sequential change diagnosis procedure of the form \eqref{general_stopping_time}-\eqref{family} should be selected so that its false alarm rate does not exceed $\alpha$, i.e.,
\begin{equation}\label{eqn:false_alarm_rate}
{\mathbb{E}}_\infty[\tau(b,h)] \geq 1/\alpha,
\end{equation}
where $\alpha$ is a user-specified number in $(0,1)$, and its worst-case conditional probability of false isolation does not exceed $\beta$, i.e.,
\begin{equation} \label{optimizers}
\max_{j \in [K]} \sup_{\nu \in \mathbb{N}_0 }{\mathbb{P}}_{\nu, j}(\widehat{\tau}(b,h) \neq j | \tau(b,h) > \nu) \leq \beta,
\end{equation}
where $\beta$ is a user-specified number in $(0,1)$. As we have argued earlier, this quantity cannot, in general, become arbitrarily small for each scheme we consider in this work, even for very large values of $b$ and $h$. Moreover, even for the schemes for which it can be controlled, the change-point for which the supremum is achieved is unknown. Therefore, in order to satisfy the second constraint, if it is even possible to do so, the conditional probability
$${\mathbb{P}}_{\nu, j}(\widehat{\tau}(b,h) \neq j | \tau(b,h) > \nu)
$$
needs to be estimated, for various values of $(b,h)$, for a sufficiently large number of $\nu$s.
Motivated by these considerations, in this section we propose a novel method for selecting the thresholds $b$ and $h$, which is more computationally efficient than the one described above, as it does not involve computation of conditional probabilities of false isolation. For this, we propose selecting $b$ and $h$ to satisfy, in addition to the false alarm constraint \eqref{eqn:false_alarm_rate},
\begin{equation}\label{eqn:delay_constraint}
\mathcal{J}_i[\tau(b,h)] \leq r \, \max_{j \in [K]} \mathcal{L}_j(\alpha), \quad \text{ for all } i \in [K],
\end{equation}
for some user-specified constant $r>1$, where
$ \mathcal{L}_j(\alpha)$ is the optimal worst-case Lorden delay in $ \mathcal{C}(\alpha) $ when $g = g_j$, i.e.,
\begin{align} \label{optimal_cusum_performance}
\mathcal{L}_j(\alpha) \equiv \inf_{(T,D) \in \mathcal{C}(\alpha)} \mathcal{J}_j[T] ={\mathbb{E}}_{j}[\sigma_j(b_j(\alpha))],
\end{align}
$b_j(\alpha)$ being the threshold $b$ for which
${\mathbb{E}}_{\infty}[\sigma_j(b)]=1/\alpha$ (\cite{moustakides1986optimal}). In other words, we propose selecting $b$ and $h$ so that the maximum Lorden delay of $ \tau(b,h)$ can increase at most by a factor $r$ relative to that of the optimal procedure that knows a priori the true post-change distribution.
More formally, given user-specified values of $ \alpha \in (0,1)$ and $r>1$, the set of thresholds $(b,h)$ for which the average time to false alarm and detection delay conditions hold can be expressed as follows:
\begin{equation}
\mathcal{S}(\alpha, r) \equiv \bigcap_{i \in [K]} \mathcal{D}_i(\alpha, r )\cap \mathcal{A}(\alpha),
\end{equation}
where $\mathcal{A}(\alpha)$ is the set of pairs for which the false alarm constraint is satisfied, i.e.,
\begin{equation}
\mathcal{A}(\alpha) \equiv \left\lbrace (b,h) : {\mathbb{E}}_{\infty}[\tau(b,h)]\geq 1 / \alpha \right\rbrace,
\end{equation}
and, for $i \in [K]$,
$\mathcal{D}_i(\alpha, r)$ is the set of pairs satisfying \eqref{eqn:delay_constraint}, i.e.,
\begin{align}
\mathcal{D}_i(\alpha, r) &\equiv \left\lbrace (b,h) : \mathcal{J}_i[\tau(b,h)] \leq r \max_{j \in [K]} \mathcal{L}_j(\alpha)\right\rbrace.
\end{align}
Once $\mathcal{S}(\alpha, r)$ is computed, a rather natural and intuitive selection for $b$ and $h$ in $\mathcal{S}(\alpha, r)$ is to \textit{select $h$ as large as possible within this region, and to subsequently select $b$ as large as possible given this choice of $h$.} While this selection is not guaranteed to optimize \eqref{optimizers}, our simulations studies in the next section suggest that it can, at least, provide a good approximation to the true optimizer, whose computation is not feasible.
\subsection{Computational considerations}
The implementation of the proposed design requires, first of all, the computation, for every $i \in [K]$, of the optimal Lorden delay when the post-change is $g_i$, namely $\mathcal{L}_i(\alpha)$, defined in \eqref{optimal_cusum_performance}.
This task can be performed easily using Monte Carlo simulation, or using existing approximations, such as Siegmund's corrected Brownian approximations (\cite{siegmund1979corrected}).
Second, it requires the computation of the expected time to false alarm and the Lorden delay of the scheme of interest, ${\mathbb{E}}_\infty[\tau(b,h)]$ and $\mathcal{J}_{i}[\tau(b,h)]$, for every ${i \in [K]}$ and various values of $b, h$. This task is particularly simple for procedures that satisfy
\eqref{worst case at 0}, i.e., for which
\begin{equation}
\mathcal{J}_{i}[\tau(b,h)]= {\mathbb{E}}_i[\tau(b,h)], \quad i \in [K],
\end{equation}
such as the min-CuSum, the Matrix CuSum and the proposed scheme. Indeed, for these schemes, one simply needs to simulate paths of the sequence $(X_n)$ under ${\mathbb{P}}_\infty$ and ${\mathbb{P}}_i$, $i \in [K]$, which allow
the estimation of ${\mathbb{E}}_\infty[\tau(b,h)]$ and ${\mathbb{E}}_i[\tau(b,h)]$, for ${i \in [K]},$
simultaneously for a grid of values of $ (b,h)$.
\iffalse
\begin{algorithm}[b]
\caption{Designing a Change Diagnosis Procedure}
\begin{algorithmic}[1]
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\REQUIRE $ \alpha \in (0,1), r > 1 $
\ENSURE $(b,h)$
\FOR {$i = 1$ to $K$}
\STATE Compute $ {\mathbb{E}}_i[\sigma_i(b_i(\alpha))] $\\
(Monte Carlo or theoretical approximation)
\STATE Compute $ \mathcal{D}_i(\alpha, r) $ (Monte Carlo)
\ENDFOR
\STATE Compute $ \mathcal{A}(\alpha) $ (Monte Carlo)
\STATE Select $(b,h) \in \mathcal{S}(\alpha, r) $ \\
(select $h$ as large as possible, or do further simulation)
\RETURN $(b,h)$
\end{algorithmic}
\label{algo}
\end{algorithm}
\fi
On the other hand, for procedures that do not satisfy \eqref{worst case at 0}, such as the Generalized CuSum, its window-limited version, and the Vector CuSum, the Lorden delay can be impossible to estimate, and only a biased, low estimate of the Pollak delay may be obtainable in practice. Thus, it is not in general possible to have a fair comparison between a procedure that satisfies \eqref{worst case at 0} and a procedure that does not.
In view of this, in our simulation studies in the next section we evaluate, for simplicity, the expected delay for these procedures only when $\nu=0$, keeping in mind that this can be much smaller than the actual Lorden delay for procedures that do not satisfy \eqref{worst case at 0}.
\section{Simulation studies}\label{sec:simulation}
In this section we compare the min-CuSum, the Matrix CuSum, the Vector CuSum, the window-limited Generalized CuSum, and the proposed procedure in the multichannel setup of Subsection \ref{subsec:multichannel}. We focus on the case that there are two channels $(d=2)$, with Gaussian pre-change and post-change distributions, specifically
\begin{equation}\label{eqn:p_and_q}
p_i \equiv \mathcal{N}(0,1), \quad q_i \equiv \mathcal{N}(1,1),
\end{equation}
and we consider two setups for this problem. In the first one the change can happen in only one of the two channels, i.e., \eqref{multichannel_single} holds with ${K = d = 2}$. In the second, the change can also happen in both channels simultaneously, in which case $K=3$ and \eqref{multichannel_simultaneous} holds.
\subsection{Design}
In both setups, we design each scheme according the method described in the previous section. Specifically, we fix $ \alpha = 1 \%$ and consider several values of $r$. We consider $b$ in increments of $0.01$ starting from $0$ and $h$ in increments of $ 0.05$ starting from $ 0.05$.
To implement each of the above schemes, we first estimate, for each $i \in [K]$, the optimal Lorden delay in \eqref{optimal_cusum_performance}, with $b_i(\alpha)$ replaced by the smallest $b$ in our grid for which the estimate of
${\mathbb{E}}_\infty[\sigma_i (b)]$ exceeds $1/\alpha$. The estimated thresholds and detection delays are provided in Table \ref{table:traditional_multichannel_table} and Table \ref{table:multichannel_table}.
Subsequently, for each diagnosis procedure, we estimate the regions $ \mathcal{D}_1(\alpha, r), \mathcal{D}_3(\alpha, r), $ and $\mathcal{A}(\alpha)$, thereby obtaining $ \mathcal{S}(\alpha, r)$ for various values of $r$.
This process is illustrated with figures for the proposed procedure and for the Matrix CuSum when $r=1.3$ and when $r=2$ in Appendix G.
For the Monte Carlo estimation of each of these quantities, i.e., ${\mathbb{E}}_{\infty}[\sigma_i (b)]$, ${\mathbb{E}}_{\infty}[\tau(b,h)]$, and ${\mathbb{E}}_i[\tau(b,h)], i \in [K],$ we simulate $ {L=5 \times 10^4} $ paths of relevant statistics under $ {\mathbb{P}}_1 $ and $ {\mathbb{P}}_3$ and $ 0.1\cdot L$ paths under $ {\mathbb{P}}_{\infty}$. We do not need to simulate any paths under $ {\mathbb{P}}_2 $ due to its symmetry with $ {\mathbb{P}}_1$. The standard error of the estimate of ${\mathbb{E}}_\infty[\sigma_i (b)]$ is less than $ 1.4 \%$ in each case.
\begin{table}[b]
\caption{Multichannel problem with single faults - optimal worst-case detection delay:\\ Estimate (Standard Error)}
\label{table:traditional_multichannel_table}
\begin{center}
\begin{tabular}{ cccc }
& $b_i(\alpha)$ &$ \mathcal{J}_{i}[\sigma_i(b_i(\alpha)]$ \\
\hline
$i =1$ & {2.85} & 6.0797 (0.0166)\\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[bt]
\caption{Multichannel problem with simultaneous faults - optimal worst-case detection delay: Estimate (Standard Error)}
\label{table:multichannel_table}
\begin{center}
\begin{tabular}{ cccc }
& $b_i(\alpha)$ &$ \mathcal{J}_{i}[\sigma_i(b_i(\alpha))]$ \\
\hline
$i =1, 2$ & {2.85} & 6.0965 (0.0165) \\
$i = 3$ & 3.04 & 3.7450 (0.0097) \\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Comparisons}
Since it is not feasible to compute the worst-case conditional probability of misidentification in \eqref{optimizers}, we compare the various schemes based on their worst-case conditional probabilities of misidentification when $ {\nu \in \lbrace 0, 10, \ldots, 50 \rbrace}$, i.e.,
\begin{equation}\label{eqn:simulated_probabilities}
\max_{j \in [K]} \max_{\nu \in \lbrace 0, 10, \ldots, 50 \rbrace}{\mathbb{P}}_{\nu, j}(\widehat{\tau}(b,h) \neq j | \tau(b,h) > \nu)
\end{equation}
For each scheme, we compute this quantity in two ways as far as it concerns the selection of $b$ and $h$:
\begin{itemize}
\item using the proposed method in the previous section, i.e., the largest value of $h$ in $\mathcal{S}(\alpha, r)$, and subsequently the largest value of $b$ given this choice of $h$,
\item using the values of $b$ and $h$ that optimize \eqref{eqn:simulated_probabilities} in $\mathcal{S}(\alpha,r)$.
\end{itemize}
The first method requires no additional simulation, after computing $ \mathcal{S}(\alpha, r)$, whereas for the second task we simulate $L$ paths of relevant statistics for the Monte Carlo estimation of each of the probabilities in \eqref{eqn:simulated_probabilities}.
The two different methods for selecting $b$ and $h$ provide indistinguishable results in all cases apart from that of the Matrix CuSum in the setup of a single fault, where \eqref{eqn:simulated_probabilities} is optimized by
a smaller $h$ and a larger $b$ compared to the ones obtained by the method of the previous section. Although the difference is not dramatic, we present the results with the second choice of thresholds, making sure that all comparisons are fair for the Matrix CuSum.
In Fig. \ref{fig:stability} we plot, for each scheme, the worst-case probability of false isolation
$$ \max_{j \in [K]} {\mathbb{P}}_{\nu, j}(\widehat{\tau}(b,h) \neq j | \tau(b,h) > \nu) $$
against ${ \nu \in \lbrace 0, 10,\ldots, 50 \rbrace}$, when $r=2$, for the single fault and the simultaneous fault setup, respectively. These graphs suggest that
\eqref{eqn:simulated_probabilities} is a reasonable proxy for \eqref{optimizers} for all schemes, apart from the Matrix CuSum. This means that using \eqref{eqn:simulated_probabilities} instead of \eqref{optimizers} in our comparisons is favorable to the Matrix CuSum.
\begin{figure}%
\centering
\subfloat[\centering Without simultaneous faults]{{\includegraphics[width=6.8cm]{multichannel_stability.eps} }}%
\qquad
\subfloat[\centering With simultaneous faults]{{\includegraphics[width=6.8cm]{multichannel_with_third_stability.eps} }}%
\caption{Worst-case probabilities of false isolation (with respect to the type of change) plotted against the change-point $\nu$ with $ \alpha = 1 \% $ and $ r= 2 $ in the multichannel problem without and with simultaneous faults, respectively. Standard errors of probability estimates provided as error bars.}%
\label{fig:stability}%
\end{figure}
\iffalse
\begin{figure}[bt]
\begin{center}
{\includegraphics[scale=1]{multichannel_stability.eps}}
\caption{Worst-case probabilities of false isolation (with respect to the type of change) plotted against the change-point $\nu$ with $ \alpha = 1 \% $ and $ r= 2 $ in the multichannel problem without simultaneous faults. Standard errors of probability estimates provided as error bars.}
\label{fig:stability_multichannel}
\end{center}
\end{figure}
\begin{figure}[bt]
\begin{center}
{\includegraphics[scale=1]{multichannel_with_third_stability.eps}}
\caption{Worst-case probabilities of false isolation (with respect to the type of change) plotted against the change-point $\nu$ with $ \alpha = 1 \% $ and $ r = 2 $ in the multichannel problem with simultaneous faults. Standard errors of probability estimates provided as error bars.}
\label{fig:stability_orthogonal}
\end{center}
\end{figure}
\fi
\subsection{Results}
In Fig. \ref{fig:comparison} we plot, for each scheme, the worst-case conditional probabilities of misidentification in \eqref{eqn:simulated_probabilities} against $r$ for the single fault and the simultaneous faults setup, respectively. As explained in the previous section, we compare separately the schemes that satisfy \eqref{worst case at 0}, i.e., the proposed procedure, Matrix CuSum, and min-CuSum, and those that do not, i.e., the Vector CuSum and the Generalized CuSum, since the constraints have different interpretations between the two groups.
\iffalse
\begin{figure}
\begin{center}
\subfloat[Proposed]{
\includegraphics[scale=1]{A1_300dpi.eps}
\label{fig:proposed_threshold_regions_1}
}
\subfloat[Matrix CuSum]{
\includegraphics[scale=1]{M1_300dpi.eps}
\label{fig:matrix_cusum_threshold_regions_1}
}
\captionsetup{justification=raggedright, singlelinecheck=false}
\caption{Computing the region $ \mathcal{S}(1 \%, 1.3)$}
\label{fig:threshold_regions1}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\subfloat[Proposed]{
\includegraphics{A2_300dpi.eps}
\label{fig:proposed_threshold_regions_2}
}
\subfloat[Matrix CuSum]{
\includegraphics{M2_300dpi.eps}
\label{fig:matrix_cusum_threshold_regions_2}
}
\captionsetup{justification=raggedright, singlelinecheck=false}
\caption{Computing the region $ \mathcal{S}(1 \%, 2)$}
\label{fig:threshold_regions2}
\end{center}
\end{figure}
\fi
From these graphs we observe that, in both setups and for all values of $r$, the proposed scheme performs as well or better than the min-CuSum, and the latter performs better than the Matrix CuSum. However, in the single fault setup the differences between these schemes vanish as $r$ increases. On the other hand, in the simultaneous fault setup the worst-case probability of the Matrix CuSum is very close to 1 even for large values of $r$, whereas for the two other schemes it is not much larger than $0.2$ even for small values of $r$.
Moreover, we observe that, in both setups and for all values of $r$, the Vector CuSum performs better than the window-limited Generalized CuSum, independently of the choice of window size, $m$, in the latter. However, in the single fault setup the choice of $m$ does not make any practical difference, whereas the difference between the two schemes vanishes as $r$ increases. On the other hand, in the simultaneous fault setup a larger $m$ leads to substantially worse misidentification probability for the Generalized CuSum and, even with a smaller value of $m$,
the Generalized CuSum performs substantially worse than the
Vector CuSum.
\iffalse
\begin{figure}
\begin{center}
{\includegraphics[scale=1]{multichannel_comparison.eps}}
\caption{Worst-case probability of false isolation with respect to both the change-point and the type of change as a function of the detection delay allowance factor $r$ in the multichannel problem with single faults. Standard errors of probability estimates provided as error bars.}
\label{fig:comparison_multichannel}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
{\includegraphics[scale=1]{multichannel_with_third_comparison.eps}}
\caption{Worst-case probabilities with respect to the change-point and post-change density as a function of the detection delay allowance factor $r$ in the multichannel problem with simultaneous faults. Standard errors of probability estimates provided as error bars.}
\label{fig:comparison_orthogonal}
\end{center}
\end{figure}
\fi
\begin{figure}%
\centering
\subfloat[\centering Without simultaneous faults]{{\includegraphics[width=6.8cm]{multichannel_comparison.eps} }}%
\qquad
\subfloat[\centering With simultaneous faults]{{\includegraphics[width=6.8cm]{multichannel_with_third_comparison.eps} }}%
\caption{Worst-case probabilities with respect to the change-point and post-change density as a function of the detection delay allowance factor $r$ in the multichannel problem without and with simultaneous faults, respectively. Standard errors of probability estimates provided as error bars.}%
\label{fig:comparison}%
\end{figure}
\section{Conclusion}\label{sec:conclusion}
In this paper we revisit the problem of sequential change diagnosis and we propose a novel procedure that enjoys the various statistical and practical advantages of existing approaches in the literature. We analyze it theoretically and propose a method for its design that does not require the estimation of (small) probabilities, nor the simulation of quantities over multiple change-points. Both this theoretical analysis and this practical design are formulated for a more general family of sequential change diagnosis procedures in the literature.
A natural follow-up to this work is an investigation into our conjecture that the proposed scheme guarantees uniform error control over all possible change-points. Other directions include extensions of the proposed method to non-iid setups and/or composite (non-discrete) post-change scenarios, as well as the application of the proposed adaptive window to the Generalized CuSum algorithm. Finally, the numerical studies in this work seem to support the use of a pure sequential change detection algorithm, namely the min-CuSum, for the sequential change-diagnosis problem, complementing certain existing works in the literature (\cite{warnercusum, han2007detection}). This is a topic we plan to consider in more detail in the future.
\begin{appendices}
\section{Properties of CuSum Statistics}\label{app:lemmas}
Throughout all following appendices we use the following notation for the sums of log-likelihood ratios for $n \in \mathbb{N}$:
\begin{align}\label{def:Z}
\begin{split}
Z_i(n) &\equiv \sum_{k=1}^n \ell_i(k), \\
Z_{ij}(n) &\equiv \sum_{k = 1}^n \log \ell_{ij}(k).
\end{split}
\end{align}
and we further denote partial sums by
\begin{align}\label{def:Z_partial}
\begin{split}
Z_i(n,m) &\equiv Z_i(n) - Z_i(m), \\
Z_{ij}(n,m) &\equiv Z_{ij}(n) - Z_{ij}(m).
\end{split}
\end{align}
By defining the above quantities, we may conveniently express the CuSum in the two following ways,
\begin{equation}
Y_i(n) = \max_{0 \leq k \leq n} Z_i(n,k),
\end{equation}
and
\begin{equation} \label{eqn:cusum_min_form}
Y_i(n) = Z_i(n) - \min_{0 \leq k \leq n} Z_i(k),
\end{equation}
with similar expressions obtained for $ Y_{ij}$ by replacing $ Z_i $ with $ Z_{ij}$.
\begin{lemma} \label{lem:vector_cusum_smaller_than_matrix}
For each $i, j \in [K]$ with $i \neq j$,
\begin{equation}\label{eqn:vector_cusum_smaller_than_matrix}
Y_i(n)-Y_j(n)\leq Y_{ij}(n), \quad \forall \; n \in \mathbb{N}.
\end{equation}
\end{lemma}
\begin{IEEEproof}
By \eqref{eqn:cusum_min_form} we have
\begin{align*}
Y_i(n)-Y_j(n)
&= \max_{0 \leq k \leq n} Z_i(n,k) - \max_{0 \leq k \leq n} Z_j(n,k) \\
& \leq\max_{0 \leq k \leq n} Z_{ij}(n,k)= Y_{ij}(n).
\end{align*}
\end{IEEEproof}
\begin{lemma} \label{lem:new}
For each $i,j \in [K]$ with $i \neq j$, for all $x \geq 0$ and $n \in \mathbb{N}$ we have
$$ {\mathbb{P}}_{\infty}(Y_i (n) \geq x) \leq e^{-x}.$$
If also $\psi_{ij}$ exists around zero and
\begin{equation}\label{assume_negative_drift}
{\mathbb{E}}_\infty[ \ell_{ij}(1)] <0,
\end{equation} then
$$ {\mathbb{P}}_{\infty}(Y_{ij} (n) \geq x) \leq e^{-r_{ij} x}$$
where $r_{ij}$ is the positive root of $ \psi_{ij}.$
\end{lemma}
\begin{IEEEproof}
For all $x > 0$ and $n \in \mathbb{N}$
\begin{align*}
{\mathbb{P}}_{\infty}(Y_i (n) \geq x)
&= {\mathbb{P}}_{\infty} \left( \max_{0 \leq m < n} Z_i(n,m) \geq x \right) \\
&= {\mathbb{P}}_{\infty} \left( \max_{0 < m \leq n} Z_i(m) \geq x \right) \leq e^{-x},
\end{align*}
where the second equality follows by the random walk property and the inequality holds by Ville's supermartingale inequality and the fact that $\{ \exp [ Z_i(n) ], n \in \mathbb{N}\}$ is a ${\mathbb{P}}_{\infty}$-martingale with mean 1.
By \eqref{assume_negative_drift} it follows that
$ \left\lbrace \exp [ r_{ij} Z_{ij}(n) ], n \in \mathbb{N} \right\rbrace$ is a positive martingale with expectation 1 under ${\mathbb{P}}_{\infty}$,
where $ r_{ij}$ is the positive root of $\psi_{ij}$, and observe that
\begin{align*}
{\mathbb{P}}_{\infty} \left( Y_{ij}(n) \geq x \right)
&= {\mathbb{P}}_{\infty} \left( \max_{0 \leq s < n} Z_{ij}(n,s) \geq x \right) \\
&= {\mathbb{P}}_{\infty} \left( \max_{0 < s \leq n} Z_{ij}(s) \geq x \right) \leq e^{-r_{ij}x},
\end{align*}
where the second equality follows from the random walk property the inequality follows from Ville's supermartingale inequality.\\
\end{IEEEproof}
For the next lemma, we introduce the first regeneration time of $Y_{i}$ for each $ i \in [K]$:
\begin{equation} \label{def: eta}
\eta_i \equiv \inf\{n \in \mathbb{N}:Y_i(n) = 0\}
\end{equation}
and we note that
\begin{equation} \label{def: a}
a_i \equiv \sup_{\theta \in (0,1)} |\psi_i (\theta)|>0
\end{equation}
for each $i \in [K]$, where $ \psi_i$ is the cumulant generating function of $ \ell_i(1)$ under $ {\mathbb{P}}_{\infty}$, i.e.,
$$ \psi_{i}(\theta) \equiv \log \left( {\mathbb{E}}_{\infty} \left[ \exp [ \theta \, \ell_{i}(1) ] \right] \right), \quad \theta \in \mathbb{R}.$$
\begin{lemma} \label{lemma: regeneration bounded by expo}
For each $i \in [K]$, the first regeneration time $\eta_i $ is small under ${\mathbb{P}}_\infty$ in the sense that $$ {\mathbb{P}}_{\infty}(\eta_i > n) \leq e^{-a_in} \quad \forall n \in \mathbb{N},$$ and consequently
\begin{equation}\label{eqn:moment_bound}
{\mathbb{E}}_{\infty}[(\eta_i)^m] \leq \frac{e^{a_i}}{a_i^{m}} m! \quad \text{for all} \; m \in \mathbb{N}.
\end{equation}
\end{lemma}
\begin{IEEEproof}
For every $n \in \mathbb{N}$, \begin{align*}
\lbrace \eta_i > n \rbrace &= \lbrace Y_i(m) > 0 \; \text{for all} \; m \in [n] \rbrace \\
&= \lbrace Z_i(m) > 0 \; \text{for all} \;m \in [n] \rbrace \\
&\subseteq \lbrace Z_i(n) > 0 \rbrace.
\end{align*}
The first equality follows from definitions. The second equality can be easily verified by writing $ Y_i(m) $ in the form \eqref{eqn:cusum_min_form}, and the inequality is obvious.
Since $ \lbrace \ell_i(n), \, n \in \mathbb{N} \rbrace$ is an iid sequence with negative expectation under ${\mathbb{P}}_\infty$ and $ \psi_i(1) = 0$, by the Chernoff bound we have
$$ {\mathbb{P}}_{\infty}(\eta_i > n) \leq {\mathbb{P}}_{\infty}(Z_i(n) > 0) \leq e^{-a_i n} \quad \forall \; n \in \mathbb{N}.$$
This condition implies that $\eta_i$ is stochastically smaller than a Geometric random variable with success probability~$ 1 - e^{-a_i}.$ Specifically, if $G$ is distributed as Geom$(1-e^{-a_i})$, then $${\mathbb{P}}_{\infty}(\eta_i > n) \leq {\mathbb{P}}_{\infty}(G > n), \text{ for all } n \in \mathbb{N}_0,$$ from which it follows that $${\mathbb{E}}_{\infty}[\eta_i^m] \leq {\mathbb{E}}_{\infty}[G^m] \text{ for all } m \in \mathbb{N}. $$ Moreover, by routine calculation, \begin{align*}
{\mathbb{E}}_{\infty}[G^m] &= m \, \int_0^{\infty} t^{m-1} {\mathbb{P}}_{\infty}(G > t) \, dt \\
&= m \, \int_0^{\infty} t^{m-1} e^{-a_i \lfloor t \rfloor} \, dt \\
&\leq m \, \int_0^{\infty} t^{m-1} e^{-a_i(t-1)} \, dt \\
&= \frac{m e^{a_i}}{a_i^m} \int_0^{\infty} u^{m-1} e^{-u} du = \frac{e^{a_i} m!}{a_i^m},
\end{align*}
and in particular, $$ {\mathbb{E}}_{\infty}[\eta_i^m] \leq \frac{e^{a_i} m!}{a_i^m}.$$
\end{IEEEproof}
\section{Properties of CuSum Stopping Times}
Throughout this Appendix, we fix $b, h>0$ and ${i, j \in [K]}$ such that $i \neq j$. Moreover, we introduce the following quantity:
$$ \omega_{ij} \equiv \sup_{t \geq 0} {\mathbb{E}}_j \left[ \ell_{ij}(1) - t | \ell_{ij}(1) \geq t \right].$$
That is, $\omega_{ij}$ is a bound for the expected overshoot of the random walk $Z_{ij}$ above a threshold under ${\mathbb{P}}_j$.
We recall from Section \ref{sec:CUSUM} that $\sigma_{i}(b)$ is the CuSum stopping time, with threshold $b$, for detecting a change from $f$ to $g_i$,
and we introduce the CuSum stopping time, with threshold $h$, for detecting a change from $g_j$ to $g_i$, i.e.,
\begin{align*}
\sigma_{ij}(h) &\equiv \inf \lbrace n \in \mathbb{N} : Y_{ij}(n) \geq h \rbrace.
\end{align*}
We denote by $U_{ij}(x;h)$ the expectation of $\sigma_{ij}(h)$ under ${\mathbb{P}}_j$ when $ Y_{ij}$ is initialized from some $x \geq 0 $, i.e.,
\begin{equation}\label{U}
U_{ij}(x;h) \equiv {\mathbb{E}}_j[\sigma_{ij}(h) \,|\, Y_{ij}(0) = x], \quad x \geq 0.
\end{equation}
Clearly $U_{ij}(\cdot; h)$ is a non-increasing function, so that
$$ U_{ij}(x;h) \leq U_{ij}(0;h) \equiv {\mathbb{E}}_j[\sigma_{ij}(h)], \quad \text{ for all } x \geq 0.$$
By standard properties of the CuSum test
(see \cite[Section 8.2.6]{tartakovsky2014sequential}) it follows that
\begin{align}
U_{ij}(0;h) ={\mathbb{E}}_j[\sigma_{ij}(h)] &\geq e^h, \label{ARL_CUSUM_LB}
\end{align}
and that
\begin{align} \label{u_ineq}
U_{ij}(x;h) &\geq u_{ij}(x; h), \quad x \in [0,h],
\end{align}
where the function $u_{ij}$ is defined as follows:
\begin{equation} \label{def:u}
u_{ij}(x ;h) \equiv \frac{x - e^{-(h - x)}(h+ \omega_{ij})}{I_{ji}} + (1 - e^{-(h - x)}) \, U_{ij}(0; h) , \quad x\geq 0.
\end{equation}
Note that $u_{ij}$ is a differentiable function with derivative
\begin{equation} \label{derivative2}
-u_{ij}'(x ;h) = \frac{-1}{I_{ji}} + \frac{h + \omega_{ij}}{I_{ji}}e^{x-h} + U_{ij}(0;h) e^{x-h}.
\end{equation}
Moreover, it follows that \begin{equation}\label{eqn:upper_bound_on_uprime}
-u_{ij}'(x ;h) \leq \left( \frac{h + \omega_{ij}}{I_{ji}} + U_{ij}(0;h) \right) \, e^{x-h}, \quad x \in [0, h].
\end{equation}
For $h\geq 1$,
\begin{equation} \label{inequality}
U_{ij}(x ; h) \mathds{1} ( \lbrace x \leq h \rbrace) \geq u_{ij}(x;h) \quad \text{ for all } x \geq 0.
\end{equation}
Indeed, since $ -e^x \leq -(x + 1)$ for every $x \in \mathbb{R}$, we have
$$ u_{ij}(x;h) \leq \frac{(x - h)(1 - h)}{I_{ji}}, \quad x \geq h,$$
and, consequently, for $h \geq 1$,
$$ u_{ij}(x;h) \leq 0 \quad \text{ for all } x \geq h.$$
\section{}\label{app:delay}
\begin{IEEEproof}[Proof of Lemma \ref{lem:new_lemma_for delay}]
Fix $\delta>0$. Set
$$D(b,h) \equiv \max \lbrace b / I_{i}, h / I^*_i \rbrace(1 + \delta),$$
and observe that
\begin{align} \label{bound}
{\mathbb{E}}_i &[\tau_i(b,h)] = \sum_{n = 0}^{\infty} {\mathbb{P}}_i \left( \tau_i(b,h) > n \right) \nonumber\\
&\leq \lceil D (b,h) \rceil + \sum_{n \geq \lceil D (b,h) \rceil} {\mathbb{P}}_i \left( \tau_i(b,h) > n \right).
\end{align}
For every $n \in \mathbb{N}$ we have
\begin{align*}
& \left\lbrace \tau_i(b,h)> n \right\rbrace \\
&= \bigcap_{m = 1}^n \left\lbrace Y_i(m) < b \quad \text{or} \quad \min_{j \in [K]: j \neq i} W_{ij}(m) < h \right\rbrace \\
&\subseteq \left\lbrace Y_i(n) < b \quad \text{ or } \quad \min_{j \in [K]: j \neq i} W_{ij}(n) < h \right\rbrace \\
&= \bigcup_{\stackrel{j \in [K]}{j \neq i}} \left\lbrace W_{ij}(n) < h \right\rbrace \cup \left\lbrace Y_i(n) < b \right\rbrace.
\end{align*}
For $n>D(b,h) $ we have
$$ b < n \frac{I_{i}}{1 + \delta} \quad \& \quad h < n\frac{I_i^*}{1 + \delta} \leq n\frac{I_{ij}}{1 + \delta} $$
for every $j \in [K]$ such that $j\neq i$, and consequently,
\begin{align*}
& \left\lbrace \tau_i(b,h) > n \right\rbrace \\
& \quad \quad \subseteq \bigcup_{\stackrel{j \in [K]}{j \neq i}} \left\lbrace W_{ij}(n) < n \frac{ I_{ij} }{1 + \delta} \right\rbrace \cup \left\lbrace Y_i(n) < n \frac{ I_{i}}{1 + \delta} \right\rbrace
\end{align*}
for every $n > D(b,h)$. Combining this with \eqref{bound} we obtain
\begin{align} \label{bound2}
\begin{split}
{\mathbb{E}}_{i}[\tau_i(b,h)] &\leq \lceil D(b,h) \rceil \\
&+ \sum_{n =1 }^\infty {\mathbb{P}}_i \left( Y_i(n) < n I_{i} (1 + \delta)^{-1} \right) \\
& + \sum_{n =1 }^\infty \sum_{\stackrel{j \in [K]}{j \neq i}} {\mathbb{P}}_i \left( W_{ij}(n) < n I_{ij} (1 + \delta)^{-1} \right).
\end{split}
\end{align}
The third term in the upper bound converges by assumption. Moreover, since $Y_i(n) \geq Z_i(n)$ for every
$ n \in \mathbb{N}$ and by the Chernoff bound it follows that ${\mathbb{P}}_i \left( Z_{i}(n) \leq\rho n \right)$ is an exponentially decaying sequence for every $\rho < I_i$, we conclude that
${\mathbb{P}}_i \left( Y_{i}(n) \leq \rho n \right)$ is also an exponentially decaying sequence for every $\rho < I_i$. Therefore, the second series in the upper bound converges, and this completes the proof. \\
\end{IEEEproof}
\begin{IEEEproof}[Proof of Lemma \ref{lem:det_delay_bound}]
For every $n \in \mathbb{N}$ we have
\begin{align*}
\min_{R_i(n) \leq k \leq n} Z_{ij}(k)
&\leq Z_{ij}(R_i(n))\\
&=\sum_{m = 1}^{R_i(n) }\ell_{ij}(m)\leq\sum_{m = 1}^{R_i(\infty) } |\ell_{ij}(m)|,
\end{align*}
where $R_i(\infty) $ is the final regeneration time of $Y_i$, i.e.,
$$R_i(\infty) \equiv \sup \lbrace n \in \mathbb{N}
: Y_{i}(n) = 0 \rbrace, $$
and $Z_i$ is defined in \eqref{def:Z}. Consequently,
\begin{align*}
Y^{\prime}_{ij}(n) &= \max_{R_i(n) \leq k \leq n} Z_{ij}(n,k) \\
&= Z_{ij}(n) - \min_{R_i(n) \leq k \leq n} Z_{ij}(k)\\
&\geq Z_{ij}(n)- \sum_{m = 1}^{R_i(\infty) } |\ell_{ij}(m)|.
\end{align*}
Let $\rho \in (0, I_{ij})$. Then there is a $\delta>0$ such that $\rho+\delta< I_{ij}$, and as a result we have
\begin{align*}
{\mathbb{P}}_i \left( Y^{\prime}_{ij}(n) \leq \rho n \right) &\leq {\mathbb{P}}_i \left( Z_{ij}(n) \leq (\rho + \delta ) n \right)\\
&+ {\mathbb{P}}_i \left( \sum_{m=1}^{R_i(\infty) }| \ell_{ij}(m)| \geq n \delta \right).
\end{align*}
By the Chernoff bound, the first term in the upper bound goes to 0 exponentially fast in $n$. Thus, it remains to show that the second term is summable, for which it suffices to show that
$${\mathbb{E}}_i \left[ \sum_{m=1}^{R_i(\infty) } |\ell_{ij}(m)| \right] < \infty.$$
By the first assumption of the lemma we have
$$ || \ell_{ij}(1) ||_p \equiv \sqrt[p]{{\mathbb{E}}_i \left[ |\ell_{ij}(1)|^p \right]} <\infty,$$
and as a result
\begin{align*}
{\mathbb{E}}_i \left[ \sum_{m=1}^{R_i(\infty) } |\ell_{ij}(m)| \right] &= \sum_{m = 1}^{\infty} \mathbb{E}_i \left[ |\ell_{ij}(m)| \; \mathds{1} ( \lbrace R_i(\infty) \geq m \rbrace ) \right] \\
&\leq || \ell_{ij}(1) ||_p \; \sum_{m= 1}^{\infty}
\sqrt[q]{{\mathbb{P}}_i \left( R_i(\infty) \geq m \right) }
\end{align*}
where $ q \equiv p / (p-1), $ the equality follows by Tonelli's theorem, and the inequality by H{\"o}lder's inequality. Therefore, it suffices to show that the series in the upper bound converges. By the second assumption of the lemma there is an $\epsilon>0$ such that
$$ {\mathbb{E}}_i \left[ \left( \ell_{i}^-(1) \right)^{2 + \epsilon +q} \right] < \infty,$$
and by \cite[Theorem 1]{janson1986moments}
it follows that
$${\mathbb{E}}_i[(R_i(\infty) )^{1 + \epsilon +q}] < \infty,$$
or equivalently
$$\sum_{m= 1}^{\infty} m^{\epsilon +q} \, {\mathbb{P}}_i(R_i(\infty) \geq m) < \infty.$$ As a result, there is some $ s \in \mathbb{N} $ such that
$$ m^{\epsilon + q} \, {\mathbb{P}}_i(R_i(\infty) \geq m) < 1 \quad \text{for all} \quad m \geq s,$$
and this further implies that
$$ \sum_{m = s}^{\infty} \sqrt[q]{{\mathbb{P}}_i \left( R_i(\infty) \geq m \right) } \leq \sum_{m = s}^{\infty} m^{-(1 + \epsilon /q)} <\infty,$$
which completes the proof. \\
\end{IEEEproof}
\section{}\label{app:false_alarm_bound}
\begin{IEEEproof}[Proof of Lemma \ref{lem:false_alarm_bound}]
We fix $b, h>0$ and observe that the assumption of the Lemma implies that, for all $i, j \in [K], j \neq i$,
$$ \sup_{n \in \mathbb{N}_0} {\mathbb{P}}_{\infty}(W_{ij}(n) \geq h) \leq Q_{ij} \, e^{-q_{ij}h}. $$
As a result, for every $i \in [K]$ we have
$$ \sup_{n \in \mathbb{N}_0} {\mathbb{P}}_{\infty} \left( \min_{j \in [K] : j \neq i} W_{ij}(n) \geq h \right) \leq G_i(h), $$
where
$$ G_i(h) \equiv \min_{j \in [K]: j \neq i} \left\lbrace Q_{ij} e^{-q_{ij} h} \right\rbrace. $$
Since also
\begin{align*}
\tau(b,h) &\geq \inf \left\{ n \in \mathbb{N} : \max_{i \in [K]} \min_{j \in [K]: j \neq i} W_{ij}(n) \geq h \right\},
\end{align*}
for every $n \in \mathbb{N}$ we have
\begin{align*}
{\mathbb{P}}_{\infty}(\tau(b,h) \leq n) &\leq \sum_{m = 1}^n \sum_{i = 1}^K {\mathbb{P}}_{\infty} \left( \min_{j \in [K]: j \neq i} W_{ij}(m) \geq h \right) \\
&\leq \sum_{m = 1}^n \sum_{i = 1}^K G_i(h) \equiv n G(h),
\end{align*}
and, consequently,
$$ {\mathbb{P}}_{\infty}(\tau(b,h) > n) \geq \left( 1 - n G(h)\right)^+. $$
As a result,
\begin{align*}
{\mathbb{E}}_{\infty} \left[ \tau(b,h) \right] &= \sum_{n = 0}^{\infty} {\mathbb{P}}_{\infty}(\tau(b,h) > n) \\
&\geq \sum_{n = 0}^{\infty}\left( 1 - n G(h) \right)^+\\
&=\sum_{n = 0}^{\lfloor 1/G(h) \rfloor} \left( 1 - n G(h) \right) \geq \frac{1}{2G(h)},
\end{align*}
which is what we wanted to show. \\
\end{IEEEproof}
\section{}\label{app:uniform_exponential_bound}
\begin{IEEEproof}[Proof of Theorem \ref{thm:uniform_exponential_bound}]
Throughout the proof, we fix arbitrary $x>0 $ and $n \in \mathbb{N}_0 $. Since, by definition, $Y^{\prime}_{ij} \leq Y_{ij}$, we have
\begin{align*}
{\mathbb{P}}_{\infty} \left( Y^{\prime}_{ij}(n) \geq x \right) \leq {\mathbb{P}}_{\infty} \left( Y_{ij}(n) \geq x \right),
\end{align*}
and the result follows from Lemma \ref{lem:new} when ${\mathbb{E}}_\infty[ \ell_{ij}(1)]<0$. Thus, in the remainder of the proof we focus on the case that ${\mathbb{E}}_\infty[ \ell_{ij}(1)] \geq 0$. By the definition of $Y'_{ij}$ and the total probability law, for any $ \delta > 0 $ we have
\begin{align} \label{very_first_bound}
\begin{split}
{\mathbb{P}}_{\infty} \left( Y^{\prime}_{ij}(\nu) \geq x \right) & \leq {\mathbb{P}}_{\infty} \left( \max_{\nu - \delta x < s \leq \nu} Z_{ij}(\nu, s) \geq x \right) \\
& \quad + {\mathbb{P}}_{\infty}\left( \nu - R_i(\nu) \geq \delta x \right).
\end{split}
\end{align}
We start by upper bounding the first term in the right-hand side of \eqref{very_first_bound}. Since ${ \{ Z_{ij}(n), n \in \mathbb{N} \} }$ is a random walk under ${\mathbb{P}}_\infty$,
\begin{align*}
{\mathbb{P}}_{\infty} \left( \max_{\nu - \delta x < s \leq \nu} Z_{ij}(\nu,s) \geq x \right)
&= {\mathbb{P}}_{\infty} \left( \max_{1 \leq s \leq \delta x} Z_{ij}(s) \geq x \right) .
\end{align*}
By assumption, $ \psi_{ij} $ is finite around zero, so $\psi_{ij}(\theta)<\infty$ for $\theta>0$ small enough. For any such $\theta$,
$$\{\exp [ \theta Z_{ij}(n) ], n \in \mathbb{N}\}$$
is a positive submartingale under ${\mathbb{P}}_{\infty}$, as can be seen by combining Jensen's inequality with the fact that $ {\mathbb{E}}_{\infty}[\ell_{ij}(1)] \geq 0$, and by Doob's submartingale inequality it follows that
\begin{align} \label{first_bound_old}
{\mathbb{P}}_{\infty} \left( \max_{1 \leq s \leq \delta x} Z_{ij}(s) \geq x \right) \leq \exp [- \theta x + \lfloor \delta x \rfloor \psi_{ij}(\theta)].
\end{align}
Since $ {{\mathbb{E}}_{\infty}[\ell_{ij}(1)] \geq 0}$, by the strict convexity of $\psi_{ij}$ it follows that
$\psi_{ij}(\theta) > 0$ for $ \theta > 0$ small enough. Therefore, since also $ \lfloor x \delta \rfloor > 0$, we can rewrite \eqref{first_bound_old} as
\begin{equation}\label{first_bound}
{\mathbb{P}}_{\infty} \left( \max_{1 \leq s \leq \delta x} Z_{ij}(s) \geq x \right) \leq \exp [- ( \theta - \delta \psi_{ij}(\theta)) \, x].
\end{equation}
We continue with the second term in the right-hand side of \eqref{very_first_bound}. By Markov's inequality, for any $\theta > 0$,
\begin{align*}
& {\mathbb{P}}_{\infty}\left( \nu - R_i(\nu) \geq \delta x \right) \\
&\leq e^{-\theta \delta x} \, {\mathbb{E}}_{\infty} \left[ e^{\theta ( \nu - R_i(\nu))} \right] \\
&= e^{-\theta \delta x} \, \sum_{m = 0}^{\infty} \frac{ \theta^m}{m!} \; {\mathbb{E}}_{\infty}[ (\nu - R_i(\nu))^{m}],
\end{align*}
where the equality follows from a Taylor series expansion and the Monotone Convergence Theorem.
Since $ \nu - R_i(\nu)$ is the age at time $\nu$ of a discrete renewal process whose events are the regenerations of $ \lbrace Y_i(n), n \in \mathbb{N} \rbrace$, by Lorden's inequality (see \cite[Theorem 3]{lorden1970excess} and the comments therein),
for every $m \in \mathbb{N}$ and $\nu \in \mathbb{N}$ we have:
$${\mathbb{E}}_{\infty} \left[ (\nu - R_i(\nu))^m \right] \leq \frac{m+2}{m+1} \; \frac{{\mathbb{E}}_{\infty} [\eta_i^{m+1}]}{{\mathbb{E}}_{\infty}[\eta_i]},
$$
where $\eta_i$ is the first regeneration time of the sequence ${ \{Y_{i}(n), n \in \mathbb{N} \} }$, defined in \eqref{def: eta}.
By Lemma \ref{lemma: regeneration bounded by expo} we then conclude that
$$ {\mathbb{P}}_{\infty}(\nu - R_i(\nu) \geq \delta x) \leq \frac{e^{-\theta \delta x} e^{a_i}}{ {\mathbb{E}}_{\infty} [\eta_i] a_i} \, \sum_{m=0}^{\infty} \left(\theta/ a_i \right)^m (m+2), $$
where $a_i$ is defined in \eqref{def: a}, which means that, for every $ \theta < a_i$, there is a constant $K_{\theta}>0$, which does not depend on $x$ or $\delta$, such that
\begin{equation} \label{second_bound}
{\mathbb{P}}_{\infty}(\nu - R_i(\nu) \geq \delta x) \leq K_{\theta} \, e^{-\theta \delta x}.
\end{equation}
Combining \eqref{first_bound} and \eqref{second_bound} we conclude that, for every ${\theta \in (0, a_i)}$ such that $\psi_{ij}(\theta)<\infty$,
$${\mathbb{P}}_{\infty} \left( Y^{\prime}_{ij}(\nu) \geq x \right) \leq e^{-x(\theta - \delta \psi_{ij}(\theta))} + K_{\theta}\, e^{-\theta \delta x}.$$
The above inequality implies the desired result once we select a sufficiently small $\theta \in (0, a_i)$ so that
$\psi_{ij}(\theta)<\infty$ and then a sufficiently small $\delta>0$ so that $\theta - \delta \psi_{ij}(\theta) > 0$, or equivalently $ \delta \in (0, \theta / \psi_{ij}(\theta) )$, recalling that $ \psi_{ij}(\theta) > 0 $ for $ \theta >0$ by strict convexity. \\
\end{IEEEproof}
\section{}\label{app:family_misspecification_bound_theorem}
We present a lemma to help us in the proof of the Theorem.
\begin{lemma}\label{lem:prob_to_expectation}
Let $ T_1, T_2 $ be $ \mathbb{N}$-valued random variables on a probability space with a probability measure $ {\mathbb{P}} $ and finite expectation $ {\mathbb{E}}[T_i] < \infty, i = 1,2.$ If $ {\mathbb{P}}(T_1 > T_2 ) > 0, $ then $$ {\mathbb{P}} \left( T_1 > T_2\right) = \frac{{\mathbb{E}} \left[ T_1 - (T_2 \wedge T_1) \right] }{{\mathbb{E}} \left[ T_1 - T_2 \, | \, T_1 > T_2\right]}. $$
\end{lemma}
\begin{IEEEproof}
It suffices to observe that
\begin{equation}
{\mathbb{E}} \left[ T_1 - (T_2 \wedge T_1) \right] = {\mathbb{E}} \left[ T_1 - (T_2 \wedge T_1)\, | \, T_1 > T_2 \right] \times {\mathbb{P}}(T_1 > T_2)
\end{equation} and also that $$
{\mathbb{E}} \left[ T_1 - (T_2 \wedge T_1)\, | \, T_1 > T_2) \right] = {\mathbb{E}} \left[ T_1 - T_2 \, | \, T_1 > T_2 \right].
$$
\end{IEEEproof}
Now, proceeding with the proof of the Theorem, we follow similar steps as in the proof of \cite[Theorem 2]{nikiforov2000simple}.
\begin{IEEEproof}[Proof of Theorem \ref{th:family_misspecification_bound_theorem}]
Throughout this proof, for simplicity we write $ \tau_i, \tau, \widehat{\tau}$, instead of $\tau_i(b,h), \tau(b,h)$, $\widehat{\tau}(b,h)$, respectively, but it is important to keep this dependence in mind. By the definition of $\tau$ and $\widehat{\tau}$ in \eqref{family},
$$\{\widehat{\tau}=i\} \subseteq
\{\tau_{i} \leq \tau_j \}.
$$
Moreover, since $\tau\leq \tau_i$, on the event $\{\tau>\nu\}$ we have
\begin{align*}
\tau_i &= \inf \lbrace n > \nu : Y_i \geq b, \, W_{ik}(n) \geq h, \; \; \forall \, k \in [K], k \neq i \rbrace \\
&\geq \inf \lbrace n > \nu : W_{ij}(n) \geq h \rbrace\\
&\geq \inf \lbrace n > \nu : W^\nu_{ij}(n) \geq h \rbrace = \tau^\nu_{ij} ,
\end{align*}
where the last inequality holds because, by
the assumption that \eqref{pathwise bound} holds and the definition of $W_{ij}^\nu$ in \eqref{W_nu},
\begin{align} \label{implication}
W_{ij} (n) \leq W^{\nu}_{ij}(n) \quad \text{for all} \quad n \in \mathbb{N},
\end{align}
and the last equality is simply the definition of $\tau^\nu_{ij}$ in \eqref{tau_nu}. Therefore,
\begin{align*}
\{\widehat{\tau}=i, \tau>\nu \}
&\subseteq
\{ \tau_i \leq \tau_j, \tau>\nu \} \subseteq \{ \tau^\nu_{ij} \leq \tau_j, \tau>\nu \},
\end{align*}
and consequently
\begin{align}\label{eqn:1111}
{\mathbb{P}}_{\nu, j}( \widehat{\tau}=i | \; \tau > \nu)
&\leq {\mathbb{P}}_{\nu, j} ( \tau_{ij}^{\nu} \leq \tau_j |\; \tau > \nu).
\end{align}
Using assumption \eqref{condition technical} of Theorem \ref{th:family_misspecification_bound_theorem} and applying
Lemma \ref{lem:prob_to_expectation} with ${\mathbb{P}}\equiv {\mathbb{P}}_{\nu, j}(\cdot \, | \, \tau > \nu)$,
we have
\begin{align}\label{eqn:1}
\begin{split}
& {\mathbb{P}}_{\nu, j} ( \tau_{ij}^{\nu} \leq \tau_j \, |\, \tau > \nu)
= 1- {\mathbb{P}}_{\nu, j} ( \tau_{ij}^{\nu} > \tau_j \,| \, \tau > \nu)\\
&=1- \frac{{\mathbb{E}}_{\nu, j}[\tau^{\nu}_{ij} - \tau_j \wedge \tau^\nu_{ij} \, | \, \tau > \nu]}{{\mathbb{E}}_{\nu, j}[\tau^{\nu}_{ij} - \tau_j \, |\, \tau > \nu, \tau^{\nu}_{ij} > \tau_j]}.
\end{split}
\end{align}
We proceed by further lower bounding the denominator.
Since $ \tau \leq \tau_j, $ on the event $\lbrace \tau > \nu, \tau^{\nu}_{ij} > \tau_j \rbrace$ we have
\begin{itemize}
\item $\nu < \tau \leq \tau_j < \tau_{ij}^{\nu},$
\item $\tau_{ij}^{\nu} - \tau_j $ is a function of $W_{ij}^{\nu}(\tau_j)$ and $
\{X_{n}, n > \tau_j \}$
\end{itemize}
and, as a result,
\begin{equation}
{\mathbb{E}}_{\nu, j}[ \tau_{ij}^{\nu} - \tau_j \, |\, \tau > \nu, \tau_{ij}^{\nu} > \nu, \mathcal{F}_{\tau_j}] = U_{ij}(W_{ij}^{\nu}(\tau_j) ; h) \cdot \mathds{1} ( \{\tau > \nu, \tau_{ij}^{\nu} > \nu\} )
\end{equation}
where $U_{ij}$ is defined as in \eqref{U}.
Since $ W_{ij}^{\nu}(\tau_j) < h $ on $ \lbrace \tau_{ij}^{\nu} > \tau_j > \nu \rbrace$ and the function $U_{ij}(\cdot\, ; h) $ is non-increasing, we further obtain
\begin{align*}
{\mathbb{E}}_{\nu, j}[ \tau_{ij}^{\nu} - \tau_j \, |\, \tau > \nu, \tau_{ij}^{\nu} > \nu, \mathcal{F}_{\tau_j}]
&\leq U_{ij}(0; h)= {\mathbb{E}}_j[\sigma_{ij}],
\end{align*}
and, by the law of iterated expectation, we conclude that:
\begin{equation}\label{eqn:3}
{\mathbb{E}}_{\nu, j}[ \tau_{ij}^{\nu} - \tau_j \, | \, \tau > \nu, \tau_{ij}^{\nu} > \nu] \leq {\mathbb{E}}_j[\sigma_{ij}].
\end{equation}
Combining \eqref{eqn:1111}-\eqref{eqn:3}, we obtain
\begin{align*}
{\mathbb{P}}_{\nu, j}&(\widehat{\tau} = i | \tau > \nu) \leq 1 - \frac{{\mathbb{E}}_{\nu, j}[\tau^{\nu}_{ij} - \tau_j \wedge \tau^\nu_{ij} | \; \tau > \nu]}{{\mathbb{E}}_{j}[\sigma_{ij}]} \\
&\leq \frac{{\mathbb{E}}_{j}[\sigma_{ij}] - {\mathbb{E}}_{\nu, j}[\tau^{\nu}_{ij} - \nu | \; \tau > \nu]}{{\mathbb{E}}_{j}[\sigma_{ij}]} + \frac{{\mathbb{E}}_{\nu, j}[ \tau_j - \nu | \; \tau > \nu]}{{\mathbb{E}}_{j}[\sigma_{ij} ]},
\end{align*}
and applying \eqref{ARL_CUSUM_LB} we arrive at
\begin{align*}
{\mathbb{P}}_{\nu, j}(\widehat{\tau} = i | \tau > \nu)
&\leq \frac{{\mathbb{E}}_{j}[\sigma_{ij}] - {\mathbb{E}}_{\nu, j}[\tau^{\nu}_{ij} - \nu | \; \tau > \nu]}{{\mathbb{E}}_{j}[\sigma_{ij}]} \\
&+ e^{-h} \, {\mathbb{E}}_{\nu, j}[ \tau_j - \nu | \; \tau > \nu].
\end{align*}
Comparing with \eqref{show}, it is clear that it suffices to show that
$$ \frac{{\mathbb{E}}_{j}[\sigma_{ij}] - {\mathbb{E}}_{\nu, j}[\tau^{\nu}_{ij} - \nu | \; \tau > \nu]}{{\mathbb{E}}_{j}[\sigma_{ij}]} \leq \frac{C_{ij}}{1 - c_{ij}}e^{-c_{ij}h} \left( 1 + \phi_{ij}(h) \right),
$$
where $\phi_{ij}$ is a function that goes to $0$ as $h \to \infty$.
In order to do so, we add and subtract $u_{ij}(0;h)$ (defined in \eqref{def:u}) on the left-hand side,
\begin{align}\label{eqn:star1}
\begin{split}
{\mathbb{E}}_j[\sigma_{ij}] - &{\mathbb{E}}_{\nu, j}[\tau^{\nu}_{ij} - \nu| \tau > \nu] \\
= &\; U_{ij}(0;h) - u_{ij}(0;h) + u_{ij}(0;h) - {\mathbb{E}}_{\nu, j}[\tau^{\nu}_{ij} - \nu\,| \, \tau > \nu].
\end{split}
\end{align}
We also obtain from \eqref{def:u} that
\begin{align} \label{eqn:star2}
U_{ij}(0;h) - u_{ij}(0;h) = e^{-h} \left( \frac{h + \omega_{ij}}{I_{ji}} + U_{ij}(0;h) \right).
\end{align}
Moreover, since $ \tau^{\nu}_{ij} - \nu$ is a function of only $W_{ij}(\nu) $ and $ \lbrace X_n, n > \nu \rbrace, $
\begin{align*}
{\mathbb{E}}_{\nu, j} \left[ \tau^{\nu}_{ij} - \nu | \mathcal{F}_{\nu} , \tau>\nu \right] = U_{ij}(W_{ij}(\nu) ; h) \cdot \mathds{1} ( \lbrace \tau > \nu \rbrace )
\end{align*}
and, by the law of iterated expectation,
\begin{align*}
{\mathbb{E}}_{\nu, j}[\tau^{\nu}_{ij} - \nu \, | \, \tau > \nu ]
& = {\mathbb{E}}_{\nu, j}[ U_{ij}(W_{ij}(\nu) ; h) \, |\, \tau > \nu].
\end{align*}
We conclude that
\begin{align*}
& {\mathbb{E}}_{\nu, j}[ U_{ij}(W_{ij}(\nu) ; h) \, |\, \tau > \nu] \\
&\geq {\mathbb{E}}_{\nu, j}[U_{ij}(W_{ij}(\nu) \wedge h ; h) \cdot \mathds{1} ( \lbrace W_{ij}(\nu) \leq h \rbrace )| \tau > \nu] \\
&\geq {\mathbb{E}}_{\infty}[u_{ij}(W_{ij}(\nu) \wedge h ; h) | \tau > \nu],
\end{align*}
where the first inequality holds because
$U(\cdot \, ; h)$ is non-negative and the last one, when $ h \geq 1$, by
\eqref{inequality}. Putting these together, we obtain
\begin{align*}
\begin{split}
& u_{ij}(0;h) - {\mathbb{E}}_{\nu, j}[\tau^{\nu}_{ij} - \nu| \tau > \nu] \\
&\leq {\mathbb{E}}_{\infty}[u_{ij}(0;h) - u_{ij}(W_{ij}(\nu) \wedge h ; h) \, | \, \tau > \nu].
\end{split}
\end{align*}
Since the function $u_{ij}(\cdot \, ;h)$ is differentiable in $[0,h]$, by the Fundamental Theorem of Calculus and inequality \eqref{eqn:upper_bound_on_uprime} we obtain
\begin{align*}
u_{ij}(0;h) - u_{ij}(W_{ij}(\nu) \wedge h ; h) &= - \int_0^{h} u'_{ij}(x ; h) \cdot \mathds{1} ( \lbrace W_{ij}(\nu) \geq x \rbrace ) \; dx \\
&\leq \left( \frac{h + \omega_{ij}}{I_{ji}} + U_{ij}(0;h) \right) e^{-h} \; \int_0^{h} e^{x} \; \mathds{1} ( \lbrace W_{ij}(\nu) \geq x \rbrace ) \; dx.
\end{align*}
By Tonelli's theorem and assumption \eqref{condition on the change-point} we further obtain
\begin{align*}
{\mathbb{E}}_{\infty}\left[\int_0^{h} e^{x} \; \mathds{1}( \lbrace W_{ij}(\nu) \geq x \rbrace ) \; dx \, | \, \tau > \nu \right] &\leq \int_0^{h} e^{x} \, {\mathbb{P}}_{\infty}(W_{ij}(\nu) \geq x \, \big| \, \tau > \nu) \, dx \\
&\leq \int_0^{h} \ e^{x} \; C_{ij} e^{-c_{ij}x} \; dx \leq \frac{C_{ij}}{1 - c_{ij}} \, e^{(1 - c_{ij})h}
\end{align*}
and, consequently,
\begin{equation}\label{eqn:star3}
u_{ij}(0;h) - {\mathbb{E}}_{\nu, j}[\tau^{\nu}_{ij} - \nu| \tau > \nu] \leq \frac{C_{ij}}{1 - c_{ij}} \, e^{-c_{ij}h} \, \left( U_{ij}(0;h) + \frac{h + \omega_{ij}}{I_{ji}} \right).
\end{equation}
By \eqref{eqn:star1}-\eqref{eqn:star3} we obtain
\begin{align}\label{eqn:star4}
\begin{split}
&{\mathbb{E}}_j[\sigma_{ij}] - {\mathbb{E}}_{\nu, j}[\tau^{\nu}_{ij} - \nu| \tau > \nu] \\
& \quad \leq e^{-h} \left( \frac{h + \omega_{ij}}{I_{ji}} + U_{ij}(0;h) \right) + \frac{C_{ij}}{1 - c_{ij}} \, e^{-c_{ij}h} \, \left( U_{ij}(0;h) + \frac{h + \omega_{ij}}{I_{ji}} \right).
\end{split}
\end{align}
Dividing both sides by $ {\mathbb{E}}_{j}[\sigma_{ij}]$, which is equal by definition to $ U_{ij}(0;h)$, and then applying inequality \eqref{ARL_CUSUM_LB} completes the proof. \\
\end{IEEEproof}
\begin{IEEEproof}[Proof of Corollary \ref{coro: h_choice}]
By Theorem \ref{thm:delay_bound} it is clear that, for any given $\alpha \in (0,1)$, the upper bound in \eqref{show} goes to 0 as $h \to \infty$. Therefore, for any given
$\alpha, \beta \in (0,1)$, there is an $h_{\alpha, \beta}$ so that the worst-case probability of false isolation does not exceed $\beta$. Therefore, to prove the corollary it is enough to show that, for all $ \nu \in N_{\boldsymbol{C}}$,
$$ {\mathbb{P}}_{\nu, j}(\tau_{ij}^{\nu}(h) > \tau_j(b,h) \,|\, \tau(b,h) > \nu) > 0,$$
at least for $h$ large enough and $ b = b_{\alpha}$,
where $ \tau_{ij}^{\nu}$ is defined in \eqref{tau_nu}. In the rest of the proof, we fix $\nu \in N_{\boldsymbol{C}}$, and for simplicity we suppress $b$ and $h$ in the notation, except where necessary.
First of all, we observe that
\begin{align} \label{dec}
\begin{split}
{\mathbb{P}}_{\nu, j} ( \tau_{ij}^{\nu} > \tau_j \,| \, \tau > \nu) &= {\mathbb{P}}_{\nu, j} ( \tau_{ij}^{\nu} > \tau_j \, | \, Y'_{ij}(\nu) \leq h/2, \tau > \nu ) \\
&\cdot {\mathbb{P}}_{\nu, j} ( Y'_{ij}(\nu) \leq h/2 \, | \, \tau > \nu).
\end{split}
\end{align}
Since $\nu \in N_{\boldsymbol{C}}$, by \eqref{condition on the change-point} it follows that
\begin{equation*}
{\mathbb{P}}_{\infty} (Y'_{ij}(\nu) \leq h/2 | \tau > \nu) \geq 1 -C_{ij}e^{-c_{ij}h/2},
\end{equation*}
where the lower bound is positive for $h > 2 \log (C_{ij}) / c_{ij}$. Therefore, it remains to show that the first factor in \eqref{dec} is positive at least for large $h$. To this end, we first show that it is bounded below by the probability that
$\sigma_{ij}$ is larger than $\tau_j$ when its statistic $Y_{ij}$ is initialized from $h/2$, i.e.,
\begin{align} \label{dec2}
{\mathbb{P}}_{\nu, j} ( \tau_{ij}^{\nu} > \tau_j \, | \, Y'_{ij}(\nu) \leq h/2, \tau > \nu ) &\geq {\mathbb{P}}_{j} \big( \sigma_{ij} > \tau_j \, | \, Y_{ij}(0) = h/2 \big).\end{align}
Indeed, by the law of iterated expectation,
\begin{align}\label{eqn:total_exp}
\begin{split}
&{\mathbb{P}}_{\nu, j} ( \tau_{ij}^{\nu} > \tau_j \, | \, Y'_{ij}(\nu) \leq h / 2, \tau > \nu ) = \\
&{\mathbb{E}}_{\nu,j} \Big[ \zeta(Y'_{ij}(\nu), Y_j(\nu), Y'_{j1}(\nu), \ldots, Y'_{jK}(\nu) \big) \Big| \tau > \nu, Y'_{ij}(\nu) \leq h / 2 \Big],
\end{split}
\end{align}
where
\begin{align*}
&\zeta(y'_{ij}, y_j, y'_{j1}, \ldots, y'_{jK}) \\
&\equiv
{\mathbb{P}}_{\nu, j} \big( \tau_{ij}^{\nu} > \tau_j \, | \, Y'_{ij}(\nu)=y'_{ij}, Y_j(\nu)= y_j, Y'_{jk}(\nu)=y'_{jk}, 1 \leq k \leq K \big).
\end{align*}
The stopping time $\tau_{ij}^{\nu}$ depends on~$\mathcal{F}_\nu$ only through $Y'_{ij}(\nu)$, whereas $\tau_j$ depends on $\mathcal{F}_\nu$ only through
$$Y_j(\nu), Y'_{j1}(\nu), \ldots, Y'_{jK}(\nu),$$
and $\zeta$ is increasing in its first argument and decreasing in each of the other arguments. Moreover, $\tau^\nu_{ij}-\nu$ has the same distribution as $\sigma_{ij}$ when the latter is initialized from $Y'_{ij}(\nu)$. As a result, we conclude that, for all values of $(y'_{ij}, y_j, y'_{j1}, \ldots, y'_{jK})$ in $\lbrace \tau > \nu, Y'_{ij}(\nu) \leq h / 2 \rbrace$,
\begin{align*}
\zeta(y'_{ij}, y_j, y'_{j1}, \ldots, y'_{jK}) &\geq
\zeta(h/2,0, \ldots, 0) = {\mathbb{P}}_{j} \big( \sigma_{ij} > \tau_j \, | \, Y_{ij}(0) = h/2 \big),
\end{align*}
which proves \eqref{dec2}.
It remains to show that lower bound in \eqref{dec2} is positive. For this, it suffices to show that
\begin{equation}\label{dec3}
{\mathbb{P}}_{j} \big( \sigma_{ij}(h) > \tau_j(b,h) \, | \, Y'_{ij}(0) = h/2 \big) \geq {\mathbb{P}}_j \big( \sigma_{ij}(h/2) > \tau_j(b,h) \big)
\end{equation}
since the latter is greater than zero.
Indeed, by \eqref{ARL_CUSUM_LB} and Theorem \ref{thm:delay_bound} it follows that ${\mathbb{E}}_j[\sigma_{ij}(h/2)]$ has an exponential lower bound in~$h$, and $ {\mathbb{E}}_j[\tau_j(b_{\alpha}, h)]$ a linear upper bound in~$b_{\alpha}$ and~$h$ under~${\mathbb{P}}_j$, which implies that the lower bound is positive, at least for~$h$ large enough.
To prove \eqref{dec3} it suffices to show that $ \sigma_{ij}(h)$ initialized from $Y_{ij}(0) = h/2$ is stochastically larger than $ \sigma_{ij}(h/2) $ initialized from $ Y_{ij}(0) = 0.$ Indeed, the first one corresponds to the first time $Y_{ij}$ crosses the threshold~$h$ when initialized from $h/2$, whereas the second to the first time $Y_{ij}$ crosses the threshold~$h/2$ when initialized from $0$. In both cases, $Y_{ij}$ needs to increase by $h/2$, but in the first case it can fall below its initialization point, whereas in the second it is reflected at its initialization point and, as a result, it is pathwise closer to the threshold than in the first case.
\end{IEEEproof}
\begin{IEEEproof}[Proof of Theorem \ref{th:optimality}]
By Theorem \ref{thm:delay_bound} it follows that for every $\delta>0$ there is constant $C_\delta>0$ so that
\begin{align} \label{delay under 00}
\mathcal{J}_i \left[ \tau_i(b,h) \right] \leq (1+\delta) \left( \max \lbrace b / I_{i}, h / I^*_i \rbrace + C_\delta \right).
\end{align}
By this and Theorem \ref{th:family_misspecification_bound_theorem}, there are constants $ {C > 0}, {c \in (0,1)}, $ and a function $ \phi(h)$ that goes to zero as $h $ goes to infinity so that
\begin{align}\label{eqn:exact_upper_bound_thm8}
\begin{split}
& \max_{j \in [K]} \sup_{\nu \in N_{\boldsymbol{C}}} {\mathbb{P}}_{\nu, j}(\widehat{\tau}(b,h) \neq j | \tau(b,h) > \nu) \\
&\leq C \left( e^{-c h} (1 + \phi(h) ) + e^{-h} (\max\{ b_{\alpha}, h \} + 1) \right)
\end{split}
\end{align}
Therefore, if we set
$$
h_\beta \equiv \left( |\log \beta|+\log C \right)/c, \quad \beta \in (0,1),
$$
then by \eqref{eqn:exact_upper_bound_thm8} it follows that
$$ \max_{j \in [K]} \sup_{\nu \in N_{\boldsymbol{C}}} {\mathbb{P}}_{\nu, j}(\widehat{\tau}(b_\alpha, h_\beta) \neq j | \tau(b,h) > \nu) \lesssim \beta
$$
as $\alpha, \beta \to 0$ so that $| \log \beta | \gg \log |\log \alpha|$
and by \eqref{delay under 00} we conclude that
$$ \mathcal{J}_i[\tau(b_{\alpha}, h_{\beta})] \lesssim
\frac{|\log \alpha|}{I_i}$$
as $\alpha, \beta \to 0$ so that $ \log |\log \beta| \ll
\log |\log \alpha|$, which completes the proof.
\end{IEEEproof}
\newpage
\section{}
Illustration of computing threshold regions:
\begin{figure}[h!]
\begin{center}
\subfloat[Proposed]{
\includegraphics[scale=1]{A1_300dpi.eps}
\label{fig:proposed_threshold_regions_1}
}
\subfloat[Matrix CuSum]{
\includegraphics[scale=1]{M1_300dpi.eps}
\label{fig:matrix_cusum_threshold_regions_1}
}
\captionsetup{justification=raggedright, singlelinecheck=false}
\caption{Computing the region $ \mathcal{S}(1 \%, 1.3)$}
\label{fig:threshold_regions1}
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\subfloat[Proposed]{
\includegraphics{A2_300dpi.eps}
\label{fig:proposed_threshold_regions_2}
}
\subfloat[Matrix CuSum]{
\includegraphics{M2_300dpi.eps}
\label{fig:matrix_cusum_threshold_regions_2}
}
\captionsetup{justification=raggedright, singlelinecheck=false}
\caption{Computing the region $ \mathcal{S}(1 \%, 2)$}
\label{fig:threshold_regions2}
\end{center}
\end{figure}
\end{appendices}
\newpage
\bibliographystyle{IEEEtranN}
|
3,212,635,537,442 | arxiv | \section{Introduction}
Fine-grained opinion mining is an important field in natural language processing (NLP). It comprises various tasks, such as aspect term extraction (ATE) \cite{DBLP:conf/emnlp/LiuXZ12,DECNN,Li2018,MaLWXW19}, opinion term extraction (OTE) \cite{DBLP:conf/naacl/FanWDHC19,Wu2020}, and aspect-level sentiment classification (ASC) \cite{DBLP:conf/ijcai/MaLZW17,DBLP:conf/emnlp/SunZMML19}.
Existing studies generally solve these tasks individually or couple two of them as aspect and opinion terms co-extraction task \cite{DBLP:journals/tkde/LiuXZ15,Wang2016,DBLP:conf/aaai/WangPDX17,DBLP:conf/acl/DaiS19}, aspect term-polarity co-extraction task \cite{LuoLLZ19,LiBLL19}, and aspect-opinion pair extraction task \cite{SDRN,SpanMIT}. However, none of these studies can identify aspects, opinion expressions, and sentiments in a complete solution.
To deal with this problem, the latest literature \cite{DBLP:conf/aaai/PengXBHLS20} presents aspect sentiment triplet extraction (ASTE) task, which aims to identify triplets such as (\textit{food, delicious, positive}) in Figure \ref{intro}.
\begin{figure}
\centering
\includegraphics[width=0.44\textwidth]{pic/intro.eps}
\caption{An example of ASTE task. The aspects, opinion expressions, and sentiments are marked with red, blue, and green, respectively.}\label{intro}
\end{figure}
Although these studies have achieved great progress, there are still several challenges existing in fine-grained opinion mining.
\textbf{First}, aspects and opinion expressions generally appear together in a review sentence and have explicit corresponding relations. Hence, how to adequately learn the association between ATE and OTE and make them mutually beneficial is a challenge.
\textbf{Second}, the corresponding relations between aspects and opinion expressions can be complicated, such as one-to-many, many-to-one, and even overlapped and embedded. Thus, it is challenging to flexibly and exactly detect these relations.
\textbf{Third}, each review sentence may contain multiple sentiments. For example, given the review in Figure \ref{intro}, the sentiments of \textit{price} and \textit{food} are negative and positive, respectively.
These sentiments are generally guided by the corresponding relations between aspects and opinion expressions.
Thus, how to properly introduce these relations to sentiment classification task is another challenge.
To address the aforementioned challenges, we deal with ASTE task and formalize it as a machine reading comprehension (MRC) task.
Given a query and a context, MRC task aims to capture the interaction between them and extract specific information from the context as the answer.
Different from the general MRC task, we further devise multi-turn queries to identify aspect sentiment triplets due to the complexity of ASTE. Specially, we define this formalization as multi-turn machine reading comprehension (MTMRC) task.
By introducing the answers to the previous turns into the current turn as prior knowledge, the associations among different subtasks can be effectively learned.
For example, given the review in Figure \ref{intro}, we can identify the aspect \textit{food} in the first turn and introduce it into the second turn query \textit{What opinions given the aspect food?} to jointly identify the opinion expression \textit{delicious} and the relation between \textit{food} and \textit{delicious}. Then, we can use the aspect \textit{food} and the opinion expression \textit{delicious} as the prior knowledge of the third turn query to predict that the sentiment of \textit{food} is \textit{positive}. According to these turns, we can flexibly capture the association between ATE and OTE, detect complex relations between opinion entities\footnote{In this paper, we briefly note aspects and opinion expressions as opinion entities.}, and utilize these relations to guide sentiment classification.
Based on MTMRC, we propose a bidirectional machine reading comprehension (BMRC) framework\footnote{https://github.com/NKU-IIPLab/BMRC.} in this paper.
Specifically, we design three-turn queries to identify aspect sentiment triplets.
In the first turn, we design \textit{non-restrictive extraction} queries to locate the first entity of each aspect-opinion pair.
Then, \textit{restrictive extraction} queries are designed for the second turn to recognize the other entity of each pair based on the previously extracted entity.
In the third turn, \textit{sentiment classification} queries are proposed to predict aspect-oriented sentiments based on the extracted aspects and their corresponding opinion expressions.
Since there is no intrinsic order when extracting aspects and opinion expressions, we further propose a bidirectional structure to recognize the aspect-opinion pairs. In one direction, we first utilize a non-restrictive extraction query to identify aspects such as $\left\{food, price\right\}$ in Figure \ref{model}. Then, given the specific aspect like \textit{food}, the second-turn query looks for its corresponding opinion expressions such as $\left\{delicious\right\}$ in Figure \ref{model} via a restrictive extraction query. Similarly, the other direction extracts opinion expressions and their corresponding aspects in a reversed order.
To verify the effectiveness of BMRC, we make comprehensive analyses on four benchmark datasets. The experimental results show that our approach substantially outperforms the existing methods. In summary, our contributions are three-fold:
\begin{itemize}
\item We formalize aspect sentiment triplet extraction (ASTE) task as a multi-turn machine reading comprehension (MTMRC) task. Based on this formalization, we can gracefully identify aspect sentiment triplets in a unified framework.
\item We propose a bidirectional machine reading comprehension (BMRC) framework. By devising three-turn queries, our model can effectively build the associations among opinion entity extraction, relation detection, and sentiment classification.
\item We conduct extensive experiments on four benchmark datasets. The experimental results demonstrate that our model achieves state-of-the-art performances.
\end{itemize}
\section{Related Work}
In this paper, we transform the aspect sentiment triplet extraction task into a multi-turn machine reading comprehension task. Thus, we introduce the related work from two parts, including fine-grained opinion mining and machine reading comprehension.
\subsection{Fine-grained Opinion Mining}
Fine-grained opinion mining consists of various tasks, including aspect term extraction (ATE) \cite{Wang2016, DBLP:conf/acl/HeLND17, Li2018, DECNN, DBLP:conf/emnlp/LiL17}, opinion term extraction (OTE) \cite{liu2015fine, poria2016aspect, DECNN, Wu2020}, aspect-level sentiment classification (ASC) \cite{DBLP:conf/acl/DongWTTZX14, DBLP:conf/emnlp/TangQL16, DBLP:conf/aaai/LiW0Z019, DBLP:conf/acl/HeLND18, DBLP:conf/naacl/HazarikaPVKCZ18, DBLP:conf/emnlp/NguyenS15, DBLP:conf/acl/LiuCWMZ18}, etc. The studies solve these tasks individually and ignore the dependency between them.
To explore the interactions between different tasks, recent studies gradually focus on the joint tasks such as aspect term-polarity co-extraction \cite{he2019interactive, DBLP:conf/emnlp/MitchellAWD13, DBLP:conf/aaai/LiL17, LiBLL19}, aspect and opinion terms co-extraction \cite{DBLP:journals/tkde/LiuXZ15,Wang2016, DBLP:conf/aaai/WangPDX17,DBLP:conf/acl/DaiS19}, aspect category and sentiment classification \cite{DBLP:conf/emnlp/HuZZCSCS19}, and aspect-opinion pair extraction \cite{SDRN,SpanMIT}. Besides, there are also a lot of studies \cite{RACL2020,he2019interactive} solving multiple tasks with a multi-task learning network. However, none of these studies could identify aspects, opinion expressions and sentiments in a unified framework.
To deal with this issue, Peng et al. \shortcite{DBLP:conf/aaai/PengXBHLS20} proposed a two-stage framework to solve aspect sentiment triplet extraction (ASTE) task, which aims to extract triplets of aspects, opinion expressions and sentiments. However, the model suffers from error propagation due to its two-stage framework. Besides, separating the extraction and pairing of opinion entities means that the associations between different tasks are still not adequately considered.
\subsection{Machine Reading Comprehension}
Machine reading comprehension (MRC) aims to answer specific queries based on a given context. Recent researches have proposed various effective architectures for MRC, which adequately learn the interaction between the query and context. For example, BiDAF \cite{DBLP:conf/iclr/SeoKFH17} employs a RNN-based sequential framework to encode queries and passages, while QANet \cite{DBLP:conf/iclr/YuDLZ00L18} employs both convolution and self-attention. Several MRC systems \cite{DBLP:conf/naacl/PetersNIGCLZ18, radford2019language} adopt context-aware embedding as well and obtain comparable results, especially BERT-based MRC model \cite{DBLP:conf/naacl/DevlinCLT19}.
Recently, there is a tendency to apply MRC on many NLP tasks, including named entity recognition \cite{li2019unified}, entity relation extraction \cite{DBLP:conf/acl/LiYSLYCZL19,DBLP:conf/conll/LevySCZ17}, and summarization \cite{DBLP:journals/corr/abs-1806-08730}, etc.
Due to the advantages of MRC framework, we naturally transform ASTE into a multi-turn MRC task to better construct the associations among aspects, opinions, aspect-opinion relations and sentiments through well-designed queries.
Different from the existing methods \cite{li2019unified,DBLP:conf/acl/LiYSLYCZL19}, we innovatively propose a bidirectional framework, which can identity triplets more comprehensively by making the two directions complement each other. This framework can be further extended to other tasks such as entity relation extraction.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{pic/model.eps}
\caption{The bidirectional machine reading comprehension (BMRC) framework.}\label{model}
\end{figure}
\section{Problem Formulation}
Given a review sentence $X=\left\{x_{1}, x_{2},..., x_{N}\right\}$ with $N$ tokens, ASTE task aims to identify the collection of triplets $T=\left\{\left(a_{i}, o_{i}, s_{i}\right)\right\}_{i=1}^{\left | T\right |}$, where $a_{i}$, $o_{i}$, $s_{i}$, and $\left | T\right |$ represent the aspect, the opinion expression, the sentiment, and the number of triplets\footnote{The $\left |*\right |$ represents the number of elements in the collection $*$.}, respectively.
To formalize ASTE task as a multi-turn MRC task, we construct three types of queries\footnote{We use superscripts $\mathcal{N}$, $\mathcal{R}$, and $\mathcal{S}$ to denote the query types.}, including non-restrictive extraction queries $Q^{\mathcal{N}}=\left\{q_{i}^{\mathcal{N}}\right\}_{i=1}^{\left | Q^{\mathcal{N}}\right |}$, restrictive extraction queries $Q^{\mathcal{R}}=\left\{q_{i}^{\mathcal{R}}\right\}_{i=1}^{\left | Q^{\mathcal{R}}\right |}$ and sentiment classification queries $Q^{\mathcal{S}}=\left\{q_{i}^{\mathcal{S}}\right\}_{i=1}^{\left | Q^{\mathcal{S}}\right |}$.
Concretely, in the first turn, each non-restrictive extraction query $q_{i}^{\mathcal{N}}$ aims to extract either aspects $A=\left\{a_{i}\right\}_{i=1}^{\left | A\right |}$ or opinion expressions $O=\left\{o_{i}\right\}_{i=1}^{\left | O\right |}$ from the review sentence to trigger aspect-opinion pairs.
In the second turn, given the opinion entities recognized by $q_{i}^{\mathcal{N}}$, each restrictive extraction query $q_{i}^{\mathcal{R}}$ aims to identify either the corresponding aspects or the corresponding opinion expressions. To be more specific, given each aspect $a_{i}$ extracted by $q_{i}^{\mathcal{N}}$, the restrictive extraction query extracts its corresponding opinion expressions $O_{a_{i}}=\left\{o_{a_{i},j}\right\}_{j=1}^{\left | O_{a_{i}}\right |}$.
In the final turn, each sentiment classification query $q_{i}^{\mathcal{S}}$ predicts the sentiment $s_{a_{i}}\in \left\{\text{Positive}, \text{Negative}, \text{Neutral}\right\}$ for each aspect $a_{i}$.
\section{Methodology}
\subsection{Framework}
To deal with ASTE task, we propose a bidirectional machine reading comprehension (BMRC) framework. The overall framework is illustrated in Figure \ref{model}. Concretely, we first design non-restrictive extraction queries and restrictive extraction queries to extract aspect-opinion pairs. Considering that each pair can be triggered by an aspect or an opinion expression, we further construct a bidirectional structure.
In one direction, the aspects are first extracted via non-restrictive extraction queries, and then the corresponding opinion expressions for each aspect are identified via restrictive extraction queries. We define the above process as A$\rightarrow$O direction. Similarly, in O$\rightarrow$A direction, the framework recognizes opinion expressions and their corresponding aspects in a reversed order.
After that, we design sentiment classification queries to predict the sentiment polarity for each aspect. Furthermore, our model jointly learns to answer the above queries to make them mutually beneficial.
Besides, we adopt BERT as the encoding layer for richer semantics representations. During inference, the model fuses the answers to different queries and forms the triplets.
\subsection{Query Construction}
In BMRC, we adopt a template-based approach to construct queries.
Specifically, we first design the non-restrictive extraction query and the restrictive extraction query in the A$\rightarrow$O direction as follows:
\begin{itemize}
\item \textbf{A$\rightarrow$O non-restrictive extraction query} $q_{A\rightarrow O}^{\mathcal{N}}$: We design query `\textit{What aspects?}' to extract the collection of aspects $A=\left\{a_{i}\right\}_{i=1}^{\left | A\right |}$ from the given review sentence $X$.
\item \textbf{A$\rightarrow$O restrictive extraction query} $q_{A\rightarrow O}^{\mathcal{R}}$: We design query `\textit{What opinions given the aspect $a_{i}$?}' to extract the corresponding opinions $O_{a_{i}}=\left\{o_{a_{i},j}\right\}_{j=1}^{\left | O_{a_{i}}\right |}$ for each aspect $a_{i}$ and form aspect-opinion pairs.
\end{itemize}
Reversely, the O$\rightarrow$A direction extraction queries are constructed as follows:
\begin{itemize}
\item \textbf{O$\rightarrow$A non-restrictive extraction query} $q_{O\rightarrow A}^{\mathcal{N}}$: We use query `\textit{What opinions?}' to extract the collection of opinion expressions $O=\left\{o_{i}\right\}_{i=1}^{\left | O\right |}$.
\item \textbf{O$\rightarrow$A restrictive extraction query} $q_{O\rightarrow A}^{\mathcal{R}}$: We design query `\textit{What aspect does the opinion $o_{i}$ describe?}' to recognize the corresponding aspects $A_{o_{i}}=\left\{a_{o_{i},j}\right\}_{j=1}^{\left | A_{o_{i}}\right |}$ for each opinion expression $o_{i}$.
\end{itemize}
With the above queries, opinion entity extraction and relation detection are naturally fused, and the dependency between them is gracefully learned via the restrictive extraction queries.
Then, we devise sentiment classification queries to classify the aspect-oriented sentiments as follows:
\begin{itemize}
\item \textbf{Sentiment Classification query} $q^{\mathcal{S}}$: We design query `\textit{What sentiment given the aspect $a_{i}$ and the opinion $o_{a_{i},1}/.../o_{a_{i},\left | O_{a_{i}}\right |}$?}' to predict sentiment polarity $s_{a_{i}}$ for each aspect $a_{i}$.
\end{itemize}
With sentiment classification queries, the semantics of aspects and their corresponding opinion expressions can be adequately considered during sentiment prediction.
\subsection{Encoding Layer}
Given the review sentence $X=\left\{x_{1}, x_{2},..., x_{N}\right\}$ with $N$ tokens and each query $q_{i}=\{q_{i,1},q_{i,2},...,q_{i,\left|q_{i}\right|}\}$ with $\left|q_{i}\right|$ tokens, the encoding layer learns the context representation for each token. Inspired by the successful practice on many NLP tasks, we adopt BERT as the encoder.
Formally, we first concatenate the query $q_{i}$ and the review sentence $X$ to obtain the combined input $I=\{[{\rm CLS}],q_{i,1},q_{i,2},...,q_{i,\left|q_{i}\right|},[{\rm SEP}],x_{1}, x_{2},..., x_{N}\}$, where $[{\rm CLS}]$ and $[{\rm SEP}]$ are the beginning token and the segment token. The initial representation $\mathbf{e}_{i}$ for each token is constructed by summing its word embedding $\mathbf{e}_{i}^{w}$, position embedding $\mathbf{e}_{i}^{p}$, and segment embedding $\mathbf{e}_{i}^{g}$. Then, BERT is used to encode the initial representation sequence $E=\left\{\mathbf{e}_{1}, \mathbf{e}_{2},...,\mathbf{e}_{\left|q_{i}\right|+N+2}\right\}$ as the hidden representation sequence $H=\left\{\mathbf{h}_{1}, \mathbf{h}_{2},...,\mathbf{h}_{\left|q_{i}\right|+N+2}\right\}$ with the stacked Transformer blocks.
\subsection{Answer Prediction}
\subsubsection{Answer for Extraction Query}
For non-restrictive and restrictive extraction queries, the answers could be multiple opinion entities extracted from the review sentence $X$. For example, given the review in Figure \ref{model}, the aspects \textit{food} and \textit{price} should be extracted as the answer to the A$\rightarrow$O non-restrictive extraction query $q_{A\rightarrow O, 1}^{\mathcal{N}}$. Thus, we utilize two binary classifiers to predict the answer spans. Specifically, based on the hidden representation sequence $H$, one classifier predicts whether each token $x_{i}$ is the start position of the answer or not, and another predicts the possibility that each token is the end position:
\begin{equation}
p\left(y_{i}^{start}|x_{i}, q\right) = {\rm softmax}(\mathbf{h}_{\left|q\right|+2+i}W_{s}),
\end{equation}
\begin{equation}
p\left(y_{i}^{end}|x_{i}, q\right) = {\rm softmax}(\mathbf{h}_{\left|q\right|+2+i}W_{e}),
\end{equation}
where $W_{s}\in \mathbf{R}^{d_{h}\times 2}$ and $W_{e}\in \mathbf{R}^{d_{h}\times 2}$ are model parameters, $d_{h}$ denotes the dimension of hidden representations in $H$, and $\left|q\right|$ is the query length.
\subsubsection{Answer for Sentiment Classification Query}
Following existing work \cite{DBLP:conf/naacl/DevlinCLT19}, the answer to sentiment classification query is predicted with the hidden representation of $\left[{\rm CLS}\right]$. Formally, we append a three-class classifier to BERT for predicting the sentiment $y^{\mathcal{S}}$ as follows:
\begin{equation}
p\left(y^{\mathcal{S}}|X, q\right) = {\rm softmax}(\mathbf{h}_{1}W_{c}),
\end{equation}
where $W_{c}\in \mathbf{R}^{d_{h}\times 3}$ is model parameter.
\subsection{Joint Learning}
To jointly learn the subtasks in ASTE and make them mutually beneficial, we fuse the loss functions of different queries. For non-restrictive extraction queries in both two directions, we minimize the cross-entropy loss as follows:
\begin{small}
\begin{equation}
\begin{aligned}
\mathcal{L}_{\mathcal{N}}=-\sum_{i=1}^{\left | Q^{\mathcal{N}}\right |}\sum_{j=1}^{N}[p\left(y_{j}^{start}|x_{j}, q_{i}^{\mathcal{N}}\right){\rm log}\hat{p}\left(y_{j}^{start}|x_{j}, q_{i}^{\mathcal{N}}\right)\\+p\left(y_{j}^{end}|x_{j}, q_{i}^{\mathcal{N}}\right){\rm log}\hat{p}\left(y_{j}^{end}|x_{j}, q_{i}^{\mathcal{N}}\right)],
\end{aligned}
\end{equation}
\end{small}
where $p\left(*\right)$ represents the gold distribution, and $\hat{p}\left(*\right)$ denotes the predicted distribution.
Similarly, the loss of restrictive extraction queries in both two directions is calculated as follows:
\begin{small}
\begin{equation}
\begin{aligned}
\mathcal{L}_{\mathcal{R}}=-\sum_{i=1}^{\left | Q^{\mathcal{R}}\right |}\sum_{j=1}^{N}[p\left(y_{j}^{start}|x_{j}, q_{i}^{\mathcal{R}}\right){\rm log}\hat{p}\left(y_{j}^{start}|x_{j}, q_{i}^{\mathcal{R}}\right)\\+p\left(y_{j}^{end}|x_{j}, q_{i}^{\mathcal{R}}\right){\rm log}\hat{p}\left(y_{j}^{end}|x_{j}, q_{i}^{\mathcal{R}}\right)].
\end{aligned}
\end{equation}
\end{small}
For the sentiment classification queries, we minimize the cross-entropy loss function as follows:
\begin{small}
\begin{equation}
\mathcal{L}_{\mathcal{S}} = -\sum_{i=1}^{\left | Q^{\mathcal{S}}\right |}p\left(y^{\mathcal{S}}|X,q_{i}^{\mathcal{S}}\right){\rm log}\hat{p}\left(y^{\mathcal{S}}|X,q_{i}^{\mathcal{S}}\right).
\end{equation}
\end{small}
Then, we combine the above loss functions to form the loss objective of the entire model:
\begin{equation}
\mathcal{L}\left(\theta\right) = \mathcal{L}_{\mathcal{N}}+\mathcal{L}_{\mathcal{R}}+\mathcal{L}_{\mathcal{S}}.
\end{equation}
The optimization problem in Eq.(7) can be solved by any gradient descent approach. In this paper, we adopt the AdamW \cite{DBLP:journals/corr/abs-1711-05101} approach.
\subsection{Inference}
During inference, we fuse the answers to different queries to obtain triplets. Specifically, in the A$\rightarrow$O direction, the non-restrictive extraction query $q_{A \rightarrow O}^{\mathcal{N}}$ first identifies the aspect collection $A=\left\{a_{1}, a_{2},..., a_{\left | A\right |}\right\}$ with $\left | A\right |$ aspects. For each predicted aspect $a_{i}$, the A$\rightarrow$O restrictive query $q_{A\rightarrow O, i}^{\mathcal{R}}$ recognizes the corresponding opinion expression collection and obtains the set of predicted aspect-opinion pairs $V_{A\rightarrow O}=\left[\left(a_{k}, o_{k}\right)\right]_{k=1}^{K}$ in the A$\rightarrow$O direction. Similarly, in the O$\rightarrow$A direction, the model identifies the set of aspect-opinion pairs $V_{O\rightarrow A}=\left[\left(a_{l}, o_{l}\right)\right]_{l=1}^{L}$ in a reversed order.
Then, we combine $V_{A\rightarrow O}$ and $V_{O\rightarrow A}$ as follows:
\begin{equation}
V ={V}'\cup \left\{\left(a,o\right)|\left(a,o\right)\in {V}'',p\left(a,o\right)>\delta \right\},
\end{equation}
\begin{equation}
p\left(a,o\right) =\left\{\begin{matrix}
p\left ( a \right )p\left ( o|a \right )& if \left ( a,o \right )\in V_{A\rightarrow O}\\
p\left ( o \right )p\left ( a|o \right )& if \left ( a,o \right )\in V_{O\rightarrow A}
\end{matrix}\right.,
\end{equation}
where ${V}'$ and ${V}''$ denote the intersection and difference set of $V_{A\rightarrow O}$ and $V_{O\rightarrow A}$, respectively. Each aspect-opinion pair in ${V}''$ is valid only if its probability $p\left(a,o\right)$ is higher than the given threshold $\delta$. The probability of each opinion entity is calculated by multiplying the probabilities of its start and end positions.
Finally, we construct sentiment classification query $q_{i}^{\mathcal{S}}$ to predict the sentiment $s_{a_{i}}$ of each aspect $a_{i}$. Based on these, the triplet collection $T=\left[\left(a_{i}, o_{i}, s_{i}\right)\right]_{i=1}^{\left | T\right |}$ can be obtained.
\begin{table}[]
\centering
\scalebox{0.8}{
\begin{tabular}{c|c|c|c|c|c|c}
\hline
\multicolumn{1}{c|}{\multirow{2}{*}{Datasets}} & \multicolumn{2}{c|}{Train}& \multicolumn{2}{c|}{Dev}& \multicolumn{2}{c}{Test}\\ \cline{2-7}
&\#S & \#T &\#S & \#T&\#S & \#T\\ \hline
14-Lap \cite{DBLP:conf/semeval/PontikiGPPAM14} & 920 &1265 & 228 &337 & 339 &490\\
14-Res \cite{DBLP:conf/semeval/PontikiGPPAM14} & 1300& 2145& 323 & 524 & 496 & 862\\
15-Res \cite{DBLP:conf/semeval/PontikiGPMA15} & 593 & 923 & 148 & 238 & 318 & 455\\
16-Res \cite{DBLP:conf/semeval/PontikiGPAMAAZQ16} & 842 & 1289& 210 & 316 & 320 & 465 \\ \hline
\end{tabular}}
\caption{Statistics of datasets. \#S and \#T denote the number of sentences and triplets, respectively. }
\label{Statistic}
\end{table}
\section{Experiments}
\begin{table*}[]
\centering
\scalebox{0.75}{
\begin{tabular}{l|l|cccc|cccc|cccc|cccc}
\hline
\multicolumn{1}{c|}{\multirow{2}{*}{Evaluation}} & \multicolumn{1}{c|}{\multirow{2}{*}{Models}} & \multicolumn{4}{c|}{14-Lap} & \multicolumn{4}{c|}{14-Res} & \multicolumn{4}{c|}{15-Res} & \multicolumn{4}{c}{16-Res} \\ \cline{3-18}
& & A-S & O & P & T & A-S & O & P & T & A-S & O & P & T & A-S & O & P & T \\ \hline
\multicolumn{1}{c|}{\multirow{5}{*}{Precision}} &TSF & 63.15 & 78.22 & 50.00 & 40.40 & 76.60 & 84.72 & 47.76 & 44.18 & 67.65 & 78.07 & 49.22 & 40.97 & 71.18 & 81.09 & 52.35 & 46.76 \\
&RINANRTE+ & 41.20 & 78.20 & 34.40 & 23.10 & 48.97 & 81.06 & 42.32 & 31.07 & 46.20 & 77.40 & 37.10 & 29.40 & 49.40 & 75.00 & 35.70 & 27.10 \\
&Li-unified-R+ & 66.28 & 76.62 & 52.29 & 42.25 & 73.15 & 81.20 & 44.37 & 41.44 & 64.95 & 79.18 & 52.75 & 43.34 & 66.33 & 79.84 & 46.11 & 38.19
\\
&RACL+R & 59.75 & 77.58 & 54.22 & 41.99 & 75.57 & 82.28 & 73.58 & 62.64 & 68.35 & 76.25 & 67.89 & 55.45 & 68.53 & 82.52 & 72.77 & 60.78 \\
&Ours &\textbf{72.73} & \textbf{84.67} & \textbf{74.11} & \textbf{65.12} & \textbf{77.74} & \textbf{87.22} & \textbf{76.91} & \textbf{71.32} & \textbf{72.41} & \textbf{82.99} & \textbf{71.59} & \textbf{63.71} & \textbf{73.69} & \textbf{85.31} & \textbf{76.08} & \textbf{67.74}
\\ \hline
\multicolumn{1}{c|}{\multirow{5}{*}{Recall}} & TSF & 61.55 & 71.84 & 58.47 & 47.24 & 67.84 & 80.39 & 68.10 & 62.99 & 64.02 & 78.07 & 65.70 & 54.68 & 72.30 & 86.67 & 70.50 & 62.97 \\
&RINANRTE+ & 33.20 & 62.70 & 26.20 & 17.60 & 47.36 & 72.05 & 51.08 & 37.63 & 37.40 & 57.00 & 33.90 & 26.90 & 36.70 & 42.40 & 27.00 & 20.50 \\
&Li-unified-R+ & 60.71 & 74.90 & 52.94 & 42.78 & 74.44 & 83.18 & 73.67 & 68.79 & 64.95 & 75.88 & 61.75 & 50.73 & 74.55 & 86.88 & 64.55 & 53.47 \\
&RACL+R & \textbf{68.90} & \textbf{81.22} & \textbf{66.94} & 51.84 & \textbf{82.23} & \textbf{90.49} & 67.87 & 57.77 & \textbf{70.72} & \textbf{83.96} & 63.74 & 52.53 & \textbf{78.52} & \textbf{91.40} & 71.83 & 60.00 \\
&Ours & 62.59 & 67.18 & 61.92 & \textbf{54.41} & 75.10 & 82.90 & \textbf{75.59} & \textbf{70.09} & 62.63 & 73.23 & \textbf{65.89} & \textbf{58.63} & 72.69 & 83.01 & \textbf{76.99} & \textbf{68.56} \\ \hline
\multicolumn{1}{c|}{\multirow{5}{*}{F$_{1}$-score}} &TSF & 62.34 & 74.84 & 53.85 & 43.50 & 71.95 & 82.45 & 56.10 & 51.89 & 65.79 & 78.02 & 56.23 & 46.79 & 71.73 & 83.73 & 60.04 & 53.62\\
&RINANRTE+ & 36.70 & 69.60 & 29.70 & 20.00 & 48.15 & 76.29 & 46.29 & 34.03 & 41.30 & 65.70 & 35.40 & 28.00 & 42.10 & 54.10 & 30.70 & 23.30 \\
&Li-unified-R+ & 63.38 & 75.70 & 52.56 & 42.47 & 73.79 & 82.13 & 55.34 & 51.68 & 64.95 & 77.44 & 56.85 & 46.69 & 70.20 & 83.16 & 53.75 & 44.51 \\
&RACL+R & 64.00 & \textbf{79.36} & 59.90 & 46.39 & \textbf{78.76} & \textbf{86.19} & 70.61 & 60.11 & \textbf{69.51} & \textbf{79.91} & 65.46 & 53.95 & \textbf{73.19} & \textbf{86.73} & 72.29 & 60.39 \\
&Ours & \textbf{67.27} & 74.90 & \textbf{67.45} & \textbf{59.27} & 76.39 & 84.99 & \textbf{76.23} & \textbf{70.69} & 67.16 & 77.79 & \textbf{68.60} & \textbf{61.05} & 73.18 & 84.13 & \textbf{76.52} & \textbf{68.13} \\ \hline
\end{tabular}}
\caption{Experimental results (\%). Specifically, `A-S', `O', `P', and `T' denote aspect term and sentiment co-extraction, opinion term extraction, aspect-opinion pair extraction, and aspect sentiment triplet extraction, respectively. }\label{precision}
\end{table*}
\subsection{Datasets}
To verify the effectiveness of our proposed approach, we conduct experiments on four benchmark datasets\footnote{https://github.com/xuuuluuu/SemEval-Triplet-data} from the SemEval ABSA Challenges \cite{DBLP:conf/semeval/PontikiGPPAM14, DBLP:conf/semeval/PontikiGPMA15, DBLP:conf/semeval/PontikiGPAMAAZQ16} and list the statistics of these datasets in Table \ref{Statistic}. Specifically, the golden annotations for opinion expressions and relations are derived from \citet{DBLP:conf/naacl/FanWDHC19}. And we split the datasets as \citet{DBLP:conf/aaai/PengXBHLS20} did.
\subsection{Experimental Settings}
For the encoding layer, we adopt the \textbf{BERT-base} \cite{DBLP:conf/naacl/DevlinCLT19} model with 12 attention heads, 12 hidden layers and the hidden size of 768, resulting into 110M pretrained parameters. During training, we use AdamW \cite{DBLP:journals/corr/abs-1711-05101} for optimization with weight decay 0.01 and warmup rate 0.1. The learning rate for training classifiers and the fine-tuning rate for BERT are set to 1e-3 and 1e-5 respectively. Meanwhile, we set batch size to 4 and dropout rate to 0.1. According to the triplet extraction F$_{1}$-score on the development sets, the threshold $\delta$ is manually tuned to 0.8 in bound $[0, 1)$ with step size set to 0.1.
We run our model on a Tesla V100 GPU and train our model for 40 epochs in about 1.5h.
\subsection{Evaluation}
To comprehensively measure the performances of our model and the baselines, we use \textit{Precision}, \textit{Recall}, and \textit{F$_{1}$-score} to evaluate the results on four subtasks, including aspect term and sentiment co-extraction, opinion term extraction, aspect-opinion pair extraction, and triplet extraction.
For reproducibility, we report the testing results averaged over 5 runs with different random seeds. At each run, we select the testing results when the model achieves the best performance on the development set.
\subsection{Baselines}
To demonstrate the effectiveness of BMRC, we compare our model with the following baselines:
\begin{itemize}
\item \textbf{TSF} \cite{DBLP:conf/aaai/PengXBHLS20} is a two-stage pipeline model for ASTE. In the first stage, TSF extracts both aspect-sentiment pairs and opinion expressions. In the second stage, TSF pairs up the extraction results into triplets via an relation classifier.
\item \textbf{RINANTE+} adopts RINANTE \cite{DBLP:conf/acl/DaiS19} with additional sentiment tags as the first stage model to joint extract aspects, opinion expressions, and sentiments. Then, it adopts the second stage of TSF to detect the corresponding relations between opinion entities.
\item \textbf{Li-unified-R+} jointly identifies aspects and their sentiments with Li-unified \cite{LiBLL19}. Meanwhile, it predicts opinion expressions with an opinion-enhanced component at the first stage. Then, it also uses the second stage of TSF to predicts relations.
\item \textbf{RACL+R} first adopts RACL \cite{RACL2020} to identify the aspects, opinion expressions, and sentiments. Then, we construct the query 'Matched the aspect $a_{i}$ and the opinion expression $o_{j}$?' to detect the relations. Note that RACL is also based on BERT.
\end{itemize}
\subsection{Results}
The experimental results are shown in Table \ref{precision}.
According to the results, our model achieves state-of-the-art performances on all datasets.
Although the improvements on aspect term and sentiment co-extraction and opinion term extraction are slight, our model significantly surpasses the baselines by an average of 5.14\% F$_{1}$-score on aspect-opinion pair extraction and an average of 9.58\% F$_{1}$-score on triplet extraction.
The results indicate that extracting opinion entities and relations in pipeline will lead to severe error accumulation. By utilizing the BMRC framework, our model effectively fuses and simplifies the tasks of ATE, OTE, and relation detection, and avoids the above issue.
It is worth noting that the increase in precision contributes most to the boost of F1-score, which shows that the predictions of our model own higher reliability than those baselines. Besides, RACL+R outperforms than other baselines because BERT can learn richer context semantics. TSF and Li-unified-R+ achieve better performances than RINANTE+ because TSF and Li-unified-R+ introduce complex mechanisms to solve the issue of sentiment contradiction brought by the unified tagging schema. Different from those approaches, our model gracefully solves this issue by transforming ASTE into a multi-turn MRC task.
Considering that the datasets released by \citet{DBLP:conf/aaai/PengXBHLS20} remove the cases that one opinion expression corresponds to multiple aspects, we also conduct experiments on AFOE datasets \footnote{https://github.com/NJUNLP/GTS} \cite{Wu2020EMNLP} and report the results in Table \ref{AFOE}. The AFOE datasets, which retains the above cases, are also constructed based on the datasets of \citet{DBLP:conf/naacl/FanWDHC19} and the original SemEval ABSA Challenges. And we further compare our model with two baselines, including IMN+IOG and GTS\footnote{It worth noting that this paper has not been published when we submit our paper to AAAI 2021.} \cite{Wu2020EMNLP} . Specifically, IMN+IOG is a pipeline model which utilizes the interactive multi-task learning network (IMN) \cite{he2019interactive} as the first stage model to identity the aspects and their sentiments. Then, IMN+IOG use the Inward-Outward LSTM \cite{DBLP:conf/naacl/FanWDHC19} as the second stage model to extract the aspect-oriented opinion expressions. And GTS is a latest model which proposes a grid tagging schema to identify the aspect sentiment triplets in an end-to-end way. Particularly, GTS also utilizes BERT as the encoder and designs an inference strategy to exploit mutual indication between different opinion factors. According to the results, our model and GTS significantly outperform IMN+IOG because the joint methods can solve the error propagation problem. Compared with GTS, our model still achieves competitive performances, which verify the effectiveness of our model.
\section{Ablation Study}
To further validate the origination of the significant improvement of BMRC, we conduct ablation experiments and answer the following questions:
\begin{itemize}
\item Does the restrictive extraction query build the association between opinion entity extraction and relation detection?
\item Does the bidirectional structure promote the performance of aspect-opinion pair extraction?
\item Do the relations between aspects and opinion expressions enhance the sentiment classification?
\item How much improvement can the BERT bring?
\end{itemize}
\subsection{Effect of the Restrictive Extraction Query}
\label{dependency}
We first validate whether the restrictive extraction query could effectively capture and exploit the dependency between opinion entity extraction and relation detection for better performance. Accordingly, we construct a two-stage model similar to TSF, called `Ours w/o REQ'. In the first stage, we remove the restrictive extraction query $Q^{\mathcal{R}}$ from BMRC for only the opinion entity extraction and sentiment classification. The stage-2 model, which is responsible for relation detection, is also based on MRC with the input query `\textit{Matched
the aspect $a_{i}$ and the opinion expression $o_{j}$?}'. Experimental results are shown in Figure \ref{Thred}. Although the performances on aspect extraction and opinion extraction are comparable, the performances of `Ours w/o REQ' on triplet extraction and aspect-opinion pair extraction are evidently inferior than BMRC. The reason is that with the removal of the restrictive extraction query, the opinion entity extraction and relation detection are separated and no dependency would be captured by `Ours w/o REQ'.
This indicates the effectiveness of the restrictive query at capturing the dependency.
\begin{table}[]
\centering
\scalebox{0.95}{
\begin{tabular}{l|c|c|c|c}
\hline
Models & 14-Lap* & 14-Res* & 15-Res* & 16-Res* \\ \hline
IMN + IOG & 47.68 & 61.65 & 53.75 & - \\
GTS & 54.58 & \textbf{70.20} & 58.67 & \textbf{67.58} \\
Ours & \textbf{57.83} & 70.01 & \textbf{58.74} & 67.49 \\ \hline
\end{tabular}
}
\caption{Experimental results of aspect sentiment triplet extraction on the AFOE datasets. (\textit{F$_{1}$-score, \%}). }\label{AFOE}
\end{table}
\begin{table}[]
\centering
\scalebox{0.9}{
\begin{tabular}{c|c|c|c|c}
\hline
\multirow{2}{*}{Datasets} & \multicolumn{2}{c|}{A} & \multicolumn{2}{c}{A-S} \\ \cline{2-5}
& Ours & Our w/o REQ & Ours & Ours w/o REQ \\ \hline
14-Lap & 78.94 & \textbf{80.06} & \textbf{67.27} & 61.61 \\
14-Res & \textbf{83.31} & 82.73 & \textbf{76.39} & 66.26 \\
15-Res & 75.67 & \textbf{79.00} & \textbf{67.16} & 56.82 \\
16-Res & \textbf{83.28} & 80.60 & \textbf{73.18} & 68.82 \\ \hline
\end{tabular}
}
\caption{Experimental results of the ablation study on relation-aware sentiment classification (\textit{F$_{1}$-score, \%}). Specifically, `A' and `A-S' stand for aspect term extraction and aspect term and sentiment co-extraction, respectively. }\label{SA}
\end{table}
\begin{figure*}
\centering
\begin{minipage}{0.35\textwidth}
\centering
\includegraphics[width=1\textwidth]{pic/aspect.eps}
(a) Aspect Term Extraction\\
\end{minipage}
\begin{minipage}{0.35\textwidth}
\centering
\includegraphics[width=1\textwidth]{pic/opinion.eps}
(b) Opinion Term Extraction\\
\end{minipage}
\begin{minipage}{0.35\textwidth}
\centering
\includegraphics[width=1\textwidth]{pic/pair.eps}
(c) Aspect-Opinion Pair Extraction\\
\end{minipage}
\begin{minipage}{0.35\textwidth}
\centering
\includegraphics[width=1\textwidth]{pic/triplet.eps}
(d) Triplet Extraction\\
\end{minipage}
\caption{Experimental results of ablation study on the restrictive extraction query and the bidirectional structure.}\label{Thred}
\end{figure*}
\subsection{Effect of the Bidirectional MRC Structure}
To explore the effect of bidirectional MRC structure, we compare our model with two unidirectional models, including `Ours w/o AO' and `Ours w/o OA'. Concretely, `Ours w/o AO' extracts triplets only through O$\rightarrow$A direction, and `Ours w/o OA' extracts triplets through A$\rightarrow$O direction. As shown in Figure \ref{Thred}, `Ours w/o OA' shows inferior performance on opinion term extraction without O$\rightarrow$A direction MRC, while `Ours w/o AO' shows worse performance on aspect term extraction. This further harms the performances on aspect-opinion pair extraction and triplet extraction.
The reason is that both aspects and opinion expressions can initiate aspect-opinion pairs, and the model will be biased when relations are forced to be detected by either aspects or opinions only.
By introducing the bidirectional design, the two direction MRCs can complement each other and further improve the performance of aspect-opinion pair extraction and triplet extraction.
\subsection{Effect of Relation-Aware Sentiment Classification}
In order to examine the benefit that the relations between aspects and opinion expressions provide for the sentiment classification, we compare the performances of our model and `Ours w/o REQ'. Experimental results on aspect term extraction and aspect term and sentiment co-extraction are shown in Table \ref{SA}. Since `Ours w/o REQ' separate relation detection and sentiment classification in two stages, the detected relations cannot directly provide assistance to sentiment classification.
According to the results, although removing relation detection from joint learning does not harm the performance of aspect term extraction seriously, the performances of aspect term and sentiment co-extraction are all significantly weakened. This clearly indicates that the relations between aspects and opinion expressions can effectively boost the performance of sentiment classification.
\begin{table}[]
\centering
\begin{tabular}{l|c|c|c|c}
\hline
Models & 14-Lap & 14-Res & 15-Res & 16-Res \\ \hline
TSF & 43.50 & 51.89 & 46.79 & 53.62 \\
Ours w/o BERT & 48.15 & 63.32 & 53.77 & 63.16 \\
Ours w/o REQ & 51.40 & 57.20 & 47.79 & 61.03 \\
Ours & \textbf{59.27} & \textbf{70.69} & \textbf{61.05} & \textbf{68.13} \\ \hline
\end{tabular}
\caption{Experimental results of the ablation study on aspect sentiment triplet extraction (\textit{F$_{1}$-score, \%}), which aims to analyze the effect of BERT. }\label{BERT}
\end{table}
\subsection{Effect of BERT}
We analyze the effect of BERT and our contributions from two perspectives. First, we construct our model based on BiDAF, which is a typical reading comprehension model without BERT, and refer it to `Ours w/o BERT'. According to the results shown in Table \ref{BERT}, it significantly surpasses TSF by an average of 8.15\% F1-score on triplet extraction, which shows that our model can achieve SOTA performance without BERT. Besides, compared with `Ours w/o BERT', our model further improves 7.69\% F1-score, which is brought by BERT.
Second, we compare our model with 'Ours w/o REQ' and TSF. The ablation model 'Ours w/o REQ' can be regarded as an implementation version of TSF based on BERT and MRC framework. By comparing it with TSF, the results show that BERT based MRC model can bring an average of 5.41\% F1-score improvement on triplet extraction against the counterpart model without BERT. By further introducing the bidirectional MRC structure and three types of queries, our model further outperforms 'Ours w/o REQ' by 10.4\% F1-score. These two-fold analyses indicate that our contributions play a greater role in improving performances than BERT.
\section{Conclusion}
In this paper, we formalized the aspect sentiment triplet extraction (ASTE) task as a multi-turn machine reading comprehension (MTMRC) task and proposed the bidirectional MRC (BMRC) framework with well-designed queries. Specifically, the non-restrictive and restrictive extraction queries are designed to naturally fuse opinion entity extraction and relation detection, enhancing the dependency between them. By devising the bidirectional MRC structure, it can be ensured that either an aspect or an opinion expression can trigger an aspect-opinion pair just like human's reading behavior. In addition, the sentiment classification query and joint learning manner
are used to further promote sentiment classification with the incorporation of relations between aspects and opinion expressions. The empirical study demonstrated that our model achieves state-of-the-art performance.
\section{Acknowledgments}
This research is supported by the National Natural Science Foundation of China under grant No. 61976119 and the Major Program of Science and Technology of Tianjin under grant No. 18ZXZNGX00310.
\bibliographystyle{aaai21}
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\author[G. Sacc\`a]{Giulia Sacc\`a, with an Appendix by Claire Voisin}
\address{Mathematics Department\\
Columbia University \\
Mathematics Department\\
2990 Broadway\\
New York, NY 10027}
\title[Birational geometry of the Intermediate Jacobian fibration]{Birational geometry of the Intermediate Jacobian fibration of a cubic fourfold}
\begin{document}
\maketitle
\begin{abstract}
We show that the intermediate Jacobian fibration associated to \emph{any} smooth cubic fourfold $X$ admits a hyper--K\"ahler compactification $J(X)$ with a regular Lagrangian fibration $\pi: J \to \P^5$. This builds upon \cite{LSV}, where the result is proved for general $X$, as well as on the degeneration techniques introduced in \cite{KLSV} and the minimal model program.
We then study some aspects of the birational geometry of $J(X)$: for very general $X$ we compute the movable and nef cones of $J(X)$, showing that $J(X)$ is not birational to the twisted version of the intermediate Jacobian fibration \cite{Voisin-twisted}, nor to an OG$10$-type moduli space of objects in the Kuznetsov component of $X$; for any smooth $X$ we show, using normal functions, that the Mordell-Weil group $MW(\pi)$ of the fibration is isomorphic to the integral degree $4$ primitive algebraic cohomology of $X$, i.e., $MW(\pi) \cong H^{2,2}(X, \mathbb{Z})_0$.
\end{abstract}
\tableofcontents
\section*{Introduction}
The geometry of smooth cubic fourfolds has ties to that of K3 surfaces and, more generally, to that of higher dimensional hyper--K\"ahler manifolds. For example, with certain special cubic fourfolds one can associate a K3 surface via Hodge theoretic \cite{Hassett-special} or derived categorical \cite{Kuznetsov} methods. From a more geometric perspective, given a smooth cubic fourfold $X$, hyper--K\"ahler manifolds of K3$^{[n]}$-type are constructed geometrically, via parameter spaces of rational curves of certain degrees on $X$ \cite{Beauville-Donagi,LLSvS}, or as moduli spaces of objects in the Kuznetsov component of $X$ \cite{Lahoz-Lehn-Macri-Stellari,BLMNPS}. These constructions give rise to $20$-dimensional families of polarized hyper--K\"ahler manifolds, the maximal possible dimension of families of polarized hyper--K\"ahler manifolds of K3$^{[n]}$-type.
As the cubic fourfold becomes special, for example when it acquires more algebraic classes, the geometry of these hyper--K\"ahler manifolds also becomes more interesting. For example, when $X$ has an associated K3 surface in the sense of \cite{Hassett-special, Kuznetsov, Addington-Thomas, Huybrechts-category}, these hyper--K\"ahler manifolds become isomorphic, or birational, to moduli spaces of objects in the derived category of the corresponding K3 surface \cite{Addington,BLMNPS}.
In \cite{LSV} a Lagrangian fibered hyper--K\"ahler manifold is constructed starting from a general cubic fourfold. This hyper--K\"ahler manifold is a deformation of O'Grady's $10$-dimensional exceptional example.
More precisely, let $X \subset \P^5$ be a smooth cubic fourfold and let $\pi_U: J_U \to U \subset (\P^5)^\vee$ be the family of intermediate Jacobians of the smooth hyperplane sections of $X$. This fibration was considered by Donagi-Markman in \cite{Donagi-Markman}, where they showed that the total space has a holomorphic symplectic form. The main result of \cite{LSV} is to construct, for general $X$, a smooth projective hyper--K\"ahler compactification $J$ of $J_U$, with a flat morphism $J \to {(\P^5)^\vee}$ extending $\pi_U$, and to show that this hyper--K\"ahler $10$-fold is deformation equivalent to O'Grady's $10$-dimensional example. In \cite{Voisin-twisted}, Voisin constructs a hyper--K\"ahler compactification $J^T$ of a natural $J_U$-torsor $J_U^T$, which is non-trivial for very general $X$. The two hyper--K\"ahler manifolds $J$ and $J^T$ are birational over countably many hyperpurfaces in the moduli space of cubic fourfolds.
These two constructions give rise to two $20$-dimensional families of hyper--K\"ahler manifolds of OG$10$-type, each of which form an open subset of a codimension two locus inside the moduli space of hyper--K\"ahler manifolds in this deformation class.
If one wishes to study the geometry of these hyper--K\"ahler manifolds as the cubic fourfold becomes special, a first step is to check if a hyper--K\"ahler compactification of the fibration $J_U \to U $ can be constructed for an arbitrary smooth cubic fourfold. The starting result of this paper is that this can indeed be done.
\begin{thmintro}[Theorem \ref{hkcom}] \label{main thm 1}
Let $X \subset \P^5$ be a smooth cubic fourfold, and let $\pi_U: J_U \to U \subset (\P^5)^\vee$ be the Donagi--Markman fibration. There exists a smooth projective hyper--K\"ahler compactification $J $ of $ J_U$ with a morphism $\pi: J \to (\P^5)^\vee$ extending $\pi_U$.
\end{thmintro}
The same techniques also give the existence of a Lagrangian fibered hyper--K\"ahler compactification for the non trivial $J_U$-torsor $J_U^T \to U$ of \cite{Voisin-twisted} for any smooth $X$ (see Remark \ref{twisted}). Moreover, with little extra work, the Theorem is proved also for mildly singular cubic fourfolds such as, for example, cubic fourfolds with a simple node (see Theorem \ref{Csix}). For a general cubic fourfolds with one node, the existence of such a Lagrangian fibered hyper--K\"ahler manifold provides a positive answer to a question of Beauville \cite{Beauville-Fano-K3}(see Remark \ref{rem beau}).
We should point out that as a consequence of the ``finite monodromy implies smooth filling'' results of \cite{KLSV}, we prove in Proposition \ref{prop hk model} that $J_U$ admits projective birational model that is hyper-K\"ahler. Theorem \ref{main thm 1} shows that there exists a hyper--K\"ahler model with a Lagrangian fibration extending $\pi_U$.
There are several ingredients in the construction of the hyper--K\"ahler compactification of \cite{LSV}: a cycle-theoretic construction of the holomorphic symplectic form, the problem of the existence of so--called very good lines for any hyperplane section of $X$, a smoothness criterion for relative compactified Prym varieties, the independence of the compactification from the choice of a very good line. Here we pursed a different direction and instead rely on the existence of a hyper--K\"ahler compactification for general $X$, use the degenerations techniques introduced in \cite{KLSV}, and implement some results from birational geometry and the minimal model program, following \cite{Lai,Kollar-Elliptic}. One advantage of our method is that it opens the door to using birational geometry to compactify Lagrangian fibrations.
The second result of this paper is concerned with the hyper--K\"ahler birational geometry of $J$. We show that the relative theta divisor $\Theta$ of the fibration is a prime exceptional divisor and that for general $X$ it can be contracted after a Mukai flop.
\begin{thmintro}[Theorem \ref{NSU}] \label{main thm 2} Let $q$ be the Beauville-Bogomolov form on $H^2(J, \mathbb{Z})$. The relative theta divisor $\Theta \subset J$ is a prime exceptional divisor with $q(\Theta)=-2$.
For very general $X$, there is a unique other hyper--K\"ahler birational model of $J$, denoted by $N$, which is the Mukai flop $p: J \dashrightarrow N$ of $J$ along the image of the zero section. $N$ admits a divisorial contraction $h: N \to \bar N$, which contracts the proper transform of $\Theta$ onto an $8$-dimensional variety which is birational to the LLSvS $8$-fold $Z(X)$.
\end{thmintro}
Thus, for very general $X$, $J$ is the unique hyper--K\"ahler birational model with a Lagrangian fibration, it is not birational to $J^T$ (Corollary \ref{J not JT}), and its movable cone is the union of its nef cone and the nef cone of $N$. This answers a question by Voisin \cite{Voisin-twisted}.
As a consequence of this Theorem we show that for very general $X$, $J$ is not birational to a moduli spaces of objects in the Kuznetsov component $\mathcal K u(X)$ of $X$ (see Corollary \ref{notkuz}).
In the opposite direction, it was recently proved \cite{Pertusi-et-al}) that the twisted hyper--K\"ahler manifold $J^T$ is birational to a moduli space of objects of OG$10$-type in $\mathcal K u(X)$. By objects of OG$10$-type, we mean objects whose Mukai vector is of the form $2w$, with $w^2=2$. As a consequence, the family of intermediate Jacobian fibrations is the only known family of hyper--K\"ahler manifolds associated with cubic fourfolds whose very general point cannot be described as a moduli space of objects in the Kuznetsov component of $X$.
Given $J=J(X)$, a hyper--K\"ahler compactification of the intermediate Jacobian fibration for any smooth cubic fourfold $X$, a natural question to ask is how the geometry of $J$ changes as $X$ becomes less general. One way to answer this question is the following theorem, describing the Mordell-Weil group of $\pi$ in terms of the primitive algebraic cohomology of $X$. In Section \ref{section MW} we prove:
\begin{thmintro} \label{main thm 3} (Theorem \ref{MW})
Let $MW(\pi)$ be the Mordell-Weil group of $\pi: J \to \P^5$, i.e., the group of rational sections of $\pi$ and let $H^{2,2}(X, \mathbb Z)_0$ be the primitive degree $4$ integral cohomology of $X$.
The natural group homomorphism
\[
\phi_X: H^{2,2}(X, \mathbb Z)_0 \to MW(\pi)
\]
induced by the Abel-Jacobi map is an isomorphism.\end{thmintro}
The proof of this result uses the theory of normal functions, as developed by Griffiths and Zucker, as well as the techniques used by Voisin to prove the integral Hodge conjecture for cubic fourfolds.
A consequence of this is a geometric description of the Lagrangian fibered hyper--K\"ahler manifolds with maximal Mordell-Weil rank whose existence was proved by Oguiso in \cite{Oguiso-MW}: indeed, Oguiso's examples are (birationally) given by $J=J(X) \to \P^5$, where $X$ a smooth cubic fourfold with $H^{2,2}(X, \mathbb{Z})$ of maximal rank.
\subsection*{Plan of the paper} In Section \ref{birational section} we prove the existence of a hyper--K\"ahler compactification for $J_U$ and for $J_U^T$, in the case of any smooth, or mildly singular, $X$. This uses some results from the minimal model program, which are briefly recalled.
In Section \ref{ogtentype} we review some basic results about moduli spaces of OG$10$-type and we compute, using the Bayer-Macr\`i techniques adapted to these singular moduli spaces by Meachan-Zhang \cite{Meachan-Zhang}, the nef and movable cones of certain moduli spaces of OG$10$-type that appear as limits of the intermediate Jacobian fibration.
The main result of Section \ref{section theta} is the computation that $q(\Theta)=-2$.
Section \ref{section general X} is devoted to the proof of Theorem \ref{main thm 2} and its preparation: Given a family of cubic fourfold degenerating to the chordal cubic, we construct a certain degeneration of the intermediate Jacobian fibration and identify the limit of the corresponding degeneration of the relative Theta divisor. By the results of Section \ref{ogtentype}, the limiting theta divisor can be contracted after a Mukai flop of the zero section and we deduce the analogue result for $\Theta$.
The computation of the Mordell-Weil group occupies Section \ref{section MW}.
Finally, in the Appendix by C. Voisin, some applications to the Beauville conjecture on the polynomial relations in the Chow group of a projective hyper--K\"ahler manifold are given for $J=J(X)$, in the case of very general $J$ of Picard number $2$ or $3$. This is obtained as an application of the computation of $q(\Theta)=-2$ from Theorem \ref{main thm 2}.
\subsection*{Acknowledgements} I would like to thank J. Koll\'ar for pointing my attention to the techniques of \cite{Kollar-Elliptic} and \cite{Lai}, which are used in this paper.
It is my pleasure to thank E. Arbarello, C. Camere, G. Di Cerbo, K. Hulek, R. Laza, E. Macr\`i, C. Onorati, G. Pearlstein, A. Rapagnetta, L. Tasin, C. Voisin for useful and interesting discussions related to the topic of this paper. I also thank the anonymous referee for having read this paper very carefully and for many useful comments.
Finally, I would like to warmly thank Coll\`ege de France and \'Ecole Normal Sup\'erieure for the hospitality and the great working conditions while the final version of this manuscript was being prepared. This work is partially supported by NSF Grant DMS-1801818.
\section{A hyper--K\"ahler compactification of the intermediate Jacobian fibration for any smooth cubic fourfold}
\label{birational section}
We denote by $X \subset \P^5$ a smooth cubic fourfold, by ${(\P^5)^\vee}$ the dual projective space parametrizing hyperplane sections $Y=X \cap H \subset X$, and by $U \subset {(\P^5)^\vee}$ the open subset parametrizing smooth hyperplane sections. The dual hypersurface of $X$, parametrizing singular hyperplane sections, is denoted by $X^\vee \subset {(\P^5)^\vee}$. Its smooth locus
\[
U_1:=(\P^5)^\vee \setminus Sing(X^\vee) \subset (\P^5)^\vee
\]
parametrizes hyperplane sections of $X$ that are smooth or have one simple node and no other singularities. In what follows, we freely drop the $^\vee$ from ${(\P^5)^\vee}$ and write simply $\P^5$. From the context it will be clear if we are referring to the projective space parametrizing hyperplane sections of $X$ or the projective space containing $X$. For a smooth cubic threefold $Y$, the Griffiths' intermediate Jacobian of $Y$ will be denoted by
\[
\Jac(Y)\cong H^1(Y, \Omega_Y^2)^\vee \slash H_3(Y,\mathbb{Z}).
\]
It is a principally polarized abelian fivefold which parametrizes rational equivalence classes of homologically trivial $1$-cycles on $Y$ \cite[Thm. 6.24]{Voisin-Chow}.
Over $U$ consider the Donagi-Markman fibration
\begin{equation} \label{intjacfibr}
\pi_U: J_U=J_U(X) \to U
\end{equation}
whose fiber over a smooth hyperplane section $Y=X \cap H$ is the intermediate Jacobian $\Jac(Y)$. By \cite{Donagi-Markman}, $J_U$ is quasi-projective and admits a holomorphic symplectic form $\sigma_{J_U}$ with respect to which $\pi_U$ is Lagrangian. The main results of \cite{LSV} is the following theorem
\begin{thm}[\hspace{1sp}\cite{LSV}] \label{LSVthm}
Let $X$ be a general cubic fourfold. Then there exists a smooth projective compactification $J=J(X)$ of $J_U$, with a flat morphism $\pi: J \to {(\P^5)^\vee}$ extending $\pi_U$ which has irreducible fibers and which admits a rational zero section $s: {(\P^5)^\vee} \dashrightarrow J$. Moreover, $J$ is an irreducible holomorphic symplectic manifold, deformation equivalent to O'Grady's $10$-dimensional exceptional example.
\end{thm}
We will say that $X$ is general in the sense of LSV if the construction of \cite{LSV} works for $J_U(X)$, and we refer to $J=J(X)$ as in Theorem \ref{LSVthm} as the LSV fibration. A necessary condition for this to happen is that the hyperplane sections of $X$ are palindromic (see \cite{Brosnan}). For example, a cubic fourfold containing a plane is not general in the sense of LSV.
To extend the theorem above for any $X$, we use the existence of a hyper--K\"ahler compactifiction for general $X$, the cycle theoretic description of the holomorphic symplectic form that was given in \cite{LSV}, the degeneration results from \cite{KLSV}, and techniques from the minimal model program (following \cite{Kollar-Elliptic,Lai}). We start by recalling the construction of a natural partial compactification of $J_U$, which already appeared in \cite{Donagi-Markman, LSV}.
\begin{lemma} [\hspace{1sp}\cite{Donagi-Markman,LSV}] \label{lem Juno} For any smooth $X$, there is a canonical partial compactification $J_{U_1}=J_{U_1}(X)$ of $J_U$, with a projective morphism $\pi_{U_1}: J_{U_1} \to U_1$ with irreducible fibers extending $\pi_U$. $J_{U_1}$ is smooth and has a holomorphic symplectic form $\sigma_{J_{U_1}}$ extending $\sigma_{J_U}$.
\end{lemma}
\begin{proof} This is already proved in \cite[\S 8.5.2 and Thm. 8.18]{Donagi-Markman}. Alternatively, one can use
\cite[Cor. 2.38]{Collino-Murre-I}, and \cite[Def. 2.2 and 2.9, Prop. 1.4, and Lem. 5.2]{LSV}.
\end{proof}
Before giving an application of the cycle-theoretic construction of the holomorphic symplectic form \cite[\S 1]{LSV}, we recall the definition of symplectic variety.
\begin{defin} A normal projective variety $M$ is called symplectic if its smooth locus carries a holomorphic symplectic form which extends to a regular (i.e. holomorphic) form on any resolution of singularities of $M$.
\end{defin}
\begin{lemma} \label{lem K eff} \label{cor not uniruled}
Let $\bar J $ be a normal projective compactification of $J_U$. Then
\begin{enumerate}
\item The smooth locus of $\bar J$ admits a homolorphic two form extending $\sigma_{J_U}$. In particular, the canonical class $K_{\bar J}$ of $\bar J$ is effective and is trivial if and only if $\bar J$ is a symplectic variety.
\item $\bar J$ is not uniruled.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) The first statement is \cite[Thm. 1.2 iii)]{LSV}, while the second follows from the fact that the canonical class of $\bar J$ is the (closure of the) codimension one locus where the generically non-degenerate holomorphic two form is degenerate. (2) Let $\tilde J \to \bar J$ be a resolution of singularities. By (1), $\tilde J$ has effective canonical class and thus by \cite{Miyaoka-Mori} it is not uniruled.
\end{proof}
The following is an application of the degeneration techniques of \cite{KLSV}.
\begin{prop} \label{prop hk model}
Let $X$ be a smooth cubic fourfold and let $J_{U}=J_{U}(X)$ be as above. Then there exists a smooth projective hyper--K\"ahler manifold $M$ birational to $J_{U}$ and of OG$10$-type.
\end{prop}
\begin{proof}
Let $\mathcal X \to \Delta$ be a family of smooth cubic fourfolds with $\mathcal X_0= X$. Here $\Delta$ is an open affine subset of a smooth projective curve, or a small disk. We will use the notation $t=0$ to denote a chosen special point in $\Delta$ and $t \neq 0$ to denote any other point. Up to restricting $\Delta$ if necessary, assume that for $t \neq 0$, $\mathcal X_t$ is general in the sense of LSV. By \cite[Prop. 2.10]{LSV}, we can assume that for any $t \neq 0$ all the hyperplane sections of $\mathcal X_t$ admit a very good line (see \cite[Def. 2.9]{LSV}). Consider the open set $\mathcal V = (\P^5)^\vee \times \Delta \setminus Sing(\mathcal X_0^\vee) \times\{ 0\}$, so that $\mathcal V_t= (\P^5)^\vee$ for $ t \neq 0$ and $\mathcal V_0=U_1 \times \{0\} $ parametrizes the hyperplane sections of $\mathcal X_0= X$ that have at most one nodal point and no other singularities. The construction of \cite[\S 5]{LSV} can be carried out in families, yielding a projective morphism
\[
\mathcal J_\mathcal{V} \to \mathcal V
\]
which is fibered in compactified Prym varieties and is such that, denoting by $\mathcal J_t$ the fiber of the induced smooth quasi-projective morphism $ \mathcal J_\mathcal{V} \to \Delta$, for $t \neq 0$, $\mathcal J_t$ is the LSV fibration $J(\mathcal X_t)$, and $\mathcal J_0=J_{U_1}(X)$.
Let $\widetilde{\mathcal J} \to \Delta$ be a projective morphism extending $ \mathcal J_\mathcal{V} \to \Delta$. The central fiber $\mathcal J_0$ has a multiplicity one component which contains $J_{U_1}$ as dense open subset. By Lemma \ref{cor not uniruled}, this component is not uniruled. By \cite[Cor. 5.2]{KLSV} there is a birational model $M$ of $J_{U_1}(X)$ that is a hyper--K\"ahler manifold, deformation equivalent to the smooth fibers ${\mathcal J}_t=J(\mathcal X_t)$, $t \neq 0$.
\end{proof}
By \cite{Matsushita-Def}, given a hyper--K\"ahler manifold $M$ with a Lagrangian fibration $\pi: M \to \P^n$, the locus inside $\Def(M)$ where the Lagrangian fibration deforms is an open subset of the hypersurface where the class $\pi^* \mathcal O(1)$ stays of type $(1,1)$. However, this fact alone is not enough to imply the existence of a hyper--K\"ahler compactification of $J_{U_1}$ for \emph{any} smooth $X$.
This is what we prove in the following Theorem \ref{hkcom}, whose proof uses the mmp following Koll\'ar \cite[\S 8]{Kollar-Elliptic} and Lai \cite{Lai}. In \S \ref{subsectmmp} we will recall some basic facts about the mmp that are needed in the proof of Theorems \ref{hkcom} and \ref{deg lagr 1}. We refer to \cite{Kollar-Mori} and to \cite{Hacon-Kovacs} for the basic definitions and fundamental results.
\begin{thm} \label{hkcom}
For any smooth cubic fourfold $X$, there exists a smooth projective hyper--K\"ahler compactification $J=J(X)$ of $J_U(X)$, with a projective flat morphism $\pi: J \to \P^5$ extending $\pi_U$.
\end{thm}
\begin{proof}
Let $\bar J \to \P^5$ be any normal projective compactification of $J_{U_1}$ with a regular morphism $\bar \pi : \bar J \to \P^5$.
By Lemma \ref{lem K eff}, there is a holomorphic two form $\bar \sigma$ on the smooth locus of $\bar J$ extending $\sigma_{J_{U_1}}$, the canonical class $K_{\bar J} \ge 0$ is effective, and $K_{\bar J}=0$ if and only if $\bar J$ is a symplectic variety.
Since $K_{\bar J}$ is supported on the complement of $J_{U_1}$, $\codim \bar \pi(\Supp(K_{\bar J}) ) \ge 2$. By definition \cite[Def. 7]{Kollar-Elliptic}, this means that $K_{\bar J}$ is $\bar \pi$-exceptional, if it is non trivial. If this is the case, then by \cite[III 5.1]{Nakayama} (cf. also \cite[Lem 2.10]{Lai}), $K_{\bar J}$ is not $\bar \pi$-nef. More precisely, there is a component of $K_{\bar J}$ that is covered by curves that are contracted by $\bar \pi$ and that intersect $K_{\bar J}$ negatively.
Let $\tilde J \to \P^5$ be a smooth projective compactification of $J_{U_1}$ admitting a regular morphism $\tilde \pi : \tilde J \to \P^5$ and let $K_{\tilde J}$ be its canonical class. If the effective divisor $K_{\tilde J}$ is not trivial, we use the mmp to contract $\Supp(K_{\tilde J})$ relatively to $\P^5$. Let $H$ be a $\tilde \pi$-ample $\mathbb{Q}$-divisor such that the pair $(\tilde J, H)$ is klt and $K_{\tilde J}+H$ is relatively big and nef. The mmp with scaling over $\P^5$ (see \S \ref{subsectmmp} below) produces a sequence of birational maps
\begin{equation} \label{MMP scaling}
\tilde J=J_0 \stackrel{\psi_0}{\dashrightarrow} J_1 \stackrel{\psi_1}{\dashrightarrow} \cdots \dashrightarrow J_i \stackrel{\psi_i}{\dashrightarrow} \cdots
\end{equation}
over $\P^5$ (i.e., there are projective morphisms $\pi_i: J \to \P^5$ such that $\pi_0=\widetilde \pi$ and $\pi_i:=\pi_{i-1} \circ \psi_i^{-1}$)
and a non increasing sequence of non negative rational numbers $t_0 =1 \ge t_1 \ge \dots t_i \ge \dots\ge 0$, with the following properties
\begin{enumerate}
\item For every $i \ge 0$, $K_{J_{i }}+t_i H_i$ is $\pi_i$-big and $\pi_i$-nef.
\item For every $i \ge 0$, $J_i$ is a $\mathbb{Q}$-factorial terminal compactification of $J_{U_1}$. The fact that the birational morphisms $\psi_i$ are isomorphisms away from $J_{U_1}$ follows from the fact that the $K_{J_i}$-negative rays of the mmp correspond to rational curves that are contained in the support of $K_{J_i}$. Thus, by Lemma \ref{lem K eff} the smooth locus of $J_i$ carries a holomorphic two form $\sigma_i$ extending $\sigma_{ J_{U_1}}$;
\item $K_{J_{i }}$ is effective and, if not trivial, it has a component covered by $K_{J_{i }} $-negative curves which are contracted by $\pi_i$;
\item The process stops if and only if there exists an $i$ such that $K_{J_{i}}$ is $\pi_{i}$-nef. This holds if and only if $K_{J_{i}}=0$.
\end{enumerate}
The number of irreducible components of the support of $K_{J_{i}}$ is non increasing, since the birational maps of the mmp extract no divisors. In fact, we claim that this number is eventually strictly decreasing. By (4) above, this happens if and only if the process eventually stops.
Suppose that this is not the case. Then by Lemma \ref{lemma t goes to zero}, $\lim t_i=0$. Recall, as already observed, that if $K_{J_{k}} \neq 0$, then there exists a component that is covered by $K_{J_k } $-negative curves that are contracted by $\pi_k$.
Since we are assuming that $\lim t_i=0$, this implies that for $i \gg 0$, $t_i$ is small enough so that this component is contained in the relative stable base locus $\mathbb B ((K_{J_k}+ t_i H_k)/\P^5)$. Since by Lemma \ref{lemma sbl}, the divisorial components of $\mathbb B ((K_{J_k}+ t_i H_k)/\P^5)$ are contracted by $J_k\dashrightarrow J_i$, it follows that for $i \gg0$, the number of irreducible components of the effective divisor $K_{J_i} $ is strictly less than the number of components of $K_{J_k} $. Thus, the claim is proved and for some $i \gg 0$, the process gives a model with $K_{J_{i}}=0$. By Lemma \ref{cor not uniruled}, $\bar J:=J_i$ is a $\mathbb{Q}$-factorial terminal symplectic compactification of ${ J_{U_1}}$. Finally, by Proposition \ref{GLR} below $\bar J$ is smooth and the theorem is proved.
\end{proof}
\begin{prop}(Greb-Lehn-Rollenske) \label{GLR}
Let $\bar M$ be a $\mathbb{Q}$-factorial terminal symplectic variety. Suppose that $\bar M$ is birational to a smooth hyper--K\"ahler manifold $M$. Then $\bar{ M}$ is smooth.
\end{prop}
\begin{proof}
This is \cite[6.4]{Greb-Lehn-Rollenske}.
\end{proof}
\begin{rem}
The techniques used to prove the theorem above can be applied to similar contexts to give $\mathbb{Q}$-factorial terminal symplectic compactifications of other quasi-projective Lagrangian fibrations. We plan to come back to this in upcoming work.
\end{rem}
As a consequence of Theorem \ref{deg lagr 1} below, we will give a slightly stronger version of the theorem just proved (see Remark \ref{rem other proof}) showing that, given a family of smooth cubic fourfolds whose general fiber is general in the sense of \cite{LSV}, then up to a base change and birational transformations, the corresponding family of LSV intermediate Jacobian fibrations can be filled with a Lagrangian fibered smooth projective hyper--K\"ahler compactification of the Donagi-Markman fibration of the limiting cubic fourfold.
Another approach to Theorem \ref{main thm 1} would be to show that the rational map $M \dashrightarrow \P^5$ induced by the birational map $\phi: M \dashrightarrow J_{U_1}$ of Proposition \ref{prop hk model} is almost holomorphic (see \cite[Def.1]{Matsushita-almost}). By \cite{Matsushita-almost} this would imply the existence of a birational hyper--K\"ahler model of $M$ with a regular morphism to $\P^5$. It seems, however, that controlling the mmp of Proposition \ref{prop hk model} to ensure that $M\dashrightarrow \P^5$ is almost holomorphic is not too far from running the relative mmp as in the proof of Theorem \ref{hkcom}.
Given a smooth cubic fourfold $X$, we will refer to both the Donagi-Markman fibration $J_U$ and to any hyper--K\"ahler compactification $J$ of $J_U$ as in Theorem \ref{hkcom}, as the intermediate Jacobian fibration. Hopefully, it will be clear from the context which one we are referring to.
\begin{rem} Unlike the compactification of \cite{LSV}, the proof of Theorem \ref{hkcom} is not constructive and, for a given $X$, the hyper--K\"ahler compactification that we show to exists may not be unique.
We will return to this question in Section \ref{section general X}.
\end{rem}
\subsection{The mmp with scaling} \label{subsectmmp}
In this subsection we recall some basic tools and known results from the minimal model program (mmp) that are used to prove Theorems \ref{hkcom} and \ref{deg lagr 1}. For the basic notions and the fundamental results we refer to \cite{Kollar-Mori} and \cite{Hacon-Kovacs}. In this section, by divisor we will mean a $\mathbb{Q}$-divisor.
Let $M$ be a normal $\mathbb{Q}$-factorial variety with a projective morphism $\pi: M \to B$ to a normal quasi-projective variety $B$. Let $\Delta$ be an effective divisor on $M$ and let $H$ be a general divisor on $M$ that is ample (or big) over $B$.
We assume that the pair $(M, \Delta+H)$ is klt and that $K_M+\Delta+H$ is nef over $B$.
The mmp with scaling of $H$ \cite[\S 5.E]{Hacon-Kovacs} produces a sequence of birational maps $\psi_i: M_i \dashrightarrow M_{i+1}$ over $B$, such that: $M_0=M$, $\Delta_{i+1}=(\psi_i)_*\Delta_{i}$, $H_{i+1}=(\psi_i)_*H{i}$ and $\psi_i$ is the flip or the divisorial contraction for a $(K_{M_i}+\Delta_i)$-negative relative extremal ray $R_i$ over $B$. We let $\pi_{i}$ be the induced regular morphism $M_i \to B$. The sequence is defined inductively in the following way.
Let
\[
t_i=\inf\{ t \ge 0 \, | \, K_{M_i}+\Delta_i+ tH_i \,\, \text{ is nef over } B \}.
\]
If $t_i=0$, then $K_{M_i}+\Delta_i$ is nef over $B$ and the process stops. Otherwise, there is a $0<t' \le t_i$ such that $K_{M_i}+\Delta_i+ t'H_i$ is not nef over $B$.
By the Cone Theorem (see \cite[Ch. 3]{Kollar-Mori} or \cite[5.4]{Hacon-Kovacs}), $K_{M_i}+\Delta_i+ t_iH_i$ is nef over $B$ and there exists a $(K_{M_i}+\Delta_i)$-negative extremal ray $R_i$ over $B$ such that $(K_{M_i}+\Delta_i+ t_iH_i) \cdot R_i=0$.
Let $c_i: M_i \to Z_i$ be the extremal contraction over $B$ associated to $R_i$, which exists by the Contraction part of the Cone Theorem \cite[(5.4.3-4)]{Hacon-Kovacs}).
If $\dim Z_i < \dim M_i$, then $c_i$ is a Mori fiber space and we stop.
If $c_i$ is not a Mori fiber space then it is either a divisorial or flipping contraction.
In the first case, we let $M_{i+1}=Z_i$ and $\psi_i=c_i$. In second case, we let $\psi_i: M_i \dashrightarrow M_{i+1}$ the $(K_{M_i}+\Delta_i+t'H_i)$-flip (which exists by \cite[Cor. 5.73]{Hacon-Kovacs}). By construction, $\psi_i$ extracts no divisors, meaning that $\psi_i^{-1}$ contracts no divisors.
By the contraction part of the Cone Theorem, the divisor $K_{M_{i+1}}+\Delta_{i+1}+ t_iH_{i+1}$ is nef over $B$. The pair $(M_i,\Delta_{i+1}+ t_iH_{i+1})$ is klt (see \cite[3.42--3.44]{Kollar-Mori}) and $M_i$ is $\mathbb{Q}$-factorial (see \cite[3.18]{Kollar-Mori}). If $\Delta=0$ and $M$ is terminal, then so is $M_i$. As long as $K_{M_i}+\Delta_i$ is not $\pi_i$-nef, $t_{i+1}$ is non zero and $\Delta_{i+1}+ t_iH_{i+1}$ is big over $B$. Thus we can keep going, producing a non increasing sequence $t_i \ge t_{i+1} \ge \cdots $ of non negative rational numbers and a sequence of birational maps $\psi_i: M_i \dashrightarrow M_{i+1}$ over $B$. The process stops if there exists an $N$ such that $c_N: M_N \to Z_N$ is a Mori fiber space over $B$ or such that $K_{M_N}+\Delta_N$ is nef over $B$. Otherwise, the sequence is infinite.
The pair $(M_i,\Delta_i+t_iH_i)$ is a log terminal model (ltm) for $(M, \Delta+t_iH)$ over $B$ (see Definition 5.29 and Lemma 5.31 of \cite{Hacon-Kovacs}).
We will need the following Lemmas:
\begin{lemma} \label{lem not iso}
For any $i>j$, let $\psi_{ij}: M_j \dashrightarrow M_i$ be the induced birational morphism over $B$. Then $\psi_{ij}$ is not an isomorphism.
\end{lemma}
\begin{proof}
This is \cite[Lem 5.62]{Hacon-Kovacs}.
\end{proof}
\begin{lemma}(\hspace{1sp}\cite[Ex. 5.10]{Hacon-Kovacs})
Let $(M, \Delta)$ be a klt pair as above and suppose that $\Delta$ is big over $B$ and that $K_M+\Delta$ is nef over $B$. Then $K_M+\Delta$ is semiample over $B$, i.e., there exists a projective morphism $f: M \to Z$ over $B$ and an ample divisor $L$ on $B$ such that $K_M+\Delta \sim_{\mathbb{Q},B} f^*L$.
\end{lemma}
\begin{proof}
Since $\Delta$ is big over $B$, we can write $\Delta\sim_{\mathbb{Q},B} A+C$, where $A$ is ample over $B$ and $C \ge 0$. Choose an $0 < \epsilon \ll 1$, such that $(M,\Delta')$ is klt, where $\Delta'=(1-\epsilon)\Delta+\epsilon C$.
Then
\[
(K_M+\Delta)-(K_M+\Delta')=\epsilon A,
\]
is ample over $B$. By the Basepoint free Theorem (e.g. see \cite[(5.1)]{Hacon-Kovacs}), $K_M+\Delta$ is semiample over $B$.
\end{proof}
\begin{lemma} \label{lemma sbl} Let the notation be as above and for any $i> 0$, let $\phi_i: M \dashrightarrow M_i$ as the induced birational map over $B$. Then the divisors contracted by $\phi_i$ are the divisorial components of $\mathbb B((K_{M_i}+\Delta_i+ t_iH_i)/B)$, the stable base locus over $B$ (cf. \cite[\S 2.E]{Hacon-Kovacs}). Similarly, $\psi_{ij}: M_j \dashrightarrow M_i$ contracts the divisorial components of $\mathbb B((K_{M_j}+\Delta_j+ t_iH_j)/B)$.
\end{lemma}
\begin{proof}
Since $(M_i, \Delta_i+ t_iH_i)$ is klt, $\Delta_i+ t_iH_i$ is big over $B$, and $K_{M_i}+\Delta_i+ t_iH_i$ is nef over $B$, by the lemma above, $K_{M_i}+\Delta_i+ t_iH_i$ is semiample over $B$.
Let $W$ be a smooth birational model resolving $\phi_i$, and let $p$ and $q$ be the induced birational morphisms to $M$ and $M_i$.
By \cite[Lemma 5.31]{Hacon-Kovacs} pair $(M_i,\Delta_i+t_iH_i)$ is a log terminal model for $(M, \Delta+t_iH)$ over $B$ (see \cite[Definition 5.29]{Hacon-Kovacs}).
Thus,
\begin{equation} \label{ltm}
p^*(K_M +\Delta+t_i H)=q^*(K_{M _i}+\Delta_i+t_i H_i)+ E
\end{equation}
where $E=\sum_F (a(F; M,\Delta+t_i H)-a(F; M_i,\Delta_i+t_i H)) F $ is an effective $q$-exceptional divisor whose support contains the divisors contracted by $\phi_i$. Since
$p^{-1} \mathbb B((K_M+\Delta+t_i H)/B)=\mathbb B(p^*(K_M+\Delta+t_i H)/B)=\mathbb B(q^*(K_{M _i}+\Delta_i+t_i H_i)+ E/B)=\Supp (E)$, the first statement follows. The second statement is proved in the same way, since by \cite[Lemma 5.31]{Hacon-Kovacs}, the pair $(M_i, \Delta_i+t_i H_i)$ is a log terminal model for $(M_j, \Delta_j+t_i H_j)$ over $B$ and hence the equivalent of (\ref{ltm}) holds.
\end{proof}
\begin{lemma} \label{lemma t goes to zero} Let the notation be as above. If the mmp with scaling does not terminate then
\[
\lim_{i \to \infty} t_i=0.
\]
\end{lemma}
\begin{proof}
This is \cite[Prop. 3.2]{Druel}. The only difference is the relative setting, but the proof is the same:
Suppose the mmp does not terminate and that $\lim t_i=t_\infty>0$. By \cite[Thm E]{BCHM} there are finitely many log terminal models of $(M, \Delta+(t_\infty+t) H)$, with $t \in [0,1-t_\infty]$. We have already observed that $(M_i, \Delta_i+t_i H_i)$ is a ltm for $(M, \Delta+t_i H)=(M, \Delta+t_\infty H+(t_i-t_\infty )H))$ over $B$. Thus, if the sequence is infinite there are integers $i >j$ such that the birational map $M_j \dashrightarrow M_i$ is an isomorphism. This gives a contradiction with Lemma \ref{lem not iso} above.
\end{proof}
\subsection{Variants}
In this section we give some variants of the results of the previous section. First we notice that the compactification result of Theorem \ref{hkcom} holds also for the twisted intermediate Jacobian fibration (see Remark \ref{twisted}). Then we consider the case of the intermediate Jacobian fibration associated to a mildly singular cubic fourfold (see Proposition \ref{checa} and Remark \ref{mild sing}). We then give a slightly stronger version of Theorem \ref{hkcom}, in that we show that the Lagrangian fibered hyper--K\"ahler compactification works in families (see Proposition \ref{Csix} and Theorem \ref{deg lagr 1}). As an application, we give a positive answer to a question of Beauville (see Remark \ref{rem beau}).
\begin{rem}[The twisted case] \label{twisted} In \cite{Voisin-twisted}, Voisin constructs a non trivial $J_U$-torsor $J_U^T \to U$ defined from a class in $H^1(U, \mathcal J_U[3])$, where $\mathcal J_U$ is the sheaf of holomorphic sections of $J_U \to U$ and where $\mathcal J_U[3] \subset \mathcal J_U$ is the sheaf of $3$-torsion points. The non triviality (for very general $X$) of this class corresponds to the non existence, for the universal family of hyperplanes sections of $X$, of a relative $1$-cycle of degree $1$.
The main result of the paper is to produce, for general $X$, a hyper--K\"ahler compactification $J^T=J^T(X)$ with Lagrangian fibration to $\P^5$ extending $J_U^T \to U$. This builds on the compactification of \cite{LSV}.
We will refer to this hyper--K\"ahler manifold as the twisted intermediate Jacobian fibration. This hyper--K\"ahler manifold is deformation equivalent to the non twisted version $J(X)$, as they agree as soon as $X$ has a $2$-cycle which restricts to a $1$-cycle of degree $1$ or $2$ on its hyperplane sections.
Lemma \ref{cor not uniruled}, Proposition \ref{prop hk model} and Theorem \ref{hkcom} work the same for the non-trivial torsor $J_U^T \to U$, giving a Lagrangian fibered hyper--K\"ahler $J^T=J^T(X)$ for every smooth $X$. In Section \ref{Kuz} we will return to the twisted intermediate Jacobian fibration and in Corollary \ref{J not JT} we prove that for very general $X$ these two fibrations are not birational and that on $J$ there is a unique isotropic class in the movable cone of $J$. This fact will be used in the Appendix \ref{appendix}.
\end{rem}
Finally, we show that the Lagrangian fibered hyper--K\"ahler compactification exists generically also over $\mathcal C_{6}$, the divisor in the moduli space of cubic fourfolds whose general point parametrizes cubics with one $A_1$ singularity. The following Proposition is an adaptation of \cite[\S 2]{LSV} to the case of a cubic fourfold with mild singularities.
\begin{prop} \label{checa}
Let $X_0 \subset \P^5$ be a cubic fourfold with one simple node $o \in X_0$ and no other singularities, let $U \subset \P^5$ be the open locus parametrizing smooth hyperplane sections, and let $\pi_U: J_U=J_U(X_0) \to U$ be the Donagi-Markman fibration. Then, there exits a holomorphic symplectic form $\sigma_U$ on $J_U$, which extends to a holomorphic two form on any smooth projective compactification. As a consequence, Lemma \ref{cor not uniruled} holds for $J_U$, namely
any projective compactification of $J_U$ has smooth locus admitting a generically non-degenerate holomorphic $2$-form extending $\sigma_U$ and is not uniruled.
Similarly, for the twisted intermediate Jacobian $J_U^T=J_U^T(X_0)$.
\end{prop}
\begin{proof}
Let $\widetilde X_0$ (resp. $\widetilde \P^5$) be the blow-up of $X_0$ (resp. $\P^5$) at the point $o$. Let $E \subset \widetilde \P^5$ be the exceptional divisor. Projection from $o$ determines an isomorphism $\widetilde X_0 \cong BL_S \P^4$, where $S$ is the $(2,3)$ complete intersection in $ \P^3$ parametrizing lines in $X_0$ by $o$. The surface $S$ is a smooth K3 surface and thus $H^1(\widetilde X_0, \Omega^3_{\widetilde X_0} )$ is one dimensional; let $\eta$ be a generator. The same argument as in \cite[Theorem 1.2 ]{LSV} shows that $\eta$ induces a holomorphic two form $\sigma$ on $J_U$, with respect to which the fibers of $J_U \to U$ are isotropic. To show that $\sigma$ is non-degenerate, it suffices to show that for any smooth hyperplane section $Y$ (which in particular does not pass by the point $o$), the map
\begin{equation} \label{etat}
T_{[Y]}U=H^0(Y,\mathcal O_Y(1)) \to H^1(Y, \Omega^2_Y)=H^0(J_U, \Omega^1_{J_U}),
\end{equation}
induced by $\sigma$, via the fact that the fibers of $J_U \to U$ are isotropic, is an isomorphism. By \cite[Theorem 1.2 (ii)]{LSV}, this map is given by the cup product with a class $\eta_Y \in H^1(Y, \Omega^2_Y(-1))$ defined in the following way: let $\eta_{|Y} \in H^1(Y, (\Omega^3_{\widetilde X_0})_{|Y} )$ be the restriction of $\eta$ to $Y$. Since $H^1(Y, \Omega^3_Y)=0$, the exact sequence $0 \to \Omega^2_Y(-1) \to (\Omega^3_{\widetilde X_0})_{|Y} \to \Omega^3_Y \to 0$ implies that $\eta_{|Y}$ lifts to a class $\eta_Y \in H^1( \Omega^2_Y(-1))$. By Griffiths residue theory (see \cite[ Lemma 1.7]{LSV}), $H^1( \Omega^2_Y(-1))$ is one dimensional and cup product with any non-zero element induces an isomorphism $H^0(Y,\mathcal O_Y(1)) \to H^1(Y, \Omega^2_Y)$;
more precisely, using the canonical isomorphism $\Omega^2_Y(-1)=T_Y(3)$, this space is spanned by the class of the non-trivial extension $0 \to T_Y \to (T_{\P^4})_{|Y} \to \mathcal O_Y(-3) \to 0$.
It follows that to show that (\ref{etat}) is an isomorphism, we only need to show that $\eta_Y \neq 0$, which amounts to showing that $\eta_{|Y} \neq 0$.
Under the isomorphism $ \Omega^3_{\widetilde X_0}=T_{\widetilde X_0}(-3)(2E)$, the class of a generator of $H^1(\widetilde X_0, \Omega^3_{\widetilde X_0} )$ corresponds to the class of the extension $ 0 \to T_{\widetilde X} \to (T_{\widetilde \P^5})_{|\widetilde X} \to \mathcal O_{\widetilde X}(3)(-2E) \to 0$. Restricting to $Y$ and considering the tangent bundle sequence for $Y$ in $\P^4$ we get the following diagram of short exact sequences
\[
\xymatrix{
0 \ar[r] & (T_{\widetilde X})_{|Y} \ar[r] & (T_{\widetilde \P^5})_{|Y} \ar[r] & \mathcal O_Y(3) \ar[r] & 0\\
0 \ar[r] & T_Y \ar[r] \ar[u] & (T_{ \P^4})_{|Y} \ar[u]^\alpha \ar[r] & \mathcal O_Y(3) \ar@{=}[u] \ar[r] & 0
}
\]
where the first two vertical arrows are injective. The extension class of the first row is $\eta_{|Y}$ and the second row is non split, as we already observed. Since $\coker (\alpha)=\mathcal O_Y(1)$, then $\Hom(\mathcal O_Y(3),\coker (\alpha))=0$. Thus any splitting of the first row would induce a splitting of the second row, giving a contraction.
\end{proof}
\begin{rem} \label{mild sing}
The Proposition \ref{checa} holds, more generally, for any cubic fourfold with isolated singularities, as long as a general one parameter smoothing of it has finite monodromy. This corresponds to the K3 surface $S$ of lines through one of the singular points having canonical singularities. The case of the degeneration to the chordal cubic \cite{Hassett-special}, which has finite monodromy but central fiber with $2$--dimensional singular locus, will be discussed at length in Section \ref{deg chordal}.
\end{rem}
\begin{prop} \label{Csix}
Let $X_0 \subset \P^5$ be as in Proposition \ref{checa} (or as in Remark \ref{mild sing}) and let $\pi_U: J_U \to U$ be the corresponding intermediate Jacobian fibration. Then there exists a hyper--K\"ahler compactification $J=J(X_0)$ of $J_U$, with a regular flat morphism to ${(\P^5)^\vee}$ extending $\pi_U$. Moreover, if $\mathcal X \to \Delta$ is a general family of smooth cubic fourfolds degenerating to $X_0$, then up to a base change, there exists a family of Lagrangian fibered hyper--K\"ahler manifolds
\[
\mathcal J \to \P^5_\Delta \to \Delta
\]
such that for $t \neq 0$, $\mathcal J_t=J(\mathcal X_t)$ is the LSV compactification and, for $t=0$, $\mathcal J_0$ is a hyper--K\"ahler compactification of $J_U=J_U(X_0)$. Similarly, the analogue statement holds for the twisted intermediate Jacobian.
\end{prop}
\begin{proof}
By Proposition \ref{checa} above, $J_U$ has a holomorphic symplectic form that extends to a regular form on any smooth projective compactification. As in Lemma \ref{cor not uniruled}, it follows that $J_U$ is not uniruled. Let $\mathcal X \to \Delta$ be a family of smooth cubic fourfolds degenerating to $\mathcal X_0=X_0$ with the property that for $t \neq 0$ $\mathcal X_t$ is general in the sense that of LSV. As in the beginning of Proposition \ref{hkcom}, let $\mathcal J_{\mathcal V} \to \mathcal V$ be such that the fiber over $t \neq 0$ of $\mathcal J_{\mathcal V} \to \Delta$ is the LSV compactification $J(\mathcal X_t)$ and, over $t=0$, is $J_U \to U$.
We are thus in the position of applying Theorem \ref{deg lagr 1} below, which proves the proposition.
\end{proof}
A consequence of this proposition is a positive answer to a question of Beauville \cite{Beauville-Fano-K3}, as explained in the following remark.
\begin{rem} \label{rem beau}
Given a smooth cubic threefold $Y$, let $\ell \subset Y$ be a line. In \cite{Beauville-cubicthreefolds, Druel}, it is shown that the moduli space of Ulrich bundles on $Y$ with rank $2$, $c_1=0$ and $c_2=2\ell$ is birational to the intermediate Jacobian of $Y$ (more precisely, it can be identified with the blowup of the intermediate Jacobian fibration along the Fano surface).
Now let $X_0$ be cubic fourfold with one simple node and let $S \subset \P^4$ be the $(2,3)$ complete intersection K3 surface parametrizing lines through the singular point of $X_0$. Consider the Mukai vector $v=2v_0=2(1,0,-1) \in H^*(S,\mathbb{Z})$ and let $\widetilde M_{2v_0}(S)$ be the symplectic resolution of the singular moduli space of OG$10$-type (cf. \S \ref{ogtentype}). \\
By considering the relative moduli spaces of Ulrich bundles supported on the $5$-dimensional family of cubic threefolds containing $S$ and by restricting the bundles to $S$, Beauville \cite[\S 5, Example $d=3$]{Beauville-Fano-K3} shows that there is a birational map $ J_U \dashrightarrow M_v(S)$. This induces a rational map $M_{2v_0}(S)\dashrightarrow \P^5$ and Beauville asks whether there exists a hyper--K\"ahler manifold birational to $M_v(S)$ which admits a \emph{regular} morphism to $\P^5$.
Proposition \ref{Csix} thus gives a positive answer to this question.
\end{rem}
The proof of the proposition above relies on the following Theorem, which is the Lagrangian fibration analogue of results from \cite[Thm 2.1 and Cor. 5.2]{KLSV}. Theorem \ref{deg lagr 1} will be used also in Section \ref{section general X} for the proof of Proposition \ref{3fams} (and thus also of Theorem \ref{NSU}).
As usual, $\Delta$ is an open affine subset of a smooth curve, or a small analytic disk. In both cases, we keep the notation $t=0$ to denote a chosen special point in $\Delta$ and $t \neq 0$ to denote any other point.
\begin{thm} \label{deg lagr 1} Let $\tilde f: \widetilde{\mathcal J} \to \Delta$ be a projective degeneration of hyper--K\"ahler manifolds of dimension $2n$. Suppose that there is a commutative diagram
\[
\xymatrix{
\widetilde{\mathcal J} \ar[dr]_{\tilde f} \ar[r]^{\tilde \pi} & \P^n_\Delta \ar[d]^p \\
& \Delta
}
\]
where $\widetilde{\mathcal J} \to \P^n_\Delta$ is a projective fibration such that for $t \neq 0$, $\mathcal J_t \to \P^n_t$ is a Lagrangian fibration. Assume that the central fiber $\widetilde{\mathcal J}_0= Y_0 +\sum_{i \in I} m_i Y_i$ has a reduced component $Y_0$ which is not uniruled. Suppose, furthermore, that there is an open subset of $ Y_0 \setminus \cup _{ i \ge 1} (Y_i \cap Y_0)$ such that the morphism to $\P^n_0$ is a fibration $J_{U_0} \to U_0 \subset \P^n_0$ in abelian varieties. Then
\begin{enumerate}
\item There exists a projective degeneration $\bar f: {\bar{\mc J}} \to \Delta$ of hyper--K\"ahler manifolds such that:
\begin{enumerate}
\item ${\bar{\mc J}}$ is $\mathbb{Q}$-factorial, terminal, and isomorphic to $\widetilde{\mathcal J}$ over $\Delta^*$;
\item The central fiber ${\bar{\mc J}}_0$ is a reduced, irreducible, and a normal symplectic variety with canonical singularities and admitting a symplectic resolution;
\item There is a relative Lagrangian fibration $\bar \pi: {\bar{\mc J}} \to \P^n_\Delta$ compatible, via the birational map $ {\bar{\mc J}} \dashrightarrow \widetilde{\mathcal J}$, with $\tilde \pi$ and such that, up to restricting the open set $ U_0 \subset \P^n_0$, the morphism ${\bar{\mc J}}_0 \to \P^n_0$ extends the abelian fibration $J_{U_0} \to U_0$.
\end{enumerate}
\item Up to a base change $\Delta' \to \Delta$, there exists a (non necessarily projective) family $\mathcal J \to {\Delta'}$ of hyper--K\"ahler manifolds, with a birational morphism $\mathcal J \to {\bar{\mc J}}':= {\bar{\mc J}} \times_{\Delta'} \Delta$ over $\Delta'$, which is an isomorphism away from the central fiber and in the central fiber is a symplectic resolution of ${\bar{\mc J}}_0$. Moreover, $\mathcal J$ has a family of Lagrangian fibrations $ \pi{'}: \mathcal J \to \P^n_{\Delta'}$ compatible with the base change of $\bar \pi$.
\end{enumerate}
\end{thm}
\begin{proof}
The proof follows ideas from \cite{Takayama,Kollar-Elliptic,KLSV}. Up to passing to a log resolution of the pair $(\widetilde{\mathcal J}, \widetilde{\mathcal J}_0)$, we can assume that $\widetilde{\mathcal J}_0= Y_0 +\sum_{i = 1}^k m_i Y_i$ is a normal crossing divisor. By \cite[Thm 2.1 and Cor 5.2]{KLSV}, running the mmp over $\Delta$ contracts the components $Y_i$, for $i \ge 1$, and yields a birational model of $\tilde{\mathcal J}$ with an irreducible central fiber which is a symplectic variety. In particular, $Y_0$ is the unique component of $\widetilde{\mathcal J}_0$ that is not uniruled (cf. \cite[Remark 2.2]{KLSV}). To prove the theorem we only need to show that the birational maps required to contract the other components can be preformed relatively to $\P^n_\Delta$ and, furthermore, that they induce isomorphism away from $\cup _{ i \ge 1} Y_i$. This is to ensure that the central fiber has a Lagrangian fibration extending $J_{U_0} \to U_0$ (maybe up to restricting the open subset $U_0 \subset \P^n_0$).
The canonical class $K_{\widetilde{\mathcal J}}$ is trivial over $\Delta^*$, so it is $\tilde f$-equivalent to divisor of the form $\sum_{i = 0}^k a'_i Y_i$. Following \cite[2.3 (1)]{Takayama} we set $r=\min{a'_i/m_i}$, so\footnote{For a projective morphism $f:A \to B$ and two $\mathbb{Q}$-Cartier divisors $D$ and $D'$ on $A$, we write $D=_{\mathbb{Q}, B}D'$ or $D\sim_{\mathbb{Q}, f}D'$ iff $D$ and $D'$ are $\mathbb{Q}$-linearly equivalent up to the pullback of a $\mathbb{Q}$-Cartier divisor from $B$.}
\[
K_{\widetilde{\mathcal J}}=_{\mathbb{Q}, \Delta} \sum_{i = 0}^k a_i Y_i,
\]
where $a_i=a'_i-rm_i \ge 0$ are non negative rational numbers and for at least one $i$, $a_i=0$. Let $J \subsetneq \{0, 1, \dots, k\}$ be the set of indices such that $a_i >0$ and let $J^c$ be its complement.
By \cite[Prop. 5.1]{Takayama}
\begin{enumerate}
\item For every $j \in J$, the irreducible component $Y_j$ is uniruled.
\item If $|J^c| \ge 2$, then for every $j \in J^c$, the irreducible component $Y_j$ is uniruled.
\end{enumerate}
Since $Y_0$ is not uniruled, it follows that $J=\{1, \cdots, k\}$ and thus
\[
K_{\widetilde{\mathcal J}}=_{\mathbb{Q}, \tilde \pi} \sum_{i=1}^k a_i Y_i, \quad \,\, a_i > 0.
\]
By assumption, for every $i \ge 1$, the closed subset $Y_0 \cap Y_i$ is in the complement of $J_{U_0}$ and, since the fibers of $\widetilde{\mathcal J}_0 \to \P^n_\Delta$ are connected, it follows that the induced map $Y_i \to \P^n_0$ is not dominant. Thus, the codimension of $ \tilde \pi(Y_i) $ in $\P^n_\Delta$ is greater or equal to $2$. In other words, $Y_i$ is $\tilde \pi$-exceptional.
We are in the same setting of Theorem \ref{hkcom}, namely a projective morphism from a smooth quasi-projective variety with a canonical class that is relatively $\mathbb{Q}$-linearly equivalent to an effective divisor all of whose components are relatively exceptional.
We can thus argue as in the proof of Theorem \ref{hkcom}, running the mmp over $\P^n_\Delta$ with scaling of an ample divisor in order to contract each of the $Y_i$, $i \ge 1$. This yields a birational map $\widetilde{\mathcal J} \dashrightarrow {\bar{\mc J}}$ over $\P^n_\Delta$, where ${\bar{\mc J}} \to \Delta$ has irreducible fibers and the fibration ${\bar{\mc J}} \to \P^n_\Delta$ has $\mathbb{Q}$-factorial terminal total space and is such that $K_{\bar{\mc J}}= \tilde \pi^* B $, for some $\mathbb{Q}$-divisor $B$ on $\P^n_\Delta$. Since at each step the $K$-negative rays of the mmp are contained in uniruled components of the central fiber, it follows that the birational map $\widetilde{\mathcal J} \dashrightarrow {\bar{\mc J}}$ is an isomorphism away from $\cup_{i \ge 1} Y_i$. In particular, the central fiber ${\bar{\mc J}}_0$, which is irreducible, has an open subset which is isomorphic to $ J_{U_0} $.
Since for $t\neq 0 $, $(K_{{\bar{\mc J}}})_{|\mathcal J_t}=0$, $B_{|\mathcal \P^n_t}=0$ for $t\neq 0$. In particular, $B$ is $p$-trivial, where $p: \P^n_\Delta \to \Delta$ is the projection, and thus $K_{{\bar{\mc J}}}$ is $\tilde f$-trivial. We can now argue as in the last part of the proof of \cite[Theorem 1.1]{LSV} to show that ${\bar{\mc J}}_0$ is a normal with canonical singularities. As in \cite[Corollary 4.2]{LSV} it follows that ${\bar{\mc J}}_0$ is a symplectic variety and that, up to a base change $\Delta' \to \Delta$ there exits a smooth family $\mathcal J \to \Delta'$ with a birational morphism $\mathcal J \to {\bar{\mc J}}':= {\bar{\mc J}} \times_{\Delta'} \Delta$ with the desired properties.
\end{proof}
\begin{rem} \label{rem other proof}
Theorem \ref{deg lagr 1} gives another proof of Theorem \ref{hkcom}, as well as the stronger statement of the existence of a relative intermediate Jacobian fibration $\mathcal J \to \P^5_\Delta$, associated to any family $\mathcal X \to \Delta$ of smooth cubic fourfolds for which the general fiber is general in the sense of LSV.
\end{rem}
\section{Moduli spaces of OG$10$-type} \label{ogtentype}
By \cite[Cor 6.3]{LSV} (see also \cite[\S 6.3]{KLSV}) any hyper--K\"ahler compactification $J$ of $J_U$ is deformation equivalent to O'Grady's $10$-dimensional example. We start this section by recalling the basic definitions and first properties of those singular moduli spaces of sheaves on a K3 surface whose symplectic resolutions are hyper--K\"ahler manifolds in this deformation class. Then we use the methods of Bayer-Macr\`i, as adapted by Meachan-Zhang to this class of singular moduli spaces, to study the movable cone of certain moduli spaces that appear naturally as limits of the intermediate Jacobian fibration, when the underlying cubic fourfold degenerates to the chordal cubic (cf. Section \ref{deg chordal}).
We start by recalling the following fundamental theorem.
\begin{thm}[\hspace{1sp}{\cite{Mukai,Yoshioka,OGrady99,Lehn-Sorger, Kaledin-Lehn-Sorger,Perego-Rapagnetta}}] \label{OG10} Let $(S,H)$ be a general polarized K3 surface and let $v_0 \in H^*_{alg}(S,\mathbb{Z})$ be a primitive Mukai vector which we suppose to be positive in the sense of \cite[Def. 5.1]{Bayer-Macri-proj} (cf. also \cite[Rem. 3.1.1]{dCRS}). Let $m \ge 2$ be an integer. The moduli space $M_{mv_0, H}(S)$ of $H$--semistable sheaves on $S$ with Mukai vector $mv_0$ is an irreducible normal projective symplectic variety of dimension $m^2v_0^2+2$, which admits a symplectic resolution if and only if $m=2$ and $v_0^2=2$.
When this is the case, the symplectic resolution $\widetilde M_{2v_0, H}(S) \to M_{2v_0, H}(S)$ is the blow up of the singular locus $ \Sym^2 M_{v_0, H}(S) \subset M_{2v_0, H}(S)$, with its reduced induced structure. Moreover, $\widetilde M_{2v_0, H}(S) $ is an irreducible holomorphic symplectic manifold and its deformation class is independent of $(S, H)$ and of $v_0$; in particular, $\widetilde M_{2v_0, H}(S) $ is deformation equivalent to O'Grady's original $10$-dimensional exceptional example.
\end{thm}
We will refer to a Mukai vector of the form $2 v_0$ with $v_0^2=2$ as a Mukai vector of OG$10$--type and to a hyper--K\"ahler manifold in this deformation class as a hyper--K\"ahler of OG$10$-type.
\subsection{Contracting the relative theta divisor on relative Jacobian of curves} \label{ogten}
It is known \cite{Arcara-Bertram, Bayer-Macri-MMP, Bayer-Macri-proj, Bayer-BN} that the birational geometry of moduli spaces of pure dimension one sheaves on a K3 surface is related to Brill-Noether loci. For example, on the degree $g-1$ Beauville-Mukai system of a genus $g$ linear system on a K3 surface, the relative theta divisor can be contracted, possibly after performing a finite sequence of birational transformations. This is the content of the following example.
\begin{example} [\hspace{1sp}\cite{Arcara-Bertram,Bayer-BN}]
Let $(S,C)$ be a general polarized K3 surface of genus $g$, with $\NS(S)=\mathbb{Z} C$. Set $v=(0,C,0) \in H^*(S, \mathbb{Z})$ and let $M_{v}$ be the moduli space of $C$-stable sheaves on $S$ with Mukai vector\footnote{This Mukai vector is not positive in the sense defined above, since both the first and last entry are zero. However, since for general $(S,C)$, tensoring by $C$ induces an isomorphism with $M_{v'}$, where $v'=(0,C,g-1)$, the results of \cite{Bayer-Macri-MMP} still hold. See also \cite{Perego-Rapagnetta} for other considerations about the last entry of the Mukai vector.} $v$. Since we are assuming $(S,C)$ to be general in moduli, we are suppressing the polarization from the notation (thus $M_v$ will denote the moduli space of $C$-semistable sheaves on $S$ with Mukai vector $v$; when we consider instead a Bridgeland stability condition $\sigma$, the corresponding moduli space will be denoted $M_{v, \sigma}$). This moduli space is smooth and $M_{v} \to \P^g=|C|$ is the degree $g-1$ relative compactified Jacobian of the genus $g$ linear system $|C|$ on $S$. There is a naturally defined effective, irreducible, relatively ample theta divisor $\theta \subset M_{v}$ which parametrizes sheaves with a non trivial global section and which can be realized as the zero locus of a canonical section of the determinant line bundle (see \cite[\S 2.3]{LePotier} or \cite[Thm 5.3]{Alexeev-compactified}). Recall that there is a Hodge isometry $\NS(M_{v})\cong v^\perp= \langle (0,0,1), (1,0,0)\rangle$ (see for example \cite[Theorem 3.6 ]{Bayer-Macri-MMP}).
The class $\ell:=(0,0,1)$ is the class of the isotropic line bundle inducing the Lagrangian fibration $M_{v} \to \P^g$ while the theta divisor $\theta$ corresponds to the class $-(1,0,1)=-v(\mathcal O_S)$ (see \cite[pg. 643]{LePotier} or also \cite[Prop. 7.1 and Thm 12.3]{Bayer-Macri-MMP}\footnote{Compared to \cite{Bayer-Macri-MMP}, there is a difference in a choice of sign in the isomorphism $\NS(M_{v})\cong v^\perp$.}). Since $\theta^2=-2$, the irreducible effective divisor $\theta$ is prime exceptional. By \cite{Druel-exc}, it can be contracted on a hyper--K\"ahler birational model of $M_v$. Since the rays corresponding to divisorial contractions and to Lagrangian fibrations must be in the boundary of the movable cone \cite{Huybrechts-kahler-cone}, it follows that
\[
\overline{ \Mov}(M_{v})=\mathbb{R}_{\ge 0}\ell +\mathbb{R}_{\ge 0}h
\]
where $h=(-1,0,1) \in \theta^\perp \cap v^\perp$ is a big line which is nef on some birational model of $M_{v}$ (this also follows from \cite[Thm 12.3]{Bayer-Macri-MMP}). Using \cite{Bayer-Macri-MMP}, the walls of the nef cones of the various birational models can be computed. Since we don't need this, we omit the computation.
\end{example}
\subsection{Movable cones of certain moduli spaces of OG$10$-type.} \label{movog10} If we consider a non primitive genus $g$ linear system $|mC|$, $m \ge 2$, then the relative compactified Jacobian of degree $g-1$ is singular. For singular moduli spaces of OG$10$-type, i.e. when $v=2w$ with $w^2=2$, \cite{Meachan-Zhang} have adapted the techniques of Bayer-Macr\`i \cite{Bayer-Macri-proj,Bayer-Macri-MMP} to compute the nef and movable cones of these moduli spaces. We refer to \cite{Bridgeland,Arcara-Bertram,Bayer-Macri-proj,Bayer-Macri-MMP} for the relevant definitions and main results on Bridgeland stability conditions on K3 surfaces and to \cite{Meachan-Zhang} for the results on moduli spaces of OG$10$ type.
By \cite[Thm. 7.6 (3)]{Meachan-Zhang}, all birational models of $M_{2w}=M_{2w, C}$ which are isomorphic to $M_{2w}$ in codimension one are isomorphic to a Bridgeland moduli space $M_{2w, \sigma}$, for some Bridgeland stability condition $\sigma$ on $S$. Moreover, by \cite[Cor. 2.8]{Meachan-Zhang}
\begin{equation} \label{NSsing}
\NS(M_{2w, \sigma}) \cong w^\perp.
\end{equation}
We now apply the results of \cite{Meachan-Zhang} to describe the nef and movable cones of certain singular models of OG$10$ appearing as limits of the intermediate Jacobian fibration. By \cite{Perego-Rapagnetta-factoriality}, the factoriality properties of a singular moduli space $M_{2w}$ of OG$10$ type depend on the divisibility of the primitive Mukai vector $w \in H^*_{alg}(S,\mathbb{Z})$. More precisely, by \cite[Thm 1.1]{Perego-Rapagnetta-factoriality}, $M_{2w}$ is factorial if and only if $w \cdot u \in 2\mathbb{Z}$ for every $u \in H^*_{alg}(S, \mathbb{Z})$. Otherwise, $M_{2w}$ is $2$-factorial. Since there can be different birational models with different factoriality properties (cf. Remark \ref{g12}), it is important to choose the correct model to work with.
Now let $(S,C)$ be a general K3 surface of degree $2$ and set
\begin{equation} \label{vkappa}
v_k:=(0, C, k-2),
\end{equation}
The Le Potier morphism $\pi: M_{2v_k} \to \P^5$ realizes the singular moduli space $M_{2v_k}$ as a compactification of the degree $2k$ relative Jacobian of the genus $5$ hyperelliptic linear system $|2C|$. Composing $\pi$ with the sympectic resolution $m: \widetilde M_{2v_k} \to M_{2v_k} $, we get a natural Lagrangian fibration:
\begin{equation} \label{lagrfibr}
\widetilde \pi: \widetilde M_{2v_k} \to \P^5
\end{equation}
By the result of Perego-Rapagnetta mentioned above, $M_{2v_k}$ is factorial if and only if $k$ is even. It turns out that the birational class of these moduli spaces is independent of $k$, but the isomorphism class depends on the parity of $k$ (see \cite[Proposition 3.2.7]{dCRS}): Indeed, tensoring a pure dimension one sheaf by $\mathcal O_S(C)$ determines an isomorphism
\begin{equation} \label{isokeven}
M_{2v_k} \stackrel{\sim}{\to} M_{2v_{k+2}}.
\end{equation}
\begin{rem} \label{g12}
Tensoring a line bundle supported on a smooth hyperelliptic curve of genus $5$ by the unique $g^1_2$ on the curve defines a birational morphism $M_{2v_k} \dashrightarrow M_{2v_{k+1}}$
(I thank A. Rapagnetta for pointing out this to me). As a side remark, notice that the map thus defined is \emph{not} an isomorphism in codimension one. Indeed, it can be checked that when passing to the birational morphism $\widetilde M_{2v_k} \dashrightarrow \widetilde M_{2v_{k+1}} $ between the two resolutions, which is an isomorphism in codimension $2$, the exceptional divisor of one model is exchanged with the proper transform of the locus parametrizing sheaves on reducible curves on the other model.
\end{rem}
In view of Lemma \ref{collino} below and the isomorphism (\ref{isokeven}), we will focus on the case $k=0$.
\begin{rem} \label{zerosect}
For general $(S, C)$ it is not hard to check that the structure sheaf of every curve in $|2C|$ satisfies the numerical criterion for $C$-stability and hence that the fibration $M_{2v_0} \to \P^5$ admits a regular zero section. Notice also that the image of this section is not contained in the singular locus of $M_{2v_0}$.
\end{rem}
By \cite[Cor. 2.8]{Meachan-Zhang}, $\NS(M_{2v_0}) \cong v_0^\perp= U= \langle (0,0,1), (-1,C,0)\rangle$, where, as above,
\begin{equation} \label{ell}
\ell=(0,0,1)
\end{equation}
is the line bundle inducing the Lagrangian fibration $\pi: M_{2v_0} \to \P^5=|2C|$. Under the isomorphism $M_{2v_2} \cong M_{2v_{0}}$, induced by tensoring with $\mathcal O_S(-C)$, the relative theta divisor is mapped isomorphically to the prime exceptional divisor
\begin{equation} \label{deftheta}
\theta:=-(1,- C, 2),
\end{equation}
parametrizing sheaves which receive a non trivial morphism from the spherical object $\mathcal O_S(-C)$ (see also Lemma \ref{MZ}). Indeed, the relative theta divisor in $M_{2v_2}$ parametrizes sheaves with a non trivial morphism from $\mathcal O_S$ and thus its image in $M_{2v_0}$ is exactly the divisor $\theta$.
Notice that
\begin{equation} \label{thetasq}
\theta^2=-2.
\end{equation}
For later use we highlight the following remark:
\begin{rem} \label{Alexeev}
The effective divisor $\theta \subset M_{2v_0}$ with cohomology class (\ref{deftheta}) does not contain the singular locus of $M_{2v_0}$: Using the description of $\theta$ as the zero locus of a section of the determinant line bundle \cite[Thm 5.3]{Alexeev-compactified}, which is compatible with $S$-equivalence classes, it is enough to show that the section defining $\theta$ is not identically zero on the singular locus of $M_{2v_0}$. It is therefore sufficient to show that there are $S$-equivalence classes of polystable sheaves all of whose members have a zero space of global sections. This is clear, since the generic semistable sheaf with Mukai vector $2v_0$ is an extension of two degree $1$ line bundles each supported on two distinct curves of genus $2$.
\end{rem}
The following Lemma is an application of \cite[Thms 5.1-5.2-5.3]{Meachan-Zhang} to $M_{2v_0}$ (N.B. Example 8.6 of loc. cit. is for odd $k$, so in view of Remark \ref{g12} it is concerned with a birational model of $M_{2v_0}$ which is not isomorphic in codimension one and hence we cannot immediately apply it here).
\begin{lemma} \label{MZ}
Let the notation be as above. Then
\[
\Nef(M_{2v_0})= \mathbb{R}_{\ge 0}{\ell}+\mathbb{R}_{\ge 0}h_0 , \quad \Mov(M_{2v_0,C})= \mathbb{R}_{\ge 0} \ell+\mathbb{R}_{\ge 0} h .
\]
where
\[
\ell = (0,0,1), \quad h_0=(-1,C,1), \quad h=(-1,C,0).
\]
Moreover, the wall spanned by $h_0=(-1,C,1) $ contracts the zero section of $M_{2v_0} \to \P^5$ and the class corresponding to $h=(-1,C,0)$ is big and nef on the Mukai flop of $M_{2v_0} $ along the zero section and contracts the proper transform of $\theta$.
\end{lemma}
\begin{proof} Since $\ell=\pi^* \mathcal O_{\P^5}(1)$ is nef and isotropic, it is one of the two rays of both the Nef and the Movable cones of $M_{2v_0}$. By \cite[Thm 5.3]{Meachan-Zhang} there is a divisorial contraction of BNU-type (notation as in loc. cit), determined by the spherical class $s=(1,-C,2)$, which is orthogonal to $v_0$.
The second ray of the movable cone is thus determined by $s^\perp \cap v_0^\perp$. We pick $h=(-1,C,0)$ as a generator of this ray, since $h \cdot \ell >0$.
By the same theorem in \cite{Meachan-Zhang}, the flopping walls are determined by $w^\perp \cap v_0^\perp$ for $w$ spherical and such that $w \cdot v_0=2$. There is a unique ray in $\Mov( M_{2v_0})$, that is of this form. It is determined by $w=(1,0,1)=v(\mathcal O_S)$ or, equivalently, by $w'=(-1,2C, -5)=2v_0-w$. We can choose $h_0=(-1,C,1)$ as generator of this ray.
As in \cite[Remark 8.5]{Meachan-Zhang}, we can see that this wall corresponds to the flop of the $\P^5$ corresponding to the sheaves with a morphism from $\mathcal O_S$, i.e., of the image of the zero section.
\end{proof}
\begin{rem}
It can be shown that the birational model on the other side of the wall can be identified with the Gieseker moduli space $M_{2w_0}$, where $w_0=(2,C,0)$.
Since we don't need this in the rest of the paper, we omit the proof.
\end{rem}
\begin{rem} \label{thetaamp}
The theta divisor $\theta$ is Cartier, since by \cite{Perego-Rapagnetta} $M_{2v_0}$ is factorial (see also \S \ref{movog10}). Moreover, it is relatively ample over $\P^5$, since by the description of the Nef cone of Lemma \ref{MZ} we can write $\theta$ as a sum of an ample line bundle and a multiple of $\ell=\pi^* \mathcal O_{\P^5}(1)$.
\end{rem}
\section{The relative theta divisor on the intermediate Jacobian fibration.} \label{section theta}
For any smooth cubic threefold $Y$, there is a canonically defined theta divisor in $\Jac(Y)$, which is $(-1)$-invariant and whose unique singular point lies at the origin. For the hyper--K\"ahler compactification $J=J(X) \to {(\P^5)^\vee}$ of the intermediate Jacobian fibration associated to a smooth cubic $4$-fold $X$, there is an effective relative theta divisor $\Theta \subset J$, which is defined as the closure of the union of the canonical theta divisor in the smooth fibers. More precisely, by
\cite{Clemens-Griffiths,Clemens} (see also \cite[Lem. 5.4]{LSV}), $\Theta $ can be defined
as the closure of the image of the Abel-Jacobi difference mapping
\begin{equation} \label{defTheta}
\begin{aligned}
\mathcal F \times_{{(\P^5)^\vee}} \mathcal F & \dashrightarrow J\\
(\ell, \ell', Y) & \longmapsto \phi_Y(\ell-\ell')
\end{aligned}
\end{equation}
The relative theta divisor $\Theta$ played an important role in \cite{LSV}, where it was shown that for general $X$ the divisor $\Theta$ is $\pi$-ample and $J$ is identified with the relative Proj of the sheaf of $\mathcal O_{\P^5}$-algebras associated with this divisor. Another useful way of realizing the Theta divisor is using twisted cubics \cite{Clemens}.
Let $Z=Z(X)$ be the Lehn-Lehn-Sorger-van Straten $8$fold \cite{LLSvS}. Then $Z$ is the blowdown $g: Z' \to Z$ of a smooth $10$ fold $Z'$ whose points parametrize nets of (generalized) twisted cubics. The exceptional locus of $g$ parametrizes non ACM cubics and its image in $Z$ is isomorphic to the cubic itself.
Let
\[
r: \P_{Z'} \to {Z'}
\]
be the $\P^1$-bundle over $Z'$ whose fiber over a twisted cubic $[C] \in Z'$ is the pencil $\P^1_{ C }$ of hyperplane sections of $X$ containing $\Sigma_C:=X \cap \langle C \rangle$. Here $ \langle C \rangle=\P^3$ is the linear span of the curve. By \cite[Sublemma 5.5]{LSV} (see also \cite[\S 4]{Clemens} or \cite[Prop. 6.10]{KLSV}), the Abel-Jacobi map
\begin{equation} \label{AJtwisted}
\begin{aligned}
\varphi: \P_{Z'} & \dashrightarrow J\\
(C, Y) & \longmapsto \phi_Y(C-h^2)
\end{aligned}
\end{equation}
is birational onto its image, which is precisely $\Theta$. Here $h^2$ is the class of the intersection of two hyperplanes in $Y$.
\begin{rem} \label{nonACM}
For later use, we remark the following two facts. First of all that the restriction of $\P_{Z'}$ to the locus of non-CM cubics is mapped to the zero section of $J \to \P^5$ (which lies in $\Theta$).
Second, using the Gauss map (see \cite[\S 12]{Clemens-Griffiths} or also \cite[\S 3]{Harris-Roth-Starr}), one can see that if $C $ is a twisted cubic in a smooth cubic threefold $Y$ with the property that $ \phi_Y(C-h^2)=0$ in $\Jac(Y)$, then the cubic surface $\Sigma_C= Y \cap \langle C \rangle$ is singular.
\end{rem}
For every $X$, the N\'eron-Severi group of $J=J(X)$ has at least rank $2$, since
\[
\NS(J(X)) \supset \langle L, \Theta \rangle.
\]
Here $L=\pi^* \mathcal O_{\P^5}(1)$ and $\Theta$ is, as above, the relative theta divisor obtained as the closure of the image of (\ref{defTheta}).
\begin{lemma} \label{tr lattice} For any smooth $X$, there is an isomorphism of rational Hodge structures
$H^2(J, \mathbb{Q})_{tr} \cong H^4(X, \mathbb{Q})_{tr}$. In particular, $\rho(J)=\rk H^{2,2}(X,\mathbb{Q})+1$.
\end{lemma}
\begin{proof}
The first statement was already noted in \cite{LSV}, while the second follows from the first and the fact that $b_2(J)=24$ and $b_4(X)=23$.
\end{proof}
\begin{rem} \label{section deform}
The locus, inside $\Def(J)$, parametrizing intermediate Jacobian fibrations is of codimension $2$ and corresponds to the locus where the classes $L$ and $\Theta$ stay of type $(1,1)$. By \cite[Thm 6]{Sawon}, a Lagrangian fibration with a section deforms, as Lagrangian fibration with a section, over a smooth codimension $2$ locus of the deformation space of the underlying hyper--K\"ahler manifold. Since by Theorem \ref{LSVthm} for general $X$ the LSV compactification $J(X)$ has a section, it follows that the codimension $2$ locus where $L$ and $\Theta$ stay algebraic is exactly the locus where the section deforms.
\end{rem}
We highlight the following corollary for future reference.
\begin{cor} \label{onlyJ} For very general $X$, $\rho(J)=2$. Thus,
\begin{enumerate}
\item $J$ is the only projective hyper--K\"ahler birational model of $J_U$ where $L$ is nef. In particular, any hyper--K\"ahler compatification of $J_U$ with a Lagrangian fibration extending $J_U \to U$ is isomorphic to the \cite{LSV} compactification.
\item There is at most one prime exceptional divisor on $J$.
\end{enumerate}
\end{cor}
\begin{proof}
(1) Since $\rho(J)=2$, the boundary of movable cone of $J$ has two rays, of which $L$ is one. (2) If there is a prime exceptional divisor, its class has to be orthogonal to the second extremal ray of the movable cone \cite[Thm 1.5]{Markman-Survey}. Since two prime exceptional divisors with proportional classes have to be isomorphic \cite[Cor. 3.6 (3)]{Markman-prime-exceptional}, there is at most one prime exceptional divisor.
\end{proof}
The following Lemma was communicated to me by K. Hulek and R. Laza. I thank them for sharing this observation with me and for raising the question of computing $q(\Theta)$.
\begin{lemma} \label{LTheta}
$q(L, \Theta)=1$. In particular, $\langle L, \Theta \rangle$ is a primitive sublattice of $\NS(J)$, isomorphic to the standard hyperbolic lattice $U$ of rank $2$. For very general $X$, $\NS(J)=U$.
\end{lemma}
\begin{proof} The computation of $q(L, \Theta)$ goes as in \cite[Lemma 1]{Sawon-discr}: one expands in $t$ the Fujiki equality $q(L+t \Theta)^5=c(L+t \Theta)^{10}$, where $c=945$ is the Fujiki constant \cite{Rapagnetta} and uses the fact that $\Theta^5L^5=(\Theta_{|J_{[H]}})^5=5!$. The final statement follows from Lemma \ref{tr lattice}.
\end{proof}
I thank C. Onorati for many discussions around $\Theta$ and for his interest in the following computation.
\begin{prop} \label{Thetasq} The irreducible divisor $\Theta \subset J$ is prime exceptional, in particular, it can be contracted on some projective birational hyper--K\"ahler model of $J$. Moreover, $q(\Theta)=-2$.
\end{prop}
\begin{proof}
Let $\varphi: \P_{Z'} \dashrightarrow \Theta$ be the Abel-Jacobi map as in (\ref{AJtwisted}) and let $V \subset Z'$ be a non empty open subset such that the restriction of $\varphi$ to $r^{-1}(V)=:\P_V$ is regular.
By restricting $V$ if necessary, we can assume that all twisted cubic parametrized by $V$ are such that $\Sigma_C$ is smooth and, in particular, that $C$ is ACM.
Recall that for any twisted cubic $[C] \in V$, we have set $\P^1_{C}=r^{-1}(C)$. The rational curve $\varphi(\P^1_{C}) \subset J$ is smooth because it maps to $\pi(\varphi(\P^1_{C})) \subset \P^5$, which is the pencil of hyperplane sections of $X$ that contain the curve $C$. Moreover, $(\pi \circ \varphi)(\P_V)$ intersects the dual variety $X^\vee$ in a dense open subset and, similarly, $\varphi(\P_V)$ intersects $\Theta_{X^\vee}$, the restriction of $\Theta$ to $X^\vee$, in a dense open subset (this statement follows from \cite{Clemens} and the fact, proven there, that for a cubic threefold with one $A_1$ singularity the Abel-Jacobi mapping is birational onto its image).
We start by showing that for a general $[C] \in V$, the smooth rational curve $\varphi(\P^1_{C})$ is contained in the smooth locus of $\Theta$.
The singular locus $\Theta_{sing}$ of $\Theta$ has an irreducible component that is equal to the closure of the zero section of $J_U \to U$, while any other irreducible component of the singular locus is properly contained in $\Theta_{X^\vee}$, the restriction of $\Theta$ to $X^\vee$.
Since $V$ parametrizes ACM curves, by Remark \ref{nonACM} it follows that the intersection of $\varphi(\P_V)$ with the image of the zero section of $J \to \P^5$ is contained in $J_{X^\vee}$, the restriction of $J$ to $X^\vee \subset \P^5$.
Let $B:= \varphi^{-1}(\Theta_{sing} \cap \varphi(\P_V))$ be the locus in $\P_V$ parametrizing points mapped to the singular locus of $\Theta$ and let $W_V :=(\pi \circ \varphi)^{-1}(X^\vee \cap (\pi \circ \varphi)(\P_V) )$ be the locus in $\P_V$ of pairs $(C, Y)$ such that $Y$ is singular. By what we have observed, it follows that $B \subseteq W_V$.
Notice that $W_V$ is irreducible of dimension $8$, because it maps to an open subset of $X^\vee$, with fibers parametrizing equivalence classes of twisted cubics contained in a cubic threefold with one $A_1$ singularity; these form an irreducible subset of $Z(X)$, as follows from \cite[\S 3]{Clemens}.
We have already observed that the general point of $\Theta_{X^\vee}$ is contained in the image $ \varphi(\P_V)$.
Thus, if $Y_0$ corresponds to a general point in $X^\vee$, there is a twisted cubic $C \subset Y_0$, with $[C] \in V$ and such that $\varphi(C,Y_0)=\phi_{Y_0}(C)$ lies in the smooth locus of $\Theta$.
It follows that $B$ is strictly contained in $W_V$. Since $W_V$ is irreducible, $\dim B=7$. Thus $B$ does not dominate $V$ and hence the image of the open subset $\P_{V'}:=r^{-1}(V')$, where $V':=V \setminus r(B)$ is contained in the smooth locus of $\Theta$.
For the general point in $(C, Y) \in \P_V$, let $R:=\varphi(\P^1_{C}) \subset \Theta$ be the corresponding element of the ruling.
By generic smoothness, the differential of $\varphi$ is of maximal rank at a general point $x \in R$, so by \cite[II. 3.4]{Kollar-rational-curves}, the vector bundle $(T_\Theta)_{|R}$ is globally generated at $x \in R$.
It follows that $(T_\Theta)_{|R}=\oplus \mathcal O_R(a_i)$, with $a_i \ge 0$.
By Lemma \ref{normalbundle} below, the restriction of the tangent bundle of $J$ to the smooth rational curve $R$ is of the form $\mathcal O_R^{\oplus 8} \oplus \mathcal O_R (2) \oplus \mathcal O_R(-2)$.
Using this and the fact that $R $ is contained in the smooth locus of $\Theta$, we find that $(T_\Theta)_{|R}= \mathcal O_R(2) \oplus \mathcal O_R^{\oplus 8}$ and hence that $N_{R | \Theta}=\oplus \mathcal O_R^{\oplus 8}$. In particular, $\Theta \cdot R=-2$.
Consider the lattice embedding $H^2(J, \mathbb{Z}) \subset H^2(J,\mathbb{Z})^\vee=H_2(J,\mathbb{Z})$ induced by the Beauville-Bogomolov form. We claim that under this embedding, the classes of $R$ and of $\Theta$ are equal, i.e. that $R \cdot x= q(\Theta, x)$ for every $x \in H^2(J,\mathbb{Z})$.
This immediately proves the proposition, as it implies that $q(\Theta)=\Theta \cdot R=-2$. By \cite[Cor. 3.6 (1)]{Markman-prime-exceptional} and \cite[Prop. 4.5]{Druel-exc}, the class of the ruling of a prime exceptional divisor is proportional, via a positive constant, to the class of the exceptional divisor. Thus, to prove the claim it suffices to show that $\Theta$ is prime exceptional, since the constant would have to be equal to $1$, as both $R \cdot L$ and $q(\Theta, L)$ are equal to $1$.
To prove that $\Theta$ is prime exceptional we use standard techniques on deformations of maps from rational curves to hyper--K\"ahler manifolds, following \cite[\S 5.1]{Markman-prime-exceptional} or also \cite[\S 3]{Charles-Mongardi-Pacienza}.
We include a proof because of setting of Markman is different and because the proof in \cite[\S 3]{Charles-Mongardi-Pacienza} is for projective families of hyper--K\"ahler manifolds.
Choose $R\subset \Theta$ a general element in the ruling and let $\Def(J)_R \subset \Def(J)$ be the smooth hypersurface in the deformation space of $J$ where the class of $R$ stays of Hodge type.
Let $\mathcal {H}ilb \to \Def(J)_R$ be the component of the relative Duady space containing the point $[R]$. Since $N_{R|J}=\mathcal O_R(-2) \oplus \mathcal O_R^{\oplus 8}$, then by \cite[Thm 1]{Ran} it follows that the morphism $\rho: \mathcal {H}ilb \to \Def(J)_R$ is smooth at $R$ and of relative dimension $8$.
Let $T \subset \Def(J)_R$ be a general curve containing $0$ (in particular we can assume that for very general $t \in T$, the N\'eron-Severi of the corresponding deformation $\mathcal J_t$ of $J$ is one dimensional and spanned by a line bundle whose class is proportional to $R_t$, the parallel transport of the class of $R$ to $\mathcal J_t$) and let $\rho_T: \mathcal H ilb_T \to T$ be the component of the base change to $T$ of $\mathcal H ilb \to \Def(J)_R$ that contains $[R]$.
Since $\rho$ is smooth at $[R]$, $\rho_T$ is dominant of relative dimension $8$.
Up to a base change and to restricting $T$, we can assume that $\mathcal H ilb_T \to T$ has irreducible fibers for $t \neq 0$.
Let $\mathcal J_T \to T$ be the base change of the universal family to $ T \to \Def(J)$ and let $\mathcal D \subset \mathcal J_T$ be the image of the universal family over $\mathcal H ilb_T$ under the evaluation map.
Then $\mathcal D$ is irreducible of relative codimension $1$. Moreover, $\mathcal D_t$ irreducible for $t \neq 0$, and $D:=\mathcal D_0$ is a union of effective uniruled divisors containing $\Theta$ as an irreducible component (with a given multiplicity $m\ge 1$). By the choice of $T$, for very general $t$, $\rho(\mathcal J_t)=1$. It follows that the class of $\mathcal D_t$ is proportional to the class of $R_t$ and hence that the class of $D=\mathcal D_0$ is proportional to that of $R$. Moreover, the proportionality constant is positive, as both $D$ and $R$ intersect positively with a K\"ahler class.
Hence, since $\Theta \cdot R$ is negative, so is $q(\Theta, D)$. Moreover, since the product of two distinct irreducible uniruled divisor is non negative, it follows that $q(\Theta,D) \ge m \, q(\Theta, \Theta)$.
Thus $q(\Theta, \Theta)<0$, i.e., $\Theta$ is prime exceptional. Thus, as already observed, the classes of $\Theta$ and of $R$ have to be the same and hence $q(\Theta, \Theta)=-2$.
\end{proof}
\begin{rem}
A posteriori, once we know that $\Theta$ is prime exceptional, we can use \cite[Lem 5.1]{Markman-prime-exceptional} to show that $\mathcal D_0=\Theta$.
\end{rem}
\begin{lemma} \label{normalbundle}
Let $M$ be a hyper--K\"ahler manifold of dimension $2n$ and $R \subset M$ be a smooth rational curve. Suppose $R$ is a general ruling of a uniruled divisor. Then
\[
(T_M)_{|R}=\mathcal O_R^{\oplus 2n-2} \oplus \mathcal O_R (2) \oplus \mathcal O_R(-2), \quad \text{and thus} \quad N_{R|M}=\mathcal O_R(-2) \oplus \mathcal O_R^{\oplus 2n-2}
\]
\end{lemma}
\begin{proof}
Since $T_M$ is self dual, $(T_M)_{|R}=\bigoplus_i \mathcal O_R (a_i) \oplus \bigoplus_i \mathcal O_R (-a_i)$, for some $a_i \ge 0$. Since $R$ is general and its deformations sweep out a divisor, by \cite[II. 3.4]{Kollar-rational-curves}, the rank of the evaluation map $\rk[H^0(R, (T_M)_{|R}) \otimes \mathcal O_R \to (T_M)_{|R}]$ at a general point of $R$ is equal to $2n-1$. Hence $a_2= \dots =a_n=0$ and $a_1 \ge 2$ (cf. \cite[Prop. 4.5]{Druel-exc}). Since the normal sheaf of $R $ in $M$ is torsion free and contains the quotient $\mathcal O_R(a_1) \slash T_R=\mathcal O_R(a_1) \slash \mathcal O_R(2)$, it follows that $a_1=2$.
\end{proof}
Notice that the same argument as the last part of the proof of Proposition \ref{Thetasq} shows the following.
\begin{prop}
Let $M$ be a hyper--K\"ahler manifold of dimension $2n$ and let $E \subset M$ be an irreducible uniruled divisor. Suppose that a general curve $R$ in the ruling is smooth and that $E \cdot R <0$ (e.g. if $R$ is contained in the smooth locus of $E$), then $E$ is prime exceptional and hence, under the lattice embedding $H^2(M, \mathbb{Z}) \subset H^2(M,\mathbb{Z})^\vee=H_2(M,\mathbb{Z})$ induced by the Beauville-Bogomolov form, the classes of $E$ and $R$ are proportional by a positive constant.
\end{prop}
\begin{cor} \label{J not JT}
For very general $X$, the movable cone of $J(X)$ is spanned by $L$ and $H$, where $H$ is a generator of $\Theta^\perp \subset \NS(J)$ with $q(H,L)>0$ and $q(H) >0$, i.e.
\[
\Mov (J)=\mathbb{R}_{\ge 0} L+\mathbb{R}_{\ge 0} H .
\]
In particular, there is a unique hyper--K\"ahler model of $J$ with a Lagrangian fibration and $J$ is not birational to the twisted intermediate Jacobian fibration $J^T$.
\end{cor}
\begin{proof} We already know that one of the rays of the movable cone of $J$ is spanned by $L$.
By \cite[Thm 1.5]{Markman-prime-exceptional} the closure of the movable cone is spanned by classes that intersect non-negatively with all prime exceptional divisors. Since by Proposition \ref{Thetasq} $\Theta$ is prime exceptional, the second ray of the movable cone is determined by $\Theta^\perp$, which is spanned by a class $H$ which is big and nef on some birational hyper--K\"ahler model of $J$. Thus, $q(H) >0$ and $q(H,L)>0$. In particular, the movable cone is strictly contained in the positive cone implying that the only isotropic class that is movable is $L$.
\end{proof}
In terms of the other projective hyper--K\"ahler birational models of $J$, we can actually prove something more precise. The main result of \S \ref{section general X} describes, for general $X$, on which birational model of $J$ the proper transform of $\Theta$ can be contracted.
\subsection{Induced automorphisms} \label{induced auto}
For hyper--K\"ahler manifolds of K3$^{[n]}$-type, a considerable amount of literature has been devoted to the study and classification of automorphism groups. This includes studying the automorphisms induced from a K3 surface to the moduli spaces of sheaves on it. In view of Theorem \ref{hkcom}, a natural question is to study the induced action on $J$ of the automorphism group of $X$ in relation to the Lagrangian fibration structures. I thank G. Pearlstein for asking questions that led me to the following observations.
Let $X$ be a smooth cubic fourfold and let $\tau$ be an automorphism of $X$. Then $\tau$ acts on the universal family of hyperplane sections of $X$ and thus also on the Donagi-Markman fibration $J_U \to U$ (which is identified with the relative $\Pic^0$ of the family of Fano surfaces of the hyperplane sections of $X$). By abuse of notation we denote by
\[
\tau: J \dashrightarrow J
\]
the induced birational morphism. Notice that $\tau$ preserves $\Theta$ and $L$ so the induced action of $\tau^*$ is the identity on $U= \langle L, \Theta \rangle \subset \NS(J)$.
\begin{prop} \label{auto}
Let $X$ be a smooth cubic fourfold and suppose that the fibers of $\pi: J \to \P^5$ are irreducible (by \cite{LSV} this happens for general $X$).
\begin{enumerate}
\item Then $\Theta$ is $\pi$-ample and so is any $B \in \NS(J)$ with $q(L,B)>0$.
\item Any birational automorphism $\tau: J\dashrightarrow J$ which fixes $L=\pi^* \mathcal O(1)$ extends to a regular automorphism.
\item $L$ is nef on a unique hyper--K\"ahler birational model of $J$. In other words, if $J' $ is a birational hyper--K\"ahler model of $J$, with birational map $f:J' \dashrightarrow J$, and the induced map $\pi': J' \dashrightarrow J \to \P^5$ is regular, then $f$ is an isomorphism.
\end{enumerate}
\end{prop}
\begin{proof}
(1) Let $H$ be an ample line bundle on $J$ and let $J_t$ be a smooth fiber of $J \to \P^5$. Then $[H_{|J_t}]=m [\Theta_{|J_t}]$ for a positive integer $m$ so the restrictions of $H$ and $m\Theta$ are topologically equivalent for any smooth fiber. Since the fibers of $\pi$ are irreducible, it follows that the restrictions of $H$ and $m \Theta$ to any fiber are numerically equivalent (see \cite[Lem 4.4]{Voisin-twisted}). By Nakai-Moishezon, $m \Theta$ is $\pi$-ample. Similary, if $q(B,L)>0$, then there exists positive integers $a$ and $b$ such that $aB$ and $b\Theta$ are numerically equivalent on every fiber.
(2) By assumption, $\tau^* L=L$ so $q(\tau^* \Theta, L)=q(\Theta, L)=1$. As a consequence, $\tau^* \Theta$ and $\Theta$ are topologically equivalent on the smooth fibers and hence, as above, numerically equivalent on every fiber. Thus, $\tau^* \Theta$ is $\pi$-ample. It follows that $\tau$ is a regular morphism.
(3) Let $H'$ be any ample line bundle on $J'$ and let $L'=f^{*}L={\pi'}^*\mathcal O(1)$. Then $0< q(L', H')=q(L, f^*H')$, so by (1) $ f^*H'$ is ample and $f$ is an isomorphism.
\end{proof}
In addition to birational automorphisms induced by the automorphisms of $X$, some examples of birational automorphisms which preserve $L$ are
\begin{enumerate}
\item $\iota: J \to J$ induced by the action of $(-1)$ on the smooth fibers of $J \to \P^5$.
\item $t_\alpha: J \to J$ induced by the translation of a rational section of $\alpha: \P^5 \dashrightarrow J$ (cf.\S \ref{section MW}).
\item More generally, any birational automorphism induced by an element of the automorphism group of $J_K$, the generic fiber of $J \to \P^5$.
\end{enumerate}
\begin{rem}
As already mentioned just below Theorem \ref{LSVthm}, a necessary condition for the irreducibility of the fibers of $J \to \P^5$ is given in \cite{Brosnan}. This condition is satisfied if and only if the hyperplane sections $Y$ of $X$ satisfy $d(Y):=b_2(Y)-b_4(Y)=0$, where $b_i(Y)$ denotes the $i$-th Betti number of $Y$ and where $d(Y)$ is called the defect of $Y$. It is easy to see that if $Y$ contains a plane then $d(Y)>0$.
\end{rem}
\section{Birational geometry of $J(X)$, for general $X$} \label{section general X}
To describe the birational geometry of the intermediate Jacobian fibration we degenerate the underlying cubic to the chordal cubic, following an idea already contained in \cite{KLSV}. There, it is observed that the central fiber of the corresponding family of intermediate Jacobian fibrations can be chosen to be birational to a moduli space of sheaves of OG$10$-type on a K3 surface of genus $2$. As in Section \ref{ogtentype}, by moduli space of OG$10$-type we mean a moduli space of sheaves on a K3 surface with Mukai vector $2w$, with $w^2=2$. We first refine the construction of this degeneration in order to have a central fiber that is actually \emph{isomorphic} to a certain singular moduli space of sheaves on the associated K3 surface. In this way, we can keep track of the limits of the relative theta divisor and of the line bundle inducing the Lagrangian fibration. This is done in Section \ref{deg chordal}. The results of Meachan--Zhang \cite{Meachan-Zhang}, which were recalled in Lemma \ref{MZ} imply that the central fiber of the relative theta divisor can be contracted after a Mukai flop of the zero section. For $X$ general, we then deduce the same result for $J(X)$ and, for very general $X$, we compute the nef and movable cone of $J(X)$. This is the content of Theorem \ref{NSU}.
\begin{thm} \label{NSU} Let $X$ be a smooth cubic fourfold and let $J=J(X) \to \P^5$ be a hyper--K\"ahler compactification of the intermediate Jacobian fibration as in \S \ref{birational section}.
For very general $X$,
\begin{enumerate}
\item There is a unique other hyper--K\"ahler birational model of $J$, denoted by $N$, which is the Mukai flop $p: J \dashrightarrow N$ of $J$ along the image of the zero section;
\item There is a divisorial contraction $h: N \to \bar N$ which contracts the proper transform of $\Theta$ onto an $8$-dimensional variety which is birational to the LLSvS $8$-fold $Z(X)$.
\end{enumerate}
In other words, we have $\Mov (J)=\langle L, H \rangle= \Nef(J) \cup p^* \Nef(N)$, $ \Nef(J)=\langle L, H_0 \rangle$, and where $p^* \Nef(N)=\langle H_0 , H \rangle$, were $H_0$ is a big and nef line bundle on $J$ which contracts the zero section of $J \to \P^5$ and $H$ is as in Corollary \ref{J not JT}.
For general $X$, the relative theta divisor $\Theta$ can be contracted after the Mukai flop of the zero section of $J \to \P^5$.
\end{thm}
Before the proof of the Theorem, which will be given in the following Section \S \ref{deg chordal}, we mention, as a consequence of the Theorem above, the relation between the intermediate Jacobian fibration and moduli spaces of objects in the Kuznetsov component of $X$.
\subsection{Comparison with moduli spaces of objects in the Kuznetsov component of $X$} \label{Kuz}
The recent paper \cite{BLMNPS} establishes the existence and the fundamental properties of moduli spaces of objects in the Kuznetsov component $\mathcal{K}u(X)$ of a smooth cubic fourfold $X$. We refer the reader to \S 29 of loc. cit for the relevant definitions and the precise statements of the results.
Given a smooth cubic fourfold $X$, the extended Mukai lattice $\widetilde H(\mathcal K u (X), \mathbb{Z})$ is a lattice, whose underlying group is the topological $K$-theory of $\mathcal K u (X)$ and whose Mukai pairing and weight two Hodge structure are induced from those on $X$. The only classes in $\widetilde H(\mathcal K u (X), \mathbb{Z})$ that are of type $(1,1)$ for very general $X$ are contained in a rank $2$ lattice $A_2$, which is spanned by two classes $\lambda_1$ and $\lambda_2$, that satisfy $\lambda_1^2=\lambda_2^2=2$ and $\lambda_1 \cdot \lambda_2=-1$ (see \cite[(29.1)]{BLMNPS}). A description of a full connected component of the space of Bridgeland stability conditions on $\mathcal{K}u(X)$ is also produced (Thm 29.1 of loc. cit.). It is shown that, for a primitive Mukai vector with $v^2 \ge -2$ and for a $v$-generic stability condition $\sigma$ in this component, the moduli space $M_\sigma(\mathcal{K}u(X), v)$ of Bridgeland stable objects in $\mathcal{K}u(X)$ with Mukai vector $v$ is a non empty smooth projective hyper--K\"ahler manifold of dimension $v^2+2$, deformation equivalent to a Hilbert scheme of points on a K3 surface; moreover, the formation of these moduli spaces works in families (see Thm 29.4 of loc. cit. for the precise statement).
For a Mukai vector of OG$10$-type in the $A_2$ lattice, i.e., of the form $v=2\lambda$ with $\lambda^2=2$, \cite{Pertusi-et-al} shows that for a $\lambda$-generic stability condition $\sigma$ the moduli space $M_\sigma(\mathcal{K}u(X), v)$ is an irreducible normal projective symplectic variety of dimension $10$ admitting a symplectic resolution which is deformation equivalent to a manifold of OG$10$-type. The genericity condition here means that the polystable objects with Mukai vector $v$ are the direct sum of two stable objects with Mukai vector $\lambda$. More precisely, the singular locus of $M_\sigma(\mathcal{K}u(X), v)$ is isomorphic $\Sym^2 M_\sigma(\mathcal{K}u(X), \lambda)$.
Moreover, in \cite{Pertusi-et-al} it is shown that for general $X$ the twisted intermediate Jacobian fibration $J^T(X)$ is birational to $M_\sigma(\mathcal{K}u(X), 2\lambda)$, for $\lambda^2=2$.
For the non-twisted case we have the following corollary of Theorem \ref{NSU} that goes in the opposite direction.
\begin{cor} \label{notkuz} For very general $X$, $J(X)$ is not birational to a moduli space of the form $M_\sigma(\mathcal{K}u(X),v )$. \end{cor}
\begin{proof}
First of all, by \cite[Rem 29.3]{BLMNPS}, if non empty, the dimension of a moduli space $M_\sigma(\mathcal{K}u(X), v)$ is $v^2+2$. This dimension is equal to $10$ if and only if either $v$ is primitive (hence $M_\sigma(\mathcal{K}u(X), v)$ is of K3$^{[5]}$-type and thus cannot be birational to $J(X)$) or else $v=2\lambda$ with $\lambda^2=2$.
By the results of \cite{Pertusi-et-al} cited in the remark above, for $v=2\lambda$ with $\lambda \in A_2$ and $\lambda^2=2$ and $\sigma$ a $\lambda$-generic stability condition, the singular locus of $M_\sigma(\mathcal{K}u(X), v)$ is isomorphic the second symmetric product of a hyper--K\"ahler manifold of K3$^{[2]}$--type. By dimension reasons, the symplectic resolution $\widetilde M_\sigma(\mathcal{K}u(X),v ) \to M_\sigma(\mathcal{K}u(X), v)$ is not a small contraction. Suppose by contradiction that $J(X)$ is birational to $\widetilde M_\sigma(\mathcal{K}u(X), v)$. Then by Theorem \ref{NSU}, the symplectic resolution has to coincide with $N \to \overline N$ and $M_\sigma(\mathcal{K}u(X), v) \cong \bar N$. This implies that the singular locus of $M_\sigma(\mathcal{K}u(X), v)$ has to be birational to the Lehn-Lehn-Sorger-van Straten $8$-fold $Z(X)$, which gives a contradiction. Indeed, $Z(X)$ cannot be birational to $\Sym^2 M_\sigma(\mathcal{K}u(X), \lambda)$ since, by Proposition \ref{GLR}, this would imply that the latter has a symplectic resolution. This, however, is not true because $\Sym^2 M_\sigma(\mathcal{K}u(X), \lambda)$ is a $\mathbb{Q}$-factorial sympectic variety with singular locus of codimension strictly greater than $2$ and hence does not admit a symplectic resolution (since it does not admit a semi-small resolution).
\end{proof}
\begin{rem} We expect the more general statement to hold: for very general $X$, $J(X)$ is not birational to a Bridgeland moduli space of objects on a $2$-CY category that is deformation equivalent to the derived category of a K3 surface. We present a rough sketch of the argument. Assume there is a family of Bridgeland stability conditions on the family of derived categories realizing the deformation. Then as in \cite[21.24]{BLMNPS} a relative moduli space exists as an algebraic space; by a generalization of a theorem of Mukai \cite[Thm 1.4]{Perry} the stable locus of each fiber is smooth and has a holomorphic symplectic form; the singular locus parametrizing strictly semi-stable objects of codimension $\ge 2$. One then expects such moduli spaces to be normal and irreducible. As in the proof of the projectivity in \cite[Thm 29.4]{BLMNPS} then shows that these moduli spaces are projective. Finally, a similar argument to the one above shows that the contraction $N \to \bar N$ cannot be the symplectic resolution of one of these moduli spaces.
\end{rem}
In the next subsection we construct the degeneration of the intermediate Jacobian fibration that will allow us to prove Theorem \ref{NSU}. The proof of the Theorem will be given at the end of the Section.
\subsection{Degeneration to the Chordal Cubic} \label{deg chordal}
The secant variety to the Veronese embedding of $\P^2$ in $\P^5$ is a cubic hypersurface isomorphic to $\Sym^2 \P^2$, called the chordal cubic. Such a singular cubic fourfold is unique up to the action of the projective linear group.
Given a one parameter family of cubic fourfolds degenerating to the chordal cubic, it was proved in \cite{KLSV} that, up to a base change, one can fill the corresponding degeneration of intermediate Jacobian fibrations with a smooth central fiber that is birational to $\widetilde M_{2v_0}=\widetilde M_{2v_0}(S)$, where $(S,C)$ is the degree $2$ $K3$ surface associated to the degeneration of cubic fourfolds as in \cite{Collino-fundamental,Hassett-special, Laza} and where $v_0=(0,C,-2)$ is as in (\ref{vkappa}). We will use this degeneration to study the birational properties of the intermediate Jacobian fibration, at least for general $X$. For this purpose, we need to control what happens to the line bundles $L$ and $\Theta$ under the corresponding degeneration of intermediate Jacobian fibrations. We achieve this by constructing a particular degeneration whose central fiber is precisely the singular moduli space $M_{2v_0}$ and is such that the Lagrangian fibrations of the members of this degeneration fit in a relative Lagrangian fibration. This is done in Proposition \ref{3fams}. With this degeneration, we are not only able to identify precisely the limits of $L$ and $\Theta$ (see Lemma \ref{Udeforms}), but we are also able to deform the results about the birational geometry of $M_{2v_0}$ away from the central fiber (see Proposition \ref{propmukflop}), eventually proving Theorem \ref{NSU}.
Let $\mathcal X \to \Delta$ be a one parameter family of cubic fourfolds degenerating to the chordal cubic. By this we will mean $\Delta$ is a small disk or an open affine subset in the base of a pencil of cubic fourfolds with the property that the general fiber is smooth and the central fiber is isomorphic to the chordal cubic.
Recall the following facts (proved in \cite{Hassett-special}, cf. also \cite{Laza} and \cite{KLSV}): (a) the monodromy of this family has order $2$; (b) to such a degeneration one can associate a degree $2$ polarized K3 surface $(S,C)$; (c) for a general pencil, the polarized K3 surface $(S,C)$ is general in moduli.
Suppose that for $t \neq 0$ the cubic fourfold $\mathcal X_t$ is general in the sense of LSV (i.e. in the sense that the construction of the hyper--K\"ahler compactification of \cite{LSV} works for $J_U(\mathcal X_t)$) and let $\mathcal J^* \to \Delta^*$ be the family of intermediate Jacobians associated to the smooth locus $\mathcal X^* \to \Delta^*$ of the pencil, with corresponding family of Lagrangian fibrations $\pi_{\Delta^*}:\mathcal J^* \to \P^5 _ {\Delta^*}$.
\begin{lemma}[\hspace{1sp}\cite{Collino-fundamental, KLSV}] \label{collino}
Up to a degree $2$ base change, we can extend $\pi_{\Delta^*}:\mathcal J^* \to \P^5 _ {\Delta^*}$ to a projective morphism $\pi_{\mathcal V}: \mathcal J_{\mathcal V} \to \mathcal V$, where $\mathcal V \subset \P^5 \times \Delta$ is an open subset such that $\mathcal V_t=\P^5$ for $t \neq 0$ and $ \mathcal V_0 \subset \P^5$ is non empty for $t=0$, and where $\mathcal { J}_0 \to \mathcal V_0 \subset \P^5$ is identified with the restriction of $M_{2v_0}(S) \to |2C|=\P^5$ (cf. \ref{lagrfibr}) to an open subset $V \subset |2C|$. Moreover, $\mathcal J_{\mathcal V} \to \mathcal V$ has a zero section and is polarized by a relative principal polarization.
\end{lemma}
\begin{proof} Let $H \subset \P^5$ be a general hyperplane. For the degeneration $\mathcal Y:= \mathcal X \cap (H \times \P^5)$ of a single smooth cubic $3$-fold the statement is due to Collino \cite{Collino-fundamental}. In Prop. 1.16 of loc. cit, it is also shown that the class of the limit polarization is the theta divisor of the Jacobian of the genus $5$ hyperelliptic curve, which is the limiting abelian variety. For the statement about the limit of the intermediate Jacobian fibration, this is \cite[\S 6.3 ]{KLSV}.
\end{proof}
We now compactify the projective family $\mathcal J_{\mathcal V}$ of the Lemma above to construct a family of Lagrangian fibered holomorphic symplectic varieties in such a way that the central fiber is exactly $M_{2v_0}=M_{2v_0}(S)$ (or $\widetilde M_{2v_0}=\widetilde M_{2v_0}(S)$) (cf. (\ref{lagrfibr})).
\begin{prop} \label{3fams} Let $\mathcal X \to \Delta$ be as above a general family of smooth cubic fourfolds degenerating to the chordal cubic. Suppose that for very general $t \in \Delta$, $\mathcal X_t$ is very general. Let $(S,C)$ be the corresponding K3 surface of degree $2$ as above. Then, possibly up to a base change, there are two degenerations of the corresponding intermediate Jacobian fibration, fitting in the following commutative diagram
\begin{equation} \label{diagr3fam}
\xymatrix{
\widetilde{\mathcal{M}} \ar[r]^m \ar[dr]_{\widetilde f} & \mathcal{M} \ar[d]^{f} \\
& \Delta
}
\end{equation}
where:
\begin{enumerate}
\item $\widetilde f: \widetilde{\mathcal{M}} \to \Delta$ is a family of smooth hyper--K\"ahler manifolds, with $\widetilde{\mathcal{M}}_t=J(\mathcal X_t)$ for $t \neq 0$ and $\widetilde{\mathcal{M}}_0=\widetilde M_{v_0}(S)$. The family is equipped with a relative Lagrangian fibration $ \widetilde{\mathcal{M}} \to \P^5_\Delta$, where for each $t$ the corresponding Lagrangian fibration is the obvious one.
\item $f: \mathcal M \to \Delta$ is a degeneration of hyper--K\"ahler manifolds, with ${\mathcal{M}}_t=J(\mathcal X_t)$ for $t \neq 0$ and ${\mathcal{M}}_0= M_{v_0}(S)$. The morphism $m: \widetilde{\mathcal{M}} \to \mathcal M$ is proper, birational, for $t \neq 0$ it is an isomorphism and for $t=0$ it is the natural symplectic resolution $m_0: \widetilde M_{2v_0}(S) \to M_{2v_0}(S)$ of Theorem \ref{OG10}. Moreover, there is a relative Lagrangian fibration ${\mathcal{M}} \to \P^5_\Delta$ where for each $t$ the corresponding Lagrangian fibration is the obvious one.
\end{enumerate}
\end{prop}
\begin{proof} Start from the projective morphism $\pi_{\mathcal V}:\mathcal J_{\mathcal V} \to \mathcal V$ of Lemma \ref{collino}. There is an isomorphism $\mathcal J_{\mathcal V_0}\cong (\widetilde M_{2v_0})_V$ , where $ (\widetilde M_{2v_0})_V$ is the restriction of the Lagrangian fibration $\widetilde \pi: \widetilde M_{2v_0} \to \P^5$ (cf. (\ref{lagrfibr})) to an open subset $V \subset \P^5$.
Let $\overline{\mathcal J_{\mathcal V}} \to \P^5_\Delta$ be any projective morphism extending $\pi_{\mathcal V}$. Applying Theorem \ref{deg lagr 1} (2) to $\overline{\mathcal J_{\mathcal V}} \to \P^5_\Delta$ yields, possibly up to the base change, a family $\widetilde g: \widetilde{\mathcal J} \to \Delta$ of smooth hyper--K\"ahler manifolds (projective over $(\Delta)^*$), with a relative Lagrangian fibration $\widetilde{\mathcal J} \to \P^5_{\Delta}$. Let $\mathcal L$ be the line bundle on $\widetilde{\mathcal J}$ inducing it on every fiber. Let
\begin{equation}
\phi_0: \widetilde{\mathcal J}_{0} \dashrightarrow \widetilde M_{2v_0}
\end{equation}
be the birational morphism induced by the isomorphism of open subsets $\mathcal J_{\mathcal V_0} \cong (\widetilde M_{2v_0})_V$. Then $(\phi_0)_* \mathcal L_0=\widetilde \ell:=\widetilde \pi^* \mathcal O_{\P^5}(1)$.
We now use an argument very similar to that in the proof of \cite[Thm 1.3 and 1.7]{KLSV}, to construct a family which is isomorphic to $\mathcal J$ over $\Delta$ and whose central fiber is actually \emph{isomorphic} to $ \widetilde M_{2v_0}(S)$.
Let $\Lambda$ be the OG$10$ lattice. Fixing a marking of the central fiber and trivializing the local system $R^2 \widetilde g_* \mathbb{Z}$ induces a marking $\eta_t: H^2(\widetilde{\mathcal J}_t, \mathbb{Z}) \to \Lambda$ of every fiber. Let $\mathcal D \subset \P (\Lambda \otimes_\mathbb{Z} \mathbb{C})$ be the period domain and let $\mathcal P: \Delta \to \mathcal D$ be the period mapping induced by these markings.
Let $\rho_0= \eta_0 (\phi_0)^*: H^2( \widetilde M_{2v_0}, \mathbb{Z}) \to \Lambda$ be the induced marking on $\widetilde M_{2v_0}$.
Let $\rho_t: H^2(\widetilde{\mathcal M}_t, \mathbb{Z}) \to \Lambda$ be markings induced by $\rho_0= \eta_0 (\phi_0)^*$ on fibers of the universal family over $\Def(\widetilde M)$ and let $\mathcal P_{\widetilde M}: \Def(\widetilde M) \to \mathcal D$ be the induced period mapping. Since $\mathcal P_{\widetilde M}$ is a local isomorphism, we can lift $\mathcal P$ to a map $ \xi: \Delta \to \Def(\widetilde M_{2v_0})$. Pulling back the universal family gives a family $\widetilde f: \widetilde{\mathcal M} \to \Delta$ with central fiber $\widetilde{\mathcal M}_0=\widetilde M_{2v_0}$. As in \cite{KLSV} the two families $\widetilde g: \widetilde{\mathcal J} \to {\Delta}$ and $\widetilde f: \widetilde{\mathcal M} \to \Delta$ are relatively birational over $\Delta$, since for every $t \in \Delta$, the marked pairs $(\widetilde{\mathcal J}_{t}, \eta_t)$ and $(\widetilde{\mathcal M}_t, \rho_t)$ are non separated points.
To show that the two families $\widetilde{\mathcal J}$ and $\widetilde{\mathcal M}$ are isomorphic away from the central fiber, first recall that by \cite[Thm 4.3]{Huybrechts-compact-hk} (cf. also \cite[Thm 3.2]{Markman-Survey}), for every $t$ there exists an effective cycle
\[
\Gamma_t = Z_t+ \sum W_{i,t}
\]
of pure dimension $10$ in $\widetilde{\mathcal M}_t \times \widetilde{\mathcal J}_t$ such that: $Z_t$ is the graph of a birational map; the codimension of the images of the $W_{i,t}$ in $\widetilde{\mathcal M}_t $ and in $ \widetilde{\mathcal J}_t$ are equal and positive; $[\Gamma_t]_*$ is a Hodge isometry and is equal to $\rho_t^{-1} \circ \eta_t: H^2(\widetilde{\mathcal J}_t, \mathbb{Z}) \to H^2(\widetilde{\mathcal M}_t, \mathbb{Z})$.
Let $\widetilde{\mathcal L}$ be the line bundle on $\mathcal M$ such that $\widetilde{\mathcal L}_t=\rho_t^{-1} \eta_t ({\mathcal L})=[\Gamma_t]_*(\mathcal L_t)$. Since $\widetilde{\mathcal L}_0=\widetilde \pi^* \mathcal O_{\P^5}(1)$ induces a Lagrangian fibration on $\widetilde{\mathcal M}_0=\widetilde M_{2v_0}$, by \cite{Matsushita-Def} $\widetilde{\mathcal L}$ induces a Lagrangian fibration on $\mathcal M_t$ for every $t$ (maybe up to restricting $\Delta$).
For very general $t$, $\widetilde{\mathcal L}_t=[Z_t]_*(\mathcal L_t)$, since the isotropic class $[Z_t]_*(\mathcal L_t)$ lies in the movable cone of $\widetilde{\mathcal M}_t$ and hence by Corollary \ref{J not JT} it has to be equal to $\widetilde{\mathcal L}_t$.
Corollary \ref{onlyJ} implies that for very general $t$, $Z_t$ is the graph of an \emph{isomorphism} between $\widetilde{\mathcal J}_t$ and $\widetilde{\mathcal M}_t$.
The same countability argument as in the proof of \cite[Thms 1.3 and 1.7]{KLSV} shows that there exists a component of the Hilbert scheme parametrizing graphs of such cycles $Z_t \subset \mathcal J_t \times \mathcal M_t$ that dominates $\Delta$. It follows that there is a cycle $\mathcal Z$ in the fiber product $\widetilde{\mathcal J} \times_\Delta \widetilde{\mathcal M}$ which, maybe up to restricting $\Delta$, induces an isomorphism for $t \neq 0 $.
The conclusion is that the family $\widetilde{\mathcal M} \to \Delta$ is such that central fiber is $\widetilde{\mathcal M} _0 \cong \widetilde M_{2v_0}$ while for $t \neq 0$, we have $\widetilde{\mathcal M}_t \cong J(\mathcal X_t)$.
Now we construct the second family.
By \cite[Thm 2.2]{Namikawa-def} there is a finite morphism
\[
\Xi: \Def(\widetilde M_{2v_0}) \to \Def( M_{2v_0})
\]
induced by the symplectic resolution $m_0: \widetilde M_{2v_0} \to M_{2v_0} $ and compatible with the universal families on the two deformation spaces (for more details see \S 2 of loc. cit.). Set $\nu=\Xi \circ \xi: \Delta \to \Def( M_{2v_0})$ and let
\[
\mathcal M \to \Delta
\]
be the pullback via $\nu$ of the universal family via $\nu$ on $\Def( M_{2v_0})$. Then the birational map $m: \widetilde{\mathcal M} \to \mathcal M$ over $\Delta$ induced by \cite[Thm 2.2]{Namikawa-def} has the desired properties.
Finally, the statement about the Lagrangian fibrations follows from the fact that, since the Lagrangian fibration $\widetilde M_{2v_0} \to \P^5$ in the central fiber factors via $\widetilde M_{2v_0} \to M_{2v_0}$, the morphism $\widetilde {\mathcal M} \to \P^5_{\Delta}$ factors via a morphism $\mathcal M \to \P^5_{\Delta}$.
\end{proof}
As a consequence of the last part of the proof, notice that there is a line bundle $\mathcal L_{\mathcal M}$ on ${\mathcal M}$ with
\[
m^* \mathcal L_{\mathcal M}=\widetilde{\mathcal L}
\]
and such that its restriction to the central fiber satisified $\mathcal L_{{\mathcal M}_0}=\ell$, where $\ell$ is as in (\ref{ell}).
For any $t \neq 0$, let $ \Theta_t$ be the relative Theta divisor in $ \mathcal M_t=J(\mathcal X_t)$.
\begin{lemma} \label{lemmacomp}
For $\star=\widetilde{\mathcal M} $ or $\mathcal M$, let $\Theta_\star$ be the divisor defined as the closure of $ \cup_{t \neq 0} \Theta_t$ in $\star$.
Then, $\Theta_{{\mathcal M}}$ is a Cartier divisor and hence the following compatibility conditions hold (notation as in diagram (\ref{diagr3fam})):
\begin{equation} \label{compatibility}
\Theta_{\widetilde{\mathcal M}_0} \cong (m^*\Theta_{\mathcal M})_{|\widetilde{\mathcal M}_0} =m_0^*\Theta_{\mathcal M_0}
\end{equation}
where $\Theta_{\star_0}:=(\Theta_\star)_{|0}$ is the fiber of $\Theta_\star$ over $t=0$.
\end{lemma}
\begin{proof} Let $\mathcal I_{\Theta_{\mathcal M}} \subset \mathcal O_{{\mathcal M}}$ be the ideal sheaf of $\Theta_{\mathcal M}$ in ${\mathcal M}$. Since the morphism $\Theta_{\mathcal M} \to \Delta$ is flat, it follows that the restriction $(\mathcal I_{\Theta_{\mathcal M}} )_{|\mathcal M_0}$ is the ideal sheaf of $\Theta_{\mathcal M_0}$ in ${\mathcal M_0}$. By \cite{Perego-Rapagnetta} (cf. \S \ref{movog10}), $\mathcal M_0=M_{2v_0}$ is factorial so $(\mathcal I_{\Theta_{\mathcal M}} )_{|\mathcal M_0}$ is locally free. Hence, so is $\mathcal I_{\Theta_{\mathcal M}} $. It follows that the divisors $\Theta_{\widetilde{\mathcal M}}$ and $m^*\Theta_{\mathcal M}$ agree and so do their central fibers.
\end{proof}
The next Lemma identifies the limit of $\Theta_t$ in $M_{2v_0}=\mathcal M_0$ and shows that all line bundles on $M_{2v_0}$ deform over $\mathcal M \to \Delta$. Recall first that by (\ref{NSsing}), $\NS(M_{2v_0})=U=\langle \ell, \theta \rangle$ and that for every $t$
\[
\NS(\mathcal M_t) \supseteq U_t=\langle \mathcal L_t, \Theta_t \rangle
\]
equality holding for very general $t$. Here $\Theta_0=\Theta_{\widetilde{ \mathcal M_0}}$. In particular, inside $\NS(\widetilde M_{2v_0})$ we have the following rank $2$ sublattices both of which are isomorphic to the hyeperbolic lattice $U$: The limit lattice $U_0$ spanned by the limits $\mathcal L_0=\widetilde \ell$ and $\Theta_0$ and the pullback lattice $m_0^*\NS(M_{2v_0})=\langle m_0^* \ell, m_0^*\theta \rangle$.
\begin{lemma} \label{Udeforms} Let the notation be as above. Then
\begin{enumerate}
\item The two sublattices $U_0=\langle \widetilde \ell, \Theta_{\widetilde{ \mathcal M_0}}\rangle$ and $\langle m_0^* \ell, m_0^*\theta \rangle$ of $\NS(\widetilde M_{2v_0})$ are the same.
\item The limit of the relative Theta divisor in $\mathcal M_0$ is precisely $\theta$, the relative theta divisor on $M_{2v_0}(S)$ of (\ref{deftheta}).
\end{enumerate}
\end{lemma}
\begin{proof}
By Lemma \ref{collino} and Lemma \ref{lemmacomp}, the limit theta divisor $(\Theta_{\mathcal M})_0$ is an effective line bundle on $\mathcal M_0=M_{2v_0}$ which restricts to a theta divisor on the smooth fibers of $M_{2v_0} \to \P^5$. Thus $\Theta_{\mathcal M_0}$ is linearly equivalent to an effective line bundle of the form $\theta+a \ell$ for some integer $a$. We show that $a=0$. By (\ref{compatibility}), $ \Theta_{\widetilde{\mathcal M}_0} =m_0^*\Theta_{\mathcal M_0}=m_0^*(\theta+a \ell)$ and $\widetilde{\mathcal L}_0=m_0^*\ell $. This is enough to conclude that the two sublattices
\[
U=\langle \Theta_{\widetilde{\mathcal M}_0}, \widetilde{\mathcal L}_0 \rangle \quad \text{ and } U=m_0^* \langle \theta, \ell \rangle=m_0^* \NS(M_{2v_0})
\]
of $\NS (\widetilde M_{2v_0})$ are the same.
This proves the first part of the Lemma.
By Remark \ref{Alexeev} above, $\theta$ does not contain the singular locus of $M$, thus $m_0^* \theta$ coincides with its proper transform and is irreducible.
Since it has negative Beauville-Bogomolov square (cf. (\ref{thetasq})), it is a prime exceptional divisor.
By \cite[\S 5.1]{Markman-prime-exceptional}, a prime exceptional divisor deforms where its first Chern class remains algebraic.
Thus $m_0^* \theta$ deforms to relative effective prime exceptional divisor $\theta_{\widetilde{\mathcal M} }$ on $\widetilde{\mathcal M}$.
By Corollary \ref{onlyJ} and Proposition \ref{Thetasq}, for very general $t \neq 0$, the fiber over $t$ of the two irreducible effective divisors $\theta_{\widetilde{\mathcal M} }$ and $\Theta_{\widetilde{\mathcal M}}$ have to agree since there is only one prime exceptional divisors on $\mathcal M_t$. Thus $\theta_{\widetilde{\mathcal M} }$ and $\Theta_{\widetilde{\mathcal M}}$ have to be equal for every $t$. In particular, so are their restrictions to the central fiber.
\end{proof}
\begin{cor} \label{zerosectgenX}
Let $X$ be a general cubic fourfold and let $\pi: J(X) \to \P^5$ be the intermediate Jacobian fibration of \cite{LSV}. The natural rational zero section of $\pi$ is regular.
\end{cor}
\begin{proof} Consider a degeneration of cubic fourfolds to the chordal cubic as in Proposition \ref{3fams} and let $\mathcal M \to \Delta$ be the corresponding family. By Lemma \ref{lemmacomp} the divisor $\Theta_{\mathcal M}$ is Cartier and by Remark \ref{thetaamp} it is relatively ample (up to restricting $\Delta$). Since $\Theta_{\mathcal M}$ is $-1$-invariant, it follows that the birational involution $-1$ is biregular. One component of the fixed locus of this involution has the property that its restriction to every fiber is precisely the closure of the corresponding rational zero section. Since by Remark \ref{zerosect} in the central fiber the section is regular, it follows that for general $t \in \Delta$ the corresponding rational section is also regular.
\end{proof}
Consider the family ${\mathcal M} \to \Delta$ of Proposition \ref{3fams}, with its relative theta divisor $\Theta_{{\mathcal M}}$. By Druel \cite{Druel} we know that for every $t$, the prime exceptional divisor $\Theta_t$ can be contracted on a hyper--K\"ahler projective birational model of ${\mathcal M} _t$. In the central fiber ${\mathcal M} _t=M_{2v_0}$ we have, by Lemma \ref{Udeforms}, that $\Theta_{{\mathcal M_0}}=\theta$. By Lemma \ref{MZ} this divisor can be contracted after a Mukai flop. We now show that the same is true for any $t \neq 0$, namely, that after a Mukai flop the relative theta divisor can be contracted, possibly up to restricting $\Delta$.
\begin{prop} \label{propmukflop} For general $X$, the relative theta divisor $\Theta$ on $J=J(X)$ can be contracted after the Mukai flop of the zero section.
\end{prop}
\begin{proof}
Let ${\mathcal M} \to \P^5_{\Delta}$ be as in Proposition \ref{3fams}. By Corollary \ref{zerosectgenX}, there is a relative zero section $s: \P^5_\Delta \to \mathcal M$. Let $T$ be its image. Then $T$ is contained in the smooth locus of the fibers of $\widetilde{\mathcal M} \to {\Delta}$. Let
\[
P: \mathcal M \dashrightarrow \mathcal N
\]
be the relative Mukai flop of $T$ in ${\mathcal M}$. By Proposition \ref{MZ}, the Mukai flop of the zero section in the central fiber $M_{2v_0}$ can be preformed in the projective category. Thus, the central fiber of $\mathcal N$ is projective and so are all the fibers of $g: \mathcal N \to \Delta$ (since by Lemma \ref{Udeforms} there is an ample class on the central fiber that deforms over $\Delta$). For $t \neq 0$, $\mathcal N_t$ is smooth while the central fiber $\mathcal N_0 $ has the same singularities as $\mathcal M_0=M_{2v_0}$, since they are isomorphic away from the flopped locus which does not meet the singular locus.
Via the birational morphism $P$, which is a relative isomorphism in codimension $1$, we can identify the second integral cohomology group of the fibers of the two families.
In particular, for every $t \in \Delta$ we have $P_*U_t \subset \NS(\mathcal N_t)$ with equality holding for very general $t$ and for $t=0$. In what follows we freely restrict $\Delta$, if necessary, without any mention.
As in Propostion \ref{MZ}, let $H$ be the big and nef line bundle on $\mathcal N_0$ that contracts $\theta$ (i.e. $H$ is a generator of the ray $\theta^\perp$). Since $H \in P_*U_0$, by Lemma \ref{Udeforms}, $H$ deforms to a line bundle $\mathcal H$ on $\mathcal N$. For very general $t$, its restriction $\mathcal H_t$ is a generator of the one dimensional space $(P_t)_*\Theta_t^ \perp \subset \NS(\mathcal N_t)$.
By \cite{Druel-exc}, $(P_t)_*\Theta_t$ can be contracted on a birational model of $\mathcal N_t$. We now show that it can be contracted on $\mathcal N_t$ itself. For very general $t$, the line bundle inducing the divisorial contraction has to be $\mathcal H_t$, or rather its proper transform on an appropriate birational model of $\mathcal N_t$. It follows that for very general $t$ (and thus for all $t$) $\mathcal H_t$ is big. Moreover, since $\mathcal H_0$ is big and nef and $\mathcal N_0$ has rational singularities, $H^i(\mathcal N_0, \mathcal H_0^k)=0$ for $i>0$ and any $k \ge 0$. It follows that the locally free sheaf $g_* \mathcal H^k$ satisfies base change. Since $\mathcal H_0$ is semi-ample, so is $\mathcal H_t$ for all $t$ in $\Delta$. For $k \gg 0$, the regular morphism $\Psi: \mathcal N \to \P (g^*g_* \mathcal H^k)$, relative over $\Delta$, is birational onto its image and contracts $\Theta_t$ for very general $t$ and for $t=0$. Up to further restricting $t$, we can assume that the locus contracted on $\mathcal N_t$ is irreducible, and hence that $\Psi_t$ contracts precisely $(P_t)_*\Theta_t$ for every $t$.
\end{proof}
The proof of Theorem \ref{NSU} is now complete:
\begin{proof}[Proof of Theorem \ref{NSU}] Let $X$ be general. By Proposition \ref{propmukflop}, the Mukai flop $p: J \dashrightarrow N$ of $J$ along the zero section is projective and on $J$ there exists a big and nef line bundle that contracts the zero section. For very general $X$, $H_0$ is unique, up to a positive rational multiple and $\Nef(J)=\langle L, H_0 \rangle$. Moreover, we have showed that for general $X$ there is a divisorial contraction $N \to \bar N$, contracting $p_*\Theta$. Since the divisorial contraction $N \to \overline N$ contracts the ruling of $\Theta$ (cf. Prop. \ref{Thetasq}), by (\ref{AJtwisted}) it follows that the image of $\Theta$ in $\bar N$ is birational to the LLSvS $8$-fold $Z(X)$.
For very general $X$, $\Nef(N)=\langle p_*H_0, p_*H \rangle$, where $p^*H$ is the unique (up to a positive multiple) big and nef line bundle inducing the contraction.
By \cite[Prop. 4.2]{Huybrechts-kahler-cone}, $H$ is the second ray of the movable cone of $J$, i.e. $\Mov(J)=\langle L, H \rangle$.
\end{proof}
\section{The Mordell-Weil group of $J(X)$} \label{section MW}
Let $a: \mathcal A \to B$ be a projective family of abelian varieties over an irreducible basis $B$ and suppose that $a$ admits a zero section. The Mordell-Weil group $MW(a)$ of $a: \mathcal A \to B$ is the group of rational sections of $a: \mathcal A \to B$. Equivalently, if $K$ denotes the function field of $B$, $MW(a)$ is the group of $K$-rational points of the generic fiber $\mathcal A_K$. For Lagrangian hyper--K\"ahler manifolds, the study of the Mordell-Weil group of abelian fibered hyper--K\"ahler manifolds was started by Oguiso in \cite{Oguiso-shioda,Oguiso-MW}.
The aim of this section is to prove the following theorem
\begin{thm} \label{MW}
Let $X$ be a smooth cubic fourfold and let $\pi: J=J(X) \to \P^5$ be as in Theorem \ref{hkcom}, a smooth projective hyper--K\"ahler compactification of $J_U$. Let $MW(\pi)$ be the Mordell-Weil group of $\pi$, i.e., the group of rational sections of $\pi$ and let $H^{2,2}(X, \mathbb Z)_0$ be the primitive degree $4$ integral cohomology of $X$.
The natural group homomorphism
\[
\phi_X: H^{2,2}(X, \mathbb Z)_0 \to MW(\pi)
\]
induced by the Abel-Jacobi map (cf. \ref{AbelJacobi}) is an isomorphism.
\end{thm}
\begin{cor} The group $MW(\pi)$ is torsion free.
\end{cor}
\begin{rem} \label{rem og} In \cite{Oguiso-MW} Oguiso proved the existence of Lagrangian fibered hyper--K\"ahler manifolds whose Mordell-Weil group has rank $20$. This is the maximal possible rank among all the known examples of hyper--K\"ahler manifolds, as follows from the Shioda-Tate formula of \cite{Oguiso-shioda} (see also Proposition \ref{shioda-tate} below). Oguiso considers deformations of the abelian fibration $\widetilde M_{2v_0} \to \P^5$ (cf. \ref{lagrfibr}) preserving both the Lagrangian fibration structure and the zero section; among these deformations, Oguiso shows the existence of Lagrangian fibration with rank $20$ Mordell-Weil group \cite[Thm 1.4 (2)]{Oguiso-shioda}.
The general deformation of $\widetilde M_{2v_0} \to \P^5$ for which both the Lagrangian fibration structure and the zero section are preserved (this is a codimension two condition) is, up to birational isomorphism, $J(X)$ (see Remark \ref{section deform}). By the Theorem above Lagrangian fibrations of the form $J(X)$, for $X$ with $\rk H^{2,2}(X, \mathbb Z)=22$, satisfy $\rk MW(\pi)=20$. Thus, they provide an explicit description of Oguiso's examples.
\end{rem}
The following Proposition is essentially a reformulation of results from \cite{Oguiso-shioda,Oguiso-MW}.
\begin{prop} \label{shioda-tate}
Let $\pi: M \to \P^n$ be a projective hyper--K\"ahler manifold with a fixed (rational) section. Let $K=\mathbb{C}(\P^n)$ be the function field of the base and let $M_K$ the base change of $M$ to the generic point of $\P^n$.
There is a commutative diagram,
\[
\xymatrix{
& & 0 \ar[d] & 0 \ar[d] & \\
0 \ar[r] & \mathbb{Z} L \oplus \bigoplus_i \mathbb{Z} D_{i} \ar[r] \ar@{=}[d] & L^\perp \ar[d] \ar[r]& \Pic^0(M_K) \ar[r] \ar[d] & 0 \\
0 \ar[r] &\mathbb{Z} L \oplus \bigoplus_i \mathbb{Z} D_{i} \ar[r] & \NS(M) \ar@{->>}[d]^{r_b} \ar[r]^{r_K} & \Pic(M_K) \ar[r] \ar@{->>}[d] & 0 \\
& & \mathbb{Z} \ar@{=}[r] & NS(M_K) &
}
\]
where $L=\pi^* \mathcal O_{\P^n}(1)$ and where the $D_{1}, \dots ,D_k$ are the irreducible components of the complement of the regular locus of $\pi$ that do not meet the section. In particular, $\rk (MW(\pi))=\rk (\NS(M))-\rk \mathbb{Z} L \oplus \mathbb{Z} D_{i}-1=\rk (\NS(M))-k$.
\end{prop}
\begin{proof}
The column on the left is exact by definition. By \cite{Voisin-lagr}, for $b$ in the locus $U \subset \P^n$ parametrizing smooth fibers of $\pi$, $\im[r_b: \NS(M) \to H^2(M_b)]=\mathbb{Z}$. The same argument as in Lemma \ref{LTheta} shows that a line bundle $D$ on $M$ lies in $L^\perp$ if and only if $D^n \cdot L^n=(D_{|M_b})^n=0$. Since $\rk r_b=1$, this holds if and only if $D \cdot L^n=D_{|M_b}=0$, which is equivalent to $D \in \ker r_b$. This shows that the central column is exact. The same argument of \cite[Thm 1.1]{Oguiso-MW}, which was used to show that $\rk \NS(M_K)=1$, shows that any element in $\ker(r_b)=L^\perp$ goes to zero in $\NS(M_K)$. Thus there are induced horizontal morphisms $L^\perp \to \Pic^0(M_K) $ and $\mathbb{Z} \to \NS(M_K) $. Since $\NS(M) \to \Pic(M_K) $ is surjective, the bottom horizontal morphism is an isomorphism. The natural morphism $\mathbb{Z} L \oplus \mathbb{Z} D_{i} \to \NS(M)$ is injective, since by \cite[Lem 2.4]{Oguiso-shioda} it has maximal rank over $\mathbb{Q}$ and $\NS(M)$ is torsion free. Clearly, $ \mathbb{Z} L \oplus_i \mathbb{Z} D_{i} \subset \ker(r_K)$. To show the reverse inclusion, let $D$ be any line bundle on $M$ that goes to zero in $\Pic(M_K)$. Then, by what we have already proved, for any smooth fiber we have $r_b(D)=[D_{|M_b}]=0$. It follows that $D$ is a linear combination of $L=\pi^* \mathcal O_{\P^n}(1)$ and boundary divisors, i.e., $D \in \mathbb{Z} L \oplus_i \mathbb{Z} D_{i}$. Since $\rk(MW(\pi))=\rk \Pic^0(M_K)$, the last statement also follows.
\end{proof}
\begin{rem}
The study of the Mordell-Weil group for the Beauville-Mukai system is being carried out in a joint work in progress with Chiara Camere.
\end{rem}
\begin{cor} \label{ranks}
Let $J=J(X) \to \P^5$ be a hyper--K\"ahler compactification of the intermediate Jacobian fibration. Then
\[
\rk MW(\pi)=\rk \NS(J)-2=\rk H^{2,2}(X,\mathbb{Z})_0.
\]
\end{cor}
\begin{proof}
The discriminant locus of $\pi$ is irreducible and the fibers of $\pi$ over the general point of the discriminant are also irreducible (cf. Lemma \ref{lem Juno}). Thus, in the notation of Proposition above, $\ker r_K=\mathbb{Z} L$ and the equality $\rk MW(\pi)=\rk \NS(J)-2$ follows. The remaining equality follows from Lemma \ref{tr lattice}.
\end{proof}
\begin{rem} The corollary just proven, which relies on Oguiso's Shioda-Tate formula above, is the only part of this section where we use that $J_U$ admits a hyper--K\"ahler compactification with a regular Lagrangian fibration extending $J_U \to U$. Indeed, to define the Abel-Jacobi map $\phi_X$ and to prove that it is injective (\S \ref{sec injectivity}), we don't need to assume the existence of a hyper--K\"ahler compactification. However, we will use this Corollary in the proof of the surjectivity (\S \ref{sec surjectivity}).
\end{rem}
\begin{rem} An interesting problem is to study the action on $J$ of the birational automorphisms induced by translation by a non-trivial element of $MW(\pi)$ as well as to study the automorphism group of the generic fiber $J_K$.
Notice that, as a consequence of the observations of \S \ref{induced auto}, if $J \to \P^5$ has irreducible fibers then the birational automorphisms induced by translation are regular morphisms.
\end{rem}
\subsection{The Abel-Jacobi mapping} \label{subsection AJ}
This sections uses some ingredients from the theory of normal functions (certain holomorphic sections of intermediate Jacobian fibrations), as developed and used by Griffiths \cite{Griffiths-periods,Griffiths-periods-III}, Zucker \cite{Zucker-inventiones,Zucker-HC}, and Voisin \cite{Voisin-Some-Aspects}. We refer to these papers, as well as to \cite[\S 7.2.1, 8.2.2]{Voisin2} for the relevant theory.
The first task is to define the morphism $\phi_X: H^{2,2}(X,\mathbb{Z})_0 \to MW(\pi)$. One way to do this is to use relative Deligne cohomology, which allows to define an algebraic section of the fibration $J_U \to U$. See, for example, \cite{Voisin-Some-Aspects, ElZein-Zucker}.
A more geometric way to define the morphism $\phi_X$ is in terms of algebraic cycles and Abel-Jacobi maps, which is what we use here. This is possible because the integral Hodge conjecture holds for degree $4$ Hodge class on $X$ \cite{Voisin-Some-Aspects, Zucker-HC}. It allows us to avoid, in the current presentation, defining the normal function associated with a cohomology class. The reader should keep in mind, however, that constructing an algebraic section of the intermediate Jacobian fibrations with a Hodge class on $X$ is a key ingredient in the proof of the Hodge conjecture of \cite{Voisin-Some-Aspects, Zucker-HC}, so the short cut is only at the level of our presentation.
As already mentioned, the integral Hodge conjecture holds for degree $4$ Hodge classes on $X$. In particular for every class $\alpha \in H^{2,2}(X, \mathbb{Z})$ there is an algebraic cycle $Z$ such that $[Z]=\alpha$. Let $V \subset \P^5$ be the open subset parametrizing smooth hyperplane sections of $X$ that do not contain any of the components of $Z$. If $\alpha$ is a primitive cohomology class, then for $b=[Y_b] \in V$ the $1$-cycle $Z_b$ satisfies
\[
[Z_{Y_b}]=0 \, \text{ in } \,H^4(Y_b, \mathbb{Z}) = \mathbb{Z}
\]
and hence determines a point $ \phi_{Y_b}(Z_b) \in \Jac(Y_b)$ in the intermediate Jacobian of $Y_b$. By Griffiths \cite{Griffiths-periods-III} (see also \cite[\S 7.2.1]{Voisin2}) the assignment
\[
\begin{aligned}
\sigma_Z: V & \longrightarrow J_V,\\
b & \longmapsto \phi_{Y_b}(Z_b)
\end{aligned}
\]
defines a holomorphic section of the restriction of $J$ to $V \subset \P^5$. By \cite{Zucker-inventiones}, this section is, in fact, algebraic: indeed, consider a Lefschetz pencil ${\mc Y'} \to \P^1 $ of hyperplanes of $X$ with $\P^1 \subset V$ and with the property that none of the singular points of the members of the pencil are contained in $Z$. By \cite[(4.58)]{Zucker-inventiones} the restriction of $\sigma_Z$ to the non-empty open subset $V \cap \P^1$ of the pencil extends to a holomorphic function on all of $\P^1$ and is thus algebraic (see also \cite{ElZein-Zucker}). Since a holomorphic function that is algebraic in each variable is algebraic (see for example \cite[Thm. 5 Chap. IX]{Bochner-Martin}), it follows that $\sigma_Z$ actually defines a rational function on $\P^5$, i.e.
\[
\sigma_Z \in MW(\pi).
\]
Notice that the holomorphic section $\sigma_Z$ does not depend on the algebraic cycle representing $\alpha$. Indeed, since $\CH_0(X)=\mathbb{Z}$, by \cite[Thm 6.24]{Voisin-Chow} it follows that the cycle map $\CH^2(X) \to H^{2,2}(X,\mathbb{Z})$ is injective. It follows that if $Z$ and $Z'$ are homologous, then they are rationally equivalent in $X$ and hence so are their restrictions to a general smooth hyperplane section.
The conclusion of this discussion is that the Abel-Jacobi map induces a well defined group homomorphism
\begin{equation} \label{AbelJacobi}
\begin{aligned}
\phi_X: H^{2,2}(X, \mathbb{Z})_0 & \longrightarrow MW(\pi).\\
\alpha=[Z] & \longmapsto \sigma_\alpha:=\sigma_Z
\end{aligned}
\end{equation}
We prove injectivity of $\phi_X$ in \S \ref{sec injectivity} and surjectivity in \S \ref{sec surjectivity}. Since we will restrict to general pencils in $\P^5$, we start by recalling a few standard facts about Lefschetz pencils of cubic threefolds.
\subsection{Preliminaries on Lefschetz pencils}
We start by setting up the notation. Let $\P^1 \subset (\P^5)^\vee$ be a Lefschetz pencil with base locus a smooth cubic surface $\Sigma \subset X$. We have the following diagram
\[
\xymatrix{
& \Sigma \times \P^1 \ar[dl]_{p_1} \ar@{^{(}->}[r]^i & {\mc Y'} \ar[dl]_p \ar[dr]^q & \\
\Sigma \ar@{^{(}->}[r] & X & & \P^1
}
\]
where ${\mc Y'}=Bl_\Sigma X$, where $q:{\mc Y'} \to \P^1$ is the fibration of threefolds, and where $i: \Sigma \times \P^1 \to {\mc Y'}$ is the inclusion of the exceptional divisor in ${\mc Y'}$. Let $j: {U'} \subset \P^1$ be the open subset parametrizing smooth fibers.
The following Lemma is standard. We include a proof for lack of reference.
\begin{lemma} \label{integral loc inv cycle} The homology and cohomology groups of a cubic threefold which is smooth or has one $A_1$ singularity have no torsion. Moreover, using the notation as above,
\[
R^1q_* \mathbb{Z}=0, \quad R^2q_* \mathbb{Z}=\mathbb{Z}, \quad R^3q_* \mathbb{Z}=j_* j^* R^3q_* \mathbb{Z}, \quad R^4q_* \mathbb{Z}=\mathbb{Z}.
\]
\end{lemma}
\begin{proof}
The statement about the homology groups of a cubic threefold with at most an $A_1$ singularity follow from \cite[Example 5.3 and Theorem 2.1]{Dimca-homology-compositio}; using the universal coefficient theorem, the statement on the cohomology groups then follow. From loc. cit it also follows that $H^4(Y, \mathbb{Z})=H_4(Y, \mathbb{Z})^\vee=\mathbb{Z}$, and hence $R^4q_* \mathbb{Z}=\mathbb{Z}$ follows by proper base change.
The first two statements on the higher direct images follow from Lefschetz hyperplane section theorem. The third equality, which is also known as the ``local invariant cycle'' property, can be seen as follows (it is well known to hold with $\mathbb{Q}$-coefficents, we now show it with $\mathbb{Z}$-coefficients). By adjunction, there is a natural morphism
\[
\epsilon: R^3q_* \mathbb{Z} \to j_* j^* R^3q_* \mathbb{Z}
\]
which is an isomorphism over $U$. To show $\epsilon$ is an isomorphism over any point of $B:=\P^1\setminus {U'}$ we restrict, for every $b_0 \in B$, to a small disk $\Delta$ centered at $b_0$. Then $\epsilon$ is an isomorphism around $b_0$ if and only if the specialization morphism
\[
H^3(Y_{b_0}, \mathbb{Z}) \cong H^3({\mc Y'}, \mathbb{Z}) \to H^3(Y_{b}, \mathbb{Z})^{inv}=(j_* j^* R^3q_* \mathbb{Z})_{b_0}
\]
is an isomorphism (cf. \cite[pg 439-440]{Steenbrink}), where $b \in \Delta \cap {U'}$ and $H^3(Y_{b}, \mathbb{Z})^{inv} \subset H^3(Y_{b}, \mathbb{Z})$ are the local monodromy invariants. Let $\delta \in H_3(Y_{b}, \mathbb{Z})$ be the vanishing cycle of ${\mc Y'}_\Delta$. By the Picard--Lefschetz formula $H^3(Y_{b}, \mathbb{Z})^{inv}=\mathbb{Z} \delta^\perp$, where $\perp$ is taken with respect to the intersection product (which is non--degenerate since $H^3(Y_{b}, \mathbb{Z})$ is torsion free). By \cite[Cor. 2.17]{Voisin2}, there is a short exact sequence
\[
0 \to \mathbb{Z} \delta \to H_3(Y_b, \mathbb{Z}) \to H_3({\mc Y'}_\Delta, \mathbb{Z})\cong H_3(Y_{b_0}, \mathbb{Z}) \to 0.
\]
where $0 \neq \delta \in H_3(Y_b, \mathbb{Z})$ is the class of the vanishing cycle.
Dualizing, we get a short exact sequence
\[
0 \to H^3(Y_{b_0}, \mathbb{Z})\to H^3(Y_b,\mathbb{Z}) \to (\mathbb{Z} \delta)^\vee \to 0.
\]
(recall the absence of torsion in the homology groups of $Y_b$ and $Y_{b_0}$).
Using the isomorphism $H_3(Y_b,\mathbb{Z}) \cong H^3(Y_b, \mathbb{Z})$ induced by Poincar\'e duality we make the identification $\mathbb{Z} \delta^\perp=\ker [H^3(Y_{b}, \mathbb{Z})\to \mathbb{Z} \delta^\vee]=\im[H^3(Y_{b_0}, \mathbb{Z})\to H^3(Y_b,\mathbb{Z})]$.
\end{proof}
It is well known that for a Lefschetz pencil the Leray spectral sequence with coefficients in $\mathbb{Q}$ degenerates at $E_2$. For a Lefschetz pencil of cubic threefolds, this is true also for $\mathbb{Z}$ coefficients. Again, we include a proof for lack of reference.
For the whole family of smooth hyperplane sections of $X$ the Leray spectral sequence with integers coefficients does \emph{not} degenerate at $E_2$; this is the starting point of the construction of the non trivial $J_U$-torsor of \cite{Voisin-twisted} (cf. Remark \ref{twisted}).
\begin{lemma} \label{Leray filtration} Let $q: {\mc Y'} \to \P^1$ be as above. The Leray spectral sequence with $\mathbb{Z}$ coefficients degenerates at $E_2$. In particular, the Leray filtration on $H^4({\mc Y'}, \mathbb{Z})$ is given by:
\begin{equation} \label{leray filtration}
\begin{aligned}
\mathbb{Z} =H^2(\P^1, R^2 f_*\mathbb{Z}) \subset L_1 \subset H^4({\mc Y'}, \mathbb{Z}) \twoheadrightarrow H^0(\P^1, R^4 f_*\mathbb{Z})=\mathbb{Z}\\
0 \to H^2(\P^1, R^2 f_*\mathbb{Z}) \to L_1 \stackrel{\gamma}{\to} H^1(\P^1, R^3 f_*\mathbb{Z}) \to 0.
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof} Because of the many vanishings in the $E_2$-page of the spectral sequence, the only map we need to show is trivial is $H^0(\P^1, R^4 q_*\mathbb{Z}) \to H^2(\P^1, R^3 q_*\mathbb{Z})$. For this, it is enough to show that $H^4(Y_b,\mathbb{Z}) \to H^0(\P^1, R^4 q_*\mathbb{Z})$ is surjective, which is clearly true since both groups are generated by the class of a line.
\end{proof}
Consider the decomposition
\begin{equation} \label{blowup}
H^4({\mc Y'},\mathbb{Z})=H^4(X,\mathbb{Z}) \oplus H^2(\Sigma,\mathbb{Z}) \oplus H^0(\Sigma, \mathbb{Z}),
\end{equation}
given by the blowup formula.
The inclusion of the first summand is given by the pullback $p^*$; we freely omit the symbol $p^*$ when viewing $H^4(X,\mathbb{Z}) $ as a subspace of $H^4({\mc Y'},\mathbb{Z})$. The inclusion of the second factor is given by $
H^2(\Sigma,\mathbb{Z}) \ni C \mapsto i_*(C \times \P^1) \in H^4({\mc Y'},\mathbb{Z})$. Finally,
the inclusion of the last summand is given by $H^0(\Sigma,\mathbb{Z})=H^0(\Sigma, \mathbb{Z}) \otimes H^2(\P^1,\mathbb{Z}) \ni [\Sigma]=[\Sigma \times p] \mapsto i_*([\Sigma \times p]) \in H^4({\mc Y'},\mathbb{Z})$, where $p \in \P^1$ is a point.
We highlight the following results for later use.
\begin{lemma} \label{lemmakerAJ} There is a natural isomorphism $H^0(\Sigma, \mathbb{Z}) \cong H^2(\P^1, R^2 q_* \mathbb{Z})$ which allows the identification of the inclusion $H^0(\Sigma, \mathbb{Z})\cong H^0(\Sigma, \mathbb{Z})\otimes H^2( \P^1, \mathbb{Z}) \stackrel{i_*}{\to} H^4({\mc Y'},\mathbb{Z})$ of (\ref{blowup}) with the inclusion $H^2(\P^1, R^2 q_* \mathbb{Z}) \to H^4({\mc Y'},\mathbb{Z})$ induced by the Leray filtration (\ref{Leray filtration}).
\end{lemma}
\begin{proof}
The closed embedding $i: \Sigma \times \P^1 \hookrightarrow {\mc Y'}$ determines an isomorphism $ {p_2}_* \mathbb{Z} \cong R^2 q_* \mathbb{Z}$ of constant local systems. Here $p_2: \Sigma \times \P^1 \to \P^1$ is the projection on the section factor. Since $H^2( \P^1, {p_2}_* \mathbb{Z}) = H^0(\Sigma, \mathbb{Z}) \otimes H^2(\P^1,\mathbb{Z}) $ the lemma follows.
\end{proof}
Via $p^*$, we make the identification $H^4(X,\mathbb{Z})_0 \cong L_1 \cap H^4(X,\mathbb{Z})$ and we set $ L_1^{2,2}=L_1 \cap H^{2,2}(\mathcal Y',\mathbb{Z})$. Here $L_1 \subset H^4(\mathcal Y',\mathbb{Z})$ denotes the second piece of the Leray filtration (cf. (\ref{leray filtration})).
\begin{cor} \label{periniettivita} The surjective morphism $\gamma: L_1 \to H^1(\P^1, R^3 q_* \mathbb{Z})$ of (\ref{leray filtration}) restricts to an injection
\[
\bar \gamma:L_1 \cap (H^{2,2}(X,\mathbb{Z}) \oplus H^2(\Sigma,\mathbb{Z})) \cong L_1^{2,2} \slash \ker(\gamma) \to H^1(\P^1, R^3 q_* \mathbb{Z}).
\]
\end{cor}
\begin{proof} From lemmas \ref{Leray filtration} and \ref{lemmakerAJ} above, it follows that $\ker (\gamma)=H^2(\P^1, R^2 q_* \mathbb{Z})=H^0(\Sigma, \mathbb{Z}) $. Thus, by (\ref{blowup}), it follows that $H^4(X,\mathbb{Z})\oplus H^2(\Sigma,\mathbb{Z})) \cap \ker(\gamma)=\{0\}$. Since $H^0(\Sigma, \mathbb{Z}) \subset L_1$ and
\begin{equation} \label{uffa}
L_1 \cap \Big(H^{2,2}(X,\mathbb{Z}) \oplus H^2(\Sigma,\mathbb{Z}) \oplus H^0(\Sigma, \mathbb{Z})\Big)=L_1 \cap \Big(H^{2,2}(X,\mathbb{Z}) \oplus H^2(\Sigma,\mathbb{Z}) \Big) \oplus H^0(\Sigma, \mathbb{Z})
\end{equation}
the Corollary follows.
\end{proof}
\begin{lemma} \label{fiveterm}
The restriction morphism $H^1(\P^1, R^3 q_*\mathbb{Z}) \to H^1({U'}, R^3 {q_{U'}}_*\mathbb{Z})$ is injective.
\end{lemma}
\begin{proof}
The Leray spectral sequence for the open immersion $j: {U'} \to \P^1$, applied to the sheaf $j^*R^3 q_*\mathbb{Z}= R^3 {q_{U'}}_*\mathbb{Z}$, gives a $5$-term exact sequence starting with
\[
0 \to H^1(\P^1, j_* j^*R^3 q_*\mathbb{Z}) \to H^1({U'}, R^3 {q_{U'}}_*\mathbb{Z}) \to \dots
\]
This concludes the proof, since by Lemma \ref{integral loc inv cycle}, $R^3 q_*\mathbb{Z}= j_* j^*R^3 q_*\mathbb{Z}$.
\end{proof}
\subsection{Injectivity of $\phi_X$.} \label{sec injectivity} The proof of injectivity uses the Hodge class of a normal function (cf. \cite{Zucker-inventiones} and \cite[\S 8.2.2]{Voisin2}).
For a pencil ${\mc Y'} \to \P^1$ as above, set
\[
H^{2,2}({\mc Y'}, \mathbb{Z})_0:=L_1^{2,2}=\ker[H^{2,2}({\mc Y'}, \mathbb{Z}) \to H^0(\P^1, R^4 q_*\mathbb{Z})]=L_1 \cap H^{2,2}({\mc Y'}, \mathbb{Z}).
\]
and let
\[
\pi'=J' \to \P^1, \text{ and } J_{U'} \to {U'},
\]
be the restriction of the intermediate Jacobian fibration to $\P^1$ and to ${U'}$. Choosing a set of generators for $H^{2,2}(X, \mathbb{Z})_0$, let ${\mc Y'} \to \P^1$ be a general enough pencil so that the restriction morphism
\begin{equation} \label{AJ for pencil}
\phi_X': H^{2,2}(X, \mathbb{Z})_0 \to MW(\pi')
\end{equation}
is well defined. Here, $MW(\pi')$ is the group of rational sections of $\pi'$. Similarly, we get a group homomorphism $\phi'_{\mc Y'}:H^{2,2}({\mc Y'}, \mathbb{Z})_0 \to MW(\pi')$.
Moreover, if $\alpha \in H^{2,2}(X,\mathbb{Z})_0$ then
\[
\phi'_{\mc Y'}(p^*\alpha)=\phi'_X(\alpha) \in MW(\pi').
\]
Recall the Hodge class of a normal function (cf. \cite[8.2.2]{Voisin2}, \cite[(3.9)]{Zucker-inventiones})).
Let $ \mathcal H^3= R^3{q_{U'}}_* \mathbb{Z} \otimes_\mathbb{Z} \mathcal O_{U'}$ be the Hodge bundle associated to the weight $3$ variation of Hodge structure of the pencil and let $F^* \mathcal H^3$ be the Hodge filtration. The sheaf $\mathcal J_{U'}$ of holomorphic sections of the intermediate Jacobian fibration fits in the following exact sequence
\[
0 \to R^3{{f'}_{U'}}_* \mathbb{Z} \to \mathcal H^3 \slash F^2 \mathcal H^3 \to \mathcal J_{U'} \to 0.
\]
and the coboundary morphism
\[
\begin{aligned}
H^0({U'}, \mathcal J_{U'} ) & \stackrel{cl}{\longrightarrow} H^1({U'},R^3f_* \mathbb{Z})\\
\nu & \longmapsto cl(\nu)
\end{aligned}
\]
associates to every holomorphic section $\nu$ of $J_{U'} \to {U'}$ a class $cl(\nu)$ in $H^1({U'},R^3q_* \mathbb{Z})$, called the Hodge class of $\nu$ (in the present context, this class is of Hodge type with respect to the Hodge structure on $H^1({U'},R^3q_* \mathbb{Z})$ induced from that on $H^4({\mc Y'}, \mathbb{Z})$ via the degeneracy of the Leray spectral sequence, see \cite[\S 3]{Zucker-inventiones}).
\begin{lemma} \label{injectivity} Let ${\mc Y'} \to \P^1$ be a general pencil. The homomorphism of (\ref{AJ for pencil})
\[
\phi'_X: H^{2,2}(X, \mathbb{Z})_0 \stackrel{\beta}{\longrightarrow} MW(\pi')
\]
is injective.
\end{lemma}
\begin{proof}
By \cite[ Prop. (3.9)]{Zucker-inventiones} (see also \cite[Lem 8.20]{Voisin2}), the following diagram is commutative
\begin{equation} \label{diagraminj}
\xymatrix{
H^{2,2}(X, \mathbb{Z})_0 \ar[r]^{p^*} \ar[dr]_{\phi'_X} & H^{2,2}({\mc Y'}, \mathbb{Z})_0 \ar[d]_{\phi'_{\mc Y'}} \ar[r]^\gamma & H^1(\P^1, R^3 p_* \mathbb{Z}) \ar[d]^\varepsilon \\
& H^0({U'}, \mathcal J_{U'}) \ar[r]_{cl} &H^1({U'},R^3{f'}_* \mathbb{Z})
}
\end{equation}
The map $\varepsilon$ is injective by Lemma \ref{fiveterm}, and $p^* \circ \gamma$ is injective by Lemma \ref{periniettivita}. Hence, $cl \circ \phi'_X$ is injective and thus so is $\phi'_X$.
\end{proof}
\subsection{Surjectivity of $\phi_X$.} \label{sec surjectivity}
There are three ingredients in the proof of surjectivity: the fact that $\rk MW(\pi)=\rk H^{2,2}(X, \mathbb{Z})_0$ as proved in Corollary \ref{ranks}; the restriction, once again, to Lefschetz pencils; the techniques used in \cite{Voisin-Some-Aspects,Zucker-HC} for the proof of the integral Hodge conjecture for cubic fourfolds. We remark that we use their argument in a slightly different way. To prove the Hodge conjecture one starts with a cohomology class, uses it to define a normal function, and then uses the normal function to construct an algebraic cycle representing the cohomology class (possibly up to a multiple of a complete intersection surface). See \cite{Voisin-Some-Aspects} for more details. Here we start with a rational section of the intermediate Jacobian fibration, we restrict to a general pencil, and use the same method of Voisin to construct an algebraic cycle inducing the section via the Abel-Jacobi map. Then we have to check that the cohomology class representing this cycle is primitive, that it is independent of the pencil, and that it induces, via $\phi_X$, the section we started from.
Since by Corollary \ref{ranks} the cokernel of the injection $\phi_X: H^{2,2}(X, \mathbb{Z})_0 \to MW(\pi)$ is finite, for any $\sigma \in MW(\pi)$ there is an integer $N$ and a cohomology class $ \alpha \in H^{2,2}(X, \mathbb{Z})_0$ such that
\begin{equation} \label{Nsigma}
\sigma_\alpha:=\phi_X(\alpha)=N \sigma.
\end{equation}
We will show, again using Lefschetz pencils, that given $\sigma$ and $\alpha$ as above, there exists a $\bar \beta' \in H^{2,2}(X, \mathbb{Z})_0$ such that $\alpha=N \bar \beta'$. This will give the desired surjectivity. Before we do so, let us introduce some results that we will need.
For a general pencil ${\mc Y'} \to \P^1$, let
\[
(J^T)' \to \P^1
\]
be the restriction of the intermediate Jacobian fibration $J^T \to \P^5$ of \cite{Voisin-twisted} (cf. Remark \ref{twisted}) to the pencil. For a conic $C \subset \Sigma$, consider the relative $1$-cycle of degree $2$ in ${\mc Y'} \to \P^1$ (any other degree $2$ relative $1$--cycle that comes from $\Sigma$ will do). This defines a section of $(J^T)' \to \P^1$, which trivializes the torsor $(J^T)_{U'}$ inducing an isomorphism $J'_{U'} \cong(J^T)_{U'}$. It is easily seen that this extends to an isomorphism $t_C: J' \cong(J^T)'$ over $\P^1$. For any $\sigma' \in H^0({U'}, \mathcal J_{U'})$, we may consider the induced section
\[
(\sigma^T)':=t_C \circ \sigma' \in H^0({U'}, \mathcal J^T_{U'})
\]
The following result is proved in Voisin \cite{Voisin-Some-Aspects} (see also \cite[(3.2)]{Zucker-HC}, where the result is proved over $\mathbb{Q}$).
\begin{prop} (\hspace{1sp}\cite[\S 2.3]{Voisin-Some-Aspects}) \label{lifting} For any section $\sigma' \in MW(\pi') $
there is a relative $1$-cycle $Z$ on $ {\mc Y'}$ of degree $2$, such that the cohomology class
\[
\beta'=[Z]-[C \times \P^1] \in H^{2,2}({\mc Y'}, \mathbb{Z})_0
\]
satisfies $\phi'_{{\mc Y'}}(\beta')=\sigma'$ in $MW(\pi') $.
\end{prop}
\begin{proof}
For the reader's convenience, we give a brief sketch of the argument. By a result of Markushevich-Tikhomirov \cite{Markushevich-Tikhomirov} and Druel \cite{Druel} there is a a relative birational morphism $c_2: {\mc M'}_{U'} \to J^T_{U'}$, where $ {\mc M'}_{U'} \to {U'}$ is the relative moduli space of sheaves on ${\mc Y'}_{{U'}} \to {U'}$ with $c_1=0$ and $c_2=2\ell$. The morphism associates to every sheaf corresponding to a point in $ {\mc M'}_{U'} $ the Abel-Jacobi invariant of its second Chern class. Given a section $(\sigma^T)' \in H^0({U'}, \mathcal J^T_{U'})$ as above, Voisin uses $ {\mc M'}_{U'} \to {U'}$ to construct a family $\mathcal C_{U'}$ of degree $2$ curves in the fibers of ${\mc Y'}_{U'} \to {U'}$ with the property that for every $b \in {U'}$, the curve $\mathcal C_b$ represents the $c_2$ of a sheaf over $(\sigma^T)'(b)$. By construction, letting $Z $ be the closure of $\mathcal C_{U'}$ in ${\mc Y'}$ and setting $\beta':=[Z]-[C \times \P^1] \in H^{2,2}({\mc Y'}, \mathbb{Z})_0$, we have $\phi'_{{\mc Y'}}(\beta')=\sigma' $ in $H^0({U'}, \mathcal J_{U'})$.
\end{proof}
Let $\sigma \in MW(\pi)$. For a general pencil $\P^1 \subset \P^5$, let $\sigma'=\sigma_{|\P^1}$ be the restriction of $\sigma $ to $\P^1$, and let $\beta'$ be as in the Proposition above so that $\phi'_{{\mc Y'}}(\beta')=\sigma'$. It is tempting to say that, via $\phi_X$, the class $\beta'$ induces $\sigma$ globally and not just on that pencil. This is indeed the case, though we first need to check that $\beta'$ lies in the primitive cohomology of $X$ and that $\beta'$ is independent of the pencil as well as of the chosen isomorphism $t_C: J' \cong(J^T)' $. More precisely, we need to check that $\beta'$ induces $\sigma$ over an open subset of $\P^5$ and not just on the chosen pencil. Before checking this, we have the following Proposition.
Recall that we have set $H^{2,2}({\mc Y'},\mathbb{Z})_0=L_1 \cap H^{2,2}({\mc Y'},\mathbb{Z})$.
\begin{prop}(\hspace{1sp}\cite[(4.17)]{Zucker-inventiones}) \label{propaj} The Abel-Jacobi morphism $\phi_{{\mc Y'}}: H^{2,2}({\mc Y'}, \mathbb{Z})_0 \to MW(\pi') \subset H^0({U'}, \mathcal J_{U'})$ is surjective and defines an isomorphism
\[
\bar \phi_{{\mc Y'}}: L_1 \cap (H^{2,2}(X,\mathbb{Z}) \oplus H^2(\Sigma,\mathbb{Z})) \to MW(\pi')
\]
\end{prop}
\begin{proof} By diagram (\ref{diagraminj}) and the fact that $\epsilon$ is injective, $\ker (\phi_{{\mc Y'}})=\ker \gamma$, which by Lemma \ref{Leray filtration} is equal to $H^0(\Sigma, \mathbb{Z})$. Since $\phi_{{\mc Y'}}$ is surjective by the proposition above, the induced morphism $\bar \phi_{{\mc Y'}}: H^{2,2}({\mc Y'},\mathbb{Z})_0 \slash H^0(\Sigma, \mathbb{Z}) \to MW(\pi')$ is an isomorphism. Finally, by (\ref{uffa}),
$H^{2,2}({\mc Y'},\mathbb{Z})_0 \slash H^0(\Sigma, \mathbb{Z}) \cong L_1 \cap (H^{2,2}(X,\mathbb{Z}) \oplus H^2(\Sigma,\mathbb{Z}))$.
\end{proof}
We can now end the proof of surjectivity:
For $\sigma \in MW(\pi)$, let $\alpha \in H^{2,2}(X,\mathbb{Z})_0$ be as in (\ref{Nsigma}). Restricting to a pencil ${\mc Y'} \to \P^1$,
set $\sigma'=\sigma_{|\P^1}$ and let $\beta'$ be as in Proposition \ref{lifting} such that $\phi_{\mathcal Y'}(\beta')=\sigma'$.
Finally, let $\bar \beta' $ be the projection of $\beta'$ onto $L_1 \cap (H^{2,2}(X,\mathbb{Z}) \oplus H^2(\Sigma,\mathbb{Z}))$. Notice that abuse of notation, we are omitting $p^*$ from the inclusion of $H^4(X,\mathbb{Z})$ in $H^4(\mathcal Y',\mathbb{Z})$ and we will write $\alpha$ instead of $p^* \alpha$.
We have,
\[
\phi_{\mathcal Y'}(\alpha)=(\phi_X(\alpha))_{|\P^1}=N \sigma'=N \phi_{\mathcal Y'}(\beta' )= \phi_{\mathcal Y'}(N \beta' ).
\]
By Proposition \ref{propaj}, $ \alpha=N \bar \beta' \in L_1 \cap (H^{2,2}(X,\mathbb{Z}) \oplus H^2(\Sigma,\mathbb{Z}))$. Since $ \alpha \in H^{2,2}(X,\mathbb{Z})_0 \subset L_1 \cap \Big(H^{2,2}(X,\mathbb{Z}) \oplus H^2(\Sigma,\mathbb{Z})\Big)$, it follows that $\bar \beta'$, too, has to lie in $H^{2,2}(X,\mathbb{Z})_0 \subset H^{2,2}({\mc Y'}, \mathbb{Z})$. Moreover, the class $ \bar \beta'$, which a priori depends on the chosen Lefschetz pencil is independent of the pencil. Set $\sigma_{\bar \beta'}=\phi_X(\bar \beta')$. Then, for \emph{any} sufficiently general Lefschetz pencil $\P^1 \subset \P^5$ we have an equality of sections
\[
(\sigma_{\bar \beta'})_{|\P^1} =\sigma_{|\P^1},
\]
and hence the two rational sections $\sigma_{\bar \beta'}$ and $\sigma$ coincide. This ends the proof of surjectivity.
|
3,212,635,537,444 | arxiv | \section{Introduction} \label{sec:intro}
Most of the stellar mass in galaxies was assembled between a redshift of $z=3$ to 1 (e.g., \citealt{CarilliWalter2013,MadauDickinson2014}); at $z=2$, the star-formation rate density (SFRD) of the Universe peaked, fueled by enhanced gas accretion from the intergalactic medium \citep{Keres2005,Keres2009,Genzel2008,Tacconi2010}. Statistical X-ray and infrared (IR) observations of galaxies hosting active galactic nuclei (AGN) support the coincident mass growth of central supermassive black holes (SMBHs; \citealt{Shankar2009,Aird2010,Delvecchio2014}) and possibly SMBH-galaxy co-evolution (e.g., \citealt{Ferrarese2000,Gebhardt2000,Kormendy2013,MadauDickinson2014}).
In particular, AGN feedback may be responsible for quenching star formation \citep[e.g.,][]{Springel2005,Cicone2014} or/and maintaining quenched galaxies by suppressing cooling flows \citep[e.g.,][]{Keres2005,Croton2006,Bower2006}.
AGN feedback helps cosmological theories of galaxy evolution match observations -- in particular, the number counts of massive and quenched galaxies at $z=0$ (e.g., \citealt{Bower2006,Nelson2018}).
Testing this framework against observations, particularly at high redshift, is limited by the ability to separate the AGN and star-forming components of the spectral energy distributions (SEDs) of galaxies,
as to test the `quasar mode' of AGN feedback, it is necessary to simultaneously constrain the AGN luminosity and star formation rate (SFR) at a time when the AGN should be heavily \cchb{dust-enshrouded}
\citep[e.g.,][]{DiMatteo2005,Hopkins2008}.
The far-IR (FIR)/sub-millimeter regime is thus a powerful probe of dust-obscured mass assembly.
\cch{It has long been known that \cchb{dust-enshrouded} AGN can dominate the total IR luminosity \citep[e.g.,][]{Sanders1988}, but it is
typically assumed that AGN contribute negligibly to FIR emission at wavelengths longer than rest-frame $100\,\mu$m,
as predicted by torus models \citep[e.g.,][]{Fritz2006,Nenkova2008a,Nenkova2008b}.
Consequently, cold dust emission is sometimes treated as a `safe'
SFR tracer even when an AGN dominates the bolometric luminosity of the galaxy by explicitly
converting the FIR luminosity into an SFR value
\citep[e.g.,][]{Hatziminaoglou2010,Kalfountzou2014,Azadi2015,Gurkan2015,Stacey2018,Banerji2018,Wethers2020}.}
Alternatively, the full SED may be fit using an SED
modeling code that includes an AGN component
\citep[e.g.,][]{Ciesla2015,Chang2017,Lanzuisi2017,Dietrich2018,Leja2018,Pouliasis2020,Ramos2020,Yang2020}.
\cch{Such SED modeling codes
generally assume that AGN contribute
negligibly to cold-dust emission longward of $\sim 100$
\micron, and rely
on SED model libraries generated by performing RT
on AGN torus models. The resulting SEDs
include hot dust emission from the torus
\emph{but not potential AGN-powered cold
dust emission on host-galaxy scales by construction},
even though UV/optical photons emitted by the accretion
disk or IR photons from the torus can in principle
heat ISM dust directly and/or by progressive attenuation and re-radiation at longer wavelengths.
That this possibility is realized in
some systems is suggested by results indicating that
the host galaxy can be responsible for a
non-negligible fraction of the ing column density
in some AGN (\citealt{Hickox2018}
and references therein). The origin of the far-IR emission in galaxies hosting luminous AGN has been debated over the past few decades (e.g., \citealt{Downes1998,Page2001,Franceschini2003,Ruiz2007,Kirkpatrick2012,Kirkpatrick2015}). Over the years, a growing body of literature has been making the case for AGN-powered FIR emission in QSOs (e.g., \citealt{Sanders1989,Yun1998,Nandra2007,Petric2015,Schneider2015,Symeonidis2016,Symeonidis2017,Symeonidis2018})}; however, other works find conflicting results (e.g., \citealt{Stanley2017,Shangguan2020,DiMascia2021}).
\jm{A promising method for decomposing the far-IR SEDs into AGN and SF components directly relies on mid-IR spectroscopy (e.g., \citealt{Pope2008}). \cite{Kirkpatrick2012,Kirkpatrick2015} use this technique to subtract AGN-heated dust from the SEDs of a sample of high$-z$ dusty galaxies using mid-IR spectroscopic methods assuming all of the $\sim80-90$ K dust emission is attributed to the AGN alone. These authors find that up to 75\% of L$_{\mathrm{\scriptsize IR}}$\ can be powered by AGN heating of the hot dust in galaxies with spectroscopic signatures of AGN in the mid-IR. Whether this conclusion extends to host-galaxy-scale cold dust emission at wavelengths past $100\,\mu$m is difficult to test empirically because of sparse wavelength coverage in the rest-frame far-IR/sub-mm and because dust re-processing erases information about the original energy source. \citet{Roebuck2016} applied \citeauthor{Kirkpatrick2015}'s method to attempt to recover the AGN fraction of simulated galaxies, including the one employed in this work. Although the recovered and true AGN fractions were in qualitative agreement, \cite{Roebuck2016} found that SED decomposition underestimated the AGN contribution to the IR luminosity in some cases.}
In this work, we \cch{build on previous work in which we used hydrodynamical simulations plus radiative transfer (RT) calculations to investigate the effects of galaxy-scale dust reprocessing of AGN torus emission \citep{Younger2009,Snyder2013,Roebuck2016}.
Specifically, we use a hydrodynamical simulation of an equal-mass galaxy merger post-processed with RT to} investigate the possibility that heating from a luminous AGN embedded within a dusty galaxy can power host-galaxy-scale cold dust emission \jm{at rest-frame wavelengths $\lambda \ga 100\,\mu$m}.
We create initial conditions for two identical
galaxies on a parabolic orbit.
The hydrodynamical simulation evolves the system,
including models for star formation, stellar feedback,
black hole accretion and AGN feedback.
Then, we perform dust RT calculations in
post-processing for various time snapshots to
compute spatially resolved UV-mm SEDs of the simulated galaxy merger.
By performing dust RT calculations both including and excluding the emission from the AGN, we can explicitly identify IR emission powered by the AGN on both torus and host-galaxy
scales.
\cch{This particular simulation is a massive, gas-rich,
merger designed to be analogous to $z \sim 2-3$
submillimeter galaxies (SMGs), and it is indeed
consistent with many properties of observed SMGs
\citep{Hayward2011,Hayward2012,Hayward2013}.
We chose this particular simulation because
it exhibits a high SFR, a high AGN luminosity, and high
dust ation; such systems are the most likely
candidates for AGN powering host-galaxy scale cold-dust
emission.}
\cch{We show that in this admittedly extreme system,}
\cchb{dust-enshrouded} AGN can be the dominant power source of cold-dust emission on host-galaxy scales, with cool FIR colors indistinguishable from those of
purely star-forming galaxies. We estimate the potential bias in observations of galaxy SFRs
and discuss implications for SED decomposition.
The remainder of this paper is organized as follows: in Section \ref{sec:methods}, we discuss the details of our numerical simulation and RT calculations.
Section \ref{sec:results} explains how we extract the AGN-powered dust emission and presents our main results.
We discuss the implications of these calculations for inferred trends in galaxy evolution and some limitations of this work in Section \ref{sec:discussion}. We summarize in Section \ref{sec:conclusions}. Throughout this work, we assume a Salpeter IMF and adopt a $\Lambda$CDM cosmology with $\Omega_m=0.3$, $\Omega_\Lambda=0.7$, and $H_0=70$ km s$^{-1}$ Mpc$^{-1}$.
\section{Simulation and Radiative Transfer Details\label{sec:methods}}
This work makes use of the results of RT calculations originally presented in \citet{Snyder2013}.
Our re-analysis focuses on the output from a \texttt{Gadget-2} simulation \citep{Springel2005} of an idealized (non-cosmological) major merger of two identical disk galaxies.
The initial conditions were generated following \citet{Springel2005merger}.
The initial halo and baryonic masses are $9\times10^{12}$ M$_\odot$ and $4\times10^{11}$ M$_\odot$, respectively.
The initial black hole mass is $1.4\times10^5$ M$_\odot$, and the initial gas fraction is 60\%. Star formation and stellar feedback are modeled as described in \citet{Springel2003} and \citet{Springel2005merger}.
Black holes grow via Eddington-limited Bondi-Hoyle accretion.
The RT code \texttt{SUNRISE} \citep{Jonsson2006,Jonsson2010} was used to compute SEDs for seven viewing angles every 10 Myr
throughout the simulation run.
\cchb{The outputs of the hydrodynamical simulation are used as input for the RT calculations, i.e., the former is used to specify the 3D distribution of sources of emission (stars and AGN) and dust.
The ages are metallicities of the star particles
are used to assign single-age stellar population SEDs to individual star particles.
The metal distribution determines the dust density distribution; a dust-to-metals ratio of 0.4 was assumed.} For more in-depth discussion of the simulation assumptions, sub-grid models, and numerical methods, see \cite{Snyder2013} and \cite{Hayward2011}.
For each time snapshot of the merger simulation, different RT calculations were performed;
they differ only in how the luminosity of the AGN is computed. The `fiducial' RT runs use the accretion rate from the simulation to compute the
AGN luminosity, assuming 10\% radiative efficiency.
\cchb{We use luminosity-dependent AGN SED templates derived from observations of unreddened QSOs \citep{Hopkins2007c} as the input SEDs emitted by the AGN particle(s)}; see \citet{Snyder2013} for details.
In the `AGN-off' runs, the luminosity of the AGN is artificially set to zero (i.e., only stellar emission is considered in the radiative
transfer calculations; note that AGN feedback is still included because the same hydrodynamical simulation is used as input, so these runs determine the impact of the AGN emission on the total SED \emph{all else being equal}). Finally, in the `AGN-10x' RT calculations, \cch{the luminosity of the AGN is artificially boosted by a factor of 10
(equivalent to assuming a radiative efficiency of 100\%
or an instantaneous accretion rate ten times the value
computed in the simulation). As in the `AGN-off'
runs, the AGN feedback
is not altered (the same hydrodynamical simulations
are used) because we wish to isolate the effect of
the AGN on the SED}. In other words, the input of a boosted AGN spectrum into the RT calculations is the only difference between the AGN-10x and AGN-off results.
\cite{Snyder2013} performed these calculations simply to span a larger range in the AGN fractional contribution to the bolometric luminosity,
but these should not be considered `unphysical' by any means, as the relatively crude accretion model employed and the limited resolution of the hydrodynamical simulation
may cause the BH accretion rate to be underestimated \citep[e.g.,][]{Hayward2014b,Angles-Alcazar2020}.
The Eddington ratio ($\lambda_{edd}$$\equiv \mathrm{L}_{\mathrm{AGN}}/\mathrm{L}_{edd}$) varies from zero to one in the fiducial runs and thus spans a range of $\lambda_{edd}$$\,=0-10$ in the AGN-10x runs.
There is both theoretical \citep[e.g.,][]{Begelman2002,Jiang2014} and observational \citep[e.g.,][]{Kelly2013,Shirakata2019} evidence for super-Eddington accretion.
\begin{figure}[t!]
\centering
\includegraphics[width=0.48\textwidth]{Fig1.pdf}
\caption{\footnotesize Rest-frame SED for the AGN-10x model (dashed line) at the time of peak L$_{\mathrm{AGN}}$/L$_{\mathrm{SF}}$, where the AGN dominates the bolometric luminosity and the time at which
the AGN contributes maximally to the FIR luminosity during this simulation. The unnatenuated AGN torus spectrum (L$_\lambda^{\mathrm{AGN,10x,input}}$) is shown as a dotted line, and the unnatenuated stellar spectrum is shown as a solid grey line (L$_\lambda^{\mathrm{stellar,input}}$). For the same simulation
snapshot, we show the corresponding AGN-off SED shaded in blue. The effective AGN SED, calculated by taking the difference between the AGN-10x and AGN-off SEDs, is
indicated by the solid line. This corresponds to the primary torus emission attenuated by dust \emph{plus} host-galaxy-scale thermal dust emission powered by the AGN.
Due to heavy dust obscuration experienced by the AGN, the AGN contributes negligibly to the UV-optical SED, but it dominates longward of a
few microns, including the FIR cold dust emission traditionally associated with star formation.
\label{fig:sed}}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=0.95\textwidth]{Fig2.pdf}
\caption{Emission maps of the simulated merger at a representative time following the peak SFR and peak bolometric AGN luminosity (marked by the dashed line in the top panel). The first (second) column
shows maps of the emission at rest-frame wavelengths $15\,\mu$m, $115\,\mu$m, and $370\,\mu$m for the AGN-off (AGN-10x) RT run. The third column shows difference images
between the AGN-10x and AGN-off maps, i.e., the dust emission powered directly by the AGN. The color scale and stretch is constant across all panels.
For comparison with observations, we blur the images to $100$ pc resolution, typical of spatially resolved studies of lensed systems at $z\sim2$. In the far-right column, we \jm{further blur each map to the spatial resolution achievable with \textit{JWST}/MIRI (\textit{top}) and ALMA high-frequency observations in extended array configurations} (\textit{center, bottom}) for a galaxy at $z=2$. \jm{The \textit{JWST}/MIRI beam is shown as a solid black circle, and the ALMA beams as white hatched circles.} The AGN-powered cold dust emission includes both a compact nuclear
component and an extended component spanning a few kpc.
\label{fig:extent}}
\end{figure*}
\section{Results\label{sec:results}}
For every snapshot and each of the seven viewing angles, we calculate the SED of the AGN-powered dust emission, i.e., the dust emission implicit in the input torus model SED
plus the host-galaxy dust emission powered by the AGN (rather than stars). Our \texttt{SUNRISE} calculations with and without AGN emission are otherwise identical; therefore, taking the difference
between the SEDs with and without AGN emission yields the SED of all photons of AGN origin, including those reprocessed by dust. Originally demonstrated in \cite{Roebuck2016} for our fiducial simulation
(see their Fig. 2),
this differencing technique is shown in Figure \ref{fig:sed} for the AGN-10x run at $t_{peak}$, the simulation time when the ratio of bolometric AGN luminosity (L$_{\mathrm{AGN}}$) to bolometric stellar luminosity
(L$_{\mathrm{SF}}$) reaches its maximal value of $\sim10$. At this time, the mass of the BH is $3.2\times10^8\,\mathrm{M_\odot}$, and it is accreting at a rate of $82\,\mathrm{M_\odot\,yr^{-1}}$, while SFR$\,=470\,\mathrm{M_\odot\,yr^{-1}}$. The SEDs shown in Fig.~\ref{fig:sed} are the following:
\begin{itemize}
\item $\mathrm{L_\lambda^{AGN,\,10x}}$\,-- The attenuated$+$re-radiated SED corresponding to the AGN-10x run, where the luminosity of the AGN is artifically boosted by a factor of 10.
\item $\mathrm{L_\lambda^{AGN,\,off}}$\,-- The attenuated$+$re-radiated SED corresponding to the RT run where the luminosity of the AGN is set to zero.
\item L$_{\lambda}^{\mathrm{AGN,eff}}$\,-- The ``effective AGN SED'' calculated from the difference between $\mathrm{L_\lambda^{AGN,\,10x}}$ and $\mathrm{L_\lambda^{AGN,\,off}}$, which removes stellar-heated dust emission, leaving behind only AGN photons, \cchb{including IR photons from the torus that are absorbed by dust in the ISM} re-radiated at progressively longer wavelengths.
\item L$_\lambda^{\mathrm{AGN,10x,input}}$\,-- The input AGN SED (from \citealt{Hopkins2007c}) in the AGN-10x run. \cchb{This SED is integrated from 0.1--1000 \micron} to calculate L$_{\mathrm{AGN}}$.
\item $\mathrm{L_\lambda^{stellar,\,input}}$\,-- The unattenuated stellar spectrum, which is the same across all RT runs. \cchb{This SED is integrated over 0.1--1000 \micron} to determine L$_{\mathrm{SF}}$.
\end{itemize}
L$_{\lambda}^{\mathrm{AGN,eff}}$\ peaks at $31\,\mu$m, corresponding to an effective dust temperature (from Wien's law) of $96$ K, \jm{on the warm end of dust temperatures observed in AGN-host galaxies: \cite{Kirkpatrick2015} find that the warm dust component, attributed to AGN heating, is $\sim80-90$ K for high$-z$ ultra-luminous IR galaxies (ULIRGs; L$_{\mathrm{\scriptsize IR}}$$>10^{12}$ L$_\odot$) with high AGN fractions.}
\jm{In the simulated SEDs,} most of the IR emission between 10 and 50 $\mu$m comes from re-processed AGN photons, and L$_{\lambda}^{\mathrm{AGN,eff}}$\ is greater than L$_{\lambda}^{\mathrm{AGN,off}}$\ by a factor of $\gtrsim2$ from $100-1000\,\mu$m. In this particular (extreme) case, even the coldest dust is predominantly heated by the AGN.
\cchb{The top panel of Fig.~\ref{fig:extent} shows
the time evolution of the SFR, AGN luminosity, and fractional contribution of the AGN to the IR luminosity. Near coalescence of the two galaxies,
there is a strong starburst. The SFR then rapidly decays due to gas consumption (there is no cosmological gas accretion in this idealized simulation) and AGN feedback. Approximately 50
Myr after the peak SFR, the AGN luminosity peaks.
During this period, both the newly formed stars
and AGN are deeply obscured by dust.
The AGN contributes $\ga 30\%$ of the IR luminosity
for $\sim 0.5$ Gyr, starting at the peak of the
starburst. The AGN luminosity then declines, decreasing below
10\% of its peak luminosity $\sim 100$ Myr later.}
The lower part of Fig.~\ref{fig:extent} presents maps of the \jm{5\,$\mu$m, 115\,$\mu$m and 370\,$\mu$m} dust emission in the AGN-off (first column) and AGN-10x (second column) runs $\sim150$ Myr after $t_{peak}$,
\jm{a time at which there is significant dust-obscured star formation but the \cchb{dust-enshrouded} AGN dominates the luminosity (L$_{\mathrm{AGN}}$/L$_{\mathrm{SF}}$$>1$); this time is marked by
the dashed vertical line in the top panel.
Given the short duration of the starburst and
peak in AGN luminosity, most observations of
heavily dust-obscured AGN should probe this
phase (subsequent to the peak SFR and AGN luminosity),
so these maps should be considered `representative'.
Fig.~\ref{fig:extent} also shows} the difference between the \jm{AGN-10x and AGN-off simulations} (third column), which \jm{captures} the spatial extent of the dust emission \emph{powered by the AGN}. These maps have been convolved with a Gaussian
with FWHM of 100 pc, \jm{representative} of spatially resolved studies of lensed systems at $z\sim2-3$ \jm{(e.g., \citealt{Swinbank2015,Sharda2018,Canameras2017,Massardi2018})}.
The third column shows that the cold dust emission at \jm{far-IR wavelengths $\gtrsim100\,\mu$m} powered by the AGN consists of a compact nuclear component and a lower-surface-brightness
extended component on kpc scales, clearly demonstrating that photons from the AGN heat cold dust throughout the host galaxy.
\jm{In the fourth column of Figure \ref{fig:extent}, we} show maps of the AGN-powered dust emission convolved \jm{to the spatial resolution that \textit{JWST}/MIRI (\textit{top}) and ALMA's high-frequency, extended configurations (\textit{center, bottom}) can achieve for an un-lensed galaxy at $z=2$. With ALMA, the nuclear dust heated by the \cchb{dust-enshrouded} AGN can be resolved from the extended cold dust.}
\jm{At coarser spatial resolutions, imaging with \textit{JWST}/MIRI can test for buried AGN heating host-galaxy dust by comparing central and extended emission on scales above and below $\sim1$ kpc (solid circle), nearly equal to the half-mass radius of the system of 1.1 kpc. In Figure \ref{fig:cntr_flux}, we show the ratio of the flux within a simulated \textit{JWST}/MIRI beam placed on the center of a mock observation to the total flux in the map, for MIRI filters with central wavelengths between 5 and 25$\,\mu$m. Before the simulated aperture photometry, we first convolve the maps to the spatial resolution of JWST assuming the galaxy is at $z=2$. As expected, the AGN-10x simulation exhibits more concentrated emission in all MIRI filters than then AGN-off run because of the strong, nuclear dust-heating source. The presence of this luminous \cchb{dust-enshrouded} AGN increases the fraction of the total flux within the central $\sim$kpc relative to the total flux by a factor of $\sim1.5$ in most MIRI filters.
IR imaging will be key for identifying such \cchb{dust-enshrouded} AGN because,}
as we shall see below, the cold FIR colors would preclude distinguishing this source from a compact nuclear starburst based on the broadband FIR SED alone.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Fig3.pdf}
\caption{The ratio of flux within a simulated \textit{JWST}/MIRI beam placed on the center of the galaxy relative to the total flux across the PSF-convolved map, for various MIRI filters. We show the results of such mock observations for the AGN-10x (hatched) and AGN-off (solid gray) runs at $t_{peak}$, the same snapshot used in Fig. \ref{fig:extent}. The maps have been redshifted to $z=2$ and convolved with the wavelength-dependent PSF of \textit{JWST}/MIRI. The shaded and hatched regions contain the range of simulated observations for all seven viewing angles. The presence of a powerful, dust-enshrouded AGN boosts the fraction of the total flux within the central beam by a factor of $\sim1.5$ between observed wavelengths of $\sim6-15\,\mu$m. Mid-IR imaging at \textit{JWST}'s resolution could be used to identify such \cchb{dust-enshrouded} AGN.}
\label{fig:cntr_flux}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=.9\textwidth]{Fig4.pdf}
\caption{The ratio of the IR luminosity integrated in different bands for \jm{RT calculations with AGN+host galaxy-powered dust emission (`fiducial' or `AGN 10x') relative to those with only host galaxy dust emission (`AGN off')} vs. the ratio of bolometric AGN to star formation luminosity. Data from fiducial (AGN 10x) runs are shown as red squares (black circles). In each panel, we show data for all time snapshots and viewing angles. When the luminosity of the
AGN is ten times that of star formation, it can boost the total IR (8-1000 \micron) luminosity by an order of magnitude and the far-IR (40-500 \micron) luminosity by a factor of $\sim7$. The cold-dust FIR luminosity (100-1000 \micron),
traditionally assumed to be powered exclusively by star formation, can be boosted by a factor of three. \cchb{Therefore, depending on the precise wavelength range used, the SFR inferred from the IR or far-IR luminosity will overestimate the true SFR by a factor of $3-10$.}
\label{fig:lumRatio}}
\end{figure*}
\begin{figure}
\gridline{\fig{Fig5_top.pdf}{0.45\textwidth}{}}
\vspace{-3em}
\gridline{\fig{Fig5_bottom.pdf}{0.45\textwidth}{}}
\vspace{-2em}
\caption{\textit{Top:} \textit{Herschel} PACS 100$\mu$m/70$\mu$m vs. 160$\mu$m/100$\mu$m colors (assuming $z=2$). Cyan and red points correspond to the AGN 10x and fiducial runs, respectively, \jm{and the size of each marker is proportional to L$_{\mathrm{AGN}}$/L$_{\mathrm{SF}}$}. For comparison, we show the colors of empirical templates from \cite{Kirkpatrick2015} (K15) at $f_{\mathrm{\scriptsize AGN, IR}}$$=0\%$ (diamond), $f_{\mathrm{\scriptsize AGN, IR}}$$=50\%$ (square), and $f_{\mathrm{\scriptsize AGN, IR}}$$=100\%$ (triangle). Magenta and black contours contain 95\% and 68\% of the distribution in Herschel colors spanned by the \citep{Nenkova2008a,Nenkova2008b} and \cite{Fritz2006} dusty AGN torus models, respectively. \textit{Bottom:} \textit{Herschel} SPIRE
350$\mu$m/250$\mu$m vs. 500$\mu$m/350$\mu$m colors (assuming $z=2$) following the same color scheme as the upper panel. Blue contours contain the $z\sim2$ AGN-hosting galaxy sample of \cite{Hatziminaoglou2010}. Colors from the \cite{Kirkpatrick2015} SEDs are shown as described in the \textit{Top} panel, and we also include the SPIRE colors of the AGN SED from \cite{Symeonidis2016} (green triangle). The FIR colors of the simulated galaxies are redder (cooler) than those of \emph{all} of the torus models and consistent with those of the observed galaxies, suggesting the `cold' FIR colors of these AGN hosts are not necessarily indicative of ongoing star formation; instead, the FIR emission may be predominantly powered by \cchb{dust-enshrouded} AGN.
\label{fig:colors}}
\end{figure}
Having demonstrated that the AGN can power cold-dust emission on host-galaxy scales, we now quantify how the AGN affects the IR luminosity in different bands throughout the simulation.
Taking the ratio of L$_{\mathrm{\scriptsize IR}}$\ in the AGN-on and AGN-off simulations captures the fractional change in L$_{\mathrm{\scriptsize IR}}$\ due to dust emission powered by photons of AGN origin. In Figure \ref{fig:lumRatio}, we
plot the ratio of integrated IR luminosities between the AGN-on and AGN-off simulations in four different wavelength bands versus L$_{\mathrm{AGN}}$/L$_{\mathrm{SF}}$. The total (far-)IR luminosity from $8-1000\,\mu$m ($40-500\,\mu$m) \jm{is boosted}
by a factor of $\sim10$ ($\sim7$) as L$_{\mathrm{AGN}}$/L$_{\mathrm{SF}}$~increases from 1 to $\sim10$. The rise in L$_{\mathrm{\scriptsize IR}}$\ with L$_{\mathrm{AGN}}$/L$_{\mathrm{SF}}$\ is mostly driven by increased warm dust emission at shorter wavelengths, as shown in the $10-30\,\mu$m panel of Fig.~\ref{fig:lumRatio}. However, the integrated $100-1000\,\mu$m luminosity can be increased by the AGN by as much as a factor of $\sim3$, \jm{demonstrating that even the coldest dust emission can be powered by the AGN}. This result is at odds with the notion
that cold dust emission at these FIR wavelengths is predominantly heated by young stars even when an AGN dominates the bolometric luminosity of the galaxy
\citep[e.g.,][]{Stanley2017,Shangguan2020}.
FIR colors are often used to distinguish AGN-powered and stellar-powered dust emission, as IR-selected galaxies that host AGN tend to have warmer IR colors than those that do not (e.g., \citealt{deGrijp1987,Sanders1988};
but cf. \citealt{Younger2009}). For this reason, we present \emph{Herschel} FIR color-color plots of the simulation runs in Figure \ref{fig:colors}, \jm{while also tracking the ratio of AGN luminosity relative to the star-formation luminosity (L$_{\mathrm{AGN}}$/L$_{\mathrm{SF}}$). High L$_{\mathrm{AGN}}$/L$_{\mathrm{SF}}$\ tends to result in bluer colors; however, this is modulated by changes in extinction with time and viewing angle. Higher levels of extinction ($\mathrm{A_V}$) for fixed L$_{\mathrm{AGN}}$/L$_{\mathrm{SF}}$\ yield redder FIR colors.}
For comparison with observations, we also show the empirically derived galaxy SED templates of \cite{Kirkpatrick2015}, which span a range in AGN contribution to the mid-IR emission,
as well as the $z\sim2$ Type 1 and Type 2 AGN sample of \cite{Hatziminaoglou2010}.
The simulated galaxies have FIR colors on the Wien side of the dust distribution (PACS photometry, Fig.~\ref{fig:colors}, \textit{top}) consistent with the observed galaxies. \cchb{The SPIRE colors (Fig.~\ref{fig:colors}, \textit{bottom}), which trace the dust peak, of the simulated galaxies are somewhat \cchb{bluer (warmer)} than those of the majority of the observed galaxies by a factor of $\sim2-3$ when L$_{\mathrm{AGN}}$/L$_{\mathrm{SF}}$$\,>1$,
but they are redder (colder) than those of \emph{any} of the torus models in these two widely used torus SED libraries because the torus models do not include
galaxy-scale cold dust emission.
These results suggest that the cool FIR colors of the observed AGN hosts are not necessarily evidence that their FIR cold-dust emission is powered by ongoing star formation;
rather, the AGN itself may be the dominant power source for the cold-dust emission.}
\begin{deluxetable}{lcccc}
\tablecaption{Maximal boosting of host-galaxy FIR dust emission by a \cchb{dust-enshrouded} AGN at selected wavelengths, and for integrated IR luminosities. \label{tab:tab1}}
\tabletypesize{\footnotesize}
\tablehead{$\lambda_{obs}$ & $z=0$ & $z=1$ & $z=2$ & $z=3$}
\startdata
$70\,\mu m$ & 3.8 & 10.8& 30.9 & 47.4 \\[.3ex]
$100\,\mu m$ & 2.9 & 5.8 & 11.9 & 25.0 \\[.3ex]
$160\,\mu m$ & 2.4 & 3.4 & 5.2 & 8.5 \\[.3ex]
$250\,\mu m$ & 2.2 & 2.6 & 3.3 & 4.3 \\[.3ex]
$350\,\mu m$ & 2.3 & 2.3 & 2.7 & 3.2 \\[.3ex]
$500\,\mu m$ & 2.6 & 2.2 & 2.4 & 2.6 \\[.3ex]
\hline
\hline
& Fiducial & AGN-10x & & \\[.3ex]
$\mathrm{L_{IR}(8-1000\mu m)}$ & 2.5 & 14.4 & & \\[.3ex]
$\mathrm{L_{FIR}(40-500\mu m)}$ & 2.0 & 7.2 & & \\[.3ex]
$\mathrm{L(10-30\mu m)}$& 6.7 & 142.7 & &\\[.3ex]
$\mathrm{L(100-1000\mu m)}$ & 1.5 & 3.2 & &
\enddata
\tablecomments{Maximal boosting as a function of $\lambda_{obs}$ is derived from the redshifted AGN-10x SED considering all timestamps and viewing angles. Generally, these maximum values occur around the most extreme snapshot in our simulations, where L$_{\mathrm{AGN}}$/L$_{\mathrm{SF}}$$>10$, and should be interpreted as the maximum possible correction an IR SFR would require to accurately recover the true dust-enshrouded SFR under such admittedly extreme conditions. }
\end{deluxetable}
\section{Discussion\label{sec:discussion}}
\subsection{Implications}\label{sec:implications}
Our simulations suggest that in extreme cases, AGN may dominate the FIR luminosity of a galaxy at all wavelengths.
The central AGN in our simulations heats dust at radii greater than the half-mass radius of the system at times following merger coalescence. Moreover, our simulations span a similar parameter space in FIR colors compared to observations but are redder than all dusty torus models from the CLUMPY \citep{Nenkova2008a,Nenkova2008b} and \cite{Fritz2006} AGN torus SED libraries, instead exhibiting colors that could be interpreted to indicate pure star formation. Consequently, without robust exclusion of a central deeply \cchb{dust-enshrouded} AGN, even cold-dust emission
at $\ga 100$ \micron~may \cchb{may be `contaminated' by AGN-powered dust emission and thus not yield accurate SFRs}.
\jm{Although the current consensus is that star formation dominates cold-dust emission at $\lambda \ga 100 \micron$ (see Section \ref{sec:intro}), some empirical studies have suggested that AGN play an importnat role in powering dust emission even at these wavelengths \citep[e.g.,][]{Sanders1996}. For example,} \cite{Symeonidis2016}, \citet{Symeonidis2017} and \citet{Symeonidis2018} show that optical through sub-mm emission in $z<0.18$ QSOs and $z\sim1-2$ \jm{extremely luminous IR galaxies} can be dominated by AGN-heated dust out to kpc scales in the host galaxy. \jm{To fit a library of empirically-derived IR AGN SEDs, \cite{Kirkpatrick2012} and \cite{Kirkpatrick2015} require an $\sim80-90$ K dust component, which they attribute to AGN heating of the dust.} Given that the amount of obscuration in luminous quasars correlates with cold dust emission (e.g., \citealt{Page2004,Page2011,Chen2015}), far-IR fluxes contaminated by buried AGN may be a systematic source of uncertainty when measuring star formation rates in galaxies lacking an AGN classification. \jm{As we show in Table \ref{tab:tab1}, L$_{\mathrm{\scriptsize IR}}$\ and $\mathrm{L_{FIR}}$ can be boosted by factors of $\sim10$ and $\sim7$ respectively by AGN-powered cold dust emission. This could lead to overestimates of IR SFRs by a factor of $\sim2-3$ on average for the most extreme cases of AGN-heated dust at $z\lesssim3$ in galaxies with high $\lambda_{edd}$\ and L$_{\mathrm{AGN}}$/L$_{\mathrm{SF}}$$>1$. }
\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{Fig6.pdf}
\caption
The total (\emph{dotted black line}) and effective AGN (\emph{solid black line}) SEDs already shown in Figure \ref{fig:sed}, but now compared against the dusty AGN torus models from \cite{Fritz2006}. We fit the entire model library to the AGN 10x spectrum between $5-13$~\micron and show the domain of the top 100 best-fitting models as a shaded grey region.
The torus model can reproduce our effective AGN SED's mid-IR dust emission from $\sim2-10$ \micron~but drastically underestimates the FIR emission powered by the AGN.
Thus, inferring the AGN contribution to the FIR using standard torus models would result in overestimating the SFR.\label{fig:sedFit}
}
\end{figure}
For the reasons mentioned above, indicators of dust-obscured AGN activity are especially important when \cchb{attempting to measure co-eval} stellar and SMBH mass assembly. X-ray observations can \cchb{identify} Compton-thick QSOs in dusty galaxies out to high redshifts (e.g., \citealt{Brandt2015} and references therein). \cchb{For example, using \textit{Chandra} observations, \cite{Vito2020} discovered highly obscured AGN residing within two dusty, star-forming galaxies resident in a $z\sim4$ protocluster, one of which is a Compton-thick QSO that has a luminosity comparable to the most-luminous QSOs known. The results presented in the present work suggest that these
obscured AGN may power an appreciable fraction of the cold-dust emission from these galaxies. This example
demonstrates the need for} X-ray follow-up of systems purportedly harboring high levels of obscured star-formation. Alternatively, spectroscopic methods in the mid-IR ($\sim3-30\,\mu$m) may be used to discriminate between \cchb{dust-enshrouded} AGN and star-formation using high excitation emission lines \citep{Spinoglio1992} or polycyclic aromatic hydrocarbon (PAH) features + mid-IR continuum decomposition (e.g., \citealt{Pope2008,Kirkpatrick2015}). Presently, mid-IR spectroscopic techniques are the most sensitive method for identifying \cchb{dust-enshrouded} AGN at all levels of AGN strength relative to the luminosity of the host galaxy \citep{Hickox2018}. Using, amongst other simulations, the fiducial run discussed in this work, \cite{Roebuck2016} find the fraction of $8-1000\,\mu$m luminosity attributed to AGN (the IR AGN fraction) to be a good predictor of the bolometric AGN fraction at times up to coalescence and at $A_V>1$. For such \cchb{dust-obscured} AGN phases, \cchb{quantifying the AGN fraction using mid-IR data may enable subtracting host-galaxy AGN-heated dust emission to derive accurate SFRs for AGN hosts} (e.g., \citealt{Kirkpatrick2017}).
Spectroscopic methods and spatially resolved data for more galaxies will improve our understanding of obscured mass assembly; however, a revision of SED-fitting techniques is necessary to fully leverage the power of future observatories. In particular, our results suggest that in highly
dust-obscured galaxies, the host-galaxy-scale
dust emission cannot be considered decoupled from
emission from the AGN, as is done
in community-standard SED fitting codes. As we show in Figure \ref{fig:sedFit}, simpler IR SED decomposition methods also suffer from the lack of AGN-heated cold dust, as it is common to subtract an AGN torus model from the FIR SED of a galaxy and assume the residual power arises from star formation alone. If we make this assumption for our simulation and subtract the best-fitting \cite{Fritz2006} AGN torus model from the total SED, then the residual IR emission would overestimate the true SFR by a factor of $\sim7$ in the extreme case where L$_{\mathrm{AGN}}$/L$_{\mathrm{SF}}$\ is maximal. SED fits which include an AGN may also need to consider an associated FIR cold-dust component that does not contribute to the inferred SFR.
\subsection{Limitations of this work}
We have presented a `case study' of a single galaxy
merger simulation. As emphasized above, the
simulation is representative of $z \sim 2-3$
massive, rapidly star-forming, highly
dust-obscured galaxies (SMGs), and we thus
expect our results to only apply to the bright
end of the IR luminosity function. It would be
valuable to repeat our analysis using simulations
that span a much broader range of properties and
redshifts. Moreover, our results may be sensitive
to the structure of the simulated galaxy's
ISM, which can be affected by both the
mass and spatial resolution of the simulation
and the sub-grid models used.
For this reason, it would be worthwhile to repeat
this analysis on simulations with much higher
resolution and more sophisticated treatments
of star formation, BH accretion, and stellar
and AGN feedback \citep[e.g.,][]{Angles-Alcazar2020}.
\section{Summary and Conclusions\label{sec:conclusions}}
We have analyzed synthetic SEDs of a hydrodynamical galaxy merger simulation generated via dust RT calculations to investigate the influence of \cchb{dust-enshrouded} AGN on host-galaxy-scale cold dust emission.
Our main conclusions are as follows:
\begin{enumerate}
\item \cchb{Heavily dust-enshrouded} AGN can power significant cold-dust FIR emission on host-galaxy (kpc) scales; in this particular simulation, the AGN can boost the total IR luminosity by an order of magnitude and the FIR (100-1000 \micron)
luminosity by a factor of 3 (i.e., the AGN can be the primary heating source of cold dust).
\item Our simulations have FIR colors probing the Wien side of the dust emission consistent with those of $z\sim2$ dusty galaxies, including AGN hosts and `purely star-forming' galaxies. The FIR colors of our simulations tracing the dust peak are somewhat warmer than observed galaxies when L$_{\mathrm{AGN}}$/L$_{\mathrm{SF}}$\ is high.
However, in both regimes, the FIR colors of the AGN-powered dust emission are redder than those of any of the widely used AGN torus SED models that we considered.
\item Our results
have important consequences for IR- and (sub)mm-selected galaxies, which are more likely to feature significant host-galaxy obscuration of AGN they host than are less-dust-obscured galaxies. For such systems, if a deeply \cchb{dust-enshrouded} AGN is present, the FIR emission may be powered by both young stars and the AGN. Applying standard torus models, which do not include host-galaxy-scale AGN-powered cold-dust emission, to decompose the SED may thus result in overestimating the SFR and underestimating the AGN fraction.
\end{enumerate}
This work should be considered a `case study' that demonstrates the need for caution when interpreting the FIR SEDs of galaxies that may host \cchb{dust-enshrouded} AGN.
Spatially resolved observations (from, e.g., ALMA and \textit{JWST}) and additional AGN diagnostics should be sought to confirm whether a \cchb{dust-enshrouded} AGN is present when studying such systems. Moreover,
there is a need to develop AGN SED models that account for dust obscuration and emission from both the clumpy torus \emph{and} the host galaxy rather
than treating the AGN and its host as decoupled.
\newline
\footnotesize We thank Myrto Symeonidis and Kedar Phadke for insightful and useful comments on the manuscript that improved this work. The Flatiron Institute is supported by the Simons Foundation. HAS and JRM-G acknowledge partial support from NASA Grants NNX14AJ61G and NNX15AE56G, and from SOFIA grant NNA17BF53C (08$\_$0069). LR acknowledges the SAO REU program, funded by the National Science Foundation REU and Department of Defense ASSURE programs under NSF Grant AST-1659473, and by the Smithsonian Institution. The simulations in this paper were performed on the Odyssey cluster supported by the FAS Research Computing Group at Harvard University. This research has made use of NASA’s Astrophysics Data System.
|
3,212,635,537,445 | arxiv | \section*{Keywords}
Convolution Neural Network; Document Images; Parametric-ReLU; PSNR; Optical Character Recognizer.
\section{Introduction}
Research in AI tackles the problems which humans can solve easily, but are difficult for machines to solve. Rather than hard-coding these tasks as computer instructions, a learning based approach is more sensible where computers train/learn from and adapt to the real-world examples, in a hierarchical manner, from simpler to more complex situations. This approach closely resembles the way a person acquires knowledge from the world to behave in an "expected" or "sensible" manner. An "intelligent" being perceives real world information through its senses; but it is difficult to formally provide such information to the computers in its raw analog form. Hence data is digitized and further processed to more concise and efficient-to-compute numeric-vector format (called as feature-vector or more concisely as features) to be handled by training/learning algorithms.
The choice of representation of features has significant effect on the performance of the machine-learning algorithms. So a lot of effort needs to be put in designing and hand-coding the features. Lately, multi-layer neural-network based learning techniques, also called "deep learning algorithms", are gaining popularity as one need not manipulate raw data to a great extent, and training the network itself takes care of generating features at different layers in a hierarchy of complexity and suitable to the task it is being trained for.
In this paper, we use such techniques to enhance the quality of low-resolution document-images for aiding the subsequent recognition by OCR. We train a Convolution Neural Network (CNN) to learn the mapping between low- and high- resolution example images and generate high-resolution images from the test dataset of low-resolution images. A fully connected neural network architecture does not take into account the spatial structure of images. For instance, it treats input pixels which are far apart and close together in exactly the same way~\cite{nielsen2015neural}. Therefore the usage of convolution neural network is necessary, which learns local patterns and takes into account, the spatial structure of feature maps ~\cite{nielsen2015neural}. It is also much easier to train a CNN than fully-connected dense architectures due to lesser number of model parameters. In a broader sense, convolution neural network allows computational models that are composed of multiple processing layers to learn representations of data with multiple levels of abstraction ~\cite{lecun2015deep}.
Convolution neural networks have layers of "3D filters". Each filter operates on all the outputs obtained from the filters of the previous layer corresponding to a particular neighborhood. The area of the neighborhood involved with reference to the input image increases with every subsequent layer. For example at first layer one filter may detect a single vertical line, and another might detect horizontal lines. The next layer now takes the features from the previous layer and can detect (corner) more complex features and so on. This is a basic view of CNN.
Handcrafting features is difficult in the case of document images and therefore convolution neural network is used, which is a data-driven approach to learn the right filter bank for a particular dataset. Features are learned as the training progreses from simple to more complex ones with increasing depth. Increasing depth doesn't always guarantee improved performance, but in our case the use of filters of size $[1\times 1]$ at three consecutive layers after the first hidden layer, each followed by ReLU or PReLU non-linearity, gives better performance than the same experiment performed over a three layer architecture. Convolution layers' filters of size $[1\times 1]$, described in detail in later sections, aims to further enhance the non-linearity in the model and decreases the dimensionality of the feature space.
There are various factors, which affect the accuracy of an OCR on document images the most important of them being the image resolution or spatial resolution. We often have low-resolution images at our disposal, where the high resolution counterpart is not available, or the document images which were scanned at very low resolution and the original image got destroyed or lost. When images are scanned at a low resolution and the quality is low, it affects the OCR recognition and gives bad results in terms of character or word level accuracy. This motivates us to work on the crucial stage of document image enhancement before passing it on to OCR.
There are several techniques by which high resolution images can be obtained. Performance gain is obtained even in bi-cubic interpolated images over plain low-resolution images. But with this method we can't recover high frequency components ~\cite{parker1983comparison}. Classical reconstruction based techniques require multiple low resolution images to reconstruct a higher-resolution image. Sparse representation dictionary learning based method, used for image super-resolution is very slow. This technique involves the use of an over-complete dictionary i.e. $ D^{{d*K}} $ where $d$ is the dimension of the signal and $ K \ge d $ ~\cite{yang2010image} ~\cite{yang2012coupled}.
Thus, there is a need for an algorithm or architecture, which is useful in such scenarios. We have designed a five layer convolution neural network, which learns mapping from low- to high- resolution image patches. We have created a large dataset contaning 5.1 million LR and HR patch pairs of sizes $ 16\times 16 $ and $ 10 \times 10 $ respectively, for training the model.
The rest of the paper is organized as follows. Related work is discussed in Section 2. Section 3 describes the problem formulation. Training model and dataset creation are explained in Sections 4 and 5, respectively. Section 6 summarizes the experiments performed and results obtained. Finally, conclusions are drawn and concluding remarks are drawn in Section 7.
\section{Related work}
The technique for image super-resolution (SR) in the literature can be classified broadly into two categories: a classical multiple image SR and single image SR. The former approach requires multiple images with sub-pixel alignment to obtain a high resolution image~\cite{yang2010_1_image}. The latter approach is discussed in detail in ~\cite{nasrollahi2014super} and ~\cite{yang2014single}. The sparse-coding based method and its several improvements ~\cite{yang2010image}, ~\cite{yang2012coupled} are among the state-of-the-art SR methods. Chao Dong et. al.~\cite{dong2014image} proposed deep learning based natural image super-resolution, which showed that traditional SR method can be viewed as a deep convolution neural network which is light weight and achieves state-of-the-art restoration quality. It is also fast enough for practical applications.
\section{Problem formulation}
Given a low resolution image, the objective is to increase the resolution so that the OCR recogniton accuracy improves. The problem can be mathematically formulated as follows.\\ Assume that the given low resolution images is $ y $, and its bicubic interpolated image is $ Y $. The task is to learn mapping function $J$ to recover a high resolution image from $ Y $ i.e. $J(Y)$ which gives better OCR recognition accuracy than $Y$ and also improved PSNR.
The training set is defined as $I = \{(Y_{i},X_{i}) : 1 \leq i \leq N\}$, where $Y_{i}$ is the LR image and $X_{i} \approx J_{\lambda}(Y_{i}) $ is the corresponding HR image patch of the training set.
We use a parametric model to learn a non-linear mapping function $J_{\lambda}(Y)$, $\lambda$ being the parameters of the model, which minimize the reconstruction error between $J_{\lambda}(Y)$ and the corresponding ground truth high resolution image $X$. The learned model/function should best fit training data and generalize well to the test data.
The model is represented as,
$\lambda = \{ W_{i}, b_{i} \} $ where
$ W_{i} $ = $ \big\{ W_{i}^{j} : \hspace{.1cm} 1\leq j \leq n_{i}\big\} $, and $W_{i}^{j}$ is the $j^{th}$ filter at the $i^{th}$ layer as mentioned in Table~\ref{CNNarch}
and we learn these model parameters with a training algorithm in a supervised-manner as follows. \\
Let $ * $ denotes the convolution operation and $ 0 < \alpha < 1 $ is the data dependent parameter.\\
Initialize $ J_{0} = Y $ \\
$ \boldsymbol{for} \hspace{.2cm} i=1:4 \hspace{.2cm} \boldsymbol{do} $ \\
$ \hspace{1cm} Z^{j}_{i} =W^{j}_{i}*J_{i-1} + b_{i}$\\
$ \hspace{1cm}J^{j}_{i}= \max(Z_{i}^{j} , 0)\hspace{.5cm} \hspace{1cm} \bf (for\hspace{.1cm}ReLU) $
$$\boldsymbol{or}$$
$\hspace{1cm} J_{i}^{j} = \left \{
\begin{tabular}{c}
$\hspace{.0000001cm}\alpha Z_{i}^{j} \hspace{.3cm} if \hspace{.3cm} Z_{i}^{j} \le 0 $ \\
$\hspace{.1cm} Z_{i}^{j} \hspace*{1.1cm} else $
\end{tabular}
\hspace{.5cm} \bf ( for\hspace{.1cm} PReLU)
\right\}$
$\boldsymbol{end} \hspace{.1cm} \boldsymbol{for} $
\vspace{.1cm}
$J_\lambda(Y) = W_{5} $ * $ J_{4}(Y) + b_5 $\\
The loss function is given by,
\[L(\lambda)=\frac{1}{N}\sum\limits_{i=1}^{N}\parallel J_{\lambda}(Y_i)-X_i\parallel^2_{F}\]
To minimize this loss function, back-propagation with stochastic gradient descent algorithm (SGD) with momentum is used. The momentum provides faster convergence and reduced oscillation about the minima. Stochastic gradient decent algorithm performs parameter update at each iteration and is faster than batch gradient descent(BGD). BGD performs redundant computations for large datasets, since it recomputes gradients for similar examples before each parameter update ~\cite{ruder2016overview}. SGD removes this redundancy by performing one update over each mini-batch.
\section{CNN model}
\subsection{Initializing the five layer network}
Deep neural networks have a highly non-linear architecture and the solution for the model parameters is non-convex. Initialization is one of the most important steps in designing deep neural models. If not performed with proper care, network may not perform well, and can even become dead or unresponsive in the middle of training, resulting in poorly trained model. Initialization means setting the model weights with initial values. Several initialization strategies have been proposed in the literature, some of them are : uniform initialization scaled by square root of the number of inputs i.e. W $ \sim U [\frac{-1}{\sqrt{n_{i}}} , \frac{1}{\sqrt{n_{i}}} ] $ ~\cite{backproplecun} ; Glorot uniform initialization i.e. W $ \sim U [\frac{-1}{n_{i} + n_{o}} , \frac{1}{ n_{i} + n_{o} } ] $, Glorot normal initialization i.e. W $ \sim \mathcal{N} [ 0 , \frac{1}{ n_{i} + n_{o} } ] $ ~\cite{glorot2010understanding} and He-uniform initialization i.e, W $ \sim U [\frac{-1}{ n_{i} } , \frac{1}{ n_{o} } ] $ ~\cite{he2015delving}, where $n_{i}$ and $n_{o}$ are the fan-in and fan-out of the layer W respectively.
The paper ~\cite{glorot2010understanding} gives insights on the initialization of deep neural networks. In their work, they have explained that proper initialization helps signals to reach deep into the network. If the starting weights in a network are very small, then the signal shrinks as it passes through each layer until it is too small to be useful. However, if the starting weights in a network are large, then the signal grows as it passes through each layer until it is too large to be useful. All the initialization techniques mentioned above aims at keeping the weights in a controlled range during training.
For implementation reasons, it is difficult to find out how many neurons in the next layer consume the output of the current ones. In our work, we have used initialization presented in ~\cite{he2015delving}. It started from Glorot and Bengio~\cite{glorot2010understanding} and suggest using var(W)=${2}/{n_i}$ instead of ${2}/{(n_o+n_i)}$ ~\cite{glorot2010understanding}. This makes sense because a rectifying linear unit is zero for half of its input and hence we need to double the weight variance to keep the signal's variance constant.
\subsection{Non-linearity layers' activation function}
We use non-linear layers at several places in the CNN model. Among the non-linearities proposed, the most popular ones are sigmoid and rectified linear unit (ReLU) ~\cite{glorot2011deep}. Recently, extensions of ReLU non-linearities were proposed such as Leaky ReLU~\cite{leaky}, parametric ReLU (PRELU)~\cite{prelu}, SReLU~\cite{srelu} and randomized ReLU~\cite{rrelu}. In our experiments, we have used ReLU and PReLU units as activation functions. The advantage of using ReLU instead of sigmoid is because we avoid the vanishing gradient problem in the former case. ReLU is represented mathematically as $f(x)=max(0,x)$, i.e., it clips the -ve input to zero. ReLU can be implemented simply by thresholding a matrix of activations at zero and thus it is computationally much more efficient than sigmoid or tanh. Convergence of stochastic gradient descent with ReLU is faster than for the sigmoid/tanh functions ~\cite{ruder2016overview}, due to its linear, non-saturating form. The disadvantage with ReLU is that it can be fragile during training and can "die" in the middle of training. For example, a large gradient flowing through a ReLU neuron may result in update of weights such that the neuron will never activate on any datapoint again. If this happens, then the gradient flowing through the unit will be zero forever from that point on ~\cite{li2015cs231n}. In other words, the ReLU units can irreversibly die during training since they can get knocked off the data manifold ~\cite{li2015cs231n}. This problem can be solved if the learning rate is set properly.
Other activation functions that can be used are Leaky ReLU, which reported better performance; but the results are however, not always consistent. It is the same as ReLU but the slope of the negative part is $0.01$. Parametric ReLu (PReLU) is also the same as ReLU, but the slopes of negative input are learned from the data rather than being pre-defined. In randomized ReLU (RReLU), the slopes for negative input are randomized in a given range while training, and then fixed during testing. It has been reported that RReLU can reduce over fitting due to its random nature ~\cite{xu2015empirical}.
\begin{table}[!h]
\centering
\caption{5-Layer CNN architecture}
\label{table:3}
\resizebox{0.47\textwidth}{!}
{
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\textbf{Layer} & \textbf{Feature map} & \textbf{Filter} & \textbf{stride} & \textbf{pad} & \textbf{output} \\ [0.5ex]
\hline
conv1 & $ 16\times16\times1 $ & $ 5\times5\times1\times64 $ & 1 & 0 & $ 12\times12\times64 $ \\
\hline
Act.fn. & $ 12\times12\times64 $ & $ ReLU/PReLU $ & 1 & 0 & $ 12\times12\times64 $ \\
\hline
conv2 & $ 12\times12\times64 $ & $ 1\times1\times64\times44 $ & 1 & 0 & $ 12\times12\times44 $ \\
\hline
Act. fn. & $12\times12\times44$ & $ ReLU/PReLU $ & 1 & 0 & $12\times12\times44$ \\
\hline
conv3 & $12\times12\times44 $ & $ 1\times1\times44\times24 $ & 1 & 0 & $ 12\times12\times24 $ \\
\hline
Act.fn. & $12\times12\times44 $ & $ ReLU/PReLU $ & 1 & 0 & $ 12\times12\times24 $\\
\hline
conv4 & $ 12\times12\times24 $ & $ 1\times1\times24\times14 $ & 1 & 0 & $ 12\times12\times14 $ \\
\hline
Act.fn. & $ 12\times12\times14 $ & $ ReLU/PReLU $ & 1 & 0 & $ 12\times12\times14 $\\
\hline
conv5 & $ 12\times12\times14 $ & $ 3\times3\times14\times1 $ & 1 & 0 & $ 10\times10\times1 $ \\ [.1ex]
\hline
\end{tabular}}
\label{CNNarch}
\end{table}
Figure~\ref{fig-cnn-model} gives the architecture of the 5-layer CNN used for obtaining the HR patches from the LR input patches. The first layer has 64 filters each of size $[5\times 5]$ producing 64 feature maps of size $[12\times 12]$, for a $16\times 16$ image input patch. To further embed non-linearity into the CNN model and increase depth of the network to reduce the dimensionality of feature space, we use layers with filter size $[1\times 1]$. The use of filters of size $[1\times 1]$ in 3 consecutive layer after the first hidden layers followed by ReLU/PReLU activation to provide more non-linearity, helps to obtain good features. This model gives better performance than a plain three layer architecture.
\begin{figure}[!hr]
\includegraphics[width=0.47\textwidth,height=0.32\textwidth]{arch5.png}
\caption{The 5-layer convolution neural network model with activation in between the layers.}
\label{fig-cnn-model}
\end{figure}
From experiments, we found that using 2/3 times the number of filters as that of the previous layer gives better results.
\begin{figure}[!h]
\includegraphics[width=0.49\textwidth,height=0.32\textwidth]{allfiltersoutputupto4layer1.png}
\caption{Output at each convolution layer showing the feature maps obtained at each layer for a sample input.}
\label{fig-feat-maps}
\end{figure}
Figure~\ref{fig-feat-maps} shows that the learned filters~\ref{fig-filters} are performing operations on the input image to obtain simple to more complex features at progressive layers. Since document images contain less number of features than natural images, we are reducing the number of features by using filters of size $[1\times 1]$ at three consecutive layers.
\begin{figure}[!h]
\includegraphics[width=0.47\textwidth]{filtersvisualizationateachconvlayerenglish.png}
\caption{Learned filters at each layer which are used to obtain the feature maps~\ref{fig-feat-maps}}
\label{fig-filters}
\end{figure}
The input to CNN is a low resolution image of size $ [16\times16\times1]$. The first convolution layer computes the outputs of neurons that are connected to the local regions in the input, each computing a dot product between their weights and a small region they are connected to. In our case, we have used 64 filters of size $ [5\times5] $, which result in output volume of size $[12\times12\times64]$. Similarly, the second layer has 44 filters of size $[1\times 1]$ and generates output volume of size $[12\times12\times44]$. The third layer has 24 filters of size $1\times1$ which generates output volume of size $[12\times12\times24]$; and fourth layer has 14 filters of size $1\times1$ which generates output volume of size $[12\times12\times14]$. The last layer has 1 filter of size $3\times3$ to produce the output volume $[10\times10\times1]$. Each convolution layer is followed by a ReLU/PReLU non-linearity layer. ReLU/PReLU layer applies element-wise activation function, $max(0,x)$, and leaves the size of the volume unchanged.
If the input to a convolution layer is of size $width1\times height1 \times depth1$ and at each layer we have four hyper-parameter namely number of filters (nf), filter spatial extent (se), stride (s) and amount of zero padding (zp); the output size $width2 \times height2 \times depth2$ is calculated according to the below mentioned formula~\cite{li2015cs231n}.
$$width2=(width1-se+2\times zp)/s+1$$
$$height2=(height1-se+2\times zp)/s+1$$
$$depth2=nf$$
Figure ~\ref{fig-feat-maps} shows the obtained feature maps at each convolution layer using the learned filters in Figure ~\ref{fig-filters}
\section{The training and test datasets}
\begin{figure}[!h]
\centering
\includegraphics[width=0.25\textwidth]{LR_HR1.png}
\caption{Some LR and HR pairs from the Tamil training image}
\label{fig-dataset}
\end{figure}
We create a training dataset of 5.1 million HR-LR image patch-pairs, by randomly cropping from 135 document images. We use document images having text in three different languages, viz. Tamil, Kannada and English, scanned at different resolutions of 100, 200 and 300 DPI. This enhances robustness of the trained model towards input of different languages and resolutions. The Figure ~\ref{fig-dataset} shows some training samples of Tamil images.\\ The HR image patch is of size $10 \times 10$, and cropped directly from the document images. The corresponding LR patch is of size $16 \times 16$ and cropped from the image obtained by down-sampling and up-sampling the original document image with a factor of 2. The image patch-pair sizes smaller than the selected ones do not result in any significant performance improvement.
For the testing dataset, we use different set of 18 document images and generate HR-LR image patch-pairs in the same manner.
\section{Results of super-resolution experiments}
We implement the model using theano~\cite{theano} and Keras~\cite{keras} libraries in python. We used NVIDIA TITAN GTX GeForce (12GB memory) GPU for training.
The training image patches are of size $16\times16$, which are normalized by subtracting the mean over the entire training data and range-normalized to $[0,1]$, as pre-processing step. The training dataset is also randomly shuffled to prevent the model from being trained to some arbitrary data pattern. We have used MSE loss function and SGD solver with standard back-propagation. We trained for 50 epochs with the learning rate of 0.0001 and batch-size of 32. Figure ~\ref{tc} shows the training curve obtained using input images from three languages namely English, Tamil and Kannada at 150 dpi using ReLU and PReLU activations.
\begin{figure}[!h]
\includegraphics[width=0.49\textwidth]{lcR.png}
\caption{Training curve obtained using input images from three languages namely English, Tamil and Kannada at 150 dpi using ReLU or PReLU activations.}
\label{tc}
\end{figure}
Suppose R is the reconstructed image and $ X_{h} $ is the ground truth image. We use PSNR (peak-signal-to-noise ratio) as the metric for image quality listed in Table ~\ref{PSNR}, which is calcutaed as follows:
$$E= X_{h} - R$$
$$RMSE = \sqrt{\frac{1}{N^2}\parallel E \parallel_{F}^2}$$
$$PSNR = 20 log_{10}\frac{255}{RMSE}$$
\begin{figure}[!h]
\includegraphics[width=0.49\textwidth,height=0.49\textwidth]{english.png}
\caption{A sample English image with its improvement in PSNR over bicubic interpolation}
\label{English}
\end{figure}
\begin{figure}[!h]
\includegraphics[width=0.49\textwidth,height=0.49\textwidth]{tamil.png}
\caption{A sample Tamil image with its improvement in PSNR over bicubic interpolation}
\label{Tamil_image}
\end{figure}
\begin{figure}[!h]
\includegraphics[width=0.49\textwidth,height=0.49\textwidth]{kannada.png}
\caption{A sample Kannada image with its improvement in PSNR over bicubic interpolation}
\label{Kannada_image}
\end{figure}
\begin{table}[!h]
\begin{center}
\caption{Comparison of average PSNR SR images with that of the bicubic interpolated images for each language for different input resolutions.}
\resizebox{0.47\textwidth}{!}
{
\begin{tabular}{|l|c|c|c|c|}
\hline
Language & Resolution & Bicubic PSNR(dB)& \bf CNN\_ReLU PSNR(dB) & \bf CNN\_PReLU (dB) \\
\hline\hline
English & 150 & 18.62 & \bf 26.02 & \bf 26.3 \\
\hline
Tamil & 150 & 20.34 & \bf 28.29 & \bf 28.48 \\
\hline
Kannada & 150 & 18.41 & \bf 24.92 & \bf 26.12\\
\hline
English & 100 & 16.44 & \bf 21.56 & \bf 21.64 \\
\hline
Tamil & 100 & 16.94 & \bf 22.42 & \bf 22.72 \\
\hline
Kannada & 100 & 16.24 & \bf 20.55 & \bf 21.00\\
\hline
\bf Tamil & \bf 75 & \bf 15.40 & \bf 19.20 & \bf 19.44 \\
\hline
English & 50 & 13.37 & \bf 14.81 & \bf 15.08 \\
\hline
Tamil & 50 & 13.30 & \bf 14.56 & \bf 14.97 \\
\hline
Kannada & 50 & 13.06 & \bf 14.46 & \bf 14.71\\
\hline
\end{tabular}
\label{PSNR}
}
\end{center}
\end{table}
Table~\ref{OCR_accuracy} lists the mean character level accuracy (CLA) and word level accuracy (WLA) obtained from the OCR performance as performance metrics for the quality of the HR image.
Figures [ ~\ref{English} ~\ref{Tamil_image} ~\ref{Kannada_image}] show comparison of results obtained on test images using CNN\_PReLU and ReLU over bicubic interpolated image.
\begin{table}
\begin{center}
\caption{Mean character and word level accuracies of CNN derived SR images (in \%) for 75 dpi Tamil input images.}
\resizebox{0.40\textwidth}{!}
{
\begin{tabular}{|l|c|c|c|c|}
\hline
Metric & LR\_Input & Bicubic & \bf CNN\_PReLU \\
\hline\hline
CLA & 56.3 & 91.1 & \bf 94.0 \\
\hline
WLA & 12.5 & 59.6 & \bf 62.9 \\
\hline
\end{tabular}
\label{OCR_accuracy}
}
\end{center}
\end{table}
\section{Conclusion}
A five-layer CNN has been designed that can obtain high resolution document images from single low resolution images. The network's performance scales across the three languages that it has been tested for. Further, the same trained network works on different input resolutions of 50, 100 and 150 dots per inch. The increase in the PSNR value of the output image over a bicubic interpolated image varies from 1.5 dB to 7.8 dB for input images of 50 and 150 dpi, respectively. The OCR accuracy has improved by 3\% at the word level for Tamil images with 75 dpi input resolution. Thus, within the tested input resolutions and the languages, the proposed technique is language and resolution independent.\\
As future work, the performance of dedicated networks trained on a particular language and input resolution will be tested and compared with the results reported here.
{\small
\bibliographystyle{ieee}
|
3,212,635,537,446 | arxiv | \section{Introduction}
In this paper we improve the main statements in \cite{Can-R-S} concerning a local version of Brunella's alternative for germs of codimension one holomorphic foliations.
We know that any {\em non dicritical} germ of codimension one foliation $\mathcal F$ in $({\mathbb C}^3,0)$ always has an invariant germ of analytic surface, as proved in \cite{Can-C} (the result is also true in higher ambient dimension \cite{Can-M}). Following a local version of Brunella's alternative \cite{Cer-L-R} and a conjecture of D. Cerveau \cite{Cer3} we ask whether any germ of codimension one foliation ${\mathcal F}$ over $({\mathbb C}^3,0)$ without invariant germ of surface satisfies the following property:
\begin{quote}
\em $(\star)$
There is an open neighborhood $U$ of \/ $0\in {\mathbb C}^3$ such that any leaf of ${\mathcal F}\vert_ U$ contains a germ of analytic curve at the origin.
\end{quote}
In view of the main result in \cite{Can}, any germ of codimension one foliation $\mathcal F$ in $({\mathbb C}^3,0)$ admits a reduction of singularities
\begin{equation*}
\label{eq:pi}
\pi:(M,\pi^{-1}(0))\rightarrow ({\mathbb C}^3,0).
\end{equation*}
Using the arguments in \cite{Can-C}, we see that if $\mathcal F$ is without germ of invariant surface then
there is a {\em compact dicritical component} $D$ in the exceptional divisor $E$ of $\pi$ (this means that $D$ is an irreducible surface contained in $\pi^{-1}(0)$ and transversal to the transformed foliation $\pi^*{\mathcal F}$).
We develop our study inside the class of germs of codimension one foliations of {\em Complex Hyperbolic} type, for short CH-foliations. We recall \cite{Can-R-S} that a germ of codimension one foliation $\mathcal F$ in $({\mathbb C}^n,0)$ is a CH-foliation if for any generically transversal map
$$
\phi: ({\mathbb C}^2,0)\rightarrow ({\mathbb C}^n,0)
$$
the transformed foliation $\phi^*{\mathcal F}$ has no saddle-nodes in its reduction of singularities, that is $\phi^*{\mathcal F}$ is a {\em generalized curve} in the sense of \cite{Cam-N-S}. If $n=3$, given a reduction of singularities $\pi$ of $\mathcal F$, we have a CH-foliation if and only if there are no saddle-nodes among the singularities of $\pi^*{\mathcal F}$ of dimensional type two. We borrow the terminology of D. Cerveau in \cite{Cer2}, where ``complex hyperbolic'' stands for simple singularities in dimension two that are not saddle-nodes.
Although in this paper we only consider a particular class of codimension one foliations, we believe that there are enough reasons to state the following conjecture:
\begin{quote}
\em ``Any germ $\mathcal F$ of CH-foliation on $({\mathbb C}^3,0)$ without germ of invariant analytic surface satisfies $(\star)$''.
\end{quote}
Our general strategy to prove the conjecture is to show that all the leaves ``go'' to a compact dicritical component after reduction of singularities. In fact, if $L$ is a leaf of $\pi^*{\mathcal F}$ intersecting a compact dicritical component $D$ at a point $p$, we can find a germ of analytic curve $(\tilde \gamma, p)\subset L$ and the image $(\pi(\tilde\gamma),0)$ is the desired germ of analytic curve. As we have shown in \cite{Can-R-S}, the main obstruction to following this strategy is the existence of a certain type of {\em uninterrupted nodal components}. They are a three-dimensional version of the ``nodal separators'' introduced by Mattei and Mar\'{\i}n in \cite{Mat-M}; they have also been recently considered by Camacho and Rosas \cite{Cam-R} in the study of local minimal invariant sets in dimension two. Now, the natural procedure is to prove that any {uninterrupted nodal component} goes to a compact dicritical component, carrying the leaves with it, and thus it does not produce an obstruction to property $(\star)$. Indeed, it is necessary to assume that the foliation has no invariant germ of surface. We interpret this fact after reduction of singularities by observing that all the {\em partial separatrices} also go to a compact dicritical component.
The relationship between uninterrupted nodal components and partial separatrices is the main argument we use in this paper to obtain a proof of the conjecture for a particular class of CH-foliations on $({\mathbb C}^3,0)$.
Let us explain what are the {\em uninterrupted nodal components} and the {\em partial separatrices} for a
given reduction of singularities $\pi$ of a CH-foliation
$\mathcal F$ of $({\mathbb C}^3,0)$. First of all,
we quickly recall the final situation after reduction of singularities \cite{Can,Can-C}.
The exceptional divisor $E$ of $\pi$ is a normal crossings divisor and
the singular locus $\mbox{\rm Sing}\pi^*{\mathcal F}$ is a finite union of irreducible nonsingular curves having normal crossings with $E$. Any point $p\in \mbox{\rm Sing}\pi^*{\mathcal F}$ has {\em dimensional type} $\tau_p\in \{2,3\}$, which corresponds to the number of variables needed to locally describe the foliation.
If $\tau_p=2$, there are local coordinates $(x,y,z)$ at $p$ such that $\pi^*{\mathcal F}$ is given by
\begin{equation}
\label{eq:ttype}
\frac{dy}{y}-(\lambda +\phi(x,y))\frac{dx}{x}=0,\; \phi(0,0)=0, \lambda\in {\mathbb C}\setminus {\mathbb Q}_{\geq 0}
\end{equation}
and moreover $(x=0)\subset E_{\mbox{\rm\small inv}}\subset (xy=0)$, where $E_{\mbox{\rm\small inv}}$ is the union of the invariant irreducible components of $E$. Note that $xy=0$ are invariant surfaces for $\mathcal F$ and that the singular locus
$
\mbox{\rm Sing}\pi^*{\mathcal F}
$
is
$
(x=y=0)
$
locally at $p$. The {\em transversal type} of $\pi^*{\mathcal F}$ at $p$ is the germ of foliation ${\mathcal T}_p$ in $({\mathbb C}^2,0)$ given by Equation (\ref{eq:ttype}).
Let $\Gamma$ be the only irreducible curve of
$
\mbox{\rm Sing}\pi^*{\mathcal F}
$ passing through $p$. We know that ${\mathcal T}_p={\mathcal T}_q$ for any $q\in \Gamma$ with $\tau_q=2$. Thus ${\mathcal T}_p={\mathcal T}_\Gamma$ is the {\em transversal type} of $\Gamma$. We say that $\Gamma$ is {\em nodal} if $\lambda\in {\mathbb R}_{>0}$; in this case the transversal type is linearizable of the form $d(y/x^\lambda)=0$. If $\lambda\in {\mathbb R}_{<0}$, we say that $\Gamma$ is {\em a real saddle} and if $\lambda\in {\mathbb C}\setminus{\mathbb R}$ we say that $\Gamma$ is {\em a complex saddle}.
At a point $q$ of dimensional type three, the foliation $\pi^*{\mathcal F}$ is locally given by
$$
\frac{dx}{x}+(\lambda+\phi(x,y,z))\frac{dy}{y}+(\mu+\psi(x,y,z))\frac{dz}{z}=0
$$
where $\phi(0,0,0)=\psi(0,0,0)=0$ and
$
\lambda,\mu\in {\mathbb C}\setminus {\mathbb Q}_{\leq 0},\; \mu/\lambda \in {\mathbb C}\setminus {\mathbb Q}_{\leq 0}
$.
Moreover
$$
(xy=0)\subset E_{\mbox{\small\rm inv}}\subset (xyz=0).
$$
Note that the coordinate planes $xyz=0$ are invariant surfaces and
$$
\mbox{\rm Sing}\pi^*{\mathcal F}= (x=y=0)\cup (x=z=0)\cup (y=z=0).
$$
Thus there are exactly three curves $\Gamma_1,\Gamma_2,\Gamma_3$ of $\mbox{\rm Sing}\pi^*{\mathcal F}$ arriving at $q$. Up to reordering, we have the following five possibilities:
\begin{enumerate}
\item $\Gamma_1,\Gamma_2$ are nodal curves and $\Gamma_3$ is a real saddle.
\item $\Gamma_1$ is a nodal curve and $\Gamma_2,\Gamma_3$ are complex saddles.
\item $\Gamma_1,\Gamma_2$ and $\Gamma_3$ are real saddles.
\item $\Gamma_1$ is a real saddle and $\Gamma_2,\Gamma_3$ are complex saddles.
\item $\Gamma_1,\Gamma_2$ and $\Gamma_3$ are complex saddles.
\end{enumerate}
We define an {\em uninterrupted nodal component} ${\mathcal N}\subset \mbox{\rm Sing}\pi^*{\mathcal F}$ as any connected union of nodal curves such that at each point $q$ of dimensional type three there are exactly two curves $\Gamma_1,\Gamma_2\subset {\mathcal N}$ through $q$ (we have the first case in the list above). We say that $\mathcal N$ is {\em incomplete} if it intersects the compact dicritical part of the exceptional divisor. As we have seen in \cite{Can-R-S}, if $\mathcal N$ is incomplete the leaves ``supported'' by $\mathcal N$ contain a germ of analytic curve. We have also obtained the following result:
\begin{proposition}[\cite{Can-R-S}]
\label{pro:starforincompletenodal}
Consider a CH-foliation $\mathcal F$ on $({\mathbb C}^3,0)$ without germ of ana\-lytic surface and let $\pi$ be a reduction of singularities of $\mathcal F$. If any uninterrupted nodal component $\mathcal N$ is incomplete, then $\mathcal F$ satisfies $(\star)$.
\end{proposition}
Thus, the conjecture is proved once we assure that there is a reduction of singularities such that any uninterrupted nodal component is incomplete.
Let us now introduce the concept of {\em partial separatrix}. We say that a curve $\Gamma\subset\mbox{\rm Sing}{\pi^*{\mathcal F}}$ is a {\em trace curve} if it is contained in only one invariant irreducible component of the exceptional divisor $E$. Otherwise, the curve is the intersection of two invariant irreducible components of $E$ and it is a {\em corner curve}. By definition, a {\em partial separatrix} $C$ is any connected component of the union of trace curves. We say that $C$ is {\em complete} if it does not intersect the compact dicritical part of $E$, otherwise, we say it is {\em incomplete}.
Following Cano-Cerveau's argumentations as in \cite{Can-C}, given a partial separatrix $C$ we find a germ of invariant surface $$(S,C\cap \pi^{-1}(0))\subset (M,\pi^{-1}(0))$$ supported by $C$. The inclusion above is closed if and only if $C$ is complete. In this case we find by direct image a germ of surface $(\pi(S),0)$ invariant for $\mathcal F$. Hence, we conclude:
\begin{quote}
\em
If $\mathcal F$ has no invariant germ of analytic surface, all the partial se\-paratrices are incomplete.
\end{quote}
The incomplete partial separatrices are the ``guides'' we use to take the uninterrupted nodal components to a compact dicritical component of the exceptional divisor. To do this, we need an accurate control of the transitions of the Camacho-Sad indices along the curves in the singular locus from one component of the exceptional divisor to another. This quantitative analysis focused on the partial separatrices is in contrast with the qualitative and combinatorial arguments we used in \cite{Can-R-S} to obtain the first results concerning the conjecture.
In this paper we prove the conjecture for the case of {\em special relatively isolated complex hyperbolic} germs $\mathcal F$ of codimension one foliations in $({\mathbb C}^3,0)$. We precise the definitions in the next sections, but roughly speaking, this means that we can perform a reduction of singularities by blowing-up points until we reach a situation of equireduction along non compact curves, which we resolve by blowing-up only curves. This class of foliations contains both the cases of equireduction and the foliations associated to absolutely isolated singularities of surfaces. There are previous works on absolutely isolated singularities of vector fields \cite{Cam-C-S} or on foliations desingularized by punctual blow-ups \cite{Can-C-S}; also, the results of Sancho de Salas in \cite{San} concern these conditions very closely.
The main result of this paper is:
\begin{theorem}
\label{teo:mainteo}
Any special relatively isolated CH-foliation $\mathcal F$ in $({\mathbb C}^3,0)$ without germ of invariant ana\-lytic surface satisfies property $(\star)$.
\end{theorem}
Theorem \ref{teo:mainteo} improves the results in \cite{Can-R-S}. What we know from \cite{Can-R-S} is that if we take a complete nodal component $\mathcal N$, then the projection of $\mathcal N$ contains at least one of the germs of curve of $\mbox{\rm Sing}{\mathcal F}$ in $({\mathbb C}^3,0)$. In this way we have a criterion for the non existence of complete uninterrupted nodal components by looking at generic points of the germs of curve in $\mbox{\rm Sing}{\mathcal F}$.
We prove Theorem \ref{teo:mainteo} by showing that all the uninterrupted nodal components are incomplete. The argument is based on a control of the evolution of {\em incomplete points}. They are points such that there is a ``local'' partial separatrix over them which is incomplete. At the final step of the reduction of singularities, all the points are complete. We find a contradiction with the existence of a complete uninterrupted nodal component ${\mathcal N}$ as follows. At the ``birth level'' of ${\mathcal N}$ in the sequence of reduction of singularities, we find an incomplete point in a particular situation concerning the partial separatrices through it. We prove that this situation is part of a class of scenarios which persists along the reduction of singularities. In each scenario, there is at least one incomplete point. Then ``a fortiori'' we find an incomplete point at the last step of the reduction of singularities and obtain the desired contradiction.\\
\noindent{\em Acknowledgements}. The second author is very grateful to the group ECSING and the Department of Mathematics at Universidad de Valladolid for the warm welcome on the numerous visits.\\
\noindent {\em Funding}. This work was partially supported by the Spanish research project MTM2010-15471 and the Brazilian PDE/CsF scholarship 245480/2012-9 from CNPq.
\section{Special Relatively Isolated CH-foliations}
A {\em special relatively isolated sequence ${\mathcal S}=\{\pi_k\}_{k=1}^N$ of blow-ups of $({\mathbb C}^3,0)$} is a sequence
$$
{\mathcal S}: ({\mathbb C}^3,0)=(M_0,F_0)\stackrel{\pi_1}{\leftarrow} (M_1,F_1) \stackrel{\pi_2}{\leftarrow}\cdots \stackrel{\pi_N}{\leftarrow} (M_N,F_N),\quad F_{k+1}=\pi_{k+1}^{-1}(F_k),
$$
given by blow-ups $\pi_{k+1}$ with center at closed germs
$(Y_{k},Y_{k}\cap F_k)\subset (M_{k},F_k)$,
such that for any $0\leq k\leq N-1$ we have
\begin{enumerate}
\item $Y_{k}\cap F_k$ is a single point $Y_{k}\cap F_k=\{p_k\}$.
\item $Y_{k}$ is either $\{p_k\}$ or a germ of nonsingular closed curve $(Y_k,p_k)\subset (M_k,F_k)$ having normal crossings with the exceptional divisor $E^{k}\subset M_{k}$ of
$$
\sigma_k:(M_k,F_k)\rightarrow ({\mathbb C}^3,0),
$$
where
$\sigma_k=\pi_1\circ\pi_2\circ\cdots \circ\pi_k$.
\item If $(Y_k,p_k)$ is a germ of curve, we have an {\em equireduction sequence over $(Y_k,p_k)$} in the following sense. For any $k<\ell\leq N-1$ we have one of the following situations:
\begin{enumerate}
\item $p_k\notin \pi_{k,\ell}(Y_{\ell})$, where $ \pi_{k,\ell}=\pi_{k+1}\circ\pi_{k+2}\circ\cdots\circ \pi_\ell$.
\item The center $(Y_\ell,p_\ell)$ is a germ of curve and $\pi_{k,\ell}$ induces an isomorphism $$\bar\pi_{k,\ell}:(Y_\ell,p_\ell)\rightarrow (Y_k,p_k). $$
\end{enumerate}
\end{enumerate}
Now, we say that a CH-foliation $\mathcal F$ over $({\mathbb C}^{3},0)$ is a {\em special relatively isolated CH-foliation} if there is a sequence $\mathcal S$ of blow-ups as above such that
\begin{enumerate}
\item[i)] For any $0\leq k\leq N-1$ the center $Y_{k}\subset M_{k}$ of the blow-up $\pi_{k+1}$ is contained in the locus $\mbox{\rm Sing}^*({\mathcal F}_{k},E^{k})$ of non simple points for ${\mathcal F}_k, E^k$, where ${\mathcal F}_{k}=\sigma_k^*{\mathcal F}$ is the transform of $\mathcal F$ by $\sigma_k$.
\item[ii)] All the points in $F_N$ are simple points for ${\mathcal F}_N, E^N$. (See \cite{Can})
\end{enumerate}
Since $\mathcal F$ is a CH-foliation, the simple points after reduction of singularities are without saddle-nodes. Conversely, the fact that there is a reduction of singularities without saddle-nodes in the last step is enough to assure that $\mathcal F$ is a CH-foliation.
We refer to \cite{Can-R-S} for more details on the definitions.
Now, let us introduce some useful notations and remarks. The exceptional divisor $E^k\subset M_k$ of $\sigma_k$ is a union of components
$$
E^k=E^k_1\cup E^k_2\cup\cdots\cup E^k_k
$$
where $E^k_\ell$ is the stric transform of $E^{k-1}_\ell$ for $1\leq\ell<k$ and $E^k_k=\pi_{k}^{-1}(Y_k)$ is the exceptional divisor of the last blow-up $\pi_k$. Recall that $M_k$ is a germ over the fiber $F_k=\sigma_k^{-1}(0)$.
For any $k\geq 1$ we have that $F_k\subset E^k$ and the exceptional divisor $E^k$ is a closed germ
$$
(E^k,F_k)\subset (M_k,F_k).
$$
A given irreducible component $E^k_i$ may be an {\em invariant component}, if it is invariant for ${\mathcal F}_k$, or a {\em dicritical component}, when it is generically transversal to the foliation. We denote $E_{\mbox{\sl\small inv}}^k$ the union of the invariant components and $E_{\mbox{\sl\small dic}}^k$ the union of the dicritical ones.
Recall the definition of $\pi_{k,\ell}=\pi_{k+1}\circ\pi_{k+2}\circ\cdots\circ \pi_\ell$. For certain special cases, we adopt the following simplified notations:
\begin{eqnarray*}
\sigma_k=\pi_{0,k}&:&(M_k,F_k)\rightarrow ({\mathbb C}^3,0)\\
\rho_k=\pi_{k,N}&:&(M,F)\rightarrow (M_k,F_k)\\
\pi=\pi_{0,N}&:& (M,F)\rightarrow ({\mathbb C}^3,0).
\end{eqnarray*}
We denote the final step of the reduction of singularities by $$M=M_N,\; E=E^N,\; \pi^*{\mathcal F}={\mathcal F}_N,\; F=F_N=\pi^{-1}(0).$$
From now on, we fix a special relatively isolated CH-foliation $\mathcal F$ of $({\mathbb C}^3,0)$ without germ of invariant analytic surface and a special relatively isolated sequence of blow-ups $\mathcal S$ performing a reduction of singularities of $\mathcal F$.
If the first blow-up $\pi_1$ is centered at the origin $0\in {\mathbb C}^3$, we have that the fiber $F_k$ is the union $E^k_{\mbox{\sl\small c}}$ of the compact components of $E^k$, for any $1\leq k\leq N$. As we shall explain later, the case when the first blow-up is centered in a germ of curve is of no interest to us, since in this situation the foliation $\mathcal F$ has invariant surfaces.
Thus, we also suppose along the paper that $\pi_1$ is a blow-up centered at the origin and hence $F_k=E^k_{\mbox{\sl\small c}}$.
\section{Partial separatrices}
\label{sec:partialseparatrices}
Here we do a revision, adapted to our case, of the partial separatrices introduced in \cite{Can-R-S} and we briefly recall the global picture of a reduction of singularities (see \cite{Can} for more details).
\begin{definition} A {\em partial separatrix $C$ for ${\mathcal F}, {\pi}$} is any connected component of the union $T$ of the trace curves of $\mbox{\rm Sing}(\pi^*{\mathcal F})$. We say that $C$ is {\em complete} if it does not intersect the union $E_{\mbox{\sl\small c,dic}}$ of the compact dicritical components of $E$.
We say that $C$ is {\em incomplete} if it does intersect $E_{\mbox{\sl\small c,dic}}$.
\end{definition}
A partial separatrix $C$ must be considered as a connected component of the germ $(T,T\cap F)$. So it is also a germ $(C,C\cap F)$. For shortness, we write $C$ to denote the partial separatrix if there is no risk of confusion. The {\em compact part} $C\cap F$ of a partial separatrix is also connected and it is the union of the compact curves in $C$ or just a single point.
\begin{example}
Let us recall Darboux-Jouanolou's example \cite{Jou}. It is the conic foliation in $({\mathbb C}^3,0)$ given by the 1-form
$$
\omega= (z^{m+1}-x^my)dx+(x^{m+1}-y^{m}z)dy+(y^{m+1}-z^m x)dz.
$$
The reduction of singularities consists of an initial dicritical blow-up of the origin followed by $m^2+m+1$ blow-ups centered at each of the lines of the singular locus
$$
z^{m+1}-x^my= x^{m+1}-y^{m}z= y^{m+1}-z^m x=0.
$$
We find $2(m^2+m+1)$ partial separatrices, all of them incomplete. Each one is a single non compact curve $(C^{(i)},p_i)$, $i=1,2,\ldots, 2(m^2+m+1)$ and hence the compact part $C^{(i)}\cap F$ is just the point $p_i$.
\end{example}
Let $C$ be a partial separatrix and take a point $p\in C\cap F$. Recalling that the final singularities are complex hyperbolic, depending on the dimensional type $\tau=\tau(\pi^*{\mathcal F};p)$ we find two situations:
\begin{enumerate}
\item If $\tau=2$, there are coordinates $x,y,z$ at $p$ such that $E_{\mbox{\sl\small inv}}=(x=0)$, $E_{\mbox{\sl\small dic}}\subset (z=0)$ and
\begin{eqnarray*}
C=(x=y=0)=
\mbox{\rm Sing}(\pi^{*}{\mathcal F}).
\end{eqnarray*}
Moreover $S=(y=0)$ is the only invariant germ of surface for $\pi^*{\mathcal F}$ at $p$ not contained in $E$.
\item If $\tau=3$, there are coordinates $x,y,z$ at $p$ such that $E=E_{\mbox{\sl\small inv}}=(xy=0)$,
\begin{eqnarray*}
C=(x=z=0)\cup (y=z=0),
\end{eqnarray*}
and $\mbox{\rm Sing}(\pi^{*}{\mathcal F})=C\cup (x=y=0)$.
Moreover $S=(z=0)$ is the only invariant germ of surface for $\pi^*{\mathcal F}$ at $p$ not contained in $E$.
\end{enumerate}
Gluing these situations along $C$ as in \cite{Can-C}, we find a germ of surface $(S_C,C\cap F)$ invariant for $\pi^*{\mathcal F}$ and not contained in $E$. Moreover, the inclusion of germs
$$
(S_C,C\cap F)\subset (M,F)
$$
is a closed immersion if and only if $S_C\cap F=C\cap F$. On the other hand, we have
$$
S_C\cap F=C\cap F \Leftrightarrow C\cap E_{\mbox{\sl\small c,dic}}=\emptyset.
$$
That is, we obtain a closed immersion exactly when $C$ is a complete partial separatrix. In this case, by Grauert's Theorem of the direct image under a proper morphism, we obtain a germ of surface $(\pi(S_C),0)$ invariant for $\mathcal F$. We conclude:
\begin{proposition}
\label{pro:incompletitudpartialsep}
If $\mathcal F$ has no invariant germ of surface, then all the partial separatrices are incomplete.
\end{proposition}
To finish this section, we give a result that justifies our assumption on the first blow-up being centered at the origin.
\begin{proposition} If the first blow-up is centered at a germ of curve $\gamma$, then $\mathcal F$ has a germ of invariant surface.
\end{proposition}
\begin{proof} Note that we have equireduction along $\gamma$ and thus the fiber $F=\pi^{-1}(0)$ is a union of compact curves.
We have the following possible cases:
\begin{enumerate}
\item $F$ contains a non invariant curve.
\item There is a dicritical component in the exceptional divisor, but all the curves in $F$ are invariant.
\item All the components of the exceptional divisor are invariant and there is a curve $\Gamma$ in $F$ contained in the singular locus of $\pi^*{\mathcal F}$.
\item All the components of the exceptional divisor are invariant and the curves in $F$ are not contained in the singular locus of $\pi^*{\mathcal F}$.
\end{enumerate}
If there is a non invariant curve $\Gamma\subset F$, then $\Gamma$ is necessarily contained in a dicritical component of $E$. Taking a generic point $p\in \Gamma$, the foliation $\pi^*{\mathcal F}$ is non singular at $p$ and transversal to $\Gamma$. Thus, we find a germ of invariant surface $(\tilde S,p)$ that gives a closed immersion into $(M,F)$ and hence it projects onto
a germ of surface $(S,0)$ invariant for $\mathcal F$.
If all the curves of $F$ are invariant but there is a dicritical component $E_i$ of $E$, we consider the curve $\Gamma=E_i\cap F$. At a generic point $p$ of $\Gamma$, there is a germ of surface $(\tilde S,p)$ not contained in $E$ such that $(\Gamma,p)=(\tilde S\cap E_i,p)$. By an extension of the argument of Cano-Cerveau \cite{Can-C} also used in \cite{Reb-R} we can prolong $(\tilde S,p)$ over the fiber $F$ to find a closed immersion of a germ of invariant surface $(\tilde S,F)$ in $(M,F)$. Finally, we project it by $\pi$ to obtain a germ of surface $(S,0)$ invariant for $\mathcal F$. This argument is also valid for the case $(3)$.
Suppose now that all the irreducible components of the exceptional divisor $E$ are invariant and the curves in $F$ are not contained in the singular locus of $\pi^*{\mathcal F}$. This gives a non dicritical equireduction along $\gamma$ in the sense of \cite{Can2, Can-M}. In those papers it is proved that the reduction of singularities is given by the one of
${\mathcal F}\vert_{\Delta}$, where $\Delta$ is a plane transverse to $\gamma$. Then, any Camacho-Sad separatrix $\Sigma$ of ${\mathcal F}\vert_{\Delta}$ induces a germ of surface $(S,0)$ invariant for $\mathcal F$.
\end{proof}
\section{Partial separatrices at intermediate steps}
Let us give some remarks and definitions concerning the behavior of partial separatrices at an intermediate step $(M_k,F_k)$ of the sequence $\mathcal S$ of reduction of singularities, with $0\leq k\leq N$.
\begin{notation} If $C$ is a partial separatrix, we denote $C_k=\rho_{k}(C)$. Let us remark that $C_k\cap F_k$ is a connected nonempty compact set and $\rho_k(C\cap F)=C_k\cap F_k$.
\end{notation}
Consider an irreducible compact curve $\Gamma\subset M_k$ in the singular locus $\mbox{\rm Sing}{\mathcal F}_k$. We have that $\Gamma\subset F_k$. By the properties of the sequence $\mathcal S$, only finitely many points of $\Gamma$ will be modified in the further blow-ups $\pi_{k+1},\pi_{k+2},\ldots,\pi_N$. Thus, there is a well defined strict transform $\Gamma'\subset M$ of $\Gamma$ under $\rho_k$ with $\Gamma'\subset \mbox{\rm Sing}\pi^*{\mathcal F}$. Moreover, at a generic point $p\in \Gamma$ we have
a point $p'\in \Gamma'$ where $\rho_k$ induces an isomorphism
$$
(M,p')\rightarrow (M_k,p).
$$
In particular the pair ${\mathcal F}_k, E_k$ has a simple singularity at such points $p$.
We say that $\Gamma$ is a {\em trace curve}, respectively a {\em corner curve}, if and only if $\Gamma'$ is so.
If $\Gamma$ is a trace curve, there is exactly one partial separatrix $C$ such that $\Gamma'\subset C$. We say that $C$ is the {\em partial separatrix asociated to $\Gamma$} and we denote it by $C_\Gamma$.
Let us note that $C=C_\Gamma$ if and only if $\Gamma\subset C_k$.
\begin{definition} Consider a point $p\in F_k$ and a partial separatrix $C$ where $p\in C_k$. We say that $C$ is {\em complete at $p$} if
for any dicritical component $E_i$ of $E$
such that
$E_i\subset \rho_k^{-1}(p)$ we have $E_i\cap C = \emptyset$. Otherwise we say that $C$ is {\em incomplete at $p$}.
\end{definition}
\begin{remark}
\label{rk:sepparcialcompleta}
If $C$ is complete at $p$, we find a closed immersion
$$
(S_{C}, C\cap \rho_k^{-1}(p))\subset (M,\rho_k^{-1}(p) )
$$
where $(S_{C}, C\cap \rho_k^{-1}(p))$ is a finite union of germs of surface invariant for $\pi^*{\mathcal F}$. Taking the image by $\rho_k$, we obtain a finite union
$$
(\rho_k(S_C),p)\subset (M_k,p)
$$
of germs of surface at $p$ invariant for ${\mathcal F}_k$.
\end{remark}
\begin{remark} A partial separatrix $C$ is complete, as stated in the introduction, if and only if it is complete at the origin $0\in {\mathbb C}^3$. On the other hand, any partial separatrix $C$ is complete at the points $p\in C\cap F$ in the final step of the reduction of singularities, even if $p$ belongs to a compact dicritical component $E_i$ of $E$.
\end{remark}
\begin{remark}
\label{rk:completitudequirreduction}
Let $p_k\in F_k$ be a point such that the center $Y_k$ of $\pi_{k+1}$ is a germ of curve with $p_k\in Y_k$. In view of the equireduction properties of the sequence of reduction of singularities, we have that $\rho_k^{-1}(p_k)$ is a union of compact curves an hence it does not contain any component of $E$. Then any partial separatrix is complete at $p_k$.
\end{remark}
\begin{remark}
\label{rk:completoestable}
We have $C_k=\pi_{k+1}(C_{k+1})$. If the partial separatrix $C$ is complete at a point $p\in C_k\cap F_k$ then it is
complete at all the points in $C_{k+1}\cap \pi_{k+1}^{-1}(p)$.
Moreover, assume that $\pi_{k+1}$ satisfies one of the following conditions:
\begin{enumerate}
\item The center $Y_k$ of $\pi_{k+1}$ does not contain $p$.
\item The center $Y_k$ is a germ of curve.
\item The blow-up $\pi_k$ is non dicritical.
\end{enumerate}
Then the partial separatrix $C$ is complete at $p\in C_k\cap F_k$ if and only if it is complete at all the points $p'\in C_{k+1}\cap \pi_{k+1}^{-1}(p)$.
\end{remark}
\begin{proposition}
\label{pro:puntoscompletos}
Let $C$ be a partial separatrix complete at $p\in C_k\cap F_k$. We have:
\begin{enumerate}
\item[a)] If $\pi_{k+1}$ is centered at $p$, then $\pi_{k+1}$ is a non-dicritical blow-up.
\item[b)] If $p\in E^k_i$, where $E^k_i$ is compact invariant, there is a compact trace curve $\Gamma\subset C_k\cap E^k_i$, with $p\in \Gamma$.
\item[c)] If $p\in E^k_j$, where $E^k_j$ is compact dicritical, then $C\cap E_j^N\ne \emptyset$.
\end{enumerate}
\end{proposition}
\begin{proof} We will do induction on $N-k$ to prove statements a), b) and c) in this order. If $k=N$, we are done. Assume that $k<N$. Since it is a local problem at $p$, if $p\notin Y_k$ we conclude by induction. Thus, we assume that $p\in Y_k$.
{\em First case: the center of $\pi_{k+1}$ is a germ of curve $(Y_k,p)$}. We only have to prove b) and c). Note that for any compact component $E^{k}_s$ such that $p\in E^k_s$ we have $$\pi_{k+1}^{-1}(p)\subset E^{k+1}_s$$
and there is a point $p'\in C_{k+1}\cap \pi_{k+1}^{-1}(p)$. Assume b),
we know that $C$ is complete at $p'$ and by induction hypothesis on $p'\in E^{k+1}_i$ we conclude that
there is a trace compact curve $\Gamma'\subset E^{k+1}_i\cap C_{k+1}$ with $p'\in \Gamma'$. Now, it is enough to consider $\Gamma=\pi_{k+1}(\Gamma')$. Assume c),
we know that $C$ is complete at $p'$ and by induction hypothesis on $p'\in E^{k+1}_j$ we conclude that $C\cap E^N_j\ne \emptyset$.
{\em Second case: the center of $\pi_{k+1}$ is the point $p$}. We first prove a). If $\pi_{k+1}$ is a dicritical blow-up, there is a point $p'\in C_{k+1}\cap E^{k+1}_{k+1}$. We know that $C$ is complete at $p'$ and by induction hypotesis we apply c) at $p'$ to obtain that $C\cap E^N_{k+1}\ne\emptyset$. This contradicts the fact that $C$ is complete at $p$.
Now we prove b) and c) already assuming that $\pi_{k+1}$ is non-dicritical. Take a point $p'\in C_{k+1}\cap E^{k+1}_{k+1}$, we know that $C$ is complete at $p'$. Since $E^{k+1}_{k+1}$ is compact and invariant, by induction hypothesis on $p'\in E^{k+1}_{k+1}$, we find a trace compact curve $\Gamma''\subset C_{k+1}\cap E^{k+1}_{k+1}$. Note that for any compact component $E^k_s$ such that $p\in E^k_s$ we have that
$$
E^{k+1}_s\cap E^{k+1}_{k+1}
$$
is a projective line in the projective plane $E^{k+1}_{k+1}$. In particular there is at least one point $p''_s\in \Gamma''\cap E^{k+1}_s$. We know that $C$ is complete at the point $p''_s$. Assume b), we apply induction hypothesis on $p''_i\in E^{k+1}_{i}$ to find a trace compact curve $\Gamma'\subset E^{k+1}_i\cap C_{k+1}$ such that $p''_i\in \Gamma'$. We conclude by taking $\Gamma=\pi_{k+1}(\Gamma')$. Assume c), we apply induction hypothesis on $p''_j\in E^{k+1}_{j}$ to find that $C\cap E^{N}_j\ne \emptyset$.
\end{proof}
\begin{remark} Proposition \ref{pro:puntoscompletos} can also be proved by invoking the germ of surface $(S_{C},p)$ obtained in
Remark \ref{rk:sepparcialcompleta}
and considering the intersections with the corresponding compact component of $E^k$. We have used the inductive arguments because of the general style of the paper.
\end{remark}
\begin{proposition}
\label{pro:incompletodicriticoono}
Let $C$ be an incomplete partial separatrix and consider an index $0\leq k\leq N$. Then there is a point
$p\in C_k$ such that $C$ is not complete at $p$ or there is a compact dicritical component $E^k_j$ such that $C_k\cap E_j^k\not=\emptyset$.
\end{proposition}
\begin{proof} Induction on $N-k$. If $k=N$ we are done, since $C$ intersects at least one compact dicritical component of the exceptional divisor. Take $k<N$. In order to find a contradiction, assume
that $C$ is complete at any $p\in C_k$ and that it does not intersect any compact dicritical component in $E^k$. We already know that $C$ is complete at any point in $C_{k+1}$ by Remark \ref{rk:completoestable}. By Proposition \ref{pro:puntoscompletos}, we have that if $E^{k+1}_{k+1}$ is a compact component with $E^{k+1}_{k+1}\cap C_{k+1}\ne\emptyset$, then $E^{k+1}_{k+1}$ is an invariant component. This gives the desired contradiction by applying induction hypothesis.
\end{proof}
\begin{definition} We say that $p\in F_k$ is an {\em incomplete point} if and only if there is a partial separatrix $C$ such that $p\in C_k$ and $C$ is incomplete at $p$.
\end{definition}
If there are no partial separatrices $C$ such that $p\in C_k$ the point $p$ is considered to be complete.
\section{Transition of Camacho-Sad indices}
Let us consider an irreducible compact curve $\Gamma\subset F_k\cap \mbox{\rm Sing}{\mathcal F}_k$ and an invariant compact component $E^k_i$ of $E^k$ such that $\Gamma\subset E^k_i$. Note that, since $\Gamma$ is compact, there are no dicritical components containing $\Gamma$.
Consider a plane section $\Delta$ transverse to $\Gamma$ at a generic point $p\in \Gamma$. Taking appropriate local coordinates $x,y$ at $p\in \Delta$, the restricted foliation ${\mathcal F}_k\vert_\Delta$ is given by a $1$-form
$$
\omega=x\left\{(\lambda x+\mu y+\phi(x,y))\frac{dx}{x}-dy\right\}
$$
where $E^k_i\cap \Delta=(x=0)$, $\mu\ne 0$ and $\phi(x,y)$ has a zero of order at least two at the origin. The Camacho-Sad index of ${\mathcal F}_k\vert_\Delta$ at $p$ with respect to the invariant curve $x=0$ is by definition the value $1/\mu$, see \cite{Cam-S, Can-C-D}. We denote
$$
\mbox{\rm Ind}({\mathcal F},E_i;\Gamma)= \mbox{\rm Ind}({\mathcal F}_k\vert_\Delta, E^k_i\cap \Delta;p)=1/\mu.
$$
This index may be calculated in any step $k'\geq k$ of the reduction of singularities and at any point of the strict transform of $\Gamma$ of dimensional type two.
\begin{remark}
\label{rk:productoindices}
Assume that $\Gamma$ is contained in two compact invariant components $E^k_i, E^k_j$ of $E^k$. We have that
$\Gamma=E^k_i\cap E^k_j$. By the general properties of Camacho-Sad index \cite{Cam-S}, we have that
$$
\mbox{Ind}({\mathcal F},E_i;\Gamma)\mbox{Ind}({\mathcal F},E_j;\Gamma)=1.
$$
\end{remark}
\begin{definition}
\label{def:nodalsaddle}
Let $\Gamma\subset \mbox{\rm Sing}(\pi^*{\mathcal F})$ be a compact curve contained in a compact invariant component $E_i$.
\begin{enumerate}
\item $\Gamma$ is a {\em nodal curve} if and only if $\mbox{Ind}({\mathcal F},E_i;\Gamma)\in {\mathbb R}_{>0}\setminus {\mathbb Q}$.
\item $\Gamma$ is a {\em real saddle curve} if and only if $\mbox{Ind}({\mathcal F},E_i;\Gamma)\in {\mathbb R}_{<0}$.
\item $\Gamma$ is a {\em complex saddle curve} if and only if $\mbox{Ind}({\mathcal F},E_i;\Gamma)\in {\mathbb C}\setminus {\mathbb R}$.
\end{enumerate}
\end{definition}
Note that by Remark \ref{rk:productoindices} the definition above does not depend on the invariant component $E_i$ of $E$ such that $\Gamma\subset E_i$.
Take a point $p\in E^k_i$ where $E^k_i$ is a compact invariant component of $E^k$. We are interested in considering irreducible germs of curves
$$(\gamma,p)\subset (\mbox{\rm Sing}{\mathcal F}_k\cap E^k_i,p)
$$
Such a germ $(\gamma,p)$ is contained in exactly one compact curve $\Gamma\subset \mbox{\rm Sing}{\mathcal F}_k\cap E^k_i$. This allows us to put
$$
\mbox{Ind}({\mathcal F},E_i;\gamma)=\mbox{Ind}({\mathcal F},E_i;\Gamma).
$$
We denote ${\mathcal B}^k_i(p)$ the set of irreducible germs of curves $(\gamma,p)\subset (\mbox{\rm Sing}{\mathcal F}_k\cap E^k_i,p)$.
In Proposition \ref{pro:indicesymultiplicidad} we precise a relationship between the indices, counted with multiplicity, with respect to two incident compact components.
\begin{proposition}
\label{pro:indicesymultiplicidad}
Consider a point $p\in \Gamma=E^k_i\cap E^k_j$ where $E^k_i$ and $E^k_j$ are compact components of $E^k$. Assume that $E^k_i$ is an invariant component of $E^k$.
\begin{enumerate}
\item[a)] If $E^k_j$ is a dicritical component and ${\mathcal G}={\mathcal F}_k\vert_{E^k_j}$ we have
\begin{equation*}
\mbox{\rm Ind}({\mathcal G},\Gamma;p)=
\sum_{\gamma\in {\mathcal B}^k_i(p)}(\gamma,\Gamma)_p \mbox{\rm Ind}({\mathcal F},E_i;\gamma),
\end{equation*}
where $(\gamma,\Gamma)_p$ is the intersection multiplicity of $\gamma,\Gamma$ at $p$.
\item[b)] If $E^k_j$ is an invariant component and $\alpha=\mbox{\rm Ind}({\mathcal F},E_i;\Gamma)$, we have
\begin{equation*}
\sum_{\gamma\in {\mathcal B}^k_i(p)\setminus \{\Gamma\}}(\gamma,\Gamma)_p \mbox{\rm Ind}({\mathcal F},E_i;\gamma)=-\alpha
\sum_{\delta\in {\mathcal B}^k_j(p)\setminus\{\Gamma\}}(\delta,\Gamma)_p \mbox{\rm Ind}({\mathcal F},E_j;\delta).
\end{equation*}
\end{enumerate}
\end{proposition}
\begin{proof} We do induction on $N-k$. Let us consider first the case $k=N$:
\begin{enumerate}
\item If $p$ is non singular, it belongs to at most one invariant component of the divisor. We have a) with ${\mathcal B}^k_i(p)=\emptyset$. Thus we are done.
\item Assume that $p$ is of dimensional type two and $E_j$ is a dicritical component. Then the singular locus is non singular at $p$ and $E_j$ gives a section transversal to it. We are done by the definition of the generic index.
\item Assume $p$ is of dimensional type two and $E_j$ is invariant. The singular locus is $\Gamma$. In this case ${\mathcal B}^k_i(p)={\mathcal B}^k_j(p)=\{\Gamma\}$ and there is nothing to prove.
\item
\label{enumerate4}
Assume that $p$ is of dimensional type three. Then $E_j$ is necessarily invariant and there are local coordinates $x,y,z$ at $p$ such that
$$
E_i=(x=0),\; E_j=(y=0),\quad \Gamma=(x=y=0),
$$
the plane $z=0$ is invariant and the singular locus is given by $\Gamma\cup \gamma\cup \delta$, where
$$ \gamma=(x=z=0);\quad \delta= (y=z=0).$$
Moreover, the foliation ${\mathcal F}$ is given locally at $p$ by an integrable 1-form of the type
$$
\omega=\frac{dx}{x}+ (-\alpha+b(x,y,z))\frac{dy}{y}+(-\beta+c(x,y,z))\frac{dz}{z},\quad \alpha\ne 0\ne\beta.
$$
By the integrability condition $\omega\wedge dw=0$, we have
\begin{eqnarray*}
b(x,y,z)&=&xb'(x,y,z)+y b''(x,y,z) \\ c(x,y,z)&=& xc'(x,y,z)+yc''(x,y,z)
\end{eqnarray*}
and thus
$$
\frac{-\beta+c(x,y,z)}{-\alpha+b(x,y,z)}=\frac{\beta}{\alpha}+yf'(x,y,z)+zf''(x,y,z).
$$
Then, we have
\begin{eqnarray*}
\mbox{Ind}({\mathcal F},E_i;\Gamma)=\alpha,\;
\mbox{Ind}({\mathcal F},E_i;\gamma)=\beta,\;
\mbox{Ind}({\mathcal F},E_j;\delta)=-\beta/\alpha.
\end{eqnarray*}
The desired relation is
$
\beta= -\alpha (-\beta/\alpha)
$,
that is obviously satisfied.
\end{enumerate}
Now, suppose that $k<N$. If $p\notin Y_k$ we are done by induction; hence we assume $p\in Y_k$. Moreover, the center $Y_k$ of the blow-up $\pi_{k+1}$ cannot be a germ of curve, since there are two compact components of $E^k$ through $p$ and $Y_k$ should have normal crossings with $E^k$. Thus $Y_k=\{p\}$.
Let us give some remarks and fix notations.
We put
$$
\Gamma'=E^{k+1}_i\cap E^{k+1}_j,\; L'_i=E^{k+1}_i\cap E^{k+1}_{k+1},\; L'_j=E^{k+1}_j\cap E^{k+1}_{k+1},\; p'=\Gamma'\cap L'_i.
$$
In view of Noether's formula for the intersection multiplicity (see \cite{Ful} for instance), given \break $\gamma\in {\mathcal B}(i;p)\setminus \{\Gamma\}$ we have
\begin{equation*}
(\gamma,\Gamma)_p=(\gamma',\Gamma')_{p'}+\sum_{q\in L'_i} (\gamma',L'_i)_q
\end{equation*}
where $\gamma'$ stands for the strict transform of $\gamma$.
Let us also note that
$$
\mbox{Ind}({\mathcal F},E_i;\gamma)= \mbox{Ind}({\mathcal F},E_i;\gamma'),
$$
since the computations are made at generic points of $\gamma\subset E_i^k$.
On the other hand, note that $E^{k+1}_{k+1}$ is a projective plane and $L'_i,L'_j\subset E^{k+1}_{k+1}$ are both projective lines. In particular, let $\Lambda\subset E^{k+1}_{k+1}$ be a global irreducible curve $\Lambda\subset \mbox{\rm Sing}({\mathcal F}_{k+1})$ of degree $d_\Lambda$ with $\Lambda\ne L'_i,L'_j$. By Bezout's Theorem, we know that
\begin{equation*}
d_\Lambda=\sum_{q\in L'_i;\;q\in \delta\subset \Lambda} (\delta,L'_i)_q= \sum_{q\in L'_j;\;q\in \delta\subset \Lambda} (\delta,L'_j)_q ,
\end{equation*}
where $\delta$ runs over the irreducible branches of $\Lambda$ at $q$.
Now, we have four cases to consider:
\begin{enumerate}
\item[i)] $E^k_j$ is dicritical and $\pi_{k+1}$ is a dicritical blow-up.
\item[ii)] $E^k_j$ is dicritical and $\pi_{k+1}$ is a non dicritical blow-up.
\item[iii)] $E^k_j$ is invariant and $\pi_{k+1}$ is a dicritical blow-up.
\item[iv)] $E^k_j$ is invariant and $\pi_{k+1}$ is a non dicritical blow-up.
\end{enumerate}
{\em Assume first that $E^k_j$ is a dicritical component.} Let us note that
$$\Gamma\not\subset\mbox{\rm Sing}({\mathcal F}_k);\;\Gamma',L'_j\not\subset\mbox{\rm Sing}({\mathcal F}_{k+1}).$$
The induced induced foliation ${\mathcal G}'$ by ${\mathcal F}_{k+1}$ on $E^{k+1}_j$ is the transform of $\mathcal G$ by the restriction
$$
\tilde\pi_{k+1}: E^{k+1}_j\rightarrow E^k_j
$$
of the blow-up $\pi_{k+1}$. In particular, by the known properties of Camacho-Sad index (see \cite{Cam-S, Can-C-D}) we have that
\begin{equation*}
\label{eq:csindex}
\mbox{\rm Ind}({\mathcal G},\Gamma;p)= \mbox{\rm Ind}({\mathcal G}',\Gamma';p')+1.
\end{equation*}
{\em First case: $\pi_{k+1}$ is a dicritical blow-up}. Let us denote ${\mathcal G}_1$ the induced foliation by ${\mathcal F}_{k+1}$ on $E^{k+1}_{k+1}$. The self-intersection of the projective line $L'_i$ in the projective plane $E^{k+1}_{k+1}$ is equal to $+1$. Then we have
\begin{equation*}
\sum_{q\in L'_i}\mbox{\rm Ind}({\mathcal G}_1,L'_i;q)=+1,
\end{equation*}
Let us note that since $\Gamma', L'_i$ are not in the singular locus we have a bijection
$$
{\mathcal B}^k_i(p)\leftrightarrow \bigcup_{q\in L'_i} {\mathcal B}^{k+1}_i(q)
$$
given by the strict transform $\gamma\mapsto \gamma'$. Applying induction hypothesis to the points of $L'_i$ we deduce that
$$
\sum_{q\in L'_i;\gamma\in {\mathcal B}^k_i(p)}(\gamma',L'_i)_q\mbox{\rm Ind}({\mathcal F},E_i;\gamma)=\sum_{q\in L'_i}\mbox{\rm Ind}({\mathcal G}_1,L'_i;q)=+1.
$$
Applying induction hypothesis at $p'$ as well, we have
\begin{eqnarray*}
&&\sum_{\gamma\in {\mathcal B}^k_i(p)}(\gamma,\Gamma)_p \mbox{Ind}({\mathcal F},E_i;\gamma)=\\
&&=\sum_{\gamma\in {\mathcal B}^k_i(p)}
\left(
(\gamma',\Gamma')_{p'}
+\sum_{q\in L'_i}(\gamma',L'_i)_q \right)
\mbox{Ind}({\mathcal F},E_i;\gamma)=\\
&&=\mbox{Ind}({\mathcal G}',\Gamma';p')+1= \mbox{Ind}({\mathcal G},\Gamma;p).
\end{eqnarray*}
This case is ended.
{\em Second case: $\pi_{k+1}$ is a non dicritical blow-up}.
Let us denote $\tilde \alpha= \mbox{Ind}({\mathcal F},E_{i};L'_i)$. By induction hypothesis at $p'$ we have
\begin{equation*}
\mbox{Ind}({\mathcal G}',\Gamma';p')=\tilde \alpha +
\sum_{\gamma\in {\mathcal B}^k_i(p)}(\gamma',\Gamma')_{p'} \mbox{Ind}({\mathcal F},E_i;\gamma'),
\end{equation*}
and since $\mbox{Ind}({\mathcal G}',\Gamma';p')=\mbox{Ind}({\mathcal G},\Gamma;p)-1$, we can put
\begin{equation*}
\mbox{Ind}({\mathcal G},\Gamma;p)=\tilde \alpha +1+
\sum_{\gamma\in {\mathcal B}^k_i(p)}(\gamma',\Gamma')_{p'} \mbox{Ind}({\mathcal F},E_i;\gamma').
\end{equation*}
By Noether's Theorem we have
$
(\gamma',\Gamma')_{p'}=(\gamma,\Gamma)_p-\sum_{q\in L'_i}(\gamma',L'_i)_q
$.
Then
\begin{equation*}
\mbox{Ind}({\mathcal G},\Gamma;p)= \sum_{\gamma\in {\mathcal B}^k_i(p)}(\gamma,\Gamma)_p \mbox{Ind}({\mathcal F},E_i;\gamma)
+ \beta
\end{equation*}
where
$$
\beta= \tilde \alpha +1-
\sum_{q\in L'_i}\sum_{\gamma\in {\mathcal B}^k_i(p)}(\gamma',L'_i)_q \mbox{Ind}({\mathcal F},E_i;\gamma').
$$
Now, it is enough to show that $\beta=0$. By induction hypothesis in the statement b) referred to
$E^{k+1}_i$ and $E^{k+1}_{k+1}$ we have that
\begin{equation*}
\sum_{q\in L'_i;\;\gamma\in {\mathcal B}^k_i(p)}(\gamma',L'_i)_q \mbox{Ind}({\mathcal F},E_i;\gamma')=-\tilde\alpha
\sum_{\Lambda\subset E^{k+1}_{k+1},\;\Lambda\ne L'_i}d_\Lambda \mbox{Ind}({\mathcal F},E_{k+1};\Lambda).
\end{equation*}
Now, we apply induction hypothesis in the statement a) referred to $E^{k+1}_{k+1}$ and $E^{k+1}_j$ to obtain
\begin{equation*}
\sum_{q\in L'_j}\mbox{Ind}({\mathcal G}',L'_j;q)=1/\tilde \alpha+
\sum_{\Lambda\subset E^{k+1}_{k+1};\Lambda\ne L'_i}d_{\Lambda}\mbox{Ind}({\mathcal F},E_{k+1};\Lambda),
\end{equation*}
where $1/\tilde\alpha= \mbox{Ind}({\mathcal F},E_{k+1};L'_i)$. Recalling that the self intersection of $L'_j$ in $E^{k+1}_j$ is equal to $-1$, we have
$
-1= \sum_{q\in L'_j}\mbox{Ind}({\mathcal G}',L'_j;q)
$ and
we obtain
$$
-1=1/\tilde\alpha -(1/\tilde\alpha)\sum_{q\in L'_i;\;\gamma\in {\mathcal B}^k_i(p)}(\gamma',L'_i)_q \mbox{Ind}({\mathcal F},E_i;\gamma').
$$
That is
\begin{eqnarray*}
\beta=\tilde \alpha +1- \sum_{q\in L'_i;\;\gamma\in {\mathcal B}^k_i(p)}(\gamma',L'_i)_q \mbox{Ind}({\mathcal F},E_i;\gamma')=0
\end{eqnarray*}
and we are done.
{\em Let us suppose finally that $E^k_j$ is an invariant component.}
{\em First case: $\pi_{k+1}$ is a dicritical blow-up}. Let ${\mathcal G}_1$ be the induced foliation by ${\mathcal F}_{k+1}$ on $E^{k+1}_{k+1}$.
By applying induction hipothesis at $L'_i$ and $L'_j$ and recalling that the self-intersection of $L'_i,L'_j$ inside $E^{k+1}_{k+1}$ is $+1$, we have
$$
1= \sum_{q\in L'_i}\mbox{Ind}({\mathcal G}_1,L'_i;q)= \sum_{q\in L'_j}\mbox{Ind}({\mathcal G}_1,L'_j;q),
$$
and hence
\begin{eqnarray}
\label{eq:csindexdos}
1=\alpha+
\sum_{q\in L'_i}\sum_{\gamma\in {\mathcal B}^k_i(p)\setminus \{\Gamma\}}(\gamma',L'_i)_q\mbox{Ind}({\mathcal F},E_i;\gamma'),\\
\label{eq:csindextres}
1= (1/\alpha)+
\sum_{q\in L'_j}\sum_{\delta\in {\mathcal B}^k_j(p)\setminus \{\Gamma\}}(\delta',L'_j)_q\mbox{Ind}({\mathcal F},E_j;\delta').
\end{eqnarray}
Also, by induction hypothesis at $p'$ referred to $E^{k+1}_i$ and $E^{k+1}_j$ we have
\begin{equation}
\label{eq:csindexcuatro}
\sum_{\gamma\in {\mathcal B}^k_i(p)\setminus\{\Gamma\}}(\gamma',\Gamma')_{p'} \mbox{Ind}({\mathcal F},E_i;\gamma')=-\alpha
\sum_{\delta\in {\mathcal B}^k_j(p)\setminus\{\Gamma\}}(\delta',\Gamma')_{p'} \mbox{Ind}({\mathcal F},E_j;\delta').
\end{equation}
Using Noether's formula and the equalities (\ref{eq:csindexdos}, \ref{eq:csindextres}) , we have
\begin{eqnarray*}
\sum_{\gamma\in {\mathcal B}^k_i(p)\setminus\{\Gamma\}}(\gamma,\Gamma)_{p} \mbox{Ind}({\mathcal F},E_i;\gamma)= \sum_{\gamma\in {\mathcal B}^k_i(p)\setminus\{\Gamma\}}(\gamma',\Gamma')_{p'} \mbox{Ind}({\mathcal F},E_i;\gamma')+ (1-\alpha),\\
-\alpha\sum_{\delta\in {\mathcal B}^k_j(p)\setminus\{\Gamma\}}(\delta,\Gamma)_{p} \mbox{Ind}({\mathcal F},E_i;\delta)= -\alpha\sum_{\gamma\in {\mathcal B}^k_j(p)\setminus\{\Gamma\}}(\delta',\Gamma')_{p'} \mbox{Ind}({\mathcal F},E_i;\delta')+ (1-\alpha)
\end{eqnarray*}
and we are done by Equation (\ref{eq:csindexcuatro}).
{\em Second case: $\pi_{k+1}$ is a non dicritical blow-up}. Let us denote
$$
\beta=\mbox{Ind}({\mathcal F}, E_i;L'_i);\quad \rho= \mbox{Ind}({\mathcal F}, E_j;L'_j).
$$
We have
$
1/\beta=\mbox{Ind}({\mathcal F}, E_{k+1};L'_i)$ and $1/\rho= \mbox{Ind}({\mathcal F}, E_{k+1};L'_j).
$
Let us put
$$
\epsilon= \sum_{\Lambda\subset E_{k+1}^{k+1}, \Lambda\ne L'_i,L'_j}d_\Lambda \mbox{Ind}({\mathcal F}, E_{k+1};\Lambda).
$$
Now, if we take a generic plane section $\Delta$ at $p$ and we apply Camacho-Sad's equality to ${\mathcal F}_k\vert_\Delta$ after the blow-up $\pi_{k+1}$, we obtain
\begin{equation*}
\label{eq:cssection}
-1= 1/\beta +1/\rho +\epsilon.
\end{equation*}
By induction hypothesis referred to $E_i^{k+1}$ and $E_{k+1}^{k+1}$, we have the following equality
\begin{eqnarray*}
\alpha+\sum_{q\in L'_i}\sum_{\gamma\in {\mathcal B}^k_i(p)\setminus\{\Gamma\}}(\gamma',L'_i)_q \mbox{Ind}({\mathcal F}, E_i;\gamma')=
-\beta((1/\rho)+\epsilon)=\beta +1
\end{eqnarray*}
and thus
\begin{equation*}
\sum_{q\in L'_i}\sum_{\gamma\in {\mathcal B}^k_i(p)\setminus\{\Gamma\}}(\gamma',L'_i)_q \mbox{Ind}({\mathcal F}, E_i;\gamma')=-\alpha +\beta +1.
\end{equation*}
Now, applying induction referred to $E_j^{k+1}$ and $E_{k+1}^{k+1}$, we have
\begin{eqnarray*}
(1/\alpha)+\sum_{q\in L'_i}\sum_{\delta\in {\mathcal B}^k_j(p)\setminus\{\Gamma\}}(\delta',L'_i)_q \mbox{Ind}({\mathcal F}, E_i;\delta')=
-\rho((1/\beta)+\epsilon)=\rho+1
\end{eqnarray*}
and thus
\begin{equation*}
-\alpha \sum_{q\in L'_i}\sum_{\delta\in {\mathcal B}^k_j(p)\setminus\{\Gamma\}}(\delta',L'_i)_q \mbox{Ind}({\mathcal F}, E_i;\delta')=1-\alpha(\rho+1).
\end{equation*}
Applying induction hypothesis at $p'$, we have
\begin{eqnarray*}\beta+
\sum_{\gamma\in {\mathcal B}^k_i(p)\setminus\{\Gamma\}}(\gamma',\Gamma')_{p'} \mbox{Ind}({\mathcal F}, E_j;\gamma')=\\=
-\alpha\left(\rho +\sum_{\delta\in {\mathcal B}^k_j(p)\setminus\{\Gamma\}}(\delta',\Gamma')_{p'} \mbox{Ind}({\mathcal F}, E_j;\delta')\right).
\end{eqnarray*}
Thus, by Noether's equality, we only have to verify that
$$
(-\alpha+\beta+1) -\beta= (1-\alpha(\rho+1)) +\alpha\rho
$$
and this is evident.
\end{proof}
\begin{corollary}
\label{cor:indicesymultiplicidad}
Let $p\in F_k$ be a point such that $p\in E^k_i\cap E^k_j\cap E^k_\ell$ where $E^k_i$, $E^k_j$ and $E^k_\ell$ are compact invariant components of $E^k$. Let us denote
\begin{eqnarray*}
&\Gamma_\ell= E^k_i\cap E^k_j, \; \Gamma_j= E^k_i\cap E^k_\ell,\; \Gamma_i= E^k_j\cap E^k_\ell.\\
&\alpha=\mbox{\rm Ind}({\mathcal F}, E_i;\Gamma_\ell),\;\beta=\mbox{\rm Ind}({\mathcal F}, E_i;\Gamma_j), \;\rho=
\mbox{\rm Ind}({\mathcal F}, E_j;\Gamma_i).
\end{eqnarray*}
Then, we have
$
\beta=-\alpha\rho
$.
\end{corollary}
\begin{proof} Induction on $N-k$. If $k=N$, we are done (see (\ref{enumerate4}) in the proof of the case $k=N$ in Proposition \ref{pro:indicesymultiplicidad}).
Assume that $k<N$ and $p\in Y_k$ as in previous proofs. We have that $Y_k=\{p\}$. If the blow-up is non dicritical, we put
$$
\nu=\mbox{\rm Ind}({\mathcal F}, E_i;E_i\cap E_{k+1}),\; \xi=\mbox{\rm Ind}({\mathcal F}, E_\ell;E_\ell\cap E_{k+1}),\; \mu=\mbox{\rm Ind}({\mathcal F}, E_j;E_j\cap E_{k+1}).
$$
By induction hypothesis, we have
$
\nu=-\alpha\mu, \;\nu=-\xi\beta,\; \mu=-\xi\rho
$.
That is, we have $\xi\beta=\alpha\mu=-\alpha\xi\rho$ and thus $\beta=-\alpha\rho$.
Assume now that the blow-up is dicritical. We denote by $\Gamma'_\ell,\Gamma'_j,\Gamma'_i$ the strict transforms of $\Gamma_\ell,\Gamma_j,\Gamma_i$ respectively. We also denote by
\begin{equation*}
{\mathcal B}_i^*={\mathcal B}^k_i(p)\setminus \{\Gamma_\ell,\Gamma_j\};\quad
{\mathcal B}_j^*={\mathcal B}^k_j(p)\setminus \{\Gamma_\ell,\Gamma_i\};\quad
{\mathcal B}_\ell^*={\mathcal B}^k_\ell(p)\setminus \{\Gamma_i,\Gamma_j\};
\end{equation*}
and we put
$$
I_u^{(v)}=\sum_{\gamma\in {\mathcal B}_u^*}(\Gamma_v,\gamma)_p\mbox{\rm Ind}({\mathcal F},E_u;\gamma)
$$
for $u\ne v$ with $u,v\in\{i,j,\ell\}$.
Given a germ of curve $\gamma$ we denote by $\gamma'$ the strict transform of $\gamma$, as usual. Take also the following notations
$$
p'_u=\Gamma'_u\cap E_{k+1}^{k+1},\; L'_{u}= E_u^{k+1}\cap E^{k+1}_{k+1}; \quad u\in \{i,j,\ell\}
$$
and
\begin{equation*}
I'_u=\sum_{{q'\in L'_u}, {\gamma\in {\mathcal B}_u^*}}(L'_u,\gamma')_{q'}\mbox{\rm Ind}({\mathcal F},E_i;\gamma)
;\quad u\in \{i,j,\ell\}.
\end{equation*}
Finally, we put
$$
{I'}_u^{(v)}=\sum_{\gamma\in {\mathcal B}_u^*}(\Gamma'_v,\gamma')_{p'_v}\mbox{\rm Ind}({\mathcal F},E_u;\gamma')
$$
for $u\ne v$ with $u,v\in\{i,j,\ell\}$.
By Noether's formula, we have
$$
I_u^{(v)}=I'_u+{I'}_i^{(v)}, \mbox{ for } u,v\in\{i,j,\ell\}, u\ne v.
$$
Now, by applying part a) of Proposition \ref{pro:indicesymultiplicidad} to the exceptional divisor and each of three other divisors, we have
\begin{eqnarray}
\label{eq:dicritica}
\alpha+\beta+{I'}_i=\frac{1}{\beta}+\frac{1}{\rho}+I'_\ell=\frac{1}{\alpha}+\rho+I'_{j}=1.
\end{eqnarray}
By Proposition \ref{pro:indicesymultiplicidad} we have
\begin{eqnarray*}
&I_i^{(\ell)}=-\alpha I_j^{(\ell)}; \; I_i^{(j)}=-\beta I_\ell^{(j)};\; I_\ell^{(i)}=-(1/\rho) I_j^{(i)}.\\
&{I'}_i^{(\ell)}=-\alpha {I'}_j^{(\ell)}; \; {I'}_i^{(j)}=-\beta {I'}_\ell^{(j)};\; {I'}_\ell^{(i)}=-(1/\rho) {I'}_j^{(i)}.
\end{eqnarray*}
We deduce that
\begin{eqnarray*}
&I'_i=-\alpha I'_j; \; I'_i=-\beta I'_\ell;\; I'_\ell=-(1/\rho) I'_j.
\end{eqnarray*}
This implies that
$$
-\alpha I'_j=\beta(1/\rho) I'_j.
$$
Hence, if $I'_j\ne 0$ we conclude that $\beta=-\alpha\rho$ and we are done.
Assume now that $I'_j=0$ and hence $I'_i=I'_j=I'_\ell=0$. By Equation \ref{eq:dicritica} we have
$$
\alpha+\beta=(1/\beta)+(1/\rho)=(1/\alpha)+\rho=1.
$$
We deduce that $1+\alpha\rho=\alpha$, hence $1=\alpha(1-\rho)$, but $1=\alpha+\beta$. This implies that $\beta=-\alpha\rho$ as desired.
\end{proof}
Let us consider a point $p$ in a compact invariant component $E^k_i$ of $E^k$ and a partial separatrix $C$. We denote by
$
{\mathcal B}^k_i(C;p)
$
the set of germs of curve $(\gamma,p)$ such that $(\gamma,p)\subset (C_k\cap E^k_i,p)$.
\begin{corollary}
\label{cor:transitioncompleta}
Let us consider a point $p\in \Gamma= E^k_i\cap E^k_j$, where $E^k_i$ and $E^k_j$ are compact invariant components of $E^k$ and a partial separatrix $C$. Assume that $p$ is a complete point for $C$.
We have
$$
\sum_{\gamma\in{\mathcal B}_i^k(C;p)}(\gamma,\Gamma)_p\mbox{\rm Ind}({\mathcal F},E^k_i;\gamma)=-\alpha \sum_{\eta\in{\mathcal B}_j^k(C;p)}(\eta,\Gamma)_p\mbox{\rm Ind}({\mathcal F},E^k_j;\eta).
$$
where $\alpha=\mbox{\rm Ind}({\mathcal F},E^k_i;\Gamma)$.
\end{corollary}
\begin{proof} We do induction on $N-k$ as usual. If $k=N$ we are done, by the local expression at simple points. Assume that $k<N$. We suppose without loss of generality that $p\in Y_k$ and thus the next blow-up $\pi_{k+1}$ is centered at the point $p$. Since $C$ is complete at $p$, the blow-up is non-dicritical. Put
$$
\Gamma'=E^{k+1}_{i}\cap E^{k+1}_j,\; L'_i=E^{k+1}_{k+1}\cap E^{k+1}_i, \; L'_j=E^{k+1}_{k+1}\cap E^{k+1}_j
$$ and let $p'=\Gamma'\cap E^{k+1}_{k+1}$. Denote as usual by $\gamma'$ the strict transform of the germ of curve $\gamma$ and put
\begin{eqnarray*}
&I'_u(C)=\sum_{q\in L'_u}\sum_{\gamma\in {\mathcal B}_u^k(C;p)}(\gamma',L'_u)_q \mbox{\rm Ind}({\mathcal F},E^k_u;\gamma),\\
&I''_u(C)= \sum_{\gamma\in {\mathcal B}_u^k(C;p)}(\gamma',\Gamma')_{p'} \mbox{\rm Ind}({\mathcal F},E^k_u;\gamma)
\end{eqnarray*}
for $u\in\{i,j\}$.
We have that
$$
\sum_{\gamma\in{\mathcal B}_u^k(C;p)}(\gamma,\Gamma)_p\mbox{\rm Ind}({\mathcal F},E^k_u;\gamma)=I'_u(C)+I''_u(C),\;u\in\{i,j\}
$$
and by induction hypothesis we know that $I''_i(C)=-\alpha I''_j(C)$. Let us denote
$$
\beta=\mbox{\rm Ind}({\mathcal F},E^{k+1}_i;L'_i);\quad \rho=\mbox{\rm Ind}({\mathcal F},E^{k+1}_j;L'_j).
$$
Also by induction hypothesis, we have
$$
(-1/\beta)I'_i(C)=\sum_{\Lambda}d_\Lambda \mbox{\rm Ind}({\mathcal F},E^{k+1}_{k+1};\Lambda)= (-1/\rho)I'_j(C),
$$
where $\Lambda$ stands for the global irreducible curves $\Lambda \subset E^{k+1}_{k+1}$ such that $\Lambda\subset C_{k+1}$. Applying Corollary \ref{cor:indicesymultiplicidad}, we deduce that
$$
I'_i(C)=({\beta}/{\rho}) I'_j(C)=-\alpha I'_j(C)
$$
and we are done.
\end{proof}
\section{Indices of partial separatrices}
Consider a partial separatrix $C$. Here we show that it is possible to define the index $\mbox{\rm Ind}(C;E_i)$ relative to any invariant compact component $E_i$ of $E$. Given an invariant compact component $E_i$ of $E$, we denote by
$
{\mathcal B}_iC
$ the set of global irreducible curves in $C\cap E_i$. We put $\mbox{\rm Ind}(C;E_i)=0$ if there is no compact curve of $C$ contained in $E_i$. Otherwise, we shall put
$$
\mbox{\rm Ind}(C;E_i)=\mbox{\rm Ind}({\mathcal F}, E_i;\Gamma)
$$
where $\Gamma\in {\mathcal B}_iC$. Proposition \ref{pro:indexpartialseparatriz} assures that the definition is consistent.
\begin{proposition}
\label{pro:indexpartialseparatriz}
Let $C$ be a partial separatrix and $E_i$ a compact invariant component of $E$. If $\Gamma_1,\Gamma_2\in {\mathcal B}_iC$, we have
$
\mbox{\rm Ind}({\mathcal F}, E_i;\Gamma_1)=\mbox{\rm Ind}({\mathcal F}, E_i;\Gamma_2)
$.
\end{proposition}
Before giving the proof of Proposition \ref{pro:indexpartialseparatriz}, let us introduce the {\em dual graph ${\mathcal G}_N$ of the compact invariant components}. This graph has vertices corresponding to the compact invariant components; two vertices $E_i, E_j$ are joined by a wedge if and only if $E_i\cap E_j\ne \emptyset$. It is the last one of the series of dual graphs ${\mathcal G}_k$ of the compact invariant components of $E^k$.
Since each new invariant compact component is produced by the blow-up of a point, we see that given two compact invariant components $E_i$ and $E_j$ we have that either $E_i\cap E_j=\emptyset$ or $E_i\cap E_j$ is an irreducible compact curve.
The graph ${\mathcal G}_{k+1}$ is obtained from ${\mathcal G}_k$ as follows. If the blow-up $\pi_{k+1}$ is dicritical, then ${\mathcal G}_{k+1}={\mathcal G}_k$. If the center of $\pi_{k+1}$ is a curve, we also have that ${\mathcal G}_{k+1}={\mathcal G}_k$. If $\pi_{k+1}$ is non dicritical and the center is a point $p_k$, we have four possibilities:
\begin{enumerate}
\item The point $p_k$ does not belong to any invariant compact component of $E^k$. In this case, the graph ${\mathcal G}_{k+1}$ is obtained from ${\mathcal G}_k$ by adding a new connected component to ${\mathcal G}_k$ consisting in a single vertex that represents the exceptional divisor $E^{k+1}_{k+1}$. No new wedges are added.
\item The point $p_k$ belongs to a single invariant compact component $E^k_i$ of $E^k$. Then ${\mathcal G}_{k+1}$ is obtained from ${\mathcal G}_k$ by adding a new vertex that represents the exceptional divisor $E^{k+1}_{k+1}$ and a new wedge connecting it with $E^{k+1}_i$.
\item The point $p_k$ belongs to exactly two invariant compact components $E^k_i, E^k_j$ of $E^k$. Then ${\mathcal G}_{k+1}$ is obtained from ${\mathcal G}_k$ by adding a new vertex that represents the exceptional divisor $E^{k+1}_{k+1}$ and two new wedges connecting it with $E^{k+1}_i$ and $E^{k+1}_j$.
\item The point $p_k$ belongs to three invariant compact components $E^k_i, E^k_j, E^k_\ell$ of $E^k$. Then ${\mathcal G}_{k+1}$ is obtained from ${\mathcal G}_k$ by adding a new vertex that represents the exceptional divisor $E^{k+1}_{k+1}$ and three new wedges connecting it with $E^{k+1}_i$, $E^{k+1}_j$ and $E^{k+1}_\ell$.
\end{enumerate}
A {\em chain } of length $s$ in ${\mathcal G}_N$ is any sequence
\begin{equation}
c=(E_{i_0},w_1, E_{i_1},w_2, E_{i_2},\ldots,w_{s-1},E_{i_{s-1}},w_s, E_{i_s})
\end{equation}
such that $w_{n}=E_{i_{n-1}}\cap E_{i_n}$ is a wedge for $n=1,2,\ldots,s$. If we have another chain
$$
c_1=(E_{i_s},w_{s+1}, E_{i_{s+1}},w_{s+2}, E_{i_{s+2}},\ldots,w_{t-1},E_{i_{t-1}},w_t, E_{i_t})
$$
starting at $E_{i_s}$, we can compose the two chains to obtain
$$
c*c_1=(E_{i_0},w_1, E_{i_1},w_2, E_{i_2},\ldots,w_{t-1},E_{i_{t-1}},w_t, E_{i_t}).
$$
Let us consider a complex number $\mu\ne 0$. The {\em transformed number } $c(\mu)$ of by the chain $c$ is defined as follows. If $s=0$ we put $c(\mu)=\mu$. Put
$$
c=c_{s-1}*(E_{i_{s-1}},w_s, E_{i_s})
$$
where $c_{s-1}$ has length $s-1$.
For
$
\alpha=\mbox{\rm Ind}({\mathcal F}, E_{i_{s}}; w_{s})
$,
we define $$c(\mu)=-\alpha c_{s-1}(\mu).$$
Let us denote $c^{-1}$ the chain obtained by reversing the order in $c$. By Remark \ref{rk:productoindices} we have that
\begin{equation}
\label{eq:reversible}
c^{-1}(c(\mu))=\mu;\; c(c^{-1}(\mu))=\mu.
\end{equation}
\begin{lemma}
\label{lema:cadenacircular}
Consider a (circular) chain
\begin{eqnarray*}
c=(E_{i_0},w_1, E_{i_1},w_2, E_{i_2},\ldots,w_{s-1},E_{i_{s-1}},w_s, E_{i_s})
\end{eqnarray*}
such that $E_{i_0}=E_{i_s}$. For any $\mu\ne 0$ we have
$
c(\mu)=\mu.
$
\end{lemma}
\begin{proof} In view of Equation \ref{eq:reversible} the result is true if and only if it is true for one of the shifted chains
$$
c_j=(E_{i_j},w_j, E_{i_{j+1}},w_{j+1}, E_{i_{j+2}},\ldots,w_{s},E_{i_{s}}=E_{i_0},w_1, E_{i_1},\ldots,E_{i_{j-1}},w_{j-1}, E_{i_j}).
$$
We do induction on the number of vertices of the graph ${\mathcal G}_N$ and the length of $c$. If we have only one vertex, we are done. Let $v$ be the last vertex incorporated to the construction of ${\mathcal G}$. If this vertex $v$ does not appear in $c$, we are done by induction. Assume that $v$ appears in $c$.
If $v$ is a connected component of $\mathcal G$, we have only one vertex in $c$ and we are done. If $v$ is not isolated, we have three possibilities:
\begin{enumerate}
\item $v$ is connected with exactly one vertex $v_1$ with a wedge $w'_1$.
\item $v$ is connected with two vertices $v_1,v_2$ by means of respective wedges $w'_1,w'_2$. In this case $v_1$ and $v_2$ are connected by wedge $\tilde w_{12}$.
\item $v$ is connected with three vertices $v_1,v_2,v_3$ by means of respective wedges $w'_1,w'_2,w'_3$. In this case $v_1,v_2,v_3$ are connected two by two by wedges $\tilde w_{12},\tilde w_{13},\tilde w_{23}$.
\end{enumerate}
In case (1), up to performing a shift of $c$, we may assume that $c$ has the form
$$
c=(v,w'_1,v_1,w_2,\ldots,w_{s-1},v_1,w'_1,v)
$$
and we are done by induction applied to $c'=(v_1,w_2,\ldots,w_{s-1},v_1)$ as follows. Let $\alpha= \mbox{\rm Ind}({\mathcal F},v;w'_1)$, put
$
\mu'=-\alpha\mu
$, then we have
$$
c(\mu)=(-1/\alpha)c'(\mu')=(-1/\alpha)\mu'=\mu.
$$
In case (2), up to interchanging the role of $v_1$ and $v_2$ the appearance of $v$ may be in one of the following two forms,
\begin{eqnarray*}
c&=&c_1*(v_1,w'_1,v,w'_1,v_1)*c_2, \\
c&=& c_1*(v_1,w'_1,v,w'_2,v_2)*c_2.
\end{eqnarray*}
The first one is treated as in the previous case. Assume we have the second one. Let us denote
$$
\alpha=\mbox{\rm Ind}({\mathcal F}, v_1;\tilde w_{12});\;\beta=\mbox{\rm Ind}({\mathcal F}, v_1;\tilde w'_1);\;
\rho =\mbox{\rm Ind}({\mathcal F}, v_2;\tilde w'_1).
$$
We know that $\beta=-\alpha\rho$. Consider the circular chain
$$
\tilde c=c_1*(v_1,\tilde w_{12},v_2)*c_2.
$$
In view of the fact that
$$
(v_1,\tilde w_{12},v_2)(\tilde\mu)=-\alpha\tilde\mu=(-\beta)(-1/\rho)\tilde\mu= (v_1,w'_1,v,w'_2,v_2)(\tilde \mu),
$$
we deduce that $\tilde c(\mu)=c(\mu)$ and we are done since by induction we have $\tilde c(\mu)=\mu$.
Case (3) is treated as the previous one.
\end{proof}
Now we go to the proof of Proposition \ref{pro:indexpartialseparatriz}. Since the compact part of the partial separatrix $C$ is connected, we can join a generic point $p_1$ in $\Gamma_1$ with a generic point $p_2$ in $\Gamma_2$ by a real path $\gamma$. Moreover $\gamma$ may be chosen in such a way that it produces only finitely many changes of irreducible curves in $C$. The connected change of (trace) irreducible curves of $C$ gives a transition
of invariant compact component of the divisor. In this way, we obtain a circular chain
$$
c=(E_{i_0},w_1, E_{i_1},w_2, E_{i_2},\ldots,w_{s-1},E_{i_{s-1}},w_s, E_{i_s}=E_{i_0})
$$
such that if $\mu$ is the index for $\Gamma_1$ then $c(\mu)$ is the index for $\Gamma_2$. By Lemma \ref{lema:cadenacircular} we have that $c(\mu)=\mu$ and the proof is ended.
\begin{remark} Corollary \ref{cor:transitioncompleta} may now be reformulated by stating that
$$
\left(\sum_{\gamma\in{\mathcal B}_iC}(\gamma,\Gamma)_p\right)\mbox{\rm Ind}(C;E_i)=-\alpha \left(\sum_{\eta\in{\mathcal B}_jC}(\eta,\Gamma)_p\right)\mbox{\rm Ind}(C;E_j).
$$
\end{remark}
\section{Real saddles at incomplete points}
Here we give a result relating incomplete points and real saddle curves. This is a key point in the proof of Theorem \ref{teo:mainteo}.
\begin{proposition}
\label{pro:notallrealsaddles}
Let $p$ be an incomplete point belonging to a compact invariant component $E^k_i$. There is a compact curve $\Gamma\subset\mbox{\rm Sing}{\mathcal F}_k$ with $\Gamma\subset E^k_i$ such that $\Gamma$ is not a real saddle.
\end{proposition}
\begin{proof} As usual we do induction on $N-k$. If $k=N$ there is nothing to prove, since $p$ is a complete point. Assume that $k<N$. We assume without loss of generality that $p\in Y_k$. Moreover, since $p$ is an incomplete point, we necessarily have that $Y_k=\{p\}$ in view of Remark \ref{rk:completitudequirreduction}. Now, it is enough to find $\gamma\in {\mathcal B}^k_i(p)$ such that
$$
\mbox{\rm Ind}({\mathcal F},E_{i};\gamma)\notin {\mathbb R}_{<0}.
$$
We assume by contradiction that all $\gamma\in {\mathcal B}^k_i(p)$ are real saddle curves.
{\em First case: $\pi_{k+1}$ is a dicritical blow-up}. We apply Proposition \ref{pro:indicesymultiplicidad} to the dicritical component $E^{k+1}_{k+1}$ to see that
\begin{equation}
\label{eq:dicincompleto}
\sum_{q\in L}\sum_{\gamma\in {\mathcal B}^k_i(p)}(\gamma',L)_q \mbox{\rm Ind}({\mathcal F},E_i;\gamma)=\sum_{q\in L}
\mbox{\rm Ind}({\mathcal F}\vert_{E^{k+1}_{k+1}},L;q)
\end{equation}
where $L=E^{k+1}_{k+1}\cap E^{k+1}_i$. The left hand side of Equation \ref{eq:dicincompleto} is a negative number but the right hand side coincides with the self-intersection of $L$ in $E^{k+1}_{k+1}$, that is, it has the value $+1$. This is the desired contradiction.
{\em Second case: $\pi_{k+1}$ is a non dicritical blow-up}. Put $L=E^{k+1}_{k+1}\cap E^{k+1}_i$ as before and
$
\alpha= \mbox{\rm Ind}({\mathcal F},E_i;L)
$. Let us consider a generic plane $\Delta$ at $p$ and ${\mathcal G}={\mathcal F}_k\vert_\Delta$. The blow-up $\pi_{k+1}$ induces a blow-up
$
\tilde\Delta\rightarrow \Delta
$
and the transform of $\mathcal G$ by this blow-up is $\tilde{\mathcal G}={\mathcal F}_{k+1}\vert_{\tilde\Delta}$. By the properties of the indices of Camacho-Sad we have
$$
\sum_{q\in \tilde\Delta\cap E^{k+1}_{k+1}} \mbox{\rm Ind}(\tilde{\mathcal G},\tilde\Delta\cap E^{k+1}_{k+1};q)=-1.
$$
Moreover, we have
$$
\sum_{q\in \tilde\Delta\cap E^{k+1}_{k+1}} \mbox{\rm Ind}(\tilde{\mathcal G},\tilde\Delta\cap E^{k+1}_{k+1};q)=
\mbox{\rm Ind}({\mathcal F}, E_{k+1};L)+
\sum_{\Lambda\subset E^{k+1}_{k+1},\Lambda \ne L} d_{\Lambda}\mbox{\rm Ind}({\mathcal F},E_{k+1};\Lambda).
$$
We know that
$
\mbox{\rm Ind}({\mathcal F}, E_{k+1};L)= 1/\alpha
$
and by Proposition \ref{pro:indicesymultiplicidad} we have
$$
\sum_{\Lambda\subset E^{k+1}_{k+1},\Lambda \ne L} d_{\Lambda}\mbox{\rm Ind}({\mathcal F},E_{k+1};\Lambda)=
-\frac{1}{\alpha}\left\{
\sum_{q\in E^{k+1}_{k+1}}\sum_{\gamma\in {\mathcal B}^k_i(p)}(\gamma,L)_q \mbox{\rm Ind}({\mathcal F},E_i;\gamma)
\right\}.
$$
That is
$$
-1= \frac{1}{\alpha}-\frac{r}{\alpha}
$$
with $r<0$. Thus $\alpha=(r-1)$ is a negative real number and $L$ is a real saddle. By induction hypothesis, all the points in $L$ must be complete, since otherwise the non real saddle in $E^{k+1}_i$ is not $L$ and projects to a non real saddle in $E^k_i$. Moreover, since the blow-up is non dicritical, there is at least one incomplete point $q\in E^{k+1}_{k+1}$. Let $\Theta$ be a curve through $q$ that is not a real saddle and consider a point $p'\in \Theta\cap L$. If $\Theta$ is a trace curve, by the transition of indices at complete points given in Corollary \ref{cor:transitioncompleta} we deduce the existence of a trace curve $\tilde\Theta\subset E^{k+1}_i$ such that $\tilde\Theta\subset E^{k+1}_i$ is not a real saddle. If $\Theta$ is contained in the intersection of two divisors, by Corollary \ref{cor:indicesymultiplicidad} we find a non real saddle $\tilde\Theta\subset E^{k+1}_i$ with $\tilde\Theta\not\subset E^{k+1}_{k+1}$. The projection of $\tilde\Theta$ gives the desired contradiction.
\end{proof}
\section{Uninterrupted Nodal Components}
Let us recall the notion of uninterrupted nodal component introduced in \cite{Can-R-S}. By definiton, an {\em uninterrupted nodal component} of ${\mathcal F}_N, E^N$ is a connected union $\mathcal N$ of irreducible curves $\Gamma\subset \mbox{\rm Sing}{\mathcal F}_N$ satisfying the following conditions:
\begin{enumerate}
\item Each $\Gamma\subset {\mathcal N}$ is a {\em nodal} curve (see Definition \ref{def:nodalsaddle}).
\item The component ${\mathcal N}$ is {\em uninterrupted} in the sense that there are exactly two curves $\Gamma_1$ and $\Gamma_2$ in ${\mathcal N}$ through any point $p\in {\mathcal N}$ of dimensional type three.
\end{enumerate}
Recall that an uninterrupted nodal component $\mathcal N$ is {\em incomplete} if and only if it intersects at least one compact dicritical component of the exceptional divisor $E^N$.
Otherwise, we say that $\mathcal N$ is {\em complete}.
The next result shows the compatibility between the uninterrupted nodal components and the partial separatrices, in the last step of the reduction of singularities.
\begin{proposition}[Global trace transitions]
\label{pro:globaltransitions} Let $\mathcal N$ be a uninterrupted nodal component. Consider
a partial separatrix $C$ and a compact invariant component $E_i$ of the exceptional divisor $E$.
If there is $\Gamma_0\in {\mathcal B}_iC$ with $\Gamma_0\subset {\mathcal N}$ then $\Gamma\subset {\mathcal N}$ for any $\Gamma\in {\mathcal B}_iC$.
\end{proposition}
\begin{proof} The proof is similar to the proof of Proposition \ref{pro:indexpartialseparatriz}. Let us consider the dual graph ${\mathcal G}_N$ as in Proposition \ref{pro:indexpartialseparatriz}. Take two curves $\Gamma_0,\Gamma_1\in {\mathcal B}_iC$. We can connect $\Gamma_0,\Gamma_1$ by a circular chain
$$
c=(E_i=E_{i_0},w_1,E_{i_1},w_2,E_{i_2},\ldots,w_{s-1},E_{i_{s-1}},w_s,E_{i_s}=E_i)
$$
as in Lemma \ref{lema:cadenacircular}. Now, let us recall that at a point of dimensional type three we have either no curves of ${\mathcal N}$ or exactly two of them. In this way, we have the following rule of behavior for the curves $\Gamma_{i_j}\subset E_{i_j}\cap C$ that we are considering in the chain $c$:
\begin{enumerate}
\item If $\Gamma_{i_{j-1}}\subset {\mathcal N}$ and $w_j\not\subset {\mathcal N}$, then $\Gamma_{i_{j}}\subset {\mathcal N}$.
\item If $\Gamma_{i_{j-1}}\subset {\mathcal N}$ and $w_j\subset {\mathcal N}$, then $\Gamma_{i_{j}}\not\subset {\mathcal N}$.
\item If $\Gamma_{i_{j-1}}\not\subset {\mathcal N}$ and $w_j\subset {\mathcal N}$, then $\Gamma_{i_{j}}\subset {\mathcal N}$.
\item If $\Gamma_{i_{j-1}}\not\subset {\mathcal N}$ and $w_j\not\subset {\mathcal N}$, then $\Gamma_{i_{j}}\not\subset {\mathcal N}$.
\end{enumerate}
Let us denote $\epsilon(w_{i_j})=-1$ if $w_{i_j}\subset {\mathcal N}$ and $\epsilon(w_{i_j})=1$ otherwise. Now, it is enough to prove that
$$
\epsilon(w_{i_1})\epsilon(w_{i_2})\cdots \epsilon(w_{i_s})=1.
$$
This can be done by the same arguments as in the proof of Lemma \ref{lema:cadenacircular}.
\end{proof}
Now, we consider an intermediate step $(M_k,F_k)$ of the reduction of singularities and we will study the transition properties
of a {\em fixed complete uninterrupted nodal component $\mathcal N$} at this level $k$ (see also \cite{Can-R-S}).
We put $${\mathcal N}_k=\rho_k({\mathcal N}).$$
Note that ${\mathcal N}_k\cap F_k$ is either a single point or a finite union of compact curves.
\begin{proposition}[Triple points transitions]
\label{pro:tripetransition}
Let $p\in F_k$ be a point belonging to three compact components $E^k_i,E^k_j$ and $E^k_\ell$ of $E^k$. Assume that $\Gamma_\ell=E^k_i\cap E^k_j\subset {\mathcal N}_k$. Then $E^k_i,E^k_j$ and $E^k_\ell$ are invariant and
$$
\Gamma_j\subset {\mathcal N}_k \Leftrightarrow \Gamma_i \not\subset {\mathcal N}_k,
$$
where $\Gamma_j=E^k_\ell\cap E^k_i$ and $\Gamma_i=E^k_\ell\cap E^k_j$.
\end{proposition}
\begin{proof} We do induction on $N-k$. If $k=N$ we are done by the definition of complete uninterrupted nodal component.
Assume that $k<N$ and $p\in Y_k$ as usual. Since $p$ is in the intersection of three compact components then $Y_k=\{p\}$. Denote
$$
p'_u=E^{k+1}_{k+1}\cap \Gamma'_u,\quad L'_u= E^{k+1}_{k+1}\cap E^{k+1}_u; \quad u\in\{i,j,\ell\}.
$$
By induction on $p'_\ell$ we have that $E^{k+1}_i, E^{k+1}_j$ and $E^{k+1}_{k+1}$ are invariant. In particular $\pi_{k+1}$ is non dicritical. We also have that either $L'_i$ or $L'_j$ are contained in ${\mathcal N}_{k+1}$. Now by induction on $p'_j$ or $p'_i$ respectively, we deduce that $E^{k+1}_\ell$ and hence $E^{k}_\ell$ is invariant.
Assume now that $L'_i\subset {\mathcal N}_{k+1}$ and hence $L'_j\not \subset {\mathcal N}_{k+1}$. By induction on $p'_j$ we have two possibilities:
\begin{enumerate}
\item $\Gamma'_j\subset {\mathcal N}_{k+1}$ and $L'_\ell\not\subset {\mathcal N}_{k+1}$. It is not possible to have that $\Gamma'_i\subset {\mathcal N}_{k+1}$ since at $p'_i$ we have the two other corner curves not in ${\mathcal N}_{k+1}$.
\item $\Gamma'_j\not\subset {\mathcal N}_{k+1}$ and $L'_\ell\subset {\mathcal N}_{k+1}$. Then we have that $\Gamma'_i\subset {\mathcal N}_{k+1}$ since $L'_j\not\subset {\mathcal N}_{k+1}$.
\end{enumerate}
We conclude in the same way in the case that $L'_j\subset {\mathcal N}_{k+1}$ and $L'_i\not \subset {\mathcal N}_{k+1}$.
\end{proof}
\begin{proposition}[Trace transitions]
\label{pro:tracetransitions} Let $C$ be a partial separatrix. Consider a point $p\in C_k$ complete for $C$ and belonging to a compact invariant component $E^k_i$.
Suppose that there is $\gamma\in {\mathcal B}_i^k(C;p)$ with $\gamma\subset {\mathcal N}_k$ and that $p$ belongs to another compact component $E^k_j$.
We have:
\begin{enumerate}
\item $E^k_j$ is an invariant component.
\item Put $\Gamma=E^k_i\cap E^k_j$. Then one of the following statements holds:
\begin{enumerate}
\item $\Gamma\subset {\mathcal N}_k$ and any $\tilde\gamma\in {\mathcal B}^k_j(C;p)$ is a real saddle.
\item $\Gamma$ is a real saddle and for any $\tilde\gamma\in {\mathcal B}^k_j(C;p)$ we have $\tilde\gamma\subset {\mathcal N}_k$.
\end{enumerate}
\end{enumerate}
\end{proposition}
\begin{proof} As usual, we do induction on $N-k$. If $k=N$, we are done. Indeed,
$p$ does not belong to any dicritical compact component, since $p\in {\mathcal N}$ and $\mathcal N$ is complete. Moreover, the alternative in (2) means that $\mathcal N$ is uninterrupted.
Assume now that $k<N$ and $p\in Y_k$ as usual.
If $Y_k$ is a curve, there is only one compact component of $E^{k}$ through $p$ and $E^k_j$ does not exist. We assume thus that $Y_k=\{p\}$. Since $p\in C_k$ is a complete point for $C$, then $\pi_{k+1}$ is non dicritical. Let us denote
$$
L'_i=E^{k+1}_{k+1}\cap E^{k+1}_{i},\; L'_j=E^{k+1}_{k+1}\cap E^{k+1}_{j}; \Gamma'= E^{k+1}_i\cap E^{k+1}_{j}
$$
and let $p'$ be the intersection point $E^{k+1}_{k+1}\cap \Gamma'$.
Let us see that $E^{k+1}_j$ is invariant.
If $L'_i\subset{\mathcal N}$, then $E^{k+1}_j$ is invariant by Proposition \ref{pro:tripetransition} applied at $p'$.
If $L'_i\not\subset{\mathcal N}$, the strict transform $\gamma'$ of $\gamma$ intersects $L'_i$ at some points. Let $q$ be one of such points. By induction hypothesis on $q$ there is a curve $\gamma^*\subset E^{k+1}_{k+1}$ with $\gamma^*\subset {\mathcal N}_{k+1}$ corresponding to the same partial separatrix $C$. We conclude that $E^{k+1}_j$ is invariant by induction hypothesis applied at the points of $\gamma^*\cap E^{k+1}_j$.
Now, assume that $\Gamma\subset {\mathcal N}_k$. Hence $\Gamma'\subset {\mathcal N}_{k+1}$. By Proposition \ref{pro:tripetransition}, we have two possible situations:
\begin{enumerate}
\item [i)] $L'_i\subset {\mathcal N}_{k+1}$ and $L'_j\not\subset {\mathcal N}_{k+1}$.
\item [ii)] $L'_j\subset {\mathcal N}_{k+1}$ and $L'_i\not\subset {\mathcal N}_{k+1}$.
\end{enumerate}
Assume we have i). Consider a point $q\in\gamma'\cap L'_i$. By induction hypothesis at $q$, there is a curve $\gamma^*\in {\mathcal B}^{k+1}_{k+1}(C;q)$ such that $\gamma^*\not\subset {\mathcal N}_{k+1}$. We consider a point $q'\in \gamma^*\cap L'_j$. Also by induction hypothesis at $q'$ there is $\tilde\gamma'\in {\mathcal B}^{k+1}_j(C;q')$ such that $\tilde\gamma'\not\subset {\mathcal N}_{k+1}$. Now it is enough to take $\tilde\gamma$ the image of $\tilde\gamma'$ by $\pi_{k+1}$. In the case ii) we do a similar argumentation.
Also, if
$\Gamma\not\subset {\mathcal N}_k$ we have two possibilities:
\begin{enumerate}
\item [i)] $L'_i\subset {\mathcal N}_{k+1}$ and $L'_j\subset {\mathcal N}_{k+1}$.
\item [ii)] $L'_j\not\subset {\mathcal N}_{k+1}$ and $L'_i\not\subset {\mathcal N}_{k+1}$.
\end{enumerate}
By the same kind of argumentations we find $\tilde\gamma\in {\mathcal B}^k_j(C;p)$ with $\tilde\gamma\subset {\mathcal N}_k$.
The statements relative to the real saddles are consequence of Proposition \ref{pro:indicesymultiplicidad}.
\end{proof}
\section{Incompleteness of uninterrupted nodal components}
As explained in the Introduction, Theorem \ref{teo:mainteo} is a consequence of the following result:
\begin{theorem}
\label{teo:nodalcomponents} Any uninterrupted
nodal component $\mathcal N$ of ${\mathcal F}_N,E^N$ is incomplete.
\end{theorem}
In this section we provide a proof for Theorem \ref{teo:nodalcomponents}. We assume that ${\mathcal F}$ has no germ of invariant surface and that ${\mathcal N}$ is a {\em complete} uninterrupted nodal component. We shall find a contradiction with the fact that ${\mathcal N}$ is complete.
Let $b>0$ be the {\em date of birth of the compact part of }
$\mathcal N$, that is we assume that ${\mathcal N}_k\cap F_k$ is a single point for $0\leq k<b$ and that ${\mathcal N}_b$ contains at least one compact curve. Note that ${\mathcal N}$ contains at least one compact curve, because $\pi_1$ is the blow-up centered at the origin and hence the fiber $F=\pi^{-1}(0)$ is the union of the compact components of $E$. If we take a point $q\in {\mathcal N}\cap F$, the compact components of $E$ through $q$ are invariant, by the completeness of $\mathcal N$. If the dimensional type of $q$ is two, the singular locus of ${\mathcal F}_N$ coincides locally with ${\mathcal N}$ and it is contained in the invariant components of $E$ through $p$. If the dimensional type is three, we have two curves of $\mathcal N$, one of them is necessarily contained in a compact invariant component. As a consequence of this we find that $1\leq b\leq N$.
\begin{lemma}
\label{lema:nomonoidal}
The blow-up $\pi_b$ is not centered at a germ of curve.
\end{lemma}
\begin{proof}
Suppose by contradiction that $\pi_b$ is the (monoidal) blow-up centered at a germ of curve $(Y_{b-1},p)$. The point $p\in F_{b-1}$ is contained in a compact component $E^{b-1}_i$ of $E^b$ transversal to $Y_{b-1}$. Now, we have equireduction along $Y_{b-1}$. We consider all the blow-ups we do over $Y_{b-1}$ and we reach a desingularized situation over the point $p$. The fiber of $p$ contains a maximal connected union of compact curves in ${\mathcal N}$, say
$$
\Gamma_{j_1}\cup\Gamma_{j_2}\cup \cdots\cup \Gamma_{j_s}, \quad \Gamma_{j_\ell}\cap \Gamma_{j_{\ell+1}}\ne\emptyset, \; \ell=1,2,\ldots,s-1.
$$
Each $\Gamma_{j_\ell}$ is of the form $\Gamma_{j_\ell}=E_i\cap E_{j_\ell}$ where $E_{j_\ell}$ is non compact. Moreover, by the fact that $\mathcal N$ is uninterrupted, we have two possibilities:
\begin{enumerate}
\item The curves $\Gamma_{j_\ell}$ represent all the components of $E$ contained in the inverse image of $Y_{b-1}$.
\item There are two noncompact curves $\gamma_1=E_{j_1}\cap E_{j_0}$ and $\gamma_s= E_{j_s}\cap E_{j_{s+1}}$ such that $\gamma_1,\gamma_s\subset{\mathcal N}$ and none of the curves $E_{j_\ell}\cap E_{j_{\ell+1}}$ are in ${\mathcal N}$ for $\ell=1,2,\ldots, s-1$.
\end{enumerate}
Moreover the concerned divisors are non dicritical. Now, we can apply the refined Camacho-Sad Theorem \cite{Ort-B-V} to a transversal plane section at a generic point of $Y_{b-1}$ near $p$. In this way, we find a non compact trace curve of generic index not in ${\mathbb R}_{>0}$ that cuts one of the compact curves $\Gamma_{j_\ell}$. Since ${\mathcal N}$ is uninterrupted there is a compact trace curve of ${\mathcal N}$ contained in $E_i^{b-1}$, this is the desired contradiction.
\end{proof}
In view of Lemma \ref{lema:nomonoidal} we suppose that $\pi_b$ is a (quadratic) blow-up centered at the point $p$. We also know that $\pi_b$ is non dicritical, since there is a compact curve $\Gamma\subset E^b_b\cap {\mathcal N}_b$. Moreover, the point $p$ belongs only to compact components of $E^{b-1}$, otherwise the blow-up should be monoidal.
We consider separately the cases $b=1$ and $b>1$.
Assume that $b=1$ and consider the exceptional divisor $E^1_1=\pi_1^{-1}(0)$. The curve $\Gamma\subset {\mathcal N}_1\cap E^1_1$ is a compact trace curve and thus there is a partial separatrix $C=C_\Gamma$ with $\Gamma\subset C_1$. Consider the set $C_1\cap E^1_1$. In view of Proposition \ref{pro:globaltransitions}, any irreducible component $\Gamma'$ of $C_1\cap E^1_1$ satisfies $\Gamma'\subset{\mathcal N}_1$. Recall that $C$ is an incomplete partial separatrix in view of Proposition \ref{pro:incompletitudpartialsep} and thus there is at least a point $q\in C_1\cap E^1_1$ such that $q$ is incomplete for $C$ by Proposition \ref{pro:incompletodicriticoono}. Therefore, we have the following situation for $k=1$:
\begin{quote}
{\bf A($k$):} There is a compact trace curve $\Gamma\subset {\mathcal N}_k$ and a point $q\in \Gamma$ incomplete for the partial separatrix $C_\Gamma$.
\end{quote}
Assume now that $b>1$. In this case $p$ belongs to at least one compact component of $E^{b-1}$. Recall that all the components of $E^{b-1}$ through $p$ are compact. We discuss case by case.
(1). We have only one component $E^{b-1}_i$ of $E^{b-1}$ through $p$, which may be dicritical or invariant.
(1-a). Assume that $E^{b-1}_i$ is dicritical. We perform the blow-up $\pi_b$ and obtain a trace curve \break $\Gamma\subset {\mathcal N}_b\cap E^{b}_b$. By Proposition \ref{pro:tracetransitions}, the points in $\Gamma\cap E^{b}_i$ are incomplete for $C_\Gamma$. We arrive to situation {\bf A($k$)} for $k=b$.
(1-b). Let us suppose now that $E^{b-1}_i$ is invariant. Consider the projective line \break
$L=E^b_i\cap E^b_b$, then either $L\subset {\mathcal N}_b$ or not.
(1-b-1). Assume that $L\subset {\mathcal N}_b$. By taking a generic plane section at $p$ and by Camacho-Sad's argument on the sum of indices as in the proof of Proposition \ref{pro:notallrealsaddles}, we find a compact trace curve $\Theta\subset E^b_b$ such that the index of $\Theta$ is not in ${\mathbb R}_{> 0}$. We consider a point $q$ of intersection of $L$ and $\Theta$. If $q$ is a complete point for $C_\Theta$, by Proposition \ref{pro:tracetransitions} we should obtain a trace curve in $E^b_i$ contained in ${\mathcal N}_b$; this is not possible since $b$ is the date of birth of $\mathcal N$. Thus $q$ is an incomplete point. We obtain the following situation for $k=b$:
\begin{quote}
{\bf B($k$):} There are a compact curve $\Gamma\subset {\mathcal N}_k$ such that $\Gamma=E^k_i\cap E^k_j$ is the intersection of two invariant compact components and an incomplete point $q\in \Gamma$.
\end{quote}
(1-b-2).
Assume that $L\not\subset {\mathcal N}_b$. Then there is a trace curve $\Gamma\subset {\mathcal N}_b\cap E^b_b$. Consider a point $q\in \Gamma\cap L$. If $q$ is complete for $C_\Gamma$, we apply the trace transitions of Proposition \ref{pro:tracetransitions} and this contradicts the fact that $b$ is the date of birth of $\mathcal N$. Thus the point $q$ is incomplete for $C_\Gamma$ and we arrive to situation {\bf A($k$)} for $k=b$.
(2). There are two components $E^{b-1}_i, E^{b-1}_j$ of $E^{b-1}$ through $p$.
(2-a). If both components are dicritical, we do an argument as in case (1-a) to obtain {\bf A($k$)} for $k=b$.
(2-b). If $E^{b-1}_i$ in invariant and $E^{b-1}_j$ is dicritical, we have two possibilities:
(2-b-1). There is a trace curve $\Gamma\subset {\mathcal N}_b\cap E^b_b$. Take a point $q\in \Gamma\cap E^b_j$, in view of Proposition \ref{pro:tracetransitions}, the point $q$ must be incomplete for $C_\Gamma$. We obtain {\bf A($k$)} for $k=b$.
(2-b-2). The other case is that $L=E^b_i\cap E^b_b$ is contained in $\mathcal N$. By Proposition \ref{pro:tripetransition} this is not possible.
(2-c). Both $E^{b-1}_i$ and $E^{b-1}_j$ are invariant.
(2-c-1). If $E^{b}_i\cap E^b_b\subset {\mathcal N}_b$, by Proposition \ref{pro:tripetransition} we have that $E^{b}_j\cap E^b_b\subset {\mathcal N}_b$. By the already used argument on the sum of indices for a generic plane section at $p$, we find a trace curve $\Theta\subset E^b_b$ of index not in ${\mathbb R}_{>0}$. The trace transitions of $\Theta$ described in Proposition \ref{pro:tracetransitions} will produce curves of ${\mathcal N}_{b-1}$ previously existing in $E^{b-1}_i\cup E^{b-1}_j$, unless we have incomplete points in $E^{b}_i\cap E^{b}_b$ and $E^{b}_j\cap E^{b}_b$ at the intersections with $\Theta$. Thus we obtain {\bf B($k$)} for $k=b$.
(2-c-2). Assume now that $E^{b}_i\cap E^b_b\not\subset {\mathcal N}_b$. By Proposition \ref{pro:tripetransition} we also have that $E^{b}_j\cap E^b_b\not\subset {\mathcal N}_b$, since otherwise we should have $E^{b-1}_i\cap E^{b-1}_j\subset {\mathcal N}_{b-1}$. The other possible curves in $E^b_b$ are of trace type and thus the curve $\Gamma\subset {\mathcal N}_b$ that appears after $\pi_b$ is a trace curve. We obtain {\bf A($k$)} for $k=b$ as in (1-b-2).
(3). There are three components $E^{b-1}_i, E^{b-1}_j$ and $E^{b-1}_\ell$ of $E^{b-1}$ through $p$. We use the same kind of argumentation as in the cases (1) and (2) to reach one of the situations {\bf A($k$)} or {\bf B($k$)} for $k=b$.
\\
We have identified two situations {\bf A($k$)} and {\bf B($k$)} such that one of them appears in the birth level of the uninterrupted nodal component $\mathcal N$. We would like to show the persistency of this phenomenon at further levels of the reduction of singularities. However, another situation must be considered, which is the following:
\begin{quote}
{\bf C($k$):} There is a compact invariant component $E^{k}_i$, a compact curve $\Gamma\subset E^k_i\cap {\mathcal N}_k$ and an incomplete point $q\in \Gamma$ such that the following holds: every global irreducible curve $\Theta\subset\mbox{\rm Sing}{\mathcal F}_k\cap E^k_i$ with $q\in \Theta$ is either in ${\mathcal N}_k$ or a real saddle.
\end{quote}
\begin{proposition}[Persistency]
\label{prop:persistency}
Assume that there is an index $1\leq k<N$, a global curve $\Gamma\subset {\mathcal N}_k$ and an incomplete point $q\in \Gamma$ in one of the situations {\bf A($k$)}, {\bf B($k$)} or {\bf C($k$)}. Then there is a global curve $\Gamma'\subset {\mathcal N}_{k+1}$ and an incomplete point $q'\in \Gamma'$ in one of the situations {\bf A($k+1$)}, {\bf B($k+1$)} or {\bf C($k+1$)}.
\end{proposition}
\begin{proof} If $\pi_{k+1}$ is centered at $Y_k$ with $q\notin Y_k$, we obviously reach {\bf A($k+1$)}, {\bf B($k+1$)} or {\bf C($k+1$)} at the ``same'' point $q$. Thus, we assume $q\in Y_k$. Moreover, since $q$ is incomplete, we have $Y_k=\{q\}$. Indeed, if $Y_k$ is a germ of curve, the point $q$ is complete. Then $E^{k+1}_{k+1}=\pi_k^{-1}(q)$ is a projective plane.
(a). Assume that we have {\bf A($k$)}. Let $E^k_i$ be the compact invariant component such that $\Gamma\subset E^k_i$. We consider two cases:
(a-1). {\em The blow-up $\pi_{k+1}$ is dicritical}. We consider the strict transform $\Gamma'$ of $\Gamma$ and a point \break $q'\in \Gamma'\cap E^{k+1}_{k+1}$. In view of Proposition \ref{pro:tracetransitions} the point $q'$ must be incomplete for $C_{\Gamma}$ and we recover the situation {\bf A($k+1$)}.
(a-2). {\em The blow-up $\pi_{k+1}$ is non dicritical}. Let us put $L=E^{k+1}_{k+1}\cap E^{k+1}_i$.
(a-2-1). Assume first that $L\subset {\mathcal N}_{k+1}$. If there is an incomplete point $q'\in L$ we obtain {\bf B($k+1$)}. Thus we assume that all the points in $L$ are complete. We find an incomplete point $q'\in E^{k+1}_{k+1}$. By Proposition \ref{pro:notallrealsaddles}, there is a global irreducible curve $\Gamma'\subset {E^{k+1}_{k+1}}$ with $q'\in \Gamma'$ that is not a real saddle. We consider a (complete) point $p'\in L\cap \Gamma'$. Now, by Proposition \ref{pro:tracetransitions} or Proposition \ref{pro:tripetransition} we see that $\Gamma'$ must be contained in ${\mathcal N}_{k+1}$.
This argument also works for all non real saddle curves through $q'$. Hence we find {\bf C($k+1$)} or {\bf B($k+1$)} at $q'$.
(a-2-2).
It remains to consider the case that $L\not\subset {\mathcal N}_{k+1}$. If there is a point $q'\in L\cap \tilde \Gamma$ incomplete for $C_\Gamma$, where $\tilde\Gamma$ is the strict transform of $\Gamma$, we obtain {\bf A($k+1$)} at $q'$. If not, we consider the transitions given in Proposition \ref{pro:tracetransitions} to see that $C_\Gamma\cap E^{k+1}_{k+1}$ is contained in ${\mathcal N}_{k+1}$. Moreover, there exists a point $q'\in C_\Gamma\cap E^{k+1}_{k+1}$ incomplete for $C_\Gamma$. We recover {\bf A($k+1$)} at $q'$.
(b). Assume we have {\bf B($k$)}. Put $L_i=E^{k+1}_{k+1}\cap E^{k+1}_i$ and $L_j=E^{k+1}_{k+1}\cap E^{k+1}_j$. Let $p'=L_i\cap L_j$. By Proposition \ref{pro:tripetransition} we know that $\pi_{k+1}$ is a non dicritical blow-up. Moreover, we have that $$L_i\subset{\mathcal N}_{k+1} \Leftrightarrow L_j\not\subset{\mathcal N}_{k+1}.$$
To fix ideas, suppose that $L_i\subset{\mathcal N}_{k+1}$ and $L_j\not\subset{\mathcal N}_{k+1}$. If there is an incomplete point at $L_i$ we have {\bf B($k+1$)} at such a point. So we assume that all the points in $L_i$ are complete. This means that there is an incomplete point $q'\in E^{k+1}_{k+1}\setminus L_i$. We repeat at this point the previous argument for the case (a-2-1)
and we recover {\bf C($k+1$)} at $q'$.
(c). Let us assume finally that we have {\bf C($k$)}. We also suppose that we are not in the situations {\bf A($k$)} or {\bf B($k$)} already studied and hence $\Gamma$ is a trace curve and $q$ is complete for $C_\Gamma$.
As in case (b), we may assume that $\pi_{k+1}$ is non dicritical, otherwise we obtain {\bf A($k+1$)} at the strict transform of $\Gamma$. Let us put $L=E^{k+1}_{k+1}\cap E^{k+1}_i$.
(c-1). Assume first that $L\subset {\mathcal N}_{k+1}$. We may assume that all the points in $L$ are complete, otherwise we get {\bf B($k+1$)}. All the global irreducible curves $\Theta\subset E^{k+1}_{k+1}$ with $\Theta \ne L$ are either real saddles or curves in ${\mathcal N}_{k+1}$ in view of Propositions \ref{pro:tracetransitions} and \ref{pro:tripetransition}. On the other hand, we necessarily have an incomplete point $q'\in E^{k+1}_{k+1}$. The non real saddle passing through $q'$ given by Proposition \ref{pro:notallrealsaddles} is then contained in ${\mathcal N}_{k+1}$, as well as any other non real saddle curve. Thus, we recover {\bf C($k+1$)} at $q'$.
(c-2). Let us assume that $L\not\subset {\mathcal N}_{k+1}$. Let $\Gamma'\subset{\mathcal N}_{k+1}$ be the strict transform of $\Gamma$ and take a complete point $p\in \Gamma'\cap L$. By the transition rules in Proposition \ref{pro:tracetransitions} we obtain that $L$ is a real saddle. If there is an incomplete point $q'\in L$ we are done, since it satisfies {\bf A($k+1$)}. We suppose that all the points in $L$ are complete and we take an incomplete point $q'\in E^{k+1}_{k+1}\setminus L$. Let us see that all the global irreducible curves $\Theta\subset E^{k+1}_{k+1}\cap \mbox{\rm Sing}{\mathcal F}_{k+1}$ are either real saddles or contained in ${\mathcal N}_{k+1}$. In this way we obtain {\bf C($k+1$)} at $q'$ and we are done.
We look at the transitions through $L$ at $\Theta\cap L$ described in Corollary \ref{cor:transitioncompleta}. Recalling that the curves in $E^{k+1}_{i}\cap \mbox{\rm Sing}{\mathcal F}_{k+1}$ arriving at $L$ are either real saddle curves or in ${\mathcal N}_{k+1}$, we see that $\Theta$ is also in ${\mathcal N}_{k+1}$ or a real saddle curve.
\end{proof}
As a consequence of Proposition \ref{prop:persistency} we arrive to {\bf A,B} or {\bf C} in the final step, which is not possible since all the points in the final step are complete points. This is the desired contradiction. Thus, the only possibility is that there are no complete uninterrupted nodal components. This ends the proof of Theorem \ref{teo:nodalcomponents}.
|
3,212,635,537,447 | arxiv | \section{Introduction}
The overwhelming majority of stars in the Universe display
absorption lines in their visible range spectrum.
Emission lines in stellar spectra are therefore notable exceptions
which betray fascinating properties such as the presence of
unusually strong chromospheric activity, rapid rotation or in
the case of interest to this paper, strong mass loss and high
luminosity. Nearby giant H~{\sc ii} regions are hosts to an
interesting zoo of emission-line objects (Walborn \& Fitzpatrick
2000), most of them being post-main sequence massive stars
of notable interest for our understanding of stellar evolution
at the top end of the initial mass function. They also offer
important testbeds to understand the more distant,
unresolved starbursts: individual stars can be counted and
spectroscopically classified, allowing
a direct comparison with the modelisation of the spatially integrated
properties of their ionizing cluster (see Vacca et al. 1995
for such a comparison in 30 Doradus, or Bruhweiler, Miskey \&
Smith Neubig 2003 in NGC 604).
Among massive stars with emission lines, those of Wolf-Rayet (WR)
type are the easiest to detect and
classify because of their strong and broad emission lines in the
visible part of the spectrum. Thanks to surveys in Local Group
galaxies (Massey \& Johnson 1998)
and improvements in theoretical models (Meynet \& Maeder 2005),
the evolutionary status of WR stars is now understood well enough
to use them as diagnostics to infer the properties of starburst
regions. For instance, in unresolved clusters or starburst knots
of distant galaxies, the equivalent width of the
`WR-bumps' are good indicators of the age and upper mass limit of the
stellar population (Pindao et al. 2002).
The small spiral galaxy M33 is host to four giant H{\sc ii} regions
bright enough to have their own NGC number: NGC 604, the second most
luminous starburst cluster in the Local Group; then, in decrasing
order of H$\alpha$ luminosity, NGC 595, NGC 592 and NGC 588. Despite their different galactocentric distances, these four regions have very similar metallicities, with $12 +$ log O/H $= 8.4 - 8.5$ (Magrini et al. 2007).
Two papers published in the fall of 1981 presented
the spectroscopic discovery of WR stars in these
clusters: D'Odorico \& Rosa (1981) derived a
surprisingly large (50) number of WR stars in NGC 604,
while Conti \& Massey
(1981) noticed that some WR stars in the four regions
were excedingly luminous. Both studies however suffered from a
lack of spatial resolution. More WR candidates were identified
by interference filter imagery (taking advantage of their strong
HeII emission) and spectroscopically confirmed by
Massey \& Conti (1983), Armandroff \& Massey (1985, 1991) and Massey,
Conti \& Armandroff (1987). The most detailed catalogue of WR
stars (with spectral classification) and WR candidates in
M33 is presented in Massey \& Johnson (1998).
Drissen, Moffat \& Shara (1990, 1993) identified more WR candidates based on high
resolution CCD images with interference filters; but until now however,
none of these were spectroscopically confirmed.
In the first paper (Abbott et al. 2004) of this series dedicated
to WR stars in M33, we presented new spectra of one Of, 14 WN,
one transition-type WN/WC and 26 WC stars in the field of M33.
In this second paper, we present spectra of known WR stars and
most of the WR candidates in the giant HII regions NGC 604, NGC
595 and NGC 592.
\section{Observations}
The spectroscopic data were obtained with the
Multi-Object Spectrograph (MOS) attached
to the 3.6-m Canada-France-Hawaii Telescope (CFHT) in 2000
September and 2001 October. Prior to spectroscopy, a high
quality image of each region was obtained in order to identify
the targets and position the slitlets with sub-acrsecond
accuracy. The seeing during both observing runs ranged from 0.7$''$ to 1.0$''$, except when the observations of NGC 592 were obtained (1.5$''$ at high air mass; see below). After the spectra were
obtained, a superposition of these images with the resulting 2D
spectral images ensured a correct {\it a-posteriori} identification of the stars.
We used the B600 grating, which,
combined with the slit width of 1.5 arcsec, provided a spectral
resolution of $\sim 9$ \AA .
Exposure times were 900s for NGC 592, 2700s for NGC 604 and 2700s
for NGC 595. In the case of NGC 592, the observations were
obtained at the end of the night with a long and wide (5$''$)
slit, which reduced the spectral resolution.
The data were then reduced using standard procedures in IRAF.
To complement these spectroscopic observations, we have used archival images obtained with the
Wide Field and Planetary Camera 2 (WFPC2) and the Advanced Camera for Surveys (ACS) on board the {\it Hubble Space Telescope} to
correctly identify emission-line stars in all regions.Table 1 lists
the names and properties of the imaging datasets that we have used
for this research. The data were extracted from the Canadian Astronomy Data Centre's web interface.
\begin{table*}
\begin{minipage}{170mm}
\caption{Archival HST images used in this study \label{tbl01}}
\begin{tabular}{@{}lccc@{}}
\hline
Region & Program ID & Dataset & Exp. time \\
Filter & & (CADC) & (sec) \\
\hline
NGC 588: & & & \\
F170W & 5384 & U2C60701B & 700 \\
F336W & " & U2C60703B & 320 \\
F439W & " & U2C60801B & 360 \\
F547M & " & U2C60803B & 200 \\
F469N & " & U2C60805B & 600 \\
& & & \\
NGC 592: & & & \\
F170W & 9127 & U6DJ010DB & 1560 \\
F255W & " & U6DJ0109B & 1440 \\
F336W & " & U6DJ0107B & 520 \\
F439W & " & U6DJ0103B & 600 \\
F555W & " & U6DJ0101B & 320 \\
& & & \\
NGC 595: & & & \\
F170W & 5384 & U2C60901B & 700 \\
F336W & " & U2C60903B & 320 \\
F439W & " & U2C60A01B & 360 \\
F547M & " & U2C60A03B & 200 \\
& & & \\
NGC 604: & & & \\
F336W & 5237 & U2AB0207B & 1200 \\
F555W & " & U2AB0201B & 400 \\
F547M & 5773 & U2LX0307B & 1000 \\
F656N & " & U2LX0301B & 2200 \\
F673N & " & U2LX0303B & 2200 \\
F220W & 10419 & J96Y11010 & 600 \\
F250W & " & J96Y11020 & 800 \\
\hline
\end{tabular}
\end{minipage}
\end{table*}
\section{Results}
Spectrograms of Of and WN stars in the spectral range of the strongest emission lines are shown in Figure 1, while those of the three WC stars are shown in Figure 2. Table~\ref{tbl02} summarizes the photometric and spectroscopic properties of all the emission-line stars detected in these regions, from this and from previous work.
The equivalent widths listed in this table as EW (WR) refer to the entire emission lines in the ''WR bump''
between 4630 and 4700 \AA\ .
\begin{table*}
\centering
\begin{minipage}{170mm}
\caption{Properties of emission-line stars in M33's giant HII regions.\label{tbl02}}
\begin{tabular}{@{}lcccccrrll@{}}
\hline
Star\footnote{Newly confirmed emission-line stars appear in bold characters.} & RA & Dec & B & M$_{B}$\footnote{Indicative only: the absolute magnitude was determined assuming a distance of 850 kpc (distance modulus = 24.6) and a uniform extinction A$_B$ = 0.5 mag.} & EW(WR)\footnote{Equivalent width, in \AA\ , of the emission lines in the 4600 - 4700 \AA\ region.} &Sp. (old) & Sp (new) & Alt. names\footnote{CM: Conti \& Massey 1981; MC: Massey \& Conti 1983; MCA: Massey, Conti \& Armandroff 1987; MJ: Massey \& Johnson 1998; AM: Armandroff \& Massey 1985, 1991.}\\
\hline
N604 - WR1 & 01:34:32.37 & +30:47:00.9 & 17.7 & -7.4 & ----- &WCE & ----- & CM11, MC74, MJ-WR135 \\
N604 - WR2 & 01:34:32.50 & +30:47:00.3 & 17.1 & -8.0 & ----- &WN & ----- & CM11, MC74, MJ-WR135 \\
N604 - WR3 & 01:34:32.69 & +30:47:05.4 & 17.8 & -7.3 & ----- &WN & ----- & CM12, MJ-WR136 \\
N604 - WR4 & 01:34:32.65 & +30:47:07.1 & 17.1 & -8.0 & ----- &WN & ----- & CM12, MJ-WR136\\
N604 - WR5 & 01:34:32.83 & +30:47:04.6 & 19.0 & -6.1 & 450 &WC & WC6 & CM12 \\
N604 - WR6 & 01:34:33.68 & +30:47:05.8 & 17.9 & -7.2 & 35 &WN & WNL & CM13 \\
N604 - {\bf WR7} & 01:34:33.87 & +30:46:57.6 & 20.1 & -5.0 & 160 &WR? & WC4 & \\
N604 - {\bf WR8} & 01:34:32.17 & +30:47:07.0 & 18.5 & -6.6 & 18 &WR? & WN6 & \\
N604 - {\bf WR10} & 01:34:32.74 & +30:46:56.5 & 18.5 & -6.6 & 18 &WR? & WN6 & \\
N604 - {\bf WR11} & 01:34:34.06 & +30:46:56.2 & 20.7 & -4.4 & 6 &WR? & WNE & \\
N604 - {\bf WR12} & 01:34:32.55 & +30:47:04.4 & 18.1 & -7.0 & 15 &WR? & WN10 & \\
N604 - {\bf WR13} & 01:34:35.15 & +30:47:05.8 & ----- & ----- & 5 &----- & O6.5Iaf & \\
N604 - {\bf V1} & 01:34:32.30 & +30:47:03.9 & 17.8 & -7.3 & 13 &WR? & Of/WNL & \\
& & & & & & & & & \\
N595 - WR1 & 01:33:34.22 & +30:41:38.1 & 18.3 & -6.8 & 47 &WNL & WN7 & CM6, MC32, AM6, MJ-WR49 \\
N595 - WR2a & 01:33:33.76 & +30:41:34.0 & 19.7 & -5.4 & ----- &WNL? & ----- & CM5, MC31, MJ-WR47 \\
N595 - WR2b & 01:33:33.72 & +30:41:34.2 & 18.2 & -6.9 & ----- &WNL? & ----- & CM5, MC31, MJ-WR47 \\
N595 - WR3 & 01:33:33.81 & +30:41:29.8 & 20.5 & -4.6 & 300 &WC & WC6 & MC29, AM5 \\
N595 - WR4 & 01:33:32.95 & +30:41:36.2 & 18.1 & -7.0 & ----- &WN & ----- & AM4, MJ-WR43 \\
N595 - WR5 & 01:33:32.80 & +30:41:46.2 & 18.2 & -6.9 & 80 &WNL & WN7h & AM3, MJ-WR42 \\
N595 - WR6 & 01:33:32.61 & +30:41:27.3 & 19.1 & -6.0 & ----- &WN & ----- & MC28, AM2, MJ-WR41 \\
N595 - WR7 & 01:33:34.28 & +30:41:30.5 & 20.3 & -4.8 & ----- & WN & ----- & AM7,MJ-WR51 \\
N595 - WR8 & 01:33:34.02 & +30:41:17.2 & 20.3 & -4.8 & ----- &WN & ----- & MCA4, MJ-WR48 \\
N595 - {\bf WR9} & 01:33:33.28 & +30:41:29.8 & 20.5 & -4.6 & 27 &WR? & WN7 & \\
& & & & & & & & & \\
N592 - WR1 & 01:33:11.81 & +30:38:53.1 & 18.1 & -7.0 & ----- &WN & ----- & CM3,MC17,MJ-WR25 \\
N592 - {\bf WR2} & 01:33:12.42 & +30:38:48.4 & 19.7 & -5.4& 25 &WR? & WN4 & \\
N592 - MC19 & 01:33:15.00 & +30:38:07.4 & 19.8 & -5.3 & ----- &WCE & ----- & MJ-WR30 \\
N592 - MCA2 & 01:33:10.70 & +30:39:00.6 & 20.8 & -4.3 & ----- &WN & ----- & MJ-WR24 \\
& & & & & & & & & \\
N588 - UIT008 & 01:32:45.33 & +30:38:58.4 & 17.6 & -7.5 & ----- &Ofpe/WN9 & ----- & MJ-WR5, J1 \\
N588 - MC3 & 01:32:45.66 & +30:38:54.4 & 18.1 & -7.0 & ----- &WN & ----- & CM1, MC3,MJ-WR6,J2\\
\hline
\end{tabular}
\end{minipage}
\end{table*}
\begin{figure}
\includegraphics[width=84mm]{fig1.eps}
\caption{Rectified spectra of Of and WN stars in our sample, in the 4600 \AA\ region. We should note that much of the strong, narrow emission line at 4471 \AA\ in the spectra of N604-WR8 and WR10 is of nebular origin.}
\end{figure}
\begin{figure}
\includegraphics[width=84mm]{fig2.ps}
\caption{Rectified spectra of WC stars in our sample, in the wavelength ranges of the most prominent lines.}
\end{figure}
\begin{figure}
\includegraphics[width=84mm]{fig3.eps}
\caption{Finding chart for emission-line stars in NGC 604, from an ultraviolet HST/ACS image (combination of F220W and F250W filters). Star WR13 is outside this field of view, and is shown in Figure 6. The field of view is 23$'' \times 20''$ (80 pc $\times 70$ pc), with North to the top and East to the left.}
\end{figure}
\subsection{NGC 604}
Drissen, Moffat \& Shara (1993; hereafter DMS93) presented finding charts for the W-R candidates in NGC 604, from aberrated
WF/PC images. We update this chart in Figure 3 with a much better quality image obtained from a series of archival HST/ACS images (filters F220W and F250W).
Bruhweiler et al. (2003) were able to derive the spectral type of
40 stars in the central region of NGC 604 based on ultraviolet
spectral imagery with HST's Imaging Spectrograph (STIS). UV
spectroscopic classification is not hampered by the very strong
nebular emission lines present in the visible region; this
survey is therefore the most complete stellar census so far
in this cluster. We now describe the spectra of the newly observed stars.
WR5 was first identified as a WR by Conti \& Massey (1981), but
their spectrogram also included neighboring WR4 and WR3; it was
thus classified as a WN star. However, Drissen, Moffat \& Shara (1990) tentatively classified it as WC based on their narrow-band photometry.
Our spectrogram (Figure 2) is unambiguous:
with very strong C~{\sc iv} lines at $\lambda$ 4650 and 5808
\AA\ , and weaker but still conspicuous lines of C~{\sc iii} $\lambda$ 5696
and O~{\sc iii}-{\sc v} $\lambda$ 5590, WR5 is a WC6 star,
according to the classification criteria of Crowther et al.
(1998).
One of the most intriguing stars in the sample is WR6, which
shows broad Balmer and HeI lines unlike most typical late-type WN stars.
An excellent spectrogram of this star, obtained nine years before
ours, is shown by Terlevich et al. (1996; see their Figure 7).
These authors call attention to the strong spectroscopic
variability of WR6's emission lines between 1980 and 1992. Our
spectrogram (Figure 4) is virtually identical to the one shown by Terlevich
et al., including line widths and He / H line ratios; this suggests a (temporary?) stagnation in the rapid spectroscopic evolution of this star observed recently. We must note that our spectrogram includes the strong continuuum from the star 0.6$''$ to the west of WR6, but our 2D spectrogram clearly shows that all the broad emission lines come
from the faintest of the pair.
Two WR stars, WR7 and WR11, are located in an area with high extinction,
according to the extinction map of Ma\'{\i}z-Apell{\'a}niz (2004; compare his Figure 6 with
our Figure 3). Their continuum and emission-line flux was therefore found to
be low in the previous imaging work. But the spectrum of WR7 is unambiguous
with very strong lines of C {\sc iv} at 4650 \AA\ and 5812 \AA\ but no evidence of
C {\sc iii} 5696, indicating a WC4
classification. WR11, first noticed as an emission-line star by DMS93, is not correctly identified in the original finding chart (their Figure 6). A careful superposition of the 1991 emission-line image from the Canada-France-Hawaii telescope, which was originaly used to identify WR candidates, with the more recent HST/ACS images, as well as an analysis of the location of the emission line in our more recent 2D spectra, clearly identifies WR11 as the brightest star in a very tight group of a half-dozen stars separated by less than 2$''$; its correct identification is now shown in Figure 3. The only emission feature visible in the spectrum of WR11 is a broad (FWHM = 37 \AA\ ) He {\sc ii} 4686 line. Because the group of stars in which WR11 stands is unresolved in ground-based spectrum, the true equivalent width of the line is certainly
much higher, making WR11 a genuine WN star. However, we
find no evidence for He {\sc ii} 5411 nor C {\sc iv} 5800 in its
spectrum.
WR8 and WR10 were first detected in the CFHT images of DMS90, and Figure 1 shows
that their spectra are very similar. With well-resolved emission lines of He{\sc ii} 4686
(FWHM = 15 \AA ), N {\sc iii} 4634, 40 and C {\sc iv} 5808, as well as absorption features of N{\sc v} 4604 and 4620 \AA , the spectrum of these two stars is very similar to that of
the Galactic WN6 star HD 93162 (WR25; see Figure 1 in Walborn \& Fitzpatrick 2000).
WR12 was identified as having a weak He II excess by DMS93
in their ground-based CFHT images, but not in their pre-CoStar
HST images because of the low S/N ratio. WFPC2 ultraviolet
(F170W filter) images shown by Bruhweiler et al. (2003)
clearly separate WR12 into two stars separated by less than
0.3$''$, labeled 690A (the faintest of the pair, classified as O5 III according to STIS UV
spectra) and 690B (B0 Ib).
Based on this UV classification, one
should not expect to see He II 4686 in emission in either star (Walborn \&
Fitzpatrick 1990). Our slit
includes both stars, and the resulting spectrum (Figures 1 and 5) clearly shows a
broad emission bump (total W$_e$ = 15 \AA )
which includes the He II line (W$_e$ = 5
\AA\ confirming the early diagnostic based on imagery) accompanied
by He{\sc i} 4713, N{\sc iii} 4634,40 and possibly C{\sc iv} 4650. We also detect
strong P Cygni profiles of He {\sc i} at 4388, 4471 and 4921 \AA\
as well as an absorption line of He {\sc ii} 5411;
a careful examination of the 2D spectroscopic
images along the trace of WR12 clearly shows that these
lines are of stellar origin and are not significantly contaminated by nebular
emission. The more extended spectrogram of WR12 shown in
Figure 5 bears striking similarities with that of the WN10 star Sk$-66^{o}40$. Although our optical spectrum is a composite
of two stars, it is more likely that most of the emission lines
come from the brightest one (690B).
\begin{figure}
\includegraphics[width=84mm]{fig4.eps}
\caption{Spectrum of NGC 604-WR6. The emission lines are diluted by the continuum from the bright star
0.6$''$ to the west of WR6. }
\end{figure}
\begin{figure}
\includegraphics[width=84mm]{fig5.eps}
\caption{Spectrum of NGC 604-WR12.}
\end{figure}
To confirm this, we have
analyzed WFPC2 images (program 5773) obtained with the nebular filters F656N
(H$\alpha$) and F673N (centered on the [SII] $\lambda\lambda$ 6717,31
doublet). Since stellar spectra do not show the presence of these
sulfur lines, and since the two nebular filters are close enough
in terms of wavelength (reducing possible extinction effects),
we have used the F673N image as a continuum reference (Figure 6) and subtracted it from
the H$\alpha$ image. Selected fields from the resulting
continuum-subtracted image are shown in Figures 7 and 8. Figure 7 demonstrates the power of this
technique on a field centered on NGC 604-WR6, which is known to be
a strong H$\alpha$ emitter (see Figure 4). Figure 8 shows the field around WR12. In addition to WR12, He {\sc ii} emission-line stars WR3, WR8 and V1 clearly show up as H$\alpha$ emitters. A comparison of the upper and lower panels of Figure 8
also confirms that, as expected, star 690B is a strong H$\alpha$ emitter whereas 690A does not
show up.
\begin{figure}
\includegraphics[width=84mm]{fig6.eps}
\caption{WFPC2 F673N image of NGC 604, showing the location of WR13 and the two regions enlarged in Figures 7 and 8. North is at the top, east to the left ($30'' \times 30''$).}
\end{figure}
\begin{figure}
\includegraphics[width=84mm]{fig7.eps}
\caption{WFPC2 images of the region surrounding NGC 604-WR6 in "continuum" (upper panel, F673N filter)
and in ``pure H$\alpha$" (lower panel; F656N - F673N). The field of view is 3.1$'' \times 2.4''$, with North up and East to the left.}
\end{figure}
\begin{figure}
\includegraphics[width=84mm]{fig8.eps}
\caption{WFPC2 images of the region surrounding NGC 604-WR3, WR12, WR8 and V1 in "continuum" (upper panel, F673N filter) and in ``pure H$\alpha$'' (F656N - F673N). Note the elongation of WR12 in the upper panel, corresponding to stars 690B (lower, brighter component: WR12 itself) and 690A (fainter, upper component) in
Bruhweiler et al. (2003). Only star 690B shows an excess of H$\alpha$. The field of view is $9.3'' \times 7.5''$, with North up and East to the left.}
\end{figure}
\begin{figure}
\includegraphics[width=84mm]{fig9.eps}
\caption{Comparison between the 1992 and 2001 spectra of NGC 604 - V1.}
\end{figure}
N604-V1 was found to be photometrically variable (on a timescale of one night, possibly eclipsing?) by
Drissen, Moffat \& Shara (1990). DMS93 allude to a spectrogram, obtained in October 1992, showing He {\sc ii} in emission, but did not
publish it. The more recent spectrogram (October 2001) shows this line (FWHM = 14 \AA ),
as well as N {\sc iii} 4640 in emission, absorption lines of He {\sc ii} 4541 and 5411,
and P Cygni profiles for the He {\sc i} lines
at 4471 and 5876 \AA . We also detect a weak (W$_{e}$ = 1.0 \AA ) emission line of C {\sc iii} $\lambda$ 5696.
A comparison between spectra obtained in
1992 and 2001 shows a significant weakening (by a factor of 2.5) of the He
{\sc ii} 4686 line (Figure 9). The 1992 spectrum did not show the He {\sc ii} absorption lines (perhaps filled
with emission), nor the C {\sc iii} $\lambda$ 5696 emission.
Finally, WR13 was never identified as a WR candidate but since it is the brightest star
inside a prominent ionized arc in the northeastern part of the nebula (see Figure 6), we took advantage of the
MOS capabilities and superposed a slitlet on it to satisfy our curiosity.
Its spectrum shows very weak emission lines of He{\sc ii}
4686 (W$_e$ = 1.2 \AA ) and N{\sc iii} 4640 (W$_e$ = 3.3 \AA ),
as well as absorption lines of He{\sc ii} 4541 and 5411 and
He{\sc i} 4471. A comparison with stars in the atlas of Walborn
\& Fitzpatrick (1990) suggests a spectral type O6.5 Iaf, similar
to the galactic star HD 163758.
\subsection{NGC 595}
\begin{figure}
\includegraphics[width=84mm]{fig10.eps}
\caption{Finding chart for emission-line stars in NGC 595 from HST/WFPC2 images (combination of F336W, F439W and F547M). The field of view is $30'' \times 35''$, with North at the top and East to the left. }
\end{figure}
An updated finding chart for W-R stars in NGC 595 is shown in Figure 10.
Among all W-R candidates in NGC 595, only WR2 (components a and b), WR9 and WR11 lacked a clear, individual, spectroscopic identification, althought spectrograms including both WR2ab and WR11 (both objects, separated by less than 2$''$, are themselves multiple as shown by WFPC2 images) as well as WR9 clearly show the presence of at least one WNL star (Conti \& Massey 1981, Massey et al. 1996). But a re-examination of the HST/WFPC images discussed by DMS93 shows that the excess of light at 4686 \AA\ at the location of the candidate W-R star WR11 is marginal at best, and no emission excess could be detected for WR10. Indeed, Massey \& Johnson (1998) question the presence of emission lines in WR10.
Moreover, Royer et al. (2003) unambiguously detect WR2 and WR9 as W-R candidates in their images obtained with a set of narrow-band filters, but also fail to detect WR10 and WR11. So we decided to remove both objects from our list of W-R candidates and therefore they do not appear in Figure 10.
The only new spectroscopic confirmation from this paper is that of NGC 595-WR9. Its spectrum (Figure 1) is comparable to that of the Galactic star WR22 (HD 92740; see Figure 1 in Walborn \& Fitzpatrick 2000), with W$_{e}$ = 18 \AA\ for He {\sc ii} 4686 and 9 \AA\ for N {\sc iii} 4634, 40. We also detect an emission line of He {\sc ii} 5411 (W$_{e}$ = 4.5 \AA ).
The first spectrogram of WR3 (MC29), published by Massey \& Conti (1983), allowed a crude WC classification. Line ratios (C {\sc iv} 5808, C {\sc iii} 5696 and O {\sc iii,v} 5590) in our spectrogram (Figure 2) allows us to refine the classification to WC6. It is interesting to note that WR3 is the only WC star in NGC 595.
NGC 595-WR1 and NGC 595-WR5 have very similar spectra in terms of line ratio, and both can be classified as WN7. The equivalent width of all lines are larger by a factor of two for WR5; this could be due, at least in part, to some contamination by the continuum of the bright star 1.3$''$ south of WR1. NGC595-WR5 is in a region of relatively weak nebular emission, and we were able to subtract the nebular component from its spectrum, which is shown in its entirety in Figure 11. The relative strengths of the pure helium lines (4200 \AA , 4541 \AA , and 5411 \AA ) to those for which helium and hydrogen both contribute (4340 \AA\ and 4860 \AA ) clearly indicate the presence of hydrogen in the atmosphere of NGC 595-WR5.
\begin{figure}
\includegraphics[width=84mm]{fig11.eps}
\caption{Spectrum of NGC 595-WR5. The (weak) surrounding nebular component has been subtracted, leaving only stellar lines.}
\end{figure}
\subsection{NGC 592 and NGC 588}
\begin{figure}
\includegraphics[width=84mm]{fig12.eps}
\caption{Composite visible image of NGC 592 from HST/WFPC2}
\end{figure}
NGC 592 and NGC 588 are much less luminous and rich in WR stars than NGC 595 and NGC 604, by factors of 3 to 10, but they nevertheless include a fair population of young, massive stars.
We defer the analysis of their stellar content, based on HST archives, to another paper (\'Ubeda, Drissen \& Crowther, in preparation), but we present here a spectrum and some images relevant to their emission-line star content.
NGC 592 is ten times less luminous
than NGC 604, both in terms of H$\alpha$ luminosity
(Bosch et al. 2002) and in UV continuum (as measured
with the large FUSE aperture at 1150~\AA; see Pellerin 2006),
and until now very little is known about its stellar content.
It includes four WR stars: two in the core (identified as WR1 and WR2
following DMS90), and two more in the outskirts: MC19 and MCA2. Their positions are labeled in a composite
visible image in Figure 12 as WR1, WR2, MC19 and MCA2.
Despite the factor ten smaller in H$\alpha$ luminosity compared to NGC 604, Figure 12 shows that the ionizing cluster of NGC 592 still contains a fair number of massive stars.
A spectrogram of NGC~592--WR1 (WNL)
is shown in Conti \& Massey (1981, CM3).
NGC~592--WR1 (also known as MC17,
or WR25 in Massey \& Johnson 1998) was originally misidentified with the brighter
but slightly redder companion 1$''$ to the north-east
(star B67 in Humphreys \& Sandage 1980), but it was already
strongly suggested by the images presented in Drissen et al. (1990) that the WR star
was not the brightest component of the tight pair; it is
now obvious in the WFPC2 images. NGC~592--WR1 is also the cluster's brightest UV source
in the F170W image.
NGC~592--WR2, first identified as a WR candidate by Drissen et al. (1990), lacked spectral confirmation
until now. Our spectrogram of this star is
shown in Figure 13, and is that of a WN4 star as deduced from the weakness of
the He~{\sc i} 5876 line and the absence of N~{\sc iii} 4640. A comparison of our spectrum
with those of a dozen Galactic WNE stars shows that the red spectrum of NGC~592--WR2
most closely resembles that of the Galactic WN4 star WR44,
analysed by Hamann \& Koesterke (1998).
A careful
comparison of the WFPC2 image and the CFHT 2D spectra clearly
shows that WR2 is the brightest of a close pair separated by
0.6$''$.
The spectrum of MC19 (WCE) was first published by
Massey \& Conti (1983), while MCA2 is a WN star whose spectrum has been published by Massey, Conti \& Armandroff (1987).
\begin{figure}
\includegraphics[width=84mm]{fig13.eps}
\caption{Full spectrogram of NGC 592-WR2, compared with that of
the Galactic WN4 star WR44 (LSS 2289). Note that all nebular
lines have been removed from the spectrogram of NGC 592 - WR2.}
\end{figure}
NGC 588 was not part of the imaging survey by Drissen, Moffat \& Shara (1990).
However, two emission-line stars are known in this cluster: MC3, a WNL, detected with narrow-band imagery by Conti \& Massey (1981) and spectroscopically confirmed by Massey \& Conti (1983); and UIT-008 , a transition Of/WN9 star (see Massey \& Johnson 1998 for a visible spectrogram and Bianchi, Bohlin \& Massey 2004 for a UV spectrogram) selected for follow-up spectroscopy because of its high UV luminosity. NGC 588 is, like the other giant HII regions discussed in this paper, a crowded place and high-resolution images are required to properly identify the stars. Fortunately, NGC 588 was imaged with WFPC2 in many bands in 1994 (Figure 14), including F469N,
a narrowband filter centered on the He{\sc ii} 4686 emission line. Only MC3 clearly stands out in the F469N image;
it is 0.5 mag brighter than in the F439W image. UIT-008 is marginally brighter in the F469N image than in the continuum F439W, by 0.1 mag; this is consistent with the weakness of its emission lines.
\begin{figure}
\includegraphics[width=84mm]{fig14.eps}
\caption{Composite visible image of NGC 588 from HST/WFPC2.}
\end{figure}
\section{Conclusions}
We have spectroscopically confirmed the presence of emission lines in a sample of WR candidates selected by interference filter imagery. Most of them are genuine WR stars, but we have also detected transition-type evolved O stars, with an He II equivalent width as low as 5 \AA\ . As demontrated here and in previous publications, this technique is very efficient, especially in crowded fields (see also Hadfield \& Crowther 2007), with a very high success rate and relatively few false detections. The census of the WR population in NGC 588, NGC 592, NGC 595 and NGC 604 is now essentially complete: only three stars (NGC 604-WR2, NGC 604-WR4 and NGC 595-WR2), which are members of very dense and barely resolved groups, lack a clear identification although their WR nature is not in doubt.
We have also obtained good quality spectrograms of previously known WR stars in these regions, allowing a better spectral type identification. The four giant HII region studied here harbour about 20\% of the entire known Wolf-Rayet population of M33, with a WC/WN ratio significantly lower (0.25) than that of the field (0.4; Massey \& Johnson 1998).
\section*{Acknowledgements}
We thank the referee for a carefull reading of the manuscipt and suggestions.
This paper is based on observations obtained at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii.
LD is grateful to the Canada Research Chair program, the Natural Sciences and Engineering Research Council of Canada, and the Fonds Qu\'eb\'ecois de la Recherche sur la Nature et les Technologies (FQRNT) for financial support. LU acknowledges FQRNT for a postdoctoral fellowship.
PAC wishes to thank the Royal Society for providing financial assistance through their wonderful University Research Fellowship scheme for the past eight years.
|
3,212,635,537,448 | arxiv | \section{Introduction}
Stochastic models describing evolution of certain arrays of particles or biological species (genes, bacteria, infected individuals) are of theoretical interest and also important for various applications, \cite{New_14}, \cite{Bansaye_15}, \cite{Mel_16}, \cite{Pardoux_16}, \cite{KV_17}. Investigation of time-dependence of population size in probabilistic terms ascends to introduction of the classical Galton-Watson branching process in 1874, see \cite{Jagers_11} for historical overview. A rigorous analysis of the spread of a new dominant gene can be traced back to the well-known paper by \cite{KPP_37}. Later a plenty of models involving reproduction, death, and movement of particles have appeared. The probabilistic ones include the so-called branching random walks (BRW) and are linked with other models in mathematics, physics, and informatics. Most of BRW models are space-homogeneous as in \cite{Lifshits_14}, \cite{Shi_LNM_15}, \cite{Mallein_16}, and references therein. We consider a non-homogeneous process, called \emph{catalytic branching random walk} (CBRW), defined for any time $t\geq 0$ on a multidimensional lattice $\mathbb{Z}^d$, $d\in\mathbb{N}$. Particles give offspring or die at specified locations called \emph{catalysts}. The catalysts take fixed positions on $\mathbb{Z}^d$. Their number is arbitrary finite. Outside the catalysts the particles move randomly until hitting a catalyst.
So CBRW models incorporate two stochastic mechanisms: the particles randomly move in space and, moreover, each of them could give
a random number of offsprings in the presence of catalysts only. These issues are discussed further in detail for the populations
initiated by a single specimen. Our main goal is to examine the time-evolution of the moving front, separating, in a sense,
the once populated area and its empty environment. Whenever the locations of particles are normalized in appropriate manner by
continuous in time functions, the rescaled front is a surface in $\mathbb{R}^d$. We study the spread of population for $t\to\infty$
when the distributions of particles jumps have heavy tails. The exact limiting front surface in $\mathbb{R}^d$, called the limiting shape of the front, is identified with probability one. The proofs combine analysis of non-linear integral equations, multidimensional renewal theorems, the Laplace transform, large deviation theory for heavy-tailed distributions, the coupling method, and other probabilistic-analytic techniques.
Likewise the classical branching processes (\cite{Sev_74}), CBRW is classified as supercritical, critical, and subcritical depending
on the characteristics of the reproduction and of the random walk. In the most general framework the classification is given
by \cite{B_TPA_15} by means of the Perron eigenvalue of a certain matrix. The particles population survives globally and locally
with positive probability if and only if CBRW is supercritical, \cite{B_Doklady_15}. Exponential growth of the total and local particles numbers occurs only in supercritical CBRW as established in \cite{B_TPA_15} and \cite{B_Doklady_15}. This is the reason to consider the rate of population spread just for supercritical CBRW.
\cite{Carmona_Hu_14} study for CBRW on $\mathbb{Z}$ the strong (that is almost sure) limit behavior of the maximum, being the location of the right-most particle on $\mathbb{Z}$, under assumption of light distribution tails of the random walk. \cite{B_SPA_18} extends analysis to CBRW on the lattice of arbitrary dimension. Until now the investigation of the maximum of CBRW with heavy distribution tails of the random walk has been tackled only in \cite{B_Arxiv_18}, where the distribution tail is a regularly varying function. It follows from \cite{Carmona_Hu_14} that the unrescaled population front propagates linearly in time in case of light distribution tails whereas according to \cite{B_Arxiv_18} it spreads exponentially fast in case of regularly varying tails.
We focus here on a novel assumption for CBRW models that the distribution tails of the random walk are semi-exponential. Such distribution belongs to a fundamental class of heavy-tailed distributions including the Weibull one (\cite{Borovkov_Borovkov_08}, Ch.~5). It follows that the non-normalized population front on $\mathbb{Z}^d$ for CBRW, with independent components of the random walk jump distributed semi-exponentially, propagates in a power way and faster than a linear in time function.
An equation of the form $H({\bf z})=\nu$, ${\bf z}\in \mathbb{R}^d$, is obtained, describing the surface of the normalized front limit (limiting shape of the front) for the model under consideration, where $\nu$ is a Maltusian parameter, $H$ is an explicit function depending only on ${\bf z}$ and parameters of the semi-exponential distributions of the jumps components. The growth rate of the normalizing factors also depends exceptionally on these parameters.
Surprisingly, in the case of semi-exponential tails the limiting front shape is a surface of non-convex set which contrasts with a convex one in the case of light tails. Our results for CBRW on $\mathbb{Z}$ agree with those for \emph{homogeneous} branching random walk on the real line proved in \cite{Gantert_00}. One can refer to \cite{Maillard_16}, \cite{GMV_2017},
and others on progress in studying the spread of the population for homogeneous branching random walk with regularly varying jump distribution tails ``heavier'' than the semi-exponential ones.
We found only the paper \cite{Gantert_00} devoted to homogeneous branching random walk on the real
line with semi-exponential increments. There are no investigations of such model in a
multidimensional case. Possibly, it is explained by the absence of quite general results
on large deviations of vectors with semi-exponential distribution. In its turn, according
to \cite{Borovkov_Borovkov_08}, p.~400, the latter is a hard problem.
It seems that our work opens the study of multidimensional BRW models with semi-exponential jumps distributions.
Observe that CBRW captures the nature of the intermittency phenomenon for random walk in random potential (\cite{Konig_16}, p.10): the main contribution to the total population size is due to few small remote islands, called intermittent islands. In CBRW the counterparts of the intermittent islands are the catalysts points. Investigations in mathematical theory of intermittency and applications to magnetic and temperature fields of turbulent flows, chemical kinetics, hydrodynamics, and biological models are performed in \cite{Gartner_Molchanov_90}, \cite{Zeldovich_14}, \cite{Ortgiese_Roberts_16}, \cite{Chernousova_Molchanov_18}.
We refer to the works by \cite{AB_00}, \cite{Molchanov_Yarovaya_12}, \cite{HTV_12}, \cite{Topchii_Vatutin_13}, \cite{DR_13}, and \cite{Platonova_Ryadovkin_17}, and also to references therein, for analysis of other aspects of CBRW or its modifications. Most of them are devoted to long-time behavior of total and local particles numbers. The exception is \cite{Molchanov_Yarovaya_12}, where the population front of symmetric CBRW with binary splitting and light-tailed increments was defined and studied from the viewpoint of moments boundedness of local particles numbers. Remarkably, the notions of the propagation front in \cite{Molchanov_Yarovaya_12} and \cite{B_SPA_18} are different but lead to the same growth rate. However, our approach seems more powerful due to the strong convergence results under milder restrictions on CBRW.
In Section~\ref{s:main_results_semiexp} we introduce necessary notation and formulate the main results in multidimensional setting.
Theorems~\ref{T:main_result_semiexp_lattice} and \ref{T:one_point_semiexp} characterize the front propagation almost surely.
In Section~\ref{s:proof_2_semiexp} we provide the proofs of these theorems. To simplify exposition we establish 6 lemmas and partition the proofs into several steps. At first we consider the case of a single catalyst and then extend the obtained results to the case of an arbitrary finite number of catalysts. Illustrative examples are gathered in Section~\ref{s:examples_semiexp}.
Section Conclusion completes the exposition.
\section{Model Description and Main Results}
\label{s:main_results_semiexp}
All random elements are defined on a complete probability space $(\Omega,\mathcal{F},{\sf P})$, letter $\omega$ stands
for a point of $\Omega$. The index ${\bf z}$ in expressions of the form ${\sf E}_{\bf z}$ and ${\sf P}_{\bf z}$ marks the starting point of either CBRW or the random walk ${\bf S}$, depending on the context. Bold font of ${\bf z}$ emphasizes that ${\bf z}$ is a multidimensional vector whereas the symbol $z$ means that $z$ is a real number.
Recall the description of CBRW on $\mathbb{Z}^d$, $d\in\mathbb{N}$ (in our setting given in \cite{B_SPA_18}). At the initial time ${t=0}$ there is a single particle that moves on $\mathbb{Z}^d$ according to a continuous-time Markov chain ${\bf S}=\{{\bf S}(t),t\geq0\}$ generated by the infinitesimal matrix ${Q=(q({\bf x},{\bf y}))_{{\bf x},{\bf y}\in\mathbb{Z}^d}}$. Assume that ${\bf S}$ is irreducible and space-homogeneous, with the conservative matrix $Q$, that is,
$Q$ has finite elements and
\begin{equation}\label{condition1}
q({\bf x},{\bf y})=q({\bf x}-{\bf y},{\bf 0})
\quad\mbox{and}\quad
\sum\limits_{{\bf y}\in\mathbb{Z}^d}{q({\bf x},{\bf y})}=0,
\end{equation}
where $q({\bf x},{\bf y})\geq0$, for ${\bf x}\neq{\bf y}$, and $q:=-q({\bf x},{\bf x})\in(0,\infty)$, for any ${\bf x},{\bf y}\in\mathbb{Z}^d$. Stress that, contrary to Platonova and Ryadovkin (2017) and Yarovaya (2017), we do not restrict ourselves to the case of symmetric generator $Q$.
When this particle hits a finite set of catalysts $W=\{{\bf w}_1,\ldots,{\bf w}_N\}\subset\mathbb{Z}^d$, say at the point ${\bf w}_k$, it spends there random time, distributed exponentially with parameter $\beta_k>0$. The particle either branches there with probability $\alpha_k$ or leaves the point ${\bf w}_k$ with probability $1-\alpha_k$ ($0\leq\alpha_k<1$). If the particle branches, it produces a random non-negative integer number $\xi_k$ of offsprings, located at the same point ${\bf w}_k$, and dies instantly. Whenever the particle leaves ${\bf w}_k$, it jumps to the point ${\bf y}\neq{\bf w}_k$ with probability $-(1-\alpha_k)q({\bf w}_k,{\bf y})q({\bf w}_k,{\bf w}_k)^{-1}$ and resumes its motion governed by the Markov chain ${\bf S}$. All the newly born particles are supposed to behave as independent copies of their parent.
Denote by $f_k(s):={\sf E}{s^{\xi_k}}$, $s\in[0,1]$, the probability generating function of $\xi_k$, $k=1,\ldots,N$. Employ the standard assumption of existence of a finite derivative $f_k'(1)$, that is the finiteness of $m_k:={\sf E}{\xi_k}$, for any $k=1,\ldots,N$.
We forget for a while about the catalysts and consider only the motion of a particle on $\mathbb{Z}^d$ according to the Markov chain ${\bf S}$ with the generator $Q$ and the starting point ${\bf x}$. The conditions imposed on the elements $q({\bf x},{\bf y})$, ${\bf x},{\bf y}\in\mathbb{Z}^d$, allow us to use an explicit construction of the random walk on $\mathbb{Z}^d$ with generator $Q$ by Theorem~1.2 in \cite{Bremaud_99}, Ch.~9, Sec.~1. Whence ${\bf S}$ is a regular jump process with right continuous trajectories and, for transition times of the process, $\tau^{(0)}:=0$ and $\tau^{(n)}:=\inf\left\{t\geq\tau^{(n-1)}:{\bf S}(t)\neq {\bf S}(\tau^{(n-1)})\right\}$, $n\ge1$, the following property is valid. The random variables $\left\{\tau^{(n+1)}-\tau^{(n)}\right\}_{n=0}^{\infty}$ are independent and each of them has exponential distribution with parameter $q$. Denote by $\Pi=\{\Pi(t),t\geq0\}$ the Poisson process constructed as the renewal process with the interarrival times $\tau^{(n+1)}-\tau^{(n)}$, $n\in\mathbb{Z}_+$, (\cite{Feller_71}, Ch.~1, Sec.~4), that is, $\Pi$ is a Poisson process with constant intensity $q$. Let ${\bf Y}^i=\left(Y^i_1,\ldots,Y^i_d\right)$ be the value of the $i$th jump of the random walk ${\bf S}$ ($i=1,2,\ldots$). In view of Theorem~1.2 in \cite{Bremaud_99}, Ch.~9, Sec.~1, the random vectors ${\bf Y}^1,{\bf Y}^2,\ldots$ are i.i.d., have distribution ${\sf P}({\bf Y}^1={\bf y})=q({\bf 0},{\bf y})/q$, ${\bf y}\in\mathbb{Z}^d$, ${\bf y}\neq{\bf 0}$, and do not depend on the sequence $\{\tau^{(n+1)}-\tau^{(n)}\}_{n=0}^{\infty}$. We can write (for a version of ${\bf S}$)
\begin{equation}\label{S(t)=representation}
{\bf S}(t)={\bf x}+\sum_{i=1}^{\Pi(t)}{\bf Y}^i,
\end{equation}
where ${\bf x}$ is the initial state of the Markov chain ${\bf S}$ and $\sum_{i\in\varnothing}{\bf Y}^i:={\bf 0}$. Equality~(\ref{S(t)=representation}) implies that ${\bf S}$ is a process with independent increments. In what follows we consider the version of the process ${\bf S}$ constructed in such a way. It is called a compound Poisson process.
We employ various stopping times (with respect to the natural filtration of ${\bf S}$).
For ${\bf x}\in \mathbb{Z}^d$, set
$$\tau_{\bf x}:=\mathbb{I}({\bf S}(0)={\bf x})\inf\{t\geq0:{\bf S}(t)\neq{\bf x}\},$$
that is, the stopping time $\tau_{\bf x}$ is the time of the first exit from the starting point ${\bf x}$ of the random walk. As usual,
$\mathbb{I}(A)$ stands for the indicator of a set $A\in\mathcal{F}$. Clearly, ${\sf P}_{\bf x}(\tau_{\bf x}\leq t)=1-e^{-qt}=:G_0(t)$, ${\bf x}\in\mathbb{Z}^d$, $t\geq0$. Let
$$_T\overline{\tau}_{{\bf x},{\bf y}}:=\mathbb{I}({\bf S}(0)={\bf x})\inf\{t\geq0:{\bf S}(t+\tau_{\bf x})={\bf y},{\bf S}(u)\notin T,\tau_{\bf x}\leq u< t+\tau_{\bf x}\}$$
be the time elapsed from the exit moment of this Markov chain (in other terms, particle) from the starting point
${\bf x}$ till the moment of the first hitting point ${\bf y}$, whenever the particle trajectory does not pass the set $T\subset\mathbb{Z}^d$. If there is no such finite $t$, we put ${_T\overline{\tau}_{{\bf x},{\bf y}}=\infty}$. An extended random variable $_T\overline{\tau}_{{\bf x},{\bf y}}$ is called \emph{hitting time} of state ${\bf y}$ \emph{under taboo} on set $T$ after exit out of starting state ${\bf x}$ (\cite{Chung_60}, Ch.~2, Sec.~11, and \cite{B_SPL_14}). Denote by $_T\overline{F}_{{\bf x},{\bf y}}(t)$, $t\geq0$, the improper c.d.f. of this extended random variable and let $_T\overline{F}_{{\bf x},{\bf y}}(\infty):=\lim_{t\to\infty}{_T\overline{F}_{{\bf x},{\bf y}}(t)}$. If the taboo set $T$ is empty, expressions $_{\varnothing}\overline{\tau}_{{\bf x},{\bf y}}$ and $_{\varnothing}\overline{F}_{{\bf x},{\bf y}}$ are shortened as $\overline{\tau}_{{\bf x},{\bf y}}$ and $\overline{F}_{{\bf x},{\bf y}}$. Mainly we will be interested in the situation when $T=W_k$, where $W_k:=W\setminus\{{\bf w}_k\}$, $k=1,\ldots,N$.
We also use some auxiliary functions. Here and further let
$$F^{\ast}(\lambda):=\int\nolimits_{0-}^{\infty}{ e^{-\lambda t}\,d{F(t)}},\quad\lambda\geq0,$$
denote the Laplace transform of a c.d.f. $F(t)$, $t\geq0$, with support on non-negative semi-axis. For $j,k=1,\ldots,N$, ${\bf x},{\bf y}\in\mathbb{Z}^d$, and $t\geq0$, set
\begin{equation}\label{G-i,G-ij=}
G_j(t):=1-e^{-\beta_j t},\quad G_{j,k}(t):=G_j\ast{_{W_k}\overline{F}_{{\bf w}_j,{\bf w}_k}(t)},\quad{_T F_{{\bf x},{\bf y}}(t)}:=G_0\ast{_T\overline{F}_{{\bf x},{\bf y}}(t)},
\end{equation}
where $\ast$ between c.d.f. stands for their convolution. By definition the function ${_T F_{{\bf x},{\bf y}}(\cdot)}$ is a c.d.f. of the variable $_T\tau_{{\bf x},{\bf y}}:=\tau_{\bf x}+{_T\overline{\tau}_{{\bf x},{\bf y}}}$ called \emph{hitting time} of state ${\bf y}$ \emph{under taboo} on set $T$ when the starting state is ${\bf x}$.
As in \cite{B_TPA_15}, consider a matrix function $D(\lambda)=\left(d_{i,j}(\lambda)\right)_{i,j=1}^N$, $\lambda\geq0$, taking values in the set of irreducible matrices of size $N\times N$, with elements defined by way of
$$d_{i,j}(\lambda)=\delta_{i,j}\alpha_im_iG^{\ast}_i(\lambda)+(1-\alpha_i)G^{\ast}_{i,j}(\lambda),$$ where $\delta_{i,j}$ is the Kronecker delta. According to Definition~$1$ in \cite{B_TPA_15} CBRW is called {\it supercritical} if the Perron root (that is, positive eigenvalue being the spectral radius) $\rho(D(0))$ of the matrix $D(0)$ is greater than $1$. In view of monotonicity of
all elements of the matrix function $D(\cdot)$ there exists the solution $\nu>0$ of the equation $\rho(D(\lambda))=1$. On account of Theorem $1$ in \cite{B_TPA_15} just this positive number $\nu$ specifies the rate of exponential growth of the mean total and local particles numbers (in the literature devoted to population dynamics and classical branching processes one traditionally speaks of {\it Malthusian parameter}). In the sequel we consider the supercritical CBRW on $\mathbb{Z}^d$.
Let $N(t)\subset\mathbb{Z}^d$ be the (random) set of particles existing in CBRW at time ${t\geq0}$. For a particle $v\in N(t)$, denote by ${\bf X}^v(t)=\left(X^v_1(t),\ldots,X^v_d(t)\right)$ its position at time $t$. Consider the set
$$
\mathcal{I}:=\left\{\omega:\limsup_{t\to\infty}\{v\in N(t):{\bf X}^v(t)\in W\}\neq\varnothing\right\}\in\mathcal{F}.
$$
To avoid operations with a continuum number of sets $\left\{A_t\right\}_{t\geq0}$, we just put $\limsup_{t\to\infty}A_t:=\cap_{m=1}^{\infty}\cup_{k=1}^{\infty}\cap_{n=k}^{\infty}A_{n/2^m}$, that is, we deal with the binary rational values of the parameter $t$ only instead of its all non-negative values. For each $\omega\in\mathcal{I}$, there is an increasing to infinity sequence of \emph{binary rational} values $t_l(\omega)$, $l\in\mathbb{N}$, such that at each time $t_l(\omega)$ there are particles at $W$. The event consisting of $\omega$ for which there exists a similar sequence of \emph{any} (not only binary rational) values $t_l(\omega)$ is of the same probability ${\sf P}(\mathcal{I})$, and we may call $\mathcal{I}$ {\it the event of infinite number of visits of catalysts}. The behavior of CBRW on the set complement $\overline{\mathcal{I}}$ is a.s. trivial. Indeed, for $t\geq t_0(\omega)$ large enough either CBRW dies out or CBRW constitutes the system of some random walks (without branching) starting respectively from ${\bf X}^v(\omega,t_0)$, $v\in N(t_0)$, at time $t_0$. The supercritical regime of CBRW guarantees that ${\sf P}(\mathcal{I})>0$ (Theorem~4 of \cite{B_Doklady_15}).
Assumptions made are of the same type as in the previous papers \cite{Carmona_Hu_14} (with discrete time on $\mathbb{Z}$), \cite{B_SPA_18}, and \cite{B_Arxiv_18} devoted to the study of the population spread in CBRW.
The following hypothesis corresponds to the new case under consideration.
Let the components of the random walk jumps be semi-exponentially distributed,
that is, for any $i=1,\ldots,d$ and $y\in\mathbb{Z}_+$, one has by \cite{Borovkov_Borovkov_08}, p.~29,
\begin{eqnarray}
{\sf P}(Y^1_i>y)&=&L^{(1,+)}_i(y)\exp\left\{-y^{\gamma^{+}_i}L^{(2,+)}_i(y)\right\}:=R^{+}_i(y),\label{assumption:tails_right_semiexp}\\
{\sf P}(Y^1_i<-y)&=&L^{(1,-)}_i(y)\exp\left\{-y^{\gamma^{-}_i}L^{(2,-)}_i(y)\right\}:=R^{-}_i(y).\label{assumption:tails_left_semiexp}
\end{eqnarray}
Symbol ``$+$'' marks the right distribution tail whereas symbol ``$-$'' refers to the left one. For each $i=1,\ldots,d$ and $\kappa\in\{+,-\}$, functions $L^{(1,\kappa)}_i(y)$ and $L^{(2,\kappa)}_i(y)$, ${y\in\mathbb{Z}_+}$, are slowly varying, while parameters $\gamma^{\kappa}_i$ are taken from $(0,1)$. Recall that
$$
{\sf P}\left(Y^1_i>y\right)={q^{-1}\sum_{{\bf x}:\,x_i>y}{q({\bf 0},{\bf x})}},
$$
where ${\bf x}=\left(x_1,\ldots,x_d\right)\in\mathbb{Z}^d$.
It follows from (\ref{assumption:tails_right_semiexp}) and (\ref{assumption:tails_left_semiexp}) that $-\ln R^{\kappa}_i(y)$, $y\in\mathbb{Z}_+$, is a regularly varying function of index $\gamma^{\kappa}_i$. In accordance with \cite{Seneta_76}, Ch.~1, Sec.~5, property $5^\circ$, there exists an asymptotically uniquely determined inverse function $R^{-1,\,\kappa}_i(s)$, $s\geq0$, in the sense that $-\ln{R^{\,\kappa}_i\left(R^{-1,\,\kappa}_i(y)\right)}\sim y$, $R^{-1,\,\kappa}_i\left(-\ln{R^{\,\kappa}_i(y)}\right)\sim y$, as $y\to\infty$, $y\in\mathbb{Z}_+$, and
$$
R^{-1,\kappa}_i(s)=s^{1/\gamma^{\kappa}_i}L^{(3,\kappa)}_i(s),
$$
where $L^{(3,\kappa)}_i(s)$, $s\geq0$, is a slowly varying function at infinity.
Functions $R^{-1,\kappa}_i(\cdot)$, $i=1,\ldots,d$, play an important role in normalization of
coordinates of particles of $N(t)$. Emphasize that we have to use notation involving
$\kappa$ since the normalization of a particle position, in general, depends on
the orthant (one among $2^d$) of $\mathbb{R}^d$ containing this particle.
Unlike the random walks with either light or regularly varying distribution tails, a diversity of large deviations zones is
inherent in case of walks with semi-exponential increments. They are \emph{Cr\'{a}mer deviation} zone, \emph{intermediate} zone, and \emph{maximum jump approximation} zone, \cite{Borovkov_Borovkov_08}, p.~238. We deal with two latter ones. Writing in a compact form, assume that, for each fixed ${\bf x}=\left(x_1,\ldots,x_d\right)\in\mathbb{R}^d$, ${\bf x}\neq{\bf 0}$, one has
\begin{eqnarray}\label{assumptions:large_deviations_tails_right}
& &{\sf P}_{\bf 0}\left(\sgn({\bf x}){\bf S}(u)/{\bf R}^{-1,\kappa({\bf x})}(t)\in\left[\left|{\bf x}\right|,+\infty\right)\right)\nonumber\\
&=&h(u)\left(1+\delta(u,t)\right)\prod_{i=1}^d\left({\sf P}\left(\sgn(x_i)Y_i\geq|x_i| R^{-1,\kappa(x_i)}_i(t)\right)\right)^{\left(1+\varepsilon_i(t)\right)},
\end{eqnarray}
where $h(u)$, $u\geq0$, is a positive non-decreasing function such that $h(u)\sim c u^d$, $u\to\infty$, for
a constant $c>0$, the functions $\varepsilon_i(t)=\varepsilon_i(t,{\bf x})\to0$, as $t\to\infty$, and $\delta(u,t)=\delta(u,t,{\bf x})\to0$, as $u,t\to\infty$, $u\leq t$, $i=1,\ldots,d$. In relation (\ref{assumptions:large_deviations_tails_right}) the notation $\sgn({\bf x}){\bf S}(u)/{\bf R}^{-1,\kappa({\bf x})}(t)$ means the vector in $\mathbb{R}^d$ with $i$th component $\sgn(x_i)S_i(t)/R^{-1,\kappa(x_i)}_i(t)$, $i=1,\ldots,d$, and $[|{\bf x}|,+\infty):=[|x_1|,+\infty)\times\ldots\times[|x_d|,+\infty)$. Here $\kappa(x_i)=$``$+$'' if $x_i\geq0$ and $\kappa(x_i)=$``$-$'' if $x_i<0$. As a precaution, we put $\sgn(x_i)S_i(t)/R^{-1,\kappa(x_i)}_i(t):=0$ whenever $R^{-1,\kappa(x_i)}_i(t)=0$. Recall that $\sgn(y)=1$, for $y>0$, and $\sgn(y)=-1$, for $y<0$, whereas $\sgn(0)=0$. Provided that the components of the random walk jumps are independent, broad sufficient conditions for the validity of (\ref{assumptions:large_deviations_tails_right}) are gathered, e.g., in Theorem~5.4.1~(i), (ii) of \cite{Borovkov_Borovkov_08}.
Define the following sets in $\mathbb{R}^d$
\begin{equation}\label{def_O_semiexp}
\mathcal{O}_{\varepsilon}:=\left\{{\bf x}\in\mathbb{R}^d:\sum_{i=1}^d\left|x_i\right|^{\gamma^{\kappa(x_i)}_i}>\nu+\varepsilon\right\},\quad\varepsilon\geq0,\quad\mathcal{O}:=\mathcal{O}_{0},
\end{equation}
\begin{equation}\label{def_Q_semiexp}
\mathcal{Q}_{\varepsilon}:=\left\{{\bf x}\in\mathbb{R}^d:\sum_{i=1}^d\left|x_i\right|^{\gamma^{\kappa(x_i)}_i}<\nu-\varepsilon\right\},\;
\varepsilon\in[0,\nu),\;\mathcal{Q}:=\mathcal{Q}_0,
\end{equation}
\begin{equation}\label{def_P_semiexp}
\mathcal{P}:=\partial\mathcal{O}=\partial\mathcal{Q}=\left\{{\bf x}\in\mathbb{R}^d:\sum_{i=1}^d\left|x_i\right|^{\gamma^{\kappa(x_i)}_i}=\nu\right\},
\end{equation}
where $\partial\mathcal{U}$ stands for the boundary of a set $\mathcal{U}$. Stipulate that ${\bf X}^v(u)/{\bf R}^{-1,\kappa}(t)$ is a vector in $\mathbb{R}^d$ with $i$th coordinate equal to $X^v_i(u)/R^{-1,\kappa(X^v_i(u))}_i(t)$, $u,t\geq0$, $i=1,\ldots,d$.
\begin{Thm}\label{T:main_result_semiexp_lattice}
Let assumptions (\ref{condition1}), (\ref{assumption:tails_right_semiexp})-(\ref{assumptions:large_deviations_tails_right}) be satisfied for supercritical CBRW on $\mathbb{Z}^d$ with Malthusian parameter $\nu$. Then, for each starting point ${\bf z}\in\mathbb{Z}^d$, the following relations are valid.
\begin{equation}\label{T:assertion_1_semiexp}
{\sf P}_{\bf z}\!\left(\omega:\forall\varepsilon>0\;\exists t_1=t_1(\omega,\varepsilon)\;\mbox{s.t.}\;\forall t\geq t_1\;\mbox{and}\;\forall v\in N(t),\;{\bf X}^v(t)/{\bf R}^{-1,\kappa}(t)\notin\mathcal{O}_{\varepsilon}\right)=1,
\end{equation}
\begin{equation}\label{T:assertion_2_semiexp}
{\sf P}_{\bf z}\!\left(\left.\omega:\!\forall\varepsilon\!\in(0,\nu)\,\exists t_2=t_2(\omega,\varepsilon)\;\mbox{s.t.}\;\forall t\geq t_2\;\exists v\in N(t),\;{\bf X}^v(t)/{\bf R}^{-1,\kappa}(t)\notin\mathcal{Q}_{\varepsilon}\right|\mathcal{I}\right)=1,
\end{equation}
where the sets $\mathcal{O}_{\varepsilon}$ and $\mathcal{Q}_{\varepsilon}$ are defined in formulas (\ref{def_O_semiexp}) and (\ref{def_Q_semiexp}).
\end{Thm}
{\bf Remark 1}.
Theorem~\ref{T:main_result_semiexp_lattice} means that, for almost all $\omega$, for any time large enough and any $\varepsilon>0$, there are no particles with properly normalized positions outside the surface $\partial\mathcal{O}_{\varepsilon}$ and, for almost all $\omega\in\mathcal{I}$, there are always such particles outside the surface $\partial\mathcal{Q}_{\varepsilon}$. In other words, for $\omega\in\mathcal{I}$, the most distant particles (``front'' of the population spread) after normalization are located for any time large enough in the interlayer between $\partial\mathcal{O}_{\varepsilon}$ and $\partial\mathcal{Q}_{\varepsilon}$ with $\varepsilon$ small enough. For almost all $\omega\notin\mathcal{I}$, the limit of the normalized particles positions is trivial, that is, equals ${\bf 0}$ (in \cite{B_Doklady_15} one can find necessary and sufficient conditions to guarantee that ${\sf P}(\mathcal{I})=1$). It is natural to call \emph{the limiting shape of the front} the surface $\mathcal{P}=\partial\mathcal{O}=\partial\mathcal{Q}$. Equivalently one can reformulate Theorem~\ref{T:main_result_semiexp_lattice} describing the neighborhoods of $\mathcal{P}$ and $\mathcal{Q}$ in terms of Euclidean distances (instead of taking $\mathcal{O}_{\varepsilon}$ and $\mathcal{Q}_{\varepsilon}$ for $\varepsilon >0$).
\vskip0.2cm
The next result asserts that each point of $\mathcal{P}$ can be considered as a limiting point for the normalized particles positions in CBRW, that is, the surface $\mathcal{P}$ is minimal in a sense.
\begin{Thm}\label{T:one_point_semiexp}
Let conditions of Theorem~\ref{T:main_result_semiexp_lattice} be satisfied. Then, for each ${\bf z}\in\mathbb{Z}^d$ and ${\bf y}\in\mathcal{P}$, one has
$${\sf P}_{\bf z}\left(\left.\omega:\forall t\geq0\;\exists v_{\bf y}=v_{\bf y}(t,\omega)\in N(t)\;\mbox{such that}\;\lim_{t\to\infty}\frac{{\bf X}^{v_{\bf y}}(t)}{{\bf R}^{-1,\kappa}(t)}={\bf y}\right|\mathcal{I}\right)=1.$$
\end{Thm}
{\bf Remark 2}.
Stress that we dot not give a rigorous definition of the front
(but only limiting shape of the front), since it can vary
being dependent on the studied features of the cloud $N(t)$
and basic assumptions. For example, in our framework we could write the definition
of the front as follows. The front of the population propagation is a cloud of particles at time
$t$ such that upon the normalization of particles positions as in
Theorem~\ref{T:main_result_semiexp_lattice} and letting $t\to\infty$,
the almost sure limit points of the particles from the cloud form the surface
$\mathcal{P}$ in (\ref{def_P_semiexp}) called the limiting shape of the front.
However, in other framework such as \cite{B_Arxiv_18} there is a \emph{random limit}
of the properly normalized positions of the most distant (from the origin) particles
in the sense of \emph{weak} convergence. So, the possible mentioned definition of the front
is not suitable for \cite{B_Arxiv_18}. Whenever we talk about the propagation
front for CBRW on $\mathbb{Z}^d$ we mean the generalization to the multidimensional case
of the maximum and the minimum bounding the population on an integer line.
Nevertheless, an attractive possible definition of the front as the set of particles at time $t$
being the most distant on each ray from the origin seems also inconvenient since
continuum of rays will not contain any particle.
\section{Proofs}\label{s:proof_2_semiexp}
To establish Theorems \ref{T:main_result_semiexp_lattice} and \ref{T:one_point_semiexp} we derive a system of non-linear integral equations and estimate its solution, use renewal theory, Laplace transform, coupling, theory of large deviations for the random walk with semi-exponential distributions of jumps, and other probabilistic and analytic technique.
Divide the proof of Theorem~\ref{T:main_result_semiexp_lattice} into 5 Steps. Within Steps~1,~2, and 3 we consider the case of a single catalyst ${\bf w}_1$ located, without loss of generality, at the origin, that is $W=\{{\bf w}_1\}$ with ${\bf w}_1={\bf 0}$, and the starting point of CBRW being ${\bf 0}$. Within steps 4 and 5 we turn to the general case. In fact, proving
Theorem~\ref{T:main_result_semiexp_lattice} we simultaneously obtain the statement of Theorem~\ref{T:one_point_semiexp}.
\vskip0.2cm
\emph{Step 1.} At this stage we prove statement (\ref{T:assertion_1_semiexp}) in the case of a single catalyst located at ${\bf 0}$ and the starting point ${\bf 0}$ as well.
Let $E(t;\mathcal{U}):={\sf P}_{\bf 0}\left(\exists v\in N(t):{\bf X}^v(t)\in\mathcal{U}\right)$ for a set $\mathcal{U}\subset\mathbb{R}^d$. The following result is a multidimensional counterpart of Lemma~1 in \cite{B_Arxiv_18}
providing an integral equation for the probability $E(t;\mathcal{U})$.
\begin{Lm}\label{L:equation_multi}
Let condition (\ref{condition1}) be valid. Then the probability $E(t;\mathcal{U})$, $t\geq0$, $\mathcal{U}\subset\mathbb{R}^d$, ${\bf 0}\notin\mathcal{U}$, satisfies the non-linear integral equation of convolution type
$$
E(t;\mathcal{U})=\alpha_1\int\nolimits_0^t{ \left(1-f_1\left(1-E(t-s;\mathcal{U})\right)\right)\,dG_1(s)}
$$
\begin{equation}\label{E(t;S)_equation}
+(1-\alpha_1)\int\nolimits_0^t{ E(t-s;\mathcal{U})\,dG_{1,1}(s)}+I\left(t;\mathcal{U}\right),
\end{equation}
where we set
$$I(t;\mathcal{U}):=\sum_{{\bf y}\neq{\bf 0}}{(1-\alpha_1)\frac{q({\bf 0},{\bf y})}{q}\int\nolimits_0^t{ {\sf P}_{\bf y}\left({\bf S}(t-s)\in\mathcal{U},\tau_{{\bf y},{\bf 0}}>t-s\right)\,dG_1(s)}}.$$
\end{Lm}
{\sc Proof.} Similar to the proof of Lemma~1 in \cite{B_Arxiv_18}, consider all the possible evolutions of the parent particle in CBRW. Namely, after time, distributed exponentially with parameter $\beta_1$, it may either produce $k\in\mathbb{Z}_+$ offsprings with probability $\alpha_1{\sf P}(\xi_1=k)$, or jump to the point ${\bf y}\neq{\bf 0}$ with probability $(1-\alpha_1)q({\bf 0},{\bf y})/q$ and first return to the origin in time $\tau_{{\bf y},{\bf 0}}$. If the parent particle does not return to the origin until time $t$, it performs an ordinary random walk ${\bf S}$ starting from ${\bf y}$. At last, it might occur that the parent particle has not undergone changes by time $t$. Summarizing all the above and taking into account (\ref{G-i,G-ij=}), we can write the following formula, for any $\mathcal{U}\subset\mathbb{R}^d$, ${\bf 0}\notin\mathcal{U}$,
\begin{eqnarray*}
1-E(t;\mathcal{U})&=&\alpha_1\sum_{k=0}^{\infty}{\sf P}(\xi_1=k)\int\nolimits_0^t{ \left(1-E(t-s;\mathcal{U})\right)^k\,dG_1(s)}+(1-G_1(t))\\
&+&\sum_{{\bf y}\neq{\bf 0}}{(1-\alpha_1)\frac{q({\bf 0},{\bf y})}{q}\int\nolimits_0^t{ \left(1-E(t-s;\mathcal{U})\right)\,d\left(G_1\ast F_{{\bf y},{\bf 0}}(s)\right)}}\\
&+&\sum_{{\bf y}\neq{\bf 0}}{(1-\alpha_1)\frac{q({\bf 0},{\bf y})}{q}\int\nolimits_0^t{ {\sf P}_{\bf y}\left({\bf S}(t-s)\notin\mathcal{U},\,\tau_{{\bf y},{\bf 0}}>t-s\right)\,dG_1(s)}}.
\end{eqnarray*}
Rewriting the latter equation with respect to unknown function $E(t;\mathcal{U})$ and taking into account the obvious identity
$${\sf P}_{\bf y}\left({\bf S}(s)\in\mathcal{U},\tau_{{\bf y},{\bf 0}}>s\right)=1-F_{{\bf y},{\bf 0}}(s)-{\sf P}_{\bf y}\left({\bf S}(s)\notin\mathcal{U},\tau_{{\bf y},{\bf 0}}>s\right),\quad s\geq0,$$
we get the desired assertion.
Lemma~\ref{L:equation_multi} is proved.
The next lemma provides a convenient form for the function $I$ expressed in terms of the probability ${\sf P}_{\bf 0}\left({\bf S}(t)\in\mathcal{U}\right)$ when ${\bf 0}\notin\mathcal{U}$. Its proof follows the proof of Lemma~2 in \cite{B_Arxiv_18} and is omitted here.
\begin{Lm}\label{L:J-1(t;a)=_semiexp}
Let condition (\ref{condition1}) be satisfied. Then, for any $t\geq0$ and $\mathcal{U}\subset\mathbb{R}^d$, ${\bf 0}\notin\mathcal{U}$, the following identity holds true
\begin{eqnarray}\label{J-1(t;a)=_semiexp}
\frac{q I(t;\mathcal{U})}{(1-\alpha_1)\beta_1}&=&{\sf P}_{\bf 0}\left({\bf S}(t)\in\mathcal{U}\right)-\int\nolimits_0^t{ {\sf P}_{\bf 0}\left({\bf S}(t-s)\in\mathcal{U}\right)\,d F_{{\bf 0},{\bf 0}}(s)}\\
&-&\frac{\beta_1-q}{\beta_1}\int\nolimits_0^t{ {\sf P}_{\bf 0}\left({\bf S}(t-s)\in\mathcal{U}\right)\,d\left(G_1(s)-G_1\ast F_{{\bf 0},{\bf 0}}(s)\right)}.\nonumber
\end{eqnarray}
\end{Lm}
The definition of supercritical regime of CBRW implies that $\alpha_1m_1+(1-\alpha_1)F_{{\bf 0},{\bf 0}}(\infty)>1$
and $\alpha_1m_1G_1^{\ast}(\nu)+(1-\alpha_1)\,G_1^{\ast}(\nu)\overline{F}^{\,\ast}_{{\bf 0},{\bf 0}}(\nu)=1$.
In terms of the function $G(t):=\alpha_1m_1G_1(t)+(1-\alpha_1)\,G_1\ast\overline{F}_{{\bf 0},{\bf 0}}(t)$, $t\geq0$, it means that $G^{\ast}(\nu)=1$.
\begin{Lm}\label{L:E(t;)_estimate_semiexp}
Let conditions (\ref{condition1}), (\ref{assumption:tails_right_semiexp}), and (\ref{assumptions:large_deviations_tails_right}) be satisfied. Fix ${\bf x}=(x_1,\ldots,x_d)$ from the set $\partial\mathcal{O}_{\varepsilon}\cap\left\{{\bf x}\in\mathbb{R}^d:x_i\geq0,i=1,\ldots,d\right\}=\left\{{\bf x}\in\mathbb{R}^d_+:\sum_{i=1}^d x_i^{\gamma^+_i}=\nu+\varepsilon\right\}=:\partial\mathcal{O}^+_{\varepsilon}$. Then there exists $\varepsilon_0=\varepsilon_0(\nu,\varepsilon)>0$ such that
\begin{equation}\label{E(t;U)_estimate_semiexp}
E(t;\Delta({\bf x};t))\leq C e^{-\varepsilon_0t},\quad t\geq0,
\end{equation}
where $\Delta({\bf x};t):=\left[x_1R^{-1,+}_1(t),+\infty\right)\times\ldots\times\left[x_dR^{-1,+}_d(t),+\infty\right)\subset\mathbb{R}^d$ and $C$ is a positive constant.
\end{Lm}
{\sc Proof.} For any $\mathcal{U}\subset\mathbb{R}^d$, ${\bf 0}\notin\mathcal{U}$, by mean value theorem on $f_1$, equation (\ref{E(t;S)_equation}) entails the inequality
$$E\left(t;\mathcal{U}\right)\leq\int\nolimits_0^t{ E\left(t-s;\mathcal{U}\right)\,d G(s)}+I\left(t;\mathcal{U}\right).$$
Iterating this inequality $k$ times we get
$$E\left(t;\mathcal{U}\right)\leq\int\nolimits_0^t{ E\left(t-s;\mathcal{U}\right)\,d G^{\ast (k+1)}(s)}+\int\nolimits_0^t{ I\left(t-s;\mathcal{U}\right)\,d\sum_{j=0}^k{G^{\ast j}(s)}}.$$
For any fixed $t$, one has $G^{\ast k}(t)\to 0$, as $k\to\infty$, for example, due to Lemma~22 in \cite{Vatutin_book_09}. Hence, the term $\int\nolimits_0^t{ E\left(t-s;\mathcal{U}\right)\,dG^{\ast(k+1)}(s)}$ is negligibly small for large $k$. The latter inequality can be rewritten as follows
\begin{equation}\label{E(t;U)_inequality_semiexp}
E\left(t;\mathcal{U}\right)\leq\int\nolimits_0^t{ I\left(t-s;\mathcal{U}\right)\,d\sum_{j=0}^{\infty}{G^{\ast j}(s)}}.
\end{equation}
It follows from (\ref{J-1(t;a)=_semiexp}) that
$$I\left(t;\mathcal{U}\right)\leq\frac{(1-\alpha_1)\beta_1}{q}{\sf P}_{\bf 0}\left({\bf S}(t)\in\mathcal{U}\right)+\frac{(1-\alpha_1)\left|\beta_1-q\right|}{q}\int\nolimits_0^t{ {\sf P}_{\bf 0}\left({\bf S}(t-s)\in\mathcal{U}\right)\,dG_1(s)}.$$
Combining this inequality with (\ref{E(t;U)_inequality_semiexp}) we get
$$E\left(t;\mathcal{U}\right)\leq\int\nolimits_{0}^{t}{ {\sf P}_{\bf 0}({\bf S}(t-s)\in\mathcal{U})\,d\!\left(\frac{(1-\alpha_1)\beta_1}{q}\sum_{k=0}^{\infty}G^{\ast k}(s)\!+\!\frac{(1-\alpha_1)\left|\beta_1-q\right|}{q}G_1\ast\sum_{k=0}^{\infty}G^{\ast k}(s)\right)}\!.$$
Consider $\mathcal{U}=\Delta({\bf x};t)$, where ${\bf x}\in\partial\mathcal{O}^+_{\varepsilon}$. Then by virtue of assumptions (\ref{assumption:tails_right_semiexp}), (\ref{assumptions:large_deviations_tails_right}), and Theorem~25 in \cite{Vatutin_book_09}, p.~30, for any $\delta_1$ and $\delta_2$ from $(0,1)$, there exists $T=T(\delta_1,\delta_2)$ such that, for any $t\geq T$, one has
$$E\left(t;\Delta\left({\bf x};t\right)\right)\leq C_1\int\nolimits_{0}^{t}{ h(t-s)\,d\sum_{k=0}^{\infty}G^{\ast k}(s)}\;\prod_{i=1}^d{\sf P}\left(Y_i^1\geq x_iR^{-1,+}_i(t)\right)^{1-\delta_1}$$
$$\leq C_2\,e^{\nu t}\frac{\int\nolimits_0^{ \infty}h(s)e^{-\nu s}\,ds}{\int\nolimits_0^{\infty}{se^{-\nu s}\,d G(s)}}\prod_{i=1}^d{R^+_i\left(x_iR^{-1,+}_i(t)\right)^{1-\delta_1}}\leq C_3\,e^{\nu t}\prod_{i=1}^d{\exp\left\{-(1-\delta_1)x^{\gamma^+_i}_i t(1+o(1))\right\}}$$
$$\leq C_3\,\exp\left\{\nu t-(1-\delta_1)t\sum_{i=1}^d{x^{\gamma^+_i}_i}+\delta_2t\right\}\leq C_3\exp\left\{-\left((1-\delta_1)(\nu+\varepsilon)-\nu-\delta_2\right)t\right\},$$
for some positive constants $C_1$, $C_2$, and $C_3$. One can take $\delta_1,\delta_2\in(0,1)$ in such a way that $(1-\delta_1)(\nu+\varepsilon)-\nu-\delta_2=\varepsilon_0>0$. Lemma~\ref{L:E(t;)_estimate_semiexp} is proved.
\begin{Lm}\label{L:T:assertion_1_semiexp}
Let conditions (\ref{condition1}), (\ref{assumption:tails_right_semiexp}) and (\ref{assumptions:large_deviations_tails_right}) be valid. Then the following relation holds true
\begin{equation}\label{Step1:T:assertion_1_semiexp}
{\sf P}_{\bf 0}\!\left(\omega:\forall\varepsilon>0\;\exists t_3=t_3(\omega,\varepsilon)\;\mbox{s.t.}\;\forall t\geq t_3\;\mbox{and}\;\forall v\in N(t),\;{\bf X}^v(t)/{\bf R}^{-1,\kappa}(t)\notin\mathcal{O}_{\varepsilon}\cap\mathbb{R}^d_+\right)=1.
\end{equation}
\end{Lm}
{\sc Proof.} Fix $j\in\mathbb{N}$ and ${\bf x}\in\partial\mathcal{O}^+_{\varepsilon+1/j}$.
Set $A_t:=\{\omega:\forall v\in N(t)\;\mbox{one
has}\;{\bf X}^v(t)\notin\Delta({\bf x};t)\}$, $t\geq 0$.
As usual, $\overline{A}$ stands for the complement of a set $A$ and $\{A_n\;\mbox{infinitely often}\;\}=\{A_n\;\mbox{i.o.}\}=\cap_{k=1}^{\infty}\cup_{n=k}^{\infty}A_n$, for a sequence of sets $A_n$. By virtue of Borel-Cantelli lemma the estimate (\ref{E(t;U)_estimate_semiexp}) entails ${\sf P}_{\bf
0}\left(\overline{A}_{n/2^m}\;\mbox{i.o.}\right)=0$, for any fixed
$m\in\mathbb{N}$. Consequently, ${\sf P}_{\bf
0}\left(\cap_{m=1}^{\infty}\cup_{k=1}^{\infty}\cap_{n=k}^{\infty}A_{n/2^m}\right)=1$.
It means that, for almost all $\omega\in\Omega$ and for any
$m\in\mathbb{N}$, there exists a positive integer $k_1=k_1(m,\omega)$ such
that, for any $n\geq k_1$ and every $v\in N(n/2^m)$, one has
${\bf X}^v(n/2^m)\notin\Delta({\bf x};n/2^m)$. Since the
set of binary rational numbers is dense in $\mathbb{R}$ and the
sojourn time of a particle $v\in N(t)$ in a set $\Delta({\bf x};t)$ (conditioned on the event that the particle has hit the set) contains non-zero interval with probability $1$, we
conclude that
\begin{equation}\label{P(O_gamma,epsilon)=1_semiexp}
{\sf P}_{\bf 0}\left(\omega:\exists t_4(\omega)\;\mbox{such that}\;\forall t\geq t_4(\omega)\,\mbox{and}\,\forall v\in N(t),\;{\bf X}^v(t)\notin\Delta({\bf x};t)\right)=1.
\end{equation}
Unfix ${\bf x}\in\partial\mathcal{O}^+_{\varepsilon+1/j}$. If the set $\partial\mathcal{O}^+_{\varepsilon+1/j}$ is finite (it occurs when $d=1$), put $\Upsilon_j=\partial\mathcal{O}^+_{\varepsilon+1/j}$. Otherwise, let $\Upsilon_j$ be an everywhere dense set in $\partial\mathcal{O}^+_{\varepsilon+1/j}$ joined with points ${\bf x}\in\mathbb{R}^d_+$ with $x_i=0$, $i=1,\ldots,d$, $i\neq l$, for each $l=1,\ldots,d$. For instance, let $\Upsilon_j$ be the set of vectors ${\bf x}$ from $\partial\mathcal{O}^+_{\varepsilon+1/j}$ with rational coordinates $x_i$, $i=1,\ldots,d$, $i\neq l$, for each $l=1,\ldots,d$.
Unfix $j\in\mathbb{N}$. Consider the domain $\mathcal{O}_{\varepsilon}\cap\mathbb{R}^d_+=\cup_{j=1}^{\infty}\cup_{{\bf x}\in\Upsilon_j}[x_1,+\infty)\times\ldots\times[x_d,+\infty)$. Take into account that the relation ${\bf X}^v(t)\notin\Delta({\bf x};t)$ is equivalent to ${\bf X}^v(t)/{\bf R}^{-1,\kappa}(t)\notin[x_1,+\infty)\times\ldots\times[x_d,+\infty)$. Then formula \eqref{P(O_gamma,epsilon)=1_semiexp} entails the equality
$${\sf P}_{\bf 0}\left(\omega:\exists t_5(\omega)\;\mbox{such that}\;\forall t\geq t_5(\omega)\,\mbox{and}\,\forall v\in N(t),\;{\bf X}^v(t)/{\bf R}^{-1,\kappa}(t)\notin\mathcal{O}_{\varepsilon}\cap\mathbb{R}^d_+\right)=1,$$
valid for each $\varepsilon>0$. Hence the latter relation implies the assertion (\ref{Step1:T:assertion_1_semiexp}). Lemma~\ref{L:T:assertion_1_semiexp} is proved.
When $d=1$, Lemma~\ref{L:T:assertion_1_semiexp} states that $\limsup\nolimits_{t\to\infty}M_t/(t^{1/\gamma^+_1}L^{(3,+)}_1(t))\leq\nu^{1/\gamma^+_1}$ a.s. Thus, we obtain the upper estimate for the maximum $M_t$ in the case of CBRW with a single catalyst at $0$ and the starting point $0$.
Let $d\in\mathbb{N}$. In Lemma~\ref{L:T:assertion_1_semiexp} we consider the particles propagation in the positive orthant $\mathbb{R}^d_+$. Now trace the spread of particles with growing time in other directions. Without loss of generality, we deal with $\mathbb{R}_{-}\times\mathbb{R}^{d-1}_+$. Reflect the lattice $\mathbb{Z}^d$ with particles in CBRW on it at each time $t$ with respect to plane $x_1=0$. We get a new CBRW on $\mathbb{Z}^d$ and may apply
to it Lemma~\ref{L:T:assertion_1_semiexp}. Consequently,
\begin{equation}\label{reformulation_Lemma_semiexp}
{\sf P}_{\bf 0}\!\left(\omega:\forall\varepsilon>0\;\exists t_6=t_6(\omega,\varepsilon)\;\mbox{s.t.}\;\forall t\geq t_6\;\mbox{and}\;\forall v\in N(t),{\bf X}^v(t)/{\bf R}^{-1,\kappa}(t)\notin\mathcal{O}_{\varepsilon}\!\cap\!\left(\mathbb{R}_{-}\!\times\!\mathbb{R}^{d-1}_+\right)\right)\!=\!1.
\end{equation}
In the same manner reformulation of Lemma~\ref{L:T:assertion_1_semiexp} for other orthants in $\mathbb{R}^d$ combined with (\ref{Step1:T:assertion_1_semiexp}) and (\ref{reformulation_Lemma_semiexp}) leads to the first assertion of Theorem~\ref{T:main_result_semiexp_lattice} in the case of CBRW with a single catalyst at ${\bf 0}$ and the starting point ${\bf 0}$.
\vskip0.2cm
\emph{Step 2.}
We prove statement (\ref{T:assertion_2_semiexp}) whenever there is a single catalyst located at ${\bf 0}$ and the starting point is ${\bf 0}$ as well. We temporarily assume that ${\sf E}\xi^2_1<\infty$ and follow the ideas of \cite{Carmona_Hu_14}, Sect.5.2.
\begin{Lm}\label{L:lower_estimate_semiexp}
Let conditions (\ref{condition1}), (\ref{assumption:tails_right_semiexp}), and (\ref{assumptions:large_deviations_tails_right}) be satisfied. Choose function $r=r(t)$ in such a way that $r(t)\leq t$, $r(t)\nearrow+\infty$, $t-r(t)\nearrow+\infty$ and $t-r(t)=o(t)$, as $t\to\infty$ (for example, we can put $r(t)=t-\ln{t}$). Fix both $\varepsilon\in(0,\nu)$ and ${\bf x}\in\partial\mathcal{Q}^+_{\varepsilon}:=\partial\mathcal{Q}_{\varepsilon}\cap\mathbb{R}^d_+$. Then, for some positive constant $C_4$, one has
\begin{equation}\label{E(t;U)_lower_estimate_semiexp}
{\sf P}_{\bf 0}\left({\bf X}^v(t)\notin\Delta({\bf x};t)\mbox{\;for any\;}v\in N(t),\mu(r;{\bf 0})\geq C_4 e^{\nu r}\right)\leq\exp\left\{-e^{\varepsilon t+o(t)}\right\},\quad t\to\infty.
\end{equation}
\end{Lm}
{\sc Proof.} In view of Theorem~4 in \cite{B_Doklady_15}, on the set $\mathcal{I}$ at time $r$, $0<r<t$, there are at least $[C_4e^{\nu r}]$ particles at ${\bf 0}$ for some positive constant $C_4$ (as usual, $[r]$ stands for the integer part of a number $r\in\mathbb{R}_+$). If these particles move according to the random walk ${\bf S}$ such that ${\bf S}(u)\neq{\bf 0}$ for each $u\in[\tau_{\bf 0},t-r]$, then remote particles in CBRW at time $t$ are not less far than $[C_4e^{\nu r}]$ i.i.d. copies of ${\bf S}(t-r)$ with ${\bf S}(u)\neq{\bf 0}$, for each $u\in[\tau_{\bf 0},t-r]$.
For a set $\mathcal{U}\subset\mathbb{R}^d$, ${\bf 0}\notin\mathcal{U}$, and $t\geq0$, the following identity is valid
$${\sf P}_{\bf 0}\left({\bf S}(t)\in\mathcal{U},\tau_{{\bf 0},{\bf 0}}>t\right)={\sf P}_{\bf 0}\left({\bf S}(t)\in\mathcal{U}\right)-\int\nolimits_0^t{ {\sf P}_{\bf 0}\left({\bf S}(t-s)\in\mathcal{U}\right)\,d F_{{\bf 0},{\bf 0}}(s)}.$$
Then assumptions~(\ref{assumption:tails_right_semiexp}) and (\ref{assumptions:large_deviations_tails_right}) imply that
$${\sf P}_{\bf 0}\left({\bf S}(t-r)\in\Delta({\bf x};t),\tau_{{\bf 0},{\bf 0}}>t-r\right)$$
$$={\sf P}_{\bf 0}\left({\bf S}(t-r)\in\Delta({\bf x};t)\right)-\int\nolimits_0^{t-r}{ {\sf P}_{\bf 0}\left({\bf S}(t-r-s)\in\Delta({\bf x};t)\right)\,d F_{{\bf 0},{\bf 0}}(s)}$$
$$=\left(h(t-r)-\int\nolimits_0^{t-r}{ h(t-r-s)\,dF_{{\bf 0},{\bf 0}}(s)}+o\left(h(t-r)\right)\right)\prod_{i=1}^d{\sf P}\left(Y_i^1\geq x_iR^{-1,+}_i(t)\right)^{1+\varepsilon^{+}_i(t)}$$
$$=\left(h(t-r)\left(1-F_{{\bf 0},{\bf 0}}(t-r)\right)+\int_0^{t-r}{ \left(h(t-r)-h(t-r-s)\right)\,dF_{{\bf 0},{\bf 0}}(s)}+o\left(h(t-r)\right)\right)$$
$$\times\prod_{i=1}^d{R^+_i\left(x_iR^{-1,+}_i(t)\right)^{1+\varepsilon^+_i(t)}}=\exp\left\{-t\sum_{i=1}^d{x_i^{\gamma_i^{+}}}+o(t)\right\}=\exp\left\{-(\nu-\varepsilon)t+o(t)\right\},\quad t\to\infty.$$
The latter relation leads to the estimate
$${\sf P}_{\bf 0}\left({\bf X}^v(t)\notin\Delta({\bf x};t)\mbox{\;for any\;}v\in N(t),\mu(r;{\bf 0})\geq C_4 e^{\nu r}\right)$$
$$\leq\left(1-{\sf P}_{\bf 0}\left({\bf S}(t-r)\in\Delta({\bf x};t),\tau_{{\bf 0},{\bf 0}}>t-r\right)\right)^{\left[C_4 e^{\nu r}\right]}$$
$$\leq\exp\left\{-\left[C e^{\nu r}\right]e^{-(\nu-\varepsilon)t+o(t)}\right\}=\exp\left\{-e^{\nu r-(\nu-\varepsilon)t+o(t)}\right\},\quad t\to\infty.$$
The assertion of Lemma~\ref{L:lower_estimate_semiexp} now follows from our choice of $r=r(t)$. Lemma~\ref{L:lower_estimate_semiexp} is proved.
\begin{Lm}\label{L:T:assertion_2_semiexp}
Let conditions (\ref{condition1}), (\ref{assumption:tails_right_semiexp}), and (\ref{assumptions:large_deviations_tails_right}) be valid. Then the following relation holds true
\begin{equation}\label{Step2:T:assertion_2_semiexp}
{\sf P}_{\bf 0}\!\left(\left.\omega:\forall\varepsilon\in(0,\nu)\;\exists t_7=t_7(\omega,\varepsilon)\;\mbox{s.t.}\;\forall t\geq t_7\;\exists v\in N(t),\;{\bf X}^v(t)/{\bf R}^{-1,\kappa}(t)\notin\mathcal{Q}_{\varepsilon}\cap\mathbb{R}^d_+\right|\mathcal{I}\right)=1.
\end{equation}
\end{Lm}
{\sc Proof.} Fix ${\bf x}\in\partial\mathcal{Q}^+_{\varepsilon}$.
Denote by $B_t$ the event $\{\omega:\exists v\in N(t)\;\mbox{such that}\;{\bf X}^v(t)\in\Delta({\bf x};t)\}$. By virtue of Borel-Cantelli lemma and Theorem~4 in \cite{B_Doklady_15} the estimate (\ref{E(t;U)_lower_estimate_semiexp}) entails ${\sf P}_{\bf
0}\left(\left.\overline{B}_{n/2^m}\;\mbox{i.o.}\right|\mathcal{I}\right)=0$, for any fixed
$m\in\mathbb{N}$. Consequently, ${\sf P}_{\bf
0}\left(\left.\cap_{m=1}^{\infty}\cup_{k=1}^{\infty}\cap_{n=k}^{\infty}B_{n/2^m}\right|\mathcal{I}\right)=1$.
It means that for almost all $\omega\in\Omega$ and for any
$m\in\mathbb{N}$ there exists positive integer $k_2=k_2(m,\omega)$ such
that, for any $n\geq k_2$, one can find $v\in N(n/2^m)$ such that
${\bf X}^v(n/2^m)\in\Delta({\bf x};n/2^m)$. Since the
set of binary rational numbers is dense in $\mathbb{R}$ and the
sojourn time of a particle $v\in N(t)$ in a set $\Delta({\bf x};t)$ (conditioned on the event that the particle has hit the set) contains non-zero interval with probability $1$, we
conclude that
\begin{equation}\label{P(O_gamma,epsilon)=1_semiexp_lower}
{\sf P}_{\bf 0}\left(\left.\omega:\exists t_8(\omega)\;\mbox{such that}\;\forall t\geq t_8(\omega)\,\mbox{one has}\,\exists v\in N(t),\;{\bf X}^v(t)\in\Delta({\bf x};t)\right|\mathcal{I}\right)=1.
\end{equation}
Unfix ${\bf x}\in\partial\mathcal{Q}^+_{\varepsilon}$. If the set $\partial\mathcal{Q}^+_{\varepsilon}$ is finite (it occurs when $d=1$), put $\Upsilon=\partial\mathcal{Q}^+_{\varepsilon}$. Otherwise, let $\Upsilon$ be an everywhere dense set in $\partial\mathcal{Q}^+_{\varepsilon}$ joined with points ${\bf x}\in\mathbb{R}^d_+$ with $x_i=0$, $i=1,\ldots,d$, $i\neq l$, for each $l=1,\ldots,d$. For instance, let $\Upsilon$ be the set of vectors ${\bf x}$ from $\partial\mathcal{Q}^+_{\varepsilon}$ with rational coordinates $x_i$, $i=1,\ldots,d$, $i\neq l$, for each $l=1,\ldots,d$.
Consider the domain $\mathcal{Q}_{\varepsilon}\cap\mathbb{R}^d_+=\mathbb{R}^d_+\setminus\cup_{{\bf x}\in\Upsilon}[x_1,+\infty)\times\ldots\times[x_d,+\infty)$. Take into account that the relation ${\bf X}^v(t)\in\Delta({\bf x};t)$ is equivalent to ${\bf X}^v(t)/{\bf R}^{-1,\kappa}(t)\in[x_1,+\infty)\times\ldots\times[x_d,+\infty)$. Then formula \eqref{P(O_gamma,epsilon)=1_semiexp_lower} entails the equality
$${\sf P}_{\bf 0}\left(\left.\omega:\exists t_9(\omega)\;\mbox{such that}\;\forall t\geq t_9(\omega)\,\exists v\in N(t),\;{\bf X}^v(t)/{\bf R}^{-1,\kappa}(t)\notin\mathcal{Q}_{\varepsilon}\cap\mathbb{R}^d_+\right|\mathcal{I}\right)=1,$$
valid for each $\varepsilon\in(0,\nu)$. Unfix $\varepsilon\in(0,\nu)$. Then the latter relation implies the assertion (\ref{Step2:T:assertion_2_semiexp}). Lemma~\ref{L:T:assertion_2_semiexp} is proved.
When $d=1$, Lemma~\ref{L:T:assertion_2_semiexp} states that $\liminf\nolimits_{t\to\infty}M_t/(t^{1/\gamma^+_1}L^{(3,+)}_1(t))\geq\nu^{1/\gamma^+_1}$ a.s. on the set $\mathcal{I}$. Thus, we obtain the lower estimate for the maximum $M_t$ in the case of CBRW with a single catalyst at $0$ and the starting point $0$ under the additional assumption ${\sf E}\xi^2_1<\infty$.
Let $d\in\mathbb{N}$. In Lemma~\ref{L:T:assertion_2_semiexp} we consider the particles propagation in the positive orthant $\mathbb{R}^d_+$. Similarly to discussion at the end of \emph{Step~1}, we may reformulate Lemma~\ref{L:T:assertion_2_semiexp} for other orthants in $\mathbb{R}^d$. Reformulation of Lemma~\ref{L:T:assertion_2_semiexp} for other orthants in $\mathbb{R}^d$ combined with (\ref{Step2:T:assertion_2_semiexp}) leads to the second assertion of Theorem~\ref{T:main_result_semiexp_lattice} in the case of CBRW with a single catalyst at ${\bf 0}$ and the starting point ${\bf 0}$ whenever ${\sf E}\xi^2_1<\infty$.
Combination of the proved in \emph{Step 1} assertion (\ref{T:assertion_1_semiexp}) and relation (\ref{P(O_gamma,epsilon)=1_semiexp_lower}), valid for each ${\bf x}\in\partial\mathcal{Q}_{\varepsilon}$, implies the statement of Theorem~\ref{T:one_point_semiexp} for the case of a single catalyst at ${\bf 0}$ and the starting point ${\bf 0}$ whenever ${\sf E}\xi^2_1<\infty$.
\vskip0.2cm \emph{Step 3.}
Assume that $W=\{{\bf w}_1\}$ with ${\bf w}_1={\bf 0}$ and the starting point of CBRW is
${\bf 0}$ whereas now ${\sf E}\xi^2_1=\infty$. To verify assertion (\ref{P(O_gamma,epsilon)=1_semiexp_lower}) (and, as a consequence, (\ref{T:assertion_2_semiexp})) under such assumptions one can follow the
proof scheme proposed in \cite{Carmona_Hu_14}, Sec.~5.3, based on
a coupling. In contrast to \cite{Carmona_Hu_14} we employ Theorem~3 of \cite{B_Doklady_15}
devoted to the strong convergence of the total and local particles
numbers in supercritical CBRW instead of using properties of a
fundamental martingale as in \cite{Carmona_Hu_14}. Moreover, here
we exploit function $g(u)=\alpha_1
f_1\left(q_{esc}+(1-q_{esc})u\right)+(1-\alpha_1)q_{esc}-u$,
${u\in[0,1]}$, where $q_{esc}={\sf P}_{\bf 0}\left(\overline{\tau}_{{\bf 0},{\bf 0}}=\infty\right)=1-\overline{F}_{{\bf 0},{\bf 0}}(\infty)$ is the
escape probability of the random walk ${\bf S}$. Other details of
the Step~3 proof can be omitted.
\vskip0.2cm \emph{Step 4.} Now we deal with $N>1$ and ${\bf x}\in W$, say ${\bf x}={\bf w}_i$. Let us discuss here the main differences between the case of single and multiple catalysts and sketch the subsequent proof omitting cumbersome details. In the multiple setting the counterpart of the probability $E(t;\mathcal{U})$ is the vector ${\bf E}(t;\mathcal{U}):=\left(E_{{\bf w}_1}(t;\mathcal{U}),\ldots,E_{{\bf w}_N}(t;\mathcal{U})\right)$ with $E_{{\bf w}_i}(t;\mathcal{U}):={\sf P}_{{\bf w}_i}\left(\exists v\in N(t):{\bf X}^v(t)\in\mathcal{U}\right)$, $i=1,\ldots,N$, $t\geq0$, for $\mathcal{U}\subset\mathbb{R}^d$. Similarly to Lemma~\ref{L:equation_multi}, it satisfies the following system of non-linear integral equations of convolution type
\begin{eqnarray*}
E_{{\bf w}_i}(t;\mathcal{U})&=&\alpha_i\int\nolimits_0^t{ \left(1-f_i\left(1-E_{{\bf w}_i}(t-s;\mathcal{U})\right)\right)\,dG_i(s)}\\
&+&(1-\alpha_i)\sum_{j=1}^N\int\nolimits_0^t{ E_{{\bf w}_j}(t-s;\mathcal{U})\,dG_{i,j}(s)}+I_{{\bf w}_i}\left(t;\mathcal{U}\right),
\end{eqnarray*}
where
$$I_{{\bf w}_i}(t;\mathcal{U}):=\sum_{{\bf y}\notin W}{(1-\alpha_i)\frac{q({\bf w}_i,{\bf y})}{q}\int\nolimits_0^t{ {\sf P}_{\bf y}\left({\bf S}(t-s)\in\mathcal{U},{_{W_k}\tau_{{\bf y},{\bf w}_k}}>t-s,k=1,\ldots,N\right)\,dG_i(s)}},$$
$t\geq0$, $i=1,\ldots,N$, $\mathcal{U}\subset\mathbb{R}^d$, $W\cap\mathcal{U}=\varnothing$.
A multiple setting counterpart of Lemma~\ref{L:J-1(t;a)=_semiexp} states that
\begin{eqnarray*}
& &\frac{q I_{{\bf w}_i}(t;\mathcal{U})}{(1-\alpha_i)\beta_i}={\sf P}_{{\bf w}_i}\left({\bf S}(t)\in\mathcal{U}\right)-\sum_{k=1}^N\int\nolimits_0^t{ {\sf P}_{{\bf w}_k}\left({\bf S}(t-s)\in\mathcal{U}\right)\,d\,{_{W_k}F_{{\bf w}_i,{\bf w}_k}(s)}}\\
&-&\frac{\beta_i-q}{\beta_i}\int\nolimits_0^t{ {\sf P}_{{\bf w}_i}\left({\bf S}(t-s)\in\mathcal{U}\right)\,d G_i(s)}+\sum_{k=1}^N\frac{\beta_i-q}{\beta_i}\int\nolimits_0^t{ {\sf P}_{{\bf w}_k}\left({\bf S}(t-s)\in\mathcal{U}\right)\,d G_i\ast{_{W_k}F_{{\bf w}_i,{\bf w}_k}(s)}}.
\end{eqnarray*}
It follows that
$$I_{{\bf w}_i}(t;\mathcal{U})\leq\frac{(1-\alpha_i)\beta_i}{q}{\sf P}_{{\bf w}_i}\left({\bf S}(t)\in\mathcal{U}\right)+\frac{(1-\alpha_i)\left|\beta_i-q\right|}{q}\int\nolimits_0^t{ {\sf P}_{{\bf w}_i}\left({\bf S}(t-s)\in\mathcal{U}\right)\,d G_i(s)}=:K_i(t;\mathcal{U}).$$
While proving Lemma~\ref{L:E(t;)_estimate_semiexp} we obtained inequality (\ref{E(t;U)_inequality_semiexp}). Likewise we derive the following vector inequality, valid coordinate-wise,
$${\bf E}(t;\mathcal{U})\leq\sum_{k=0}^{\infty}{\bf G}^{\ast k}\ast{\bf K}(t;\mathcal{U}).$$
Here a multiple setting counterpart of function $G$ is a matrix ${\bf G}(t)=\left(G^{\,(N)}_{i,j}(t)\right)_{i,j=1}^N$ with entries $G^{\,(N)}_{i,j}(t):=\delta_{i,j}\alpha_i m_i G_i(t)+(1-\alpha_i)G_i\ast{_{W_j}\overline{F}_{{\bf w}_i,{\bf w}_j}(t)}$, $t\geq0$, whereas ${\bf K}(t;\mathcal{U})$, $t\geq0$, is the vector-column with $i$th coordinate $K_i(t;\mathcal{U})$. The element $d_{i,j}(\lambda)$ of matrix $D(\lambda)$, $\lambda\geq0$, is just the Laplace transform of $G^{\,(N)}_{i,j}$.
Recall that the operation ``$\ast$'' of convolution of matrices is defined exactly as matrix multiplication except that we convolve elements rather than multiply them.
As for validating Lemma~\ref{L:E(t;)_estimate_semiexp} for $N=1$, in case $N>1$ we inspect the asymptotic behavior of $\sum_{k=0}^{\infty}{\bf G}^{\ast k}\ast{\bf K}(t;\mathcal{U})$ when $\mathcal{U}=\Delta({\bf x};t)$ with ${\bf x}\in\partial\mathcal{O}^+_{\varepsilon}$ and $t\to\infty$. Employing Corollary~3.1, item (i), in \cite{Crump_70}, we deduce the same estimate as (\ref{E(t;U)_estimate_semiexp}) after replacing $E(t;\Delta({\bf x};t))$ by $E_{{\bf w}_i}(t;\Delta({\bf x};t))$. The rest of the proofs of Theorems~\ref{T:main_result_semiexp_lattice} and \ref{T:one_point_semiexp} in case of CBRW with general catalysts set $W$ and the starting point from $W$ is implemented similar to the arguments of Steps~1 -- 3.
\vskip0.2cm {\it Step 5.} Turning to a supercritical CBRW on $\mathbb{Z}^d$ with a finite catalysts set $W$ and the starting
point ${\bf z}\notin W$, we supplement the catalysts set $W$ with ${\bf w}_{N+1}={\bf x}$ and put $\alpha_{N+1}=0$, $m_{N+1}=0$,
$G_{N+1}(t)=1-e^{-qt}$, $t\geq0$. According to Lemma~3 in \cite{B_TPA_15} a new CBRW with catalysts set $\{{\bf w}_1,\ldots,{\bf w}_{N+1}\}$ is supercritical whenever the underlying CBRW is supercritical and the Malthusian parameters in these CBRW coincide. Then one can apply the proved part of Theorems~\ref{T:main_result_semiexp_lattice} and \ref{T:one_point_semiexp} to the new CBRW and obtain the desired assertions of those theorems for CBRW with an arbitrary starting point.
\vskip0.2cm The proof of Theorems~\ref{T:main_result_semiexp_lattice} and \ref{T:one_point_semiexp} is complete.
\vskip0.2cm
{\bf Remark 3}.
Within the proofs we track the evolution of the particles ``at the front''. As for CBRW with regularly varying tails in \cite{B_Arxiv_18}, a particle ``at the front'' at time $t$ was born at time $t-o(t)$ and then reached the front within time $o(t)$.
This significantly differs from the case of ``light'' tails.
It follows from the proof of Theorem~1.1 in \cite{Carmona_Hu_14}
and Theorem~1 in \cite{B_SPA_18} treating CBRW with light tails that
a particle ``at the front'' at time $t$ was born at time $\theta t$ for
specified $\theta\in(0,1)$ and then walked only until time $t$.
\section{Examples}\label{s:examples_semiexp}
According to formula (\ref{S(t)=representation}) the random walk ${\bf S}$ is a jump process with increments ${\bf Y}^j$. In this section we assume that the coordinates of each jump ${\bf Y}^j=\left(Y^j_1,\ldots,Y^j_d\right)$, $j\in\mathbb{N}$, are independent. Without loss of generality, consider ${\bf x}\in\mathbb{R}^d_+$, ${\bf x}\neq{\bf 0}$. Then
\begin{eqnarray}\label{example1}
{\sf P}_{\bf 0}\left({\bf S}(u)\in\Delta({\bf x};t)\right)&=&\sum_{k=1}^{\infty}{\sf P}\left(\Pi(u)=k\right){\sf P}\left(\sum_{j=1}^k{{\bf Y}^j}\in\Delta({\bf x};t)\right)\nonumber\\
&=&\sum_{k=1}^{\infty}{\sf P}\left(\Pi(u)=k\right)\prod_{i=1}^d{\sf P}\left(\sum_{j=1}^kY^j_i\geq x_iR^{-1,+}_i(t)\right).
\end{eqnarray}
Write $Y=Y^+-Y^-$, where $Y^+:=Y\mathbb{I}\{Y\geq0\}$ and $Y^-:=-Y\mathbb{I}\{Y<0\}$.
Consider $Y^j_i=Y^{(j,+)}_i-Y^{(j,-)}_i$, where $Y^{(j,+)}_i$, $Y^{(j,-)}_i\geq0$ are defined in the mentioned way. Set ${\sf P}\left(Y^{(j,\kappa)}_i=0\right)=1-L^{(1,\kappa)}_i$, $\kappa\in\{+,-\}$, $j=1,2,\ldots$. Then $L^{(1,+)}_i+L^{(1,-)}_i=1$, $i=1,\ldots,d$. Let also $Y^{(j,\kappa)}_i$ conditioned to be strictly positive have a discrete Weibull distribution, \cite{FKZ_11}, p.~10, with parameters $\gamma^{\kappa}_i$ and $\left(L^{(2,\kappa)}_i\right)^{-1/\gamma^{\kappa}_i}$, where $\gamma^{\kappa}_i\in(0,1)$ and $L^{(1,\kappa)}_i$, $L^{(2,\kappa)}_i$ are positive constants. In other words,
for, $y=1,2,\ldots$,
$${\sf P}\left(Y^{(j,\kappa)}_i=y\right)=L^{(1,\kappa)}_i\left(\exp\left\{-L^{(2,\kappa)}_i
(y-1)^{\gamma^{\kappa}_i}\right\}- \exp\left\{-L^{(2,\kappa)}_iy^{\gamma^{\kappa}_i}\right\}\right),$$
or equivalently
\begin{equation}\label{example_tails}
{\sf P}\left(Y^{(j,\kappa)}_i>y\right)=L^{(1,\kappa)}_i\exp\left\{-L^{(2,\kappa)}_iy^{\gamma^{\kappa}_i}\right\},\quad y\in\mathbb{Z}_+.
\end{equation}
These formulas are particular cases of (\ref{assumption:tails_right_semiexp}) and (\ref{assumption:tails_left_semiexp}) with functions $L^{(1,\kappa)}_i(y)=L^{(1,\kappa)}_i$ and $L^{(2,\kappa)}_i(y)=L^{(2,\kappa)}_i$ for any $y\in\mathbb{Z}_+$, each $\kappa\in\{+,-\}$ and $i=1,\ldots,d$.
We verify the validity of assumption (\ref{assumptions:large_deviations_tails_right}), without loss of generality, for ${\bf x}\in\mathbb{R}^d_+$, ${\bf x}\neq{\bf 0}$. Choose parameters $L^{(1,\kappa)}_i$, $\kappa\in\{+,-\}$, for each fixed $i=1,\ldots,d$, in such a way that ${\sf E}Y^j_i=0$. Then relation (\ref{example1}) and Theorem 5.4.1 in \cite{Borovkov_Borovkov_08} imply
$${\sf P}_{\bf 0}\left({\bf S}(u)\in\Delta({\bf x};t)\right)=\sum_{k=1}^{\infty}{\sf P}\left(\Pi(u)=k\right)\prod_{i=1}^d k\left({\sf P}\left(Y^j_i\geq x_iR^{-1,+}_i(t)\right)\right)^{\left(1+\varepsilon_i(t)\right)}(1+\delta_i(k,t))$$
$$=\left(\sum_{k=1}^{\infty}k^d{\sf P}\left(\Pi(u)=k\right)\prod_{i=1}^d\left(1+\delta_i(k,t)\right)\right)\prod_{i=1}^d{\left({\sf P}\left(Y^j_i\geq x_iR^{-1,+}_i(t)\right)\right)^{\left(1+\varepsilon_i(t)\right)}},$$
where $\varepsilon_i(t)\to0$, as $t\to\infty$, and $\delta_i(k,t)\to0$, as $k,t\to\infty$. The latter equality entails the desired formula (\ref{assumptions:large_deviations_tails_right}), where $h(u)(1+\delta(u,t))=e^{-qu}\sum_{k=1}^{\infty}k^d(qu)^k/k!\prod_{i=1}^d{\left(1+\delta_i(k,t)\right)}\sim (qu)^d$, as $u,t\to\infty$, $u\leq t$.
Thus, whenever hypothesis (\ref{example_tails}) holds true for supercritical CBRW, all the conditions of Theorem~\ref{T:main_result_semiexp_lattice} are satisfied. As mentioned above, in this case we assume that $L^{(1,+)}_i+L^{(1,-)}_i=1$ and ${\sf E}Y^j_i=0$. It means that parameters $L^{(1,\kappa)}_i$, $\kappa\in\{+,-\}$, satisfy, for each $i=1,2$, the following equations system
$$\left\{
\begin{aligned}
&L^{(1,+)}_i+L^{(1,-)}_i=1,\\
&L^{(1,+)}_i\sum_{y=0}^{\infty}{\exp\left\{-L^{(2,+)}_iy^{\gamma^+_i}\right\}}
-L^{(1,-)}_i\sum_{y=0}^{\infty}{\exp\left\{-L^{(2,-)}_iy^{\gamma^-_i}\right\}}=0.
\end{aligned}
\right.
$$
For each $i=1,2$, we have two unknown variables $L^{(1,\kappa)}_i$, $\kappa\in\{+,-\}$,
and two relations involving them in the latter system.
For example, focusing on the case $d=2$ and setting $\gamma^+_1=3/4$, $\gamma^+_2=1/2$, $\gamma^-_1=1/3$, $\gamma^-_2=1/4$, $L^{(2,+)}_1=1$, $L^{(2,+)}_2=2$, $L^{(2,-)}_1=3$ and $L^{(2,-)}_2=4$, we solve the corresponding systems with the help of packet Wolfram Mathematica and find $L^{(1,+)}_1\approx 0.382737$, $L^{(1,-)}_1\approx 0.617263$ and $L^{(1,+)}_2\approx 0.450655$, $L^{(1,-)}_2\approx 0.549345$.
Nevertheless, the limiting shape $\mathcal{P}$ of the front of the particles population described in (\ref{def_P_semiexp}) is determined exceptionally by parameters $\gamma^{\kappa}_i$, $\kappa\in\{+,-\}$, $i=1,\ldots,d$, and the Malthusian parameter $\nu$. So, to compare different forms of $\mathcal{P}$, we do no need to specify other parameters such as $L^{(1,\kappa)}_i$ and $L^{(2,\kappa)}_i$, $\kappa\in\{+,-\}$, $i=1,\ldots,d$.
\vskip0.2cm {\it Example 1.} Let $d=2$ and put $\gamma^+_1=\gamma^+_2=\gamma^-_1=\gamma^-_2=1/2$, $\nu=2$. Then the plot of the limiting shape $\mathcal{P}$ of the front is drawn on Figure~\ref{Example_123} to the left.
\vskip0.2cm {\it Example 2.} Consider now non-symmetric limiting shape $\mathcal{P}$ of the front from the example above with $d=2$, $\gamma^+_1=3/4$, $\gamma^+_2=1/2$, $\gamma^-_1=1/3$, $\gamma^-_2=1/4$, and $\nu=1$. Its plot is represented on Figure~\ref{Example_123} at the middle.
\vskip0.2cm {\it Example 3.} For $d=3$ and $\gamma^{\kappa}_i=3/4$, $\kappa\in\{+,-\}$, $i=1,2,3$, $\nu=1$, the plot of $\mathcal{P}$ is drawn on Figure~\ref{Example_123} to the right.
{\bf Remark 4}.
These Figures illustrate the fact that for CBRW on $\mathbb{Z}^d$ with semi-exponential increments the surface $\mathcal{P}$ is a boundary of a \emph{star shape set} in $\mathbb{R}^d$ with the center at ${\bf 0}$. The set is non-convex, though radially-convex. This arises from our additional assumption (\ref{assumptions:large_deviations_tails_right}) implicitly comprising
the condition of independence of the random walk jump vector coordinates. As a consequence,
to reach a distant set along the semiaxes is more probable than to reach a distant set, say,
diagonal-wise. Indeed, in the first case it is enough to perform one ``big jump'' whereas in
the second case we need to perform several ``big'' jumps along different semiaxes. However,
in the case of light tails large deviations are due to many ``small'' jumps rather than one or few ``big'' jumps, \cite{Borovkov_Borovkov_08}, p.XX.
\begin{center}
\begin{figure}
\includegraphics[width=17cm]{Example123}
\caption[]{The plots of $\mathcal{P}$ for Examples~1, 2 and 3.}\label{Example_123}
\end{figure}
\end{center}
\section{Conclusion}\label{s:conclusion}
Theorem~\ref{T:main_result_semiexp_lattice} is a counterpart of Theorem 1.1 in \cite{Carmona_Hu_14} for $d=1$ and Theorem~1 in \cite{B_SPA_18} for $d>1$ describing the population front in CBRW on $\mathbb{Z}^d$ in the case of light-tailed
jump distribution of the random walk. The novelty of our results is the following.
\begin{enumerate}
\item The normalization of the vector ${\bf X}^v(t)$, determining the almost sure limiting shape of the front, in general is defined component-wise. It
explicitly depends both on the sign of the component
and the jump distribution of the random walk in the corresponding direction, whereas in \cite{B_SPA_18} the normalizing
factor is the same for all components of ${\bf X}^v(t)$ and just equals $t$.
\item The asymptotic behavior of the normalizing factors for the front of CBRW on $\mathbb{Z}^d$ with semi-exponential distribution tails takes an intermediate position between CBRW with light and regularly varying tails
since the normalization of each component grows as a regularly varying function of index exceeding $\mathrm{1}$. Hence the growth is faster than linear (as for light tails). In the case of regularly varying tails
the normalizing factor grows exponentially fast as shown in \cite{B_Arxiv_18}.
The reason is that semi-exponential distribution tails take an intermediate position between the light and the regularly varying tails.
\item Pictures in Section~\ref{s:examples_semiexp}
illustrate that the limiting shape $\mathcal{P}$ of the front is a boundary of a nonconvex star shape set in $\mathbb{R}^d$ which sharply contrasts to
limiting shape of the front in the case of light tails (where one has a boundary of a convex set). The nature of this effect
is explained in Remark 4.
\end{enumerate}
We have demonstrated that for the new CBRW model it is possible to determine the limiting shape of the front for the appropriately rescaled positions of particles population. So, one can imagine that the spread of the cloud of particles is described in time by a surface $\mathcal{P}$ of points with their coordinate components multiplied by the explicitly indicated normalizing factors (in general, different for each component). It would be interesting to study the fluctuations of
particles around this moving surface in $\mathbb{R}^d$ ($d>1$) or, equivalently, it means the analysis of the convergence rate (in a sense) of the normalized particles positions around the surface $\mathcal{P}$.
|
3,212,635,537,449 | arxiv | \section{Introduction}\label{sec-intro}
An $[n,\ell,d]$ linear code over the finite field ${\mathbb{F}}_p$ is an $\ell$-dimensional subspace of ${\mathbb{F}}_p^n$
with minimum (Hamming) distance $d$, where $p$ is a prime. Let $A_i$ denote the number of codewords
with Hamming weight $i$ in a code $\mathcal{C}$ of length $n$. The weight enumerator of $\mathcal{C}$
is defined by
\begin{eqnarray*}
1+A_1z+A_2z^2+\cdots+A_nz^n.
\end{eqnarray*}
The sequence $(A_1,A_2,\cdots,A_{n})$ is called the weight distribution of the code. Clearly, the weight
distribution gives the minimum distance of the code, and thus the error correcting capability. In addition, the
weight distribution of a code allows the computation of the error probability of error detection and correction
with respect to some error detection and error correction algorithms \cite{Klov}. Thus the study of the weight
distribution of a linear code is important in both theory and applications.
An $[n,k]$ linear code ${\mathcal C}$ over ${\mathbb{F}}_p$ is called {cyclic} if
$(c_0,c_1, \cdots, c_{n-1}) \in {\mathcal C}$ implies $(c_{n-1}, c_0, c_1, \cdots, c_{n-2})$
$\in {\mathcal C}$.
By identifying any vector $(c_0,c_1, \cdots, c_{n-1}) \in {\mathbb{F}}_p^n$
with
$$
c_0+c_1x+c_2x^2+ \cdots + c_{n-1}x^{n-1} \in {\mathbb{F}}_p[x]/(x^n-1),
$$
any code ${\mathcal C}$ of length $n$ over ${\mathbb{F}}_p$ corresponds to a subset of ${\mathbb{F}}_p[x]/(x^n-1)$.
The linear code ${\mathcal C}$ is cyclic if and only if the corresponding subset in ${\mathbb{F}}_p[x]/(x^n-1)$
is an ideal.
It is well known that every ideal of ${\mathbb{F}}_p[x]/(x^n-1)$ is principal. Let ${\mathcal C}=\langle g(x) \rangle$,
where $g(x)$ is monic and has the least
degree. Then $g(x)$ is called the generator polynomial and
$h(x)=(x^n-1)/g(x)$ is referred to as the parity-check polynomial of
${\mathcal C}$. A cyclic code is called irreducible
if its parity-check polynomial is irreducible over ${\mathbb{F}}_p$. Otherwise, it is called reducible.
The weight distributions of both irreducible and reducible cyclic codes have been interesting subjects
of study for many years and are very hard problems in general. For information on the weight distribution
of irreducible cyclic codes, the reader
is referred to the recent survey \cite{Ding12}. Information
on the weight distribution of reducible cyclic codes could be found in \cite{YCD}, \cite{Feng07}, \cite{Luo081},
\cite{Luo082}, \cite{Zeng10}, \cite{Ma11}, \cite{Ding11}, \cite{Wang12}.
For the duals of the known cyclic codes whose weight distributions were established, most of them have at most two zeros (see \cite{YCD,Feng07,Luo081,Luo082,Ma11,Ding11,Wang12,Xiong1,Feng12,Ding12}),
only a few of them have three or more zeros (see \cite{Feng07,Luo081,Zeng10,Li12}).
The objective of this paper is to settle the weight distribution of a family of five-weight cyclic codes
whose duals have three zeros.
This paper is organized as follows. Section \ref{sec-code} defines the family of cyclic codes.
Section \ref{sec-prelim} presents results on quadratic forms which will be needed in subsequent
sections. Section \ref{sec-wtds} solves the
weight distribution problem for the family of cyclic codes. Section \ref{sec-conclusion} concludes
this paper and makes some comments.
\section{The family of cyclic codes}\label{sec-code}
In this section, we introduce the family of cyclic codes to be studied in the sequel.
Before doing this, we first give some notations which will be fixed throughout the paper unless otherwise stated.
Let $p$ be an odd prime and $q=p^m$, where $m$ is odd and $m\geq 5$.
Let $d_1=(p^{2k}+1)/2$ and $d_2=(p^{4k}+1)/2$, where $k$ is any positive integer with $\gcd(m,k)=1$.
Let $\pi$ be a generator of the finite field ${\mathbb{F}}_{q}$, and let $h_i(x)$ denote the minimal polynomial of $\pi^{-i}$ over ${\mathbb{F}}_p$
for any integer $i$. It is easy to check that $h_1(x)$, $h_{d_1}(x)$ and $h_{d_2}(x)$ have
degree $m$ and are pairwise distinct. Define
\begin{eqnarray}\label{eqn-parity-check}
h(x)=h_1(x)h_{d_1}(x)h_{d_2}(x).
\end{eqnarray}
Then $h(x)$ has degree $3m$ and is a factor of $x^{q-1}-1$.
Let ${\mathcal C}_{(p,m,k)}$ be the cyclic code with parity-check polynomial $h(x)$. Then
${\mathcal C}_{(p,m,k)}$ has length $q-1$ and dimension $3m$. Using the well-known Delsarte's
Theorem \cite{Delsarte}, one can prove that
\begin{eqnarray}\label{eqn-def-code-C}
{\mathcal C}_{(p,m,k)}=\{{\bf{c}}_{\Delta}: \Delta=(\delta_0,\delta_1,\delta_2) \in {\mathbb{F}}^3_{q}\}
\end{eqnarray}
where the codeword
\begin{eqnarray*}
{\bf{c}}_{\Delta}=\left({\rm Tr}(\delta_0 \pi^i+\delta_1 \pi^{id_1}+\delta_2 \pi^{id_2})\right)_{i=0}^{q-2}
\end{eqnarray*}
and ${\rm Tr}$ denotes the absolute trace from ${\mathbb{F}}_q$ to ${\mathbb{F}}_p$.
Let $h'(x)=h_1(x)h_{d_1}(x)$ and ${\mathcal C}_{(p,m,k)}'$ be the cyclic code with parity-check polynomial $h'(x)$.
Then ${\mathcal C}_{(p,m,k)}'$ is a subcode of ${\mathcal C}_{(p,m,k)}$ with dimension $2m$. Trachtenberg \cite{Trachtenberg} proved
that ${\mathcal C}_{(p,m,k)}'$ has three nonzero weights and determined its weight distribution.
The objectives of this paper are to show that ${\mathcal C}_{(p,m,k)}$ have five nonzero weights
and settle the weight distribution of this class of cyclic codes ${\mathcal C}_{(p,m,k)}$.
\section{Mathematical foundations}\label{sec-prelim}
In this section, we give a brief introduction to quadratic forms
over finite fields which will be useful in the sequel. Quadratic forms have
been well studied (see the monograph \cite{Niddle} and the references therein), and have applications
in sequence design (\cite{Trachtenberg}, \cite{Klapperodd}, \cite{Tang}), and coding theory (\cite{Feng07},
\cite{Luo081}, \cite{Luo082}, \cite{Zeng10}).
\begin{definition}
Let $x=\sum_{i=1}^mx_i\alpha_i$ where $x_i\in {\mathbb{F}}_p$ and
$\{\alpha_1,\cdots,\alpha_m\}$ is a basis for ${\mathbb{F}}_{q}$ over ${\mathbb{F}}_p$.
Then a function $Q(x)$ from ${\mathbb{F}}_{q}$ to ${\mathbb{F}}_p$ is a quadratic
form over ${\mathbb{F}}_{p}$ if it can be represented as
\begin{eqnarray*}
Q(x)=Q\left(\sum_{i=1}^m x_i\alpha_i\right)=\sum_{i=1}^m\sum_{j=1}^mb_{i,j}x_ix_j
\end{eqnarray*}
where $b_{i,j}\in {\mathbb{F}}_p$. That is, $Q(x)$ is a homogeneous
polynomial of degree 2 in the ring ${\mathbb{F}}_p[x_1,x_2,\cdots,x_m]$.
\end{definition}
The rank of the
quadratic form $Q(x)$ is defined as the codimension of the ${\mathbb{F}}_p$-vector space
\begin{eqnarray*}
V=\{z\in {\mathbb{F}}_{q}: Q(x+z)-Q(x)-Q(z)=0 \textrm{~for~all~}x\in {\mathbb{F}}_{q}\}.
\end{eqnarray*}
That is $|V|=p^{m-r}$ where $r$ is the rank of $Q(x)$.
In order to determine the weight distribution of the aforementioned code ${\mathcal C}_{(p,m,k)}$,
we need to deal with the
exponential sum of the following form:
\begin{eqnarray}\label{eqn_s_f}
S_{f}=\sum_{x\in {\mathbb{F}}_{q}}\zeta_p^{f(x)}
\end{eqnarray}
where $\zeta_p$ is a complex primitive $p$-th root of unity, and $f(x)$ is a
function from ${\mathbb{F}}_q$ to ${\mathbb{F}}_p$ satisfying
\begin{enumerate}
\item $f(yx)=yf(x)$ for all $y\in {\mathbb{F}}_p$;
\item $Q(x)=f(x^2)$ is a quadratic form over ${\mathbb{F}}_p$.
\end{enumerate}
Note that any nonsquare in ${\mathbb{F}}_p$
is also a nonsquare in ${\mathbb{F}}_q$ since $m$ is odd. It is easy to verify that
\begin{eqnarray}\label{eqn_oneform_twoform}
2S_f=\sum_{x\in {\mathbb{F}}_q}\zeta_p^{Q(x)}+\sum_{x\in {\mathbb{F}}_q}\zeta_p^{\lambda Q(x)}
\end{eqnarray}
where $\lambda$ is a fixed nonsquare in ${\mathbb{F}}_p$.
The following result can be traced back to Trachtenberg \cite{Trachtenberg}
whose proof is based on (\ref{eqn_oneform_twoform}) and the classification
of quadratic forms over finite fields in odd characteristic.
For more details, we refer the reader to Pages 30-36 in
\cite{Trachtenberg} and Lemma 4 in \cite{Tang}.
\begin{lemma}\label{Lemma-berg}
Let $S_f$ be defined
by (\ref{eqn_s_f}) and $r$ be the rank of the quadratic form $Q(x)=f(x^2)$.
Then $S_f=0$ if $r$ is odd, and $S_f=\pm p^{m-{r/2}}$ otherwise.
\end{lemma}
\section{The Weight distribution of the family of cyclic codes}\label{sec-wtds}
In this section, we shall establish the weight distribution
of the code ${\mathcal C}_{(p,m,k)}$ of (\ref{eqn-def-code-C}) defined in Section \ref{sec-code}.
To this end, we need a series of lemmas. Before introducing them, for any $\Delta=(\delta_0,\delta_1,\delta_2)\in {\mathbb{F}}_{q}^3$, we define
\begin{eqnarray}\label{eqn-f-Delta}
f_{\Delta}(x)={\rm Tr}(\delta_0 x+\delta_1 x^{d_1}+\delta_2 x^{d_2}), ~ x\in {\mathbb{F}}_q
\end{eqnarray}
and
\begin{eqnarray}\label{eqn-S-f-delta}
S_{f_{\Delta}}=\sum_{x\in {\mathbb{F}}_q}\zeta_p^{f_{\Delta}(x)}.
\end{eqnarray}
\begin{lemma}\label{lemma-rankmain}
Let $f_{\Delta}(x)$ be defined by (\ref{eqn-f-Delta}).
Then $f_{\Delta}(yx)=yf_{\Delta}(x)$ for any $y\in {\mathbb{F}}_p$.
And for any $\Delta\neq (0,0,0)$, the quadratic form
$Q_{\Delta}(x)=f_{\Delta}(x^2)$ has rank $m-i$ for some $0\leq i\leq 4$.
\end{lemma}
\begin{proof}
Recall that $d_1=(p^{2k}+1)/2$ and $d_2=(p^{4k}+1)/2$.
Thus $y^{d_1}=y$ and $y^{d_2}=y$ for any $y\in {\mathbb{F}}_p$.
This together with the linear properties of the trace function
means that $f_{\Delta}(yx)=yf_{\Delta}(x)$ for any $y\in {\mathbb{F}}_p$.
Clearly, $Q_{\Delta}(x)=f_{\Delta}(x^2)={\rm Tr}(\delta_0 x^2+\delta_1 x^{p^{2k}+1}+\delta_2 x^{p^{4k}+1})$
is a quadratic form over ${\mathbb{F}}_p$.
We now calculate the rank of $Q_{\Delta}(x)$.
Note that
\begin{eqnarray*}
Q_{\Delta}(x+z)-Q_{\Delta}(x)-Q_{\Delta}(z)={\rm Tr}(z L_{\Delta}(x))
\end{eqnarray*}
where
\begin{eqnarray*}
L_{\Delta}(x)=2\delta_0 x+\delta_1 x^{p^{2k}}+\delta_1^{p^{-2k}}x^{p^{-2k}}+\delta_2 x^{p^{4k}}+\delta_2^{p^{-4k}} x^{p^{-4k}}.
\end{eqnarray*}
We need to calculate the number of roots of the linearized polynomial $L_{\Delta}(x)$. Let $H_{\Delta}(x)=(L_{\Delta}(x))^{p^{4k}}$. Then
\begin{eqnarray}\label{eqn_g(z)}
H_{\Delta}(x)=\delta_2^{p^{4k}} x^{p^{8k}}+\delta_1^{p^{4k}} x^{p^{6k}}+\delta_1^{p^{2k}}x^{p^{2k}}+2\delta_0^{p^{4k}}x^{p^{4k}}+\delta_2 x.
\end{eqnarray}
Clearly, $L_{\Delta}(x)$ has the same number of roots in ${\mathbb{F}}_{p^m}$ as
$H_{\Delta}(x)$.
Fix an algebraic closure ${\mathbb{F}}_{p^\infty}$ of ${\mathbb{F}}_p$, then all
roots of $H_{\Delta}(x)$ form a vector space over ${\mathbb{F}}_{p^{2k}}$ of dimension at most
$4$ since its degree is at most $p^{8k}=(p^{2k})^4$ for any $(\delta_0,\delta_1,\delta_2)\neq (0,0,0)$.
Note that $\gcd(m,2k)=1$, it is straightforward (see Lemma 4,
\cite{Trachtenberg}) to verify that elements in ${\mathbb{F}}_{p^m}$
that are linearly independent over ${\mathbb{F}}_{p}$ are also linearly
independent over ${\mathbb{F}}_{p^{2k}}$. Therefore, the roots of $H_{\Delta}(x)$
in ${\mathbb{F}}_{p^m}$ form a vector space over ${\mathbb{F}}_{p}$ of
dimension at most $4$. Thus the rank of $Q_{\Delta}(x)$ is at
least $m-4$ for any $\Delta\neq (0,0,0)$. This completes the proof.
\end{proof}
\vspace{2mm}
\begin{lemma}\label{lem-N2}
Let $\mathfrak{N}_2$ denote the number of solutions $(x,y) \in {\mathbb{F}}_q^2$
of the following system of equations
\begin{eqnarray}\label{eqn-N2}
\left\{
\begin{array}{l}
x+y=0 \\
x^{d_1}+y^{d_1}=0 \\
x^{d_2}+y^{d_2}=0.
\end{array}
\right.
\end{eqnarray}
Then $\mathfrak{N}_2=q.$
\end{lemma}
\begin{proof}
The conclusion follows directly from the observation that $(x,y)$
is a solution of (\ref{eqn-N2}) if and only if $y=-x$.
\end{proof}
\begin{lemma}\label{lem-N3}
Let $\mathfrak{N}_3$ denote the number of solutions $(x,y,u) \in {\mathbb{F}}_q^3$
of the following system of equations
\begin{eqnarray}\label{eqn-N3}
\left\{
\begin{array}{l}
x+y+u=0 \\
x^{d_1}+y^{d_1}+u^{d_1}=0 \\
x^{d_2}+y^{d_2}+u^{d_2}=0.
\end{array}
\right.
\end{eqnarray}
Then $\mathfrak{N}_3=qp+q-p.$
\end{lemma}
\begin{proof}
We distinguish between the following two cases to calculate the number of solutions $(x,y,u) \in {\mathbb{F}}_q^3$
of (\ref{eqn-N3}).
{\textit{Case A}}, when $u=0$: In this case, by Lemma \ref{lem-N2},
the number of solutions of (\ref{eqn-N3}) is equal to $q$.
{\textit{Case B}}, when $u\neq 0$: In this case, for each $u\in {\mathbb{F}}^*_q$,
the equation system (\ref{eqn-N3}) has the same number of solutions $(x,y)\in {\mathbb{F}}_q^2$ of
\begin{eqnarray*}
\left\{
\begin{array}{l}
1+x+y=0 \\
1+x^{d_1}+y^{d_1}=0 \\
1+x^{d_2}+y^{d_2}=0
\end{array}
\right.
\end{eqnarray*}
which has the same number of solutions $x\in {\mathbb{F}}_q$ of
\begin{eqnarray}\label{eqn-N3-2}
\left\{
\begin{array}{l}
1+x^{d_1}=(1+x)^{d_1} \\
1+x^{d_2}=(1+x)^{d_2}.
\end{array}
\right.
\end{eqnarray}
By performing square on both sides of each equation in (\ref{eqn-N3-2}), we have
\begin{eqnarray*}
\left\{
\begin{array}{l}
x(x^{(p^{2k}-1)/2}-1)=0 \\
x(x^{(p^{4k}-1)/2}-1)=0 \\
\end{array}
\right.
\end{eqnarray*}
which implies that $x\in {\mathbb{F}}_p$ since $\gcd(m,2k)=\gcd(m,4k)=1$. Conversely,
for any $x\in {\mathbb{F}}_p$, it is clear that $x$ is a solution to (\ref{eqn-N3-2})
since $x^{d_i}=x$ and $(1+x)^{d_i}=1+x$ for each $i=1,2$. Thus (\ref{eqn-N3-2})
has exactly $p$ solutions.
Summarizing the results of the two cases above, we have that $\mathfrak{N}_3=q+(q-1)p=qp+q-p$. This completes the proof.
\end{proof}
\vspace{2mm}
The following lemma is
the key to establishing the weight distribution
of the proposed code ${\mathcal C}_{(p,m,k)}$. Its proof is lengthy and is
presented in Appendix I.
\vspace{2mm}
\begin{lemma}\label{lem-mainA}
Let $\mathfrak{N}_4$ denote the number of solutions $(x,y,u,v) \in {\mathbb{F}}_q^4$
of the following system of equations
\begin{eqnarray}\label{eqn-mainlemmaA}
\left\{
\begin{array}{l}
x+y+u+v=0 \\
x^{d_1}+y^{d_1}+u^{d_1}+v^{d_1}=0 \\
x^{d_2}+y^{d_2}+u^{d_2}+v^{d_2}=0.
\end{array}
\right.
\end{eqnarray}
Then $\mathfrak{N}_4=q(qp+q-p).$
\end{lemma}
\begin{proof}
See Appendix I.
\end{proof}
\vspace{2mm}
\begin{theorem}\label{theorem-ds-S}
Let $S_{f_{\Delta}}$ be defined by (\ref{eqn-S-f-delta}). Then, as $\Delta$ runs
through ${\mathbb{F}}_q^3$, the value distribution of $S_{f_{\Delta}}$ is given by Table
\ref{Tb_dis-S}.
\end{theorem}
\begin{table*}[!t]
\renewcommand{\arraystretch}{2}
\centering
\begin{threeparttable}
\caption{Value Distribution of $S_{f_{\Delta}}$}\label{Tb_dis-S}
\begin{tabular}{|l|l|}
\hline
Value& Frequency\\
\hline
\hline
$p^m$ & $1$ \\
\hline
$0$ & $(p^m-1)(p^{2m}-p^{2m-1}+p^{2m-4}+p^m-p^{m-1}-p^{m-3}+1)$\\
\hline
$p^{(m+1)/2}$ & $\frac{(p^{m+1}+p^{(m+3)/2})(p^{2m}-p^{2m-2}-p^{2m-3}+p^{m-2}+p^{m-3}-1)}{2(p^2-1)}$\\
\hline
$-p^{(m+1)/2}$ & $\frac{(p^{m+1}-p^{(m+3)/2})(p^{2m}-p^{2m-2}-p^{2m-3}+p^{m-2}+p^{m-3}-1)}{2(p^2-1)}$\\
\hline
$p^{(m+3)/2}$ & $\frac{(p^{m-3}+p^{(m-3)/2})(p^{m-1}-1)(p^m-1)}{2(p^2-1)}$\\
\hline
$-p^{(m+3)/2}$ & $\frac{(p^{m-3}-p^{(m-3)/2}(p^{m-1}-1)(p^m-1)}{2(p^2-1)}$\\
\hline
\end{tabular}
\begin{tablenotes}
\end{tablenotes}
\end{threeparttable}
\end{table*}
\begin{proof}
It is clear that $S_{f_{\Delta}}=p^m$ if $\Delta=(0,0,0)$.
Otherwise, by Lemmas \ref{lemma-rankmain} and \ref{Lemma-berg}, we have
\begin{eqnarray*}
S_{f_{\Delta}}\in \{0,\pm p^{(m+1)/2}, \pm p^{(m+3)/2}\}.
\end{eqnarray*}
To determine the distribution of these values, we define
\begin{eqnarray*}
n_{1,i}&=&\# \{\Delta \in {\mathbb{F}}_q^3\setminus \{(0,0,0)\}: ~~S_{f_{\Delta}}=(-1)^i p^{(m+1)/2}\},\\
n_{2,i}&=&\# \{\Delta \in {\mathbb{F}}_q^3\setminus \{(0,0,0)\}: ~~S_{f_{\Delta}}=(-1)^i p^{(m+3)/2}\}
\end{eqnarray*}
where $i=0,1$. Then the value distribution of $S_{f_{\Delta}}$ is as follows
\begin{eqnarray}\label{eqn-des-S}
\begin{array}{lcl}
p^m&\textrm{occurring}&1~{\textrm{time}}\\
p^{(m+1)/2}&\textrm{occurring}&n_{1,0}~{\textrm{times}}\\
-p^{(m+1)/2}&\textrm{occurring}&n_{1,1}~{\textrm{times}}\\
p^{(m+3)/2}&\textrm{occurring}&n_{2,0}~{\textrm{times}}\\
-p^{(m+3)/2}&\textrm{occurring}&n_{2,1}~{\textrm{times}}\\
0&\textrm{occurring}&p^{3m}-1-n_1-n_2~{\textrm{times}}.
\end{array}
\end{eqnarray}
By (\ref{eqn-des-S}), we immediately have
\begin{eqnarray}\label{eqn-total-S-1}
\left\{
\begin{array}{lcl}
\sum_{\Delta\in {\mathbb{F}}^3_q}S_{f_{\Delta}}&=&p^{m}+(n_{1,0}-n_{1,1})p^{(m+1)/2}+(n_{2,0}-n_{2,1})p^{(m+3)/2}\\
\sum_{\Delta\in {\mathbb{F}}^3_q}S^2_{f_{\Delta}}&=&p^{2m}+(n_{1,0}+n_{1,1})p^{m+1}+(n_{2,0}+n_{2,1})p^{m+3}\\
\sum_{\Delta\in {\mathbb{F}}^3_q}S^3_{f_{\Delta}}&=&p^{3m}+(n_{1,0}-n_{1,1})p^{(3m+3)/2}+(n_{2,0}-n_{2,1})p^{(3m+9)/2}\\
\sum_{\Delta\in {\mathbb{F}}^3_q}S^4_{f_{\Delta}}&=&p^{4m}+(n_{1,0}+n_{1,1})p^{2m+2}+(n_{2,0}+n_{2,1})p^{2m+6}.
\end{array}
\right.
\end{eqnarray}
On the other hand, applying Lemmas \ref{lem-N2}, \ref{lem-N3} and \ref{lem-mainA}, we have
\begin{eqnarray}\label{eqn-total-S-2}
\begin{array}{lcl}
\sum_{\Delta\in {\mathbb{F}}^3_q}S_{f_{\Delta}}&=&p^{3m}\\
\sum_{\Delta\in {\mathbb{F}}^3_q}S^2_{f_{\Delta}}&=&p^{4m}\\
\sum_{\Delta\in {\mathbb{F}}^3_q}S^3_{f_{\Delta}}&=&p^{3m}(p^{m+1}+p^m-p)\\
\sum_{\Delta\in {\mathbb{F}}^3_q}S^4_{f_{\Delta}}&=&p^{4m}(p^{m+1}+p^m-p).
\end{array}
\end{eqnarray}
Combining Equations (\ref{eqn-total-S-1}) and (\ref{eqn-total-S-2}) gives
$$
n_{1,0}= \frac{(p^{m+1}+p^{(m+3)/2})(p^{2m}-p^{2m-2}-p^{2m-3}+p^{m-2}+p^{m-3}-1)}{2(p^2-1)},
$$
$$
n_{1,1}= \frac{(p^{m+1}-p^{(m+3)/2})(p^{2m}-p^{2m-2}-p^{2m-3}+p^{m-2}+p^{m-3}-1)}{2(p^2-1)},
$$
$$
n_{2,0}= \frac{(p^{m-3}+p^{(m-3)/2})(p^{m-1}-1)(p^m-1)}{2(p^2-1)},
$$
$$
n_{2,1}= \frac{(p^{m-3}-p^{(m-3)/2})(p^{m-1}-1)(p^m-1)}{2(p^2-1)}.
$$
The value distribution of $S_{f_{\Delta}}$ depicted in Table \ref{Tb_dis-S} then follows
from the values of $n_{1,0}, n_{1,1}, n_{2,0}$ and $n_{2,1}$, and the analysis above.
\end{proof}
The following is the main result of the paper.
\begin{theorem}\label{Th_main1}
Let ${\mathcal C}_{(p,m,k)}$ be the code in (\ref{eqn-def-code-C}). Then ${\mathcal C}_{(p,m,k)}$ is a
cyclic code over ${\mathbb{F}}_p$ with parameters
$$
[p^m-1,3m,(p-1)(p^{m-1}-p^{(m+1)/2})].
$$
Furthermore, the weight distribution of ${\mathcal C}_{(p,m,k)}$ is given by Table \ref{Tb_wd}.
\end{theorem}
\begin{proof}
The length and dimension of the code follow directly from the definition of ${\mathcal C}_{(p,m,k)}$. We only need
to determine its minimal weight and weight distribution.
In terms of exponential sums, the weight of the codeword ${\bf{c}}_{\Delta}$ in ${\mathcal C}_{(p,m,k)}$ is given by
\begin{eqnarray}\label{eqn-weight}
{\textrm{WT}}({\bf c}_{\Delta})&=&\#\{x\in {\mathbb{F}}^*_q: ~~{\rm Tr}(\delta_0 x+ \delta_1 x^{d_1}+ \delta_2 x^{d_2})\neq 0\} \nonumber\\
&=&q-1-\#\{x\in {\mathbb{F}}^*_q: ~~{\rm Tr}(\delta_0 x+ \delta_1 x^{d_1}+ \delta_2 x^{d_2})= 0\} \nonumber\\
&=&q-1-{1\over p}\sum_{x\in {\mathbb{F}}_q^*}\sum_{y\in {\mathbb{F}}_p}\zeta_p^{y{\rm Tr}(\delta_0 x+ \delta_1 x^{d_1}+ \delta_2 x^{d_2})} \nonumber\\
&=&p^{m}-p^{m-1}-{1\over p}\sum_{y\in {\mathbb{F}}^*_p}\sum_{x\in {\mathbb{F}}_q}\zeta_p^{{\rm Tr}(\delta_0 yx+ \delta_1 yx^{d_1}+ \delta_2 yx^{d_2})}\nonumber \\
&=&(p-1)p^{m-1}-{1\over p}\sum_{y\in {\mathbb{F}}^*_p}\sum_{x\in {\mathbb{F}}_q}\zeta_p^{{\rm Tr}(\delta_0 yx+ \delta_1 (yx)^{d_1}+ \delta_2 (yx)^{d_2})} \nonumber\\
&=&(p-1)p^{m-1}-{1\over p}\sum_{y\in {\mathbb{F}}^*_p}\sum_{x\in {\mathbb{F}}_q}\zeta_p^{{\rm Tr}(\delta_0 x+ \delta_1 x^{d_1}+ \delta_2 x^{d_2})}\nonumber \\
&=&(p-1)p^{m-1}-{p-1\over p}S_{f_{\Delta}}
\end{eqnarray}
where $S_{f_{\Delta}}$ is given by (\ref{eqn-S-f-delta}) and in the fifth identity we used the fact that
$y^{d_i}=y$ for any $y\in {\mathbb{F}}_p$. The minimal weight and weight distribution of ${\mathcal C}_{(p,m,k)}$ then follow from (\ref{eqn-weight})
and the value distribution of the exponential sum $S_{f_{\Delta}}$ depicted in Table \ref{Tb_dis-S}.
\end{proof}
\vspace{2mm}
\begin{table*}[!t]
\renewcommand{\arraystretch}{2}
\centering
\begin{threeparttable}
\caption{Weight Distribution of the Code ${\mathcal C}_{(p,m,k)}$ in Theorem \ref{Th_main1}}\label{Tb_wd}
\begin{tabular}{|l|l|}
\hline
Hamming Weight& Frequency\\
\hline
\hline
$0$ & $1$ \\
\hline
$(p-1)p^{m-1}$ & $(p^m-1)(p^{2m}-p^{2m-1}+p^{2m-4}+p^m-p^{m-1}-p^{m-3}+1)$\\
\hline
$(p-1)(p^{m-1}-p^{(m-1)/2})$ & $\frac{(p^{m+1}+p^{(m+3)/2})(p^{2m}-p^{2m-2}-p^{2m-3}+p^{m-2}+p^{m-3}-1)}{2(p^{2}-1)}$\\
\hline
$(p-1)(p^{m-1}+p^{(m-1)/2})$ & $\frac{(p^{m+1}-p^{(m+3)/2})(p^{2m}-p^{2m-2}-p^{2m-3}+p^{m-2}+p^{m-3}-1)}{2(p^{2}-1)}$\\
\hline
$(p-1)(p^{m-1}-p^{(m+1)/2})$ & $\frac{(p^{m-3}+p^{(m-3)/2})(p^{m-1}-1)(p^m-1)}{2(p^{2}-1)}$\\
\hline
$(p-1)(p^{m-1}+p^{(m+1)/2})$ & $\frac{(p^{m-3}-p^{(m-3)/2})(p^{m-1}-1)(p^m-1)}{2(p^{2}-1)}$\\
\hline
\end{tabular}
\begin{tablenotes}
\end{tablenotes}
\end{threeparttable}
\end{table*}
\begin{example}
Let $p=3$, $m=5$ and $k=1$. Then the code ${\mathcal C}_{(p,m,k)}$ is
a $[242,15,108]$ code over ${\mathbb{F}}_3$ with the weight enumerator
\begin{eqnarray*}
1+14520 z^{108}+ 2548260 z^{144}+9740258 z^{162}+ 2038608 z^{180}+7260 z^{216}
\end{eqnarray*}
which confirms the weight distribution in Table \ref{Tb_wd}.
\end{example}
\begin{example}
Let $p=3$, $m=7$ and $k=2$. Then the code ${\mathcal C}_{(p,m,k)}$ is
a $[2186,21,1296]$ code over ${\mathbb{F}}_3$ with the weight enumerator
\begin{eqnarray*}
1+8951670 z^{1296}+ 1732767876 z^{1404}+7102473578 z^{1458}+ 1608998742 z^{1512}+7161336 z^{1620}
\end{eqnarray*}
which confirms the weight distribution in Table \ref{Tb_wd}.
\end{example}
\begin{example}
Let $p=5$, $m=5$ and $k=1$. Then the code ${\mathcal C}_{(p,m,k)}$ is
a $[3124,15,2000]$ code over ${\mathbb{F}}_3$ with the weight enumerator
\begin{eqnarray*}
1+1218360 z^{2000}+ 3147430000 z^{2400}+24462797524 z^{2500}+ 2905320000 z^{2600}+812240 z^{3000}
\end{eqnarray*}
which confirms the weight distribution in Table \ref{Tb_wd}.
\end{example}
\section{Summary and concluding remarks}\label{sec-conclusion}
In this paper, we studied a family of five-weight cyclic codes. The duals of the cyclic codes have three zeros.
The weight distribution of this family of cyclic codes is completely determined.
Finally we mention that the weight distribution of ${\mathcal C}_{(p,m,k)}$ can also be settled in a more general case
where $m/\gcd(m,k)$ is odd. In what follows we only report the conclusion. The proof is similar to that of
Theorem \ref{Th_main1}.
\begin{theorem}\label{Th_main2}
Let $\gcd(m,k)=e$, $m/e$ be odd, and $m/e\geq 5$. Let ${\mathcal C}_{(p,m,k)}$ be the code in (\ref{eqn-def-code-C}). Then ${\mathcal C}_{(p,m,k)}$ is a
cyclic code over ${\mathbb{F}}_p$ with parameters
$$
[p^m-1,3m,(p-1)(p^{m-1}-p^{(m+3e-2)/2})].
$$
Furthermore, the weight distribution of ${\mathcal C}_{(p,m,k)}$ is given by Table \ref{Tb_wd2}.
\end{theorem}
\begin{table*}[!t]
\renewcommand{\arraystretch}{2}
\centering
\begin{threeparttable}
\caption{Weight Distribution of the Code ${\mathcal C}_{(p,m,k)}$ in Theorem \ref{Th_main2}}\label{Tb_wd2}
\begin{tabular}{|l|l|}
\hline
Hamming Weight& Frequency\\
\hline
\hline
$0$ & $1$ \\
\hline
$(p-1)p^{m-1}$ & $(p^m-1)(p^{2m}-p^{2m-e}+p^{2m-4e}+p^m-p^{m-e}-p^{m-3e}+1)$\\
\hline
$(p-1)(p^{m-1}-p^{(m+e-2)/2})$ & $\frac{(p^{m+e}+p^{(m+3e)/2})(p^{2m}-p^{2m-2e}-p^{2m-3e}+p^{m-2e}+p^{m-3e}-1)}{2(p^{2e}-1)}$\\
\hline
$(p-1)(p^{m-1}+p^{(m+e-2)/2})$ & $\frac{(p^{m+e}-p^{(m+3e)/2})(p^{2m}-p^{2m-2e}-p^{2m-3e}+p^{m-2e}+p^{m-3e}-1)}{2(p^{2e}-1)}$\\
\hline
$(p-1)(p^{m-1}-p^{(m+3e-2)/2})$ & $\frac{(p^{m-3e}+p^{(m-3e)/2})(p^{m-e}-1)(p^m-1)}{2(p^{2e}-1)}$\\
\hline
$(p-1)(p^{m-1}+p^{(m+3e-2)/2})$ & $\frac{(p^{m-3e}-p^{(m-3e)/2})(p^{m-e}-1)(p^m-1)}{2(p^{2e}-1)}$\\
\hline
\end{tabular}
\begin{tablenotes}
\end{tablenotes}
\end{threeparttable}
\end{table*}
\section*{Appendix I}\label{Appendix-I}
{\em Proof of Lemma \ref{lem-mainA}}:
For any $(\bar{a}, \bar{b},\bar{c}) \in {\mathbb{F}}_q^3$, let $\bar{N}_{(\bar{a}, \bar{b},\bar{c})}$ denote the number of
solutions $(x,y,u,v) \in {\mathbb{F}}_q^4$
of the following system of equations
\begin{eqnarray}\label{eqn-mainlemmaA2}
\left\{
\begin{array}{l}
x+y=\bar{a} \\
x^{d_1}+y^{d_1}=\bar{b} \\
x^{d_2}+y^{d_2}=\bar{c} \\
u+v=-\bar{a} \\
u^{d_1}+v^{d_1}=-\bar{b} \\
u^{d_2}+v^{d_2}=-\bar{c}.
\end{array}
\right.
\end{eqnarray}
It is then obvious that
$$
\mathfrak{N}_4=\sum_{(\bar{a}, \bar{b},\bar{c}) \in {\mathbb{F}}_q^3} \bar{N}_{(\bar{a}, \bar{b},\bar{c})}.
$$
\par
For any $(\bar{a}, \bar{b},\bar{c}) \in {\mathbb{F}}_q^3$, let $\hat{N}_{(\bar{a}, \bar{b},\bar{c})}$ denote the number of
solutions $(x,y) \in {\mathbb{F}}_q^2$
of the following system of equations
\begin{eqnarray}\label{eqn-mainlemmaA3}
\left\{
\begin{array}{l}
x+y=\bar{a} \\
x^{d_1}+y^{d_1}=\bar{b} \\
x^{d_2}+y^{d_2}=\bar{c}.
\end{array}
\right.
\end{eqnarray}
Since $d_1$ and $d_2$ are odd, $\bar{N}_{(\bar{a}, \bar{b},\bar{c})}=\left(\hat{N}_{(\bar{a}, \bar{b},\bar{c})}\right)^2$, we have
\begin{eqnarray}\label{eqn-N41}
\mathfrak{N}_4=\sum_{(\bar{a}, \bar{b},\bar{c}) \in {\mathbb{F}}_q^3} \left(\hat{N}_{(\bar{a}, \bar{b},\bar{c})}\right)^2.
\end{eqnarray}
We distinguish among the following three cases to calculate $\hat{N}_{(\bar{a}, \bar{b},\bar{c})}$.
{\textit{Case A}}, when $\bar{a}=\bar{b}=\bar{c}=0$: In this case, $\hat{N}_{(0, 0,0)}=q$
since $(x,y)$ is a solution of (\ref{eqn-mainlemmaA3}) if and only if $y=-x$. Thus $(\hat{N}_{(0, 0,0)})^2=q^2$.
{\textit{Case B}}, when $\bar{a}\neq 0$, and ($\bar{b}=0$ or $\bar{c}=0$): In this case,
it is clear that $\hat{N}_{(\bar{a}, \bar{b},\bar{c})}=0$.
{\textit{Case C}}, when $\bar{a} \ne 0$, $\bar{b} \ne 0$ and $\bar{c} \ne 0$. In this case,
for any given $\bar{a}\neq 0$, Equation System (\ref{eqn-mainlemmaA3}) has the same number of solutions as
\begin{eqnarray}\label{eqn-mainlemmaB}
\left\{
\begin{array}{l}
x+y =1 \\
x^{d_1} + {y}^{d_1} =b \\
x^{d_2} + {y}^{d_2} =c
\end{array}
\right.
\end{eqnarray}
where $b={\bar{b}}/{\bar{a}^{d_1}}$ and $c={\bar{c}}/{\bar{a}^{d_2}}$.
Clearly, $(b,c)$ runs over ${\mathbb{F}}^*_q\times {\mathbb{F}}^*_q$ as $(\bar{b}, \bar{c})$ does.
By Lemma \ref{lem-mainB}, we have
\begin{eqnarray*}
\sum_{(\bar{a}, \bar{b},\bar{c}) \in ({\mathbb{F}}^*_q)^3}\left(\hat{N}_{(\bar{a}, \bar{b},\bar{c})}\right)^2 =(q-1) \left[p^2+(p+1)^2 \frac{q-p}{2(p+1)} + (p-1)^2 \frac{q-p}{2(p-1)} \right]=q(qp-p).
\end{eqnarray*}
Summarizing all the cases above, we have
\begin{eqnarray*}
\mathfrak{N}_4=q^2 + q(qp-p)=q(qp+q-p).
\end{eqnarray*}
This completes the proof.
\mbox
\begin{lemma}\label{lem-mainB}
Let $\mathsf{N}_{(b,c)}$ denote the number of solutions $(x,y) \in {\mathbb{F}}_q^2$
of (\ref{eqn-mainlemmaB}),
where $(b, c) \in {\mathbb{F}}^*_q\times {\mathbb{F}}^*_q$. Then we have the following conclusions.
\begin{itemize}
\item[B1] $\mathsf{N}_{(1,1)}=p$.
\item[B2] When $(b,c)$ runs over ${\mathbb{F}}_q^* \times {\mathbb{F}}_q^* \setminus \{(1,1)\}$,
\begin{eqnarray*}
\mathsf{N}_{(b,c)} = \left\{
\begin{array}{ll}
p+1 & \mbox{ for } \frac{q-p}{2(p+1)} \mbox{ times} \\
p-1 & \mbox{ for } \frac{q-p}{2(p-1)} \mbox{ times} \\
0 & \mbox{ for the rest.}
\end{array}
\right.
\end{eqnarray*}
\end{itemize}
\end{lemma}
The proof of Lemma \ref{lem-mainB} is lengthy and technical. We first prove some auxiliary results.
\section*{Auxiliary results for proving Lemma \ref{lem-mainB}}
We prove Lemma \ref{lem-mainB} only for the case that $p \equiv 3 \pmod{4}$. The proof
for the case $p \equiv 1 \pmod{4}$ is similar and omitted. Hence we assume that $p \equiv 3 \pmod{4}$
from now on.
\subsection{Case 1}
In (\ref{eqn-mainlemmaB}) we substitute $(x, y)$ with $(x_1^2, -y_1^2)$ and obtain the
following system of equations
\begin{eqnarray}\label{eqn-mainlemmaB11}
\left\{
\begin{array}{l}
x_1^2-y_1^2=1 \\
x_1^{2d_1}-y_1^{2d_1}=b \\
x_1^{2d_2}-y_1^{2d_2}=c
\end{array}
\right.
\end{eqnarray}
where $b, c \in {\mathbb{F}}_q^*$. Our task is to compute the number $\mathtt{N}_{(b,c)}$ of solutions $(x_1, y_1) \in {\mathbb{F}}_q^2$
of (\ref{eqn-mainlemmaB11}). To this end, we first compute the number $\mathtt{N}_{b}$ of solutions
$(x_1, y_1) \in {\mathbb{F}}_q^2$ of the
following system of equations
\begin{eqnarray}\label{eqn-mainlemmaB12}
\left\{
\begin{array}{l}
x_1^2-y_1^2=1 \\
x_1^{2d_1}-y_1^{2d_1}=b
\end{array}
\right.
\end{eqnarray}
where $b \in {\mathbb{F}}_q^*$.
\begin{lemma}\label{lem-case11}
Let symbols and notations be the same as before. As for Equation (\ref{eqn-mainlemmaB12}), we have
\begin{eqnarray*}
\mathtt{N}_{b} = \left\{
\begin{array}{ll}
p-1 & \mbox{ if } b=1 \\
2(p-1) & \mbox{ for } \frac{q-p}{2(p-1)} \mbox{ elements } b \ne 1 \\
0 & \mbox{ for the rest } b \neq 1.
\end{array}
\right.
\end{eqnarray*}
\end{lemma}
\begin{proof}
Let $(x_1,y_1)$ be a solution of the first equation in (\ref{eqn-mainlemmaB12}).
It is clear that $x_1\neq y_1$.
Let $\theta=x_1-y_1$. It then follows that $\theta\in {\mathbb{F}}^*_{q}$ and
\begin{eqnarray}\label{theta-case11}
x_1=\frac{\theta+\theta^{-1}}{2}, \ y_1=\frac{\theta^{-1}-\theta}{2}.
\end{eqnarray}
Thus $(x_1,y_1)$ is uniquely determined by $\theta$.
Substituting (\ref{theta-case11}) into the second equation of (\ref{eqn-mainlemmaB12}), we obtain
\begin{eqnarray}\label{eqn-w-1}
\theta^{p^{2k}-1} + \theta^{1-p^{2k}} =2b.
\end{eqnarray}
Let $w=\theta^{p^{2k}-1}$. Then (\ref{eqn-w-1}) is equivalent to
\begin{equation}\label{eqn-quadratic-w1}
w^2-2{b} w+1=0.
\end{equation}
If (\ref{eqn-quadratic-w1}) has no solution, i.e., $b^2-1$ is not a square in ${\mathbb{F}}_q^*$, then $\mathtt{N}_{b}=0$.
Otherwise, suppose that $w_1$ and $w_2=w_1^{-1}$ are two solutions of (\ref{eqn-quadratic-w1}).
We then have
\begin{eqnarray}\label{eqn-wtheta-1}
\theta^{p^{2k}-1}=w_1
\end{eqnarray}
or
\begin{eqnarray}\label{eqn-wtheta-2}
\theta^{p^{2k}-1}=w_1^{-1}.
\end{eqnarray}
Clearly, (\ref{eqn-wtheta-1}) and (\ref{eqn-wtheta-2}) have the same number of solutions $\theta\in {\mathbb{F}}_q$.
Note that $\gcd(p^{2k}-1, q-1)=p-1$. Thus both (\ref{eqn-wtheta-1}) and (\ref{eqn-wtheta-2})
have no solution or exactly $p-1$ solutions. If $w_1=w_1^{-1}$, then $w_1=\pm 1$ and $b=\pm 1$.
However $-1$ is not a square, thus, $w_1=1$ and $b=1$. In this case, (\ref{eqn-wtheta-1}) and (\ref{eqn-wtheta-2})
become the same equation and have $p-1$ solutions.
If $w_1\neq w_1^{-1}$, then (\ref{eqn-wtheta-1}) and (\ref{eqn-wtheta-2}) have distinct solutions.
Based on above analysis, we conclude
$$
\mathtt{N}_{1}=p-1 {\textrm{~and~}} \mathtt{N}_{b}=0 {~\textrm{or}~2(p-1)} \textrm{~for~}b\neq 1.
$$
Define
\[T=\#\{b\in {\mathbb{F}}_q: N_{b}=2(p-1)\}.\]
Note that the first equation in (\ref{eqn-mainlemmaB12}) has $q-1$ solutions in ${\mathbb{F}}_q$ thanks to
Lemma 6.24 in \cite{Niddle}.
When $(x,y)$
runs through all these solutions, the second equation in (\ref{eqn-mainlemmaB12}) will
give a $2(p-1)$-to-$1$ correspondence
\[(x,y)\mapsto b=x^{p^{2k}+1}-y^{p^{2k}+1}\]
if $N_{b}=2(p-1)$. Therefore
\[(p-1)+2(p-1)T=q-1\]
which leads to
\[T=\frac{q-p}{2(p-1)}.\]
This completes the proof.
\end{proof}
\begin{lemma}\label{lem-case12}
Let symbols and notations be the same as before. As for Equation System (\ref{eqn-mainlemmaB11}), we have
\begin{eqnarray*}
\mathtt{N}_{(b,c)} = \left\{
\begin{array}{ll}
p-1 & \mbox{ if } (b,c)=(1,1) \\
2(p-1) & \mbox{ for } \frac{q-p}{2(p-1)} \mbox{ pairs } (b,c) \ne (1,1) \\
0 & \mbox{ for the rest pairs} (b,c) \in ({\mathbb{F}}_q^* )^2 \setminus \{(1,1)\}.
\end{array}
\right.
\end{eqnarray*}
\end{lemma}
\begin{proof}
Let $(x_1, y_1)$ be any solution of (\ref{eqn-mainlemmaB11}). Let $\theta=x_1-y_1$. It then follows from
the first equation in (\ref{eqn-mainlemmaB12}) that
\begin{eqnarray}\label{theta-case1}
x_1=\frac{\theta+\theta^{-1}}{2}, \ y_1=\frac{\theta^{-1}-\theta}{2}.
\end{eqnarray}
Using the second and third equations in (\ref{eqn-mainlemmaB11}), we obtain
\begin{eqnarray*}
\left\{
\begin{array}{l}
b =\frac{1}{2} \left( \theta^{p^{2k}-1} + \theta^{1-p^{2k}} \right) \\
c =\frac{1}{2} \left( \theta^{p^{4k}-1} + \theta^{1-p^{4k}} \right).
\end{array}
\right.
\end{eqnarray*}
Let $w=\theta^{p^{2k}-1}$ and
\begin{eqnarray}\label{eqn-Case1D1}
\left\{
\begin{array}{l}
\tilde{b}= \left( \theta^{p^{2k}-1} + \theta^{1-p^{2k}} \right)=w+w^{-1} \\
\tilde{c}= \left( \theta^{p^{4k}-1} + \theta^{1-p^{4k}} \right)=w^{p^{2k}+1}+(w^{-1})^{p^{2k}+1}.
\end{array}
\right.
\end{eqnarray}
Then $w^2-\tilde{b} w +1=0$ and
$$
w=\frac{\tilde{b}}{2} \pm \sqrt{ \left(\frac{\tilde{b}}{2}\right)^2 -1 }.
$$
It follows from the first equation in (\ref{eqn-Case1D1}) that
\begin{eqnarray}\label{eqn-Case1D2}
\tilde{b}^{p^{2k}}=w^{p^{2k}}+(w^{-1})^{p^{2k}}.
\end{eqnarray}
Combining the first equation in (\ref{eqn-Case1D1}) and (\ref{eqn-Case1D2}), we obtain
$$
\tilde{b}^{p^{2k}+1}=\tilde{c} + w^{p^{2k}-1}+(w^{-1})^{p^{2k}-1}.
$$
Whence,
\begin{eqnarray}\label{eqn-426}
\tilde{c} = \tilde{b}^{p^{2k}+1} - \left( w^{p^{2k}-1}+(w^{-1})^{p^{2k}-1} \right).
\end{eqnarray}
Note that
$$
w=\frac{\tilde{b}}{2} \pm \sqrt{ \left(\frac{\tilde{b}}{2}\right)^2 -1 }
$$
if and only if
$$
w^{-1}=\frac{\tilde{b}}{2} \mp \sqrt{ \left(\frac{\tilde{b}}{2}\right)^2 -1}.
$$
By (\ref{eqn-426}), $\tilde{c}$ is uniquely determined by $\tilde{b}$.
Therefore, $c$ is uniquely determined by $b$.
In addition, it is easily seen that $\tilde{c}=2$ if and only if $\tilde{b}=2$.
Hence the number of solutions of (\ref{eqn-mainlemmaB11}) is the same as that
of (\ref{eqn-mainlemmaB12}). The desired conclusions then follow from Lemma
\ref{lem-case11}.
\end{proof}
\begin{lemma}\label{lem-case13}
Let $\mathtt{M}_{(b,c)}$ denote the number of solutions $(x, y)$ of (\ref{eqn-mainlemmaB})
such that $x$ is a square and $y$ is a nonquare or $y=0$. Then
\begin{eqnarray*}
\mathtt{M}_{(b,c)} = \left\{
\begin{array}{ll}
\frac{p+1}{4} & \mbox{ if } (b,c)=(1,1) \\
\frac{p-1}{2} & \mbox{ for } \frac{q-p}{2(p-1)} \mbox{ pairs } (b,c) \ne (1,1) \\
0 & \mbox{ for the rest pairs} (b,c) \in ({\mathbb{F}}_q^* )^2 \setminus \{(1,1)\}.
\end{array}
\right.
\end{eqnarray*}
\end{lemma}
\begin{proof}
Consider now the solutions of (\ref{eqn-mainlemmaB11}). If $(x_1, y_1)$ is a solution
of (\ref{eqn-mainlemmaB11}), so are $(-x_1, y_1)$, $(x_1, -y_1)$ and $(-x_1, -y_1)$.
If $y_1 \ne 0$, they are indeed four different solutions of (\ref{eqn-mainlemmaB11}),
but give only one solution of (\ref{eqn-mainlemmaB}).
Since $-1$ is a quadratic nonresidue in ${\mathbb{F}}_q$, $x_1 \ne 0$. However, it is possible that $y_1=0$.
If $y_1=0$, then $(b,c)=(1,1)$. In this case, we have two special solutions $(\pm1, 0)$
of (\ref{eqn-mainlemmaB11}). They give only one solution of (\ref{eqn-mainlemmaB}).
It then follows from Lemma \ref{lem-case12} that
$$
\mathtt{M}_{(1,1)} = \frac{\mathtt{N}_{(1,1)}-2}{4}+1=\frac{p-3}{4} +1=\frac{p+1}{4}
$$
and
\begin{eqnarray*}
\mathtt{M}_{(b,c)} &=& \frac{\mathtt{N}_{(b,c)}}{4} \\
&=& \left\{
\begin{array}{ll}
\frac{p-1}{2} & \mbox{ for } \frac{q-p}{2(p-1)} \mbox{ pairs } (b,c) \ne (1,1) \\
0 & \mbox{ for the rest pairs} (b,c) \in ({\mathbb{F}}_q^* )^2 \setminus \{(1,1)\}.
\end{array}
\right.
\end{eqnarray*}
The proof is then completed.
\end{proof}
\subsection{Case 2}
\begin{lemma}\label{lem-case23}
Let $\mathtt{M}_{(b,c)}$ denote the number of solutions $(x, y)$ of (\ref{eqn-mainlemmaB})
such that $y$ is a square and $x$ is a nonsquare or $x=0$. Then
\begin{eqnarray*}
\mathtt{M}_{(b,c)} = \left\{
\begin{array}{ll}
\frac{p+1}{4} & \mbox{ if } (b,c)=(1,1) \\
\frac{p-1}{2} & \mbox{ for } \frac{q-p}{2(p-1)} \mbox{ pairs } (b,c) \ne (1,1) \\
0 & \mbox{ for the rest pairs} (b,c) \in ({\mathbb{F}}_q^* )^2 \setminus \{(1,1)\}.
\end{array}
\right.
\end{eqnarray*}
\end{lemma}
This case is symmetric to Case 1. Hence the proof of this lemma is similar to that of Lemma \ref{lem-case13}
and is omitted.
\subsection{Case 3}
In (\ref{eqn-mainlemmaB}) we substitute $(x, y)$ with $(x_1^2, y_1^2)$ and obtain the
following system of equations
\begin{eqnarray}\label{eqn-mainlemmaB31}
\left\{
\begin{array}{l}
x_1^2+y_1^2=1 \\
x_1^{2d_1}+y_1^{2d_1}=b \\
x_1^{2d_2}+y_1^{2d_2}=c
\end{array}
\right.
\end{eqnarray}
where $b, c \in {\mathbb{F}}_q^*$. Our task is to compute the number $N_{(b,c)}$ of solutions $(x_1, y_1) \in {\mathbb{F}}_q^2$
of (\ref{eqn-mainlemmaB31}). To this end, we first compute the number $\mathtt{N}_{b}$ of solutions
$(x_1, y_1) \in {\mathbb{F}}_q^2$ of the
following system of equations
\begin{equation}\label{origin}
\left\{
\begin{array}{l}
x^2+y^2=1 \\
x^{p^{2k}+1}+y^{p^{2k}+1}=b
\end{array}
\right.
\end{equation}
where $b \in {\mathbb{F}}_q^*$.
\begin{lemma}\label{lem-case31}
Let symbols and notations be the same as before. As for Equation (\ref{origin}), we have
\[
N_{b}= \left\{
\begin{array}{ll}
p+1 &\text{if}\; b=1 \\
2(p+1) & \text{for}\;\frac{q-p}{2(p+1)}\;\text{elements}\; b\neq 1 \\
0 &\text{for the rest}\; b\neq 1.
\end{array}
\right.
\]
\end{lemma}
\begin{proof}
Choose $t\in {\mathbb{F}}_{p^2}$ such that $t^2=-1$. From
\begin{equation}\label{first}
x^2+y^2=1
\end{equation}
we can assume
\begin{equation}\label{theta}
x=\frac{\theta+\theta^{-1}}{2},\qquad y=\frac{t(\theta-\theta^{-1})}{2}
\end{equation}
with $\theta\in {\mathbb{F}}_{q^2}^*$. It is easy to see that all the
solutions $(x,y)\in {\mathbb{F}}^2_q$ of (\ref{first}) can be expressed as in
(\ref{theta}) with a unique $\theta\in {\mathbb{F}}_{q^2}^*$.
Substituting (\ref{theta}) into
\begin{equation}\label{second}
x^{p^{2k}+1}+y^{p^{2k}+1}=b,
\end{equation}
we obtain
\begin{equation}\label{b}
\theta^{p^{2k}-1}+\theta^{1-p^{2k}}=2b.
\end{equation}
Denote by $w=\theta^{p^{2k}-1}$. Then (\ref{b}) is equivalent to
\begin{equation}\label{quadratic}
w^2-2b w+1=0.
\end{equation}
Let $w_1$ and $w_2=w_1^{-1}$ be two solutions of (\ref{quadratic}).
Then we have $w_1\in {\mathbb{F}}_{q^2}^*$.
From (\ref{theta}) and $x\in {\mathbb{F}}_q$ we have
$$
\theta+\theta^{-1}=(\theta+\theta^{-1})^q=\theta^q+\theta^{-q}
$$
which implies
\[\theta^{q+1}=1\;\text{or}\;\theta^{q-1}=1.\]
\begin{itemize}
\item If $\theta^{q+1}=1$, then $y^q=\frac{t^q(\theta^q-\theta^{-q})}{2}=\frac{t(\theta-\theta^{-1})}{2}=y$ since $t^q=-t$. It follows that $y\in {\mathbb{F}}_q$.
For a fixed $b$, recall that $w_1$ and $w_2=w_1^{-1}$ are two solutions of
(\ref{quadratic}). Then we have
\begin{equation}\label{end1}
\theta^{p^{2k}-1}=w_1,\quad \theta^{q+1}=1
\end{equation}
or
\begin{equation}\label{end1'}
\theta^{p^{2k}-1}=w_1^{-1},\quad \theta^{q+1}=1.
\end{equation}
If $\theta_1$ and $\theta_2$ are two solutions of (\ref{end1}),
then $(\theta_1/\theta_2)^{p^{2k}-1}=(\theta_1/\theta_2)^{q+1}=1$
which is equivalent to $(\theta_1/\theta_2)^{p+1}=1$. As a
consequence,
if (\ref{end1}) has solutions, then it has exactly $p+1$
solutions.
If $w_1=w_1^{-1}$, then (\ref{end1'}) is the same with
(\ref{end1}) and apparently it gives no more solutions. In this
case $w_1=\pm 1$ and $b=\pm 1$. But $b=-1$ can be excluded
since, otherwise, $w_1=-1$, then
$\theta^{p^{2k}-1}=-1$ which contradicts to $\theta\in {\mathbb{F}}_{p^{2m}}$.
The remaining case is $b=1$ which corresponds to $w_1=1$. In
this case we have $p+1$ solutions of $\theta$ which gives
exactly the same number of solutions of (\ref{origin}).
If $w_1\neq w_1^{-1}$, then (\ref{end1'}) has the same number
of solutions as (\ref{end1}) and moreover, their solutions are
distinct. Therefore (\ref{end1}) and (\ref{end1'}) both have
$p+1$ solutions or no solutions in ${\mathbb{F}}_{q^2}$.
\item If $\theta^{q-1}=1$ and $\theta^{q+1}\neq 1$, then $\theta\in {\mathbb{F}}_q^*$. Note that $t\notin{\mathbb{F}}_q^*$,
$y=\frac{t(\theta-\theta^{-1})}{2}$ is not in ${\mathbb{F}}_q^*$ except for
$\theta=\theta^{-1}=\pm 1$. But the exception case will not occur
since $\theta^{q+1}\neq 1$.
\end{itemize}
Summarizing up, we conclude
\[N_1=p+1 \textrm{~and~} N_{b}=0\;\text{or}\;2(p+1)\;\text{for}\; b\neq 1.\]
Define
\[T=\#\{b\in {\mathbb{F}}_q: N_{b}=2(p+1)\}.\]
Note that (\ref{first}) has $q+1$ solutions in ${\mathbb{F}}_q$ thanks to
Lemma 6.24 in \cite{Niddle}. When $(x,y)$
runs through all these solutions, the equation (\ref{second}) will
give a $2(p+1)$-to-$1$ correspondence
\[(x,y)\mapsto b=x^{p^{2k}+1}+y^{p^{2k}+1}\]
if $N_{b}=2(p+1)$. Therefore
\[(p+1)+2(p+1)T=q+1\]
which implies
\[T=\frac{q-p}{2(p+1)}.\]
The proof is now finished.
\end{proof}
\begin{lemma}\label{lem-case32}
Let symbols and notations be the same as before. As for Equation System (\ref{eqn-mainlemmaB31}), we have
\[
N_{(b, c)}= \left\{
\begin{array}{ll}
p+1 &\text{if } (b, c)=(1,1) \\
2(p+1) & \text{for~} \frac{q-p}{2(p+1)}\;\text{ pairs } (b, c) \neq (1,1) \\
0 &\text{for the rest } (b, c) \neq (1,1).
\end{array}
\right.
\]
\end{lemma}
\begin{proof}
The proof of this lemma is similar to that of Lemma \ref{lem-case12} and is derived from Lemma
\ref{lem-case31}. The details of the proof is omitted here.
\end{proof}
\begin{lemma}\label{lem-case33}
Let $M_{(b,c)}$ denote the number of solutions $(x, y)$ of (\ref{eqn-mainlemmaB})
such that both $x$ and $y$ are squares. Then
\begin{eqnarray*}
M_{(b,c)} = \left\{
\begin{array}{ll}
\frac{p+5}{4} & \mbox{ if } (b,c)=(1,1) \\
\frac{p+1}{2} & \mbox{ for } \frac{q-p}{2(p+1)} \mbox{ pairs } (b,c) \ne (1,1) \\
0 & \mbox{ for the rest pairs} (b,c) \in ({\mathbb{F}}_q^* )^2 \setminus \{(1,1)\}.
\end{array}
\right.
\end{eqnarray*}
\end{lemma}
\begin{proof}
Consider now the solutions of (\ref{eqn-mainlemmaB31}). If $(x_1, y_1)$ is a solution
of (\ref{eqn-mainlemmaB31}), so are $(-x_1, y_1)$, $(x_1, -y_1)$ and $(-x_1, -y_1)$.
If $x_1y_1 \ne 0$, they are indeed four different solutions of (\ref{eqn-mainlemmaB31}),
but give only one solution of (\ref{eqn-mainlemmaB}).
However, it is possible that $x_1y_1=0$.
If $(b,c)=(1,1)$, Equation (\ref{eqn-mainlemmaB31}) has four special solutions $(\pm 1, 0)$
and $(0, \pm 1)$. They give only two solutions of (\ref{eqn-mainlemmaB}).
It then follows from Lemma \ref{lem-case32} that
$$
M_{(1,1)} = \frac{N_{(1,1)}-4}{4}+2=\frac{p+5}{4}.
$$
If $(b,c) \ne (1,1)$, then the four distinct solutions $(\pm x_1, \pm y_1)$ give only one
solution of (\ref{eqn-mainlemmaB}). In this case, it then follows from Lemma \ref{lem-case32} that
\begin{eqnarray*}
M_{(b,c)} &=& \frac{N_{(b,c)}}{4} \\
&=& \left\{
\begin{array}{ll}
\frac{p+1}{2} & \mbox{ for } \frac{q-p}{2(p+1)} \mbox{ pairs } (b,c) \ne (1,1) \\
0 & \mbox{ for the rest pairs} (b,c) \in ({\mathbb{F}}_q^* )^2 \setminus \{(1,1)\}.
\end{array}
\right.
\end{eqnarray*}
The proof is then completed.
\end{proof}
\subsection{Case 4}
\begin{lemma}\label{lem-case43}
Let $M_{(b,c)}$ denote the number of solutions $(x, y)$ of (\ref{eqn-mainlemmaB})
such that both $x$ and $y$ are either nonsquares or zero. Then
\begin{eqnarray*}
M_{(b,c)} = \left\{
\begin{array}{ll}
\frac{p-3}{4} & \mbox{ if } (b,c)=(1,1) \\
\frac{p+1}{2} & \mbox{ for } \frac{q-p}{2(p+1)} \mbox{ pairs } (b,c) \ne (1,1) \\
0 & \mbox{ for the rest pairs} (b,c) \in ({\mathbb{F}}_q^* )^2 \setminus \{(1,1)\}.
\end{array}
\right.
\end{eqnarray*}
\end{lemma}
\begin{proof}
The proof of this lemma is similar to that of Lemma \ref{lem-case33} and is omitted here.
\end{proof}
\section*{The proof of Lemma \ref{lem-mainB}}
Note that the solutions $(1, 0)$ and $(0,1)$ of (\ref{eqn-mainlemmaB}) are counted more
than once in Cases 1, 2, 3 and 4. By analyzing the proofs of Lemmas \ref{lem-case13},
\ref{lem-case23}, \ref{lem-case33} and \ref{lem-case43}, we have
$$
\mathtt{N_{(1,1)}}=\frac{p-3}{4} + \frac{p-3}{4} + \frac{p+1}{4} + \frac{p-3}{4} + 2 =p.
$$
When $(b, c) \ne (1,1)$, $\mathtt{N_{(b,c)}}$ is the sum of the solutions given in Lemmas
\ref{lem-case13}, \ref{lem-case23}, \ref{lem-case33} and \ref{lem-case43}. This completes
the proof.
|
3,212,635,537,450 | arxiv | \section{#1}}
\newcommand{\addtocounter{section}{1} \setcounter{equation}{0}{\addtocounter{section}{1} \setcounter{equation}{0}
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\begin{flushright}
IC/95/105 \\
\end{flushright}
\vspace*{3.0cm}
\begin{center}
{\bf A DYNAMICAL LEFT--RIGHT SYMMETRY BREAKING MODEL }
\footnote{Talk given at XXXth Rencontres de Moriond ``Electroweak
Interactions and Unified Theories'', Les Arcs, France, March 11-18, 1995.}\\
\vspace*{0.7cm}
{\large E.Kh. Akhmedov} \\
\vspace*{0.5cm}
{\em International Centre for Theoretical Physics,\\
Strada Costiera 11, I-34100, Trieste, Italy \\
and \\
National Research Center ``Kurchatov Institute'',\\
123182 Moscow, Russia }\\
\end{center}
\vspace*{2.5cm}
\begin{center}
ABSTRACT
\end{center}
Left--right symmetry breaking in a model with composite Higgs scalars is
discussed. It is assumed that the low--energy degrees of freedom are just
fermions and gauge bosons and that the Higgs bosons are generated dynamically
through a set of gauge-- and parity--invariant 4-fermion operators. It is
shown that in a model with composite bi-doublet and two triplet scalars there
is no parity breaking at low energies, whereas in the model with two doublets
instead of two triplets parity is broken automatically regardless of the
choice of the parameters of the model. For phenomenologically allowed values
of the right--handed scale the tumbling symmetry breaking mechanism is realized
in which parity breaking at a high scale $\mu_R$ propagates down and eventually
causes the electroweak symmetry breaking at the scale $\mu_{EW}\sim 100$ GeV.
The model exhibits a number of low and intermediate mass Higgs bosons with
certain relations between their masses. In particular, the $SU(2)_L$ Higgs
doublet $\chi_L$ is a pseudo--Goldstone boson of the accidental (approximate)
$SU(4)$ symmetry of the Higgs potential and therefore is expected to be
relatively light.
\renewcommand{\thefootnote}{\arabic{footnote}}
\setcounter{footnote}{0}
\newpage
Several years ago, a very interesting approach to electroweak symmetry breaking was
put forward, so called ``top condensate'' model$^{1)-4)}$. In this model no
fundamental Higgs boson is present;
instead, it is assumed that there is a strong attractive interaction between
the top quarks, which can lead to the formation of the $t\bar{t}$ bound state
playing a role of the Higgs scalar. This interaction is assumed
to result from a new physics at some high--energy scale $\Lambda$, the origin
and precise nature of which are not specified. At low energies this new
physics would manifest itself through non-renormalizable interactions between
usual fermions and gauge bosons. At the energies $E \ll \Lambda$ the lowest
dimensional operators are most important, which are just the four--fermion
(4-f) operators. The simplest gauge--invariant 4-f operator which includes
the heaviest top quark is
\begin{equation}
{\cal L}_{4f}=G(\bar{Q}_{Li}t_R)(\bar{t}_R Q_{Li})\,,
\label{4f1}
\end{equation}
where $Q_L$ is the left--handed doublet of third generation quarks, $G$ is
the dimensionful coupling constant, $G \sim \Lambda^{-2}$, and it is
implied that the colour indices are summed over within each bracket.
This 4-f interaction can be studied analytically in the large $N_c$
(number of colours) limit in the fermion bubble approximation. For large
enough $G$ ($G>G_{cr}=8\pi^2/N_c\Lambda^2$) the electroweak symmetry gets spontaneously
broken, $W^\pm{}$ and $Z^0$ bosons and top quark acquire masses, and a
composite Higgs scalar doublet $H\sim \bar{t}_R Q_L$ is formed.
The top condensate approach reproduced correctly the structure of the
low--energy effective Lagrangian of the standard model and demonstrated how the electroweak
symmetry breaking can result from a high--energy dynamics. The question
naturally arises as to whether a similar approach can work in more complicated
cases, i.e. whether the Higgs sectors and symmetry breaking patterns of
more complicated models can always be successfully described starting from a
set of gauge--invariant 4-f operators. In this talk I will report the main
results of the analysis of dynamical symmetry breaking in the left--right
symmetric models done in collaboration with M. Lindner, E. Schnapka and
J.W.F. Valle$^{5)}$.
We studied dynamical symmetry breaking in the left--right symmetric (LR)
models based on the gauge group $SU(2)_L \times SU(2)_R \times U(1)_{B-L}$
following the BHL approach to the standard model $^{3)}$.
The Higgs sector of the most popular LR model consists of a
bi-doublet $\phi \sim (2,2,0)$ and two triplets, $\Delta_L \sim (3,1,2)$
and $\Delta_R \sim (1,3,2)$. Assuming that these scalars are composite, their
fermionic content is
$$\phi_{ij} \sim \alpha (\bar{Q}_{Rj}Q_{Li}) + \beta (\tau_2 \bar{Q}_L Q_R
\tau_2)_{ij}+\mbox{leptonic terms}~, $$
\begin{equation}
\vec{\Delta}_L \sim (\Psi_L^T C \tau_2 \vec{\tau}\Psi_L),\;\;
\vec{\Delta}_R\sim (\Psi_R^T C \tau_2 \vec{\tau}\Psi_R)~.
\label{phiij}
\end{equation}
Here $Q_L,\Psi_L$ ($Q_R,\Psi_R$) are left--handed (right--handed) doublets
of quarks and leptons, respectively; $i$ and $j$ are isospin indices.
In models with Higgs bosons generated by 4-f operators the composite scalars
are, roughly speaking, ``square roots'' of these 4-f operators. One can
therefore obtain the above composite Higgs bosons starting from the 4-f
operators which are ``squares'' of the expressions in eq.~(\ref{phiij}).
A convenient way to study models with composite Higgs bosons is the
auxiliary field technique, in which one introduces the static auxiliary
scalar fields (with appropriate quantum numbers) with Yukawa couplings and
mass terms but no kinetic terms and no quartic couplings.
One can use the equations of motion for these fields to express them in
terms of the fermionic degrees of freedom and recover the initial 4-f
structures.
The static auxiliary fields can acquire gauge invariant kinetic terms and
quartic self--interactions through the radiative corrections and become
physical propagating scalar fields at low energies provided that the
corresponding gap equations are satisfied$^{3)}$. The kinetic terms and
mass corrections can be derived from the 2--point Green function, whereas
the quartic couplings are given by the 4--point functions. Given the Yukawa
couplings of the scalar fields, one can readily calculate these functions in
the fermion bubble approximation, in which they are given by the corresponding
1-fermion--loop diagrams.
It is usually assumed that the Higgs potential of the LR model is exactly
symmetric with respect to the discrete parity symmetry. However, parity can
be spontaneously broken by $\langle \Delta_R \rangle > \langle \Delta_L
\rangle$ provided $\lambda_2 >
\lambda_1$ where $\lambda_1$ and $\lambda_2$ are the coefficients of the
$[(\Delta_L^\dagger \Delta_L)^2+(\Delta_R^\dagger \Delta_R)^2]$ and
$2(\Delta_L^\dagger \Delta_L)(\Delta_R^\dagger \Delta_R)$ quartic couplings
in the Higgs potential. In the conventional approach,
$\lambda_1$ and $\lambda_2$ are free parameters and one can always choose
$\lambda_2>\lambda_1$. On the contrary, in the composite Higgs approach
based on a certain set of the effective 4-f couplings, the
parameters of the effective Higgs potential are not arbitrary: they are
all calculable in terms of the 4-f couplings $G_a$ and the scale of new
physics $\Lambda$ $^{3)}$. In particular, in the fermion bubble
approximation at one loop level the quartic couplings $\lambda_1$ and
$\lambda_2$ are induced through the Majorana--like Yukawa couplings
$f(\Psi_L^T C \tau_2 \vec{\tau}\vec{\Delta}_L \Psi_L + \Psi_R^T C
\tau_2 \vec{\tau}\vec{\Delta}_R \Psi_R) + h.c.$.
It is easy to see that to induce the $\lambda_2$ term one needs the
$\Psi_L$--$\Psi_R$ mixing in
the fermion line in the loop, i.e. the lepton Dirac mass term
insertions. However, the Dirac mass terms are generated by the VEVs of the
bi-doublet $\phi$; they are absent at the parity breaking scale which is
supposed to be higher than the electroweak scale. Even if parity and electroweak symmetry
are broken simultaneously (which is hardly a phenomenologically viable
scenario), this would not save the situation since $\lambda_2$ is finite in
the limit $\Lambda\to \infty$ whereas the diagrams contributing to $\lambda_1$
are logarithmically divergent and so the inequality $\lambda_2>\lambda_1$
cannot be satisfied.
One is therefore led to consider a model with a different composite
Higgs content. The simplest LR model includes two doublets, $\chi_L
\sim (2,1,-1)$ and $\chi_R \sim (1,2,-1)$, instead of the triplets
$\Delta_L$ and $\Delta_R$.
As we shall see, the model with composite doublets will automatically lead
to the correct pattern of the dynamical breaking of parity.
In conventional LR
models, one can have the doublet Higgs bosons without introducing any other
new particles. In our model all scalars are composite, thus
we must introduce additional singlet fermions -- otherwise it would not
be possible to build up the composite $\chi_L$ and $\chi_R$ fields. We
assume that in addition to the usual quark and lepton doublets there is a
gauge--singlet fermion $S_L \sim (1,\;1,\;0)$.
To maintain the discrete parity symmetry one needs a right--handed
counterpart of $S_L$. This can be either another particle, $S_R$, or
the right--handed antiparticle of $S_L$, $(S_L)^c\equiv C\bar{S}_L^T =
S^c_R$. The latter choice is
more economical and, as we shall see, leads to the desired symmetry
breaking pattern. We therefore assume that under parity operation
$S_L \leftrightarrow S^c_R$.
With this new singlet
and usual quark and lepton doublets
one can introduce
the following gauge--invariant 4-f interactions$^5$:
\begin{eqnarray}
{\cal L}_{4f}'&=&G_1(\bar{Q}_{Li}Q_{Rj})(\bar{Q}_{Rj}Q_{Li})+
[G_2(\bar{Q}_{Li}Q_{Rj})(\tau_2\bar{Q}_{L}Q_{R}\tau_2)_{ij}
+h.c.]\nonumber \\
& &+G_3(\bar{\Psi}_{Li}\Psi_{Rj})(\bar{\Psi}_{Rj}\Psi_{Li})+
[G_4(\bar{\Psi}_{Li}\Psi_{Rj})(\tau_2\bar{\Psi}_{L}\Psi_{R}\tau_2)_{ij}
+h.c.]\nonumber \\
& &+[G_5(\bar{Q}_{Li}Q_{Rj})(\bar{\Psi}_{Rj}\Psi_{Li})+h.c.]+
[G_6(\bar{Q}_{Li}Q_{Rj})(\tau_2\bar{\Psi}_{L}\Psi_{R}\tau_2)_{ij}
+h.c.] \nonumber \\
& &+G_7[(S_L^T C \Psi_L)(\bar{\Psi}_L C \bar{S}_L^T)+
(\bar{S}_L\Psi_R)(\bar{\Psi}_R S_L)]+G_8 (S_L^T C S_L)(\bar{S}_L C
\bar{S}_L^T)~.
\label{L4f}
\end{eqnarray}
These interactions are not only gauge invariant, but also (for hermitean
$G_2$, $G_4$, $G_5$ and $G_6$) symmetric with respect to the discrete parity
symmetry.
The composite Higgs scalars which can be induced by these 4-f couplings
include, in addition to the bi-doublet $\phi$ of the structure given in
eq.~(\ref{phiij}), two doublets $\chi_L$ and $\chi_R$ and also a singlet
$\sigma$:
\begin{equation}
\chi_L \sim S_L^T C \Psi_L, \;\;\;\; \chi_R \sim \bar{S}_L\Psi_R =
(S^c_R)^T C \Psi_R, \;\;\;\; \sigma \sim \bar{S}_L C \bar{S}_L^T.
\label{composite}
\end{equation}
Under parity $\chi_L \leftrightarrow \chi_R$, $\sigma \leftrightarrow
\sigma^\dagger$.
In the auxiliary field formalism the scalars $\chi_L$, $\chi_R$, $\phi$ and
$\sigma$ have the following bare mass terms and Yukawa couplings:
\begin{eqnarray}
{\cal L}_{aux}&=&-M_0^2(\chi_L^\dagger \chi_L+\chi_R^\dagger \chi_R)
-M_1^2 \mathop{\rm tr}} \def\Tr{\mathop{\rm Tr}{(\phi^\dagger \phi)}-\frac{M_2^2}{2}\mathop{\rm tr}} \def\Tr{\mathop{\rm Tr}{(\phi^\dagger
\tilde{\phi}+h.c.)}-M_3^2 \sigma^\dagger\sigma \nonumber \\
& & -\left[\bar{Q}_L(Y_1\phi+Y_2\tilde{\phi})Q_R +
\bar{\Psi}_L(Y_3\phi+Y_4\tilde{\phi})\Psi_R + h.c.\right] \nonumber \\
& &
-\left[
Y_5(\bar{\Psi}_L \chi_L S^c_R+\bar{\Psi}_R \chi_R S_L)
+Y_6 (S_L^T C S_L)\sigma + h.c.
\right]
\label{Laux}
\end{eqnarray}
where the field $\tilde{\phi}\equiv \tau_2\phi^*\tau_2$ has the same
quantum numbers as $\phi$: $\tilde{\phi}\sim (2,\;2,\;0)$.
By integrating out the auxiliary scalar fields one can reproduce the 4-f
structures of eq.~(\ref{L4f}).
Consider now parity breaking in the present LR model.
Using the Yukawa couplings of the doublets $\chi_L$ and $\chi_R$, one can
calculate the fermion-loop
contributions to the quartic couplings $\lambda_1 [(\chi_L^\dagger
\chi_L)^2+(\chi_R^\dagger \chi_R)^2]$ and $2\lambda_2 (\chi_L^\dagger
\chi_L)(\chi_R^\dagger \chi_R)$ in the effective Higgs potential.
One can easily make sure that the fermion--loop diagrams yield
$\lambda_1=\lambda_2$. Recall that one needs $\lambda_2 > \lambda_1$ to have
spontaneous parity breaking in the LR models. As we shall see, taking into
account the gauge-boson loop contributions to $\lambda_1$ and $\lambda_2$
will automatically secure this relation.
Both $\lambda_1$ and $\lambda_2$ obtain corrections from $U(1)_{B-L}$ gauge
boson loops, whereas only $\lambda_1$ is corrected by diagrams with $W^i_L$
or $W^i_R$ loops.
Since all these contributions have a relative minus sign compared
to the fermion loop ones, one finds $\lambda_2>\lambda_1$ irrespective of
the values of the Yukawa or gauge couplings or any other parameter of the
model, provided that the $SU(2)$ gauge coupling $g_2\neq 0$.
Thus the condition for spontaneous parity breaking is automatically
satisfied in our model.
We have a very interesting situation here. In a model with
composite triplets $\Delta_L$ and $\Delta_R$ parity
is never broken, i.e. the model is not phenomenologically viable. At the
same time, in the model with two composite doublets $\chi_L$ and $\chi_R$
instead of two triplets (which requires introduction of an additional singlet
fermion $S_L$) parity is broken automatically. This means that, unlike in
the conventional LR models, in the composite Higgs approach {\em whether or
not parity is spontaneously broken depends on the particle content of the
model rather than on the choice of the parameters of the Higgs potential}.
Although there are no triplet Higgs bosons in the present version of the
model, a modified seesaw mechanism is operative which ensures the smallness
of the masses of neutrinos taking part in the usual electroweak interactions. This
is because the neutrinos mix with the gauge--singlet fermion $S_L$.
Minimization of the effective Higgs potential shows that for the choice of
the 4-f couplings resulting in the dynamical breaking of the LR gauge
symmetry, the $\chi_L$ and $\sigma$ fields do not develop VEVs; for
this reason the lightest neutrinos remain massless in our model.
Analysis of the vacuum structure of the model shows that for
the right--handed scale $v_R=\langle \chi_R^0\rangle$ to lie in the
phenomenologically allowed domain, the effective (mass)$^2$ term of the
composite bi-doublet $\phi$ must always be positive; the electroweak symmetry is
broken because of the mixing of $\phi$ with $\chi_R$.
Thus we have a tumbling scenario where the breakdown of parity and $SU(2)_R$
occurring at the scale $\mu_R$ causes the electroweak symmetry breaking at a
lower scale $\mu_{EW}$.
The physical Higgs boson sector of the model includes 4 charged scalars,
4 neutral CP--even and 2 CP--odd scalars. Two of neutral CP--even bosons,
$H_1$ and $H_2$, are directly related to the two steps
of symmetry breaking, $SU(2)_R\times U(1)_{B-L}\rightarrow U(1)_Y$ and
$SU(2)_L\times U(1)_Y \rightarrow U(1)_{em}$. In the bubble approximation
the mass of $H_2$, whose properties are similar to the properties of the
standard model Higgs boson, is approximately $2 m_t$. This coincides with the
top--condensate prediction$^{1)-4)}$ and reflects the fact that this scalar
is the $t\bar{t}$ bound state. Analogously, $H_1$ is the bound state of
heavy neutrinos with its mass being approximately twice the heavy neutrino
mass $M \sim \mu_R$.
In conventional LR models only one scalar, which is the analog of the standard model
Higgs boson, is light (at the electroweak scale), all the others have their masses
of the order of the right--handed scale $M$. In our
case, the masses of those scalars are also proportional to $M$, but all
of them except the mass of $H_1$ have some suppression factors.
The masses of $\chi_L$ are suppressed because of their
pseudo--Goldstone nature. It was already mentioned that at the
fermion--bubble level $\lambda_1=\lambda_2$ which means that the $(\chi_L,
\chi_R)$ sector of the Higgs potential depends on $\chi_L$ and $\chi_R$
only through the combination $(\chi_L^\dagger\chi_L+\chi_R^\dagger \chi_R)$.
This, in turn, implies that the Higgs potential has a global $SU(4)$
symmetry which is bigger than the original gauge symmetry.
In fact, the origin of this $SU(4)$ symmetry can be traced back to the 4-f
operators of eq.~(\ref{L4f}). It is an accidental symmetry
resulting from the gauge invariance and parity symmetry of the $G_7$ term.
Note that no such symmetry occurs in conventional LR models. After
$\chi_R^0$ acquires a VEV, $SU(4)$ symmetry is broken down to $SU(3)$, and
the components of $\chi_L$ are the corresponding Goldstone bosons. The $SU(4)$
symmetry of the Higgs potential is not exact: it is broken by the $SU(2)$
gauge--boson loop contributions which make $\lambda_2>\lambda_1$ (and also
by $\phi$--dependent terms). As a result, $\chi_L$ are pseudo--Goldstone
bosons with their mass vanishing in the limit $g_2\rightarrow 0$,
$m_{\tau}\to 0$. In fact, though the $SU(2)$ gauge coupling constant $g_2$
is smaller than the typical Yukawa constants in our model, it is not too
small; estimates of the $\chi_L$ mass give $M_{\chi_L} \sim 10^{-1} M$.
The bi-doublet $\phi$ can be viewed as consisting of two doublets, $\phi_1$
which develops a VEV and is similar to the standard model Higgs doublet, and the
orthogonal field $\phi_2$ which is VEVless. In the conventional LR models
$\phi_2$ is heavy, $M_{\phi_2}\sim M$. In our case the mass of its charged
components $\phi_2^\pm$ is suppressed by the factor $m_{\tau}/m_t$ and
is therefore of the order $10^{-2} M$. The masses of the neutral CP--even
and CP--odd components are even smaller; they are related to the masses of
$\phi_2^\pm$ and the standard model Higgs boson $H_2$ by
\begin{equation}
M_{\phi_{2r}^0}^2=M_{\phi_{2i}^0}^2=M_{\phi_2^\pm}^2-\frac{M_{H_2}^2}{2}
=\frac{2}{3}M^2\frac{m_{\tau}^2}{m_t^2}-\frac{M_{H_2}^2}{2}.
\end{equation}
This equation imposes an upper limit on the
standard model Higgs boson mass $M_{H_2}$ (for a given $M$) or a lower limit on the
right--handed mass $M$ (for a given $M_{H_2}$). These limits follow from the
requirement that $M_{\phi_2^\pm}^2$ be positive, i.e. from the vacuum
stability condition. For example, for $M_{H_2} \approx 200$ GeV we find
$M\buildrel > \over {_{\sim}} 17$ TeV.
Assuming that only one of the neutral components of the bi-doublet develops
a non-vanishing VEV, one can readily relate the top quark mass to the scale
of new physics $\Lambda$ (on which it depends logarithmically) and the electroweak
VEV. For example, for $\Lambda\simeq 10^{15}$ GeV, one finds $m_t\simeq 165$
GeV. However, this only holds in the bubble approximation; the renormalization group improved
result is substantially higher, 220--230 GeV. In this respect the predictions
of the model are similar to those of BHL$^{3)}$. If one assumes
that both neutral components of the bi-doublet develop non-vanishing VEVs,
one can easily get an acceptable value of $m_t$. However, the top quark mass
is adjusted rather than predicted, i.e. one looses predictivity in the
fermion sector in this case (although keeps interesting predictions in the
Higgs boson sector).
To summarize, we have a successful dynamical LR model in which a tumbling
symmetry breaking mechanism is operative. The model exhibits
a number of low and intermediate scale Higgs bosons and predicts the
relations between masses of various scalars and
between fermion and Higgs boson masses which are in principle testable.
If the right--handed scale $\mu_R$ is of the order of a few tens of TeV,
the neutral $CP$--even and $CP$--odd scalars $\phi_{2r}^0$ and $\phi_{2i}^0$
can be even lighter than the electroweak Higgs boson. In fact, they can be as light
as $\sim 50$ GeV and so might be observable at LEP2. Such light $\phi_{2r}^0$
and $\phi_{2i}^0$ can also provide a positive contribution to
$R_b=\Gamma(Z\to b\bar{b})/\Gamma(Z\to hadrons)$ which is necessary to
account for the discrepancy between the LEP observations and the standard model
predictions. \\
\centerline{REFERENCES}
\vspace*{0.2cm}
\baselineskip=.3truecm
\noindent
1. Y. Nambu, in {\em New Theories in Physics, Proc. XI Int. Symposium on
Elementary Particle \hspace*{0.4truecm} Physics}, eds. Z. Ajduk, S. Pokorski
and A. Trautman (World Scientific, Singapore, 1989) \hspace*{0.4truecm} and
EFI report No. 89-08 (1989), unpublished. \\
2. A. Miransky, M. Tanabashi, K. Yamawaki, Mod. Phys. Lett. A4 (1989) 1043;
Phys. Lett. \hspace*{0.4truecm} B221 (1989) 177. \\
3. W.A. Bardeen, C.T. Hill, M. Lindner, Phys. Rev. D41 (1990) 1647. \\
4. W.J. Marciano, Phys. Rev. Lett. 62 (1989) 2793. \\
5. E.Kh. Akhmedov, M. Lindner, E. Schnapka, J.W.F. Valle, to be published.\\
\end{document}
|
3,212,635,537,451 | arxiv | \section{Introduction}
After the discovery that
perfect hydrodynamics can be valid for describing the phenomenology
of Relativistic Heavy Ion Collider (RHIC)
at Brookhaven National Laboratory
\cite{qcd001, qcd002, qcd003},
people are now interested in
relativistic hydrodynamics for \textit{dissipative} systems;
see the recent excellent review articles
\cite{Hirano:2008hy,Romatschke:2009im}.
Recently,
Tsumura, Kunihiro (the present authors) and Ohnishi (abbreviated as TKO)
\cite{TKO}
derived generic covariant hydrodynamic equations
for a viscous fluid from the relativistic Boltzmann equation
in a systematic manner with no heuristic arguments
on the basis of the so-called renormalization group (RG) method
\cite{rgm001,env001,env004,HK02,env006}.
Although the hydrodynamic equations they derived are
the so-called first-order ones,
the equations have remarkable aspects:
The generic equation derived by TKO can produce
a relativistic dissipative hydrodynamic equation
in any frame with an appropriate choice
of a macroscopic flow vector $\Ren{a}^{\mu}$
($\mu = 0,\, 1,\, 2,\, 3$),
which defines the coarse-grained space and time;
the resulting equation in the energy frame
coincides with that of Landau and Lifshitz
\cite{hen002},
while that in the particle frame is similar to,
but slightly different from, the Eckart equation
\cite{hen001}.
Let $\delta T^{\mu\nu}$ and $\delta N ^{\mu}$
be the dissipative term
of the symmetric energy-momentum tensor
and the particle-number vector, respectively.
Owing to the ambiguity in the separation of the energy and the mass
inherent in relativistic theories,
one must choose the
local rest frame (LRF)
where the flow velocity
$u^\mu$ with $u^\mu \, u_\mu = 1$ is
defined:
One of the typical frame is the energy (Landau) frame in which
$\delta T^{\mu\nu} \, u_{\mu} \, \Delta_{\nu\rho} = 0$
with $\Delta^{\mu\nu} \equiv g^{\mu\nu} - u^\mu \, u^\nu$
and $g^{\mu\nu} = \mathrm{diag}(+1,\,-1,\,-1,\,-1)$,
i.e. there is no dissipative energy flow.
On the other hand, another typical frame is the particle (Eckart)
frame in which
$\delta N ^{\mu} \, \Delta_{\mu\nu} = 0$,
i.e. there is no dissipative particle flow.
Both in the energy and particle frames,
the dissipative terms of the energy-momentum tensor
and the particle-number vector
are usually assumed to satisfy the constraints,
\begin{eqnarray}
\label{eq:PT-E}
u_{\mu} \, \delta T^{\mu\nu} \, u_{\nu} = 0,
\end{eqnarray}
and $u_{\mu} \, \delta N^{\mu} = 0$.
These phenomenological ansatz have been employed as the matching conditions
even in the subsequent ``derivations''
of the so-called second-order equations
\cite{hen003,mic004,mic001,Betz:2008me};
note that
even in the Grad's moment method \cite{grad},
some ansatz are needed to
$\delta T^{\mu\nu}$ and $\delta N^{\mu}$ as the matching conditions,
for which different proposals exist
\cite{marle:69,mic004}.
Here we emphasize that
the matching conditions touch on the fundamental but not yet fully understood problem
how to define the LRF in the relativistic fluid dynamics for a viscous system.
The way how to define the LRF or equivalently to fix the matching condition is unsolved yet,
and remains a nontrivial and fundamental problem
in the field of nonequilibrium relativistic dynamics,
although there have been no serious consideration on this difficult
problem in the literature.
Actually,
we shall argue that these phenomenological ansatz, especially Eq.(\ref{eq:PT-E}),
can be false and actually is not compatible with
the underlying kinetic equation.
In fact, it is found that
the TKO equation in the particle frame
derived from the relativistic Boltzmann equation satisfies
\begin{eqnarray}
\label{eq:PT-TKO}
\delta T^{\mu}_{\,\,\,\mu} = 0,
\end{eqnarray}
but does not satisfy Eq.(\ref{eq:PT-E}).
One should here note that
the derived condition (\ref{eq:PT-TKO}) is identical to a matching condition
postulated by Marle
\cite{marle:69}
and advocated by Stewart
\cite{mic003}
in the derivation of the relativistic hydrodynamics
from the relativistic Boltzmann equation
with use of the Grad's moment method.
In their paper
\cite{TKO},
TKO proved that
the Eckart constraint (\ref{eq:PT-E}) in the particle frame
cannot be compatible with the underlying relativistic Boltzmann equation
for the first-order hydrodynamic equation.
In spite of the first-order one,
the TKO equation in the particle frame
is free from the pathological properties
\cite{Tsumura:2007wu}
in contrast to the original Eckart equation
with which the thermal equilibrium
becomes unstable for a small perturbation
\cite{hyd002}.
One may naturally ask if the Eckart
constraint (\ref{eq:PT-E})
should be replaced with (\ref{eq:PT-TKO})
even for the so-called second-order equation like Israel-Stewart one.
And are any modifications needed to the constraints in the Landau frame?
A purpose of this Letter
\cite{QM09}
is to answer these questions both by phenomenological and microscopic analyses.
We shall see that
the Eckart constraint should be replaced
with the new one even in the second-order equation,
while no modification is necessary
for the constraints in the energy frame.
We shall derive
the second-order dissipative relativistic hydrodynamic equations
in a generic frame with a continuous parameter $\theta$
from the relativistic Boltzmann equation.
We shall derive the relaxation terms
for a generic frame with the new constraint,
and present explicitly those in the energy and particle frames.
We shall show that
the viscosities are frame-independent
but the relaxation times are generically frame-dependent
in accordance with the observation by Betz et al.
\cite{Betz:2008me},
although the constraint to $\delta T^{\mu\nu}$ is quite different.
\setcounter{equation}{0}
\section{
A general phenomenological derivation
of relativistic dissipative hydrodynamic equations;\,
existence of possible extra terms in the dissipative terms
}
Let $T^{\mu\nu}$ and $N^\mu$ be
the symmetric energy-momentum tensor and the particle-number vector
of the system we consider, respectively.
The total number of independent variables is fourteen,
and the dynamical evolution of these variables
are governed by the respective balance equations;
\begin{eqnarray}
\label{eq:2-001}
\partial_\mu T^{\mu\nu} &=& 0,\\
\label{eq:2-002}
\partial_\mu N^\mu &=& 0.
\end{eqnarray}
With use of an arbitrary four vector $u^\mu$ with $u^\mu \, u_\mu = 1$,
$T^{\mu\nu}$ and $N^\mu$ can be cast into the tensor-decomposed forms,
\begin{eqnarray}
\label{eq:2-003}
T^{\mu\nu} &=& (e + \delta e) \, u^\mu
\, u^\nu
- (p + \delta p) \,
\Delta^{\mu\nu} + q^\mu \, u^\nu + q^\nu \, u^\mu +
\pi^{\mu\nu},\\
\label{eq:2-004}
N^\mu &=& (n + \delta n) \, u^\mu + \nu^\mu,
\end{eqnarray}
respectively.
Here,
$e+\delta e$, $p+\delta p$, and $n+\delta n$
are the internal energy, pressure,
and particle-number density in the dissipative system;
$e + \delta e \equiv T_{ab} \, u^a \, u^b$,
$p + \delta p \equiv -1/3 \, T_{ab} \, \Delta^{ab}$,
and $n + \delta n \equiv N_a \, u^a$,
with
$e = e(T,\,\mu)$,
$p = p(T,\,\mu)$,
and $n = n(T,\,\mu)$
being the corresponding quantities
in the local equilibrium state
characterized by the temperature $T$ and the chemical potential $\mu$.
Note that we have made it explicit by
$\delta e$, $\delta p$, and $\delta n$ that
the dissipations may cause corrections
to all these quantities,
although only the correction to the pressure has been considered in the literature;
$\delta p$ is identified with the bulk pressure $\Pi$.
We emphasize that
there is no persuading reasoning that
only the pressure acquires corrections due to the dissipative process.
The dissipative parts of the energy-momentum tensor and
particle-number vector are identified as
$\delta T^{\mu\nu} = \delta e \, u^\mu \, u^\nu + \delta p \, \Delta^{\mu\nu} +
q^\mu \, u^\nu + q^\nu \, u^\mu + \pi^{\mu\nu}$ and
$\delta N^{\mu} = \delta n \, u^\mu + \nu^\mu$, respectively.
The energy flow relative to $u^\mu$ is denoted by $q^\mu$,
$\nu^\mu$ is the flow of particle number relative to $u^\mu$,
and finally
$\pi^{\mu\nu}$ is the shear stress tensor;
$q^\mu \equiv T_{ab} \, u^a \, \Delta^{b\mu}$,
$\nu^\mu \equiv N_a \, \Delta^{a\mu}$,
and $\pi^{\mu\nu} \equiv T_{ab} \, \Delta^{ab\mu\nu}$.
Here
the space-like, symmetric and traceless tensor
$\Delta^{\mu\nu\rho\sigma} \equiv
1/2 \, (\Delta^{\mu\rho} \, \Delta^{\nu\sigma} + \Delta^{\mu\sigma} \,
\Delta^{\nu\rho} - 2/3 \, \Delta^{\mu\nu} \, \Delta^{\rho\sigma})$
is introduced.
One can easily confirm that
$q^\mu \, u_\mu = 0$, $\nu^\mu \, u_\mu = 0$,
$\pi^{\mu\nu} = \pi^{\nu\mu}$,
and $u_\mu \, \pi^{\mu\nu} = \pi^\mu_{\,\,\,\mu} = 0$.
This implies that
the total number of independent components
of $q^\mu$, $\nu^\mu$, and $\pi^{\mu\nu}$ is eleven.
Since $T^{\mu\nu}$ and $N^\mu$
have the fourteen components in total,
$\delta e$, $\delta p$, and $\delta n$ have
only one independent component other than $T$ and $\mu$.
We take $\delta p= \Pi$ as the independent component as a natural choice,
then $\delta e$ and $\delta n$ can be expressed as
$\delta e = f_e \, \Pi$ and $\delta n = f_n \, \Pi$,
where $f_e$ and $f_n$ are functions of $T$ and $\mu$;
$f_e = f_e(T,\,\mu)$ and $f_n = f_n(T,\,\mu)$.
Here we have assumed that the dissipative order of
$\delta e$ and $\delta n$ are the same as
that of $\delta p$ at most.
We remark that
although $f_e$ and $f_n$ may take finite values generically,
the functional forms of
$f_e$ and $f_n$ cannot be determined by the phenomenological theory,
as those of $e$, $p$, and $n$ can not, either.
All the previous analyses assumed
$f_e = f_n = 0$, which has not been recognized so far.
Now we shall show that
the just usual phenomenological derivation
of the hydrodynamic equations
in which the second law of thermodynamics is utilized
allows the existence of $\delta e$ and $\delta n$, i.e.,
finite values of $f_e$ and $f_n$,
in the relativistic dissipative hydrodynamic equations.
It is found that the essential point of the proof
is the same for the first- and second-order equations
where the entropy current $S^\mu$ is
at most linear and bilinear with respect to
$\Pi$, $q^\mu$, $\nu^\mu$, and $\pi^{\mu\nu}$, respectively,
although the resulting mathematical expressions
are much more complicated in the second-order one
\cite{next002}.
Thus we here take the first-order equation, for the sake of simplicity.
The second-order equations with finite $f_e$ and $f_n$
will be derived microscopically later in this article.
So the entropy current is given by
\begin{eqnarray}
\label{eq:2-015}
T \, S^\mu = p \, u^\mu + u_\nu \, T^{\mu\nu} -
\mu \, N^\mu.
\end{eqnarray}
The second law of thermodynamics
reads $\partial_\mu S^\mu \ge 0$.
The divergence of $S^\mu$ is found to take the form
\begin{eqnarray}
\label{eq:2-016}
\partial_\mu S^\mu = \Pi \, \Bigg[
f_e\,D\frac{1}{T} - \frac{1}{T} \, \nabla^\mu u_\mu
- f_n\,D\frac{\mu}{T}
\Bigg]
+ q^\mu \, \Bigg[ \frac{1}{T} \, Du_\mu + \nabla_\mu
\frac{1}{T} \Bigg] - \nu^\mu \,
\nabla_\mu \frac{\mu}{T}
+ \pi^{\mu\nu} \, \frac{1}{T} \, \nabla_\mu u_\nu,
\end{eqnarray}
where
$D \equiv u^a \, \partial_a$
and
$\nabla^\mu \equiv \Delta^{\mu a} \, \partial_a$.
Here, we have used
the conservation laws, Eq.'s (\ref{eq:2-001}) and (\ref{eq:2-002}),
and the first law of thermodynamics,
$D(p/T) + e \, D(1/T) - n \, D(\mu/T) = 0$.
The frames define the flow velocity $u^{\mu}$ of the fluid:
The flow velocity in the particle frame
and the energy frame are
defined by setting
$u^\mu = N^\mu / \sqrt{N^\nu \, N_\nu}$
and
$u^\mu = T^{\mu a} \, u_a / \sqrt{T^{\nu b} \, u_b \, T_{\nu c} \, u^c}$, respectively
\cite{mic001}.
By these settings,
a closed system of the relativistic dissipative
hydrodynamic equations is obtained.
Note that
$u^\mu = N^\mu / \sqrt{N^\nu \, N_\nu}$
($u^\mu = T^{\mu a} \, u_a / \sqrt{T^{\nu b} \, u_b \, T_{\nu c} \, u^c}$)
is equivalent to $\nu^\mu = 0$
($q^\mu = 0$).
In the particle frame where $\nu^\mu = 0$,
Eq.(\ref{eq:2-016}) is reduced to
\begin{eqnarray}
\label{eq:2-019}
\partial_\mu S^\mu = \Pi \, \Bigg[
f_e\,D\frac{1}{T} - \frac{1}{T} \, \nabla^\mu u_\mu
- f_n\,D\frac{\mu}{T}
\Bigg]
+ q^\mu \, \Bigg[ \frac{1}{T} \, Du_\mu + \nabla_\mu \frac{1}{T} \Bigg]
+ \pi^{\mu\nu} \, \frac{1}{T} \, \nabla_\mu u_\nu.
\end{eqnarray}
It is found that
the following constitutive equations,
\begin{eqnarray}
\label{eq:2-021}
\Pi &=& \zeta \, T \, \Bigg[
f_e\,D \frac{1}{T} - \frac{1}{T} \, \nabla^\mu u_\mu
- f_n\,D\frac{\mu}{T} \Bigg],\\
\label{eq:2-022}
q^\mu &=& - \lambda \, T^2 \, \Bigg[ \frac{1}{T} \, Du^\mu +
\nabla^\mu \frac{1}{T} \Bigg],\\
\label{eq:2-023}
\pi^{\mu\nu} &=& 2 \, \eta \, \Delta^{\mu\nu\rho\sigma} \,
\nabla_\rho u_\sigma,
\end{eqnarray}
guarantees the second law of thermodynamics,
$\partial_\mu S^\mu \ge 0$,
with $\zeta$, $\lambda$, and $\eta$
being the bulk viscosity, heat conductivity, and shear viscosity, respectively.
This is because the divergence $\partial_\mu S^\mu$ now becomes positive
semi-definite;
\begin{eqnarray}
\label{eq:2-020}
\partial_\mu S^\mu
= \frac{\Pi^2}{\zeta T}
- \frac{q^\mu q_\mu}{\lambda T^2}
+ \frac{\pi^{\mu\nu}\pi_{\mu\nu}}{2\eta T} \ge 0.
\end{eqnarray}
Thus we realize that
there is nothing wrong
with the resultant relativistic dissipative hydrodynamic equations
with finite $f_e$ and $f_n$, or equivalently
finite $\delta e$ and $\delta n$,
which is compatible with the second law of thermodynamics.
Eq.'s (\ref{eq:2-021})-(\ref{eq:2-023}) with
a restricted condition $f_e = f_n = 0$
are identical to the constitutive equations proposed by Eckart
that are commonly used.
In the energy frame where $q^\mu = 0$,
we can obtain the constitutive equations
in the same way as the particle-frame case with
$f_e$ and $f_n$ being kept finite.
The resultant equations are given by Eq.'s (\ref{eq:2-021}), (\ref{eq:2-023}), and
\begin{eqnarray}
\label{eq:2-024}
\nu^\mu = \lambda \, \hat{h}^{-2} \,
\nabla^\mu \frac{\mu}{T},
\end{eqnarray}
with $\hat{h} \equiv (e + p)/n\,T$ being the enthalpy.
It is noted that
these equations are reduced to
the constitutive equations by Landau
if we can set $f_e = f_n = 0$.
By applying the above argument
to the entropy current at most bilinear
with respect to $\Pi$, $q^\mu$, $\nu^\mu$, and $\pi^{\mu\nu}$,
we can obtain the relaxation equations
with $f_e$ and $f_n$ being finite,
which make up the so-called second-order
relativistic dissipative hydrodynamic equations
together with the conservation laws
in Eq.'s (\ref{eq:2-001}) and (\ref{eq:2-002})
\cite{next002}.
Now the dissipative part of the energy-momentum tensor satisfies
$u_{\mu} \, \delta T^{\mu\nu} \, u_{\nu} = \delta e = f_e \, \Pi$
and
$\delta T^{\mu}_{\,\,\,\mu} = \delta e - 3 \, \delta p = (f_e - 3) \, \Pi$.
As emphasized before,
the values of $f_e$ and $f_n$ can be determined only from a microscopic theory.
The phenomenological theory cannot proceed further
because no such logic to determine them is implemented in the theory.
In the following section,
we shall show that the microscopic theory
gives $f_e = 3$ together with $f_n = 0$
in the particle frame while
$f_e = f_n = 0$ in the energy frame, and hence
$\delta T^{\mu}_{\,\,\,\mu} = 0$
but $u_{\mu} \, \delta T^{\mu\nu} \, u_{\nu} = 3 \, \Pi \neq 0$
in the particle frame.
This fact tells us that the usual constraint employed for the particle frame
must be abandoned,
and all the analyses based on this constraint should be redone.
\setcounter{equation}{0}
\section{
Derivation of second-order equations
as long wavelength and low frequency
limit of relativistic Boltzmann equation
}
The argument so far is in the stage of thermodynamics
where the argument is robust
but the parameters such as $f_e$ and $f_n$
as well as the equations of state $e$, $p$ and $n$ appearing in the theory
remain undetermined.
The problem which we encounter is how to reduce a dynamical equation
to a slower one described with fewer dynamical variables.
For this purpose,
we will investigate the infrared limit of
the relativistic Boltzmann equation
with use of a powerful reduction method,
the ``RG method''
\cite{rgm001,env001,env004,HK02,env006}.
The RG method is a systematic reduction theory of the dynamics
leading to the coarse-graining of temporal and spatial scales.
The full presentation of the reduction
of the relativistic Boltzmann equation
to the second-order hydrodynamic equation is technical and involved.
So we here only present main results with key several equations,
leaving the detailed account to another publication
\cite{next002},
although the derivation of a wide class of the first-order equations
is presented
in Ref.\cite{TKO}.
We start with the simple relativistic Boltamann equation,
\begin{eqnarray}
\label{eq:2.1.1}
p^\mu \, \partial_\mu \, f_p(x) = C[f]_p(x),
\end{eqnarray}
where $f_p(x)$ denotes the one-particle distribution function
defined in the phase space $(x^{\mu} \,,\, p^{\mu})$
with $p^\mu$ being the four momentum of the on-shell particle.
The right-hand side of Eq.(\ref{eq:2.1.1}) is the collision integral,
$C[f]_p(x) \equiv \frac{1}{2!} \, \sum_{p_1} \, \frac{1}{p_1^0} \,
\sum_{p_2} \, \frac{1}{p_2^0} \, \sum_{p_3} \, \frac{1}{p_3^0} \,
\omega(p \,,\, p_1|p_2 \,,\, p_3) \,
( f_{p_2}(x) \, f_{p_3}(x) - f_p(x) \, f_{p_1}(x) )$,
where $\omega(p \,,\, p_1|p_2 \,,\, p_3)$ denotes
the transition probability owing to the microscopic two-particle interaction.
We are interested in the hydrodynamical regime
where the time- and space-dependence of the physical quantities are small.
In another word,
the time and space entering the hydrodynamic equation are
the ones coarse-grained from those in the kinetic equation.
Thus we are lead to introduce a macroscopic Lorentz vector,
$\Ren{a}^\mu_p(x)$
which specifies the covariant coordinate system
and we call the \textit{macroscopic-frame vector}.
With use of $\Ren{a}^\mu_p(x)$,
we define the macroscopic covariant coordinate system $(\tau \,,\, \sigma^\mu)$ as
$\mathrm{d}\tau \equiv \Ren{a}_p^\mu(x) \, \mathrm{d}x_\mu$
and
$\varepsilon^{-1} \, \mathrm{d}\sigma^\mu
\equiv ( g^{\mu\nu} - \Ren{a}_p^\mu(x)\Ren{a}_p^\nu(x)/\Ren{a}_p^2(x) )
\, \mathrm{d}x_\nu
\equiv \Ren{\Delta}_p^{\mu\nu}(x) \, \mathrm{d}x_\nu$.
We note that the small quantity $\varepsilon$ has been introduced to tag
that the space derivatives are small for the system we are interested in.
$\varepsilon$ may be identified with the ratio of the average
particle distance over the mean free path,
i.e., the Knudsen number.
In this coordinate system, Eq.(\ref{eq:2.1.1}) can be cast into
\begin{eqnarray}
\label{eq:2.1.6}
\frac{\partial}{\partial \tau} f_p(\tau \,,\, \sigma)
= \frac{1}{p \cdot \Ren{a}_p(\tau \,,\, \sigma)} \, C[f]_p(\tau \,,\, \sigma)
- \varepsilon \, \frac{1}{p \cdot \Ren{a}_p(\tau \,,\, \sigma)} \,
p \cdot \Ren{\nabla} f_p(\tau \,,\, \sigma),
\end{eqnarray}
where $\Ren{a}_p^\mu(\tau \,,\, \sigma) \equiv \Ren{a}_p^\mu(x)$,
$\Ren{\Delta}_p^{\mu\nu}(\tau \,,\, \sigma) \equiv \Ren{\Delta}_p^{\mu\nu}(x)$,
and $f_p(\tau \,,\, \sigma) \equiv f_p(x)$.
Since $\varepsilon$ appears in front of
$\Ren{\nabla}^\mu \equiv \Ren{\Delta}_p^{\mu\nu}(\tau \,,\, \sigma)
\, \frac{\partial}{\partial \sigma^\nu}$,
Eq.(\ref{eq:2.1.6}) has a form to which the perturbative expansion
with respect to $\varepsilon$ can be applied.
In the perturbative expansion, we shall take the coordinate system
where $\Ren{a}_p^\mu(\tau \,,\, \sigma)$ has no $\tau$ dependence,
i.e., $\Ren{a}_p^\mu(\tau \,,\, \sigma) = \Ren{a}_p^\mu(\sigma)$.
The zeroth-order approximate solution we construct
is a stationary solution,
which is identical to a local equilibrium distribution function
given by the Juetner function
$f^{\mathrm{eq}}_p \equiv (2\,\pi)^{-3} \, \exp [(\mu - p \cdot u)/T]$.
Note that this solution
contains five would-be integration constants,
$T$, $\mu$, and $u^\mu$ with $u^\mu \, u_\mu = 1$,
which can be identified with
the temperature, the chemical potential,
and the fluid velocity, respectively.
The collision integral is expanded around the zeroth-order solution
and is reduced to the linear operator
$A_{pq} \equiv (p \cdot \Ren{a}_p)^{-1} \, \frac{\partial}{\partial
f_q} C[f^\mathrm{eq}]_p$.
Furthermore, it is found to be convenient to
convert $A_{pq}$ to
$L_{pq} \equiv f^{\mathrm{eq}-1}_p \, A_{pq} \, f^{\mathrm{eq}}_q =
[ f^{\mathrm{eq}-1} \, A \, f^{\mathrm{eq}} ]_{pq}$,
with the diagonal matrix
$f^\mathrm{eq}_{pq} \equiv f^\mathrm{eq}_p \delta_{pq}$.
We also define the inner product between arbitrary vectors
$\varphi_p$ and $\psi_p$ by
\begin{eqnarray}
\label{eq:3.2.11}
\langle \, \varphi \,,\, \psi \,
\rangle \equiv \sum_{p} \, \frac{1}{p^0} \, (p \cdot \Ren{a}_p) \,
f^{\mathrm{eq}}_p \, \varphi_p \, \psi_p.
\end{eqnarray}
With this inner product, we can define a normed linear space.
Now the first-order solution is given
in terms of the five zero modes of $L$,
$\varphi^\alpha_{0p} = (1,\,p^\mu)$.
The corresponding variables are
just $T$, $\mu$, and $u^\mu$ with $u_{\mu}u^{\mu} = 1$.
The zero modes span a linear space $\mathrm{P}_0$,
which is an invariant manifold for the asymptotic dynamics
of the relativistic Boltamann equation
in the terminology in the dynamical systems
\cite{env004,inv-manifold}.
Then the second-order solution is given by incorporating
the next slow modes, which span a linear space $\mathrm{P}_1$.
We naturally require $\mathrm{P}_1$ is orthogonal to
$\mathrm{P}_0 $, that is, $\mathrm{P}_0 \perp \mathrm{P}_1$.
We find that $\mathrm{P}_1$ is expanded by the bilinear forms of momenta;
$\varphi^{\mu\nu}_{1p} \equiv [ Q_0 \, \tilde{\varphi}^{\mu\nu} ]_p$,
where $\tilde{\varphi}^{\mu\nu}_p \equiv p^\mu \, p^\nu$, and
$Q_0$ is the projection to complement to $\mathrm{P}_0$.
By definition,
$\langle \, \varphi^{\mu\nu}_1 \,,\, \varphi^\alpha_0 \, \rangle =
0$ is satisfied.
Note that the dimension of $\varphi^{\mu\nu}_{1p}$ is nine,
which correspond to
the number of the new would-be integration constants,
$\Pi$, $J^\mu$ with $J^\mu \, u_\mu = 0$,
and $\pi^{\mu\nu}$ with
$\pi^{\mu\nu} = \pi^{\nu\mu}$
and
$\pi^{\mu\nu} \, u_\nu = \pi^\mu_{\,\,\,\mu} = 0$.
A generic choice of the macroscopic frame vector is
$\Ren{a}^\mu_p = ((p\cdot u) \, \cos \theta + m \,
\sin\theta)/(p\cdot u) \, u^\mu$,
where $\theta$ is a parameter defining the frame.
For example, $\theta = 0$ ($\theta = \pi/2$) gives
the energy (particle) frame.
The resultant generic relaxation equations
of the second-order hydrodynamic equation with
$\theta$ being kept are
\begin{eqnarray}
\Pi &=& X_\Pi - \tau_\Pi \, D\Pi - \ell_{\Pi J} \, \nabla^a J_a
+ X_{\Pi\Pi} \, \Pi + X_{\Pi J}^a \, J_a + X_{\Pi\pi}^{ab} \,
\pi_{ab},\\
J^\mu &=& X^\mu_J - \tau_J \, \Delta^{\mu a} \, DJ_a - \ell_{J\Pi}
\, \nabla^\mu \Pi - \ell_{J\pi} \, \Delta^{\mu abc} \, \nabla_a
\pi_{bc} + X_{J\Pi}^\mu \, \Pi + X_{JJ}^{\mu a} \, J_a +
X_{J\pi}^{\mu ab} \, \pi_{ab},\\
\pi^{\mu\nu} &=& X^{\mu\nu}_\pi - \tau_\pi \, \Delta^{\mu\nu ab}
\, D \pi_{ab} - \ell_{\pi J} \, \Delta^{\mu\nu ab} \, \nabla_a J_b
+ X_{\pi\Pi}^{\mu\nu} \, \Pi + X_{\pi J}^{\mu\nu a} \, J_a +
X_{\pi\pi}^{\mu\nu ab} \, \pi_{ab}.
\end{eqnarray}
Here, $X_\Pi$, $X^\mu_J$, and $X^{\mu\nu}_\pi$
are the thermodynamic forces;
their simple forms retaining only
$X_\Pi$, $X^\mu_J$, and $X^{\mu\nu}_\pi$
are the usual constitutive equations.
The relaxation equations of
$\Pi$, $J^\mu$, and $\pi^{\mu\nu}$ are characterized by
the relaxation times $\tau_\Pi$, $\tau_J$, and $\tau_\pi$,
while $\ell_{\Pi J}$, $\ell_{J \Pi}$, $\ell_{J \pi}$, and $\ell_{\pi J}$
mean the relaxation lengths.
The correction to the thermodynamic forces
$X_\Pi$, $X^\mu_J$, and $X^{\mu\nu}_\pi$ are given by
$X_{\Pi\Pi}$, $X_{\Pi J}^a$, $X_{\Pi \pi}^{ab}$,
$X_{J \Pi}^\mu$, $X_{J J}^{\mu a}$, $X_{J \pi}^{\mu ab}$,
$X_{\pi\Pi}^{\mu\nu}$, $X_{\pi J}^{\mu\nu a}$, and $X_{\pi \pi}^{\mu\nu ab}$.
The continuity equations of the second-order equation
in the energy frame
is found to be given by setting $\theta = 0$
as in the first-order case
\cite{TKO}
and read $\partial_\mu T^{\mu\nu} = 0$
and
$\partial_\mu N^\mu = 0$,
where
\begin{eqnarray}
T^{\mu\nu} &=& e \, u^\mu \, u^\nu - (p + \Pi) \, \Delta^{\mu\nu}
+ \pi^{\mu\nu},\\
N^\mu &=& n \, u^\mu + J^\mu.
\end{eqnarray}
The thermodynamic forces are
$X_\Pi = - \zeta \, \nabla^a u_a$,
$X^\mu_J = \lambda \, \hat{h}^{-2} \, \nabla^\mu (\mu/T)$,
and
$X^{\mu\nu}_\pi = 2 \, \eta \, \Delta^{\mu\nu ab} \, \nabla_a u_b$,
which clearly show that $f_n = 0$ and $f_e=0$ as was anticipated.
The energy-momentum tensor and particle-number vector
in the particle frame with $\theta = \pi/2$ read
\begin{eqnarray}
T^{\mu\nu} &=& (e + 3 \, \Pi) \, u^\mu \, u^\nu - (p + \Pi) \, \Delta^{\mu\nu}
+ u^\mu \, J^\nu + u^\nu \, J^\mu + \pi^{\mu\nu},\\
N^\mu &=& n \, u^\mu,
\end{eqnarray}
and
$X_\Pi = - \zeta \, (3 \, \gamma - 4 )^{-2} \,
(\nabla^a u_a - 3 \, T \, DT^{-1})$,
$X^\mu_J = - \lambda \, T^2 \, (\nabla^\mu T^{-1} + T^{-1} \, D u^\mu )$,
and
$X^{\mu\nu}_\pi = 2 \, \eta \, \Delta^{\mu\nu ab} \, \nabla_a u_b$,
where
$\gamma \equiv 1 + (z^2 - \hat{h}^2 + 5\,\hat{h} - 1)^{-1}$ is the ratio
of the specific heats.
Thus we find that $f_e = 3$ with $f_n = 0$, as we announced.
Although we have obtained the relaxation equations
for the dissipative forces $\Pi$, $J^{\mu}$, and $\pi ^{\mu \nu}$
for arbitrary $\theta$
\cite{next002},
we shall only write down them for two typical frames,
i.e., the energy ($\theta=0$)
and the particle ($\theta=\pi/2$) frames for the sake of the space.
(A) In the energy frame ($\theta=0$):
\begin{eqnarray}
\label{eq:EF_Pi}
\Pi &=& - \zeta \, \nabla^a u_a - \tau_\Pi \, D\Pi - \ell_{\Pi J}
\, \nabla^a J_a\nonumber\\
&&{}- \frac{1}{2} \, \tau_\Pi \, \Bigg\{
\kappa_\Pi \, \nabla^a u_a
+ \frac{\zeta \, T}{\tau_\Pi} \,
\partial_a \Big(\frac{\tau_\Pi}{\zeta \, T} \, u^a\Big)
\Bigg\} \,
\Pi\nonumber\\
&&{}- \frac{1}{2} \, \ell_{\Pi J} \, \Bigg\{
\kappa^{(0)}_{\Pi J} \, \nabla^a \frac{\mu}{T}
-
\kappa^{(1)}_{\Pi J} \, D u^a
+ \frac{\zeta \, T}{\ell_{\Pi J}} \,
\partial_b \Big(\frac{\ell_{\Pi J}}{\zeta\,T} \,
\Delta^{bc}\Big)\,\Delta_c^{\,\,\,a}
\Bigg\} \, J_a\nonumber\\
&&{}-\frac{1}{2} \, \ell_{\Pi\pi} \, \Bigg\{
- \kappa_{\Pi\pi} \, \Delta^{abcd} \, \nabla_c u_d
\Bigg\} \, \pi_{ab},\\
\label{eq:EF_J}
J^\mu &=& \lambda \, \hat{h}^{-2} \, \nabla^\mu
\frac{\mu}{T} - \ell_{J\Pi}
\, \nabla^\mu \Pi
- \tau_J \, \Delta^{\mu a} \, DJ_a
- \ell_{J\pi} \, \Delta^{\mu abc} \, \nabla_a
\pi_{bc}\nonumber\\
&&{}- \frac{1}{2} \, \ell_{J\Pi} \, \Bigg\{
\kappa_{J\Pi}^{(0)} \, \nabla^\mu \frac{\mu}{T}
-
\kappa_{J\Pi}^{(1)} \, Du^\mu
+ \frac{\lambda \, \hat{h}^{-2}}{\ell_{J\Pi}} \,
\Delta^\mu_{\,\,\,a} \,
\partial_b \Big(
\frac{\ell_{J\Pi}}{\lambda \, \hat{h}^{-2}} \, \Delta^{ab}
\Big)
\Bigg\} \,
\Pi\nonumber\\
&&{}- \frac{1}{2} \, \tau_J \, \Bigg\{
\Delta^{\mu a} \, \Bigg[
\kappa_J^{(0)} \, \nabla^b u_b
+ \frac{\lambda \, \hat{h}^{-2}}{\tau_J} \,
\partial_b \Big(\frac{\tau_J}{\lambda \, \hat{h}^{-2}} \, u^b\Big)
\Bigg]
- 2 \, \kappa_J^{(1)} \, \Delta^{\mu abc} \, \nabla_b u_c
- 2 \, \omega^{\mu a}
\Bigg\} \, J_a\nonumber\\
&&{}- \frac{1}{2} \, \ell_{J\pi} \, \Bigg\{
\Delta^{\mu cab} \, \Big(
\kappa^{(0)}_{J\pi} \, \nabla_c \frac{\mu}{T}
-
\kappa^{(1)}_{J\pi} \, D u_c
\Big)
+ \frac{\lambda\,\hat{h}^{-2}}{\ell_{J\pi}} \, \Delta^\mu_{\,\,\,c} \,
\partial_d \Big(\frac{\ell_{J\pi}}{\lambda\,\hat{h}^{-2}} \, \Delta^{cdef}\Big) \,
\Delta_{ef}^{\,\,\,\,\,\,ab}
\Bigg\} \, \pi_{ab},\\
\label{eq:EF_pi}
\pi^{\mu\nu} &=& 2 \, \eta \, \Delta^{\mu\nu ab} \, \nabla_a
u_b - \ell_{\pi J} \, \Delta^{\mu\nu ab} \, \nabla_a J_b
- \tau_\pi \, \Delta^{\mu\nu ab}
\, D \pi_{ab}\nonumber\\
&&{}- \frac{1}{2} \, \ell_{\pi\Pi} \, \Bigg\{
- \kappa_{\pi\Pi} \, \Delta^{\mu\nu ab} \,
\nabla_a u_b
\Bigg\} \, \Pi\nonumber\\
&&{}- \frac{1}{2} \, \ell_{\pi J} \, \Bigg\{
\Delta^{\mu\nu ba} \, \Big(
\kappa^{(0)}_{\pi J} \, \nabla_b \frac{\mu}{T}
-
\kappa^{(1)}_{\pi J} \, D u_b
\Big)
+ \frac{\eta\,T}{\ell_{\pi J}} \,
\Delta^{\mu\nu}_{\,\,\,\,\,\,bc} \, \partial_d
\Big(\frac{\ell_{\pi J}}{\eta\,T} \,
\Delta^{bcde}\Big) \, \Delta_e^{\,\,\,a}
\Bigg\} \, J_a\nonumber\\
&&{}- \frac{1}{2} \, \tau_\pi \, \Bigg\{
\Delta^{\mu\nu ab} \,
\Bigg[
\kappa^{(0)}_{\pi} \, \nabla^c u_c
+ \frac{\eta\,T}{\tau_\pi} \,
\partial_c \Big(\frac{\tau_\pi}{\eta\,T} \, u^c\Big)
\Bigg]
- 4 \, \kappa^{(1)}_\pi \, \Delta^{\mu\nu ce} \,
\Delta_e^{\,\,\,dab} \, \Delta_{cd}^{\,\,\,\,\,\,fg} \,
\nabla_f u_g
- 4 \, \Delta^{\mu\nu ce} \, \Delta_{e}^{\,\,\,dab}
\, \omega_{cd}
\Bigg\} \, \pi_{ab},\nonumber\\
\end{eqnarray}
where $\omega^{\mu\nu} \equiv (\nabla^\mu u^\nu - \nabla^\nu
u^\mu)/2$ is the vorticity.
(B) In the particle frame ($\theta=\pi/2$):
\begin{eqnarray}
\label{eq:PF_Pi}
\Pi &=& - \zeta \, (3 \, \gamma - 4 )^{-2} \,
\Big(\nabla^a u_a - 3 \, T \, D\frac{1}{T}\Big) - \tau_\Pi \, D\Pi - \ell_{\Pi J}
\, \nabla^a J_a\nonumber\\
&&{}- \frac{1}{2} \, \tau_\Pi \, \Bigg\{
\kappa_\Pi \, \nabla^a u_a
+ \frac{\zeta \, (3 \, \gamma - 4 )^{-2} \, T}{\tau_\Pi} \,
\partial_a \Big(\frac{\tau_\Pi}{\zeta \, (3 \, \gamma - 4 )^{-2} \, T} \, u^a\Big)
\Bigg\} \,
\Pi\nonumber\\
&&{}- \frac{1}{2} \, \ell_{\Pi J} \, \Bigg\{
\kappa^{(0)}_{\Pi J} \, \nabla^a \frac{\mu}{T}
-
\kappa^{(1)}_{\Pi J} \, D u^a
+ \frac{\zeta \, (3 \, \gamma - 4 )^{-2} \, T}{\ell_{\Pi J}} \,
\partial_b \Big(\frac{\ell_{\Pi J}}{\zeta\,(3 \, \gamma - 4 )^{-2} \, T} \,
\Delta^{bc}\Big)\,\Delta_c^{\,\,\,a}
\Bigg\} \, J_a\nonumber\\
&&{}-\frac{1}{2} \, \ell_{\Pi\pi} \, \Bigg\{
- \kappa_{\Pi\pi} \, \Delta^{abcd} \, \nabla_c u_d
\Bigg\} \, \pi_{ab},\\
\label{eq:PF_J}
J^\mu &=&
- \lambda \, T^2 \, \Big(\nabla^\mu
\frac{1}{T} + \frac{1}{T} \, D u^\mu \Big) - \ell_{J\Pi}
\, \nabla^\mu \Pi
- \tau_J \, \Delta^{\mu a} \, DJ_a
- \ell_{J\pi} \, \Delta^{\mu abc} \, \nabla_a
\pi_{bc}\nonumber\\
&&{}- \frac{1}{2} \, \ell_{J\Pi} \, \Bigg\{
\kappa_{J\Pi}^{(0)} \, \nabla^\mu \frac{\mu}{T}
-
\kappa_{J\Pi}^{(1)} \, Du^\mu
+ \frac{\lambda \, T^{2}}{\ell_{J\Pi}} \,
\Delta^\mu_{\,\,\,a} \,
\partial_b \Big(
\frac{\ell_{J\Pi}}{\lambda \, T^{2}} \, \Delta^{ab}
\Big)
\Bigg\} \,
\Pi\nonumber\\
&&{}- \frac{1}{2} \, \tau_J \, \Bigg\{
\Delta^{\mu a} \, \Bigg[
\kappa_J^{(0)} \, \nabla^b u_b
+ \frac{\lambda \, T^{2}}{\tau_J} \,
\partial_b \Big(\frac{\tau_J}{\lambda \, T^{2}} \, u^b\Big)
\Bigg]
- 2 \, \kappa_J^{(1)} \, \Delta^{\mu abc} \, \nabla_b u_c
- 2 \, \omega^{\mu a}
\Bigg\} \, J_a\nonumber\\
&&{}- \frac{1}{2} \, \ell_{J\pi} \, \Bigg\{
\Delta^{\mu cab} \, \Big(
\kappa^{(0)}_{J\pi} \, \nabla_c \frac{\mu}{T}
-
\kappa^{(1)}_{J\pi} \, D u_c
\Big)
+ \frac{\lambda\,T^{2}}{\ell_{J\pi}} \, \Delta^\mu_{\,\,\,c} \,
\partial_d \Big(\frac{\ell_{J\pi}}{\lambda\,T^{2}} \, \Delta^{cdef}\Big) \,
\Delta_{ef}^{\,\,\,\,\,\,ab}
\Bigg\} \, \pi_{ab},
\end{eqnarray}
and Eq.(\ref{eq:EF_pi}).
Note that the effective bulk viscosity $\zeta_\mathrm{eff} \equiv
\zeta \, (3 \, \gamma - 4 )^{-2}$ \cite{Tsumura:2007wu}
appears in Eq.(\ref{eq:PF_Pi}).
Here we have introduced
the new coefficients,
$\ell_{\Pi\pi}$, $\ell_{\pi\Pi}$,
$\kappa_\Pi$,
$\kappa^{(0)}_{\Pi J}$,
$\kappa^{(1)}_{\Pi J}$,
$\kappa_{\Pi\pi}$,
$\kappa^{(0)}_{J \Pi}$,
$\kappa^{(1)}_{J \Pi}$,
$\kappa^{(0)}_J$,
$\kappa^{(1)}_J$,
$\kappa^{(0)}_{J\pi}$,
$\kappa^{(1)}_{J\pi}$,
$\kappa_{\pi\Pi}$,
$\kappa^{(0)}_{\pi J}$,
$\kappa^{(1)}_{\pi J}$,
$\kappa^{(0)}_\pi$,
and
$\kappa^{(1)}_\pi$,
which are complicated functions of $T$ and $\mu$ \cite{next002}.
We have found that the relaxation times are frame dependent except for $\tau_\pi$
while the transport coefficients
such as the viscosities and the thermal conductivity are frame independent.
For a demonstration of the frame-dependence of the relaxation times,
we show in FIG.\ref{fig:1} the frame ($\theta$) dependence of
$\tau_\Pi$ and $\tau_J$,
which tends to increase when the frame changes
from the energy to particle frame.
\begin{figure}
\begin{center}
\begin{minipage}{.45\linewidth}
\includegraphics[width=\linewidth]{fig_tau_PI.eps}
\end{minipage}
\begin{minipage}{.45\linewidth}
\includegraphics[width=\linewidth]{fig_tauJ.eps}
\end{minipage}
\end{center}
\caption{
\label{fig:1}
The $\theta$ dependence of $\tau_\Pi$ and $\tau_J$
at $m/T = 0.5$ and $\mu/T = 0.0$.
We normalized the relaxation times
by the corresponding transport coefficients.
The energy and particle frames correspond to
$\theta=0$ and $\pi/2$.
}
\end{figure}
\section{
Brief summary
}
In summary, we have derived
the second-order dissipative relativistic hydrodynamic equations
in a generic frame with a continuous parameter $\theta$;
the generic frame is reduced
to the energy and particle frame with the parameter choice
$\theta=0$ and $\pi/2$, respectively.
A notable point of our result is that
the dissipative part of the symmetric energy-momentum tensor
$\delta T^{\mu\nu}$ in the \textit{particle} frame
satisfies the equality
$\delta T^{\mu}_{\,\,\,\mu} = 0$,
in contrast to the usual choice
$u_{\mu} \, \delta T^{\mu\nu} \, u_{\nu} = 0$,
while $\delta T^{\mu\nu}$ of our derived equation
in the \textit{energy} frame
satisfies the usual constraint $u_{\mu} \, \delta T^{\mu\nu} \, u_\nu = 0$.
We emphasize that
this novel equality in the particle frame
is a consequence of the derivation
based on the renormalization-group method,
a powerful method for the reduction of dynamical systems.
We note that
the same constraints were also derived
for the first-order dissipative relativistic hydrodynamic equation
\cite{TKO,Tsumura:2007wu}.
We have also shown that
the phenomenological derivation
based on the second law of thermodynamics allows that
$u_{\mu} \, \delta T^{\mu\nu} \, u_{\nu}$ can be proportional
to the bulk pressure $\Pi$
and non-vanishing in the particle frame.
Indeed, our microscopic derivation shows that
$u_{\mu} \, \delta T^{\mu\nu} \, u_{\nu} = 3 \, \Pi$.
We have presented the relaxation equations
in the energy and particle frames,
explicitly as typical examples,
although we have obtained the microscopic expressions
for them in a more generic frame
\cite{next002}.
We have shown that
the viscosities are frame-independent
but the relaxation times are generically frame-dependent,
as depicted in FIG.\ref{fig:1}.
The detailed derivation of the equations
and discussions on the phenomenological consequences
of the hydrodynamic equations thus obtained
will be discussed in forthcoming papers
\cite{next002}.
\section*{acknowledgment}
We thank Tetsu Hirano for his interest in this work and discussions.
T.K. thanks Dirk Rischke for his interest in our work.
This work was partially supported
by a Grant-in-Aid for Scientific Research by the Ministry of Education,
Culture, Sports, Science and Technology (MEXT) of Japan (No.20540265),
by Yukawa International Program for Quark-Hadron Sciences,
and by the Grant-in-Aid for the global COE program
``The Next Generation of Physics, Spun from Universality and Emergence''
from MEXT.
|
3,212,635,537,452 | arxiv | \section{Introduction}
Knowledge about thermalization in strongly coupled systems is important for us to understand the results from relativistic heavy-ion collisions at RHIC and LHC and some non-equilibrium processes in the early universe. The AdS/CFT correspondence\cite{Maldacena,Witten} provides us with an elegant tool for studying the strongly coupled large $N_C$ super Yang-Mills (CFT). The gravity dual of static plasma in 4-dimensional Minkowski space($M^4$) is the AdS black brane(black hole) solution in the Poincare patch of $AdS_{5}$. And studying gravitational collapse of matter fields in the bulk of $AdS_5$ helps understand the thermalization process on the CFT side.
To study thermalization by AdS/CFT, one needs first to choose some far-from-equilibrium states. Such states can be prepared by injecting energy into the CFT vacuum. This can be done by turning on sources for some boundary operators\cite{holographicReconstruction}. Examples for such boundary sources are the boundary metric coupled to the boundary stress energy tensor\cite{CY01} or scalar sources coupled to marginal scalar composite operators\cite{Minwalla}. Another kind of initial states are obtained by reconstructing some bulk metric based on the knowledge of the CFT data\cite{holographicReconstruction}. One of this kind is the double shockwave metric\cite{JanikPeschanski2005,Yuri:2008, GrumillerPaul, Gubser:2008pc,AlvarezGaume:2008fx, Lin:2009pn, Yuri:2009,Kovchegov:2009du, CY02, withPaul,KiritsisTaliotis}, motivated by understanding thermalization in heavy-ion collisions(see \cite{CasalderreySolana:2011us} for a recent review). Of course one can also start from the gravity side by constructing consistent initial conditions for Einstein's equations. For this kind of initial conditions, the interested reader is referred to \cite{Heller:2011} for boost-invariant cases and \cite{Bizon01,Bizon02,Garfinkle01,Garfinkle02,Gubser:2012} for the global patch of $AdS_{d+1}$. Given initial conditions and suitable boundary conditions, thermalization in strongly coupled CFT may be studied by solving Einstein's equations in the bulk of $AdS_5$.
In this paper, we study thermalization of a spatially homogenous system in CFT\cite{Esko:1999,LinShuryak, Minwalla, holographicThermal,Galante:2012pv,Erdmenger:2012xu}. The energy is injected into the system by turning on a spatially homogenous scalar source $\phi_0(t)$ for a marginal scalar composite operator $\hat{O}$. The cases of weak fields were first tackled by the authors of Ref. \cite{Minwalla} using perturbative techniques. We will deal with more general cases using a numerical method similar to that used in the global patch of $AdS_{d+1}$\cite{Bizon01,Bizon02,Garfinkle01,Garfinkle02}. In CFT the Lagrangian of the system in general takes the following form
\begin{equation}
S = \underbrace{S_{CFT} + \int d^4x \phi_0(t) \hat{O}(t)}_{\text{thermalization}} + \underbrace{\int d^4x A_{0}^{a} J_a}_{\text{probes}},\label{equ:actionCFT}
\end{equation}
where $S_{CFT}$ is the CFT action, $A_{0}^a$ and $J_a$ denote all the other external sources and currents(operators). On the gravity side, the bulk metric is determined by solving the gravitational equations coupled to the massless scalar field $\phi$ corresponding to $\phi_0$. And the back-reaction of all the other fields $A^a$ corresponding to the boundary sources $A_0^{a}$ is assumed to be negligible. The expectation values of $J_a$ and their correlators can be calculated by solving the equations of motion of those weak fields in the bulk metric\cite{Witten}. They are also essential for understanding the details of thermalization in a coordinate-independent way\cite{holographicThermal}.
In the introduction we summarize our results. The metric in Schwarzschild(Poincare) coordinates can be written in the for
\begin{equation}\label{equ:FeffermanGraham}
ds^2 = \frac{1}{u^2}\left( -f e^{-2 \delta} dt^2 + f^{-1} du^2 + d\vec{x}^2 \right),
\end{equation}
where $f$ and $\delta$ are functions of $t$ and $u$ only. And the AdS black hole metric is given by
\begin{equation}
f_{bh} = 1-\frac{u^4}{u_0^4},\text{ and } \delta = 0,\label{equ:adsstaticbh}
\end{equation}
where $u_0=\frac{1}{\pi T}$ and $T$ is the thermal equilibrium(Hawking) temperature. The energy is injected into the system according to\cite{holographicReconstruction}
\begin{equation}
{{\dot{T}}^{ (4) } }_{00} = \left< \hat{O}\right> \dot{\phi_0},\label{equ:energyConservation}
\end{equation}
where $ \left< \hat{O}\right>$ is the expectation value of $\hat{O}$. The source $\phi_0(t)$ explicitly takes the following form
\begin{equation}
\phi_0(t) = \frac{\epsilon}{a} e^{-a t^2},\label{equ:phi0t}
\end{equation}
where $\epsilon$ and $a\equiv \frac{1}{\Delta t^2}$ are two parameters and $\Delta t$ characterizes the duration of the source being turned on. However, our qualitative conclusions should hold regardless of the source's shape. The source induces an ingoing wave of the scalar field in the bulk, which will eventually collapse to form a black hole. We will not define the thermalization time $t_T$ in terms of the formation time of the apparent horizon because it is coordinate-dependent\cite{WaldGR,Hubeny:2011}. Instead, we define $t_T$ as the moment when the AdS black hole metric is established in the most part of the bulk that is causally connected to the boundary. $t_T$ can be interpreted as the scale-dependent thermalization time\cite{holographicThermal} defined by a spacelike Wilson loop $\left< W(l)\right>$ with $l\sim \frac{1}{T}$ in CFT. With such a definition, our results respect the following scaling invariance
\begin{equation}
(a,\epsilon, t_T, T) \to (\lambda^2 a, \lambda^2 \epsilon, t_T/\lambda, \lambda T)
\end{equation}
with $\lambda >0$, which gives
\begin{equation}
t_T = \frac{g_t}{T}.\label{equ:tTg}
\end{equation}
$g_t$ is scale-dependent\cite{holographicThermal} and in this paper we only discuss thermalization with $l \sim \frac{1}{T}$. In the following, we will show that $g_t=O(1)$ in this case.
\begin{figure}
\begin{center}
\includegraphics[width=16cm]{topdown}
\end{center}
\caption{Narrow wave($\Delta t \lesssim \frac{1}{T}$). At $t \sim \Delta t$, the source on the boundary CFT is turned off and a narrow ingoing wave with a width $\Delta u \sim \Delta t $ is induced near the boundary $u=0$. The wave starts to propagate at the speed of light($du_s/dt\simeq1$) in the bulk and leaves behind a submanifold$(u<u_s)$ equipped with the AdS black hole metric. It propagates more slowly in the deeper interior of $AdS_5$. At $t \sim \frac{1}{T}$, $du_s/dt\simeq0$ and the AdS black hole metric is established in the most part of the bulk that is causally connected to the boundary.
}\label{fig:topdown}
\end{figure}
In the cases with $\frac{\epsilon}{a}\lesssim 1$, the boundary source induces a narrow wave with $\Delta t \equiv \frac{1}{\sqrt{a}} \lesssim \frac{1}{T}$ in the bulk. Our results are illustrated in Fig. \ref{fig:topdown}. We find that $g_t\lesssim 1$ for all the narrow waves and $g_t \simeq 0.73$ if $\Delta t \lesssim \frac{0.02}{T}$. In CFT, the interpretation of our results is as follows: the system, after the source is turned off, thermalizes in a time scale $t_T \sim \frac{1}{T}$ in a top-down manner\cite{LinShuryak,holographicThermal}. This is in sharp contrast with bottom-up thermalization in perturbative QCD\cite{bottomup,Kurkela:2011}, in which soft gluons equilibrate more quickly than hard gluons. In CFT the interaction is equally efficient for soft and hard modes because of the vanishing beta function. At $t\sim \Delta t$, only time-like high momentum($\omega\sim \frac{1}{\Delta t} = \sqrt{a}$) modes are present in the system(see \cite{energylossandpT} for a similar interpretation in the case of a classical string and \cite{HattaIancuAl} for an $R$ current). Those high $\omega$ modes split into low $\omega$ modes very rapidly\cite{HattaIancuAl}. In the meanwhile the interaction is so efficient that the (remaining) high $\omega$ modes equilibrate in a time $\sim\frac{1}{\omega}$. Such a splitting-equilibration continues from higher $\omega$ to lower $\omega$ modes. At $t \sim \frac{1}{T}$ all the modes with $\omega \gtrsim T$ achieve thermal equilibrium.
\begin{figure}
\begin{center}
\includegraphics[height=6cm]{pulse}
\includegraphics[height=6cm]{sinpulse}
\end{center}
\caption{Two-stage and multi-stage thermalization. Fig. (a): The energy is injected into the CFT vacuum by the two pulses of $\dot{\phi}$. If $\Delta t \gtrsim \frac{1}{T_L}$, the system first thermalizes at a lower temperature $T_L$ at $t \simeq 0$. Then, the second pulse heats up the plasma to a higher temperature $T_H$. Fig. (b) shows a source for a possible multi-stage thermalization.
}\label{fig:2stagesThermalization}
\end{figure}
In the cases with $\frac{\epsilon}{a}\gtrsim 1$, the induced waves are broad and we do not evaluate the exact value of $g_t$ in (\ref{equ:tTg}). The typical thermalization time $t_T$ shows up in a way illustrated in Fig. \ref{fig:2stagesThermalization}. The system is expected to thermalize in a time $t_T\sim \frac{1}{T}$. Therefore, if the source is turned on and off several times with period $\Delta t \sim \frac{1}{T}$, the system should achieve thermal equilibrium within each time interval $\Delta t$. We refer to such a thermalization pattern as a two-stage(Fig. \ref{fig:2stagesThermalization}(a)) or multiple-stage(Fig. \ref{fig:2stagesThermalization}(b)) thermalization. This is the interpretation of the double- or multiple-collapse solutions that we find in the bulk. According to eq. (\ref{equ:energyConservation}), the energy is injected into the system by the two pulses of $\dot{\phi}$ as illustrated in Fig. \ref{fig:2stagesThermalization}(a). After the source is turned off at $t \gtrsim 2 \Delta t$, the scalar field will eventually collapse. If $\frac{\epsilon}{a} \gtrsim 3.0$, at $t\simeq0$ the falling of the scalar field results in a submanifold defined by $u\leq 0.98 u_0$, which is equipped with the AdS black hole metric with the Hawking temperature $T=T_L$. We interpret it as an intermediate thermal equilibrium state with $T=T_L$ in CFT. And it is called the first collapse even though the trapped region has not formed yet(in Schwarzschild coordinates) at this time. Then the final collapse at $t\sim 2 \Delta t$ is naturally interpreted as the second stage of thermalization(heating-up process) in CFT. The criteria for obtaining such double-collapse solutions is $\Delta t \gtrsim \frac{1}{T_L}$. If $\Delta t > \frac{1}{T_L}$, the system thermalizes in a time $\sim \Delta t > \frac{1}{T_L}$ simply because the thermal equilibrium is destroyed by the continuous injection of high $\omega$ modes from top down before $t=0$. A multiple-collapse solution can be defined in a similar way and we find that the above criteria is also parametrically true for obtaining multiple-collapse solutions induced by a periodic source(see Fig. \ref{fig:2stagesThermalization}(b)). Therefore, we conclude that in such a strongly coupled system the typical thermalization time is $t_T \sim \frac{1}{T}$.
This paper is organized as follows. In Sec. \ref{sec:eom}, we derive the equations of motion for a massless scalar field coupled to gravity in the Poincare patch of $AdS_5$. The numerical schemes in addition to initial conditions and boundary conditions are discussed in Sec. \ref{sec:numerics}. In this section, we also define the thermalization time $t_T$.
Our numerical results are presented in Sec. \ref{sec:results}. In Sec. \ref{sec:discussion}, we briefly conclude. In Appendix \ref{app:EddingtonFinkelstein}, we give the equations of motion in Eddington-Finkelstein coordinates. The details of our numerical methods are presented in Appendix \ref{app:numerics}.
\section{ Einstein-Klein-Gordon equations}
\label{sec:eom}
On the AdS side, we need to calculate the back-reaction of a massless scalar field to the bulk geometry. The bulk action in $AdS_{d+1}$ corresponding to the first two terms on the right-hand side of (\ref{equ:actionCFT}) is given by
\begin{equation}
S=\frac{1}{2\kappa_{d+1}^2}\left\{ \int
d^{d+1}x\sqrt{-g}\left\{ R-2\Lambda - 2 \left(\partial \phi \right)^2 \right\} + 2 \int_{\partial M} d^dx \sqrt{\gamma} K \right\},\label{equ:action}
\end{equation}
where $\Lambda=-{d(d-1)\over 2L^2}$ for $AdS_{d+1}$, $\kappa_5^2 = \frac{4 \pi^2 L^3}{N_c^2}$, $L$ is a parameter of dimension of length, $\gamma$ is the induced metric on the boundary and $K$ is the trace of the extrinsic curvature of the boundary.
We need to solve the following Einstein-Klein-Gordon equations
\begin{eqnarray}
&&\partial_a\left(\sqrt{-g} g^{ab} \partial_b \phi \right)=0,\\
&&R_{ab}-\frac{1}{2} g_{ab} R - \frac{d(d-1)}{2L^2} g_{ab} = T_{ab},
\end{eqnarray}
where $T_{ab}$ is the stress tensor of the scalar field, which is given by
\begin{equation}
T_{ab}= 2 \partial_a \phi \partial_b \phi - g_{ab} \left(\partial \phi \right)^2.
\end{equation}
In the following, we take $L=1$ and $d=4$.
In this paper, the scalar sources are assumed to be spatially homogeneous on the boundary $M^4$. In Schwarzschild coordinates, one needs to solve the following equations of motion
\begin{subequations}\label{equ:eom}
\begin{eqnarray}
&&\dot{V} = u^3 \left(\frac{f e^{-\delta} P}{u^3}\right)^\prime,\label{equ:Vdot}\\
&&\dot{P} = \left( f e^{-\delta} V \right)^\prime,\label{equ:Pdot}\\
&&\dot{f} = \frac{4}{3} u f^2 e^{-\delta} V P,\label{equ:fdot}\\
&&\delta^\prime=\frac{2}{3} u \left( V^2 + P^2 \right),\label{equ:deltap}\\
&&f' = \frac{2}{3} u f \left(V^2+ P^2\right) +\frac{4}{u}\left( f-1\right),\label{equ:fp}
\end{eqnarray}
\end{subequations}
where the derivatives with respect to $t$ and $u$ are denoted respectively by overdots and primes, $P\equiv \phi^\prime$, $V\equiv f^{-1} e^\delta \dot{\phi}$ and we take (\ref{equ:fp}) as a constraint equation. If $V$ and $P$ vanish in a submanifold near the boundary of $AdS_5$, the only solution to (\ref{equ:deltap}) and (\ref{equ:fp}) is
\begin{equation}
f = 1 - \frac{u^4}{u_0^4},~~\delta = 0,~~V=0~~\mbox{and}~~P=0,\label{equ:adsbhasy}
\end{equation}
which is Birkhoff's theorem\cite{WaldGR} in such a spatially homogeneous case in $AdS_5$. The equations of motion above are invariant under the following scaling transformation
\begin{eqnarray}\label{equ:dilation}
&&\phi(x^a) \rightarrow \tilde{\phi}=\phi(\lambda^{-1} \tilde{x}^a), ~~V\rightarrow \lambda \tilde{V},~~P\rightarrow \lambda \tilde{P},\nonumber\\\
&&x^a \rightarrow \tilde{x}^a=\lambda x^a,~~f(x^a)\rightarrow \tilde{f}(\tilde{x}^a)=f(\lambda^{-1}\tilde{x}^a),~~\delta(x^a)\rightarrow \tilde{\delta}(\tilde{x}^a)=\delta(\lambda^{-1}\tilde{x}^a).\end{eqnarray}
\section{Gravitational collapse of massless scalar fields}\label{sec:numerics}
In this paper, we study the response of the vacuum/plasma to an external scalar source in CFT. On the gravity side, the scalar source provides the boundary conditions and the vacuum AdS/the AdS black hole metric provides the initial conditions for solving eq. (\ref{equ:eom}). We aim to see how a submanifold near the boundary of $AdS_5$ equipped with the AdS black hole metric in (\ref{equ:adsstaticbh}) forms by gravitational collapse of a massless scalar field.
%
\subsection{Initial conditions}
%
Before the scalar source is turned on, the scalar field is assumed to vanish in the bulk. In this case, by Birkhoff's theorem one has
\begin{equation}
f_{bh}=1-\frac{u^4}{u_0^4},~~\delta=0,~~V=0~~\mbox{and}~~P=0,\label{equ:adsbh}
\end{equation}
which is the AdS black hole metric in eq. (\ref{equ:adsstaticbh}), the gravity dual of static plasma. By holographic renormalization, the stress tensor of the boundary CFT is given by\cite{Kraus}
\begin{equation}
{T^{(4)\mu}}_{\nu} = \frac{3}{8 } N_c^2 \pi^2 T^4 \mbox{diag} \{-1,1/3,1/3,1/3 \}.
\end{equation}
Taking $T\to 0$, one gets the vacuum AdS metric, dual to the CFT vacuum, as follows
\begin{eqnarray}
f_{vac}=1,~~\delta=0,~~V=0~~\mbox{and}~~P=0.\label{equ:initial}
\end{eqnarray}
Both (\ref{equ:initial}) and (\ref{equ:adsbh}) will be used as initial conditions for solving (\ref{equ:eom}) in this paper.
\subsection{Boundary conditions}
The boundary condition for solving (\ref{equ:deltap}) is given by
\begin{equation}
\delta(t,0)=0.\label{equ:deltabc}
\end{equation}
The boundary conditions for solving (\ref{equ:Vdot}) and (\ref{equ:Pdot}) are given by the scalar source in eq. (\ref{equ:phi0t}) in the boundary CFT, which is rewritten in the following form
\begin{equation}
V(t,0)=-2 t \epsilon e^{-a t^2}\text{ and }P(t,\infty) = 0.\label{equ:phi0}
\end{equation}
The above boundary conditions give a unique solution in the bulk. Another solution can be obtained by replacing $(a, \epsilon)$ with $ (\lambda^2 a, \lambda^2 \epsilon)$ in (\ref{equ:phi0}). These two solutions are related to each other by the scaling transformation in eq. (\ref{equ:dilation}). In the following, we denote such an equivalence briefly by
\begin{equation}
(a, \epsilon) \cong (\lambda^2 a, \lambda^2 \epsilon).\label{equ:dilationEqual}
\end{equation}
As a result, we only need to study the dependence of solutions either on $a$ or $\epsilon$(see Fig. \ref{fig:dilationPoincare} for an example).
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{dilationPoincare}
\includegraphics[width=7.8cm]{dilationDPoincare}
\end{center}
\caption{Usage of the scaling transformation in (\ref{equ:dilation}). Here, we show two numerical solutions with boundary conditions respectively given by $(a, \epsilon)=(2.5, 0.125)$ and $(a, \epsilon) = (10, 0.5)$. In these two figures the dashed curves show the metric functions $f$ and $\delta$ of the solution with $(a, \epsilon)=(10, 0.5)$ under the scaling transformation($\lambda = 2$). They are the same as those with $(a, \epsilon)=(2.5, 0.125)$. This example illustrates that for the equivalent solutions in (\ref{equ:dilationEqual}) we need only to calculate one of them and get the rest by the scaling transformation in (\ref{equ:dilation}).
}\label{fig:dilationPoincare}
\end{figure}
\subsection{Numerical scheme}
The equations of motion in eq. (\ref{equ:eom}) can be solved numerically either by the Chebyshev pseudo-spectral method\cite{withPaul} or the finite difference method\cite{MortonMayers}(see Appendix \ref{app:numerics} for a detailed description of our numerical methods). Given $f$, $\delta$, $V$ and $P$ at $t = t_n$, we first calculate $V$, $P$ and $f$ at the next time step $t_{n+1}$ by solving (\ref{equ:Vdot}), (\ref{equ:Pdot}) and (\ref{equ:fdot}). We use the third order Adams-Bashforth method as our time-marching scheme. Then, $\delta$ at $t_{n+1}$ is obtained by solving eq. (\ref{equ:deltap}). Given the initial conditions in (\ref{equ:initial})/(\ref{equ:adsbh}), the bulk metric at late times can be calculated by repeating the above two steps.
In numerical simulations, one needs only to study the evolution of the scalar field in the bulk region $0\leq u\leq u_{max}$ with $u_{max}$ being some bulk cutoff. $V = 0$ and $P = 0$ in the bulk region in which the scalar field has not reached yet. From (\ref{equ:eom}), it is easy to show that $f$ is independent of $t$ and $\delta$ is independent of $u$ in this region. The geometry in this region can not influence the propagation of the scalar in the bulk. Therefore, one can arbitrarily choose $u_{max}$ to be any point in this region. In this case, the boundary condition at $u=\infty$ in (\ref{equ:phi0}) is replaced by $P(t, u_{max}) = 0$. On the other hand, it helps to save computation time by choosing $u_{max}$ close to the deepest region that the scalar field can reach within the thermalization time $t_T$. Using trial and error, we find that it is sufficient to choose $u_{max} \simeq 2 u_0$ or $u_{max}\simeq 3 \Delta t$ respectively for narrow waves or broad waves discussed in the next section.
\subsection{Thermalization time}
In this paper, we define a thermalization time $t_T$ by the first time when\footnote{As discussed below, $t_T$ is coordinate-independent but this simple form in the definition of $t_T$ is coordinate-dependent. See Appendix \ref{app:EddingtonFinkelstein} for the discussion in Eddington-Finkelstein coordinates.}
\begin{equation}
f(t_T, u)\geq f_{min}~~\mbox{and}~~f(t_T,u_{min}) = f_{min},\label{equ:tT}
\end{equation}
where $f_{min}$ is chosen to be 0.01 for the narrow waves. A similar definition of the black hole formation time is also used in the global patch by the authors of Ref.s \cite{Garfinkle01,Garfinkle02}. We also require that $f(t_T, u)$ at $u< u_{min}$ is that of the AdS black hole metric in (\ref{equ:adsbh}). The goodness-of-fit is evaluated by
\begin{equation}
\sigma = \frac{\sqrt{\sum\limits_{i=1}^n\left(f_{bh}(u_i)/f_i - 1\right)^2}}{n},\label{equ:sigma}
\end{equation}
where $f_i$ is our numerical result at $u_i$ and $n$ is the number of the grid points $u_i < u_{min}$. In the following we shall show that the energy density of the scalar field narrowly peaks around $u=u_{min}$ when $u_{min} \gtrsim 0.98 u_0$. Before encountering the coordinate singularity at the apparent horizon(given by $f=0$), we will refer to such a state as a collapse.
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{uvir}
\includegraphics[width=8cm]{wilsonloop}
\end{center}
\caption{Submanifold probed by a spacelike Wilson loop. Fig. (a) shows u(0) as a function of $l$ in the vacuum AdS(Vacuum) and the AdS black hole metric(AdS BH). Because of the scaling invariance in the vacuum AdS, $u(0)$ is a linear function of $l$, i.e., $u(0)\simeq0.835 l$(note that a logarithmic $l$-axis is used in Fig. (a)). Fig. (b) shows the spacelike geodesics $u$ as a function of $x$ with boundary conditions $u(0) = 0.5, 0.9$ and $0.997$. Here, we take $u_0 = \frac{1}{\pi T}=1$.
}\label{fig:wilsonloop}
\end{figure}
The thermalization time $t_T$ has a coordinate-independent interpretation on the CFT side. It is the thermalization time defined by a non-local operator $\left< O(l)\right>$ at scale $l \sim \frac{1}{T}$. And the $f_{min}$ dependence of $t_T$ corresponds to the scale dependence of the thermalization time discussed in Ref. {\cite{holographicThermal}}. Here we only consider the rectangular spacelike Wilson loop $\left<W(l)\right>$\cite{Maldacena:wilsonloop, holographicThermal}(see Ref. \cite{holographicThermal} for the discussion of other non-local operators). $\left<W(l)\right>$ is a functional of spacelike geodesics which satisfy the following equations
\begin{eqnarray}
&&d_x \left( \frac{f e^{-2 \delta} d_x t}{G u^2 } \right)+\frac{1}{2 G u^2 } \left( 2 \dot{\delta} f e^{-2 \delta} d_x t^2 - \dot{f} e^{-2\delta} d_x t^2 - f^{-2} \dot{f} d_x u^2 \right)=0,\\
&&d_x \left( \frac{f^{-1} d_x u}{G u^2} \right)+\frac{2G}{u^3} - \frac{1}{2 G u^2}\left( 2 \delta' f e^{-2\delta} d_x t^2 - f' e^{-2\delta} d_x t^2 - f^{-2} f' d_x u^2 \right) = 0,
\end{eqnarray}
where $G\equiv \sqrt{ 1 - f e^{-2\delta} d_x t^2 + f^{-1} d_x u^2} $, $d_x \equiv d/dx$ and the boundary conditions are given by $u(-\frac{l}{2}) = 0 = u(\frac{l}{2})$ and $t(-\frac{l}{2})=t=t(\frac{l}{2})$. As we shall see in the next section, at $t\sim t_T$ there is a sharp transition between the submanifold with vanishing $V$ and $P$ near the boundary and the rest of the bulk. In this submanifold, $\delta'$, $\dot{\delta}$ and $\dot{f}$ vanish. Therefore, it is a good approximation for us to solve the geodesic equation in the AdS black hole metric instead, which takes the following form
\begin{equation}
\frac{d^2}{dx^2}u + \frac{2 \left(1-2 u^4+u^8+ d_x u^2\right)}{u \left(1-u^4\right)}=0,\label{equ:u}
\end{equation}
where $t(x) = t$ and we have taken $u_0=1$. Eq. (\ref{equ:u}) can be easily solved numerically by the shooting method starting from $u(x=0)=u(0)$ and $u'(0)=0$. As showed in Fig. \ref{fig:wilsonloop}(a), $u(0)$ is a monotonically increasing function of $l$. As a result, to probe high momentum modes($\omega\sim \frac{1}{l} \gg T$) one only needs to know the bulk metric at $u\lesssim l$ while for low momentum modes($\omega\sim \frac{1}{l} \sim T$) all the information of the bulk metric at $u\lesssim u_{min}\sim u_0$ is needed. In our definition of $t_T$, $u_{min} = 0.997 u_0$. Solving eq. (\ref{equ:u}) with $u(0) = u_{min}$ gives $l = 3.16 u_0\simeq\frac{1.0}{T}$. Therefore, at $t\simeq t_T$ the expectation values of $\left<W(l)\right>$ with $l\lesssim \frac{1}{T}$ all reduce to thermal results(see Fig. \ref{fig:wilsonloop}(b)). In this paper, we only talk about thermalization in this sense and focus on the gravity side. A quantitative analysis like that in Ref. \cite{holographicThermal} on the CFT side is also essential for obtaining more detailed information but we leave it to future studies.
\section{Results}\label{sec:results}
In this section, we discuss results with boundary sources of different values of $(a, \epsilon)$ in (\ref{equ:phi0}). According to (\ref{equ:dilationEqual}), we need only to study those parameter sets with different ratios of $\frac{\epsilon}{a}$. In the following, we first fix either $\epsilon$ or $a$ for convenience of our numerical calculation. Then, general results are obtained by the scaling transformation in (\ref{equ:dilation}). We study both narrow ($\frac{\epsilon}{a}\lesssim1$) and broad($\frac{\epsilon}{a}\gtrsim1$) waves.
\subsection{Narrow waves$\left(\Delta t = \frac{1}{\sqrt{a}} \lesssim \frac{1}{T}\right)$: $\frac{\epsilon}{a}\lesssim1$}
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{energyPoincare}
\includegraphics[width=8cm]{lightSpeedPoincare}
\end{center}
\caption{An ingoing narrow wave. This is a detailed example of the falling shell illustrated in Fig. \ref{fig:topdown}. Fig. (a): $f^2(V^2+P^2) $ and the metric function $f$ at different times. Fig. (b): $f^2(V^2+P^2) $ and $\frac{du}{dt} = f e^{-\delta}$, the speed of light-like geodesics, at different times. Here, $(a, \epsilon) = (400, 0.5)$.
}\label{fig:e001a10}
\end{figure}
In the limit $\frac{\epsilon}{a}\ll1$, the source induces a narrowly peaked ingoing wave propagating in the bulk of $AdS_5$, which has been illustrated in Fig. \ref{fig:topdown}. Let us take for example a narrow wave with $\Delta t = 0.05$ and $u_0 = 1.00$. Fig. \ref{fig:e001a10} shows the energy density of the scalar field\footnote{The energy density of the scalar field for an observer with $n^a=\frac{1}{\sqrt{-g_{00}} } \delta^a_0$ is $\rho = n^a n^b T_{ab} = f \left( V^2 + P^2 \right)$. In this paper, we use $f^2 \left( V^2 + P^2 \right)$ instead in order to show the energy density at different times in the same figure.}, $f$ and the speed of light-like geodesics of the solution. At $t \simeq 2 \Delta t$, the source on the boundary CFT is turned off and a narrow ingoing wave is induced near the boundary. The wave starts to propagate with the group velocity $du_s/dt\simeq1$ in the bulk. Its group velocity becomes slower in the deeper interior of $AdS_5$. At $t = 2.29\simeq \frac{0.73}{T}$ the AdS black hole metric is established in the submanifold defined by $u<0.997u_0$ and the speed of lightlike geodesics $\frac{du}{dt} = f e^{-\delta} = 0.0063$ at $u\gtrsim u_0$. This is interpreted as the gravity dual of the thermalized system defined by the spacelike Wilson loop $\left< W(l\simeq \frac{1}{T})\right>$ in CFT. At each time $t\gtrsim 2 \Delta t$, the bulk metric is composed of three parts: (I) the AdS black hole metric behind the wave, (II) the Vacuum AdS metric with a time dilation, i.e., $\delta> 0$, before the wave and (III) the transition over a region $\Delta u \lesssim 4 \Delta t$ between (I) and (II) across the wave. Therefore, in this case the system in the boundary CFT thermalizes in a top-down manner\cite{holographicThermal}. Fig. \ref{fig:e001a10}(b) shows that the speed of light becomes slower at later times in region (II).
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{u0a05}
\includegraphics[width=8cm]{u0Poincare}
\end{center}
\caption{The dependence of $u_0$ on $a$ for $\epsilon = 0.5$(Fig. (a)) and $f$ for $a = 10$ but different $\epsilon$(Fig. (b)). At $a\gtrsim5.0$, $u_0$ is almost independent of $a$. This fact and the scaling transformation in (\ref{equ:dilation}) allow us to conclude the relation in (\ref{equ:TPoincare}), which is verified in Fig. (b).
}\label{fig:aPoincare}
\end{figure}
Let us first fix $\epsilon = 0.5$. As showed in Fig. \ref{fig:aPoincare}(a), $u_0$ only weakly depends on $a$ as $a\gtrsim5$. Using the scaling transformation in eq. (\ref{equ:dilation}) and neglecting such a weak dependence on $a$, one can conclude that as long as the condition \begin{equation}\frac{\epsilon}{a}\lesssim 0.1\label{equ:condPoincare}\end{equation} holds, $u_0$ is (almost) independent of $a$ and proportional to $\epsilon^{-\frac{1}{2}}$, that is,
\begin{equation}
u_0 = \frac{u_0^{(0.5)}}{\sqrt{2\epsilon}} \simeq 0.699 \epsilon^{-\frac{1}{2}},~~\mbox{or}, ~~T = \frac{\sqrt{2\epsilon}}{\pi u_0^{(0.5)}}\simeq 0.455 \epsilon^{\frac{1}{2}},\label{equ:TPoincare}
\end{equation}
where $u_0^{(0.5)}$ is our numerical result with $(a,\epsilon)=(a^{(0.5)}, 0.5)$ and $a^{(0.5)}$ denotes the value of $a$ with fixe $\epsilon = 0.5$. The above relation $T\propto \epsilon^{\frac{1}{2}}$ is also obtained by perturbative techniques in the limit $\frac{\epsilon}{a}\ll1$ in Ref. \cite{Minwalla}. Therefore, the perturbative techniques should be applicable when eq. (\ref{equ:condPoincare}) is satisfied. The condition (\ref{equ:condPoincare}) can be further rewritten in the following form
\begin{equation}
\Delta t\equiv \frac{1}{\sqrt{a}} \lesssim \frac{0.14}{T}.\label{equ:narrowWave}
\end{equation}
As a verification of (\ref{equ:TPoincare}), Fig. \ref{fig:aPoincare}(b) shows the metric function $f $ with $\epsilon = 0.1, 0.05$ and $0.01$ and fixed $a = 10$ at two different times. In region (I) they all agree with the AdS black hole metric with $u_0$ given by (\ref{equ:TPoincare}).
\begin{figure}
\begin{center}
\includegraphics[height=6.5cm]{tTa05}
\includegraphics[height=6.5cm]{g}
\end{center}
\caption{$t_T$ and $u_0$ for $\epsilon = 0.5$(Fig. (a)) and $g_t$ as a function of $\frac{a}{2\epsilon}$(Fig. (b)).
}\label{fig:g}
\end{figure}
Fig. \ref{fig:g}(a) shows the thermalization time $t_{T}$ for $\epsilon = 0.5$ and different values of $a$. Again, the scaling transformation in (\ref{equ:dilation}) allows us to make conclusions for the cases with $(a, \epsilon)=(\lambda^2 a^{(0.5)}, 0.5 \lambda^2)\cong (a^{(0.5)}, 0.5)$, that is,
\begin{equation}
t_T = \frac{t_T^{(0.5)}}{\pi u_0^{(0.5)}}\frac{1}{T}\equiv \frac{g_t\left( \frac{a}{2\epsilon} \right)}{T},\label{equ:narrowWavetT}
\end{equation}
where $g_t\simeq 0.73$ for $\frac{a}{2\epsilon}\gtrsim 200$ or, equivalently, $\Delta t \lesssim \frac{0.02}{T}$ and $g_t\lesssim 1.0$ for $\frac{\epsilon}{a}\lesssim 0.5$(see Fig. \ref{fig:g}(b)). Note that the thermalization time $t_T$ is physically different from the coordinate-dependent formation time of the black brane $t_{BH}\sim \Delta t$ in ingoing Eddington-Finkelstein coordinates reported in Ref. \cite{Minwalla}. We discuss this in detail in Appendix \ref{app:EddingtonFinkelstein}.
In the limit $a\to \infty$, one may make a simple estimate as follows: the source induces a three-peak wave in the bulk as showed in Fig. \ref{fig:e001a10}. The last(leftmost) peak does not contribute so much to the bending of the spacetime but does determine the location of the turning point between (I) and (II). Since the leftmost piece of this peak propagates in an approximate AdS black hole background, the thermalization time $t_T$ may be estimated by
\begin{equation}
\Delta t_T \simeq \int_{0}^{u_{min}}\frac{du}{f_{bh}} = \frac{1}{4} u_{0} \left( 2 \arctan\frac{u_{min}}{u_{0}} + \log \frac{u_0+u_{min}}{u_0-u_{min}} \right)\simeq 2.02 u_0\simeq\frac{0.64}{T}.\label{equ:narrowWavetTest}
\end{equation}
At worst, this is the lower bound for $t_T$, which guarantees that the system thermalizes without violating causality. We take this as another justification for defining $t_T$ by eq. (\ref{equ:tT}). Taking $a = 400$, our numerical calculation gives $t_T\simeq 2.29$ and the estimated thermalization time is given by $(\Delta t_T + 2 \Delta t )\simeq$ 2.12.
\subsection{Broad waves($\Delta t \gtrsim \frac{1}{T}$): $\frac{\epsilon}{a}\gtrsim1$}
In this subsection we discuss solutions with $\frac{\epsilon}{a}\gtrsim 1$, or, $\Delta t\gtrsim \frac{1}{T}$ by eq. (\ref{equ:narrowWave}). As illustrated in Fig. \ref{fig:2stagesThermalization}(a), the energy is injected into the CFT vacuum by the two pulses of $\dot{\phi}$. Based on the discussion in the previous subsection, the system is expected to achieve thermalization in a time scale $t_T \sim \frac{1}{T}$. Therefore, if $\Delta t \gtrsim \frac{1}{T}$, one can expect a two-stage thermalization. Similarly, if $\dot{\phi}_0$ is periodic with period $\Delta t \gtrsim \frac{1}{T}$, a multi-stage thermalization, as illustrated in Fig. \ref{fig:2stagesThermalization}(b), may also be expected. We shall show how the results on the gravity side live up to such an expectation.
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{e01a01}
\includegraphics[width=8cm]{e03a01}
\end{center}
\caption{Single-collapse(Fig. (a)) and double-collapse(Fig. (b)) solutions. In both figures we show $f$ as a function of $u$ from $t = -3.0$(top right) to 7.0(bottom left) with time steps equal to 1. In Fig. (a), $f$ at $t=0$ is far from that of the AdS black hole metric. In Fig. (b), $f$ at $t = 0$ is that of the AdS black hole metric at $u\leq u_{min}= 0.98 u_0$.
}\label{fig:twostageCollapse}
\end{figure}
In the cases with $\frac{\epsilon}{a}\gtrsim 1$, it is more convenient to study solutions with $a$ fixed at a smaller value. We choose $a = 0.1$($\Delta t = 3.16$). Solutions with $1.0\geq \epsilon\geq 0.05$ are obtained by numerically solving (\ref{equ:eom}). We observe a transition from a single collapse to a double collapse in those solutions at $\epsilon \simeq (0.2-0.3)
$\footnote{Here, the first collapse at $t=0$ is determined by the following requirements: (a) $f(0,u)\geq f_{min}$ for all $u$, (b) $f(0,u_{min}) = f_{min}$ and (c) $f$ is that of the AdS black hole metric at $u\leq u_{min}$. The exact transition value of $\epsilon$ depends on the choice of $f_{min}$ or, equivalently, the scale $l$ for $\left< W(l) \right>$. For the solution with $\frac{\epsilon}{a}=3.0$ we have $l\simeq\frac{0.7}{T}$ or, equivalently, $u_{min} \simeq 0.98 u_0$ for the first collapse. At $t = 0.61$, min$\{f\}\simeq 0.01$. However, $f$ is not that of the AdS black hole metric near the boundary. Therefore, we do not interpret it as being in thermal equilibrium.}.
Fig. \ref{fig:twostageCollapse} shows the metric function $f$ of a single-collapse solution($\epsilon = 0.1$) and a double-collapse solution($\epsilon=0.3$). In the single-collapse solution the scalar field collapses once at $u_0 = 1.76$ and $t = 6.61\sim 2\Delta t$. In contrast, in the double-collapse solution the scalar field collapses twice respectively at $u\simeq u_L = 0.73$ and $t\simeq 0$
\footnote{Here, $u_L$ is obtained by the least-squared fit of $f_{bh}$ with $u_0 = u_L$ to our numerical results for $u<u_{min}$.}
and at $u \simeq u_H=0.48$ and $t=5.47\sim 2\Delta t$. For the first collapse, $\frac{1}{T} = \pi u_L = 2.29 < \Delta t$ while for the single-collapse solution $2 \Delta t>\frac{1}{T} = 5.53 > \Delta t$. The scalar fields induced by the source with $\epsilon \gtrsim 0.3$ all undergo such a double collapse.
\begin{figure}
\begin{center}
\includegraphics[width=5cm]{e03a01E0}
\includegraphics[width=5cm]{e03a01E3p5}
\includegraphics[width=5cm]{e03a01E6}
\includegraphics[width=5cm]{e03a01c0}
\includegraphics[width=5cm]{e03a01c3p5}
\includegraphics[width=5cm]{e03a01c6}
\end{center}
\caption{The energy density of the scalar field($f^2(V^2+P^2)$), $f$ and the speed of light($du/dt=f e^{-\delta}$) for the double-collapse solution with $(a, \epsilon) = (0.1, 0.3)$.
}\label{fig:Ee03a01}
\end{figure}
Let us understand better the double-collapse solution. In Fig. \ref{fig:Ee03a01} we plot the energy density of the scalar field, $f$ and the speed of lightlike geodesics($du/dt = f e^{-\delta}$) of the solution with $\epsilon = 0.3$. At $t\geq 5.47$, two peaks are observed in the energy density. They correspond to the two collapses at $u\sim u_L$ and $u\sim u_H$. Also the speed of light drops dramatically in the bulk region $u\gtrsim u_H$. The peak at $u\sim u_L$ is resulted from the first collapse at $t\simeq0$ when the speed of light starts to drop below $10^{-6}$ at $u\gtrsim u_L$. From $t = 0$ to $5.47$, the scalar field mainly accretes at $u\sim u_H$, which only significantly modifies $f$ at $u\lesssim u_L$ and results in the second peak of the energy density.
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{e03a01bh}
\includegraphics[width=8cm]{e03a01bhE}
\end{center}
\caption{Collapse of a massless scalar field on the AdS black hole background. In both figures, the dotted curves(Black Hole) show the results of turning on the scalar source at $t\geq0$ on the AdS black hole background. The solid curves(Double Collapse) show the results of the double-collapse solution with $(a, \epsilon) = (0.1, 0.3)$. In the left figure, $f$ is showed as a function of $u$ from $t = 0$(top right) to 6.0(bottom left) with time steps equal to 1.
}\label{fig:responsebh}
\end{figure}
As another support for regarding the state at $t=0$ as being in (approximate) thermal equilibrium, we study the response of static plasma with $T = T_L$ to the scalar source which is turned on only after $t=0$. On the AdS side, we solve eq. (\ref{equ:eom}) using the AdS black hole metric in (\ref{equ:adsbh}) as the initial conditions. Here, we only discuss the solution with $T = T_L = 0.73$ and $(a,\epsilon)=(0.1,0.3)$. Fig. \ref{fig:responsebh} shows the metric function $f$ at different times and the energy density at $t = 6.0$ of this solution. Their small difference from those of the double-collapse solution with $(a,\epsilon)=(0.1,0.3)$ justifies that an intermediate thermal equilibration is established at $t\simeq0$.
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{uL}
\end{center}
\caption{$u^{(0.1)}_L$ as a function of $\epsilon/10a$.
}\label{fig:uL}
\end{figure}
Boundary conditions with $\frac{\epsilon}{a}\gtrsim 3$ all give such double-collapse solutions\footnote{We have not calculated solutions with even larger $\frac{\epsilon}{a}>10$ because it is numerically too time-consuming. However, based on the physical argument given in the paper, it is reasonable to expect that the conclusion should also apply to the cases with $\frac{\epsilon}{a} > 10$.} by the scaling transformation in (\ref{equ:dilation}). In general we conclude that the criteria for the system to achieve thermalization induced by the first pulse of $\dot{\phi}$ at $t\simeq0$ is $\Delta t \gtrsim \frac{1}{T_L}$. This is exactly what we expected on the CFT side. The intermediate thermalization temperature $T_L$ for arbitrary $(a, \epsilon)$ is given by
\begin{equation}
T_L =\frac{1}{\pi u_L}= \frac{\sqrt{10 a}}{\pi u_L^{(0.1)}\left( \frac{\epsilon}{10a} \right)},
\end{equation}
where $u_L^{(0.1)}\left( \frac{\epsilon}{10a} \right)$, showed in Fig. \ref{fig:uL}, is our numerical results with $a=0.1$. At larger $\epsilon$, $u_L^{(0.1)}$ scales according to the following power law
\begin{equation}
u_L^{(0.1)}(\epsilon) \simeq 0.109 \epsilon^{-\beta}\label{equ:uL}
\end{equation}
with $\beta \simeq 1.74$. The second collapse(the heating-up process) can also be studied by turning on the source at $t\geq0$ and starting from the initial conditions in (\ref{equ:adsbh}) with $u_0 =u_L = \frac{1}{\pi T_L}$, as we did for the case with $(a, \epsilon) = (0.1, 0.3)$.
\begin{figure}
\begin{center}
\includegraphics[height=6cm]{sin}
\includegraphics[height=6cm]{sinE}
\end{center}
\caption{A multiple-collapse solution. Fig. (a) shows $f$ and $f_{bh}$ at different times from $t=\pi$(right) to $t=10\pi$(left) with time intervals equal to $\pi$. Here $u_0$ in $f_{bh}$ is obtained by performing least-square fit of the AdS black hole metric to our results at $u\leq 0.505$ with $\sigma \sim 10^{-4}$. Fig. (b) shows the energy density of the scalar field ($f \left( V^2 + P^2 \right)$) at $t = 10 \pi$. Each spike here is the result of the collapse around $t = \pi, 2 \pi, \cdot\cdot\cdot, 10\pi$.
}\label{fig:sin}
\end{figure}
The qualitative conclusion above is independent of the source's shape. As a justification, Fig. \ref{fig:sin} shows a multiple-collapse solution. The periodic source is given by
\begin{equation}
\dot{\phi}_0 = \epsilon \sin(t) \theta(t)\label{equ:sin}
\end{equation}
with $\epsilon=0.3$. Here, one naturally takes $\Delta t = \pi$. Fig. \ref{fig:sin}(a) shows $f$ at $t = \pi, 2\pi,\cdot\cdot\cdot, 10\pi$. At $t=\pi$, $f$ is different\footnote{Here, we define $u_{min}$ by $\left|\frac{f_{min}}{f_{bh}(u_{min})}-1\right|=0.1$ with $f_{min}$ being our numerical result at $u=u_{min}$.} from that of the AdS black hole metric at $u\gtrsim u_{min}= 0.93 u_0$ and $\frac{1}{T} = \pi u_0 = 1.52 \pi > \Delta t$. In contrast, at $t = 10\pi$, $f$ is that of the AdS black hole metric at $u\lesssim 0.99 u_0$ and $\frac{1}{T} = 0.74 \pi$. The energy density of the scalar field at $t=10 \pi$ is showed in Fig. \ref{fig:sin}(b). Each spike in the figure is the result of the collapse around $t = \pi, 2 \pi, \cdot\cdot\cdot, 10\pi$. Therefore, the parametric criteria for a system driven by a periodic source to achieve thermal equilibrium within each time interval $\Delta t$ is also $\Delta t \gtrsim \frac{1}{T}$ with $T$ being the intermediate thermal equilibrium temperature.
\section{Discussions}\label{sec:discussion}
In this paper, we focus on the bulk metric resulted from gravitational collapse of a massless scalar field in the Poincare patch of $AdS_5$. Mostly, we aim to understand the typical thermalization time scale $t_T$ of its CFT dual. We find that thermalization in such a strongly coupled system is rapid in the sense that \begin{equation} t_T \simeq \frac{O(1)}{T},\end{equation}
where the coefficient of $O(1)$ depends on the scale $l\sim \frac{1}{T}$ of nonlocal operators and on the boundary source's shape. Such a rapid thermalization time seems to be typical of the strongly coupled CFT\cite{CY01,Heller:2012km}.
We leave many unanswered but intriguing questions for future studies. There are still more details about such a thermalization process that can be understood only by evaluating non-local operators\cite{holographicThermal} or other relevant probes\cite{energylossandpT,Baier:2012a,Baier:2012b}. Moreover, in this paper we only study gravitational collapse of massless scalars. It is interesting to know what is the typical thermalization time $t_T$ in the cases of massive and tachyonic scalars, which are respectively dual to irrelevant and relevant operators in the boundary CFT\cite{Witten}.
\section*{Acknowledgements}
The author would like to thank R. Baier, Y. Kovchegov, P. Romatschke, S. Stricker and A. Vuorinen for reading this manuscript and providing illuminating comments. This work is supported by the Humboldt foundation through its Sofja Kovalevskaja program.
|
3,212,635,537,453 | arxiv |
\section{Introduction}\label{sec:intro}
The solution of large sparse linear systems of equations
\begin{equation}\label{eq:linsys}
Ax = b,\ b\in \mathbb{R}^{n},\ A \in \mathbb{R}^{n \times n} \text{\ symmetric positive definite,}
\end{equation} that arise in the discretization of partial differential equations, typically makes up the bulk of computations in modern scientific computing. It is thus of utmost importance to come up with efficient algorithms to solve these systems of equations. By exploiting a separation of scales multigrid methods can achieve optimal linear complexity for this task, but heavily rely of the availability of expert knowledge about the particular partial differential equation that the linear systems originates from as well as the employed discretization scheme; cf.~\cite{BrenScot2008}.
Due to the fact that this information might not readily be available or that there is no known geometric multigrid construction, the concept of algebraic multigrid methods has been introduced in~\cite{Bran1986,BranMcCoRuge1985,RugeStue1987,Stue1983,XuZika2017}. Efficiency in algebraic multigrid methods is achieved by pairing a simple iterative scheme, the \textit{smoother}, with a variational coarse grid correction. Assuming a smoother is defined by $M \approx A^{-1}$, the error propagator of a two-grid algebraic multigrid method with Galerkin coarse grid construction is given by
\begin{equation} \label{eq:eprop}
E_{2g} = (I-MA) (I-P (P^T A P)^{-1} P^T A) (I-MA).
\end{equation} Due to the variational construction of coarse grid correction the whole setup of an algebraic multigrid method can be reduced to the definition of the interpolation operator $P$. In particular, the dimension of the coarse space, $n_{c}$, the interpolation relations, i.e., the sparsity pattern of $P\in \mathbb{R}^{n\times n_{c}}$ and its entries need to be defined. Typically these tasks are split into two parts. Finding $n_{c}$ and the sparsity pattern of $P$ is often referred to as the \textit{coarsening} problem, while determining the entries of $P$ is known as the \textit{interpolation} problem.
In the first algebraic multigrid methods~\cite{BranMcCoRuge1985,RugeStue1987} operator based approaches have been suggested to solve both problems. In case $A$ has $M$-matrix structure, e.g., as a particular discretization of an elliptic partial differential equation, it can be shown that these approaches lead to methods with fast convergence. However, these early approaches rely heavily on assumptions about the underlying problem and therefore cannot be extended significantly beyond the $M$-matrix case. A huge step in overcoming this limitation has been the introduction of adaptivity in algebraic multigrid methods~\cite{BranBranKahlLivs2011,BrezFalgMacLMantMcCoRuge2006,BrezFalgMacLMantMcCoRuge2004}.
Common in all adaptive approaches in algebraic multigrid methods is the idea to guide the definition of interpolation, posed in terms of the coarsening and interpolation problem either by using spectral information about $A$ and/or the smoothing iteration. Due to the fact that explicit calculation of (partial) spectra is prohibitively expensive these methods rely on an iterative approximation process that makes use of the emerging multigrid hierarchy. In contrast to the interpolation problem, where several approaches showed promosing performance~\cite{BranBranKahlLivs2015b,BranBranKahlLivs2011,MandBrezVane1999,OlsoSchrTumi2011,WanChanSmit1999,XuZika2004}, the coarsening problem turned out to be harder to tackle.
Many of the approaches that have been tried to solve the coarsening problem in adaptive algebraic multigrid methods revolve around the idea of strength of connection, a concept introduced in classical algebraic multigrid. This includes approaches based on binary variable relations such as~\cite{BranBranKahlLivs2015b,BranChenKrauZika2013,BranChenZika2012,OlsoSchrTumi2011} but approaches that take relations of more than two variables into account such as~\cite{KahlRott2018}.
The detection of strongly connected pairs of variables is also found in aggregation-based approaches such as~\cite{BranChenKrauZika2013,LivnBran2012,NapoNota2016,Nota2010}.
In some sense the idea of compatible relaxation, introduced in~\cite{BranFalg2010}, comes closest to general applicability, but is hard to integrate and mesh with typical solutions to the interpolation problems, i.e., the definition of the entries of $P$, in adaptive algebraic multigrid approaches.
In this paper we propose a new way of solving the coarsening problem which resembles the least squares interpolation approach of the bootstrap algebraic multigrid framework~\cite{BranBranKahlLivs2011,KahlRott2018}. By considering the test vectors of the bootstrap framework as instances of a Gaussian process~\cite{adler2010geometry,bogachev1998gaussian,vanmarcke2010random} we are able to apply techniques from machine learning,
especially the concept of conditional (a posteriori) distributions. To calibrate this statistical model to the data, we use parametric
semivariogram models to fit the covariance structure to the data provided by algebraically smooth test vectors.
This enables us to efficiently solve the coarsening and interpolation problem at the same time. The idea of Kriging interpolation is to view the values of test vectors at coarse grid variables as partial observations of a Gaussian process. Based on these observed values at the coarse grid variables, expected values and errors at the fine grid variables can be derived by computing conditional expectations and variances. This method originally stems from spatial interpolation in geostatistics, see e.g. \cite{christakos2012random}
but has been widely used in various machine learning and engineering tasks in the past \cite{forrester2008engineering,kleijnen2009kriging}. Its main advantage lies in its statistical properties as best linear unbiased predictor, given the data of the field on the 'observed locations' -- the coarse grid -- and the spatial correlation structure of the data. Coincidentially, the Kriging iterpolator bears some resemblence to the least-squares formulation of interpolation introduced in~\cite{BranBranKahlLivs2011} and more specifically the operator-based modification of it found in~\cite{MantMcCoParkRuge2010}.
Inheriting from the spatial correlation structure of the Gaussian process, the conditional variance given the observations on certain points, is low at points with sufficient observations in the neighborhood and high elsewhere. This additional piece of information can now be used if one has to decide on where to make the next observation. In this paper we apply a greedy optimization procedure, picking the point of highest conditional variance given coarse grid results, to adaptively refine the coarse grid by subsequently adding fine grid points to the coarse grid until the conditional variance on all remaining fine grid points is small.
A statistical view on the adaptive setup in algebraic multigrid methods is not completely new. It has been used in~\cite{LivnBran2012} to motivate a definition of strength of connection based on a measure of correlation present in test vectors, but this construction lacks the framing of Gaussian processes.
Other related work in the context of the integration of ideas from stochastics into multigrid method has been presented in~\cite{owhadi2017multigrid}. While this work also contains a deep mathematical analysis of convergence, there are several points where our work takes a different route. First of all, our work is purely algebraic and only uses structures that can be derived from the matrix $A$. Hence we do not use any structures that stem from the spatial structures of the underlying PDE and thus we do not have techniques at our disposal that use spectral equivalence and other techniques based on harmonic analysis. Instead, we only use distance notions between 'nodes' that can intrinsically be derived from $A$. Also, we resort to data driven estimation of correlation structures instead of an analytic derivation of these structures from the matrix $A$. This empirical approach largely reduces the computational cost in the choice of priors.
Furthermore, while the prior chosen in \cite{owhadi2017multigrid} is supported by error analysis in the $A$-norm, it has the disadvantage of producing a generalized random field with distributional paths \cite{gelfand1964generalized}, which is a disputable choice for the prior belief on the solution to a PDE. The price we pay is to use a more experimental and less deeply founded approach.
In~\cref{sec:multigrid} we give a short introduction into the construction of algebraic multigrid methods and highlight how adaptivity can be used in order to capture the nature and underlying structure of the problem at hand, especially in terms finding suitable coarsenings. Then we give an overview on Gaussian processes in~\cref{sec:gaussian}, where we motivate the connection to the adaptive setup process and discuss interpolation in the context of these processes. This leads us directly to the formulation of Kriging interpolation in~\cref{sec:linear}, which we discuss and compare to the least squares interpolation approach in~\cref{app:kriging-ls}. Finally, using additional heuristics we present our new adaptive coarsening approach in~\cref{sec:coarsening} before closing with numerical tests in~\cref{sec:numerics} demonstrating the potential of our approach and some final remarks in~\cref{sec:conclusion}.
\section{Adaptive algebraic multigrid methods}\label{sec:multigrid}
The efficiency of multigrid methods lies in the complementarity of the smoothing iteration and the coarse grid correction. Algebraic multigrid methods construct complementarity without relying on knowledge of the underlying problem or the employed discretization strategy. Assuming that $A$ is symmetric positive definite, it is common to consider a Galerkin construction for the coarse grid correction error propagator
\[
E_{\rm cgc} = I - P \left(P^{T}AP\right)^{-1}P^{T}A.
\] Thus it is completely determined by the definition of the interpolation operator $P: \mathbb{R}^{n_c} \rightarrow \mathbb{R}^{n}$. Due to the fact that in this case $E_{\rm cgc}$ corresponds to the $A$-orthogonal projection onto the space $A$-orthogonal to $\operatorname{range}(P)$, we can assume for simplicity sake and a better intuition of the construction of $P$, that it can be represented in the following form
\begin{equation}\label{eq:canonical_interpolation}
P = \begin{bmatrix} P_{\mathcal{F}\!,\mathcal{C}}\\ I \end{bmatrix}.
\end{equation} This particular form of $P$ can be assumed due to the fact that $E_{\rm cgc}$ remains unchanged when transforming $P \rightarrow PX$ for any non-singular $X$, i.e., it depends solely on $\operatorname{range}(P)$ and not on the basis representation of this subspace. Using this form of $P$, we observe, that the index set of all variables $\mathcal{V} = \{1,2,\ldots,n\}$ can be split into two disjoint sets. The set $\mathcal{C}$ of variables that define the coarse grid and the remaining set $\mathcal{F} = \mathcal{V}\setminus\mathcal{C}$ of variables solely being present in the fine grid system as depicted in~\cref{fig:variable_splitting}.
\begin{figure}
\begin{center}
\begin{tikzpicture}
\node at (0,0) {\includegraphics[width=.9\textwidth]{./figs/coarseningExample.pdf}};
\end{tikzpicture}
\end{center}
\caption[Splitting of the set of variables $\mathcal{V}$ into a set of coarse variables $\mathcal{C}$ and a set of fine variables $\mathcal{F}$.]{Splitting of the set of all variables $\mathcal{V}$ into a set of coarse variables $\mathcal{C}$ depicted by \raisebox{-.1em}{\resizebox{.75em}{!}{\tikz{\node[circle,fill=black,minimum width=1em] at (0,0) { };}}}
and a set of fine variables $\mathcal{F}$ depicted by \raisebox{-.1em}{\resizebox{.75em}{!}{\tikz{\node[circle,fill=white,draw=black,minimum width=1em] at (0,0) { };}}}
.}\label{fig:variable_splitting}
\end{figure}
In that sense~\eqref{eq:canonical_interpolation} defines interpolation from the variables with $\mathcal{C}$ indices, which are kept identical when moving from coarse to fine variables, to $\mathcal{F}$ variables with interpolation weights found in $P_{\rm \mathcal{F}\!,\mathcal{C}}$. As already mentioned in the introduction the definition of interpolation now reduces to three questions. First, which coarse variable set $\mathcal{C} \subset \mathcal{V}$ to choose, in particular this also amounts to determining the coarsening ratio $\tfrac{|\mathcal{C}|}{n}$. Second, for each variable $i$ in $\mathcal{F}$, a set of interpolatory variables $\mathcal{C}_i \subset \mathcal{C}$ has to be defined, which corresponds to the sparsity pattern of $P_{\rm \mathcal{F}\!,\mathcal{C}}$. Last, the entries of $P_{\rm \mathcal{F}\!,\mathcal{C}}$ have to be defined such that complementarity of the smoothing iteration and the coarse grid correction is achieved. In addition, as an implicit requirement, the sparsity of the coarse system of equations, given by $P^{T}AP$ has to be guaranteed in order to be able to apply the construction recursively and thus achieve optimal linear complexity.
Based on the findings in~\cite{BranCaoKahlFalgHu2018} the complementarity of the smoothing iteration and the coarse-grid correction is equivalent to the requirement that $\operatorname{range}(P)$ approximates the space spanned by eigenvectors of the error propagator of the smoother corresponding to eigenvalues close to $1$, i.e., components that are slow to converge--also known as algebraically smooth error components~\cite{RugeStue1987}.
The adaptive construction of algebraic multigrid methods can thus be interpreted as the generation of a low dimensional, sparse representation of the space of algebraically smooth error. In the context of the bootstrap algebraic multigrid framework and its extensions~\cite{BranBranKahlLivs2011,KahlRott2018,MantMcCoParkRuge2010} this is facilitated by the use of a set of algebraically smooth test vectors $\{v^{(1)},\ldots,v^{(K)}\}$ which are obtained by smoothing initially random test vectors with entries that stem from a normal distribution. By drawing on a connection to Gaussian processes we are now going to modify the construction of interpolation in this setting using ideas that originate in statistical geophysics.
\section{Gaussian Processes}\label{sec:gaussian}
In this section, we introduce Gaussian processes \cite{adler2010geometry} and the Kriging predictor \cite{bivand2008applied,sherman2011spatial} to interpolate from coarse to fine grid points. We discuss various approaches for modeling the covariance of the underlying Gaussian process, including semivariogram estimation based on test vectors.
\subsection{\label{sec:stochProc} Gaussian Stochastic Processes and Kriging}
A stochastic process with the index set $\mathcal{I}$ is a collection of real valued random variables $X=\{X_i\mid i\in \mathcal{I}\}$ on a common probability space $(\Omega,\Sigma,\mathbb{P})$, where $\mathbb{E}[\cdot]$ denotes the expected value with respect to $\mathbb{P}$. Here $\Omega$ is the event set, $\Sigma$ the sigma field and $\mathbb{P}$ the probability measure. $X$ is a Gaussian process, if the distribution of any finite collection $X_\mathcal{H}=(X_{i_1},\ldots,X_{i_q})^T$ for finite subsets $\mathcal{H}=\{i_1,\ldots,i_q\}\subseteq \mathcal{I}$ is a multivariate Gaussian distribution $N(\mu_\mathcal{H},{C}_\mathcal{H})$ with probability density
\begin{equation}
\label{eq:densGauss}
f_\mathcal{H}(x_\mathcal{H})=\frac{1}{\sqrt{2\pi}^q|{C}_\mathcal{H}|^{\frac{1}{2}}}e^{-\frac{1}{2}(x_\mathcal{H}-\mu_\mathcal{H})^T{C}_\mathcal{H}^{-1}(x_\mathcal{H}-\mu_\mathcal{H})},~~x_\mathcal{H}\in\mathbb{R}^m,
\end{equation}
where $|\cdot|$ denotes the determinant, $\mu_\mathcal{H}\in\mathbb{R}^q$ is the expected value and ${C}_\mathcal{H}\in \mathbb{R}^{q\times q}$ is a positive definite covariance matrix. Consistency of the finite dimensional distributions in the sense of Kolmogrov \cite{tao2011introduction} implies that there exist functions
\[
\mu:\mathcal{I}\to\mathbb{R} \text{\ and\ } {C}:\mathcal{I}\times\mathcal{I}\to \mathbb{R}
\] such that $\mu_\mathcal{H}=(\mu_{i_1},\ldots,\mu_{i_q})^T$ and ${C}_\mathcal{H}=({C}_{i,j})_{i,j\in \mathcal{H}}$.
In the given context, we interpret $\mathcal{I}$ as the computational domain, and consider $\mathcal{V} \subseteq \mathcal{I}$ as a finite discretization of $\mathcal{I}$. $X_i$ represents the epistemic uncertainty about the solution of $\eqref{eq:linsys}$ at the grid point $i\in\mathcal{V}$.
Let us suppose we gathered partial information on $X_{\mathcal{V}}$, e.g., by solving $\eqref{eq:linsys}$ on a coarse subset of variables $\mathcal{C} \subseteq \mathcal{V}$ and we would like to infer about the solution on the whole grid $\mathcal{V}$. By construction, any prediction $\widehat X_i, i\in \mathcal{V}$ can only depend on information from $X_\mathcal{C}$ and therefore has to be measurable with respect to the sigma field $\sigma_\mathcal{C}\subseteq\Sigma$
associated with $X_{\mathcal{C}}$. We thus wish to find an optimal solution to the problem of making a prediction based on the values of $X_{\mathcal{C}}$ that minimizes the expected squared error, also called mean square error (\emph{MSE}), for $X_i, i \in \mathcal{V}$
\begin{equation}
\label{eq:minError}
\mathbb{E}\left[\left(X_i-\widehat X_i\right)^2\right]\to \min \text{\quad s.t.\quad } \widehat X_i \text{\ is\ } \sigma_\mathcal{C}\text{-measurable}.
\end{equation}
Let $L^2(\Omega,\Sigma,\mathbb{P})$ be the space of square integrable random variables, then $L^2(\Omega,\sigma_{\mathcal{C}},\mathbb{P})\linebreak\subseteq L^2(\Omega,\sigma,\mathbb{P})$ of $\sigma_\mathcal{C}$-measurable $L^2$ functions is a closed subspace. Thus, the problem \eqref{eq:minError} is uniquely solved by the conditional expected value $\widehat X_i \in \mathbb{E}[X_i|X_\mathcal{C}]$ defined as the $L^2$-projection of $X_i\in L^2(\Omega,\Sigma,\mathbb{P})$ to $L^2(\Omega,\sigma_{\mathcal{C}},\mathbb{P})$. This definition immediately implies the interpolation property $\widehat X_i=X_i$ for all $i\in \mathcal{C}$ as $X_i$ in this case is $\sigma_{\mathcal C}$ measurable itself.
To calculate the conditional expected value $\mathbb{E}[X_i|X_\mathcal{C}]$ along with the minimum expected squared error \eqref{eq:minError} on the remaining set of variables $\mathcal{F} = \mathcal{V}\setminus\mathcal{C}$, we first compute the conditional distribution of $X_{\mathcal{F}}$ given $X_\mathcal{C} = x_\mathcal{C}$ via its density as
\begin{align*}
\begin{split}
f_{\mathcal{F}|\mathcal{C}}(x_{\mathcal{F}}|x_\mathcal{C})
& = \frac{f_{\mathcal{V}}\left(x_{\mathcal{F}},x_\mathcal{C}\right)}{f_\mathcal{C}(x_\mathcal{C})}\\
& = \frac{1}{\sqrt{2\pi}^q|{C}_{\mathcal{F}|\mathcal{C}}|^{\frac{1}{2}}}e^{-\frac{1}{2}(x_{\mathcal{F}}-\mu_{\mathcal{F}}(x_\mathcal{C}))^T{C}_{\mathcal{F}|\mathcal{C}}^{-1}(x_{\mathcal{F}}-\mu_{\mathcal{F}}(x_{\mathcal{C}}))},
\end{split}
\end{align*}
where from \eqref{eq:densGauss} we get from straight forward calculation
\begin{subequations}
\begin{align}\label{eq:kriging}
\mu_{\mathcal{F}}(x_{\mathcal{C}})
&= \mu_{\mathcal{F}}+{C}_{\mathcal{F}\!,\mathcal{C}}{C}_\mathcal{C}^{-1}\!\left(x_\mathcal{C}-\mu_{\mathcal{C}}\right),\\
\label{eq:krigVar}
{C}_{\mathcal{F}|\mathcal{C}}
&= {C}_{\mathcal{C}} - {C}_{\mathcal{F}\!,\mathcal{C}}{C}_{\mathcal{C}}^{-1}{C}_{\mathcal{F}\!,\mathcal{C}}^{T},
\end{align}
\end{subequations}
where we introduced the notation ${C}_{\mathcal{H}\!,\mathcal{J}}=(C_{i,j})_{i\in \mathcal{H}, j \in \mathcal{J}}$ for any finite $\mathcal{H}\!,\mathcal{J}\subseteq \mathcal{I}$.
Using \eqref{eq:kriging} and \eqref{eq:krigVar} we see that for any $i \in \mathcal{F}$ the prediction $\widehat X_i = \mu_{i}(X_{\mathcal{C}})$, i.e., the conditional expected value, minimizes the MSE and its conditional variance is given by
\begin{align*}
\begin{split}
\sigma^2_{i|\mathcal{C}} &= {C}_{i}-{C}_{i,\mathcal{C}}{C}_{\mathcal{C}}^{-1}{C}_{i,\mathcal{C}}^{T}\\
&= \min_{\widehat X_
}\mathbb{E}\left[\left(X_i-\widehat{X}_i\right)^2\right].
\end{split}
\end{align*}
\subsection{\label{sec:linear}Linear Interpolation from Kriging}
Note that \eqref{eq:kriging} provides an affine-linear interpolation rule and not a linear one as required for the construction of the matrix $P$ in \eqref{eq:canonical_interpolation}. This problem can be dealt with in two ways: First, we can set the expected value $\mu_\mathcal{\mathcal{V}}=0$. This is consistent with the estimation of the data $\mu_\mathcal{V}$ and $C_\mathcal{V}$ defining the Gaussian process on the entire fine grid $\mathcal{V}$ from test vectors, see subsection \ref{sec:covMod}. Alternatively, assuming that $\sigma_\mathcal{V}$ is fixed or already estimated and that $\mu_{i}\cong \mu$ is constant, we obtain the value of $\mu$ as the best linear unbiased predictor \emph{BLUP} from the data $x_\mathcal{C}$ on $\mathcal{C}$. In fact, suppose that we estimate $\mu$ linearly by $\widehat \mu=w_{\mathcal{C}}^T x_{\mathcal{C}}$. The requirement that this estimator is unbiased results in
\begin{align*}
\begin{split}
\mathbb{E}_\mu\left[w^T_\mathcal{C}X_\mathcal{C}\right] &= w^T_\mathcal{C}\mathbb{E}_\mu\left[X_\mathcal{C}\right]=w^T_\mathcal{C}\mathbbm{1}_\mathcal{C}\mu\\
&\Rightarrow\ w^T_\mathcal{C}\mathbbm{1}_\mathcal{C}=1,
\end{split}
\end{align*}
where $\mathbbm{1}_\mathcal{C}$ is the vector of all ones, i.e., $(\mathbbm{1}_{\mathcal{C}})_i=1, i\in\mathcal{C}$ and $\mathbb{E}_\mu$ stands for the expected value for the Gaussian process with constant mean $\mu$. The optimal set of weights $w_{\mathcal{C}}$, given $C_\mathcal{V}$ and hence by restriction $C_\mathcal{C}$, is obtained by the following constrained optimization problem
\begin{equation}\label{eq:krigingMeana}
w_\mathcal{C} \in \operatorname{argmin} \left\{\mathbb{E}_\mu\left[(w^TX_\mathcal{C}-\mu)^2\right],\ w^T\mathbbm{1}_\mathcal{C}=1\right\}.
\end{equation} Using a standard Lagrangian approach, we can reformulate this constrained quadratic optimization problem to the following set of equations
\begin{equation}\label{eq:krigingMeanb}
\frac{\partial L(w,\lambda)}{\partial w}=0 \text{\ and\ } \frac{\partial L(w,\lambda)}{\partial \lambda}=0,
\end{equation} where $L(w,\lambda) = w^TC_\mathcal{C}w-\lambda(w^T\mathbbm{1}_\mathcal{C}-1)$.
It is now easily seen that the solution $w_{\mathcal{C}}$ of the equations \eqref{eq:krigingMeanb} is given by
\begin{equation*}
w = \frac{\mathcal{C}^{-1}_\mathcal{C}\mathbbm{1}_\mathcal{C}}{\mathbbm{1}_\mathcal{C}^TC_\mathcal{C}^{-1}\mathbbm{1}_\mathcal{C}^T}\ \Longrightarrow\ \widehat\mu=\frac{\mathbbm{1}_\mathcal{C}^T\mathcal{C}^{-1}_\mathcal{C}X_\mathcal{C}}{\mathbbm{1}_\mathcal{C}^TC_\mathcal{C}^{-1}\mathbbm{1}_\mathcal{C}^T}
\end{equation*}
Inserting $\mu_\mathcal{C}=\frac{\mathbbm{1}_\mathcal{C}^T\mathcal{C}^{-1}_\mathcal{C}x_\mathcal{C}}{\mathbbm{1}_\mathcal{C}^TC_\mathcal{C}^{-1}\mathbbm{1}_\mathcal{C}^T}\mathbbm{1}_{\mathcal{C}}$ into \eqref{eq:kriging} results in a prediction that depends linearly on the observed data.
Thus this predictor corresponds to the construction of an interpolation matrix $P=P_{\mathcal{F}\!,\mathcal{C}}$ in \eqref{eq:canonical_interpolation}.
From both alternatives described here, we follow the second, estimating the value of $\mu$ from the coarse grid data $x_\mathcal{C}$ and not setting it to zero. The reason is that, while for test vectors $\mu=0$ is a natural choice (see subsection \ref{sec:covMod}), this is not necessarily the case for the problem, the multigrid solver is applied to. A more thorough comparison of both approaches goes beyond the scope of this initial study. Note, that so far, we have only considered the calculation of the entries of $P_{\mathcal{F}\!,\mathcal{C}}$ given a subset $\mathcal{C}$ and neglecting the sparsity requirement of $P$ for now. The choice of $\mathcal{C}$ and the construction of localized, i.e., sparse, interpolation will be discussed next.
\subsection{\label{sec:locKrig} Local Kriging}
The computational cost of \eqref{eq:kriging} and \eqref{eq:krigVar} in many cases is prohibitive, due to the fact that $C_\mathcal{C}$ in general is not sparse. However, we can localize~\eqref{eq:kriging} and~\eqref{eq:krigVar} in the following sense. Assuming that there exists a (pseudo) distance $d_\mathcal{V}(i,j)$ on $\mathcal{V}$ and that the correlation
\[
\varrho(i,j)=\frac{C_{i,j}}{\sqrt{{C}_{i,i}{C}_{j,j}}}
\] decreases sufficiently fast when $d_\mathcal{V}(i,j)$ grows, we can neglect the effect of observations in far away points in $\mathcal{C}$ on the prediction $\widehat X_i$ of $X_i$. In many cases, it is therefore sufficient to choose a subset $\mathcal{C}_i$ of $\mathcal{C}$ containing the $q_{\rm max}$ points in $\mathcal{C}$ that are closest to $i\in \mathcal{F}$ for the calculation of a suitable predictor. The number of neighbours $q_{\rm max}$ can be chosen independently of the size of $\mathcal{C}$ and $\mathcal{V}$ and is referred to as the \emph{caliber} of interpolation.
We note that \eqref{eq:kriging} and \eqref{eq:krigVar} remain valid when replacing $\mathcal{C}$ with $\mathcal{C}_i$ and $\mathcal{F}$ with $i$. Also, the estimates \eqref{eq:krigingMeana} and \eqref{eq:krigingMeanb} can be localized accordingly. The complexity of computing the predictions on $\mathcal{F}$ therefore acquires the optimal linear growth in the size of this set, provided the construction $\mathcal{C}_i$ is either negligible for practical purposes or can be carried out with optimal complexity as well. Due to the applied localization this can be guaranteed by employing efficient graph based techniques.
\subsection{\label{sec:covMod} Non-Parametric Covariance Estimation}
The selection of the expected value $\mu_\mathcal{V}$ and the covariance $C_\mathcal{V}$ has to essentially capture the correlation structure of the problem at hand. Assuming that a set of test vectors $V = \begin{bmatrix}v^{(1)}&\mid &\cdots&\mid&v^{(K)}\end{bmatrix}$ is given on $\mathcal{V}$, we define the average value and the empirical covariance matrix of these test vectors by
\begin{subequations}
\begin{align}
\label{eq:average}
\widehat{\mu}_\mathcal{V}&=\frac{1}{K}V\mathbbm{1}_K,\\
\label{eq:empCov}
\widehat C&= \frac{1}{K}\left(V-\frac{1}{K}V\mathbbm{1}_K\mathbbm{1}_K^T\right) \left(V-\frac{1}{K}V\mathbbm{1}_K\mathbbm{1}_K^T\right)^T ,
\end{align}
\end{subequations}
where $\mathbbm{1}_K\in\mathbb{R}^K$ is the column vector with value $1$ in every entry. By construction the rank of the empirical covariance matrix is bounded from above by $K$, the number of test vectors, so that it is in general not possible to simply replace the theoretical covariance $C$ in \eqref{eq:kriging} and \eqref{eq:krigVar} by the empirical counterparts. As long as $K<|\mathcal{C}|$, i.e., the number of test vectors is smaller than the number of coarse grid variables, $\widehat{C}_\mathcal{C}^{-1}$ does not exist.
There are two ways to fix this problem. First, we can regularize the calculation by replacing $\widehat C_\mathcal{C}\to \widehat C_\mathcal{C}+\varepsilon I$ where $\varepsilon > 0$ stands for a substitution $X\to X+N$ where $N=\{N_i\}, i\in \mathcal{I}$ is spatially uncorrelated Gaussian white noise with zero mean and variance $\varepsilon$.
Second, in the context of local Kriging described in subsection \ref{sec:locKrig} it is only required that $\widehat C_{\mathcal{C}_i}$ is positive definite for all $i\in\mathcal{F}$. As we can choose $|C_{\mathcal{C}_i}| = q_{\rm max}\leq K$, this guarantees the non-singularity of $\widehat C_{\mathcal{C}_i}$ for almost all sets of randomly generated testvectors $V$ so that in this case an $\varepsilon$-regularisation is not required.
Note that in many cases, the test vectors $v^{(j)}$ will be constructed by the application of one or a few smoothing, e.g., Gauss-Seidel, steps applied to a vector with noise data on $\mathcal{V}$, which is statistically centred around $0$. In such cases, $\widehat\mu_{\mathcal{V}}=0$ by theoretical considerations and we can replace the statistical estimation in \eqref{eq:average}. Likewise, in this case we could as well simplify \eqref{eq:empCov} by omission of the terms $\frac{1}{K}V\mathbbm{1}_K\mathbbm{1}_K^T$.
There exist several choices for the distance function in local Kriging. The first option is to use the coordinate distance $d^c(i,j)$, if an embedding of $\mathcal{V}$ in $\mathbb{R}^d$ is known and the underlying continuum problem is isotropic. This will not always be the case, especially in the context of algebraic multigrid. Instead we use the graph distance $d^A(i,j)$, which measures the shortest path in the undirected graph associated with the system matrix $A$ of~\eqref{eq:linsys} over $\mathcal{V}$ assuming that the length of an edge is defined as the inverse of the corresponding matrix entry, i.e., edge $\{i,j\}$ has length $\frac{1}{|A_{i,j}|}$.
In the context of local Kriging, it is not necessary to assemble the full matrix $\widehat C$, but only $n_{\mathcal{F}} = |\mathcal{F}|$ submatrices $\widehat C_{\mathcal{C}_{i}}$ of size $q_{\rm max}\times q_{\rm max}$ and $n_{\mathcal{F}}$ matrices $\widehat C_{i,\mathcal{C}_{i}}$ of size $1\times q_{\rm max}$. Each entry of these matrices requires flops proportional to $K$, which gives linear complexity in $n_{\mathcal{F}}$, provided the search for $\mathcal{C}_{i}$ is either negligible in terms of compute time on the relevant problem sizes or is implemented with optimal complexity in the sense that the search for $\mathcal{C}_{i}$ has bounded complexity.
\subsection{Parametric Semivariogram Estimation}\label{sec:semiVar}
The disadvantage in the procedure described above lies in the fact that a relatively large number $K$ of test vectors $v^{(j)}$ is required in order to obtain a reasonable estimate $\widehat C$ for the underlying covariance structure $C$. Given that each test vector is calculated with a computational cost proportional to $n$, the generation of up to $K\approx 100$ test vectors can be a significant computational burden.
Also, the large number $n$ of estimates of matrices $\widehat C_{\mathcal{C}_i}$ increases the probability that there is at least one $i\in\mathcal{\mathcal{F}}$ for which the estimate for $\widehat C_{\mathcal{C}_i}$ is poor, making it more difficult to obtain guarantees for the estimation of $\widehat X_i$ for all $i\in\mathcal{F}$.
To reduce $K$ down to numbers in the range $1$ -- $10$ and to stabilise the estimation of single elements in $C_{\mathcal{C}_i}$, we follow methods from geostatistics that allow, under appropriate assumptions, a more efficient estimate.
To this purpose, we assume that the underlying continuous problem has some kind of translation invariance. This can either be caused by a strict invariance of the underlying operator which is discretized by $A$ (neglecting the effect of boundary conditions) or an invariance in some statistical sense, where the local inhomogeneity is statistically the same around all points in the computational domain. In both cases it is legitimate, to work with translation invariant models for the a priori distribution of the Gaussian process.
A stochastic process $X=\{X_i\}, i\in \mathcal{I}$ with the special choice $\mathcal{I}=\mathbb{R}^d$ is stationary, if the process $X_h=\{X_{i+h} \mid i\in \mathcal{I}\}$ has the same finite dimensional distributions as $X$ for all $h\in\mathbb{R}^d$. By \eqref{eq:densGauss} for Gaussian processes this amounts to a constant $\mu_i$ independent of $i$ and $C_{i+h,j+h}=C_{i,j}$ for all $i,j,h\in\mathbb{R}^d$. Consequently, the covariance function $C_{i,j} = C_{0,i-j} = :C(i-j)$ defines a function $C:\mathbb{R}^d\to \mathbb{R}$. Furthermore, the process is called isotropic, if $C(h)=C(\Lambda h)$ for any rotation matrix $\Lambda\in SO(d)$ and $h\in \mathbb{R}^d$. This can be justified if the underlying operator in the continuum (approximately) shares this property. In this case, we can model (with a slight abuse of notation) $C(h)=C(|h|)$ where only a function $C:\mathbb{R}_+\to\mathbb{R}$ has to be estimated from the test vectors $V$. This is a standard task in geostatistics \cite{bivand2008applied,sherman2011spatial}, which we briefly review.
In geostatistics it is customary to fit semivariograms instead of the covariance function. Both are connected via
\begin{align}
\begin{split}\label{eq:semiVar}
\gamma(|h|)&=\frac{1}{2}\mathbb{E}\left[(X(0)-X(h))^2\right]\\
&=\frac{1}{2}\mathbb{E}\left[(X(0)-\mu)^2+2(X(0)-\mu)(X(h)-\mu)+(X(h)-\mu)^2\right]\\
&=C(0)-C(|h|)
\end{split}
\end{align}
and contain the same information, as the asymptotic sill value $C(0)$ can be obtained by letting $h\to\infty$ and thereby $C(|h|)\to 0$ as correlations decline at large distances. Note, that in \eqref{eq:semiVar} we made use of the stationarity assumption.
In the next step we generate the empirical semivariogram based on the test vectors $V$ on $\mathcal{V}$ and the coordinate distance function $d^c(i,j)$. To this purpose, we collect all pairs of values $(d^c(i,j),(v_i^{(\ell)}-v_j^{(\ell)})^2), i,j\in \mathcal{V}$ and $\ell=1,\ldots,K$ in the so-called variogram cloud. Discretising the range of all values $d^c(i,j)$ into bins of width $\Delta$, we obtain the empirical semivariogram as
\begin{equation}
\label{eq:empSemivar}
\hat\gamma (h)=\frac{1}{K\# \{(i,j):|h|-\frac{\Delta}{2}\leq d^c(i,j)<|h|+\frac{\Delta}{2}\}}\sum_{\{(i,j):|h|-\frac{\Delta}{2}\leq d^c(i,j)<|h|+\frac{\Delta}{2}\}\atop \ell=1,\ldots,K}(v_i^{(\ell)}-v_j^{(\ell)})^2.
\end{equation}
In the following, parametric semivariogram functions $\gamma_\theta$ are fitted to $\hat\gamma(|h|)$, mostly using weighted least squares \cite{bivand2008applied}. There are many known families of semivariograms, but in this work we use the exponential family
\begin{equation}\label{eq:spherical_model}
\gamma_\theta(|h|)=\sigma^2\left(1-e^{-\left(\frac{|h|}{\eta}\right)}\right),~~\theta=(\sigma^2,\eta)\in\mathbb{R}_+^2
\end{equation}
and the spherical family defined by
\begin{equation}\label{eq:exponential_model}
\gamma_\theta(|h|)=\left\{\begin{array}{cc}\sigma^2\left(\frac{3|h|}{2a}-\frac{1}{2}\left(\frac{h}{\eta}\right)^3\right)&\mbox{for } |h|<\eta\\\sigma^2&\mbox{for }|h|\geq \eta\end{array}\right. , ~~\theta=(\sigma^2,\eta)\in\mathbb{R}_+^2.
\end{equation}
Empirical semivariograms and fitted semivariograms of exponential and spherical type can be seen in Figures \ref{fig:semivarCircle} and \ref{fig:semivarSquare} below. For further models we refer to \cite{bivand2008applied,sherman2011spatial}. From the fitted semivariogram one then computes the covariance function $C_\theta(|h|)$ which can be used to compute $C_\mathcal{H}$.
Provided that the correlation length $\eta$ (also called range in the geostatistical literature) is much smaller than the size of the underlying domain, it is often enough to work with just a few or even just one test vector, $K=1$: If spatial correlations quickly decrease, the random field effectively contains many resamplings of its statistics in just one realization, i.e., in one test vector over a sufficiently extended grid $\mathcal{V}$.
\subsection{\label{sec:graphDist} Parametric Semivariograms for Inhomogeneity and Anisotropy}
In most situations where homogeneity and isotropy of $X$ cannot be expected, we replace the coordinate metric $d^c(i,j)$ with the graph metric $d^A(i,j)$ introduced in subsection \ref{sec:locKrig}. This is very much in the spirit of algebraic multigrid approaches, where the knowledge of coordinate lists of the variables cannot be guaranteed and the use of the graph distance dates back at least to the definition of strength-of-connection in classical AMG~\cite{RugeStue1987}.
This pragmatic approach however comes with a conceptional problem. It is not clear if the weighted graph obtained from $\mathcal{V}$ and $A$ with distance function $d^A(i,j)$ can be isometrically embedded to some space $\mathbb{R}^{d'}$. In fact, this is possible if and only if the condition
\begin{equation}
\label{eq:embedding}
w^T[d^A(i,j)^2]_{i,j\in\mathcal{V}}w\leq 0, \text{\ for all\ } w\in \mathbb{R}^n \mbox{ s.t. }\mathbbm{1}_n^Tw=0
\end{equation}
holds, see \cite{deza2009geometry}. If this is true, then the positivity of the covariance matrix $C$ follows from the fact that for many families of parametric semivariograms $C_\theta(|h|)$ defines a positive definite function in \emph{any} dimension $d$ and in particular in the dimension $d'$ of the isometric embedding, which can differ from the dimension $d$ of the underlying continuum problem.
If the condition \eqref{eq:embedding} is violated, formula \eqref{eq:kriging} still defines an interpolator, as is easily checked, but the Kriging variance \eqref{eq:krigVar} needs no longer to be non-negative and the probabilistic interpretation of the Gaussian process in gone. In fact, we observe this in the examples presented below, despite observing high correlation between coordinate distance and graph distance.
In this situation we still can carry out the algebraic manipulations from both formulae and we use the Gaussian process in the probabilistic sense simply as a source of inspiration.
Note however that local models of Gaussian processes on $\mathcal{C}_i\cup\{i\}$, $i\in \mathcal{F}$ with metric $d^A(i,j)$ may well exist as the embedding problem for such smaller graphs is much alleviated. In fact, in the numerical examples given below we do not observe any non-positive covariance metrics $\widehat C_{\mathcal{C}_i\cup\{i\}}$ for moderate size of $q_{\rm max}$. We therefore suggest that the calculation of local Kriging predictors and local Kriging variances can still be used and still give reasonable results, even though the (global) probabilistic interpretation has to be used with caution and strictly speaking the localized version of \eqref{eq:embedding} should be checked, at least if the method proposed in the following section does not show the expected performance. If this condition is violated, it seems to be reasonable to lower $q_{\rm max}$ at least locally, to obtain smaller local graphs which are more easily embedded.
\section{Adaptive Coarsening using Gaussian Processes}\label{sec:coarsening}
The fact that algebraically smooth error can be interpreted as instances of a spatial Gaussian process allows us to use the methodology of Gaussian processes and the Kriging interpolation developed in~\cref{sec:gaussian} to solve the coarsening problem of algebraic multigrid methods. The calculation of a splitting of the variables $\mathcal{V}$ into a set of coarse variables $\mathcal{C}$ and the remaining fine variables $\mathcal{F} = \mathcal{V}\setminus \mathcal{C}$ as well as the computation of interpolation suited to this setting is carried out in three consecutive steps.
First, we need to determine the covariance structure $C$ of algebraically smooth error when viewed as instances of a spatial Gaussian process. In order to do so, we start with a number, $K$, of test vectors $v^{(1)},\ldots,v^{(K)}$, each with entries that are normally distributed with mean zero and variance $1$. These initially random vectors are then subject to a number, $\nu$ of smoothing iterations with right-hand-side zero, where we employ the smoother that is going to be used in the algebraic multigrid method~\eqref{eq:eprop} as well. These test vectors are then fed into the calculation of the covariance structure. In our tests we compare localized non-parametric approaches (cf.~\cref{sec:covMod}), which use a sizeable number of test vectors, with parametric models (cf.~\cref{sec:semiVar}) that make use of only a small number of test vectors.
Once the covariance structure is determined we can use Kriging interpolation to tackle the coarsening problem. The set of coarse variables $\mathcal{C}$ has to be chosen in such a way that interpolation of information from these variables to the remaining variables $\mathcal{F}$ is as accurate as possible for algebraically smooth error. In accordance with the interpretation of Kriging interpolation as the interpolation that minimizes the MSE under the assumption that algebraically smooth error can be interpreted as instances of a Gaussian field, we can use the variance of the Kriging estimator~\eqref{eq:krigVar} in order to define the coarse variable set. Starting with $\mathcal{C} = \emptyset$ and using the fact that any variable that ends up in $\mathcal{C}$ during the coarsening has zero variance after interpolation we proceed to add those variables to $\mathcal{C}$ with largest variance. In case there is a tie, we choose the first occurrence, but other selection strategies are possible as well.
Exploiting the fact that the correlation distance can be used to limit the reach of the Kriging interpolation, we can actually add multiple variables to $\mathcal{C}$ at the same time if they are spaced so far apart that interpolation between them is not considered due to the localization of the Kriging interpolation. After variables are added to $\mathcal{C}$, we update the Kriging interpolation of all affected variables. By using one of the pseudo-distances we first determine for each $i\in \mathcal{F}$ the set of interpolatory variables $\mathcal{C}_i \subseteq \mathcal{C}$, again respecting the limitation of reach due to a finite correlation distance. Based on these sets, the Kriging estimator and the corresponding variances or MSE are calculated/updated. This process of adding variables to $\mathcal{C}$ based on the uncertainty with which we can predict the value at the respecting variable and updating Kriging interpolation is repeated until a prescribed tolerance on either the size of the coarse variable set $\mathcal{C}$, typically relative to the total number of variables $\mathcal{V}$, or the largest remaining uncertainty of the Kriging estimator is reached. The resulting process is roughly summarized in~\cref{alg:coarsening}.
\begin{algorithm2e}[ht]
\SetAlgorithmStyle
\caption{Coarsening based on Kriging interpolation}\label{alg:coarsening}
\KwData{}\medskip
Initialize $\mathcal{C} = \emptyset$, $\mathcal{F} = \mathcal{V}$\;
\While{$|\mathcal{C}| < n_c$}{
Choose $i \in \mathcal{F}$ with largest variance $\mu_i$\;
Add $i$ to $\mathcal{C}$, remove $i$ from $\mathcal{F}$\;
\For{$j\in \mathcal{F}$ with $i \in \mathcal{C}_j$}{
Update Kriging interpolation with new $\mathcal{C}_j$ set\;
Compute corresponding updated variances $\mu_j$ (set $\mu_i = 0$)\;
}
}
\end{algorithm2e}
\section{Numerical Case Studies}\label{sec:numerics}
In order to gauge the efficiency of the new coarsening scheme based on Gaussian fields and Kriging interpolation we consider the general diffusion problem
\begin{equation*}\label{eq:poisson_univ}
- \left( c_1 \frac{\partial^2}{\partial x^2} + c_2 \frac{\partial^2}{\partial y^2} + c_3 \left( \frac{\partial}{\partial x} \frac{\partial}{\partial y} + \frac{\partial}{\partial y} \frac{\partial}{\partial x} \right) \right) u = f \, ,
\end{equation*} for constant and anisotropic coefficients by choosing $c_1,c_2$ and $c_3$ accordingly. We further choose the computational domain as the unit square $(0,1)^2$, employing a finite difference discretization on a regular mesh and the unit circle $U(1)$, where the discretization is defined by linear finite elements on a triangularization generated in MATLAB. We considered $4$ parameter combinations in our tests collected in~\cref{tab:parameters}.
\begin{table}[ht]\small
\centering
\begin{tabular}{c||c|c|c|c}
name & s-iso & s-aniso & c-iso & c-aniso \\\hline
domain & square & square & circle & circle \\
\textbf{$n$} & $2025$ & $2025$ & $2521$ & $2521$\\
$\begin{bmatrix} c_1 & c_3 \\ c_3 & c_2\end{bmatrix}$ & $\begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}$ & $\begin{bmatrix} 1 & 0 \\ 0 & 10^{-2}\end{bmatrix}$ & $\begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}$ & $\begin{bmatrix} 1 & 0 \\ 0 & 10^{-2}\end{bmatrix}$
\end{tabular}
\caption{Parameter choices of the considered test cases.}
\label{tab:parameters}
\end{table}
The overall aim of these case studies is to get a first impression of the performance of the new approach of constructing coarse variable sets and interpolation. Thus we compare methods with respect to the underlying covariance model, i.e., the empirical covariance function vs.~semivariogram models (spherical and exponential). In addition, we test the robustness of the approach with respect to the number of test vectors used to construct the covariance model. In all tests we employ localization in the calculation of Kriging interpolation, which is an inevitable technique to guarantee overall linear complexity and well-posedness of the Kriging interpolation as discussed in~\cref{sec:locKrig} especially for the empirical construction.
The test vectors are generated by applying just one iteration of a colored Gauss-Seidel iteration to white noise vectors and we use the same iterative method for the smoother of our resulting two-grid method as well. That is, in the reported results we run a $V(1,1)$-cycle two-grid method with a direct solve for the coarse grid system of equations.
\paragraph{Pseudo-distances}
The use of pseudo-distances is an important aspect when it comes to the independence from problem specific knowledge as is required by a true algebraic multigrid approach. Thus we first compare the correlation of the graph pseudo-distance and the true geometric distances based on the coordinates of the unknowns. In this we found a correlation of the distances for the cases c-iso of $97.68\%$ and for s-iso of $99.65\%$, which leads us to believe that using the algebraic graph distance as a pseudo distance in our Gaussian field analysis is viable.
\paragraph{Covariance models}
In the following we fit the exponential and the shperical semivariogram models to the empirical semivariogram generated from 1, 10 and 100 test vectors, see equations \eqref{eq:exponential_model}, \eqref{eq:spherical_model} and \eqref{eq:empSemivar}. This is done for the isotropic and the anisotropic case, both for the circle (Figure \ref{fig:semivarCircle}) and the square grid (Figure \ref{fig:semivarSquare}). The fits and models are performed using the R library \texttt{gstat}. The fits expose reasonable quality and not much variation caused by the number of test vectors used.
\begin{figure}[h]
\centerline{
\includegraphics[width=.45\textwidth]{vgmFitExp_circ_iso.pdf}\hfill
\includegraphics[width=.45\textwidth]{vgmFitSph_circ_iso.pdf}
}
\centerline{
\includegraphics[width=.45\textwidth]{vgmFitExp_circ_aniso.pdf}\hfill
\includegraphics[width=.45\textwidth]{vgmFitSph_circ_aniso.pdf}
}
\caption{\label{fig:semivarCircle} Fitted semivariogram models for the unstructured mesh on the circle: The top row displays the isotropic and the bottom row the anisotropic case. On the left the exponential semivariogram model is used and on the right the spherical.}
\end{figure}
\begin{figure}[h]
\centerline{
\includegraphics[width=.45\textwidth]{vgmFitExp_square_iso.pdf}\hfill
\includegraphics[width=.45\textwidth]{vgmFitSph_square_iso.pdf}
}
\centerline{
\includegraphics[width=.45\textwidth]{vgmFitExp_square_aniso.pdf}\hfill
\includegraphics[width=.45\textwidth]{vgmFitSph_square_aniso.pdf}
}
\caption{\label{fig:semivarSquare} Fitted semivariogram models for the structured mesh on the square: The top row displays the isotropic and the bottom row the anisotropic case. On the left the exponential semivariogram model is used and on the right the spherical.}
\end{figure}
\paragraph{Coarsening and two-grid results}
Finally, after analyzing the components of the metrics and fits underlying the Gaussian process framing of the coarse grid construction we are ready to apply our coarsening algorithm and present two-grid results using the Kriging interpolation. In this we use the test cases described in~\cref{tab:parameters} combined with
\begin{itemize}
\item an empirical construction of the covariance structure, which we term \emph{emp}-$K$,
\item a spherical covariance model based on a semivariogram fit~\eqref{eq:spherical_model}, termed \emph{sph}-$K$,
\item an exponential covariance model based on a semivariogram fit~\eqref{eq:exponential_model}, termed \emph{exp}-$K$.
\end{itemize} In this naming convention $K$ represents the number of test vectors used in the construction of the model or semivariogram, respectively.
In~\cref{tab:twogrid-iso} we collect results of the resulting two-grid methods for the isotropic test cases s-iso and c-iso. The quantities we report are the asymptotic convergence rates, $\rho$ of the two-grid methods, which are indicative of the overall compatibility of the smoother and coarse-grid correction, and iteration numbers, $k$, of the conjugate gradients method preconditioned with the two-grid method that are required to reduce the initial residual by a factor of $10^8$. This latter quantity gives insight into the flaws of the two-grid construction. Oftentimes, as also explored in~\cite{BranBranKahlLivs2011,BranBranKahlLivs2015a,KahlRott2018}, the two-grid method might show bad asymptotic convergence rates even though the preconditioned conjugate gradients iteration converges rapidly. This typically corresponds to the presence of a few outliers in the spectrum of the preconditioned matrix and indicates that the two-grid construction provides better complementarity of coarse grid construction and smoothing than the asymptotic convergence rate suggests.
\begin{table}[ht]
\resizebox{\textwidth}{!}{
\begin{tabular}{c}
\begin{tabular}{c|c|c|c|c|c|c|c|c}
\textbf{s-iso} & emp-$10$ & emp-$100$ & sph-$1$ & sph-$10$ & sph-$100$ & exp-$1$ & exp-$10$ & exp-$100$ \\\hline
$\rho$ & $.387$ & $.302$ & $.256$ & $.251$ & $.253$ & $.224$ & $.225$ & $.222$\\
$k$ & $10$ & $9$ & $9$ & $9$ & $10$ & $9$ & $8$ & $9$\\
\end{tabular}\\ \\
\begin{tabular}{c|c|c|c|c|c|c|c|c}
\textbf{c-iso} & emp-$10$ & emp-$100$ & sph-$1$ & sph-$10$ & sph-$100$ & exp-$1$ & exp-$10$ & exp-$100$ \\\hline
$\rho$ & $.563$ & $.319$ & $.275$ & $.27$ & $.273$ & $.314$ & $.294$ & $.303$\\
$k$ & $14$ & $10$ & $10$ & $10$ & $10$ & $11$ & $10$ & $10$\\
\end{tabular}
\end{tabular}
}
\caption{Asymptotic convergence rates $\rho$ of the two-grid V($1,1$) cycle and number of iterations $k$ of the conjugate gradients method preconditioned with the two-grid method to reduce the initial residual by a factor of $10^{8}$ for both isotropic test cases. All approaches generate a coarse variable set with $n_c = \tfrac{n}{4}$ variables, use a localization of distance $4$ and a caliber of $4$, i.e., $|\mathcal{C}_i|\leq 4$ for all $i\in \mathcal{F}$.}
\label{tab:twogrid-iso}
\end{table}
Taking into account that all results of~\cref{tab:twogrid-iso} are generated with test vectors that are smoothed by only a single iteration of colored Gauss-Seidel and that a $V(1,1)$-cycle is employed the results are surprisingly good; cf.~\cite{BranBranKahlLivs2011}. In part this can be explained by the implicit preservation of the constant vector in the Kriging interpolation, a modification that has been shown to be particularly effective in improving the performance of bootstrap AMG in~\cite{BranBranKahlLivs2011,BranBranKahlLivs2015b,MantMcCoParkRuge2010}. The most interesting observation is the fact that the model based constructions of Kriging interpolation are competitive when using only a single test vector. Due to the fact that the descriptive power of the semivariogram approach should get better the more points are available for sampling, i.e., the finer the discretization, this finding should scale well with respect to the problem size. In~\cref{fig:corsening-iso} we collected some sections of the resulting coarsenings of the methods considered in~\cref{tab:twogrid-iso}.
The coarsenings do not show any particularly interesting features. As only sections in the bulk are shown and the coarsening ratio of the depicted section is smaller than the preset $\tfrac{1}{4}$, a mild agglomeration of coarse grid variables at the boundary takes place. It remains to be seen if this poses a problem when recursing on the construction in a multigrid fashion.
\begin{figure}[ht]
\begin{center}\resizebox{.8\textwidth}{!}{
\begin{tikzpicture}
\begin{scope}
\node at (0,0) {\includegraphics[width=.45\textwidth]{./figs/coarseningSquareIsoEmp.pdf}};
\end{scope}
\begin{scope}[xshift = .5\textwidth]
\node at (0,0) {\includegraphics[width=.45\textwidth]{./figs/coarseningCircleIsoEmp.pdf}};
\end{scope}
\begin{scope}[yshift = -.35\textwidth]
\node at (0,0) {\includegraphics[width=.45\textwidth]{./figs/coarseningSquareIsoSph.pdf}};
\end{scope}
\begin{scope}[xshift=.5\textwidth,yshift=-.35\textwidth]
\node at (0,0) {\includegraphics[width=.45\textwidth]{./figs/coarseningCircleIsoSph.pdf}};
\end{scope}
\begin{scope}[yshift=-.7\textwidth]
\node at (0,0) {\includegraphics[width=.45\textwidth]{./figs/coarseningSquareIsoExp.pdf}};
\end{scope}
\begin{scope}[xshift=.5\textwidth,yshift=-.7\textwidth]
\node at (0,0) {\includegraphics[width=.45\textwidth]{./figs/coarseningCircleIsoExp.pdf}};
\end{scope}
\end{tikzpicture}}
\end{center}
\caption{Illustration of the variable splittings obtained by emp-$10$ (top), sph-$1$ (mid) and exp-$1$ (bottom) for the isotropic test cases on the square (left) and circle (right). In order to avoid cluttering of the illustrations we just show a section of the bulk of the domains (i.e., $[.35,.4]\times[.65,.6]$ and $[-.3,-.2]\times[.3,.2]$ for square and circle, resp.).}\label{fig:corsening-iso}
\end{figure}
The more interesting test cases arise when anisotropy is present in the model, especially for discretization on unstructured grids, where a canonical coarsening that follows the anisotropy is not available. Testing automatic coarsening approaches on the simple grid-aligned case in order to gauge their robustness and ability to reproduce the canonical coarsenings has been common in the past~\cite{BranBranKahlLivs2015b,KahlRott2018}. Analogous to the results for the isotropic test cases we report in~\cref{tab:twogrid-aniso} asymptotic convergence rates and iterations counts of the preconditioned conjugate gradients method.
\begin{table}[ht]
\resizebox{\textwidth}{!}{
\begin{tabular}{c}
\begin{tabular}{c|c|c|c|c|c|c|c|c}
\textbf{s-aniso} & emp-$10$ & emp-$100$ & sph-$1$ & sph-$10$ & sph-$100$ & exp-$1$ & exp-$10$ & exp-$100$ \\\hline
$\rho$ & $.463$ & $.305$ & $.06$ & $.06$ & $.154$ & $.224$ & $.225$ & $.222$\\
$k$ & $9$ & $8$ & $6$ & $6$ & $6$ & $9$ & $8$ & $9$\\
\end{tabular}\\ \\
\begin{tabular}{c|c|c|c|c|c|c|c|c}
\textbf{c-aniso} & emp-$10$ & emp-$100$ & sph-$1$ & sph-$10$ & sph-$100$ & exp-$1$ & exp-$10$ & exp-$100$ \\\hline
$\rho$ & $.648$ & $.533$ & $.69$ & $.684$ & $.681$ & $.704$ & $.71$ & $.685$\\
$k$ & $19$ & $15$ & $22$ & $21$ & $21$ & $22$ & $24$ & $22$\\
\end{tabular}
\end{tabular}
}
\caption{Asymptotic convergence rates $\rho$ of the two-grid V($1,1$) cycle and number of iterations $k$ of the conjugate gradients method preconditioned with the two-grid method to reduce the initial residual by a factor of $10^{8}$. All approaches generate a coarse variable set with $n_c = \tfrac{n}{2}$ variables, use a localization of distance $4$ in graph distance and a caliber of $2$ and $3$ for all $i\in \mathcal{F}$ for the test cases formulated on the square and circle, respectively.}
\label{tab:twogrid-aniso}
\end{table} The grid aligned anisotropy in the square test case s-aniso does not pose any difficulty for any of the model based approaches and again a single test vector is sufficient to obtain a good enough statistics for a suitable fit of the model as already suggested by~\cref{fig:semivarSquare,fig:semivarCircle}. The results suggest that the spherical model is better suited for this problem than the exponential model and we see a clear advantage of the model based approaches over the empirical approach, showing extremely fast convergence. Similar to the isotropic case we see a notable improvement when increasing the number of the test vectors for the empirical model from $10$ to $100$, which is quite frankly an unfeasible number of test vectors, but serves the illustrative purpose quite well. As can be seen in~\cref{fig:corsening-aniso} the coarsenings obtained by the different approaches for the square test case show that the anisotropy has been clearly detected and the coarsening constructed accordingly. When it comes to the results for the anisotropic problem on the circular domain and unstructured grid the results are comparable to results reported, e.g., in~\cite{BranBranKahlLivs2015b} for non-grid aligned anisotropies. Whilst all approaches yield good preconditioners for the conjugate gradients method with comparable iteration numbers, the asymptotic convergence rates are considerably worse at around $.7$ compared to the grid-aligned case for the model based approaches. Interestingly the empirical construction is able to cope with the unstructured grid better than the model based approaches. Taking into account that the model based approaches implicitly assume shift invariance of the problem, which might be violated more strongly in this test case compared to the circular isotropic problem, this does not come as a large surprise.
\begin{figure}[ht]
\begin{center}\resizebox{.8\textwidth}{!}{
\begin{tikzpicture}
\begin{scope}
\node at (0,0) {\includegraphics[width=.45\textwidth]{./figs/coarseningSquareAnisoEmp.pdf}};
\end{scope}
\begin{scope}[xshift = .5\textwidth]
\node at (0,0) {\includegraphics[width=.45\textwidth]{./figs/coarseningCircleAnisoEmp.pdf}};
\end{scope}
\begin{scope}[yshift = -.35\textwidth]
\node at (0,0) {\includegraphics[width=.45\textwidth]{./figs/coarseningSquareAnisoSph.pdf}};
\end{scope}
\begin{scope}[xshift=.5\textwidth,yshift=-.35\textwidth]
\node at (0,0) {\includegraphics[width=.45\textwidth]{./figs/coarseningCircleAnisoSph.pdf}};
\end{scope}
\begin{scope}[yshift=-.7\textwidth]
\node at (0,0) {\includegraphics[width=.45\textwidth]{./figs/coarseningSquareAnisoExp.pdf}};
\end{scope}
\begin{scope}[xshift=.5\textwidth,yshift=-.7\textwidth]
\node at (0,0) {\includegraphics[width=.45\textwidth]{./figs/coarseningCircleAnisoExp.pdf}};
\end{scope}
\end{tikzpicture}}
\end{center}
\caption{Illustration of the variable splittings obtained by emp-$10$ (top), sph-$1$ (mid) and exp-$1$ (bottom) for the anisotropic test cases on the square (left) and circle (right). In order to avoid cluttering of the illustrations we just show a section of the bulk of the domains (i.e., $[.35,.4]\times[.65,.6]$ and $[-.3,-.2]\times[.3,.2]$ for square and circle, resp.).}\label{fig:corsening-aniso}
\end{figure}
The coarsenings depicted in~\cref{fig:corsening-aniso} demonstrate the capability of the approach to detect the anisotropy and construct coarsenings that are suitable to construct an efficient two-grid method. Even though the resulting two-grid methods for the model based approaches yield worse asymptotic convergence rates for the non grid-aligned anisotropy in c-aniso this cannot be traced to a defect in the coarsening structure. The respective sections of the coarsenings show clearly that the approach detected the anisotropy, which leads us to believe that the poor performance is more due to a poor choice of interpolatory set or interpolation weights or both rather than a poor coarsening structure.
\section{Conclusion and Outlook}\label{sec:conclusion}
In this paper we presented a new approach to adaptive algebraic multigrid construction using ideas from geostatistics and statistical learning theory. Based on the resemblance of algebraically smooth error to instances of spatial Gaussian fields we develop an empirical and semivariogram based approach to recover the covariance structure of the unknown, underlying Gaussian process. Once the covariance structure is known, efficient interpolation can be formulated by Kriging interpolation. Further exploiting the interpretation of the local interpolation error as the variance of the representation, we are able to formulate a coarsening approach that is seamlessly integrated into the determination of interpolation. Finally, by using graph distance and assuming shift invariance of the operator with respect to this pseudo-distance we are able to obtain good statistics for the semivariogram fit using only a single test vector. This is due to the fact that the semivariogram collects information at all variables to formulate a covariance function that depends solely on the distance between variables. Combined with the observation that the correlation distance of the underlying Gaussian process is very small, a single vector provides enough information about the short range correlation of values. In addition, the short correlation distance allows us to strictly localize all calculations which preserves the linear complexity of the whole process.
One apparent aim of future development is the integration of the Gaussian process approach into a multigrid setup and potentially a bootstrap type setup that is able to generate additional information about the underlying process on coarse scales as well. In line with~\cite{owhadi2017multigrid} we plan to investigate the connection between the partial differential operator, its discretization and the smoothing scheme with the resulting covariance structure of the Gaussian fields. To some extend this development can be seen in line with the investigation of optimal interpolation in algebraic multigrid methods in~\cite{BranCaoKahlFalgHu2018}, where an explicit influence of the smoother on the optimal construction of interpolation has been shown. Insight into this might allow us to translate the demonstrated potential for efficient adaptive algebraic multigrid constructions using a minimal amount of test vectors to more complex problems.
|
3,212,635,537,454 | arxiv | \section{Introduction and Overview}\label{intro}
Recall that, given a function $\varphi\colon\mathbb{R}^n\to\mathbb{R}$, which is twice continuously differentiable (${\cal C}^2$-smooth) around some point $\bar{x}\in\mathbb{R}^n$, the {\em classical Newton method} to solve the nonlinear {\em gradient system}
\begin{equation}\label{gra}
\nabla\varphi(x)=0
\end{equation}
constructs the iterative procedure
\begin{equation}\label{clas-newton}
x^{k+1}:=x^k+d^k\;\mbox{ for all}\;k\in\mathbb{N}:=\big\{1,2,\ldots\big\},
\end{equation}
where $x^0\in\mathbb{R}^n$ is a given starting point, and where $d^k$ is a solution to the linear system
\begin{equation}\label{newton-iter}
-\nabla\varphi(x^k)=\nabla^2\varphi(x^k)d^k,\quad k=0,1,\ldots,
\end{equation}
written in terms of the Hessian matrix $\nabla^2\varphi(x^k)$ of $\varphi$ at $x^k$. As known in classical optimization, Newton's algorithm in \eqref{clas-newton} and \eqref{newton-iter} is well-defined (i.e., the equations in \eqref{newton-iter} are solvable for $d^k$), and the sequence of its iterations $\{x^k\}$ superlinearly (even quadratically) converges to a solution $\bar{x}$ of \eqref{gra} if $x^0$ is chosen sufficiently close to $\bar{x}$ and if the Hessian matrix $\nabla^2\varphi(\bar{x})$ is positive-definite. Note also that, besides being a necessary condition for local minimizers of $\varphi$, the gradient system \eqref{gra} is important for its own sake and holds not only for local minimizers and local maximizers of $\varphi$. Furthermore, a counterpart of the classical Newton method has been developed for solving more general nonlinear equations of the type $f(x)=0$, where $f\colon\mathbb{R}^n\to\mathbb{R}^m$ is a continuously differentiable (${\cal C}^1$-smooth) mapping, and where $\nabla^2\varphi(x^k)$ in \eqref{newton-iter} is replaced by the Jacobian matrix of $f$ at the points in question. We are not going to deal with the latter method and its extensions in this paper while being fully concentrated on the gradient systems \eqref{gra} and their appropriate subgradient counterparts.
Concerning the gradient systems of type \eqref{gra} where $\varphi$ may not be ${\cal C}^2$-smooth around $\bar{x}$, we mention that the enormous literature has been devoted to developing various versions of the (generalized) Newton method; see, e.g., the books by Dontchev and Rockafellar \cite{Donchev09}, Facchinei and Pang \cite{JPang}, Izmailov and Solodov \cite{Solo14}, Klatte and Kummer \cite{Klatte}, Ulbrich \cite{Ul}, and the references therein. The vast majority of such extensions deals with functions $\varphi$ in \eqref{gra} of class ${\cal C}^{1,1}$ (or ${\cal C}^{1+}$ in the notation of Rockafellar and Wets \cite{Rockafellar98}) around $\bar{x}$, which consists of continuously differentiable functions with locally Lipschitzian derivatives. The most popular generalized Newton method to solve \eqref{gra} for functions of this type is known as the {\em semismooth Newton method} initiated independently by Kummer \cite{Kummer} and by Qi and Sun \cite{LQi}. In the semismooth Newton method, the Hessian matrix of $\varphi$ in \eqref{newton-iter} is replaced by the (Clarke) {\em generalized Jacobian} (collection of matrices) of the gradient mapping $\nabla\varphi$. Then the corresponding Newton iterations are well-defined in a neighborhood of $\bar{x}$ and exhibit a local superlinear convergence to the solution $\bar{x}$ of \eqref{gra} provided that each matrix from the generalized Jacobian is {\em nonsingular} and that the gradient mapping $\nabla\varphi$ is {\em semismooth} around $\bar{x}$ in the sense of Mifflin \cite{Mifflin}. The latter property has been well investigated and applied in variational analysis and optimization, not only in connection with the semismooth Newton method. Besides the aforementioned books and papers, we refer the reader to Burke and Qi \cite{Burke}, Henrion and Outrata \cite{ho}, and Meng et al. \cite{msz} among many other publications on the theory and applications of such functions.\vspace*{0.03in}
In the case of ${\cal C}^{1,1}$ functions $\varphi$, our Newton-type algorithm proposes replacing \eqref{newton-iter} by
\begin{equation}\label{newtonC11}
-\nabla\varphi(x^k)\in\partial^2\varphi(x^k)(d^k),\quad k=0,1,\ldots,
\end{equation}
where $\partial^2\varphi$ stands for {\em second-order subdifferential/generalized Hessian} of $\varphi$ introduced by Mordukhovich \cite{m92} for arbitrary extended-real-valued functions. This construction reduces to the classical Hessian operator for ${\cal C}^2$-smooth functions while maintaining key properties of the latter for important classes of functions in broad generalities; see below. In what follows we obtain efficient conditions ensuring the solvability of the inclusions in \eqref{newtonC11} and superlinear convergence of iterates $\{x^k\}$ if the starting point $x^0$ is sufficiently close to $\bar{x}$. As shown in this paper, the obtained conditions allow us to use the proposed algorithm \eqref{newtonC11} to solve systems \eqref{gra} with ${\cal C}^{1,1}$ functions $\varphi$ in the situations where the semismooth Newton method cannot be applied.
Observe that algorithm \eqref{newtonC11} has been recently introduced and developed, in an equivalent form, in the paper by Mordukhovich and Sarabi \cite{BorisEbrahim} to find {\em tilt-stable local minimizers} for functions $\varphi$ of class ${\cal C}^{1,1}$. We'll discuss tilt stability of local minimizers, the notion introduced by Poliquin and Rockafellar \cite{Poli}, in the corresponding place below with a detailed comparison of the results obtained in this paper and in the paper by Mordukhovich and Sarabi. Note that here we do not assume that $\bar{x}$ is a local minimizer of $\varphi$, not even talking about its tilt stability. Observe also that both the latter paper and the current one employ the {\em semismooth$^*$ property} of $\nabla\varphi$ that has been recently introduced and developed by Gfrerer and Outrata \cite{Helmut} as an improvement of the standard semismoothness used before.
The main thrust of this paper is on developing a generalized Newton method to solve {\em subgradient inclusions} of the following type:
\begin{equation}\label{subgra-inc}
0\in\partial\varphi(x),
\end{equation}
where $\varphi\colon\mathbb{R}^n\to\Bar{\mathbb{R}}:=(-\infty,\infty]$ is an extended-real-valued function belonging to a broad class of {\em prox-regular} and subdifferentially continuous functions, which overwhelmingly appear in variational analysis and optimization. The subdifferential operator used in \eqref{subgra-inc} is understood as the (Mordukhovich) {\em limiting subdifferential} of extended-real-valued functions that agrees with the classical gradient for ${\cal C}^1$-smooth functions and the subdifferential of convex analysis when $\varphi$ is convex. In very general settings, the limiting subdifferential enjoys comprehensive calculus rules that can be found in the books by Mordukhovich \cite{Mordukhovich06,Mor18} and by Rockafellar and Wets \cite{Rockafellar98}. If $\varphi$ is smooth around $\bar{x}$, then \eqref{subgra-inc} clearly reduces to \eqref{gra}.
The generalized Newton method, which is proposed in this paper to solve the subgradient inclusion \eqref{subgra-inc}, is also based on the second-order subdifferential $\partial^2\varphi$ with replacing \eqref{newtonC11} by
\begin{equation}\label{subgra-prox}
-v^k\in\partial^2\varphi(x_k-\lambda v^k,v^k)(\lambda v^k+d^k)\;\mbox{ with }\;v^k:=\frac{1}{\lambda}\Big(x^k-{\rm Prox}_\lambda\varphi(x^k)\Big),
\end{equation}
where ${\rm Prox}_\lambda\varphi(x)$ stands for the {\em proximal mapping} of $\varphi$ corresponding to a constructive choice of the parameter $\lambda>0$. This form is shown to be closely related to the Newton-type algorithm developed by Mordukhovich and Sarabi \cite{BorisEbrahim}, in terms of Moreau envelopes with somewhat different choice of parameters, to find tilt-stable minimizers of prox-regular and subdifferentially continuous functions $\varphi$. The latter is not an ultimate framework of \eqref{subgra-inc}.
Here we develop a new approach to solvability of systems \eqref{subgra-prox} with respect to the directions $d^k$ and to local superlinear convergence of iterates $x^k\to\bar{x}$ under certain {\em metric regularity} and {\em subregularity} properties of the subdifferential mapping $\partial\varphi$. In particular, all our assumptions hold if $\partial\varphi$ is semismooth$^*$ at the reference point and {\em strongly metrically regular} around it. As shown by Drusvyatskiy and Lewis \cite{dl} for prox-regular and subdifferentially continuous functions, the latter property is equivalent to tilt stability of $\bar{x}$ required by Mordukhovich and Sarabi \cite{BorisEbrahim} {\em provided that} $\bar{x}$ is a local minimizer of $\varphi$, which is not assumed here.
The developed generalized Newton algorithm for subgradient inclusions is finally applied to solving a major class of nonsmooth {\em Lasso problems} that appear in practical models of statistics, machine learning, etc. For such problems we compute the second-order subdifferential and the proximal mapping from \eqref{subgra-prox} entirely in terms of the given data, derive explicit calculation formulas, and then provide a numerical implementation in a testing example.\vspace*{0.03in}
The rest of the paper is organized as follows. Section~\ref{prel} presents and discusses those notions of variational analysis and generalized differentiation, which are broadly used in the formulations and proofs of the main results obtained below.
Section~\ref{sec:solv} is devoted to {\em solvability} issues for the {\em generalized equations} given by
\begin{equation}\label{ge-cod}
-\bar{v}\in D^*F(\bar{x},\bar{v})(d)\;\mbox{ for }\;d\in\mathbb{R}^n,
\end{equation}
where $D^*F(\bar{x},\bar{v})(\cdot)$ is the {\em coderivative} of a set-valued mapping $F\colon\mathbb{R}^n\rightrightarrows\mathbb{R}^n$ that is associated with the limiting subdifferential $\partial\varphi$ as defined in Section~\ref{prel}. The framework of \eqref{ge-cod} encompasses all the versions \eqref{newton-iter}, \eqref{newtonC11}, and \eqref{subgra-prox} of the Newton-type algorithms discussed above. Using well-developed calculus rules of limiting generalized differentiation allows us to prove that \eqref{ge-cod} is solvable for $d$ if the mapping $F$ is {\em strongly metrically subregular} at the reference point $(\bar{x},\bar{v})$. Furthermore, the {\em strong metric regularity} of $F$, in particular, ensures the solvability of \eqref{ge-cod} and the {\em compactness} of the feasible directions therein {\em around} the point in question.
The solvability results established in Section~\ref{sec:solv} for coderivative inclusions are applied to solvability issues for generalized Newton systems of types \eqref{newtonC11} and \eqref{subgra-prox} involving the second-order subdifferential $\partial^2\varphi$. In this way we identify broad classes of functions $\varphi\colon\mathbb{R}^n\to\Bar{\mathbb{R}}$ for which the required assumptions on $F=\partial\varphi$ are satisfied.
Section~\ref{sec:newtonC11} presents a generalized Newton algorithm to solve the gradient equations \eqref{gra} with functions $\varphi$ of class ${\cal C}^{1,1}$ according to the iteration procedure \eqref{newtonC11}. The main result of this section establishes a local {\em superlinear convergence} of iterates \eqref{newtonC11} to a designated solution $\bar{x}$ of \eqref{gra} under the semismoothness$^*$ of the gradient mapping $\nabla\varphi$ at $\bar{x}$ and under merely its {\em metric regularity} around this point. Even in the case of tilt-stable local minimizers of $\varphi$, the obtained result improves the one from Mordukhovich and Sarabi \cite{BorisEbrahim}. We also compare the new algorithm with some other generalized Newton methods and show, in particular, that our algorithm is well-defined and exhibits a superlinear convergence of iterates when the semismooth Newton algorithm cannot be even constructed.
Section~\ref{sec:prox} is the culmination of the paper. It describes and justifies a new Newton-type algorithm to solve the subgradient inclusions \eqref{subgra-inc} for {\em prox-regular} and subdifferentially continuous functions $\varphi\colon\mathbb{R}^n\to\Bar{\mathbb{R}}$ that is based on the iteration procedure in \eqref{subgra-prox}. Note that such an extended-real-valued framework of $\varphi$ incorporates problems of {\em constrained optimization} for which inclusion \eqref{subgra-inc} provides a necessary condition for local minimizers. The results obtained here justifies a constructive and well-defined algorithm, with a verifiable choice of the starting point, that superlinearly converges to the solution $\bar{x}$ of \eqref{subgra-inc} under about the same assumptions as in Section~\ref{sec:newtonC11}, but being now addressed to $\partial\varphi$ instead of $\nabla\varphi$. In fact, the proofs of the main results in this section are based on the reduction to the ${\cal C}^{1,1}$ case by using Moreau envelopes and the machinery of variational analysis taken from Rockafellar and Wets \cite{Rockafellar98}.
In Section~\ref{lasso} we develop applications of the Newton-type algorithm of Section~\ref{sec:prox} to solving a major class of {\em Lasso problems} with the complete calculation of all the algorithm parameters in terms of the given problem data and presenting an example of the numerical implementation.
The concluding Section~\ref{conc} summarizes the major contributions of the paper and discusses some topics of the future research. Our notation is standard in variational analysis and optimization and can be found in the aforementioned books. Recall that $\mathbb{B}_r(\bar{x}):=\big\{x\in\mathbb{R}^n\;\big|\;\|x-\bar{x}\|\le r\big\}$ stands for the closed ball with center $\bar{x}$ and radius $r>0$.\vspace*{-0.1in}
\section{Variational Analysis: Preliminaries and Discussions}\label{prel}
Here we present the needed background material from variational analysis and generalized differentiation by following the books of Mordukhovich \cite{Mordukhovich06,Mor18} and Rockafellar and Wets \cite{Rockafellar98}.
Given a set $\Omega\subset\mathbb{R}^s$ with $\bar{z}\in\Omega$, the (Bouligand-Severi) {\em tangent/contingent cone} to $\Omega$ at $\bar{z}$ is
\begin{equation}\label{tan}
T_\Omega(\bar{z}):=\big\{w\in\mathbb{R}^s\;\big|\;\exists\,t_k\downarrow 0,\;w_k\to w\;\mbox{ as }\;k\to\infty\;\mbox{ with }\;\bar{z}+t_k w_k\in\Omega\big\}.
\end{equation}
The (Fr\'echet) {\em regular normal cone} to $\Omega$ at $\bar{z}\in\Omega$ is defined by
\begin{equation}\label{rnc}
\widehat{N}_\Omega(\bar{z}):=\Big\{v\in\mathbb{R}^s\;\Big|\;\limsup_{z\overset{\Omega}{\rightarrow}\bar{z}}\frac{\langle v, z-\bar{z}\rangle}{\|z-\bar{z}\|}\le 0\Big\},
\end{equation}
where the symbol $z\overset{\Omega}{\rightarrow}\bar{z}$ indicates that $z\to\bar{z}$ with $z\in\Omega$. It can be equivalently described via a duality correspondence with \eqref{tan} by
\begin{equation}\label{dua}
\widehat N_\Omega(\bar{z})=T^*_\Omega(\bar{z}):=\big\{v\in\mathbb{R}^s\;\big|\;\langle v,w\rangle\le 0\;\mbox{ for all }\;w\in T_\Omega(\bar{z})\big\}.
\end{equation}
The (Mordukhovich) {\em limiting normal cones} to $\Omega$ at $\bar{z}\in\Omega$ is defined by
\begin{equation}\label{lnc}
N_\Omega(\bar{z}):=\big\{v\in\mathbb{R}^s\;\big|\;\exists\,z_k\stackrel{\Omega}{\to}\bar{z},\;v_k\to v\;\text{ as }\;k\to\infty\;\text{ with }\;v_k\in\widehat{N}_\Omega(z_k)\big\}.
\end{equation}
Note that the regular normal cone \eqref{rnc} is always convex, while the limiting one \eqref{lnc} is often nonconvex (e.g., for the graph of $|x|$ at $\bar{z}=(0,0)$), and hence it cannot be obtained by the duality correspondence of type \eqref{dua} from any tangential approximation of $\Omega$ at $\bar{z}$. Nevertheless, the normal cone \eqref{lnc} as well as the coderivative and subdifferential constructions for mappings and functions generated by it (see below) enjoy comprehensive {\em calculus rules} that are based on {\em variational/extremal principles} of variational analysis.\vspace*{0.03in}
Given further a set-valued mapping $F\colon\mathbb{R}^n\rightrightarrows\mathbb{R}^m$ with the graph
\begin{equation*}
\mbox{\rm gph}\, F:=\big\{(x,y)\in\mathbb{R}^n\times\mathbb{R}^m\;\big|\;y\in F(x)\big\},
\end{equation*}
the {\em graphical derivative} of $F$ at $(\bar{x},\bar{y})\in\mbox{\rm gph}\, F$ is defined via \eqref{tan} by
\begin{equation}\label{gra-der}
DF(\bar{x},\bar{y})(u):=\big\{v\in\mathbb{R}^m\;\big|\;(u,v)\in T_{{\rm gph}\,F}(\bar{x},\bar{y})\big\},\quad u\in\mathbb{R}^n.
\end{equation}
The coderivative constructions for $F$ at $(\bar{x},\bar{y})\in\mbox{\rm gph}\, F$ are defined via the regular normal cone \eqref{rnc} and the limiting normal cone \eqref{lnc} to the graph of $F$ at this point. They are, respectively, the {\em regular coderivative} and the {\em limiting coderivative} of $F$ at $(\bar{x},\bar{y})$ given by
\begin{equation}\label{reg-cod}
\widehat D^*F(\bar{x},\bar{y})(v):=\big\{u\in\mathbb{R}^n\;\big|\;(u,-v)\in\widehat N_{{\rm gph}\,F}(\bar{x},\bar{y})\big\},\quad v\in\mathbb{R}^m,
\end{equation}
\begin{equation}\label{lim-cod}
D^*F(\bar{x},\bar{y})(v):=\big\{u\in\mathbb{R}^n\;\big|\;(u,-v)\in N_{{\rm gph}\,F}(\bar{x},\bar{y})\big\},\quad v\in\mathbb{R}^m.
\end{equation}
In the case where $F(\bar{x})=\{\bar{y}\}$, we omit $\bar{y}$ in the notation of \eqref{gra-der}--\eqref{lim-cod}. Note that if $F\colon\mathbb{R}^n\to\mathbb{R}^m$ is ${\cal C}^1$-smooth around $\bar{x}$, then
\begin{equation*}
DF(\bar{x})=\nabla F(\bar{x})\;\mbox{ and }\;\widehat{D}^*F(\bar{x})=D^*F(\bar{x})=\nabla F(\bar{x})^*,
\end{equation*}
where $\nabla F(\bar{x})^*$ is the adjoint operator of the Jacobian $\nabla F(\bar{x})$.\vspace*{0.05in}
Before considering the first-order and second-order subdifferential constructions for extended-real-valued functions, which are of our main use in this paper and are closely related to the limiting normals and coderivatives, we formulate the {\em metric regularity} and {\em subregularity} properties of set-valued mappings that are highly recognized in variational analysis and optimization. These properties are strongly employed in what follows, while being completely characterized via the limiting coderivative and graphical derivative defined above.
To proceed, recall that the {\em distance function} associated with a nonempty set $\Omega\subset\mathbb{R}^s$ is
\begin{equation*}
{\rm dist}(x;\Omega):=\inf\big\{\|w-x\|\;\big|\;w\in\Omega\big\},\quad x\in\mathbb{R}^s.
\end{equation*}
A mapping $\widehat{F}\colon\mathbb{R}^n\rightrightarrows\mathbb{R}^m$ is a {\em localization} of $F\colon\mathbb{R}^n\rightrightarrows\mathbb{R}^m$ at $\bar{x}$ for $\bar{y}\in F(\bar{x})$ if there exist neighborhoods $U$ of $\bar{x}$ and $V$ of $\bar{y}$ such that
\begin{equation*}
\mbox{\rm gph}\,\widehat{F}=\mbox{\rm gph}\, F\cap(U\times V).
\end{equation*}
\begin{Definition}[\bf metric regularity and subregularity of mappings]\label{met-reg} Let $F\colon\mathbb{R}^n\rightrightarrows\mathbb{R}^m$ be a set-valued mapping, and let $(\bar{x},\bar{y})\in\mbox{\rm gph}\, F$. We say that:\\[1ex]
{\bf(i)} $F$ is {\sc metrically regular} around $(\bar{x},\bar{y})$ with modulus $\mu>0$ if there exist
neighborhoods $U$ of $\bar{x}$ and $V$ of $\bar{y}$ such that
\begin{equation*}
{\rm dist}\big(x;F^{-1}(y)\big)\le\mu\,{\rm dist}\big(y;F(x)\big)\;\text{ for all }\;(x,y)\in U\times V,
\end{equation*}
where $F^{-1}(y):=\{x\in\mathbb{R}^n\;|\;y\in F(x)\}$ is the inverse mapping of $F$. If in addition $F^{-1}$ has a single-valued localization around $(\bar{y},\bar{x})$, then $F$ is {\sc strongly metrically regular} around $(\bar{x},\bar{y})$ with modulus $\mu>0$.\\[1ex]
{\bf(ii)} $F$ is {\sc metrically subregular} at $(\bar{x},\bar{y})$ with modulus $\mu>0$ if there exist neighborhoods $U$
of $\bar{x}$ and $V$ of $\bar{y}$ such that
\begin{equation*}
{\rm dist}\big(x;F^{-1}(\bar{y})\big)\le\mu\,{\rm dist}\big(\bar{y};F(x)\big)\;\text{ for all }\;x\in U.
\end{equation*}
If in addition we have $F^{-1}(\bar{y})\cap U=\{\bar{x}\}$, then $F$ is {\sc strongly metrically subregular} at $(\bar{x},\bar{y})$ with modulus $\mu>0$.
\end{Definition}
\medskip
The following remark summarizes relationships between the above regularity properties and presents generalized differential characterizations of the major ones used in this paper.
\begin{Remark}[\bf on metric regularity and subregularity]\label{Morcri} {\rm Observe the following:\\[1ex]
{\bf(i)} We obviously have that the strong metric regularity of a set-valued mapping implies its both metric regularity and strong metric subregularity, and also that a strongly
metrically subregular mapping is metrically subregular. However, metric regularity and strong metric subregularity are generally incomparable. For example, the mapping $F(x):=|x|$ for all $x\in\mathbb{R}$ is strongly metrically subregular at $(0,0)$, but it is not metrically regular around this point. On the contrary, the mapping $F\colon\mathbb{R}\rightrightarrows\mathbb{R}$ defined by $F(x):=[x,\infty)$ is metrically regular around $(0,0)$ while not being strongly metrically subregular at this point.\\[1ex]
{\bf(ii)} Simple examples show that a mapping may exhibit the strong metric subregularity property at some point while not being strongly metrically regular and even merely metrically regular around
the reference point. Indeed, consider the simplest nonsmooth function $F(x):=|x|$ with $(\bar{x},\bar{y})=(0,0)$ discussed in the previous part of this remark.\\[1ex]
{\bf(iii)} A major advantage of the generalized differential constructions defined above is the possibility to get in their terms complete {\em pointwise characterizations} of the
metric regularity and strong metric subregularity properties of general set-valued mappings. Namely, a mapping $F\colon\mathbb{R}^n\rightrightarrows\mathbb{R}^m$, which graph is locally closed around $(\bar{x},\bar{y})\in\mbox{\rm gph}\, F$, is {\em metrically regular} around this point if and only if we have the kernel coderivative condition
\begin{equation}\label{cod-cr}
\mbox{\rm ker}\, D^*F(\bar{x},\bar{y}):=\big\{v\in\mathbb{R}^m\;\big|\;0\in D^*F(\bar{x},\bar{y})(v)\big\}=\{0\}
\end{equation}
established by Mordukhovich \cite[Theorem~3.6]{Mordu93} via his limiting coderivative \eqref{lim-cod} and then labeled as the {\em Mordukhovich criterion} in Rockafellar and Wets \cite[Theorem~7.40, 7.43]{Rockafellar98}. Broad applications of this result are based on robustness and full calculus available for the limiting coderivative; see the aforementioned book by Rockafellar and Wets as well as the books by Mordukhovich \cite{Mordukhovich06,Mor18} with their extensive commentaries and bibliographies.
A parallel characterization of the (nonrobust) {\em strong metric subregularity} property of a (locally) closed-graph mapping $F\colon\mathbb{R}^n\rightrightarrows\mathbb{R}^m$ at $(\bar{x},\bar{y})\in\mbox{\rm gph}\, F$ is given by $\mbox{\rm ker}\, DF(\bar{x},\bar{y})=\{0\}$ via the (nonrobust) graphical derivative \eqref{gra-der} of $F$ at $(\bar{x},\bar{y})$ and is known as the {\em Levy Rockafellar criterion}. We refer the reader to the book by Dontchev and Rockafellar \cite[Theorem~4E.1]{Donchev09} with the references and discussions therein.}
\end{Remark}\vspace*{-0.05in}
Finally in this section, we recall the limiting first-order and second-order subdifferential constructions for extended-real-valued functions that are used for describing the subgradient inclusions \eqref{subgra-inc} and the Newton-type algorithms \eqref{newtonC11} and \eqref{subgra-prox} to compute their solutions. Given an extended-real-valued function $\varphi\colon\mathbb{R}^n\to\Bar{\mathbb{R}}$, consider its domain and epigraph
\begin{equation*}
\mbox{\rm dom}\,\varphi:=\big\{x\in\mathbb{R}^n\;\big|\;\varphi(x)<\infty\big\}\;\mbox{ and }\;\mbox{\rm epi}\,\varphi:=\big\{(x,\alpha)\in\mathbb{R}^{n+1}\;\big|\;\alpha\ge\varphi(x)\big\}.
\end{equation*}
Then for a fixed point $\bar{x}\in\mbox{\rm dom}\,\varphi$ we define the {\em basic/limiting subdifferential} and the {\em singular subdifferential} of $\varphi$ at $\bar{x}$ by, respectively,
\begin{equation}\label{lim-sub}
\partial\varphi(\bar{x}):=\big\{v\in\mathbb{R}^n\;\big|\;(v,-1)\in N_{{\rm\small epi}\,\varphi}\big(\bar{x},\varphi(\bar{x})\big)\big\},
\end{equation}
\begin{equation}\label{sin-sub}
\partial^\infty\varphi(\bar{x}):=\big\{v\in\mathbb{R}^n\;\big|\;(v,0)\in N_{{\rm\small epi}\,\varphi}\big(\bar{x},\varphi(\bar{x})\big)\big\}
\end{equation}
via the limiting normal cone \eqref{lnc} to the epigraph of $\varphi$ at $(\bar{x},\varphi(\bar{x}))$. For simplicity we use here the geometric definitions of the subdifferentials \eqref{lim-sub} and \eqref{sin-sub} while referring the reader to the aforementioned monographs on variational analysis for equivalent analytic representations. Recall that the basic subdifferential $\partial\varphi(\bar{x})$ reduces to the gradient $\{\nabla\varphi(\bar{x})\}$ if $\varphi$ is ${\cal C}^1$-smooth around $\bar{x}$ (or merely strictly differentiable at this point), and that $\partial\varphi(\bar{x})$ is the subdifferential of convex analysis if $\varphi$ is convex. On the other hand, the singular subdifferential $\partial^\infty\varphi(\bar{x})$ of a lower semicontinuous (l.s.c.) function $\varphi$ reduces to $\{0\}$ if and only if $\varphi$ is locally Lipschitzian around $\bar{x}$. Both constructions \eqref{lim-sub} and \eqref{sin-sub} enjoy in parallel full subgradient calculi in very general settings. Let us also mention the coderivative {\em scalarization formula}
\begin{equation}\label{scal}
D^*F(\bar{x})(v)=\partial\langle v,F\rangle(\bar{x})\;\mbox{ for all }\;v\in\mathbb{R}^m,
\end{equation}
which holds whenever $F\colon\mathbb{R}^n\to\mathbb{R}^m$ is locally Lipschitzian around $\bar{x}$.\vspace*{0.03in}
Now we are ready to define the {\em second-order subdifferential} of $\varphi\colon\mathbb{R}^n\to\Bar{\mathbb{R}}$ at $\bar{x}\in\mbox{\rm dom}\,\varphi$ for $\bar{v}\in\partial\varphi(\bar{x})$ in the sense of Mordukhovich \cite{m92} as the mapping $\partial^2\varphi(\bar{x},\bar{v})\colon\mathbb{R}^n\rightrightarrows\mathbb{R}^n$ such that
\begin{equation}\label{2nd}
\partial^2\varphi(\bar{x},\bar{v})(u):=\big(D^*\partial\varphi\big)(\bar{x},\bar{v})(u)\;\mbox{ for all }\;u\in\mathbb{R}^n,
\end{equation}
i.e., by applying the coderivative \eqref{lim-cod} to the first-order subgradient mapping \eqref{lim-sub}. This second-order construction appears in the Newton-type iterations \eqref{newtonC11} and \eqref{subgra-prox} for ${\cal C}^{1,1}$ and prox-regular functions, respectively, which both go back to the classical Newton algorithm \eqref{clas-newton} for ${\cal C}^2$-smooth functions due to the relationship
\begin{equation}\label{C2_Case}
\partial^2\varphi(\bar{x})(u)=\big\{\nabla^2\varphi(\bar{x})u\big\}\;\mbox{ whenever }\;u\in\mathbb{R}^n
\end{equation}
in the ${\cal C}^2$-smooth case. If $\varphi$ is of class ${\cal C}^{1,1}$ around $\bar{x}$, then the computation of $\partial^2\varphi(\bar{x})$ reduces to the computation of the limiting subdifferential \eqref{lim-sub} of the gradient mapping $\nabla\varphi$ by the scalarization formula \eqref{scal}. Besides well-developed second-order calculus for \eqref{2nd}, variational analysis achieves constructive computations of the second-order subdifferential, entirely in terms of the given data, for major classes if nonsmooth functions arising in important problems of constrained optimization, bilevel programming, optimal control, operations research, mechanics, economics, statistics, machine learning, etc. Among many other publications, we refer the reader to Colombo et al. \cite{chhm}, Ding et al. \cite{dsy}, Dontchev and Rockafellar \cite{dr}, Henrion et al. \cite{hmn,hos}, Mordukhovich \cite{Mordukhovich06,Mor18}, Mordukhovich and Outrata \cite{BorisOutrata}, Mordukhovich and Rockafellar \cite{mr}, Outrata and Sun \cite{os}, Yao and Yen \cite{yy}, and the bibliographies therein. The new computation of this type is provided in Section~\ref{lasso} for the Lasso problem.\vspace*{-0.1in}
\section{Solvability of Coderivative Inclusions}\label{sec:solv}\vspace*{-0.05in}
A crucial step in the design and justification of numerical algorithms is to establish their {\em well-posedness}, i.e., the {\em solvability} of the corresponding iterative systems. In the case of the classical Newton method to solve $\nabla\varphi(x)=0$, we have the equation for $d\in\mathbb{R}^n$ written as
\begin{equation}\label{newton-eq}
-\nabla\varphi(x)=\nabla^2\varphi(x)d,
\end{equation}
which is solvable if the Hessian matrix $\nabla^2\varphi(x)$ is invertible. In the case of the generalized Newton algorithm for nonsmooth functions discussed in Section~\ref{intro}, we extend \eqref{newton-eq} in the following way. Given a subgradient $v\in\partial\varphi(x)$, consider the inclusion
\begin{equation}\label{newton-inc}
-v\in\partial^2\varphi(x,v)(d)
\end{equation}
and find conditions ensuring the solvability of \eqref{newton-inc} with respect to $d\in\mathbb{R}^n$. Due to \eqref{2nd}, inclusion \eqref{newton-inc} can be written as
\begin{equation}\label{cod-inc}
-v\in D^*F(x,v)(d)
\end{equation}
with $F:=\partial\varphi$, where $D^*$ stands for the limiting coderivative \eqref{lim-cod}.\vspace*{0.05in}
The main goal of this section is to investigate the {\em solvability} of the {\em coderivative inclusion} \eqref{cod-inc} with respect to the direction $d\in\mathbb{R}^n$. In the next section we proceed with the study of solvability of the generalized Newton systems \eqref{newton-inc} and establish appropriate conditions on functions $\varphi$, which allow us to efficiently apply the solvability results obtained for \eqref{cod-inc} to the case of systems \eqref{newton-inc} of our main interest.\vspace*{0.03in}
The first theorem here verifies the solvability of \eqref{cod-inc} for $d$ at any point $(\bar{x},\bar{v})\in\mbox{\rm gph}\, F$ where the mapping $F$ is {\em strongly metrically subregular}. The proof of this result is based on major calculus rules for the limiting generalized differential constructions.
\begin{Theorem}[\bf solvability of coderivative inclusions]\label{mainre} Let $F\colon\mathbb{R}^n\rightrightarrows\mathbb{R}^n$ be a set-valued mapping which graph is closed around a point $(\bar{x},\bar{v})\in\mbox{\rm gph}\, F$. If $F$ is strongly metrically subregular at $(\bar{x},\bar{v})$, then there exists $d\in\mathbb{R}^n$ satisfying the inclusion
\begin{equation}\label{cod-inc1}
-\bar{v}\in D^*F(\bar{x},\bar{v})(d).
\end{equation}
\end{Theorem}
\begin{proof} It follows from Theorem~3I.3 in Dontchev and Rockafellar \cite{Donchev09} that the assumed strong metric subregularity of $F$ around $(\bar{x},\bar{v})$ is equivalent to the existence of neighborhoods $U$ of $\bar{x}$ and $V$ of $\bar{v}$ together with a constant $\ell>0$ such that $F^{-1}(\bar{v})\cap U=\{\bar{x}\}$ and
\begin{equation}\label{isolate}
\|x-\bar{x}\|\le\ell\|v-\bar{v}\|\;\mbox{ for all }\;x\in F^{-1}(v)\cap U\;\mbox{ and }\;v\in V.
\end{equation}
Passing to appropriate subsets, we suppose for convenience that the sets $U$ and $V$ are closed and bounded. Consider further the set-valued mapping $G\colon\mathbb{R}^n\rightrightarrows\mathbb{R}^n$ given by
\begin{equation*}
G(v):=F^{-1}(v)\cap U\;\mbox{ for all }\;v\in\mathbb{R}^n
\end{equation*}
and then define the {\em marginal/optimal value function} $\mu\colon\mathbb{R}^n\to\Bar{\mathbb{R}}$ and the corresponding argminimum mapping $M\colon\mathbb{R}^n\rightrightarrows\mathbb{R}^n$ by, respectively,
\begin{equation}\label{marg}
\mu(v):=\inf\big\{\varphi(v,x)\;\big|\;x\in G(v)\big\}\;\mbox{ and }\;M(v):= \big\{x\in G(v)\;\big|\;\varphi(v,x)=\mu(v)\big\},\quad v\in\mathbb{R}^n,
\end{equation}
where $\varphi(v,x):=\langle\bar{v},x\rangle$ for all $(v,x)\in\mathbb{R}^n\times\mathbb{R}^n$. It is clear that the set $\mbox{\rm gph}\, G$ is locally closed around $(\bar{v},\bar{x})$ and that $(\bar{v},\bar{x})\in\mbox{\rm gph}\, M$ and that the marginal function in \eqref{marg} is l.s.c.\ around $\bar{v}$. To show next that the mapping $M$ is locally bounded around $\bar{v}$, observe that for each $v\in V$ we have the relationships
\begin{equation*}
M(v)\subset G(v)= F^{-1}(v)\cap U\subset U,
\end{equation*}
which yield that the image set $M(V)$ is bounded, i.e., the mapping $M$ is locally bounded around $\bar{v}$. To evaluate now the limiting subdifferential \eqref{lim-sub} of the marginal function \eqref{marg} by Theorem~4.1(ii) from Mordukhovich \cite{Mor18}, it remains to check the qualification condition
\begin{equation*}
\partial^\infty\varphi(\bar{v},\bar{x})\cap\big(-N_{\text{gph}\,G}(\bar{v},\bar{x})\big)=\{0\}
\end{equation*}
therein, which automatically holds due to the Lipschitz continuity of the function $\varphi$ around $(\bar{x},\bar{v})$. Since we have in \eqref{marg} that $M(\bar{v})=\{\bar{x}\}$, $\nabla_v\varphi(\bar{v},\bar{x})=0$, and $\nabla_x\varphi(\bar{v},\bar{x})=\bar{v}$, the aforementioned results from Mordukhovich \cite[Theorem 4.1(ii)]{Mor18} tells us that
\begin{equation}\label{inc}
\partial\mu(\bar{v})\subset\nabla_v\varphi(\bar{v},\bar{x})+D^* G(\bar{v},\bar{x})\big(\nabla_x\varphi(\bar{v},\bar{x})\big)=D^*G(\bar{v},\bar{x})(\bar{v}).
\end{equation}
To proceed further, consider the mapping $H(v)\equiv U$ on $\mathbb{R}^n$ and observe that $N_{\text{gph}\,H}(\bar{v},\bar{x})=\{0\}$ by $(\bar{v},\bar{x})\in\text{int}(\text{gph}\,H)$. Thus $N_{\text{gph}H}(\bar{v},\bar{x})=\{0\}$ and the qualification condition
\begin{equation*}
N_{\text{gph}\,H}(\bar{v},\bar{x})\cap\big(-N_{\text{gph}\,F^{-1}}(\bar{v},\bar{x})\big)=\{0\}
\end{equation*}
is satisfied. This allows us to apply the normal cone intersection rule from Mordukhovich \cite[Theorem~2.16]{Mor18} and get the relationships
\begin{equation*}
N_{\text{gph}\,G}(\bar{v},\bar{x})\subset N_{\text{gph}\,F^{-1}}(\bar{v},\bar{x})+N_{\text{gph}\,H}(\bar{v},\bar{x})=N_{\text{gph}\,F^{-1}}(\bar{v},\bar{x}).
\end{equation*}
Combining the latter with \eqref{inc} gives us the inclusions
\begin{equation}\label{mu-inc}
\partial\mu(\bar{v})\subset D^*G(\bar{v},\bar{x})(\bar{v})\subset D^*F^{-1}(\bar{v},\bar{x})(\bar{v}).
\end{equation}
Let us now show that $\partial\mu(\bar{v})\ne\emptyset$. Recall from Mohammadi et al. \cite[Proposition~2.1]{mms} that if a function $\psi\colon\mathbb{R}^n\to\Bar{\mathbb{R}}$ is l.s.c.\ at $\bar{v}\in\mbox{\rm dom}\,\psi$ and satisfies the ``lower calmness" property
\begin{equation}\label{calm}
\psi(v)\ge\psi(\bar{v})-\eta\|v-\bar{v}\|\;\mbox{ for all }\;v\in V
\end{equation}
with some $\eta\ge 0$ and a neighborhood $V$ of $\bar{v}$, then $\partial\psi(\bar{v})\ne\emptyset$. To establish \eqref{calm} for $\psi:=\mu$, we verify in what follows the fulfillment of the estimate
\begin{equation}\label{calmness}
\mu(v)-\mu(\bar{v})\ge-\ell\|\bar{v}\|\cdot\|v-\bar{v}\|\;\mbox{ whenever }\;v\in V.
\end{equation}
Observe first that if $G(v)=\emptyset$ for some $v\in V$, then $\mu(v)=\infty$, and therefore estimate \eqref{calmness} is obviously satisfied. In the remaining case where $G(v)\ne\emptyset$ for a fixed vector $v\in V$, pick any $x\in G(v)$ and deduce from \eqref{isolate} that
\begin{equation*}
\langle\bar{v},x\rangle-\langle\bar{v},\bar{x}\rangle\ge-\|\bar{v}\|\cdot\|x-\bar{x}\|\ge-\ell\|\bar{v}\|\cdot\|v -\bar{v}\|.
\end{equation*}
Indeed, the condition $M(\bar{v})=\{\bar{x}\}$ clearly yields \eqref{calmness}, and hence we have $\partial\mu(\bar{v})\ne\emptyset$. It justifies by \eqref{mu-inc} the existence of $u\in\mathbb{R}^n$ satisfying $u\in D^*F^{-1} (\bar{v},\bar{x})(\bar{v})$. To complete the proof of the theorem, recall the well-known coderivative relationship
\begin{equation}\label{inverse}
z\in D^*F(\bar{x},\bar{y})(u)\Longleftrightarrow-u\in\big(D^*F^{-1}\big)(\bar{y},\bar{x})(-z),
\end{equation}
which readily ensures the fulfillment of \eqref{cod-inc1} and thus finishes the proof.
\end{proof}
The following example shows that the strong metric subregularity assumption of Theorem~\ref{mainre} cannot be replaced by metric subregularity of $F$ at $(\bar{x},\bar{v})$, or even by metric regularity of $F$ around this point in order to guarantee the solvability of the coderivative inclusion \eqref{cod-inc1}.
\begin{Example}[\bf insolvability of coderivative inclusions under metric regularity]\label{insolv} {\rm Consider the set-valued mapping $F\colon\mathbb{R}\rightrightarrows\mathbb{R}$ defined by
\begin{equation}\label{ex1}
F(x):=[0,1]\;\mbox{ for all }\;x\in\mathbb{R}.
\end{equation}
Then the graphical set $\mbox{\rm gph}\, F=\mathbb{R}\times[0,1]$ is convex, and we easily calculate the limiting normal cone to the graph of \eqref{ex1} as follows:
\begin{equation*}
N_{\text{gph}\,F}(x,y)=\begin{cases}
\{(0,0)\}&\text{if}\qquad y\in(0,1),\\
\{0\}\times\mathbb{R}_{+}&\text{if}\qquad y=1,\\
\{0\}\times\mathbb{R}_{-}&\text{if}\qquad y=0.\\
\end{cases}
\end{equation*}
Pick $(\bar{x},\bar{v}):=\left(1,\frac{1}{2}\right)\in\mbox{\rm gph}\, F$ and show that $F$ is metrically regular around $(\bar{x},\bar{v})$. Indeed, taking $u\in\mathbb{R}^n$ with $0\in D^*F(\bar{x},\bar{v})(u)$, we have
\begin{equation*}
(0,-u)\in N_{\text{gph}\,F}(\bar{x},\bar{v})=\{(0,0)\},
\end{equation*}
and hence $u=0$. It follows from the Mordukhovich criterion \eqref{cod-cr} that $F$ is metrically regular around $(\bar{x},\bar{v})$. However, it is easy to see that there exists no $d\in\mathbb{R}$ which solves \eqref{cod-inc1}.}
\end{Example}\vspace*{0.05in}
The next example shows that strong metric subregularity, being a nonrobust property, does not ensure the {\em robust solvability} of the coderivative inclusion \eqref{cod-inc}\, i.e., its solvability in a neighborhood of the reference point. Given $(x,v)\in\mbox{\rm gph}\, F$, consider the set of feasible solutions of \eqref{cod-inc} at this point that defined by
\begin{equation}\label{newton-dir}
\Gamma_F(x,v):=\big\{d\in\mathbb{R}^n\;\big|\;-v\in D^*F(x,v)(d)\big\}.
\end{equation}
\begin{Example}[\bf failure of robust solvability under strong metric subregularity]\label{ex-nonrob} {\rm Consider the set-valued mapping $F\colon\mathbb{R}\rightrightarrows\mathbb{R}$ defined by
\begin{equation}\label{ex2}
F(x):=\begin{cases}
\{0\}\cup[1,\infty)&\text{if}\quad x=0,\\
[1,\infty)&\text{if}\quad x\ne 0.
\end{cases}
\end{equation}
It is clear that $F$ is strongly metrically subregular at $(0,0)$ and the graph of $F$ is closed around this point. Let us show that for any neighborhood $U\times V$ of the origin in $\mathbb{R}^2$ and nonzero pair $(x,v)\in\mbox{\rm gph}\, F\cap(U\times V)$ there exists no $d\in\mathbb{R}$ satisfying the coderivative inclusion \eqref{cod-inc}. To proceed, observe that for $\mbox{\rm gph}\, F=(0,0)cup\mathbb{R}\times[1,\infty)$. Then we get by the direct computation that the normal cone to the graph of $F$ at $(x,v)\in\mbox{\rm gph}\, F$ is
\begin{equation*}
N_{\text{gph}\,F}(x,v)=\begin{cases}
\mathbb{R}^2&\text{if}\quad x =0,\;v =0,\\
\big\{(0,\lambda)\;\big|\;\lambda\le 0\big\}&\text{if}\quad v\ge 1.
\end{cases}
\end{equation*}
Hence we have the following formula for the limiting coderivative of the mapping $F$ from \eqref{ex2} at any $(x,v)\in\mbox{\rm gph}\, F$ in a neighborhood of $(0,0)$:
\begin{equation*}
D^*F(x,v)(u)=\begin{cases}
\mathbb{R}&\text{if}\quad (x,v)=(0,0),\\
\{0\}&\text{if}\quad(x,v)\ne(0,0),\;u\ge 0,\\
\emptyset&\text{otherwise}.
\end{cases}
\end{equation*}
It easily follows from this formula that there exists no $d\in\mathbb{R}$ satisfying \eqref{cod-inc} for any $(x,v)\in\mbox{\rm gph}\, F$ except $(x,v)=(0,0)$. Observe also that the set $\Gamma_F(0,0)=\mathbb{R}$ from \eqref{newton-dir} is not bounded.}
\end{Example}
Now we show that the replacement of the strong metric subregularity of $F$ at $(\bar{x},\bar{v})$ in Theorem~\ref{mainre} by its {\em robust} counterpart, which is the {\em strong metric regularity} of $F$ around $(\bar{x},\bar{v})$, leads us to robust solvability of the coderivative inclusion \eqref{cod-inc} and thus ensures the well-posedness of generalized Newton iterations.
\begin{Theorem}[\bf robust solvability of coderivative inclusions]\label{strongsol} Let $F\colon\mathbb{R}^n\rightrightarrows\mathbb{R}^n$ be a set-valued mapping which graph is closed around $(\bar{x},\bar{v})\in\mbox{\rm gph}\, F$. If $F$ is strongly metrically regular around this point, then there is a neighborhood $U\times V$ of $(\bar{x},\bar{v})$ such that for each $(x,v)\in\mbox{\rm gph}\, F\cap(U\times V)$ there exists a direction $d\in\mathbb{R}^n$ satisfying the coderivative inclusion \eqref{cod-inc}. Moreover, the set-valued mapping $\Gamma_F$ from \eqref{newton-dir} is compact-valued for all $(x,v)\in\mbox{\rm gph}\, F\cap(U\times V)$.
\end{Theorem}
\begin{proof} Since $F$ is strongly metrically regular around $(\bar{x},\bar{v})$, it follows from Definition~\ref{met-reg}(i) that the inverse mapping $F^{-1}\colon\mathbb{R}^n\rightrightarrows\mathbb{R}^n$ admits a single-valued localization $\vartheta\colon V\to U$ around $(\bar{v},\bar{x})$, which is locally Lipschitzian around $\bar{v}$. This implies, together with the scalarization formula \eqref{scal}, that for each $(x,v)\in\mbox{\rm gph}\, F\cap(U\times V)$ we have the representations
\begin{equation}\label{scar1}
D^*F^{-1}(v,x)(v)=D^*\vartheta(v)(v)=\partial\langle v,\vartheta\rangle(v).
\end{equation}
It is well known (see, e.g., Mordukhovich \cite[Theorem~1.22]{Mor18}) that the limiting subgradient set of a locally Lipschitzian function is nonempty and compact. Hence it follows from \eqref{scar1} that the set $D^*F^{-1}(v,x)(-v)$ is nonempty and compact in $\mathbb{R}^n$. Taking any $u\in D^*F^{-1}(v,x)(-v)$, we deduce from \eqref{inverse} that $-v\in D^*F(x,v)(d)$ with $d:=-u$, which verifies the robust solvability of the coderivative inclusion \eqref{cod-inc} around $(\bar{x},\bar{v})$. The claimed compactness of \eqref{newton-dir} follows from the compactness of $\partial\langle v,\vartheta\rangle(v)$. This completes the proof of the theorem.
\end{proof}
Note that the strong metric regularity assumption of Theorem~\ref{strongsol} is {\em just a sufficient condition} for robust solvability of the coderivative inclusion \eqref{cod-inc} and its second-order subdifferential specifications studied in the next section. As we see below, the required robust solvability is exhibited even in the case of gradient systems \eqref{newton-iter} generated by semismooth functions $\varphi$ without the strong metric regularity assumption on $\nabla\varphi$.\vspace*{-0.1in}
\section{Solvability of Generalized Newton Systems}\label{sec:solvN}\vspace*{-0.05in}
In this section we consider the second-order subdifferential inclusions \eqref{newton-inc} generated by extended-real-valued functions $\varphi\colon\mathbb{R}^n\to\Bar{\mathbb{R}}$. Such systems appear in the novel generalized Newton algorithms, which were discussed in Section~\ref{intro} and will be fully developed in the subsequent sections. As mentioned, the second-order subdifferential systems \eqref{newton-inc} are specifications of the coderivative ones \eqref{cod-inc} for $F:=\partial\varphi$, while the subdifferential structure of $F$ creates strong opportunities to efficiently implement and improve the general assumptions of Theorems~\ref{mainre} and \ref{strongsol} for important classes of functions $\varphi$ that are overwhelmingly encountered in the broad territories of finite-dimensional variational analysis and optimization.\vspace*{0.03in}
There are two groups of assumptions in both Theorems~\ref{mainre} and \ref{strongsol}: one on the closed graph of $F$, and the other on the strong metric subregularity and regularity properties. Let us start with the first one: {\em when is the limiting subgradient mapping $\partial\varphi$ of $($locally$)$ closed graph}?\vspace*{0.03in}
It is well known and easily follows from definitions \eqref{lnc} and \eqref{lim-sub} that the limiting subgradient mapping $\partial\varphi\colon\mathbb{R}^n\rightrightarrows\mathbb{R}^n$ is closed-graph around $(\bar{x},\bar{v})\in\mbox{\rm gph}\,\partial\varphi$ if $\varphi$ is {\em continuous} around $\bar{x}$. However, this important and broad setting does not encompass functions that are locally extended-real-valued around $\bar{x}$, while such functions are the most interesting for applications, e.g., to constrained optimization. This is the reason for the following definition taken from Rockafellar and Wets \cite[Definition~13.28]{Rockafellar98}.
\begin{Definition}[\bf subdifferentially continuous functions]\label{sub-cont} A function $\varphi\colon\mathbb{R}^n\to\Bar{\mathbb{R}}$ is {\sc subdifferentially continuous} at $\bar{x}\in\mbox{\rm dom}\,\varphi$ for $\bar{v}\in\partial\varphi(\bar{x})$ if for any sequence $(x_k,v_k)\to(\bar{x},\bar{v})$ with $v_k\in\partial\varphi(x_k)$ we have $\varphi(x_k)\to\varphi(\bar{x})$ as $k\to\infty$. If this holds for all $\bar{v}\in\partial\varphi(\bar{x})$, then $\varphi$ is called to be subdifferentially continuous at $\bar{x}$.
\end{Definition}
Note that $\varphi$ is obviously subdifferentially continuous at any point $\bar{x}\in\mbox{\rm dom}\,\varphi$ where $\varphi$ is continuous merely relative to its domain. Furthermore, it easily follows from the subdifferential construction of convex analysis that any convex extended-real-valued function is subdifferentially continuous at every $\bar{x}\in\mbox{\rm dom}\,\varphi$. As has been realized in variational analysis, the class of subdifferentially continuous functions is much broader and includes, in particular, {\em strongly amenable} functions, {\em lower-${\cal C}^2$} functions, etc.; see Rockafellar and Wets \cite[Chapter~13]{Rockafellar98}.\vspace*{0.05in}
The next theorem on solvability and robust solvability of generalized Newton systems is a direct consequence of Theorems~\ref{mainre}, \ref{strongsol} and Definition~\ref{sub-cont}.
\begin{Theorem}[\bf solvability and robust solvability of generalized Newton systems]\label{solvability} Let $\varphi\colon\mathbb{R}^n\to\Bar{\mathbb{R}}$ be subdifferentially continuous around some $\bar{x}\in\mbox{\rm dom}\,\varphi$. Then the following hold:\\[1ex]
{\bf(i)} Given $\bar{v}\in\partial\varphi(\bar{x})$, assume that the subgradient mapping $\partial\varphi\colon\mathbb{R}^n\rightrightarrows\mathbb{R}^n$ is strongly metrically subregular at $(\bar{x},\bar{v})$. Then there exists $d\in\mathbb{R}^n$ satisfying the second-order subdifferential inclusion \eqref{newton-inc} for $(x,v):=(\bar{x},\bar{v})$.\\[1ex]
{\bf(ii)} Given $\bar{v}\in\partial\varphi(\bar{x})$, assume that the subgradient mapping $\partial\varphi$ is strongly metrically regular around $(\bar{x},\bar{v})$. Then there is a neighborhood $U\times V$ of $(\bar{x},\bar{v})$ such that for each $(x,v)\in\mbox{\rm gph}\,\partial\varphi\cap(U\times V)$ there exists a direction $d\in\mathbb{R}^n$ satisfying the second-order subdifferential inclusion \eqref{newton-inc}. Moreover, the set-valued mapping $\Gamma_{\partial\varphi}$ from \eqref{newton-dir} is compact-valued for all $(x,v)\in\mbox{\rm gph}\,\partial\varphi\cap(U\times V)$.
\end{Theorem}
\begin{proof} It is easy to check that the imposed subdifferential continuity assumption on $\varphi$ ensures that the graph of $\partial\varphi$ is locally closed around $(\bar{x},\bar{v})\in\mbox{\rm gph}\,\partial\varphi$. Then the claimed assertions (i) and (ii) follow from Theorem~\ref{mainre} and Theorem~\ref{strongsol}, respectively.
\end{proof}
Observe that Example~\ref{insolv}, which shows that the strong metric subregularity assumption on the mapping $F$ in \eqref{cod-inc} cannot be replaced by the metric regularity and hence by the metric subregularity ones in the conclusion of Theorem~\ref{mainre}, still works for Theorem~\ref{solvability}(i) dealing with mappings $F=\partial\varphi$ of the subdifferential type. Indeed, it is shown by Wang \cite[Theorem~4.6]{wang} that there exists a Lipschitz continuous function $\varphi\colon\mathbb{R}\to\mathbb{R}$ such that $\partial\varphi(x)=[0,1]$ for all $x\in\mathbb{R}$. Thus we get from Example~\ref{insolv} that the subgradient mapping $\partial\varphi$ for this function is metrically regular around the point $(0,1/2)\in\mbox{\rm gph}\,\partial\varphi$, while the second-order subdifferential inclusion \eqref{newton-inc} is not solvable for $d$ at this point.\vspace*{0.05in}
Let us further reveal the class of functions where the {\em metric regularity} of $\partial\varphi$ can replace the strong metric subregularity assumption in the solvability result of Theorem~\ref{solvability}(i). This issue is very appealing since metric regularity is a robust property, which is fully characterized---via the Mordukhovich criterion \eqref{cod-cr}---by the robust limiting coderivative (and hence by the second-order subdifferential \eqref{2nd} for the subgradient systems \eqref{subgra-inc}) enjoying comprehensive calculus rules and computation formulas discussed above. To proceed in this direction, we significantly use the subdifferential structure of \eqref{subgra-inc} with $F=\partial\varphi$ in \eqref{cod-inc}. The following class of functions was introduced by Poliquin and Rockafellar \cite{Poliquin}.
\begin{Definition}[\bf prox-regular functions]\label{prox-reg} A function $\varphi\colon\mathbb{R}^n\to\Bar{\mathbb{R}}$ is {\sc prox-regular} at a point $x\in\mbox{\rm dom}\,\varphi$ for a subgradient $\bar{v}\in\partial\varphi(\bar{x})$ if $\varphi$ is l.s.c.\ around $\bar{x}$ and there exist $\varepsilon>0$ and $r>0$ such that for all $x,u\in\mathbb{B}_\varepsilon(\bar{x})$ with $|\varphi(u)-\varphi(\bar{x})|<\varepsilon$ we have
\begin{equation}\label{prox}
\varphi(x)\ge\varphi(u)+\langle v,x-u\rangle-\frac{r}{2}\|x-u\|^2\;\mbox{ whenever }\;v\in\partial\varphi(u)\cap\mathbb{B}_\varepsilon(\bar{v}).
\end{equation}
If this holds for all $\bar{v}\in\partial\varphi(\bar{x})$, $\varphi$ is said to be prox-regular at $\bar{x}\in\mbox{\rm dom}\,\varphi$.
\end{Definition}
In follows we say that $\varphi$ is {\em continuously prox-regular} at $\bar{x}$ for $\bar{v}$ (and just at $\bar{x})$ if it is simultaneously prox-regular and subdifferentially continuous according to Definitions~\ref{sub-cont} and \ref{prox-reg}. It is easy to see that that if $\varphi$ is continuously prox-regular at $\bar{x}$ for $\bar{v}\in\partial\varphi(\bar{x})$, then the condition $|\varphi(u)-\varphi(\bar{x})|<\varepsilon$ the definition of prox-regularity above could be omitted. Furthermore, in this case the subdifferential graph $\mbox{\rm gph}\,\varphi$ is locally closed around $(\bar{x},\bar{v})$.
As discussed in the book by Rockafellar and Wets \cite{Rockafellar98}, the class of continuously prox-regular functions is fairly broad containing, besides $\mathcal{C}^2$-smooth functions, functions of class $\mathcal{C}^{1,1}$, convex l.s.c.\ functions, lower-$\mathcal{C}^2$ functions, strongly amenable functions, etc. This class plays a central role in second-order variational analysis and its applications; see the books by Rockafellar and Wets \cite{Rockafellar98} and by Mordukhovich \cite{Mor18} with the commentaries and references therein.\vspace*{0.03in}
To establish the desired solvability theorem for the second-order subdifferential inclusions \eqref{newton-inc} with continuously prox-regular functions $\varphi$, we need to recall yet another notion of generalized second-order differentiability taken from Rockafellar and Wets \cite[Chapter~13]{Rockafellar98}.
Given $\varphi\colon\mathbb{R}^n\to\Bar{\mathbb{R}}$ with $\bar{x}\in\mbox{\rm dom}\,\varphi$, consider the family of second-order finite differences
\begin{equation*}
\Delta^2_\tau\varphi(\bar{x},v)(u):=\frac{\varphi(\bar{x}+\tau u)-\varphi(\bar{x})-\tau\langle v,u\rangle}{\frac{1}{2}\tau^2}
\end{equation*}
and define the {\em second subderivative} of $\varphi$ at $\bar{x}$ for $v\in\mathbb{R}^n$ and $w\in\mathbb{R}^n$ by
\begin{equation*}
d^2\varphi(\bar{x},v)(w):=\liminf_{\tau\downarrow 0\atop u\to w}\Delta^2_\tau\varphi(\bar{x},v)(u),
\end{equation*}
Then $\varphi$ is said to be {\em twice epi-differentiable} at $\bar{x}$ for $v$ if for every $w\in\mathbb{R}^n$ and every choice $\tau_k\downarrow 0$ there exists a sequence $w^k\to w$ such that
\begin{equation*}
\frac{\varphi(\bar{x}+\tau_k w^k)-\varphi(\bar{x})-\tau_k\langle v,w^k\rangle}{\frac{1}{2}\tau_k^2}\to d^2\varphi(\bar{x},v)(w)\;\mbox{ as }\;k\to\infty.
\end{equation*}
Twice epi-differentiability has been recognized as an important property in second-order variational analysis with numerous applications to optimization; see the aforemention monograph by Rockafellar and Wets and the recent papers by Mohammadi et al. \cite{mms,mms1,ms}. In particular, the latter papers develop a systematic approach to verify epi-differentiability via {\em parabolic regularity}, which is a major second-order property of extended-real-valued functions that goes far beyond the class of {\em fully amenable} functions investigated in Rockafellar and Wets \cite{Rockafellar98}.\vspace*{0.05in}
Now we are ready to establish solvability of the second-order subdifferential inclusion \eqref{newton-inc} under merely metric regularity of the limiting subgradient mappings $\partial\varphi$ for the class of continuously prox-regular and twice epi-differentiable functions.
\begin{Theorem}{\bf(solvability of generalized Newton systems under metric regularity).}\label{twiceepi} Let $\varphi\colon\mathbb{R}^n\to\Bar{\mathbb{R}}$ be continuously subdifferentially prox-regular continuous at $\bar{x}$ for some $\bar{v}\in\partial\varphi(\bar{x})$. Suppose in addition that the subgradient mapping $\partial\varphi$ is metrically regular around $(\bar{x},\bar{v})$ and that one of two following properties holds:\\[1ex]
{\bf(i)} $\varphi$ is a univariate function, i.e., $n=1$.\\[1ex]
{\bf(ii)} $\varphi$ is twice epi-differentiable at $\bar{x}$ for $\bar{v}$.\\[1ex]
Then there exists $d\in\mathbb{R}^n$ satisfying the second-order subdifferential system \eqref{newton-inc} at $(\bar{x},\bar{v})$.
\end{Theorem}
\begin{proof} As mentioned above, the graph of the subgradient mapping $\partial\varphi$ is locally closed around $(\bar{x},\bar{v})$. Let us show now that the imposed assumptions ensure that the mapping $\partial\varphi$ is strongly metrically subregular at $(\bar{x},\bar{v})$. We are going to employ the Levy-Rockafellar criterion for the latter property telling us that it is equivalent to the condition
\begin{equation}\label{lev-rock}
0\in\big(D\partial\varphi\big)(\bar{x},\bar{v})(u)\Longrightarrow u=0
\end{equation}
discussed in Remark~\ref{Morcri}(iii). To verify \eqref{lev-rock}, pick any $u\in\mathbb{R}^n$ such that $0\in(D\partial\varphi)(\bar{x},\bar{v})(u)$. Then the continuous prox-regularity of $\varphi$ and each of the imposed assumptions (i) and (ii) ensure the following relationship between the graphical derivative and limiting coderivative of $\partial\varphi$:
\begin{equation}\label{der-cod}
\big(D\partial\varphi\big)(\bar{x},\bar{v})(u)\subset\big(D^*\partial\varphi\big)(\bar{x},\bar{v})(u)\;\mbox{ for all }\;u\in\mathbb{R}^n.
\end{equation}
In the univariate case (i) inclusion \eqref{der-cod} was proved by Rockafellar and Zagrodny \cite[Theorem~4.1]{rock-zag}, while the twice epi-differentiable case (ii) was done in the equivalent form in Theorem~1.1 of the latter paper; see also Rockafellar and Wets \cite[Theorem~13.57]{Rockafellar98}. Since the subdifferential mapping $\partial\varphi$ is assumed to be metrically regular at $(\bar{x},\bar{v})$, we get by using \eqref{der-cod} and the Mordukhovich criterion \eqref{cod-cr} that
\begin{equation*}
0\in\big(D\partial\varphi\big)(\bar{x},\bar{v})(u)\Longrightarrow 0\in\big(D^*\partial\varphi\big)(\bar{x},\bar{v})(u)\Longrightarrow u=0,
\end{equation*}
which ensures by \eqref{lev-rock} that $\partial\varphi$ is strongly metrically subregular at $(\bar{x},\bar{v})$. Using finally the result of Theorem~\ref{solvability}(i), we arrive at the claimed solvability and thus complete the proof.
\end{proof}
Note that Theorem~\ref{twiceepi} concerns solvability of the second-order subdifferential inclusion \eqref{newton-inc} {\em at} the chosen point $(\bar{x},\bar{v})\in\mbox{\rm gph}\,\partial\varphi$. What about {\em robust} solvability of \eqref{newton-inc} {\em around} the reference point in the line of Theorem~\ref{solvability}(ii)? This is discussed in the following remark.
\begin{Remark}[\bf robust solvability under metric regularity]\label{robust-newton} {\rm Theorem~\ref{solvability}(ii) tells us that the {\em strong} metric regularity of $\partial\varphi$ around $(\bar{x},\bar{v})$ ensures the robust solvability of \eqref{newton-inc} around this point. But it has been recognized that the strong metric regularity of subgradient mappings $\partial\varphi$ is {\em equivalent} to merely the metric regularity of them for major subclasses of continuously prox-regular functions $\varphi\colon\mathbb{R}^n\to\Bar{\mathbb{R}}$ with the {\em conjecture} that it holds for the {\em entire class} of such functions at {\em local minimizers} of $\varphi$; see Drusvyatskiy et at. \cite[Conjecture~4.7]{dmn}. This is largely discussed in the mentioned paper by Drusvyatskiy et el., and now we recall some results from that paper. Indeed, the equivalence clearly holds (and not only for local minimizers of $\varphi$) for ${\cal C}^2$-smooth functions and for l.s.c.\ convex functions due the fundamental Kenderov theorem on maximal monotone operators \cite{kender}. The claimed equivalence is also valid for a broad class of functions given by $\varphi(x)=\varphi_0(x)+\delta_\Omega(x)$, where $\varphi_0$ is a ${\cal C}^2$-smooth function, and where $\delta_\Omega$ is the indicator function of a polyhedral convex set; see Dontchev and Rockafellar \cite{dr}. Yet another large setting of such an equivalence is revealed in Drusvyatskiy et al. \cite[Theorem~4.13]{dmn} for continuously prox-regular functions $\varphi$ with $0\in\partial\varphi(\bar{x})$ under the additional condition that the second-order subdifferential $\partial^2\varphi(\bar{x},0)$ is {\em positive-semidefinite} in the sense that
\begin{equation*}
\langle v,u\rangle\ge 0\;\mbox{ for all }\;v\in\partial^2\varphi(\bar{x},0)(u),\quad u\ne 0.
\end{equation*}
Note that the requirement that $\bar{x}$ is a local minimizer of $\varphi$ is {\em essential} for the validity of this conjecture even for twice epi-differentiable functions of class ${\cal C}^{1,1}$ with piecewise linear and directionally differentiable gradients; see Example~\ref{ex-kummer} in the next section.}
\end{Remark}\vspace*{-0.2in}
\section{Generalized Newton Method for ${\cal C}^{1,1}$ Gradient Equations}\label{sec:newtonC11}\vspace*{-0.05in}
In this section we propose and justify a generalized Newton algorithm to solve gradient systems of type \eqref{gra}, where $\varphi\colon\mathbb{R}^n\to\mathbb{R}$ is a function of class ${\cal C}^{1,1}$ around a given point $\bar{x}$. To begin with, let us formulate the {\em semismooth$^*$ property} of set-valued mappings $F\colon\mathbb{R}^n\rightrightarrows\mathbb{R}^m$ introduced recently by Gfrerer and Outrata \cite{Helmut}. This property is used here for the justification of local superlinear convergence of our Newton-type algorithm to solve gradient equations \eqref{gra} and then to solve subgradient inclusions \eqref{subgra-inc} in the next section.
To formulate the semismooth$^*$ property of set-valued mappings, recall first the notion of the {\em directional limiting normal cone} to a set $\Omega\subset\mathbb{R}^s$ at $\bar{z}\in\Omega$ in the direction $d\in\mathbb{R}^s$ introduced by Ginchev and Mordukhovich \cite{gin-mor} as
\begin{equation}\label{dir-nc}
N_\Omega(\bar{z};d):=\big\{v\in\mathbb{R}^s\;\big|\;\exists\,t_k\downarrow 0,\;d_k\to d,\;v_k\to v\;\mbox{ with }\;v_k\in\widehat{N}_\Omega(\bar{z}+t_k d_k)\big\}.
\end{equation}
It is obvious that \eqref{dir-nc} reduces to the limiting normal cone \eqref{lnc} for $d=0$. Given a set-valued mapping $F\colon\mathbb{R}^n\rightrightarrows\mathbb{R}^m$ and a point $(\bar{x},\bar{y})\in\mbox{\rm gph}\, F$, the {\em directional limiting coderivative} of $F$ at $(\bar{x},\bar{y})$ in the direction $(u,v)\in\mathbb{R}^n\times\mathbb{R}^m$ is defined by Gfrerer \cite{g} as
\begin{equation*}
D^*F\big((\bar{x},\bar{y});(u,v)\big)(v^*):=\big\{u^*\in\mathbb{R}^n\;\big|\;(u^*,-v^*)\in N_{\text{gph}\,F}\big((\bar{x},\bar{y});(u,v)\big)\big\}\;\mbox{ for all }\;v^*\in\mathbb{R}^m
\end{equation*}
by using the directional normal cone \eqref{dir-nc} to the graph of $F$ at $(\bar{x},\bar{y})$ in the direction $(u,v)$. The aforementioned semismooth$^*$ property of $F$ is now formulated as follows.
\begin{Definition}[\bf semismooth$^*$ property of set-valued mappings]\label{semi*} A mapping $F\colon\mathbb{R}^n\rightrightarrows\mathbb{R}^m$ is {\sc semismooth$^*$} at $(\bar{x},\bar{y})\in\mbox{\rm gph}\, F$ if whenever $(u,v)\in\mathbb{R}^n\times\mathbb{R}^m$ we have the equality
\begin{equation*}
\langle u^*,u\rangle=\langle v^*,v\rangle\;\mbox{ for all }\;(v^*,u^*)\in\mbox{\rm gph}\, D^*F\big((\bar{x},\bar{y});(u,v)\big)
\end{equation*}
via the graph of the directional limiting coderivative of $F$ at $(\bar{x},\bar{y})$ in all the directions $(u,v)$.
\end{Definition}
Semismooth$^*$ mappings are largely investigated in Gfrerer and Outrata \cite{Helmut}, where this property is verified for any mapping $F\colon\mathbb{R}^n\rightrightarrows\mathbb{R}^m$ with the graph represented as a union of finitely many closed and convex sets, for normal cone mappings generated by convex polyhedral sets. If $F\colon\mathbb{R}^n\to\mathbb{R}^m$ is locally Lipschitzian around $\bar{x}$ and directionally differentiable at this point, then its semismooth$^*$ property reduces to the classical semismoothness. Other equivalent descriptions and properties of semismooth$^*$ mappings are given in Mordukhovich and Sarabi \cite{BorisEbrahim}.\vspace*{0.05in}
Now we are ready to present and discuss the major assumptions used in the rest of the paper for the design and justification of our generalized Newton algorithms to solve the gradient and subgradient systems. The following assumptions are formulated for the general subgradient inclusions \eqref{subgra-inc} at a given point $\bar{x}$ satisfying \eqref{subgra-inc}.\\[1ex]
{\bf(H1)} Given a subgradient $\bar{v}\in\partial\varphi(\bar{x})$, the second-order subdifferential inclusion \eqref{newton-inc} is robustly solvable around $(\bar{x},\bar{v})$, i.e., there is a neighborhood $U\times V$ of $(\bar{x},\bar{v})$ such that for every $(x,v)\in\mbox{\rm gph}\,\partial\varphi\cap(U\times V)$ there exists a direction $d\in\mathbb{R}^n$ satisfying \eqref{newton-inc}.\\[1ex]
{\bf(H2)} The subgradient mapping $\partial\varphi$ is metrically regular around $(\bar{x},\bar{v})$.\\[2ex]
{\bf(H3)} The subgradient mapping $\partial\varphi$ is semismooth$^*$ at $(\bar{x},\bar{v})$.\vspace*{0.05in}
Observe that in the case where $\varphi$ is of class ${\cal C}^{1,1}$ around $\bar{x}$ we have $v=\nabla\varphi(x)$ and the second-order subdifferential system \eqref{newton-inc} is written as
\begin{equation}\label{2ndC11}
-\nabla\varphi(x)\in\partial^2\varphi(x)(d).
\end{equation}
The robust solvability assumption (H1) has been discussed in Section~\ref{sec:solvN} with presenting sufficient conditions for its fulfillment; see Theorems~\ref{solvability}(ii), \ref{twiceepi} and Remark~\ref{robust-newton}. Note that the strong metric regularity of $\partial\varphi$ around $(\bar{x},\bar{v})$ for subdifferentially continuous functions $\varphi$ ensures that both assumptions (H1) and (H2) are satisfied. However, this is just a sufficient condition for the validity of (H1) and (H2). The following example borrowed from Klatte and Kummer \cite[Example~BE.4]{Klatte}, where it was constructed for different purposes, presents a function $\varphi\colon\mathbb{R}^2\to\mathbb{R}$ of class ${\cal C}^{1,1}$ (i.e., certainly being continuously prox-regular), which is twice epi-differentiable on the entire space $\mathbb{R}^2$ with the semismooth, metrically regular, but not strongly metrically regular gradient mapping $\nabla\varphi$ around the point in question. It is worth mentioning that the given example illustrates that assuming $\bar{x}$ to be a local minimizer of $\varphi$ is {\em essential} to the validity of Conjecture~4.7 from Drusvyatskiy et al. \cite{dmn}; see Remark~\ref{robust-newton}.
\begin{Example}[\bf all assumptions hold without strong metric regularity]\label{ex-kummer} {\rm Let $z:=(x,y)\in\mathbb{R}^2$ be written in the polar coordinates $(r,\theta)$ by
\begin{equation*}
z=r(\cos\theta+i\sin\theta).
\end{equation*}
We now describe the function $\varphi\colon\mathbb{R}^2\to\mathbb{R}$ and its partial derivatives on the eight cones
\begin{equation*}
C(k):=\left\{z:=(r\cos\theta,r\sin \theta)\;\Big|\;\theta\in\left[(k-1)\frac{\pi}{4},k\frac{\pi}{4}\right],\;r\ge 0\right\},\quad k=1,\ldots,8.
\end{equation*}
The analytic expressions of $\varphi$, $\nabla_x\varphi$, and $\nabla_y\varphi$ are collected in the table:
\begin{center}
\begin{tabular}{ || p{1em} | c | c| c| c| | }
\hline
$k$ & $C(k)$ & $\varphi(z)$ & $\nabla_x\varphi(z)$ & $\nabla_y\varphi(z)$\\[0.5em] \hline\hline
1 & $C(1)$ & $y(y-x)$ & $-y$ & $2y-x$\\
2 & $C(2)$ & $x(y-x)$ & $-2x+y$ & $x$\\
3 & $C(3)$ & $x(y+x)$ & $2x+y$ & $x$ \\
4 & $C(4)$ & $-y(y+x)$ & $-y$ & $-2y-x$\\
5 & $C(5)$ & $y(y-x)$ & $-y$ & $2y-x$\\
6 & $C(6)$ & $x(y-x)$ & $-2x+y$ & $x$\\
7 & $C(7)$ & $x(y+x)$ & $2x+y$ & $x$\\
8 & $C(8)$ & $-y(y+x)$ & $-y$ & $-2y-x$\\
\hline
\end{tabular}
\end{center}
The function $\varphi$ and its gradient have the following properties:\\[1ex]
{\bf(a)} The function $\varphi$ is of {\em class ${\cal C}^{1,1}$} with $\nabla\varphi$ being {\em piecewise linear} on $\mathbb{R}^2$. This is an obvious consequence of the definition. Thus $\varphi$ is
{\em continuously prox-regular} on $\mathbb{R}^2$.\\[1ex]
{\bf(b)} The mapping $\nabla\varphi$ is {\em metrically regular} around $(0,0)$. Indeed, it is observed by Klatte and Kummer \cite[Example~BE.4]{Klatte} that the inverse mapping $(\nabla\varphi)^{-1}$ is Lipschitz-like (pseudo-Lipschitz, Aubin) around $(\bar{x},\bar{v})$ with $\bar{x}=(0,0)$ and $\bar{v}=(0,0)$. As well known (see, e.g., Mordukhovich \cite[Theorem~3.2(ii)]{Mor18}), the latter property is equivalent to the metric regularity of the mapping $\nabla\varphi$ around $\bar{x}$.\\[1ex]
{\bf(c)} The mapping $\nabla\varphi$ is {\em directionally differentiable} on $\mathbb{R}^2$, which follows from Lemma~4.6.1 in Facchinei and Pang \cite{JPang}. Hence $\varphi$ is {\em twice epi-differentiable} on $\mathbb{R}^2$ as follows from Rockafellar and Wets \cite[Theorems~9.50(b) and 13.40]{Rockafellar98}.\\[1ex]
{\bf(d)} The mapping $\nabla\varphi$ is {\em semismooth} on $\mathbb{R}^2$ due to its piecewise linearity. This fact can be found, e.g., in Ulbrich \cite[Proposition~2.26]{Ul}.\\[1ex]
{\bf(e)} The mapping $\nabla\varphi$ is {\em not strongly metrically regular} around $(\bar{x},\bar{v})$ with $\bar{x}=(0,0)$ and $\bar{v}=(0,0)$. To verify it, we proceed accordingly to Definition~\ref{met-reg}(i) and
let $\vartheta$ be an arbitrary localization of $(\nabla\varphi)^{-1}$ around $\bar{v}$ for $\bar{x}$, i.e.,
\begin{equation*}
\mbox{\rm gph}\,\vartheta=\mbox{\rm gph}\,(\nabla\varphi)^{-1}\cap(V \times U)
\end{equation*}
for some neighborhoods $V$ of $\bar{v}$ and $U$ of $\bar{x}$. Find $\varepsilon>0$ and $\gamma>0$ such that $\mathbb B_\varepsilon(\bar{x})\subset U$ and $\mathbb B_\gamma(\bar{v})\subset V$ and then pick $t\in\mathbb{R}$ such that $0<t<{\rm min}\{\gamma,\varepsilon/\sqrt{5}\}$. This shows that $(t,0)\in\mathbb B_\gamma(\bar{v})\subset V$ with the representation
\begin{equation*}
(0,t)=\left(r\cos\theta,r\sin\theta\right)\;\mbox{ for }\;r=t\;\mbox{ and }\;\theta=\frac{\pi}{2},
\end{equation*}
and thus $\nabla\varphi(0,t)=(t,0)$. Furthermore, we have
\begin{equation*}
(-2t,-t)=\left(r\cos\theta,r\sin\theta\right)\;\mbox{for}\;r=\sqrt{5}t\;\mbox{and}\;\theta\in\Big[\pi,\frac{5\pi}{4}\Big],\;\cos\theta=-\frac{2}{\sqrt{5}},\;\sin\theta=-\frac{1}{\sqrt{5}},
\end{equation*}
which tell us that $\nabla\varphi(-2t,-t)=(t,0)$ and $(0,t),(-2t,-t)\in\mathbb B_\varepsilon(\bar{x})\subset U$. This yields
\begin{equation*}
\big((t,0),(0,t)\big),\big((t,0),(-2t,-t)\big)\in\mbox{\rm gph}\,(\nabla\varphi)^{-1}\cap\big(\mathbb B_\gamma(\bar{v})\times\mathbb B_\varepsilon(\bar{x})\big)\subset\mbox{\rm gph}\,\vartheta.
\end{equation*}
The latter means that there exists no localization $\vartheta$ of $(\nabla \varphi)^{-1}$ around $(\bar{v},\bar{x})$ which
is single-valued, and hence the mapping $\nabla\varphi$ is not strongly metrically regular around $(\bar{x},\bar{v})$.\vspace*{0.03in}
Remembering finally that the metric regularity is a robust property and therefore holds for all points in some neighborhood of $(0,0)$, we deduce from (a)--(d) and Theorem~\ref{twiceepi} that all the imposed assumptions (H1)--(H3) are satisfied without the fulfillment of the strong metric regularity of $\nabla\varphi$ around $(0,0)$ as shown in (e). It is easy to see that $\bar{x}=(0,0)$ is a stationary point of $\varphi$ while not its local minimizer. Thus this example does not contradict the conjecture from Drusvyatskiy et al. \cite{dmn} discussed in Remark~\ref{robust-newton}.}
\end{Example}
Now we are ready to formulate a generalized Newton algorithm to solve the gradient equation $\nabla\varphi(x)=0$, labeled as \eqref{gra} in Section~\ref{intro}, where $\varphi$ is of class ${\cal C}^{1,1}$ around the reference point. This algorithm is based on the second-order subdifferential \eqref{2nd} of the function $\varphi$ in question.
\begin{Algorithm}[\bf Newton-type algorithm for ${\cal C}^{1,1}$ functions]\label{NM} {\rm Do the following:\\[1ex]
{\bf Step~0:} Choose a starting point $x^0\in\mathbb{R}^n$ and set $k=0$.\\[1ex]
{\bf Step~1:} If $\nabla\varphi(x^k)=0$, stop the algorithm. Otherwise move on Step~2.\\[1ex]
{\bf Step~2:} Choose $d^k\in\mathbb{R}^n$ satisfying
\begin{equation}\label{alC11}
-\nabla\varphi(x^k)\in\partial\big\langle d^k,\nabla\varphi\big\rangle(x^k).
\end{equation}
{\bf Step~3:} Set $x^{k+1}$ given by
\begin{equation*}
x^{k+1}:=x^k+d^k\;\mbox{ for all }\;k=0,1,\ldots.
\end{equation*}
{\bf Step~4:} Increase $k$ by $1$ and go to Step~1.}
\end{Algorithm}
The main step and novelty of Algorithm~\ref{NM} is the generalized Newton system \eqref{alC11} expressed in terms of the limiting subdifferential of the scalarized gradient mapping. Due to the coderivative scalarization formula \eqref{scal} and the second-order construction \eqref{2nd} we have
\begin{equation*}
\partial\big\langle d^k,\nabla\varphi\big\rangle(x^k)=\big(D^*\nabla\varphi\big)(x^k)(d^k)=\partial^2\varphi(x^k)(d^k),
\end{equation*}
i.e., the iteration system \eqref{alC11} agrees with the second-order subdifferential inclusion \eqref{newton-inc} which solvability was discussed in Section~\ref{sec:solvN}; see Theorem~\ref{twiceepi}. The computation of the second-order subdifferential for ${\cal C}^{1,1}$ functions reduces to that of the (first-order) limiting subdifferential of the classical gradient mapping; significantly simplifies the numerical implementation. Note also that, according to definition \eqref{lim-cod}, the direction $d^k$ in \eqref{alC11} can be found from
\begin{equation*}
\big(-\nabla\varphi(x^k),-d^k\big)\in N\big((x^k,\nabla\varphi(x^k));\mbox{\rm gph}\,\nabla\varphi\big).
\end{equation*}
The main goal for the rest of this section is to show that the metric regularity and the semismooth$^*$ properties of $\nabla\varphi$ imposed in (H2) and (H3) ensure the {\em convergence} of iterates $x^k\to\bar{x}$ with {\em superlinear rate}. To proceed in this way, we present the following three lemmas of their own interest. The first lemma gives us a necessary and sufficient condition for the metric regularity of continuous single-valued mappings $F\colon\mathbb{R}^n\to\mathbb{R}^m$ via their limiting coderivatives. Since $F$ is single-valued, we are talking about its metric regularity around $\bar{x}$ instead of $(\bar{x},F(\bar{x}))$.
\begin{Lemma}[\bf yet another characterization of metric regularity]\label{metricforF} Let $F\colon\mathbb{R}^n\to\mathbb{R}^m$ be continuous around $\bar{x}$. Then it is metrically regular around this point if and only if there exists $c>0$ and a neighborhood $U$ of $\bar{x}$ such that we have the estimate
\begin{equation}\label{H2metrical}
\|v\|\ge c\|u\|\;\mbox{ for all }\;v\in D^*F(x)(u),\;x\in U,\;\mbox{ and }\;u\in\mathbb{R}^m.
\end{equation}
\end{Lemma}
\begin{proof} If $F$ is metrically regular and continuous around $\bar{x}$, then it follows from the book by Mordukhovich \cite[Theorem~1.54]{Mordukhovich06} that there are $c>0$ and an (open) neighborhood $U$ of $\bar{x}$ with
\begin{equation}\label{Frechet}
\|v\|\ge c\|u\|\;\mbox{ for all }\;v\in\widehat{D}^*F(x)(u),\;x\in U,\;\mbox{ and }\;u\in\mathbb{R}^m
\end{equation}
in terms of the regular coderivative \eqref{reg-cod}. Fix any $x\in U$, $u\in\mathbb{R}^m$, and an element $v\in D^*F(x)(u)$ from the limiting coderivative \eqref{lim-cod}. The continuity of $F$ ensures that the graphical set $\mbox{\rm gph}\, F$ is closed around $(x,F(x))$. Then using Corollary~2.36 from the aforementioned book gives us sequences $x_k\to x$, $u_k\to u$, and $v_k\to v$ as $k\to\infty$ such that $v_k\in\widehat{D}^*F(x_k)(u_k)$, and thus $x_k\in U$ for all $k\in{\rm I\!N}$ sufficiently large. It follows from \eqref{Frechet} that
\begin{equation*}
\|v_k\|\ge c\|u_k\|\;\mbox {for large }\;k\in{\rm I\!N}.
\end{equation*}
Letting $k\to\infty$ in the above inequality, we arrive at the estimate $\|v\|\ge c\|u\|$. On the other hand, the fulfillment of \eqref{H2metrical} immediately yields the metric regularity of $F$ around $\bar{x}$ by the coderivative criterion \eqref{cod-cr}. This completes the proof of the lemma.
\end{proof}
The next lemma presents an equivalent description of semismoothness$^*$ for Lipschitzian gradient mappings. This result follows from the combination of Proposition~3.7 in Gfrerer and Outrata \cite{Helmut} and Theorem~13.52 in Rockafellar and Wets \cite{Rockafellar98} due to the symmetry of generalized Hessian matrices; see Mordukhovich and Sarabi \cite[Proposition~2.4]{BorisEbrahim} for more details.
\begin{Lemma}[\bf equivalent description of semismoothness$^*$]\label{equisemi} Let $\varphi\colon\mathbb{R}^n\to\mathbb{R}$ be of class ${\cal C}^{1,1}$ around $\bar{x}$. Then the gradient mapping $\nabla\varphi$ is semismooth$^*$ at $\bar{x}$ if and only if
\begin{equation*}
\nabla\varphi(x)-\nabla\varphi(\bar{x})-\partial^2\varphi(x)(x-\bar{x})\subset o(\|x-\bar{x}\|),
\end{equation*}
which means that for every $\varepsilon>0$ there exists $\delta>0$ such that
\begin{equation*}
\|\nabla\varphi(x)-\nabla\varphi(\bar{x})-v\|\le\varepsilon\|x-\bar{x}\|\;\mbox{ for all }\;v\in\partial^2\varphi(x)(x-\bar{x})\;\mbox{ and }\;x\in\mathbb B_\delta(\bar{x}).
\end{equation*}
\end{Lemma}
The last lemma establishes a useful subadditivity property of the limiting coderivative of single-valued and locally Lipschitzian mappings.
\begin{Lemma}[\bf subadditivity of coderivatives]\label{phanracoderivative} Let $F\colon\mathbb{R}^n\to\mathbb{R}^m$ be locally Lipschitzian around $\bar{x}$. Then we have the inclusion
\begin{equation}\label{cod-subadd}
D^*F(\bar{x})(u_1+u_2)\subset D^*F(\bar{x})(u_1)+D^*F(\bar{x})(u_2)\;\mbox{ for all }\;u_1,u_2\in\mathbb{R}^m.
\end{equation}
\end{Lemma}
\begin{proof} It follows from the scalarization formula \eqref{scal} that
\begin{equation*}
D^*F(\bar{x})(u_1+u_2)=\partial\langle u_1+u_2,F\rangle(\bar{x}).
\end{equation*}
Employing the subdifferential sum rule from Mordukhovich \cite[Theorem~2.33]{Mordukhovich06} and the scalarization formula again gives us
\begin{equation*}
\partial\langle u_1+u_2,F\rangle(\bar{x})\subset\partial\langle u_1,F\rangle(\bar{x})+\partial\langle u_2,F\rangle(\bar{x})=D^*F(\bar{x})(u_1)+D^*F(\bar{x})(u_2),
\end{equation*}
which readily verifies \eqref{cod-subadd} and thus completes the proof of the lemma.
\end{proof}
Now we are ready to derive the main result of this section that establishes the {\em local superlinear convergence} of Algorithm~\ref{NM} under the imposed assumptions.
\begin{Theorem}{\bf(local convergence of the Newton-type algorithm for ${\cal C}^{1,1}$ functions).}\label{localconverge} Let $\bar{x}$ be a solution to \eqref{gra} for which assumptions {\rm(H1)--(H3)} are satisfied. Then there exists a neighborhood $U$ of $\bar{x}$ such that for all $x^0\in U$ Algorithm~{\rm\ref{NM}} is well-defined and generates a sequence $\{x^k\}$ that converges superlinearly to $\bar{x}$, i.e.,
\begin{equation*}
\lim_{k\to\infty}\frac{\|x^{k+1}-\bar{x}\|}{\|x^k-\bar{x}\|}=0.
\end{equation*}
\end{Theorem}
\begin{proof} Define the set-valued mapping $G_\varphi\colon\mathbb{R}^n\times\mathbb{R}^n\rightrightarrows\mathbb{R}^n$ by
\begin{equation*}
G_\varphi(x,u):=-\big(D^*\nabla\varphi\big)(x)(-u)=-\partial^2\varphi(x)(-u)\;\mbox{ for all }\;x,u\in\mathbb{R}^n.
\end{equation*}
Assumption (H1) allows us to construct the sequence of iterates $\{x^k\}$ in Algorithm~\ref{NM}. Using (H2) and its characterization from Lemma~\ref{metricforF}, we find $c>0$ and a neighborhood $U$ of $\bar{x}$ with
\begin{equation*}
\|v\|\ge c\|u\|\;\mbox{ for all }\;v\in G_\varphi(x,u),\;x\in U,\;\mbox{ and }\;u\in\mathbb{R}^n.
\end{equation*}
Let us now verify the inclusion
\begin{equation}\label{G-inc}
\nabla\varphi(x)-\nabla\varphi(\bar{x})+G_\varphi(x,u)\subset G_\varphi(x,x+u-\bar{x})+o(x-\bar{x})\mathbb B
\end{equation}
for the above vectors $x,u$. Indeed, taking any $v_1\in G_\varphi(x,u)$, i.e., $-v_1\in\partial^2\varphi(x)(-u)$, and using the subadditivity inclusion from Lemma~\ref{phanracoderivative} lead us to
\begin{equation*}
\partial^2\varphi(x)(-u)\subset\partial^2\varphi(x)(-x-u+\bar{x})+\partial^2\varphi(x)(x-\bar{x})
\end{equation*}
and thus ensure the existence of $v_2\in\partial^2\varphi(x)(-x-u+\bar{x})$ such that $-v_1-v_2\in\partial^2\varphi(x)(x-\bar{x})$. The semismoothness$^*$ assumption (H3) and its equivalent description in Lemma~\ref{equisemi} tell us that
\begin{equation*}
\lim_{x\to\bar{x}}\frac{\|\nabla\varphi(x)-\nabla\varphi(\bar{x})+v_1+v_2\|}{\|x-\bar{x}\|}=0,
\end{equation*}
which therefore verifies \eqref{G-inc}. All of this allows us to proceed similarly to the proof of Theorem~10.7 in Klatte and Kummer \cite{Klatte} and find a neighborhood $U$ of $\bar{x}$ such that, whenever the starting point $x^0\in U$ is selected, Algorithm~\ref{NM} generates a well-defined sequence of iterates $\{x^k\}$, which converges superlinearly to $\bar{x}$. This completes proof of the theorem.
\end{proof}
The assumption (H1) has been already discussed and cannot be removed or relaxed; otherwise Algorithm~\ref{NM} is not well-defined. Now we present two examples showing that assumptions (H2) and (H3) are essential for the convergence (not even talking about its superlinear rate) of Algorithm~\ref{NM}. Let us start with the semismoothness$^*$ assumption (H3).
\begin{Example}[\bf semismooth$^*$ property is essential for convergence]\label{semi-conv} {\rm Consider the Lipschitz continuous function given by
\begin{equation*}
\psi(x):=\left\{\begin{array}{ll}
\displaystyle x^2\sin\frac{1}{x}+2x&{\rm if}\;\;x\ne 0,\\
0&{\rm if}\;\;x=0,
\end{array}
\right.
\end{equation*}
which was used in Jiang et al. \cite{defeng} to show that the semismooth Newton method for solving the equation $\psi(x)=0$ fails to locally converge to $\bar{x}:=0$. Consider further the ${\cal C}^{1,1}$ function
\begin{equation*}
\varphi(x):=\int_{0}^{x}\psi(t)dt,\quad x\in\mathbb{R},
\end{equation*}
with $\nabla\varphi(x)=\psi(x)$ on $\mathbb{R}$, and hence $\nabla\varphi(\bar{x})=0$. As shown in Mordukhovich and Sarabi \cite[Example~4.5]{BorisEbrahim}, the mapping $\nabla\varphi$ is not semismooth$^*$ at $\bar{x}$ and iterations \eqref{alC11} constructed to compute tilt-stable local minimizers of $\varphi$ (see below) do not locally converge to $\bar{x}$.}
\end{Example}
The next example reveals that assumption (H2) cannot be improved by relaxing the metric regularity property to the metric subregularity or even to the strong metric subregularity of the gradient mapping $\nabla\varphi$ at $\bar{x}$ in order to keep the convergence of iterates in Algorithm~\ref{NM}.
\begin{Example}[\bf convergence failure under strong metric subregularity]\label{smr-fail} {\rm Consider the function $\varphi\colon\mathbb{R}^2\to\mathbb{R}$ given by the formula
\begin{equation}\label{phC11}
\varphi(x,y):=\frac{1}{2}x|x|+\frac{1}{2}y|y|\;\mbox{ for all }\;(x,y)\in\mathbb{R}^2.
\end{equation}
It is clear that the function $\varphi$ is of class ${\cal C}^{1,1}$ around $\bar{z}:=(0,0)$ with $\nabla\varphi(x,y)=(|x|,|y|)$ for all $z:=(x,y)\in\mathbb{R}^2$ and $\nabla\varphi(\bar{z})=0$. The simple computation tells us that
\begin{equation*}
\big(D\nabla\varphi\big)(x,y)(u_1,u_2)=\begin{cases}
\big\{\big(|u_1|,|u_2|\big)\;\big|\;u_1,u_2\in\mathbb{R}\big\}&\text{if}\quad x=0,\;y=0,\\
\big\{\big(u_1\,\text{sgn}(x),|u_2|\big)\;\big|\;u_1,u_2\in\mathbb{R}\big\}&\text{if}\quad x\ne 0,\;y=0,\\
\big\{\big(|u_1|,u_2\,\text{sgn}(y)\big)\big|\;u_1,u_2\in\mathbb{R}\big\}&\text{if}\quad x=0,\;y \ne 0,\\
\big\{\big(u_1\,\text{sgn}(x),u_2\,\text{sgn}(y)\big)\;\big|\;u_1,u_2\in\mathbb{R}\big\}&\text{if} \quad x\ne 0,\;y\ne 0.
\end{cases}
\end{equation*}
It follows from the Levy-Rockafellar criterion that the mapping $\nabla\varphi$ is strongly metrically subregular at any point $(x,y)\in\mbox{\rm gph}\,\nabla\varphi$. Furthermore, $\nabla\varphi$ is semismooth$^*$ at $\bar{z}$ since it is piecewise linear on $\mathbb{R}^2$, and thus assumption (H3) is satisfied. The fulfillment of (H1) is proved in Theorem~\ref{solvability}. Let us now show that the sequence of iterates $\{z^k\}$ generated by Algorithm~\ref{NM} does not converge to $\bar{z}$. Indeed, fix any $r>0$ and pick an arbitrary starting point $z^0$ in the form $z^0:=(0,r)\in\mathbb B_r(\bar{z})$. To run the algorithm, we need to find $d^0\in\mathbb{R}^2$ such that
\begin{equation}\label{Step1Ex}
-\nabla\varphi(z^0)\in\partial^2\varphi(z^0)(d^0).
\end{equation}
Using the second-order subdifferential \eqref{2nd} for the function $\varphi$ from \eqref{phC11} gives us
\begin{equation*}
\partial^2\varphi(z^0)(u_1,u_2)=\begin{cases}
\big\{(\alpha u_1,u_2)\;\big|\;\alpha\in[-1,1]\big\}&\text{if}\qquad u_1\ge 0,\;u_2\in\mathbb{R},\\
\big\{(\alpha u_1,u_2)\;\big|\;\alpha\in\{-1,1\big\}&\text{if}\qquad u_1<0,\;u_2\in\mathbb{R}.
\end{cases}
\end{equation*}
It shows that the direction $d^0=(1,-r)$ satisfies inclusion \eqref{Step1Ex}. Put further $z^1:=z^0+d^0=(1,0)$ and find a direction $d^1\in\mathbb{R}^2$ such that
\begin{equation}\label{Step2Ex}
-\nabla\varphi(z^1)\in\partial^2\varphi(z^1)(d^1).
\end{equation}
Computing again the second-order subdifferential brings us to the expression
\begin{equation*}
\partial^2\varphi(z^1)(u_1,u_2)= \begin{cases}
\big\{(u_1,\alpha u_2)\;\big|\;\alpha\in[-1,1]\big\}&\text{if}\qquad u_1\in\mathbb{R},\;u_2\ge 0,\\
\big\{(u_1,\alpha u_2)\;\big|\;\alpha\in\{-1,1\}\big\}&\text{if}\qquad u_1\in\mathbb{R},\;u_2<0
\end{cases}
\end{equation*}
and then verify that the direction $d^1=(-1,r)$ satisfies \eqref{Step2Ex}. Thus $z^2:=z^1+d^1=(0,r)$. Continue this process, we construct the sequence of iterates $\{z^k\}$ such that $z^{2k}=z^0$, $z^{2k+1}=z^1$ for all $k\in{\rm I\!N}$. It is obvious that $\{z^k\}$ does not converge to $\bar{z}$.}
\end{Example}
There are several Newton-type methods to solve Lipschitzian equations $f(x)=0$ that apply to gradient systems \eqref{gra} with $f(x)=\nabla\varphi(x)$, where $\varphi$ is of class ${\cal C}^{1,1}$. Such methods are mainly based of various {\em generalized directional derivatives} and can be found, e.g., in Klatte and Kummer \cite{Klatte}, Pang \cite{p90}, Hoheisel et al. \cite{HungBoris}, Mordukhovich and Sarabi \cite{BorisEbrahim}, and the references therein. It is beyond the scope of this paper to discuss their detailed relationships with Algorithm~\ref{NM}. However, let us compare the proposed algorithm with the {\em semismooth Newton method} to solve equation \eqref{gra}, which is based on the generalized Jacobian of Clarke \cite{cl} for the mapping $f=\nabla\varphi$.
\begin{Remark}{\bf(comparing Algorithm~\ref{NM} with the semismooth Newton method).}\label{semialg} {\rm In the setting of \eqref{gra}, the semismooth Newton method constructs the iterations
\begin{equation}\label{iterationN}
x^{k+1}=x^{k}-(A^k)^{-1}\nabla\varphi(x^k),\quad k=0,1,\ldots,
\end{equation}
where for each $k$ a nonsingular matrix $A^k$ is taken from the {\em generalized Jacobian} of $\nabla\varphi$ at $x=x^k$, which is expressed for functions $\varphi$ of class ${\cal C}^{1,1}$ around $x$ via the convex hull $A^k\in{\rm co}\overline\nabla^2\varphi(x^k)$ of the {\em limiting Hessian} matrices defined by
\begin{equation}\label{lim-hes}
\overline\nabla^2\varphi(x):=\Big\{\lim_{m\to\infty}\nabla^2\varphi(u_m)\;\Big|\;u_m\to x,\;u_m\in Q_\varphi\Big\},\quad x\in\mathbb{R}^n,
\end{equation}
where $Q_\varphi$ stands for the set on which $\varphi$ is twice differentiable. It follows from the classical Rademacher theorem that \eqref{lim-hes} is a nonempty compact in $\mathbb{R}^{n\times n}$. Observe that
\begin{equation*}
{\rm co}\,\big[\partial\langle u,\nabla\varphi\rangle(x)\big]=\mbox{\rm co}\,\partial^2\varphi(x)(u)=\big\{A^*u\;\big|\;A\in{\rm co}\overline\nabla^2\varphi(x)\big\},\quad u\in\mathbb{R}^n,
\end{equation*}
which tells us that, in contrast to our algorithm \eqref{alC11}, the semismooth method \eqref{iterationN} employs the {\em convex hull} of the corresponding set. A serious disadvantage of \eqref{iterationN} is that it requires the {\em invertibility} of all the matrices from ${\rm co}\overline\nabla^2\varphi(x^k)$; otherwise the semismooth Newton algorithm \eqref{iterationN} is simply {\em not well-defined}. Observe that nothing like that is required to run our Algorithm~\ref{NM}. Indeed, the invertibility assumption is even more restrictive than the strong metric regularity of $\nabla\varphi$ around $\bar{x}$, which is also not required in the imposed assumptions (H1)--(H3) that ensure the well-posedness and local superlinear convergence of Algorithm~\ref{NM}.}
\end{Remark}
Next we consider a particular case of the gradient stationary equation \eqref{gra}, where $\bar{x}$ is a {\em local minimizer} of $\varphi$. Moreover, our attention is paid to the remarkable class of local minimizers exhibiting the property of {\em tilt stability} introduced by Poliquin and Rockafellar \cite{Poli}. In the case of tilt-stable minimizers, Algorithm~\ref{NM} was developed by Mordukhovich and Sarabi \cite{BorisEbrahim}. The results established below improve those from the latter paper. First we recall the notion of tilt-stable local minimizers for the general case of extended-real-valued functions $\varphi\colon\mathbb{R}^n\to\Bar{\mathbb{R}}$.
\begin{Definition}[\bf tilt-stable local minimizers]\label{def:tilt} Given $\varphi\colon\mathbb{R}^n\to\Bar{\mathbb{R}}$, a point $\bar{x}\in\mbox{\rm dom}\,\varphi$ is a {\sc tilt-stable local minimizer} of $\varphi$ if there exists a number $\gamma>0$ such that the mapping
\begin{equation}\label{tilt}
M_\gamma\colon v \mapsto{\rm argmin}\big\{\varphi(x)-\langle v,x\rangle\;\big|\;x \in\mathbb B_\gamma(\bar{x})\big\}
\end{equation}
is single-valued and Lipschitz continuous on some neighborhood of $0\in\mathbb{R}^n$ with $M_\gamma(0)=\{\bar{x}\}$.
\end{Definition}
This notion was largely investigated and characterized in second-order variational analysis with various applications to constrained optimization. Besides the seminal paper by Poliquin and Rockafellar \cite{Poli}, we refer the reader to Chieu et al. \cite{ChieuNghia}, Drusvyatskiy and Lewis \cite{dl}, Drusvyatskiy et al. \cite{dmn}, Gfrerer and Mordukhovich \cite{gm}, Mordukhovich \cite{Mor18}, Mordukhovich and Nghia \cite{MorduNghia}, Mordukhovich and Rockafellar \cite{mr} with the bibliographies therein. Some of these characterizations are used in the proof of the following theorem.
\begin{Theorem}[\bf Newton-type method for tilt-stable minimizers of ${\cal C}^{1,1}$ functions]\label{thm:tilt} Let $\varphi\colon\mathbb{R}^n\to\mathbb{R}$ be of class ${\cal C}^{1,1}$ around a given point $\bar{x}$, which is a tilt-stable local minimizer of $\varphi$. Then there is a neighborhood $U$ of $\bar{x}$ such that the following assertions hold:\\[1ex]
{\bf(i)} For any $x\in U$ there exists a direction $d\in\mathbb{R}^n$ satisfying the inclusion
\begin{equation}\label{directiondescent}
-\nabla\varphi(x)\in\partial\langle d,\nabla\varphi\rangle(x).
\end{equation}
Furthermore, we have that $\langle\nabla\varphi(x),d\rangle<0$ whenever $x\ne\bar{x}$ and that for each $c\in(0,1)$ there is $\delta>0$ ensuring the fulfillment of the inequality
\begin{equation}\label{linesearch}
\varphi(x+td)\le\varphi(x)+ct\langle\nabla\varphi(x),d\rangle\;\mbox{ for all }\;t\in(0,\delta).
\end{equation}
{\bf(ii)} If in addition the gradient mapping $\nabla\varphi$ is semismooth$^*$ at $(\bar{x},0)$, then
Algorithm~{\rm\ref{NM}} is well-defined for any starting point $x^0\in U$ and generates a sequence $\{x^k\}$ that superlinearly converges to $\bar{x}$, while the sequence of the function values $\{\varphi(x^k)\}$ superlinearly converges to $\varphi(\bar{x})$, and the sequence of the gradient values $\{\nabla\varphi(x^k)\}$ superlinearly converges to $0$.
\end{Theorem}
\begin{proof} To verify (i), deduce from Drusvyatskiy and Lewis \cite[Proposition~3.1]{dl} that the imposed tilt stability of the local minimizer $\bar{x}$ implies that the gradient mapping $\nabla\varphi$ is strongly metrically regular around $(\bar{x},0)$. Then Theorem~\ref{strongsol} tells us that there exists a neighborhood $U_1$ of $\bar{x}$ such that for all $x\in U_1$ we can find a direction $d\in\mathbb{R}^n$ satisfying \eqref{directiondescent}. Furthermore, it follows from Chieu et al. \cite[Theorem~4.7]{ChieuLee17} that there exists another neighborhood $U_2$ of $\bar{x}$ such that $\varphi$ is strongly convex on $U_2$, and we have
\begin{equation}\label{pd}
\langle z,u\rangle>0\;\mbox{ for all }\;z\in\partial^2\varphi(x)(u)\;\mbox{ and }\;x\in U_2,\;u\ne 0.
\end{equation}
Denote $U:=U_1\cap U_2$ and fix any $x\in U$ with $x\ne\bar{x}$, which gives us a direction $d\in\mathbb{R}^n$ satisfying \eqref{directiondescent}. To show now that $d\ne 0$, assume on the contrary that $d =0$ and then get from \eqref{directiondescent} that $\nabla\varphi(x)=0$. Hence it follows from the strong convexity of $\varphi$ that $x$ is a strict global minimizer of $\varphi$ on $U$, which clearly contradicts the tilt stability of $\bar{x}$ by Definition~\ref{def:tilt}, and thus $d\ne 0$. Combining the latter with \eqref{directiondescent} and \eqref{pd}, we get $\langle\nabla \varphi(x),d\rangle<0$ and hence conclude that \eqref{linesearch} holds by using, e.g., Lemma~2.19 from Izmailov and Solodov \cite{Solo14}.\vspace*{0.03in}
Next we verify assertion (ii). As follows from Theorem~\ref{solvability}(ii), the imposed strong metric regularity of $\nabla\varphi$ around $\bar{x}$ ensures the fulfillment of assumption (H1) and of course (H2). The additional semismoothness$^*$ assumption on $\nabla\varphi$ at $(\bar{x},0)$ is (H3) in this setting, and hence we deduce the well-posedness and local superlinear convergence of iterates $\{x^k\}$ in Algorithm~\ref{NM} from Theorem~\ref{localconverge}. To show now that the sequence of values $\{\varphi(x^k)\}$ converges superlinearly to $\varphi(\bar{x})$, conclude by the ${\cal C}^{1,1}$ property of $\varphi$ around $\bar{x}$ and Lemma~A.11 from Izmailov and Solodov \cite{Solo14} that there exists $\ell>0$ ensuring the estimate
\begin{equation*}
|\varphi(x^{k+1})-\varphi(\bar{x})|\le\frac{\ell}{2}\|x^{k+1}-\bar{x}\|^2\;\mbox{ for sufficiently large }\;k\in{\rm I\!N}.
\end{equation*}
Furthermore, the second-order growth condition that follows from tilt stability of $\bar{x}$ (see, e.g., Mordukhovich and Nghia \cite[Theorem~3.2]{MorduNghia}) gives us $\kappa>0$ such that
\begin{equation*}
|\varphi(x^k)-\varphi(\bar{x})|\ge\varphi(x^k)-\varphi(\bar{x})\ge\frac{1}{2\kappa}\|x^k-\bar{x}\|^2\;\mbox{ for large }\;k\in{\rm I\!N}.
\end{equation*}
Combining the two estimates above produces the inequality
\begin{equation*}
\frac{|\varphi(x^{k+1})-\varphi(\bar{x})|}{|\varphi(x^k)-\varphi(\bar{x})|}\le\ell\kappa\frac{\|x^{k+1}-\bar{x}\|}{\|x^k-\bar{x}\|},
\end{equation*}
which deduces the claimed superlinear convergence of $\{\varphi(\bar{x})\}$ from the one established for $\{x^k\}$.\vspace*{0.03in}
To finish the proof, it remains to to show that the sequence $\{\nabla\varphi(x^k)\}$ superlinearly converges to $0$. Indeed, the Lipschitz continuity of $\nabla\varphi$ around $\bar{x}$ gives us a constant $\ell>0$ such that
\begin{equation*}
\|\nabla\varphi(x^{k+1})\|=\|\nabla\varphi(x^{k+1})-\nabla\varphi(\bar{x})\|\le\ell\|x^{k+1}-\bar{x}\|\;\mbox{ for large }\;k\in{\rm I\!N}.
\end{equation*}
Furthermore, the strong local monotonicity of $\nabla\varphi$ around $(\bar{x},0)$ under the tilt stability of $\bar{x}$ (see, e.g., Poliquin and Rockafellar \cite[Theorem~1.3]{Poli} for a more general result) tells us that there exists a constant $\kappa>0$ ensuring the inequality
\begin{equation*}
\langle\nabla\varphi(x^k)-\nabla\varphi(\bar{x}),x^k-\bar{x}\rangle\ge\kappa\|x^k-\bar{x}\|^2\;\mbox{ for large }\;k\in{\rm I\!N},
\end{equation*}
and hence $\|\nabla\varphi(x^k)\|\ge\kappa\|x^k-\bar{x}\|$ for such $k$. Thus we arrive at the estimate
\begin{equation*}
\frac{\|\nabla\varphi(x^{k+1})\|}{\|\nabla\varphi(x^k)\|}\le\frac{\ell}{\kappa}\frac{\|x^{k+1}-\bar{x}\|}{\|x^k-\bar{x}\|},
\end{equation*}
which verifies that the gradient sequence $\{\nabla\varphi(x^k)\}$ superlinearly converges to 0 as $k\to\infty$ due to such a convergence of $x^k\to\bar{x}$ obtained above. This completes the proof of the theorem.
\end{proof}\vspace*{-0.03in}
We conclude this section by comparing the results of Theorem~\ref{thm:tilt} with the recent ones established by Mordukhovich and Sarabi \cite{BorisEbrahim}.
\begin{Remark}[\bf comparison with known results under tilt stability]\label{comp-tilt} {\rm The aforementioned paper \cite{BorisEbrahim} developed Algorithm~\ref{NM}, written in an equivalent form, for computing tilt-stable minimizers $\bar{x}$ of $\varphi\in{\cal C}^{1,1}$. As follows from Drusvyatskiy and Lewis \cite[Theorem~3.3]{dl} (see also Drusvyatskiy et al. \cite[Proposition~4.5]{dmn} for a more precise statement), that the tilt-stability of $\bar{x}$ for $\varphi$ is equivalent to the strong metric regularity of $\nabla\varphi$ around $(\bar{x},0)$ provided that $\bar{x}$ is a local minimizer of $\varphi$. The latter requirement is essential as trivially illustrated by the function $\varphi(x):=-x^2$ on $\mathbb{R}$, where $\bar{x}=0$ is not a tilt-stable local minimizer while $\nabla\varphi$ is strongly metrically regular around $(\bar{x},0)$. Note also that the solvability/well-posedness of Algorithm~\ref{NM} holds under weaker assumptions than the strong metric regularity; see Section~\ref{sec:solvN}. The local superlinear convergence of $\{x^k\}$ to a tilt-stable minimizer $\bar{x}$ in Theorem~\ref{thm:tilt}(ii) follows from Mordukhovich and Sarabi \cite[Theorem~4.3]{BorisEbrahim} under the semismoothness$^*$ of $\nabla\varphi$ at $(\bar{x},0)$. Besides the local superlinear convergence of $\varphi(x^k)\to\varphi(\bar{x})$ and $\nabla\varphi(x^k)\to 0$ in Theorem~\ref{thm:tilt}(ii), the new statements of Theorem~\ref{thm:tilt}(i) include the {\em descent property} $\langle\nabla\varphi(x^k),d^k\rangle<0$ of the algorithm and the {\em backtracking line search} \eqref{linesearch} at each iteration.}
\end{Remark}\vspace*{-0.2in}
\section{Generalized Newton Algorithm for Prox-Regular Functions}\label{sec:prox}\vspace*{-0.05in}
This section is devoted to the design and justification of a generalized Newton algorithm to solve subgradient inclusions of type \eqref{subgra-inc}, where $\varphi\colon\mathbb{R}^n\to\Bar{\mathbb{R}}$ is a continuously prox-regular function. We have already considered this remarkable class of functions in Section~\ref{sec:solvN} concerning solvability of the second-order subdifferential systems \eqref{newton-inc}, which play a crucial role in the design of the generalized Newton algorithm to find a solution of \eqref{subgra-inc} in this section. The approach developed here is to reduce the subgradient inclusion \eqref{subgra-inc} to a gradient one of type \eqref{gra} with the replacement of $\varphi$ from the class of continuously prox-regular functions by its Moreau envelope, which is proved to be of class ${\cal C}^{1,1}$. This leads us to the well-defined and implementable generalized Newton algorithm expressed in terms of the second-order subdifferential of $\varphi$ and the single-valued, monotone, and Lipschitz continuous proximal mapping associated with this function. We show that the proposed algorithm exhibits local superlinear convergence under the standing assumptions imposed and discussed above.\vspace*{0.03in}
First we formulate the notions of Moreau envelopes and proximal mappings associated with extended-real-valued functions. Recall that $\varphi\colon\mathbb{R}^n\to\Bar{\mathbb{R}}$ is {\em proper} if $\mbox{\rm dom}\,\varphi\ne\emptyset$.
\begin{Definition}[\bf Moreau envelopes and proximal mappings]\label{def:moreau} Given a proper l.s.c.\ function $\varphi\colon\mathbb{R}^n\to\Bar{\mathbb{R}}$ and a parameter value $\lambda>0$, the {\sc Moreau envelope} $e_\lambda\varphi$ and the {\sc proximal mapping} $\textit{\rm Prox}_\lambda\varphi$ are defined by the formulas
\begin{equation}\label{Moreau}
e_\lambda\varphi(x):=\inf\left\{\varphi(y)+\frac{1}{2\lambda}\|y-x\|^2\;\Big|\;y\in\mathbb{R}^n\right\},
\end{equation}
\begin{equation}\label{Prox}
\textit{\rm Prox}_\lambda\varphi(x):={\rm argmin}\left\{\varphi(y)+\frac{1}{2\lambda}\|y-x\|^2\;\Big|\;y\in\mathbb{R}^n\right\}.
\end{equation}
If $\lambda=1$, we use the notation $e\varphi(x)$ and $\text{\rm Prox}_{\varphi}(x)$ in \eqref{Moreau} and \eqref{Prox}, respectively.
\end{Definition}
Both Moreau envelopes and proximal mappings have been well recognized in variational analysis and optimization as efficient tools of regularization and approximation of nonsmooth functions. This has have done particularly for convex functions and more recently for continuously prox-regular functions; see Rockafellar and Wets \cite{Rockafellar98} and the references therein. Proximal mappings and the like have been also used in numerical algorithms of the various types revolving around computing proximal points; see, e.g., the book by Beck \cite{Beck} and the paper by Hare and Sagastiz\'abal \cite{Hare} among many other publications. In what follows we are going to use the proximal mapping \eqref{Prox} for designing a generalized Newton algorithm to solve subgradient inclusions \eqref{subgra-inc} for continuously prox-regular functions with applications to a class of Lasso problems.\vspace*{0.03in}
Here is our basic Newton-type algorithm to solve subgradient inclusions \eqref{subgra-inc} generated by prox-regular functions $\varphi$. Note that this algorithm constructively describes the area of {\em choosing a starting point} that depends on the {\em constant of prox-regularity}. The {\em subproblem} of this algorithm at each iteration is to find a {\em unique} solution to the optimization problem in \eqref{Prox}, which is a {\em regularization} of $\varphi$ by using {\em quadratic penalties}.
\begin{Algorithm}[\bf Newton-type algorithm for subgradient inclusions]\label{NM3} {\rm Let $r>0$ be a constant of prox-regularity of $\varphi\colon\mathbb{R}^n\to\Bar{\mathbb{R}}$ from \eqref{prox}.\\[1ex]
{\bf Step~0:} Pick any $\lambda\in(0,r^{-1})$, choose a starting point $x^0$ by
\begin{equation}\label{start-point}
x^0\in U_\lambda:=\text{rge}(I+\lambda\partial\varphi),
\end{equation}
and set $k:=0$.\\[1ex]
{\bf Step~1:} If $0\in\partial\varphi(x^k)$, then stop. Otherwise compute
\begin{equation}\label{subprob}
v^k:=\frac{1}{\lambda}\Big(x^k-\text{\rm Prox}_\lambda\varphi(x^k)\Big).
\end{equation}
{\bf Step~2}: Choose $d^k\in\mathbb{R}^n$ such that
\begin{equation}\label{prox-dir}
-v^k\in\partial^2\varphi(x^k-\lambda v^k,v^k)(\lambda v^k+d^k).
\end{equation}
{\bf Step~3:} Compute $x^{k+1}$ by
\begin{equation*}
x^{k+1}:=x^k+d^k,\quad k=0,1,\ldots.
\end{equation*}
{\bf Step 4:} Increase $k$ by $1$ and go to Step~1.}
\end{Algorithm}
Note that Algorithm~\ref{NM3} does not include computing the Moreau envelope \eqref{Moreau} while requiring to solve subproblem \eqref{subprob} built upon the proximal mapping \eqref{Prox}. By definition of the second-order subdifferential \eqref{2nd} and the limiting coderivative \eqref{lim-cod}, the implicit inclusion \eqref{prox-dir} for $d^k$ can be rewritten explicitly as
\begin{equation}\label{prox-dir1}
(-v^k,-\lambda v^k-d^k)\in N_{\text{gph}\,\partial\varphi}(x^k-\lambda v^k,v^k).
\end{equation}
Observe finally that for convex functions $\varphi\colon\mathbb{R}^n\to\Bar{\mathbb{R}}$ we can choose $\lambda$ in Step~0 arbitrarily from $(0,\infty)$ with $U_\lambda=\mathbb{R}^n$ in \eqref{start-point}. This is due a well-known result of convex analysis, which is reflected in the following lemma that plays a crucial role in the justification of Algorithm~\ref{NM3}. Recall that $I$ stands for the identity operator, and that $\varphi$ is {\em prox-bounded} if it is bounded from below by a quadratic function on $\mathbb{R}^n$.
\begin{Lemma}{\bf(Moreau envelopes and proximal mappings for prox-regular functions).}\label{rela} Let $\varphi\colon\mathbb{R}^n\to\Bar{\mathbb{R}}$ be prox-bounded on $\mathbb{R}^n$ and continuously prox-regular at $\bar{x}$ for $\bar{v}\in\partial\varphi(\bar{x})$ with the positive parameters $r,\varepsilon$ in \eqref{prox}. Then we have the following assertions, where $\lambda$ can be chosen arbitrarily from $(0,\infty)$ with $U_\lambda=\mathbb{R}^n$ if $\varphi$ is convex:\\[1ex]
{\bf(i)} The Moreau envelope $e_\lambda\varphi$ is of class ${\cal C}^{1,1}$ on the set $U_\lambda$ taken from \eqref{start-point}, which contains a neighborhood of $\bar{x}+\lambda\bar{v}$ for all $\lambda\in(0,r^{-1})$. Furthermore, $\bar{x}$ is a solution to
inclusion \eqref{subgra-inc} if and only if $\nabla e_\lambda\varphi(\bar{x})=0$.\\[1ex]
{\bf(ii)} The proximal mapping $P_\lambda\varphi$ is single-valued, monotone, and Lipschitz continuous on $U_\lambda$ and satisfies the
condition $P_\lambda\varphi(\bar{x}+\lambda\bar{v})=\bar{x}$.\\[1ex]
{\bf(iii)} The gradient of $e_\lambda\varphi$ is calculated by
\begin{equation*}
\nabla e_\lambda\varphi(x)=\frac{1}{\lambda}\Big(x-\text{\rm Prox}_\lambda\varphi(x)\Big)=\big(\lambda I+\partial\varphi^{-1}\big)^{-1}(x)\;\mbox{ for all }\;x\in U_\lambda.
\end{equation*}
\end{Lemma}
\begin{proof} Denoting $\varphi_{\bar{v}}(x):=\varphi(x)-\langle\bar{v},x\rangle$, observe that $\varphi_{\bar{v}}$ satisfies all the assumptions of Proposition~13.37 from Rockafellar and Wets \cite{Rockafellar98}, which therefore yields assertions (i) and (ii) of this lemma. The results for convex functions $\varphi$ follow from Theorem~2.26 of the aforementioned book. Assertion (iii) is taken from Poliquin and Rockafellar \cite[Theorem~4.4]{Poliquin}.
\end{proof}
The next simple lemma is also needed for the proof of the main result of this section.
\begin{Lemma}[\bf second-order subdifferential graph]\label{2gph} For any $\lambda\in(0,r^{-1})$, $x\in U_\lambda$, and $v=\nabla e_\lambda\varphi(x)$ in the notation above we have the equivalence
\begin{equation*}
(v^*,x^*)\in{\rm gph}\big(D^*\nabla e_\lambda\varphi\big)(x,v)\iff(v^*-\lambda x^*,x^*)\in\mbox{\rm gph}\,\partial^2\varphi (x-\lambda v,v).
\end{equation*}
\end{Lemma}
\begin{proof} The relationship in \eqref{inverse} tells us that $(v^*,x^*)\in\mbox{\rm gph}\,(D^*\nabla e_\lambda\varphi)(x,v)$ if and only if
\begin{equation*}
-v^*\in D^*(\lambda I+\partial\varphi^{-1})(v,x)(-x^*).
\end{equation*}
Elementary operations with the limiting coderivative yield
\begin{equation}\label{inverseofMoreau}
D^*(\lambda I+\partial\varphi^{-1})(v,x)(-x^*)=-\lambda x^*+(D^*\partial\varphi^{-1})(v,x-\lambda v)(-x^*).
\end{equation}
This ensures the equivalence of \eqref{inverseofMoreau} to the inclusion
\begin{equation*}
\lambda x^*-v^*\in(D^*\partial\varphi^{-1})(v,x-\lambda v)(-x^*),
\end{equation*}
which implies by \eqref{inverse} that $x^*\in\partial^2\varphi(x-\lambda v,v)(v^*-\lambda x^*)$ and thus completes the proof.
\end{proof}
Now we are ready to formulate and prove our major result on the well-posedness and superlinear convergence of the proposed algorithm for prox-regular functions.
\begin{Theorem}{\bf(local superlinear convergence of Algorithm~\ref{NM3}).}\label{localNM3} In addition to the standing assumption {\rm(H1)--(H3)} with $\bar{v}=0$ therein, let $\varphi\colon\mathbb{R}^n\to\Bar{\mathbb{R}}$ be prox-bounded on $\mathbb{R}^n$ and continuously prox-regular at $\bar{x}$ for $0\in\partial\varphi(\bar{x})$ with constants $r,\varepsilon>0$ from \eqref{prox}. Then there exists a neighborhood $U$ of $\bar{x}$ such that for all starting points $x^0\in U$ we have that Algorithm~{\rm\ref{NM3}} is well-defined and generates a sequence of iterates $\{x^k\}$, which converges superlinearly to the solution $\bar{x}$ of \eqref{subgra-inc} as $k\to\infty$.
\end{Theorem}
\begin{proof} It follows from Lemma~\ref{rela}(i) that solving the subgradient inclusion \eqref{subgra-inc} for the class of prox-regular functions under consideration is equivalent to solving the gradient system $\nabla e_\lambda\varphi(x)=0$ with the ${\cal C}^{1,1}$ function $e_\lambda\varphi$ under the indicated choice of parameters. To apply Algorithm~\ref{NM} to the equation $\nabla e_\lambda\varphi(x)=0$, we need to check that assumptions (H1)--(H3) imposed in this theorem on $\partial\varphi$ are equivalent to the corresponding assumptions on $\nabla e_\lambda\varphi$ imposed in Theorem~\ref{localconverge}.
Let us first check that assumption (H1) of the theorem yields its fulfillment of its counterpart for $e_\lambda\varphi$. Since $U_\lambda$ from \eqref{start-point} contains a neighborhood of $\bar{x}$ by Lemma~\ref{rela}(i), it suffices to show that there is a neighborhood $\widetilde U$ of $\bar{x}$ such that for each $x\in\widetilde U$ there exists $d\in\mathbb{R}^n$ satisfying
\begin{equation}\label{solforenve}
-\nabla e_\lambda\varphi(x)\in\big(D^*\nabla e_\lambda\varphi\big)(x)(d).
\end{equation}
Indeed, assumption (H1) gives us neighborhoods $U$ of $\bar{x}$ and $V$ of $\bar{v}=0$ for which inclusion \eqref{newton-inc} holds. Since $\nabla e_\lambda\varphi$ and $I-\nabla e_\lambda\varphi$ are continuous around $\bar{x}$ and since
\begin{equation*}
\big((I-\lambda\nabla e_\lambda\varphi)(\bar{x}),\nabla e_\lambda\varphi(\bar{x})\big)=(\bar{x},0)\in(U\cap U_\lambda)\times V,
\end{equation*}
there exists a neighborhood $\widetilde U$ of $\bar{x}$ such that $\widetilde U\subset(U\cap U_\lambda)$ and
\begin{equation*}
(I-\lambda\nabla e_\lambda\varphi)(\widetilde U)\times\nabla e_\lambda\varphi(\widetilde U)\subset(U\cap U_\lambda)\times V.
\end{equation*}
Fix now any $x\in\widetilde U$ and put $y:=\nabla e_\lambda\varphi(x)$. Performing elementary coderivative transformations brings us to the equalities
\begin{equation*}
\big(D^*\nabla e_\lambda\varphi\big)^{-1}(y,x)(y)=D^*\big(\lambda I+\partial\varphi^{-1}\big)(y,x)(y)=\lambda y+\big(D^*\partial\varphi^{-1}\big)(y,x-\lambda y)(y).
\end{equation*}
It follows from \eqref{newton-inc} that there exists $-\widetilde{d}\in(D^*\partial\varphi^{-1})(y,x-\lambda y)(y)$. Denoting finally $d:=\lambda y+\widetilde{d}$, we arrive at the inclusion
\begin{equation*}
-d\in\big(D^*\nabla e_\lambda\varphi\big)^{-1}(y,x)(y),
\end{equation*}
which tells us by \eqref{inverse} that $-\nabla e_\lambda\varphi(x)\in(D^*\nabla e_\lambda\varphi)(x)(d)$ and thus verifies \eqref{solforenve}.\vspace*{0.03in}
Next we show that the the metric regularity assumption (H2) on $\partial\varphi$ around $(\bar{x},0)$ is equivalent to the metric regularity of the Moreau envelope gradient $\nabla e_\lambda\varphi$. To proceed, pick any $\lambda\in(0,r^{-1})$ from Step~0 of the algorithm and then get by Lemma~\ref{2gph} that
\begin{equation*}
0\in\big(D^*\partial\varphi\big)(\bar{x},0)(u)\iff 0\in\big(D^*\nabla e_\lambda\varphi\big)(\bar{x},0)(u).
\end{equation*}
Then the claimed equivalence between the metric regularity properties follows directly from the Mordukhovich coderivative criterion \eqref{cod-cr} applied to both mappings $\partial\varphi$ and $\nabla e_\lambda\varphi$.
The equivalence between the semismoothness$^*$ of $\partial\varphi$ at $(\bar{x},0)$ and of $\nabla e_\lambda\varphi$ around this point is verified in the proof of Theorem~6.2 in Mordukhovich and Sarabi \cite{BorisEbrahim}.
Thus we can apply the above Theorem~\ref{localconverge} to the gradient system $\nabla e_\lambda\varphi(x)=0$ and complete the proof of this theorem by using Lemma~\ref{rela}.
\end{proof}
Now we present a direct consequence of Theorem~\ref{localNM3} to computing {\em tilt-stable local minimizers} of continuously prox-regular functions.
\begin{Corollary}{\bf(computing tilt-stable minimizers of prox-regular functions).}\label{Newtonfortilt} Let $\varphi\colon\mathbb{R}^n\to\Bar{\mathbb{R}}$ be continuous prox-regular at $\bar{x}$ for $\bar{v}:=0\in\partial\varphi(\bar{x})$ with $r>0$ taken from \eqref{prox}, and let $\bar{x}$ be a tilt-stable local minimizer of $\varphi$. Assume that the subgradient mapping $\partial\varphi$ is semismooth$^*$ at $(\bar{x},\bar{v})$. Then whenever $\lambda\in(0,r^{-1})$ there exists a neighborhood $U$ of $\bar{x}$ such that Algorithm~{\rm\ref{NM3}} is well-defined for all $x^0\in U$ and generates a sequence of iterates $\{x^k\}$, which superlinearly converges to $\bar{x}$ as $k\to\infty$.
\end{Corollary}
\begin{proof} Observe first that, in the case of minimizing $\varphi$ under consideration, the prox-boundedness assumption on $\varphi$ imposed in Theorem~\ref{localNM3} can be dismissed without loss of generality. Indeed, we can always get this property by adding to $\varphi$ the indicator function of some compact set containing a neighborhood of $\bar{x}$, which makes $\varphi$ to be prox-bounded without changing the local minimization. Furthermore, it follows from Drusvyatskiy and Lewis \cite[Theorem~3.3]{dl} that the subgradient mapping $\partial\varphi$
is strongly metrically regular around $(\bar{x},\bar{v})$. Thus assumption (H2) holds automatically, which (H1) follows from Theorem~\ref{solvability}(ii). The fulfillment of (H3) is assumed in this corollary, and so we deduce its conclusions from Theorem~\ref{localNM3}.
\end{proof}
Note that the well-posedness and local superlinear convergence of the coderivative-based generalized Newton algorithm of computing tilt-stable local minimizers of the Moreau envelope $e_\lambda\varphi$ under the semismoothness$^*$ assumption on $\partial\varphi$ has been recently obtained by Mordukhovich and Sarabi \cite[Theorem~6.2]{BorisEbrahim}. As follows from the discussions above, the latter algorithm is equivalent to Algorithm~\ref{NM3} in the case of tilt-stable local minimizers. However, the explicit form of Algorithm~\ref{NM3} seems to be more convenient for implementations, which is demonstrated below. Moreover, in Algorithm~\ref{NM3} we specify the area of starting points $x^0$ in Step~0 and also verify the choice of the prox-parameter $\lambda$ ensuring the best performance of the algorithm.\vspace*{0.05in}
Let us illustrate Algorithm~\ref{NM3} by the following example of solving the subgradient inclusion \eqref{subgra-inc} generated by a nonconvex, nonsmooth, and continuously prox-regular function $\varphi\colon\mathbb{R}\to\Bar{\mathbb{R}}$. This function is taken from Mordukhovich and Outrata \cite[Example~4.1]{BorisOutrata} and relates to the modeling of some mechanical equilibria.
\begin{Example}[\bf illustration of computing by Algorithm~\ref{NM3}] {\rm Consider the function
\begin{equation*}
\varphi(x):=|x|+\frac{1}{2}\big(\max\{x,0\}\big)^2-\frac{1}{2}\big(\max\{0,-x\}\big)^2+\delta_\Gamma(x),\quad x\in\mathbb{R},
\end{equation*}
where $\delta_\Gamma$ is the indicator function of the set $\Gamma:=[-1,1] $. We can clearly write $\varphi$ in the form
\begin{equation}\label{ph0}
\varphi(x)=\vartheta(x)+\delta_\Gamma(x),\quad x\in\mathbb{R},
\end{equation}
via the continuous, nonconvex, nonsmooth, piecewise quadratic function $\vartheta\colon\mathbb{R}\to\mathbb{R}$ given by
\begin{equation*}
\vartheta(x):=\begin{cases}
-x-\frac{1}{2}x^2&\text{if}\quad x\in[-1,0],\\
x+\frac{1}{2}x^2&\text{if}\quad x\in[0,1].
\end{cases}
\end{equation*}
Then we get by the direct calculation that
\begin{equation*}
\partial\varphi(x)=\begin{cases}
(-\infty,0]&\text{if}\quad x=-1, \\
\{-1-x\}&\text{if}\quad x\in(-1,0),\\
[-1,1]&\text{if}\quad x=0,\\
\{1 + x\}&\text{if}\quad x\in(0,1),\\
[2,\infty)&\text{if}\quad x=1.
\end{cases}
\end{equation*}
It is not hard to check that the subgradient mapping $\partial\varphi$ is strongly metrically regular around $(\bar{x},0)$ with $\bar{x}=0$ and semismooth$^*$ at this point, and thus all the assumptions of Theorem~\ref{localNM3} hold. Moreover, we have the lower estimate
\begin{equation*}
\varphi(x)\ge\varphi(u)+v(x-u)-\frac{1}{2}\|x-u\|^2\;\mbox{ for all }\;(x,u)\in\mbox{\rm gph}\,\partial\varphi\cap(U\times V),
\end{equation*}
where $U=[-1,1]$ and $V=\mathbb{R}$. Choosing $\lambda:=\frac{1}{2}\in(0,1)$ and $x^0:=\frac{1}{3}\in{\rm rge}(I+\frac{1}{2}\partial\varphi)$, we run Algorithm~\ref{NM3} with the starting point $x^0$.
\begin{equation*}
\text{\rm Prox}_\lambda\varphi(x^0)=\text{\rm argmin}_{y\in\mathbb{R}}\big\{\varphi(y)+(y-x^0)^{2}\big\}=0.
\end{equation*}
Thus $v^0$ in Step~1 of the algorithm is calculated by $v^0=\frac{1}{\lambda}(x^0-P_\lambda\varphi(x^0))=\frac{2}{3}$. To find $d^0\in\mathbb{R}$ in Step~2 of the algorithm, we have by \eqref{prox-dir1} that
\begin{equation*}
(-v^0,-\lambda v^0-d^0)\in N_{\text{gph}\,\partial\varphi}(x^0-\lambda v^0,v^0)=N_{\text{gph}\,\partial\varphi}\left(0,2/3\right).
\end{equation*}
The second-order calculations in Mordukhovich and Outrata [41, Equation (4.7)] yield
\begin{equation*}
N_{\text{gph}\,\partial\varphi}\left(0,2/3\right)=\big\{(\omega,z)\in\mathbb{R}^2\;\big|\;z=0\big\}.
\end{equation*}
This tells us that $-\lambda v^0-d^0=0$, and hence $d^0=-\frac{1}{3}$. Letting $x^1:=x^0+d^0=0$ by Step~3, we arrive at $0\in\partial\varphi(x^1)$, and thus solve the subgradient inclusion \eqref{subgra-inc} for $\varphi$ from \eqref{ph0}.}
\end{Example}
We conclude this section with the brief discussion on some other Newton-type methods to solve set-valued inclusions involving the subgradient systems \eqref{subgra-inc}.
\begin{Remark}{\bf(comparison with other Newton-type methods to solve inclusions).}\label{comp-inc} {\rm Let us make the following observations:\\[1ex]
{\bf(i)} Josephy \cite{josephy} was the first to propose a Newton-type algorithm to solve {\em generalized equations} in the sense of Robinson \cite{rob} written by
\begin{equation}\label{ge}
0\in f(x)+F(x),
\end{equation}
where $f\colon\mathbb{R}^n\to\mathbb{R}^n$ is a smooth single-valued mapping, and where $F\colon\mathbb{R}^n\rightrightarrows\mathbb{R}^n$ is a set-valued one, which originally was considered as the normal cone to a convex set while covering in this case classical variational inequalities and nonlinear complementarity problems. The Josephy-Newton method constructs the next iterate $x^{k+1}$ by solving the linearized generalized equation
\begin{equation}\label{lin-ge}
0\in f(x^k)+\nabla f(x^k)(x-x^k)+F(x)
\end{equation}
with the same set-valued part $F(x)$ as in \eqref{ge}. Note that in this method we must have a nonzero $f$ in \eqref{ge}; otherwise algorithm \eqref{lin-ge} stops at $x^1$. There are several results on the well-posedness and superlinear convergence of the Josephy-Newton method under appropriate assumptions; see, e.g., the books by Facchinei and Pang \cite{JPang} and by Izmailov and Solodov \cite{Solo14} with the references therein. The major assumption for \eqref{ge}, which is the most related to our paper, is the strong metric regularity of $f+F$ around $(\bar{x},0)$ imposed in Dontchev and Rockafellar \cite[Theorem~6C.1]{Donchev09}. In the case of subgradient systems \eqref{subgra-inc}, we actually have $f=\nabla\varphi$ and $F=0$ in \eqref{ge}, which gives us the strong metric regularity of $\nabla\varphi$. As discussed in Sections~\ref{sec:solvN} and \ref{sec:newtonC11}, the latter assumption is more restrictive that our standing ones imposed in Theorem~\ref{localconverge}. A similar strong metric regularity assumption is imposed in the semismooth Newton method of solving generalized equations (without changing $F$ in iterations) presented in Theorem~6F.1 of the aforementioned book by Dontchev and Rockafellar.\\[1ex]
{\bf(ii)} Gfrerer and Outrata \cite{Helmut} have recently introduced the new {\em semismooth$^*$ Newton method} to solve the generalized equation $0\in F(x)$. In contrast to the Josephy-Newton and semismooth Newton methods for generalized equations, the new method approximates the set-valued part $F$ of \eqref{ge} by using a certain linear structure inside the limiting coderivative graph under the nonsingularity of matrices from the mentioned linear structure. Local superlinear convergence of the proposed algorithm is proved there under the additional semismooth$^*$ assumptions on $F$. Further specifications of this algorithm have been developed in the very fresh paper by Gfrerer at al. \cite{go} for the generalized equation \eqref{ge} with a smooth function $f$ and $F=\partial\varphi$, where $\varphi$ is an l.s.c.\ convex function. The metric regularity assumption on $f+F$ imposed therein is equivalent to the strong metric regularity of this mapping due to Kenderov's theorem mentioned in Remark~\ref{robust-newton}.\\[1ex]
{\bf(iii)} Another Newton-type method to solve the inclusions $0\in F(x)$, with the verification of well-posedness and local superlinear convergence, was developed by Dias and Smirnov \cite{ds} based on {\em tangential approximations} of the graph of $F$.
The main assumption therein is the metric regularity of $F$ and the main tool of analysis is the Mordukhovich criterion \eqref{cod-cr}. Observe that the well-posedness result of the suggested algorithm requires the Lipschitz continuity of $F$, which is rarely the case for subgradient mappings associated with nonsmooth functions.}
\end{Remark}\vspace*{-0.2in}
\section{Applications to Lasso Problems}\label{lasso}\vspace*{-0.05in}
This section is devoted to the application of our main Algorithm~\ref{NM3} to solving the following class of unconstrained {\em Lasso problems}, where Lasso stands for Least Absolute Shrinkage and Selection Operator. The Lasso, known also as the {\em $\ell^1$-regularized least square optimization problem}, was described by Tibshirani \cite{Tibshirani} and since that has been largely applied to various issues in statistics, machine learning, image processing, etc. This problem as defined by
\begin{eqnarray}\label{Lasso}
\text{minimize }\;\varphi(x):=\frac{1}{2}\|Ax-b\|_2^2+\mu\|x\|_1,\quad x\in\mathbb{R}^n,
\end{eqnarray}
where $A$ is an $m\times n$ matrix, $\mu>0$, $b\in\mathbb{R}^m$, and where
\begin{equation*}
\|x\|_1:=\sum_{i=1}^n|x_i|\;\mbox{ and }\;\|x\|_2:=\Big(\sum_{i}^n|x_i|^2\Big)^{1/2}\;\mbox{ for }\;x=(x_1,\ldots,x_n).
\end{equation*}
The Lasso problem \eqref{Lasso} is convex and always has an optimal solution, which is fully characterized by the subgradient inclusion
\begin{equation}\label{sumLasso}
0\in\partial\varphi(x)\;\mbox{ with }\;\varphi\;\mbox{ from }\;\eqref{Lasso}.
\end{equation}
In order to apply Algorithm~\ref{NM3} to solving the subgradient system \eqref{sumLasso}, we first provide {\em explicit calculations} of the subdifferential, proximal mapping, and second-order subdifferential of $\varphi$.
\begin{Proposition}[\bf calculation of subdifferential]\label{calc-sub} For $\varphi$ from \eqref{Lasso} we have
\begin{equation}\label{norm1}
\partial\varphi(x)=A^*(Ax-b)+\mu F(x)\;\mbox{ for all }\;x\in\mathbb{R}^n,
\end{equation}
where the set-valued mapping $F\colon\mathbb{R}^n\rightrightarrows\mathbb{R}^n$ is calculated by
\begin{equation*}
F(x)=\left\{v\in\mathbb{R}^n\;\bigg|\;
\begin{array}{@{}cc@{}}
v_j=\text{\rm sgn}(x_j),\;x_j\ne 0,\\
v_j\in[-1,1],\;x_j=0
\end{array}\right\}.
\end{equation*}
\end{Proposition}
\begin{proof} We get by the direct calculation that $\partial(\|\cdot\|_1)(x)= F(x)$ for all $x\in\mathbb{R}^n$. Thus the subdifferential formula \eqref{norm1} follows from the sum and chain rules of convex analysis.
\end{proof}
Next we compute the second-order subdifferential of the cost function in the Lasso problem.
\begin{Proposition}[\bf second-order subdifferential calculation]\label{secondordercalculus} For the function $\varphi$ from \eqref{Lasso} and any $(x,y)\in\mbox{\rm gph}\,\partial\varphi$ we have the second-order subdifferential formula
\begin{equation*}
\partial^2\varphi(x,y)(v)=\Big\{w\in\mathbb{R}^n\;\Big|\;\Big(\frac{1}{\mu}(w-A^*Av)_i,-v_i\Big)\in G\Big(x_i,\frac{1}{\mu}\big(y-A^*(Ax-b)\big)_i\Big),\;i=1,\ldots,n\Big\}
\end{equation*}
valid for all $v=(v_1,\ldots,v_n)\in\mathbb{R}^n$, where the mapping $G\colon\mathbb{R}^2\rightrightarrows\mathbb{R}^2$ is defined by
\begin{equation}\label{G}
G(t,p):=\begin{cases}
\{0\}\times\mathbb{R}&\text{\rm if}\quad t\ne 0,\;\in\{-1,1\},\\
\mathbb{R}\times\{0\}&\text{\rm if}\quad t=0,\;p\in(-1,1),\\
(\mathbb{R}_{+}\times\mathbb{R}_{-})\cup(\{0\}\times\mathbb{R})\cup(\mathbb{R}\times\{0\})&\text{\rm if}\quad t=0\;p=-1,\\
(\mathbb{R}_{-}\times\mathbb{R}_{+})\cup(\{0\}\times\mathbb{R})\cup(\mathbb{R}\times\{0\})&\text{\rm if}\quad t=0,\;p=1,\\
\emptyset&\text{\rm otherwise}.
\end{cases}
\end{equation}
\end{Proposition}
\begin{proof} Fix any $(x,y)\in\mbox{\rm gph}\,\partial\varphi$. Applying the second-order subdifferentials sum rule from Mordukhovich \cite[Proposition~1.121]{Mordukhovich06} to the summation function \eqref{Lasso} yields
\begin{equation*}
\partial^2\varphi(x,y)(v)=A^*Av+\mu\partial^2(\|\cdot\|_1)\Big(x,\frac{1}{\mu}\big(y-A^*(Ax-b)\big)\Big)(v)\;\mbox{ for all }\;v\in\mathbb{R}^n.
\end{equation*}
This tells us that $w\in\partial^2\varphi(x,y)(v)$ if and only if
\begin{equation}\label{second1}
\frac{1}{\mu}(w-A^*Av)\in\partial^2(\|\cdot\|_1)\Big(x,\frac{1}{\mu}\big(y-A^*(Ax-b)\big)\Big)(v),\quad v\in\mathbb{R}^n.
\end{equation}
Consider further the function $\psi(t):=|t|$ for $t\in\mathbb{R}$ and observe that
\begin{equation*}
\mbox{\rm gph}\,\partial\psi=\big((\mathbb{R}\setminus\{0\})\times\{-1,1\}\big)\cup\big(\{0\}\times[-1,1]\big)\;\mbox{ and }\;\|x_1\|_1=\sum_{i=1}^n\psi(x_i),\quad x\in\mathbb{R}^n.
\end{equation*}
Furthermore, we see that $N_{\text{gph}\,\partial\psi}=G$ for the mapping $G$ defined in \eqref{G}. It follows from Mordukhovich and Outrata \cite[Theorem~4.3]{BorisOutrata} that
\begin{equation*}
\partial^2(\|\cdot\|_1)\Big(x,\frac{1}{\mu}\big(y-A^*(Ax-b)\big)\Big)(v)=\Big\{u\in\mathbb{R}^n\;\Big|\;(u_i,-v_i)\in N_{\text{gph}\,\partial\psi}\Big(x_i,\frac{1}{\mu}\big(y-A^*(Ax-b)\big)_i\Big)\Big\}.
\end{equation*}
Combining the latter with \eqref{second1}, we arrive at the claimed formula for $\partial^2\varphi(x,y)$.
\end{proof}
The following proposition explicitly computes the proximal mapping \eqref{Prox} of the Lasso function $\varphi$ from \eqref{Lasso} in the setting where the matrix $A$ is diagonal. For simplicity, confine ourselves to the case where $\lambda=1$ in \eqref{Prox}.
\begin{Proposition}[\bf computation of the proximal mapping]\label{proxcalculus} Let $A$ in \eqref{Lasso} be a diagonal $n\times n$ matrix $A:=\text{\rm diag}(a_1,\ldots,a_n)$ with $a_i>0$ for all $i=1,\ldots,n$, and let $b=(b_1,\ldots,b_n)\in\mathbb{R}^n$. Then the proximal mapping $\text{\rm Prox}_{\varphi}\colon\mathbb{R}^n\to\mathbb{R}^n$ is computed by
\begin{equation}\label{proximalofvarphi}
\text{\rm Prox}_{\varphi}(x)=\displaystyle\prod_{i=1}^n\left[\left|\frac{x_i+a_ib_i}{a_i^2+1}\right|-\frac{\mu}{a_i^2+1}\right]_{+}\text{\rm sgn}\left(\frac{x_i+a_ib_i}{a_i^2+1}\right).
\end{equation}
\end{Proposition}
\begin{proof} The diagonality of $A$ allows us to represent the function $\varphi$ from \eqref{Lasso} in the form
\begin{equation*}
\varphi(x)=\sum_{i=1}^{n}\varphi_i(x_i)+\frac{1}{2}\sum_{i=1}^{n}b_i^2,
\end{equation*}
where the univariate functions $\varphi_i\colon\mathbb{R}\to\mathbb{R}$ are defined by
\begin{equation*}
\varphi_i(t):=\frac{1}{2}a_i^2t^2-a_i b_i t+\mu|t|\;\mbox{ for all }\;t\in\mathbb{R}\;\mbox{ and }\;i=1,\ldots,n.
\end{equation*}
The proximal mapping formula for separable functions taken from Beck \cite[Theorem~6.5]{Beck} yields
\begin{equation}\label{prox1}
\text{\rm Prox}_{\varphi}(x)=\text{\rm Prox}_{\varphi_1}(x_1)\times\ldots\times\text{\rm Prox}_{\varphi_n}(x_n).
\end{equation}
Applying now Lemma~6.5 and Theorem~6.13 from the aforementioned book by Beck gives us the following representations for all $i=1,\ldots,n$:
\begin{equation*}
\text{\rm Prox}_{\varphi_i}(t)=\text{\rm Prox}_{\left(\frac{\mu}{a_i^2+1}|\cdot|\right)}\left(\frac{t+a_ib_i}{a_i^2+1}\right)=\left[\left|\frac{t+a_ib_i}{a_i^2+1}\right|-\frac{\mu}{a_i^2+1}\right]_{+}\text{sgn}\left(\frac{t+a_ib_i}{a_i^2+1} \right).
\end{equation*}
Combining the latter with \eqref{prox1}, we arrive at \eqref{proximalofvarphi} and thus complete the proof.
\end{proof}
The next theorem provides a verifiable condition on the matrix $A$ under which all the assumptions of Theorem~\ref{localNM3} hold for the Lasso problem \eqref{Lasso}, and hence we can run Algorithm~\ref{NM3}, with computing its parameters entirely in terms of the given data of \eqref{Lasso}, to determine an optimal solution to the Lasso problem.
\begin{Theorem}[\bf solving Lasso]\label{lasso-solv} If the matrix $Q:=A^*A$ is positive-definite, then all the assumptions of Theorem~{\rm\ref{localNM3}} are satisfied at the reference optimal solution $\bar{x}$ to the Lasso problem \eqref{Lasso}, which is equivalently described by the subgradient inclusion \eqref{sumLasso}. Thus Algorithm~{\rm\ref{NM3}} with $U_\lambda=\mathbb{R}^n$ in Step~$0$ and with the data $\partial\varphi$, $\partial^2\varphi$, and ${\rm Prox}_{\varphi}$ explicitly computed in terms of $(A,b,\mu)$ by the formulas in Propositions~{\rm\ref{calc-sub}}--{\rm\ref{proxcalculus}}, respectively, is well-defined around $\bar{x}$, and the sequence of its iterates $\{x^k\}$ superlinearly converges to $\bar{x}$ as $k\to\infty$.
\end{Theorem}
\begin{proof} By taking into account the solvability results of Section~\ref{sec:solvN}, we see that all the assumptions of Theorem~\ref{localNM3} hold for the Lasso problem \eqref{Lasso} if the subgradient mapping $\partial\varphi$ is strongly metrically regular around $(\bar{x},0)$ and semismooth$^*$ at this point. Now we show more: both these properties are satisfied for $\varphi$ from \eqref{Lasso} at {\em every point} in the subdifferential graph $\mbox{\rm gph}\,\partial\varphi$.
Indeed, it follows from the positive-definiteness of $Q$ that $\varphi$ is strongly convex on $\mathbb{R}^n$, and hence the mapping $\partial\varphi$ is strongly maximal monotone. Then it follows from Lemma~3.3 in Mordukhovich and Nghia \cite{MorduNghia1} (see also Mordukhovich \cite[Theorem~5.13]{Mor18}) that $\partial\varphi$ is strongly metrically regular at every point of the subdifferential graph $\mbox{\rm gph}\,\partial\varphi$. Furthermore, we get from the formula for the mapping $F$ in Proposition~\ref{calc-sub} that the graph of $F$ is the union of finitely many closed convex sets, and hence $F$ is semismooth$^*$ at all the points in its graph. Since the function $x\mapsto A^*(Ax-b)$ is continuously differentiable on $\mathbb{R}^n$, it follows from the subdifferential formula \eqref{norm1} and Proposition~3.6 in Gfrerer and Outrata \cite{Helmut} that the mapping $\partial\varphi$ is semismooth$^*$ at every point of its graph. All of this allows us to deduce the conclusions of the theorem from those in Theorem~\ref{localNM3}, where the specifications of $\lambda$ and $U_\lambda$ are due to the convexity of $\varphi$.
\end{proof}
To simplify the subsequent numerical implementations of Algorithm~\ref{NM3}, consider further the Lasso problem \eqref{Lasso} with $A$ being such an $m\times n$ matrix that
\begin{equation}\label{diagonalmatrix}
Q:=A^*A =\text{diag}(q_1,\ldots,q_n),
\end{equation}
where $q_i>0$ for all $i=1,\ldots,n$, $b\in\mathbb{R}^m$, and $\mu>0$. Denote
\begin{equation*}
\widetilde{A}:=\sqrt{Q}:=\text{diag}(a_1,\ldots,a_n)\quad\text{and}\quad\widetilde{b}:=(A\widetilde{A}^{-1})^*b
\end{equation*}
with $a_i:=\sqrt{q_i}$ for all $i=1,\ldots,n$. It is easy to see that the Lasso problem \eqref{Lasso} is equivalent the following problem of unconstrained convex optimization:
\begin{eqnarray}\label{Lassotransform}
\text{minimize}&\text{}&\widetilde{\varphi}(x):=\frac{1}{2}\|\widetilde{A}x-\widetilde{b}\|_2^2+\mu\|x\|_1,\quad x\in\mathbb{R}^n.
\end{eqnarray}
To apply Algorithm~\ref{NM3} to computing an optimal solution to \eqref{Lassotransform}, take $\lambda=1$ and $U_\lambda=\mathbb{R}^n$ due to the convexity of $\widetilde{\varphi}$. Consider the sequence $\{x^k\}$, $\{v^k\}$, and $\{d^k\}$ generated by Algorithm~\ref{NM3} for \eqref{Lassotransform}. The results above give us the following explicit calculation formulas for all $i=1,\ldots,n$:
\begin{eqnarray}\label{v}
v^k_i&=&x^k_i-\text{\rm Prox}_\varphi(x^k_i)\nonumber\\
&=&x^k_i-\left[\left|\frac{x^k_i+a_ib_i}{a_i^2+1}\;\right|\;-\frac{\mu}{a_i^2+1}\right]_{+}\text{\rm sgn}\left(\frac{x^k_i+a_ib_i}{a_i^2+1}\right),
\end{eqnarray}
\begin{equation}\label{d}
d^k=\begin{cases}
-v^k_i-\displaystyle\frac{v^k_i}{a_i^2}&\text{if}\quad x^k_i-v^k_i\ne 0,\\
-v^k_i&\text{if}\quad x^k_i-v^k_i=0.
\end{cases}
\end{equation}\vspace*{0.05in}
Finally, let us illustrate the obtained calculation formulas by the following example.
\begin{Example}\label{calNMLasso}\rm Consider the Lasso problem \eqref{Lasso} with $m=4$, $n=3$,
\begin{equation*}
A:=\begin{pmatrix}
4/7 & 3 & 6\\
12/7& 2 & -3\\
6/7 & -6 & 2\\
0 & 24 & 0\\
\end{pmatrix},\quad
b:=\left(\frac{104}{49},\frac{347}{49},-\frac{649}{49},0\right)^*,\;\mbox{ and }\;\mu:=\frac{1}{3}.
\end{equation*}
Then we have that the quadratic matrix
\begin{equation*}
Q:=A^*A=\begin{pmatrix}
4 & 0 & 0\\
0 & 625 & 0\\
0 & 0 & 49
\end{pmatrix}
\end{equation*}
is diagonal and positive-definite. The data of the equivalent problem \eqref{Lassotransform} are
\begin{equation*}
\widetilde{A}=\begin{pmatrix}
2 & 0 & 0\\
0 & 25 & 0\\
0 & 0 &7
\end{pmatrix}\quad\text{and}\quad\widetilde{b}=(1,4,-5).
\end{equation*}
Using \eqref{v} and \eqref{d}, we collect the calculations for $v^k$, $d^k$, and $x^k$ in the following table:
\begin{center}
\begin{tabular}{ || p{1em} | c | c| c| c| | }
\hline
$k$ & $x^k$ & $v^k$ & $\|v^k\|$ & $d^k$ \\[0.5em]\hline\hline
0 & $(-2,0,0)$ & $(-2,-0.1592,0.6933)$ & $2.1227$ & $(2,0.1592,-0.6933)$\\
1 & $(0, 0.1595, -0.7075)$ & $(-0.3333,0,0)$ & $0.3333$ & $(0.4167,0,0)$\\
2 & $(0.4167, 0.1595, -0.7075)$ & $(0,0,0)$ & $0$ & Stop the algorithm\\
\hline
\end{tabular}
\end{center}
\end{Example}\vspace*{-0.05in}
\section{Concluding Remarks}\label{conc}\vspace*{-0.05in}
This paper proposes and develops a generalized Newton method to solve gradient and subgradient systems by using second-order subdifferentials of ${\cal C}^{1,1}$ and extended-real-valued prox-regular functions, respectively, together with the appropriate tools of second-order variational analysis. The suggested two algorithms for gradient and subgradient systems are comprehensively investigated with establishing verifiable conditions for their well-posedness/solvability and local superlinear convergence. Applications to solving a major class of nonsmooth Lasso problems are given and employed in computation.
The results of this paper concern far-going nonsmooth extensions of the basic Newton method for ${\cal C}^2$-smooth functions. Besides further implementations of the obtained results and their applications to practical models, in our future research we intend to develop other, more advanced versions of nonsmooth second-order algorithms of the Newton and quasi-Newton types with establishing their local and global convergence as well as efficient specifications for important classes of variational inequalities and constrained optimization.\\[1ex]
{\bf Acknowledgements.} Research of the second and third authors was partly supported by the US National Science Foundation under grants DMS-1512846 and DMS-1808978, and by the US Air Force Office of Scientific Research under grant \#15RT0462. The research of the second author was also supported by the Australian Research Council under Discovery Project DP-190100555.\vspace*{-0.1in}
\small
|
3,212,635,537,455 | arxiv | \section{Introduction}
Understanding the center of the enveloping algebra of a finite dimensional Lie algebra
in characteristic $p>0$ is of fundamental importance in the modular representation theory of Lie algebras.
Many such Lie algebras arise from a base change from a Lie algebra in characteristic 0.
Let $\mathfrak{g}$ be a Lie algebra of an algebraic group over $S\subset\mathbb{C}$-a finitely generated ring,
let $S\to \bold{k}$ be a base change to a characteristic $p$ field. Then the center of the enveloping algebra $U(\mathfrak{g}_{\bold{k}})$
contains two distinguished subalgebras: the $p$-center $Z_p$ which is generated by elements of the form $g^p-g^{[p]}, g\in \mathfrak{g}_{\bold{k}},$
and the image of $Z(U(\mathfrak{g}))$ in $Z(U(\mathfrak{g}_{\bold{k}})),$ to be denoted by $Z_{\text{HC}},$ (the Harish-Chandra part of the center).
Thus, it is a very natural to ask whether for large enough $p$ the center of $U(\mathfrak{g}_{\bold{k}})$
is generated by $Z_p$ and $Z_{\text{HC}}$ (stated as a conjecture in \cite{K}).
Unfortunately the answer is already negative
for indecomposable 3-dimensional solvable Lie algebras (see Remark \ref{remark}). For the case of a semi-simple $\mathfrak{g}$, the answer is positive
as follows from the classical theorem of Veldkamp \cite{V}.
The main result of this paper can be seen as a generalization of Veldkamp's theorem for a class of Lie algebras.
\begin{theorem}\label{center}
Let $\mathfrak{g}$ be an algebraic Lie algebra over a finitely generated ring $S\subset\mathbb{C}$,
such that $\Sym \mathfrak{g}$ has no nontrivial $\mathfrak{g}$-semi-invariants, and
$\Sym(\mathfrak{g})^{\mathfrak{g}}=S[f_1,\cdots, f_n]$ is a polynomial algebra.
Let $g_i\in Z(U(\mathfrak{g}))$ be the symmetrization of $f_i, 1\leq i\leq n.$
Then for all primes $p\gg 0$ and a base change $S\to \bold{k}$ to a field of characteristic $p,$ the center
of $U(\mathfrak{g}_{\bold{k}})$ is a
free $Z_p$-module
with a basis $\lbrace g^{\alpha}=g_1^{\alpha_1}\cdots g_n^{\alpha_n}, 0\leq\alpha_i<p\rbrace.$
Moreover $Z(U(\mathfrak{g}_{\bold{k}}))$ is a complete intersection ring and
$$Z(U(\mathfrak{g}_{\bold{k}}))\cong Z_p\otimes_{Z_p\cap Z_{\HC}} Z_{\HC}, \quad Z_{\HC}=U(\mathfrak{g})^{G_{\bold{k}}}.$$
\end{theorem}
This result was obtained in \cite{T1} under the addition assumption that
the coadjoint action of $G$ (algebraic group corresponding to $\mf{g}$) on $\spec(\Sym(\mathfrak{g})/(f_1,\cdots, f_n))\subset \mathfrak{g}^*$
has an open dense orbit.
\begin{remark}\label{remark}
Let $\mathfrak{g}$ be the following
3-dimensional Lie algebra: $\mathfrak{g}=\mathbb{C}x\oplus \mathbb{C}y\oplus\mathbb{C}z$
with the bracket $$[z,x]=nx, [z,y]=my, [x, y]=0, n, m\in\mathbb{N}.$$
Then for large enough $p$ and a characteristic $p$ field $\bold{k}$, we have that $Z_{\HC}=\mathbf{k}.$
Let $1\leq a, b< p$ be such that $na=bm.$ Then $x^ay^{p-b}\in Z(U(\mathfrak{g}_{\bold{k}}))\setminus Z_p.$
Therefore $Z(U(\mathfrak{g}_{\bold{k}}))$ is not generated by $Z_p, Z_{\text{HC}}.$ On the other hand, $x, y\in \Sym(\mathfrak{g})$ represent
nontrivial semi-invariants.
\end{remark}
\begin{remark}\label{examples}
Let us discuss known examples of Lie algebras $\mf{g}$ satisfying the assumption in Theorem \ref{center}
that $\Sym(\mathfrak{g})$ has no nontrivial $\mathfrak{g}$-semi-invariants and
$\Sym(\mathfrak{g})^{\mathfrak{g}}$ is a polynomial algebra in detail.
Let $\mathfrak{g}$ be a semi-simple Lie algebra, then for the corresponding multiparameter Takiff algebras (also known as truncated multicurrent algebras)
$\mathfrak{g}_m=\mathfrak{g}\otimes_{\mathbb{C}}\mathbb{C}[t_1,\cdots, t_n]/(t_1^{m_1},\cdots, t_n^{m_n})$ it is known that
$\Sym(\mathfrak{g}_m)^{\mathfrak{g}_m}$ is a polynomial algebra \cite{MS} (see also \cite{PY}). Since $\mathfrak{g}_m$ is a perfect Lie algebra,
it follows that multi-parameter Takiff algebra satisfy the assumption in Theorem \ref{center}.
There are a lot of results in the literature whether $L=\mathfrak{g}\ltimes V$ has the property that $\Sym(L)^L$ is a polynomial algebra, where
$\mathfrak{g}$ is a simple Lie algebra and $V$ its representation (as an incomplete list, see \cite{K1}, \cite{K2}, \cite{P}, \cite{PY}, \cite{Y}). Just few examples to name, the following
perfect algebras satisfy the assumption:
$\mf{sl}_n\ltimes \mathbb{C}^n, \mf{so}_n\ltimes\mathbb{C}^n, \mf{sp}_{2n}\ltimes \mathbb{C}^{2n}.$
Since nilpotent Lie algebras have no nontrivial semi-invariants, any nilpotent Lie algebra $\mathfrak{g}$
with the property that $\Sym (\mathfrak{g})^{\mathfrak{g}}$ is a polynomial algebra satisfies assumptions of Theorem \ref{center}.
The class of such Lie algebras is quite large: all but 3 indecomposable nilpotent Lie algebras of dimension $\leq 7$ \cite{O1}, nilpotent Lie algebras of index at most 2 \cite{O2}, nilradical of a borel subalgebra of a semi-simple Lie algebra [\cite{O2}, Corollary 32].
\end{remark}
The proof of Theorem \ref{center} crucially relies on the proof of the first Kac-Weisfeler conjecture for very
large primes given in \cite{T2}.
Next we apply Theorem \ref{center} to the isomorphism problem of enveloping algebras-a well-known open problem in ring theory.
Recall that it asks whether a $\mathbb{C}$-algebra isomorphism between enveloping
algebras of Lie algebras implies an isomorphism of the underlying Lie algebras.
For the background and the detailed discussion of this problem we refer the reader to the survey article by Usefi \cite{U}.
Very recently, the solution of the isomorphism problem of enveloping algebras for nilpotent Lie algebras was presented in \cite{CPNW},
using techniques of higher category theory.
In analogy with the derived isomorphism problem for rings of differential operators
for smooth affine varieties, in \cite{T1} we considered the following natural generalization.
\begin{conjecture}
Let $\mathfrak{g}, \mathfrak{g'}$ be finite dimensional Lie algebras over $\mathbb{C}.$
If the derived categories of bounded complexes of $U(\mathfrak{g})$-modules and $ U(\mathfrak{g'})$-modules are equivalent,
then $\mathfrak{g}\cong \mathfrak{g'}.$
\end{conjecture}
In our approach to the above conjecture we follow the well-known blueprint of "dequantization"
by reducing to the modulo large prime $p$ technique, which allows a translation of questions
about various quantizations to the ones about Poisson algebras, inspired largely by the proof of Belov-kanel an
Kontsevich of equivalence between the Jacobian and Dixmier conjectures.
Using this approach, we proved in \cite{T1} that certain class of Lie algebras that includes Frobenius Lie algebras
are derived invariants
of their enveloping algebras.
As an application of Theorem \ref{center} to the derived equivalence problem
we have the following.
\begin{theorem}\label{iso}
Let $\mathfrak{g}, \mathfrak{g'}$ be algebraic Lie algebras satisfying assumptions in Theorem \ref{center}. If $U(\mathfrak{g})$
is derived equivalent to $U(\mathfrak{g'})$ then $\mathfrak{g}/Z(\mathfrak{g})\cong \mathfrak{g'}/Z(\mathfrak{g'}).$
\end{theorem}
\section{The first Kac-Weisfeiler conjecture}
In this section we recall some results associated with the first Kac-Weisfeiler conjecture that are used
in proof of our main results.
Recall that the first Kac-Weisfeiler conjecture asserts that for a $p$-restricted Lie algebra $\mathfrak{g}$
over
$\bold{k}$, the maximal possible dimension of an irreducible $\mathfrak{g}$-module is
$p^{\frac{1}{2}(\dim (\mathfrak{g})- \ndex(\mathfrak{g}))}$. Equivalently
the rank of $U(\mathfrak{g})$ over its center
equals $p^{\dim (\mathfrak{g})-\ndex(\mathfrak{g})}.$
Let $\mathfrak{g}$ be a Lie algebra of an algebraic group $G$ defined over a finitely generated
ring $S\subset \mathbb{C}.$ Then for all $p\gg 0$ and a base change $S\to\mathbf{k}$ to an algebraically
closed field of characteristic $p,$ the first Kac-Weisfeiler conjecture was established for $\mathfrak{g}_{\bold{k}}$
by Martin, Stewart, and Topley. Also proved independently by the author.
Namely we make use of the following result.
\begin{theorem}[\cite{T2}, Theorems 3.8 and 3.9]\label{KW}
Let $\mathfrak{g}$ be an algebraic Lie algebra over a finitely generated ring $S\subset\mathbb{C}.$
Then for all $p\gg 0$ and a base change $S\to\mathbf{k}$ to an algebraically closed field of characteristic $p$,
the fraction field of $Z(U(\mathfrak{g}_{\bold{k}}))$
is generated by the image of the center of $\Frac(U(\mathfrak{g}))$ and $\Frac(Z_p(\mathfrak{g}_{\bold{k}})).$
Also, the following equality of degrees of field extensions holds:\
$$[\Frac(Z(U(\mathfrak{g}_{\bold{k}}))): \Frac(Z_p(\mathfrak{g}_{\bold{k}}))]=p^{\ndex(\mathfrak{g})}=
[\Frac(\Sym(\mathfrak{g}_{\bold{k}})^{\mathfrak{g}_{\bold{k}}}): \Frac(\Sym(\mathfrak{g}_{\bold{k}})^p)]$$
\end{theorem}
\begin{remark}
The equality
$$[\Frac(\Sym(\mathfrak{g}_{\bold{k}})^{\mathfrak{g}_{\bold{k}}}): \Frac(\Sym(\mathfrak{g}_{\bold{k}}))^p]=p^{\ndex(\mathfrak{g})}$$
follows from the fact that on the one hand (as proved in [\cite{T2} Theorem 3.8, 3.9])
$$[\Frac(\Gr Z(U(\mathfrak{g}_{\bold{k}})): \Frac(\Sym(\mathfrak{g}_{\bold{k}}))^p]\geq p^{\ndex(\mathfrak{g})},$$
and on the other hand it was proved in \cite{PS} that
$$[\Frac(\Sym\mathfrak{g}_{\bold{k}}): \Frac(\Sym(\mathfrak{g}_{\bold{k}})^{\mathfrak{g}_{\bold{k}}})]\geq p^{\dim (\mathfrak{g})-\ndex(\mathfrak{g})}.$$
\end{remark}
We also need to recall the following simple result from commutative algebra (for a proof see [\cite{T2} Lemma 3.11]. )
\begin{lemma}\label{rank}
Let $S\subset\mathbb{C}$ be a finitely generated ring.
Let $A$ be a finitely generated commutative algebra over $S$ such that $A_{\mathbb{C}}$ is a domain.
Let $B\subset A$ be a finitely generated $S$-subalgebra.
Then for all $p\gg 0$ and a base change $S\to\bold{k}$ to an algebraically closed field $\bold{k}$ of characteristic $p$
the rank of $A_{\bold{k}}$ over $B_{\bold{k}}A_{\bold{k}}^p$ is $p^{\dim(A)-\dim(B)}.$
\end{lemma}
\section{The proof of Theorem \ref{center}}
We crucially rely on the following result from Panyushev-Yakimova [\cite{PY}, Remark 1.3]
(see also [\cite{PY}, Proposition 1.2], [\cite{JS} Proposition 5.2]).
\begin{prop}\label{codim2}
Let $\mathfrak{g}$ be a an algebraic Lie algebra over $\mathbb{C}$ such that
$\Sym(\mathfrak{g})$ has no nontrivial $\mathfrak{g}$-semi-invariants.
Let $\mathcal{O}=\Sym (\mathfrak{g})^{\mathfrak{g}}$ be finitely generated.
Put $Y=\spec \mathcal{O}.$ We have the quotient map $\pi:\mathfrak{g}^*=\spec \Sym(\mathfrak{g})\to Y.$
Denote by $Y_{sm}$ the smooth locus of $Y.$
Let $U=\lbrace x\in \mathfrak{g}^*, \pi(x)\in Y_{sm}, d\pi_{x} \text{is onto}\rbrace.$
Then $\mathfrak{g}^*\setminus U$
has codimension $\geq 2$ in $\mathfrak{g}^*.$
\end{prop}
\begin{proof}[Proof of Theorem \ref{center}]
Since $\mathfrak{g}$ has no nontrivial semi-invariants in $\Sym (\mathfrak{g})$, it follows easily that
$$(\Frac(\Sym (\mathfrak{g})))^{\mathfrak{g}}=\Frac(\Sym (\mathfrak{g})^{\mathfrak{g}}).$$
Put
$$\mathcal{O}=\bold{k}[f_1,\cdots, f_n], \quad A=\Sym(\mathfrak{g}_{\bold{k}}),\quad B=\Sym(\mathfrak{g}_{\bold{k}})^p\mathcal{O},\quad
B'=(\Sym \mathfrak{g}_{\bold{k}})^{\mathfrak{g}_{\bold{k}}}.$$
Clearly $B\subset B'.$ Let $K=\Frac(A)$ and $K_0=\Frac(B).$
We first claim that $$[K_0:K^p]=p^n.$$
Indeed, this follows at once since the degree of $K$ over $K_0$ is $p^{\dim(\mathfrak{g})-n}$ by Lemma \ref{rank},
and
$$[K: K^p]=p^{\dim \mathfrak{g}}.$$
Next we argue that $B$ a normal domain.
Indeed, it follows from Proposition \ref{codim2} that for all $p\gg 0$ and a base change $S\to \mathbf{k},$ the compliment of
$$U_{\bold{k}}=\lbrace x\in \mathfrak{g}_{\bold{k}}^*, d\pi_{x} \text{ is onto}\rbrace.$$
in ${\mathfrak{g}}_{\bold{k}}^*$ has codimension $\geq 2.$
We claim that the multiplication map $\phi$ induces an isomorphism
$$ \phi:A^p\otimes_{\mathcal{O}^p} \mathcal{O}\cong B.$$
Indeed, $\phi$ is a surjective map from a free $A^p$-module of rank $p^n$ onto a $A^p$-module of rank $p^n.$ Thus $\phi$ must be an isomorphism.
If $I\in U_{\bold{k}},$ put $I'=I\cap\mathcal{O}.$ It follows easily that $A^p_{I}\otimes_{\mathcal{O}_{I}^p} \mathcal{O}_{I'}$ is a regular ring.
Hence $B$ is a Cohen-Macaulay ring regular in codimension 1. Therefore it is a normal domain by Serre's criterion of normality.
It follows that $B'$ and $B$ have the same field of fractions
as they are both extensions of degree $p^n$ of $A^p$ by Theorem \ref{KW} and
$B\subset B'$. Thus, normality of $B$ yields that $B=B'.$ Hence
$$\Sym (\mathfrak{g}_{\bold{k}})^{\mathfrak{g}_{\bold{k}}}=\Sym (\mathfrak{g}_{\bold{k}})^p[f_1,\cdots,f_m],\quad
Z(U(\mathfrak{g}_{\bold{k}}))=Z_p[g_1,\cdots, g_m].$$
Next we show that $B$ is a free $A^p$-module with basis $\lbrace f^{\alpha}=\prod_{i=1}^n f_i^{\alpha_i}, \alpha_i<p\rbrace.$
Indeed, since $[K_0:K^p]=p^n,$ it follows that $\lbrace f^{\alpha}, \alpha_i<p\rbrace$
are linearly independent over $K^p.$ In particular they form a basis of $B$ over $A^p.$
Hence, $Z(U(\mathfrak{g}_{\bold{k}}))$ is a free module over $Z_p$ with a basis $\lbrace g^{\alpha}, \alpha_i<p\rbrace$ as desired.
Thus we obtain that
$$ \Gr Z(U(\mathfrak{g}_{\bold{k}}))=B=(\Sym (\mathfrak{g}_{\bold{k}}))^{\mathfrak{g}_\bold{k}}.$$
We have the natural surjective homomorphism (that restricts to the Frobenius homomorphism on $A$ and sends $x_i$ to $f_i^p$)
$$A[x_1,\cdots, x_n]/(x_1^p-f_1,\cdots, x_n^p-f_n)\to B,$$
which must be an isomorphism since both rings are free $A^p$-modules of rank $p^n.$
Since $(x_1^p-f_1,\cdots, x_n^p-f_n)$
is a regular sequence in $A[x_1,\cdots, x_n],$ it follows that $B$ is a complete intersection.
Therefore $Z(U(\mathfrak{g}_{\bold{k}}))$ is also a complete intersection ring.
Our next goal is show that $\Sym (\mathfrak{g}_{\bold{k}})^{G_{\bold{k}}}=\bold{k}[f_1,\cdots, f_n],$
which implies $U(\mathfrak{g})^{G_{\bold{k}}}=Z_{\HC}.$
Let $x\in \Sym (\mathfrak{g}_{\bold{k}})^{G_{\bold{k}}}\subset B.$
As $B$ is a free $\Sym (\mathfrak{g}_{\bold{k}})^p$ module with basis $\lbrace f^{\alpha}=\prod_{i=1}^n f_i^{\alpha_i}, \alpha_i<p\rbrace,$
we may write
$$x=\sum_{\alpha}x_{\alpha}f^{\alpha}, \quad x_{\alpha}\in (\Sym \mathfrak{g}_{\bold{k}})^p, \alpha_i<p.$$
Then $$x_{\alpha}\in (\Sym (\mathfrak{g}_{\bold{k}})^{G_{\bold{k}}})^p.$$
Replacing $x$ by $x_{\alpha}^{\frac{1}{p}}$ and continuing in this manner, we obtain that
$$x\in \bigcap_{n}(\Sym (\mathfrak{g}_{\bold{k}}))^{p^n}\mathcal{O}=\mathcal{O}.$$
We also get that
$$Z_p\cap Z_{\HC}=Z_p^{G_{\bold{k}}}.$$
So $$\Gr(Z_p\cap Z_{\HC}) =\bold{k}[f_1^p,\cdots, f_n^p].$$
Hence $Z_{\HC}$ is a free
$Z_p\cap Z_{\HC}$-module of rank $p^n.$ Therefore the natural ring homomorphism
$$Z_p\otimes_{Z_p\cap Z_{\HC}}Z_{\HC}\to Z(U(\mathfrak{g}_{\bold{k}}))$$
is a surjective homomorphism of free $Z_p$-modules of rank $p^{n}.$ Thus it is an isomorphism as desired.
\end{proof}
\section{Applications to rigidity of enveloping algebras}
Recall that given an associative flat $\mathbb{Z}$-algebra $R$ and a prime number $p,$
then the center $Z(R/pR)$ of its reduction modulo $p$ acquires the natural Poisson bracket,
defined as follows. Given central elements $a, b\in Z(R/pR)$, let $z, w\in R$ be their lifts respectively.
Then the Poisson bracket $\lbrace a, b \rbrace$ is defined to be $$\frac{1}{p}[z, w] \mod p\in Z(R/pR).$$
\noindent In particular, given a a Lie algebra $\mathfrak{g}$ over $S\subset\mathbb{C}$--a finitely generated ring
such that $\mathfrak{g}$ is a finite free $S$-module, then for a prime $p>0,$ the center
$Z(U(\mathfrak{g}_p))$
of the enveloping algebra of $\mathfrak{g}_{p}=\mathfrak{g}/p\mathfrak{g}$ is equipped with the natural $S/pS$-linear Poisson bracket
defined as above.
The significance of understanding $Z(U(\mathfrak{g}_p))$ (as a Poisson algebra)
in regards with the derived isomorphism problem above lies in its derived invariance: if $U(\mathfrak{g})$
and $U(\mathfrak{g'})$ are derived equivalent, then $Z(U(\mathfrak{g}_{p}))\cong Z(U(\mathfrak{g'}_{p}))$
as Poisson $S/pS$-algebras for $p\gg 0.$ This easily follows from the derived invariance of the Hochschild cohomology and
the Gersenhaber bracket (see [\cite{T1} Lemma 4]).
Recall that if $I\subset Z(U(\mathfrak{g}_p))$ is a Poisson ideal,
then the Poisson bracket induces Lie bracket on $I/I^2.$ In particular, if $\mf{m}$ is a Poisson ideal such that $Z(U(\mathfrak{g}_p))/\mf{m}=S/pS,$
then $\mf{m}/\mf{m}^2$ is a finite $S/pS$-Lie algebra.
Therefore, the collection of isomorphisms classes of $S/pS$-Lie algebras $\mf{m}/\mf{m}^2,$
as $\mf{m}$ ranges over maximal Poisson ideals of $Z(U(\mathfrak{g}_p))$ (so $Z(U(\mathfrak{g}_p))/\mf{m}=S/pS$)
is a derived invariant of $U(\mathfrak{g})$ (for $p\gg 0$). The significance of this derived invariant of $\mathfrak{g}$ is highlighted by
the fact that given a Poisson ideal $\mathfrak{m}\subset Z(U(\mathfrak{g}_p))$ as above, as the key computation by Kac and Radul shows (see Lemma \ref{key} below),
there is a canonical Lie algebra
homomorphism $\mathfrak{g}_{p}\to \mf{m}/\mf{m}^2$ (when $\mathfrak{g}$ is an algebraic Lie algebra).
There is one distinguished such maximal Poisson ideal--the augmentation ideal
$\mathfrak{m}(\mathfrak{g}_p)=Z(U(\mathfrak{g}_p))\cap \mathfrak{g}_{p}U(\mathfrak{g}_{p}).$
Nex we need to recall a key computation of the Poisson bracket for restricted Lie
algebras due to Kac and Radul \cite{KR}.
Let $R$ be a commutative reduced ring of characteristic $p>0.$
Let $\mathfrak{g}$ be a restricted Lie algebra over $R$
with the restricted structure map $x\to x^{[p]}, x\in \mathfrak{g}.$
It is well-known that the map $x\to x^{p}-x^{[p]}$ induces
homomorphism of $R$-algebras $$i:\Sym (\mathfrak{g})\to Z_p(\mathfrak{g}),$$
where $Z_p(\mathfrak{g})$ is viewed as an $R$-algebra via the Frobenius map $F:R\to R.$
The homomorphism $i$ is an isomorphism when $R$ is perfect.
Recall also that
the Lie algebra bracket on $\mathfrak{g}$ defines the Kirillov-Kostant Poisson bracket on the symmetric algebra $\Sym(\mathfrak{g}).$
The following is the above mentioned key result from \cite{KR}.
\begin{lemma}\label{key}
Let $S$ be a finitely generated integral domain over $\mathbb{Z}.$
Let $\mathfrak{g}$ be an algebraic Lie algebra over $S.$
Then $Z_p(\mathfrak{g}_p)$ is a Poisson subalgebra of $Z(U(\mathfrak{g}_p)),$ moreover the induced Poisson
bracket coincides with the negative of the Kirillov-Kostant bracket:
$$\lbrace a^p-a^{[p]}, b^p-b^{[p]}\rbrace=-([a, b]^p-[a,b]^{[p]}),\quad a\in \mathfrak{g}_p, b\in \mathfrak{g}_p.$$
\end{lemma}
We record the following simple result that illustrates usefulness of "dequantizing" $U(\mathfrak{g})$
to $\Sym(\mathfrak{g}_{\bold{k}})$ in regards with the isomorphism problem for enveloping algebras.
\begin{lemma}\label{trivial}
Let $\mathcal{L}$ be a finite dimensional Lie algebra over a field $\bold{k}.$
Let $\mf{m}$ be a Poisson ideal of $\Sym(\mathcal{L})$ such that $\Sym(\mathcal{L})/\mf{m}=\bold{k}.$
Then $\mf{m}/\mf{m}^2\cong \mathcal{L}$ as Lie algebra. In particular,
if $\mathcal{L}'$ is another Lie algebra over $\bold{k}$ such that
$\Sym(\mathcal{L})\cong \Sym(\mathcal{L'})$ as Poisson $\bold{k}$-algebras,
then $\mathcal{L}\cong\mathcal{L'}.$
\end{lemma}
\begin{proof}
Let $\mf{m}$ be a maximal Poisson ideal. Then $\mf{m}=(g-\chi(g), g\in \mathcal{L})$ for some
$\chi\in \mathcal{L}^{*}.$ It follows that $\chi$ must be a character of $\mathcal{L}$, hence
the homomorphism $g\to g-\chi(g)$ defines a Lie algebra isomorphism
$\mathcal{L}\to \mf{m}/\mf{m}^2$ as desired.
\end{proof}
\begin{proof}[Proof of Theorem \ref{iso}]
We may assume that Lie algebras $\mathfrak{g}, \mathfrak{g'}$ and the corresponding
derived isomorphism are defined over a finitely generated ring $S\subset\mathbb{C}.$
Let $Z(U(\mathfrak{g}))=S[f_1,\cdots, f_n]$ and $Z(U(\mathfrak{g'}))=S[f'_1,\cdots, f'_n].$
Thus we have an $S$-algebra isomorphism $S[f_1,\cdots, f_n]\cong S[f'_1,\cdots, f'_n]$ and a Poisson
algebra isomorphism
$Z(U(\mathfrak{g}_{\bold{k}}))\cong Z(U(\mathfrak{g'}_{\bold{k}})).$
Moreover these isomorphisms are compatible with reduction modulo $p$ maps
so that the following diagram commutes:
\[\begin{tikzcd}
S[f_1,\cdots,f_n] \arrow{r}{} \arrow[swap]{d}{} & S[f_1',\cdots, f_n'] \arrow{d}{} \\
Z(U(\mathfrak{g}_{\bold{k}})) \arrow{r}{} & Z(U(\mathfrak{g}'_{\bold{k}})).
\end{tikzcd}
\]
Let $\mf{m}$ be a maximal Poisson ideals in $Z(U(\mathfrak{g}_{\bold{k}}))$,
and let $\mf{n}\subset Z(U(\mathfrak{g'}_{\bold{k}}))$ be the corresponding maximal Poisson ideal under the above isomorphisms.
Hence we get an isomorphism of Lie algebras $\mf{m}/\mf{m}^2\cong \mf{n}/\mf{n}^2.$
Put
$$\mathfrak{m}'=\mathfrak{m}\cap Z_p(\mathfrak{g}_{\bold{k}}),\quad \mathfrak{m}''=\mathfrak{m}\cap Z_{\HC}(\mathfrak{g}_{\bold{k}}), \mathfrak{m}_1=\mathfrak{m}'\cap \mathfrak{m}''.$$
Then Theorem \ref{center} implies the following Lie algebra isomorphism
$$\mathfrak{m}/\mathfrak{m}^2\cong \mathfrak{m}'/\mathfrak{m}'^2\times_{\mf{m}_1/\mf{m}_1^2} \mf{m''}/\mf{m}''^2.$$
On the other hand, since the Poisson bracket vanishes on $Z_{\HC},$
we conclude that $\mf{m}/\mf{m}^2$ is a trivial central extension of the image of $\mf{m}'/\mf{m'}^2\cong \mathfrak{g}_{\bold{k}}$
and the kernel of the natural homomorphism $\mf{m'}/\mf{m'}^2\to \mf{m}/\mf{m}^2 $ is central. On the other hand, if $g\in Z(\mf{m}'/\mf{m'}^2)$, then
$g=f^p$ with $f\in \mf{m}_1$, hence the image of $g$ in $\mf{m}/\mf{m}^2$ is 0.
Thus $\mf{m}/\mf{m}^2$ is a trivial central extension of $\mathfrak{g}_{\bold{k}}/Z(\mathfrak{g}_{\bold{k}}).$
Similarly $\mf{n}/\mf{n}^2$ is isomorphic to
a trivial central extension of $\mathfrak{g'}_{\bold{k}}/Z(\mathfrak{g'}_{\bold{k}}).$
Since we have the compatible isomorphism
$\bold{k}[g_1,\cdots, g_n]\cong \bold{k}[g'_1,\cdots, g'_n$] we get that
$$\mathfrak{g}_{\bold{k}}/Z(\mathfrak{g}_{\bold{k}})\oplus V\cong \mathfrak{g}_{\bold{k}}/Z(\mathfrak{g'}_{\bold{k}})\oplus V'$$
with $V\cong V'$ being abelian $\bold{k}$-Lie algebras.
Thus $\mathfrak{g}_{\bold{k}}/Z(\mathfrak{g}_{\bold{k}})\cong \mathfrak{g'}_{\bold{k}}/Z(\mathfrak{g'}_{\bold{k}}).$
Hence $\mathfrak{g}/Z(\mathfrak{g})\cong\mathfrak{g'}/Z(\mathfrak{g'}).$
\end{proof}
\begin{acknowledgement}
I am very grateful to Lewis Topley for making numerous useful suggestions.
\end{acknowledgement}
|
3,212,635,537,456 | arxiv | \section{#1}\vspace{-2pt}}
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\begin{document}
\title{HEMlets PoSh: Learning Part-Centric Heatmap Triplets for 3D Human Pose and Shape Estimation}
\author{Kun~Zhou,
Xiaoguang~Han,~\IEEEmembership{Member,~IEEE,}
Nianjuan~Jiang,~\IEEEmembership{Member,~IEEE,}
Kui~Jia,~\IEEEmembership{Member,~IEEE,}
and~Jiangbo~Lu,~\IEEEmembership{Senior~Member,~IEEE
\IEEEcompsocitemizethanks{\IEEEcompsocthanksitem K.~Zhou, N.~Jiang and J.~Lu are with SmartMore Co., Ltd., Shenzhen, China. (Corresponding email:~jiangbo.lu@gmail.com)\protec
\IEEEcompsocthanksitem X.~Han is with Shenzhen Institute of Big Data, The Chinese University of Hong Kong (Shenzhen), Shenzhen, China
\IEEEcompsocthanksitem K.~Jia are with South China University of Technology, Guangzhou, China.}
\IEEEtitleabstractindextext{
\begin{abstract}
Estimating 3D human pose from a single image is a challenging task. This work attempts to address the uncertainty of lifting the detected 2D joints to the 3D space by introducing an intermediate state - Part-Centric Heatmap Triplets ({\emph{HEMlets}}), which shortens the gap between the 2D observation and the 3D interpretation. The HEMlets utilize three joint-heatmaps to represent the relative depth information of the end-joints for each skeletal body part. In our approach, a Convolutional Network~(ConvNet) is first trained to predict HEMlets from the input image, followed by a volumetric joint-heatmap regression. We leverage on the integral operation to extract the joint locations from the volumetric heatmaps, guaranteeing end-to-end learning. Despite the simplicity of the network design, the quantitative comparisons show a significant performance improvement over the best-of-grade methods (e.g. $20\%$ on Human3.6M). The proposed method naturally supports training with ``in-the-wild'' images, where only weakly-annotated relative depth information of skeletal joints is available. This further improves the generalization ability of our model, as validated by qualitative comparisons on outdoor images. Leveraging the strength of the HEMlets pose estimation, we further design and append a shallow yet effective network module to regress the SMPL parameters of the body pose and shape. We term the entire HEMlets-based human pose and shape recovery pipeline {\emph{HEMlets PoSh}}. Extensive quantitative and qualitative experiments on the existing human body recovery benchmarks justify the state-of-the-art results obtained with our HEMlets PoSh approach.
\end{abstract}
\begin{IEEEkeywords}
3D human pose estimation, deep Learning, heatmaps, human body mesh recovery
\end{IEEEkeywords}}
\maketitle
\IEEEdisplaynontitleabstractindextext
\IEEEpeerreviewmaketitle
\IEEEraisesectionheading{\section{Introduction}\label{sec:introduction}}
\IEEEPARstart{H}{uman} pose estimation from a single image is an important problem in computer vision,
because of its wide applications, e.g., video surveillance and human-computer interaction.
Given an image containing a single person, 3D human pose inference aims to predict
3D coordinates of the human body joints. Recovering 3D information of human poses
from a single image faces several challenges. The challenges are at least threefold: 1) reasoning about 3D human poses from a single image is by itself
very difficult due to the inherent ambiguities; 2) for such a regression task, how to effectively bridge the gap between the 2D image and the target 3D human pose is actually challenging yet important; 3) for ``in-the-wild" images, both 3D capturing and manual labeling require
a lot of effort to obtain high-quality 3D annotations, making the
training data extremely scarce.
For 2D human pose estimation, almost all best performing methods are detection
based~\cite{SH-NET,ke2018multi,wei2016convolutional}. Detection-based approaches essentially
divide the joint localization task into local image classification tasks. The latter
is easier to train, because it effectively reduces the feature and target dimensions
for the learning system~\cite{sun2017integral}.
Existing 3D pose estimation methods often use detection as an intermediate
supervision mechanism as well. A straightforward strategy is to
use volumetric heatmaps to represent the likelihood map of each 3D
joint location~~\cite{pavlakos2017coarse}. Sun~{\it et al}.~\cite{sun2017integral}
further proposed a differentiable soft-argmax operator that unifies the joint detection
task and the regression task into an end-to-end training framework.
This significantly improves the state-of-the-art 3D pose estimation accuracy.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{teaser-eps-converted-to.pdf}
\caption{Overview of the HEMlets-based 3D pose estimation. (a)~Input RGB image. Our algorithm encodes (b)~the 2D locations for the joints $p$ and $c$, but also (c)~their relative depth relationship for each skeletal part $\vv{pc}$ into HEMlets. (d) Output 3D human pose.}
\label{fig:teaser}
\end{figure}
In this work, we propose a novel effective intermediate representation for 3D pose estimation -
\emph{Part-Centric Heatmap Triplets (HEMlets)} (as shown in Fig.~\ref{fig:teaser}).
The key idea is to polarize the 3D volumetric space around each
distinct skeletal part, which has the two end-joints kinematically connected.
Different from
\cite{pavlakos2018ordinal}, our relative depth information is represented as three polarized heatmaps,
corresponding to the different state of the local depth ordering of the part-centric joint pairs. Intuitively, HEMlets encodes the co-location
likelihoods of pairwise joints in a dense per-pixel manner with the coarsest
discretization in the depth dimension. Instead of considering arbitrary joint pairs, we focus on kinematically connected
ones as they possess semantic correspondence with the input image, and are thus a more effective target for the subsequent learning.
In addition, the encoded relative depth information is strictly local for the part-centric joint pairs and suffers less from potential inconsistent data annotation.
The proposed network architecture is shown in Fig.~\ref{fig:framework}. A ConvNet is first
trained to learn the HEMlets and 2D joint heatmaps, which are then fed together with the high-level image features to another ConvNet to produce a volumetric
heatmap for each joint. We leverage on the soft-argmax regression~\cite{sun2017integral}
to obtain the final 3D coordinates of each joint. Significant improvements
are achieved compared to the best competing methods quantitatively and qualitatively.
Most notably, our HEMlets method achieves a MPJPE of 39.9mm on Human3.6M~\cite{ionescu2014human3}, yielding about $20\%$ improvement over one best-of-grade method~\cite{sun2017integral}.
The merits of the proposed method lie in three aspects:
\begin{itemize}
\item
\textbf{\textit{Learning strategy. }}
Our method takes on a progressive learning strategy, and
decomposes a challenging 3D learning task into a sequence of easier sub-tasks with mixed
intermediate supervisions, i.e., 2D joint detection and HEMlets learning. HEMlets
is the key bridging and learnable component leading to 3D heatmaps, and is much easier to
train and less prone to over-fitting. Its training can also take advantage of existing labeled
datasets of relative depth ordering~\cite{pavlakos2018ordinal,shi2018fbi}.
\item
\textbf{\textit{Representation power.}}~HEMlets is based on 2D per-joint heatmaps, but extends them by a couple of additional heatmaps
to encode local depth ordering in a dense per-pixel manner. It builds on top of 2D heatmaps
but unleashes the representation power, while still allowing leveraging the soft-argmax
regression~\cite{sun2017integral} for end-to-end learning.
\item
\textbf{\textit{Simple yet effective. }}
The proposed method features a simple network architecture design, and it is easy to train and
implement. It achieves state-of-the-art 3D pose estimation results validated by the evaluations over all standard benchmarks.
\end{itemize}
A preliminary version of this work on 3D human pose estimation was published in the IEEE/CVF Conference on Computer Vision (ICCV) 2019~\cite{hemletsiccv}. This paper makes a few major contributions and extensions over the initial conference version as follows. First, we extend the proposed HEMlets pose framework to further recover the human body model from the given input image. We design a simple body model regression network connected to the preceding HEMlets pose network to recover a SMPL human body mesh from a single color image. In addition, we also describe in detail the proposed method, the training process as well as the model architecture. Our code will be made publicly available at the project website. Second, we introduce and present a new weakly-annotated FBI dataset and elaborate its advantages in obtaining weak annotations for the relative depth relationship between a pair of skeletal joints. The comparison of the FBI dataset and the recent Ordinal dataset\cite{pavlakos2018ordinal} is also provided. Lastly, we conduct thorough experiments including more ablation studies, as well as quantitative and qualitative evaluations for the recovered human body shape and pose. Extensive experiments justify the state-of-the-art performance of the proposed HEMlets-based pose and shape estimation method (termed as {\it HEMlets PoSh}) on all mainstream benchmark datasets.
\section{Related Work}\label{sec:relatedWork}
In this section, we review the approaches that are based on deep ConvNets for 3D human pose estimation and 3D body model recovery from a single color image.
\subsection{3D Body Pose Estimation}
We first conduct the literature review of 3d pose estimation in the following four aspects.
\textbf{Direct Encoder-Decoder.} With the powerful feature extraction capability of deep ConvNets, many approaches~\cite{li20143d,tekin2016structured,park20163d} learn end-to-end
\textit{Convolutional Neural Networks}~(CNNs) to infer human poses directly from the images. Li and Chen~\cite{li20143d} are
the first who used CNNs to estimate 3D human pose via a multi-task framework. Tekin~{\it et al}.~\cite{tekin2016structured} designed an auto encoder to model the joint dependencies in a high-dimensional feature space. Park~{\it et al}.~\cite{park20163d} proposed fusing 2D joint locations with high-level image features to boost the estimation of 3D human pose. However, these single stage
methods are limited by the availability of 3D human pose datasets and cannot take advantage of large-scale 2D pose datasets that are vastly available.
\textbf{Transition with 2D Joints.} To avoid collecting 2D-3D paired data, a large number of works~\cite{ronchi2018s,zhou2017towards,yang20183d,martinez2017simple,fang2018learning,shi2018fbi} decompose the task of 3D pose estimation into two independent stages by: 1) firstly inferring 2D joint locations using well-studied 2D pose estimation methods, such as~\cite{zhou2017towards,ronchi2018s}; 2) and then learning a mapping to lift them into the 3D space. These approaches mainly focus on tackling
the second problem. For example, a simple fully connected residual network is proposed by Martinez~{\it et al}.~\cite{martinez2017simple} to directly recover 3D human pose from its 2D projection. Fang~{\it et al}.~\cite{fang2018learning} considered prior knowledge of human body configurations and proposed human pose grammar, leading to better recovery of the 3D pose from only 2D joint locations. Yang~{\it et al}.~\cite{yang20183d} adopted an adversarial learning scheme to ensure the anthropometrical validity of the output pose and further improved the performance. Recently, by involving a reprojection mechanism, the proposed method in~\cite{wandt2019repnet} shows insensitivity to overfitting and
accurately predicts the result from noisy 2D poses. Though promising results have been achieved by these two-stage methods,
a large gap exists between the 3D human pose and its 2D projections due to inherent ambiguities.
\textbf{3D-Aware Intermediate States.} To further bridge the gap between the 2D image and the target 3D human pose under estimation, some recent works~\cite{pavlakos2017coarse,shi2018fbi,pavlakos2018ordinal,sun2017integral} proposed to involve 3D-aware states for intermediate supervisions. Namely, a network is firstly trained to map the input image to these 3D-aware states, and then another network is trained to convert those states to the 3D joint locations. Finally, these two networks are combined and optimized jointly. A volumetric representation for 3D joint-heatmaps is proposed in ~\cite{pavlakos2017coarse}, with which the 3D pose is regressed in a coarse-to-fine manner. However, regressing a probability grid in the 3D space globally is also a very challenging task. It usually suffers from quantization errors for the joint locations. To address this issue, Sun~{\it et al}.~\cite{sun2017integral} exploited a soft-argmax operation and proposed
an end-to-end training scheme for the 3D volumetric regression,
achieving by far the best performance on 3D pose estimation.
Inspired by~\cite{pons2014posebits} that the relative depth ordering
across joints is helpful for resolving pose ambiguities, Pavlakos~{\it et al}.~\cite{pavlakos2018ordinal} adopted a ranking loss for pairwise ordinal depth to train the 3D human pose predictor explicitly. A similar scheme of relative depth supervision is utilized in the work of~\cite{ronchi2018s}. Forward-or-Backward Information~(FBI), proposed in ~\cite{shi2018fbi}, is another kind of relative depth information but focuses more on the bone orientations. Recently, Sharma~{\it et al}.~\cite{Sharma_2019_ICCV} proposed to train a deep conditional variational autoencoder to map 2D poses to 3D poses by learning ordinal maps. In this work, we propose HEMlets, a novel representation that encodes both
2D joint locations and the part-centric relative depth ordering simultaneously. Experiments justify that this representation reaches by far the best balance between representation efficiency and learning effectiveness.
\textbf{``In-the-Wild'' Adaptation.}
All the aforementioned approaches are mainly trained on the datasets collected under indoor settings, due to the difficulty of annotating 3D joints for ``in-the-wild'' images ~\cite{bourdev2009poselets}.
Thus, many strategies are developed to make domain adaptation.
By exploiting graphics techniques, previous works~\cite{varol2017learning,chen2016synthesizing} have synthesized a large ``faked'' dataset mimicking real images. Though these data benefit 3D pose estimation, they are still far from realistic, making the applicability limited. Recently, both Pavlakos~{\it et al}.~\cite{pavlakos2018ordinal} and Shi~{\it et al}.~\cite{shi2018fbi} proposed to label the relative depth relationship across joints
instead of the exact 3D joint coordinates. This weak annotation scheme
not only makes building large-scale ``in-the-wild'' datasets
feasible but also provides 3D-aware information
for training the inference model in a weakly-supervised manner.
With HEMlets representation, we can readily use these weakly annotated ``in-the-wild'' data for domain adaptation.
\subsection{3D Body Model Recovery}
In the recent years, 3D full body models have become popular, which are typically represented with a parametric human body space, such as SMPL~\cite{SMPL:2015}. The advantage is that a human body mesh can be easily generated from a set of body shape and pose parameters, so this turns the task of recovering a 3D body model from a single image into a problem of solving for a set of parameters. As a pioneering work in this arena, a two-stage framework~\cite{bogo2016keep} is proposed. It firstly infers 2D skeleton joints from the input image with a CNN-based model, and then searches the optimal parameters of SMPL to fit the joints with an optimization approach. Due to the depth ambiguity, the second stage tends to converge to a local minimum. To better address this problem, many works~\cite{kanazawa2018end,pavlakos2018learning,kolotouros2019convolutional} proposed to build an end-to-end pipeline to map images into the parametric space using deep regression models.
However, the key challenge to train regression-based models is the lack of paired data, due to the inherent difficulty of annotating or capturing a groundtruth 3D model for a person instance. Existing approaches address this challenge roughly along three main directions. First, some works directly tackled the issue by putting efforts on constructing target datasets. The work of~\cite{varol2017learning} firstly built a synthetic dataset using graphics techniques, however, training only on this dataset is still difficult to produce a model that is applicable to real images. Lassner~{\it et al}.~\cite{lassner2017unite} proposed to apply the algorithm of~\cite{bogo2016keep} to obtain 3D body models for real images and then manually sift out the reasonable results, to build the final human body dataset. Unfortunately, the obtained 3D human shapes are still non-ideal and contain erroneous body part results.
For the second category, several works attempted to directly add extra constraints on the output parameters to ease the training process. For example, the strategy of adversarial learning was utilized in~\cite{kanazawa2018end}, where a discriminator is exploited to constrain the regressed parameters against a reasonable distribution. Given the regressed SMPL models, Kolotouros~{\it et al}.~\cite{kolotouros2019spin} further adopted the approach of~\cite{bogo2016keep} to obtain better parameters to fit the input images. They then directly set an extra objective for the regression model to enforce its output parameters equal to the optimized ones.
Lately, a large body of works proposed to use 2D intermediate representations, such as silhouettes~\cite{tan2018indirect,pavlakos2018learning} and densepose~\cite{alp2018densepose}-based representation~\cite{xu2019denserac,guler2019holopose}, which leverage on the idea of self-supervised training. Specifically, other than the main branch of mapping input images to SMPL meshes, these methods also constructed a novel branch to convert the input images to the proposed intermediate representations. A differentiable mesh renderer was then utilized to render the output meshes, which are compared with the learnt intermediate representations. This brings extra supervisions to guide the training process.
More recently, some other 3D representations, such as voxels, mesh and UV-maps, have been used for building generative neural networks to infer 3D models from images. The work of~\cite{varol2018bodynet}, as the first attempt of this kind, proposed an approach to generate a voxel representation of a 3D body from a single image using 3D ConvNets. By taking a template human body mesh as an extra input and treating the mesh as a graph representation, Kolotouros~{\it et al}.~\cite{kolotouros2019convolutional} trained a Graph ConvNet to learn the deformation of the template model for fitting both the target pose and shape. The algorithm of DenseBody~\cite{yao2019densebody} represented the 3D body model with a parameterized UV-map, and then turned the task of geometry inference into a problem of image synthesis. This method further advances body mesh reconstruction accuracy.
In this work, we find that the impact of 3D pose estimation on the accuracy of the final recovered body model is much more significant than the regressed shape parameters. Based on the estimated 3D pose obtained with our HEMlets pose approach, we show that a simple body regression method for SMPL model inference outperforms all the afore-discussed approaches.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{mergeBody-eps-converted-to.pdf}
\caption{Part-centric heatmap triplets $\{{\bf T}^{-1}_k,{\bf T}^{0}_k,{\bf T}^{+1}_k\}$ where $p$ and $c$ are the parent joint and the child joint. (a,~b) Joints and skeletal parts. We locate the parent joint $p$ of the $k$-th skeletal part $B_k$ at the zero polarity heatmap ${\bf T}^{0}_k$~(c-e). The child joint $c$ is located, according to relative depth of $p$ and $c$, in the positive~(c), zero~(d) and negative polarity heatmap~(e), respectively. }
\label{fig:tri_heatmaps}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[width=16.5cm]{network1-eps-converted-to.pdf}
\caption{The network architecture of our proposed approach. It consists of
four major modules: (a) A ResNet-50 backbone for image feature extraction. (b) A ConvNet for image feature upsampling. (c) Another ConvNet for HEMlets learning and 2D joint detection.
(d) A 3D pose regression module adopting a soft-argmax operation for 3D human pose estimation. (e) Details of the HEMlets learning module. ``Feature concatenate" denotes concatenating the feature maps from the HEMlets learning branch and the upsampling branch together.}
\label{fig:framework}
\end{figure*}
\section{HEMlets Pose Estimation} \label{sec:method}
We propose a unified representation of heatmap triplets to model the local information of body skeletal parts, i.e., kinematically connected joints, whereas the corresponding 2D image coordinates and relative depth ordering are considered. By such a representation, images annotated with relative depth ordering of skeletal parts can be treated equally with images annotated with 3D joint information. While the latter is usually very scarce, the former is relatively easy to obtain~\cite{shi2018fbi,pavlakos2018ordinal}. In this section, we first present the proposed part-centric heatmap triplets and its encoding scheme. Then, we elaborate a simple network architecture that utilizes the part-centric heatmap triplets for 3D human pose estimation.
\subsection{Part-Centric Heatmap Triplets}
\label{sec:constructHemlets}
We divide the full body skeleton consisting of $N=18$ joints into $K=14$ parts as shown in Fig.~\ref{fig:tri_heatmaps}(a).
Specifically, we use $B$ to denote the set of skeletal parts,
where $B = \left\{ B_1,B_2,\ldots,B_K \right\}$. For each part,
we denote the two associated joints as $(p, c)$, with $p$ being the parent node and $c$ being the child node. The relative depth ordering, denoted as $r(z_p, z_c)$, can be then described as a tri-state function~\cite{pavlakos2018ordinal,shi2018fbi}:
\begin{equation}
r(z_p,z_c) =
\begin{cases}
1& \text{$z_p - z_c>\epsilon$}\\
0& \text{$\left| z_p - z_c \right|<\epsilon$} \\
-1& \text{$z_p - z_c<-\epsilon$}
\end{cases} ,
\label{eq:polarity}
\end{equation}
\noindent where $\epsilon$ is used to adjust the sensitivity of the function to the relative depth difference. The absolute depths of the two joints $p$ and $c$ are denoted by $z_p$ and $z_c$, respectively.
We argue that directly using the discretized label as an intermediate state for learning the
3D pose from a 2D joint heatmap, as was done in \cite{pavlakos2018ordinal,shi2018fbi}, is not
as effective. Since this abstraction tends to lose some important features encoded in
the joints' spatial domain. Instead of elevating the
problem straight away to the 3D volumetric space, we utilize an
intermediate representation of the 3D-aware relationship of the parent joint $p_k$
and the child joint $c_k$ of a skeletal part $B_k$. Provided with the supervision signals, we define
polarized target heatmaps where a pair of normalized Gaussian peeks
corresponding to the 2D joint locations are
placed accordingly across three heatmaps (see Figure~\ref{fig:tri_heatmaps}).
We term them as the \textit{negative polarity heatmap} ${\bf T}^{-1}_k$, the \textit{zero polarity heatmap} ${\bf T}^{0}_k$ and the
\textit{positive polarity heatmap} ${\bf T}^{+1}_k$ with respect to the function value in
Eq.~(\ref{eq:polarity}). The parent joint $p_k$ is always placed in the
zero polarity heatmap ${\bf T}^{0}_k$. The child joint $c_k$ will appear in the negative/positive
polarity heatmap, if its depth is larger/smaller than that of the parent joint $p_k$~(i.e., $|r(z_p,z_c)| \neq 0$).
Both parent and child joints are co-located in the zero polarity heatmap if their
depths are roughly the same~(i.e., $r(z_p,z_c)=0$).
Formally, we denote the heatmap triplets of the skeletal part $B_k$ as the stacking
of three heatmaps ${\bf T}^{-1}_k,{\bf T}^{0}_k,{\bf T}^{+1}_k$:
\begin{equation}
{\bf T}_k = \operatorname{Stack}[{\bf T}^{-1}_k,{\bf T}^{0}_k,{\bf T}^{+1}_k] ,
\end{equation}
Given 3D groundtruth coordinates of all joints,
we can readily compute the heatmap triplets of each skeletal part. For
easy reference, we shall refer to the part-centric heatmap triplets
${\bf T}_k$ as \textit{HEMlets}, and use it afterwards.\\
{\bf Discussions.}
Here we provide some understandings of HEMlets from a few perspectives. First, different from a joint-specific 2D heatmap that models the detection likelihood for each intended joint on the $(x, y)$ plane, HEMlets models part-centric pairwise joints' co-location likelihoods on the $(x, y)$ plane simultaneously with their ordinal depth relations. This helps to learn geometric constraints (e.g., bone lengths) implicitly. Second, by augmenting a 2D heatmap to a triplet of heatmaps, HEMlets learns and evaluates the co-location likelihood for a pair of connected joints $(p,c)$ by the joint probability distribution $P(x_p,y_p,x_c,y_c,r(z_p,z_c))$ in a locally-defined volumetric space. In contrast, Pavlakos~{\it et al}.~\cite{pavlakos2018ordinal} relaxed the learning target and marginalized the 3D probability distributions independently for the $(x, y)$ plane i.e., $P(x_p,y_p), P(x_c,y_c)$ and the $z$-dimension, with the latter supervised independently by $r(z_p,z_c)$ based on a ranking loss. Third, by exploiting the available supervision signals to a larger extent, HEMlets brings the benefit of making the knowledge more explicitly expressed and easier to learn, and bridges the gap in learning the 3D information from a given 2D image.
\subsection{3D Pose Inference}
\noindent
{\bf Network architecture.}
We employ a fully convolutional network to predict the 3D human pose as
illustrated in Figure~\ref{fig:framework}. A ResNet-50~\cite{ghiasi2014occlusion}
backbone architecture is adopted for basic feature extraction.
One of the two upsampling branches is used to learn the HEMlets
and the 2D heatmaps of skeletal joints, and the other one
is used to perform upsampling of the learned features to the same resolution
as the output heatmaps. Both HEMlets and the 2D joint heatmaps are then
encoded jointly by a 2D convolutional operation to form a latent global
representation. Finally these global features are joined with the convolutional features
extracted from the original image to predict a 3D feature map for each joint.
We perform a soft-argmax operation~\cite{sun2017integral} to aggregate
information in the 3D feature maps to obtain the 3D joint estimations. \\
\label{sec:training}
{\bf HEMlets loss.}
Let us denote with ${\bf T}^{\rm gt}$ the groundtruth HEMlets of all skeletal parts
and with ${\bf \hat{T}}$ the corresponding prediction. We use a standard $L_2$
distance between ${\bf T}^{\rm gt}$ and ${\bf \hat{T}}$ to compute the HEMlets loss as follows:
\begin{equation}
{\mathcal{L}}^{\rm HEM} = {\lVert ({\bf T}^{\rm gt} - {\bf \hat{T}}) \odot {\bf \Lambda} \rVert}_2^2 ,
\label{eq:tri-state-loss}
\end{equation}
where $\odot$ denotes an element-wise multiplication, and ${\bf \Lambda}$ is a binary tensor
to mask out missing annotations.\\
{\bf Auxiliary 2D joint loss.}
As HEMlets essentially contains heatmap responses of 2D joint locations, we
adopt a heatmap-based 2D joint detection scheme to facilitate HEMlets prediction.
The $L_2$ loss of 2D joint prediction is computed as:
\begin{equation}
\mathcal{L}^{\rm 2D} = \sum_{n=1}^N{\lVert {\bf H}^{\rm gt}_n - {\bf \hat{H}}_n \rVert}_2^2,
\label{eq:heat-loss}
\end{equation}
where ${\bf H}^{\rm gt}_n$ is the groundtruth 2D heatmap of the $n$-th 2D joint
and ${{\bf{\hat{H}}}_n}$ is the corresponding network prediction.\\
{\bf Soft-argmax 3D joint loss.}
To avoid quantization errors and allow end-to-end learning, Sun~{\it et al}.~\cite{sun2017integral}
suggested soft-argmax regression for 3D human pose estimation. Given learned
volumetric features ${\bf F}_n$ of size $(h \times w \times d)$ for the $n$-th joint,
the predicted 3D coordinates are given as:
\begin{equation}
[\hat{x}_n,\hat{y}_n,\hat{z}_n] = \int_{\mathbf{v}}{ \mathbf{v} \cdot \operatorname{Softmax}({\bf F}_n)},
\label{eq:Integral}
\end{equation}
where $\mathbf{v}$ denotes a voxel in the volumetric feature space of ${\bf F}_n$. For robustness, we employ the $L_1$ loss for the regression of 3D joints. Specifically, the loss is defined as:
\begin{equation}
\mathcal{L}^{\rm 3D}_{\lambda} = \sum_{n=1}^N{ ( \left| x_n^{\rm gt} - {\hat{x}}_n\right| + \left| y_n^{\rm gt} - {\hat{y}}_n\right| + \lambda \left| z_n^{\rm gt} - {\hat{z}}_n\right| )},
\label{eq:3d-loss-lambda}
\end{equation}
where the groundtruth 3D position of the $n$-th joint is given as $(x_n^{\rm gt},y_n^{\rm gt},z_n^{\rm gt})$. We use the same 2D and 3D mixed training strategy in~\cite{sun2017integral}~($\lambda \in \{0,1\}$): $\lambda$ in Eq.~(\ref{eq:3d-loss-lambda}) is set to $1$ when the training data is from 3D datasets, and $\lambda = 0$ when the data is from 2D datasets. \\
{\bf Training strategy.}
For HEMlets prediction, We combine $\mathcal{L}^{\rm HEM}$ and $\mathcal{L}^{\rm 2D}$ for the intermediate supervision.
The loss function is defined as:
\begin{equation}
\mathcal{L}^{\rm int} = \mathcal{L}^{\rm HEM} + \mathcal{L}^{\rm 2D}.
\label{eq:3d-loss}
\end{equation}
By using $\mathcal{L}^{\rm HEM}$ and $\mathcal{L}^{\rm 2D}$ jointly as supervisions, we allow training the
network using images with 2D joint annotations and 3D joint annotations. By
3D joint annotation, we refer to annotations with exact 3D joint coordinates or
relative depth ordering between part-centric joint pairs.
The end-to-end training loss $\mathcal{L}^{\rm tot}$ is defined by combining $\mathcal{L}^{\rm int}$ with $\mathcal{L}^{\rm 3D}_{\lambda}$:
\begin{equation}
\mathcal{L}^{\rm tot} = \alpha * \mathcal{L}^{\rm int} + \mathcal{L}^{\rm 3D}_{\lambda},
\label{eq:mix-loss}
\end{equation}
where $\alpha=0.05$ in all our experiments.\\
\subsection {\bf Implementation Details}
Now we present a few implementation details in the proposed method. As different human pose datasets may have different definitions for body joints, we choose to accommodate this difference from different supervision sources. The purpose is to take advantage of more human pose annotation sources, when using the 2D and 3D mixed training strategy~\cite{sun2017integral}. Figure~\ref{fig:JointTraining} illustrates the joint structures defined by the Human3.6M~\cite{ionescu2014human3} and MPII~\cite{Andriluka} datasets, as well as our joint structure definition. We take the union of these two sets of joint definitions to form a 18-joint set as the regression target. Suppose performance evaluation is conducted on the Human3.6M dataset, then only those estimated joints used by Human3.6M will be evaluated, as in Martinez~{\it et al. }'s work~\cite{martinez2017simple}.
To prepare the human bounding-box input for the proposed network, we crop from the original input image a square-shaped region based on the ground-truth bounding-box, and then resize it proportionally to $256 \!\times\! 256$. To obtain the final metric scale prediction from the network output (in voxel/pixel space), we resort to the average body bone length learned during the training phase to enable this prediction mapping. We did not use the ground-truth (depth) information during the test phase, e.g. the distance to the root/pelvis for obtaining the scaling factor.
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{jointMerge.pdf}
\caption{A unified body joint definition adopted in our method by merging the joints defined by the Human3.6M and MPII datasets.}
\label{fig:JointTraining}
\end{figure}
We implement our method in PyTorch. The model is trained in an end-to-end manner
using both images with 3D annotations (e.g., Human3.6M~\cite{ionescu2014human3} or
HumanEva-I~\cite{sigal2010humaneva}), and 2D annotations~(MPII~\cite{Andriluka}).
In our experiments, we adopt an adaptive value of $\epsilon$ in
Eq.~(\ref{eq:polarity}) for each skeletal part:
${\epsilon}_k = 0.5 {\lVert B_k \rVert}$~($ {\lVert B_k \rVert}$ is the
3D Euclidean distance between the two end joints of the skeletal part $B_k$).
The training data is further augmented with rotation ($\pm30^{\circ}$),
scale ($0.75\!-\!1.25$), horizontal flipping (with a probability of $0.5$) and color distortions.
By using a batch size of $64$, a learning rate of $0.001$ and Adam optimization, the
training took $100$K iterations to converge. It took about a few days~($2\!-\!4$) with
four NVIDIA GTX 1080 GPUs to train the HEMlets pose estimation model.
\section{HEMlets Body Model Regression} \label{sec:shape_method}
So far, we have presented the proposed HEMlets pose estimation method in detail. It is natural to consider whether the proposed method can be extended also to recover human body models from the given input color images. To this end, we design and append a shallow yet effective network module to the preceding HEMlets pose network, which leverages the 3D pose estimation accuracy to regress the parameters of the body shape and pose. In this work, we employ the popular 3D body SMPL model~\cite{SMPL:2015}, where a human body mesh is parameterized by a 3D body shape parameter $\beta \in \mathbb{R}^{10}$ and a pose parameter $\theta \in \mathbb{R}^{24 \times 3}$.
As shown in Fig.~\ref{fig:shapeModule}, the newly added body model regression module is very simple. It takes the predicted 3D joint coordinates from the early stage as input, together with the high-level image features extracted from the given color image. This regression module is trained to regress the SMPL shape and pose parameters as final outputs. It is worth noting that we do not perform explicit human image segmentation, but instead use the high-level image features as implicit cues for shape regression.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{shape_Reg.pdf}
\caption{HEMlets-based parametric 3D human body regression from a single color image. We append a shallow yet effective SMPL body mesh regression network to the preceding HEMlets pose estimation network, which is trained end-to-end to regress the SMPL shape and pose parameters $\{\beta, \theta\}$.}
\label{fig:shapeModule}
\end{figure}
In our implementation, the additional regression module is trained together with the 3D pose network in an end-to-end manner. Similar to recent works~\cite{xu2019denserac,kolotouros2019convolutional}, the SMPL pose parameter $\theta$ is converted into 24 rotation matrices for pose regression, which avoids the known singularity problem of the axis-angle representation. Following the similar strategy, the SMPL pose loss $\mathcal{L}_{\theta}$ is defined as:
\begin{equation}
\mathcal{L}_{\theta} = \sum_{i=1}^{24} {\lVert R_i^{\rm gt} - \hat{R_i} \rVert} \;,
\label{eq:pose loss}
\end{equation}
where $R_i$ denotes the rotation matrix corresponding to the $i$-th joint. The SMPL shape regression loss $\mathcal{L}_{\beta}$ is simply computed using the $L_1$ loss as,
\begin{equation}
\mathcal{L}_{\beta} = \sum_{i=1}^{10} {\lVert {\beta}_i^{\rm gt} - \hat{{\beta}_i} \rVert} \;.
\label{eq:beata loss}
\end{equation}
Finally, the end-to-end training loss ${L}_{\rm mesh}$ for the parametric 3D human body regression is given by
\begin{equation}
\mathcal{L}_{\rm mesh} = \mathcal{L}_{\theta} + \mathcal{L}_{\beta} + \mathcal{L}^{\rm tot} \;,
\label{eq:shape loss}
\end{equation}
where $\mathcal{L}^{\rm tot}$ is the total loss defined for pose estimation in Eq.~(\ref{eq:mix-loss}).
\vspace{-2pt}
\section{Weakly-Annotated FBI Dataset} \label{sec:fbi}
{
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{FBI_annotation_single.pdf}
\caption{User annotation interface for obtaining the weakly-annotated FBI dataset. An annotator is asked to assign a label of either ``\textit{Backward}", ``\textit{Forward}" or ``\textit{Unknown}" to a given skeletal part.}
\label{fig:annotation}
\end{figure}
In this section, we introduce a new Forward-or-Backward Information (FBI) dataset, and elaborate its advantages in obtaining weak annotations for the relative depth relationship between a pair of skeletal joints. To prepare this FBI dataset, 12K images are randomly drawn from the MPII dataset~\cite{Andriluka}, for which only 2D joint annotations are available. Then, each body part is assigned with a label of either ``\textit{Backward}", ``\textit{Forward}" or ``\textit{Unknown}". We designed a simple user interface to facilitate the annotation. As shown in Fig.~\ref{fig:annotation}, an annotator was presented with one image at a time with the native 2D skeleton overlaid over the input image. The annotator was asked to assign ``\textit{Backward}" or ``\textit{Forward}" labels to only a subset of the body parts for which she/he is confident with. The rest of the body parts are assigned with the ``\textit{Unknown}" labels by default.
\subsection{Comparison with The Ordinal Dataset~\cite{pavlakos2018ordinal}}
At the first glance, both FBI and Ordinal~\cite{pavlakos2018ordinal} annotation schemes aim at annotating the depth ordering between two body joints. However, the FBI scheme simplifies the annotation objective and reduces the annotation complexity with a good reason. The Ordinal scheme tries to annotate the relative depth information between every pair of joints. For each image, there are $\binom{14}{2}=91$ questions that need to be answered by the annotator. This annotation requirement is not only time-consuming, but also prone to human errors. The FBI scheme, on the other hand, only requires the annotator to answer at most 14 questions for each image. Furthermore, the annotator only needs to tell the relative depth ordering of two kinematically connected joints, which is intuitive and less prone to human errors, as illustrated in Fig.~\ref{fig:annotation_comparison}. Empirically, we observe that body parts with ambiguous relative depth ordering, namely, near-equal-depth joints are difficult to annotate with good accuracy. Therefore, the FBI annotation scheme only asks for ``confident" annotations from annotators. Joint pairs can be skipped and retain an ``\textit{Unknown}" label by default.
\begin{figure}
\centering
\includegraphics[width= 0.90\columnwidth]{Ordinal_FBI_comparison_single.pdf}
\caption{A simple illustration of the difference between the FBI and Ordinal annotation schemes. (a) Global relative depth ordering between disconnected joints (e.g., $(q_0,q_1)$ in the top-left image, $(q_0,q_1)$ and $(q_0,q_2)$ in the bottom-left image) need to be annotated in the Ordinal scheme, which are however challenging to annotate correctly. (b) In contrast, only local relative depth ordering between connected joints (e.g., $(p_i,c_i)$ and $(p_j,c_j)$ in the right-side images) need to be annotated in the FBI scheme.}
\label{fig:annotation_comparison}
\end{figure}
\subsection{FBI Annotation Quality and Speed}
In order to assess the annotation quality of using the FBI scheme, 1000 images with 3D ground truth are randomly selected from the Human3.6M dataset~\cite{ionescu2014human3}. Then, they are mixed with 12K in-the-wild images for user annotations. We briefed ten first-time annotators about the FBI scheme, and collected their annotations on a total number of 13K images. For evaluation, we retrieve the annotations of all the images from the Human3.6M dataset and compare them against the ground-truth relative depth relations. We find that when the ground truth tilt angle of the skeletal bone $B_k$ with respect to the image plane is greater than $30^{\circ}$, the percentage of annotation errors is only $7.4\%$ and the percentage of skipped annotations is less than $10\%$. However, when this tilt angle is below $20^{\circ}$, both the rates of annotation errors and skipped annotations increase noticeably. This experimental study agrees with our conjecture that the body parts with small tilt angles (hence with ambiguous relative depth ordering) are much harder to annotate.
Regarding the annotation time, on average each image takes less than 20 seconds to annotate using the FBI scheme, while the Ordinal scheme needs roughly 1 minute per image.
}
\section{Experiments} \label{sec:Experimental-Result}
In this section, we evaluate the proposed HEMlets-based human pose and shape estimation methods by conducting comprehensive experiments over the main benchmark datasets.
\begin{table*}[ht]
\setlength{\tabcolsep}{2pt}
\small
\resizebox{\textwidth}{50mm}{
\begin{tabular}{lcccccccccccccccc}
\vspace{4pt}
\\ \hline
\textbf{Protocol \#1} & Direct & Discuss & Eating & Greet & Phone & Photo & Pose & Purch. & Sitting & SittingD. & Smoke & Wait & WalkD. & Walk & WalkT. & \textbf{Avg} \\ \hline
LinKDE~{\it et al}.~\cite{ionescu2014human3} &132.7 &183.6 &132.3 &164.4 &162.1 &205.9 &150.6 &171.3 &151.6 &243.0 &162.1 &170.7 &177.1 &96.6 &127.9 &162.1 \\
Tome~{\it et al}.~\cite{tome2017lifting} &65.0 &73.5 &76.8 &86.4 &86.3 &110.7 &68.9 &74.8 &110.2 &173.9 &85.0 &85.8 &86.3 &71.4 &73.1 &88.4 \\
Rogez~{\it et al}.~\cite{rogez2017lcr} &76.2 &80.2 &75.8 &83.3 &92.2 &105.7 &79.0 &71.7 &105.9 &127.1 &88.0 &83.7 &86.6 &64.9 &84.0 &87.7 \\
Tekin~{\it et al}.~\cite{tekin2017learning} &54.2 &61.4 &60.2 &61.2 &79.4 &78.3 &63.1 &81.6 &70.1 &107.3 &69.3 &70.3 &74.3 &51.8 &74.3 &69.7 \\
Martinez~{\it et al}.~\cite{martinez2017simple}&53.3 &60.8 &62.9 &62.7 &86.4 &82.4 &57.8 &58.7 &81.9 &99.8 &69.1 &63.9 &67.1 &50.9 &54.8 &67.5 \\
Fang~{\it et al}.~\cite{fang2018learning} &50.1 &54.3 &57.0 &57.1 &66.6 &73.3 &53.4 &55.7 &72.8 &88.6 &60.3 &57.7 &62.7 &47.5 &50.6 &60.4 \\
Pavlakos~{\it et al}.~\cite{pavlakos2018ordinal} &48.5 &54.4 &54.4 &52.0 &59.4 &65.3 &49.9 &52.9 &65.8 &71.1 &56.6 &52.9 &60.9 &44.7 &47.8 &56.2 \\
S{\'a}r{\'a}ndi~{\it et al}.~\cite{sarandi2018robust}&51.2 &58.7 &51.7 &53.4 &56.8 &59.3 &50.7 &52.6 &65.5 &73.2 &56.8 &51.4 &56.6 &47.0 &42.4 &55.8 \\
Sun~{\it et al}.~\cite{sun2017integral} &47.5 &47.7 &49.5 &50.2 &51.4 &55.8 &43.8 &46.4 &58.9 &65.7 &49.4 &47.8 &49.0 &38.9 &43.8 &49.6 \\
Sharma~{\it et al}.~\cite{Sharma_2019_ICCV} &48.6 &54.5 &54.2 &55.7 &62.6 &72.0 &50.5 &54.3 &70.0 &78.3 &58.1 &55.4 &61.4 &45.2 &49.7 &58.0 \\
Chen~{\it et al}.~\cite{chen2019weakly} &41.1 &44.2 &44.9 &45.9 &46.5 &\textbf{39.3} &41.6 &54.8 &73.2 &\textbf{46.2} &48.7 &\textbf{42.1} &\textbf{35.8} &46.6 &38.5 &46.3 \\
\hline
Ours*~(w/o MPII 2D) &38.5 & 45.8 & 40.3 & 54.9 & 39.5 & 45.9 &39.2 & 43.1 &49.2 &71.1 & 41.0 &53.6 &44.5 & 33.2 &34.1 &45.1\\
Ours & \textbf{34.4} & \textbf{42.4} & \textbf{36.6} & \textbf{42.1} & \textbf{38.2} & 39.8 & \textbf{34.7} & \textbf{40.2} & \textbf{45.6} & 60.8 & \textbf{39.0} & 42.6 &42.0 & \textbf{29.8} & \textbf{31.7} & \textbf{39.9}\\
\hline
\textbf{Protocol \#2} & Direct & Discuss & Eating & Greet & Phone & Photo & Pose & Purch. & Sitting & SittingD. & Smoke & Wait & WalkD. & Walk & WalkT. & \textbf{Avg} \\ \hline
Nie~{\it et al}.~\cite{nie2017monocular} &90.1 &88.2 &85.7 &95.6 &103.9 &92.4 &90.4 &117.9 &136.4 &98.5 &103.0 &94.4 &86.0 &90.6 &89.5 &97.5 \\
Chen~{\it et al}.~\cite{chen20173d} &53.3 &46.8 &58.6 &61.2 &56.0 &58.1 &41.4 &48.9 &55.6 &73.4 &60.3 &45.0 &76.1 &62.2 &51.1 &57.5 \\
Martinez~{\it et al}.~\cite{martinez2017simple} &39.5 &43.2 &46.4 &47.0 &51.0 &56.0 &41.4 &40.6 &56.5 &69.4 &49.2 &45.0 &49.5 &38.0 &43.1 &47.7 \\
Fang~{\it et al}.~\cite{fang2018learning} &38.2 &41.7 &43.7 &44.9 &48.5 &55.3 &40.2 &38.2 &54.5 &64.4 &47.2 &44.3 &47.3 &36.7 &41.7 &45.7 \\
Pavlakos~{\it et al}.~\cite{pavlakos2018ordinal} &34.7 &39.8 &41.8 &38.6 &42.5 &47.5 &38.0 &36.6 &50.7 &56.8 &42.6 &39.6 &43.9 &32.1 &36.5 &41.8 \\
Yang~{\it et al}.~\cite{yang20183d} & \textbf{26.9} &\textbf{30.9} &36.3 &39.9 &43.9 &47.4 &28.8 &\textbf{29.4} &\textbf{36.9} &58.4 &41.5 &\textbf{30.5} &\textbf{29.5} &42.5 &32.2 &37.7 \\
Sharma~{\it et al}.~\cite{Sharma_2019_ICCV} &35.3 &35.9 &45.8 &42.0 &40.9 &52.6 &36.9 &35.8 &43.5 &51.9 &44.3 &38.8 &45.5 &29.4 &34.3 &40.9 \\ \hline
Ours*~(w/o MPII 2D) &30.8 &36.7 &31.7 &37.5 &32.5 &36.5 &29.4 &34.8 &38.5 &50.5 &33.1 &35.2 &35.9 &25.5 &27.7 &34.2 \\
Ours &29.1 & 34.9 & \textbf{29.9} & \textbf{32.6} & \textbf{31.2} & \textbf{32.3} & \textbf{27.0} & 33.3 & 37.6 & \textbf{45.9} & \textbf{32.2} &31.5 &34.5 & \textbf{22.9} & \textbf{25.9} & \textbf{32.1}\\ \hline
\hline
\textbf{PA MPJPE} & Direct & Discuss & Eating & Greet & Phone & Photo & Pose & Purch. & Sitting & SittingD. & Smoke & Wait & WalkD. & Walk & WalkT. & \textbf{Avg} \\ \hline
Yasin~{\it et al}.~\cite{yasin2016dual} &88.4 &72.5 &108.5 &110.2 &97.1 &81.6 &107.2 &119.0 &170.8 &108.2 &142.5 &86.9 &92.1 &165.7 &102.0 &108.3 \\
Sun~{\it et al}.~\cite{sun2017integral} &36.9 &36.2 &40.6 &40.4 &41.9 &34.9 &35.7 &50.1 &59.4 &40.4 &44.9 &39.0 &\textbf{30.8} &39.8 &36.7 &40.6 \\
Dabral~{\it et al}.~\cite{dabral2018learning} &28.0 &30.7 &39.1 &34.4 &37.1 &44.8 &28.9 &32.2 &39.3 &60.6 &39.3 &31.1 &37.8 &25.3 &28.4 &36.3 \\ \hline
Ours*~(w/o MPII 2D) &25.3 &29.1 &30.9 &30.1 &27.7 &32.7 &26.1 &28.3 &29.6 &41.9 &30.6 &26.4 &31.8 &21.7 &23.5 &29.1 \\
Ours & \textbf{21.6} & \textbf{27.0} & \textbf{29.7} & \textbf{28.3} & \textbf{27.3} & \textbf{32.1} & \textbf{23.5} & \textbf{30.3} & \textbf{30.0} & \textbf{37.7} & \textbf{30.1} & \textbf{25.3} & 34.2 &\textbf{19.2} & \textbf{23.2} & \textbf{27.9}\\ \hline
\end{tabular}
}
\vspace{2pt}
\caption{Quantitative comparisons of the mean per-joint position error (MPJPE) on Human3.6M~\cite{ionescu2014human3} under Protocol~\#1 and Protocol~\#2, as well as using PA MPJPE as the evaluation metric. Similar to most of the competing methods (e.g.,~\cite{sun2017integral,pavlakos2018ordinal,yang20183d,dabral2018learning,tekin2017learning,fang2018learning}), our models were trained on the Human3.6M dataset and used also the extra MPII 2D pose dataset~\cite{Andriluka}. We also report the results without using the extra MPII 2D pose dataset.}
\label{table:protocols}
\end{table*}
\subsection{3D Human Pose Estimation}
We perform quantitative evaluation on three benchmark datasets: Human3.6M~\cite{ionescu2014human3}, HumanEva-I~\cite{sigal2010humaneva} and MPI-INF-3DHP~\cite{mehta2017monocular}.
Ablation study is conducted to evaluate our design choices.
We demonstrate that the proposed method shows superior generalization ability to
in-the-wild images.
\subsubsection{Datasets and Evaluation Protocols}
{\bf Human3.6M.}
Human3.6M~\cite{ionescu2014human3} contains 3.6 million RGB images captured by a MoCap System in
an indoor environment, in which 7 professional actors were performing
15 activities such as walking, eating, sitting, making a phone call and
engaging in a discussion, etc.
We follow the standard protocol as in~\cite{martinez2017simple, pavlakos2017coarse},
and use 5 subjects (S1, S5, S6, S7, S8) for training and the rest 2 subjects (S9, S11)
for evaluation (referred to as Protocol~\#1).
Some previous works reported their results with 6 subjects
(S1, S5, S6, S7, S8, S9) used for training and only S11 for evaluation~\cite{yasin2016dual,sun2017integral,dabral2018learning} (referred to as Protocol~\#2). Despite {\it not} using S9 also as training data,
we compare our results with these methods.
{\bf HumanEva-I.}
HumanEva-I~\cite{sigal2010humaneva} is one of the early datasets for evaluating 3D human poses.
It contains fewer subjects and actions compared to Human3.6M.
Following~\cite{Bo2010Twin}, we train a single model on
the training sequences of Subject 1, 2 and 3, and evaluate on
the validation sequences.
{\bf MPI-INF-3DHP.}
This is a recent 3D human pose dataset which includes both indoor and outdoor scenes~\cite{mehta2017monocular}.
Without using its training set, we evaluate our model trained from Human3.6M only
on the test set. The results are reported using the 3DPCK and the AUC metric~\cite{Andriluka,mehta2017monocular,pavlakos2018ordinal}.
{\bf Evaluation metric.}
We follow the standard steps to align the 3D pose prediction with the groundtruth by
aligning the position of the central hip joint, and use
the \textit{Mean Per-Joint Position Error}~(MPJPE) between
the groundtruth and the prediction as evaluation metrics.
In some prior works~\cite{yasin2016dual,sun2017integral,dabral2018learning},
the pose prediction was further aligned with the
groundtruth via a rigid transformation. The resulting MPJPE is termed as \textit{Procrustes Aligned}~(PA) MPJPE.
\subsubsection{Results and Comparisons}
\label{sec:stoa}
{\bf Human3.6M.}
We compare our method against state-of-the-art under three protocols, and the quantitative
results are reported in Table~\ref{table:protocols}. As can be seen,
our method outperforms all competing methods on nearly all action subjects
for the protocols used. It is worth mentioning that our method
makes considerable improvements on some challenging actions for 3D pose
estimation such as {\it Sitting} and {\it Walking}.
Thanks to HEMlets learning, our method demonstrates a clear advantage for handling
complicated poses.
With a simple network architecture and little parameter
tuning, we produce the most competitive results compared to previous
works with carefully designed networks powered by e.g., adversarial training schemes
or prior knowledge. On average, we improve the 3D pose prediction accuracy
by $20\%$ than that reported in Sun~{\it et al}.~\cite{sun2017integral} under Protocol \#1.
We also report our performance using PA MPJPE as the evaluation metric, and
compare with these methods that make use of S9 as additional training data. We
still outperform all of them across all action subjects, even {\it without}
utilizing S9 for training. {For fair comparison with the competing methods in Table~\ref{table:protocols}, our models were trained similarly on the Human3.6M dataset and used also the extra MPII 2D pose dataset. We have also trained our method by additionally using the FBI dataset, and the 3D pose prediction accuracy obtained further improves as the MPJPE goes from 39.9mm down to 36.5mm.}
\begin{table}[t]
\small
\setlength{\tabcolsep}{2pt}
\centering
\begin{tabular}{l|ccc|ccc|c}
\hline
\multirow{2}{*}{Approach}&
\multicolumn{3}{c|}{Walking}&
\multicolumn{3}{c|}{Jogging} & \multirow{2}{*}{Avg}\\ \cline{2-7}
&S1 &S2 &S3 &S1 &S2 &S3 \\ \hline
Simo-Serra~{\it et al}.~\cite{simo2013joint} &65.1 &48.6 &73.5 &74.2 &46.6 &32.2 &56.7 \\
Moreno-Noguer~{\it et al}.~\cite{moreno20173d} &19.7 &13.0 &24.9 &39.7 &20.0 &21.0 &26.9 \\
Martinez~{\it et al}.~\cite{martinez2017simple} &19.7 &17.4 &46.8 &26.9 &18.2 &18.6 &24.6 \\
Fang~{\it et al}.~\cite{fang2018learning} &19.4 &16.8 &37.4 &30.4 &17.6 &16.3 &22.9 \\
Pavlakos~{\it et al}.~\cite{pavlakos2018ordinal} &18.8 &12.7 &29.2 &\textbf{23.5} &15.4 &14.5 &18.3 \\ \hline
Ours &\textbf{13.5} &\textbf{9.9} &\textbf{17.1} &24.5 &\textbf{14.8} &\textbf{14.4} &\textbf{15.2} \\ \hline
\end{tabular}
\vspace{4pt}
\caption{Detailed results on the validation set of HumanEva-I~\cite{mehta2017monocular}. }
\label{table:HumanEva-I}
\end{table}
{\bf HumanEva-I.}
With the same network architecture where {\it only} the HumanEva-I dataset is used for training, our results are reported in Table~\ref{table:HumanEva-I} under the popular protocol~\cite{simo2013joint,moreno20173d,martinez2017simple,fang2018learning,pavlakos2018ordinal}. Different from these approaches~\cite{pavlakos2018ordinal,moreno20173d,martinez2017simple,fang2018learning} which used extra 2D datasets~(e.g., MPII) or pre-trained 2D detectors~(e.g., CPM~\cite{wei2016convolutional}), our method still outperforms previous approaches.
\begin{table}[t]
\small
\setlength{\tabcolsep}{2pt}
\centering
\begin{tabular}{lccccc}
\hline
\multirow{3}{*}{Approach}
&Studio & Studio &
\multirow{2}{*}{Outdoor }&
\multirow{2}{*}{All }&
\multirow{2}{*}{All } \\
&GS &no GS \\
\cline{2-6}
&3DPCK &3DPCK &3DPCK &3DPCK &AUC\\
\hline
Mehta~{\it et al}.~\cite{mehta2017monocular} &70.8 &62.3 &58.8 &64.7 &31.7\\
Zhou~{\it et al}.~\cite{zhou2017towards} &71.1 &64.7 &72.7 &69.2 &32.5\\
Pavlakos~{\it et al}.~\cite{pavlakos2018ordinal} &\textbf{76.5} &63.1 &77.5 &71.9 &35.3\\
\hline
Ours &75.6 &\textbf{71.3} &\textbf{80.3} &\textbf{75.3} &\textbf{38.0}\\
\hline
\end{tabular}
\vspace{4pt}
\caption{Detailed results on the test set of MPI-INF-3DHP~\cite{mehta2017monocular}. No training data from this dataset was used to train our model.}
\label{table:3dhpResults}
\end{table}
{\bf MPI-INF-3DHP.}
We evaluate our method on the MPI-INF-3DHP dataset using two metrics, the PCK and AUC. The results are generated by the model we trained for Human3.6M. In Table~\ref{table:3dhpResults}, we compare with three recent methods which are not trained on this dataset. Our result of ``Studio GS" is one percentage lower than~\cite{pavlakos2018ordinal}. But our method outperforms all these methods with particularly large margins for the ``Outdoor" and ``Studio no GS" sequences.
\subsubsection{Ablation Study}
\label{sec:ablation}
We study the influence on the final estimation performance of different choices made in our network design and training procedure.
{\bf Alternative intermediate supervision.}
First, We examine the effectiveness of using HEMlets supervision.
We evaluate the model trained without any intermediate supervision (Baseline),
with 2D heatmap supervision only, with HEMlets supervision only, and
with both 2D heatmap supervision and HEMlets supervision (Full).
All of these design variants are evaluated with the same experimental
setting (including training data, network architecture and $\mathcal{L}^{\rm 3D}_{\lambda}$ loss
definition) under Protocol \#1 on Human3.6M.
\begin{table}[t]
\small
\renewcommand\tabcolsep{3.0pt}
\begin{center}
\begin{tabular}{llcc}
\hline
Method & Supervision & H3.6M \#1 & H3.6M \#1$^*$ \\
\hline
Baseline & $\mathcal{L}^{\rm 3D}_{\lambda}$& 47.1 &55.3\\
w/ 2D heatmaps & $\mathcal{L}^{\rm 3D}_{\lambda} + \mathcal{L}^{\rm 2D}$ & 44.2 & 49.9\\
w/ HEMlets & $\mathcal{L}^{\rm 3D}_{\lambda} + \mathcal{L}^{\rm HEM}$& 42.6 & 46.0\\
Full & $\mathcal{L}^{\rm 3D}_{\lambda} + \mathcal{L}^{\rm HEM} + \mathcal{L}^{\rm 2D}$ & 39.9 & 45.1\\
\hline
\end{tabular}
\end{center}
\caption{Ablative study on the effects of alternative intermediate supervision evaluated on Human3.6M using Protocol~\#1. The last column~$^*$ reports the results using only the Human3.6M dataset for training (without using the extra MPII 2D pose dataset).}
\label{table:ablationStudy}
\end{table}
The detailed results are presented in Table~\ref{table:ablationStudy}. Using 2D heatmaps supervision for training, the prediction error is reduced by 3.0mm compared to the baseline. The HEMlets supervision provided 1.7mm lower mean error compared to the 2D heatmaps supervision. This validates the effectiveness of the intermediate supervision. By combining all these choices, our approach using HEMlets with 2D heatmap supervision achieves the lowest error. Without using the extra MPII 2D pose dataset, we repeated this study. Similar conclusions can still be drawn. But the gap between w/ HEMlets (excluding $\mathcal{L}^{\rm 2D}$, 46.0mm) and Full (45.1mm) shrinks, suggesting the strength of the HEMlets representation in encoding both 2D and (local) 3D information.
To further illustrate the effectiveness of HEMlets representation, we provide a visual comparison in Fig.~\ref{fig:example}. Though the 2D joint errors of the two estimations are quite close, the method with HEMlets learning significantly improves the 3D joint estimation result and fixes the gross limb errors.
\begin{figure}[t]
\centering
\includegraphics[width = \columnwidth ]{vizAblation-eps-converted-to.pdf}
\caption{An example image with the detected joints overlaid and shown from a novel view, using different methods: (a) $\mathcal{L}^{\rm 3D}_{\lambda} + \mathcal{L}^{\rm 2D}$ (2D error:~15.2; 3D joint error:~{\bf81.3mm}). (b) $\mathcal{L}^{\rm 3D}_{\lambda} + \mathcal{L}^{\rm 2D} + \mathcal{L}^{\rm HEM}$ (2D error:~13.0; 3D error:~{\bf41.2mm}). (c) Ground-truth. HEMlets learning helps fixing local part errors, see the blue skeletal part in~(a) versus the red skeletal part in~(b).}
\label{fig:example}
\end{figure}
Regarding the runtime, tested on a NVIDIA GTX 1080 GPU, our full model (with a total parameter number of 47.7M) takes 13.3ms for a single forward inference, while the baseline model (with 34.3M parameters) takes 8.5ms.
{\bf Variants of HEMlets.}
We next experimented with some variants of HEMlets on Human3.6M and MPII 2D pose datasets. In the first variant, we use five-state heatmaps,
referred to as \textit{5s-HEM}, where the child joint is placed to different layers of the
heatmaps according to the angle of the associated skeletal part with respect to the imaging plane.
Specifically, we
define the five states corresponding to the $(-90^\circ,-60^\circ)$, $(-60^\circ, -30^\circ)$,
$(-30^\circ, 30^\circ)$, $(30^\circ, 60^\circ)$ and $(60^\circ, 90^\circ)$ range, respectively.
In the second variant, we place a pair of joints in the negative and positive polarity
heatmaps respectively according to their depth ordering (i.e., the closer/farther joint will appear in the positive/negative polarity heatmap). If their depths are roughly the same, they are co-located in the zero polarity heatmap. We refer to this variant as \textit{2s-HEM}. We trained 5s-HEM, 2s-HEM and HEMlets with the Human3.6M dataset only. A comparison on the MPJPE of the validation set is given in Fig.~\ref{fig:Fiv-tri-state result}.
The other two variants produce inferior convergence compared to HEMlets under the same experiment setting.
\begin{figure}[h]
\centering
\includegraphics[width=0.80\columnwidth]{11.pdf}
\caption{The MPJPE of the validation set of 5s-HEM, 2s-HEM and HEMlets, respectively. All are trained with the Human3.6M dataset. }
\label{fig:Fiv-tri-state result}
\end{figure}
\begin{table}[h]
\begin{center}
\begin{tabular}{lcc}
\hline
Dataset &3DPCK\\
\hline
Base & 75.3 \\
w/ Ordinal~\cite{pavlakos2018ordinal} & 76.1 \\
w/ FBI~\cite{shi2018fbi} & \textbf{76.9} \\
w/ FBI~\cite{shi2018fbi} + Ordinal~\cite{pavlakos2018ordinal} & 76.5 \\
\hline
\end{tabular}
\end{center}
\caption{Evaluation of 3DPCK scores by adding different augmenting datasets that provide relative depth ordering annotations. Base denotes using the base datasets~(Human3.6M and MPII). }
\label{table:DataSetEffectiveness}
\end{table}
{\bf Augmenting datasets.}
Many state-of-the-art approaches use a mixed training strategy for 3D human pose estimation.
In addition to exploiting Human3.6M and MPII datasets, we study the effect of using
augmenting datasets such as Ordinal~\cite{pavlakos2018ordinal} and FBI~\cite{shi2018fbi}
for training. Firstly, we adapt the annotations of Ordinal and FBI datasets to the required form of HEMlets.
Then we train our model using different combinations of these additional datasets.
The comparisons on the MPI-INF-3DHP dataset~\cite{mehta2017monocular} are reported in Table~\ref{table:DataSetEffectiveness}. We find augmenting datasets
slightly increase the 3DPCK score for the trained model. Interestingly, training with FBI annotations
attains a better 3DPCK score than Ordinal annotations. We suspect this is
due to the amount of manual annotation errors related to different annotation schemes. In Fig.~\ref{fig:DifDataset result}, we also provide some visual examples to compare the effectiveness of different augmenting datasets. One can find that the model fine-tuned with the FBI dataset produces better predictions than the ones trained additionally with Ordinal~\cite{pavlakos2018ordinal}.
\begin{figure}[h]
\centering
\includegraphics[width=0.95\columnwidth]{difDataset_compose1.png}
\caption{The qualitative results for some examples of MPI-INF-3DHP~\cite{mehta2017monocular}, using different additional datasets. For each example, we present the input RGB image, the 3D human pose predicted by three different models. The groundtruth pose is shown in dashed line.}
\label{fig:DifDataset result}
\end{figure}
\begin{figure*}[ht]
\centering
\includegraphics[width=0.95\linewidth]{wild_3d__d6_cut.png}
\caption{Qualitative results on different validation datasets: the first two columns are from the test dataset of 3DHP~\cite{mehta2017monocular}. The other columns are from Leeds Sports Pose~(LSP)~\cite{johnson2010clustered}. Our approach produces visually correct results even on challenging poses~(last column). }
\label{fig:wild}
\end{figure*}
{\bf Generalization.}
For an evaluation of in-the-wild images from Leeds Sports Pose~(LSP)~\cite{johnson2010clustered} and the validation set of MPI-INF-3DHP~\cite{mehta2017monocular}, we list some visual results predicted by our approach. As shown in Fig.~\ref{fig:wild}, even for challenging data~(e.g., self-occlusion, upside-down), our method yields visually correct pose estimations for these images.
\subsection{3D Human Body Model Recovery}
In this part, we evaluate the proposed human body recovery method of regressing the SMPL parameters on three public datasets i.e., SURREAL~\cite{varol2017learning}, UP-3D~\cite{lassner2017unite} and 3DPW\cite{von2018recovering}. Before the experimental studies, we first give an introduction to the datasets and related evaluation protocols.
\subsubsection{Datasets and Evaluation Protocols}
\label{sec:dataset}
{\bf SURREAL.} SURREAL~\cite{varol2017learning} contains 6M frames from 1,964 video sequences of 115 subjects, where the images are photo-realistic renderings of people under large variations in shape, texture, viewpoint and pose. Because these synthetic bodies are created using SMPL body models, the corresponding model parameters are used as groundtruth for training a human body regression model.
{\bf UP-3D.}
To construct the UP-3D dataset~\cite{lassner2017unite}, the authors have collected a large number of real images mostly from the 2D human pose datasets~(i.e., MPII~\cite{Andriluka} and LSP~\cite{johnson2010clustered}). There are two main steps to produce the final dataset. First, a SMPLify optimization~\cite{bogo2016keep} is applied to obtain the 3D human body mesh results. Then, those inaccurate body fitting results were inspected and discarded manually by humans. As a result, 8,515 images with the fitted SMPL body parameters are obtained, among which 7,818 images are used for training and 1,389 for testing.
{\bf 3DPW.} Recently, the work of~\cite{von2018recovering} presented a new dataset which is captured under in-the-wild environment. Specifically, a moving hand-held camera is used for recording RGB frames while IMUs are attached on actors to capture poses. In total, 60 video sequences (more than 51,000 frames) of 5 subjects are captured, where 7 actors with 18 different clothing styles are asked to perform different activities, such as walking, playing golf and etc.
{\bf Evaluation metric.} We follow the standard protocols, as detailed in~\cite{yao2019densebody} to conduct evaluations. When dealing with the datasets of SURREAL and UP-3D, to measure the accuracy of the inferred body mesh, the average per-vertex Euclidean distance between it and the groundtruth is used (which is referred to as ``surface''). We also report the accuracy of the output 3D pose, where the average per-joint Euclidean distance between the estimated pose (with the hip joint aligned) and the groundtruth is used (which is referred to as ``joint''). For the dataset of 3DPW, we follow the works of~\cite{kanazawa2018end,kolotouros2019spin} to evaluate the reconstruction error of 3D poses, which is noted as ``Rec. Error''.
Basically, ``Rec. Error'' is computed as follows in SPIN~\cite{kolotouros2019spin}: it measures the average per-joint Euclidean distance between the predicted 3D human pose and the groundtruth after a global alignment post-process, where the predicted pose is regressed from the inferred 3D body mesh. In addition, following the work of~\cite{xu2019denserac}, the recovered 3D meshes are also projected onto a 2D image plane for evaluating the accuracy of the mask and part segmentation. By doing so, mIoU and F1 scores are reported.
\begin{table}[t]
\footnotesize
\setlength{\tabcolsep}{2pt}
\centering
\begin{tabular}{l|cc|cc|cc|c}
\hline
\multirow{2}{*}{Approach}&
\multicolumn{2}{|c|}{Human3.6M}&
\multicolumn{2}{|c|}{SURREAL}&
\multicolumn{2}{|c|}{UP-3D}&
3DPW\\ \cline{2-8}
&Pro.\#1 &Pro. \#2 &surface &joint &surface &joint &Rec. Error \\ \hline
Pavlakos~{\it et al}.~\cite{pavlakos2018learning} & - &75.9 & - &- &117.7 &- &- \\
HMR~\cite{kanazawa2018end} & 88.0 &59.1 & - &- &- &- &81.3 \\
BodyNet ~\cite{varol2018bodynet} & - &- &73.6 &- &- &- &- \\
SMPLR~\cite{madadi2018smplr} & 56.5 & 46.3 &74.5 &46.1 &- &- &- \\
DenseRaC~\cite{xu2019denserac} & 76.8 &- &- &- &- &- &- \\
TexturePose~\cite{pavlakos2019texturepose}& 51.3 & 49.7 &- &- &- &- &- \\
DenseBody~\cite{yao2019densebody} & 47.3 &38.1 &54.2 &40.1 &91.7 &71.4 &- \\
SPIN (SPIN*)~\cite{kolotouros2019spin }&- & 41.1 &- &- &- &-
&66.3 (59.2*) \\ \hline
\strut Ours & \textbf{39.9} &\textbf{32.1} &\textbf{53.3} &\textbf{37.7}
&\textbf{79.8} &\textbf{67.5} &\textbf{58.8} \\ \hline
\end{tabular}
\vspace{5pt}
\caption{Quantitative comparisons of fully body model recovery results over different datasets. * denotes the version that also applies the SMPLify optimization~\cite{bogo2016keep} as post-processing. All the numbers listed are in mm.
}
\label{table:Shape}
\end{table}
\begin{table}
\small
\setlength{\tabcolsep}{3pt}
\centering
\begin{tabular}{l|c|c|c|c }
\hline
\multirow{2}{*}{Approach}&
\multicolumn{2}{|c|}{FB Seg.}&
\multicolumn{2}{|c}{Part Seg.} \\ \cline{2-5}
&Accuracy &F1 &Accuracy &F1 \\ \hline
SMPLify oracle~\cite{lassner2017unite} & 92.17 &0.88 &88.82 &0.67 \\
SMPLify~\cite{bogo2016keep} & 91.89 &0.88 &87.71 &0.64 \\
HMR~\cite{kanazawa2018end} & 91.67 &0.87 &87.12 &0.60 \\
SPIN~\cite{kolotouros2019spin } &91.07 &0.86 &88.48 &0.65 \\
SPIN*~\cite{kolotouros2019spin} &91.83 &0.87 &89.41 &0.68 \\
BodyNet ~\cite{varol2018bodynet} & \textbf{92.80} &0.84 &- &- \\
DenseRaC~\cite{xu2019denserac} & 92.40 &0.88 &87.90 &0.64 \\
\hline
Ours & 92.30 &0.88 &\textbf{90.18} &\textbf{0.71} \\
Ours* &\textbf{93.67} &\textbf{0.90} &\textbf{91.19} &\textbf{0.74} \\ \hline
\end{tabular}
\vspace{4pt}
\caption{Quantitative comparisons between our method and existing ones on foreground and part segmentation of the recovered full body mesh on the UP-3D dataset. * denotes the version that also applies the SMPLify optimization~\cite{bogo2016keep} as post-processing.}
\label{table:SegEva}
\end{table}
\subsubsection{Results and Comparisons}
Next, we report the evaluation results and also compare them with state-of-the-art methods both quantitatively and qualitatively.
{\bf Quantitative comparisons.} In Table~\ref{table:Shape}, we numerically compare our method to existing leading approaches on the evaluation metrics presented in Sect.~\ref{sec:dataset}. As can be seen, our method produces the best accuracy for both the output skeleton joints and the generated body mesh. Table~\ref{table:SegEva} also lists the accuracy of the foreground and the part segmentation, given the generated body mesh. Our proposed method again gives the best performance. It is noteworthy that the part segmentation F1 score of our method evaluated on the UP-3D dataset exceeds 0.70 for the first time.
{\bf Qualitative comparisons.} We also conduct qualitative comparisons between our method and some of existing methods, as shown in Fig.~\ref{fig:shapecomp}. Here, HMR~\cite{kanazawa2018end} and SPIN~\cite{kolotouros2019spin} are selected as two representative body mesh recovery approaches. Given an input image, the output body mesh of each method is shown in two views. It can be observed that our method performs better than HMR and SPIN, even when the human pose is challenging.
{\color{blue}
\begin{figure*}[t]
\centering
\includegraphics[width=16.5cm]{imgcompose_hmradds.png}
\caption{Qualitative comparisons of our method with some existing ones on human body model recovery. For each example, the input image is first shown, which is followed by the results of HMR~\cite{kanazawa2018end}, SPIN~\cite{kolotouros2019spin} and ours. For each resulting body mesh, two views are provided for visualization.
}
\label{fig:shapecomp}
\end{figure*}
}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{multiperson.pdf}
\caption{The results of the proposed approach on multi-person scenarios.}
\label{fig:multi-shape}
\end{figure}
{
{\bf Ablation study.}
For the ablative analysis of 3D human body shape regression, we trained two alternative models to evaluate the necessity of each branch of the proposed HEMlets-based body regression method depicted in Fig.~\ref{fig:shapeModule}: 1) by only using the 3D joints branch in Fig. 5, and 2) by only using the high-level feature extraction branch in Fig.~\ref{fig:shapeModule}. The results are reported in Table~\ref{table:shape}. It can be observed that the 3D joints regression component plays a more important role for the task of 3D body shape regression.
\begin{table}[t]
\begin{center}
\begin{tabular}{lc}
\hline
Method &Rec. Error measured on 3DPW\\
\hline
w/ only high-level features &{63.1} \\
w/ only the 3D joints &{60.3} \\
Full &\bf{58.8} \\
\hline
\end{tabular}
\end{center}
\caption{Evaluating the impact of each branch in Fig.~\ref{fig:shapeModule} on human body estimation. }
\label{table:shape}
\end{table}
\subsubsection{Extended Studies}
\begin{table}[t]
\begin{center}
\begin{tabular}{lc}
\hline
Method &Rec. Error measured on 3DPW\\
\hline
Proposed model &58.8 \\
w/ groundtruth shape $\beta^{\rm gt}$
&57.2 \\
w/ groundtruth pose $\theta^{\rm gt}$ &\bf{9.4} \\
\hline
\end{tabular}
\end{center}
\caption{Evaluation of the impact of learning $\theta$ and $\beta$ on human body estimation. }
\label{table:Shape_abalation}
\end{table}
\begin{figure}[t]
\center
\includegraphics[width=0.96\linewidth]{failureCase.png}
\caption{Some failure cases of the proposed approach.}
\label{fig:failurecase}
\end{figure}
We make a few extended studies to further understand the proposed HEMlets-based human pose and shape estimation method.
\indent{\bf How does pose accuracy affect body recovery?} As the accuracy of the output body mesh relies on both the estimated pose and the shape parameters. An interesting question is which factor affects more. To reach an answer, we run two alternative versions of our full model on the 3DPW dataset: 1) replacing its estimated shape parameter with the groundtruth shape, and 2) replacing the estimated pose parameter with the groundtruth pose. The results are reported in Table~\ref{table:Shape_abalation}. As one can see, the accuracy of pose estimation has a greater impact. This suggests 3D pose estimation is critical and provides more significant contributions to the task of human body mesh recovery from a single color image.
\indent{\bf Multi-person 3D pose and shape.} Fig.~\ref{fig:multi-shape} shows our method can also work well for the multi-person scenarios. To do that, we firstly employ the code of OpenPose~\cite{cao2016realtime} to detect person instances. Each instance is then cropped, to which the proposed HEMlets PoSh approach is applied for individual 3D body model inference.
{\bf Failure cases.} Our method tends to fail for some complicated scenarios, e.g., poor lighting, severe occlusions and background interference. Some of such failure cases are shown in Fig.~\ref{fig:failurecase}.
More supplementary materials including demo videos are available at the project website:~\url{https://sites.google.com/site/hemletspose/}. We will make our code publicly available for research uses and also link it at the same project website.
\section{Conclusion}
In this paper, we proposed a simple and highly effective HEMlets-based 3D pose estimation method from a single color image. HEMlets is an easy-to-learn intermediate representation encoding the relative forward-or-backward depth relation for each skeletal part's joints, together with their spatial co-location likelihoods. It is proved very helpful to bridge the input 2D image and the output 3D pose in the learning procedure. We demonstrated the effectiveness of the proposed method tested over the standard benchmarks, yielding a relative accuracy improvement of about 20\% over one best-of-grade method~\cite{sun2017integral} on the Human3.6M benchmark. Good generalization ability is also witnessed for the presented approach. Extending the HEMlets pose estimation network, we further designed a simple parametric 3D human body regression network to estimate the SMPL body shape and pose from the input color image. Extensive experiments have shown the state-of-the-art performance of the proposed HEMlets PoSh method both quantitatively and qualitatively. Specifically, it has achieved the best human body recovery results across different benchmark datasets and evaluation metrics, and notably, obtained the lowest ``surface'' errors for the SURREAL and UP-3D datasets.
We believe the proposed HEMlets idea is actually general, which may potentially benefit other 3D regression problems e.g., scene depth estimation. Future directions also include an optimized real-time system that detects and tracks multiple persons robustly, and reconstructs their 3D poses and body shapes.
\ifCLASSOPTIONcompsoc
\section*{Acknowledgments}
\else
\section*{Acknowledgment}
\fi
This work is supported in part by the National Natural Science Foundation of China (Grant No.: 61771201), the Program for Guangdong Introducing Innovative and Enterpreneurial Teams (Grant No.: 2017ZT07X183), the Pearl River Talent Recruitment Program Innovative and Entrepreneurial Teams in 2017 (Grant No.: 2017ZT07X152), the Shenzhen Fundamental Research Fund (Grants No.: KQTD2015033114415450 and ZDSYS201707251409055), and Department of Science and Technology of Guangdong Province Fund (2018B030338001). The authors would like to thank Yulong Shi and Kaiqi Wang for assisting in some early experiments. This work was mainly done when Kun, Nianjuan and Jiangbo were working in Shenzhen Cloudream Technology Co., Ltd.
{\small
\bibliographystyle{ieee_fullname}
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3,212,635,537,457 | arxiv | \section{Introduction}\label{I}
At the beginning of the 19th century, the General Theory of Relativity brought the revolution to the modern cosmology proposed by Albert Einstein. The Riemannian space-time formulates this theory based on the Levi-Civita connection, a torsion-free, and metric compatibility connection. It also helps us to understand the geodesic structure of the Universe. Later on, it faced problems like fine-tuning, cosmic co-incidence, initial singularity, cosmological constant, and flatness \cite{sahni/2000}, since modern cosmology is growing by a prominent number of accurate observations. Besides, the cosmological observation, such as type Ia supernovae \cite{Perlmutter/1999,Riess/1998}, cosmic microwave background (CMB) radiation \cite{Spergel/2003,Komatsu/2011}, large scale structure \cite{Tegmark/2004,Seljak/2005}, baryon acoustic oscillations \cite{Eisenstein/2005}, and weak lensing \cite{Jain/2003} confirms that currently, our Universe is going through an accelerated expansion phase that happens because of the highly negative pressure produced by the unknown form of matter and energy, called dark energy and dark matter. To overcome the above issues, researchers started to modify the Einstein's theory of relativity, and they ended up with several modified theories of gravity such as $f(R)$ gravity \cite{nojiri/2006}, $f(R,T)$ gravity \cite{harko/2011}, $f(T)$ gravity \cite{cai/2016}, $f(Q)$ gravity \cite{jimenez/2018}, $f(Q,T)$ gravity \cite{xu/2019}, etc. As a result, cosmologists found many interesting results such as Yousaf et al., studied the self-gravitating structures \cite{yousaf/2020}, gavastars \cite{yousaf/2020a}, Sahoo et al. \cite{sahoo/2020} studied the wormhole geometry, bouncing cosmology, and accelerated expansion of the universe using modified theories of gravity. The main advantage of the modified theories of gravity is that it successfully describes the late-time cosmic acceleration and the early time inflation. In the early stage of the Universe, there is a possibility of particle creation. In this study, we are focusing on particle production in the teleparallel gravity.
\newline
The $f(T)$ theories of gravity are the generalization of the Teleparallel Equivalent of General Relativity (TEGR), where $T$ is the torsion scalar \cite{Cai/2016}. TEGR was first presented by Einstein. In the context of $f(T)$ theories, the Reimann-Cartan space-time requires the torsional curvature to vanish. Furthermore, this type of space-time is constructed by the Weitzenb\"ock connection \cite{weit/1923,vinckers/2020}. TEGR is equivalent to GR; the reason is that both cases' action is the same except the surface term in TEGR. But, the physical interpretation is different from each other. The construction of gravitational Lagrangian in TEGR formulation was done in \cite{maluf/1994,andrade/1997}. TEGR is Lorentz invariant theories, whereas the modified $f(T)$ theories are not Lorentz invariant. Also, the motion equations in $f(T)$ gravity are not necessarily Lorentz invariant, because the certainty is that this theory's property explains the recent interests \cite{tamanini/2012,meng/2011,ferraro/2011,cruz/2014}. Moreover, the modification of TEGR was motivated by the $f(R)$ gravity theory. In the teleparallel gravity theory, we use the Weitzenb\"ock connection instead of the Levi-Civita connection, which uses $f(R)$ gravity to vanish the non-zero torsion curvature. Here, we would like to mention that the $f(T)$ gravity does not need the Equivalence principle because the Weitzenb\"ock connection describes its gravitational interaction. It is a simple modified theory compared to other modified theories because the torsion scalar $T$ contains only the first-order derivatives of the vierbeins. In contrast, Ricci scalar $R$ contains the second-order derivatives of the metric tensors. Recently, Mandal et al., studied the acceleration expansion of the Universe using the parametrization technique with presuming exponential and logarithmic form of $f(T)$ \cite{mandal/2020} in $f(T)$ gravity. They also studied a complete cosmological scenario of the Universe in $f(T)$ gravity, where they discussed the difference between General Relativity and Teleparallel gravity \cite{mandal/2020a} in $f(T)$ gravity. M. Sharif and S. Rani studied the dynamical instability ranges in Newtonian as well as post-Newtonian regimes considering power-law $f(T)$ model with anisotropic fluid in $f(T)$ gravity \cite{sharif/2014}. Cai et al., studied the matter bounce cosmology using perturbation technique and they found a scale-invariant power spectrum, which is consistent with cosmological observations in $f(T)$ gravity \cite{cai/2011}. In \cite{sharif/2013}, the wormhole solutions with non-commutative geometry have been studied assuming power-law $f(T)$ model and a particular shape function in teleparallel gravity. Inflationary universe studied using power-law $f(T)$ function and logamediate scale factor in \cite{kazem/2017}, and constant-roll inflation studied in \cite{awad/2017}.
\newline
In the early stage of the universe, the possibility of particle creation has been discussed for curved space-time by Schrodinger \cite{schrodinger/1939}, Dewitt \cite{dewitt/1953}, Imamura \cite{imamura/1960}. Later, the first ever particle creation was treated by an external gravitational field by Parkar \cite{parkar/1968, parkar/1969}. In flat space-time, the unique vacuum state is identifying by the guidance of Lorentz invariance. Moreover, we do not have Lorentz symmetry in curved space-time. In general, there are more than one vacuum state exists in a curved space-time. Therefore, the particle creation idea becomes open to discuss, but it's physical interpretation becomes more difficult \cite{birrell/1982,ford/1997}. The interaction between the dynamical external gradients causes the particle creation from the vacuum. The particle creation produces negative pressure, so it is considered to explain the accelerated expansion of the universe and got some unexpected outcomes. Also, it might play the role of unknown gradients of the universe. In \cite{zimdahl/2001,qiang/2007} studied the particle creation with SNe Ia data. Singh \cite{csingh/2012}, and Singh and Beesham \cite{singh/2011,singh/2012} studied the particle creation with some kinematical tests in FLRW cosmology. The continuous creation of particle predicts the assumptions of standard Big Bang cosmology.\\
The thermodynamical study of black hole gives the fundamental relation between thermodynamics and gravitation \cite{bardeen/1973,bekenstein/1973,hawking/1975,gibbons/1977,bamba/2011c}. In GR, the relation between the entropy and the horizon area with the Einstein equation derives from the Clausius relation in thermodynamics \cite{jacobson/1995,elizalde/2008}. This idea is also used for other theories, mainly, the generalized thermodynamics laws and modified theories of gravity which are derived from the GR \cite{miaoa/2011,karami/2012}. Among the modified theories of gravity, $f(R)$ gravity got more attention on this framework. Thereby, one can obtained the gravitational field equation through the non-equilibrium feature of thermodynamics by using the Clausius approach. There are some work have been done in the thermodynamics of particle creation in $f(T)$ gravity theory \cite{setare/2013,bamba/2012,csingh/2014,salako/2013,bamba/2011c}.\\
In this work, we study the theoretical significance of particle creation in $f(T)$ gravity theory considering a flat FRW model. Assuming $f(T)$ as the sum of torsion scalar $T$ and an arbitrary function of torsion scalar $T$, we studied the thermodynamics of particle creation with $f(T)=0$ is a simple teleparallel gravity, $f(T)=A(-T)^q$ as power law gravity and $f(T)=A(1-e^{-qT})$ as exponential gravity. After that we discussed the behaviour of supplementary pressure $p_c$, particle number density $n$, and the particle creation rate $\psi$ for three models. Also we compared the effect of the cosmological pressure $p_m$ with the supplementary pressure $p_c$ for different values of equation of state parameter $\omega$ on particle creation.\\
This work is organised as follows. In Sec. \ref{II}, we discussed the thermodynamics of particle creation, which is followed by the overview of $f(T)$ gravity and it's field equations in Sec. \ref{III}. In Sec. \ref{IV}, we discussed three $f(T)$ gravity models. Finally, the results are summarized in Sec. \ref{VI}
\section{Thermodynamics of particle creation}\label{II}
If we assume the total number of particles in the universe to be conserved, the laws of thermodynamics can be expressed as
\begin{equation}\label{3a}
d Q = d (\rho_{m}V)+p_{m}dV
\end{equation}
and
\begin{equation}\label{3b}
T dS =p_{m}dV + d(\rho_{m}V)
\end{equation}
where $p_{m}$, $\rho_{m}$, $V$, $T$ and $S$ denote respectively the cosmological pressure, density, volume, temperature and entropy. Also, $dQ$ represent the heat exchange in the time interval $dt$. From (\ref{3a}) and (\ref{3b}), we further obtain,
\begin{equation}\label{3c}
d Q=TdS
\end{equation}
Eq. (\ref{3c}) reflects the fact that the entropy is a conserved quantity, since for lose adiabatic system $dQ=0$. We now consider a scenario in which the total number of particles in the universe is not constant. Under this condition, Eq (\ref{3a}) gets modified to \cite{cpfrt}
\begin{equation}\label{3d}
d Q = d (\rho_{m}V)+p_{m}dV + (h/n) d(nV)
\end{equation}
where $N=nV$, $n$ being the number density of the particles and $h=(p_{m}+\rho_{m})$ the enthalpy per unit volume of the system. For an adiabatic system where $dQ=0$, (\ref{3d}) reads \cite{cpfrt}
\begin{equation}\label{3e}
d (\rho_{m}V)+p_{m}dV = (h/n) d(nV)
\end{equation}
In \cite{cpfrt}, the authors stated that in cosmology this change in the total number of particles in the universe can be understood as a transformation of gravitational field energy to the matter. \\
For an open thermodynamic system, Eq. (\ref{3e}) can be expressed as \cite{cpfrt}
\begin{equation}\label{3f}
d (\rho_{m}V) = -\left(p_{m}+p_{c} \right) dV
\end{equation}
where
\begin{equation}\label{3g}
p_{c} = -(h/n)(dN/dV)
\end{equation}
represents supplementary pressure associated with the creation of particles \cite{cpfrt}. Note that negative $p_{c}$ indicate production of particles whereas positive $p_{c}$ implies particle annihilation and finally for $p_{c}=0$ the total number of particles is constant. Using Eq (\ref{3b}) and (\ref{3e}), it can also be shown that \cite{cpfrt}
\begin{equation}\label{3h}
S =S_{0}\left( \frac{N}{N_{0}}\right)
\end{equation}
where $S_{0}$ and $N_{0}$ represent current values of these quantities. \\
Additionally, we assume the particles follow a barotropic equation of state and therefore can be written as
\begin{equation}\label{3i}
p_{m}=(\omega)\rho_{m}
\end{equation}
where $-1\leq \omega \leq 1$ is the EoS parameter. The number density of particles is related to the density $\rho_{_{m}}$ as \cite{singh/2012}
\begin{equation}\label{3j}
n = n_{0} \left(\frac{\rho_{m}}{\rho_{0}} \right)^{\frac{1}{1+\omega}}
\end{equation}
where $\rho_{0} \geq 0$ and $n_{0} \geq 0$ are the present values of density and particle number density respectively.\\
We now consider the matter creation rate to be defined as \cite{21}
\begin{equation}\label{3k}
\psi(t)=3 \beta n H
\end{equation}
where $0 \leq \beta \leq 1$ is assumed to be a constant and $\psi(t)$ represent the rate of particle creation and has a dimension of $t^{-1}$. $\psi$ can either be positive or negative depending on the creation or annihilation of particles. $\psi = 0$ indicate particle number being conserved in the universe. For cosmological matter following barotropic equation of state (Eq. \ref{3i}), the supplementary pressure $p_{c}$ can be expressed as \cite{cpfrt}
\begin{equation}\label{3l}
p_{c}=-\beta (\omega +1) \rho_{m}
\end{equation}
\section{Overview of $f(T)$ Gravity}\label{III}
Let us consider the extension of Einstein-Hilbert Lagrangian of $f(T)$ theory of gravity (which is similar to $f(R)$ gravity extension from the Ricci scalar $R$ to $R+f(R)$ in the action), namely the teleparallel gravity term $T$ to $T+f(T)$, where $f(T)$ is an arbitrary function of $T$ as
\begin{equation}
\label{a}
S=\frac{1}{16\pi G}\int[T+f(T)]e d^4x,
\end{equation}
where $e=det(e^i_\mu)=\sqrt{-g}$ and $G$ is the gravitational constant. Assume $k^2=8\pi G=M_p^{-1}$, where $M_p$ is the Planck mass.The gravitational field is defined by the torsion one as
\begin{equation}
\label{b}
T^{\gamma}_{\mu \nu}\equiv e^{\gamma}_i(\partial_{\mu} e^i_{\nu}-\partial_{\nu} e^i_{\mu}).
\end{equation}
The contracted form of the above torsion tensor is
\begin{equation}
\label{c}
T\equiv \frac{1}{4}T^{\gamma \mu \nu}T_{\gamma \mu \nu}+\frac{1}{2}T^{\gamma \mu \nu}T_{\nu \mu \gamma}-T^{\gamma}_{\gamma \mu}T^{\nu \mu}_{\nu}.
\end{equation}
By the variation of the total action $S+L_m$, here $L_m$ is the matter Lagrangian gives us the field equation for $f(T)$ gravity as
\begin{multline}
e^{-1}\partial_{\mu}(ee^{\gamma}_i S^{\mu \nu}_{\gamma})(1+f_T)-(1+f_T)e^{\lambda}_i T^{\gamma}_{\mu \lambda}S^{\nu \mu}_{\gamma}\\
+e^{\gamma}_i S^{\mu \nu}_{\gamma}\partial_{\mu}(T)f_{TT}+\frac{1}{4}e^{\nu}_i[T+f(T)]=\frac{k^2}{2} e^{\gamma}_iT^{(M)\nu}_{\gamma},
\label{d}
\end{multline}
where $f_T=df(T)/dT$, $ f_{TT}=d^2f(T)/dT^2$, the "superpotential `` tensor $S^{\mu \nu}_{\gamma}$ written in terms of cotorsion $K^{\mu \nu}_{\gamma}=-\frac{1}{2}(T^{\mu \nu}_{\gamma}-T^{\nu \mu}_{\alpha}-T^{\mu \nu}_{\alpha})$ as $S^{\mu \nu}_{\gamma}=\frac{1}{2}(K^{\mu \nu}_{\gamma}+\delta^{\mu}_{\gamma}T^{\alpha \nu}_{\alpha}-\delta^{\nu}_{\gamma}T^{\alpha \mu}_{\alpha})$ and $T^{(M)\nu}_{\gamma}$ represents the energy-momentum tensor to the matter Lagrangian $L_m$.
Now we consider a flat FLRW universe with the metric as
\begin{equation}
\label{e}
ds^2=dt^2-a^2(t)dx^{\mu} dx^{\nu},
\end{equation}
where $a(t)$ is the scale factor, which gives us
\begin{equation}
\label{f}
e^{i}_{\mu}=diag(1,a,a,a).
\end{equation}
Using equation (\ref{f}) into the field equation (\ref{d}), we get the modified field equation as follows
\begin{equation}\label{4a}
H^{2}=\frac{8 \pi G}{3}\rho_{m} -\frac{f}{6} + \frac{T f_{T}}{3},
\end{equation}
\begin{equation}\label{4b}
\dot{H}=-\left[\frac{4 \pi G (\rho_{m}+p_{m}+p_{c})}{1+f_{T}+2 T f_{T T}} \right] ,
\end{equation}
where $H\equiv \dot{a}/a$ be the Hubble parameter and "dot`` represents the derivative with respect to $t$. Here, $\rho_m$ and $p_m$ be the energy density and pressure of the matter content, $p_c$ be the supplementary pressure. Also, we have used
\begin{equation}
\label{g}
T=-6H^2,
\end{equation}
which holds for a FLRW Universe according to equation (\ref{c}).
\section{$f(T)$ gravity models}\label{IV}
In this section we shall investigate the temporal evolution of particle production in radiation ($\omega=1/3$) and dust universe ($\omega=0$) for various $f(T)$ gravity models with model parameters constrained from cosmological observations related to gravitational baryogenesis.\\
For the purpose of analysis, we shall assume a power law evolution of scale factor of the form
\begin{equation}\label{5a}
a(t) = a_{0} t^{\left[ \frac{2}{3(1+\omega)}\right] }
\end{equation}
where $a_{0}>0$ is a constant.
\subsection{Simple Teleparallel gravity}
In simple teleparallel equivalent of general relativity \cite{12}, where $f(T)=0$ , for a universe composed of perfect fluid, the field equations (\ref{4a}) and (\ref{4b}) becomes
\begin{equation}\label{6a}
H^{2}=\frac{8 \pi G}{3}\rho_{m}
\end{equation}
\begin{equation}\label{6b}
\dot{H}=-4 \pi G (\rho_{m}+p_{m}+p_{c})
\end{equation}
Substituting (\ref{5a}) in (\ref{6a}), we obtain the expression of density $\rho_{m}$ as
\begin{equation}\label{6c}
\rho_{m}=\frac{4}{3}\left( \frac{1}{t^{2}(1+\omega)^{2}}\right)
\end{equation}
The expression of supplementary pressure $p_{c}$, particle number density $n$ and particle creation rate $\psi$ are obtained respectively as
\begin{equation}\label{6d}
p_{c}=\frac{4}{3} \left( \frac{\beta (1+\omega)}{t^{2}(1+\omega)^{2}}\right)
\end{equation}
\begin{equation}\label{6e}
n=\left[ \frac{4}{3}\left( \frac{1}{t^{2}(1+\omega)^{2}}\right)\right] ^{\frac{1}{(1+\omega)}}
\end{equation}
\begin{equation}
\psi = 3 \beta \left[ \frac{2}{3 t(1+\omega)}\right] \times \left[ \frac{4}{3}\left( \frac{1}{t^{2}(1+\omega)^{2}}\right)\right] ^{\frac{1}{(1+\omega)}}
\end{equation}
\begin{figure}[H]
\centering
\includegraphics[width=11 cm]{p1.pdf}
\caption{The behaviour of supplementary pressure $p_c$ with respect to cosmic time $t$ for $\omega=\frac{1}{3}, \omega=0$ and $\beta=1$.}
\label{f1}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=11 cm]{n1.pdf}
\caption{The behaviour of particle number density $n$ with respect to cosmic time $t$ for $\omega=\frac{1}{3}, \omega=0$ and $\beta=1$.}
\label{f2}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=11 cm]{psi1.pdf}
\caption{The behaviour of particle creation rate $\psi$ with respect to cosmic time $t$ for $\omega=\frac{1}{3}, \omega=0$ and $\beta=1$.}
\label{f3}
\end{figure}
\subsection{Power Law Gravity}
The power law model of Bengochea and Ferraro \cite{16} reads
\begin{equation}\label{7a}
f(T)= A(-T)^{q}
\end{equation}
where $A$ is a constant and $q>1$. In \cite{oiko}, the authors reported viable baryon-to-entropy ratio for $A=-10^{-7} \texttt{or} -10^{-6}$ and $q\gtrsim 4.8$. However, other values of the model parameters could also yield viable estimates of baryon-to-entropy ratio. Nonetheless, we restrict ourselves to the values $A=-10^{-7}$ and $q=5$ for the present analysis. Substituting (\ref{5a}) and (\ref{7a}) in (\ref{4a}) and (\ref{4b}), the expression of density $\rho_{m}$ reads
\begin{equation}\label{7b}
\rho_{m}=A6^{(q-1)}(1-18q)t^{-2q}\left[ \frac{2}{3(1+\omega)}\right]^{2q}
\end{equation}
The expression of supplementary pressure $p_{c}$, particle number density $n$ and particle creation rate $\psi$ for the power law gravity are obtained respectively as
\begin{equation}\label{7c}
p_{c}=-\beta (1+\omega)A6^{(q-1)}(1-18q)t^{-2q}\left[ \frac{2}{3(1+\omega)}\right]^{2q}
\end{equation}
\begin{equation}\label{7d}
n=\left[ A6^{(q-1)}(1-18q)t^{-2q}\left[ \frac{2}{3(1+\omega)}\right]^{2q}\right] ^{1/(1+\omega)}
\end{equation}
\begin{equation}\label{7e}
\psi = 3 \beta \left[ \frac{2}{3 t(1+\omega)}\right] \times \left[ A6^{(q-1)}(1-18q)t^{-2q}\left[ \frac{2}{3(1+\omega)}\right]^{2q}\right] ^{1/(1+\omega)}
\end{equation}
\begin{figure}[H]
\centering
\includegraphics[width=11 cm]{p2.pdf}
\caption{The behaviour of supplementary pressure $p_c$ with respect to cosmic time $t$ for $\omega=\frac{1}{3}, \omega=0$ and $\beta=1, q=5, A=-10^{-7}$}
\label{f4}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=11 cm]{n2.pdf}
\caption{The behaviour of particle number density $p_c$ with respect to cosmic time $t$ for $\omega=\frac{1}{3}, \omega=0$ and $\beta=1, q=5, A=-10^{-7}$.}
\label{f5}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=11 cm]{psi2.pdf}
\caption{The behaviour of particle creation rate $\psi$ with respect to cosmic time $t$ for $\omega=\frac{1}{3}, \omega=0$ and $\beta=1, q=5, A=-10^{-7}$.}
\label{f6}
\end{figure}
\subsection{Exponential Gravity}
The exponential $f(T)$ model given in \cite{23} reads
\begin{equation}\label{8a}
f(T)=A (1-e^{-q T})
\end{equation}
where $A$ and $q$ are model parameters. In \cite{oiko} the authors reported a wide range of values of $A$ and $q$ for which a viable baryon-to-entropy ratio could be realized. However, we shall work with $A=1$ and $q=10^{-10}$ as these values were used in \cite{oiko} to fit the baryon-to-entropy ratio with observations. Substituting (\ref{5a}) and (\ref{8a}) in (\ref{4a}) and (\ref{4b}), the expression of density $\rho_{m}$ reads
\begin{equation}\label{8b}
\rho_{m} =\left[ \frac{18 A \left[ \frac{2}{3(1+\omega)}\right]^{2}q e^{ \frac{6\left[ \frac{2}{3(1+\omega)}\right]^{2}q}{t^{2}}}}{t^{2}}\right]
\end{equation}
The expression of supplementary pressure $p_{c}$, particle number density $n$ and particle creation rate $\psi$ for the exponential gravity are obtained respectively as
\begin{equation}\label{8c}
p_{c}=-\beta (1+\omega) \left[ \frac{18 A \left[ \frac{2}{3(1+\omega)}\right]^{2}q e^{ \frac{6\left[ \frac{2}{3(1+\omega)}\right]^{2}q}{t^{2}}}}{t^{2}}\right]
\end{equation}
\begin{equation}\label{8d}
n=\left[ \frac{18 A \left[ \frac{2}{3(1+\omega)}\right]^{2}q e^{ \frac{6\left[ \frac{2}{3(1+\omega)}\right]^{2}q}{t^{2}}}}{t^{2}}\right]^{1/(1+\omega)}
\end{equation}
\begin{equation}\label{8e}
\psi = 3 \beta \left[ \frac{2}{3 t(1+\omega)}\right] \times \left[\left( \frac{18 A \left[ \frac{2}{3(1+\omega)}\right]^{2}q e^{ \frac{6\left[ \frac{2}{3(1+\omega)}\right]^{2}q}{t^{2}}}}{t^{2}}\right)^{1/(1+\omega)} \right]
\end{equation}
\begin{figure}[H]
\centering
\includegraphics[width=11 cm]{p3.pdf}
\caption{The behaviour of supplementary pressure $p_c$ with respect to cosmic time $t$ for $\omega=\frac{1}{3}, \omega=0$ and $\beta=1, q=10^{-10}, A=1$.}
\label{f7}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=11 cm]{n3.pdf}
\caption{The behaviour of particle number density $n$ with respect to cosmic time $t$ for $\omega=\frac{1}{3}, \omega=0$ and $\beta=1, q=10^{-10}, A=1$.}
\label{f8}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=11 cm]{psi3.pdf}
\caption{The behaviour of particle creation rate $\psi$ with respect to cosmic time $t$ for $\omega=\frac{1}{3}, \omega=0$ and $\beta=1, q=10^{-10}, A=1$.}
\label{f9}
\end{figure}
\section{Discussion of Outcomes and Conclusions}\label{VI}
In this article, we have studied the thermodynamics of an open system with particle creation of a flat FLRW universe in $f(T)$ theory of gravity. We have constructed three cosmological models by assuming suitable functions for $f(T)$ as $f(T)=0, f(T)=A(-T)^q , f(T)=A(1-e^{-qT})$ and the particle creation rate $\psi$. To analyze our models, we have considered the power law evolution of the scale factor and studied the behaviour of physical quantities (i.e. the supplementary pressure $p_c$, particle number density $n$, and the particle creation rate $\psi$) through their graphical representations with respect to cosmic time $t$ and some fixed values of $\beta$ in various phase of the evolution of the universe. And, details of our cosmological models discussed in the following.
In our simple Teleparallel gravity model, we have considered the minimal coupling between matter and geometry. In Fig. \ref{f3}, profiles of $\psi$ have been shown. From Fig. \ref{f3}, one can easily observe that the rate of particle creation is high in the early time and tends to zero when $t$ tends to infinity. But, the number of particle in the universe increases with cosmic time $t$ shown in Fig. \ref{f2}. The supplementary pressure $p_c$ has higher negative which shows that the particle production is high during the early stage and tends to zero when $t$ tends infinity, in Fig. \ref{f1}. From this model we have concluded that the evolution of the universe depends on the contribution of the particle production.
In power law gravity and exponential gravity models, we have considered the non-minimal coupling between matters. The profiles of $\psi, n$ and $p_c$ have been shown for the corresponding models. In Fig. \ref{f6},\ref{f9}, the particle creation rate $\psi$ is high in the early stage and it tends to zero as cosmic time $t$ tends to infinity. Also, the particle number density n in Fig. \ref{f5},\ref{f8} goes to zero as cosmic time $t$ goes to infinity which concluded that the expansion rate overcomes the rate particle creation as the supplementary pressure in Fig. \ref{f4},\ref{f7} is negative throughout the evolution of the universe in different phases. The density parameters in three models shows that the universe is open in the presence of particle creation in $f(T)$ theory of gravity.
In summary, we have studied the cosmological models with particle production in $f(T)$ theory of gravity to explore the current accelerated phenomenon of the universe. We have found that the particle creation produces negative pressure which may derive the accelerated expansion of the universe and play the role of unknown matter called ``dark energy" in $f(T)$ theory of gravity. We may expect that the particle creation process be a constraint for the unexpected observational outcomes. The new fact about this article is that the particle creation is studied by the thermodynamics approach in $f(T)$ theory of gravity.
\textbf{Acknowledgements}\ S.M. acknowledges Department of Science \& Technology (DST), Govt. of India, New Delhi,
for awarding Junior Research Fellowship (File No.
DST/INSPIRE Fellowship/2018/IF180676). PKS acknowledges DST, New Delhi, India for providing facilities through DST-FIST lab, Department of Mathematics, BITS-Pilani, Hyderabad Campus where a part of this work was done. The authors thank S. Bhattacharjee for stimulating discussions. We are very much grateful to the honorable referee and the editor for the illuminating suggestions that have significantly improved our work in terms of research quality and presentation.
|
3,212,635,537,458 | arxiv | \section{Introduction}
Many high mass accretion rate AGN show X--ray spectra which rise
smoothly below 1~keV above the extrapolated 2--10~keV emission
\cite{rf:1}, equivalent to a fixed temperature of $\sim
0.1-0.2$~keV. This is far too high a temperature to be simply the high
energy tail of the accretion disc emission, and its lack of relation
to the underlying disc temperature argues strongly against it being
Compton scattered disc emission\cite{rf:2}. The apparently
fixed temperature is much easier to explain if it arises from atomic
rather than continuum processes. One potential physical association is
with the large increase in opacity between 0.7--3~keV due to
OVII/OVIII and Fe L shell absorption. However, the soft excess is
observed to be fairly featureless, so if it is atomic in origin then
there must be a strong velocity shear in order to Doppler smear the
characteristic atomic features into a pseudo--continuum.
Partially ionised material with a strong velocity shear can produce
the soft excess in two different geometries, one where the material is
optically thick and out of the line of sight, seen via reflection
(e.g. from an accretion disc). Alternatively, the material can be
optically thin and in the line of sight, seen in absorption (e.g. a
wind above the disc). We discuss each of these possiblities in detail
below, but show that the model degeneracies mean that {\em both} can
fit the data even of the archetypal 'reflection dominated' AGN
MCG--6--30--15.
Instead we use physical arguments on the 'fine tuning' of the
ionisation parameter to show that the absorption model is strongly
favoured, and that reflection from a hydrostatic disc cannot produce
the soft X--ray excess\cite{rf:3}
\section{Reflection}
The increase in absorption opacity between 0.7--3~keV means a decrease
in reflection between these energies, or equivalently, a rise in the
reflected emission below 0.7~keV, producing a soft excess. This
continuum reflection is enhanced by emission lines from the partially
ionised material, especially OVII/VIII Ly $\alpha$ at 0.6--0.7~keV as
well as Ly$\alpha$ lines from C,N and Fe L transitions\cite{rf:4}.
Such optically thick reflection from material in the
inner disc is strongly smeared by the disc velocity field and such
models can match the shape of the soft excess. Importantly the
parameters required for the relativistic smearing of the soft excess
can be the {\em same} as those required to produce the associated iron
K$\alpha$ line emission, though objects with the highest
signal--to--noise require {\em multiple} reflection components to fit
the spectra\cite{rf:5}\tocite{rf:8}.
\begin{figure}[b]
\epsfxsize = 0.95\textwidth
\centerline{\epsfbox{chris_done_fig1.eps}}
\caption{The spectra in the left and right hand panels show
the deconvolved XMM-Newton data from MCG--6--30--15 using the
reflection and absorption models, respectively. The middle panel shows
sketch geometries for a reflection dominated system (lightbending and disc
fragmentation) compared to absorption in an outflow from the disc.}
\label{fig:1}
\end{figure}
However, the parameters derived from these models can be uncomfortably
extreme, with the amount of smearing implying extreme Kerr spacetime,
with perhaps also extraction of the spin energy of the black hole
\cite{rf:8,rf:9}. Also, the size of the soft
X-ray excess can be much larger than expected for a reflection origin
with isotropic illumination\cite{rf:10}. The
objects with these large soft X--ray excesses then require anisotropic
illumination models, e.g. where the X--ray source is extremely close
to the black hole so that lightbending suppresses the observed direct
continuum flux and enhances the disc illumination\cite{rf:7}.
Alternatively, the disc might fragment into inhomogeneous
regions which hide a direct view of the intrinsic source flux\cite{rf:5,rf:6}.
These reflection geometries are sketched in the
middle panel of Fig 1.
MCG--6--30--15 is the archetypal object which shows all these
features. The left panel of Fig 1 shows the XMM data for this object,
deconvolved with an intrinsic power law, its reflection from two
different ionisation and velocity smeared reflectors both with twice
solar iron abundance, an additional narrow neutral iron line, and
three narrow warm absorber systems to account for the complex
absorption seen at low energies\cite{rf:11}. This
fits the data ($\chi^2=2327.2/1874$) but requires that the intrinsic
power law (with $\Gamma=2.17$) is not visible, and that one of the
reflectors has extreme smearing ($r_{in}=1.28$, $r_{out}=3.5$, with
highly centrally concentrated emissivity)\cite{rf:9}.
\section{Absorption}
The same physical process of the opacity increase between 0.7--3~keV
can also produce the soft excess via absorption, plausibly from a wind
from the accretion disc\cite{rf:2}, as sketched in Fig
1. Again, relativistic velocity shear is required to smear out the
characteristic atomic features into a pseudo--continuum but the
difference between here is that these motions
are no longer Keplarian, so cannot be used to simply infer the inner
disc radius (and hence black hole spin).
The right hand panel of Fig 1 shows this model fit to the XMM-Newton
data. The model description is the same as for the reflection fits,
except that one of the smeared reflectors is replaced with the smeared
absorption model, and there is no need for an additional narrow iron
line. This gives a similarly good fit to the data
($\chi^2=2215.6/1877$) but now the intrinsic power law is seen
($\Gamma=2.31$), and the remaining reflector has $\Omega/2\pi=0.5$ and
is not extremely smeared ($R_{in}=25$, with emissivity consistent with
the expected gravitational energy release i.e. $\propto r^{-3}$).
Plainly the 0.3--10~keV spectral fits alone cannot distinguish between
a reflection and absorption origin, not even when variability is
included\cite{rf:7,rf:12}. Nor can
data at higher energies as the two models also make very similar
predictions for the 10--30~keV flux of $\sim 3\times 10^{-11}$ ergs
cm$^{-2}$ s$^{-1}$ from the observed 2--10~keV flux of $4.5\times
10^{-11}$ ergs cm$^{-2}$ s$^{-1}$. This is not the case in other
objects, such as 1H0707-496 and especially PG~1211+104,
where Suzaku HXD data may break the model
degeneracies\cite{rf:10}.
The difference here is the additional complexity due to
the narrow warm absorber systems at low energies which gives more
freedom in deconvolving the underlying spectrum.
\section{Reflection from a disc}
We can try instead to break the model degeneracies using physical
plausibility arguments as opposed to observational data. Both
absorption and reflection require the same basic ionization conditions
i.e. partially ionised Oxygen to produce the big jump in opacity at
0.7~keV (equivalently $\xi\sim 10^3$ where $\xi=L/nr^2$ is the
photoionisation parameter). This
may arise rather naturally in an absorption geometry if the material
is in some sort of pressure balance\cite{rf:13}.
The front of the cloud is heated
by the X--ray illumination, so expands, so its density is low and
ionisation is high. Further into the cloud the heating is less intense
so the material is cooler, so must be denser to be in constant
pressure. The lower ionisation finally allows ion species to exist,
dramatically enhancing the cooling and hence increasing the density.
This rapid transition means the cloud has to contract, which may mean
this neutral material is strongly clumped. A line of sight though the
cloud includes only the highly ionized front edge (invisible) and the
partially ionized transition region, which has an average value of
$\log\xi\sim 3$ across a region with column of $\sim 10^{22-23}$
cm$^{-2}$\cite{rf:13}.
However, the {\em same} rapid change in ionisation state in an X--ray
illuminated, hydrostatic disc produces a similarly stratified vertical
structure but has very different observational consequences. Again
the rapid transition from completely ionised to mostly neutral occurs
over a column of $10^{22-23}$ cm$^{-2}$, i.e. an optical depth of $\le
0.01$. Thus the zone with the 'correct' ionisation parameter to
produce the soft excess is only a very small fraction of the total
disc photosphere ($\tau=0\to 1$), so the soft excess is very much
smaller than that produced from constant density reflection models
which can arbitrarily set $\xi=10^3$ over the entire photosphere (see
Fig 2). Since even the constant density models require a reflection
dominated geometry to match the strongest soft excesses seen, this
means that reflection from a hydrostatic disc simply cannot be the
origin of these soft excesses\cite{rf:3}.
\begin{figure}[t]
\begin{tabular}{cc}
\epsfxsize = 0.45\textwidth \epsfbox{fxfd01.ps} &
\epsfxsize = 0.45\textwidth \epsfbox{tau_xi_t_fdfx10.ps}
\end{tabular}
\caption{The left hand panel shows the reflected spectra produced from
a disc in hydrostatic equilibrium at two different mass accretion
rates (dashed red, dotted blue) compared to that from a constant
density slab (solid magenta). The right hand panel shows the corresponding ionisation
parameter. }
\label{fig:1}
\end{figure}
\section{Conclusions}
The rapid transition from complete ionisation to mostly neutral
material is characteristic of any pressure balance condition. This
means that the partially ionised zone required for both atomic models
of the soft X--ray excess is limited to a column of $10^{22-23}$
cm$^{-2}$. This strongly supports the
optically thin, absorption geometry as the origin
for the soft X--ray excess, and this more messy picture of these high
mass accretion rate AGN means they are probably not good places to
test GR.
|
3,212,635,537,459 | arxiv | \section{Introduction}\label{sec:introduction}}
Cloud computing is widely adopted in scenarios that involve Internet of Things (IoT) devices and mobile devices. It plays an essential role in improving performance and in reducing the energy consumption of local devices by offloading computing tasks from local devices to clouds~\cite{MEC_offloading_challenges}. It also empowers the system to process large amounts of data when the number of devices grows massively~\cite{Big_IoT_Data_cloud}. Furthermore, many other technologies like Edge Computing~\cite{edge_computing} and Fog Computing~\cite{fog_computing} are adopted to further enhance the Quality of Service (QoS) in different scenarios.
However, in cloud/edge/fog computing with monolithic applications, the increasingly complex business logic and variety of user requirements leads to a decline in DevOps performance. Thus, microservice and container technologies have been widely adopted to facilitate service management~\cite{docker_edge, osmotic_computing}. In addition, they ensure continuous delivery and deployment by dividing monolithic applications into several independent microservices and deploying them in containers~\cite{MicroDevOps}. Although the microservice design pattern constitutes an approach to software and systems architecture based on the concept of modularization and boundaries~\cite{Microservices}, the requests between microservices make dependencies between services more complex~\cite{sdg_dependency,version_based}. Moreover, the microservice-based applications in microservice systems share the same microservice set for similar functions, which makes the dependencies between services even more complex.
\figurename~\ref{fig:scheme_example} shows an example of services with complex dependencies, where different applications are highlighted using different colors. Because microservices can have many Application Programming Interfaces (API) with different functions, they could be called by both other services and users, like service 10 and 2 in \figurename~\ref{fig:scheme_example}. Services can also request different services according to different user requirements. For example, service 10 might call service 8 if service 10 is called by users directly, and it might call service 11 instead when called by service 7.
\begin{figure}[!t]
\centering
\includegraphics[width=\linewidth]{figures/complex-services-example-eps-converted-to.pdf}
\caption{An example of services with complex dependencies}
\label{fig:scheme_example}
\end{figure}
In operation, the scale of the microservice system can increases as users and services join, and user requirements also change frequently with time. The changes in user requirements can lead to a decline in QoS without evolving the system on time. The evolution represents the change of a system from a previous state to a new state by different technologies. Because re-deploying services according to new requirements at runtime is a common method of the evolution, an efficient online service deployment algorithm is needed to keep the QoS stable provided by the microservice system. The algorithm aims to generate an optimized service deployment scheme based on the current system status, and the system can execute the scheme as the evolution. While there already is some research focussing on the multi-component application placement algorithm for providing the best execution cost or QoS with resource constraint~\cite{TMC_deployment,GMCAPP,LDSPP}, some of the primary features in a microservice system are not fully considered:
\begin{itemize}
\item \textbf{Multiple Applications with complex dependencies}: A microservice system typically consists of multiple microservice-based applications with complex dependencies, while most studies only consider one application. With multiple applications, the system has different categories of user requirements, while the users only request one function in the system with one application.
\item \textbf{Multiple Instances}: Multiple instances of each service can be deployed on different servers for load balancing in the microservice system, while some research only assign each service to one server.
\end{itemize}
It is essential to provide the best QoS by deploying service instances to servers considering the complex dependencies with support for multiple applications and instances in microservice systems. It should be noted that the server could be a physical one or a virtual machine. Moreover, the algorithm execution time should at least not exceed the container boot time. Considering the average boot time is 1.5s in~\cite{docker_performance}, although the average boot time depends on the hardware, the algorithm execution time should not exceed 5~10s in order to avoid degrading the user experience.
We call the problem of deploying services in such a system for QoS the Service Placement Problem in Microservice Systems (SPPMS). This paper selects the average response time as the main optimization goal because it plays a vital role in cloud and edge computing.
Our main contributions are as follows:
\begin{itemize}
\item We define the Service Placement Problem in Microservice Systems (SPPMS) considering the complex service dependencies with support for multiple applications and instances. We formulate the SPPMS as a Fractional Polynomial Problem (FPP) without any restrictions on the topology of servers and services.
\item We further prove a theorem that helps to rewrite the problem formulation to a Quadratic Sum-of-Ratios Fractional Problem (QSRFP), which can significantly reduce computational complexity compared to FPP. Two efficient greedy algorithms are designed to solve QSRFP based on the theorem, and the optimal algorithm based on the lower bound function and binary search is also introduced for reference.
\item Experiments are conducted to evaluate the performance of the proposed algorithms in different situations against existing algorithms~\cite{GMCAPP,LDSPP}.
The results show that our algorithms outperform existing approaches in both quality and speed. Discussions about the selection of proposed algorithms and methods to improve the performance are also presented.
\end{itemize}
The rest of the paper is organized as follows. Section~\ref{sec:related_work} presents the related work on the deployment problems in cloud and edge computing. Section~\ref{sec:problem_formulation} introduces SPPMS and formulates it as an FPP. Section~\ref{sec:algorithm} shows the conversion from FPP to QSRFP and details the optimal algorithm and our proposed algorithms. Section~\ref{sec:experiments} presents the experiments and the analysis. Section~\ref{sec:conclusion} concludes the paper and outlines possible future work.
\section{Related Work}\label{sec:related_work}
There have already been studies on the service placement problem in cloud/edge/fog computing, and they mainly focus on two different scenarios: the service placement problem without and with dependencies between services. The former assumes the services in the system are independent of each other, and there is no request between them; the latter investigates the placement problem of the multi-component applications considering the dependencies between components.
For the service placement problem without dependencies, works~\cite{QoS_Fog_placement}, \cite{follow_me_at_edge}, \cite{autonomic}, and~\cite{fog_ga} propose algorithms that generate deployment schemes that provide better QoS or lower cost with resource constraints based on greedy or Markov approximation algorithms. The dependencies between services are always overlooked, and multiple instances for each service are not allowed. For example, the work~\cite{space_air_ground} modeled the mobility-aware joint service placement problem as a Mixed Integer Linear Programming (MILP) optimization, and the work~\cite{winning_at_starting} aimed to minimize the total delay in Mobile Edge Computing (MEC) by deploying each service to one server only. In contrast to this, the work~\cite{User_allocation} proposed a stochastic user allocation algorithm to assign users with different categories of requirements to different servers, which also means desired services should be deployed on servers. In work~\cite{deep_reinforcement_algo} and \cite{hani_2021_deep_reinforcement_fog}, deep reinforcement learning was used to seek an optimized deployment scheme on resource-constrained edges for better QoS, like lower service response time. Multiple instances for each service are allowed, but dependencies are overlooked. Lastly, the work~\cite{predictive_placement} solved the problem considering the prediction of user mobility based on a frame-based design for lower long-term time-average service delay based on Lyapunov optimization. In summary, even though these studies do well on some scenarios, the dependencies between services and multiple instances support should also be considered when solving the deployment problem in microservice systems.
In terms of the multi-component application placement problem, many studies focused on power saving, cost-saving, or better QoS, where the application consists of multiple components that can be deployed on different servers~\cite{GMCAPP,LDSPP,badep,Bahreini_2017_placement,optimal_application_deployment}. There, each component is only deployed on one server at the same time. The deployment schemes are mostly modeled as a matrix $ X $ where $ x_{i,j} \in \{0, 1\} $ stands for component $ i $ should be deployed on server $ j $ or not, and the problems are formulated as an Integer Linear Programming (ILP) problem or Mixed-ILP. The work~\cite{elitism_based} employed a Genetic Algorithm (GA) to solve the problem for minimizing multiple optimization goals, including service time, energy consumption, and service cost with resource constraints. Meanwhile, the work~\cite{wang_2017_edge_online_placement} treated both the application and the physical computing system as two graphs. Online approximation algorithms were proposed based on graph-to-graph placement, and the mapping from the application graph to the physical graph is treated as the deployment scheme. The work~\cite{stochastic_optimization} developed a stochastic optimization approach to maximize QoS in MEC with Markov Decision Processes. In summary, even though the dependencies between components are considered, the constraint that each component can only be deployed on one server at the same time makes these methods less effective in microservice systems, and the missing support of multiple applications overlooks the complex dependencies between services in the microservice systems.
The work~\cite{deng_2020_optimal} considered both the dependencies and multiple instances support in the service placement problem in resource-constrained distributed edges, while aiming to minimize running cost with QoS constraints. However, the proposed algorithm is based on the relaxed ILP problem within the branch and bound method, making it unsuitable for the online system.
In our opinion, it is essential to take the support for service dependencies and multiple instances into consideration in the service placement problem in microservice systems, while being computationally efficient. In a microservice system, most user requirements should be satisfied by a service set instead of one service, making the dependencies necessary at run-time. Multiple instances support for one service also plays an essential role in a microservice system for load balancing. Thus, those two factors must be considered for an online deployment algorithm in the placement problem.
\section{Problem Formulation}\label{sec:problem_formulation}
In this section, we formulate the Service Placement Problem in Microservice Systems (SPPMS), and we use the average response time as the optimization goal. A microservice system consists of a set of \textit{services}, and there are many instances deployed on \textit{servers} for every service. Each of the services has many functions which are exposed as APIs, and dependencies exist due to the requests between services.
\subsection{Microservice System Model}
This section details the definition of \textit{Service}, \textit{Dependency}, \textit{Function Chain}, \textit{Server}, and \textit{Deployment Scheme} in this work.
\begin{sloppypar}
\textbf{Definition 1 (Service)} The service set in the system is described as $ S $, and a service $ s_i $ is defined as a tuple ${ s_i = <\mathcal{F}_i, \mu_i, r^s_i>, s_i \in S }$, where:
\begin{itemize}
\item $ \mathcal{F}_i = \{f_{i,1}, f_{i,2}, ..., f_{i,n}\} $ denotes a set of functions offered by $ s $. Each of them is described as a pair ${ f_{i,j} = <d^{in}_{i,j}, d^{out}_{i,j}> }$ where $ d^{in}_{i,j} $ and $ d^{out}_{i,j} $ stand for the input and output data size of $ f_{i, j} $.
\item $ \mu_i $ stands for the processing capacity of $ s_i $ denoting the number of requests the service instance can process per unit time.
\item $ r^s_i $ represents the set of computing resources used for $ s_i $ to achieve $ \mu_i $, such as CPU, RAM, and hard disk storage.
\end{itemize}
\end{sloppypar}
It should be noted that the APIs of each service $ s_i $ are treated as the function set $ \mathcal{F}_i $. Thus, every API is mapped to one unique $ f_{i, j} $, and the requests between APIs are treated as dependencies between services.
\textbf{Definition 2 (Dependency)} The dependencies between services are represented as the call graph $ DG = <F, E> $ of functions provided by each service. $ DG $ is a directed graph, where:
\begin{itemize}
\item $ F $ denotes the set of functions provided by all the services.
\item $ E $ stands for the edges between functions. The weight of each edge represents the Average Call Frequency Coefficient (ACFC) between two functions. An edge from $ f_{i,j} $ to $ f_{m,n} $ means $ f_{m,n} $ is called by $ f_{i,j} $, and ACFC($f_{i,j}$, $f_{m,n}$) shows how many times $ f_{m,n} $ should be called by $ f_{i,j} $ when $ f_{i,j} $ is called once.
\end{itemize}
Note that we do not consider cyclic dependencies in this work, as they are considered a bad practice in other architectures~\cite{Auto_Detection_Smells}, and they can be hard to maintain or reuse in isolation~\cite{Definition_MBS}.
Thus, $ DG $ is a directed acyclic graph. Furthermore, due to the complex logic, ACFC should be calculated according to the request history of the whole service system. With service requests in the if-else clause at the source code level, it is difficult to estimate the ACFC because the probabilities of executing the if-clause and else clauses cannot be predicted accurately.
\textbf{Definition 3 (Function Chain)} The function chain of $ f_{i,j} $ is defined as $ L_{i,j} = <f_{i,j}, ..., f_{m, n}> $. It is used to describe the requests between functions after the system receives a request to $ f_{i,j} $. $ L_{i,j}^m \in L_{i,j} $ stands for the \textit{m}-th function in $ L_{i,j} $.
When users or IoT devices send a request to any API in the system, a set of $ f_{i,j} $ are called, and the calling path between them is a subgraph of $ DG $. In SPPMS, the non-linear calling path can be converted to a function chain as shown in \figurename~\ref{fig:chain_conversion}. By adding a new virtual call between $ f_{3,2} $ and $ f_{2,3} $, which is in red, without input and output data, the calling path graph in the upper part of the figure is converted to the equivalent function chain in the lower part. Besides, the ACFC should also be set to zero which means $ f_{3,2} $ does not call $ f_{2,3} $ in real.
\begin{figure}[!t]
\centering
\includegraphics[width=\linewidth]{figures/chain_conversion-eps-converted-to.pdf}
\caption{Conversion from function calling graph to chain}
\label{fig:chain_conversion}
\end{figure}
As the non-linear calling path can be converted to a function chain as detailed above, it does not impose additional constraints on service dependencies.
\textbf{Definition 4 (Server)} The set of servers in the system is denoted as $ N $. For $ \forall n_i \in N $, $ n_i = <r_i^n> $, where $ r_i^n $ describes the computing resources of server $ n_i $. The delay and bandwidth between $ n_i $ and $ n_j $ are defined as $ d_{i,j} $ and $ b_{i,j} $, respectively.
Because not all nodes are fully connected to others, the total delay and the lowest bandwidth are used according to the routing path of any two nodes in the real system, which are determined by the routing rules.
\textbf{Definition 5 (Deployment Scheme)} The deployment scheme is described as a matrix $ X = \{x_{i,j} | x_{i,j} \in \mathbb{N}, 1 \leq i \leq |N|, 1 \leq j \leq |S|\} $, where $ x_{i, j} $ stands for the instance count of service $ s_i $ on server $ n_j $.
Table~\ref{tab:notation} summarizes all the symbols and notations in this paper for convenience.
\begin{table}
\caption{Notation}
\label{tab:notation}
\centering
\begin{tabular}{ll}
\hline
Notation & Description \\
\hline
$ s_i $ & Microservice $ s_i, s_i \in S $ \\
$ \mathcal{F}_i $ & Function set of $ s_i $ \\
$ f_{i, j} $ & One of the function of service $ s_i $, $ f_{i,j} \in \mathcal{F}_i $ \\
$ d^{in}_{i,j},\ d^{out}_{i,j} $ & Input and output data size of $ f_{i, j} $ \\
$ \mu_i $ & Processing capacity of $ s_i $ \\
$ r^s_i $ & Computing resource set that used by $ s_i $ \\
$ F $ & All the function set provided by all services \\
$ DG $ & Dependency graph of the service system \\
ACFC($f_{i,j}$, $f_{m,n}$) & \begin{tabular}[c]{@{}l@{}}Average call frequency coefficient between\\$ f_{i,j} $ and $ f_{m,n} $\end{tabular} \\
$ L_{i,j} $ & Function chain of $ f_{i,j} $ \\
$ L_{i,j}^m $ & The m-th function of $ L_{i,j} $ \\
$ n_i $ & Server node $ n_i, n_i \in N $ \\
$ d_{i,j},\ b_{i,j} $ & Delay and bandwidth between $ n_i $ and $ n_j $ \\
$ X $ & Deployment scheme \\
$ x_{i,j} $ & Instance count of $ s_i $ on $ n_j $ \\
$ h_{i,j} $ & \begin{tabular}[c]{@{}l@{}}A response server path for $ L_{i,j} $, $ h_{i,j} \in H_{i,j} $ \end{tabular} \\
$ \lambda_{i,j}^k $ & Request rate of $ f_{i,j} $ on $ n_k $ \\
$ T $ & The average response time of the system \\
$ C_{max} $ & Maximum deployment cost \\
$ \gamma_i^u $ & Total request rate of $ s_i $ from users \\
$ \gamma_i^s $ & Total request rate of $ s_i $ from services \\
\hline
\end{tabular}
\end{table}
\subsection{Problem Definition}
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.9\linewidth]{figures/overview_request_processing-eps-converted-to.pdf}
\caption{An overview of the request routing for any $ f_{i,j} $}
\label{fig:routing_overview}
\end{figure*}
This section provides the formulation of the Service Placement Problem in Microservice Systems (SPPMS) with the optimization goal and constraints.
\subsubsection{Average Response Time}
When the microservice system responds to a request to $ f_{i,j} $, it must select one service instance for each function in $ L_{i,j} $ due to the dependencies between services. $ H_{i,j} $ denotes all the possible response server paths, and $ h_{i,j} \in H_{i,j} $ stands for one selected server path for processing $ L_{i,j} $. $ h^m_{i,j} \in h_{i,j} $ is used to denote the server where the service instance that receives the request to $ L^m_{i,j} $ locates. Also, $ |h_{i,j}| = |L_{i,j}| $.
There are many load balancing algorithms used in microservices system with multiple service instances support, such as Round Robin, Weighted, and Random. Note that we consider the commonly used Round Robin routing in our SPPMS formulation; an adaptation to many other load balancing algorithms is straightforward.
The primary process of the request routing in a microservice system is shown in \figurename~\ref{fig:routing_overview}. The transmission between users and servers is not considered as it depends on many factors outside of the system, and we also assume the requests from the users are sent to the nearest servers first, as the servers in the red rectangle shown in \figurename~\ref{fig:routing_overview}. It should be noted that the \figurename~\ref{fig:routing_overview} is based on synchronous requests. However, it is also suitable for asynchronous requests by setting the output data size of each function to 0.
Under the Round Robin routing algorithm, all the service instances of service $ s_i $ in the system have the same probability of receiving the request about the function $ f_{i,j} $ when there is a sufficiently large number of requests. In this paper, we assume that the number of requests is large enough and the probability is the same. Thus, the probability of processing the request about $ f_{i,j} $ on $ n_{k} $ is:
\begin{equation}
Prob(f_{i,j}, n_k) = \frac{x_{k,i}}{\sum_{m=1}^{|N|} x_{m,i}}
\end{equation}
For any given selected response service path $ h_{i,j} $, the probability of selecting it from $ H_{i,j} $ is:
\begin{equation}\label{eqa:prob_path}
Prob(h_{i,j}) = \prod_{k=1}^{|L_{i,j}|} Prob(L^k_{i,j}, h^k_{i,j})
\end{equation}
The probability of the request of $ f_{i,j} $ from server $ n_k $ is:
\begin{equation}
Prob(\lambda_{i,j}^k) = \frac{\lambda_{i,j}^k}{\sum_{m=1}^{|N|} \lambda_{i,j}^m}
\end{equation}
When calculating the average response time, the request processing time after receiving the request is not included to reduce the impact of each microservice implementation. Only the delay and data transmission time are included in this paper. The response time of the request of $ f_{i,j} $ from users near server $ n_k $, which is denoted as $ t_{h_{i,j},k} $, consists of two parts (see \ding{172} and \ding{173} in \figurename~\ref{fig:routing_overview}). The first part is the response time between the request received server and the first of the selected response server path, and the second part is the response time for processing the function chain. $ t_{h_{i,j},k} $ is formulated as:
\begin{equation}\label{eqa:t_p_k}
\begin{aligned}
t_{h_{i,j},k} = \sum_{l_{v,w}^m, m=2}^{|L|} \left(\frac{d^{in}_{v,w} + d^{out}_{v,w}}{b_{p^{m-1}_{i,j}, p^m_{i,j}}} + d_{p^{m-1}_{i,j}, p^m_{i,j}}\right) \\ + \left(\frac{d^{in}_{i,j} + d^{out}_{i,j}}{b_{k, p^1_{i,j}}} + d_{k, p^1_{i,j}}\right)
\end{aligned}
\end{equation}
Thus, the average response time of $ f_{i,j} $ in the whole system is described as $ T_{i,j} $:
\begin{equation}\label{eqa:T_i_j}
T_{i,j} = \sum_{k=1}^{|N|} \sum_{h_{i,j}\in H_{i,j}} Prob(\lambda_{i,j}^k) \times Prob(h_{i,j}) \times t_{h_{i,j},k}
\end{equation}
Consequently, the system's average response time is defined as $ T $ when giving the deployment scheme $ X $:
\begin{equation} \label{eqa:Tx}
T(X) = \sum_{f_{i,j} \in F} \frac{\sum_{m=1}^{|N|} \lambda_{i,j}^m}{ \sum_{f_{v,w} \in F} \sum_{m=1}^{|N|} \lambda_{v,w}^m } \times T_{i,j}
\end{equation}
\subsubsection{Constraints}
There are three types of constraints in SPPMS: the computing resource constraint of servers, the deployment cost constraint, and the service capability constraint.
For the computing resources, the used resources by service instances should not exceed the resources provided by each server:
\begin{equation} \label{eqa:c1}
\sum_{i=1}^{|S|} x_{j,i} \times r^s_i \leq r^n_j, \forall n_j \in N
\end{equation}
The deployment cost should also not exceed the given maximum cost $ C_{max} $, and the monetary cost is considered as the deployment cost here. Because most of the cloud providers support to charge based on resource usage, the deployment constraint is defined as follows, where $ c $ is the cost of one unit used resource:
\begin{equation} \label{eqa:c2}
c \times \sum_{i=1}^{|S|} \sum_{j=1}^{|N|} r^s_{i} \times x_{j,i} \leq C_{max}
\end{equation}
The last constraint is the service capability constraint, which makes sure that there are sufficient service instances to process the requests from the users and other services. The total desired capability of $ s_{i} $ contains desired capability from the users denoted as $ \gamma^u_{i} $ and from the services denoted as $ \gamma^s_{i} $. For the user part, it can be calculated by:
\begin{equation}
\gamma^u_{i} = \sum_{f_{j,k} \in \mathcal{F}_{i}} \sum_{m=1}^{|N|} \lambda_{j,k}^m
\end{equation}
The service part $ \gamma^s_{i} $ is raised by the requests between services due to dependencies, and it can be calculated directly with $ L_{i,j} $ and $ DG $. For example, assuming $ L_{1,1} = <f_{1,1}, f_{2,1}, f_{3,1}> $, $ ACDC(f_{1,1}, f_{2,1}) = 2 $, $ ACDC(f_{2,1}, f_{3,1}) = 1.5 $ and $ \lambda_{1,1} = 5 $, then $ \gamma^s_{2} = \lambda_{1,1} \times ACDC(f_{1,1}, f_{2,1}) = 10 $, and $ \gamma^s_{3} = \lambda_{1,1} \times ACDC(f_{1,1}, f_{2,1}) \times ACDC(f_{2,1}, f_{3,1}) = 15 $.
The service capability constraint is formulated as:
\begin{equation} \label{eqa:c3}
\mu_i \times \sum_{j=1}^{|N|} x_{j, i} \geq \gamma^u_{i} + \gamma^s_{i} , \forall s_i \in S
\end{equation}
\subsubsection{Optimization Problem Definition}
By combining Equations~\ref{eqa:Tx}, \ref{eqa:c1}, \ref{eqa:c2}, and \ref{eqa:c3}, we can define the optimization problem.
Given the service set $ S $, the dependency graph $ DG $, the server set $ N $, user requirements about each function on every server $ \lambda_{i,j}^k, \forall f_{i,j} \in F, \forall n_k \in N $, and maximum cost $ C_{max} $, the goal is to find a deployment scheme $ X $ that satisfies:
\begin{subequations}
\begin{gather}
min \ T(X), \ X \in \mathbb{N}_{|N| \times |S|} \\
s.t.\left\{
\begin{array}{lr}
\sum_{i=1}^{|S|} x_{j,i} \times r^s_i \leq r^n_j, & \forall n_j \in N \\
c \times \sum_{i=1}^{|S|} \sum_{j=1}^{|N|} r^s_{i} \times x_{j,i} \leq C_{max} \\
\mu_i \times \sum_{j=1}^{|N|} x_{j, i} \geq \gamma^u_{i} + \gamma^s_{i}, & \forall s_i \in S
\end{array}
\right.
\end{gather}
\end{subequations}
Because both the denominator and the numerator of $ T(X) $ are multivariate polynomials~\cite{solving_fpp}, the problem is a Fractional Polynomial Problem (FPP).
\section{Algorithms}\label{sec:algorithm}
In the following, we show how to convert our FPP to a Quadratic Sum-of-Ratios Fractional Problem (QSRFP), which can significantly reduce the computational complexity comparing to FPP. Let us assume that the average length of function chain is $ \hat{l} $. Then, to evaluate the average response time of any function, FPP needs to iterate over all the possible response path, whose size is $ |N|^{\hat{l}} $, while QSRFP does not need to. QSRFP is the problem whose constraint functions and the denominator and numerator of the optimization goal are all quadratic functions, which are possibly not convex~\cite{solving_qsrfp}. Two efficient greedy-based algorithms are detailed to solve QSRFP, and the optimal algorithm for solving QSRFP is also presented for reference. In general, a Quadratic Sum-of-Ratios Fractional Programs (QSRFP) problem has the following structure~\cite{solving_qsrfp}:
\begin{subequations} \label{eqa:fpp}
\begin{gather}
{\rm min} \ H_0(y) = \sum_{i=1}^p \frac{f_i(y)}{g_i(y)} \\
s.t.\left\{
\begin{array}{lr}
H_m(y) \leq 0, m = 1, ..., M, \\
y \in Y^0 = \{y \in R^n : \underline{y}^0 \leq y \leq \overline{y}^0\} \subset R^n
\end{array}
\right.
\end{gather}
\end{subequations}
where $ p \leq 2 $; $ f_i(y) $, $ g_i(y) $, and $ H_m(y) $ are all quadratic functions, i.e. $ f_i(y) = \sum_{j=1}^n \sum_{k=1}^n \delta^i_{jk} y_i y_k + \sum_{k=1}^n c_k^i y_k + \overline{\delta}_i $.
\subsection{Converting FPP to QSRFP}\label{subsec:converting}
Although there are already some optimization approaches for the FPP~\cite{solving_fpp,optimization_fpp,global_minimization_fpp}, it is difficult to apply them as online algorithms due to the high computational complexity of $ T(X) $.
Assuming the average length of function chain is $ \hat{l} $, there are $ |N|^{\hat{l}} $ possible response server paths for each function, and Equation~\ref{eqa:prob_path} must be calculated for $ |F| \times |N|^{\hat{l}} $ times, which is unaffordable for an online algorithm. Thus, we cannot consider FPP as defined above. To simplify the problem, we prove the following theorem:
\begin{theorem}\label{theorem}
Given ${ <f_1, f_2, ..., f_{k-1}, f_{k}, ..., f_{n}> }$ as the target function chain, let $ T_{i \rightarrow j} $ denote the average response time from $ f_i $ to $ f_j $. Then we have:
\begin{equation}
T_{1 \rightarrow n} = \sum_{i=1}^{n-1}T_{i \rightarrow i+1}
\end{equation}
\end{theorem}
\begin{proof}
Define $ H_{i \rightarrow j} $ as all the possible server paths from $ f_{i} $ to $ f_{j} $ and $ h_{i \rightarrow j} $ as a server path $ \forall h_{i \rightarrow j} \in H_{i \rightarrow j} $, $ P_{h_{i \rightarrow j}} $ denotes the probability of $ h_{i \rightarrow j} $, and $ t_{h_{i \rightarrow j}} $ represents the response time of $ h_{i \rightarrow j} $. Then $ \forall k \in [1, n] $, we have :
\begin{equation}\label{eqa:proof_T}
T_{1 \rightarrow n} = \sum_{h_{1 \rightarrow n} \in H_{1 \rightarrow n}} P_{h_{1 \rightarrow n}}t_{h_{1 \rightarrow n}}
\end{equation}
\begin{equation}\label{eqa:proof_P}
\sum_{h_{1 \rightarrow n} \in H_{1 \rightarrow n}} P_{h_{1 \rightarrow n}} = \sum_{h_{1 \rightarrow k} \in H_{1 \rightarrow k}} \sum_{h_{k \rightarrow n} \in H_{k \rightarrow n}} P_{h_{1 \rightarrow k}} P_{h_{k \rightarrow n}}
\end{equation}
\begin{equation}\label{eqa:proof_t}
t_{h_{1 \rightarrow n}} = t_{h_{1 \rightarrow k}} + t_{h_{k \rightarrow n}}
\end{equation}
After applying Equation \ref{eqa:proof_P} and \ref{eqa:proof_t} to \ref{eqa:proof_T}, $ \forall k \in [1, n] $:
\begin{equation}\label{eqa:proof}
\begin{aligned}
T_{1 \rightarrow n} & = \sum_{h_{1 \rightarrow n} \in H_{1 \rightarrow n}} P_{h_{1 \rightarrow n}}t_{h_{1 \rightarrow n}} \\
& = \sum_{h_{1 \rightarrow n} \in H_{1 \rightarrow n}} P_{h_{1 \rightarrow n}}t_{h_{1 \rightarrow k}} + \sum_{h_{1 \rightarrow n} \in H_{1 \rightarrow n}} P_{h_{1 \rightarrow n}}t_{h_{k \rightarrow n}} \\
& = \sum_{h_{k \rightarrow n} \in H_{k \rightarrow n}} \left(\sum_{h_{1 \rightarrow k} \in H_{1 \rightarrow k}} P_{h_{1 \rightarrow k}} t_{h_{1 \rightarrow k}}\right) P_{h_{k \rightarrow n}} \\
& \qquad + \sum_{h_{1 \rightarrow k} \in H_{1 \rightarrow k}} \left(\sum_{h_{k \rightarrow n} \in H_{k \rightarrow n}} P_{h_{k \rightarrow n}} t_{h_{k \rightarrow n}}\right) P_{h_{1 \rightarrow k}} \\
& = \sum_{h_{k \rightarrow n} \in H_{k \rightarrow n}} P_{h_{k \rightarrow n}} T_{1 \rightarrow k} + \sum_{h_{1 \rightarrow k} \in H_{1 \rightarrow k}} P_{h_{1 \rightarrow k}} T_{k \rightarrow n} \\
& = T_{1 \rightarrow k} + T_{k \rightarrow n}
\end{aligned}
\end{equation}
After applying Equation \ref{eqa:proof} repeatedly, we have:
\begin{equation}
\begin{aligned}
T_{1 \rightarrow n} & = T_{1 \rightarrow 2} + T_{2 \rightarrow n} \\
& = T_{1 \rightarrow 2} + T_{2 \rightarrow 3} + T_{3 \rightarrow n} \\
& = ... \\
& =T_{1 \rightarrow 2} + T_{2 \rightarrow 3} + ... + T_{n-1 \rightarrow n} \\
&= \sum_{i=1}^{n-1}T_{i \rightarrow i+1}
\end{aligned}
\end{equation}
\end{proof}
To make the formula more concise, the users can be treated as $ L_{i,j}^0 $, and $ L_{i,j} $ is abbreviated as $ L $. Then, using Theorem~\ref{theorem}, $ T_{i,j} $ in Equation~\ref{eqa:T_i_j} can be represented as:
\begin{equation}\label{eqa:q_t_i_j}
\begin{aligned}
T_{i,j} = \sum_{k=0}^{|L| - 1} \underbrace{\sum_{v}^{|N|} \sum_{w}^{|N|} (\overbrace{\frac{x_{v,L^k}}{\sum_{m}^{|N|} x_{m,L^k}} \times \frac{x_{w,L^{k+1}}}{\sum_{m}^{|N|} x_{w,L^{k+1}}}}^{prob \ of \ L^k \ on \ n_j \ and \ L^{k+1} \ on \ n_w}
\times t_{v,w,k+1})}_{avgTime\ between\ L^k\ and\ L^{k+1}\ as\ a\ quadratic\ fraction}
\end{aligned}
\end{equation}
where $ t_{v,w,k+1} = \frac{d^{in}_{L^{k+1}} + d^{out}_{L^{k+1}}}{b_{v,w}} + d_{v,w} $ denotes the response time between $ L^{k} $ on $ n_w $ and $ L^{k+1} $ on $ n_w $. By defining $ \hat{\lambda}_{i,j} = \frac{\sum_{m=1}^{|N|} \lambda_{i,j}^m}{ \sum_{f_{v,w} \in F} \sum_{m=1}^{|N|} \lambda_{v,w}^m } $, $ T(X) $ can be described as
\begin{equation}\label{eqa:TX_qsrfp}
T(X) = \sum_{f_{i,j} \in F} \hat{\lambda}_{i,j} \times T_{i,j}
\end{equation}
It should be noted that $ T_{i,j} $ is a quadratic sum-of-rations fraction, and the same as $ T(X) $ because $ \hat{\lambda}_{i,j} $ is a constant. Thus, we have converted the FPP to a QSRFP. Compared with the FPP in Equation~\ref{eqa:fpp}, there is no need to iterate over all possible response paths for each function chain, which reduces the computational cost significantly.
\subsection{Greedy-based Algorithms}
There are two greedy-based algorithms\footnote{https://github.com/HIT-ICES/AlgoDeployment} we proposed: B-QSRFP in Algorithm~\ref{algo:bfs_placement} and D-QSRFP in Algorithm~\ref{algo:dfs_placement}. They depend on two sub-algorithms~\ref{algo:best_server} and~\ref{algo:deploy} to find the best server for each service and deploy service instances.
The greedy-based algorithms are inspired by Equation~\ref{eqa:q_t_i_j}: based on it, the best placement of each service $ s_i $ is only related to the services that call $ s_i $ and the services called by $ s_i $, and the response server paths $ H $ has no influence on $ T_{i,j} $. Algorithm~\ref{algo:best_server} uses this and selects the best server for new instances of $ s_j $ based on Equation~\ref{eqa:q_t_i_j}.
This algorithm only considers the servers that have enough computing resources for at least one instance of $ s_j $ as shown in lines 3--4. For each server $ n $, the algorithm evaluates the average response time based on Equation~\ref{eqa:T_i_j} when a new instance of $ s_j $ is deployed on $ n $ as shown in lines 5--11, where $ DG.{\rm pred}(s_j) $ and $ DG.{\rm succ}(s_j) $ denotes the predecessors and successors of $ s_j $ in $ DG $. The response time between $ s_j $ and $ s_p $ equals 0 if there is no instance of $ s_p $ during the evaluation. Lines 12--14 selects the server with the smallest average response time as the best server for $ s_j $.
When deploying an instance of $ s_i $, the best servers for the predecessors and successors are changed according to Equation~\ref{eqa:q_t_i_j}, which means the algorithm needs to replace the instances of the predecessors and successors, as detailed in Algorithm~\ref{algo:deploy}.
\begin{algorithm}[H]
\caption{bestServer($s_j $, $ X $, $ N $, $ DG $) Algorithm}\label{algo:best_server}
\textbf{Input:} $ s_j $: service wanted to deploy \\
\hspace*{6ex}$ X $: current deployment scheme \\
\hspace*{6ex}$ N $: server list \\
\hspace*{6ex}$ DG $: dependency graph \\
\textbf{Output:} best node to deploy $ s_j $
\begin{algorithmic}[1]
\State $ s \gets None $
\State $ c \gets +\infty $
\For{$ n $ in $ N $}
\If{ $ n $ has enough resource for $ s_j $ }
\State $ currC \gets 0 $
\For{ $ s_p $ in $ DG.{\rm pred}(s_j) $ }
\State $ currC \gets currC + {\rm avgTime}(s_j, s_p, n, X) $ \Comment{avgTime: The avgTime between $ s_j $ and $ s_p $ based on Equation~\ref{eqa:T_i_j} after deploying an instance of $s_j$ on $n$.}
\EndFor
\For{ $ s_s $ in $ DG.{\rm succ}(s_j) $ }
\State $ currC \gets currC + {\rm avgTime}(s_j, s_s, n, X) $
\EndFor
\If{ $ currC < c $ }
\State $ s,\ c \gets n,\ currC $
\EndIf
\EndIf
\EndFor
\State \Return $ s $
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[H]
\caption{DeploySpread($s_i$, $k$, $ X $, $ N $, $ DG $) Algorithm}\label{algo:deploy}
\textbf{Input:} $ s_j $: service wanted to deploy \\
\hspace*{6ex}$ k $: how many instances wanted to deploy \\
\hspace*{6ex}$ X $: current deployment scheme \\
\hspace*{6ex}$ N $: server list \\
\hspace*{6ex}$ DG $: dependency graph
\begin{algorithmic}[1]
\While{ $ k > 0 $ }
\State $ n \gets {\rm bestServer}(s_j, X, N, DG) $
\State $ c \gets {\rm min}(\lfloor \frac{{\rm leftRes}(n)}{r_j^s} \rfloor, k ) $
\State $ {\rm deploy}(s_i, n, c) $
\State $ k \gets k - c $
\State $ S_r \gets DG.{\rm pred}(s_j) \cup DG.{\rm succ}(s_j) $
\State $ S_p \gets \{ s_j \} $
\State $ X_{copy} \gets X $
\While{ len$ (S_r) \ne 0 $ }
\State $ s_r \gets S_r.{\rm pop}() $
\State $ k_r \gets {\rm instNum}(s_r, X) $
\State delete all the instances of $ s_r $ from X
\While{ $ {\rm instNum}(s_r, X) < k_r $ }
\State $ n_r \gets {\rm bestServer}(s_r, X, N, DG) $
\State $ c_r \gets {\rm min}(\lfloor \frac{{\rm leftRes}(n_r)}{r_r^s} \rfloor, k_r - {\rm instNum}(s_r, X) ) $
\EndWhile
\If{ $ X \ne X_{copy} $ }
\State $ S_r \gets S_r \cup DG.{\rm pred}(s_r) \cup DG.{\rm succ}(s_r) \setminus S_p $
\State $ S_p \gets S_p \cup \{ s_r \} $
\EndIf
\EndWhile
\EndWhile
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[h]
\caption{BFS Placement Algorithm: B-QSRFP}\label{algo:bfs_placement}
\textbf{Input:} $ \Gamma^s $: $ \{ \gamma_{i}^s | \forall s_{i} \in S \} $ \\
\hspace*{6ex}$ \Gamma^u $: $ \{ \gamma_{i}^u | \forall s_{i} \in S \} $ \\
\hspace*{6ex}$ N $: server list \\
\hspace*{6ex}$ DG $: dependency graph \\
\textbf{Output:} Deployment Scheme
\begin{algorithmic}[1]
\State $ X \gets \mathbf{0} $
\While{ not $ DG.{\rm empty}() $ }
\State $ S_c \gets \{ s_i | s_i \in S, {\rm len}(DG.{\rm pred}(s_{i})) = 0 \} $
\State $ s_c \gets {\rm argmin}_{s_i \in S_c} \frac{\mu_i}{r_i^s} $
\State $ {\rm DeploySpread}(s_i, \lceil \frac{\gamma_i^s + \gamma_i^u}{\mu_i} \rceil, X, N, DG) $
\State $ DG.{\rm remove}(s_i) $
\EndWhile
\State \Return $ X $
\end{algorithmic}
\end{algorithm}
Algorithm~\ref{algo:deploy} is responsible for deploying $ k $ instances of $ s_i $ in the system considering the dependencies between services. After finding the best server for $ s_i $ with Algorithm~\ref{algo:best_server}, instances are deployed on the server as many as needed, as shown in lines 2--4. Lines 6--21 re-deploy instances of predecessors and successors. The instances of them are deleted and re-deployed based on Algorithm~\ref{algo:best_server} again in lines 12--16. The re-deployment is continued for the services whose deployment is changed after re-deployment in lines 17--20. The re-deployment is stopped when all the affected services are processed.
\begin{algorithm}[H]
\caption{DFS Placement Algorithm: D-QSRFP}\label{algo:dfs_placement}
\textbf{Input:} $ L_u $: $ \{ L(f_{i,j}) | \sum_m^{|N|} \lambda_{i,j}^m > 0 \} $ \\
\hspace*{6ex}$ \Lambda $: $ \{ \lambda_{i,j}^k | \forall f_{i,j} \in F, \forall k \in N \} $ \\
\hspace*{6ex}$ N $: server list \\
\hspace*{6ex}$ DG $: dependency graph \\
\textbf{Output:} Deployment Scheme
\begin{algorithmic}[1]
\State $ X \gets \mathbf{0} $
\State $ \lambda_{solved} \gets {\rm a\ dict\ whose\ values\ are\ 0}\ \forall s_i \in S $
\While{ len$(L_u) \ne 0 $ }
\State $ l_u \gets {\rm maximum\ data\ transmission\ chain\ from\ L_u} $
\State $ coe \gets 1 $
\State $ f_{prev} \gets None $
\For{ $ f_{i,j} $ in $ l_u $ }
\If{ $ f_{prev} \ne None $ }
\State $ coe \gets coe \times {\rm ACDC}(f_{prev}, f_{i,j}) $
\EndIf
\State $ f_{prev} = f_{i,j} $
\State $ \lambda_c = coe \times \sum_m^{|N|}\lambda_{i,j}^m $
\State $ \lambda_d = \lambda_{solved}[s_i] + \lambda_c - {\rm instNum}(s_i) \times \mu_i $
\If{ $ \lambda_d > 0 $ }
\State $ {\rm DeploySpread}(s_i, \lceil \frac{\lambda_d}{\mu_i} \rceil, X, N, DG) $
\State $ \lambda_{solved}[s_i] \gets \lambda_{solved}[s_i] + \lambda_c $
\EndIf
\EndFor
\State $ L_u \gets L_u \setminus \{ l_u \} $
\EndWhile
\State \Return $ X $
\end{algorithmic}
\end{algorithm}
Based on Algorithm~\ref{algo:deploy}, we propose two greedy algorithms based on depth-first search (DFS) and breadth-first search (BFS); see Algorithms~\ref{algo:bfs_placement} and~\ref{algo:dfs_placement}. The difference between them is the order of the service deployment. For BFS Placement Algorithm~\ref{algo:bfs_placement}, the service that has the best service ability with minimum computing resources and without predecessors is selected as shown in lines 3--4. Line 5 deploys the minimum number of instances which can satisfy the requests from users and services. After deploying $ s_i $, it is removed from $ DG $ in line 6, and the algorithm stops when all services are deployed as line 2.
The DFS Placement Algorithm~\ref{algo:dfs_placement}, on the other hand, deploys the function chain with maximum data transmission size first as line 4, and the services are deployed following the order of the function chain $ l_u $. Then, the algorithm calculates the request rate $ \lambda_c $ of each service in $ l_u $ and deploys the minimum number of instances to satisfy the desired ability $ \lambda_d $ in lines 8--17. Finally, it ends when all $ f_{i,j} $ that the users request are processed.
\subsection{The Optimal Algorithm}
This section briefly introduces the optimal algorithm for solving QSRFP, which mainly comes from~\cite{solving_qsrfp}.
It serves as a reference only, because its high computational cost prevents us from using it as an online algorithm
The basic idea of the solution in~\cite{solving_qsrfp} is constructing a parametric relaxation linear programming problem (PRLP) of QSRFP in $ Y $:
\begin{subequations}
\begin{gather}
{\rm min} \ H_0^L(y) = \sum_{i=1}^p \frac{f_i^L(y)}{\overline{g}_i^U} \\
s.t.\left\{
\begin{array}{lr}
H^L_m(y) \leq 0, m = 1, ..., M, \\
y \in Y^0 = \{y \in R^n : \underline{y}^0 \leq y \leq \overline{y}^0\} \subset R^n
\end{array}
\right.
\end{gather}
\end{subequations}
where $ H_0^L \leq H_0 $, $ H_m^L \leq H_m $, $ f_i^L \leq f_i $, and $ \overline{g_i}^U = {\rm max}_{y \in Y} g_i^U $. For more details about how constructing PRLP, please refer to Section~2 in~\cite{solving_qsrfp}. With Theorems~1 and~2 in~\cite{solving_qsrfp}, $ H_0^L \rightarrow H_0(y) $ as $ ||\overline{y} - \underline{y}|| \rightarrow 0 $.
The branching technique is also used for searching the optimal solution by iteratively subdividing the rectangle $ Y^k $ into two sub-rectangles. For any selected sub-rectangle $ Y^k = [\underline{y}, \overline{y}] \subseteq Y^0$, The sub-rectangles $ Y^{k,1} $ and $ Y^{k,2} $ are generated by dividing $ y_\theta $ into $ \left[\underline{y}_\theta, \frac{\underline{y}_\theta + \overline{y}_\theta}{2}\right]$ and $ [\frac{\underline{y}_\theta + \overline{y}_\theta}{2}, \overline{y}_\theta ]$, where $ \theta \in {\rm argmax}\{ \overline{y}_j - \underline{y}_j: j = 1, ..., n \} $. The Theorem~3 in~\cite{solving_qsrfp} shows it is possible to apply reducing technique by solving the PRLP in $ Y^{k,1} $ and $ Y^{k,2} $ to determine whether the optimal solution exists in $ Y^{k,1} $ and $ Y^{k,2} $.
The branch-and-bound algorithm is detailed as Algorithm~\ref{algo:bnb}, and $ LB $ and $ UB $ are the lower bound and upper bound of the optimal solution. With bisection in line 10, the algorithm reduces the search space with PRLP (reducing) in line 12 and updates $ LB $ and $ UB $ (bounding) in lines 13-24. The branch-and-bound search is used to search integer solutions as $ y^k $ is a non-integer solution in line 31. For more details, we refer the interested reader to Section~3 in~\cite{solving_qsrfp}.
\begin{algorithm}[t]
\caption{QSRFP Branch-and-Bound Algorithm}\label{algo:bnb}
\textbf{Input:} $ \epsilon $: termination error \\
\hspace*{6ex}$ Y^0 $: initial solution space
\begin{algorithmic}[1]
\State $ UB_0 = +\infty $
\State $ y^0, LB_0 \gets {\rm solving \ PRLP}(Y^0) $
\If{ $ y^0 $ is feasible to QSRFP }
\State $ UB_0 \gets {\rm min}\{ H_0(y_0), UB_0 \} $
\EndIf
\If{ $ UB_0 - LB_0 > \epsilon $ }
\State $ \Pi_0 \gets \{ Y^0 \}, \ F \gets \emptyset, \ y^k \gets y^0 $
\For{ $ k $ in 1,2,... }
\State $ UB_k \gets UB_{k-1}, \ F \gets F \cup \{ Y^{k-1} \} $
\State $ Y^{k,1}, Y^{k,2} \gets {\rm subdivides}\ Y^{k-1} $
\For{ $ t $ in $ \{ 1, 2\} $}
\State $ Y^{k,t} \gets {\rm reducing} \ Y^{k,t} $
\State $ y^{k,t}, LB^{k,t} \gets {\rm solving \ PRLP}(Y^{k,t}) $
\If{ $ y^{mid} $ of $ Y^{k,t} $ is feasible to QSRFP }
\State $ UB_k \gets {\rm min}\{ H_0(y^{mid}), UB_k \}, y^k \gets y^{mid} $
\EndIf
\If{ $ y^{k,t} $ is feasible to QSRFP }
\State $ UB_k \gets {\rm min}\{ H_0(y^{k,t}), UB_k \}, y^k \gets y^{k,t} $
\EndIf
\If{ $ UB_k \leq LB(Y^{k,t}) $ }
\State $ F \gets F \cup \{ Y^{k,t} \} $
\EndIf
\State $ \Pi_k = \{Y|Y \in \Pi_{k-1} \cup \{ Y^{k,t}\}, Y \notin F\} $
\State $ LB_k \gets {\rm min}\{ LB(Y) | Y \in \Pi_k \} $
\EndFor
\If{$ UB_k - LB_k \leq \epsilon $}
\State break
\EndIf
\EndFor
\EndIf
\State $ y \gets $ branch-and-bound searching for integer solution with $ y^k $
\State \Return $ y $
\end{algorithmic}
\end{algorithm}
\section{Experiments and Analysis}\label{sec:experiments}
This section details the experiments conducted with respect to user count, user requirement category count, server average computing resources, and different system scales to investigate the algorithms' performance in different situations. In addition, to further study the algorithms' performance, this section also presents the algorithms' computing complexity and appropriate system size for our algorithms, which severely impacts the performance. Furthermore, suggestions for speeding up the algorithms are described to help the algorithm be better applied to real environments. Lastly, we present some experiments on the minimum deployment to study whether it affects the algorithms' performance or not.
\subsection{Experimental Setup}
All the services and servers are randomly generated in all experiments. The input/output data size ranged from 0--2000 KB. Service abilities ranged from 100--400, and computing resources ranged 1--3 units, where the computing resources of services with minimum computing resources were treated as 1 unit computing resources. The dependency graph $ DG $ was also generated randomly in each experiment, whose function chain length ranged 1--7 without cycles. First, several functions are selected uniformly at random as the functions requested by the users. After that, the function chain of each function is generated. During the chain generation, the patterns in previously generated chains are followed when generating new chains to avoid cycles. For example, once the chain $ A \rightarrow B \rightarrow C $ is generated, if $ B $ is selected as the successor of $ A $ in another chain, then $ C $ is automatically chosen as the successor of $ B $. \figurename~\ref{fig:sdg_example} shows the service dependency graph generated randomly in one experiment as an example. A circle with -1 stands for the users, and circles with other numbers denote the different services. The arrows between circles represent the respective call dependency.
\begin{figure}[!t]
\centering
\includegraphics[width=\linewidth]{figures/sdg-example.png}
\caption{Service dependency graph generated randomly with 48 services}
\label{fig:sdg_example}
\end{figure}
The servers' computing resources ranged between different experiments, and the delay and bandwidth ranged 1--10ms and 50--1000 MB/s, respectively. In addition, the user requirements were assigned to each server randomly, and the user count varied across different experiments.
\begin{figure*}[!t]
\centering
\subfloat[Average response time w.r.t. user count in Experiment~1.1]{
\includegraphics[width=0.46\linewidth]{experiments/rwt_users-eps-converted-to.pdf}\label{fig:experiment_1_a5}
}
\subfloat[Execution time w.r.t. user count in Experiment~1.1]{
\includegraphics[width=0.46\linewidth]{experiments/rwt_users_time-eps-converted-to.pdf}\label{fig:experiment_1_t5}
}
\\
\subfloat[Average response time with w.r.t. user count in Experiment~1.2]{
\includegraphics[width=0.46\linewidth]{experiments/rwt_users_2-eps-converted-to.pdf}\label{fig:experiment_1_a10}
}
\subfloat[Execution time with w.r.t. user count in Experiment~1.2]{
\includegraphics[width=0.46\linewidth]{experiments/rwt_users_2_time-eps-converted-to.pdf}\label{fig:experiment_1_t10}
}
\caption{Experiment results w.r.t. user count in Experiment~1}\label{fig:experiment_1}
\end{figure*}
There were three kinds of our algorithms: BFS based algorithm B-QSRFP, DFS based algorithm D-QSRFP, and the combination of B-QSRFP and D-QSRFP denoted as BD-QSRFP. BD-QSRFP runs both B-QSRFP and D-QSRFP separately, and the best result is returned.
The following algorithms were chosen to be compared with our proposed algorithms:
\begin{itemize}
\item LDSSP: A lightweight decentralized service placement policy for multi-component application in fog computing proposed in~\cite{LDSPP} which places most popular services as close to the users as possible for lower latency and better network usage.
\item GMCAPP: An efficient multi-component application online placement algorithm in MEC introduced in~\cite{GMCAPP} which aims to minimize the total cost of running the applications. It was modified with Theorem~\ref{theorem} as the optimization goal, and the deployment cost was changed to be updated after deploying an instance.
\item GA: Genetic Algorithm is a widely used evolutionary algorithm~\cite{whitley1994genetic} that can solve the placement problem. The main operators of a GA are selection, crossover, and mutation. A feasible deployment scheme $ X $ was used as an individual in our experiments, and the two-point crossover was applied. Randomly delete/add/move an instance were implemented as the mutation with the same probability. 400 population size and a maximum of 400 rounds were used based on preliminary experiments. The tournament selection was adopted for crossover, and mutation probability was set to 0.3. We picked these parameters considering the performance and execution time: higher values lead to long execution times, and lower values make the algorithm hard to converge even on small-size data.
\item Random: Randomly generating a deployment scheme. It runs for 100 times, and the average result was used as the final result to establish a baseline.
\end{itemize}
It should be noted that due to the high computing complexity of FPP, the problem was converted as QSRFP in all algorithms, which means Equation~\ref{eqa:T_i_j} was used instead.
All the algorithms were implemented in Python 3.6.8 without parallelism and run on the computer with Intel Xeon Gold 5120, 160GB, and CentOS 7. The average response time (ms) and the algorithm execution time (ms) were picked as the metrics. The Random algorithm is for performance comparison only, and its execution time is not included in all experiments. We conducted four experiments to investigate our algorithms' performance under different situations: user count, user requirement category count, server average computing resources, and different system scales.
\subsection{Experiment~1: User Count}
User count is one factor that could affect the deployment scheme due to the service capability of each service. When the number of users increases, the system needs to deploy more instances to satisfy the increasing request frequency. After taking the service dependencies into consideration, the system is also required to deploy more instances for services called by other services.
In Experiment~1, two experiments with different numbers of servers and services were conducted. Experiment~1.1 had 5 servers and 23 services, and Experiment~1.2 had 10 servers and 50 services. The user count ranges between 500--1600 and 500--1500 in Experiments~1.1 and 1.2, respectively. There is no 1600 user count in Experiment~1.2 because the server's computing resource is insufficient even deploying each service with the minimum number.
The results are shown in \figurename~\ref{fig:experiment_1}. For Experiment~1.1, it could be seen that D-QSRFP and B-QSRFP had similar performance overall compared to GMCAPP and GA in \figurename~\ref{fig:experiment_1_a5}, while D-QSRFP and B-QSRFP took less execution time than GMCAPP. BD-QSRFP got better performance than other algorithms but with more execution time than other algorithms except for GA in \figurename~\ref{fig:experiment_1_t5}. It is worth pointing out that GA had much more execution time than other algorithms because it spends lots of time calculating the fitness of each individual with Equation~\ref{eqa:TX_qsrfp}. Because GA is the evolutionary algorithm, it must calculate the fitness after generating new individuals after crossover and mutation. Moreover, GA also needs to repair the infeasible individuals after crossover and mutation because the individuals after crossover and mutation may violate constraints, which also incurs execution cost.
In Experiment~1.2, D-QSRFP performed similar to GMCAPP with less execution time as shown in \figurename~\ref{fig:experiment_1_a10} and~\ref{fig:experiment_1_t10}. B-QSRFP and BD-QSRFP always performed best, and the execution time of BD-QSRFP was less than GMCAPP. The GA results were worse than GMCAPP, D-QSRFP, and B-QSRFP, and the execution time of GA also increased a lot compared to Experiment~1.1, which was caused by more services and servers in Experiment~1.2. With more services and servers, GA had to spend more time on fitness evaluation, and 400 rounds were not enough to find better solutions than other algorithms.
For LDSPP and Random algorithms, their performance was stable all the time in both Experiments~1.1 and~1.2. Though LDSPP had the lowest execution time, it performed worse than others except for Random. The results of Random were stable in the whole experiment due to the same service set. Even though the instance count of each service differs between different user counts, the Random algorithm always tries to evenly deploy the instances of each service across each server. As a result, the evaluated value of Equation~\ref{eqa:TX_qsrfp} is a constant, and the Random algorithm gets similar results. It should be noted that the execution time of all algorithms except for GA increased slightly with user count because more instances needed to be deployed.
From Experiment~1, it could be seen that GA is not suitable as an online algorithm for this problem because it gets harder to find a better deployment scheme within acceptable execution time than other algorithms. B-QSRFP, D-QSRFP, and GMCAPP can get similar performance, while B-QSRFP and Q-QSRFP need less execution time. In order to further study the algorithms' performance, experiments w.r.t. user requirement category count was conducted.
\begin{figure*}[!t]
\centering
\subfloat[Average response time w.r.t. user requirement category count]{
\includegraphics[width=0.46\linewidth]{experiments/rwt_svc-eps-converted-to.pdf}\label{fig:experiment_2_a}
}
\subfloat[Execution time w.r.t. user requirement category count]{
\includegraphics[width=0.46\linewidth]{experiments/rwt_svc_time-eps-converted-to.pdf}\label{fig:experiment_2_t}
}
\caption{Experiment results w.r.t. user requirement category count in Experiment~2}\label{fig:experiment_2}
\end{figure*}
\begin{figure*}[!t]
\centering
\subfloat[Average response time w.r.t. average server resources]{
\includegraphics[width=0.46\linewidth]{experiments/rwt_resources-eps-converted-to.pdf}\label{fig:experiment_3_a}
}
\subfloat[Execution time w.r.t. average server resources]{
\includegraphics[width=0.46\linewidth]{experiments/rwt_resources_time-eps-converted-to.pdf}\label{fig:experiment_3_t}
}
\caption{Experiment results w.r.t. average server resources in Experiment~3}\label{fig:experiment_3}
\end{figure*}
\subsection{Experiment~2: User Requirement Category Count}
The user requirement category count shows how many different functions are called by users directly, and each has a function chain with an average length of 4. The user requirement category count varies from 5 to 18, and the service dependency also changes. The number of services increases with the user requirement category count and ranges between 15--48, while the number of servers and users keeps the same all the time.
The experiment results are shown in \figurename~\ref{fig:experiment_2}. We can see that the performance of B-QSRFP and D-QSRFP were better than other algorithms with 7, 8, 10, 11, 13, and 15--18 user requirement category count. In other cases, one of B-QSRFP and D-QSRFP got worse results than others, but the other could get the best performance, and as a result, BD-QSRFP was always better than everybody else. GA could get similar results as BD-QSRFP initially, and the results became worse as the number of services increases; this might be due to the increase in dimensionality and an insufficient computational budget.
In \figurename~\ref{fig:experiment_2_t}, all the algorithms got higher execution time with larger count of user requirement category. Less execution time was needed for D-QSRFP and B-QSRFP compared with other algorithms except for LDSPP. The execution time of G-MCAPP was lower than BQ-QSRFP at the beginning and higher than BQ-QSRFP as the number of services increases. Random got different results as the number of services change due to the different service set in this experiment.
The results of Experiment~2 show that B-QSRFP and D-QSRFP can outperform the other approaches in both time in quality, and BD-QSRFP is the best while requiring the highest execution time. Considering the servers' computing resources keep the same in this experiment, we conducted more experiments w.r.t. server computing resources to study the algorithms' performance under different system computing resource usages.
\subsection{Experiment~3: Average Server Resources}
\begin{figure*}[!t]
\centering
\subfloat[Average response time w.r.t. system scale in Experiment~4.1]{
\includegraphics[width=0.46\linewidth]{experiments/rwt_scale-eps-converted-to.pdf}\label{fig:experiment_4_1_a}
}
\subfloat[Execution time w.r.t. system scale in Experiment~4.1]{
\includegraphics[width=0.46\linewidth]{experiments/rwt_scale_time-eps-converted-to.pdf}\label{fig:experiment_4_1_t}
}
\caption{Experiment results w.r.t. average server resources in Experiment~4.1}\label{fig:experiment_4_1}
\end{figure*}
\begin{figure*}[!t]
\centering
\subfloat[Average response time w.r.t. system scale in Experiment~4.2]{
\includegraphics[width=0.46\linewidth]{experiments/rwt_performance_50-eps-converted-to.pdf}\label{fig:experiment_4_2_a}
}
\subfloat[Execution time w.r.t. system scale in Experiment~4.2]{
\includegraphics[width=0.46\linewidth]{experiments/rwt_performance_50_time-eps-converted-to.pdf}\label{fig:experiment_4_2_t}
}
\\
\subfloat[Average response time with w.r.t. system scale in Experiment~4.3]{
\includegraphics[width=0.46\linewidth]{experiments/rwt_performance_100-eps-converted-to.pdf}\label{fig:experiment_4_3_a}
}
\subfloat[Execution time with w.r.t. system scale in Experiment~4.3]{
\includegraphics[width=0.46\linewidth]{experiments/rwt_performance_100_time-eps-converted-to.pdf}\label{fig:experiment_4_3_t}
}
\caption{Experiment results w.r.t. system scale in Experiment~4.2 and 4.3}\label{fig:experiment_4_23}
\end{figure*}
In this experiment, the average server resources are represented in the number of units, which stands for the computing resources used by the service with the minimal resource requirement. The number of servers keeps the same, and the average server resources range from 175 units to 265 units. The user count and the user requirement category count keep the same, and 175 units are the minimal average server resources that can hold enough instances to meet user requirements. As the average server resources increase, the system can deploy more service instances on each server.
Based on the results shown in in \figurename~\ref{fig:experiment_3}, B-QSRFP and D-QSRFP outperformed other algorithms in most situations in \figurename~\ref{fig:experiment_3_a}, and their execution time was lower than others except for LDSPP, while LDSPP got worse performance than them. Due to the small size of servers and services, GA got better results than GMCAPP in this experiment with the most execution time, which is still unacceptable as an online algorithm. BD-QSRFP still had the best performance in the whole experiment, and the performance of Random nearly kept the same as explained in Experiment~1.
It should be noted that the execution time of all algorithms except for GA is generally stable, but there are still some differences as the average server resources change in \figurename~\ref{fig:experiment_3_t}. It is because the instance number of each service keeps the same due to the same service set and servers. But with different average server resources, the instance number that can be run on each server differs, and the algorithm needs to spend more time to find an adequate server for each instance again when the resources of the server used before are insufficient. Thus the execution time varies.
The results in this experiment show that our algorithm B-QSRFP, D-QSRFP, and BD-QSRFP outperform others with better average response time and lower execution time. To investigate their performance with more servers and services, we conducted more experiments with different system scale.
\subsection{Experiment~4: System Scale} \label{subsec:experiment_4}
In this experiment, we enlarged the number of servers and services to check the algorithms' performance. Experiment~4.1 was conducted with 10 servers and 10-50 user requirement category count, and the maximal service size was 130, while the user count kept the same. The results are shown in \figurename~\ref{fig:experiment_4_1}.
The results in \figurename~\ref{fig:experiment_4_1_a} and~\ref{fig:experiment_4_1_t} are similar to our previous experiments. B-QSRFP and D-QSRFP could get the best or similar results compared to others with less execution time, and BD-QSRFP had the best performance all the time, while its execution time was higher. GA performed worse than other experiments due to the larger size of services and servers with more execution time. GMCAPP's execution time increased a lot when the user requirement category count increased.
To further study the algorithms' performance, Experiments~4.2 and~4.3 were conducted. There were 50 servers and 100 servers in Experiments~4.2 and~4.3, respectively. The user requirement category count ranged between 50--100 in both experiments, and the service size ranged between 150-320. Because GA performed worse than others, GA and Random were not used in Experiments~4.2 and~4.3. The results are shown in \figurename~\ref{fig:experiment_4_23}.
Based on the results in \figurename~\ref{fig:experiment_4_2_a} and~\ref{fig:experiment_4_3_a}, B-QSRFP and D-QSRFP also got better results than GMCAPP and LDSPP, and BD-QSRFP had the best performance even with large scale system. The execution time of B-QSRFP, D-QSRFP, and BD-QSRFP were lower than GMCAPP in \figurename \ref{fig:experiment_4_2_t} and~\ref{fig:experiment_4_3_t}. It should be pointed out that the maximum execution times of BD-QSRFP were about 11s and 52s in Experiments~4.2 and~4.3, respectively, which is too costly for an online algorithm. However, 100 servers are too many for the microservice system, and other technologies could improve the algorithms' speed. For more details, please refer to Section~\ref{subsubsec:system_size} and~\ref{subsubsec:speed_up}.
\subsection{Analysis}
In this section, we analyse the algorithms' computing complexity and discuss appropriate system sizes. We suggest ways for speeding up the algorithms and investigate the performance impact of a minimum instance deployment.
\subsubsection{Computing Complexity Analysis} \label{subsubsec:complexity}
The complexity of Algorithm~\ref{algo:best_server} is $ O(2|N|^3) $. In Algorithm~\ref{algo:deploy}, it would be $ \Omega(2|N|^3) $ in the ideal situation, which means sufficient computing resources and no dependency at all. In the worst situation, it would not exceed $ O(2 + 2|S|)|N|^4 $: all the services are needed to be re-deployed, and Algorithm~\ref{algo:best_server} must run $ |N| $ times to deploy all instances. For B-QSRFP and D-QSRFP, it should be $ O(2 + 2|S|)|S||N|^4 $.
\subsubsection{Appropriate System Size} \label{subsubsec:system_size}
According to Section~\ref{subsubsec:complexity}, the server size has an essential impact on our algorithms' performance, which means the microservice system should have an appropriate server size so that the execution time taken by our algorithms is affordable as an online algorithm.
Based on the results in Section~\ref{subsec:experiment_4}, 100+ servers are too many for our algorithms even after adopting the suggestions in Section~\ref{subsubsec:speed_up} to speed up. In our opinion, the microservice system should not hold too many servers at the same time during the service placement or system evolution. On the one hand, the cost of collecting the running information from all servers can be huge, i.e., the bandwidth and the evolution delay due to the data transmission. On the other hand, as a distributed system, the stability of the distributed system should be ensured during the placement or evolution, and centralized algorithms should not be used for too many servers.
Thus, for a large microservice system with thousands of servers, we suggest dividing it into several small groups whose server size does not exceed 100 to reduce the cost during the information collection and keep the distributed system's stability while makes the algorithms' execution time is affordable as online algorithms.
\subsubsection{Suggestions on Speeding up} \label{subsubsec:speed_up}
There are two methods to speed up our algorithms. One is implementing the algorithms in a high-performance programming language like C++. Because our algorithms were implemented in pure Python, the speed should be improved a lot due to the low performance of Python compared to C++.
Another is adopting multi-threads technology. It could be seen that Algorithm~\ref{algo:best_server} plays an important role in the algorithms' performance. The algorithm calculates the average time for each server by assuming deploying one instance on each server, and it is safe to do it with $ |N| $ threads, which also can speed up the algorithms.
Using B-QSRFP or D-QSRFP instead of BD-QSRFP in a large-scale system also helps to reduce execution time. Though B-QSRFP and D-QSRFP cannot get the best results, their execution time is lower than BD-QSRFP.
\subsubsection{Impact of Minimum Deployment on Performance}
The strategy used in our algorithms only deploy instances that just satisfy the total request rate from users and services. For example, assuming the total request rate of $ s_i $ is $ \lambda_i $, then $ \lceil \frac{\lambda_i}{\mu_i} \rceil $ instances are deployed in the system. According to Equation~\ref{eqa:T_i_j}, it is possible to improve the average response time by deploying more instances. To investigate the performance improvement with more instances, some experiments were conducted.
At the end of Algorithm~\ref{algo:bfs_placement} and~\ref{algo:dfs_placement}, the algorithms deploy more instances in order to lower average response time. They search the service $ s_i $ and the server $ n_j $ that can make the average response time decrease the most when deploying an instance of $ s_i $ on $ n_j $, and deploy $ s_i $ on $ n_j $ until the constraints are satisfied. The dataset is the same as Experiment~2, and the results are shown in Table~\ref{tab:push_bound}. D-QSRFP and B-QSRFP are abbreviated as D and B, and D' and B' stand for the algorithms that deploy more instances for improvement until meeting constraints.
\begin{table}[htbp]
\caption{Results with/without minimal instances} \label{tab:push_bound}
\centering
\begin{adjustbox}{width=\linewidth}
\begin{tabular}{c|cccc|cccc}
\hline
\multirow{2}{*}{Count} & \multicolumn{4}{c|}{Average Response Time} & \multicolumn{4}{c}{Execution Time} \\
\cline{2-9}
& D & D' & B & B' & D & D' & B & B' \\
\hline
5 & 13.96 & 13.96 & 10.74 & 10.74 & 8.87 & 9.34 & 4.53 & 5.22 \\
6 & 9.97 & 9.97 & 6.43 & 6.43 & 12.80 & 15.13 & 5.20 & 6.95 \\
7 & 4.68 & 4.68 & 4.80 & 4.80 & 7.97 & 9.25 & 5.76 & 7.58 \\
8 & 8.54 & 8.54 & 8.01 & 8.01 & 9.21 & 11.31 & 6.81 & 9.00 \\
9 & 8.54 & 8.54 & 11.21 & 11.21 & 10.72 & 12.68 & 6.92 & 9.04 \\
10 & 10.55 & 10.55 & 9.55 & {\cellcolor[rgb]{0.753,0.753,0.753}}9.53 & 14.66 & 17.15 & 9.10 & 27.79 \\
11 & 8.31 & 8.31 & 9.00 & 9.00 & 15.65 & 18.59 & 11.65 & 14.36 \\
12 & 13.58 & 13.58 & 9.82 & {\cellcolor[rgb]{0.753,0.753,0.753}}9.78 & 16.22 & 18.64 & 8.03 & 164.42 \\
13 & 9.19 & {\cellcolor[rgb]{0.753,0.753,0.753}}9.17 & 10.24 & 10.24 & 21.76 & 61.40 & 11.84 & 15.77 \\
14 & 11.11 & {\cellcolor[rgb]{0.753,0.753,0.753}}10.71 & 7.75 & 7.75 & 13.57 & 332.13 & 11.11 & 14.05 \\
15 & 6.24 & 6.24 & 6.13 & 6.13 & 14.94 & 18.00 & 12.94 & 16.18 \\
16 & 9.01 & 9.01 & 6.71 & 6.71 & 22.92 & 28.32 & 13.33 & 17.86 \\
17 & 13.13 & 13.13 & 10.22 & 10.22 & 26.52 & 30.38 & 17.96 & 22.72 \\
18 & 7.37 & 7.37 & 6.98 & 6.98 & 23.47 & 27.23 & 14.18 & 18.02 \\
\hline
\end{tabular}
\end{adjustbox}
\end{table}
The results with gray color show that deploying more instances can improve the average response time at the cost of a much higher execution time. Thus, it should be used only when the execution time is acceptable.
\section{Conclusion and Future Work}\label{sec:conclusion}
This paper has formulated the service placement problem in microservice systems with complex dependencies as a FPP considering the dependencies between services and support for multiple service instances. Due to the high computing complexity of FPP, we have proven that we can convert the FPP to QSRFP, which results in a significant decrease in computational complexity. Based on the QSRFP, we have proposed the greedy-based algorithms B-QSRFP, D-QSRFP, and BD-QSRFP. The experimental evaluation has shown that our algorithms outperform other approaches in both computation time and quality. We have discussed ways of speeding up our approaches and discussed appropriate system sizes.
Possible future work includes the programming framework and system infrastructure supporting self-adaptive microservice system evolution in multiple groups: the large-scale microservice system must be divided into several groups to maintain the stability of the distributed microservice system. The cost of system information collection and evolution time can also be reduced with groups, and each group can evolve and cooperate with other groups.
\appendices
\ifCLASSOPTIONcompsoc
\section*{Acknowledgments}
\else
\section*{Acknowledgment}
\fi
Research in this paper is partially supported by the National
Key Research and Development Program of China (No
2018YFB1402500), the National Science Foundation of China
(61832014, 61772155, 61832004), as well as by the Australian Research Council (DP200102364, DP210102670).
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
|
3,212,635,537,460 | arxiv | \section{Introduction}
For a finite set $V$ and a positive integer $r$ we denote by ${{V \choose r}}$ the family of all $r$-subsets of $V$. An {\em $r$-uniform graph} ({\em $r$-graph}) $G$ is a set $V(G)$ of vertices together with a set $E(G) \subseteq {V(G) \choose r}$ of edges. The {\em density} of $G$ is defined by $d(G) = {\vert E(G)\vert\over \vert{V(G) \choose r}\vert}$.
For a family ${\cal F}$ of $r$-graphs, an $r$-graph $G$ is called $\cal F$-free if it does not contain an isomorphic copy of any $r$-graph of $\cal F$. For a fixed positive integer $n$ and a family of $r$-graphs $\cal F$, the {\em Tur\'an number} of $\cal F$, denoted by $ex(n,\cal F)$, is the maximum number of edges in an $\cal F$-free $r$-graph on $n$ vertices. An averaging argument in \cite{KNS} by Katona, Nemetz, and Simonovits shows that the sequence ${ex(n, \cal F)\over {n\choose r}}$ is non-increasing. Hence $\lim_{n\to\infty}{ex(n, \cal F)\over {n\choose r}}$ exists. The {\em Tur\'{a}n density} of $\cal F$ is defined as $$\pi(\cal F)=\lim_{n\rightarrow\infty}{ex(n, \cal F) \over {n \choose r }}.$$
If $\cal F$ consists of a single $r$-graph $F$, we simply write $ex(n, \{F\})$ and $\pi(\{F\})$ as $ex(n, F)$ and $\pi(F)$. Denote
$$\Pi_{\infty}^{r}=\{ \pi(\cal F): \cal F {\rm \ is \ a \ family \ of \ } r{\rm-uniform \ graphs} \}, $$
$$\Pi_{fin}^{r}=\{\pi(\cal F): \cal F {\rm \ is \ a \ finite \ family \ of \ } r{\rm-uniform \ graphs}, \}$$
and
$$\Pi_{t}^{r}=\{ \pi(\cal F): \cal F {\rm \ is \ a \ family \ of \ } r{\rm-uniform \ graphs \ and }\ \vert \cal F \vert\le t \}. $$
Clearly,
$$\Pi_{1}^{r}\subseteq \Pi_{2}^{r}\subseteq \cdots \subseteq\Pi_{fin}^{r}\subseteq \Pi_{\infty}^{r}.$$
Finding good estimation for Tur\'an densities in hypergraphs ($r\ge 3$) is believed to be one of the most challenging problems in extremal combinatorics. The following concept concerns the accumulation points of the set $\Pi_{\infty}^{r}$.
\begin{defi}
A real number $\alpha\in [0, 1)$ is a {\em jump} for an integer $r\ge 2$ if there exists a constant $c>0$ such that for any $\epsilon>0$ and any integer $m \ge r$, there exists an integer $n_0(\epsilon, m)$ such that any $r$-uniform graph with $n > n_0(\epsilon, m)$ vertices and density $\ge \alpha + \epsilon$ contains a subgraph with $m$ vertices and density $\ge \alpha + c$.
\end{defi}
This concept describes where the set of `jumps' is closely related to Tur\'an densities. It was shown in \cite{FR84} that $\alpha$ is a jump for $r$ if and only if there exists $c>0$ such that $\Pi_{\infty}^r \cap (\alpha, \alpha+c)=\emptyset$. So every non-jump is an accumulation point of $\Pi_{\infty}^{r}$.
For 2-graphs, Erd\H{o}s-Stone-Simonovits \cite{ESi,ES} determined the Tur\'an numbers of all non-bipartite graphs asymptotically. Their result implies that $$\Pi_{\infty}^{2}=\Pi_{fin}^{2}=\Pi_{1}^{2}=\{0, {1 \over 2}, {2 \over 3}, ..., {l-1 \over l}, ...\}.$$
This implies that every $\alpha\in [ 0, 1)$ is a jump for $r=2$. For $r\geq 3$, Erd\H{o}s \cite{E64} proved that every $\alpha\in [0, r!/r^r)$ is a jump. Furthermore, Erd\H{o}s proposed the {\it jumping constant conjecture}: Every $\alpha\in [0, 1)$ is a jump for every integer $r \geq 2$. In \cite{FR84}, Frankl and R\"{o}dl disproved the Conjecture by showing that $\displaystyle{1-\frac{1}{l^{r-1}}}$ is not a jump for $r$ if $r\ge 3$ and $l>2r$. However, there are still a lot of unknowns on whether a number is a jump for $r\ge 3$. A well-known open question of Erd\H{o}s is whether $r!/r^r$ is a jump for $r\ge 3$ and what is the smallest non-jump? Another question raised in \cite{FPRT} is whether there is an interval of non-jumps for some $r\ge 3$? Both questions seem to be very challenging. Frankl-Peng-R\"{o}dl-Talbot \cite{FPRT} showed that ${5r! \over 2 r^r}$ is a non-jump for $r\ge 3$. Baber and Talbot \cite{BT0} showed that for $r=3$ every $\alpha\in[0.2299, 0.2316)\cup [0.2871, \frac{8}{27})$ is a jump. Pikhurko \cite{Pikhurko2} showed that $\Pi_{\infty}^r$ has cardinality of the continuum for $r\ge 3$. However, whether $\frac{r!}{r^r}$ is a jump remains open. Regarding the first question, we determine a non-jump smaller than ${5r! \over 2 r^r}$ for $r\ge 3$.
\begin{theo}\label{theo}
$\frac{12}{25}$ is not a jump for $r=3$.
\end{theo}
In \cite{jumpgeneral}, a way to generate non-jumps for every $p\ge r$ based on a non-jump for $r$ was given. The following result was shown there.
\begin{theo}\label{resultg}\cite{jumpgeneral}
Let $p\ge r\ge 3$ be positive integers. If $\alpha\cdot {r! \over r^r} $ is a non-jump for $r$, then $\alpha \cdot {p! \over p^p}$ is a non-jump for $p$.
\end{theo}
Combining Theorems \ref{theo} and \ref{resultg}, we will get
\begin{coro}
${54r! \over 25r^r}$ is a non-jump for $r\ge 3$.
\end{coro}
Chung and Graham \cite{FG} proposed the conjecture that every element in $\Pi_{fin}^{r}$ is a rational number. Baber and Talbot \cite{BT}, and Pikhurko \cite{Pikhurko2} disproved this conjecture independently by showing that there is an irrational number in $\Pi_{fin}^{r}$. Baber and Talbot asked that whether there is an irrational number in $\Pi_1^r$. Recently, Yan and Peng \cite{YP} showed that there is an irrational number in $\Pi_1^3$ and Wu-Peng \cite{WP} showed that there is an irrational number in $\Pi_1^4$. Pikhurko \cite{Pikhurko2} showed that $\Pi_{\infty}^r$ is closed which implies that every non-jump is a Tur\'an density (a Tur\'an density may not be a non-jump). Brown and Simonovits \cite{BS} showed that the Lagrangian of an $r$-uniform hypergraph is in $\Pi_{\infty}^r$ also indicating the existence of irrational numbers in $\Pi_{\infty}^r$. No irrational non-jump has been previously given. In this paper, we will give a infinite sequence of irrational non-jumps for $r=3$.
\begin{theo}\label{theo1}
Let $k\ge 2$ be an integer. Then $\alpha_k=\frac{2k-6k^3+4k^4-k\sqrt{4k - 1}+4k^2\sqrt{4k - 1}}{(2k^2+1)^2}$ is not a jump for $r=3$.
\end{theo}
Combining Theorem \ref{theo1} and Theorem \ref{resultg}, we can also get corresponding non-jumps for $r\ge 3$.
The proof of Theorem \ref{theo} and \ref{theo1} will be given in Section \ref{prooftheo} and Section \ref{prooftheo1}, respectively. Both proofs applied an approach developed by Frankl and R\"odl in \cite{FR84}. The crucial part in our proof is to give a `proper' construction. In the following section, we will introduce some preliminary results and sketch the idea of the proof.
\section{Preliminaries and Sketch of the proof}
\subsection{Karush-Kuhn-Tucker Conditions}
Let us consider the optimisation problem:
\begin{flushleft}
\quad\quad\quad\quad\quad maximise\ $f(x)$\\
\quad\quad\quad\quad\quad subject to $g_i(x)\leq 0$, $i=1,\dots,m,$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (3.1)
\end{flushleft}
where $x\in \mathbb{R}^n$ and $f$ and $g_i$ are differentiable functions from $\mathbb{R}^n$ to $\mathbb{R}$ for all $i$. Let $\nabla{f(x)}$ be the gradient of $f$ at $x$ i.e., the vector in $\mathbb{R}^n$ whose $i$th coordinate is ${\partial\over\partial{x_i}}f(x)$. We say that KKT conditions hold at $x^*\in \mathbb{R}^n$ if there exist $\lambda_1,\dots\lambda_m\in \mathbb{R}$ such that
\begin{enumerate}
\item $\nabla{f(x^*)}=\sum_{i=1}^m\lambda_i\nabla{g_i(x^*)},$
\item $\lambda_i\geq 0$ for $i=1,\dots,m,$
\item $\lambda_ig_i(x^*)=0$ for $i=1,\dots,m.$
\end{enumerate}
We call the constraints linear if $g_1,\dots,g_m$ are all affine functions.
\begin{theo}\label{KKT}(\cite{BV},\cite{Jenssen})
If the constraints of (3.1) are linear, then any optimal solution to (3.1) must satisfy the KKT conditions.
\end{theo}
\subsection{Properties of the Lagrangian function}
In this section we will give the definition of the Lagrangian of an $r$-uniform graph, which is a helpful tool in our proof.
\begin{defi}
For an $r$-uniform graph $G$ with vertex set $\{1,2,\ldots,n\}$, edge set $E(G)$ and a vector $\vec{x}=(x_1,\ldots,x_n) \in R^n$, define the Lagrangian fuction
$$\lambda (G,\vec{x})= \sum_{\{i_1,\ldots,i_r\}\in E(G)}x_{i_1}x_{i_2}\ldots x_{i_r}.$$
\end{defi}
Let $S=\{\vec{x}=(x_1,x_2,\ldots ,x_n): \sum_{i=1}^{n} x_i =1, x_i \ge 0 {\rm \ for \ } i=1,2,\ldots , n \}$. The Lagrangian of $G$, denoted by $\lambda (G)$, is defined as
$$\lambda (G) = \max \{\lambda (G, \vec{x}): \vec{x} \in S \}.$$
A vector $\vec{x}\in S$ is called a {\em feasible vector} on $G$, and $x_i$ is called the {\em weight} of the vertex $i$.
A feasible vector is called {\em optimal} if $\lambda (G, \vec{y})=\lambda(G)$.
\begin{fact}\label{mono}
If $G_1\subseteq G_2$, then $$\lambda (G_1) \le \lambda (G_2).$$
\end{fact}
\begin{fact}(\cite{FR84})\label{fact2}
Let $G$ be an $r$-graph on $[n]$. Let $\vec{x}=(x_1,x_2,\dots,x_n)$ be an optimal vector on $G$. Then
$$ \frac{\partial \lambda (G, \vec{x})}{\partial x_i}=r\lambda(G)$$
for every $i \in [n]$ satisfying $x_i>0$.
\end{fact}
Given an $r$-graph $G$, and $i, j\in V(G),$ define $$L_G(j\setminus i)=\{e: i\notin e, e\cup\{j\}\in E(G)\:and\: e\cup\{i\}\notin E(G)\}.$$
\begin{fact}\label{symmetry}(\cite{FR84})
Let $G$ be an $r$-graph on $[n]$. Let $\vec{x}=(x_1,x_2,\dots,x_n)$ be a feasible vector on $G$, and $i,j\in [n]$, $i\neq j$ satisfy $L_G(i \setminus j)=L_G(j \setminus i)=\emptyset$. Let $\vec{y}=(y_1,y_2,\dots,y_n)$ be defined by letting $y_\ell=x_\ell$ for every $\ell \in [n]\setminus \{i,j\}$ and $y_i=y_j={1 \over 2}(x_i+x_j)$.
Then $\lambda(G,\vec{y})\geq \lambda(G,\vec{x})$. Furthermore, if the pair $\{i,j\}$ is contained in an edge of $G$, $x_i>0$ for each $1\le i\le n$, and $\lambda(G,\vec{y})=\lambda(G,\vec{x})$, then $x_i=x_j$.
\end{fact}
We also note that for an $r$-graph $G$ with $n$ vertices, if we take $\vec{u}=(u_1, \ldots, u_n)$, where each $u_i={1\over n}$, then
$$\lambda(G)\ge \lambda(G, \vec{u})={\vert E(G)\vert \over n^r} \ge {d(G) \over r!}-\epsilon$$ for $n\ge n'(\epsilon)$, where $n'(\epsilon)$ is a sufficiently large integer. On the other hand, the blow-up of an $r$-uniform graph $G$ will allow us to construct $r$-uniform graphs with large number of vertices and density close to $r!\lambda (G)$.
\begin{defi}
Let $G$ be an $r$-uniform graph with $V(G) =\{1,2,\ldots,t\}$ and $(n_1, \ldots, n_t)$ be a positive integer vector. Define the $(n_1, \ldots, n_t)$ blow-up of $G$, $(n_1, \ldots, n_t)\otimes G$ as a $t$-partite $r$-uniform graph with vertex set $V_1\cup \ldots \cup V_t, |V_i|=n_i, 1\leq i\leq t$, and edge set $E((n_1, \ldots, n_t)\otimes G)=\{\{v_{i_1}, v_{i_2},\ldots, v_{i_r}\}, { \ \rm where \ } \{i_1,i_2,\ldots, i_r\} \in E(G) {\ \rm and \ } v_{i_k} \in V_{i_k} {\rm \ for} \ 1\le k\le r \}$.
\end{defi}
\begin{remark} (\cite{FR84})\label{remarkblow}
Let $G$ be an $r$-uniform graph with $t$ vertices and $\vec{y}=(y_1, \ldots, y_t)$ be an optimal vector for $\lambda(G)$. Then for any $\epsilon >0$, there exists an integer $n_1(\epsilon)$, such that for any integer $n\ge n_1(\epsilon)$,
\begin{equation}\label{blowden}
d((\lceil ny_1\rceil, \lceil ny_2\rceil, \ldots, \lceil ny_t\rceil)\otimes G)\ge r!\lambda(G)-\epsilon.
\end{equation}
\end{remark}
Let us also state a fact which follows directly from the definition of the Lagrangian.
\begin{fact}(\cite{FR84})\label{lblow}
For every $r$-uniform graph $G$ and every positive integer $n$, $\lambda((n, n, \ldots,n)\otimes G) =\lambda (G)$ holds.
\end{fact}
Lemma \ref{arrow} in \cite{FR84} gives a necessary and sufficient condition for a number $\alpha$ to be a jump.
\begin{lemma}\label{arrow}(\cite{FR84})
The following two properties are equivalent.
\begin{enumerate}
\item $\alpha$ is a jump for $r$.
\item There exists some finite family $\cal F$ of $r$-uniform graphs satisfying $\pi({\cal F})\le \alpha$ and $\displaystyle{\lambda (F)> \frac{\alpha}{r!}}$ for all $F \in \cal F$.
\end{enumerate}
\end{lemma}
\subsection{Sketch of the proofs of Theorem \ref{theo} and \ref{theo1}}
The general approach in proving Theorem \ref{theo} and Theorem \ref{theo1} is sketched as follows: Let $\alpha$ be a number to be proved to be a non-jump for $r=3$. Assuming that $\alpha$ is a jump for $r=3$, we will derive a contradiction by the following steps.
Step1. Construct a `proper' $3$-uniform graph
$G^*(t)$ with the Lagrangian at least ${\alpha \over 6}+\epsilon$ for some $\epsilon>0$. Then we `blow up' it to a $3$-uniform graph, say $\vec{m}\otimes G^*(t)$ with large enough number of vertices and density $\ge \alpha+\epsilon$ (see Remark \ref{remarkblow}). If $\alpha$ is a jump, then by Lemma \ref{arrow}, there exists some finite family ${\mathcal F}$ of $3$-uniform graphs with Lagrangians $>{\alpha \over 6}$ and $\pi(\cal{F})\le \alpha$. So $\vec{m}\otimes G^*(t)$ must contain some member of ${\mathcal F}$ as a subgraph.
Step 2. We show that any subgraph of $G^*(t)$ with the number of vertices not greater than $\max\{\vert V(F)\vert, F \in {\mathcal F}\}$ has the Lagrangian $\le {\alpha \over 6}$ and derive a contradiction.
\bigskip
The crucial part is to construct an $r$-uniform graph satisfying the properties in both Steps 1 and 2. Generally, whenever we find such a construction, we can obtain a corresponding non-jump. This method was first developed by Frankl and R\"odl in \cite{FR84}. The technical part in the proof is to show that the construction satisfies the property in Step 2.
\section{Proof of Theorem \ref{theo}} \label{prooftheo}
{\em Proof.} Suppose that ${12\over 25}$ is a jump for $r=3$. By Lemma \ref{arrow}, there exists a finite collection $\cal F$ of $3$-uniform graphs satisfying the following:
\begin{enumerate}
\item[i)] $\displaystyle \lambda (F) > {2 \over 25} $ for all $F \in \cal F$, and
\item[ii)] $\pi ({\cal F})\le {12\over 25}$.
\end{enumerate}
Let $G(t)=(V, E)$ be the 3-uniform defined as follows. The vertex set $V=V_1\cup V_2\cup V_{3}$, where $|V_1|=|V_2|=\frac{2t}{5}$ and $|V_{3}|=\frac{t}{5}$ and the value of $t$ will be determined later. The edge set of $G(t)$ is
$$\bigg(V_1 \times V_2 \times V_3\bigg) \bigcup \bigg( {V_1\choose 2}\times V_2\bigg) \bigcup \bigg({V_2\choose 2}\times V_3\bigg),$$
i.e., the edges consisting of one vertex from each $V_1, V_2$ and $V_3$, or two vertices from $V_1$ and one vertex from $V_2$, or two vertices from $V_2$ and one vertex from $V_3$. Then
\begin{eqnarray}\label{eg1}
\vert E(G(t))\vert&=&\frac{2t^3}{25}-\frac{3t^2}{25}.
\end{eqnarray}
We will apply the following lemma from \cite{FR84}.
\begin{lemma}\label{add}\cite{FR84}
For any $c\ge 0$ and any integer $s\ge r$, there exists $t_0(s, c)$ such that for every $t\ge t_0(s, c)$, there exists an $r$-uniform graph $A=A(s, c, t)$ satisfying:
\begin{enumerate}
\item $|V(A)|=t,$
\item $|E(A)|\geq ct^{r-1},$
\item For all $V_0 \subset V(A), r\leq |V_0| \leq s$, we have $|E(A)\cap {V_0 \choose r}| \leq |V_0|-r+1$.
\end{enumerate}
\end{lemma}
Set $s= {\rm max}_{F \in {\cal F}} |V(F)|$ and $c=1$. Let $r=3$ in Lemma \ref{add}, $t_0(s, 1)$ be given as in Lemma \ref{add} and $\frac{2t}{5}\ge t_0(s, 1)$. The $3$-uniform graph $G^*(t)$ is obtained by adding $A(s, 1, \frac{2t}{5})$ to the $3$-uniform hypergraph $G(t)$ in $V_1$. Then
$$\lambda(G^*(t))\ge\lambda(G^*(t), (\frac{1}{t}, \frac{1}{t}, \dots, \frac{1}{t}))=\frac{\big\vert E(G^*(t))\big\vert}{t^3}.$$
In view of the construction of $G^*(t)$ and equation (\ref{eg1}), we have
\begin{eqnarray}\label{egAl}
\frac{\big\vert E(G^*(t))\big\vert}{t^3}&\ge&\frac{\big\vert E(G(t))\big\vert}{t^3}+\bigg(\frac{2t}{5}\bigg)^2/{t^3} \nonumber \\
&\ge& \frac{2}{25}+\frac{1}{25t}.
\end{eqnarray}
Now suppose $\vec{y}=(y_1, y_2, ..., y_t)$ is an optimal vector of $\lambda(G^*(t))$. Let $n$ be large enough. By Remark \ref{remarkblow}, $3$-uniform graph $S_n=(\lfloor ny_1\rfloor, \ldots, \lfloor ny_{n}\rfloor)\otimes G^*(t)$ has density at least${12\over 25}+\frac{1}{50t}.$ Since $\pi({\cal F})\le {12\over 25}$, some member $F$ of $\cal F$ is a subgraph of $S_n$ for $n$ sufficiently large. For such $F\in \cal F$, there exists a subgraph $M$ of $G^*(t)$ with $|V(M)|\le |V(F)|\leq s$ so that $F\subset (s, s, \ldots, s) \otimes M$. By Fact \ref{mono} and Fact \ref{lblow}, we have
\begin{equation}\label{lambdasmall0}
\lambda(F)\overset{Fact \ref{mono}}{\le}\lambda ((s, s, \ldots, s) \otimes M)\overset{Fact \ref{lblow}}{=} \lambda (M).
\end{equation}
The following lemma will be proved in Section \ref{prooflemmaresult01}.
\begin{lemma}\label{lemmaresult01}
Let $M$ be any subgraph of $G^*(t)$ with $|V(M)| \leq s$. Then
\begin{equation}
\lambda (M) \leq \frac{2}{25}
\end{equation}
holds.
\end{lemma}
Assuming that Lemma \ref{lemmaresult01} is true and applying Lemma \ref{lemmaresult01} to (\ref{lambdasmall0}), we have $$\lambda(F) \le {2 \over 25}$$ which contradicts our choice of $F$, i.e., contradicts that $\displaystyle \lambda(F) >{2 \over 25}$ for all $F \in \cal F$. \hspace*{\fill}$\Box$\medskip
\bigskip
To complete the proof of Theorem \ref{theo1}, what remains is to show Lemma \ref{lemmaresult01}.
\subsection{Proof of Lemma \ref{lemmaresult01}}\label{prooflemmaresult01}
By Fact \ref{mono}, we may assume that $M$ is an induced subgraph of $G^*(t)$. Let $$U_i=V(M)\cap V_i=\{v_1^i, v_2^i, \cdots, v_{s_i}^i\}.$$ So $s=s_1+s_2+s_3$. Let $\vec{z}=(z_1, z_2, ..., z_s)$ be an optimal vector for $M$. Without loss of generality, assume that $v_1^1$, $v_2^1$, $\cdots$, $v_{s+2}^1$ have the $s+2$ largest weights. Then replacing the $s$ edges in $M[U_1]$ by $v_1^1v_2^1v_3^1, v_1^1v_2^1v_4^1, \dots, v_1^1v_2^1v_{s+2}^1$ doesn't decrease the Lagrangian. So we have the following claim similar to Claim 4.4 in \cite{FR84}.
\begin{claim}\label{reduce0}
If $N$ is the $3$-uniform graph formed from $M$ by removing the edges contained in $U_{1}$ and inserting the edges $v^1_{1}v^1_2v^1_{j}$, where $3\leq j \leq s_1$, then $\lambda(M)\leq \lambda(N)$.
\end{claim}
By Claim \ref{reduce0}, the proof of Lemma \ref{lemmaresult01} will be completed if we show that $\lambda(N)\leq {2 \over 25}$. By Lemma \ref{symmetry}, we can obtain an optimal vector $\vec{z}$ of $\lambda(N)$ such that
\begin{equation} \label{weights}
w(v_1^1)=w(v_2^1)\stackrel{\rm\scriptscriptstyle def}{=}\frac{a}{2}, \ \ w(v_3^1)=w(v_4^1)=\cdots =w(v^1_{s_1}) \stackrel{\rm\scriptscriptstyle def}{=}\frac{b}{s_1-2},
\end{equation}
where $w(v)$ denotes the component of $\vec{z}$ corresponding to vertex $v$.
Let $c$, $d$ be the sum of the components of $\vec{z}$ corresponding to all vertices in $U_2$ and $U_3$, respectively. Note that
$$a+b+c+d=1.$$
Then
\begin{eqnarray*}
\lambda(N)\le\bigg(\frac{a^2}{4}+ab+\frac{b^2}{2}\bigg)c+(a+b)cd+\frac{c^2d}{2}+\frac{a^2}{4}b=\lambda(a, b, c, d).
\end{eqnarray*}
From now on, we assume that $(a, b, c, d)$ is an optimal vector for $\lambda(a, b, c, d)$.
If $c=0$, then $$\lambda(a, b, c, d)=\frac{a^2b}{4}\leq \frac{1}{8}\bigg(\frac{a+a+2b}{3}\bigg)^3\leq \frac{1}{27}.$$
So we may assume that $c>0$.
If $a=0$, then $$\lambda(a, b, c, d)=\frac{b^2c}{2}+bcd+\frac{c^2d}{2}\triangleq\lambda.$$ If $b=0$, then $\lambda=\frac{c^2d}{2}\leq \frac{2}{27}.$ Similarly we have $d>0$. So we may assume that $b, c, d>0$ in this case. By Theorem \ref{KKT}, we have
$$\frac{\partial\lambda}{\partial b}=\frac{\partial\lambda}{\partial c}=\frac{\partial\lambda}{\partial d},$$ so
$$bc+cd=\frac{b^2}{2}+bd+cd=\frac{c^2}{2}+bc.$$
Combining with $b+c+d=1$, we have $b=c=2d=0.4$, and $\lambda=\frac{2}{25}$. So we may assume that $a>0$.
If $b=0$, then $$\lambda(a, b, c, d)=\frac{a^2c}{4}+acd+\frac{c^2d}{2}<\frac{a^2c}{2}+acd+\frac{c^2d}{2}\leq \frac{2}{25}$$
as we have shown that $\frac{b^2c}{2}+bcd+\frac{c^2d}{2}\le \frac{2}{25}$. So we may assume that $b>0$.
We will prove that $d>0$ next. If $d=0$, then $$\lambda(a, b, c, d)=\bigg(\frac{a^2}{4}+ab+\frac{b^2}{2}\bigg)c+\frac{a^2}{4}b\triangleq\lambda.$$
By Theorem \ref{KKT}, we have
$$\frac{\partial\lambda}{\partial a}=\frac{\partial\lambda}{\partial b}.$$ So
$$\bigg(\frac{a}{2}+b\bigg)c+\frac{ab}{2}=(a+b)c+\frac{a^2}{4},$$
i.e., $a=2b-2c$. Since $a+b+c=1$, then $c=3b-1$ and $a=2-4b$. So
\begin{eqnarray*}
\lambda&\le&\bigg(\frac{a^2}{4}+ab+\frac{b^2}{2}\bigg)c+\frac{a^2}{4}b \\
&=&\frac{11b^3}{2}-\frac{21b^2}{2}+6b-1=f(b). \\
f'(b)&=&\frac{33b^2}{2}-21b+6.
\end{eqnarray*}
Since $a, c>0$, then $\frac{1}{3}\leq b\leq \frac{1}{2}$. Therefore $f(b)$ is increasing in $[\frac{1}{3}, \frac{7-\sqrt5}{11}]$ and decreasing in $[\frac{7-\sqrt5}{11}, \frac{1}{2}]$. Then $\lambda<f(\frac{7-\sqrt5}{11})<0.076.$
So we assume that $a, b, c, d>0$, then we have
\begin{eqnarray*}
\frac{\partial\lambda(a, b, c, d)}{\partial a}&=&\bigg(\frac{a}{2}+b\bigg)c+cd+\frac{ab}{2}, \\
\frac{\partial\lambda(a, b, c, d)}{\partial b}&=&(a+b)c+cd+\frac{a^2}{4}, \\
\frac{\partial\lambda(a, b, c, d)}{\partial c}&=&\frac{a^2}{4}+ab+\frac{b^2}{2}+ad+bd+cd, \\
\frac{\partial\lambda(a, b, c, d)}{\partial d}&=&ac+bc+\frac{c^2}{2}, \\
d&=&1-a-b-c.
\end{eqnarray*}
By Theorem \ref{KKT}, we have $$\frac{\partial\lambda(a, b, c, d)}{\partial a}=\frac{\partial\lambda(a, b, c, d)}{\partial b},$$
and we get $a=2b-2c$. Therefore $d=1-a-b-c=1-3b+c$.
By $$\frac{\partial\lambda(a, b, c, d)}{\partial b}=\frac{\partial\lambda(a, b, c, d)}{\partial d},$$
we get $\frac{c^2}{2}=cd+\frac{a^2}{4}=c-3bc+c^2+b^2-2bc+c^2,$ so $$c=\frac{5b-1\pm \sqrt{19b^2-10b+1}}{3}.$$
By $$\frac{\partial\lambda(a, b, c, d)}{\partial b}=\frac{\partial\lambda(a, b, c, d)}{\partial c},$$ we get $$c=\frac{13b^2-6b}{8b-4}.$$
Therefore, $$\frac{13b^2-6b}{8b-4}=\frac{5b-1\pm \sqrt{19b^2-10b+1}}{3}.$$
By direct calculation, we have $(-b^2+10b-4)^2=\bigg(\pm(8b-4)\sqrt{19b^2-10b+1}\bigg)^2.$ Simplifying, we get $$9b(5b-2)(9b-4)(3b-2)=0.$$
If $b=\frac{2}{5}$, then $c=\frac{13b^2-6b}{8b-4}=\frac{2}{5}$ and $a=2b-2c=0$, a contradiction. \\
If $b=\frac{4}{9}$, then $c=\frac{13b^2-6b}{8b-4}=\frac{2}{9}$, $a=2b-2c=\frac{4}{9}$ and $d=-\frac{1}{9}$, a contradiction. \\
If $b=\frac{2}{3}$, then $c<\frac{1}{3}$ and $a=2b-2c>\frac{2}{3}$ and $d<0$, a contradiction.
\hspace*{\fill}$\Box$\medskip
\bigskip
\section{Proof of Theorem \ref{theo1}} \label{prooftheo1}
Let $B(2k, n)$ be the 3-graph with vertex set $[n]$ and edge set $E(B(2k, n))=\{e\in {[n]\choose 3}: e\cap [2k]\not=\emptyset\}$. Let $\alpha_k=\frac{2k-6k^3+4k^4-k\sqrt{4k - 1}+4k^2\sqrt{4k - 1}}{(2k^2+1)^2}$. We first show that $\alpha_k=6\lim_{n\to\infty}\lambda(B(2k, n))$.
Let $\vec{x}=\{x_1,x_2,\dots,x_n\}$ be an optimal vector of $\lambda(B(2k, n))$. Let $x_1+x_2+\cdots+x_{2k}=a$ and $b=1-a$. Then
\begin{eqnarray*}
\lim_{n\to\infty}\lambda(B(2k, n))&=&\bigg(\frac{a}{2k}\bigg)^3{2k\choose 3}+\bigg(\frac{a}{2k}\bigg)^2{2k\choose 2}(1-a)+a\frac{(1-a)^2}{2}=f(a)\\
f'(a)&=&(\frac{1}{4k^2}+\frac{1}{2})a^2-(\frac{1}{2k}+1)a+\frac{1}{2}.
\end{eqnarray*}
Note that $f(a)$ is increasing in $[0, \frac{2k^2+k-k\sqrt{4k-1}}{2k^2+1}]$ and decreasing in $[\frac{2k^2+k-k\sqrt{4k-1}}{2k^2+1}, 1]$. Therefore
\begin{eqnarray*}
f(\frac{2k^2+k-k\sqrt{4k-1}}{2k^2+1})&=&\frac{2k-6k^3+4k^4-k\sqrt{4k - 1}+4k^2\sqrt{4k - 1}}{6(2k^2+1)^2}\\
&=&\frac{\alpha_k}{6}.
\end{eqnarray*}
Since $k\ge 1$ and $4k-1$ is not a square number (a square number is 0 or 1 mod(4)), then $\alpha_k$ is an irrational number.
{\em Proof of Theorem \ref{theo1}.} Suppose that $\alpha_k$ is a jump. By Lemma \ref{arrow}, there exists a finite collection $\cal F$ of $3$-uniform graphs satisfying the following:
\begin{enumerate}
\item[i)] $\displaystyle \lambda (F) > {\alpha_k \over 6} $ for all $F \in \cal F$, and
\item[ii)] $\pi ({\cal F})\le \alpha_k$.
\end{enumerate}
Let $G(t)=(V, E)$ be the 3-uniform defined as follows. The vertex set $V=V_1\cup V_2\cdots\cup V_{2k}\cup V_{2k+1}$, where $|V_1|=|V_2|=\cdots=|V_{2k}|=\frac{2k+1-\sqrt{4k-1}}{4k^2+2}t$ and $|V_{2k+1}|=\frac{k\sqrt{4k-1}+1-k}{2k^2+1}t$. The edge set of $G(t)$ is
$$\bigcup_{1\le i_1<i_2<i_3\le 2k} (V_{i_1}\times V_{i_2} \times V_{i_3})\bigcup_{1\le i_1<i_2\le 2k} (V_{i_1}\times V_{i_2} \times V_{2k+1})\bigcup_{1\le i_1\le 2k} (V_{i_1}\times{V_{2k+1}\choose 2}).$$
Then
\begin{eqnarray}\label{egl}
\vert E(G(t))\vert&=&\frac{2k-6k^3+4k^4-k\sqrt{4k - 1}+4k^2\sqrt{4k - 1}}{6(2k^2+1)^2}t^3 \nonumber \\
&+&\frac{-3k-6k^2+18k^3+3k\sqrt{4k - 1}-6k^2\sqrt{4k - 1}-6k^3\sqrt{4k - 1}}{6(2k^2+1)^2}t^2. \nonumber
\end{eqnarray}
Let $\vec{u}=(u_1, \ldots,u_{t})$, where $u_i=\frac{1}{t}$ for $1\le i\le t$,
then
\begin{eqnarray}\label{egl}
\lambda(G(t))&\ge&\lambda(G(t),\vec{u})={\vert E(G)\vert \over t^3} \nonumber \\
&=&\frac{2k-6k^3+4k^4-k\sqrt{4k - 1}+4k^2\sqrt{4k - 1}}{6(2k^2+1)^2} \nonumber \\
&+&\frac{-3k-6k^2+18k^3+3k\sqrt{4k - 1}-6k^2\sqrt{4k - 1}-6k^3\sqrt{4k - 1}}{6(2k^2+1)^2t}\\
&=&\frac{\alpha_k}{6}-\frac{c_0}{t}, \nonumber
\end{eqnarray}
where $c_0=\frac{3k+6k^2-18k^3-3k\sqrt{4k - 1}+6k^2\sqrt{4k - 1}+6k^3\sqrt{4k - 1}}{6(2k^2+1)^2}>0$.
Set $s= {\rm max}_{F \in {\cal F}} |V(F)|$ and $c=k$. Let $r=3$ in Lemma \ref{add} and $t_0(s, k)$ be given as in Lemma \ref{add}. Take an integer $t>\frac{2k^2+1}{k\sqrt{4k-1}+1-k}t_0$. The $3$-uniform graph $G^*(t)$ is obtained by adding $A(s, k)$ to the $3$-uniform hypergraph $G(t)$ in $V_{2k+1}$. Then
$$\lambda(G^*(t)) \ge \frac{\big\vert E(G^*(t))\big\vert}{t^3}.$$
In view of the construction of $G^*(t)$ and equation (\ref{egl}), we have
\begin{eqnarray}\label{egAl}
\frac{\big\vert E(G^*(t))\big\vert}{t^3}&\ge&\frac{\big\vert E(G(t))\big\vert}{t^3}+\frac{k(\frac{k\sqrt{4k-1}+1-k}{2k^2+1}t)^2}{t^3} \nonumber \\
&{(\ref{egl}) \atop =}&\frac{2k-6k^3+4k^4-k\sqrt{4k - 1}+4k^2\sqrt{4k - 1}}{6(2k^2+1)^2} \nonumber \\
&+&\frac{3k-18k^2+18k^3+24k^4+3k\sqrt{4k - 1}+6k^2\sqrt{4k - 1}-18k^3\sqrt{4k - 1}}{6(2k^2+1)^2t} \nonumber \\
&\ge & {1 \over 6}(\frac{2k-6k^3+4k^4-k\sqrt{4k - 1}+4k^2\sqrt{4k - 1}}{(2k^2+1)^2})+{c_1 \over t}={\alpha_k \over 6}+{c_1 \over t}
\end{eqnarray}
where $t$ is a sufficiently large integer and $c_1=\frac{3k-18k^2+18k^3+24k^4+3k\sqrt{4k - 1}+6k^2\sqrt{4k - 1}-18k^3\sqrt{4k - 1}}{6(2k^2+1)^2} >0$.
Now suppose $\vec{y}=(y_1, y_2, ..., y_t)$ is an optimal vector of $\lambda(G^*(t))$. Let $n$ be large enough. By Remark \ref{remarkblow}, $3$-uniform graph $S_n=(\lfloor ny_1\rfloor, \ldots, \lfloor ny_{t}\rfloor)\otimes G^*(t)$ has density at least $\alpha_k+\frac{c_1}{2t}.$ Since $\pi({\cal F})\le \alpha_k$, some member $F$ of $\cal F$ is a subgraph of $S_n$ for $n$ sufficiently large. For such $F\in \cal F$, there exists a subgraph $M$ of $G^*(t)$ with $|V(M)|\le |V(F)|\leq s$ so that $F\subset (n, n, \ldots, n) \otimes M$. By Fact \ref{mono} and Fact \ref{lblow}, we have
\begin{equation}\label{lambdasmall}
\lambda(F) {{\rm Fact} \ \ref{mono} \atop \leq }\lambda ((n, n, \ldots, n) \otimes M){{\rm Fact} \ \ref{lblow} \atop =} \lambda (M).
\end{equation}
Theorem \ref{theo1} will follow from the following lemma to be proved in Section \ref{prooflemmaresult1}.
\begin{lemma}\label{lemmaresult1}
Let $M$ be any subgraph of $G^*(t)$ with $|V(M)| \leq s$. Then
\begin{equation}
\lambda (M) \leq \frac{1}{6}\alpha_k
\end{equation}
holds.
\end{lemma}
Assuming that Lemma \ref{lemmaresult1} is true and applying Lemma \ref{lemmaresult1} to (\ref{lambdasmall}), we have $$\lambda(F) \le {1 \over 6}\alpha_k$$ which contradicts our choice of $F$, i.e., contradicts that $\displaystyle \lambda(F) >{1 \over 6}\alpha_k$ for all $F \in \cal F$. \hspace*{\fill}$\Box$\medskip
\bigskip
To complete the proof of Theorem \ref{theo1}, what remains is to show Lemma \ref{lemmaresult1}.
\subsection{Proof of Lemma \ref{lemmaresult1}}\label{prooflemmaresult1}
By Fact \ref{mono}, we may assume that $M$ is an induced subgraph of $G^*(t)$. Let $$U_i=V(M)\cap V_i=\{v_1^i, v_2^i, \cdots, v_{s_i}^i\}.$$ So $s=s_1+\cdots+s_{2k+1}$.
Similar to Claim \ref{reduce0}, we have
\begin{claim}\label{reduce} If $N$ is the $3$-uniform graph formed from $M$ by removing the edges contained in $U_{2k+1}$ and inserting the edges $v^{2k+1}_{1}v^{2k+1}_2v^{2k+1}_j$, where $3\leq j \leq s_{2k+1}$, then $\lambda(M)\leq \lambda(N)$.
\end{claim}
By Claim \ref{reduce}, the proof of Lemma \ref{lemmaresult1} will be completed if we show that $\lambda(N)\leq {\alpha_k \over 6}$.
By Lemma \ref{symmetry}, there exists an optimal vector $\vec{z}$ of $\lambda(N)$ such that
\begin{equation} \label{weights}
w(v_1^{2k+1})=w(v_2^{2k+1})\stackrel{\rm\scriptscriptstyle def}{=}\frac{a}{2}, \ \ w(v_3^{2k+1})=w(v_4^{2k+1})=\cdots =w(v^{2k+1}_{s_{2k+1}}) \stackrel{\rm\scriptscriptstyle def}{=}\frac{b}{s_{2k+1}-2},
\end{equation}
where $w(v)$ denotes the component of $\vec{z}$ corresponding to vertex $v$. Let $w_1$ be the sum of the components of $\vec{z}$ corresponding to all vertices in $\cup_{i=1}^{2k}U_i$. Then
\begin{eqnarray*}
&&\lambda(N) \le \bigg(\frac{w_1}{2k}\bigg)^3{2k\choose 3}+\bigg(\frac{w_1}{2k}\bigg)^2{2k\choose 2}(1-w_1)+w_1\bigg(\frac{a^2}{4}+ab+\frac{b^2}{2}\bigg)+\frac{a^2}{4}b,
\end{eqnarray*}
where $w_1+a+b=1.$
Note that if $b\le w_1$ or $a=0$ or $w_1\ge \frac{1}{2}$, then
\begin{eqnarray*}
\lambda(N)&\le&\bigg(\frac{w_1}{2k}\bigg)^3{2k\choose 3}+\bigg(\frac{w_1}{2k}\bigg)^2{2k\choose 2}(1-w_1)+w_1\bigg(\frac{a^2}{2}+ab+\frac{b^2}{2}\bigg)\\
&=&\bigg(\frac{w_1}{2k}\bigg)^3{2k\choose 3}+\bigg(\frac{w_1}{2k}\bigg)^2{2k\choose 2}(1-w_1)+w_1\frac{(1-w_1)^2}{2}\\
&\leq &\lim_{n\to\infty}\lambda(B(2k, n))={\alpha_k\over 6}.
\end{eqnarray*}
So we may always assume that $w_1<\frac{1}{2}$. Since $b=1-w_1-a$, then
\begin{eqnarray*}
\lambda(N)&\le&\bigg(\frac{w_1}{2k}\bigg)^3{2k\choose 3}+\bigg(\frac{w_1}{2k}\bigg)^2{2k\choose 2}(1-w_1)\\
&+&w_1\bigg(\frac{a^2}{4}+a(1-w_1-a)+\frac{(1-w_1-a)^2}{2}\bigg)+\frac{a^2}{4}(1-w_1-a)\triangleq f(a),
\end{eqnarray*}
where $w_1+a<1.$
\begin{eqnarray*}
f'(a)&=&w_1\bigg(\frac{a}{2}+(1-w_1-a)-a-(1-w_1-a)\bigg)+\frac{a}{2}(1-w_1-a)-\frac{a^2}{4}\\
&=&-\frac{3a^2}{4}+\frac{a}{2}-aw_1.
\end{eqnarray*}
Note that $w_1<\frac{1}{2}$, then $f(a)$ is increasing in $[0, \frac{2-4w_1}{3}]$ and decreasing in $[\frac{2-4w_1}{3}, 1]$.
So
\begin{eqnarray*}
f(a)&\le&f(\frac{2-4w_1}{3})\\
&=&\bigg(\frac{w_1}{2k}\bigg)^3{2k\choose 3}+\bigg(\frac{w_1}{2k}\bigg)^2{2k\choose 2}(1-w_1)+\frac{11w_1^3}{54}-\frac{5w_1^2}{9}+\frac{5w_1}{18}+\frac{1}{27}=g(w_1).
\end{eqnarray*}
Then $g'(w_1)=(\frac{1}{4k^2}-\frac{7}{18})w_1^2-(\frac{1}{2k}+\frac{1}{9})w_1+\frac{5}{18}.$ Solving $g'(w_1)=0$, we obtain that $w_1=\frac{\pm\sqrt{\frac{36}{81}+\frac{1}{9k}-\frac{1}{36k^2}}-\frac{1}{9}-\frac{1}{2k}}{\frac{7}{9}-\frac{1}{2k^2}}.$
Note that $\frac{-\sqrt{\frac{36}{81}+\frac{1}{9k}-\frac{1}{36k^2}}-\frac{1}{9}-\frac{1}{2k}}{\frac{7}{9}-\frac{1}{2k^2}}<0$ and $\frac{1}{4k^2}-\frac{7}{18}<0.$ We will show that $\frac{\sqrt{\frac{36}{81}+\frac{1}{9k}-\frac{1}{36k^2}}-\frac{1}{9}-\frac{1}{2k}}{\frac{7}{9}-\frac{1}{2k^2}}>\frac{1}{2}$. It's sufficient to show that $\sqrt{\frac{36}{81}+\frac{1}{9k}-\frac{1}{36k^2}}>\frac{1}{2}+\frac{1}{2k}-\frac{1}{4k^2}$. Note that $$\sqrt{\frac{36}{81}+\frac{1}{9k}-\frac{1}{36k^2}}>\frac{2}{3}>\frac{23}{36}\ge \frac{1}{2}+\frac{1}{2k}-\frac{1}{4k^2}$$
holds for $k\ge 3$. As for $k=2$, we have
$$\sqrt{\frac{36}{81}+\frac{1}{9k}-\frac{1}{36k^2}}=\frac{\sqrt{165}}{18}>\frac{11}{16}=\frac{1}{2}+\frac{1}{4}-\frac{1}{36}.$$
Since earlier discussion allows us to assume that $w_1<\frac{1}{2}$, therefore $g(w_1)$ is increasing in $[0, \frac{1}{2}]$. Note that
$$\frac{11w_1^3}{54}-\frac{5w_1^2}{9}+\frac{5w_1}{18}+\frac{1}{27}\bigg|_{w_1=\frac{1}{2}}=\frac{1}{16}=w_1\frac{(1-w_1)^2}{2}\bigg|_{w_1=\frac{1}{2}}.$$
Then
\begin{eqnarray*}
\lambda(N)&\le& g(\frac{1}{2})\\
&=&\bigg(\frac{w_1}{2k}\bigg)^3{2k\choose 3}+\bigg(\frac{w_1}{2k}\bigg)^2{2k\choose 2}(1-w_1)+\frac{11w_1^3}{54}-\frac{5w_1^2}{9}+\frac{5w_1}{18}+\frac{1}{27}\bigg|_{w_1=\frac{1}{2}}\\
&=&\bigg(\frac{w_1}{2k}\bigg)^3{2k\choose 3}+\bigg(\frac{w_1}{2k}\bigg)^2{2k\choose 2}(1-w_1)+w_1\frac{(1-w_1)^2}{2}\bigg|_{w_1=\frac{1}{2}}\\
&\leq& \lim_{n\to\infty}\lambda(B(2k, n))={\alpha_k\over 6}.
\end{eqnarray*}
\hspace*{\fill}$\Box$\medskip
\bigskip
|
3,212,635,537,461 | arxiv | \section{Introduction}
A large amount of experimental and theoretical studies have been
devoted to studying the unusual properties of high temperature
cuprate superconductors in the past thirty years \cite{Orenstein00,
Lee06, Keimer15, Tsuei00, Damascelli03, Norman05, Uemura04,
Norman11, Scalapino12}. Although some consensuses have been reached,
many fundamental problems are still in debate, including the
microscopic pairing mechanism \cite{Orenstein00, Lee06, Uemura04,
Norman11, Scalapino12}, the origin of pseudogap \cite{Lee06,
Norman05}, and the description of non-Fermi liquid behaviors of the
normal state \cite{Orenstein00,Lee06,Keimer15}. In the past decade,
there have been accumulating experimental evidences for the
existence of a strong anisotropy in many of the physical properties
of YBa$_{2}$Cu$_{3}$O$_{6+\delta}$ (YBCO) \cite{Ando02, Hinkov08,
Daou10} and Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta}$ (BSCCO)
\cite{Lawler10, Fujita14}. Such an anisotropy is widely believed to
be driven by the formation of a novel electronic nematic order
\cite{Kivelson98, Kivelson03, Vojta09,Fradkin10,Fradkin15}, which
spontaneously breaks the $C_4$ symmetry down to a $C_2$ symmetry. In
case the nematic transition line goes across the superconducting
transition line and penetrates into the superconducting dome, there
exists a zero-temperature nematic quantum critical point (QCP). The
nematic quantum phase transition and the associated quantum critical
behaviors have been investigated extensively in recent years
\cite{Vojta00A, Vojta00B, Vojta00C, Kim08, Huh08, Xu08, Fritz09,
Wang11, Liu12,Wang2013NJP1, Wang2013NJP2, She15}.
From a theoretical perspective, there are two widely studied
scenarios to induce an electronic nematic order. First, the nematic
order can be generated by melting a stripe order that spontaneously
breaks both translational and rotational symmetry \cite{Kivelson98,
Kivelson03, Vojta09, Fradkin10, Fradkin15}. The other way is related
to Pomeranchuk instability which refers to the deformation of the
shape of the Fermi surface of a metal due to Coulomb interaction
\cite{Kivelson98, Kivelson03, Vojta09, Fradkin10, Fradkin15,
Pomeranchuk58, Yamase00A, Yamase00B, Halboth00}. In the simplest
case, Pomeranchuk instability occurs when the circular Fermi surface
of a two-dimensional metal becomes ellipse-like via quadrupolar distortion.
\cite{Kivelson98, Kivelson03, Vojta09, Fradkin10,
Fradkin15}. The Hubbard model defined on square lattices
\cite{Halboth00, Zanchi96, Maier05, Gull13, Huscroft01, Macridin06,
Gull10} provides a pertinent platform to investigate the electronic
nematic order. Halboth and Metzner \cite{Halboth00} studied a
two-dimensional Hubbard model using functional renormalization group
(RG) method, and revealed Pomeranchuk instability and nematic order.
More recently, the square lattice Hubbard model is studied by
variational cluster approximation \cite{Fang13, Fang15} and found to
display a local nematic phase under certain circumstances, which
might be applied to understand the intra-unit-cell electronic
nematicity observed in the scanning tunneling spectroscopy (STS)
measurements by Lawler \emph{et al} \cite{Lawler10}.
Another context of studying nematic order is provided by the
superconducting dome of cuprate superconductors. For a pure
$d_{x^2-y^2}$-wave superconductor, the discrete $C_{4}$ symmetry is
preserved. However, when superconductivity coexists with a nematic
order, the gap nodes are shifted from their original positions and
the $C_{4}$ symmetry is broken down to $C_{2}$ \cite{Vojta00A,
Vojta00C, Kim08, Huh08}. Such an anisotropic superconducting state
is physically equivalent to a $d_{x^2-y^2}+s$-wave superdconducting
state \cite{Vojta00A, Vojta00C}. At the nematic QCP, the massless
fermions excited from the $d_{x^2 - y^2}$-wave gap nodes couple
strongly to the quantum critical fluctuation of nematic order
parameter, which can be effectively described by a (2+1)-dimensional
field theory \cite{Vojta00A, Vojta00B, Vojta00C, Kim08, Huh08, Xu08,
Fritz09, Wang11, Liu12, Wang2013NJP1, Wang2013NJP2, She15}. This
model was first analyzed by Vojta \emph{et al.} \cite{Vojta00A,
Vojta00B, Vojta00C}, who made an $\epsilon$-expansion and argued
that the nematic phase transition is turned to first order. Kim
\emph{et al.} \cite{Kim08} later tackled the same model by means of
$1/N$-expansion, where $N$ is a large fermion flavor, and concluded
that the transition remains continuous. Huh and Sachdev \cite{Huh08}
performed a renormalization group (RG) analysis, and showed that the
fermion velocity ratio $v_{\Delta}/v_{F} \rightarrow 0$ in
the lowest energy limit, where $v_{F}$ is the Fermi velocity of
nodal fermions and $v_{\Delta}$ the gap velocity \cite{Vojta00A,
Vojta00B, Vojta00C, Kim08, Huh08, Xu08, Fritz09, Wang11, Liu12,
Wang2013NJP1, Wang2013NJP2, She15}. The unusual velocity
renormalization leads to significant changes of a number of
quantities, including density of states (DOS) \cite{Xu08}, specific
heat \cite{Xu08}, low-$T$ thermal conductivity \cite{Fritz09},
superfluid density \cite{Liu12}, and London penetration depth
\cite{She15}.
In this article, we revisit the issue of unusual physical properties
caused by the strong interaction between massless nodal fermions and
critical nematic fluctuation. We will apply the powerful RG approach
to calculate the fermion damping rate
$\left|\mathrm{Im}\Sigma^R(\omega)\right|$, where
$\mathrm{Im}\Sigma^R(\omega)$ is the imaginary part of retarded
fermion self-energy, and the corresponding quasiparticle residue
$Z_f$. Kim \emph{et al.} \cite{Kim08} have previously computed the
fermion self-energy and spectral function. Interestingly, it will
become clear below that the RG analysis give rise to different
results once the singular renormalization of fermion velocities is
taken into account. In particular, we will show that the fermion
damping rate vanishes upon approaching the Fermi level more rapidly
than the energy $\omega$, namely $\lim_{\omega \rightarrow
0}\left|\mathrm{Im} \Sigma^R(\omega)\right|/\omega \rightarrow 0$.
According to the conventional notion of quantum many-particle
physics, one would expect the system to behave as a normal Fermi
liquid. However, by analyzing the RG results, we find that the
quasiparticle residue vanishes, i.e., $Z_f \rightarrow 0$, in the
limit $\omega \rightarrow 0$. Therefore, the system is actually a
non-Fermi liquid that represents an even weaker violation of Fermi
liquid theory comparing to a marginal Fermi liquid (MFL). To the
best of our knowledge, this type of unconventional non-Fermi liquid
behavior has not been reported previously.
In realistic materials, there are always certain amount of
disorders, which may play an important role. As demonstrated earlier
by Nersesyan \emph{et al.} \cite{Nersesyan94, Nersesyan95}, the most
important disorder in cuprates behaves like a randomly distributed
gauge potential. Thus, we will mainly study the influence of random
gauge potential on the physical properties of nodal fermions and
also the stability of nematic QCP. In the absence of nematic order,
the coupling between nodal fermions and random gauge potential has
attracted considerable interest \cite{Nersesyan94, Nersesyan95,
Altland02}. Here, we consider the case in which nodal fermions
couple to both the nematic order and random gauge potential, and
then study the interplay of these two interactions by means of RG
method. We find that the effective strength of gauge potential
disorders tends to diverge at the lowest energy. This behavior
signals the emergence of a finite zero-energy DOS $\rho(0)$ and the
happening of quantum phase transition from an unconventional
non-Fermi liquid to a disorder dominated diffusive state. The nodal
fermions acquire a finite scattering rate $\gamma$, which in turn
affects the thermodynamic and spectral behaviors of nodal fermions.
The RG results for the self-energy of nodal fermions can be used to
understand a number of experimental facts observed in cuprate
superconductors. We will show that the RG results are qualitatively consistent
with some recent measurements of specific heat, fermion damping
rate, and temperature dependence of fermion velocities.
The rest of the paper will be organized as follows. We first present
the effective low-energy field theory for the interaction between
nematic order and nodal fermions in section~\ref{Sec:ModelDerivation}.
The random gauge potential is also introduced in this section. In
section~\ref{Sec:RGAnalysisClean}, we make detailed theoretical
analysis for the self-consistent RG equations of the fermion
velocities in the clean case. Based on the RG solutions, we proceed
to compute the fermion damping rate, quasiparticle residue, and
other physical quantities. We will show that the quantum critical
fluctuation of nematic order leads to unconventional non-Fermi
liquid behaviors of nodal fermions. We consider the influence of
random gauge potential in section~\ref{Sec:RGAnalysisDisorder} and find
that the effective disorder strength flows to the strong coupling
regime at low energies. Therefore, even weak disorders play a
significant role and drive the system to enter into a disorder
controlled diffusive state. In section~\ref{Sec:CompaisionExperiments},
we discuss the possible application of the RG results to
some experimental findings of cuprates. In section~\ref{Sec:Summary},
we briefly summarize our main results.
\section{Effective field theory\label{Sec:ModelDerivation}}
We start from an effective action $S = S_{\Psi} + S_{\phi} +
S_{\Psi\phi}$. The free action for the nodal fermions is
\cite{Vojta00A,Vojta00B,Vojta00C,Kim08, Huh08}
\begin{eqnarray}
S_{\Psi} &=& \int\frac{d\omega d^{2}\mathbf{k}}{(2\pi)^{3}}
\Psi^{\dagger}_{1a}(-i\omega + v_{F}k_{x} \tau^{z} +
v_{\Delta}k_{y}\tau^x)\Psi_{1a} \nonumber \\
&+& \int\frac{d\omega d^{2}\mathbf{k}}{(2\pi)^{3}}
\Psi^{\dagger}_{2a}(-i\omega + v_{F}k_{y}\tau^{z} +
v_{\Delta}k_{x}\tau^{x})\Psi_{2a},
\end{eqnarray}
where $\tau^{(x,y,z)}$ denote the Pauli matrices. The two component
Nambu spinors are defined by
$\Psi_{1a}=(f_{1a},\epsilon_{ab}f_{3b}^{\dag})^{T}$ and
$\Psi_{2a}=(f_{2a},\epsilon_{ab}f_{4b}^{\dag})^{T}$ where
$\epsilon_{ab}=-\epsilon_{ba}$. $f_{1a}$, $f_{2a}$, $f_{3a}$ and
$f_{4a}$ represent fermions excited from the nodal points $(K,K)$,
$(-K,K)$, $(-K,-K)$, and $(K,-K)$ respectively \cite{Vojta00A,
Vojta00B, Vojta00C, Kim08, Huh08}. The repeated spin index $a$ is
summed from 1 to $N$, where $N$ is the number of fermion spin
components with the physical value being $2$. The action $S_{\phi}$
describes the nematic order parameter, which is expanded in real
space as
\begin{eqnarray}
S_{\phi} = \int d\tau
d^2\mathbf{x}\left[\frac{1}{2}(\partial_{\tau}\phi)^2 +
\frac{c^2}{2}(\nabla\phi)^2 +
\frac{r}{2}\phi^2+\frac{u_0}{24}\phi^4\right],
\end{eqnarray}
where $\tau$ is imaginary time and $c$ velocity of field $\phi$. It
is convenient to choose $c=1$. Mass parameter $r$ tunes a quantum
phase transition from $d_{x^2-y^2}$-wave superconducting state to a
state where the superconducting and nematic orders coexist, with $r
= 0$ defining the QCP. Moreover, $u_0$ is the quartic
self-interaction strength. The nematic order couples to fermions via
a Yukawa-coupling term
\begin{eqnarray}
S_{\Psi\phi} = \lambda_0 \int d\tau d^2\mathbf{x}
\phi\left(\Psi^{\dagger}_{1a}\tau^{x}\Psi_{1a} +
\Psi^{\dagger}_{2a}\tau^{x}\Psi_{2a}\right),
\end{eqnarray}
where $\lambda_{0}$ is the coupling constant. In addition to the
nematic order, the nodal fermions also couple to gauge-potential
type disorders, which is described by
\begin{eqnarray}
S_{\mathrm{dis}} = v_{\Gamma}\int d^2\mathbf{x}A(\mathbf{x})
\left(\Psi_{1a}^{\dag}\Gamma \Psi_{1a} + \Psi_{2a}^{\dag}\Gamma
\Psi_{2a}\right).\label{Eq:ActionDisorder}
\end{eqnarray}
with $\Gamma=(\tau^{z},\tau^{x})$ and
$v_{\Gamma}=(v_{\Gamma1},v_{\Gamma2})$. Here, the random gauge
potential $A(\mathbf{x})$ is assumed to be a Gaussian white noise,
defined by
\begin{eqnarray}
\left<A(\mathbf{x})\right> = 0, \quad
\left<A(\mathbf{x})A(\mathbf{x}')\right> = g\delta^2(\mathbf{x} -
\mathbf{x}'),
\end{eqnarray}
where $g$ is the impurity concentration and $v_{\Gamma}$ measures
the strength of a single impurity.
We will follow Huh and Sachdev \cite{Huh08} and perform RG analysis
by employing $1/N$-expansion. The inverse of free propagator of
$\phi$ behaves as $q^2 + r$. After including the polarization, there
will be an additional linear-in-$q$ term. The $q$-term dominates at
small $q$ over $q^2$-term, so the $q^2$-term can be neglected. Near
the nematic QCP, we keep only the mass term and rescale $\phi
\longrightarrow \phi/\lambda_0$ and $r\longrightarrow N_f r
\lambda^2_0$, leading to
\begin{eqnarray}
S &=& S_{\Psi} + \int d \tau d^2\mathbf{x}
\left\{\frac{N_fr}{2}\phi^2 + \phi\left[\Psi^{\dagger}_{1a}M_{1}
\Psi_{1a} + \Psi^{\dagger}_{2a}M_{2}\Psi_{2a}\right]\right\}.
\end{eqnarray}
The free propagators for fermions $\Psi_{1a}$ and $\Psi_{2a}$ are
written as
\begin{eqnarray}
G_{1a}^{0}(\omega,\mathbf{k}) &=& \frac{1}{-i\omega +
v_{F}k_{x}\tau^{z} + v_{\Delta}k_{y}\tau^{x}}, \\
G_{2a}^{0}(\omega,\mathbf{k}) &=& \frac{1}{-i\omega +
v_{F}k_{y}\tau^{z} + v_{\Delta}k_{x}\tau^{x}},
\end{eqnarray}
respectively. At the nematic QCP, $r=0$, the propagator for the
nematic order field $\phi$ is
\begin{eqnarray}
D(\Omega,\mathbf{q}) = \frac{1}{\Pi(\Omega,\mathbf{q})},
\end{eqnarray}
where the polarization $\Pi(\Omega,\mathbf{q})$, to the leading
order of $1/N$-expansion, is given by
\begin{eqnarray}
\Pi(\Omega,\mathbf{q}) &=& \sum_{i=1,2}\sum_{a=1}^{N}
\int\frac{d\omega}{2\pi}\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}
\mathrm{Tr}\left[\tau^{x} G^{0}_{ia}(\mathbf{k},
\omega)\tau^{x} G^{0}_{ia}(\mathbf{k+q},\omega +
\Omega)\right] \nonumber \\
&=& \frac{N}{16v_{F}v_{\Delta}}
\frac{\Omega^{2}+v_{F}^{2}q_{x}^{2}}{\sqrt{\Omega^2 +
v_{F}^{2}q_{x}^{2}+v_{\Delta}^{2}q_{y}^{2}}} + (q_{x}\leftrightarrow
q_{y}).
\end{eqnarray}
After carrying out RG calculations \cite{Huh08, Wang11}, we find
that all the physical parameters flow with a varying length scale
$l$ as follows
\begin{eqnarray}
\frac{dv_{F}}{dl} &=& \left(C_{1} - C_{2} -
C_{g}\right)v_{F},\label{Eq:VRGVF}
\\
\frac{dv_{\Delta}}{dl} &=& \left(C_{1} - C_{3} -
C_{g}\right)v_{\Delta},\label{Eq:VRGVDelta}
\\
\frac{d(v_{\Delta}/v_{F})}{dl} &=&
(C_{2}-C_{3})\frac{v_{\Delta}}{v_{F}},\label{Eq:VRGVRatio}
\\
\frac{dv_{\Gamma1}}{dl} &=& \left(C_{1} - C_{2} -
C_{g}\right)v_{\Gamma1},\label{Eq:VRGVGammaGP1}
\\
\frac{dv_{\Gamma2}}{dl} &=& \left(C_{1} - C_{3} -
C_{g}\right)v_{\Gamma2},\label{Eq:VRGVGammaGP2}
\end{eqnarray}
where we have introduced a quantity
\begin{eqnarray}
C_g = \frac{v_{\Gamma1}^{2} + v_{\Gamma2}^{2}}{2\pi
v_{F}v_{\Delta}}g.
\end{eqnarray}
The expressions of $C_{1,2,3}$ can be found in the \ref{App:C1C2C3}.
We emphasize that the impact of disorders is characterized by the
quantity $C_{g}$, rather than $v_{\Gamma i}$ with $i=1,2$. It will be shown below
that $C_{g}$ flows to strong coupling at large $l$, leading to
remarkable physical properties.
\section{Unconventional non-Fermi liquid behaviors of nodal fermions
\label{Sec:RGAnalysisClean}}
It is well established that conventional metals can be described by
the standard Fermi liquid theory, which states that the fermionic
excitations of a normal Fermi liquid must have a sufficiently long
lifetime and exhibit a sharp quasiparticle peak in their spectral
peak despite the existence of Coulomb interaction \cite{Giuliani05}.
The conventional notion is that the fermionic quasiparticles
constitute a normal Fermi liquid if their zero-$T$ damping rate
$\left|\mathrm{Im}\Sigma^R(\omega)\right|$ vanishes more rapidly
than $\omega$ in the limit $\omega \rightarrow 0$. This criterion
can be mathematically expressed as
\begin{eqnarray}
\lim_{\omega \rightarrow 0}\frac{\mathrm{Im}
\Sigma^{R}(\omega)}{\omega} \rightarrow 0.
\label{Eq:DampingTradi}
\end{eqnarray}
Another criterion to identify normal Fermi liquid is to define an
important quantity: the quasiparticle residue, also called
renormalization factor, $Z_f$. The residue $Z_f$ is usually
calculated through the definition
\begin{eqnarray}
Z_{f}(\omega) = \frac{1}{\left|1 - \frac{\partial}{\partial\omega}
\mathrm{Re}\Sigma^{R}(\omega)\right|},\label{Eq:ZDefA}
\end{eqnarray}
where the real part of retarded fermion self-energy
$\mathrm{Re}\Sigma^{R}(\omega)$ is related to
$\mathrm{Im}\Sigma^{R}(\omega)$ via the Kramers-Kronig (K.-K.)
relationship. The residue $Z_f$ is finite in a normal Fermi liquid,
but vanishes in a non-Fermi liquid. Generically, the fermion damping
rate in an interacting fermion system can be formally written as
\begin{eqnarray}
\mathrm{Im}\Sigma^{R}(\omega) = C_F |\omega|^x,
\end{eqnarray}
where $C_F$ is a constant. With the help of K.-K. relation, the
corresponding real part of the fermion self-energy, in the low
energy limit, is given by
\begin{eqnarray}
\mathrm{Re}\Sigma^{R}(\omega)=\left\{\begin{array}{lll}
C_F\mathrm{sgn}(\omega)|\omega|^{x}I(x), &\mathrm{if}& 0 < x < 1, \\
C_F\frac{2}{\pi}\omega\ln\left(\frac{\omega_0}{|\omega|}\right), &
\mathrm{if} & x=1 ,\\
C_F\frac{2}{\pi}\frac{|\omega_0|^{x-1}}{x-1}\omega, & \mathrm{if} &
x > 1,
\end{array}\right.
\end{eqnarray}
where $\omega_{0}$ is a cutoff, and $I(x)$ is a function that
depends only on $x$.
It can be easily checked that $Z_f = 0$ for $0 < x \leq 1$ and $Z_f
\neq 0$ for $x > 1$. Therefore, the above two criteria are actually
equivalent because $Z_f$ automatically takes a finite value whenever
equation (\ref{Eq:DampingTradi}) is fulfilled.
\begin{figure}[htbp]
\center
\includegraphics[width=2.8in]{VRGVF.eps}
\includegraphics[width=2.8in]{VRGVDelta.eps}
\includegraphics[width=2.8in]{VRGVRatio.eps}
\caption{(a), (b), and (c) show how $v_{F}$, $v_{\Delta}$, and
$v_{\Delta}/v_{F}$ flow with a varying length scale $l$ respectively
in the clean case. The initial values of velocity ratio are
$v_{\Delta0}/v_{F0} = 0.075, 0.2, 1, 2$.\label{Fig:VRGVelocity}}
\end{figure}
However, in the present nodal fermion system, the above two criteria
are no longer equivalent. We will show by explicit calculations in
this section that the damping rate of nodal fermions vanishes more
rapidly than the energy, but the residue $Z_f$ vanishes, i.e., $Z_f
\rightarrow 0$.
\subsection{Fermion damping rate and quasiparticle residue}
Now we calculate the fermion damping rate and the residue $Z_f$
utilizing the solutions of the RG equations
(\ref{Eq:VRGVF})-(\ref{Eq:VRGVGammaGP2}). The unusual
renormalization of fermion velocities need to be taken into account
in an appropriate manner. We only present the results obtained in
the clean limit $C_{g} = 0$ in this section, and include the effect
of random gauge potential in the next section.
After solving RG equations (\ref{Eq:VRGVF})-(\ref{Eq:VRGVRatio})
self-consistently, we show the $l$-dependence of fermion velocities
$v_{F}$, $v_{\Delta}$ and ratio $v_{\Delta}/v_{F}$ obtained
at different initial values of ratio $v_{\Delta0}/v_{F0}$
in figures~\ref{Fig:VRGVelocity}(a), (b) and (c) respectively. All
the quantities $v_{F}$, $v_{\Delta}$, and $v_{\Delta}/v_{F}$
decrease with growing $l$ and flow eventually to zero as $l
\rightarrow +\infty$, but apparently $v_{F}$ decreases much more
slowly than $v_{\Delta}$. If we use the ratio $v_{\Delta}/v_{F}$ to
characterize the velocity anisotropy, it is clear that the nematic
order drives an extreme velocity anisotropy $v_{\Delta}/v_{F}
\rightarrow 0$. For later use, it is helpful to extract an
approximate analytical expressions for the velocities. Considering
the leading and sub-leading terms, the solution for
$v_{\Delta}/v_{F}$ is given by
\begin{eqnarray}
\frac{v_{\Delta}}{v_{F}} \sim \frac{\pi^2N}{8}\frac{1}{l\ln(c_{s}l)}=
\frac{\pi^2N}{8}\frac{1}{l\left[\ln(l)+\ln\left(c_{s}\right)\right]},
\end{eqnarray}
with $c_{s} \approx 0.3809/N$, which is consistent with the
expression in reference~\cite{Huh08}. In the long wavelength limit,
$l\rightarrow \infty$, $\ln(c_{s})/\ln(l)\rightarrow 0$, so the
sub-leading term can be ignored. Retaining only the leading term,
the asymptotic behavior of velocity ratio can be well approximated
by
\begin{eqnarray}
\frac{v_{\Delta}}{v_{F}} \sim \frac{\pi^2N}{8}\frac{1}{l\ln(l)}.
\label{Eq:VRatioApproximate}
\end{eqnarray}
Substituting the asymptotic form of $v_{\Delta}/v_{F}$
into equation~(\ref{Eq:VRGVF}), we obtain
\begin{eqnarray}
\frac{dv_{F}}{dl} \sim - \frac{\pi^2 c_{1}}{8}\frac{v_{F}}{l\ln(l)}
\end{eqnarray}
with $c_{1} \approx 0.078$ for large $l$. Solving this equation, we
express the renormalized $v_{F}$ as
\begin{eqnarray}
v_{F}(l) \sim \frac{1}{(\ln l)^{c_{2}}}\label{Eq:VFAsoBehavior}
\end{eqnarray}
with $c_{2} =\frac{\pi^2 c_{1}}{8}$. It can also be found that
$v_{\Delta}$ at large $l$ behaves approximately as
\begin{eqnarray}
v_{\Delta}(l) \sim \frac{\pi^2N}{8}\frac{1}{l\left[\ln(l)\right]^{1
+ c_{2}}}.\label{Eq:VDeltaAsoBehavior}
\end{eqnarray}
According to equations~(\ref{Eq:VFAsoBehavior}) and
(\ref{Eq:VDeltaAsoBehavior}), both $v_{F}$ and $v_{\Delta}$ vanish
as $l \rightarrow +\infty$.
To examine the impact of singular fermion velocity renormalization
and extreme velocity anisotropy on the properties of nodal fermions,
we will compute a number of physical quantities. Coming first is the
quasiparticle residue $Z_f$. Apart from the widely used definition
given by equation~(\ref{Eq:ZDefA}), the residue $Z_f$ can also be
calculated within the RG framework. The interaction induced
renormalization of fermion field $\Psi$ is encoded in the residue
$Z_f$, which exhibits the following $l$-dependence
\begin{eqnarray}
Z_{f} = e^{\int_{0}^{l}\left(C_{1}-C_{g}\right)dl'}
\end{eqnarray}
or alternatively
\begin{eqnarray}
\frac{dZ_{f}}{dl} = \left(C_{1}-C_{g}\right)Z_{f}.\label{Eq:ZFlowDef2}
\end{eqnarray}
\begin{figure}[htbp]
\center
\includegraphics[width=2.8in]{ZA.eps}
\includegraphics[width=2.8in]{ZB.eps}
\caption{$l$-dependence of $Z_f$ in the clean case is presented in
(a), and of $Z_{f}l$ in (b). The initial values of velocity ratio
are $v_{\Delta0}/v_{F0} = 0.075, 0.2, 1, 2$. \label{Fig:VRGZf}}
\end{figure}
The $l$-dependence of $Z_{f}$ can be easily obtained from the above
equation, and is presented in figure~\ref{Fig:VRGZf}(a). We find that
$Z_{f}$ flows to zero in the limit $l \rightarrow +\infty$, which
indicates the breakdown of normal Fermi liquid and the absence of
well-defined Landau quasiparticles. However, $Z_{f}$ decreases very
slowly with growing $l$, thus the deviation of the system from a
normal Fermi liquid ought to be quite weak. To see this point, we
plot the $l$-dependence of $Z_{f}l$ in figure~\ref{Fig:VRGZf}(b), and
find that
\begin{eqnarray}
\lim_{l\rightarrow +\infty} Z_{f}l \rightarrow +\infty.
\end{eqnarray}
Therefore, the residue $Z_{f}$ obtained at nematic QCP vanishes with
growing $l$ more slowly than that of a MFL \cite{Varma}, where
$Z_{f} \sim 1/l$. According to equation~(\ref{Eq:ZDefA}), one can
speculate that
\begin{eqnarray}
\lim_{\omega \rightarrow 0} \frac{\mathrm{Re}
\Sigma_{\mathrm{nem}}^{R}(\omega)}{\omega \ln(\omega_{0}/\omega)}
\rightarrow 0,
\end{eqnarray}
since a large length scale $l$ corresponds to a low energy $\omega$.
This speculation in turn implies that the imaginary part of the
retarded fermion self-energy
$\mathrm{Im}\Sigma_{\mathrm{nem}}^{R}(\omega)$ should display the
following low-energy behavior,
\begin{eqnarray}
\lim_{\omega \rightarrow 0}
\frac{\mathrm{Im}\Sigma_{\mathrm{nem}}^{R}(\omega)}{\omega}
\rightarrow 0.
\end{eqnarray}
To make the above discussions more quantitative, we are now going to
make a detailed analysis of the asymptotic behavior of $Z_{f}$. At
large running scale $l$, the equation of $Z_{f}$ can be approximated
by
\begin{eqnarray}
\frac{dZ_{f}}{dl} \sim -\frac{c_{4}}{N}\frac{v_{\Delta}}{v_{F}}Z_{f}
\sim -\frac{c_{5}}{l\ln l}Z_{f},
\end{eqnarray}
where $c_{4} \approx 0.426$ and $c_{5} = \frac{\pi^2 c_{4}}{8}\approx0.523$.
Solving this equation leads to the asymptotic behavior
\begin{eqnarray}
Z_{f} \sim \frac{1}{\left(\ln l\right)^{c_{5}}} \rightarrow 0
\end{eqnarray}
in the limit $l \rightarrow +\infty$. To proceed, it proves
convenient to utilize the relationship between $\omega$ and $l$:
\begin{eqnarray}
\omega = \omega_{\mathrm{0}}e^{-l},
\end{eqnarray}
where $\omega_{\mathrm{0}}$ is a cutoff. Then the real part of
retarded self-energy is approximated by
\begin{eqnarray}
\mathrm{Re}\Sigma_{\mathrm{nem}}^{R}(\omega) &\sim&
\omega\left\{\ln\left[\ln\left(\frac{\omega_{0}}{\omega}
\right)\right]\right\}^{c_{5}}\label{Eq:RealPartSelfEnergyNematicQCP}
\end{eqnarray}
as $\omega \rightarrow 0$. Using the K.-K. relation, we obtain the
imaginary part of self-energy
\begin{eqnarray}
\mathrm{Im}\Sigma_{\mathrm{nem}}^{R}(\omega) &\sim&
\frac{\pi}{2}\frac{|\omega|}{\ln\left(\frac{\omega_{0}}{\omega}\right)
\left\{\ln\left[\ln\left(\frac{\omega_{0}}{\omega}
\right)\right]\right\}^{1-c_{5}}}. \label{Eq:DampingNematicQCPCleanAppro}
\end{eqnarray}
From equation (\ref{Eq:DampingNematicQCPCleanAppro}), it is easy to
verify that this fermion damping rate is smaller than that in a MFL
and manifests the asymptotic behavior $\lim_{\omega \rightarrow 0}
\mathrm{Im}\Sigma_{\mathrm{nem}}^{R}(\omega)/\omega \rightarrow 0$,
confirming the above analysis based on numerical results. According
to the conventional notion of quantum many-body physics, one would
expect the system to behave like a normal Fermi liquid. However, in
the low energy limit $\omega\rightarrow0$, the residue $Z_{f}$
actually vanishes:
\begin{eqnarray}
Z_{f}\sim\frac{1}{\left\{\ln\left[\ln\left(\frac{\omega_{0}}{\omega}
\right)\right]\right\}^{c_{5}}}\rightarrow 0,
\label{Eq:ZNematicQCPCleanAppro}
\end{eqnarray}
which clearly implies that the system under consideration is
actually a non-Fermi liquid and the nodal fermions do not have a
well-defined quasiparticle peak in the spectral function. We see
that the above residue $Z_{f}$ flows to zero at a lower speed than
that of a MFL, i.e., $Z_{f} \sim
\frac{1}{\ln(\omega_{0}/\omega)}$. Therefore, the
quantum critical fluctuation of nematic order gives rise to an even
weaker violation of ordinary Fermi liquid theory than a MFL. To the
best of our knowledge, this sort of unconventional non-Fermi liquid
state has not been reported previously.
It is now interesting to compare this unconventional non-Fermi
liquid state with graphene. In graphene, early perturbative
calculations revealed that Dirac fermions exhibit MFL behavior with
damping rate $\mathrm{Im}\Sigma^{R}(\omega) \sim \omega$ and
vanishing $Z_f$ \cite{Gonzalez99}. Nevertheless, subsequent careful
RG studies \cite{Gonzalez99, Kotov12, WangLiu14, Hofmann14} have
found that, the fermion damping rate actually depends on energy as
$\mathrm{Im}\Sigma^{R}(\omega) \propto \omega/\ln^2\omega$ at zero
temperature due to the long-range Coulomb interaction, whereas the
corresponding $Z_{f}$ flows to a finite value as $\omega \rightarrow
0$. Therefore, graphene is a normal Fermi liquid. In contrast, the
massless nodal fermions constitute a non-Fermi liquid at the nematic
QCP, because $Z_{f}$ vanishes in the lowest energy limit. The
crucial difference between these two cases can be understood as
follows. At nematic QCP, the fermion velocities are driven to vanish
by the nematic order, so the effective fermion-nematic interaction
is significantly pronounced. In graphene, however, the fermion
velocity is dramatically enhanced at low energies by Coulomb
interaction, which then weakens the effective Coulomb interaction
and guarantees the validity of the Fermi liquid description. These
two interacting Dirac fermion systems provide interesting new
insight on the effects of strong electron correlations and also on
the criterion of non-Fermi liquid states.
Moreover, an important lesson one can learn from the research
experience of graphene is that, RG may lead to qualitatively
different spectral properties of fermions from that obtained by
ordinary perturbative expansion approach. Indeed, this is the main
motivation that has promoted us to make an extensive RG analysis of
the spectral properties of nodal fermions at the nematic QCP.
\begin{figure}[htbp]
\center
\includegraphics[width=2.85in]{RhoClA.eps}
\includegraphics[width=2.8in]{RhoClB.eps}
\caption{Fermion DOS $\rho(\omega)$ in the clean limit at different
initial values $v_{\Delta0}/v_{F0}=0.075, 0.2, 1, 2.$
\label{Fig:RhoCl}}
\end{figure}
\subsection{Density of states and specific heat}
We have showed in the last subsection that the nodal fermions
exhibit unconventional non-Fermi liquid behaviors at the nematic
QCP. In this subsection, we will compute two important quantities,
namely DOS and specific heat, on the basis of the RG solutions with
the goal to gain a better understanding of the non-Fermi liquid
state.
The fermion DOS can calculated from the retarded Green functions of
nodal fermions via the definition \cite{Xu08}
\begin{eqnarray}
\fl \rho(\omega)&=&\sum_{a=1}^{N}\int\frac{dk_{x}dk_{y}}{(2\pi)^{2}}
\frac{1}{\pi}\mathrm{Tr}\left[\mathrm{Im}G_{1a}^{R}
(v_{F}k_{x},v_{\Delta}k_{y},\omega) +
\mathrm{Im}G_{2a}^{R}(v_{F}k_{x},v_{\Delta}k_{y},\omega) \right]
\nonumber \\
\fl &=& \frac{1}{v_{\Delta}v_{F}}\sum_{a=1}^{N}
\int\frac{dk_{x}'dk_{y}'}{(2\pi)^{2}}
\frac{1}{\pi}\mathrm{Tr}\left[\mathrm{Im}G_{1a}^{R}(k_{x}',k_{y}',\omega)+
\mathrm{Im}G_{2a}^{R}(k_{x}',k_{y}',\omega) \right].
\end{eqnarray}
In the absence of interactions, the fermion DOS is well-known to be
linear in $\omega$, namely $\rho(\omega)\propto \omega$. This linear
behavior will be changed once the interaction effects are
considered. After including the fermion velocity renormalization,
employing the method presented in \cite{Xu08}, the flow equation for
$\rho(\omega)$ is given by
\begin{eqnarray}
\frac{d\ln\rho}{d\ln\omega} &=&
\left\{\begin{array}{ll}
\frac{1 + C_{1} - C_{2} - C_{3} -
C_{g}}{1 - C_{1} + C_{2} + C_{g}}, & \mathrm{if}\quad v_{\Delta}<v_{F},
\\
\\
\frac{1 + C_{1} - C_{2} - C_{3} - C_{g}}{1 - C_{1} + C_{3} + C_{g}},
& \mathrm{if}\quad v_{\Delta} > v_{F}.
\end{array}\right.
\label{Eq:RhoRGEq}
\end{eqnarray}
In cuprates, the initial value of velocity ratio
$v_{\Delta0}/v_{F0}$ is known to be much smaller than $1$
\cite{Chiao00}. Additionally, due to the quantum fluctuation of
nematic order, $v_{\Delta}/v_{F}$ decreases monotonously as the
energy scale is lowering. Therefore, for a given initial value
$v_{\Delta0}/v_{F0} < 1$, we only need to consider the case
$v_{\Delta} < v_{F}$. In order to show that the conclusion is
independent of the condition $v_{\Delta0}/v_{F0} < 1$, we also plot
the curves for the case $v_{\Delta0}/v_{F0} = 2$ in figures \ref{Fig:RhoCl} and \ref{Fig:CvCl}.
The RG equations of DOS and specific heat are also given with a
generalized form. In the clean limit, we plot the results for
$\rho(\omega)$ in figure~\ref{Fig:RhoCl}. We see from
figure~\ref{Fig:RhoCl}(a) that $\rho(\omega)$ is apparently not
linear in $\omega$, but displays the asymptotic behavior
\begin{eqnarray}
\lim_{\omega \rightarrow 0} \frac{\rho(\omega)}{\omega} \rightarrow
+\infty.
\end{eqnarray}
\begin{figure}[htbp]
\center
\includegraphics[width=2.85in]{CvClA.eps}
\includegraphics[width=2.8in]{CvClB.eps}
\caption{Specific heat $C_{V}(T)$ in the clean limit at different
initial values $v_{\Delta0}/v_{F0}=0.075, 0.2, 1, 2.$
\label{Fig:CvCl}}
\end{figure}
On the other hand, figure \ref{Fig:RhoCl}(b) implies that
\begin{eqnarray}
\lim_{\omega \rightarrow 0}
\frac{\ln(\rho(\omega)/\rho_{0})}{\ln(\omega/\omega_{0})}
\rightarrow 1.
\end{eqnarray}
This DOS $\rho(\omega)$ is qualitatively different from the power
law function $\rho(\omega)\sim\omega^{1-\alpha}$, where $\alpha$ is
a small finite value, obtained previously in reference \cite{Xu08}.
In the non-interacting limit, the fermion specific heat depends on
$T$ as $C_{V}(T)\propto T^2$. As shown in \ref{App:SpecificHeat},
including the influence of the quantum fluctuation of nematic order
and random gauge potential leads us to
\begin{eqnarray}
\frac{d \ln C_{V}}{d\ln T} &=& \left\{\begin{array}{ll} 2 +
\frac{2C_{1} - C_{2} - C_{3} - 2C_{g}}{1 - C_{1} + C_{2} + C_{g}}, &
\mathrm{if}\quad v_{\Delta}<v_{F},
\\
\\
2 + \frac{2C_{1} - C_{2} - C_{3} -
2C_{g}}{1 - C_{1} + C_{3} + C_{g}}, & \mathrm{if}\quad v_{\Delta}>v_{F}.
\end{array}\right.\label{Eq:CvRGEq}
\end{eqnarray}
At low $T$, $C_{V}(T)$ behaves as
\begin{eqnarray}
\lim_{T \rightarrow 0} \frac{C_{V}(T)}{T^{2}} \rightarrow +\infty,
\end{eqnarray}
which is visualized in figure \ref{Fig:CvCl}(a). Apparently, the
original quadratic $T$-dependence of $C_{V}(T)$ obtained in the
non-interacting limit is significantly altered. According to figure
\ref{Fig:CvCl}(b), we can express $C_{V}(T)$ at very small $T$ as
\begin{eqnarray}
\lim_{T \rightarrow 0}\frac{\ln(C_{V}(T)/C_{V0})}{\ln(T/T_{0})}
\rightarrow 2,
\end{eqnarray}
which is also distinct from the power law $T$-dependence
$C_{V}(T)\sim T^{2-\beta}$, where $\beta$ is a small finite
constant, obtained previously \cite{Xu08}.
Simple analysis reveal that the three parameters $C_{1}$, $C_{2}$,
and $C_{3}$ appearing in the RG equations (\ref{Eq:RhoRGEq}) and
(\ref{Eq:CvRGEq}) all flow to zero in the lowest energy limit
\cite{Huh08}. This asymptotic behavior makes it impossible to
express $\rho(\omega)$ and $C_V(T)$ by power law functions. As given
in \ref{App:ApproDOSCv}, the low energy behavior of $\rho(\omega)$
and the low-$T$ behavior of $C_V(T)$ can be approximately expressed
as
\begin{eqnarray}
\fl \rho(\omega)&\sim&\left(\frac{\omega}{\omega_{0}}\right)
\ln\left(\frac{\omega_{0}}{\omega}\right)
\left(\ln\ln\left(\frac{\omega_{0}}{\omega}\right)\right)^{a_{\rho}}\exp
\left[\frac{1}{2}\left(\ln\ln\ln
\left(\frac{\omega_{0}}{\omega}\right)\right)^{2}\right],
\\
\fl C_{V}(T) &\sim& \left(\frac{T}{T_{0}}\right)^{2}
\ln\left(\frac{T_{0}}{T}\right)
\left(\ln\ln\left(\frac{T_{0}}{T}\right)\right)^{a_{C}} \exp
\left[\frac{1}{2}\left(\ln\ln\ln
\left(\frac{T_{0}}{T}\right)\right)^{2}\right],
\end{eqnarray}
where $a_{\rho}$ and $a_{C}$ are two negative constants. At $N=2$,
$a_{\rho} \approx -1.896$ and $a_{C} \approx -1.466$.
We notice that our RG equation for $C_V(T)$ (\ref{Eq:CvRGEq}) is not
exactly the same as that obtained by Xu \emph{et al.} \cite{Xu08}.
This difference does not affect our conclusion that $C_V(T)$ is not
a power law function at low $T$. Indeed, if we start from the RG
equation of $C_V(T)$ presented in \cite{Xu08}, we reach the same
conclusion. A more detailed discussion is presented in
\ref{App:SpecificHeat} and \ref{App:ApproDOSCv}.
\section{Impact of Random gauge potential\label{Sec:RGAnalysisDisorder}}
In this section, we investigate the impact of random gauge potential
on the quantum critical behaviors near the nematic QCP. We will
first show the RG solutions obtained in the presence of random gauge
potential, and then re-calculate the fermion DOS and specific heat
after considering the influence of disorder scattering. To this end,
we need to retain a nonzero $C_g$ in the RG equations
(\ref{Eq:VRGVF})-(\ref{Eq:VRGVGammaGP2}), (\ref{Eq:ZFlowDef2}),
(\ref{Eq:RhoRGEq}), and (\ref{Eq:CvRGEq}), and then solve these RG
equations numerically.
\begin{figure*}[htbp]
\center
\includegraphics[width=2.7in]{VRGVFGP.eps}
\hspace{1.5cm}
\includegraphics[width=2.7in]{VRGVDeltaGP.eps}
\includegraphics[width=2.7in]{VRGVRatioGP.eps}
\hspace{1.5cm}
\includegraphics[width=2.7in]{ZfGP.eps}
\includegraphics[width=2.7in]{RhoGP.eps}
\hspace{1.5cm}
\includegraphics[width=2.7in]{CvGP.eps}
\caption{Flows of the quantities $v_{F}$, $v_{\Delta}$,
$v_{\Delta/v_{F}}$, $Z_{f}$, $\rho$, $C_{V}$ are shown in (a), (b),
(c), (d), (e), and (f) respectively at the initial value
$v_{\Delta0}/v_{F0}=0.075$ in the presence of random gauge
potential. We have chosen $v_{\Gamma0}^{2}g/v_{F0}^{2} = 0,006,
0.008, 0.01$ and $v_{\Gamma10} = v_{\Gamma20} = v_{\Gamma0}$ for
random gauge potential. \label{Fig:VRGGP}}
\end{figure*}
\subsection{Flow of effective disorder strength}
In order to make a direct comparison to the clean case in which $C_g
= 0$ and to explicitly see the influence of a nonzero $C_g$ on the
running behaviors of various parameters, we plot the $l$-dependence
of $v_{F}$, $v_{\Delta}$, $v_{\Delta}/v_{F}$, and $Z_{f}$ in
figures~\ref{Fig:VRGGP}(a)-(d). Moreover, we show the
$\omega$-dependence of $\rho(\omega)$ in figure~\ref{Fig:VRGGP}(e) and
the $T$-dependence of $C_{V}$ in figure~\ref{Fig:VRGGP}(f),
respectively. We will discuss under what circumstances these RG
results are modified in the next subsection.
Comparing figures~\ref{Fig:RhoCl}(a) and (b) with figures
\ref{Fig:VRGGP}(a) and (b), we see that the detailed $l$-dependence
of $v_{F}$ and $v_{\Delta}$ are both altered dramatically by the
random gauge potential. As shown in figure~\ref{Fig:RhoCl}(c) and
figure~\ref{Fig:VRGGP}(c), the velocity ratio $v_{\Delta}/v_{F}$
exhibits exactly the same $l$-dependence in the clean and disordered
cases, which originates from the fact that $C_g$ does not enter into
the RG equation of velocity ratio $v_{\Delta}/v_{F}$. According to
figure~\ref{Fig:VRGZf}(a) and figure~\ref{Fig:VRGGP}(d), the
renormalization factor $Z_{f}$ flows to zero in the disordered case
much more rapidly than the clean case.
As shown in figure~\ref{Fig:VRGGP}(e), the DOS $\rho(\omega)$ is
divergent in the lowest energy limit due to random gauge potential.
This is completely different from the behaviors of clean case
presented in figure~\ref{Fig:RhoCl}. In figure~\ref{Fig:VRGGP}(f), we
plot the ratio between $C_{V}^{\mathrm{Dis}}/C_{V}^{\mathrm{Cl}}$,
where $C_{V}^{\mathrm{Dis}}$ and $C_{V}^{\mathrm{Cl}}$ are the
specific heat obtained in disordered and clean cases, respectively.
Since the solutions of the RG equations about DOS and specific heat
are modified substantially in the disordered case, it turns out that
random gauge potential is a relevant perturbation in the present
system. To verify the relevance of random gauge potential, we should
appeal to the RG analysis of the effective disorder strength.
In the action term $S_{\mathrm{dis}}$ given in equation
(\ref{Eq:ActionDisorder}), the parameters that characterize the
fermion-disorder coupling seem to be $v_{\Gamma1}$ and
$v_{\Gamma2}$. It can be seen from the RG equations that
$v_{\Gamma1}$ flows in precisely the same way as $v_{F}$, and that
$v_{\Gamma2}$ as $v_{\Delta}$. Thus, both $v_{\Gamma1}$ and
$v_{\Gamma2}$ are strongly renormalized and driven to vanish as $l
\rightarrow +\infty$. However, this does not mean that the disorders
can be simply neglected. Indeed, whether disorders are important is
determined by the ratio between the interaction energy given by
$S_{\mathrm{dis}}$ and the fermion energy $E_{k} \propto
\left(v_{F}^{2}k_{x}^{2} + v_{\Delta}^{2}k_{y}^{2}\right)^{1/2}$. We
can see that this ratio is defined by $C_{g}$, which enters into the
RG equations for the parameters $v_{F}$, $v_{\Delta}$,
$v_{\Gamma1}$, and $v_{\Gamma2}$. The effective strength of random
gauge potential should be measured by $C_{g}$, rather than
$v_{\Gamma1}$ and $v_{\Gamma2}$. Recall that $C_{g}$ is a function
of five parameters, i.e., $v_{F}$, $v_{\Delta}$, $v_{\Gamma1}$,
$v_{\Gamma2}$, and $g$. Among these parameters, $g$ is assumed to be
a dimensionless constant, but the other four parameters flow
strongly with the varying $l$. Detailed RG analysis revealed that
$C_g$ goes to infinity, namely
\begin{eqnarray}
C_g &=& \frac{g}{2\pi}\left(\frac{v_{\Gamma10}^{2}}{v_{F0}^{2}}
\frac{v_{F}}{v_{\Delta}} + \frac{v_{\Gamma20}^{2}}{v_{\Delta0}^{2}}
\frac{v_{\Delta}}{v_{F}}\right) \rightarrow +\infty,
\end{eqnarray}
in the limit $l \rightarrow +\infty$ as a consequence of the
singular renormalization of the fermion velocities ratio
$v_{\Delta}/v_{F}$. More quantitatively, the large scale behavior of
$C_{g}$ can be described by
\begin{eqnarray}
C_g \sim c_{3}l\ln l,
\end{eqnarray}
where $c_{3} = \frac{4g}{\pi^3 N}
\frac{v_{\Gamma10}^{2}}{v_{F0}^{2}}$.
It is necessary to explain here why the effective strength of random
gauge potential should be represented by $C_{g}$, rather than solely
by the coefficients $v_{\Gamma i}$ with $i = 1,2$, and why $C_{g}
\rightarrow \infty$ in the lowest energy limit. In an interacting
fermion system, the effective strength of the interaction is
characterized by the ratio between the interaction energy scale and
the kinetic energy of fermions. This ratio is widely used in
condensed matter physics to judge whether an interacting fermion
system can be defined as a strongly correlated system or not. For
instance, the normal metal with a high density of itinerant
electrons is believed to be a weakly interacting system since the
energy scale of Coulomb potential is much smaller than the Fermi
energy. In a massless Dirac fermion system (such as graphene), the
effective strength of long-range Coulomb interaction is defined as
$\alpha\sim\frac{e^2}{v_{F}}$, where $e^2$ appears in the action of
the Coulomb interaction as a coupling coefficient and the fermion
velocity $v_{F}$ reflects the order of the kinetic energy
\cite{Kotov12, Hofmann14}. Another example comes from the effective
BCS model of Dirac fermion systems, where the effective strength of
pairing interaction is characterized by $g\sim \frac{u}{v_{F}}$ with
$u$ being the coupling coefficient of pairing interaction and
$v_{F}$ fermion velocity \cite{Kopnin08, Nandkishore13}.
This criterion also applies to disordered systems. When massless
Dirac fermions couple to random gauge potential, the effective
strength of random gauge potential should be defined by $C_{g} \sim
\frac{v_{\Gamma i}^2}{v_{F}v_{\Delta}}$, rather than the coefficient
of fermion-disorder coupling $v_{\Gamma i}$ \cite{Nersesyan94,
Nersesyan95}. When $v_{\Gamma i}$ flows to zero in the lowest energy
limit, $C_{g}$ does not necessarily vanish since there is a
possibility that $v_{F}$ and $v_{\Delta}$ may vanish more (or
equally) rapidly than $v_{\Gamma i}$. When the nodal fermions couple
to both the quantum fluctuation of nematic order and random gauge
potential, the four parameters $v_{F}$, $v_{\Delta}$, $v_{\Gamma1}$,
and $v_{\Gamma2}$ all flow to zero in the lowest energy limit, but
the effective strength of random gauge potential becomes very large,
namely $C_{g} \rightarrow \infty$. This originates from the fact
that $\frac{v_{\Gamma1}}{v_{F}} = \frac{v_{\Gamma10}}{v_{\Gamma20}}$
and $\frac{v_{\Gamma2}}{v_{\Delta}} =
\frac{v_{\Gamma20}}{v_{\Delta0}}$ are constants, while at the same
time the velocity ratio $v_{\Delta}/v_{F} \rightarrow 0$. The
behavior $C_{g}\rightarrow \infty$ at low energies indicates that
random gauge potential is relevant. To see this point, we neglect
the nematic order and consider only the coupling of nodal fermions
to random gauge potential, which results in simplified RG equations:
\begin{eqnarray}
\fl \frac{dv_{F}}{dl} = -C_{g}v_{F},\,\frac{dv_{\Delta}}{dl} =
-C_{g}v_{F},\,\frac{d(v_{F}/\Delta)}{dl} =
0,\,\frac{dv_{\Gamma1}}{dl} = -C_{g}v_{F},\, \frac{dv_{\Gamma2}}{dl}
= -C_{g}v_{F}.
\end{eqnarray}
Using the two relations $\frac{v_{\Gamma1}}{v_{F}} =
\frac{v_{\Gamma10}}{v_{\Gamma20}}$ and
$\frac{v_{\Gamma2}}{v_{\Delta}} = \frac{v_{\Gamma20}}{v_{\Delta0}}$,
it is easy to show that
\begin{eqnarray}
C_{g} = \frac{g}{2\pi}\frac{v_{\Gamma1}^{2} +
v_{\Gamma2}^{2}}{v_{F}v_{\Delta}} =
\frac{g}{2\pi}\frac{v_{\Gamma10}^{2} +
v_{\Gamma20}^{2}}{v_{F0}v_{\Delta0}} = C_{g0}.
\end{eqnarray}
In this case, $v_{F}$, $v_{\Delta}$, $v_{\Gamma1}$, and
$v_{\Gamma2}$ depend on length scale $l$ as
\begin{eqnarray}
\frac{v_{F}}{v_{F0}} = \frac{v_{\Delta}}{v_{\Delta0}} =
\frac{v_{\Gamma1}}{v_{\Gamma10}} = \frac{v_{\Gamma2}}{v_{\Gamma20}}
= e^{-C_{g0}l},
\end{eqnarray}
which all vanish rapidly in the limit $l \rightarrow \infty$. Since
$C_{g}$ is a constant, random gauge potential is marginal. From the
above calculations, we can conclude that the behavior $C_{g}
\rightarrow \infty$ obtained in the presence of both nematic
fluctuation and random gauge potential is directly related to the
extreme velocity anisotropy $v_{\Delta}/v_{F} \rightarrow 0$ induced
by the quantum fluctuation of nematic order. The flow of $C_{g}$
towards strong coupling regime is a clear signature that random
gauge potential should have substantial physical effects on the
low-energy behaviors of nodal fermions, which will be discussed in
the next subsection.
We notice other interesting correlated electron models in which the
coupling coefficient of an interaction vanishes at low energies, but
the interaction is not negligible due to the more (or equally) rapid
decrease of the kinetic energy of electrons. Recently, Sur and Lee
studied the influence of quantum fluctuations of an
antiferromagnetic (AF) order at an AF quantum critical point in a
metal supporting one-dimensional Fermi surface \cite{Sur15}. In
particular, they showed that the coupling coefficient of the
interaction flows to zero at low energies. However, the fermion
velocity also vanishes, thus the interaction cannot be simply
neglected. Actually, Sur and Lee found that the interaction drives
the system to become a so-called quasi-local strange metal that is
apparently qualitatively different from the free fermion system.
\subsection{Physical effects of random gauge potential}
How should we understand the divergence of $C_{g}$? In order to
answer this question, we first consider the non-interacting system
that contains only nodal fermions and random gauge potential. In
this case, the RG equation of fermion DOS becomes
\begin{eqnarray}
\frac{d\ln\rho}{d\ln\omega} = \frac{1 - C_{g0}}{1 +
C_{g0}},\label{Eq:RhoRGEqNoInteraction}
\end{eqnarray}
with
\begin{eqnarray}
C_{g0} = \frac{g}{2\pi}\frac{v_{\Gamma10}^{2} +
v_{\Gamma20}^{2}}{v_{F0}v_{\Delta0}}.\label{Eq:CgNoInteraction.}
\end{eqnarray}
If $C_{g0} < 1$, we have
\begin{eqnarray}
\rho(\omega)\sim \omega^{\alpha}\quad \mathrm{with}\quad \alpha =
\frac{1 - C_{g0}}{1 + C_{g0}},\label{Eq:DOSOnlyDisorder}
\end{eqnarray}
where $\alpha$ satisfies $0 < \alpha < 1$. In this case, the RG
equation for specific heat is given by
\begin{eqnarray}
\frac{d\ln C_{V}}{d\ln T}=2-\frac{2C_{g0}}{1+C_{g0}}.
\end{eqnarray}
The solution of this equation is given by
\begin{eqnarray}
C_{V}(T)\sim T^{\beta}\quad \mathrm{with}\quad \beta = 2 -
\frac{2C_{g0}}{1 + C_{g0}}.
\label{Eq:SpecificHeatOnlyDisorder}
\end{eqnarray}
If $C_{g} > 1$, it is easy to get
\begin{eqnarray}
\rho(\omega) \rightarrow \infty
\end{eqnarray}
in the limit $\omega \rightarrow 0$. The divergence $\rho(\omega)$
indicates the emergence of a disorder dominated diffusive state, in
which a finite disorder scattering rate $\gamma$ and a finite
zero-energy DOS $\rho(0)$ are generated \cite{Altland02,
Chakravarty, Roy}. The value $C_{g} = 1$ defines a QCP for the
quantum phase transition between the ballistic and diffusive states
of nodal fermions. Therefore, a weak random gauge potential gives
rise to power law behavior of $\rho(\omega)$, whereas a sufficiently
strong disorder can trigger the quantum phase transition to a
diffusive state.
In the presence of both nematic critical fluctuation and random
gauge potential, the fact that $C_{g}\rightarrow \infty$ in the
lowest energy limit signals the development of a disorder dominated
diffusive state and the generation of a finite $\gamma$ and a finite
$\rho(0)$ even in the case of weak random gauge potential. Although
the perturbative RG method provides a powerful tool to judge whether
and when a phase transition takes place, it cannot be used to
compute the precise value of $\gamma$. To calculate $\gamma$, one
usually needs to construct a self-consistent equation for the
retarded fermion self-energy by properly considering both the
fermion-disorder scattering and the influence of quantum critical
fluctuation of nematic order \cite{Balatsky06}. This is an
interesting yet complicated issue, which is beyond the scope of the
present work and subjected to future research. Here, we use the RG
method to make a rough estimation for the energy scale of $\gamma$.
As shown in Fig.~\ref{Fig:VRGGP}(e), for small given values of
$C_{g0}$ and $v_{\Delta0}/v_{F0}$, the solution of the RG equation
of DOS should have the following properties: $\rho(\omega)$
decreases as the varying energy scale decreases, but tends to
increase when the energy scale exceeds a critical value
$E_{c}(C_{g0},v_{\Delta0}/v_{F0})$, which is a function of $C_{g0}$
and $v_{\Delta0}/v_{F0}$. The magnitude of $\gamma$ should be an
increasing function of $E_{c}$. In the following calculations of
$\rho(0)$ and $C_V(T)$, we will regard $\gamma$ as an undetermined
constant. Fortunately, the qualitative behaviors of $\rho(0)$ and
$C_{V}(T)$ in the low energy regime is independent of the precise
value of $\gamma$.
The imaginary part of retarded fermion self-energy can be
generically written as
\begin{eqnarray}
\mathrm{Im}\Sigma^R(\omega) \approx \gamma +
\mathrm{Im}\Sigma_{\mathrm{nem}}^R(\omega), \label{Eq:DampingAll}
\end{eqnarray}
where $\mathrm{Im}\Sigma_{\mathrm{nem}}^R(\omega)$ is the
contribution induced solely by the nematic order. The
disorder-induced scattering rate $\gamma$ represents a
characteristic energy scale. At energies higher than $\gamma$,
namely $\omega > \gamma$,
$\mathrm{Im}\Sigma_{\mathrm{nem}}^R(\omega)$ dominates over $\gamma$
and all the RG results for $v_{F}$, $v_{\Delta}$, $Z_{f}$,
$\rho(\omega)$, and $C_{V}(T)$ shown in figures~\ref{Fig:VRGGP}(a)-(f)
are still applicable. At $\omega <\gamma$, the diffusive motion of
nodal fermions and its interplay with critical nematic fluctuation
govern the low-energy properties of the system.
Once a finite $\gamma$ is generated, the renormalized velocities
$v_{F}$ and $v_{\Delta}$ no more vanish at low energies, which is
apparently different from the clean case. Instead, as the energy
scale decreases, both $v_{F}$ and $v_{\Delta}$ are saturated to
certain finite values, denoted by $v_{F0}'$ and $v_{\Delta0}'$,
below the energy scale set by $\gamma$. Hence, there is no extreme
velocity anisotropy in the diffusive state. The fermion DOS and
specific heat also exhibit different behaviors comparing to the
clean case. To demonstrate the difference in DOS, we write the
retarded propagators of nodal fermions in the forms:
\begin{eqnarray}
G_{1a}^{R}(\omega,\mathbf{k}) &=& \frac{1}{-(\omega+i\gamma) +
v_{F0}'k_{x}\tau^{z} + v_{\Delta0}'k_{y}\tau^{x}},
\\
G_{2a}^{R}(\omega,\mathbf{k}) &=& \frac{1}{-(\omega+i\gamma) +
v_{F0}'k_{y}\tau^{z} + v_{\Delta0}'k_{x}\tau^{x}}.
\end{eqnarray}
Calculations find that the fermion DOS depends on $\omega$ as
\begin{eqnarray}
\fl \rho(\omega) &=& \frac{N\gamma}{2\pi v_{F0}'v_{\Delta0}'}
\left[\ln \left(\frac{\left(\gamma^2 - \omega^2 +
v_{F0}'v_{\Delta0}'\Lambda^{2}\right)^2 +
4\omega^2\gamma^2}{\left(\gamma^2-\omega^2\right)^2 +
4\omega^2\gamma^2}\right)\right.\nonumber \\
\fl &&\left.+\frac{2\omega}{\gamma} \arctan\left(\frac{\gamma^2 -
\omega^2 + v_{F0}'v_{\Delta0}'\Lambda^2}{2\omega\gamma}\right) -
\frac{2\omega}{\gamma}\arctan\left(\frac{\gamma^2 - \omega^2}
{2\omega\gamma}\right)\right].
\end{eqnarray}
In the case $\sqrt{v_{F0}'v_{\Delta0}'}\Lambda \gg \gamma$, we have
\begin{eqnarray}
\rho(0)\approx&\frac{2N\gamma}{\pi v_{F0}'v_{\Delta0}'}
\ln\left(\frac{\sqrt{v_{F0}'v_{\Delta0}'}\Lambda}{\gamma}\right).
\end{eqnarray}
In order to compute the specific heat, it is convenient to invoke
the standard Matsubara formalism of fermion propagators, i.e.,
\begin{eqnarray}
G_{1a}(\omega_{n},\mathbf{k}) &=& \frac{1}{-i(\omega_{n} +
\gamma\mathrm{sgn}(\omega_{n})) + v_{F0}'k_{x}\tau^{z} +
v_{\Delta0}'k_{y}\tau^{x}},
\nonumber \\
G_{2a}(\omega_{n},\mathbf{k}) &=& \frac{1}{-i(\omega_{n} +
\gamma\mathrm{sgn}(\omega_{n})) + v_{F0}'k_{y}\tau^{z} +
v_{\Delta0}'k_{x}\tau^{x}}, \nonumber
\end{eqnarray}
where $\omega_{n}=(2n+1)\pi T$ with $n$ being integer.
The fermion free energy is given by
\begin{eqnarray}
F = -\frac{2NT}{v_{F0}'v_{\Delta0}'} \sum_{\omega_{n}}\int
\frac{d^2k}{(2\pi)^2}\ln\left[\left(\omega_{n} +
\gamma\mathrm{sgn}(\omega_{n})\right)^2+k^2\right].\nonumber
\end{eqnarray}
Summing over $\omega_n$ leads to
\begin{eqnarray}
\fl F &\approx& -\frac{4N}{\pi v_{F0}'v_{\Delta0}'}
\int\frac{d^2k}{(2\pi)^2}\left\{k\arctan\left(\frac{k}{\pi T +
\gamma}\right)+\frac{\gamma}{2}\ln\left[\left(\pi T + \gamma\right)^{2} +
k^{2}\right]\right\}.
\end{eqnarray}
Now the fermion specific heat in the low energy regime can be approximately given by
\begin{eqnarray}
C_{V} &\approx& \frac{6NT}{v_{F0}'v_{\Delta0}'}\gamma
\ln\left(\frac{\Lambda}{\gamma}\right),\label{Eq:CvDisorder}
\end{eqnarray}
which depends on $T$ linearly. We can see that DOS and specific heat
obtained in the diffusive state exhibit entirely different behaviors
from the unconventional non-Fermi liquid state below the energy
scale $\gamma$.
We finally make a brief remark on the behavior of the system staying
away from the nematic QCP. Suppose the system stays at a distance
$r$ to the QCP, the RG results are still valid and the fermion
velocity ratio is still renormalized at energies higher than the
scale corresponding to $r$. However, the renormalization terminates
at certain low energy scale. Therefore, the effective strength of
random gauge potential is moderately enhanced compared its bare
value, though $C_g$ does not diverge.
\section{Comparison with experiments\label{Sec:CompaisionExperiments}}
In this section, we address the possible connection between the
theoretical results obtained in the last sections and the
phenomenology of cuprate superconductors. We are particularly
interested in three existing experimental findings about some of the
unusual properties of the superconducting dome.
\subsection{Anomalous residual linear-$T$ term of specific heat in cuprates}
We first discuss the residual specific heat in cuprates. Due to the
line nodes of $d$-wave gap, the specific heat in the superconducting
phase of cuprates is expected to exhibit a $T^2$ behavior, i.e.,
$C_V(T)\propto T^2$, at low $T$. This expectation is generically
consistent with experiments \cite{Fisher07}. In the lowest $T$
limit, experiments have observed a residual linear $T$ term of $C_V(T)$
\cite{Fisher07, Revaz98, Wright99, Riggs11}, which is usually
attributed to the finite fermion DOS generated by disorder
scattering and is also well consistent with the result given by
equation (\ref{Eq:CvDisorder}). There is, however, an unexpected
experimental finding \cite{Fisher07, Riggs11} that the residual
specific heat of YBCO is obviously larger in magnitude than that of
La$_{2-x}$Sr$_x$CuO$_4$ (LSCO), although the former material is
known to be cleaner than the latter. Apparently, disorder scattering
alone is not capable of accounting for this experimental fact.
Recently, coexistence of $d$-wave superconductivity with a loop
current order was proposed to give a possible explanation
\cite{Allais12, Kivelson12, WangVafek13} for the large residual
linear-$T$ term of specific heat in YBCO. In this scenario, when
$d$-wave superconductivity coexists with a loop current order, two
of four nodal points are converted to finite Fermi pockets of
Bougoliubov quasiparticles, which then generates a finite $\rho(0)$
and a residual linear-$T$ term of specific heat \cite{Allais12,
Kivelson12, WangVafek13, Berg08}. The recent ultrasound measurements
showed the possible evidence for the existence of loop current
order in YBCO\cite{Shekhter13,Zaanen13}.
Here we propose an alternative explanation for the above anomalous
experimental results of residual specific heat. Our RG analysis
found that the effective strength of random gauge potential, being
the most relevant disorder to cuprates \cite{Nersesyan95}, is
strongly enhanced by the critical nematic fluctuation, which in turn
increases the residual value of the specific heat. To understand the
role of quantum nematic fluctuation, we firstly consider only the
coupling of nodal fermions to random gauge potential. In this
special case, $C_{g}$ does not flow and thus remains a constant,
namely $C_{g} = C_{g0}$. If $C_{g0}$ is very small, which means the
system is only slightly disordered, the behavior of $\rho(\omega)$
and specific heat would be governed by equations
(\ref{Eq:DOSOnlyDisorder}) and (\ref{Eq:SpecificHeatOnlyDisorder}),
respectively. In this case, the system does not develop a finite
$\rho(0)$ and there is no residual liner-$T$ term of specific heat.
If the system is quite disordered such that $C_{g0} > 1$, it enters
a diffusive state with a finite scattering rate $\gamma$, which
induces a finite $\rho(0)$ and a residual linear-$T$ term of
specific heat. Apparently, the residual specific heat is larger in
more disordered systems, which is not consistent with the
aforementioned experiments of residual specific heat. If we consider
both the quantum fluctuation of nematic order and random gauge
potential, $C_{g}$ is significantly enhanced and flows to strong
coupling at low energies even if $C_{g0}$ takes an arbitrarily small
value. This implies that a cleaner compound might acquire a larger
amount of $\gamma$ and naturally a larger $\rho(0)$.
To make a more careful comparison between theories and experiments,
we now briefly discuss the doping dependence of our results. We use
$\delta$ to denote the doping concentration and $\delta_c$ the
nematic QCP. At zero temperature, the mass of nematic field $\phi$
is proportional to the difference between $\delta$ and $\delta_c$,
namely $r \sim |\delta - \delta_{c}|$. When the cuprate is at a
distance $r$ away from the nematic QCP, the quantum fluctuation of
nematic order is not critical, but remains important for small $r$.
At the energy scales larger than $r$, the fermion velocity
anisotropy is still considerably enhanced, which leads $C_{g}$ to
flow to larger values. For a given small value of $C_{g0}$, $C_{g}$
can flow to a sufficiently large value to induce a diffusive state
and generate a finite scattering rate $\gamma$, provided that $r$ is
made sufficiently small. In this case, the quantum fluctuation of
nematic order can result in a finite $\rho(0)$ and a finite residual
linear-$T$ term of specific heat. If the bare value $C_{g0}$ is
large enough, random gauge potential itself suffices to generate a
finite $\gamma$. Including the quantum fluctuation of nematic order
leads to larger values of both $C_{g}$ and $\gamma$. In any case,
the quantum fluctuation of nematic order tends to amplify $\gamma$,
which naturally increase $\rho(0)$ and the residual specific heat.
A number of recent experiments provided strong evidence supporting
the existence of nematic order in YBCO, whereas there is little
evidence for nematicity in LSCO. Although YBCO is cleaner than LSCO,
the quantum fluctuation of nematic order in the former
superconductor can drive $C_{g}$ to flow to much larger values. As a
consequence, the residual linear-in-$T$ term of specific heat of
YBCO would be larger than that of LSCO. We emphasize that, disorder
itself cannot explain the anomalous behaviors of the residual
specific heat observed in YBCO and LSCO, and it is necessary to
consider the interplay of quantum nematic fluctuation and random
gauge potential.
For the above elaboration, we know that the roles played by the
quantum nematic fluctuation and random gauge potential depends on
doping $\delta$. To simplify discussion, we assume the bare value
$C_{g0}$ displays only a weak $\delta$-dependence. Since the quantum
fluctuation of nematic order is most pronounced at the nematic QCP,
the scattering rate $\gamma$ and consequently the coefficient of the
residual liner-$T$ term of specific heat are maximal at the QCP, and
decrease as the system moves away from this QCP. This doping
dependence is observable, and can be examined by experiments. Within
the loop current order scenario, both the zero-energy DOS $\rho(0)$
and the residual linear-$T$ term of specific heat are proportional
to the order parameter of the loop current order, which decreases
with the growing doping in the underdoped region.
Early experiments \cite{Fisher07, Revaz98, Wright99} found that the
coefficient of linear-$T$ term of specific heat at optimally doped
YBCO is roughly $2$ mJ$\cdot$mol$^{-1}$$\cdot$K$^{-2}$. More recent
measurements performed in underdoped YBCO \cite{Riggs11} revealed
that this coefficient is $1.85\pm 0.06$
mJ$\cdot$mol$^{-1}$$\cdot$K$^{-2}$. We feel that the currently
available experimental data about the doping dependence of this
coefficient are still quite limited. We expect more extensive
measurements would be performed in the future to extract a more
quantitative doping dependence of the coefficient, which could help
to judge whether the scenario proposed in this paper works.
\subsection{Strong damping rate of nodal fermions in optimally doped BSCCO}
We next apply the RG results to understand the observed damping rate
of nodal fermions in cuprates. Valla \emph{et al.} \cite{Valla99}
have performed extensive angle resolved photoemission spectroscopy
measurements in optimally doped
Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta}$ (BSCCO). Their main
discovery is that the nodal fermions exhibit a MFL-type damping rate
in the normal state above $T_c$, which is in general consistency
with the observed linear resistivity. They further found
\cite{Valla99} that the linear damping rate is not sensitive to the
onset of superconductivity and persists well below $T_c$. This was
out of expectation since previous BCS weak coupling analysis
\cite{Orenstein00,Titov95} had predicted a quite weak damping rate,
i.e., $\mathrm{Im}\Sigma^R(\omega,T) \propto
\max(\omega^{3},T^{3})$, in the superconducting phase. Several
scenarios \cite{Orenstein00,Vojta00A, Khveshchenko01} were proposed
to account for the nearly MFL behavior. In particular, Vojta
\emph{et al.} \cite{Vojta00A} and Khveshchenko and Paaske
\cite{Khveshchenko01} have argued that the strong fermion damping
may arise from a secondary phase transition from a
$d_{x^2-y^2}$-wave superconducting state to a $d_{x^2-y^2}+is$ or
$d_{x^2-y^2}+id_{xy}$-wave superconducting state.
Experimentally, the existence of a nematic phase was observed in
BSCCO by Lawler \emph{et al.} \cite{Lawler10}. More recent
experimental studies of Fujita \emph{et al.} provided further
evidence pointing towards the existence of a nematic QCP in the
vicinity of optimal doping in BSCCO \cite{Fujita14, Fradkin15}.
Therefore, it seems natural to account for the experimental finding
of Valla \emph{et al.} by considering the quantum critical
fluctuation of nematic order. We have showed through RG analysis
that the nodal fermion damping rate caused by critical nematic
fluctuation, described by $\mathrm{Im}
\Sigma_{\mathrm{nem}}^{R}(\omega)$, is slightly weaker than that of
a MFL. In realistic experiments, it is difficult to distinguish this
non-Fermi liquid state from a MFL state. Therefore, presence of a
nematic QCP provides an alternative scenario for the nearly MFL
behavior observed in optimally doped BSCCO. However, the nearly MFL
behavior occurs only at energies higher than the value
$\omega_{\gamma}$ set by disorder scattering rate $\gamma$. Indeed,
at $\omega < \omega_{\gamma}$, $\gamma$ dominates over
$\mathrm{Im}\Sigma_{\mathrm{nem}}^{R}(\omega)$, thus a finite
zero-energy DOS $\rho(0)$ is generated. To summarize, our RG results
are qualitatively consistent with the quasiparticle self-energy
$\mathrm{Im}\Sigma(\omega,T) = \gamma+\max(\omega,T)$ observed in
reference \cite{Valla99}.
\subsection{Temperature dependence of fermion velocity in BSCCO}
Another experiment is the observation of Plumb \emph{et al.}
\cite{Plumb10} that the fermion velocity $v_{F}$ along the nodal
directions increases as $T$ grows in BSCCO. We know from RG results
that the critical nematic fluctuations drive the fermion velocities
to vanish in the lowest energy limit. This means the velocities must
increase as the energy scale is growing. Therefore, the nematic
quantum fluctuation will induce increment of fermion velocities when
the thermal energy increases with growing $T$, which is
qualitatively well consistent with the observation of Plumb \emph{et
al}. Technically, one can translate the $l$-dependence of fermion
velocity to a $T$-dependence by making the transformation $T = T_0
e^{-l}$ \cite{She15, WangLiu15}. With the help of this
transformation, it is easy to show that the fermion velocity $v_F$
is an increasing function of $T$. Therefore, the singular
renormalization of fermion velocities of nodal fermions induced by
the quantum fluctuation of namatic order, which is first discovered
by Huh and Sachdev \cite{Huh08}, agrees with the $T$-dependence of
$v_{F}$ observed in reference \cite{Plumb10}.
\section{Summary\label{Sec:Summary} and discussions}
In summary, we have found that the nodal fermions of $d$-wave
superconductors constitute an unconventional non-Fermi liquid, which
exhibits a weaker violation of Fermi liquid description than a MFL,
due to the quantum critical fluctuation of nematic order. This
unusual state represents a novel quantum state of matter that cannot
be well captured by the traditional classification of (non-) Fermi
liquids and thus enriches our knowledge of strong electron
correlation effects. We also have calculated the fermion DOS and
specific heat after incorporating the unusual renormalization of
fermion velocities. When a gauge-potential-type disorder is added to
the system, we have analyzed its interplay with the quantum
fluctuation of nematic order, and found that the effective disorder
strength flows to strong coupling, leading to diffusive motion of
nodal fermions. Therefore, even a weak random gauge potential can
drive a quantum phase transition from a unconventional non-Fermi
liquid state to a disorder dominated diffusive state. However, the
unusual fermion damping induced by the nematic order is more
important than the disorder scattering at high temperatures, where
the nodal fermions still display the unconventional non-Fermi liquid
behaviors. We finally have discussed the connection between our
theoretic results and a number of interesting experiments in the
context of some cuprate superconductors.
We now would like compare our work to the existing extensive works
on the non-Fermi liquid behaviors in two-dimensional metals produced
by the quantum critical fluctuation of nematic order. At the QCP of
Pomeranchuk instability in two-dimensional metals, the quantum
fluctuation of nematic order can lead to very strong fermion damping
\cite{Oganesyan01, Metzner03, DellAnna06, Rech06,
Garst10,Metlitski10}. To the leading order, it is found
\cite{Oganesyan01, Metzner03, DellAnna06, Rech06, Garst10,
Metlitski10} the fermion damping rate behaves as
$\mathrm{Im}\Sigma(\omega) \sim \omega^{2/3}$ and the quasiparticle
residue $Z_{f}\sim\omega^{1/3}$. Since $Z_f$ vanishes in the limit
$\omega \rightarrow 0$, this QCP exhibits non-Fermi liquid behavior.
In this paper, we have considered the interaction between the
quantum fluctuation of nematic order and massless nodal fermions in
the superconducting dome of cuprate superconductors. It is apparent
that the quantum critical fluctuation of nematic order gives rise to
a stronger fermion damping effect in metals than in the
superconducting dome of cuprates. This difference should be owing
to the different forms of the kinetic energies of fermionic
excitations. In the context of cuprates, the kinetic energy of the
massless Dirac fermions excited from the superconducting gap nodes
is $E = \sqrt{v_{F}^{2}k_{x}^{2} + v_{\Delta}^{2}k_{y}^{2}}$. In
contrast, in a two-dimensional metal with a finite Fermi surface,
the kinetic energy of fermions can be written as $E = v_{F}k_{x} +
\frac{k_{y}^{2}}{2m}$, where $k_{x}$ is the momentum component
perpendicular to the Fermi surface and $k_{y}$ is the momentum
component along the tangential direction \cite{Oganesyan01,
Metzner03, DellAnna06, Rech06, Garst10, Metlitski10}. In the
low-energy regime, the latter kinetic energy is smaller that the
former for the same given values of momenta, which indicates that
the interaction plays a more important role in the latter system
than the former. To further demonstrate this difference, we consider
the different roles of the long-range Coulomb interaction in a
two-dimensional Dirac semimetal and a two-dimensional semi-Dirac
semimetal. In a Dirac semimetal, the kinetic energy of Dirac
fermions is simply $E = v_{F}k$ with $k = \sqrt{k_{x}^{2} +
k_{y}^{2}}$. RG analysis showed that the residue $Z_{f}$ approaches
a finite value at low energies, so the system is a normal Fermi
liquid\cite{Gonzalez96, Gonzalez99, Kotov12, WangLiu14, Hofmann14}.
In a semi-Dirac semimetal, the kinetic energy of fermions is written
as $E = \sqrt{v_{F}^{2}k_{x}^{2} + \frac{k_{y}^{2}}{4m^2}}$
\cite{Isobe16, Cho16}. In this case, the long-range Coulomb
interaction drives $Z_f$ to vanish in the lowest energy limit, i.e.,
$Z_{f} \rightarrow 0$, which apparently implies the breakdown of
Fermi liquid behavior \cite{Isobe16}. Once again, we see that the
ratio between the interaction energy scale and the kinetic energy is
a crucial quantity to determine the low-energy behaviors of an
interacting fermion system.
The coupling of nodal fermions to the quantum fluctuation of
antiferromagnetic (AF) order is also an interesting problem
\cite{Uemura04, Pelissetto08}. Uemura \cite{Uemura04} considered the
coupling of nodal fermions to the $(\pi,\pi)$ AF fluctuation, and
suggested a possibility that the $(\pi,\pi)$ AF fluctuation can
connect two nodal charges in different hole pockets and then
generate a bound state of two nodal charges. More recently,
Pelissetto \emph{et al.} studied a number of different couplings
between nodal fermions and AF fluctuations using RG method
\cite{Pelissetto08}. An interesting results is that, though most of
these couplings are irrelevant, there emerges a nearly marginal
coupling between nodal fermions and an effective, AF-order induced
nematic fluctuation. This nearly marginal coupling is found
\cite{Pelissetto08} to results in a fermion damping rate that is
nearly linear in energy or temperature.
The electronic nematic state has been observed not only in some
cuprate superconductors, but also in a number of iron-based
superconductors \cite{Chubukov15A}: 122 family, such as hole doped
Ba$_{1-x}$K$_{x}$Fe$_{2}$As$_{2}$, electron doped
Ba(Fe$_{1-x}$Co$_{2}$)$_{2}$As$_{2}$, and isovalent-doped
BaFe$_{2}$(As$_{1-x}$P$_{x}$)$_{2}$; 111 family, such as NaFeAs;
1111 family, such as LaFeAsO; 11 family, such as FeSe
\cite{Chubukov15A, Fernandes14}. In most of these compounds, the
nematic order emerges in accompany with a spin density wave (SDW)
order. However, there are also exceptions. For instance, the nematic
order is observed in FeSe without any evidence for SDW order
\cite{Chubukov15A, Fernandes14, Bohmer15, Baek15, Chubukov15B, Yu15,
Glasbrenner15, WangFa15, Jiang16}. Whether the nematic order
observed in iron-based superconductors is generated by the
fluctuation of SDW order or the orbital degrees of freedom is still
in fierce debate \cite{Chubukov15A, Fernandes14, Bohmer15, Baek15,
Chubukov15B, Yu15, Glasbrenner15, WangFa15, Jiang16, Lee09}. Recent
experimental studies have unambiguously showed that there is a QCP
in the superconducting dome at the optimal doping of
BaFe$_{2}$(As$_{1-x}$P$_{x}$)$_{2}$. This QCP may correspond to the
critical point for a SDW order or nematic QCP \cite{Shibauchi14,
Dioguardi16}, and is expected to exhibit rich quantum critical
phenomena. Moreover, there are clear evidences that the
superconducting gap of BaFe$_{2}$(As$_{1-x}$P$_{x}$)$_{2}$ has nodal
line points \cite{Shibauchi14}. Since the quantum fluctuation of
nematic order is peaked at zero momentum \cite{She15, Fernandes14,
Fernandes12}, the nodal fermions excited from the nodal line points
might couple strongly to the quantum fluctuation of nematic order at
the nematic QCP \cite{She15}. This coupling is physically analogous
to the model considered in this work, and it would be interesting to
study this coupling by means of RG method. The RG analysis performed
in this work could be generalized to study the possible non-Fermi
liquid behavior and disorder effects in the context of
BaFe$_{2}$(As$_{1-x}$P$_{x}$)$_{2}$ and other iron based
superconductors, where the multi-band effects and different gap
symmetry need to be seriously taken into account.
\ack{G.Z.L. and J.R.W. would like to thank Dr. Jing Wang for very
helpful discussions. We acknowledge the support by the National
Natural Science Foundation of China under Grants No.11274286,
No.11574285, No.11504379, and No.U1532267.}
|
3,212,635,537,462 | arxiv | \section*{Introduction}
The theoretical understanding of a first principles theory for the
glass transition is still missing. Despite of great advances in the
understanding of some generic features associated to the glass
transition (such as those predicted by the mode-coupling
theory) still some questions remain largely unknown. Going beyond the
schematic mode-coupling theory seems to be an enormous task so an
alternative way of looking at the glass transition may be useful. In
this direction, the study of the topological properties of both the
potential or free energy landscape may yield further information on
the mechanisms responsible for the anomalous viscosity of the glassy
phase.
The idea that topological aspects of the potential or free energy
landscape are the ultimate reason for the glass transition goes back to
Goldstein \cite{GO} and more recently Stillinger and Weber
\cite{SW,SW2}. This approach has been recently applied to the study of
hard-spheres \cite{DV}, monoatomic as well as binary Lennard-Jones glasses
\cite{SH} or mean-field models of glasses \cite{CR1}.
Here we propose an alternative dynamical approach to study the
topological properties of the potential energy landscape. We will
concentrate on the study of the stability local properties of the
configurations visited by the system during its dynamical
evolution. This is directly achieved through the study of how dynamical
trajectories, which evolve following the same stochastic noise, depart
from each other in the presence of a potential energy saddle point or a
maximum which may induce a negative Lyapunov exponent. The simplest way
to study this problem is through damage spreading (DS) techniques to be
describe later on in some detail. Although DS was introduced almost two
decades ago as an alternative way to consider thermodynamic phase
transitions, the initial enthusiasm on this problem strongly decayed
when it was realized that DS transitions are not universal and not
necessarily related to thermodynamic singularities.
Despite of this result here we will show that these transitions have
an added interest in that they may be used as a direct way to
investigate the local free energy landscape properties by measuring
the largest Lyapunov exponent associated with the Hamming distance (to
be defined later). In what follows I will explain in more detail why
DS is a good way of looking at the rugged properties of the potential
energy landscape. Later on I will discuss the analytical results
obtained for the schematic mode-coupling theory and finally discuss
how to extend these ideas to the study of real glasses. Some
preliminary results are shown for the case of binary soft-sphere
purely repulsive glasses.
\section{Why damage spreading?}
Consider two systems evolving under a Langevin dynamics each one
described by a set of $N$ variables $x_i,y_i;1\le i\le N$ evolving in
a potential energy landscape ${\cal V}$ under the same stochastic noise
$\eta_t$ with $\langle
\eta_i(t)\eta_j(s)\rangle=2T\delta_{ij}\delta(t-s)$. Although the
present discussion can be generalized for different stochastic noises
here we will concentrate on the simplest case (for a more detailed
discussion see \cite{HR}). The equations of motion read,
\bea
\dot{x}_i(t)=F_i(\lbrace x\rbrace)+\eta_i(t)\label{1}\\
\dot{y}_i(t)=F_i(\lbrace y\rbrace)+\eta_i(t)\label{2}
\eea
\noindent
where $F_i(\lbrace x\rbrace)=-\frac{\partial {\cal V}}{\partial x_i}$.
Note that both trajectories described by the systems $x$ and $y$ never
cross in phase space so two identical configurations such that
$x_i(t)=y_i(t)$ remain identical forever (and where identical in the
past). The equation for the difference variables $z_i=x_i-y_i$ reads,
\be
\dot{z}_i(t)=F_i(\lbrace x_i\rbrace)-F_i(\lbrace y_i\rbrace)~~~.
\label{3a}
\ee
\noindent
If the $z_i$ are small we can expand (\ref{3a}) around $z_i=0$
obtaining,
\be
\dot{z}_i(t)=\sum_j \frac{\partial F_i(\lbrace y\rbrace)}{\partial
y_j}z_j=-\sum_j \frac{\partial^2 V(\lbrace y\rbrace)}{\partial
y_i\partial y_j}z_j \label{3b}
\ee
\noindent
which may be written in a simplified form,
\be
\dot{z}_i=H_{ij}(\lbrace y\rbrace)z_j
\label{4}
\ee
\noindent
where $H_{ij}$ is the Hessian matrix evaluated at the configuration
$y$. Always within the linear approximation the dynamical evolution of
the distance between configurations $z_i$ will increase or decrease
according whether the spectrum of eigenvalues of the Hessian matrix
contains positive eigenvalues. In this sense, DS probes the spectrum of
eigenvalues of the matrix and shows instabilities whenever the matrix
develops positive eigenvalues. A more precise condition is given by the
maximum Lyapunov exponent defined through,
\be
\lambda_{max}=\lim_{t\to\infty}\frac{\log(D(t))}{t}
\label{5}
\ee
where $D(t)=\frac{1}{N}\sum_i z_i^2$ which should be positive whenever
$z_i=0$ is dynamically unstable. Note that the Hessian depends on time
through the time evolution of the generic configuration $y$. This may be
an equilibrium or an off-equilibrium configuration. So in principle the
maximum Lyapunov exponent depends on time through the time evolution of
the systems $x$ and $y$. We will see later that, in general, the type of initial
condition (as well as the initial distance) are not relevant parameters
for the DS transition. In this sense DS probes the temperature at which
the lowest accessible configurations in the potential energy landscape
develop unstable modes being a direct check of the corrugated properties
of the free energy landscape. Again, we must insist on the
non-universal properties of the DS dynamics. The present discussion
on the stability properties of the Hessian matrix and its connection
with the DS transition is valid in the framework of Langevin
dynamics. For other type of dynamics (such as Monte Carlo
dynamics or Glauber) the situation may be different and the physical
meaning of DS phenomena more difficult. In some sense, Langevin dynamics
is an appropriate tool to explore the topological properties of the
potential energy landscape.
\section{DS in mode-coupling theory}
Insight on the previous problem can be obtained through a careful study
of the DS equations in the case of ideal mode-coupling theory. It is
known since the seminal work by Kirkpatrick, Thirumalai and Wolyness
\cite{KTW} that mode coupling equations can be obtained in the framework
of exactly solvable $p$-spin glass models. Due to their mean-field
character, in this class of models it is possible to unambiguously define
concepts such as the configurational entropy or complexity and the
mode-coupling transition temperature $T_c$. The description of this type
of models is possible in the framework of the TAP analysis \cite{CS}
where it is possible to show that they contain a large number of
metastable states (exponentially large with $N$) as
well as a threshold energy where the system gets trapped in an
asymptotic aging state and the fluctuation-dissipation theorem is
violated in a peculiar way \cite{BCKM}.
Spherical p-spin models (contrarily to Ising spins) have the clear
advantage of being exactly solvable so it is convenient to do analytical
computations in that case. The potential energy in this model is defined
by,
\be
{\cal V}=-\sum_{(i_1<i_2<...<i_p)}\,J_{i_1,i_2,i_3,..,i_p}\s_{i_1}\s_{i_2}\s_{i_3}..\s_{i_p}~~~.
\label{6}
\ee
where the spins $\s_i$ are real valued spins which satisfy the spherical
constraint $\sum_{i=1}^N\s_i^2=N$. The
$J_{i_1,i_2,i_3,..,i_p}$ are quenched random variables with zero mean and
variance $p!/(2N^{p-1})$. The Langevin dynamics of the model is given by,
\be
\frac{\partial \s_i}{\partial t}=F_i(\lbrace\s\rbrace)-\mu\s_i+\eta_i
\label{7}
\ee
\noindent
where $\mu$ is a Lagrange multiplier which ensures that the spherical
constraint is satisfied at all times and the noise $\eta$ satisfies the
fluctuation-dissipation relation $\langle\eta_i(t)\eta_j(s)\rangle=2T
\delta(t-s)\delta_{ij}$ where $\langle...\rangle$ denotes the noise
average. $F_i$ is the force acting on the spin $\s_i$ due to the
interaction with the rest of the spins,
\be
F_i=-\frac{\partial V}{\partial s_i}=\frac{1}{(p-1)!}
\sum_{(i_2,i_3,...,i_p)}\,J_{i_1,i_2,..,i_p}\s_{i_2}\s_{i_3}..\s_{i_p}~~~.
\label{8}
\ee
We define the overlap between two configurations of the spins
$\s,\tau$ by the relation $Q=\frac{1}{N}\sum_{i=1}^N\s_i\tau_i$ so
a distance between these two configurations is,
\be
D=\frac{1-Q}{2}
\label{9}
\ee
\noindent
in such a way that identical configurations have zero distance and
opposite configurations have maximal distance $D=1$. Then we consider
two copies of the system $\lbrace\s_i,\tau_i\rbrace$ which evolve under
equation (\ref{7}) with the same statistical noise and start from random
initial configurations.
The final equations are \cite{HR},
\bea
\hspace{-2truecm}
\frac{\partial C(t,s)}{\partial t}+\mu(t)C(t,s)=
\frac{p}{2}\int_0^s du R(s,u)C^{p-1}(t,u)+\nonumber\\
\frac{p(p-1)}{2}
\int_0^t du R(t,u)C(s,u)C^{p-2}(t,u)\label{10}\\
\hspace{-2truecm}
\frac{\partial R(t,s)}{\partial t}+\mu(t)R(t,s)=\delta(t-s)
+\frac{p(p-1)}{2}
\int_s^t du R(t,u)R(u,s)C^{p-2}(t,u)\label{11}\\
\hspace{-2truecm}
\frac{\partial Q(t,s)}{\partial t}+\mu(t)Q(t,s)=
\frac{p}{2}\int_0^s du R(s,u)Q^{p-1}(t,u)
\nonumber\\+\frac{p(p-1)}{2}
\int_0^t du R(t,u)Q(u,s)C^{p-2}(t,u)\label{12}
\eea
\noindent
The dynamical equations involve the two times correlation, response
and overlap function $C(t,s), R(t,s), Q(t,s)$ defined by (in what
follows we take $t>s$),
\bea
C(t,s)=(1/N)\sum_{i=1}^N\langle\s_i(t)\s_i(s)\rangle=(1/N)\sum_{i=1}^N\langle\tau_i(t)\tau_i(s)\rangle
\label{13}\\
R(t,s)=(1/N)\sum_{i=1}^N\frac{\partial\langle\s_i\rangle}{\partial h^{\s}_i}=
(1/N)\sum_{i=1}^N\frac{\partial\langle\tau_i\rangle}{\partial h^{\tau}_i}\label{14}\\
Q(t,s)=(1/N)\sum_{i=1}^N\langle\s_i(t)\tau_i(s)\rangle\label{15}
\eea
\noindent
where $<..>$ denotes the average over dynamical histories and
$h^{\s}_i,h^{\tau}_i$ are fields coupled to the spins $\s_i,\tau_i$
respectively. These equations are complemented with the appropriate
boundary conditions $C(t,t)=1, Q_d(t)=Q(t,t), R(s,t)=0, \lim_{t\to
(s)^+} R(t,s)=1$ and the relations,
\bea
\mu(t)=T+\frac{p^2}{2}\int_0^tdu R(t,u)C^{p-1}(t,u)\label{16}\\
\frac{1}{2}\frac{\partial Q_d(t)}{\partial t}+\mu(t)Q_d(t)=T+
\frac{p}{2}\int_0^t du R(t,u)Q^{p-1}(t,u)
\nonumber\\
+\frac{p(p-1)}{2}\int_0^t du R(t,u)Q(t,u)C^{p-2}(t,u)\label{17}~~~~~~.
\eea
These equations can be analyzed in detail
using different methods. Here we summarize the main results
obtained \cite{HR},
\begin{itemize}
\item{Existence of a dynamical transition $T_0$}
There is a temperature $T_0$ such that $D(t)=0$ (or $Q_d(t)=1$, see eq.(\ref{9}))
is a stable fixed point for $T>T_0$ becoming unstable below $T_0$.
Because of the non-monotonic character of $D(t)$ it is very difficult to
derive analytically $T_0$. Nevertheless, it is possible to obtain an upper
and a lower bound. One gets,
\be
\sqrt{\frac{p-2}{2}} \le T_0\le \sqrt{\frac{p}{2}}
\label{18}
\ee
Direct numerical integration of the equations of motion yields
$T_0(p=3)=1.04 \pm 0.02$ with and $T_0(p=4)=1.13 \pm 0.02$. The value of
$T_0$ is well above the mode-coupling temperature $T_c$ and the TAP
temperature $T_{TAP}$ below which there is an exponentially large (with
the system size) number of metastable states.
\begin{figure}[hbt]
\epsfxsize=7cm\epsfysize=7cm \epsfbox{dam.fig1.eps}
\caption{Asymptotic distance $D_{\infty}$ for $p=3$
($\alpha=1,{\cal K}=1$) obtained from the Pade analysis of the series
expansions for different initial conditions $D_0=1$ (circles), $D_0=0.5$
(triangles), $D_0=0.25$ (stars). Typical error bars are shown for the
last case.
}
\label{fig1}
\end{figure}
\item{Independence of initial conditions}
The asymptotic damage $D(\infty)=\lim_{t\to\infty}\lim_{N\to\infty}
D(t)$ is independent on the value of the initial damage $D(0)$ or the
class of initial conditions (for instance, random or
thermalized). This independence stresses the fact that DS is a true
dynamical transition and the asymptotic damage $D(\infty)$ is a
dynamical order parameter.
\begin{figure}[hbt]
\epsfxsize=7cm\epsfysize=7cm \epsfbox{dam.fig2.eps}
\caption{$Q_d(t)$ for $p=3$ ($\alpha=1,{\cal K}=1$) at
temperatures $T=0.1,0.5$ (from bottom to top at large times) for three
different values of the initial overlap $Q_d(0)=-1,0,0.5$ as a
function of time. The continuous lines are the numerical integrations
with time step $\Delta t=0.01$.
}
\label{fig2}
\end{figure}
\item{$T_0$ is the lowest DS temperature}
The DS problem can be suitably generalized for the case of correlated
noises such that $\langle\eta_i(t)\xi_j(s)\rangle=2T {\cal
K}(Q(t,s))\delta(t-s)\delta_{ij}$ where $\eta$ and $\xi$ are the noises
acting on the systems $\s$ and $\tau$ respectively. The function ${\cal
K}$ satisfies ${\cal K}(1)=1$ so both noises are identical if the two
configurations coincide. This implies that $Q_d(t)=1$ is a fixed point
of the dynamics. It can be shown that for any possible function ${\cal
K}\le 1$ (with ${\cal K}(1)=1$) there is a finite temperature damage
spreading transition $T_0$ only if ${\cal K'}(1)\le 1$. The case
discussed previously ${\cal K}=1$ (identical noises at all times) yields
the lowest damage spreading transition temperature.
\item{$T_0$ is the endpoint of a dynamical critical line}
The DS problem can be also generalized to the case ${\cal
K}(Q)\le\lambda$ with $\lambda\le 1$ and ${\cal
K}(1)=\lambda$. Obviously for $\lambda=1$ dynamical trajectories of
both systems may cross. In this case it is possible to show that the
function ${\cal K}(Q)=\lambda$ yields the lowest DS transition
temperature $T_0(\lambda)$ among the set of possible functions ${\cal
K}$ (${\cal K}(Q)\le\lambda, {\cal K}(1)=\lambda$). $T_0(\lambda)$ is
monotonic increasing function function of $\lambda$ which for
$\lambda=0$ coincides with the mode-coupling transition temperature
$T_c$ and finishes in a critical endpoint $T_0(\lambda=1)=T_0$. So
there exists a line of dynamic critical points which connect the
mode-coupling temperature $T_c$ with the DS temperature $T_0$.
\item{$T_0$ is not universal.}
The temperature $T_0$ is not universal. As it depends on the set of
correlations of the noises it also depends on the type of dynamics
(molecular dynamics, Monte Carlo with Metropolis, heat-bath or Glauber).
This is a well known result which finds its natural explanation on the
physical origin of the DS transition. For a general dynamics it is not
possible to map the DS transition with the local properties of the
potential energy landscape. Only for the case of Langevin dynamics or
molecular dynamics this is possible. Other dynamics (such as Monte Carlo
with heat-bath dynamics) use random numbers in the dynamics which
introduce complex correlations between the noises. This yields a DS
transition (related to the $T_0(\lambda)$ discussed in the previous
paragraph for the Langevin case) which is probably related with the
mode-coupling transition temperature but this issue still needs to be
further investigated.
\end{itemize}
\section{Application to binary soft-sphere glasses}
In this section we apply the previous ideas derived in the framework of
mode-coupling theory to the case of structural glasses. We consider the
binary soft-spheres model introduced in \cite{SP} and recently studied
in \cite{CMPV}. For sake of simplicity we consider a gas of $N$
particles such that half of them have diameter $\sigma_1$ and the other
half $\sigma_2$. The particles interact through a two particle purely
repulsive potential, the energy of the system being defined by
\be
{\cal V}=\sum_{i<j}\bigl ( \frac{\sigma_{ij}}{r_{ij}}\bigr )^{12}
\label{19}
\ee
The choice $\sigma_{ij}=\frac{\sigma_{i}+\s_{j}}{2}$ supposes that
diameters are additive during the collision process. The advantage of
this potential is that the thermodynamic properties depend on the
density $\rho=N/V$ and the temperature $T$ only through the constant
$\Gamma=\rho/T^{\frac{1}{4}}$. For the particular case
$\frac{\s_1}{\s_2}=1.2$ crystallization is strongly inhibited and the
glass transition (where dynamics is strongly slowed down) appears in
the vicinity of $\Gamma=1.45$. Larger values of $\Gamma$ correspond to
the glass phase while lower values correspond to the liquid phase.
The Langevin dynamics for the soft-sphere model is defined by,
\be
\dot{\vec{r}_i}=-\sum_{j\ne i}^N \nabla_iV_{ij}(r_{ij})+\vec{\eta_i}
\label{20}
\ee
\noindent
with
$\langle\eta_i^k(t)\eta_j^l(t')\rangle=2T\delta_{ij}\delta_{kl}\delta(t-t')$
where the superindex in the noise indicate the different Cartesian
components of the vector noise $\vec{\eta}(t)$. The pairwise potential
is given by $V_{ij}(r)=(\frac{\s_{ij}}{r})^{12}$.
We now consider two systems described by the variables
$\vec{r_i},\vec{s_i}$ governed by (\ref{20}) and evolving under the same
realization of the noise. We define the Euclidean distance
\be
D(t)=\frac{1}{N}\sum_{i=1}^N(\vec{r_i}-\vec{s_i})^2
\label{21}
\ee
which vanishes if the two configurations coincide. If we want to extend
the previous ideas for the spherical $p$-spin model to this system now we
must take into account the fact that at very high temperatures a gas
diffuses so $D=0$ may not be a fixed point of the dynamics.
There are two strategies to deal with this problem which are discussed below.
\begin{itemize}
\item{Particles contained in a box}
This is the most natural choice. To simulate a purely repulsive system
one must confine the particles in a cubic box of side L such that
$\rho=\frac{N}{L^3}$. In this case one may numerically solve (\ref{19})
with two different class of boundary conditions. With periodic boundary
conditions particles leave one side of the box and enter the opposite
side. This resets completely the coordinates of the particle so the
distance (\ref{21}) is discontinuous if particles cross the
boundaries. Concerning one system quantities (such as the energy or the
pair correlation function) this is not a problem because the relevant
quantity is the distance between the particles which may be taken as the
minimum value between $r_{ij}$ and $L-r_{ij}$. A similar procedure can
be used to define the distance between the two copies. Everything can be
easily solved considering free boundary conditions so particles are not
allowed to cross the boundaries. In this case, it is possible to show
that $D=0$ is asymptotically stable for the purely diffusive case
($\Gamma=0$).
Preliminary results show that the DS transition temperature $T_0=\infty$
so two configurations never coincide at finite temperature. Still both
configurations retain some correlation (so $\langle
\vec{r_i}(t)\vec{s_i}(t)\rangle > 0$) and the asymptotic damage is a
non trivial function of the temperature.
\begin{figure}
\epsfxsize=7cm\epsfysize=7cm \epsfbox{dam.fig3b.eps}
\caption{Damage $D(t)$ as a function of time for $N=32$ starting from
two different initial conditions and three different temperatures
(from top to bottom) $\Gamma=0.8,0.4,0.2$}
\label{fig3}
\end{figure}
\item{Introducing an spherical constraint}
For the purpose of studying the local properties of the potential energy
landscape we may impose the global constraint $\sum_i
\vec{r_i}^2=N(\frac{N}{\rho})^{\frac{2}{3}}$ on the particles in such a
way that the average distance between the particles is finite when $N$
goes to infinity. Because the
spherical constraint shifts the Hessian matrix (\ref{4}) by a constant
(a Lagrange multiplier) the transition with the spherical constraint
may give information on the transition for the unconstrained
case. That Lagrange multiplier can be simply obtained from the potential
energy $<{\cal V}>$ and the temperature.
The advantage of such a constraint is that now there
is no box and $D=0$ is a fixed point of the dynamics for $\Gamma=0$. The
inconvenience is that the simplicity of the original model is lost and
the thermodynamics of the new model depends on both density and
temperature instead of a unique parameter $\Gamma$.
Again, preliminary results show that $T_0=\infty$ in this case so
$D=0$ is asymptotically stable strictly only for $\Gamma=0$. Although this
approach is more involved it is probably the best way to relate the DS
transition to the ruggedness of the free energy landscape.
\end{itemize}
\section{Conclusions}
The study of the free energy landscape may yield valuable information on
the glass transition phenomena. A promising description of the glass
transition is through the Stillinger and Weber projection of the
partition function in terms on inherent structures. That method directly
looks at the potential energy landscape described in terms of basins of
attraction explored by the system during its dynamical evolution
\cite{CR1}. An alternative approach studies the dynamical properties of
the free energy landscape directly looking at the largest Lyapunov
exponent of the Hessian matrix of the potential energy landscape
weighted by the size of the basins of attraction visited by the system
during its dynamical evolution.
Exact results for the mode coupling theory reveal that there is a
transition $T_0$ which separates two well defined regime depending on
the value of the asymptotic distance. Below $T_0$ the asymptotic damage
is non zero and independent of the initial distance as well as the class
of initial conditions. Above $T_0$ the damage vanishes. We argue that
the precise value of $T_0$ is related to the vanishing of the largest
Lyapunov exponent defined in (\ref{5}). Although such an explicit
connection needs still to be done it is quite probable that DS is a
precise tool to investigate the chaotic properties of the free energy
landscape. A result in this direction has been recently obtained by
Biroli through the study of the instantaneous normal modes spectra of
the $p$-spin model \cite{BIROLI}. Whether this transition has
experimental relevance in the study of real glasses is still an open
question. Our preliminary studies of soft-sphere binary mixtures show
that $T_0$ is extremely large. Because liquids are always diffusive at
large temperatures (a feature which is directly encoded in the
wave-vector dependence of correlation functions, a general feature of
liquids) one must be careful when extending the results obtained for the
spherical $p$-spin model to real structural models of glasses. Although
a better understanding of the extension of DS to diffusive systems is
needed we can point out other interesting open problems. One the one
hand it could be very interesting to analyze the DS transition for
molecular dynamics. In that case, there is no stochasticity in the
dynamical equations so the {\em effective} source of noise comes out
directly from the mixing property of the dynamics. The analog of
equation (\ref{4}) should be very similar except for the presence of
oscillations. Still the general argument would be the same and $T_0$
expected to be identical. Such an analysis would be
welcome. Finally it would be very interesting to look at the other
endpoint of the dynamic critical line. Our present discussion was
centered on the case of identical noises. For completely uncorrelated
noises the dynamical transition temperature is expected to coincide with
the mode-coupling transition temperature. This is true in the framework
of the aforementioned exact calculations in the spherical model and
could be also analyzed for real glasses.
\ack I warmly thank G. Biroli, A. Crisanti, S. Franz, M. Heerema,
J. Kurchan and I. Pagonabarraga for useful discussions. This work has
been supported by the Spanish Ministery of Education (PB97-0971).
\section*{References}
|
3,212,635,537,463 | arxiv | \section{Introduction}
Meeting Tthe tremendous growth in demand
for cellular data~\cite{CiscoVNI:latest}
will require
new technologies that can provide orders of magnitude increases in wide-area
wireless capacity.
With the severe shortage of spectrum in traditional UHF and microwave
bands below 3~GHz, there has been considerable interest
in so-called millimeter wave (mmW) frequencies
between 30 and 300 GHz where vast amounts of essentially virgin
spectrum are still widely available
~\cite{KhanPi:11-CommMag,PietBRPC:12,rappaportmillimeter,RanRapE:14,BocHLMP:14,Rappaport2014-mmwbook}.
However, a significant challenge for using mmW for wide-area,
cellular-type coverage is range.
Due to Friis' Law ~\cite{Rappaport:02}, the high frequencies
of mmW signals result in large isotropic path loss.
Fortunately, the very small wavelengths of mmW signals combined
with advances in low-power CMOS RF circuits enable large numbers
($\geq$ 32 elements) of miniaturized antennas to be placed in small
dimensions thereby providing high beamforming gains that can
theoretically more than compensate for the increase in isotropic path loss
\cite{AkdenizCapacity:14}.
However, spatial channel estimation needed to support
beamforming presents several challenges in the mmW range:
\begin{itemize}
\item \emph{High-dimensional arrays:}
Since current mobile devices typically have one to four antennas,
the array sizes in the mmW range -- which may be 16 or 32 elements
even at the mobile -- will represent an significant
increase in the dimension of the antenna processing.
In particular, a much larger number of parameters
will need be tracked at the receiver for channel estimation.
A system with $N_{rx}$ receive antennas must
estimate $N_{rx}$ channels per transmit
stream for instantaneous beamforming
and $N_{rx}^2$ parameters for the
receive-side spatial covariance matrix used in long-term beamforming.
\item \emph{Rapid channel variations:}
The high frequencies of the mmW bands implies that the
coherence time of the channel may be very small,
meaning that each of the channels to be tracked can be
varying rapidly.
Channel tracking for small-scale fading can be avoided
by long-term beamforming~\cite{Lozano:07}, and simulations
based on experimental
measurements in \cite{AkdenizCapacity:14}
suggest that the long-term
beamforming introduces only a 1 to 2 dB loss in the mmW range.
However, since mmW signals are extremely susceptible to
blocking \cite{Rappaport2014-mmwbook},
even the large-scale channel characteristics may change
rapidly.
For example, a change in the orientation of the mobile
device, movement of a hand holding the device
or appearance of a wall would all change the channel
significantly.
Thus, channel statistics must be estimated with a limited
number of measurements.
\item \emph{Analog beamforming}:
Due to the high bandwidths and large number
of antenna elements in the mmW range,
it may not be possible from a power consumption perspective
for the mobile receiver to obtain high rate digital samples
from all antenna elements~\cite{KhanPi:11}.
Most proposed designs perform beamforming in analog (either in RF or IF)
prior to the A/D conversion
\cite{KohReb:07,KohReb:09,GuanHaHa:04,Heath:partialBF}
-- see Fig.\ \ref{fig:analogBF}. A key limitation for these
architectures is that they permit the
mobile to ``look" in only one or a small number of directions at a time. This feature significantly reduces the information
in each measurement, further complicating the channel estimation
process.
\end{itemize}
\begin{figure}
\centering {\includegraphics[trim=5.5cm 4.5cm 3cm 4.5cm ,clip=true, width=1\linewidth]{paperFigures/beamforming3.pdf}}
\caption{\textbf{Analog vs.\ digital beamforming.}
\textbf{Bottom panel}:
Front-ends at conventional frequencies typically digitize the signals from
each antenna separately. This fully digital architecture offers the greatest
flexibility. However, power consumption may be prohibitive in the mmW range
when the bandwidth and number of antennas is large. \textbf{Top panel}:
To reduce power consumption, mmW front-end receivers
may need to perform beamforming in analog via phase shifters.
A consequence of this architecture for channel estimation is that each measurement
provides information in only one direction at a time.
}
\label{fig:analogBF}
\end{figure}
In this paper, we consider the problem of estimating
the long-term receive-side spatial covariance of a channel
on a high-dimensional array from a limited number of
analog measurements.
Key to our methodology is that the mmW channels
will likely have a low-rank structure relative to the
number of antenna elements.
For example, extensive measurements at 28 and 73~GHz in New York City
\cite{Rappaport:12-28G,Samimi:AoAD,rappaportmillimeter}
-- a dense, urban environment similar to likely initial deployments
for mmW systems --
have shown that the mmW channel energy is often concentrated
in a small number of relatively narrow-beam clusters.
Analysis of this data in \cite{AkdenizCapacity:14}
have revealed that the channel is often well approximated
by a rank three or four channel, typically much smaller
than the antenna dimension.
Similar findings can be found in \cite{molisch2014propagation}.
This low-rank property implies
that the spatial covariance matrix can be characterized by
a relatively small number of
parameters for the purpose of channel estimation.
Of course, the use of low-rank spatial structure is widely-used
in array processing and underlies many classic
channel estimation for wireless systems
\cite{martone1998adaptive,ottersten1996array,wang1998blind}.
The contribution in this work is to consider
the use of low-rank channel estimation
from analog measurements. As we describe below,
each measurement from an array with analog phase shifting
provides a power measurement in a single angular direction.
We show that maximum likelihood (ML)
reconstruction of the channel covariance matrix
from a collection of such measurements made at random angles is similar to a
low-rank matrix completion problem that has been used
widely in machine learning and image processing.
There are now several algorithms to solve low-rank matrix reconstruction
--- most are either based on nuclear or trace norm regularization
\cite{wright2009robust,lin2010augmented,koltchinskii2011nuclear}
or message passing techniques \cite{keshavan2010matrix,rangan2012iterative}.
A recent work \cite{chenrobust} has also considered low-rank recovery
problem specifically for covariance matrix estimation.
In this paper, we adapt a simple iterative soft thresholding algorithm (ISTA) method
\cite{BeckTeb:09} originally used in sparse recovery problems,
but also used for matrix completion \cite{cai2010singular}.
The method here is modified to account for the non-Gaussian nature of the power
measurements in the ML objective. It is shown that the proposed ISTA-based
algorithm converges to the global maxima of the likelihood.
Unfortunately, similar to the original work \cite{cai2010singular},
the thresholding step in each iteration of the proposed ISTA method requires an eigenvalue
decomposition of the current covariance matrix estimate. This computation
may be preclude implementation for real-time system. We thus propose an alternate
approximate ML estimate, where the search is performed over a appropriately
chosen finite subspace. We show that the resulting optimization for the
approximate ML estimate is equivalent to an inference problem
for a generalized linear model (GLM) \cite{NelWed:72} with non-negative
components.
An similar ISTA method can be used to solve this GLM-type optimization,
using simple scalar thresholding avoiding all eigenvalue decompositions.
Both the exact and approximate ML algorithms are tested in both single-path
and multi-path models. The channels for the multipath test scenarios
are from \cite{AkdenizCapacity:14} based on 28~GHz New York City data mentioned above
\cite{Rappaport:12-28G,Rappaport:28NYCPenetrationLoss,Samimi:AoAD,rappaportmillimeter}.
It is shown that the exact ML method offers excellent performance in a relatively
small number of iterations and the approximate ML method is only slightly worse.
\section{Problem Formulation}
The problem is to estimate the second-order spatial
statistics between a transmitter (TX) and receiver (RX).
We assume that the TX sends data from a single antenna, or equivalently,
from multiple TX antennas with a fixed beamforming vectors. The RX has $N$ antennas,
and makes $L$ measurements.
In each measurement $\ell$, $\ell=1,\ldots,L$,
the TX sends $D$ waveforms, $p_{\ell d}(t)$, $d=1,\ldots,D$,
potentially at the same time, but at different frequencies.
An example transmission scheme is illustrated in Fig.~\ref{fig:PSSsignal}.
In this example, the transmissions are separated in time as would occur for periodic
synchronization signals such as those proposed for the Primary Synchronization Signal in
\cite{barati2014directional}. However, the method proposed here would equally apply
to measurements from a continuous sequence of time slots such as cell
reference signals.
We assume the received complex baseband
signal across the $N$ antenna from the transmission is given by
the vector $\mathbf{r}_\ell(t) \in {\mathbb{C}}^N$, where
\begin{equation} \label{eq:rell}
\mathbf{r}_\ell(t) = \frac{1}{\sqrt{D}}\sum_{d=1}^D \mathbf{h}_{\ell d}p_{\ell d}(t) + \mathbf{v}_\ell(t),
\end{equation}
where $\mathbf{h}_{\ell d}$ is the channel gain vector for the signal
$p_{\ell d}(t)$ and $\mathbf{v}_\ell(t)$ is complex AWGN with noise PSD $N_0$ Watts/Hz.
Implicitly, we assume in this model that each $p_{\ell d}(t)$ is transmitted
in a sufficiently small time and frequency region that the channel is flat across
the transmission. The factor $1/\sqrt{D}$ is used to normalized the power.
\begin{figure}
\centering {\includegraphics[trim=5cm 16cm 5cm 4.5cm ,clip=true, width=1\linewidth]{paperFigures/PSS.pdf}}
\caption{Model for the synchorization signals from which the
spatial channel must be estimated. The signal is transmitted $L$ time slots,
and, for frequency diversity, the signal may be transmitted in $D$ different frequencies
in each time slot. We will evaluate the estimation performance as a function
of $D$ and $L$.}
\label{fig:PSSsignal}
\end{figure}
We will assume a standard \emph{correlated Rayleigh fading model} \cite{TseV:07}, where
the instantaneous channel gains $\mathbf{h}_{\ell d}$ have complex Gaussian distributions
\begin{equation} \label{eq:hgauss}
\mathbf{h}_{\ell d} \sim {\mathcal CN}(0,\mathbf{Q}), \quad
\mathbf{Q} = \mathbb{E}(\mathbf{h}_{\ell d}\mathbf{h}_{\ell d}^*),
\end{equation}
for some spatial covariance matrix $\mathbf{Q}$.
In addition, we will assume that in each measurement $\ell$,
the channel is independently faded across the different
transmissions, $\mathbf{h}_{\ell d}$, $d=1,\ldots,D$. We thus call the parameter $D$
the \emph{diversity order}.
In this paper, we do not consider the problem of predicting
the \emph{instantaneous} channel gains $\mathbf{h}_{\ell d}$. Instantaneous channel tracking across a large
number of antennas may be difficult in the mmW regime due to the low coherence time
and the limitation that the channel can be observed in only direction at a time
due to analog beamforming~\cite{RanRapE:14,Rappaport2014-mmwbook}.
Instead, in this paper, we thus consider only the problem of tracking the
second-order spatial statistics, namely the matrix $\mathbf{Q}$.
As described in \cite{TseV:07}, the covariance matrix
$\mathbf{Q}$ is determined by the angles of arrivals of different paths from the transmitter,
their relative average powers and
the response of the receive antenna array to the each of these paths.
Unlike the instantaneous channel gains $\mathbf{h}_{\ell d}$ which will vary due to small scale motion
(on the order of a wavelength), the long-term statistics such as $\mathbf{Q}$
depend only on the macro-layer
scattering environment and are thus a relatively constant over much longer periods of time and frequency.
In particular, in this study, we will assume that $\mathbf{Q}$ is constant over all measurements $\ell=1,\ldots,L$.
Once the spatial covariance matrix is estimated, one can perform a number of long-term
beamforming techniques~\cite{Lozano:07}. For example,
the long-term beamforming vector that maximizes the average signal energy
can be determined from the maximal eigenvector of $\mathbf{Q}$.
Similarly, if one estimates spatial covariance matrix $\mathbf{Q}_{sig}$
of a desired signal and the covariance matrix $\mathbf{Q}_{int}$ of the interference plus noise, the
maximal eigenvector of $\mathbf{Q}_{int}^{-1/2}\mathbf{Q}_{sig}\mathbf{Q}_{int}^{-1/2}$
is the direction that maximizes the signal-to-inference plus noise (SINR).
As mentioned in the Introduction, simulations in \cite{AkdenizCapacity:14} suggest that
the loss from optimal long-term beamforming in the mmW range relative
to instantaneous beamforming is on the order of 1~dB.
In estimating the spatial covariance matrix $\mathbf{Q}$,
our key problem assumption is that the RX does not have direct digital samples of the
components of the vector $\mathbf{r}_\ell(t)$ from the different antennas.
Instead, in each measurement $\ell$,
the RX must apply some beamforming vector $\mathbf{u}_\ell \in {\mathbb{C}}^N$ in analog and then perform
the estimation from the weighted signal
$\mathbf{u}_\ell^*\mathbf{r}_\ell(t)$. To perform the estimation, we assume that the RX
performs a match filter with each of the signals $p_{\ell d}(t)$ to yield complex scalar outputs,
\begin{equation} \label{eq:zelld}
z_{\ell d} = \frac{1}{\sqrt{E_s}\|\mathbf{u}_\ell\|} \int p_{\ell d}^*(t) \mathbf{u}_\ell^*\mathbf{r}_\ell(t)dt,
\quad E_s = \int |p_{\ell d}(t)|^2dt
\end{equation}
where $E_s$ is the energy in the transmitted signal.
We assume that $E_s$ is the same for all $p_{\ell d}(t)$.
\section{Maximum Likelihood Estimation and Matrix Completion}\label{sec:MLE}
\subsection{Maximum Likelihood Estimation}
The problem is to estimate the spatial covariance matrix $\mathbf{Q}$
from the measurements $z_{\ell d}$. We will assume that noise level $N_0$
is known.
We will also assume that the signals $p_{\ell d}(t)$ are orthogonal, and the channel gains
$\mathbf{h}_{\ell d}$ are independently faded across $\ell$ and $d$ and independent of the noise $\mathbf{v}(t)$.
Under these assumptions, it can be verified that the accumulated energies
\begin{equation} \label{eq:yell}
y_\ell = \sum_{d=1}^D |z_{\ell d}|^2,
\end{equation}
provide a sufficient statistic for the unknown parameters
$\mathbf{Q}$ and $N_0$. Moreover, under the independence assumptions, the
random variables $y_\ell$ will be distributed as
\[
Y_\ell = \frac{\lambda_\ell}{2D} V_\ell,
\]
where $V_{\ell}$ is a chi-squared random variable with $2D$ degrees of freedom, and
$\lambda_\ell$ is the energy
\begin{equation} \label{eq:lamell}
\lambda_\ell(\mathbf{Q}) = \mathbf{u}_\ell^*\left[\mathbf{Q} + \gamma^{-1} \mathbf{I}\right]\mathbf{u}_\ell,\quad
\gamma = \frac{E_s}{N_0}.
\end{equation}
See similar calculations in \cite{shiu2000fading}.
If we let $\mathbf{y} = (y_1,\ldots,y_L)$ be the vector of the received powers in the $L$ measurements.
then the negative log likelihood of $\mathbf{y}$ given $\mathbf{Q}$ is
\begin{equation} \label{eq:logp}
-\log p(\mathbf{y}|\mathbf{Q}) = C + \sum_{\ell=1}^L \left[ D \log(\lambda_\ell(\mathbf{Q}))
+ \frac{D y_\ell}{\lambda_\ell(\mathbf{Q})} \right],
\end{equation}
where $\lambda_\ell(\mathbf{Q})$ is given in \eqref{eq:lamell}
and $C$ is some constant that does not depend on $\mathbf{Q}$ (although it may depend on $\mathbf{y}$).
Thus, we have the ML estimation of $\mathbf{Q}$ is given by
\begin{equation} \label{eq:JQopt}
\widehat{\mathbf{Q}} = \mathop{\mathrm{arg\,min}}_{\mathbf{Q}} J(\mathbf{Q}) \mbox{ s.t. } \mathbf{Q} \geq \mathbf{0},
\end{equation}
where
\begin{equation} \label{eq:JQ}
J(\mathbf{Q}) := \sum_{\ell=1}^L \left[ \log(\lambda_\ell(\mathbf{Q}))
+ \frac{y_\ell}{\lambda_\ell(\mathbf{Q})} \right].
\end{equation}
\subsection{Connections to Matrix Completion}
An arbitrary $N \times N$ matrix $\mathbf{Q}$ has $N^2$ unknowns, and a Hermetian matrix $\mathbf{Q}=\mathbf{Q}^*$
has $N(N+1)/2$ unknowns. Thus, one may think that one would need at least $L \geq N(N+1)/2$
measurements to fully reconstruct $\mathbf{Q}$.
However, a key property of the covariance matrix $\mathbf{Q}$ in the mmW range
is that it is typically ``almost" low-rank, meaning that the most of the energy
of the channel gains $\mathbf{h}_{\ell d}$ is concentrated in a low-dimensional
subspace. For wireless channels, the rank of the receive-side spatial covariance matrix $\mathbf{Q}$
is determined by the number of distinct angles of arrival of paths from the
transmitter \cite{TseV:07}.
In the mmW range, analysis in \cite{AkdenizCapacity:14}
of the 28 and 73~GHz measurements in New York City in
\cite{Rappaport:12-28G,Rappaport:28NYCPenetrationLoss,Samimi:AoAD,rappaportmillimeter},
revealed that when low-power transmitters were placed in microcellular
type deployments, receivers in most
street-level locations saw only two to three dominant path clusters,
each with a relatively small angular spread.
The clustering of the paths into small, relatively narrowbeam clusters
causes the spatial covariance matrix to be low-rank.
For example, simulations in \cite{AkdenizCapacity:14}
assuming a 4x4 uniform linear array with the NYC channels showed that,
most of the energy would be likely concentrated to three to four-dimensional space
-- much lower than the 16 dimensions of the antenna array.
This low-rank property can be exploited to recover the matrix $\mathbf{Q}$ from
less than $N^2$ measurements. To understand how this is possible,
consider the problem of
low-rank matrix completion used in machine vision and ranking systems
\cite{wright2009robust,lin2010augmented,koltchinskii2011nuclear}.
In the matrix completion problem,
one is to reconstruct a low-rank matrix $\mathbf{A}$ from a small number of entries $A_{ij}$.
If an $M \times N$ matrix $\mathbf{A}$ has rank $r$, it has only $O(r(M+N)$ degrees of freedom.
When the rank $r$ is small, this number of degrees of freedom
may be significantly less than the $MN$ parameters needed to describe a general matrix.
Low-rank matrix completion methods impose the low-rank rank constraint
In the ML estimation problem considered here, each measurement $y_\ell$ in \eqref{eq:lamell}
has an average value $\lambda_\ell$ in \eqref{eq:lamell} which is a linear
function of the unknown matrix $\mathbf{Q}$. Hence, the ML estimation problem can be considered
a ``noisy" matrix completion problem where we attempt to attempt to reconstruct
a matrix an $N \times N$ low-rank matrix $\mathbf{Q}$ from $L$ noisy linear measurements of $\mathbf{Q}$.
The difference between the ML estimation problem considered here in contrast to
the estimation problems here
\subsection{Sparsity Regularization} \label{sec:sparseReg}
Placing a low-rank constraint on $\mathbf{Q}$ in the
optimization \eqref{eq:JQopt} will, in general, result in a non-convex
optimization.
Most matrix completion methods such as
\cite{wright2009robust,lin2010augmented,koltchinskii2011nuclear}
thus impose the low-rank constraint
indirectly by adding a regularization term of the form
$\|\mathbf{Q}\|_*$ to the objective
where $\|\mathbf{Q}\|_*$ is the so-called nuclear norm, which is the sum of the singular values
of $\mathbf{Q}$.
In our problem, the matrix $\mathbf{Q}$ is positive semi-definite,
so the nuclear norm is simply the trace: $\|\mathbf{Q}\|_* = \mathop{\mathrm{Tr}}(\mathbf{Q})$.
We thus consider the regularized optimization
\begin{equation} \label{eq:JQoptMu}
\widehat{\mathbf{Q}} = \mathop{\mathrm{arg\,min}}_{\mathbf{Q}} J_{\mu}(\mathbf{Q}) \mbox{ s.t. } \mathbf{Q} \geq \mathbf{0},
\end{equation}
where
\begin{equation} \label{eq:JQmu}
J_{\mu}(\mathbf{Q}) := J(\mathbf{Q}) + \mu\mathop{\mathrm{Tr}}(\mathbf{Q}),
\end{equation}
and $\mu > 0$ is a regularization parameter.
When $\mu=0$, the objective function \eqref{eq:JQmu} agrees with the
original un-regularized ML objective \eqref{eq:JQ}.
Using $\mu > 0$ tends to enforce the requirement that $\mathbf{Q}$ is
lower rank by penalizing the eigenvalues of $\mathbf{Q}$.
In analogy with compressed sensing, the parameter $\mu > 0$ is
often considered a \emph{sparsity} regularizer since
the resulting eigenvalues of the optimal solution $\widehat{\mathbf{Q}}$
in \eqref{eq:JQoptMu} tend to have a sparse set of eigenvalues,
meaning that many will be zero
\cite{wright2009robust,lin2010augmented,koltchinskii2011nuclear}.
Interestingly, in the simulations below, we will see that $\mu > 0$
appears to not improve the performance over $\mu=0$. This phenomena is
significantly different than the standard matrix
completion problem where using $\mu > 0$ is essential.
However, the ML objective \eqref{eq:JQopt}
already imposes a positivity constraint $\mathbf{Q} > 0$.
It is known that in related problems \cite{lee2001algorithms},
that non-negativity constraints already tends to result in sparse solutions with many
zero values -- so it is not surprising that using $\mu > 0$ does not help.
\section{Optimization Methods} \label{sec:optim}
\subsection{ISTA Algorithm}\label{sec:optimISTA}
The objective function $J(\mathbf{Q})$ in \eqref{eq:JQ}
in the optimization \eqref{eq:JQopt},
or the sparsity-regularized version $J_{\mu}(\mathbf{Q})$ in \eqref{eq:JQmu},
are both convex and therefore can be minimized via a number of methods.
We will first consider a simple ISTA approach \cite{BeckTeb:09}
used commonly in compressed sensing. We will describe the ISTA algorithm
for the sparsity-regularized objective function $J_{\mu}(\mathbf{Q})$
in \eqref{eq:JQmu}. The algorithm for minimizing $J(\mathbf{Q})$ can be realized by
taking $\mu=0$.
For the optimization \eqref{eq:JQoptMu},
the ISTA algorithm produces a sequences of estimates $\mathbf{Q}_k$, $k=0,1,2,\ldots$
with updates that can be described as follows:
At each iteration $k$, we find an $\alpha_k > 0$ such that
\begin{eqnarray}
\lefteqn{ J_{\mu}(\mathbf{Q}) \leq \overline{J}_\mu(\mathbf{Q},\mathbf{Q}_k) := J_\mu(\mathbf{Q}_k) } \nonumber \\
&& +
\frac{\partial J_\mu(\mathbf{Q}_k)}{\partial \mathbf{Q}} \cdot
(\mathbf{Q} - \mathbf{Q}_k) + \frac{1}{2 \alpha_k}\|\mathbf{Q} - \mathbf{Q}_k\|^2_F, \label{eq:Jbardef}
\end{eqnarray}
for all possible $\mathbf{Q} \geq 0$. We will discuss how to select such a value $\alpha_k$
momentarily. Thus, $\overline{J}_\mu(\mathbf{Q},\mathbf{Q}_k)$ represents a quadratic
upper bound on the true objective $J_\mu(\mathbf{Q})$ that matches the true objective
at $\mathbf{Q}=\mathbf{Q}_k$. In the case of the objective function \eqref{eq:JQ}, it is easy to
check that the derivative in any direction $\Delta$ is given by
\begin{equation} \label{eq:derivSk}
\frac{\partial J_\mu(\mathbf{Q}_k)}{\partial \mathbf{Q}} \cdot \Delta = \mathop{\mathrm{Tr}}(\mathbf{S}_k^* \Delta),
\end{equation}
where
\begin{equation} \label{eq:Skdef}
\mathbf{S}_k = \sum_{\ell=1}^L \left[ \frac{1}{\lambda_\ell(\mathbf{Q}_k)} -
\frac{y_\ell}{\lambda_\ell^2(\mathbf{Q}_k)} \right] \mathbf{u}_\ell\mathbf{u}_\ell^* + \mu\mathbf{I}.
\end{equation}
The concept in the ISTA algorithm is, at each iteration $k$, to minimize the
upper bound $\overline{J}_\mu(\mathbf{Q},\mathbf{Q}_k)$ instead of the true objective $J_\mu(\mathbf{Q})$:
\begin{eqnarray}
\lefteqn{ \mathbf{Q}_{k\! + \! 1} = \mathop{\mathrm{arg\,min}}_{\mathbf{Q} \geq 0} \overline{J}_\mu(\mathbf{Q},\mathbf{Q}_k) } \nonumber \\
&\stackrel{(a)}{=}&
\mathop{\mathrm{arg\,min}}_{\mathbf{Q} > 0} \mathop{\mathrm{Tr}}(\mathbf{S}_k^*\mathbf{Q}) + \frac{1}{2\alpha_k}\|\mathbf{Q}-\mathbf{Q}_k\|^2_F \nonumber \\
&\stackrel{(b)}{=}& \mathop{\mathrm{arg\,min}}_{\mathbf{Q} \geq 0} \|\mathbf{Q} - \mathbf{Q}_k + \alpha_k \mathbf{S}_k\|^2_F \nonumber \\
&\stackrel{(c)}{=}& T_+\left( \mathbf{Q}_k - \alpha_k \mathbf{S}_k \right),
\end{eqnarray}
where in step (a) we substituted the definition of $\overline{J}_\mu(\cdot)$ in \eqref{eq:Jbardef}
and derivative \eqref{eq:derivSk} and removed terms that do not depend on $\mathbf{Q}$;
step (b) follows from rearranging the quadratic and in step (c) the operator
$T_+(\mathbf{P})$ is called the \emph{proximal operator} and is given by
\begin{equation} \label{eq:Tplus}
T_+(\mathbf{P}) := \mathop{\mathrm{arg\,min}}_{\mathbf{Q} > 0} \|\mathbf{Q}-\mathbf{P}\|^2_F.
\end{equation}
It is shown in \cite{cai2010singular}
that this minimization can be computed easily via an eigenvalue
decomposition. Specifically, when $\mathbf{P}=\mathbf{P}^*$, we know that
$\mathbf{P} = \mathbf{U}\mathbf{D}\mathbf{U}^*$ for some unitary $\mathbf{U}$ and real diagonal
$\mathbf{D} = \mathop{\mathrm{diag}}(\mathbf{d})$. In this case, the proximal operator is
\[
T_+(\mathbf{P}) = \mathbf{U} \mathop{\mathrm{diag}}\left[ \max(\mathbf{d},0) \right] \mathbf{U}^*,
\]
which simply thresholds the eigenvalues of the matrix.
The resulting algorithm is shown in Algorithm~\ref{algo:ISTA}.
\begin{algorithm}
\caption{ML Estimation via ISTA}
\begin{algorithmic}[1] \label{algo:ISTA}
\REQUIRE{
Matrix search directions $\mathbf{u}_\ell$, power values $y_\ell$, $\ell=1,\ldots,L$,
and SNR $\gamma$.
}
\STATE{ $t \gets 0$ }
\STATE{ Initialize $\mathbf{Q}_0 \geq 0$ }
\REPEAT
\STATE{ $\lambda_\ell \gets \mathbf{u}^*_\ell(\mathbf{Q}_k + \gamma^{-1}\mathbf{I})\mathbf{u}_\ell$, $\forall \ell$}
\STATE{ Compute the gradient $\mathbf{S}_k$ from \eqref{eq:Skdef} }
\STATE{ Select step size $\alpha_k > 0$ }
\STATE{ $\mathbf{Q}_{k\! + \! 1} \gets T_+(\mathbf{Q}_k - \alpha_k\mathbf{S}_k)$ }
\UNTIL{Terminated}
\end{algorithmic}
\end{algorithm}
A key property of the ISTA algorithm is that the objective function monotonically decreases
for a sufficiently small step-size.
\begin{proposition} \label{prop:ISTAConv}
Consider the output of the ISTA algorithm, Algorithm \eqref{algo:ISTA}, generated for
a set of measurement vectors $\mathbf{u}_\ell$, power measurements $y_\ell$ and SNR value $\gamma > 0$.
Then, there exists a minimum step size $\overline{\alpha} > 0$ such that
if $\alpha_k < \overline{\alpha}$ for all $k$, the objective $J_\mu(\mathbf{Q})$ monotonically
decreases.
\end{proposition}
\begin{IEEEproof}
From Taylor's Theorem, we know that the bound \eqref{eq:Jbardef} will be satisfied
if
\begin{equation} \label{eq:JHessBnd}
\frac{\partial^2}{\partial \mathbf{Q}^2} J_{\mu}(\mathbf{Q}) \leq \frac{1}{2\alpha_k} \mathbf{I}.
\end{equation}
Now, since $\mathbf{Q} \geq \mathbf{0}$, the power levels $\lambda_\ell(\mathbf{Q})$
in \eqref{eq:lamell} will be bounded
below by
\[
\lambda_\ell(\mathbf{Q}) \geq \frac{\|\mathbf{u}_\ell\|^2}{\gamma}.
\]
Using this bound, one can verify that there is a global upper bound on the Hessian in
the left-hand side of \eqref{eq:JHessBnd}. This implies that there exists an $\overline{\alpha} > 0$
such that if $\alpha_k < \overline{\alpha}$ then \eqref{eq:JHessBnd} will be satisfied and therefore
so will the bound \eqref{eq:Jbardef}.
We therefore have that at each iteration $k$,
\begin{equation}
J_\mu(\mathbf{Q}_{k\! + \! 1}) \stackrel{(a)}{\leq} \overline{J}_\mu(\mathbf{Q}_{k\! + \! 1},\mathbf{Q}_k)
\stackrel{(b)}{\leq} \overline{J}_\mu(\mathbf{Q}_k,\mathbf{Q}_k)
\stackrel{(c)}{=} J_\mu(\mathbf{Q}_k),
\end{equation}
where step (a) follows from quadratic upper bound approximation in \eqref{eq:Jbardef},
step (b) is based on monotonically decreasing behavior of the cost function when we are applying ISTA algorithm,
and in step (c) we substituted the $\mathbf{Q}_k$ in \eqref{eq:Jbardef}.
This shows that for sufficiently small step sizes, the objective function decreases monotonically.
\end{IEEEproof}
A more refined analysis along the lines of \cite{BeckTeb:09} will show additionally
that the $J_\mu(\mathbf{Q}_k)$ converges to a local minima, which will also be a global minima
since the function is convex.
\subsection{Adaptive Step-size Selection} \label{sec:adaptStep}
Proposition~\ref{prop:ISTAConv} guarantees that for a sufficiently small step-size $\overline{\alpha}$,
the algorithm is guaranteed to converge. However, selecting a single step-size that works for all $\mathbf{Q}_k$
may require a very small value, slowing the rate of the algorithm.
We thus use a simple, standard backtracking adaptive step-size method
\cite{nocedal2006numerical} as follows.
In each iteration of our algorithm, we attempt
a candidate step size $\alpha_k > 0$ and compute a test estimate
$\tilde{\mathbf{Q}}_{k\! + \! 1}$. We know that, from a first-order approximation
of the objective,
\[
J_\mu(\tilde{\mathbf{Q}}_{k\! + \! 1}) \approx J_\mu(\mathbf{Q}_k) +
\mathop{\mathrm{Tr}}(\mathbf{S}_k^*(\tilde{\mathbf{Q}}_{k\! + \! 1}-\mathbf{Q}_{k})).
\]
We thus test the condition
\begin{equation} \label{eq:JdecCond}
J_\mu(\tilde{\mathbf{Q}}_{k\! + \! 1}) < J_\mu(\mathbf{Q}_k) + \rho
\mathop{\mathrm{Tr}}(\mathbf{S}_k^*(\tilde{\mathbf{Q}}_{k\! + \! 1}-\mathbf{Q}_{k})),
\end{equation}
for some parameter $\rho \in (0,1)$. Typically, we take $\rho = 1/2$.
If the condition \eqref{eq:JdecCond} is met, we accept the
candidate by setting $\mathbf{Q}_{k\! + \! 1} = \tilde{\mathbf{Q}}_{k\! + \! 1}$ and
increase the step-size $\alpha_{k\! + \! 1} = 2\alpha_k$. On the
other hand, if the condition \eqref{eq:JdecCond} fails,
we discard the candidate by setting $\mathbf{Q}_{k\! + \! 1} = \mathbf{Q}_k$ and
decrease the step-size $\alpha_{k\! + \! 1} = \alpha_k/2$.
\subsection{Approximate ML Estimation via a GLM}\label{sec:optimGLM}
The ISTA algorithm, Algorithm~\ref{algo:ISTA},
is conceptually simple. But, the optimization may not be feasible
for real-time implementations. The main challenge is the thresholding step $T_+(\cdot)$.
Each thresholding step requires a eigenvalue decomposition. As we will see in the simulations,
the algorithm often needs 100 iterations.
The key insight is given by the following lemma:
\begin{lemma} \label{lem:optEquiv}
Given a set of measurement vectors $\mathbf{u}_\ell$ define the set
\begin{equation} \label{eq:Gdef}
\mathbf{G} = \Big\{\mathbf{q}\Big|\mathbf{Q} = \sum_{\ell=1}^L q_\ell \mathbf{u}_\ell \mathbf{u}_\ell^*+q_0\mathbf{I}\geq0\Big\}.
\end{equation}
Then, the optimization \eqref{eq:JQoptMu} can be rewritten as
an optimization over $\mathbf{q} \in \mathbf{G}$
via the equivalence
\begin{equation} \label{eq:Proof}
\min_{\mathbf{Q} \geq \mathbf{0}} \, J_\mu(\mathbf{Q}) = \min_{\mathbf{q} \in \mathbf{G}} \,
f(\mathbf{A}\mathbf{q}),
\end{equation}
where $\mathbf{A}$ is the matrix with components
\begin{equation} \label{eq:Adefine}
A_{\ell j} = \begin{cases}
N & \mbox{ if } j = 0, \ell = 0 \\
\|\mathbf{u}_j\|^2 & \mbox{ if } j = 0, \ell > 0 \\
\|\mathbf{u}_\ell\|^2 & \mbox{ if } \ell = 0, j > 0 \\
|\mathbf{u}_\ell^*\mathbf{u}_j|^2 & \mbox{ if } j,\ell=1,\ldots,L,
\end{cases}
\end{equation}
and $f(\mathbf{z}) = \sum_{\ell=0}^L f_\ell(z_\ell)$, and
\begin{subequations} \label{eq:fdef}
\begin{eqnarray}
f_0(z_0) &=& \mu z_0 \\
f_\ell(z_\ell) &=& \log\left(z_\ell+\frac{1}{\gamma}\|\mathbf{u}_\ell\|^2\right)+
\frac{y_\ell}{z_\ell+\frac{1}{\gamma}\|\mathbf{u}_\ell\|^2}, \nonumber \\
& & ~ \ell=1,\ldots,L.
\end{eqnarray}
\end{subequations}
\end{lemma}
\begin{IEEEproof} See Appendix \ref{sec:appendix}. \end{IEEEproof}
To understand the purpose of this lemma, recall that the chief computational difficulty
in the ISTA algorithm arises from the thresholding step to impose the positivity constraint
on $\mathbf{Q} \geq \mathbf{0}$.
The above lemma shows that the optimization \eqref{eq:JQoptMu}
over $N \times N$ matrices $\mathbf{Q} \geq \mathbf{0}$ can be replaced
by an optimization over an $(L+1)$-dimensional vector $\mathbf{q} \in \mathbf{G}$.
Unfortunately, the constraint in $\mathbf{G}$ in \eqref{eq:Gdef} still requires
a positivity constraint on the resulting matrix $\mathbf{Q}$.
However, this problem
can be approximately circumvented as follows:
We know that if $\mathbf{q} > 0$ then
\[
\mathbf{Q} = \sum_{\ell=1}^L q_\ell \mathbf{u}_\ell \mathbf{u}_\ell^*+q_0\mathbf{I}\geq \mathbf{0}
\Rightarrow \mathbf{q} \in \mathbf{G}.
\]
The converse is not true. That is, it is not the case that $\mathbf{q} \in \mathbf{G}$ implies
that $\mathbf{q} \geq 0$. However, instead of searching over all $\mathbf{q} \in \mathbf{G}$, we can
replace the optimization in \eqref{eq:Proof} with the approximate ML optimization
\begin{equation} \label{eq:JQopt1}
\widehat{\mathbf{q}} = \mathop{\mathrm{arg\,min}}_{\mathbf{q} \geq 0} f(\mathbf{A}\mathbf{q}).
\end{equation}
The resulting optimization \eqref{eq:JQopt1}
has a particularly simple structure.
First, the vector $\mathbf{q}$ has only componentwise constraints:
$\mathbf{q} \geq 0$.
Second, for any $\mathbf{q}$,
the objective function in \eqref{eq:JQopt1} can be evaluated via
a linear transform $\mathbf{z} = \mathbf{A}\mathbf{q}$ and followed by a sum
of convex functions \eqref{eq:fdef} on the components $z_\ell$.
This type of optimization described by a separable function of a linear
transform of the vector arises in a wide range of applications.
The most common application is in inference problems for
so-called generalized linear models (GLMs)~\cite{NelWed:72}.
\subsection{An ISTA Algorithm for the Approximate ML}
As before, we can apply an ISTA-type approach to the optimization
\eqref{eq:JQopt1} as follows.
Let $\mathbf{q}_k$ be the estimate at the $k$-th iteration and
be its $\mathbf{z}_k = \mathbf{A}\mathbf{q}_k$ be its transform.
We then find a $\alpha_k > 0$ such that,
\begin{eqnarray}
\lefteqn{ f(\mathbf{A}\mathbf{q}) \leq \overline{f}(\mathbf{q},\mathbf{q}_k) := f(\mathbf{A}\mathbf{q}_k) } \nonumber \\
&& +
\frac{\partial f(\mathbf{A}\mathbf{q}_k)}{\partial \mathbf{q}} \cdot
(\mathbf{q} - \mathbf{q}_k) + \frac{1}{2 \alpha_k}\|\mathbf{q} - \mathbf{q}_k\|^2_2, \label{eq:fbardef}
\end{eqnarray}
for all possible $\mathbf{q}\geq 0$.
So, $\overline{f}(\mathbf{q},\mathbf{q}_k)$ is a
quadratic upper bound on the true objective $f(\mathbf{A}\mathbf{q})$ that
matches the true objective at the current estimate $\mathbf{q}=\mathbf{q}_k$.
Also, the derivative in \eqref{eq:fbardef} is given by
\begin{equation} \label{eq:derivq}
\frac{\partial f(\mathbf{A}\mathbf{q})}{\partial \mathbf{q}}\Bigr|_{\mathbf{q}=\mathbf{q}_k} = \mathbf{s}_k^*,
\quad
\mathbf{s}_k := \mathbf{A}^*\frac{\partial f(\mathbf{z}_k)}{\partial \mathbf{z}}.
\end{equation}
Then, as before, at each iteration $k$, the ISTA algorithm minimizes the
upper bound $\overline{f}(\mathbf{q},\mathbf{q}_k)$ instead of the true objective
$f(\mathbf{A}\mathbf{q})$:
\begin{eqnarray}
\lefteqn{\mathbf{q}_{k\! + \! 1} = \mathop{\mathrm{arg\,min}}_{\mathbf{q} \geq 0} \overline{f}(\mathbf{q},\mathbf{q}_k) }
\nonumber \\
&\stackrel{(a)}{=}&
\mathop{\mathrm{arg\,min}}_{\mathbf{q} > 0} \mathbf{s}_k^*\mathbf{q} + \frac{1}{2\alpha_k}\|\mathbf{q}-\mathbf{q}_k\|^2_2+\mu\mathbf{v}^T\mathbf{q} \nonumber \\
&\stackrel{(b)}{=}& \mathop{\mathrm{arg\,min}}_{\mathbf{q} \geq 0} \|\mathbf{q}- \mathbf{q}_k + \alpha_k \mathbf{s}_k\|^2_2+\mu\mathbf{v}^T\mathbf{q} \nonumber \\
&\stackrel{(c)}{=}& T_+\left( \mathbf{q}_k - \alpha_k \mathbf{s}_k \right),
\end{eqnarray}
where in step (a) we substituted the definition of
$\overline{f}(\cdot)$ in \eqref{eq:fbardef} and derivative \eqref{eq:derivq} and
removed terms that do not depend on $\mathbf{q}$;
step (b) follows from rearranging the quadratic and
in step (c) the operator $T_+(\mathbf{p})$ is the proximal operator given by
\begin{equation} \label{eq:Tplus1}
T_+(\mathbf{p}) := \mathop{\mathrm{arg\,min}}_{\mathbf{q} > 0} \|\mathbf{q}-\mathbf{p}\|^2_2.
\end{equation}
which is simply the quadratic approximation and
removing the negative components.
It is easily checked that the proximal operator \eqref{eq:Tplus1}
is given by a simple thresholding operation
\begin{equation} \label{eq:Tplus2}
T_+(\mathbf{p}) := \max\{\mathbf{p}, 0\},
\end{equation}
which simply sets all negative components of $\mathbf{p}$ to zero.
The resulting algorithm is shown in Algorithm~\ref{algo:ISTAGLM}.
\begin{algorithm}
\caption{Approximate ML Estimation via ISTA}
\begin{algorithmic}[1] \label{algo:ISTAGLM}
\REQUIRE{
Matrix search directions $\mathbf{u}_\ell$ and power values
$y_\ell$, $\ell=1,\ldots,L$, and SNR $\gamma$.
}
\STATE{ Construct $\mathbf{A}$ in \eqref{eq:Adefine}}
\STATE{ $k \gets 0$ }
\STATE{ Initialize $\mathbf{q}_0 \geq 0$ }
\REPEAT
\STATE{ $\mathbf{z}_k \gets \mathbf{A}\mathbf{q}_k$}
\STATE{ $\mathbf{s}_k \gets \mathbf{A}^*\partial f(\mathbf{z}_k)/\partial \mathbf{z}$}
\STATE{ Select step size $\alpha_k > 0$ }
\STATE{ $\mathbf{q}_{k\! + \! 1} \gets \max\{0, \mathbf{q}_k-\alpha_k \mathbf{s}_k \}$ }
\UNTIL{Terminated}
\end{algorithmic}
\end{algorithm}
The main advantage of the Approximate ML algorithm,
Algorithm~\ref{algo:ISTAGLM} in comparison to Algorithm~\ref{algo:ISTA}
is its complexity. Each iteration involves only multiplications by $\mathbf{A}$
and $\mathbf{A}^*$ as well as simple scalar derivatives.
In particular, unlike Algorithm~\ref{algo:ISTA},
there are no eigenvalue value decompositions needed for the thresholding
step.
Also, as with Algorithm~\ref{algo:ISTA},
the objective function monotonically decreases
with sufficiently small step-sizes $\alpha_k$.
Specifically, suppose that at some time $k$,
$\alpha_k$ is sufficiently small that
the bound \eqref{eq:fbardef} is satisfied for all $\mathbf{q}$.
Then, we have
\begin{equation}
\overline{f}(\mathbf{q}_{k\! + \! 1}) \stackrel{(a)}{\leq} \overline{f}(\mathbf{q}_{k\! + \! 1},\mathbf{q}_k)
\stackrel{(b)}{\leq} \overline{f}(\mathbf{q}_k,\mathbf{q}_k)
\stackrel{(c)}{=} f(\mathbf{q}_k),
\end{equation}
where step (a) follows from quadratic upper bound approximation
in \eqref{eq:fbardef}, step (b) is based on monotonically
decreasing behavior of the cost function when we are applying
ISTA algorithm, and in step (c)
we substituted the $\mathbf{q}_k$ in \eqref{eq:fbardef}.
As in Section~\ref{sec:adaptStep}, we can adapt the step-size
with a backtracking type rule.
\section{Numerical Simulation}
\subsection{Single-path Channel}
To assess the performance of the proposed estimators,
we first consider a theoretical single path channel.
Specifically, we assume that, in each measurement $\ell$
and transmission $d$, the
single-input multi-output (SIMO) channel is given by
\begin{equation} \label{eq:hsing}
\mathbf{h}_{\ell d} = \alpha_{\ell d} \mathbf{v}(\theta, \phi),
\end{equation}
where $\theta$ and $\phi$ are the horizontal and vertical
angles of arrival (AoAs) of the path,
$\mathbf{v}(\theta,\phi)$ is the vector antenna response
to the path, and $\alpha_{\ell d}$ is a complex scalar representing
the small scale fading -- see \cite{TseV:07} for details.
The parameters $\theta$ and $\phi$ are determined by
the large-scale path directions and are thus assumed to be constant.
However, we assume that the small-scale parameter $\alpha_{\ell d}$
is independently Rayleigh faded across different measurements
$\ell$ and $d$. Under this single-path model,
the average spatial covariance is then given by the rank one matrix
\begin{equation} \label{eq:Qsing}
\mathbf{Q} = \mathbb{E}\left[ \mathbf{h}_{\ell d}\mathbf{h}_{\ell d}^* \right] =
\mathbb{E}|\alpha|^2 \mathbf{v}(\theta, \phi)\mathbf{v}^*(\theta, \phi).
\end{equation}
We assume the power is normalized so that $\mathbb{E}|\alpha|^2=1$.
Following \cite{AkdenizCapacity:14}, we assume
a two-dimensional $4 \times 4$ $\lambda/2$ uniform linear array.
This array size can be easily accommodated in a mobile in the mmW range.
For example, at 28~GHz, the array would be only approximately 1.5 cm$^2$.
We set the SNR to 10 dB per antenna.
We then simulate the algorithm through 1000 Monte Carlo trials.
In each trial, we generate random AoAs $(\theta,\phi)$
and random search directions $\mathbf{u}_\ell$ for the $L$ measurements.
The number of measurements $L$ is varied.
The random search directions $\mathbf{u}_\ell$ are taken as the
antenna response along random angles that are generated in an i.i.d.\
manner.
Following \cite{barati2014directional},
we also take the diversity order $D=4$.
We then run the ML and approximate ML algorithm to compute
estimates $\widehat{\mathbf{Q}}$ of the true spatial covariance matrix $\mathbf{Q}$.
To evaluate the accuracy of the estimate $\widehat{\mathbf{Q}}$, we measure the loss
in beamforming resulting from the estimation errors.
In general, given the true spatial covariance matrix $\mathbf{Q}$,
the optimal long-term beamforming vector $\mathbf{w}_{opt}$ is the unit
vector directed along the maximal eigenvector of $\mathbf{Q}$.
The optimal long-term beamforming gain is then
\[
G_{opt} := \mathbf{w}_{opt}^*\mathbf{Q}\mathbf{w}_{opt} = \lambda_{max}(\mathbf{Q}),
\]
where $\lambda_{max}(\mathbf{Q})$ is the maximal eigenvalue of $\mathbf{Q}$.
For a rank-one single-path channel, the optimal beamforming
vector is simply the vector aligned to the receive spatial signature,
$\mathbf{w}_{opt} \propto \mathbf{v}(\theta,\phi)$. Also, assuming the
spatial covariance matrix is normalized to unity $\mathop{\mathrm{Tr}}(\mathbf{Q})=1$,
the optimal beamforming gain
is simply $N$, the dimension of the antenna array.
See \cite{Lozano:07} for details.
To evaluate the loss from channel estimation errors, we suppose that
the receiver applies a beamforming gain from the estimated
covariance matrix $\widehat{\mathbf{Q}}$. That is, we compute $\widehat{\mathbf{w}}$ from the
maximal eigenvector of $\widehat{\mathbf{Q}}$ and then compute the actual gain,
\[
G := \widehat{\mathbf{w}}^*\mathbf{Q}\widehat{\mathbf{w}}.
\]
The loss is then given by
\[
\mbox{loss} = 10\log_{10}(G_{opt}/G),
\]
which is the loss (in dB) due to the channel estimation error.
Fig.~\ref{fig:LossvsNwSingle} plots the mean value of the loss
as a function of the number of measurements $L$.
There are several points to observe.
First, we observe that with $L$ around 60 measurements,
the exact ML algorithm
obtains a loss of less than 0.5 dB. Second, tt should be pointed out that
since the antenna array has dimension $N=16$, the Hermetian matrix $\mathbf{Q}$
has $N(N+1)/2 = 136$ unknowns. Hence, the low-rank method is successful
in estimating the matrix well even though the number of measurements is below
the number of free parameters. This property is precisely the value
of the non-negative constraints.
Third, the approximate ML method is only slightly inferior to the exact
method. As mentioned above, the approximate ML method is
significantly simpler to implement and thus the small additional loss
may justify its use.
\begin{figure}
\centering {\includegraphics[trim=0.1cm 6cm 0.1cm 6cm ,clip=true, width=1\linewidth]{paperFigures/SinglePathn.pdf}}
\caption{\textbf{Algorithm performance on a single-path channel.}
To estimate the performance of the algorithms,
we computed the optimal beamforming vector for the estimated
channel covariance matrix and then measured the loss in beamforming
gain from applying the estimated vector relative to the optimal vector.
The loss is plotted as a function of the number of power measurements $L$
for different estimation algorithms assuming an ideal single
path channel and a per antenna SNR of 10~dB. }
\label{fig:LossvsNwSingle}
\end{figure}
Finally, as a point of comparison, Fig.~\ref{fig:LossvsNwSingle},
plots the beamforming gain from a simple algorithm based on
selecting the beamforming direction that resulted in the maximum power.
Interestingly, this simple and intuitive algorithm performs considerably
worse than the proposed method.
For example, at $L=60$ measurements, the loss is 1.5 dB, about 1 dB worse
than the proposed ML estimation method.
\subsection{Multi-path Channel using NYC Measurements}
An important and surprising finding of the mmW measurements in
New York City reported in
\cite{Rappaport:12-28G,Samimi:AoAD,rappaportmillimeter} is
that in urban micro-cellular type deployments,
mmW signals are likely to propagate via multiple paths to
the receiver. Although mmW signals are blocked by many
materials, many street-level locations were able to see base stations
at 100 to 200m via diffuse scattering and reflections, even
when situated in non-line-of-sight (NLOS) locations.
It is precisely this phenomena that enables mmW pico and micro-cellular
type deployments.
To validate the channel estimation
algorithms in these scenarios, we
next simulated the algorithms with the spatial covariance
matrix $\mathbf{Q}$ generated from the model \cite{AkdenizCapacity:14}
derived from the New York City measurements
\cite{Rappaport:12-28G,Samimi:AoAD,rappaportmillimeter}
made at 28~GHz. The model in \cite{AkdenizCapacity:14}
follows a similar form to the standard 3GPP / ITU
model~\cite{3GPP36.814,ITU-M.2134} with parameters fit to the
mmW measurements. Specifically, the channel is composed of a random
number of clusters, each cluster having some random angular spread and power.
Based on data analysis in \cite{AkdenizCapacity:14},
the mmW channel typically has one to three clusters
with a small angular spread in each clusters. Details can be found in
\cite{AkdenizCapacity:14}.
Fig. ~\ref{fig:LossvsNwMulti} plots the loss for different
number of directions, $L \in \{20,60,80,100\}$. In comparison
to the single path case, we see we need slightly more measurements.
For example, for a 0.5 dB loss, we need $L=100$ measurements.
This number is still less than the number of free parameters.
However, the other features remain the same.
Specifically, the approximate ML results in only a small loss
relative to the exact ML and both methods are considerably better than
the simple strongest power method.
\begin{figure}
\centering {\includegraphics[trim=0.1cm 6cm 0.1cm 6cm ,clip=true, width=1\linewidth]{paperFigures/MultiPathn.pdf}}
\caption{\textbf{Algorithm performance on a realistic multi-path channel.}
Details are identical to Fig.~\ref{fig:LossvsNwSingle}
except we use a multi-path channel model from \cite{AkdenizCapacity:14}
based on the real NYC measurements at 28~GHz~\cite{rappaportmillimeter}.}
\label{fig:LossvsNwMulti}
\end{figure}
\subsection{Tuning the sparsity factor $\mu$}
In the simulations up to now, we have set the sparsity regularization
parameter $\mu=0$. That is, we have used the unregularized
ML objective \eqref{eq:JQ} instead of the regularized
objective \eqref{eq:JQmu}.
However, given the low-rank nature of the channel, one may expect that
adding a regularization term to force sparsity in the singular values
of $\mathbf{Q}$ would improve the estimation.
To test this hypothesis, we evaluated the beamforming loss
as a function of $\mu$. Fig.~\ref{fig:SparsityFactor} shows
the average beamforming loss at $L=50$ measurements as a function of the
iteration number for three different values of $\mu$.
It can be seen that using a non-zero value of $\mu$ only makes the
performance worse.
Similar results were found at different values of $L$ as well.
Also, for the multipath channel, using $\mu > 0$ was even worse
since the channel is inherently higher rank.
As described in Section~\ref{sec:sparseReg}, the fact that $\mu=0$
is optimal is not entirely surprising. The optimization \eqref{eq:JQopt}
already imposes the positivity constraint $\mathbf{Q} \geq \mathbf{0}$.
Similar to \cite{lee2001algorithms},
non-negative constraints tend to naturally impose sparsity,
so it is possible that additional sparsity regularization is not necessary.
\begin{figure}
\centering {\includegraphics[trim=0.1cm 6cm 0.1cm 6cm ,clip=true, width=1\linewidth]{paperFigures/muChangen.pdf}}
\caption{Calculated loss for single path channel versus iterations that ISTA has been run for 50 different beamforming directions and different sparsity factors, $\mu = \{0,\,0.5,\,1\}$}
\label{fig:SparsityFactor}
\end{figure}
\section*{Conclusions}
Millimeter wave systems rely centrally on directional transmissions.
Due to the rapid variations in the channel and need for low-latency
communication, algorithms for fast spatial channels will thus be
key for the successful deployment of these technologies.
In this work, we have considered the estimation of the long-term
receiver-side spatial covariance of the channel from analog beamformed
power measurements. ML estimation is shown to be equivalent to
an optimization that appears as a noisy, non-negative matrix
completion problems. Fast algorithms were developed to
solve this optimization and were demonstrated on both ideal
single path channels as well as channel models derived from
real measurements in urban deployments.
The algorithms show relatively fast convergence (~100 iterations)
and can provide good tracking with significantly less number of
measurements than unknowns.
Several future avenues of work are possible. First,
we have considered only analog beamforming.
Low-bit, fully digital, as proposed in
\cite{Madhow:ADC,Madhow:largeArray}, may offer significantly
improved performance and should be investigated. Also,
the current algorithms assumes the long-term statistics
are constant. Future work may also consider tracking of these
parameters. Finally, the number of iterations for convergence
is still somewhat long. Other approaches including Fast ISTA
\cite{Nesterov:07} and approximate message passing methods
\cite{Rangan:11-ISIT} may also be considered.
|
3,212,635,537,464 | arxiv | \section{Introduction}
Macros have long been studied in AI planning~\cite{fn71,korf85}.
Many domain-dependent applications of macros have been exhibited
and studied~\cite{iba89,js01,h01}; also, a number of domain-independent methods
for learning, inferring, filtering, and applying macros have been
the topic of research continuing up to the present~\cite{bems05,cs07,nlfl07}.
In this paper, we present a domain-independent algorithm that
computes macros in a novel way.
Our algorithm computes macros ``on-the-fly'' for a given set of states
and does not require previously learned or inferred information, nor does it
need any prior domain knowledge.
We exhibit the power of our algorithm by using it to define
new domain-independent tractable classes of classical planning that strictly
extend previously defined such classes~\cite{act-local},
and can be proved to include \emph{Blocksworld-arm} and
\emph{Towers of Hanoi}.
We believe that this is notable as
theoretically defined, domain-independent
tractable classes have generally struggled to
incorporate construction-type
domains such as these two.
We hence give theoretically grounded evidence of the computational
value of macros in planning.
\paragraph{\bf Our algorithm.}
Consider the following reachability problem:
given an instance of planning and a
set $S$ of states, compute the ordered pairs of states
$(s, t) \in S \times S$ such that the second state $t$ is reachable
from the first state $s$. (By \emph{reachable}, we mean that
there is a sequence of operators that transforms the first state into
the second.) This problem is clearly hard in general, as
deciding if one state is reachable from another captures the complexity
of planning itself.
A natural--albeit incomplete--algorithm for solving this reachability problem
is to first compute the pairs $(s, t) \in S \times S$ such that
the state $t$ is reachable from the state $s$ by application of a single
operator, and then to compute the transitive closure of these pairs.
This algorithm is well-known to run in polynomial time
(in the number of states and the size of the instance)
but will only discover pairs for which the reachability is evidenced
by plans staying within the set of states $S$:
the algorithm is efficient but incomplete.
The algorithm that we introduce is a strict generalization of
this transitive closure algorithm for the described reachability problem.
We now turn to a brief, high-level description of our algorithm.
Our algorithm begins by computing the pairs connected by a single operator,
as in the just-described algorithm, but each pair is labelled with
its connecting operator.
The algorithm then continually applies two types of transformations to the
current set of pairs until a fixed point is reached.
Throughout the execution of the algorithm, every pair has an associated
label which is either a single operator or a macro derived by
combining existing labels.
The first type of transformation
(which is similar to the transitive closure)
is to take pairs of states
having the form $(s_1, s_2)$, $(s_2, s_3)$ and to add the
pair $(s_1, s_3)$ whose new label is the macro
obtained by ``concatenating''
the labels of the pairs $(s_1, s_2)$ and $(s_2, s_3)$.
If the pair $(s_1, s_3)$ is already contained in the current set,
the algorithm replaces the label of $(s_1, s_3)$ with the
new label if the new label is ``more general'' than the old one.\footnote{
For the precise definitions of ``concatenation'' and ``more general'',
please refer to the technical sections of the paper.
}
The second type of transformation is to take a state $s \in S$ and
a label of an existing pair, and to see if the label applied to $s$
yields a state $t \in S$; if so, the pair $(s, t)$ is introduced,
and the same replacement procedure as before is invoked if the pair
$(s, t)$ is already present.
Our algorithm, as with the transitive closure, operates in
polynomial time
(as proved in the paper) and is incomplete.
We want to emphasize that it can, in general, identify pairs
that are not identified by the transitive closure algorithm.
Why is this? Certainly, some state pairs $(s, t)$
introduced by the first type of
transformation have macro labels that,
if executed one operator at a time, would stay within
the set $S$, and hence are pairs that are discovered by the
transitive closure algorithm.
However, the second type of transformation may apply such a macro
to other states to discover pairs $(s, t) \in S \times S$
that would \emph{not}
be discovered by the transitive closure: this occurs when
a step-by-step execution
of the macro, starting from $s$, would leave the set $S$
before arriving to $t$.
Indeed, these two transformations depend on and feed off of each other:
the first transformation introduces increasingly powerful macros,
which in turn can be used by the second to increase the set of pairs,
which in turn permits the first to derive yet more powerful macros,
and so forth.
We now describe two concrete results to offer the reader a feel for
the power of our algorithm.
Let $s$ be any state of a \emph{Blocksworld-arm} instance,
and let $S$ be the set $H(s,4)$ of states
within Hamming distance $4$ of $s$.\footnote{
The Hamming distance between two states is defined as the number of
variables at which they differ.
}
Let us use the term \emph{subtower}
to refer to a sequence of blocks
stacked on top of one another such that the top is clear.
We prove that our algorithm, given the set $S$, will discover macros
that move any subtower of $s$ onto the ground
(preserving the subtower structure).
As another result, let $s$ be the initial state of
the \emph{Towers of Hanoi} problem, for any number of discs;
and, let $S$ be the set $H(s,7)$ of states
within Hamming distance $7$ of $s$.
We prove that our algorithm, given the set $S$, will
discover macros that, starting from the state $s$,
move any subtower of discs from the initial peg to either of the other pegs.
In particular, our algorithm will report that the goal state
is reachable from the initial state $s$.
Note that, in the case of \emph{Blocksworld-arm}, the constant $4$
is independent of the state $s$, and in particular is independent of
the height of subtowers; likewise, in \emph{Towers of Hanoi},
the constant $7$ is independent of the number of discs.
Note also that, as can be proved, the transitive closure algorithm
does not detect either of these reachability conditions, even when
$S = H(s,k)$ for an arbitrarily large constant $k$.\footnote{
In the case of \emph{Towers of Hanoi}, this follows immediately from
the known exponential lower bound on the length of a plan transforming
the initial state to the goal state. For a fixed $k \geq 1$,
when given the initial state and $H(s,k)$,
the transitive closure algorithm ``stays within the set'' $H(s,k)$,
which is of polynomial $O(n^k)$ size, and will not discover pairs
$(v, v')$ which are not linked by polynomial length plans.
}
We emphasize again that our new algorithm is fully domain-independent.
Our algorithm not only returns pairs of states, but also returns,
for each state pair $(s, t)$, a succinct representation of a plan
from $s$ to $t$, as in~\cite{jonsson07}. Note that our algorithm
may discover pairs $(s, t)$ for which the shortest plan from
$s$ to $t$ is of exponential length, when measured in terms of the
original operators, as in the \emph{Towers of Hanoi} domain.
\paragraph{\bf Towards a tractability theory of domain-independent planning.}
Many of the benchmark domains--such as \emph{Blocksworld-arm}, \emph{Gripper},
and \emph{Logistics}--can now be handled
effectively and simultaneously by domain-independent planners,
as borne out by empirical evidence~\cite{hn01}.
This \emph{empirically observed} domain-independent tractability
of many common benchmark domains naturally calls for a
\emph{theoretical explanation}.
By a theoretical explanation, we mean the formal
definition of tractable classes of planning instances, and
formal proofs that domains of interest fall into the classes.
Clearly, such an explanation could bring to the fore
structural properties shared by these benchmark domains.
To the best of our knowledge, research proposing tractable
classes has generally had other foci, such as
understanding syntactic restrictions on the operator
set~\cite{bylander94,bn95,ens95},
studying restrictions of the causal graph,
as in~\cite{bd03-unary,bd06,helmert06-fd,jonsson07},
or empirical evaluation of simplification rules~\cite{haslum07}.
Aligned with the present aims is
the work of Hoffmann~\shortcite{hoffmann-utilizing} that
gives proofs that certain benchmark domains are solvable by local search
with respect to various heuristics.
To demonstrate the efficacy of our algorithm, we use it to extend
previously defined tractable classes.
In particular, previous work~\cite{act-local}
presented a complexity measure called
\emph{persistent Hamming width (PH width)}, and demonstrated that
any set of instances having bounded PH width--PH width $k$ for some
constant $k$--is
polynomial-time tractable.
It was shown that both
the \emph{Gripper} and \emph{Logistics} domains have bounded
PH width, giving a uniform explanation for their tractability.
In the present paper, we show that an extension of this measure
yields a tractable class containing
both the \emph{Blocksworld-arm} and \emph{Towers of Hanoi} domains,
and we therefore obtain a single tractable class which captures
all four of these domains.
As mentioned, we believe that this is significant as
theoretical treatments have generally had limited coverage of
construction-type
domains such as \emph{Blocksworld-arm} and \emph{Towers of Hanoi}.
We want to emphasize that our objective here is \emph{not}
to simply establish tractability of the domains under discussion:
in them, plan generation is already well-known to
be tractable on an individual, domain-dependent basis.
Rather,
our objective is to give a \emph{uniform}, \emph{domain-independent}
explanation
for the tractability of these domains.
Neither is our goal to prove that these domains have low time complexity;
again, our primary goal is to present a simple, domain-independent
algorithm for which we can establish
tractability of these domains with respect to the heavily-studied
and mathematically robust concept of polynomial time.
\paragraph{\bf Previous work on macros.}
Macros have long been studied in planning~\cite{fn71}.
Early work includes \cite{minton85}, which developed filtering
algorithms for discovered macros, and
\cite{korf85}, which demonstrated the ability of macros to
exponentially reduce the size of the search space.
Macros have been thoroughly applied in domain-specific scenarios
such as puzzles and other games. To name some examples,
there has been work on the sliding tile puzzle~\cite{iba89},
Sokoban~\cite{js01}, and Rubik's cube~\cite{h01}.
Some recent research on integrating macros into domain-independent
planning systems is as follows.
\emph{Macro-FF}~\cite{bems05} is an extension of FF that has the ability
to automatically learn and make use of macro-actions.
\emph{Marvin}~\cite{cs07} is
a heuristic search planner
that can form so-called macro-actions
upon escaping from plateaus that can be reused for future escapes.
Both of these planners participated in the
International Planning Competition (IPC).
A method for learning macros given an arbitrary planner and example
problems from a domain is given in~\cite{nlfl07}.
A more theoretical approach was taken by~\cite{jonsson07},
who studied the use of macros in conjunction with causal graphs.
This work gives tractability results, and in particular shows that
domain-independent planners can cope with exponentially long plans
in polynomial time, which is also a feature of the present work.
The use of macros in this paper contrasts with that of most works
in that macros are generated and applied not over a domain or
even over an instance, but
with respect to a ``current state'' $s$ and
a (small) set of related states $S$. This ensures that the
macros generated
are tailored to the state set $S$, and no filtering
due to over-generation of macros is necessary.
\section{Preliminaries}
An instance of the planning problem is a tuple
$\Pi = (V, \mathsf{init}, \mathsf{goal}, A)$
whose components are described as follows.
\begin{itemize}
\item $V$ is a finite set of variables, where each
variable $v \in V$ has an associated
finite domain $D(v)$.
Note that variables are not necessarily propositional, that is,
$D(v)$ may have any finite size.
A \emph{state} is a mapping $s$ defined on the variables $V$
such that $s(v) \in D(v)$ for all $v \in V$.
A \emph{partial state} is a mapping $p$ defined on a subset
$\mathsf{vars}(p)$ of the variables $V$ such that for all $v \in \mathsf{vars}(p)$,
it holds that $p(v) \in D(v)$.
\item $\mathsf{init}$ is a state called the \emph{initial state}.
\item $\mathsf{goal}$ is a partial state.
\item $A$ is a set of \emph{actions}.
An action
$a$ consists of a
\emph{precondition} $\mathsf{pre}(a)$, which is a partial state,
as well as a \emph{postcondition} $\mathsf{post}(a)$, also a partial state.
We sometimes denote an action $a$ by $\langle \mathsf{pre}(a); \mathsf{post}(a) \rangle$.
\end{itemize}
Note that when $s$ is a state or partial state,
and $W$ is a subset of the variable set $V$,
we will use $(s \upharpoonright W)$ to denote the partial state resulting from
restricting $s$ to $W$.
We say that a state $s$ is a \emph{goal state} if
$(s \upharpoonright \mathsf{vars}(\mathsf{goal})) = \mathsf{goal}$.
We say that an action $a$ is \emph{applicable} at a state $s$
if
$(s \upharpoonright \mathsf{vars}(\mathsf{pre}(a))) = \mathsf{pre}(a)$.
We define a \emph{plan} to be
a sequence of actions $P = a_1, \ldots, a_n$.
We will always speak of actions and plans
relative to some planning instance $\Pi = (V, \mathsf{init}, \mathsf{goal}, A)$,
but we want to emphasize that when speaking (for example) of an action,
the action need not be an element of $A$; we require only that
its precondition and postcondition are partial states over $\Pi$.
Starting from a state $s$, we define the state resulting from $s$
by applying a plan $P$, denoted by $s[P]$, inductively as follows.
For the empty plan $P = \epsilon$, we define $s[\epsilon] = s$.
For non-empty plans $P$, denoting $P = P' , a$, we define
$s[P' , a]$ as follows.
\begin{itemize}
\item If $a$ is applicable at $s[P']$, then $s[P', a]$ is
the state equal to $\mathsf{post}(a)$ on variables $v \in \mathsf{vars}(\mathsf{post}(a))$,
and equal to $s[P']$ on variables $v \in V \setminus \mathsf{vars}(\mathsf{post}(a))$.
\item Otherwise, $s[P', a] = s[P']$.
\end{itemize}
We say that a state $s$ is \emph{reachable} (in an instance $\Pi$)
if there exists a plan $P$ such that $s = \mathsf{init}[P]$.
We are concerned with the problem of \emph{plan generation}:
given an instance
$\Pi = (V, \mathsf{init}, \mathsf{goal}, A)$ obtain a plan $P$ that \emph{solves} it,
that is, a plan $P$ such that $\mathsf{init}[P]$ is a goal state.
Note that sometimes we will use the representation of a partial function
$f$
as the relation
$\{ (a, b): f(a) = b \}$.
\section{Macro Computation Algorithm}
In this section, we develop
our macro computation algorithm.
This algorithm makes use of
a number of algorithmic subroutines.
In particular, we will present the two macro-producing operations
discussed in the introduction, $\mathsf{apply}$ and $\mathsf{transitive}$.
First, we define the notion of \emph{action graph}, the data structure
on which these operations work.
\begin{definition}
An \emph{action graph} is a directed graph $G$
whose vertex set, denoted by $V(G)$, is a set of states,
and whose edge set, denoted by $E(G)$, consists of labelled edges
that are actions;
we denote the label of an edge $e$ by $l_G(e)$
(or $l(e)$ when $G$ is clear from context).
Note that for every ordered pair of vertices $(s, s')$, there may be
at most one edge $(s, s')$ in $E(G)$,\footnote{
That is, an action graph is not a multigraph.}
and each edge has exactly one label.
\end{definition}
We now define three functions which will themselves be used as
subroutines in $\mathsf{apply}$ and $\mathsf{transitive}$.
\begin{definition}
We define the algorithmic function
$\mathsf{better}(a, (s, s'), G)$
as follows.
Type-wise, the function
$\mathsf{better}(a, (s, s'), G)$ requires that
$a$ is an action,
$G$ is an action graph, and $s$ and $s'$ are vertices in $G$.
The pseudocode for $\mathsf{better}(a, (s, s'), G)$ is as follows:
\begin{tiny}
\begin{verbatim}
better(a, (s, s'), G) returns boolean
{
if((s, s') not in E(G))
return TRUE;
if(pre(a) strictly contained in pre(l(s, s')) AND
post(a) contained in post(l(s, s')))
return TRUE;
if(pre(a) contained in pre(l(s, s')) AND
post(a) strictly contained in post(l(s, s')))
return TRUE;
return FALSE;
}
\end{verbatim}
\end{tiny}
\end{definition}
\begin{definition}
We define the algorithmic function
$\mathsf{addlabel}(G, s, s', a)$
as follows.
Type-wise, the function
$\mathsf{addlabel}(G, s, s', a)$ requires that
$G$ is an action graph, $s$ and $s'$ are vertices in $G$,
and $a$ is an action.
The pseudocode for $\mathsf{addlabel}(G, s, s', a)$ is as follows:
\begin{tiny}
\begin{verbatim}
addlabel(G, s, s', a) returns G'
{
G' := G;
if((s, s') not in E(G))
{
place (s, s') in E(G');
}
l_{G'}(s, s') := a;
return G';
}
\end{verbatim}
\end{tiny}
\end{definition}
We remark that in our pseudocode, the assignment operator $:=$
is intended to be a value copy
(as opposed to a reference copy, as in some programming languages).
\begin{definition}
We define the algorithmic function
$\mathsf{combine}(a, a')$ as follows.
Type-wise, the function
$\mathsf{combine}(a, a')$ requires that $a$ and $a'$ are actions.
We remark that in all cases where we use the function
$\mathsf{combine}(a, a')$, there will exist states $s_1, s_2$
such that $a$ is applicable at state $s_1$,
$s_1[a] = s_2$, and $a'$ is applicable at state $s_2$.
The pseudocode for $\mathsf{combine}(a, a')$ is as follows:
\begin{tiny}
\begin{verbatim}
combine(a, a') returns action a''
{
R := vars(pre(a)) setminus vars(post(a));
s := post(a) union (pre(a) | R);
O := vars(post(a)) setminus vars(post(a'));
pr := pre(a) union (pre(a') - s);
pos := post(a') union (post(a) | O);
return <pr; pos setminus pr>;
}
\end{verbatim}
\end{tiny}
Here, the pipe symbol $|$ should be interpreted as function restriction, and
the subtraction symbol in $(\mathsf{pre}(a') - s)$
should be interpreted as a set difference, where
the partial functions $\mathsf{pre}(a')$ and $S$ are viewed as relations.
Intuitively,
the partial state $s$ represents what we know about a state
if all we are told is that the action $a$ has just been
successfully executed.
\end{definition}
\longversion{
\begin{example}
\label{ex:combine}
Let $b_1, b_2$ be distinct blocks in the Blocksworld-arm.
We consider the computation of
$\mathsf{combine}(\mathsf{unstack}_{b_1, b_2}, \mathsf{putdown}_{b_1})$.
We have $R = \emptyset$, since all variables in
the precondition of $\mathsf{unstack}_{b_1, b_2}$ appear in the postcondition.
We have $s = \mathsf{post}(\mathsf{unstack}_{b_1, b_2})$.
And, we have $O = \{ \clear{b_2} \}$.
The returned value is thus
$\langle \clear{b_1} = \mathsf{T}, \on{b_1} = b_2, \mathsf{arm} = \empty;
\on{b_1} = \mathsf{table}, \clear{b_2} = \mathsf{T} \rangle$.
\end{example}
}
The following propositions identify key properties of the
$\mathsf{combine}$ function.
\begin{prop}
Let $a$, $a'$ be actions and let $s$ be a state.
The action $\mathsf{combine}(a, a')$ is applicable at $s$
if and only if $a$ is applicable at $s$ and $a'$ is applicable at $s[a]$.
When this occurs, $s[\mathsf{combine}(a, a')]$ is equal to
$s[a, a']$.
\end{prop}
\begin{prop}
\label{prop:combine-associative}
The function $\mathsf{combine}$ is associative.
That is,
the action
$\mathsf{combine}(\mathsf{combine}(a_1, a_2), a_3)$ is equal to the action
$\mathsf{combine}(a_1, \mathsf{combine}(a_2, a_3))$,
assuming that there exists a state $s$ such that
$a_1$ is applicable in $s$,
$a_2$ is applicable in $s[a_1]$, and
$a_3$ is applicable in $s[a_1, a_2]$.
\end{prop}
We may now define the promised macro-producing operations.
\begin{definition}
We define two algorithmic functions
$\mathsf{apply}(G, A, a, s)$ and $\mathsf{transitive}(G, s_1, s_2, s_3)$.
Type-wise, the function
$\mathsf{apply}(G, A, a, s)$ requires that $G$ is an action graph,
$A$ is a set of actions, $a$ is an action, and $s$ is a vertex of $G$.
The pseudocode for $\mathsf{apply}(G, A, a, s)$ is as follows:
\begin{tiny}
\begin{verbatim}
apply(G, A, a, s) returns G'
{
G' := G;
if( a in A OR a appears as a label in G' ) {
if( s[a] != s AND s[a] in V(G)) {
if( better(a, (s, s[a]), G) {
G' := addlabel(G, s, s[a], a);
}
}
}
return G';
}
\end{verbatim}
\end{tiny}
Type-wise, the function $\mathsf{transitive}(G, s_1, s_2, s_3)$
requires that $G$ is an action graph, and that
$s_1$, $s_2$, and $s_3$ are vertices in $G$.
The pseudocode for $\mathsf{transitive}(G, s_1, s_2, s_3)$ is as follows.
\begin{tiny}
\begin{verbatim}
transitive(G, s_1, s_2, s_3) return G'
{
G' := G;
if((s_1, s_2) in E(G) and
(s_2, s_3) in E(G)) {
a := l(s_1, s_2);
a' := l(s_2, s_3);
a'' := combine(a, a');
if( better(a'', (s_1, s_3), G) {
G' := addlabel(G, s_1, s_3, a'');
}
}
return G';
}
\end{verbatim}
\end{tiny}
Within the function $\mathsf{transitive}$, in the case that the
$\mathsf{addlabel}$ function is called and returns a graph
$G'$ that is different from the input graph $G$, we say that the
transition $(s_1, a'', s_3)$
(where $s_1, s_3, a''$ are the arguments passed to the $\mathsf{addlabel}$ function)
is \emph{produced} by the function.
\end{definition}
In general, we use the term \emph{transition} to refer to
a triple $(s, a, s')$ consisting of states $s, s'$ and an action $a$
such that $a$ is applicable at $s$ and $s[a] = s'$.
\begin{definition}
An \emph{action graph program} over a set of states $S$ and a set of actions
$A$ is a sequence of commands
$\Sigma = \sigma_1, \ldots, \sigma_n$
of the form
$\mathsf{apply}(G, A, a, s)$, with $s \in S$,
or $\mathsf{transitive}(G, s_1, s_2, s_3)$, with $s_1, s_2, s_3 \in S$.
The execution of an action graph program takes place as follows.
First, $G$ is initialized to be the action graph with $S$ as vertices
and no edges.
Then, the commands of $\Sigma$ are executed in order;
for each $i$, after $\sigma_i$ is executed, $G$ is replaced with the
returned value.
\end{definition}
\longversion{
\begin{example}
We give an example action graph program over
the Blocksworld-arm domain with block set $B = \{ b_1, b_2 \}$.
The program is over the set of all actions $A$ with respect to this
block set, and the state set $S$ containing the following
three states:
$s_1$, where $b_1$ is on top of $b_2$;
$s_2$, where $b_1$ is in the arm and $b_2$ is on the table; and,
$s_3$ where both $b_1$ and $b_2$ are on the table.
Formally, we define
$s_1 = \{ \mathsf{arm} = \empty, \on{b_1} = b_2, \clear{b_1} = \mathsf{T}, \on{b_2} = \mathsf{table}, \clear{b_2} = \mathsf{F} \}$,
$s_2 = \{ \mathsf{arm} = b_1, \on{b_1} = \mathsf{arm}, \clear{b_1} = \mathsf{F}, \on{b_2} = \mathsf{table}, \clear{b_2} = \mathsf{T} \}$, and
$s_3 = \{ \mathsf{arm} = \empty, \on{b_1} = \mathsf{table}, \clear{b_1} = \mathsf{T}, \on{b_2} = \mathsf{table}, \clear{b_2} = \mathsf{T} \}$.
Consider the program
$\Sigma = \sigma_1, \sigma_2, \sigma_3, \sigma_4$
with four commands where
\begin{align*}
\sigma_1 & = \mathsf{apply}(G, A, \mathsf{unstack}_{b_1, b_2}, s_1 )\\
\sigma_2 & = \mathsf{apply}(G, A, \mathsf{pickup}_{b_2}, s_2 )\\
\sigma_3 & = \mathsf{apply}(G, A, \mathsf{putdown}_{b_1}, s_2 )\\
\sigma_4 & = \mathsf{transitive}(G, s_1, s_2, s_3).
\end{align*}
Let us consider the execution of the program.
Initially, $G$ is set to be the action graph with vertex set
$S = \{ s_1, s_2, s_3 \}$ and no edges.
Let us use $G_i$ to denote the graph returned by the command
$\sigma_i$.
After the execution of $\sigma_1$, we have
$G = G_1$ where $G_1$ is equal to $G$, but with a directed edge
from $s_1$ to $s_2$ with the action $\mathsf{unstack}_{b_1, b_2}$ as label.
When $\sigma_2$ executes, inside this command we have
$s[a] = s$ (since $s_2[\mathsf{pickup}_{b_2}] = s_2$) and thus the same
graph that was passed in is returned.
Thus, after $\sigma_2$, we have
$G = G_2 = G_1$.
The execution of $\sigma_3$ will produce a transition, and the
returned graph $G_3$ will be equal to $G_1 = G_2$,
but with an edge from $s_2$ to $s_3$ with label $\mathsf{putdown}_{b_1}$.
Finally, the execution of $\sigma_4$ will also produce a transition,
and the returned graph $G_4$ will be equal to $G_3$,
but with an edge from $s_1$ to $s_3$ with label $a_c$ where
$a_c$ is the combined action from Example~\ref{ex:combine}.
\end{example}
}
The following is our macro computation algorithm.
As input, it takes a set of states $S$ and a set of actions $A$.
The running time can be bounded by
$O(n |S|^3( |A| + |S|^2 ))$, where $n$ denotes the number of variables.
\begin{tiny}
\begin{verbatim}
compute_macros(S, A) returns G, M
{
M := empty;
V(G) := S;
E(G) := empty set;
do {
A' := (A union l(E(G)));
for all: a in A', s in V(G) {
G := apply(G, A, a, s);
}
for all s1, s2, s3 in V(G) {
G := transitive(G, s1, s2, s3);
if(transitive produces a transition) {
append "l(s1, s3) = l(s1, s2), l(s2, s3)" to M;
}
}
}
while(some change was made to G)
return (G, M);
}
\end{verbatim}
\end{tiny}
\paragraph{Understanding compute\_macros.}
By a \emph{combination} over $A$, we mean an action in $A$ or an action
that can be derived from actions in $A$ by (possibly multiple) applications
of the $\mathsf{combine}$ function.
\begin{definition}
We say that a transition $(s, a, s')$
is \emph{condition-minimal} with respect to a set of actions $A$ if for
any combination $a'$ over $A$, if $s[a'] = s'$ then
$\mathsf{pre}(a) \subseteq \mathsf{pre}(a')$ and $\mathsf{post}(a) \subseteq \mathsf{post}(a')$
(when $\mathsf{pre}(a)$, $\mathsf{pre}(a')$, $\mathsf{post}(a)$, and $\mathsf{post}(a')$
are viewed as relations).
\end{definition}
Having defined the notion of a \emph{condition-minimal} transition,
we can now naturally define the notion of a
\emph{condition-minimal} program.
\begin{definition}
Relative to a planning instance $\Pi$,
let $S$ be a set of states, and let $A$, $A'$ be sets of actions.
An \emph{$A$-condition-minimal-program}
(for short, $A$-CM-program)
over states $S$ and actions $A'$
is an action graph program over $S$ and $A$
such that when executed,
$\mathsf{apply}$ is only passed pairs $(a, s)$ such that $(s, a, s[a])$
is condition-minimal with respect to $A$, and
the $\mathsf{transitive}$ commands produce
only transitions that are condition-minimal with respect to $A$.
\end{definition}
We now define a notion of \emph{derivable} action. This notion is
defined recursively. Roughly speaking,
derivable actions are actions that will provably be discovered
as macros by the algorithm.
\begin{definition}
Relative to a planning instance $\Pi$, let $S$ be a set of states, and
let $A$ be a set of actions. We define the set of $(S, A)$-derivable
actions recursively, as the smallest set satisfying: any action of a
transition produced by an $A$-CM-program over states $S$ and the set
of actions that are $(S, A)$-derivable or in $A$, is $(S, A)$-derivable.
\end{definition}
\begin{lemma}
\label{lemma:discovery}
Relative to a planning instance $\Pi$ with action set
$A$, let $s$ be a state.
Any $(H(s,k), A)$-derivable action is discovered
by a call to the function \verb1compute_macros1 with the first two
arguments $H(s, k)$ and $A$, by which we mean that any such an action
will appear as an edge label in the graph output by \verb1compute_macros1.
\end{lemma}
We emphasize that, in the \verb1compute_macros1 procedure,
labels of edges are merely actions, which (as defined) are
precondition-postcondition pairs that need not appear in the
original set of actions $A$. When new edge labels are introduced,
they are always obtained from existing labels or from $A$
via the \verb1combine1 procedure, which permits the general
applicability of edge labels.
\begin{pf-sketch}
Let $\Sigma = \sigma_1, \ldots, \sigma_n$ be an
$A$-CM-program over $H(s, k)$ and actions that are discovered by
\verb1compute_macros1, and let $H$ be the graph returned by
\verb1compute_macros1; we prove the result by induction.
We consider the execution of the program $\Sigma$ with graph $G$.
We prove by induction on $i \geq 1$ that after the command
$\sigma_i$ is executed and returns graph $G_i$,
for every edge
$(s, s') \in E(G_i)$, it holds that
$(s, s') \in E(H)$ and $l_{G_i}(s, s') = l_H(s, s')$.
If $\sigma_i$ is an $\mathsf{apply}$ command (with arguments $s$ and $a$)
that effects a change in the graph,
then the input action must be in $l(E(G_i))$.
The command $\sigma_i$ can be successfully applied
at $H$.
Since $H$ is a fixed point over all $\mathsf{apply}$ and $\mathsf{transitive}$
commands, the action $a$ passed to $\mathsf{apply}$
or one that is better
(according to the function \verb1better1) must appear in $H$
at $l_H(s, s[a])$.
By condition-minimality of $(s, a, s[a])$,
we have that $a = l_H(s, s[a])$.
If $\sigma_i$ is a $\mathsf{transitive}$ command that produces a transition
$(s, a, s')$, then the actions $a'$ and $a''$ (from within the
execution of the command), by induction hypothesis, appear in $H$.
Since $H$ is a fixed point over all $\mathsf{apply}$ and $\mathsf{transitive}$ commands,
the action $\mathsf{combine}(a, a')$ or one that is better
must appear in $H$ at $l_H(s, s')$. By condition-minimality of
$(s, \mathsf{combine}(a, a'), s')$, we have that
$\mathsf{combine}(a, a') = l_H(s, s')$.
\end{pf-sketch}
\section{Examples}
\paragraph{Blocksworld-arm.}
We will present results with respect to the following formulation of the
Blocksworld-arm domain, which is based strongly on the
propositional STRIPS formulation.
We choose this
formulation primarily to lighten the presentation, and remark that
it is straightforward to verify that our proofs and results apply to the
propositional formulation.
\longversion{
We give the description of the used formulation right now, as we will use it
throughout the paper to present examples of introduced notions.}
\begin{domain} (Blocksworld-arm domain)
We use a formulation of this domain where there is an arm.
Formally, in an instance $\Pi=(V, \mathsf{init}, \mathsf{goal}, A)$
of the Blocksworld-arm domain, there is a set of blocks $B$,
and the variable set $V$ is defined as
$\{ \mathsf{arm} \} \cup \{ \on{b}: b \in B \} \cup \{ \clear{b}: b \in B \}$ where
$D(\mathsf{arm}) = \{ \empty \} \cup B$ and
for all $b \in B$,
$D(\on{b}) = \{ \mathsf{table}, \mathsf{arm} \} \cup B$
and
$D(\clear{b}) = \{ \mathsf{T}, \mathsf{F} \}$.
The $\on{b}$ variable tells what the block $b$ is on top of, or whether
it is being held by the arm, and the $\clear{b}$ variable tells
whether or not the block $b$ is clear.
\oldshortversion{
There are four kinds of actions:
$\mathsf{pickup}_b$, $\mathsf{putdown}_b$, $\mathsf{unstack}_{b,c}$ and $\mathsf{stack}_{b,c}$.
For example,
$\forall b \in B$, $\mathsf{pickup}_b =
\langle \clear{b} = \mathsf{T}, \on{b} = \mathsf{table}, \mathsf{arm} = \empty;
\clear{b} = \mathsf{F}, \on{b} = \mathsf{arm}, \mathsf{arm} = b \rangle$.
Also,
$\forall b, c \in B$, $\mathsf{unstack}_{b, c} =
\langle \clear{b} = \mathsf{T}, \on{b} = c, \mathsf{arm} = \empty;
\clear{b} = \mathsf{F}, \on{b} = \mathsf{arm}, \mathsf{arm} = b, \clear{c} = \mathsf{T} \rangle$.\end{domain}
}
\proofsversion{
There are four kinds of actions.
\begin{itemize}
\item $\forall b \in B$, $\mathsf{pickup}_b =
\langle \clear{b} = \mathsf{T}, \on{b} = \mathsf{table}, \mathsf{arm} = \empty;
\clear{b} = \mathsf{F}, \on{b} = \mathsf{arm}, \mathsf{arm} = b \rangle$
\item $\forall b \in B$, $\mathsf{putdown}_b =
\langle \mathsf{arm} = b;
\mathsf{arm} = \empty, \clear{b} = \mathsf{T}, \on{b} = \mathsf{table} \rangle$
\item $\forall b, c \in B$, $\mathsf{unstack}_{b, c} =
\langle \clear{b} = \mathsf{T}, \on{b} = c, \mathsf{arm} = \empty;
\clear{b} = \mathsf{F}, \on{b} = \mathsf{arm}, \mathsf{arm} = b, \clear{c} = \mathsf{T} \rangle$
\item $\forall b, c \in B$, $\mathsf{stack}_{b, c} =
\langle \mathsf{arm} = b, \clear{c} = \mathsf{T};
\mathsf{arm} = \empty, \clear{c} = \mathsf{F}, \clear{b} = \mathsf{T}, \on{b} = c \rangle$
\end{itemize}
\longversion{
We say that a state $s$ is \emph{consistent} if it satisfies the following restrictions.
\begin{itemize}
\item $\forall b'\in B, s(\clear{b'}) = \mathsf{T} \Rightarrow | \{b \in B| s(\on{b})=b'\} | = 0$.
\item $\forall b'\in B, s(\clear{b'}) = \mathsf{F} \Rightarrow | \{b \in B| s(\on{b})=b'\} | = 1$.
\item $\forall b\in B, s(\mathsf{arm}) = b \Leftrightarrow s(\on{b}) = \mathsf{arm}$.
\end{itemize}
We define the planning domain Blocksworld-arm as the set of those planning
instances $\Pi$ such that the goal state is total and the states
$\mathsf{init}$ and $\mathsf{goal}$ are consistent. With a little more effort our results
extend to the case where the $\mathsf{goal}$ may be an arbitrary partial state.
It is easy to prove that in any Blocksworld-arm planning instance a state
$s$ is reachable if and only if $s$ is consistent. In particular, all
planning instances with consistent $\mathsf{goal}$ state are solvable.
}
\end{domain}
}
\newcommand{\pbottom}[1]{\mathsf{bottom}(#1)}
\newcommand{\ptop}[1]{\mathsf{top}(#1)}
\longversion{
We show in this section that the planning domain Blocksworld-arm has
MPH width $10$. This implies that the domain-independent planner
described in the previous section solves any Blocksworld-arm instance
in polynomial time.
We should remark that we believe a closer analysis will yield a
lower value for the MPH width of Blocksworld-arm; our main focus here is
simply showing \emph{bounded} MPH width.
}
\longversion{We introduce the following definitions.}
\begin{definition}
Relative to an instance $\Pi$ of Blocksworld-arm and a reachable state
$s$ of $\Pi$,
a \emph{pile} $P$ of $s$ is a non-empty sequence of blocks $(b_1,
\ldots, b_k)$ such that $s(\on{b_i})=b_{i+1}$ for all $i\in[1,k-1]$.
The \emph{top} of the pile $P$ is the block
$\ptop{P}=b_1$, and the \emph{bottom} of the pile is
the block $\pbottom{P}=b_k$. The \emph{size} of $P$ is $|P|=k$.
A \emph{sub-tower} of $s$ is a pile $P$ such that
$s(\clear{\ptop{P}})=\mathsf{T}$; a tower is a sub-tower such that
$s(\on{\pbottom{P}})=\mathsf{table}$.
We use the notation $P_{\geq}(b)$ (respectively,
$P_{>}(b)$, $P_{\leq}(b)$, $P_{<}(b)$) to denote the sub-tower
with bottom block $b$ (respectively, the sub-tower stacked on $b$,
and the piles supporting $b$, either including $b$ or not.)
\end{definition}
\begin{definition}
Let $\Pi$ be a planning instance of Blocksworld-arm. Let $P=(b_1, \ldots,
b_k)$ be a sequence of blocks, and $b$ and $b'$ two different blocks
not in $P$. Let $S$ be the partial state $\{\clear{b_1}=\mathsf{T},
\mathsf{arm}=\empty, \on{b_1}=b_2, \ldots, \on{b_{k-1}}=b_{k}\}$. We define
several actions with $S$ as common precondition.
\begin{itemize}
\item The action $\mathsf{subtow\mbox{-}table}_{P, b} = \langle S, \on{b_k}=b;
\on{b_k}=\mathsf{table}, \clear{b}=\mathsf{T} \rangle$ moves a sub-tower $P$ from a
block $b$ to the table.
\item The action $\mathsf{subtow\mbox{-}block}_{P, b, b'} = \langle S, \on{b_k}=b,
\clear{b'}=\mathsf{T}; \on{b_k}=b', \clear{b}=\mathsf{T}, \clear{b'}=\mathsf{F}
\rangle$ moves a sub-tower $P$ from a block $b$ onto a block $b'$.
\item The action $\mathsf{tow\mbox{-}block}_{P, b'} = \langle S, \on{b_k}=\mathsf{table},
\clear{b'}=\mathsf{T}; \on{b_k}=b', \clear{b'}=\mathsf{F} \rangle$ moves a
tower $P$ onto a block $b'$.
\end{itemize}
\longversion{We remind the reader
that these actions do not belong to the set of actions $A$ of $\Pi$.}
\end{definition}
\begin{theorem} \label{th:move-towers}
Let $\Pi$ be a planning instance of Blocksworld-arm, and let $s$ be a
reachable state with $s(\mathsf{arm})=\empty$.
\begin{itemize}
\item If $P$ is a sub-tower of $s$ and $s(\on{b_k})=b$, then
$\mathsf{subtow\mbox{-}table}_{P,b}$ is
$(H(s,4),A)$-derivable.
\item If $P$ is a sub-tower of $s$, $s(\on{b_k})=b$ and
$s(\clear{b'})=\mathsf{T}$, then $\mathsf{subtow\mbox{-}block}_{P,b,b'}$ is
$(H(s,5),A)$-derivable.
\item If $P$ is a tower of $s$, $s(\on{b_k})=\mathsf{table}$ and
$s(\clear{b'})=\mathsf{T}$, then $\mathsf{tow\mbox{-}block}_{P,b'}$ is
$(H(s,4),A)$-derivable.
\end{itemize}
\end{theorem}
\begin{pf-sketch}
The proof has two parts.
First, we show that the aforementioned
actions are condition-minimal. Then, we describe how to obtain an
$A$-CM-program that produces the actions inside $H(s,5)$. We consider
the case $a=\mathsf{subtow\mbox{-}block}_{P,b,b'}$;
the remaining actions
admit similar proofs that only require Hamming distance $4$.
To prove condition-minimality of action $a$ we consider any
combination $C=(a_1, \ldots, a_t)$ of primitive actions from $A$ such
that $s[C]=s[a]$. We must show that the actions $\mathsf{unstack}_{b_1, b_2},
\ldots, \mathsf{unstack}_{b_k, b}, \mathsf{stack}_{b_k, b'}$ appear in $C$ in the
given relative order, and that no matter what are the remaining
actions of $C$, this already implies that $\mathsf{pre}(a)\subseteq \mathsf{pre}(C)$
and $\mathsf{post}(a)\subseteq \mathsf{post}(C)$. We remark that the proof is not
straight-forward, since $\mathsf{pre}(C)$ and $\mathsf{post}(C)$ are the result of
applying the $\mathsf{combine}$ subroutine to several actions not yet determined.
To prove that there exists an $A$-CM-program that produces actions
$\mathsf{subtow\mbox{-}table}$ and $\mathsf{tow\mbox{-}block}$ inside $H(s,4)$
we use a mutual induction; we omit the proof here. We then use these
results for $\mathsf{subtow\mbox{-}block}$, the proof for which we sketch here.
Precisely, we now show that $\mathsf{subtow\mbox{-}block}_{P,b,b'}$ is
$(H(s,5),A)$-derivable.
When $|P|=1$, we derive $\mathsf{subtow\mbox{-}block}_{P,b,b'}$ by combining
actions $a_1=\mathsf{unstack}_{b_1,b}$ and $a_2=\mathsf{stack}_{b_1, b'}$. The states
$s[a_1]$ and $s[a_1,a_2]$ differ from $s$ respectively $4$ and $3$
variables, so both states lie inside $H(s,5)$. When $|P|=k$, let
$P'=P_{>}(b_k)$ in state $s$. We use the derivable actions
$a_1=\mathsf{subtow\mbox{-}table}_{P', b_k}$, $a_2=\mathsf{unstack}_{b_k, b}$,
$a_3=\mathsf{stack}_{b_k, b'}$ and $a_4=\mathsf{tow\mbox{-}block}_{P', b_k}$. It is easy
to check that the state $s[a_1, a_2, a_3]$ is the one that is furthest
from $s$, differing at the $5$ variables $\clear{b}$,
$\on{b_{k-1}}$, $\clear{b_k}$, $\on{b_k}$ and $\clear{b'}$.
\end{pf-sketch}
\paragraph{Towers of Hanoi.}
\newcommand{\mathsf{move}}{\mathsf{move}}
\shortversion{
We study the formulation
of \emph{Towers of Hanoi}
where, for every disk $d$, a variable stores
the position (that is, the disk or the peg) the disk $d$ is on.
Formally, in an instance $\Pi=(V, \mathsf{init}, \mathsf{goal}, A)$ of the Towers of
Hanoi domain, there is an ordered set of disks $D=\{d_1, \ldots,
d_k\}$ and a partially ordered set of positions $P=D\cup\{p_1, p_2,
p_3\}$, where $d_i<p_j$ for every $i$ and $j$. The set of variables
$V$ is defined as $\{\on{d}: d \in D\} \cup \{\clear{x}: x \in P\}$,
where $D(\on{d})=P$ and $D(\clear{x})=\{\mathsf{T}, \mathsf{F}\}$.
The only actions in Towers of Hanoi are movement actions that move a
disk $d$ into a position $x$, provided that both $d$ and $p$ are clear and
$d<x$.
\begin{itemize}
\item $\forall d \in D$, $\forall x, x'\in P$, if $d<x$, then define
$\mathsf{move}_{d, x', x} = \langle \clear{d} = \mathsf{T}, \clear{x}=\mathsf{T},
\on{d} = x'; \clear{x} = \mathsf{F}, \clear{x'} = \mathsf{T}, \on{d}=
x\rangle$
\end{itemize}
We define this planning domain as the set of those planning
instances $\Pi$ such that the $\mathsf{init}$ and $\mathsf{goal}$ are certain
predetermined total states. Namely, in both states $\mathsf{init}$ and $\mathsf{goal}$
it holds $\on{d_i}=d_{i+1}$ for all $i\in[1, \ldots, k-1]$,
$\clear{d_1}=\mathsf{T}$, $\clear{d_i}=\mathsf{F}$ for all $i\in[2,k]$
and $\clear{p_2}=\mathsf{T}$. They only differ in three variables:
$\mathsf{init}(\on{d_k})=p_1$, $\mathsf{init}(\clear{p_1})=false$ and
$\mathsf{init}(\clear{p_3})=\mathsf{T}$, but $\mathsf{goal}(\on{d_k})=p_3$,
$\mathsf{goal}(\clear{p_1})=\mathsf{T}$ and $\mathsf{goal}(\clear{p_3})=\mathsf{F}$.
\begin{definition}
Let $\Pi$ be a planning domain instance of Towers of Hanoi. Let $i$ be
an integer $i\in[1,k]$. Let $x=\mathsf{init}(\on{d_i})$ and $x'\in\{p_2,
p_3\}$. We define the action $\mathsf{subtow\mbox{-}pos}_{i, x, x'} =
\langle \clear{d_1}=\mathsf{T}, \on{d_1}=d_2, \ldots, \on{d_{i-1}}=d_i,
\on{d_i}=x, \clear{x'}=\mathsf{T}; \on{d_i}=x', \clear{x}=\mathsf{T},
\clear{x'}=\mathsf{F}\rangle$, that is, the action that moves the tower
of depth $i$
from $x$ to $x'$.
\end{definition}
\begin{theorem}
The actions $\mathsf{subtow\mbox{-}pos}_{i, x, x'}$ are
$(H(\mathsf{init},7),A)$-derivable.
\end{theorem}
}
We prove this by induction on $i$, the height of the subtower.
To derive actions of the form $\mathsf{subtow\mbox{-}pos}_{i+1, x, x'}$ from
the actions of the form $\mathsf{subtow\mbox{-}pos}_{i, x, x'}$,
we make use of the classical recursive solution to Towers of Hanoi;
an analysis shows that this recursive step stays within Hamming distance
$7$ of the initial state.
\longversion{
\begin{domain} (Towers of Hanoi domain)
We study the formulation where, for every disk $d$, a variable stores
the position (that is, the disk or the peg) the disk $d$ is on.
Formally, in an instance $\Pi=(V, \mathsf{init}, \mathsf{goal}, A)$ of the Towers of
Hanoi domain, there is an ordered set of disks $D=\{d_1, \ldots,
d_k\}$ and a partially ordered set of positions $P=D\cup\{p_1, p_2,
p_3\}$, where $d_i<p_j$ for every $i$ and $j$. The set of variables
$V$ is defined as $\{\on{d}: d \in D\} \cup \{\clear{x}: x \in P\}$,
where $D(\on{d})=P$ and $D(\clear{x})=\{\mathsf{T}, \mathsf{F}\}$.
The only actions in Towers of Hanoi are movement actions that move a
disk $d$ into a position $x$, provided that both $d$ and $p$ are clear and
$d<x$.
\begin{itemize}
\item $\forall d \in D$, $\forall x, x'\in P$, if $d<x$, then define
$\mathsf{move}_{d, x', x} = \langle \clear{d} = \mathsf{T}, \clear{x}=\mathsf{T},
\on{d} = x'; \clear{x} = \mathsf{F}, \clear{x'} = \mathsf{T}, \on{d}=
x\rangle$
\end{itemize}
We define this planning domain as the set of those planning
instances $\Pi$ such that the $\mathsf{init}$ and $\mathsf{goal}$ are certain
predetermined total states. Namely, in both states $\mathsf{init}$ and $\mathsf{goal}$
it holds $\on{d_i}=d_{i+1}$ for all $i\in[1, \ldots, k-1]$,
$\clear{d_1}=\mathsf{T}$, $\clear{d_i}=\mathsf{F}$ for all $i\in[2,k]$
and $\clear{p_2}=\mathsf{T}$. They only differ in three variables:
$\mathsf{init}(\on{d_k})=p_1$, $\mathsf{init}(\clear{p_1})=false$ and
$\mathsf{init}(\clear{p_3})=\mathsf{T}$, but $\mathsf{goal}(\on{d_k})=p_3$,
$\mathsf{goal}(\clear{p_1})=\mathsf{T}$ and $\mathsf{goal}(\clear{p_3})=\mathsf{F}$.
\end{domain}
We show that the following actions are $(H(\mathsf{init},7),A)$-derivable.
\begin{definition}
Let $\Pi$ be a planning domain instance of Towers of Hanoi. Let $i$ be
an integer $i\in[1,k]$. Let $x=\mathsf{init}(\on{d_i})$ and $x'\in\{p_2,
p_3\}$. We define the action $\mathsf{subtow\mbox{-}pos}_{i, x, x'} =
\langle \clear{d_1}=\mathsf{T}, \on{d_1}=d_2, \ldots, \on{d_{i-1}}=d_i,
\on{d_i}=x, \clear{x'}=\mathsf{T}; \on{d_i}=x', \clear{x}=\mathsf{T},
\clear{x'}=\mathsf{F}\rangle$, that is, the action that moves the tower
of depth $i$
from $x$ to $x'$.
\end{definition}
}
\section{Width}
In this section, we present the definition of
macro persistent Hamming width and present the width results on domains.
For a state $s$, we define $\mathsf{wrong}(s)$ to be the variables that are not
in the goal state, that is,
$\mathsf{wrong}(s) = \{ v \in \mathsf{vars}(\mathsf{goal}) ~|~ s(v) \neq \mathsf{goal}(v) \}$.
\begin{definition}
With respect to a planning instance $(V, \mathsf{init}, \mathsf{goal}, A)$,
we say that a state $s'$ is an improvement of a state $s$ if
\begin{itemize}
\item for all $v \in V$, if $v \in \mathsf{vars}(\mathsf{goal})$ and
$s(v) = \mathsf{goal}(v)$, then $s'(v) = \mathsf{goal}(v)$; and,
\item there exists $u \in \mathsf{vars}(\mathsf{goal})$
such that $u \in \mathsf{wrong}(s)$ and
$s'(u) = \mathsf{goal}(u)$.
\end{itemize}
In this case, we say that such a variable $u$ is a variable being improved.
\end{definition}
\begin{definition}
\label{def:plan-improves-state}
With respect to a planning instance $(V, \mathsf{init}, \mathsf{goal}, A)$,
we say that a plan $P$ \emph{improves} a state $s$
if $s[P]$ is a goal state, or $s[P]$ is an improvement of $s$.
\end{definition}
\longversion{
We remark that in Definition~\ref{def:plan-improves-state}, we permit
$P$ to be the empty plan $\epsilon$; in particular,
we have that the empty plan improves any goal state.
\begin{definition}
A planning instance $(V, \mathsf{init}, \mathsf{goal}, A)$
has \emph{persistent Hamming width $k$}
(for short, \emph{PH width $k$}) if no plan exists, or
for every reachable state $s$,
there exists a plan (over $A$) improving $s$ that
stays within Hamming distance $k$ of $s$.
\end{definition}
In this definition, when we say
that a plan stays within Hamming distance $k$ of a state $s$,
we mean that when the plan is executed in $s$, all intermediate states
encountered (as well as the final state) are within Hamming distance $k$
of $s$.
As the empty plan improves any goal state, to show that a
planning instance has PH width $k$ (according to the given definition),
the interesting case is to consider reachable states $s$ that are
not goal states.
}
Relative to a planning instance,
we say that a state $s$ dominates another state $s'$ if
$\{ v \in V: s(v) \neq s'(v) \} \subseteq \mathsf{vars}(\mathsf{goal})$ and
$\mathsf{wrong}(s) \subseteq \mathsf{wrong}(s')$; intuitively, $s'$ may differ from $s$
only in that it may have more variables set to their goal position.
Recall that
for a state $s$ and natural number $k \geq 0$,
we use $H(s, k)$ to denote the set of all states within Hamming distance
$k$ from $s$.
We now give the official definition of our new width notion.
\begin{definition}
A planning instance $(V, \mathsf{init}, \mathsf{goal}, A)$
has \emph{macro persistent Hamming width $k$}
(for short, \emph{MPH width $k$}) if no plan exists, or
for every reachable state $s$ dominating the initial state $\mathsf{init}$,
there exists a plan over $(H(s,k), A)$-derivable actions
improving $s$ that stays within Hamming distance $k$ of $s$.
\end{definition}
It is straightforwardly verified that if an instance has
PH width $k$, then it has MPH width $k$.
We now give a
polynomial-time algorithm for sets of planning instances
having bounded MPH width. We establish the following theorem.
\begin{theorem}
\label{thm:mph-algorithm}
Let $\mathcal{C}$ be a set of planning instances having
MPH width $k$. The plan generation problem for $\mathcal{C}$
is solvable in polynomial time via the following algorithm,
in time
$O(n^{3k+2} d^{3k} (a + (nd)^{2k}))$.
Here, $n$ denotes the number of variables,
$d$ denotes the maximum size of a domain,
and
$a$ denotes the number of actions.
\end{theorem}
\begin{tiny}
\begin{verbatim}
solve_mph((V, init, goal, A), k)
{
Q := empty plan;
M := empty set of macros;
s := init;
while( s not a goal state ) {
(G, M') := compute_macros(H(s,k), A);
append M' to M;
if(an improvement s' of s is reachable from s in G) {
s := s';
}
else {
print "?";
halt;
}
append l(s, s') to Q;
}
print M;
print Q;
}
\end{verbatim}
\end{tiny}
\longversion{By the notation $l(E(G))$, we mean the set of labels
$\{ l(e): e \in E(G) \}$.
We remark that \verb1solve_mph1 can really be viewed as an extension
of an algorithm for persistent Hamming (PH) width; one essentially
obtains an algorithm for PH width from \verb1solve_mph1 by
replacing the call to \verb1compute_macros1 with a command that
simply sets $G$ to be the directed graph with vertex set $H(s, k)$ and
an edge $(s_1, s_2)$ present if there is an action $a$ in $A$
such that $s_1[a] = s_2$.
}
\begin{pf-sketch}
Let $\Pi \in \mathcal{C}$ be a planning instance such that
there exists a plan for $\Pi = (V, \mathsf{init}, \mathsf{goal}, A)$. We want to show that
\verb1solve_mph1 outputs a plan.
During the execution of \verb1solve_mph1, the state $s$ can only
be replaced by states that are improvements of it,
and thus $s$ always dominates the initial state $\mathsf{init}$.
By definition of MPH width, then, for any $s$ encountered during
execution, there exists a plan over $(H(s, k), A)$-derivable actions
improving $s$ staying within Hamming distance $k$ of $s$.
By Lemma~\ref{lemma:discovery}, all of the actions are discovered by
\verb1compute_macros1, and thus the reachability check in
\verb1solve_mph1 will find an improvement.
We now perform a running time analysis of the algorithm.
Let $v$ denote the number of vertices in the graphs in
\verb1compute_macros1, that is, $|H(s,k)|$.
We have $v \leq {n \choose k} d^k \in O((nd)^k)$.
Let $e$ be the maximum number of edges; we have
$e = {v \choose 2} \in O((nd)^{2k})$.
The \verb1do-while1 loop in \verb1compute_macros1 will execute
at most $2n \cdot e \in O(ne)$ times, since once an edge is introduced,
its label may change at most $2n$ times, by definition of \emph{better}.
Each time this loop iterates, it uses no more than
$(a + e)v + v^3$ time:
\emph{apply} can be called on no more than
$(a+e)v$ inputs, and \emph{transitive}
can be called on no more than $v^3$ inputs.
The while loop in \verb1solve_mph1 loops at most $n$ times,
and each time, by the previous discussion,
it requires $ne((a+e)v + v^3)$
time for the call to \verb1compute_macros1,
and $(v+e)$ time for the reachability check.
The total time is thus
$O(n( ne((a+e)v + v^3) + (v + e)))$
which is
$O(n^2 e((a+e)v + v^3))$
which is
$O(n^2 e(a+e)v)$ which is
$O(n^{3k+2} d^{3k} (a + (nd)^{2k}))$.
\end{pf-sketch}
\shortversion{
\paragraph{Blocksworld.}
\begin{theorem}
\label{thm:blocksworld-10}
All instances of the Blocksworld-arm domain have MPH-width $10$.
\end{theorem}
\oldshortversion{
In the proof of this theorem, we show how to
improve any reachable state $s$ of \emph{Blocksworld-arm}.
The proof is conceptually simple:
improve $s$ just by moving around a few piles of blocks. For instance,
if $s(\on{b})=b'$ but $\mathsf{goal}(\on{b})=b''$, apply actions
$\mathsf{subtow\mbox{-}table}_{P_{>}(b''), b''}$, $\mathsf{subtow\mbox{-}block}_{P_{\geq}(b),
b', b''}$. There is, however, a technical difficulty:
we must not forget that variables that were
already in the goal state in $s$ must remain so after the
improvement. For instance, if $b$ was on top of $b'$ in $s$, then
unstacking $b$ from $b'$ will make $\clear{b'}$ change from $\mathsf{F}$
to $\mathsf{T}$.
The solution to this involves considering that
if something is placed on
top of $b'$, this movement may affect some other variable
which was already in the goal state, and so forth.
}
}
\proofsversion{
According to Theorem~\ref{th:move-towers}, at any state $s$ we may
consider our set of applicable actions enriched by this new macro-actions.
We now show how can these new actions be used to improve any reachable
state $s$. The proof is conceptually simple:
improve $s$ just by moving around a few piles of blocks. For instance,
if $s(\on{b})=b'$ but $\mathsf{goal}(\on{b})=b''$, apply actions
$\mathsf{subtow\mbox{-}table}_{P_{>}(b''), b''}$, $\mathsf{subtow\mbox{-}block}_{P_{\geq}(b),
b', b''}$. However, we must not forget that variables that were
already in the goal state in $s$ must remain so after the
improvement. For instance, if $b$ was on top of $b'$ in $s$, then
unstacking $b$ from $b'$ will make $\clear{b'}$ change from $\mathsf{F}$
to $\mathsf{T}$. We may try to solve this by placing anything whatever on
top of $b'$, but then this movement may affect some other variable
which was already in the goal state, and so forth.
The following lemma is a case-by-case analysis of the solution to the
difficulty we have described.
\begin{lemma} \label{lem:clear-is-improvable}
Let $\Pi$ be an instance of the Blocksworld-arm domain, and let $s$ be a
reachable state of $\Pi$ such that $s(\mathsf{arm})=\empty$. If a block $b$ is
such that $s(\clear{b})=\mathsf{T}$ but $\mathsf{goal}(\clear{b})=\mathsf{F}$, then
there is a plan using $(H(s,6),A)$-derivable actions that improves the
variable $\clear{b}$ in $s$.
\end{lemma}
\begin{pf-sketch}
Clearly, $b=\ptop{P_1}$ for some tower $P_1$ of $s$. Let $P_2, \ldots,
P_t$ be the remaining $t-1$ towers of $s$, and let $t'$ be the number
of towers of $\mathsf{goal}$.
The proof proceeds by cases. If there is $i$ such that
$\mathsf{goal}(\on{\pbottom{P_i}}) \neq \mathsf{table}$, we say we are in
Case~1. Otherwise, it holds that $t\leq t'$. In particular, there are
$t'$ blocks $b'$ such that $\mathsf{goal}(\clear{b'})=\mathsf{T}$ (block $b$ not
one of them), and $t$ blocks $b'\neq b$ such that
$s(\clear{b'})=\mathsf{T}$ (block $b$ being one of them). It follows that
it exists a block $b'$ such that $\mathsf{goal}(\clear{b'})=\mathsf{T}$ but
$s(\clear{b'})=\mathsf{F}$. We say we are in Case~2 if the block $b'$
belongs to the tower $P_1$, and in Case~3 if not. Throughout this
proof we say that a block $b'$ is badly placed if
$s(\on{b'})\neq\mathsf{goal}(\on{b'})$.
\textbf{Case 1}. The tower $P_i$ is wrongly placed in the table, so we
are allowed to change the value of $\on{\pbottom{P_i}}$ without worry.
\begin{itemize}
\item[(a)] If $i\neq 1$, then use $\mathsf{tow\mbox{-}block}_{P_i, b}$ to stack the tower
$P_i$ on top of $b$.
\item[(b)] If $i=1$ and a tower $P_j$ with $j>1$ has a badly
placed block $b'$, then a possible solution is to insert $P_1$ below
$b'$. That is, move the sub-tower $P_{\geq}(b')$ on top of $P_1$, and
then move the new resulting tower on top of the place where $b'$ was
in state $s$, that is, on top of $s(\on{b'})$.
\item[(c)] If $i=1$ and no tower $P_j$ with $j>1$ has badly placed blocks., then
consider the pile $P_i'$ in state $\mathsf{goal}$ that $b$ belongs to, and let
$b'=\top(P_i')$. If block $b'$ is in $P_j$ for $j>1$ in state $s$,
then $P_j$ would have some badly placed block, since $b'$ and $b$,
sharing pile $P_i'$ in the goal state, would be in different piles in
state $s$. So $b'$ is in $P_1$, $\mathsf{goal}(\clear{b'})=\mathsf{T}$ but
$s(\clear{b'})=\mathsf{F}$, since $b$ is the top of $P_1$. It follows that
the block on top of $b'$ in pile $P_1$ is badly placed. To improve
$\clear{b}$ use actions $\mathsf{subtow\mbox{-}table}_{P_{>}(b'),b'}$ and
$\mathsf{tow\mbox{-}block}_{P_\leq(b'), b}$, that is, break the tower over block
$b'$ and swap the two parts.
\end{itemize}
Note that an action like $\mathsf{tow\mbox{-}block}_{P_\leq(b'), b}$ is not
derivable from $s$ since the pile $P_\leq(b')$ is not a subtower of
$s$, but it is derivable from $s'=s[\mathsf{subtow\mbox{-}table}_{P_{>}(b'),b'}]$,
a state within distance $2$ from $s$. This fact may increase the width
required to discover the derivable actions. In our case, a careful
examination reveals that Situation~(b) requires width $5$ and
Situation~(c) requires width $4$.
\textbf{Case 2}. Note that if Case~1 does not apply then $t\leq
t'$. Let $b'$ be the highest block in $P_1$ such that
$s(\clear{b'})=\mathsf{F}$ but $\mathsf{goal}(\clear{b'})=\mathsf{T}$.
\begin{itemize}
\item[(a)]
If $t>1$ and a tower $P_j$ with $j>1$ has a badly placed block
$b''$, then we insert the pile $P_{>}(b')$ below $b''$, analogously to
Situation~(b) in Case~1. This procedure improves variables $\clear{b}$ and
$\clear{b'}$ at the same time, but it needs width $6$.
\item[(b)]
If there is a second block $b''$ in $P_1$ such that
$\mathsf{goal}(\clear{b''})=\mathsf{T}$, then swap the sub-tower $P_{>}(b')$ with the pile
between $b'$ and $b''$, the block $b''$ not including. The procedure is
similar to Situation~(c) in Case~1, but it requires width 5.
\item[(c)] If there is no second block $b''$ in $P_1$ but all the
towers $P_j$ with $j>1$ have no badly placed blocks, it follows that
either $t=1$ or all towers $P_j$ with $j>1$ are exactly as in the goal
state. Observe that, in this situation, the blocks of $P_1$ form a
tower in $s$ and in $\mathsf{goal}$, but the order of the blocks in the two
towers must differ: the pile $P'=P_{\leq}(b')$, which is such that
$\mathsf{goal}(\clear{\ptop{P'}})=\mathsf{T}$ and
$\mathsf{goal}(\on{\pbottom{P'}})=\mathsf{table}$, cannot be a pile in $\mathsf{goal}$. Hence
there is a badly placed block below $b'$. This situation is analogous
to Situation~(b) in Case~2, and it also requires width 5.
\end{itemize}
\textbf{Case 3}. There is a block $b'$ such that $s(\clear{b'})=\mathsf{F}$
but $\mathsf{goal}(\clear{b'})=\mathsf{T}$, and the block is in some tower $P_i$ other than
$P_1$. We just stack the sub-tower $P_{>}(b')$ on top of $b$.
\end{pf-sketch}
\begin{pf-sketch}
(of Theorem~\ref{thm:blocksworld-10})
Let $\Pi$ be an instance of the Blocksworld-arm domain, and let $s$ be a
reachable state of $\Pi$ that is not a goal state.
We present the case where
$s(\mathsf{arm})=\mathsf{goal}(\mathsf{arm})=\empty$.
\longversion{
Assume for the
moment that $s(\mathsf{arm})=\mathsf{goal}(\mathsf{arm})=\empty$.
The proof is by cases,
depending on the variable to improve.}
\textbf{Improving $\on{b}$}.
\begin{itemize}
\item $s(\on{b})=\mathsf{table}, \mathsf{goal}(\on{b})=b'$. If
$s(\clear{b'})=\mathsf{F}$, then move the sub-tower $P_{>}(b')$ onto the
table. (This changes the variable $\on{b''}$, where $b''$ is the
block on top of $b'$ in $s$, which was not in the goal state in
$s$.) Now the block $b'$ is clear, so we stack the tower $b$ is the
bottom of onto $b'$.
\item $s(\on{b})=b'', \mathsf{goal}(\on{b})=b'$. If $s(\clear{b'})=\mathsf{F}$
then we can swap piles $P_{>}(b'')$ and $P_{>}(b')$. Otherwise, we
stack $P_{>}(b'')$ on top of $b'$, but then $\clear{b''}$ becomes
true. This is a problem if $\mathsf{goal}(\clear{b''})=\mathsf{F}$, so we may
need to apply Lemma~\ref{lem:clear-is-improvable} at the current
state. Again, a careful examination shows that we may need width
$8$.
\item $s(\on{b})=b'', \mathsf{goal}(\on{b})=\mathsf{table}$. Move $P_{\geq}(b)$ onto
the table. As in the previous case apply
Lemma~\ref{lem:clear-is-improvable} to the current state if
$\mathsf{goal}(\clear{b''})=\mathsf{F}$. In this case we may need width 7.
\end{itemize}
\textbf{Improving $\clear{b}$}.
\begin{itemize}
\item $s(\clear{b})=\mathsf{F}, \mathsf{goal}(\clear{b})=\mathsf{T}$. Move the pile $P_{>}(b)$
onto the table, so width $4$ is enough.
\item $s(\clear{b})=\mathsf{T}, \mathsf{goal}(\clear{b})=\mathsf{F}$. Just apply
Lemma~\ref{lem:clear-is-improvable}, which requires width 6.
\end{itemize}
Under the assumption that $s(\mathsf{arm})=\mathsf{goal}(\mathsf{arm})=\empty$, there is nothing
else to show, since we have explained how to improve any variable.
The width number $10$ comes from the analysis of the other cases.
\longversion{
Now, if the arm holds some block $b$ and $s(\mathsf{arm})=b$ but
$\mathsf{goal}(\mathsf{arm})\neq b'$, we can just place the block $b$ in the
table. The values of the variables $\clear{b}$, $\on{b}$ and
$\mathsf{arm}$ change, but they were not in the goal state, with the possible
exception of $\clear{b}$. (In the formulation we have given the
variable $\clear{b}$ is $\mathsf{F}$ when the arm holds the block $b$.)
We show how to improve in state $s$ a variable $v$ other than $\mathsf{arm}$,
$\on{b'}$, $\clear{b'}$. Let $s'$ be the state after dropping the
block $b$ on the table. Note that $s(\clear{b})=\mathsf{F}$ but
$s'(\clear{b})=\mathsf{T}$; if $b$ is such that $\mathsf{goal}(\clear{b})=\mathsf{F}$
then we have to improve both $v$ and $\clear{b}$. (The other variables
that change when dropping $b$, namely $\mathsf{arm}$ and $\on{b}$, could not
be in their goal state in $s$). Now $s'(\mathsf{arm})=\empty$, so we apply
the procedures to improve $v$ and $\clear{b}$ developed in the first
part of the proof. Observe that $s'$ differs from $s$ at $3$
variables; we have shown that $v$ can be improved with width $11$, and
that $\clear{b}$ needs at most width $8$. The sum of these values is a
trivial upper bound to the width required. (The actual minimal width
is certainly smaller, since the two improvements may modify common variables.)
If $v$ is either $\mathsf{arm}$, $\on{b'}$ or $\clear{b'}$, then we also drop
$b$ on the table and improve $\clear{b}$ if needed, but then we apply
procedures that make the block $b'$ become clear and on the table. If
$b'$ was in top of $b''$ we are changing $\clear{b''}$ from $\mathsf{F}$
to $\mathsf{T}$, so we may need to use Lemma~\ref{lem:clear-is-improvable}
a second time to make $\clear{b''}$ recover the value
$\mathsf{F}$. Finally, we pick up $b'$ with the hand, so that variables
$\mathsf{arm}$, $\on{b'}$ and $\clear{b'}$ improve at the same time. So during
the course of the plan we may need to improve variables $\clear{b}$
and $\clear{b''}$ (at most width $8$ for each one) and we are modifying
variables $\mathsf{arm}$, $\on{b}$, $\clear{b}$, $\on{b'}$ and $\clear{b'}$, so
we can certainly do it with width $21$.}
\end{pf-sketch}
}
\paragraph{Towers of Hanoi.}
\begin{theorem}
All instances of the Towers of Hanoi domain have MPH-width $7$.
\end{theorem}
Each instance can be solved by a single application of
the action $\mathsf{subtow\mbox{-}pos}_{k, p_1, p_3}$.
\begin{comment}
As in the case of Blocksworld-arm, the planning problem becomes easier
when we can use macro actions like $\mathsf{subtow\mbox{-}pos}_{i, x,
x'}$. In fact, a single application of $\mathsf{subtow\mbox{-}pos}_{k, p_1,
p_3}$ would solve the Towers of Hanoi instance of $k$ disks!
But this
is
exactly what happens: macro actions $\mathsf{subtow\mbox{-}pos}_{i, x, x'}$
are $H(s,?)$-derivable.
\begin{theorem}
Let $\Pi$ be a planning instance of Towers of Hanoi, and let $s$ be
a reachable state. Let $x, x'$ be two positions such that $s such that $s(c whenever $s$ contains a pile formed by the
first $i$ disks.
\end{theorem}
\end{comment}
\begin{tiny}
\bibliographystyle{plain}
|
3,212,635,537,465 | arxiv | \section*{Introduction}
\label{section: Introduction}
We present an extension of Stone duality for Boolean algebras from classical propositional logic to classical
first-order logic. In broad strokes, the leading idea is to take the traditional logical distinction between
syntax and semantics and analyze it in terms of the classical mathematical distinction between algebra and
geometry, with syntax corresponding to algebra and semantics to geometry. Insights from category theory allow
us to recognize a certain duality between the notions of algebra and geometry. We see a first glimpse of this in
Stone's duality theorem for Boolean algebras, the categorical formulation of which states that a category of
`algebraic' objects (Boolean algebras) is the categorical dual of a category of `geometrical' objects (Stone
spaces). ``Categorically dual'' means that the one category is opposite to the other, in that it can be obtained (up to equivalence)
from the other by formally reversing the morphisms. In a more far reaching manner, this form of algebra-geometry
duality is exhibited in modern algebraic geometry as reformulated in the language of schemes in the Grothendieck
school, e.g.\ in the duality between the categories of commutative rings and the category of affine schemes.
On the other hand, we are informed by the category theoretic analysis of logic that it is
closely connected with algebra, in the sense that logical theories can be regarded as categories and suitable
categories can be presented as logical theories. For instance, Boolean algebras can be seen as classical propositional
theories, categories with finite products can be seen as equational theories, Boolean coherent categories
as theories in classical first-order logic, and elementary toposes -- e.g.\ the topos of sheaves on a space -- as theories in higher-order
intuitionistic logic. Thus the study of these algebraic objects has a logical interpretation and, vice versa,
reasoning in or about logical theories has application in their corresponding algebraic objects. With the
connection between algebra and logic in hand, instances of the algebra-geometry duality can be seen to manifest
a syntax-semantics duality between an algebra of syntax and a geometry of semantics. This notion of syntax as `dual to semantics' is, expectedly, one which ignores presentation and other features which, so to speak, models can not distinguish. In the propositional case, one passes from a propositional theory to a Boolean algebra by constructing the Lindenbaum-Tarski algebra of the theory, a construction which identifies provably equivalent formulas (and orders them by provable implication). Thus any two complete theories, for instance, are `algebraically equivalent' in the sense of having isomorphic Lindenbaum-Tarski algebras. The situation is precisely analogous to a presentation of an algebra by generators and relations: a logical theory corresponds to such a presentation, and two theories are equivalent if they present `the same' -- i.e.\ isomorphic -- algebras.
A similar construction is used to obtain, for a classical first-order theory, its `corresponding' Boolean coherent category, resulting in a similar notion of algebraic or categorical equivalence.
Given this connection between formal theories and categories, Stone duality manifests a syntax-semantics duality for propositional logic as follows. While a Boolean algebra can be regarded as a propositional theory modulo `algebraic' equivalence, on the other hand a Stone space can be seen as a space of corresponding two-valued models of such a theory. A model of a propositional theory is of course just a valuation of the propositional letters, or equivalently, a Boolean homomorphic valuation of all formulas. Thus we obtain the set of models of the theory corresponding to a Boolean algebra by taking morphisms in the category of Boolean algebras from the given algebra into the two-element Boolean algebra, 2,
\begin{eqnarray}\label{eq:stonerep}
\modcat{\cat{B}}\cong\homset{\alg{BA}}{\cat{B}}{2}.
\end{eqnarray}
And with a suitable topology in place---given in terms of the elements of the Boolean algebra $\cat{B}$---we can retrieve $\cat{B}$ from the space of models $\modcat{\cat{B}}$ by taking morphisms in the category of Stone spaces from it into the two-element Stone space, 2,
\begin{eqnarray*}
\cat{B} \cong \homset{\alg{Stone}}{\modcat{\cat{B}}}{2}
\end{eqnarray*}
Here, the two-element set, 2, is in a sense living a `dual' life, and `homming into 2' forms a contravariant adjunction between the `syntactical' category of Boolean algebras and the category of topological spaces,
which, moreover, becomes an equivalence once we restrict to the `semantical' subcategory of Stone spaces.
\[\bfig
\morphism|b|/{@{>}@/_1em/}/<1250,0>[\alg{BA}`\alg{Stone};\homset{\alg{BA}}{-}{2}]
\morphism/{@{<-}@/^1em/}/<1250,0>[\alg{BA}`\alg{Stone};\homset{\alg{Stone}}{-}{2}]
\place(625,0)[\simeq]
\efig\]
Our construction for first-order logic generalizes this set-up by, on the `syntax' side, representing first-order theories by Boolean coherent categories. On the semantical side we have, for each theory, a space of models, augmented with a space consisting of the isomorphisms between those models, such that these spaces form a \emph{topological groupoid}, that is to say, such that the composition, domain and codomain, inverse arrow and identity arrow maps are all continuous. Our `semantic' side is, accordingly, a category consisting of topological groupoids and continuous homomorphisms between them. Where in Stone Duality one considers the lattice of open sets of a space in order to recover a Boolean algebra, we consider the topos (or `generalized space') of so-called \emph{equivariant sheaves} on a topological groupoid in order to recover a Boolean coherent category. In particular, we show that the topos of equivariant sheaves on the topological groupoid of models and isomorphisms of a theory is the so-called classifying topos of (the Morleyization of) the theory, from which it is known that the theory can be recovered up to a notion of equivalence. (Here we build upon earlier results in \cite{butz:96} to the effect that any such topos can be represented by a topological groupoid constructed from its points. Our construction differs from the one given there in choosing a simpler cover which is better suited for our purpose).
Our semantic representation of this topos can also be understood from the perspective of definable sets.
Suppose we have a theory, \theory, in first order logic or some fragment of it, and that $\phi(\vec{x})$ is some formula in the language of the theory. Then $\phi(\vec{x})$ induces a definable set functor,
\[
\sem{\phi(\vec{x})} : \modcat{\theory}\to \Sets
\]
from the groupoid of \theory-models to the category of sets, which sends a model \alg{M} to the extension, $\sem{\phi(\vec{x})}^{\alg{M}}$, of $\phi(\vec{x})$ in \alg{M}. The question is, then, whether these definable set functors can somehow be characterized among all functors of the form $\modcat{\theory}\rightarrow\Sets$, so that the theory can be recovered from its models in terms of them. Notice, incidentally, that in case of a positive answer, the category of sets takes on the role of a dualizing object, in analogy with \alg{2} for Stone duality. For the models of a theory can be seen as suitable functors from the algebraic representation of the theory, \synt{C}{T}, into \Sets, so that both obtaining the models from the theory and recovering the theory from the models is done by `homming' into \Sets,
\begin{align*}
\modcat{\theory} &\simeq \homset{\!}{\synt{C}{T}}{\Sets}\\
\synt{C}{T} &\simeq\homset{\!}{\modcat{\theory}}{\Sets}
\end{align*}
Here the hom-sets must be suitably restricted from all functors to just those preserving the relevant structure, the determination of which is part of the task at hand.
Now, positive, and elegant, answers to the question of the characterization of definable set functors exist, to begin with, for certain fragments of first-order logic. For algebraic theories---axiomatized only by equations in languages with only function symbols (and equality)---the categories of models (algebras) have all limits and colimits, and \emph{Lawvere duality} tells us that an algebraic theory \theory\ can be recovered (up to splitting of idempotents) from its category of models in the form of those functors $\modcat{\theory}\to\Sets$ which preserve limits, filtered colimits, and regular epimorphisms (see \cite{lawvere:63},\cite{adameklawvererosicky:03}). Expanding from the algebraic case, recall, e.g.\ from \cite[D1.1.]{elephant1}, that the \emph{Horn} formulas over a first-order signature are those formulas which are constructed using only connectives $\top$ and $\wedge$. Allowing also existential quantification brings us to \emph{regular} formulas. A Horn (regular) theory is one which can be axiomatized using sequents involving only Horn (regular) formulas. In between, a Cartesian theory is a regular theory which can be axiomatized using only formulas that are Cartesian relative to the theory, in the sense, briefly, that existential quantification does not occur except under a certain condition. Now, the category \modcat{\theory} of models and homomorphisms of a Cartesian theory \theory\ has limits and filtered colimits (but not, in general, regular epis), and \emph{Gabriel-Ulmer duality} (see e.g.\ \cite{adamekandrosicky:94})
informs us, among other things, that the definable set functors for Cartesian formulas (relative to \theory) can be characterized as the limit and filtered colimit preserving functors $\modcat{\theory}\rightarrow\Sets$ (and that the theory can be recovered in terms of them).
If we allow for unrestricted existential quantification and pass to regular logic, then categories of models need no longer have arbitrary limits. But they still have products and filtered colimits, and, as shown by M. Makkai \cite{makkai:90}, the definable set functors for regular formulas can now be characterized as those functors $\modcat{\theory}\rightarrow\Sets$ that preserve precisely that.
Adding the connectives $\bot$ and $\vee$ to regular logic gives us the fragment known as \emph{coherent logic} (see \cite[D1.1.]{elephant1}), in which a far greater range of theories can be formulated. The theory of fields, for instance, cannot be expressed as a regular theory (since the category of fields does not have arbitrary products), but it can be expressed as a coherent theory (see \cite[D1.1.7.(h)]{elephant1}). (In fact, it is a \emph{decidable} coherent theory, where ``decidable'' means, here, that there is an \emph{inequality predicate}, in the sense of a coherent formula which is provably the complement of equality.) Moreover, any classical first-order theory can be \emph{Morleyized} to yield a coherent theory with the same category of models, see \cite[D1.5.13]{elephant1} (we take the morphisms between models of a classical first-order theory to be the elementary embeddings). Thus the categories of models of coherent theories can not, in general, be expected to have more structure than those for classical first-order theories. What they do have are ultra-products. Although ultra-products are not an intrinsic feature of categories of models (for coherent theories), in the sense that they are not a categorical invariant, Makkai \cite{makkai:87} shows that model categories and the category of sets can be equipped with a notion of ultra-product structure---turning them into so-called \emph{ultra-categories}---which allows for the characterization of definable set functors as those functors that preserve this additional structure. Moreover, this approach can be modified in the case of classical first-order theories so that only the ultra-\emph{groupoids} of models and isomorphisms, equipped with ultra-product structure, need be considered, see \cite{makkai93}.
Our approach, similarly, relies on equipping the models of a theory with external structure, but in our case the structure is topological. We, too, restrict consideration to groupoids of models and isomorphisms, instead of categories of models and homomorphisms or elementary embeddings.
We carry our construction out for decidable coherent theories, corresponding to (small) decidable coherent categories (``decidable'' meaning, in the categorical setting, that diagonals are complemented). As we remarked, the theory of fields is a notable example of such a theory, and the decidable coherent theories do include all classical first-order theories in the sense that the Morleyization of a classical theory is decidable coherent. Accordingly, our construction restricts to the classical first-order case, corresponding to Boolean coherent theories.
The first part of the construction (Section \ref{Section: Reprentation Theorem}) concerns the characterization of definable set functors for a theory and the recovery of the theory from its groupoid of models in terms of them. The idea is that definable sets can be characterized as being, in a sense, \emph{compact}; not by regarding each individual set as compact, but by regarding the definable set functor as being a compact object in a suitable category. Pretend, for a moment, that the models of a theory \theory\ form a set and not a proper class, and suppose, for simplicity, that the models are all disjoint. A definable set functor from the groupoid of \theory-models and isomorphisms,
\[\sem{\phi(\vec{x})}^{(-)} : \modcat{\theory}\to \Sets\]
can, equivalently, be considered as
a set (indexed) over the set $(\modcat{\theory})_0$ of models,
\begin{equation}\label{eq:sheafrepproj}
\coprod_{\alg{M}\vDash\theory}\sem{\phi(\vec{x})}^{\alg{M}}\to^p (\modcat{\theory})_0
\end{equation}
with $p^{-1}(\alg{M})=\sem{\phi(\vec{x})}^{\alg{M}}$, together with an action on this set by the set $(\modcat{\theory})_1$ of isomorphisms,
\begin{equation}\label{eq:sheafrepact}
\left(\modcat{\theory}\right)_1\times_{(\modcat{\theory})_0}\coprod_{\alg{M}\vDash\theory}\sem{\phi(\vec{x})}^{\alg{M}}
\to^\alpha \coprod_{\alg{M}\vDash\theory}\sem{\phi(\vec{x})}^{\alg{M}}
\end{equation}
such that for any \theory-model isomorphism, $\alg{f}:\alg{M}\rightarrow\alg{N}$, and element, $\vec{m}\in \sem{\phi(\vec{x})}^{\alg{M}}$, we have $\alpha(\alg{f},\vec{m})=\alg{f}(\vec{m})\in \sem{\phi(\vec{x})}^{\alg{N}}$. Now, if the set of \theory-models and the set of isomorphisms are topological spaces forming a topological groupoid, then we can ask for the collection $$\coprod_{\alg{M}\vDash\theory}\sem{\phi(\vec{x})}^{\alg{M}}$$ of elements of the various definable sets to be a space, in such a way that the projection function $p$ in \eqref{eq:sheafrepproj} is a local homeomorphism, and such that the action $\alpha$ in \eqref{eq:sheafrepact} is continuous. This makes definable set functors into equivariant sheaves on the groupoid, and we show that in the topos of all such sheaves they can be characterized as the compact decidable objects (up to a suitable notion of equivalence).
The second part (Section \ref{Section: Duality}) concerns the construction, based on the representation result of the first part, of a duality between the category of decidable coherent categories (representing theories in first-order logic) and the category of topological groupoids of models. Specifically, we construct an adjunction between the category of decidable coherent categories and a category of `coherent' topological groupoids, such that the counit component of the adjunction is an equivalence, up to pretopos completion. As a technical convenience, we introduce a size restriction both on theories and their models (corresponding to the pretence, above, that the collection of models of a theory forms a set). The restriction, given a theory, to a set of models large enough for our purposes can be thought of as akin to the fixing of a `monster' model for a complete theory, although in our case a much weaker saturation property is asked for, and a modest cardinal bound on the size of the models is sufficient.
In summary, we present a `syntax-semantics' duality which shows how to recover a coherent decidable or a classical first-order theory from its models. Compared with the duality theory of Makkai \cite{makkai:87, makkai93}, we give an alternative notion of external structure with which to equip the models, which in our case is topological instead of based on ultra-products. This permits the use of topos theory in establishing the main results, and in particular results in a semantic construction of the classifying topos of the theory. Finally, our construction restricts to classical Stone duality in the propositional case.
Many more details of the results contained herein can be found in the second author's doctoral dissertation \cite{phd}.
\section{The Representation Theorem
\label{Section: Reprentation Theorem
\subsection{Theories and Models
\label{Subsection: Theories and Models}
We show how to recover a classical, first-order theory from its groupoid of models and model-isomorphisms, bounded in size and equipped with topological structure. We present this from a logical perspective, that is, from the
perspective of the syntax and model theory of first-order theories. One can, of course, go back and forth between this perspective and the categorical perspective of decidable or Boolean coherent categories and
set-valued coherent functors. Section \ref{Section: Duality} briefly outlines the translation between the two, and presents a duality between the `syntactical' category of theories and a `semantical' category of model-groupoids. In categorical terms, the purpose of the current section is to show that the topos of coherent sheaves on a decidable coherent category can be represented as the topos of equivariant sheaves on a topological groupoid of `points', or set-valued coherent functors, and invertible natural transformations. This builds upon earlier results in \cite{butz:96} and \cite{butz:98b} to the effect that a coherent topos can be represented by a topological groupoid constructed from its points (our construction differs from the one given in \emph{loc.cit.} in choosing a simpler cover which is better suited for our purpose).
Let $\Sigma$ be a (first-order, possibly many-sorted) signature. Recall that a formula over $\Sigma$ is \emph{coherent} if it is constructed using only the connectives $\top$, $\wedge$, $\exists$, $\bot$, and $\vee$. We consider formulas in suitable contexts, \syntob{\vec{x}}{\phi}, where the context $\vec{x}$ is a list of distinct variables containing (at least) the free variables of $\phi$. A sequent, $\phi\vdash_{\vec{x}}\psi$---where $\vec{x}$ is a suitable context for both $\phi$ and $\psi$---is coherent if both $\phi$ and $\psi$ are coherent. Henceforth we shall not be concerned with axiomatizations, and so we consider a (coherent) \emph{theory} to be a deductively closed set of (coherent) sequents.
Let \theory\ be a coherent (alternatively first-order) theory over a signature, $\Sigma$. Recall that the \emph{syntactic category}, \synt{C}{T}, of
\theory\ has as objects equivalence classes of coherent (alt.\ first-order) formulas in context, e.g.\ \syntob{\vec{x}}{\phi}, which is
equivalent to a formula in context, \syntob{\vec{y}}{\psi}, if the contexts are $\alpha$-equivalent and \theory\
proves the formulas equivalent\footnote{See \cite[D1]{elephant1} for further details. Note that we, unlike
\cite{elephant1}, choose to identify \theory-provably equivalent formulas. The reason is that they define
exactly the same sets, i.e.\ the same definable set functors.}, i.e.\ \theory\ proves the following sequents.
\begin{align*}
\phi & \vdash_{\vec{x}}\, \psi[\vec{x}/\vec{y}]\\
\psi[\vec{x}/\vec{y}] & \vdash_{\vec{x}}\, \phi
\end{align*}
An arrow between two objects, say \syntob{\vec{x}}{\phi} and \syntob{\vec{y}}{\psi} (where we may assume that $\vec{x}$ and $\vec{y}$ are distinct), consists of a class of
\theory-provably equivalent formulas in context, say \syntob{\vec{x},\vec{y}}{\sigma}, such that \theory\ proves
that $\sigma$ is a functional relation between $\phi$ and $\psi$:
\begin{align*}
\sigma &\vdash_{\vec{x},\vec{y}}\, \phi\wedge\psi\\
\phi &\vdash_{\vec{x}}\, \fins{\vec{y}}\sigma\\
\sigma\wedge \sigma(\vec{z}/\vec{y}) &\vdash_{\vec{x},\vec{y},\vec{z}}\, \vec{y}=\vec{z}
\end{align*}
If \theory\ is a coherent theory, then \synt{C}{T} is a coherent category. If \theory, in addition, has an
inequality predicate (for each sort), that is, a formula with two free variables (of that sort), $x\neq y$, such that \theory\ proves
\begin{align*}\label{Equation: DC, Axioms of inequality}
x\neq y \wedge x=y &\vdash_{x,y}\, \bot\\
\top &\vdash_{x,y}\, x\neq y \vee x=y
\end{align*}
then \synt{C}{T} is \emph{decidable}, in the sense that for each object, $A$, the diagonal, $\Delta:A\mon A\times A$, is complemented as a subobject. We call a
coherent theory which has an inequality predicate (for each sort) a \emph{decidable} coherent theory for that reason (and with
apologies for overloading the term). Finally, if \theory\ is a first-order theory, then \synt{C}{T} is a Boolean
coherent category, i.e. a coherent category such that every subobject is complemented.
Conversely, given a coherent category, \cat{C}, one can construct the coherent \emph{theory,
$\theory_{\cat{C}}$, of} \cat{C} by having a sort for each object and a function symbol for each arrow, and
taking as axioms all sequents which are true under the canonical interpretation of this language in \cat{C} (again, see
\cite{elephant1} for details). A coherent decidable category allows for the construction of a coherent decidable
theory (including an inequality predicate for each sort), and Boolean coherent \cat{C} allows for the
construction of a first-order $\theory_{\cat{C}}$. Thus we can turn theories into categories and categories back
into theories. It is in this sense that we say that (decidable) coherent categories represent (decidable)
coherent theories, and Boolean coherent categories represent first-order theories. (Since Boolean coherent categories are, of course, coherent, building the Boolean coherent syntactical category of a classical first-order theory and then taking its coherent internal theory will produce a decidable coherent theory with the same models as the original classical one; thus yielding an alternative, but less economical, way of \emph{Morleyizing} a classical theory than the one presented in \cite[D1.5.13]{elephant1}.)
We show how to recover a
theory from its models in the sense that we recover its syntactic
category, up to pretopos completion. Roughly, the pretopos completion of a theory is the theory equipped with disjoint sums and quotients
of equivalence relations, see e.g.\ \cite{makkai93}. A theory and its pretopos completion have the same models in (the pretopos) \Sets.
The category of models and homomorphisms of a coherent theory \theory\ is equivalent to the category of coherent functors from \synt{C}{T} into the category \Sets\ of sets and functions and natural transformations between them,
\[\modcat{\theory}{}\simeq \homset{}{\synt{C}{T}}{\Sets}\]
and the same holds for models in an arbitrary coherent category, \cat{E},
\[\modcat{\theory}{(\cat{E})}\simeq \homset{}{\synt{C}{T}}{\cat{E}}\]
Indeed, this is the universal property that characterizes ${\synt{C}{T}}$. The same is true for classical first-order theories if ``homomorphism'' is replaced by ``elementary embedding'' (Note that the elementary embeddings between models of a classical first-order theory coincide with the homomorphisms between models of its Morleyization.) We pass freely between considering models traditionally as structures and algebraically as functors. In passing, we note that decidability for coherent theories can be characterized semantically:
\begin{lemma}\label{Lemma: Semantic decidability}
Let \theory\ be a coherent theory over a signature $\Sigma$, and \modcat{\theory} the category of \theory-models and homomorphisms. Then \theory\ is decidable (i.e.\ has an inequality predicate for each sort) if and only if for every \theory-model homomorphism, $\alg{f}:\alg{M}\rightarrow\alg{N}$ and every sort $A$ of $\Sigma$, the component function $f_A:\sem{A}^{\alg{M}}\rightarrow \sem{A}^{\alg{N}}$ is injective.
\begin{proof}This follows from a slight rewriting of the proof of \cite[D3.5.1]{elephant1}.
\end{proof}
\end{lemma}
Given a coherent theory \theory\ taking sheaves on \synt{C}{T} equipped with the coherent coverage (finite epimorphic covering families) results in a topos \Sh{\synt{C}{T}} with the universal property that the category of \theory-models in any topos \cat{E} is equivalent to the category of geometric morphisms from \cat{E} to \Sh{\synt{C}{T}} and geometric transformations between them,
\[\modin{\theory}{\cat{E}}\simeq\homset{\!}{\cat{E}}{\Sh{\synt{C}{T}}}\]
The topos \Sh{\synt{C}{T}} is known as the \emph{classifying topos} of \theory\ (see \cite[D3]{elephant1}).
\subsection{Stone Representation
\label{Subsection: Stone representation}
Let \theory\ be a classical first-order theory or a decidable coherent theory.
We cut down to a \emph{set} of \theory-models by choosing an regular cardinal, $\kappa$, such that \theory\ (as a deductively closed set of sequents) is of
cardinality $<\kappa$. Denote by $\Sets_{\kappa}$ the category of sets of size (hereditarily) less than $\kappa$ -- or, as we shall say briefly, $\kappa$-\emph{small} sets -- and by
$X_{\theory}$ the set of \theory-models in $\Sets_{\kappa}$. This set of models is large enough for our purposes
in that, using Deligne's Theorem (and thus the Axiom of Choice), the coherent functors from the coherent category \synt{C}{T} to $\Sets_{\kappa}$ \emph{jointly reflect covers} with respect to the coherent coverage on \synt{C}{T} and the canonical coverage on $\Sets_{\kappa}$. Precisely:
\begin{lemma}
For any family \cterm{f_i:C_i\rightarrow C}{i\in I} in \synt{C}{T}, if for all coherent functors $F:\cat{C}\to \Sets_{\kappa}$, we have that the $F(f_i)$ are jointly surjective, then there exists $i_1, \ldots, i_n$ such that $\{ f_{i_1},\ldots, f_{i_n}\}$ cover $C$ in \synt{C}{T}.
\end{lemma}
\noindent For a first-order theory, this comes to saying that for any
\theory-type $p$, there exists a model \alg{M} in $X_{\theory}$ such that \alg{M} realizes $p$. We say that
$X_{\theory}$ is a \emph{saturated set of models} for \theory.
Next, for $\syntob{\vec{x}}{\phi}\in\synt{C}{T}$, the definable set functor given by $\phi$ restricts to a functor
\begin{align*}
\csem{\vec{x}}{\phi}^{(-)} :\ & X_{\theory}\to \Sets\\
& \alg{M} \longmapsto \csem{\vec{x}}{\phi}^\alg{M}
\end{align*}
which, following the equivalence $\Sets^{X_{\theory}}\simeq\Sets/X_{\theory}$, corresponds to the set over
$X_{\theory}$:
\[
\csem{\vec{x}}{\phi}_{X_{\theory}}:=\cterm{\pair{\alg{M},\vec{b}}}{ \alg{M}\in X_{\theory}, \vec{b}\in
\csem{\vec{x}}{\phi}^{\alg{M}}}\to^{\pi_{1}} X_{\theory}
\]
Where $\pi_1$ projects out the model $\alg{M}$. Note the notation ``\sox{\vec{x}}{\phi}'' for the set on the left, which
we shall make extensive use of below. The mapping $\syntob{\vec{x}}{\phi}\mapsto
(\pi_1:\sox{\vec{x}}{\phi}\rightarrow X_{\theory})$ gives us the object part of a functor,
\[
\cat{M}_d:\synt{C}{T}\to \Sets/X_{\theory}
\]
(which sends an arrow of \synt{C}{T} to the obvious function over $X_{\theory}$).
\begin{proposition}[Stone representation for coherent categories]
\label{Proposition: Stone representation for coherent theories}
The functor
\[\cat{M}_d:\synt{C}{T}\to \Sets/X_{\theory}\]
is coherent and reflects covers with respect to the coherent coverage on \synt{C}{T} and the canonical
coverage on $\Sets/{X_{\theory}}$.
As a consequence, $\cat{M}_d$ is conservative, that is, $\cat{M}_d$ is faithful and reflects isomorphisms.
\begin{proof}
Considering each \theory-model \alg{M} as a coherent functor from \synt{C}{T} to \Sets, we have a commuting triangle:
\[\bfig
\Atriangle/>`>`/<400,400>[\synt{C}{T}`\Sets/{X_{\theory}}`\prod_{\alg{M}\in
X_{\theory}}\Sets_{\alg{M}};\cat{M}_d`\pair{\ldots, \alg{M},\ldots}`]
\place(350,0)[\simeq]
\efig\]
Then $\cat{M}_d$ is coherent since all $\alg{M}\in X_{\theory}$ are coherent, and $\cat{M}_d$ reflects covers
since the $\alg{M}\in X_{\theory}$ jointly reflect covers.
\end{proof}
\end{proposition}
Let $G_{\theory}$ be the set of isomorphisms between models in $X_{\theory}$, giving us a groupoid,
\[\bfig
\morphism<750,0>[G_{\theory}\times_{X_{\theory}} G_{\theory}`G_{\theory};c]
\morphism(750,0)|a|/@{>}@<5pt>/<750,0>[G_{\theory}`X_{\theory};s]
\morphism(750,0)|m|/@{<-}/<750,0>[G_{\theory}`X_{\theory};Id]
\morphism(750,0)|b|/@{>}@<-5pt>/<750,0>[G_{\theory}`X_{\theory};t]
\Loop(750,0)G_{\theory}(ur,ul)_{\mng{i}}
\efig\]
where $c$ is composition of arrows; $i$ sends an arrow to its inverse; $s$ sends an arrow to its source/domain and $t$ to its target/codomain; and $Id$ sends an object to its identity arrow. By equipping $X_{\theory}$ with the logical topology defined below, and then introducing continuous $G_{\theory}$-actions, we will
make the objects in the image of $\cat{M}_d$|that is, the definable set functors|compact and generating, and the
embedding full. That is, we factor $\cat{M}_d$, first, through the category of sheaves on $X_{\theory}$
(equipped with the logical topology) and, second, through the category of equivariant sheaves, or sheaves with a
continuous $G_{\theory}$-action ($u^*$ and $v^*$ are forgetful functors):
\[\bfig
\qtriangle/>`>`<-/<750,300>[\synt{C}{T}`\Sh{X_{\theory}}`\Eqsheav{G_{\theory}}{X_{\theory}};\cat{M}`\cat{M}^{\dag}`v^*]
\dtriangle(0,300)/<-`<-`>/<750,300>[\Sets/X_{\theory}`\synt{C}{T}`\Sh{X_{\theory}};\cat{M}_d`u^*`]
\btriangle(1750,300)|ara|/->>`->>`->>/<600,300>[\Sets/X_{\theory}`\Sh{X_{\theory}}`\Sh{\synt{C}{T}};u`m_d`m]
\ptriangle(1750,0)/->>`->>`<<-/<600,300>[\Sh{X_{\theory}}`\Sh{\synt{C}{T}}`\Eqsheav{G_{\theory}}{X_{\theory}};`v`m^{\dag}]
\efig\]
The diagram on the right then shows the induced geometric morphisms. Our main result of Section \ref{Section: Reprentation
Theorem} (Theorem \ref{Proposition: Multisort classtop eqsh equivalence}) is that $\cat{M}^{\dag}$ is full, faithful, and
cover reflecting, and that \synt{C}{T} generates \Eqsheav{G_{\theory}}{X_{\theory}} (as a full subcategory),
whence $m^{\dag}$ is an equivalence:
\[\Eqsheav{G_{\theory}}{X_{\theory}}\ \simeq\ \Sh{\synt{C}{T}} \]
\subsection{Definable Sets are Sheaves on a Space of Models
\label{Subsection: Definable sets are sheaves on X}
We introduce the following `logical' topology on the set $X_{\theory}$ of \theory-models.
\begin{definition}
The \emph{logical topology} on $X_{\theory}$ is defined by taking as basic open sets those of the form
\[
\bopen{\syntob{\vec{x}}{\phi},\vec{b}} :=\cterm{\alg{M}\in X_{ \theory}}{\vec{b}\in
\csem{\vec{x}}{\phi}^{\alg{M}}}\subseteq X_{\theory}\]
for $\syntob{\vec{x}}{\phi}\in \synt{C}{T}$ and $b\in \Sets_{\kappa}$, with $\vec{b}$ the same length as
$\vec{x}$.
\end{definition}
In Section \ref{Subsection: Representation theorem for decidable coherent categories} we will give a more intrinsic specification, in terms of the objects and morphisms of a decidable coherent category, rather than in terms of the formulas of a decidable coherent theory.
Next, we factor $\cat{M}_d:\synt{C}{T}\to \Sets/X_{\theory}$ through \Sh{X_{\theory}} by making each $\sox{\vec{x}}{\phi}$ into a sheaf on $X_{ \theory}$ with respect to the following topological structure. We shall use $*$ to denote concatenation of tuples, $$\pair{a_1,\ldots,a_n}*\pair{b_1,\ldots,b_m}=\pair{a_1,\ldots,a_n, b_1,\ldots,b_m}.$$
\begin{definition}\label{Definition: Logical topology on sheaves}
For an object $\syntob{\vec{x}}{\phi}$ of \synt{C}{T}, the \emph{logical topology} on the set
\[ \sox{\vec{x}}{\phi}=\cterm{\pair{\alg{M},\vec{a}}}{\alg{M}\in X_{\theory},
\vec{a}\in \csem{\vec{x}}{\phi}^{\alg{M}}} \] is given by basic opens of the form
\[
\bopen{\syntob{\vec{x},\vec{y}}{\psi}, \vec{b}}:= \cterm{\pair{\alg{M},\vec{a}}}{ \vec{a}\ast\vec{b}\in
\csem{\vec{x},\vec{y}}{\phi\wedge\psi}^{\alg{M}}}
\]
(where $\vec{b}$ is of the same length as $\vec{y}$)
\end{definition}
For any object \syntob{\vec{x}}{\phi} in \synt{C}{T}, we now have the following:
\begin{lemma}\label{Lemma: p is LH}
The projection $\pi_1:\sox{\vec{x}}{\phi}\rightarrow X_{\theory}$ is a local homeomorphism.
\begin{proof}
First, the projection
is continuous.
For let a basic open $\bopen{\syntob{\vec{y}}{\psi},\vec{b}}\subseteq X_{\theory}$ be given. Then
\[
\pi_1^{-1}\left(\bopen{\syntob{\vec{y}}{\psi},\vec{b}}\right)=
\bopen{\syntob{\vec{x},\vec{y}}{\psi}, \vec{b}}\subseteq\sox{\vec{x}}{\phi}
\]
Next, the projection
is open.
For given a basic open $\bopen{\syntob{\vec{x},\vec{y}}{\psi}, \vec{b}} \subseteq \sox{\vec{x}}{\phi}$
we have
\[
\pi_1\left(\bopen{\syntob{\vec{x},\vec{y}}{\psi}, \vec{b}}\right)= \bopen{\syntob{\vec{y}}{
\fins{\vec{x}}\phi \wedge \psi}, \vec{b}}\subseteq X_{\theory}
\]
which is open.
Finally, let $\pair{\alg{M},\vec{a}}\in \sox{\vec{x}}{\phi}$ be given. Then
\[
\pair{\alg{M},\vec{a}}\in V:= \bopen{\syntob{\vec{x},\vec{y}}{\vec{x}=\vec{y}}, \vec{a}} \subseteq
\sox{\vec{x}}{\phi}
\]
and $\pair{\alg{N},\vec{a'}}\in V$ if and only if $\vec{a}=\vec{a'}$. Thus $\pi_1\upharpoonright_V$ is
injective. We now have that $\pi_1\upharpoonright_V:
V\rightarrow \pi_1(V)$ is continuous, open, and bijective, and therefore a homeomorphism.
\end{proof}
\end{lemma}
\begin{lemma}\label{Lemma: maps are continuous} Given an arrow
\[\syntob{\vec{x},\vec{y}}{\sigma}:\syntob{\vec{x}}{\phi}\to
\syntob{\vec{y}}{\psi}\]
in \synt{C}{T}, the corresponding function $f_{\sigma}: \sox{\vec{x}}{\phi}\rightarrow \sox{\vec{y}}{\psi}$ is
continuous.
\begin{proof} Given a basic open
$\bopen{\syntob{\vec{y},\vec{z}}{\xi}, \vec{c}} \subseteq \sox{\vec{y}}{\psi}$, then
\[
f_{\sigma}^{-1}\left(\bopen{\syntob{\vec{y},\vec{z}}{\xi}, \vec{c}} \right) =
\bopen{\syntob{\vec{x},\vec{z}}{\fins{\vec{y}} \sigma \wedge \xi}, \vec{c}}
\]
\end{proof}
\end{lemma}
\begin{proposition}\label{Proposition: DC, Md factors as M}
The functor
$\cat{M}_d:\synt{C}{T}\to\Sets/X_{\theory}$
factors through the category \Sh{X_{\theory}} of sheaves as
\[
\bfig \dtriangle/<-`<-`->/<600,400>[\Sets/X_{\theory}`\synt{C}{T}`\Sh{X_{\theory}};\cat{M}_d`u^*`\cat{M}] \efig
\]
where $u^*:\Sh{X_{\theory}}\to\Sets/X_{\theory}$ is the forgetful (inverse image) functor. Moreover, \cat{M} is
coherent and reflects covers.
\begin{proof}\cat{M} is obtained by Lemma \ref{Lemma: p is LH} and Lemma \ref{Lemma: maps are
continuous}.
Since $\cat{M}_d$ is coherent and the forgetful functor $u^*$ reflects coherent structure, $\cat{M}$ is
coherent. Since $u^*$ preserves covers (being geometric) and $\cat{M}_d$ reflects them, \cat{M} reflects covers.
\end{proof}
\end{proposition}
\subsection{$\thry{G}_{\theory}$ is an Open Topological Groupoid
Consider now the set $G_{\theory}$ of \theory-model isomorphisms between the models in $X_{\theory}$. Such
an isomorphism, $\alg{f}:\alg{M}\rightarrow\alg{N}$, consists of a family of bijections,
$f_A:\csem{x:A}{\top}^{\alg{M}}\rightarrow \csem{x:A}{\top}^{\alg{N}}$, indexed by the sorts of \theory, subject
to the usual conditions ensuring that \alg{f} is an invertible homomorphism of \theory-models. We equip
$G_{\theory}$ with a topology to make the groupoid,
\[\bfig
\morphism<750,0>[G_{\theory}\times_{X_{\theory}} G_{\theory}`G_{\theory};\mng{c}]
\morphism(750,0)|a|/@{>}@<5pt>/<750,0>[G_{\theory}`X_{\theory};\mng{s}]
\morphism(750,0)|m|/@{<-}/<750,0>[G_{\theory}`X_{\theory};\mathrm{Id}]
\morphism(750,0)|b|/@{>}@<-5pt>/<750,0>[G_{\theory}`X_{\theory};\mng{t}]
\Loop(750,0)G_{\theory}(ur,ul)_{\mng{i}}
\efig\]
of \theory-models and isomorphisms a topological groupoid. (For shorter notation we write
``$\thry{G}_{\theory}$'', or ``$G_{\theory}\rightrightarrows X_{\theory}$'' if we want to display the set of
objects and the set of arrows of the groupoid.)
\begin{definition}\label{Definition: Bopen of G}
The \emph{logical topology} on $G_{\theory}$ is defined by taking as sub-basic open sets those of the form
\begin{itemize}
\item $s^{-1}\left(\bopen{\syntob{\vec{x}}{\phi},\vec{a}}\right)=\cterm{\alg{f}\in G_{\theory}}{\vec{a}\in
\csem{\vec{x}}{\phi}^{s(\alg{f})}}$
\item $\bopen{B:b\mapsto c} :=
\cterm{\alg{f}\in G_{\theory}}{ b\in \csem{x:B}{\top}^{s(\alg{f})} \wedge f_{B}(b)=c}$, where $B$ is a sort of
\theory.
\item $t^{-1}\left(\bopen{\syntob{\vec{x}}{\psi},\vec{e}}\right)=\cterm{\alg{f}\in G_{\theory}}{\vec{e}\in
\csem{\vec{x}}{\psi}^{t(\alg{f})}}$
\end{itemize}
\end{definition}
The most readable form to present a basic open set $U$ is as an array displaying the `source condition', the
`preservation condition', and the `target condition', e.g.:
\begin{align*}
&U=\left(\begin{array}{c}
\syntob{\vec{x}}{\phi},\vec{a} \\
\vec{B}:\vec{b} \mapsto \vec{c} \\
\syntob{\vec{y}}{\psi},\vec{d}
\end{array}\right)\\
&=\cterm{\alg{f}:\alg{M}\Rightarrow \alg{N}}{\vec{a}\in \csem{\vec{x}}{\phi}^{\alg{M}}
\wedge \vec{b}\in \csem{\vec{x}:\vec{B}}{\top}^{\alg{M}} \wedge f_{\vec{B}}(\vec{b})=\vec{c}
\wedge \vec{d}\in \csem{\vec{y}}{\psi}^{\alg{N}}}
\end{align*}
\begin{lemma}\label{Lemma: G is a topological groupoid }
With respect to the logical topologies on $G_\theory$ and $X_\theory$, the groupoid
\[\bfig
\morphism<750,0>[G_{\theory}\times_{X_{\theory}} G_{\theory}`G_{\theory};\mng{c}]
\morphism(750,0)|a|/@{>}@<5pt>/<750,0>[G_{\theory}`X_{\theory};\mng{s}]
\morphism(750,0)|m|/@{<-}/<750,0>[G_{\theory}`X_{\theory};\mathrm{Id}]
\morphism(750,0)|b|/@{>}@<-5pt>/<750,0>[G_{\theory}`X_{\theory};\mng{t}]
\Loop(750,0)G_{\theory}(ur,ul)_{\mng{i}}
\efig\]
is a topological groupoid (i.e.\ the source, target, identity, inverse, and composition maps are all
continuous).
\begin{proof}
Straightforward verification.
\end{proof}
\end{lemma}
It is clear that if we are presented with a basic open set
\[\bopen{\syntob{\vec{y}:\vec{B}}{\phi},\vec{b}} \subseteq X_{\theory}\ \textnormal{or}\
\bopen{\syntob{\vec{x}:\vec{A},\vec{y}:\vec{B}}{\psi}, \vec{b}} \subseteq \sox{\vec{x}:\vec{A}}{\phi}\]
we can assume without loss of generality that, for $i\neq j$, $B_i=B_j$ implies $b_i\neq b_j$. We say that
$\bopen{\syntob{\vec{y}:\vec{B}}{\phi},\vec{b}}$ is presented in \emph{reduced form} if this condition is
satisfied. It is clear that, as long as we are careful, we can replace elements in a model by switching to an isomorphic model. We write this out as a technical lemma for reference.
\begin{lemma}\label{Lemma: Star of David} Let a list of sorts $\vec{A}$ of
\theory\ and two tuples $\vec{a}$ and $\vec{b}$ of $\Sets_{\kappa}$ be given, of the same length as $\vec{A}$,
and satisfying the requirement that whenever $i\neq j$, $A_i=A_j$ implies $a_i\neq a_j$ and $b_i\neq b_j$. Then
for any $\alg{M}\in X_{\theory}$, if $\vec{a}\in \csem{\vec{x}:\vec{A}}{\top}^{\alg{M}}$, there exists an
$\alg{N}\in X_{\theory}$ and an isomorphism $\alg{f}:\alg{M}\rightarrow \alg{N}$ in $G_{\theory}$ such that
$f_{\vec{A}}(\vec{a})= \vec{b}$.
\end{lemma}
\begin{proposition}\label{Proposition: Multisort, groupoid is open}
The groupoid $\thry{G}_{\theory}$ is an open topological groupoid.
\begin{proof}
It remains (by Lemma \ref{Lemma: G is a topological groupoid }) to verify that the source map is open, from which
it follows that the target map is open as well. Let a basic open subset
\[
V=\left(\begin{array}{c}
\syntob{\vec{x}:\vec{A}}{\phi},\vec{a} \\
\vec{B}:\vec{b} \mapsto \vec{c} \\
\syntob{\vec{y}:\vec{D}}{\psi},\vec{d}
\end{array}\right)
\]
of $G_{\theory}$ be given, and suppose $\alg{f}:\alg{M}\rightarrow \alg{N}$ is in $V$. We must find an open
neighborhood around \alg{M} which is contained in $s(V)$. We claim that
\[
U=\bopen{\syntob{\vec{x}:\vec{A},\vec{y}:\vec{D},\vec{z}:\vec{B}}{\phi\wedge\psi},\vec{a}\ast
f_{\vec{D}}^{-1}(\vec{d})\ast \vec{b}}
\]
does the trick. Clearly, $\alg{M}\in U$. Suppose $\alg{K}\in U$. Consider the tuples $
f_{\vec{D}}^{-1}(\vec{d})\ast \vec{b}$ and $\vec{d}\ast \vec{c}$ together with the list of sorts $\vec{D}\ast
\vec{B}$. Since $f_{\vec{D}\ast \vec{B}}$ sends the first tuple to the second, we can assume that the conditions
of Lemma \ref{Lemma: Star of David} are satisfied (or a simple rewriting will see that they are), and so there
exists a \theory-model \alg{L} and an isomorphism $\alg{g}:\alg{K}\rightarrow \alg{L}$ such that $g\in V$. So
$U\subseteq s(V)$.
\end{proof}
\end{proposition}
\subsection{Definable Sets as Equivariant Sheaves
\label{Subsection: Definable sets as equivariant sheaves}
Recall that if \thry{H} is an arbitrary topological groupoid, which we also write as $H_1\rightrightarrows H_0$,
the topos of \emph{equivariant sheaves} (or \emph{continuous actions}) on \thry{H}, written
\Sh{\thry{H}} or \Eqsheav{H_1}{H_0}, consists of the following
\cite[B3.4.14(b)]{elephant1}, \cite{moerdijk:88}, \cite{moerdijk:90}.
An object of \Sh{\thry{H}} is a pair \pair{a:A\rightarrow H_0, \alpha}, where $a$ is a local
homeo\-morphism (that is, an object of \Sh{H_0}) and $\alpha:H_1\times_{H_0} A \rightarrow A$ is a continuous
function from the pullback (in \alg{Top}) of $a$ along the source map $s:H_1\rightarrow H_0$ to $A$ such that
\[ a(\alpha(f,x))=t(f)\]
and satisfying the axioms for an action:
\begin{enumerate}[(i)]
\item $\alpha(1_h,x)=x$ for $h\in H_0$.
\item $\alpha(g,\alpha(f,x))=\alpha(g\circ f,x)$.
\end{enumerate}
For illustration, it follows that for $f\in H_1$, $\alpha(f,-)$ is a
bijective function from the fiber over $s(f)$ to the fiber over $t(f)$.
An arrow $$h:\pair{a:A\rightarrow H_0, \alpha}\to \pair{b:B\rightarrow H_0, \beta}$$ is an arrow of
\Sh{H_0},
\[\bfig \Vtriangle<300,300>[A`B`H_0;h`a`b] \efig \]
which commutes with the actions:
\[ \bfig \square<500,400>[H_1\times_{H_0}A`A` H_1\times_{H_0}B`B; \alpha`1_{H_1}\times_{H_0}h`h`\beta] \efig \]
We now return to the definable set functors, $\csem{\vec{x}}{\phi}^{(-)}:\modcat{\theory}\to\Sets$.
Ignoring the isomorphisms between the \theory-models for the moment, we have described such a functor -- restricted to $\kappa$-small models -- first as a set and then (introducing topological structure) as a sheaf over $X_{\theory}$. The action of
the functor on the model isomorphisms can now be introduced as an action of the groupoid on the sheaf, as follows.
\begin{definition}
For each $\syntob{\vec{x}}{\phi}\in \synt{C}{T}$ the function
\begin{equation}\label{Equation: The action beta}
\theta_{\syntob{\vec{x}}{\phi}}:G_{\theory}\times_{X_{\theory}} \sox{\vec{x}}{\phi}\to \sox{\vec{x}}{\phi}
\end{equation}
is defined by $\pair{\alg{f},\pair{s(\alg{f}),\vec{a}}}\mapsto\pair{t(\alg{f}),\alg{f}(\vec{a})}$. (The subscript on
$\theta$ will usually be left implicit.)
\end{definition}
\begin{lemma}\label{Lemma: Action is continuous}
The pair \pair{\cat{M}(\syntob{\vec{x}}{\phi}), \theta} is an object of \Eqsheav{G_{\theory}}{X_{\theory}},
i.e.\ the function
\[\theta:G_{\theory}\times_{X_{\theory}} \sox{\vec{x}}{\phi}\to \sox{\vec{x}}{\phi}\]
is a continuous action of $G_{\theory}$ on \sox{\vec{x}}{\phi}.
\begin{proof} We verify that $\theta$ is continuous. Let a basic open
$$U= \bopen{\syntob{\vec{x}:\vec{A},\vec{y}:\vec{B}}{\psi}, \vec{b}}\subseteq \sox{\vec{x}:\vec{A}}{\phi}$$ be
given, and suppose $\theta(\alg{f},\pair{\alg{M},\vec{a}})= \pair{\alg{N},f_{\vec{A}}(\vec{a})}\in U$ for
$\alg{M},\alg{N}\in X_{\theory}$ and $f:\alg{M}\rightarrow \alg{N}$ in $G_{\theory}$. Then we can specify an
open neighborhood around \pair{\alg{f},\pair{\alg{M},\vec{a}}} which $\theta$ maps into $U$ as:
\[
\pair{\alg{f},\pair{\alg{M},\vec{a}}}\in \left(\begin{array}{c}
- \\
\vec{B}:f^{-1}_{\vec{B}}(\vec{b}) \mapsto \vec{b} \\
-
\end{array}\right)
\times_{X_{\theory}}
\bopen{\syntob{\vec{x}:\vec{A},\vec{y}:\vec{B}}{\psi}, f^{-1}_{\vec{B}}(\vec{b})}
\]
\end{proof}
\end{lemma}
\subsection{Stable Subsets}
For a subobject (represented by an inclusion) $\syntob{\vec{x}}{\xi}\embedd \syntob{\vec{x}}{\phi}$ in
\synt{C}{T}, the open subset $\sox{\vec{x}}{\xi}\subseteq \sox{\vec{x}}{\phi}$ is closed under the
action $\theta$ in the usual sense that $\theta(a)\in \sox{\vec{x}}{\xi}$ for any point $a\in \sox{\vec{x}}{\xi}$. For an object, \pair{A\rightarrow X_{\theory}, \alpha}, of \Eqsheav{G_{\theory}}{X_{\theory}},
we call a subset, $S\subseteq A$, that is closed under the action of $G_{\theory}$ \emph{stable}, so as to
reserve ``closed'' to mean topologically closed. We claim that the only stable opens of $\sox{\vec{x}}{\phi}$ come
from subobjects of $\syntob{\vec{x}}{\phi}$ as joins. Specifically:
\begin{lemma}\label{Lemma: Stabilization of basic open U}
Let $\syntob{\vec{x}:\vec{A}}{\phi}$ in \synt{C}{T} and $U$ a basic open subset of
$\sox{\vec{x}:\vec{A}}{\phi}$ of the form
\[
U=\bopen{\syntob{\vec{x}:\vec{A},\vec{y}:\vec{B}}{\psi}, \vec{b}}
\]
be given. Then the stabilization (closure) of $U$ under the action $\theta$ of $G_{\theory}$ on
$\bopen{\syntob{\vec{x}:\vec{A}}{\phi}}$ is a subset of the form $\sox{\vec{x}:\vec{A}}{\xi}\subseteq
\sox{\vec{x}:\vec{A}}{\phi}$.
\begin{proof} We can assume without loss that $U$ is in reduced form. Let $\varphi$ be the formula expressing the
conjunction of inequalities $y_i\neq y_j$ for all pairs of indices $i\neq j$ such that $B_i=B_j$ in $\vec{B}$.
We claim that the stabilization of $U$ is $\sox{\vec{x}:\vec{A}}{\xi}$ where $\xi$ is the formula
$\finst{\vec{y}}{\vec{B}}\phi \wedge \psi\wedge \varphi$. First, $\sox{\vec{x}:\vec{A}}{\xi}$ is a stable set
containing $U$. Next, suppose $\pair{\alg{M},\vec{a}}\in \sox{\vec{x}:\vec{A}}{\xi}$. Then there exists
$\vec{c}$ such that $\vec{a}\ast \vec{c}\in \csem{\vec{x}:\vec{A},\vec{y}:\vec{B}}{\phi \wedge \psi\wedge
\varphi}^{\alg{M}}$. Then $\vec{b}$ and $\vec{c}$ (with respect to $\vec{B}$) satisfy the conditions of Lemma
\ref{Lemma: Star of David}, so there exists a \theory-model $\alg{N}$ with isomorphism
$\alg{f}:\alg{M}\rightarrow \alg{N}$ such that $f_{\vec{B}}(\vec{c})=\vec{b}$. Then $\theta
(\alg{f},\pair{\alg{M},\vec{a}})\in U$, and hence \pair{\alg{M},\vec{a}} is in the stabilization of $U$.
\end{proof}
\end{lemma}
\begin{definition}
We call a subset of the form $\sox{\vec{x}}{\xi}\subseteq \sox{\vec{x}}{\phi}$, for a subobject
\[\syntob{\vec{x}}{\xi}\embedd \syntob{\vec{x}}{\phi}\]
in \synt{C}{T}, a \emph{definable} subset of $\sox{\vec{x}}{\phi}$.
\end{definition}
\begin{corollary}\label{Corollary: Definables the only stable opens}
Any open stable subset of $\sox{\vec{x}:\vec{A}}{\phi}$ is a union of definable subsets.
\end{corollary}
We also note the following:
\begin{lemma}\label{Lemma: Basic opens of X generate representables}
Let $\bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}}$ be a basic open of $X_{\theory}$ in reduced form. Then
there exists a sheaf $\cat{M}({\syntob{\vec{x}:\vec{A}}{\xi}})$ and a (continuous) section
\[s:\bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}} \to \sox{\vec{x}:\vec{A}}{\xi}\]
such that $\sox{\vec{x}:\vec{A}}{\xi}$ is the stabilization of the open set
$s(\bopen{\syntob{\vec{x}:\vec{A}}{\phi}})\subseteq \sox{\vec{x}:\vec{A}}{\xi}$.
\begin{proof}
Let $\varphi$ be the formula expressing the inequalities $x_i\neq x_j$ for all pairs of indices $i\neq j$ such
that $A_i=A_j$ in $\vec{A}$. Let $\xi:=\phi \wedge \varphi$ and consider the function
$s:\bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}} \to \sox{\vec{x}:\vec{A}}{\xi}$ defined by $\alg{M}\mapsto
\pair{\alg{M},\vec{a}}$. The image of $s$ is the open set
$\bopen{\syntob{\vec{x}:\vec{A},\vec{y}:\vec{A}}{\vec{x}=\vec{y}}, \vec{a}}$, so $s$ is a (continuous) section.
And by the proof of Lemma \ref{Lemma: Stabilization of basic open U}, the stabilization of
$\bopen{\syntob{\vec{x}:\vec{A},\vec{y}:\vec{A}}{\vec{x}=\vec{y}}, \vec{a}}$ is exactly
$\sox{\vec{x}:\vec{A}}{\xi}$.
\end{proof}
\end{lemma}
Consider now the topos of equivariant sheaves \Eqsheav{G_{\theory}}{X_{\theory}}. For an arrow, $f:C\rightarrow
D$, of \synt{C}{T}, clearly the function $\cat{M}(f):\cat{M}(C)\rightarrow\cat{M}(D)$ commutes with the actions
$\theta_C$ and $\theta_D$, so that, by Lemma \ref{Lemma: Action is continuous}, we have
a functor $\cat{M}^{\dag}:\synt{C}{T}\to\Eqsheav{G_{\theory}}{X_{\theory}}$ which factors
$\cat{M}:\synt{C}{T}\to\Eqsheav{G_{\theory}}{X_{\theory}}$ through \Eqsheav{G_{\theory}}{X_{\theory}}:
\begin{equation}\label{Equation: M dagger}
\bfig
\dtriangle/<-`<-`->/<600,300>[\Sh{X_{\theory}}`\synt{C}{T}`\Eqsheav{G_{\theory}}{X_{\theory}};\cat{M}`v^*`\cat{M}^{\dag}]
\efig
\end{equation}
where $v^*$ is the forgetful functor. We call the image of $\cat{M}^{\dag}$ the \emph{definable} objects and
arrows of \Eqsheav{G_{\theory}}{X_{\theory}}. Since $\cat{M}$ is coherent and the forgetful functor $v^*$
reflects coherent structure, $\cat{M}^{\dag}$ is coherent. Therefore, (\ref{Equation: M dagger}) induces a
commuting diagram of geometric morphisms:
\[
\bfig
\dtriangle/->>`->>`<<-/<600,300>[\Sh{X_{\theory}}`\classtop`\Eqsheav{G_{\theory}}{X_{\theory}};m`v`m^{\dag}]
\efig
\]
where $m^{\dag}$ is a surjection because $m$ is. We state these facts for reference:
\begin{lemma}\label{Lemma: m dager surjective}
$\cat{M}^{\dag}:\synt{C}{T}\to\Eqsheav{G_{\theory}}{X_{\theory}}$ is coherent, conservative (i.e.\ faithful and
reflects isomorphisms), and reflects covers.
\end{lemma}
The remainder of this section is devoted to establishing that the geometric morphism
$$m^{\dag}:\Eqsheav{G_{\theory}}{X_{\theory}}\to\Sh{\synt{C}{T}}$$ is an equivalence. The main remaining step is
to establish that the definable objects generate \Eqsheav{G_{\theory}}{X_{\theory}} (Corollary \ref{Corollary:
definables are generators}). First, it is a known fact that any equivariant sheaf on an open topological
groupoid has an open action (see e.g.\ \cite{moerdijk:88}):
\begin{lemma}\label{Lemma: Open projection}
For any object in \Eqsheav{G_{\theory}}{X_{\theory}},
\[\left\langle\bfig\morphism<300,0>[R`X_{\theory};r]\efig,\rho\right\rangle \]
the projection $\pi_2:G_{\theory}\times_{X_{\theory}} R\to R$ is open.
\begin{proof}
By Proposition \ref{Proposition: Multisort, groupoid is open}, since pullback preserves open maps of spaces.
\end{proof}
\end{lemma}
\begin{corollary}\label{Corollary: Action is open}
For any object \pair{r:R\rightarrow X_{\theory},\rho} in \Eqsheav{G_{\theory}}{X_{\theory}},
the action \[\rho:G_{\theory}\times_{X_{\theory}} R\to R\] is open. Consequently, the stabilization of any open
subset of $R$ is again open.
\begin{proof}
Let a basic open $V\times_{X_{\theory}} U\subseteq G_{\theory}\times_{X_{\theory}} R$ be given (so that
$U\subseteq R$ and $V\subseteq G_{\theory}$ are open). Observe that, since the inverse map $i:G_{\theory}\to
G_{\theory}$ is a homeomorphism, $i(V)$ is open, and
\[
\begin{array}{rl}
\rho(V\times_{X_{\theory}}U) &= \cterm{y\in R}{\fins{\pair{f,x}\in V\times_{X_{\theory}}U}\rho(f,x)=y}\\
&= \cterm{y\in R}{\fins{f^{-1}\in i(V)}s(f^{-1}) = r(y) \wedge \rho(f^{-1},y)\in U}\\
&= \pi_2(\rho^{-1}(U)\cap(i(V)\times_{X_{\theory}} R))
\end{array}\]
is open by Lemma \ref{Lemma: Open projection}. Finally, for any open $U\subseteq R$, the stabilization of $U$ is
$\rho(G_{\theory}\times_{X_{\theory}} U)$.
\end{proof}
\end{corollary}
\begin{lemma}\label{Lemma: Well behaved section}
For any object
$\pair{\bfig\morphism<300,0>[R`X_{\theory};r]\efig,\rho}$
in \Eqsheav{G_{\theory}}{X_{\theory}},
and any element $x\in R$, there exists a basic open $\bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}}\subseteq
X_{\theory}$ and a section $v: \bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}}\rightarrow R$ containing $x$ such
that for any $\alg{f}:\alg{M}\rightarrow \alg{N}$ in $G_{\theory}$ such that $\alg{M}\in
\bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}}$ and $f_{\vec{A}}(\vec{a})=\vec{a}$ (thus $\alg{N}$ is also in
$\bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}}$), we have $\rho(\alg{f},v(\alg{M}))= v(\alg{N})$.
\begin{proof}Given $x\in R$,
choose a section $s:\bopen{\syntob{\vec{y}:\vec{B}}{\psi},\vec{b}}\rightarrow R$ such that $x\in
s(\bopen{\syntob{\vec{y}:\vec{B}}{\psi},\vec{b}})$. Pull the open set
$s(\bopen{\syntob{\vec{y}:\vec{B}}{\psi},\vec{b}})$ back along the continuous action $\rho$,
\[
\bfig \square|allb|/->` >->`
>->`->/<600,300>[V`s(\bopen{\syntob{\vec{y}:\vec{B}}{\psi},\vec{b}})`G_{\theory}\times_{X_{\theory}} R`R;`\subseteq`
\subseteq`\rho ] \place(75,250)[\spbangle] \efig
\]
to obtain an open set $V$ containing \pair{1_{r(x)},x}. Since $V$ is open, we can find a box of basic opens
around \pair{1_{r(x)},x} contained in $V$:
\[
\pair{1_{r(x)},x}\in W:= \left(\begin{array}{c}
\syntob{\vec{z}:\vec{C}}{\xi},\vec{c} \\
\vec{K}:\vec{k} \mapsto \vec{k} \\
\syntob{\vec{z'}:\vec{C'}}{\eta},\vec{c'}
\end{array}\right)
\times_{X_{\theory}} v'(U_{\syntob{\vec{y'}:\vec{D}}{\theta},\vec{d}})\subseteq V
\]
where $v'$ is a section $v':\bopen{\syntob{\vec{y'}:\vec{D}}{\theta},\vec{d}}\rightarrow R$ with $x$ in its
image. Notice that the preservation condition of $W$ (i.e.\ $\vec{K}:\vec{k} \mapsto \vec{k} $) must have the
same sets on both the source and the target side, since it is satisfied by $1_{r(x)}$. Now, restrict $v'$ to the
subset
\[
U:=\bopen{\syntob{\vec{z}:\vec{C}, \vec{z''}:\vec{K}, \vec{z'}:\vec{C'}, \vec{y'}:\vec{D}}{\xi \wedge \eta
\wedge \theta},\vec{c}\ast \vec{k}\ast \vec{c'}\ast \vec{d}}
\]
to obtain a section $v=v'\upharpoonright_{U}:U\rightarrow R$. Notice that $x\in v(U)$. Furthermore,
$v(U)\subseteq s(\bopen{\syntob{\vec{y}:\vec{B}}{\psi},\vec{b}})$, for if $v(\alg{M})\in v(U)$, then
$\pair{1_{\alg{M}},v(\alg{M})}\in W$, and so $\rho(\pair{1_{\alg{M}},v(\alg{M})})=v(\alg{M})\in
s(\bopen{\syntob{\vec{y}:\vec{B}}{\psi},\vec{b}})$. Finally, if $\alg{M}\in U$ and $\alg{f}:\alg{M}\rightarrow
\alg{N}$ is an isomorphism in $G_{\theory}$ such that
\[ f_{\vec{C}\ast \vec{K}\ast \vec{C'}\ast
\vec{D}}(\vec{c}\ast \vec{k}\ast \vec{c'}\ast \vec{d}) =\vec{c}\ast \vec{k}\ast \vec{c'}\ast \vec{d}\]
then $\pair{\alg{f},v(\alg{M})} \in W$, and so $\rho(\alg{f},v(\alg{M}))\in
s(\bopen{\syntob{\vec{y}:\vec{B}}{\psi},\vec{b}})$. But we also have $v(\alg{N})\in v(U) \subseteq
s(\bopen{\syntob{\vec{y}:\vec{B}}{\psi},\vec{b}})$, and $r(\rho(\alg{f},v(\alg{M})))=r(v(\alg{N})$, so
$\rho(\alg{f},v(\alg{M}))=v(\alg{N})$.
\end{proof}
\end{lemma}
\begin{lemma}\label{Lemma: definables are generators}
For any object in \Eqsheav{G_{\theory}}{X_{\theory}},
\[\pair{\bfig\morphism<300,0>[R`X_{\theory};r]\efig,\rho}\]
and any element $x\in R$, there exists a morphisms of \Eqsheav{G_{\theory}}{X_{\theory}} with definable domain
and with $x$ in its image.
\begin{proof}
First, we construct a function over $X_{\theory}$ with definable domain and with $x$ in its image. Choose a
section $v:\bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}}\rightarrow R$ with the property described in Lemma
\ref{Lemma: Well behaved section} such that $x\in v(\bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}})$. We can
assume that $\bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}}$ is on reduced form. Then, by Lemma \ref{Lemma:
Basic opens of X generate representables} there exists an object $\syntob{\vec{x}:\vec{A}}{\xi}$ in \synt{C}{T}
and a section $s:\bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}}\rightarrow \sox{\vec{x}:\vec{A}}{\xi}$ such that
$\sox{\vec{x}:\vec{A}}{\xi}$ is the stabilization of $s(\bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}})$. Define
a mapping $\hat{v}:\sox{\vec{x}:\vec{A}}{\xi}\rightarrow R$ as follows: for an element
$\pair{{\alg{N}},\vec{c}}\in \sox{\vec{x}:\vec{A}}{\xi}$, there exists $\pair{{\alg{M}}, \vec{a}}\in
s(\bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}})\subseteq \sox{\vec{x}:\vec{A}}{\xi}$ and
$\alg{f}:{\alg{M}}\rightarrow {\alg{N}}$ in $G_{\theory}$ such that $f_{\vec{A}}(\vec{a})=\vec{c}$. Set
$\hat{v}(\pair{{\alg{N}},\vec{c}})=\rho(\alg{f}, v({\alg{M}}))$. We verify that $\hat{v}$ is well defined:
suppose $\pair{{\alg{M}}',\vec{a}}\in s(\bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}}) \subseteq
\sox{\vec{x}:\vec{A}}{\xi}$ and $\alg{g}:{\alg{M}}'\rightarrow {\alg{N}}$ in $G_{\theory}$ is such that
$g_{\vec{A}}(\vec{a})=\vec{c}$. Then $\alg{g}^{-1}\circ \alg{f}:{\alg{M}}\rightarrow {\alg{M}}'$ sends
$\vec{a}\in \csem{\vec{x}:\vec{A}}{\phi}^{\alg{M}}$ to $\vec{a}\in \csem{\vec{x}:\vec{A}}{\phi}^{{\alg{M}}'}$,
and so by the choice of section $v:\bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}}\rightarrow R$, we have that
$\rho(\alg{g}^{-1}\circ \alg{f},v({\alg{M}}))=v({\alg{M}}')$. But then
\[
\rho(\alg{g},v({\alg{M}}'))= \rho(\alg{g},\rho(\alg{g}^{-1}\circ \alg{f},v({\alg{M}})))
=\rho(\alg{f},v({\alg{M}}))\] so the value of $\hat{v}$ at \pair{{\alg{N}},\vec{c}} is indeed independent of the
choice of \pair{{\alg{M}},\vec{a}} and $\alg{f}$. Finally, the following triangle commutes,
\begin{equation}\label{eq: commuting triangle in generators proof}
\bfig \Vtriangle/>`<-< `<-<
/<300,300>[\sox{\vec{x}:\vec{A}}{\xi}`R`\bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}};\hat{v}`s`v] \efig
\end{equation}
and so $x$ is in the image of $\hat{v}$.
Second, we verify that the function $\hat{v}:\sox{\vec{x}:\vec{A}}{\xi}\rightarrow R$
is the underlying function of a morphism,
\[
\bfig \Vtriangle<300,300>[\sox{\vec{x}:\vec{A}}{\xi}`R`X_{\theory}; \hat{v}`p`r] \efig
\]
of \Eqsheav{G_{\theory}}{X_{\theory}}, where the action on $\sox{\vec{x}:\vec{A}}{\xi}$ is denoted $\theta$
(recall \ref{Equation: The action beta} on page \pageref{Equation: The action beta}).
The definition of $\hat{v}$ makes it straightforward to see that $\hat{v}$ commutes with the actions $\theta$
and $\rho$ of $\sox{\vec{x}:\vec{A}}{\xi}$ and $R$, respectively. Remains to show that $\hat{v}$ is continuous.
Recall the triangle (\ref{eq: commuting triangle in generators proof}). Let $y\in
\hat{v}(\sox{\vec{x}:\vec{A}}{\xi})$ be given, and suppose $U$ is a open neighborhood of $y$. By Corollary
\ref{Corollary: Action is open}, we can assume that $U\subseteq \hat{v}(\sox{\vec{x}:\vec{A}}{\xi})$. Suppose
$y=\hat{v}(\pair{{\alg{N}},\vec{c}})=\rho(\alg{f},v({\alg{M}}))$ for a $\alg{f}:{\alg{M}}\rightarrow {\alg{N}}$
such that $\theta(\alg{f},s({\alg{M}}))=\pair{{\alg{N}},\vec{c}}$. We must find an open neighborhood $W$ around
$\pair{{\alg{N}},\vec{c}}$ such that $\hat{v}(W)\subseteq U$. First, define the open neighborhood $T\subseteq
G_{\theory}\times_{X_{\theory}}R$ around \pair{\alg{f},v({\alg{M}})} by
\[
T:=\rho^{-1}(U)\cap \left( G_{\theory}\times_{X_{\theory}}v(\bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}})
\right)
\]
From the homeomorphism $v(\bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}}) \cong
s(\bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}})$ we obtain a homeomorphism
$G_{\theory}\times_{X_{\theory}}v(\bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}})\cong
G_{\theory}\times_{X_{\theory}} s(\bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}})$. Set $T'\subseteq
G_{\theory}\times_{X_{\theory}}s(\bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}})$ to be the open subset
corresponding to $T$ under this homeomorphism,
\begin{align*}
\pair{\alg{f}, v({\alg{M}})} &\in T \subseteq G_{\theory}\times_{X_{\theory}}v(\bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}})\\
& \ \ \ \ \ \ \cong\\
\pair{\alg{f},s({\alg{M}})} &\in T' \subseteq
G_{\theory}\times_{X_{\theory}}s(\bopen{\syntob{\vec{x}:\vec{A}}{\phi},\vec{a}})
\end{align*}
Then $\pair{{\alg{N}},\vec{c}}=\theta(\alg{f}, s({\alg{M}}))\in \theta(T')$, and by Corollary \ref{Corollary:
Action is open}, $\theta(T')$ is open. We claim that $\hat{v}(\theta(T'))\subseteq U$: for suppose
$\pair{\alg{g},s(\alg{P})}\in T'$. Then $\pair{\alg{g},v(\alg{P})}\in T\subseteq \rho^{-1}(U)$, and so
$\hat{v}(\theta(\alg{g},s(\alg{P})))=\rho(\pair{\alg{g},v(\alg{P})})\in U$. Thus $\theta(T')$ is the required
$W$.
\end{proof}
\end{lemma}
\begin{corollary}\label{Corollary: definables are generators}
The definable objects generate the topos \Eqsheav{G_{\theory}}{X_{\theory}}.
\end{corollary}
We are thus in a position to conclude:
\begin{theorem}\label{Proposition: Multisort classtop eqsh equivalence}
For a decidable coherent theory \theory\ with a saturated set of $\kappa$-small models $X_{\theory}$, we have an equivalence of toposes,
\[
\Eqsheav{G_{\theory}}{X_{\theory}}\simeq \classtop .
\]
\begin{proof}Since, by Corollary \ref{Corollary: definables are generators},
the definable objects form a generating set, the full subcategory of definable objects is a site for
\Eqsheav{G_{\theory}}{X_{\theory}}\ when equipped with the canonical coverage inherited from
\Eqsheav{G_{\theory}}{X_{\theory}} (see e.g.\ \cite[C2.2.16]{elephant1}). We argue first that
$\cat{M}^{\dag}:\synt{C}{T}\to\Eqsheav{G_{\theory}}{X_{\theory}}$ is full: because $\cat{M}^{\dag}$ is coherent
(Lemma \ref{Lemma: m dager surjective}), definable objects are decidable. Therefore, any graph of a morphism
between definable objects is complemented. Because $\cat{M}^{\dag}$ reflects covers and any subobject of a
definable object is a join of definable subobjects (Lemma \ref{Corollary: Definables the only stable opens}),
definable objects are compact in \Eqsheav{G_{\theory}}{X_{\theory}}\ (in the sense that any covering family of
subobjects has a finite covering subfamily). But then every complemented subobject of a definable object is a
finite join of definable subobjects, and therefore definable. Hence $\cat{M}^{\dag}$ is full. By Lemma
\ref{Lemma: m dager surjective}, $\cat{M}^{\dag}$ is also faithful. Finally, the canonical coverage inherited
from \Eqsheav{G_{\theory}}{X_{\theory}}\ coincides with the coherent coverage since $\cat{M}^{\dag}$ reflects
covers precisely with respect to the canonical coverage on \Eqsheav{G_{\theory}}{X_{\theory}}\ and the coherent
coverage on \synt{C}{T}. Therefore, \synt{C}{T} equipped with the coherent coverage is a site for
\Eqsheav{G_{\theory}}{X_{\theory}}, so $\Eqsheav{G_{\theory}}{X_{\theory}}\simeq \classtop$.
\end{proof}
\end{theorem}
\begin{remark}
An alternate proof of Theorem \ref{Proposition: Multisort classtop eqsh equivalence}, following the lines of \cite{butz:98b}, is given in \cite[Chapter 3]{phd}. It proceeds by showing that the spatial covering $$m : \Sh{X_\theory} \to \classtop$$ of Section \ref{Subsection: Stone representation} is an open surjection and thus, by results of \cite{joyal:84}, an effective descent morphism. The groupoid representation $\Eqsheav{G_{\theory}}{X_{\theory}}\simeq \classtop$ then follows from descent theory.
\end{remark}
\section{Duality
\label{Section: Duality
\subsection{Representation Theorem for Decidable Coherent Categories
\label{Subsection: Representation theorem for decidable coherent categories}
Since one can pass back and forth between coherent theories and categories by taking the theories of categories
and the syntactic categories of theories, Proposition \ref{Proposition: Stone representation for coherent
theories} translates to a representation result for decidable coherent categories,
in terms of groupoids of $\Sets_{\kappa}$-valued coherent functors and invertible natural transformations
between them. We spell this representation out, including a more direct characterization of the topology on the set
of $\Sets_\kappa$-valued coherent functors (Definition \ref{Definition: DC, Coherent topology}).
Let \cat{D} be a (small) decidable coherent category, that is, a category with finite limits, images, stable
covers, finite unions of subobjects, and complemented diagonals (\cite[A1.4]{elephant1}). For a (regular) cardinal $\kappa$, we say that \cat{D} \emph{has a saturated set of $\kappa$-small models} if the
coherent functors from \cat{D} to the category of (hereditarily) $\kappa$-small sets,
\[\cat{D}\to\Sets_{\kappa}\]
jointly reflect covers, in the sense, again, that
for any family of arrows $f_i:C_i\rightarrow C$ in \cat{D}, if for all $M:\cat{D}\to\Sets_{\kappa}$ in
$X_{\cat{D}}$
\[ \bigcup_{i\in I}\funksjon{Im}{M(f_i)}= M(C)\]
then there exists $f_{i_1}, \ldots, f_{i_n}$ such that $\funksjon{Im}{f_{i_1}}\vee\ldots\vee
\funksjon{Im}{f_{i_n}}=C$.
\begin{definition}\label{Definition: DC, DCkappa}
Let \alg{dCoh} be the category of small decidable coherent categories with coherent functors between them. For $\kappa$ a (regular) cardinal, let
\alg{dCoh_{\kappa}} be the full subcategory of those categories which have a saturated set of $\kappa$-small models, i.e.\ such that the coherent functors to
$\Sets_{\kappa}$ reflect covers.
\end{definition}
Note that any coherent category which is of cardinality $<\kappa$ is in $\alg{dCoh}_{\kappa}$, as are all
distributive lattices.
\begin{definition}\label{Definition: DC, Coherent topology}
For \cat{D} in $\alg{dCoh}_{\kappa}$:
\begin{enumerate}
\item Let $X_{\cat{D}}$ be the set of coherent functors from \cat{D} to $\Sets_{\kappa}$,
\[X_{\cat{D}}=\homset{\alg{dCoh}}{\cat{D}}{\Sets_{\kappa}}.\]
\item Let $G_{\cat{D}}$ be the set of invertible natural transformations between functors in $X_{\cat{D}}$, with
$s$ and $t$ the source and target, or domain and codomain, maps,
\[s,t:G_{\cat{D}}\rightrightarrows X_{\cat{D}}\]
Denote the resulting groupoid by $\thry{G}_{\cat{D}}$.
\item The \emph{coherent topology} on $X_{\cat{D}}$ is given by taking as a subbasis the collection of sets of the
form,
\begin{align*}
&\bopen{\vec{f},\vec{a}} =\bopen{\pair{f_1:A\rightarrow B_1,\ldots,f_n:A\rightarrow B_n},\pair{a_1,\ldots,a_n}}\\
&=\cterm{M\in X_{\cat{D}}}{\fins{x\in M(A)}M(f_1)(x)=a_1\wedge \ldots \wedge M(f_n)(x)=a_n}
\end{align*}
for a finite span of arrows
\[\bfig
\Atriangle/>`>`/<500,500>[A`B_1`B_n;f_1`f_n`]\place(250,0)[\ldots]\place(750,0)[\ldots]
\morphism(500,500)<0,-500>[A`B_i;f_i]
\efig\]
in \cat{D} and $a_1,\ldots,a_n\in \Sets_{\kappa}$. Let the \emph{coherent topology} on $G_{\cat{D}}$ be the
coarsest topology such that $s,t:G_{\cat{D}}\rightrightarrows X_{\cat{D}}$ are both continuous and all sets of
the form
\[\bopen{A,a\mapsto b}=\cterm{\alg{f}:M\rightarrow N}{a\in M(A) \wedge f_A(a)=b}\]
are open, for $A$ an object of \cat{D} and $a,b\in \Sets_{\kappa}$.
\end{enumerate}
\end{definition}
\begin{remark}\label{Remark: X, propositional case}
Note that if \cat{D} is a Boolean algebra and we require coherent functors into \Sets\ to send the terminal object to the distinguished terminal object $\{\star\}$ in \Sets, then $X_{\cat{D}}$ is the Stone space of \cat{D}.
\end{remark}
For \cat{D} in \alg{dCoh_{\kappa}}, we have the decidable coherent theory $\theory_{\cat{D}}$ of \cat{D}, and
its syntactic category, \synt{C}{\theory_{\cat{D}}} (as described in Section \ref{Subsection: Theories and Models}). Sending an object, $D$, in \cat{D} to the object \syntob{x:D}{\top} in \synt{C}{\theory_{\cat{D}}},
and an arrow $f:C\rightarrow D$ to \syntob{x:C,y:D}{f(x)=y}, defines a functor
\[ \eta_{\cat{D}}:\cat{D}\to\synt{C}{\theory_{\cat{D}}} \]
which is one half of an equivalence, the other half being the (or a choice of) canonical
$\theory_{\cat{D}}$-model in \cat{D}.
Now, any $\theory_{\cat{D}}$-model, \alg{M}, in $\Sets_{\kappa}$ can be seen as a coherent functor,
$\alg{M}:\synt{C}{\theory_{\cat{D}}}\to\Sets_{\kappa}$. Composition with $\eta_{\cat{D}}$
\[\bfig
\ptriangle/>`>`<-/<750,500>[\cat{D}`\Sets_{\kappa}`\synt{C}{\theory_{\cat{D}}};\alg{M}\circ
\eta{\cat{D}}`\eta{\cat{D}}`\alg{M}]
\efig\]
induces restriction functions
\[\bfig
\square|arrb|/>`@{>}@<5pt>`@{>}@<5pt>`>/<1000,500>[G_{\theory_{\cat{D}}}`G_{\cat{D}}`X_{\theory_{\cat{D}}}`
X_{\cat{D}}; \phi_1`t`t`\phi_0]
\square|allb|/>`@{>}@<-5pt>`@{>}@<-5pt>`>/<1000,500>[G_{\theory_{\cat{D}}}`G_{\cat{D}}`X_{\theory_{\cat{D}}}`
X_{\cat{D}}; \phi_1`s`s`\phi_0]
\efig\]
commuting with source and target (as well as composition and insertion of identities) maps.
\begin{lemma}
The maps $\phi_0$ and $\phi_1$ are homeomorphisms of spaces.
\begin{proof}
Any coherent functor $M:\cat{D}\to\Sets_{\kappa}$ lifts to a unique $\theory_{\cat{D}}$-model
$\alg{M}:\synt{C}{\theory_{\cat{D}}}\to \Sets_{\kappa}$, to yield an inverse $\psi_0:X_{\cat{D}}\rightarrow
X_{\theory_{\cat{D}}}$ to $\phi_0$. Similarly, an invertible natural transformation of functors $f:M\rightarrow
N$ lifts to a unique $\theory_{\cat{D}}$-isomorphism $\alg{f}:\alg{M}\rightarrow \alg{N}$ to yield an inverse
$\psi_1:G_{\cat{D}}\rightarrow G_{\theory_{\cat{D}}}$ to $\phi_1$. We verify that these four maps are all
continuous. For a subbasic open
\[
U=\bopen{\pair{f_1:A\rightarrow B_1,\ldots,f_n:A\rightarrow B_n},\pair{a_1,\ldots,a_n}} \subseteq X_{\cat{D}}\]
we have
\[
\phi_0^{-1}(U)= \bopen{\syntob{y_1:B_1,\ldots,y_n:B_n}{\fins{x:A}\bigwedge_{1\leq i\leq n}f_i(x)=y_i},\vec{a}}
\]
so $\phi_0$ is continuous. To verify that $\psi_0$ is continuous, there are two cases to consider, namely
non-empty and empty context. For basic open
\[\bopen{\syntob{x:A_1,\ldots, x_n:A_n}{\phi}, \pair{a_1,\ldots,a_n}}
\subseteq X_{\theory_{\cat{D}}}\]
the canonical interpretation of $\theory_{\cat{D}}$ in \cat{D} yields a subobject of a product in \cat{D},
\[\csem{x:A_1,\ldots, x_n:A_n}{\phi}\subobject A_1\times\ldots\times A_n\to^{\pi_i}A_i.\]
Choose a monomorphism $r:R\mon A_1\times\ldots\times A_n$ representing that subobject. Then
\begin{align*}
&\psi_0^{-1}(\bopen{\syntob{x:A_1,\ldots, x_n:A_n}{\phi}, \pair{a_1,\ldots,a_n}})\\
&= \bopen{\pair{\pi_1\circ r:R\rightarrow A_1,\ldots,\pi_n\circ r: R\rightarrow A_n}, \pair{a_1,\ldots,a_n}}
\end{align*}
and it is clear that this is independent of the choice of product diagram and of representing monomorphism. For
the empty context case, consider a basic open $U=\bopen{\syntob{}{\varphi}, \star}$, where $\varphi$ is a
sentence of $\theory_{\cat{D}}$ and $\star$ is the element of the distinguished terminal object of \Sets\
(traditionally $\star=\emptyset$, notice that any $\bopen{\syntob{}{\varphi},a}$ with $a\neq\star$ is
automatically empty). The canonical interpretation of $\varphi$ in \cat{D} yields a subobject of a terminal
object, $\sem{\varphi}\subobject 1$. Choose a representative monomorphism $r:R\mon 1$. Then, independently of
the choices made,
\[ \psi_0^{-1}(U)=\bigcup_{a\in \Sets_{\kappa}}\bopen{r:R\rightarrow 1,a}.\]
So $\psi_0$ is continuous. With $\phi_0$ continuous, it is sufficient to check $\phi_1$ on subbasic opens of the
form $U=\bopen{A, a\mapsto b}\subseteq G_{\cat{D}}$. But
\[
\phi_1^{-1}(U)=\left(\begin{array}{c}
- \\
\syntob{x:A}{\top}:a \mapsto b \\
-
\end{array}\right)
\]
so $\phi_1$ is continuous. Similarly, it is sufficient to check $\psi_1$ on subbasic opens of the form
\[
U=\left(\begin{array}{c}
- \\
\syntob{x:A}{\top}:a \mapsto b \\
-
\end{array}\right)
\]
but $\psi_1^{-1}(U)=\bopen{A,a \mapsto b}$, so $\psi_1$ is continuous.
\end{proof}
\end{lemma}
\begin{corollary}
Definition \ref{Definition: DC, Coherent topology} yields, for a decidable coherent category \cat{D}, a
topological groupoid $\thry{G}_{\cat{D}}$ such that
\[\thry{G}_{\cat{D}} \cong \thry{G}_{\theory_{\cat{D}}}\]
in the category \alg{Gpd}.
\end{corollary}
We can now state the main representation result of this section.
\begin{Theorem}\label{Theorem: DC, Representation result}
For a decidable coherent category with a saturated set of $\kappa$-small models, the topos of coherent sheaves on
\cat{D} is equivalent to the topos of equivariant sheaves on the topological groupoid $\thry{G}_{\cat{D}}$ of
models and isomorphisms equipped with the coherent topology,
\[ \sh{D}\simeq \Sh{\thry{G}_{\cat{D}}}.\]
\begin{proof}
The equivalence $\eta_{\cat{D}}:\cat{D}\to \synt{C}{\theory_{\cat{D}}}$ yields an equivalence
$\sh{D}\simeq\Sh{\synt{C}{\theory_{\cat{D}}}}$, whence
\[ \sh{D}\simeq \Sh{\synt{C}{\theory_{\cat{D}}}}\simeq \Sh{\thry{G}_{\theory_{\cat{D}}}}\cong \Sh{\thry{G}_{\cat{D}}}\]
by Theorem \ref{Proposition: Multisort classtop eqsh equivalence}.
\end{proof}
\end{Theorem}
\subsection{The Semantical Functor $\Mod$
\label{Subsection: DC, The Semantical Functor}
We proceed to construct a `syntax-semantics' adjunction between the category \alg{dCoh_{\kappa}} (syntax) and a
subcategory of topological groupoids (semantics). Given a coherent functor
\[
F:\cat{A}\to\cat{D}
\]
between two objects of \alg{dCoh_{\kappa}}, precomposition with $F$,
\[\bfig
\morphism[\cat{A}`\cat{D};F] \morphism(500,0)/@{>}@<7pt>/[\cat{D}`\Sets_{\kappa};M]
\morphism(500,0)|b|/@{>}@<-7pt>/[\cat{D}`\Sets_{\kappa};N] \place(700,0)[\Downarrow]
\efig\]
yields a homomorphism of (discrete) groupoids
\begin{equation}\label{Equation: BC, Composition with F gives maps of groupoids}
\bfig
\square|arrb|/>`@{>}@<5pt>`@{>}@<5pt>`>/<1000,500>[G_{\cat{D}}`G_{\cat{A}}`X_{\cat{D}}`X_{\cat{A}}; f_1`t`t`f_0]
\square|allb|/>`@{>}@<-5pt>`@{>}@<-5pt>`>/<1000,500>[G_{\cat{D}}`G_{\cat{A}}`X_{\cat{D}}`X_{\cat{A}};
f_1`s`s`f_0]
\efig\end{equation}
We verify that $f_0$ and $f_1$ are both continuous. For basic open
\[
U=\bopen{\pair{g_1:A\rightarrow B_1,\ldots,g_n:A\rightarrow B_n},\pair{a_1,\ldots,a_n}}\subseteq X_{\cat{A}},
\]
we see that
\[
f_0^{-1}(U)=\bopen{\pair{F(g_1):FA\rightarrow FB_1,\ldots,F(g_n):FA\rightarrow FB_n},\pair{a_1,\ldots,a_n}}\subseteq
X_{\cat{D}}.
\]
And for basic open $U=\bopen{C,a\mapsto b}\subseteq G_{\cat{A}}$, we see that
\[f_1^{-1}(U)=\bopen{F(C),a\mapsto b}\subseteq G_{\cat{D}}\]
Thus composition with $F$ yields a morphism of topological groupoids,
$$f:\thry{G}_{\cat{D}}\to\thry{G}_{\cat{A}},$$ and thereby we get a contravariant functor,
\[
\Mod: \alg{dCoh}^{\mng{op}}_{\kappa}\to \alg{Gpd}.
\]
which we shall refer to as the \emph{semantical} functor.
Summarizing, for any decidable coherent category \cat{D}, we take
\[
\Mod(\cat{D}) = \homset{\alg{dCoh}}{\cat{D}}{\Sets_{\kappa}},
\]
regarded as a groupoid of natural isomorphisms and equipped with the coherent topology, as in Definition \ref{Definition: DC, DCkappa}.
\subsection{The Syntactical Functor $\Form$
\label{Subsection: DC, The Syntactical Functor}
We construct an adjoint to the semantical functor \Mod\ from a subcategory of \alg{Gpd} containing
the image of \Mod. As in the propositional (distributive lattices) case, there are various
subcategories that will work for this; we choose one such that is convenient for the present purpose, namely those groupoids \thry{G} which are \emph{coherent}, in the sense that \Sh{\thry{G}} is a coherent topos, that is, has a coherent site of definition (see \cite[D3.3]{elephant1}).
Recall that an object $A$ in a topos is \emph{compact} if every covering of it (in terms of morphisms or subobjects) has a finite subcovering (\cite[D3.3.2]{elephant1}).
\begin{definition}\alg{CohGpd} is the subcategory of \alg{Gpd} consisting of coherent groupoids and those morphisms $f:\thry{G}\to\thry{H}$ which preserve compact objects, in the sense that the induced inverse image functor
$f^*:\Sh{\thry{H}}\to\Sh{\thry{G}}$ sends compact objects to compact objects.
\end{definition}
\begin{remark}\label{Remark: Compact and coherent object}
Recall (e.g.\ from \cite{elephant1}) that:
\begin{enumerate}[(i)]
\item An object $C$ in a topos \cat{E} is \emph{coherent} if (1) it is compact; and (2) it is \emph{stable}, in the sense that for any morphism $f:B\to A$ with $B$ compact, the domain $K$ of the kernel pair of $f$,
\[ K\two^{k_1}_{k_2} B\to^f A\]
is again compact.
\item In a coherent topos, \sh{C} say, with \cat{C} a small coherent category, the full subcategory, $\cat{D}\embedd \sh{C}$, of coherent objects is a pretopos. \cat{D} forms a coherent site for \sh{C}; includes \cat{C} (through the Yoneda embedding); and is a pretopos completion of \cat{C}. Thus one can recover \cat{C} from \sh{C} up to pretopos completion as the coherent objects.
\item Any compact decidable object is coherent. The full subcategory of decidable objects in a coherent category is again a coherent category. Accordingly, the full subcategory of compact decidable objects in a coherent topos is a decidable coherent category.
\end{enumerate}
\end{remark}
By Theorem \ref{Theorem: DC, Representation result}, $\Mod(\cat{D})$ is a coherent groupoid, for any \cat{D} in $\alg{dCoh}_{\kappa}$, and we can recover \cat{D} from $\Mod(\cat{D})$, up to pretopos completion, by taking the compact decidable objects in \Sh{\Mod(\cat{D})}. For arbitrary coherent groupoids, however, this procedure will yield an decidable coherent category, but not necessarily one in $\alg{dCoh}_{\kappa}$, i.e.\ not necessarily with a saturated set of smaller than $\kappa$ models. However, one can use the groupoid, $\Sets^*_{\kappa}$ of smaller than $\kappa$ sets and bijections to classify a suitable collection of objects, as we now proceed to describe.
\subsubsection{The Decidable Object Classifier
\label{Subsubsection: DC, The Decidable Object Classifier}
\begin{definition}
The topological groupoid \thry{S} consists of (hereditarily) $\kappa$-small sets with bijections between
them, equipped with topology as follows. The topology on the set of objects, $S_0$, is generated by the empty
set and basic opens of the form
\[
\bopen{a_1,\ldots,a_n}:=\cterm{A\in \Sets_{\kappa}}{a_1,\ldots,a_n\in A}
\]
while the topology on the set, $S_1$ of bijections between $\kappa$-small sets is the coarsest
topology such that the source and target maps $s,t:S_1\rightrightarrows S_0$ are both continuous, and containing
all sets of the form
\[
\bopen{a\mapsto b}:=\cterm{f:A\to^{\cong} B\ \textnormal{in}\ \Sets_{\kappa}}{a\in A\wedge f(a)=b}
\]
\end{definition}
We recognize $\thry{S}$ as the groupoid of models and isomorphisms for the decidable coherent theory, $\theory_{\neq}$, of equality and inequality (with the obvious signature and axioms).
\begin{lemma}
There is an isomorphism $\thry{S}\cong \thry{G}_{\theory_{\neq}}$ in \alg{Gpd}.
\begin{proof}
Any set $A$ in $\Sets_{\kappa}$ is the underlying set of a canonical $\theory_{\neq}$-model, and any bijection
$f:A\rightarrow B$ is the underlying function of a $\theory_{\neq}$-model isomorphism, and thereby we obtain
bijections $S_0\cong X_{\theory_{\neq}}$ and $S_1\cong G_{\theory_{\neq}}$ which commute with source, target,
composition, and embedding of identities maps. Remains to show that the topologies correspond. Clearly, any
basic open $\bopen{\vec{a}}\subseteq S_0$ corresponds to the open set $\bopen{\syntob{\vec{x}}{\top},\vec{a}}\subseteq
X_{\theory_{\neq}}$. We show that, conversely, any basic open $\bopen{\syntob{\vec{x}}{\phi},\vec{a}}\subseteq
X_{\theory_{\neq}}$ corresponds to an open set of $S_0$ by induction on \syntob{\vec{x}}{\phi}. First,
$\bopen{\syntob{}{\top},a}$ is $X_{\theory_{\neq}}$ if $a=\star$ and empty otherwise, where $\{\star\}$ is the
distinguished terminal object of $\Sets$, and $\bopen{\syntob{\vec{x}}{\top},\vec{a}}\cong \bopen{\vec{a}}$. Next,
$\bopen{\syntob{x,y}{x=y},a,b}$ corresponds to $\bopen{a}\subseteq S_0$ if $a=b$, and the empty set otherwise. Similarly,
$\bopen{\syntob{x,y}{x\neq y},a,b}$ corresponds to $\bopen{a,b}\subseteq S_0$ if $a\neq b$ and the empty set otherwise.
Now, suppose $\bopen{\syntob{\vec{x}}{\phi},\vec{a}}$ corresponds to an open set $U\subseteq S_0$. Then
$\bopen{\syntob{\vec{x},y}{\phi},\vec{a},b}\cong U\cap \bopen{b}$. Next, if $\bopen{\syntob{x,\vec{y}}{\phi},a,\vec{b}}$
corresponds to an open set $U_a\subseteq S_0$ for each $a\in \Sets_{\kappa}$, then
$\bopen{\syntob{\vec{y}}{\fins{x}\phi},\vec{b}}\cong \bigcup_{a\in\Sets_{\kappa}}U_a$. Finally, if
$\bopen{\syntob{\vec{x}}{\phi},\vec{a}}$ and $\bopen{\syntob{\vec{x}}{\psi},\vec{a}}$ correspond to open sets
$U,V\subseteq S_0$, then $\bopen{\syntob{\vec{x}}{\phi\wedge\psi},\vec{a}}\cong U\cap V$, and
$\bopen{\syntob{\vec{x}}{\phi\vee\psi},\vec{a}}\cong U\cup V$. Therefore, $S_0\cong X_{\theory_{\neq}}$ as spaces.
For the spaces of arrows, it remains only to observe that open subsets of the form $\bopen{a\mapsto b}\subseteq S_1$
correspond to open subsets of the form
\[ \left(\begin{array}{c}
- \\
a \mapsto b \\
-
\end{array}\right)\subseteq G_{\theory_{\neq}}
\]
and we can conclude that \thry{S} is a topological groupoid isomorphic to $\thry{G}_{\theory_{\neq}}$ in
\alg{Gpd}.
\end{proof}
\end{lemma}
The topos \Sh{\thry{S}} of equivariant sheaves on \thry{S}, therefore, classifies decidable objects, as
$\Sh{\thry{S}}\simeq \Sh{\thry{G}_{\theory_{\neq}}}\simeq \Sh{\synt{C}{T_{\neq}}}$, where the last equivalence is by Theorem \ref{Proposition: Multisort classtop eqsh equivalence}.
\begin{corollary}\label{Corollary: Sh(S) coh}
The groupoid $\thry{S}$ of small sets is coherent.
\end{corollary}
\begin{corollary}\label{Corollary: DC, decidable object class equivalences}
There is an equivalence of toposes,
\[\Sets^{\mng{Fin}_i}\simeq \Sh{\synt{C}{T_{\neq}}}\simeq \Sh{\thry{S}}\]
where $\mng{Fin}_i$ is the category of finite sets and injections.
\begin{proof}
$\Sets^{\mng{Fin}_i}\simeq \Sh{\synt{C}{T_{\neq}}}$ by \cite[VIII, Exc.7--9]{maclane:92}.
\end{proof}
\end{corollary}
\begin{definition}\label{Stipulation: U} We fix the generic decidable object, $\cat{U}$, in \Sh{\thry{S}} to be the definable sheaf
$\pair{\csem{x}{\top}_{X_{\theory_{\neq}}}\rightarrow X_{\theory_{\neq}}\cong S_0,\theta_{\syntob{x}{\top}}}$, which we also abbreviate as $\cat{U}=\pair{U\rightarrow S_0,\theta_U}$ (see the following remark).
\end{definition}
\begin{remark}\label{Remark: U Moerdijk descripiton}
Without reference to $\theory_{\neq}$, we can characterize \cat{U} as the following equivariant sheaf: $U\rightarrow S_0$ is the set over $S_0$ such that the fiber over a set $A\in S_0$ is the set $A$ (i.e.\ $U = \coprod_{A\in S_0}A$), and the action by the set $S_1$ of isomorphisms is just applying those isomorphisms to the fibers. Thus, forgetting the topology, \cat{U} is simply the inclusion $\thry{S}\embedd\Sets$. The topology on $U$ is the coarsest such that the projection $U\rightarrow S_0$ is continuous and such that for each $a\in \Sets_{\kappa}$ the image of the section $s_a:\bopen{a}\rightarrow U$ defined by $s_a(A)=a$ is an open set. It is straightforward to verify that this is just an alternative description of $\pair{\csem{x}{\top}_{X_{\theory_{\neq}}}\rightarrow X_{\theory_{\neq}}\cong S_0,\theta_{\syntob{x}{\top}}}$.
\end{remark}
\subsubsection{Formal Sheaves
\label{Subsubsection: Morphisms into S}
We use the groupoid $\thry{S}$ of (small) sets to recover an object in $\alg{dCoh}_\kappa$ from a coherent groupoid by considering the set $\homset{\alg{CohGpd}}{\thry{G}}{\thry{S}}$ of morphisms into \thry{S}. (Consider the analogy to the propositional case, where the algebra of clopen sets of a Stone space is recovered by homming into the discrete space $2$.) First, however, a note on notation and bookkeeping: because we shall be concerned with functors into $\Sets_{\kappa}$---a subcategory of \Sets\ which is not closed under isomorphisms---we fix certain choices \emph{on the nose}, instead of working up to isomorphism or assuming a canonical choice as arbitrarily given. Without going into the (tedious) details of the underlying book-keeping, the upshot is that we allow ourselves to treat (the underlying set over $G_0$ and action of) an equivariant sheaf over a groupoid, $\thry{G}$ as a functor $\thry{G}\to\Sets$ in an intuitive way. In particular, we refer to the definable set $\csem{\vec{x}}{\phi}^{\alg{M}}$ as the fiber of $\sox{\vec{x}}{\phi}\rightarrow X_{\theory}$ over $\alg{M}$, although that is not strictly speaking the fiber (strictly speaking the fiber is, according to our definition, the set $\{\alg{M}\}\times\csem{\vec{x}}{\phi}^{\alg{M}}$). Moreover, we chose the induced inverse image functor $f^*:\Sh{\thry{H}}\to\Sh{\thry{G}}$ induced by a morphism, $f:\thry{G}\to\thry{H}$, of topological groupoids so that, for $A\in \Sh{\thry{H}}$ the fiber over $x\in G_0$ of $f^*(A)$ is the same set as the fiber of $A$ over $f_0(x)\in H_0$. For example, and in particular, any morphism of topological groupoids $f:\thry{G}\to\thry{S}$ induces a geometric morphism $f:\Sh{\thry{G}}\to\Sh{\thry{S}}$ the inverse image part of which takes the generic decidable object \cat{U} of \ref{Stipulation: U} to an (equivariant) sheaf over \thry{G},
\[
\bfig
\square(0,0)|allb|<600,400>[ A`U=\csem{x}{\top}_{X_{\theory_{\neq}}}`G_{0}`S_{0};```f_0]
\place(100,300,)[\pbangle]
\efig
\]
such that the fiber $A_x$ over $x\in G_0$ is the same set as the fiber of $U$ over $f_0(x)$, which is the set $f_0(x)\in S_0=\Sets_{\kappa}$. We hope that this is sufficiently intuitive so that we may hide the underlying book-keeping needed to make sense of it. With this in mind, then, we make the following stipulation.
\begin{definition}\label{Definition: Form(G)}
For a coherent groupoid \thry{G}, let $\Form(\thry{G})\hookrightarrow\Sh{\thry{G}}$ be the full subcategory consisting of objects of the form $f^*(\cat{U})$ for all $f:\thry{G}\to \thry{S}$ in \alg{CohGpd}. Such objects will be called \emph{formal sheaves}.
\end{definition}
Observe that:
\begin{lemma}\label{Lemma: enough to check coh on U}
For a coherent groupoid \thry{G}, a morphism $f:\thry{G}\to \thry{S}$ of topological groupoids is in \alg{CohGpd} if (and only if) the classified object $f^*(\cat{U})\in \Sh{\thry{G}}$ is compact.
\end{lemma}
\begin{proof} Assume that $f^*(\cat{U})$ is compact. By Corollary \ref{Corollary: DC, decidable object class equivalences} we have that $\synt{C}{T_{\neq}}$ is a site for \Sh{\thry{S}}. Write \cat{S} for the image of the full and faithfull inclusion of $\synt{C}{T_{\neq}}$ into \Sh{\thry{S}}. We have that \cat{U} is in \cat{S}, as the image of the object \syntob{x}{\top} in $\synt{C}{T_{\neq}}$. Since $\cat{U}$ is decidable, so is $f^*(\cat{U})$. Therefore, since $f^*(\cat{U})$ is compact, it is coherent. Now, as an inverse image functor, $f^*$ is coherent and since \Sh{\thry{G}} is a coherent topos, this means that for any $A$ in \cat{S} we have that $f^*(A)$ is coherent, and therefore, in particular, compact. Finally, for any compact object $E$ in $\Sh{\thry{S}}$ there is a cover $e:A_1+\ldots +A_n\epi E$, where $A_1,\ldots, A_n$ are objects of \cat{S}, and $f^*$ takes this to a cover $e:f^*(A_1)+\ldots +f^*(A_n)\epi f^*(E)$, whence $f^*(E)$ is compact. Hence $f^*$ takes compact objects to compact objects, so $f:\thry{G}\to \thry{S}$ is in \alg{CohGpd}.
\end{proof}
The formal sheaves on a coherent groupoid can be characterized directly:
\begin{lemma}\label{Lemma: DC, Theta(G)}
An equivariant sheaf $\cat{A}= \pair{A\rightarrow G_0, \alpha}$ on a coherent groupoid \thry{G} is formal just in case:
\begin{enumerate}[(i)]
\item $\cat{A}$ is compact decidable;
\item
each fibre $A_x$ for $x\in G_0$ is an element of $\Sets_{\kappa}$;
\item
for each set $a\in \Sets_{\kappa}$, the set $\bopen{\cat{A},a}=\cterm{x\in G_0}{a\in A_x}\subseteq G_0$ is open, and
the function $s_{\cat{A},a}:\cterm{x\in G_0}{a\in A_x}\rightarrow A$ defined by $s(x)=a$ is a continuous
section; and
\item
for any $a,b\in \Sets_{\kappa}$, the set
\[\bopen{\cat{A},a\mapsto b}=\cterm{g:x\rightarrow y}{a\in A_{x}\wedge \alpha(g,a)=b} \subseteq G_1\]
is open.
\end{enumerate}
\begin{proof}
Let a morphism
$f:\thry{G}\to\thry{S}$ in \alg{CohGpd} be given, inducing a geometric morphism $f:\Sh{\thry{G}}\to
\Sh{\thry{S}}$ such that the inverse image preserves compact objects. Then $f^*(\cat{U})$ is a compact decidable object with fibers in $\Sets_{\kappa}$; the
set $\bopen{f^*(\cat{U}),a}=f_0^{-1}(\bopen{a})\subseteq G_0$ is open; the continuous section $\bopen{a}\rightarrow U$ defined by $M\mapsto a$ pulls back along $f_0$ to yield the required section; and the set
$\bopen{f^*(\cat{U}),a\mapsto b}=f_1^{-1}(\bopen{a\mapsto b})\subseteq G_1$ is open. So $f^*(\cat{U})$ satisfies conditions (i)--(iv).
Conversely, suppose that $\cat{A}=\pair{A\rightarrow G_0, \alpha}$ satisfies conditions (i)--(iv). Define the function $f_0:G_0\rightarrow S_0$ by $x\mapsto A_x$, which is possible since
$A_x\in\Sets_{\kappa}$ by (ii). Then for a subbasic open set $\bopen{a}\subseteq S_0$, we have
\[ f_0^{-1}(\bopen{a})=\cterm{x\in G_0}{a\in A_x}=\bopen{\cat{A},a} \]
so $f_0$ is continuous by (iii). Next, define $f_1:G_1\rightarrow S_1$ by
\[g:x\rightarrow y\ \ \mapsto\ \
\alpha(g,-):A_x\rightarrow A_y.\]
Then for a subbasic open $\bopen{a\mapsto b}\subseteq S_1$, we have
\[ f_1^{-1}(\bopen{a\mapsto b})=\cterm{g\in G_1}{a\in A_{s(g)}\wedge \alpha(g,a)=b}=\bopen{\cat{A},a\mapsto b} \]
so $f_1$ is continuous by (iv). It remains to show that $f^*(\cat{U})=\cat{A}$. First, we must verify that what is a
pullback of sets:
\[\bfig
\square[A`U`G_0`S_0;```f_0]\place(100,400)[\pbangle]
\efig\]
is also a pullback of spaces. Let $a\in A$ with $V\subseteq A$ an open neighborhood. We must find an open box
around $a$ contained in $V$. Intersect $V$ with the image of the section $s_{\cat{A},a}(\bopen{\cat{A},a})$ to
obtain an open set $V'$ containing $a$ and homeomorphic to a subset $W\subseteq G_0$. Then we can write $V'$ as
the box $W\times_{S_0}\bopen{\syntob{x,y}{x=y},a}$ for the open set $\bopen{\syntob{x,y}{x=y},a}\subseteq U$. Conversely, let a basic open $\bopen{\syntob{x,\vec{y}}{\phi},\vec{b}} \subseteq U$ be given, for $\phi$ a formula of $\theory_{\neq}$. We must show that it pulls back to an open subset of $A$. Let $a\in A_z$ be given and assume that $a$ (in the fiber over $f_0(z))$ is in
$\bopen{\syntob{x,\vec{y}}{\phi},\vec{b}}$. Now, since \cat{A} is decidable, there is a canonical interpretation of
\syntob{x,\vec{y}}{\phi} in \Sh{\thry{G}} obtained by interpreting \cat{A} as the single sort, and using the
canonical coherent structure of \Sh{\thry{G}}. Thereby, we obtain an object
\[
\cat{B}:=\csem{x,\vec{y}}{\phi}^{\cat{A}}\embedd \cat{A}\times\ldots\times\cat{A}\to^{\pi_1}\cat{A}\]
in with an underlying open subset $B\subseteq A\times_{G_0}\ldots
\times_{G_0} A\to^{\pi_1}A$. One can verify that \cat{B} satisfies conditions (i)--(iv), se the proof of Lemma \ref{Lemma: Theta G is a coherent category} below. Let $W\subseteq B$ be the image of the continuous section
$s_{\cat{B},a,\vec{b}}(\bopen{\cat{B},a,\vec{b}})$. Then the pullback of $\bopen{\syntob{x,\vec{y}}{\phi},\vec{b}}$ along $f_0$ is the image of $W$ along the projection
$\pi_1:\cat{A}\times\ldots\times\cat{A}\to\cat{A}$, which is an open subset of $A$.
\end{proof}
\end{lemma}
The logically definable objects in the category of equivariant sheaves on the groupoid of models and isomorphisms of a
theory are readily seen to be a (guiding) example of objects satisfying conditions (i)--(iv) of Lemma \ref{Lemma:
DC, Theta(G)}, so we have:
\begin{lemma}\label{Lemma: DC, Definables are firm}
For any $\synt{C}{T}$ in \alg{dCoh_{\kappa}}, the canonical interpretation functor $\cat{M}^{\dag}$ of \ref{Proposition: Multisort classtop eqsh equivalence} factors through $\Form(\thry{G}_{\theory})$,
\[\cat{M}^{\dag}:\synt{C}{T}\to\Form(\thry{G}_{\theory})\embedd \Sh{\thry{G}_{\theory}}\]
\end{lemma}
Next, we show that the formal sheaves on a coherent groupoid form a decidable coherent category.
\begin{lemma}\label{Lemma: Theta G is a coherent category}
Let \thry{G} be an object of \alg{CohGpd}. Then $\Form(\thry{G})\embedd \Sh{\thry{G}}$ is a (positive) decidable
coherent category.
\begin{proof}
We verify that $\Form(\thry{G})$ is closed under the relevant operations using the characterization of Lemma \ref{Lemma: DC, Theta(G)}. By Remark \ref{Remark: Compact and coherent object}, it suffices to show that conditions (ii)--(iv) of Lemma \ref{Lemma: DC, Theta(G)} are closed under finite limits, images, and finite coproducts.
\textbf{Initial object.} Immediate.
\textbf{Terminal object.} The canonical terminal object, write \pair{X'\rightarrow X, \alpha}, is such that the
fiber over any $x\in G_0$ is $\{\star\}\in \Sets_{\kappa}$, whence the set \cterm{x\in G_0}{a\in X_x'} is $X$ if
$a=\star$ and empty otherwise. Similarly, the set $\cterm{g:x\rightarrow y}{a\in
X_x'\wedge\alpha(g,a)=b}\subseteq G_1$ is $G_1$ if $a=\star =b$ and empty otherwise.
\textbf{Finite products.} We do the binary product $\cat{A}\times\cat{B}$. The fiber over $x\in G_0$ is the
product $A_x\times B_x$, and so it is in $\Sets_{\kappa}$. Let a set $\bopen{\cat{A}\times\cat{B},c}$ be given. We
may assume that $c$ is a pair, $c=\pair{a,b}$, or $\bopen{\cat{A}\times\cat{B},c}$ is empty. Then,
\[
\bopen{\cat{A}\times\cat{B},\pair{a,b}}=\bopen{\cat{A},a}\cap \bopen{\cat{B},b}
\]
and the function $s_{\cat{A}\times\cat{B},\pair{a,b}}:\bopen{\cat{A}\times\cat{B},\pair{a,b}}\rightarrow
A\times_{G_0}B$ is continuous by the following commutative diagram:
\[\bfig
\morphism<0,-700>[\bopen{\cat{A},a}`A;s_{\cat{A},a}]
\morphism(1800,0)<0,-700>[\bopen{\cat{B},b}`B;s_{\cat{B},b}]
\morphism(0,0)/<-^{)}/<900,0>[\bopen{\cat{A},a}`\bopen{\cat{A}\times\cat{B},\pair{a,b}};\supseteq]
\morphism(900,0)<0,-700>[\bopen{\cat{A}\times\cat{B},\pair{a,b}}`A\times_{G_0} B;s_{\cat{A}\times \cat{B},
\pair{a,b}}]
\morphism(900,0)/^{ (}->/<900,0>[\bopen{\cat{A}\times\cat{B},\pair{a,b}}`\bopen{\cat{B},b};\subseteq]
\morphism(0,-700)/<-/<900,0>[A`A\times_{G_0} B;\pi_1]
\morphism(900,-700)<900,0>[A\times_{G_0} B`B;\pi_2]
\efig\]
Similarly, the set $\bopen{\cat{A}\times B,c\mapsto d}$ is either empty or of the form \[\bopen{\cat{A}\times
B,\pair{a,b}\mapsto \pair{a',b'}}\] in which case
\[\bopen{\cat{A}\times
B,\pair{a,b}\mapsto \pair{a',b'}}= \bopen{\cat{A},a\mapsto a'}\cap \bopen{\cat{B},b\mapsto b'}.\]
\textbf{Equalizers and Images.} Let \cat{A} be a subobject of $\cat{B}=\pair{\pi_1:B\rightarrow G_0,\beta}$,
with $A\subseteq B$, and \cat{B} satisfying the properties (ii)--(iv) of Lemma \ref{Lemma: DC, Theta(G)}. Then
given a set $\bopen{\cat{A},a}$,
\[ \bopen{\cat{A},a}= \pi_1(A\cap s_{\cat{B},a}(\bopen{\cat{B},a})) \]
and we obtain $s_{\cat{A},a}$ as the restriction
\[\bfig
\dtriangle/<-`>`^{ (}->/<800,500>[B`\bopen{\cat{B},a}`G_0; s_{\cat{B},a}`\pi_1`]
\dtriangle(-800,0)/<-``^{ (}->/<800,500>[A`\bopen{\cat{A},a}` \bopen{\cat{B},a}; s_{\cat{A},a}``]
\morphism(0,500)/^{ (}->/<800,0>[A`B;] \place(-500,75)[\pbanglef]
\efig\]
Similarly, given a set $\bopen{\cat{A},a\mapsto b}\subseteq G_1$,
\[ \bopen{\cat{A},a\mapsto b}= \bopen{\cat{B},a\mapsto b}\cap s^{-1}(\bopen{\cat{A},a}) \]
where $s$ is the source map $s:G_1\rightarrow G_0$. We conclude that \funksjon{\Form}{\thry{G}} is closed under
both equalizers and images.
\textbf{Binary coproducts.} Write binary coproducts in $\Sets_{\kappa}$ as $X+Y=
\cterm{\pair{0,x},\pair{1,y}}{x\in X\wedge y\in Y}$. Then if $\bopen{\cat{A}+\cat{B},c}$ is non-empty, $c$ is a pair
$c=\pair{0,a}$ or $c=\pair{1,b}$. If the former, then $\bopen{\cat{A}+\cat{B}, \pair{0,a}}= \bopen{\cat{A},a}$, and the
section is given by composition:
\[\bfig
\square/>`<-`>`^{ (}->/<1000,500>[A`A+B`\bopen{\cat{A}+\cat{B}, \pair{0,a}}= \bopen{\cat{A},a}`G_0; p_1`s_{\cat{A},a}``]
\efig\]
The latter case is similar, and so is verifying that the set $\bopen{\cat{A}+\cat{B}, c\mapsto d}$ is open.
\end{proof}
\end{lemma}
\begin{lemma}
Let \thry{G} be a coherent groupoid. Then $\Form(\thry{G})\embedd \Sh{\thry{G}}$ has a saturated set of
$\kappa$-small models.
\begin{proof}
This follows from the fact that the coherent inclusion
\[\Form(\thry{G})\embedd \Sh{\thry{G}}\]
reflects covers, since every formal sheaf is compact, and any point, given by an element $x\in
G_0$,
\[\Sets \to \Sets/G_0 \epi \Sh{G_0} \epi \Sh{\thry{G}} \epi \Sh{\Form(\thry{G})} \]
yields a coherent functor $\Form(\thry{G})\to \Sets_{\kappa}\embedd\Sets$, since the value of the point at an
equivariant sheaf is the fiber over $x$, and formal sheaves have fibers in $\Sets_{\kappa}$.
\end{proof}
\end{lemma}
\begin{lemma}
If $f:\thry{G}\to\thry{H}$ is a morphism of \alg{CohGpd}, then the induced coherent inverse image functor
$f^*:\Sh{\thry{H}}\to\Sh{\thry{G}}$ restricts to a coherent functor $\Form(f)=F:\Form(\thry{H})\to
\Form(\thry{G})$,
\[
\bfig
\square/>`^{ (}->`^{ (}->`>/<750,400>[\Form(\thry{H})`\Form(\thry{G})` \Sh{\thry{H}}` \Sh{\thry{G}}; F```f^*]
\efig\]
\begin{proof}
If \cat{A} is an object of $\Form(\thry{H})$ classified
by $h:\thry{H}\to\thry{S}$, then $f^*(\cat{A})=F(\cat{A})$ is classified by $h\circ f:\thry{G}\to\thry{S}$ in
\alg{CohGpd}.
\end{proof}
\end{lemma}
This completes the construction of the `syntactical' functor:
\begin{definition}
The functor \[\Form:\alg{CohGpd}\to\alg{dCoh}_{\kappa}^{\mng{op}}\] is defined by sending a groupoid \thry{G} to
the decidable coherent category \[\Form(\thry{G})\embedd \Sh{\thry{G}}\] of formal sheaves, and a morphism
$f:\thry{G}\to\thry{H}$ to the restricted inverse image functor $f^*:\Form(\thry{H})\to\Form(\thry{G})$.
\end{definition}
\subsection{The Syntax-Semantics Adjunction
\label{Subsection: DC, The Syntax-Semantics Adjunction}
We now show that the syntactical functor is left adjoint to the semantical functor:
\[
\bfig
\morphism|a|/{@{>}@/^7pt/}/<800,0>[\alg{dCoh_{\kappa}}^{\mng{op}}` \alg{CohGpd}; \Mod]
\morphism|b|/{@{<-}@/^-7pt/}/<800,0>[\alg{dCoh_{\kappa}}^{\mng{op}}` \alg{CohGpd}; \Form]
\place(400,0)[\top]
\efig
\]
First, we identify a counit candidate. Given $\cat{D}$ in $\alg{dCoh_{\kappa}}$, we have the `evaluation' functor
\[\cat{Y}_{\cat{D}}:\cat{D}\to\Sh{\thry{G}_{\cat{D}}}\]
which sends an object $D$ to the `definable' equivariant sheaf which is such that the fiber of $\cat{Y}(D)$ over $F\in X_{\cat{D}}$ is the set $F(D)$, or more informatively, such that the diagram,
\[\bfig
\square<500,300>[\cat{D}`\Sh{\thry{G}_{\cat{D}}}`\cat{C}_{\theory_{\cat{D}}}`\Sh{\thry{G}_{\theory_{\cat{D}}}};\cat{Y}_{\cat{D}}`\eta_{\cat{D}}`\cong`\cat{M}^{\dag}]
\efig\]
commutes, using the map $\eta_{\cat{D}}$ and isomorphism $\thry{G}_{\cat{D}}\cong\thry{G}_{\theory_{\cat{D}}}$ from Section \ref{Subsection: Representation theorem for decidable coherent categories}.
$\cat{Y}_{\cat{D}}$ factors through $\Form(\thry{G}_{\cat{D}})$, by Lemma \ref{Lemma: DC, Definables
are firm}, to yield a coherent functor
\[\epsilon_{\cat{D}}:\cat{D}\to \Form(\thry{G}_{\cat{D}})=\Form\circ \Mod(\cat{D})\]
And if $F:\cat{A}\to\cat{D}$ is an arrow of \alg{dCoh_{\kappa}}, the square
\[\bfig
\square<750,500>[\cat{A}` \Form\circ \Mod(\cat{A})` \cat{D}` \Form\circ
\Mod(\cat{D});\epsilon_{\cat{A}}` F` \Form\circ \Mod(F)` \epsilon_{\cat{D}}]
\efig\]
commutes.
Next, we consider the unit. Let \thry{H} be a groupoid in \alg{CohGpd}. We construct a morphism
\[\eta_{\thry{H}}:\thry{H}\to \thry{G}_{\Form(\thry{H})}= \Mod(\Form(\thry{G})).\]
First, as previously noticed, each $x\in H_0$ induces a coherent functor $\alg{M}_x:\Form(\thry{H})\to
\Sets_{\kappa}$. This defines a function $\eta_0:H_0\rightarrow X_{\Form(\thry{H})}$. Similarly, any
$a:x\rightarrow y$ in $H_1$ induces an invertible natural transformation $\alg{f}_a:\alg{M}_x\rightarrow \alg{M}_y$. This defines
a function $\eta_1:H_1\rightarrow G_{\Form(\thry{H})}$, such that \pair{\eta_1,\eta_0} is a morphism of
discrete groupoids. We argue that $\eta_0$ and $\eta_1$ are continuous. Let a subbasic open
$U=(\pair{g_1:\cat{A}\rightarrow \cat{B}_1,\ldots,g_n:\cat{A}\rightarrow
\cat{B}_n},\pair{a_1,\ldots,a_n})\subseteq X_{\Form(\thry{H})}$ be given, with $g_i:\cat{A}=\pair{A\rightarrow
H_0, \alpha}\to \cat{B}_i=\pair{B_i\rightarrow H_0, \beta_i}$ an arrow of $\Form(\thry{H})$ and $a_i\in
\Sets_{\kappa}$, for $1\leq i\leq n$. Form the canonical product $\cat{B}_1\times \ldots\times \cat{B}_n$ in
\Sh{\thry{H}}, so as to get an arrow $g=\pair{g_1,\ldots,g_n}:\cat{A}\to \cat{B}_1\times \ldots \times\cat{B}_n$
in $\Form(\thry{H})$. Denote by \cat{C} the canonical image of $g$ in \Sh{\thry{H}} (and thus in
$\Form(\thry{H})$), such that the underlying set $C$ (over $H_0$) of \cat{C} is a subset of
$B_1\times_{H_0}\ldots\times_{H_0}B_n$. Then
\begin{align*}
\eta_0^{-1}(U) &= \cterm{x\in H_0}{\fins{y\in \alg{M}_x(\cat{A})}\alg{M}_x(g_i)(y)=a_i\ \textnormal{for}\ 1\leq i\leq n}\\
&= \cterm{x\in H_0}{\fins{y\in A_x}g_i(y)=a_i\ \textnormal{for}\ 1\leq i\leq n}\\
&= \cterm{x\in H_0}{\pair{a_1,\ldots,a_n}\in \alg{M}_x(\cat{C})}\\
&= \cterm{x\in H_0}{\pair{a_1,\ldots,a_n}\in C_x}
\end{align*}
which is an open subset of $H_0$ by Lemma \ref{Lemma: DC, Theta(G)} since \cat{C} is in $\Form(\thry{H})$. Thus $\eta_0$ is continuous. Next,
consider a subbasic open of $G_{\Form(\thry{H})}$ of the form $U=(\cat{A},a\mapsto b)\subseteq
G_{\Form(\thry{H})}$, for $\cat{A}=\pair{A\rightarrow H_0, \alpha}$ in $\Form(\thry{H})$. Then
\begin{align*}
\eta_1^{-1}(U) &= \cterm{g:x\rightarrow y}{a\in \alg{M}_x(\cat{A})\wedge (\alg{f}_g)_{\cat{A}}(a)=b}\subseteq H_1\\
&= \cterm{g:x\rightarrow y}{a\in A_x\wedge \alpha(g,a)=b}\subseteq H_1
\end{align*}
which is an open subset of $H_1$, since \cat{A} is in $\Form(\thry{H})$. Thus $\eta_1$ is also continuous, so
that \pair{\eta_1,\eta_0} is a morphism of continuous groupoids.
\begin{lemma}\label{Lemma: DC, unit well-behaved triangle}
The triangle
\begin{equation}\label{Equation: DC, unit well-behaved triangle}\bfig
\dtriangle/<-_{)}`<-`->/<900,400>[\Sh{\thry{H}}` \Form(\thry{H})` \Sh{\thry{G}_{\Form(\thry{H})}}; `
\eta^*_{\Form(\thry{H})}` \cat{Y}_{\Form(\thry{H})}]
\efig\end{equation}
commutes.
\begin{proof}
Let $\cat{A}=\pair{A\rightarrow H_0,\alpha}$ in $\Form(\thry{H})$ be given, and write $E_{\cat{A}}\rightarrow
X_{\Form(\thry{H})}$ for the underlying sheaf of $\cat{Y}_{\Form(\thry{H})}(\cat{A})$. Write
$a:\thry{H}\rightarrow\thry{S}$ and $a':\thry{G}_{\Form(\thry{H})}\rightarrow\thry{S}$, respectively, for the \alg{CohGpd}
morphisms classifying these objects. Then the triangle
\[\bfig
\Vtriangle[\thry{H}`\thry{G}_{\Form(\thry{H})}`\thry{S};\eta_{\Form(\thry{H})}`a`a']
\efig\]
in \alg{Gpd} can be seen to commute. Briefly, for $x\in H_0$, we have
$a(x)=A_x=\alg{M}_x(\cat{A})=(E_{\cat{A}})_{\alg{M}_x}= (E_{\cat{A}})_{\eta_0(x)}=a'(\eta_0(x))$ and similarly for elements
of $H_1$.
\end{proof}
\end{lemma}
It follows from Lemma \ref{Lemma: DC, unit well-behaved triangle} that the inverse image functor $\eta_{\Form(\thry{H})}^*$ preserves
compact objects, and so $\eta_{\Form(\thry{H})}:\thry{H}\to G_{\Form(\thry{H})}$ is indeed a
morphism of \alg{CohGpd}. It remains to verify that it is the component of a natural transformation. Given a
morphism $f:\thry{G}\to\thry{H}$ of \alg{CohGpd}, we must verify that the square
\[\bfig
\square<1000,500>[\thry{G}`\Mod\circ \Form(\thry{G})` \thry{H}` \Mod\circ \Form(\thry{H});
\eta_{\Form(\thry{G})}` f` \Mod\circ \Form(f)` \eta_{\Form(\thry{H})}]
\efig\]
commutes. Let $x\in G_0$ be given. We chase it around the square. Applying $\eta_{\Form(\thry{G})}$, we obtain
the functor $\alg{M}_x:\Form(\thry{G})\to \Sets$ which sends an object $\cat{A}=\pair{A\rightarrow G_0,\alpha}$ to
$A_x$. Composing with $\Form(f):\Form(\thry{H})\to\Form(\thry{G})$, we obtain the functor
$\Form(\thry{H})\to \Sets$ which sends an object $\pair{B\rightarrow H_0,\beta}$ to the fiber over $x$
of the pullback
\[\bfig
\square[f_0^*(B)`B`G_0`H_0;```f_0]\place(100,400)[\pbangle] \efig\]
which is the same as the fiber $B_{f_0(x)}$. And this is the same functor that results from sending $x$ to
$f_0(x)$ and applying $\eta_{\Form(\thry{H})}$. For $a:x\rightarrow y$ in $G_1$, a similar check establishes
that $\eta_1\circ f_1(a):\alg{M}_{f_0(x)}\rightarrow \alg{M}_{f_0(y)}$ equals $\eta_1(a)\circ \Form(f):M_{x}\circ
\Form(f)\rightarrow \alg{M}_{y}\circ \Form(f)$. It remains to verify the triangle identities.
\begin{lemma}
The triangle identities hold:
\[
\bfig
\btriangle/<-`<-`<-/<1200,800>[\Form(\thry{H})` \Form \circ \Mod \circ \Form(\thry{H})` \Form(\thry{H});
\Form(\eta_{\thry{H}})` 1_{\Form(\thry{H})}` \epsilon_{\Form(\thry{H})}]
\place(300,300)[=]
\btriangle(0,-1200)<1200,800>[\Mod(\cat{D})` \Mod\circ \Form \circ \Mod(\cat{D})` \Mod(\cat{D});
\eta_{\Mod(\cat{D})}` 1_{\Mod(\cat{D})}` \Mod(\epsilon_{\cat{D}})]
\place(300,-900)[=] \efig
\]
\begin{proof}
We begin with the first triangle, which we write:
\[
\bfig
\btriangle/<-`<-`<-/<1000,800>[\Form(\thry{H})` \Form(\thry{G}_{\Form(\thry{H})})` \Form(\thry{H});
\Form(\eta_{\thry{H}})` 1_{\Form(\thry{H})}` \epsilon_{\Form(\thry{H})}]
\efig\]
This triangle commutes by the definition of $\epsilon_{\Form(\thry{H})}$ and Lemma \ref{Lemma: DC, unit
well-behaved triangle}, as can be seen by the following diagram:
\[\bfig
\square/>`^{ (}->`^{ (}->`<-/<1000,500>[\Form(\thry{H})` \Form(\thry{G}_{\Form(\thry{H})})` \Sh{\thry{H}}`
\Sh{\thry{G}_{\Form(\thry{H})}}; \epsilon_{\Form(\thry{H})}` ``\eta^*_{\thry{H}}]
\morphism(0,500)<1000,-500>[\Form(\thry{H})`\Sh{\thry{G}_{\Form(\thry{H})}};\cat{Y}_{\Form(\thry{H})} ]
\place(230,120)[=]\place(610,360)[=]
\efig\]
We pass to the second triangle, which can be written as:
\[\bfig
\btriangle(0,0)<1000,800>[\thry{G}_{\cat{D}}` \thry{G}_{\Form(\thry{G}_{\cat{D}})}` \thry{G}_{\cat{D}};
\eta_{\thry{G}_{\cat{D}}}` 1_{\thry{G}_{\cat{D}}}` \Mod(\epsilon_{\cat{D}})]
\efig
\]
Let $\alg{N}:\cat{D}\to\Sets$ in $X_{\cat{D}}$ be given. As an element in $X_{\cat{D}}$, it determines a coherent
functor $\alg{M}_{\alg{N}}:\Form(\thry{G}_{\cat{D}})\to \Sets$, the value of which at $\cat{A}=\pair{A\rightarrow
X_{\cat{D}}, \alpha}$ is the fiber $A_{\alg{N}}$. Applying $\Mod(\epsilon_{\cat{D}})$ is composing with the functor
$\epsilon_{\cat{D}}:\cat{D}\to \Form(\thry{G}_{\cat{D}})$, to yield the functor $\alg{M}_{\alg{N}}\circ
\epsilon_{\cat{D}}:\cat{D}\to\Sets$, the value of which at an object $B$ in \cat{D} is the fiber over $\alg{N}$ of
$\cat{Y}_{\cat{D}}(B)$, which of course is just $\alg{N}(B)$. For an invertible natural transformation $\alg{f}:\alg{M}\rightarrow \alg{N}$ in
$G_{\cat{D}}$, the chase is entirely similar, and we conclude the the triangle commutes.
\end{proof}
\end{lemma}
\begin{theorem}\label{Proposition: DC, The adjunction}
The contravariant functors \Mod\ and $\Form$ are adjoint,
\[
\bfig
\morphism|a|/{@{>}@/^7pt/}/<800,0>[\alg{dCoh_{\kappa}}^{\mng{op}}` \alg{CohGpd}; \Mod]
\morphism|b|/{@{<-}@/^-7pt/}/<800,0>[\alg{dCoh_{\kappa}}^{\mng{op}}` \alg{CohGpd}; \Form]
\place(400,0)[\top]
\efig
\]
where \Mod\ sends a decidable coherent category \cat{D} to the semantic groupoid
\homset{\alg{dCoh}}{\cat{D}}{\Sets_{\kappa}} equipped with the coherent topology, and $\Form$ sends a
coherent groupoid \thry{G} to the full subcategory $\Form(\thry{G})\embedd \Sh{\thry{G}}$ of formal sheaves, i.e.\ those classified by the morphisms in \homset{\alg{CohGpd}}{\thry{G}}{\thry{S}}.
\end{theorem}
Notice that if \cat{D} is an object of \alg{dCoh_{\kappa}}, then the counit component
$\epsilon_{\cat{D}}:\cat{D}\to \Form\circ\Mod(\cat{D})$ is a Morita equivalence of categories, in the sense
that it induces an equivalence $\sh{D}\simeq \Sh{\Form\circ\Mod(\cat{D})}$. In the case where \cat{D} is a
pretopos, the counit is, moreover, also an equivalence of categories, since any decidable compact object in
\sh{D} is coherent and therefore isomorphic to a representable in that case. Furthermore, for any \cat{D} in
\alg{dCoh_{\kappa}}, we have that the unit component $\eta_{\thry{G}_{\cat{D}}}:\thry{G}_{\cat{D}}\to
\thry{G}_{\Form(\thry{G}_{\cat{D}})}$ is a Morita equivalence of topological groupoids, in the sense that it induces an
equivalence $\Sh{\thry{G}_{\cat{D}}}\simeq \Sh{\thry{G}_{\Form(\thry{G}_{\cat{D}})}}$. We refer to the full image of
\Mod\ in \alg{Gpd} as \alg{SemGpd}, the category of \emph{semantic groupoids}.
\begin{corollary}
The adjunction of Theorem \ref{Proposition: DC, The adjunction} restricts to an adjunction
\[
\bfig
\morphism|a|/{@{>}@/^7pt/}/<800,0>[\alg{dCoh_{\kappa}}^{\mng{op}}` \alg{SemGpd}; \Mod]
\morphism|b|/{@{<-}@/^-7pt/}/<800,0>[\alg{dCoh_{\kappa}}^{\mng{op}}` \alg{SemGpd}; \Form]
\place(400,0)[\top]
\efig
\]
with the property that the unit and counit components are Morita equivalences of categories and topological
groupoids respectively.
\end{corollary}
\subsection{Stone Duality for Classical First-Order Logic
Returning to the classical first-order logical case, we can restrict the adjunction further to the full subcategory $\alg{BCoh_{\kappa}}\embedd \alg{dCoh_{\kappa}}$ of Boolean coherent categories. Unlike in the decidable coherent case, the pretopos completion of a Boolean coherent category is again Boolean, so that \alg{BCoh_{\kappa}} is closed under pretopos completion. Since, as we mentioned in Section \ref{Subsection: Theories and Models}, completing a first-order theory so that its syntactic category is a pretopos involves only a conservative extension of the theory and does not change the category of models, it is natural to represent the classical first-order theories by the subcategory of Boolean pretoposes (see e.g.\ \cite{makkaireyes}, \cite{makkai:87b}). We shall refer to the groupoids in the image of the semantic functor \Mod\ restricted to the full subcategory of Boolean pretoposes $\alg{BPTop_{\kappa}}\embedd\alg{dCoh_{\kappa}}$, as \emph{Stone groupoids}. Thus $\alg{StoneGpd}\embedd\alg{SemGpd}$ is the full subcategory of topological groupoids of models of theories in classical, first-order logic (the morphisms are still those continuous homomorphisms that preserve compact sheaves).
\begin{corollary}\label{Cor:adj for bpt}
The adjunction of Theorem \ref{Proposition: DC, The adjunction} restricts to an adjunction
\[
\bfig
\morphism|a|/{@{>}@/^7pt/}/<800,0>[\alg{BPTop_{\kappa}}^{\mng{op}}` \alg{StoneGpd}; \Mod]
\morphism|b|/{@{<-}@/^-7pt/}/<800,0>[\alg{BPTop_{\kappa}}^{\mng{op}}` \alg{StoneGpd}; \Form]
\place(400,0)[\top]
\efig
\]
with the property that the unit and counit components are Morita equivalences of topological groupoids and equivalences of pretoposes, respectively.
\end{corollary}
Moreover, given the obvious notion of `continuous natural transformation' of topological groupoid homomorphisms, the unit components of the foregoing adjunction can also be shown to be equivalences. Thus we have our main result:
\begin{theorem}\label{thm:duality for bpt}
The adjunction of Corollary \ref{Cor:adj for bpt} is a (bi-)equivalence,
\begin{equation}\label{eq: Stone duality for bccs}
{\alg{BPTop_{\kappa}}^{\mathrm{op}}} \simeq \alg{StoneGpd}
\end{equation}
establishing a duality between the category of ($\kappa$-small) Boolean pretoposes and Stone topological groupoids.
\end{theorem}
Finally, a remark on the posetal case and classical Stone duality for Boolean algebras. By a \emph{coherent space} we mean a compact topological space such that the compact open sets are closed under intersection and form a basis for the topology. A \emph{coherent function} between coherent spaces is a continuous function such that the inverse image of a compact open is again compact. Stone duality can be obtained as a restriction of a contravariant adjunction between the category \alg{dLat} of distributive lattices and homomorphisms and the category \alg{CohSpace} of coherent spaces and coherent functions
\begin{equation}\label{eq: Stone duality for distlat}
\bfig
\morphism|a|/{@{>}@/^7pt/}/<800,0>[\alg{dLat}^{\mng{op}}` \alg{CohSpace}; ]
\morphism|b|/{@{<-}@/^-7pt/}/<800,0>[\alg{dLat}^{\mng{op}}` \alg{CohSpace}; ]
\place(400,0)[\top]
\efig
\end{equation}
where, as in Stone duality, the right adjoint is the `Spec' functor obtained by taking prime filters (or homming into the lattice 2), and the left adjoint is obtained by taking the distributive lattice of compact opens (or homming into the Sierpi\'{n}ski space, i.e.\ the set 2 with one open point). This adjunction restricts to a contravariant equivalence between distributive lattices and sober coherent spaces, and further to the full subcategory of Boolean algebras, $\alg{BA}\hookrightarrow\alg{dLat}$, and the full subcategory of Stone spaces and continuous functions, $\alg{Stone}\hookrightarrow \alg{CohSpace}$, so as to give the contravariant equivalence of classical Stone duality:
\begin{equation}\label{adj:stonedual}
\bfig
\morphism|a|/{@{>}@/^7pt/}/<800,0>[\alg{BA}^{\mng{op}}` \alg{Stone}; ]
\morphism|b|/{@{<-}@/^-7pt/}/<800,0>[\alg{BA}^{\mng{op}}` \alg{Stone}; ]
\place(400,0)[\simeq]
\efig
\end{equation}
The adjunction \eqref{eq: Stone duality for distlat} can be obtained from the adjunction of Theorem \ref{Proposition: DC, The adjunction} as follows. A poset is a distributive lattice if and only if it is a coherent category (necessarily decidable), and as we remarked after Definition \ref{Definition: DC, DCkappa}, such a poset always has enough $\kappa$-small models, so that $$\alg{dLat}\embedd\alg{dCoh_{\kappa}}$$ is the subcategory of posetal objects. On the other side, any space can be considered as a trivial topological groupoid, with only identity arrows, and it is straightforward to verify that this yields a full embedding $$\alg{CohSpace}\embedd\alg{CohGpd}.$$ Since a coherent functor from a distributive lattice \cat{L} into \Sets\ sends the top object in \cat{L} to the terminal object 1 in \Sets, and everything else to a subobject of 1, restricting the semantic functor \Mod\ to \alg{dLat} gives us the right adjoint of (\ref{eq: Stone duality for distlat}). In the other direction, applying the syntactic functor \Form\ to the subcategory $\alg{CohSpace}\embedd\alg{CohGpd}$ does not immediately give us a functor into \alg{dLat}, simply because the formal sheaves do not form a poset (for instance, by Lemma \ref{Lemma: DC, Theta(G)}, the formal sheaves on a coherent groupoid include all finite coproducts of 1). However, if we compose with the functor $\mathrm{Sub}(1):\alg{dCoh_{\kappa}}\to \alg{dLat}$ which sends a coherent category $\cat{C}$ to its distributive lattice $\mathrm{Sub}_{\cat{C}}(1)$ of subobjects of 1, then it is straightforward to verify that we have a restricted adjunction
\[
\bfig
\morphism|a|/{@{>}@/^7pt/}/<800,0>[\alg{dLat}^{\mng{op}}` \alg{CohSpace}; \Mod]
\morphism|b|/{@{<-}@/^-7pt/}/<800,0>[\alg{dLat}^{\mng{op}}` \alg{CohSpace}; \hspace{1ex}\mathrm{Form}_1]
\place(400,0)[\top]
\efig
\]
where $\mathrm{Form}_1(\cat{C}) = \mathrm{Sub}_{\mathrm{Form}(\cat{C})}(1)$. Moreover, this is easily seen to be precisely the adjunction (\ref{eq: Stone duality for distlat}), of which classical Stone duality for Boolean algebras is a special case. Indeed, again up to the reflection into $\mathrm{Sub}(1)$, the duality \eqref{adj:stonedual} is precisely the poset case of the duality \eqref{eq: Stone duality for bccs} between ($\kappa$-small) Boolean pretoposes and Stone topological groupoids.
\section*{Acknowledgements}
The second author was supported by the Eduard \v{C}ech Center for Algebra and Geometry (grant no.\ LC505) during parts of the research presented here. Both authors take the opportunity to thank Lars Birkedal and Dana Scott for their support.
\bibliographystyle{ieeetr}
|
3,212,635,537,466 | arxiv | \section{Introduction} \label{sec:level1}
Microwave investigations of classical superconductors half a century ago have proven to be a very useful tool in the determination
of intrinsic parameters in the Meissner state\cite{Glover:57,Biondi:59,Klein:94} as well as the flow resistivity in the mixed
state.\cite{Cardona:64,Rosenblum:64,Gittleman:66}
These investigations were also of large technical interest for the construction of microwave transmission lines
and resonant structures.
In the last twenty years high temperature superconductors (HTSC) have been extensively investigated by microwave techniques.
Among the greatest successes, one should
mention the determination of the temperature dependence of the penetration depth which evidenced the existence of nodes in the
superconducting gap, leading to the d-wave explanation of HTSC.\cite{Hardy:93}
Flux flow resistivity can be determined from dc measurements when the current density $J$
exceeds the critical current density $J_c$ for vortex depinning.\cite{Strnad:64}
However, in many cases of low and high temperature superconductors, the dynamics of
vortex motion is more complicated and cannot be interpreted just in terms of pinned and depinned regimes.
Thermally activated flux hopping may yield ohmic behavior even for current densities below $J_c$.\cite{Palstra:90}
However, the resistivity observed in this regime should not be identified with that of flux flow.
By increasing the current density there occurs a nonlinear
transition to the flux flow regime. At much higher current densities one may encounter
nonlinear effects in the flux flow leading ultimately to the point of instability where a dramatic
increase of the measured voltage occurs.\cite{Klein:85,Doettinger:94,Xiao:96,Ruck:97,Kunchur:02,Babic:04}
The nonlinear behavior is temperature and magnetic field dependent.
Hence, the linear flux flow regime may occur inbetween the two nonlinear regimes, and may not always be well resolved.
In contrast to dc resistivity, microwave measurements can be carried out with current densities much smaller than $J_c$, and still yield
flux flow resistivity. Namely, the vortices are not driven over the pinning potential barrier, but just
oscillate within the potential well.
Depinning frequency $\omega_0$ separates two regimes of vortex oscillation. If the driving frequency $\omega$
is much larger than $\omega_0$, the viscous drag force dominates over the restoring pinning force in the response of
vortices to the microwave Lorentz force.
This is often the case with classical superconductors which have $\omega_0$ in the MHz range.
In that case, the flux flow resistivity can be extracted from the microwave absorption curves only.
When $\omega_0 \approx \omega$, or higher, the restoring pinning force is comparable to the viscous drag force,
and the response of the vortices is more complex. This is typical of HTSC where the depinning frequency
is found to be of the order of 10 GHz,\cite{Golosovsky:94} but may be found also in low temperature superconductors
such as in very thin Nb film analyzed in this paper below.
In such cases, one needs both, microwave absorption and dispersion to determine the depinning frequency and flux flow parameters.
The analysis usually employs the models
of effective conductivity in the mixed state\cite{Coffey:91,Brandt:91,Dulcic:93} based on the Bardeen-Stephen model.\cite{Bardeen:65}
The real values of $B_{c2}$ in HTSC remain experimentally unreachable except for the narrow temperature range below $T_c$, and
the $B_{c2}$ values extracted from the effective conductivity data could not be experimentally verified.
In order to probe the effective conductivity model, we have performed a series of measurements on thin films and a single
crystal of niobium, a classical type-II superconductor. The comparison of thick niobium samples with HTSC samples is not ideal
since the upper critical fields and depinning frequencies in niobium are much lower and the vortex distance doesn't always allow
the use of effective conductivity models. However, as the film thickness is reduced, one expects considerable enhancement
of $B_{c2}$, due to the reduced coherence length, and much stronger pinning, due to the surface effects.
We have measured high quality niobium thin
films using a cavity perturbation method. The temperature and field dependence of the complex frequency shift have been
measured up to the upper critical fields in order to test the validity of the effective conductivity models. The results of
the present analysis should be taken into account when the mixed state of HTSC is investigated by the microwave methods.
\section{Samples} \label{sec:level2}
The Nb films were deposited via molecular beam epitaxy (MBE) in a commercial system from DCA instruments (Finland).
The base pressure of the system is 10$^{-9}$ Pa. The (0001) surfaces of the sapphire substrates ($\alpha$-Al$_2$O$_3$) were prepared by
sputter cleaning with Ar ions (1 keV) and subsequent annealing at 1000$^{\circ}$C in ultra-high vacuum (UHV).
This annealing procedure is necessary to remove the embedded Ar gas atoms and to recover the sapphire surface crystallography and morphology.
Details of the surface preparation procedure can be found elsewhere.\cite{Bernath:98,Gao:02,Wagner:98}
The substrate treatment resulted in unreconstructed surfaces, as revealed by reflection high energy electron diffraction (RHEED).
Niobium (4 N purity) was evaporated from an electron beam
evaporator (substrate temperature 900$^{\circ}$C; film thicknesses 10\:nm, 40\:nm, 160\:nm) and the growth rate was monitored using a quartz oscillator.
Typical growth rates were between 0.01 and 0.05 nm/s. As revealed by in-situ RHEED investigations, the Nb films grew epitaxially on the (0001) sapphire
substrate.\cite{Wagner:98} A 2\:nm thick protective layer of SiO$_2$ has been deposited onto each film
in order to prevent niobium oxidation (physical vapour deposition at room temperature).
Transmission electron microscopy (TEM) investigations have shown that the film of nominal thickness
40\:nm is actually 36\:nm thick, indicating that the other films are also slightly thinner than their nominal thickness.
The samples were cut by a diamond saw into pieces 3$\times$0.5 mm$^2$ (40\:nm and 160\:nm films),
and 2.3$\times$1 mm$^2$ (10\:nm film), suitable for cavity perturbation measurements.
High purity single crystal of niobium was purchased from Metal Crystals \& Oxides Ltd., Cambridge.
Its dimensions were 3$\times$2$\times$0.5\:mm$^3$.
Superconducting properties of thin niobium films strongly depend on their thickness, purity and preparation
conditions.\cite{Wolf:76,Kodama:83,Minhaj:94,Park:85,Gubin:05,Hsu:92}
In order to compare our samples with those measured earlier,\cite{Wolf:76,Kodama:83,Minhaj:94}
we have determined T$_c$ of each film by the so called 90\% criterion, i.e. the transition temperature was defined as the
temperature where microwave absorption $\Delta (1/2Q)$ reaches 90\% of its normal state value. For the samples of the same composition and purity
one expects that T$_c$ is reduced aproximately with the inverse thickness $d^{-1}$ due to the proximity effects,\cite{Cooper:61}
weak localization and increased residual resistivity.\cite{Minhaj:94} We show in Fig.\ref{Fig1} the comparison of
transition temperatures of our thin films with results of some other authors. The $T_c$ values of present films are similar to the films measured
by Gubin et al.,\cite{Gubin:05} and, if extrapolated to ultrathin 2 nm, with the film measured by Hsu and Kapitulnik.\cite{Hsu:92}
Obviously, the films measured in this paper are of good quality.
\begin{figure}
\includegraphics[width=8cm]{Fig1.eps}%
\caption{(Color online) Thickness dependence of the transition temperature of the measured niobium films (full squares).
The data are compared with clean samples of Wolf et al.\cite{Wolf:76} and Kodama et al.\cite{Kodama:83} (dashed line),
and with dirty samples of Minhaj et al.\cite{Minhaj:94} (circles).}
\label{Fig1}
\end{figure}
To extract various superconducting parameters one needs three quantities: $T_c$, $\rho_n$ and $S=-dB_{c2}/dT$ at $T_c$.
The $B_{c2}$ values determined also by the 90\% criterion
from our measurements
gave the $S$-values $0.47\:\rm{T/K}$, $0.11\:\rm{T/K}$
and $0.08\:\rm{T/K}$ for 10\:nm, 40\:nm and 160\:nm samples, respectively.
The resistivities of the films have been determined by standard four contact method. Their residual values at
$T=10\: \rm{K}$ were $\rho_n(10\:\rm{nm})=15.2\: \rm{\mu \Omega cm}$, $\rho_n(40\:\rm{nm})=1.4\: \rm{\mu \Omega cm}$ and
$\rho_n(160\:\rm{nm})=0.33\: \rm{\mu \Omega cm}$, respectively.
The residual resistivity of 160\:nm film was
found to be an order of magnitude lower than in the film studied by Gubin et al.,\cite{Gubin:05} and
comparable to the best bulk single crystals,\cite{Blaschke:82,Klein:94} showing that the
films grew with perfect epitaxy with virtually no defects.
The residual resistivities of thinner samples are, therefore, increased only due to the surface scattering, but the values are still below the
values reported by Gubin et al.\cite{Gubin:05} For the 10\:nm film we estimate $\kappa \approx 9.5$, for the 40\:nm film
$\kappa \approx 1.5$, and for the 160\:nm film we can take the single crystal value of $\kappa \approx 0.9$.
\section{Experimental Details} \label{sec:level3}
Microwave measurements were carried out in a high-Q elliptical cavity made of copper resonating in $_e$TE$_{111}$
mode at 9.3 GHz or in $_e$TE$_{113}$ mode at 17.5 GHz.
For both modes used, microwave magnetic field has node in the center of the cavity while microwave electric field is at maximum.
The sample was mounted on a sapphire sample holder and placed in the
center of the cavity where the microwave electric field $E_{mw}$ has its maximum in both modes.
The sample was oriented with the longest side parallel to $\boldsymbol{E}_{mw}$.
Experimental setup included Oxford Systems superconducting magnet with $\boldsymbol{B}_{dc}\perp \boldsymbol{E}_{mw}$.
While $\boldsymbol{E}_{mw}$ is always in the film plane, $\boldsymbol{B}_{dc}$ can be in the film plane or perpendicular to it.
Directly measured quantities are the $Q$-factor and the resonant
frequency $f$ of the cavity loaded with the sample.
The $Q$-factor was measured by a modulation technique described
elsewhere.\cite{Nebendahl:01} The empty cavity absorption
$(1/2Q)$ was substracted from the measured data and the presented
experimental curves are due to the samples themselves.
An automatic frequency control (AFC) system was used to set the
source frequency always in resonance with the cavity. Thus, the
frequency shift can be measured as the temperature of the sample or static magnetic field is
varied.
The two measured quantities represent the complex frequency shift
$\Delta \widetilde{\omega} / \omega = \Delta f/f + i \Delta (1/2Q)$.
\section{Effective conductivity} \label{sec:level4}
For the extraction of complex conductivity data from the measured complex frequency shift
of a thin superconducting sample in the microwave electric field, one can utilize
the general solution for the complex frequency shift by Peligrad et al.\cite{Peligrad:01}.
The shift from a perfect conductor state is given by
\begin{equation}
\displaystyle {\frac{\Delta \widetilde{\omega}_p}{\omega}}=
\frac{\Gamma}{N} \left[
1+\left(\frac{\widetilde{k}^2}{k_0^2}
\frac{\tanh (i \widetilde{k} d/2) }{i \widetilde{k} d/2} -1 \right) N
\right]^{-1}
\mbox{\ ,}
\label{eq:2}
\end{equation}
where $d$ is the thickness of the slab, $N$ is the depolarization factor and
$\Gamma$ is the dimensionless filling factor of the sample in the cavity.
The complex wave vector $\widetilde{k}$ is given by
\begin{equation}
\widetilde{k}=k_0 \sqrt{\widetilde{\mu}_r \left( \widetilde{\epsilon}_r -
i \frac{\widetilde{\sigma}}{\epsilon_0 \omega} \right) } \mbox{\ ,}
\label{eq:3}
\end{equation}
where $k_0=\omega \sqrt{\mu_0 \epsilon_0}$ is the vacuum wave vector. It
describes generally any set of material parameters. For a nonmagnetic metal one can take
$\widetilde{\mu_r}=\widetilde{\epsilon_r}=1$ and the main contribution to $\widetilde{k}$ comes from conductivity.
Using these equations one can by numerical inversion of experimentally obtained complex frequency shift
determine the complex conductivity $\widetilde{\sigma}$ of the sample.
For a superconducting sample in the mixed state, in intermediate fields one can define
the effective complex conductivity\cite{Coffey:91,Brandt:91,Dulcic:93} which is a combination
of normal conductivity in the vortex cores and the conductivity of the condensed electrons
outside the cores. The effective conductivity in an oscillating electric field is given by:
\begin{figure*}
\includegraphics[width=18cm]{Fig2.eps}%
\caption{(Color online) Measured temperature dependence of complex frequency shift of four niobium samples:
a) thin film $d=10\:\rm{nm}$; b) thin film $d=40\:\rm{nm}$; c) thin film $d=160\:\rm{nm}$; d) single crystal.
Driving frequency was 9.3\:GHz. Applied perpendicular static magnetic fields are indicated in the legends.}
\label{Fig2}
\end{figure*}
\begin{equation}
\frac{1}{\widetilde{\sigma}_{\mathrm{eff}}}= \frac{1-\frac{b}{1-i(\omega_0/\omega)}}
{(1-b)(\sigma_1-i \sigma_2)+b \sigma_n}+
\frac{1}{\sigma_n}\, \frac{b}{1-i(\omega_0/\omega)}
\label{eq:4}
\end{equation}
The first term is due to the microwave current outside the vortex cores, and the second is due to
the normal current in the cores of the oscillating vortices. The meaning of the parameter
$b$ in Eq.~(\ref{eq:4}) is the volume fraction of the sample taken by the normal vortex
cores. This parameter determines the resistivity in the flux flow regime $\rho_f / \rho_n$.\cite{Tinkham:96}
The depinning frequency $\omega_0$ may vary, depending on sample, field and temperature from
strongly pinned case ($\omega_0 \gg \omega$) to the flux flow limit ($\omega_0 \ll \omega$).
In Eq.~(\ref{eq:4}) the zero field conductivity is $\sigma_1 - i \sigma_2$, and $\sigma_n$ is
the normal state conductivity. The model is limitted to high $\kappa$-values and to magnetic fields
much lower than upper critical fields, where vortices don't overlap significantly.
From the experimentally obtained field dependent complex conductivity extracted
using Eqs.~(\ref{eq:2}) and (\ref{eq:3})
one can, by numerical inversion of Eq.~(\ref{eq:4}), determine the
values of $b$ and $\omega_0 / \omega$.
\section{Results and Discussion} \label{sec:level6}
The measured complex frequency shift of the three films and the single crystal is presented in Fig.~\ref{Fig2}
for various values of DC magnetic field and for a driving frequency 9.3\:GHz.
One may notice a huge difference in signal intensities between
thin and thick samples due to very different filling factors $\Gamma$.
\begin{figure}
\includegraphics[width=8cm]{Fig3.eps}%
\caption{(Color online) Field dependence of the complex frequency shift for the 10\:nm film at various temperatures.}
\label{Fig3}
\end{figure}
Namely, a conducting sample profoundly
changes the electric field at the centre of the cavity with respect to the empty cavity field. The sample acts as a partial
short for the electric field lines.\cite{Peligrad:98} The field outside of a conductor is stronger at
rear and front sides, and this enhancement is larger for thinner films. As a consequence, the microwave
measurements in electric field are very sensitive for thin films, while for thicker samples
the signal/noise ratio is reduced.
Looking at the zero field measurements in Fig.~\ref{Fig2}, one observes a smooth transition to the
superconducting state for the thinnest film, and sharp transitions for thicker samples. In the 10\:nm sample
the fluctuation effects become considerable since the thickness becomes lower than the coherence length.
From the measured frequency shift and known normal state resistivity $\rho_n$ one can
determine the
complex conductivity in the whole temperature range and consequently the zero temperature penetration depth.
We have
obtained
$\lambda(T=0)=285\:\rm{nm}$ for the 10\:nm film, comparable to the result of Gubin et
al.\cite{Gubin:05} for the film of the same thickness.
The filling factor $\Gamma$ was determined at $T=10\: \rm{K}$ from the measured frequency shift and known
normal state resistivity for each sample, and we keep it fixed in further analysis of a given sample.
\begin{figure}
\includegraphics[width=8cm]{Fig4.eps}%
\caption{(Color online) Parameters $b$ and $\omega_0 / \omega$ of the effective conductivity model extracted from the data of Fig.~\ref{Fig3}.
The dashed line shows the extrapolation of the linear section at $T=2.9\:\rm{K}$.}
\label{Fig4}
\end{figure}
Knowing the measured temperature dependence of the complex frequency shift in zero field, we are
interested in its field dependence at fixed temperatures. As an example, we show the field dependence of
the complex frequency shift for the 10\:nm sample in Fig.~\ref{Fig3}.
From these data one proceeds in two steps. First, the numerical inversion of complex frequency shift
gives the field dependence of complex effective conductivity. Second, from this $\sigma_{\rm{eff}}$ one
numerically determines\cite{Mathematica}
parameters $b$ and $\omega_0$ by the use of Eq.~(\ref{eq:4}). The results are given in Fig.~\ref{Fig4}.
One may notice that the depinning frequency is well above the driving frequency for the thinnest sample,
contrary to the predictions ($\omega_0 \le 100 MHz$) for thick niobium samples.\cite{Gittleman:66}
We shall return to this point later.
The parameter $b$ is shown in Fig.~\ref{Fig4}a. In the effective conductivity model it is identified
with the volume fraction of the normal cores of the vortices in the mixed state. The field dependence of
this parameter has nearly linear region at lower fields, as expected for nonoverlapping vortices.
It has a meaning of reduced field $b=B/B^*_{c2}$ in the model of Bardeen and Stephen,\cite{Bardeen:65} where
$B^*_{c2}$ represents the hypothetical upper critical field given by the equation
$B^*_{c2}=\Phi_0/(2 \pi \xi(T)^2)$. Therefore, from the linear section of its field dependence one can determine
the radii of vortices and the GL coherence length at a given temperature.
Detailed analysis of flow resistivity by Larkin and Ovchinnikov leads to the correction factor 0.9 for
the linear regime, i.e. $b=0.9 B/B^*_{c2}$.\cite{Larkin:86} The extrapolation of this linear section, shown
by the dashed line in Fig.~\ref{Fig4}a, would give $B^*_{c2}(2.9\:\rm K) =1.9 \: \rm T$,
and $\xi (2.9 \:\rm K) = 41\: \rm{nm}$.
At higher fields $b$ changes its slope and approaches the normal state value at field $B_{c2}$ lower than
$B^*_{c2}$. It is the region where vortices start to overlap and the superconducting order parameter is
reduced throughout the sample. The deviation from linearity starts roughly at the field $B=0.63\:B_{c2}$,
where the distance between vortex centers is 3.4 times the coherence length.
\begin{figure}
\includegraphics[width=8cm]{Fig5.eps}%
\caption{(Color online) Comparison of $b$ obtained by full inversion of complex frequency shift (full black circles) and
$b$ obtained from absorption measurement alone under assumption that depinning frequency is much
lower than driving frequency (red open circles): a) 10\:nm film at 2.9\:K; b) 40\:nm film at 5\:K.}
\label{Fig5}
\end{figure}
One should mention that numerically determined $b$ values don't always reach unity at $B_{c2}$.
Close to $B_{c2}$, $b$ looses its meaning as a parameter of the effective conductivity model.
There are no more well defined vortices and the conductivity is dominated by fluctuations.
One can still try to numerically obtain $b$ and $\omega_0$, but this leads to negative values of depinning
frequencies, clearly showing the failure of the effective conductivity model in that region.
The determination of flow resistivity would not be possible if only the microwave absorption
had been measured. The effective conductivity depends on two parameters ($b$ and $\omega_0$),
and one has to measure two quantities, absorption and real frequency shift, in order to determine both
parameters. If one supposed that the pinning at microwave frequencies were negligible, one could have determined the
flow resistivity from absorption measurements alone. This would lead to wrong conclusions.
We show in Fig.~\ref{Fig5}a the comparison of the two ways of reasoning for the 10\:nm film at 2.9\:K.
The full circles show the parameter $b$ determined from the complex frequency shift
with the depinning frequency as the second result of the numerical inversion. The empty circles are the result of the
numerical inversion of the microwave absorption only, under the assumption that $\omega_0 =0$. One can see that the results
differ strongly in the low field region, i.e. in the region where the pinning is strong. When the depinning frequency
falls well below the driving frequency, the two curves overlap.
A similar analysis for the 40\:nm sample is shown in Fig.~\ref{Fig5}b. The two ways of reasoning give closer results since
the depinning frequency for this sample is lower than the driving frequency
($\omega_0 \le 0.7 \omega$, see Fig.~\ref{Fig6}).
If one is dealing with the samples whose upper critical field is experimentally unreachable (as in HTSC),
one is lead to infer the upper critical field from the slope of $b$ in low fields. Taking the flow resistivity values
from microwave absorption, without taking the real frequency shift into account,
one could reach erroneous values of the upper critical field and GL coherence length.
This is especially true for samples with high values of depinning frequency, which is usually the case in HTSC.
Bulk metallic superconductors usually don't show pinning at microwave frequencies.\cite{Gittleman:66}
\begin{figure}
\includegraphics[width=8cm]{Fig6.eps}%
\caption{(Color online) Field dependence of the depinning frequency $\nu_0 = \omega_0 / 2 \pi$ obtained by
driving frequency 9.3\:GHz for three niobium
films at $T/T_c\approx 0.5$: 10\:nm film at 2.9\:K (black full circles); 40\:nm film at 5\:K (red opened triangles)
and 160\:nm film at 5\:K (green full diamonds). The depinning frequency obtained by driving frequency 17.5\:GHz
for the 10\:nm film at 2.9\:K is shown by open black circles.}
\label{Fig6}
\end{figure}
In Fig.~\ref{Fig6}
we show the depinning frequency of our three niobium films at $T\approx 0.5\:T_c$. The depinning frequency for the 10\:nm
film at low fields is slightly above 20\:GHz, and for the 40\:nm film it is approximately 7\:GHz. For the 160\:nm film it is below
1\:GHz, i.e. in the frequency range that would be better examined by lower driving frequency. Obviously, in thin films
placed in a perpendicular magnetic field, vortex pinning is dominated by surface pinning centers. Thicker films approach
bulk values of depinning frequencies, showing that the presently grown niobium films don't have bulk pinning centers,
which are absent also in the best
single crystals.
To check the consistency of depinning frequency values obtained by different driving frequencies,
we obtained the depinning frequency of 10\:nm sample from measurements in the $_e$TE$_{113}$ mode
with driving frequency of 17.5\:GHz. The result is shown by the open circles in Fig.~\ref{Fig6}, with
satisfactory agreement between the two measurement series.
The above analysis for the 10\:nm film clearly shows linear section of $b$ in moderate fields. Already for the
40\:nm film, the linearity is not perfect. For the 160\:nm film and for single crystal there were no linear sections.
It is not surprising since the simple Bardeen-Stephen model predicts a linear dependence of $\rho_f/\rho_n$ on $B$
only in the high-$\kappa$ limit.
Obviously, only the thinnest film considered here fulfills the necessary conditions for the analysis in terms of
the effective conductivity model.
This analysis is a good model for the analysis of high-$\kappa$ superconductor with high values of upper critical
fields, typically HTSC.
One can compare the values of $B_{c2}$, obtained by the 90\:\% criterion, with the $B^*_{c2}$ values obtained from the
linear section of field dependence of $b$ by taking into account the Larkin-Ovchinnikov correction $b=0.9\:B/B^*_{c2}$.
\begin{figure}
\includegraphics[width=8cm]{Fig7.eps}
\caption{(Color online) Upper critical field $B_{c2}$ of the 10\:nm film obtained by the 90\:\% criterion from the microwave absorption
measurements (black full circles). Upper critical field $B^*_{c2}$ of the same sample obtained from the linear section
of $b$ (black opened circles). The same is shown for the 40\:nm film by red triangles.}
\label{Fig7}
\end{figure}
The comparison is shown in Fig.~\ref{Fig7}. Full circles show $B_{c2}$ obtained by the 90\:\% criterion and opened circles
show $B^*_{c2}$ values obtained from linear section of $b$. One can see that $B^*_{c2}$ is always slightly higher than
$B_{c2}$ which would be measured by direct transport or magnetization
measurement of the crossover to the normal state, if experimentally reachable. The reduction of order parameter due to the
overlapping of vortices in higher fields is one of the reasons. The other reason for further suppression of the
actual $B_{c2}$ value can be found in spin paramagnetic effects and spin-orbit scattering.\cite{WHH:66}
This effect is not significant in our niobium samples since the upper critical fields are low enough, but in HTSC,
one can expect significant suppression of $B_{c2}$ in the fields of the order of 100\:T.
As a consequence,
the transition to the normal state is due to a combination of effects, and could not be related uniquely to the coherence length.
However, the microwave determination of flow resistance in low field region gives the best
estimate of the coherence length.
The comparison of $B_{c2}$ and $B^*_{c2}$ for 40\:nm film is also shown in Fig.~\ref{Fig7}. The
$B^*_{c2}$ values are significantly higher than $B_{c2}$, but one should take this result with caution since there
is no strictly linear part of $b$ field dependence. This is not surprising since $\kappa\approx 1.5$ doesn't meet
a high-$\kappa$ condition needed for $B^*_{c2}$ extraction.
\section{Conclusions} \label{sec:level7}
We have measured temperature and magnetic field dependence of the microwave response of high quality niobium films in a
perpendicular static magnetic field. The data have been analyzed by the effective conductivity model for the high-$\kappa$
type-II superconductors in the mixed state.
Our analysis shows that this model yields consistent results when applied in fields not too close to
the transition to the normal state.
Thin niobium films show strong pinning probably due to the surface pinning centers.
The depinning frequency for the 10\:nm film is $\omega_0 \approx 20\:\rm{GHz}$, and for
the 40\:nm film it is $\omega_0 \approx 7\:\rm{GHz}$. In thicker samples the depinning frequency is
lower than $1\:\rm{GHz}$.
One should measure both, the microwave absorption and the fequency shift
in order to determine flow resistivity
for samples with high values of depinning frequency, as usually found in HTSC.
Although the bulk niobium samples have GL parameter $\kappa$ of the order of unity, very thin niobium films
have higher $\kappa$-values. They can serve as model system for the upper critical field determination
of high-$\kappa$ superconducting samples.
In HTSC generally, upper critical fields are of the order of hundreds tesla, and they are unreachable by laboratory magnets.
Conventional methods for the determination of $B_{c2}$ (transport measurements, susceptibility, torque,...) are based on the
crossover to the normal state when the applied field equals to $B_{c2}$. The determination of upper critical field is, therefore,
reduced to a narrow temperature region close to $T_c$. The values for lower temperatures are usually determined by an extrapolation.
The correctness of such extrapolations depends on the model used
and there is no assurance that it holds also
in HTSC. On the other hand, pulsed methods used to reach high applied fields (up to 400 T\cite{Sekitani:04}) are nonequilibrium
methods and very sensitive to induction, critical currents etc.
In this paper microwave response
was measured up to the
upper critical field which is for niobium samples reachable by conventional magnets.
The upper critical fields are determined by the flow resistivity in low fields ($B^*_{c2}$) and by the crossover to the normal state ($B_{c2}$).
The two methods give slightly different results.
The $B_{c2}$ defined as the crossover field to the normal state has certainly great technological importance. However,
at high fields, typical for HTSC, paramagnetic effects induce depairing and reduce order parameter. If one calculates
the coherence lengths from $B_{c2}$ values obtained in this way, the values are overestimated.
For the considerations of the physical parameters of the superconducting state, it is better to determine the coherence length
in rather low magnetic fields, deeply in the superconducting state.
The microwave method tested here is certainly suitable for the determination of $B^*_{c2}$ which gives correct values of effective vortex radii
at a given temperature in the fields low enough that vortices don't overlap. Therefore, the method is relevant for the determination of
coherence lengths, especially in HTSC.
\acknowledgments
We thank to Dr. M. Basleti\'c and E. Tafra for the help in the measurements of $\rho_n$.
|
3,212,635,537,467 | arxiv | \section{INTRODUCTION}
Color confinement, the absence of colored degrees of freedom from the physical spectrum, is an essential element of strong interactions.
Understanding this fact in the framework of relativistic quantum field theory (QFT) is a fundamental issue of particle and nuclear physics.
To investigate such fundamental aspects of strong interactions, the gluon, ghost, and quark propagators in the Landau gauge have been extensively studied by both lattice and continuum methods \cite{Alkofer:2000wg, Huber:2018ned, Maas13}.
Based on this progress, there has recently been an increasing interest in the analytic structures of the gluon, ghost, and quark propagators \cite{Alkofer:2003jj, SFK12, HFP14, Siringo16a, Siringo16b, DOS17, Lowdon17, Lowdon18, Lowdon:2018mbn, CPRW18, HK2018, DORS19, KWHMS19, BT2019, LLOS20, HK2020, Fischer-Huber, Falcao:2020vyr}.
In particular, \textit{complex singularities}, which are unusual singularities invalidating the K\"all\'en-Lehmann spectral representation \cite{spectral_repr_UKKL}, attract much attention.
In old literature \cite{Gribov78, Zwanziger89, Stingl85, HKRSW90, Stingl96, Dudal:2008sp}, e.g., for models motivated by the Gribov ambiguity, it was predicted that the gluon propagator in the Landau gauge has a pair of complex poles, which is a typical example of such singularities.
The recent studies done without assuming the K\"all\'en-Lehmann spectral representation, e.g., reconstructing the propagators from Euclidean data \cite{BT2019, Falcao:2020vyr}, modeling the propagators by the gluon mass \cite{TW10,TW11,PTW14,Siringo16a, Siringo16b, HK2018, HK2020}, and the ray technique of the Dyson-Schwinger equation \cite{SFK12,Fischer-Huber} consistently indicate the existence of complex singularities of the Landau-gauge gluon propagator.
Note that complex singularities were also observed using the ray technique in other models \cite{Maris:1991cb, Maris:1994ux, Maris:1995ns}.
Since complex singularities should never appear in propagators of observable particles, we can expect that they are connected to confinement.
Thus, theoretical aspects of complex singularities are of crucial importance since they could provide some hints for a better understanding of a confinement mechanism.
Theoretical consequences of complex singularities have so far been discussed only heuristically.
For example, some argue that the appearance of complex singularities might imply nonlocality, e.g., \cite{Stingl85, HKRSW90, Stingl96}. Nevertheless, this argument is not fully convincing due to the use of the naive inverse Wick-rotation.
To our knowledge, a solid study on this subject is still lacking.
Therefore, we will scrutinize the reconstruction procedure, namely the reconstruction of Wightman functions, or the vacuum expectation values of the products of field operators, from Schwinger functions, or the Euclidean correlators, by the analytic continuation.
We consider formal aspects of complex singularities.
Figure \ref{fig:introduction} frames our study in the reconstruction procedure.
In the standard reconstruction procedure, one begins with a family of Schwinger functions satisfying Osterwalder-Schrader (OS) axioms \cite{OS73, OS75} and reconstructs a QFT based on the OS theorem and the Wightman reconstruction theorem (see Theorem 3-7 in \cite{Streater:1989vi}).
In our study, we start from two-point Schwinger functions in the presence of complex singularities violating some of the OS axioms as seen below.
We first reconstruct the Wightman function based on the holomorphy in ``the tube'' \cite{Streater:1989vi} according to the flow shown in Fig.~\ref{fig:introduction}.
We then examine general properties of the two-point Wightman function and discuss possible state-space structures.
In this paper, we illustrate a typical example of a propagator with complex singularities and give a sketch of main results, omitting mathematical subtleties.
We provide full details of their rigorous proofs and derivations in a longer version \cite{HK2021}.
\section{Setup and Main results}
We use the following notations:
The space of test functions with compact supports is denoted by $\mathscr{D}(\mathbb{R}^4)$, and that of rapidly decreasing test functions by $\mathscr{S}(\mathbb{R}^4)$.
Elements of these dual spaces $\mathscr{D}'(\mathbb{R}^4)$ and $\mathscr{S}'(\mathbb{R}^4)$
are called distributions and tempered distributions, respectively.
We use $x,y,\xi,\eta$ as elements of $\mathbb{R}^4$.
We write $\xi = (\vec{\xi}, \xi_4)$ for Euclidean space and $\xi = (\xi^0, \vec{\xi})$ for Minkowski or complexified space.
In accordance with \cite{Streater:1989vi}, $\mathbb{R}^4 - i V_+$ is called the tube, where $V_+$ denotes the (open) forward light cone
\begin{align}
V_+ := \{ (\eta^0,\vec{\eta}) \in \mathbb{R}^4 ~;~ \eta^0 > |\vec{\eta}| \}.
\end{align}
\begin{figure}[t]
\begin{center}
\begin{center}
\includegraphics[width=0.9 \linewidth]{fig1.pdf}
\end{center}
\end{center}
\caption{The standard reconstruction procedure and contents of our study consisting of ($\alpha$) and ($\beta$)}
\label{fig:introduction}
\end{figure}
For simplicity, we consider a two-point function for a scalar field $\phi(x)$.
In all the evidences for complex singularities mentioned above, the complex singularities appear in an analytically-continued Euclidean propagator on the complex squared momentum $k^2$ plane.
Thus, we shall begin with a Euclidean propagator, or a two-point Schwinger function $S (x-y) = \mathcal{S}_2(x,y) = \braket{\phi(x)\phi(y)}_{Euc.}$, assuming the ``temperedness'' condition and Euclidean invariance.
With some regularity assumption of $S(\xi)$ at $\xi = 0$, namely $S(\xi) \in \mathscr{S}'(\mathbb{R}^4)$, the Schwinger function can be expressed as
\begin{align}
S(\xi) = \int \frac{d^4 k}{(2 \pi)^4}~ e^{ik \xi} D (k^2).
\end{align}
In the usual case where the K\"all\'en-Lehmann spectral representation holds, $D (k^2)$ has singularities only on the negative real axis, which is called the \textit{timelike axis}.
We call singularities except on the timelike axis \textit{complex singularities}.
A typical example of these singularities is a pair of complex conjugate poles as illustrated in Fig.~\ref{fig:complex.pdf}.
For technical reasons, we assume: (1) boundedness of complex singularities in $|k^2|$ in the complex $k^2$ plane, (2) holomorphy of $D(k^2)$ in a neighborhood of the real axis except for the timelike singularities, (3) some regularity of discontinuity on the timelike axis \footnote{Without this condition, the discontinuity would be, in general, a hyperfunction, which is inconvenient for some necessary limiting operations. Therefore, we assume that the discontinuity can be represented as a tempered distribution on $[-\infty,0]$, namely, $D(-\sigma^2 - i \epsilon) - D(-\sigma^2 + i \epsilon) \xrightarrow{\epsilon \rightarrow +0} \operatorname{Disc} D(-\sigma^2) \in \mathscr{S}'([0,\infty]), $
where $\mathscr{S}'([0,\infty])$ is the dual space of $\mathscr{S}([0,\infty]) := $ $\left\{ f(\lambda) = g(- (1+\lambda)^{-1}) ~;~g\mathrm{~is~a~}C^\infty \mathrm{~function~on~}[-1,0] \right\}$. For details, see Sec.~A.3 in \cite{Bogolyubov:1990kw}. Note that a tempered distribution with positive support can be extended to a tempered distribution on $[0,\infty]$.}, and (4) $D(k^2) \rightarrow 0$ as $|k^2| \rightarrow \infty$.
The following list outlines the main results of our study.
\begin{enumerate}
\renewcommand{\labelenumi}{(\Alph{enumi})}
\item The reflection positivity
is violated for the Schwinger function.
\item The holomorphy of the Wightman function $W(\xi - i \eta)$ in the tube $\mathbb{R}^4 - i V_+$ and the existence of the boundary value as a distribution $W(\xi) := \lim_{\substack{\eta \rightarrow 0 \\ \eta \in V_+}} W (\xi - i \eta) \in \mathscr{D}'(\mathbb{R}^4)$ are still valid. Thus, we can reconstruct the Wightman function from the Schwinger function.
\item The temperedness and the positivity condition in $\mathscr{D}(\mathbb{R}^4)$ are violated for the reconstructed Wightman function. The spectral condition is never satisfied since it requires the temperedness as a prerequisite.
\item The Lorentz symmetry and spacelike commutativity are kept intact.
\end{enumerate}
We prove the assertions (A) -- (D) rigorously in \cite{HK2021} and provide essential ideas in this paper.
From the assertion (C), complex singularities may seem to have no physical interpretation. However, we argue:
\begin{enumerate}
\renewcommand{\labelenumi}{(\Alph{enumi})}
\setcounter{enumi}{4}
\item Complex singularities can be realized in indefinite-metric QFTs and correspond to pairs of zero-norm eigenstates of complex energies.
\end{enumerate}
To demonstrate these results, we first give sketches of proofs of the assertions (A) -- (D) for an important example in the next section. We then mention their generalization to arbitrary complex singularities and discuss a possible realization in quantum theory.
\section{Special case: one pair of complex conjugate poles}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.85 \linewidth]{CCP.pdf}
\end{center}
\caption{
Complex squared momentum $k^2$ plane of the propagator $D(k^2)$ with a pair of complex conjugate poles given in (\ref{eq:propagator_complex_poles}).
}
\label{fig:complex.pdf}
\end{figure}
We begin with a propagator $D(k^2)$ with one pair of complex conjugate simple poles, which is decomposed into the ``timelike part'' $D_{tl}(k^2) $ and ``complex-pole part'' $D_{cp}(k^2)$,
\begin{align}
D(k^2) &= D_{tl}(k^2) + D_{cp}(k^2) , \notag \\
D_{tl}(k^2) &= \int_0 ^\infty d \sigma^2 \frac{\rho(\sigma^2)}{\sigma^2 + k^2} , \notag \\
D_{cp}(k^2) &= \frac{Z}{M^2 + k^2} + \frac{Z^*}{(M^*)^2 + k^2}, \label{eq:propagator_complex_poles}
\end{align}
where $\rho(\sigma^2)$ is the spectral function and $M^2 \in \mathbb{C}$ is the complex mass squared.
Without loss of generality, we can choose $\operatorname{Im} M^2 > 0$.
Such a pair of complex conjugate poles is a typical example of complex singularities, which is suggested to appear in the transverse part of the Landau-gauge gluon propagator in many works \cite{Gribov78, Zwanziger89, Stingl85, HKRSW90, Stingl96, Dudal:2008sp, BT2019, Falcao:2020vyr,Siringo16a, Siringo16b, HK2018, HK2020,Fischer-Huber}.
Note that complex conjugate pairing of $D_{cp}(k^2)$ is necessary for a real field due to the Schwarz reflection principle.
We accordingly decompose the Schwinger function as
\begin{align}
S(\xi) &= S_{tl}(\xi) + S_{cp}(\xi), \notag \\
S_{tl}(\xi) &= \int \frac{d^4 k}{(2 \pi)^4} e^{ik \xi} D_{tl}(k^2), \notag \\
S_{cp}(\xi) &= \int \frac{d^4 k}{(2 \pi)^4} e^{ik \xi} D_{cp}(k^2).
\end{align}
We proceed to reconstruct the corresponding Wightman function as an analytic continuation of the Schwinger function by identifying Wightman function at pure imaginary time with the Schwinger function,
\begin{align}
W(- i \xi_4, \vec{\xi}) := S(\vec{\xi} , \xi_4),
\end{align}
for $\xi_4 > 0$.
\textbf{(B)} As usual, for the timelike part, we can analytically continue $W_{tl} (- i \eta^0, \vec{\xi}) = S_{tl} (\vec{\xi} , \eta^0)$ to the tube $\xi - i \eta = (\xi^0 - i \eta^0, \vec{\xi} - i \vec{\eta}) \in \mathbb{R}^4 - iV_+$.
Moreover, the limit $\eta \rightarrow 0$ in $\eta \in V_+$, or ``the boundary value,'' can be taken as a tempered distribution [in $\mathscr{S}'(\mathbb{R}^4)$],
\begin{align}
W_{tl}(\xi^0,\vec{\xi}) &:= \lim_{\substack{\eta \rightarrow 0 \\ \eta \in V_+}} W_{tl} (\xi - i \eta) \notag \\
&= \int_0^\infty d\sigma^2 ~ \rho(\sigma^2) i \Delta^+ (\xi, \sigma^2), \label{eq:timelike-Wightman}
\end{align}
which is formally a sum of the free Wightman function $i \Delta^+ (\xi, \sigma^2)$ of mass $\sigma^2$ with the weight $\rho (\sigma^2)$, where
\begin{align}
i \Delta^+ (\xi, \sigma^2) &= (2 \pi) \int \frac{d^4 k}{(2 \pi)^4} e^{-i k \xi} \theta (k_0) \delta (k^2 - \sigma^2)
\end{align}
with the Loretzian vectors $\xi = (\xi^0,\vec{\xi}), ~k = (k^0,\vec{k})$.
On the other hand, the complex-pole part $S_{cp}(\xi)$ can be expressed as
\begin{align}
S_{cp}(\vec{\xi}, \xi_4) = \int \frac{d^3 \vec{k}}{(2 \pi)^3} e^{i\vec{k} \cdot \vec{\xi}} \left[ \frac{Z}{2 E_{\vec{k}} } e^{- E_{\vec{k}} |\xi_4|} + \frac{Z^*}{2 E_{\vec{k}}^*} e^{- E_{\vec{k}}^* |\xi_4|}
\right], \label{eq:simple_complex_poles_Schwinger}
\end{align}
where $E_{\vec{k}} = \sqrt{\vec{k}^2 +M^2}$ is a branch of $\operatorname{Re} E_{\vec{k}} > 0$ and $\operatorname{Im} E_{\vec{k}} > 0$ holds from the choice $\operatorname{Im} M^2 >0$.
Similarly to the timelike part, the complex-pole part of the Wightman function,
\begin{align}
W_{cp}&(\xi - i\eta) = \int \frac{d^3 \vec{k}}{(2 \pi)^3} e^{i\vec{k} \cdot (\vec{\xi} - i \vec{\eta})} \notag \\
&\times \left[ \frac{Z}{2 E_{\vec{k}} } e^{- i E_{\vec{k}} (\xi^0 - i \eta^0)} + \frac{Z^*}{2 E_{\vec{k}}^*} e^{- i E_{\vec{k}}^* (\xi^0 - i \eta^0)}
\right]. \label{eq:simple_complex_poles_hol_Wightman}
\end{align}
is holomorphic in the tube $\mathbb{R}^4 - iV_+$, since the integrand decreases rapidly in $|\vec{k}|$ for $\xi - i \eta \in \mathbb{R}^4 - iV_+$. Note that $\operatorname{Im} M^2$ does not affect the convergence because of $E_{\vec{k}} = |\vec{k}| + O(1/|\vec{k}|)$.
We can regard the Fourier transform in (\ref{eq:simple_complex_poles_hol_Wightman}) as a tempered distribution in $\vec{\xi}$ with a smooth parameter $\xi^0$, which is a distribution in $\mathscr{D}'(\mathbb{R}^4)$. Then, the limit $\eta \rightarrow 0$ with $\eta \in V_+$ can be taken to yield the reconstructed Wightman function:
\begin{align}
W_{cp}(\xi^0, \vec{\xi}) = \int \frac{d^3 \vec{k}}{(2 \pi)^3} e^{i\vec{k} \cdot \vec{\xi}} \left[ \frac{Z}{2 E_{\vec{k}} } e^{- i E_{\vec{k}} \xi^0} + \frac{Z^*}{2 E_{\vec{k}}^*} e^{- i E_{\vec{k}}^* \xi^0}
\right].
\label{eq:simple_complex_poles_Wightman}
\end{align}
Therefore, we obtain the reconstructed Wightman function $W(\xi) = W_{tl}(\xi) + W_{cp}(\xi) $ as a distribution in $\mathscr{D}'(\mathbb{R}^4)$.
\textbf{(C)}
Due to the exponential increases of the integrand as $\xi^0 \rightarrow \pm \infty$, $W_{cp}(\xi)$ is a nontempered distribution.
Thus, the reconstructed Wightman function is nontempered in the presence of complex poles $W(\xi) \notin \mathscr{S}'(\mathbb{R}^4)$.
Next, let us examine the positivity.
While the Wightman function is not a tempered distribution, we can still consider the positivity condition in $\mathscr{D}(\mathbb{R}^4)$, namely, for any test function with compact support $f \in \mathscr{D}(\mathbb{R}^4)$,
\begin{align}
\int d^4 x d^4y~ W (y-x) f^*(x) f(y) \geq 0 .
\label{eq:W-positivity}
\end{align}
This positivity condition ({\ref{eq:W-positivity}}) is violated due to the nontemperedness.
An intuitive derivation is as follows.
Intuitively, the positivity corresponds to that of the sector $\{ \phi(x) \ket{0} \}_{x \in \mathbb{R}^4}$. Suppose that this sector has a positive metric. From the translational invariance of the two-point function, the translation operator defined on the sector: $U(a) \phi(x) \ket{0} := \phi(x + a) \ket{0}$ is unitary. Since the modulus of a matrix element of a unitary operator is not more than one in a space with a positive metric, we have an $a$-independent ``upperbound,'' i.e., $|W(a)| = |\braket{0|\phi(0) U(-a) \phi(0)|0} | \leq \braket{0|\phi(0) \phi(0)|0}$.
This upperbound will imply that $W(a)$ is tempered, which contradicts the nontemperedness.
Of course, $W(0) = \braket{0|\phi(0) \phi(0)|0}$ is in general ill-defined since $W(\xi)$ is a distribution. A more delicate analysis is therefore required. We outline a rigorous proof in Appendix {\ref{app:appendix-pos}}.
\textbf{(A)}
The reflection positivity is always violated with complex poles, since steps (a) and (b) of the OS theorem \cite{OS73} imply that the reflection positivity essentially yields the temperedness of the Wightman function.
We give a sketch of these steps in Appendix {\ref{app:appendix-pos}}.
\textbf{(D)}
We consider the Lorentz symmetry and spacelike commutativity.
The invariance of the complex-pole part (\ref{eq:simple_complex_poles_hol_Wightman}) under Lorentz boosts can be explicitly checked by an integration path deformation.
Since the spatial rotational invariance is manifest, the Wightman function is Lorentz invariant. As another derivation, one can utilize the Euclidean invariance of the Schwinger function and the holomorphy in the tube. An argument similar to Bargmann-Hall-Wightman theorem (Theorem 2-11 and its Lemma of \cite{Streater:1989vi}) yields complex Lorentz invariance of $W(\xi - i \eta)$.
The spacelike commutativity is an immediate consequence of Lorentz invariance: $W(\xi) = W(-\xi)$ for spacelike $\xi$.
\section{General cases}
Let us mention a generalization of the above assertions to arbitrary complex singularities. With general complex singularities, the spectral representation is modified as, according to the Cauchy integral theorem,
\begin{align}
D(k^2) &= \int_0 ^\infty d \sigma^2 \frac{\rho(\sigma^2)}{\sigma^2 + k^2} + \sum_{M} \oint_{\Gamma_M} \frac{d \zeta}{2 \pi i} \frac{D(\zeta)}{\zeta - k^2}, \label{eq:spec_repr_complex}
\end{align}
where $M$ is a label of a complex singularity and $\Gamma_M$ is a contour surrounding the singularity clockwise.
The contribution of each complex singularity is formally a sum of complex poles with with weight $- D(\zeta)/ 2 \pi i$ over the contour \footnote{The assumptions (1) boundedness of complex singularities in $|k^2|$ and (2) holomorphy of $D(k^2)$ in a neighborhood of the real axis except for the timelike singularities are required to justify this statement.}.
This leads to a generalization of the proof of (B) and (D).
We prove the nontemperedness of (C) as follows:
Suppose the Wightman function were tempered, then the holomorphy in the tube would essentially imply the spectral condition for the Wightman function in momentum representation.
This leads to the K\"all\'en-Lehmann spectral representation, which contradicts complex singularities.
Other claims of (A) and (C) follow from the nontemperedness as above.
For details, see \cite{HK2021}.
\section{Realization in quantum theory}
\textbf{(E)}
We discuss a possible state-space structure.
Since abandoning the positivity of the full state space is common in Lorentz covariant gauge-fixed descriptions of gauge theories, we consider a quantum theory in a state space with an indefinite metric.
For a review on indefinite-metric QFTs, see e.g., \cite{Nakanishi72}.
Our aim here is to argue the correspondence between complex singularities and relevant complex-energy spectra. To this end, we shall demonstrate (E1) necessity and (E2) sufficiency of complex spectra for complex singularities when a convenient completeness relation is applicable.
\textbf{(E1)}
We begin with the necessity of complex spectra for existence of complex singularities.
Let us consider a $(0+1)$-dimensional QFT satisfying:
\begin{enumerate}
\item completeness of denumerable eigenstates $\ket{n}$ of the Hamiltonian $H$: $1 = \sum_{n,n'} \eta^{-1}_{n,n'} \ket{n} \bra{n'}$, where $\eta_{n,n'} = \braket{n|n'}$ is the nondegenerate metric \footnote{
Even in a finite dimensional vector space, the completeness of eigenstates of a hermitian operator does not always hold. Instead, by the Jordan decomposition, ``generalized eigenstates'' defined by a sequence $\{ \ket{E^{0}},~\ket{E^{1}},~\cdots, \ket{E^{n-1}} \}$: $ (H - E) \ket{E^{0}} = E \ket{E^{1}},~ (H - E) \ket{E^{1}} = E \ket{E^{2}},~\cdots~,~ (H - E) \ket{E^{n-1}} = 0$ spans the vector space.
Nevertheless, allowing generalized eigenstates does not change the conclusion.},
\item translational covariance: $\phi(t) = e^{iHt} \phi(0) e^{-iHt}$,
\item reality of eigenvalues $E_n$ of the Hamiltonian $H$.
\end{enumerate}
Then, with an assumption that the completeness relation converges well, the Wightman function is tempered as follows:
\begin{align}
& \braket{0|\phi(t) \phi(0)|0} = \int d \omega ~ \rho (\omega) e^{-i\omega t}, \notag \\
& \rho (\omega) = \sum_{n,n'} \eta^{-1}_{n,n'} \delta (\omega - E_n) \bra{0} \phi(0) \ket{n} \bra{n'} \phi(0) \ket{0}.
\end{align}
This observation demonstrates that complex singularities do not appear without complex spectra, i.e., (E1) complex singularities require complex spectra of $H$.
\textbf{(E2)}
On the other hand, eigenstates of complex eigenvalues of a Hermitian operator $H$ appear as pairs of zero-norm states. For example, a pair of zero-norm states $\{ \ket{\alpha}, \ket{\beta} \}$ can satisfy
\begin{align}
\begin{cases}
H \ket{\alpha} = E_\alpha \ket{\alpha}, ~~~ H \ket{\beta} = E^*_\alpha \ket{\beta} \\
\braket{\alpha|\alpha} = \braket{\beta|\beta} = 0,~~ \braket{\alpha|\beta} \neq 0.
\end{cases}
\end{align}
This pair of states $\{ \ket{\alpha}, \ket{\beta} \}$ can yield a pair of complex conjugate poles. Indeed, we find
\begin{align}
W_{complex}(t) &:= \sum_{n, n' \in \{ \alpha, \beta \} } \eta^{-1}_{n,n'} e^{- i E_n t} \bra{0} \phi(0) \ket{n} \bra{n'} \phi(0) \ket{0} \notag \\
&= (\braket{\beta|\alpha})^{-1} \braket{0|\phi(0)|\alpha} \braket{\beta|\phi(0)|0} e^{- i E_\alpha t} \notag \\
& + (\braket{\alpha|\beta})^{-1} \braket{0|\phi(0)|\beta} \braket{\alpha|\phi(0)|0} e^{- i E_\alpha^* t},
\end{align}
which yields,
\begin{align}
S_{complex}(\tau) &= W_{complex} (-i |\tau|) \notag \\
&= \int \frac{dk}{2 \pi} e^{ik \tau} \left[ \frac{Z}{k^2 + E_\alpha^2} + \frac{Z^*}{k^2 + (E_\alpha^*)^2} \right],
\end{align}
with $Z := \frac{2 E_\alpha \braket{0|\phi(0)|\alpha} \braket{\beta|\phi(0)|0}}{\braket{\beta|\alpha}}$ for $\operatorname{Re} E_\alpha > 0$.
This indicates that (E2) a pair of zero-norm states $\{ \ket{\alpha}, \ket{\beta} \}$ with complex conjugate energies $\{ E_\alpha, E_\alpha^* \}$ satisfying $\operatorname{Re} E_\alpha > 0$ yields a pair of complex conjugate poles if $\{ \ket{\alpha}, \ket{\beta} \}$ are not orthogonal to $\phi(0) \ket{0}$.
These claims [(E1) and (E2)] establish the correspondence between complex singularities and pairs of zero-norm states of complex eigenvalues of the Hamiltonian.
\section{Example}
The above argument is in $(0+1)$-dimensional QFT.
We show an example of $(3+1)$-dimensional QFT yielding complex poles based on a covariant operator formulation \cite{Nakanishi72b} of the Lee-Wick model \cite{LW69}.
The Lagrangian density of a complex scalar field $\phi$ with complex (squared) mass $M^2 \in \mathbb{C}$ is given by
\begin{align}
\mathscr{L} :=& \frac{1}{2} \bigl[ (\partial_\mu \phi)(\partial^\mu \phi) + (\partial_\mu \phi)^\dagger (\partial^\mu \phi)^\dagger \notag \\
&~~~ - M^2 \phi^2 - (M^*)^2 (\phi^\dagger)^2 \bigr].
\end{align}
We expand the field operator $\phi$ as
\begin{align}
\phi(x) &= \phi^{(+)} (x) + \phi^{(-)} (x), \notag \\
\phi^{(+)} (x) &= \int \frac{d^3 \vec{p}}{(2 \pi)^3} \frac{1}{\sqrt{2 E_{\vec{p}}}} \alpha(\vec{p}) e^{i \vec{p}\cdot \vec{x} - i E_{\vec{p}}t}, \notag \\
\phi^{(-)} (x) &= \int \frac{d^3 \vec{p}}{(2 \pi)^3} \frac{1}{\sqrt{2 E_{\vec{p}}}} \beta^\dagger(\vec{p}) e^{-i \vec{p}\cdot \vec{x} + i E_{\vec{p}}t},
\end{align}
where $E_{\vec{p}} := \sqrt{M^2 + \vec{p}^2}$, $\operatorname{Re} E_{\vec{p}} \geq 0$, and $\operatorname{Re} \sqrt{E_{\vec{p}}} \geq 0$.
The canonical commutation relation implies $[\alpha(\vec{p}), \beta^\dagger(\vec{q})] =[\beta(\vec{p}), \alpha^\dagger(\vec{q})] = (2 \pi)^3 \delta(\vec{p} - \vec{q})$.
We define the vacuum $\ket{0}$ by $\phi^{(+)} (x) \ket{0} = [\phi^{(-)} (x) ]^\dagger \ket{0} = 0$, which is a Lorentz invariant state, see \cite{Nakanishi72b} for details.
Note that one can explicitly check the spacelike commutativity at least at the level of elementary fields.
The Hamiltonian reads,
\begin{align}
H = \int \frac{d^3 \vec{p}}{(2 \pi)^3} \left[ E_{\vec{p}} \beta^\dagger(\vec{p}) \alpha(\vec{p}) + E_{\vec{p}}^* \alpha^\dagger(\vec{p}) \beta(\vec{p}) \right],
\end{align}
up to some constant. The complex-energy states $\ket{\vec{p},\alpha} := \alpha^\dagger(\vec{p}) \ket{0}$ and $\ket{\vec{p},\beta} := \beta^\dagger(\vec{p}) \ket{0}$ form a pair of zero-norm states:
$\braket{\vec{p},\alpha|\vec{q},\alpha} = \braket{\vec{p},\beta|\vec{q},\beta} = 0, ~~ \braket{\vec{p},\alpha|\vec{q},\beta} = \braket{\vec{p},\beta|\vec{q},\alpha} = (2 \pi)^3 \delta(\vec{p} - \vec{q}).$
We find that the Euclidean propagator of a Hermitian combination with a constant $Z \in \mathbb{C}$, $\Phi := \sqrt{Z} \phi + \sqrt{Z^*} \phi^\dagger$,
has complex poles. Indeed, the Wightman function of the Lee-Wick model,
\begin{align}
W_\Phi (t,\vec{x}) &:= \braket{0|\Phi(x)\Phi(0)|0} \notag \\
&= \int \frac{d^3 \vec{p}}{(2 \pi)^3} \left[ \frac{Z}{ 2 E_{\vec{p}}} e^{i \vec{p}\cdot \vec{x} - i E_{\vec{p}}x^0} + \frac{Z^*}{ 2 E_{\vec{p}}^*} e^{i \vec{p}\cdot \vec{x} - i E_{\vec{p}}^* t} \right],
\end{align}
coincides with the Wightman function (\ref{eq:simple_complex_poles_Wightman}) reconstructed from a pair of simple complex conjugate poles.
\section{Concluding Remarks}
Some remarks on the nontemperedness, locality, and Wick rotation are in order.
\subsection{Nontemperedness}
The exponential growth of the Wightman function $W(\xi)$ largely affects asymptotic states, which correspond to ``$\xi^0 \rightarrow \pm \infty$ limit''. This indicates that asymptotic states of the field are ill-defined without some artificial manipulations.
Such states in the full state space are far from being identified with asymptotic particle states and should be excluded from the physical state space before taking the asymptotic limit through, e.g. the Kugo-Ojima quartet mechanism \cite{Kugo:1979gm}.
Thus, \textit{the complex singularities can be considered as a signal of confinement}.
Incidentally, note that the appearance of complex singularities in a propagator of the gluon-ghost composite operator is a necessary condition for eliminating complex-energy states in ``the one-gluon state'' from the physical state space in the Becchi-Rouet-Stora-Tyutin (BRST) formalism.
Seeking such complex gluon-ghost bound states would be interesting for future prospects.
Remarkably, the Bethe-Salpeter equation for the gluon-ghost bound state has been discussed in light of BRST quartets in {\cite{Alkofer:2011pe}}.
\subsection{Locality}
Some argue that complex singularities are associated with nonlocality. For example, it is claimed in \cite{Stingl85,Stingl96,HKRSW90} that complex poles describe short-lived excitations and that the locality is broken in short range at the level of propagators but that the corresponding $S$ matrix remains causal. However, this interpretation is different from ours.
To our knowledge, the only axiomatic way to impose locality is the spacelike commutativity.
Consequently, \textit{complex singularities themselves not necessarily lead to non-locality}, as shown in this paper.
\subsection{Wick rotation}
Our reconstruction procedure is different from the naive inverse Wick rotation in the momentum space $k_E^2 \rightarrow - k^2$. Indeed, due to the complex singularities, the time-ordered propagator cannot be Fourier transformed. On the other hand, the inverse Wick rotation makes the time-ordered propagator tempered and ruins the interpretation of Euclidean field theory as an imaginary-time formalism.
Note also that the inverse Wick rotation invalidate the Hermiticity of the Hamiltonian even in an indefinite space unlike ours.
In summary, complex singularities are beyond the standard formulation yet consistent with locality. Moreover, they can appear in indefinite-metric QFTs, e.g., gauge theories in Lorentz covariant gauges.
\section*{Acknowledgements}
We thank Taichiro Kugo, Peter Lowdon, and Lorenz von Smekal for helpful and critical comments in the early stage of this work.
Y.~H. is supported by JSPS Research Fellowship for Young Scientists Grant No.~20J20215, and K.-I.~K. is supported by Grant-in-Aid for Scientific Research, JSPS KAKENHI Grant (C) No.~19K03840.
|
3,212,635,537,468 | arxiv | \section{Introduction}
Let $\mathcal{H}$ be a Hilbert space, $\mathcal{C}$ a class of closed subspaces of $\mathcal{H}$ and $\mathcal{F}= \{f_1, \dots, f_m\}$ a finite set of elements in $\mathcal{H}$.
In this article we study the existence and show how to construct an optimal subspace $\mathcal{S}$ in the class $\mathcal{C}$ that minimizes the distance to the given data $\mathcal{F}$, in the sense that $\mathcal{S}$
minimizes the functional ${\pi}(\mathcal{F},\mathcal{S})$ over $\mathcal{C}.$ The functional is defined as
\begin{equation}\label{error}
{\pi}(\mathcal{F},\mathcal{S})= \sum_{j=1}^{m} \|f_j-P_{\mathcal{S}} f_j\|^2,
\end{equation}
where $P_{\mathcal{S}}$ denotes the orthogonal projection on the subspace $\mathcal{S}.$
The motivation to find an optimal subspace in $\mathcal{C}$ is, that in many situations one wants to choose a model for a certain
class of data. Instead of imposing some conditions on the data to fit some known model, the idea is to
define a large class of subspaces convenient for the application at hand, and find from there the one that
``best fits" the data under study.
The signals that need to be modelled are ideally low dimensional but living in a high dimensional space. However, since in applications they are often corrupted by noise, they become high dimensional, however they are close to a low dimensional subspace, which is the space one seeks.
When the Hilbert space is $L^2(\mathbb{R}^d)$ it is natural to consider as a model for our data the class of
shift invariant spaces (SIS), that is, closed subspaces of $L^2(\mathbb{R}^d)$ that are invariant under translations by integers.
These spaces have been used in approximation theory,
harmonic analysis, wavelet theory, sampling theory and signal processing (see, e.g., \cite{AG01, Gro01, HW96, Mal89} and references therein).
Often, in applications, it is assumed that the signals under study belong to some shift invariant space $V$ generated by the translations of a finite set of functions $\Phi=\{\varphi_1,\cdots,\varphi_m\},$
i.e., $V = S(\Phi)= \overline{\mbox{span}}\{T_k\varphi_i\colon k\in\mathbb{Z}^d, i=1,\cdots, m\}.$
The choice of the particular finitely generated shift invariant space typically is not deduced from a set of signals.
For example in sampling theory, a classical assumption is that the signals to be sampled are band-limited, that is, they belong to
the shift invariant space $V$ generated by $\varphi(x)= \mbox{sinc}(x).$ However, the band-limited assumption is not very realistic in many applications.
Thus, it is natural to search for a
finitely generated shift invariant space that is nearest to a set of some observed data.
In this paper we study the case when $\mathcal{H} = L^2(\mathbb{R}^d).$
For this case, we restrict the class of approximating subspaces to be shift invariant spaces that have extra-invariance, that is:
If $M$ is a subgroup of $\mathbb{R}^d$ such that $\mathbb{Z}^d \subset M$ we will say that
$S(\varphi_1,\cdots,\varphi_m)$ is $M$ extra-invariant if
$$
\overline{\mbox{span}}\{T_k\varphi_j\colon j=1,\dots,m,\; k\in\mathbb{Z}^d\}=\overline{\mbox{span}}\{T_\alpha\varphi_j\colon j=1,\dots,m,\; \alpha\in M\}.
$$
Therefore, the space $S(\varphi_1,\cdots,\varphi_m)$ is invariant under translates other than the integers,
even though it is generated by the integer translates of a finite set of functions.
Such spaces with extra-invariance
are important in applications specially in those where the jitter error is an issue.
We first consider the case when the subgroup $M$ is a proper subgroup of $\mathbb{R}^d$ that contains $\mathbb{Z}^d$. For that case we obtain one of the main contributions of this paper. We prove that for any finite set of data $\mathcal{F}= \{f_1, \dots, f_m\}\subset L^2(\mathbb{R}^d),$ for any proper subgroup $M$ containing $\mathbb{Z}^d$ and for any $\ell \in \mathbb{N}$ there always exists a
SIS $V$ of length at most $\ell$ with extra invariance $M$ whose distance (in the sense of \eqref{error})
to the data $\mathcal{F}$ is the smallest possible among all the SIS of length smaller or equal than $\ell$ that are
$M$ extra-invariant.
(Here, the length of a SIS is the cardinal of the smallest set of generators).
We construct a solution $V$ and provide a set of generators whose integer translates form a tight frame of $V.$
An expression for the exact value of the error $\mathcal{E}(\mathcal{F},V)$ between the data and the
optimal subspace is also obtained using the eigenvalues of some special matrix.
Next, we consider the approximation problem for the class of generalized Paley-Wiener spaces.
Given a measurable set $\Omega \subset \mathbb{R}^d$ (not necessarily bounded), the generalized Paley-Wiener space $PW_{\Omega}$ associate to $\Omega$ is the subspace
of $L^2(\mathbb{R}^d)$ corresponding to the functions whose Fourier transform vanished outside $\Omega.$
A generalized Paley-Wiener space is always invariant under translations by the whole group $\mathbb{R}^d.$ In particular is a SIS of $L^2(\mathbb{R}^d)$ that not necessarily need to be finitely generated.
Under the hypothesis that $PW_{\Omega}$ has a Riesz basis of integer translates, we proved that $PW_{\Omega}$ is finitely generated if and only if $\Omega$ is a multi-tile. (see Proposition \ref{propPW}).
That is $\Omega$ is a multi-tile if and only if $PW_{\Omega}$ has extra invariance $M=\mathbb{R}^d$.
We study our approximation problem for those generalized Paley-Wiener spaces.
We describe for this case how to construct a set of generators and show an interesting connection with recent results about bases of exponentials. This complete all the cases of extra invariance when $M$ is not a proper subgroup.
Finally we consider a similar problem when the Hilbert space is $\ell^2(\mathbb{Z}^d)$ and $\mathcal{C}$ is a conveniently chosen class of subspaces. We obtain for this case equivalent results
to the ones for $L^2(\mathbb{R}^d).$
The approximation problem for the discrete case is related with the continuous case in a very interesting way
that is described in Section \ref{section0}.
\subsection{Previous Work}
Let us now mention some previous related work.
The problem of approximation of a set of data by shift invariant spaces (without the extra invariance restriction) started in \cite{ACHM07} where the
authors proved the existence of a minimizer for \eqref{error} over the class of low dimensional subspaces in a Hilbert space
$\mathcal{H}$ and also over the class of shift invariant spaces in $L^2(\mathbb{R}^d)$.
In \cite{ACM08} the case of multiple subspaces was considered
in the finitely dimensional case. That is, the authors found a union of low dimensional subspaces that best fits a given set of data in $\mathbb{R}^d$ and provided an algorithm to find it. In 2011, using dimensional reduction techniques, this algorithm was improved (see \cite{AACM11}).
Further, in \cite{AT11} the authors found necessary and sufficient conditions for the existence of optimal subspaces in the general context of Hilbert spaces. However they did not provide a way to construct them.
The first result for approximation of a finite set of data using shift invariant spaces with extra-invariance constrains appears in \cite{AKTW12},
where the authors consider principal shift invariant spaces in one variable and they assume that the space has a generator with orthogonal integer translates, which is a key element in their proof.
So the techniques of this particular case do not apply to our general case.
\subsection{Organization of the paper}
The paper is organized as follows. In Section \ref{preliminaries} we set the definitions and results that we need about shift invariant spaces, extra-invariance and
the approximation problem for the case
of shift invariant spaces in $L^2(\mathbb{R}^d).$
The main results of the paper are stated and proved in Sections \ref{section1}, \ref{section2} and
\ref{section0}. In Section \ref{section1} we present the $M$ extra-invariant case for shift invariant spaces, in Section \ref{section2} the case of Paley-Wiener spaces and finally we consider a discrete case, in Section \ref{section0}.
\section{preliminaries}\label{preliminaries}
We begin with a review of the basic results and definitions that will be needed in subsequent sections. The known results are generally stated without
proofs, but we provide references where the proofs can be found. Also, we introduce some of our notational conventions. For the definitions of Riesz
bases and frames in Hilbert spaces we refer the reader to \cite{Chr03, Hei11} and the references therein.
\subsection{Shift Invariant Spaces}\label{sis}
The structure of these spaces has been deeply analyzed (see for example \cite{Bow00, dBDVR94, dBDVRI94, Hel64, RS95}).
\begin{definition}
A closed subspace $V\subset L^2(\mathbb{R}^d)$ is said to be a {\it shift invariant space} if
\[
f\in V\Longrightarrow T_kf\in V, \,\, \textrm{ for any }\,\,k\in\mathbb{Z}^d,
\]
where $T_k$ is the translation by the vector $k\in\mathbb{Z}^d,$ i.e. $T_kf(x)=f(x-k)$.
For any subset $\Phi\subset L^2(\mathbb{R}^d)$ we define \[
S(\Phi)= \overline{\mbox{span}}\{T_k\varphi\colon \varphi\in\Phi, k\in\mathbb{Z}^d\}\,\,\textrm{ and }\,\,
E(\Phi)= \{T_k\varphi\colon \varphi\in\Phi, k\in\mathbb{Z}^d\}.
\]
We call $S(\Phi)$ the shift invariant space (SIS) generated by $\Phi$.
If $V=S(\Phi)$ for some finite set $\Phi$ we say that $V$ is a {\it finitely generated} SIS, and a
{\it principal} SIS if $V$ can be generated by the integer translates of a single function.
\end{definition}
For a finitely generated SIS $V\subseteq L^2(\mathbb{R}^d)$ we define the length of $V$ as
\[
\ell(V)= \min\{n\in\mathbb{N}\colon \exists \,\,\varphi_1, \cdots,\varphi_n \in
V \textrm{ with } V=S(\varphi_1, \cdots,\varphi_n)\}.
\]
In addition to the construction of a set of generators of the optimal space for the problems considered in this paper, it will be important to estimate the error of
these approximations. In order to compute these errors we need to consider what is called the Gramian $G_{\Phi}$ for a family of functions $\Phi\subset L^2(\mathbb{R}^d).$
More precisely, given $\Phi= \{\varphi_1,\cdots,\varphi_m\}$ a finite collection of functions in $L^2(\mathbb{R}^d),$ the \emph{Gramian} $G_\Phi$ of $\Phi$
is the $m\times m$ matrix of $\mathbb{Z}^d$-periodic functions
\begin{equation} \label{gram}
[G_{\Phi}(\omega)]_{ij}
= \sum_{k \in \mathbb{Z}^d} \widehat{\varphi}_i(\omega+k) \, \overline{\widehat{\varphi}_j(\omega+k)}.
\end{equation}
The Gramian of $\Phi$ is determined a.e. by its values at any measurable set of representatives $\mathcal{U}$ of the quotient $\mathbb{R}^d/\mathbb{Z}^d$ and satisfies $G_{\Phi}(\omega)^*= G_{\Phi}(\omega)$ for a.e. $\omega\in \mathcal{U}$. We will take $\mathcal{U}=[-1/2,1/2)^d.$
For a finitely generated SIS $V$, we can express the length of $V$ in terms of the Gramian as follows (see \cite{Bow00, dBDVR94, TW12})
\begin{equation}\label{length-gramian}
\ell(V)=\mathop{essup }_{\omega \in \mathcal{U}}\big{[}\text{rk}(G_{\Phi}(\omega))\big{]}
\end{equation}
where $\text{rk}(B)$ denotes the rank of a matrix $B$ and $\Phi$ is a generator set for $V$.
One important property of the Gramian is given by the following lemma concerning the measurability of the eigenvalues and the existence of measurable eigenvectors
of a non-negative matrix with measurable entries.
\begin{lemma}[Lemma 2.3.5 of \cite{RS95}]\label{medibilidad}
Let $G(\omega)$ be an $m\times m$ self-adjoint matrix of measurable functions defined on a measurable subset $E\subset \mathbb{R}^d$ with eigenvalues $\lambda_1(\omega)\ge\dots\ge \lambda_m(\omega).$ Then the eigenvalues $\lambda_i,$ $i=1,\dots, m,$ are measurable functions on $E$ and there exists an $m\times m$ matrix of measurable functions $U(\omega)$ on $E$ such that $U(\omega)U^*(\omega)=I$ a.e. $\omega\in E$ and such that
\[
G(\omega)=U(\omega)\Lambda(\omega)U^*(\omega), \quad \text{ a.e. }\omega\in E,
\]
where $\Lambda(\omega):=\text{diag}(\lambda_1(\omega),\dots, \lambda_m(\omega)).$
\end{lemma}
In \cite{Hel64}, Helson introduced range functions and used this notion to completely characterize shift invariant spaces.
Later on, several authors have used this framework to describe and characterize frames and bases of these spaces.
See for example \cite{Bow00, CP10, dBDVR94, dBDVRI94, RS95}.
We will mention the required definitions and some known results that we need later for giving the proofs of our results.
We refer to \cite{Bow00,dBDVR94, dBDVRI94, RS95} for a complete description and the proofs.
\begin{definition}\label{U}
Let $f\in L^2(\mathbb{R}^d)$ and fix $\mathcal{U} \subset \mathbb{R}^d$ to be a measurable set of representatives of the quotient $\mathbb{R}^d/\mathbb{Z}^d.$
For $\omega\in \mathcal{U},$ the {\it fiber} $\tau f(\omega)$ of $f$ at $\omega$ is the sequence
\[
\tau f(\omega)= \{\widehat{f}(\omega+k)\}_{k\in\mathbb{Z}^d}.
\]
\end{definition}
Here $\widehat{f}$ denotes the Fourier transform of the function $f,$ that is $\widehat{f}(\omega)\!= \int_{\mathbb{R}^d}e^{-2\pi i\omega x}f(x)\, dx$ when $f\in L^1(\mathbb{R}^d).$
We observe that if $f\in L^2(\mathbb{R}^d),$ then the fiber $\tau f(\omega)$ belongs to $\ell^2(\mathbb{Z}^d)$ for almost every $\omega\in \mathcal{U}.$
If $V$ is a finitely generated SIS and $\omega\in \mathcal{U}$ we define the {\it fiber space} associated to $V$ and $\omega$ as follows
\[
J_V(\omega)=\overline{\{\tau f(\omega)\colon f\in V\}},
\]
where the closure is taken in the norm of $\ell^2(\mathbb{Z}^d).$
With the above definitions we have:
\begin{lemma}[Proposition 5.6 of \cite{ACP11}]\label{dimension}
Let $V=S(\Phi)$ be a finitely generated SIS. Then
\[
\dim(J_V(\omega))= \text{rk }(G_{\Phi}(\omega)), \, \text{ a.e. }\omega\in \mathcal{U}.
\]
\end{lemma}
\begin{lemma}\label{Lema 3.1}
If $f\in L^2(\mathbb{R}^d),$ then
\begin{enumerate}
\item[$(i)$] the sequence $\tau f(\omega)= \{\widehat{f}(\omega+k)\}_{k\in\mathbb{Z}^d}$ is a well-defined sequence in $\ell^2(\mathbb{Z}^d)$ a.e. $\omega\in\mathcal{U}.$
\item[$(ii)$] $\|\tau f(\omega)\|_{\ell^2}$ is a measurable function of $\omega$ and
\[
\|f\|^2= \|\widehat{f}\|^2= \int_{\mathcal{U}} \|\tau f(\omega)\|_{\ell^2}^2 \, d\omega.
\]
\end{enumerate}
\end{lemma}
\begin{lemma}\label{Lema 3.2}
Let $V$ be a finitely generated SIS in $L^2(\mathbb{R}^d).$ Then we have
\begin{enumerate}
\item[$(i)$] $J_V(\omega)$ is a closed subspace of $\ell^2(\mathbb{Z}^d)$ for a.e. $\omega\in \mathcal{U}.$
\item[$(ii)$] $V=\{f\in L^2(\mathbb{R}^d)\colon \tau f(\omega)\in J_V(\omega) \mbox{ for a.e. } \omega\in \mathcal{U}\}.$
\item[$(iii)$] For each $f\in L^2(\mathbb{R}^d)$ we have that $\|\tau(P_V f)(\omega)\|_{\ell^2}$ is a measurable function of the variable $\omega$ and
\[
\tau(P_V f)(\omega)= P_{J_V (\omega)}(\tau f(\omega)).
\]
\item[$(iv)$] Let $\varphi_1, \dots, \varphi_m\in L^2(\mathbb{R}^d).$ We have that
\begin{enumerate}
\item[$(a)$] $\{\varphi_1, \dots, \varphi_m\}$ is a set of generators of $V,$ if and only if $\{\tau\varphi_1(\omega), \dots,\tau\varphi_m(\omega)\}$
spans $J_{V}(\omega)$ for a.e. $\omega\in \mathcal{U}.$
\item[$(b)$] The integer translations of $\varphi_1, \dots, \varphi_m$ are a frame (resp. Riesz basis) of $V,$ if and only if
$\tau\varphi_1(\omega), \dots,\tau\varphi_m(\omega)$ are a frame (resp. Riesz basis) of $J_V(\omega)$ with the same frame (resp. Riesz) bounds, for a.e. $\omega\in \mathcal{U}.$
\end{enumerate}
\end{enumerate}
\end{lemma}
\subsection{Optimality for the class of SIS in $L^2(\mathbb{R}^d)$}
\medskip
In \cite{ACHM07} the authors give a solution for the case where the approximation class is the class of SIS in $L^2(\mathbb{R}^d).$ For this, they reduce the optimization
problem into an uncountable set of finite dimensional problems in the Hilbert space $\mathcal{H}=\ell^2(\mathbb{Z}^d).$
\begin{theorem}[Theorem 2.3 of \cite{ACHM07}]\label{Teorema-original}
Let $\mathcal{F}=\{f_1, \cdots, f_m\}$ be a set of functions in $L^2(\mathbb{R}^d).$ Let $\lambda_1(\omega)\ge\dots\ge\lambda_m(\omega)$ be
the eigenvalues of the Gramian $G_{\mathcal{F}}(\omega).$ Then, there exists $V^*\in\mathcal{V}^{\ell}=\{V\colon V \mbox{ is a SIS of length at most } \ell\}$
such that
\[
\sum_{i=1}^m\|f_i- P_{V^*} f_i\|^2\le \sum_{i=1}^m\|f_i- P_{V} f_i\|^2, \quad \forall \, V\in\mathcal{V}^{\ell}.
\]
Moreover, we have that
\begin{enumerate}
\item[(1)] The eigenvalues $\lambda_i(\omega),$ $1\le i\le m$ are $\mathbb{Z}^d-$periodic, measurable functions in $L^2(\mathcal{U})$ and the
approximation error is given by,
\[
\mathcal{E}(\mathcal{F},\ell)= \sum_{i=\ell+1}^m \int_{\mathcal{U}} \lambda_i(\omega)\, d\omega.
\]\item[(2)] Let $\theta_{i} (\omega)= \lambda_i^{-1}(\omega)$ if $ \lambda_i(\omega)$ is different from zero, and zero otherwise. Then, there
exists a choice of measurable left eigenvectors $Y^1(\omega), \cdots, Y^{\ell}(\omega)$ with $Y^i= (y^i_1,\cdots, y^i_m)^t,$ $i=1, \cdots, \ell,$
associated with the first $\ell$ largest eigenvalues of $G_{\mathcal{F}}(\omega)$ such that the functions defined by
\[
\widehat{\varphi}_i(\omega)= \theta_i (\omega)\sum_{j=1}^m y^i_j(\omega) \widehat{f_j}(\omega), \quad i=1, \cdots, \ell, \, \omega\in\mathbb{R}^d
\]
are in $L^2(\mathbb{R}^d).$ Furthermore, the corresponding set of functions $\Phi=\{\varphi_1, \cdots, \varphi_{\ell}\}$ is a generator
set for the optimal subspace $V^*$ and the set $\{\varphi_i(\cdot -k), k\in \mathbb{Z}^d, i=1, \cdots, \ell\}$ is a Parseval frame for $V^*.$
\end{enumerate}
\end{theorem}
\subsection{Extra invariance}\label{EI}
We will need some definitions and known results concerning extra-invariance for shift invariant spaces. These are described in this subsection.
\begin{definition}
Let $V\subset L^2(\mathbb{R}^d)$ be a SIS. We define the {\it invariance set} as follows
\[
M:=\{x\in\mathbb{R}^d\colon T_x f\in V, \,\forall f\in V\}.
\]
\end{definition}
In \cite{ACHKM10} (see also \cite{ACP11}), the authors proved that the invariance set of a shift invariance space $V\subset L^2(\mathbb{R}^d)$ is a closed additive subgroup of $\mathbb{R}^d$ that contains $\mathbb{Z}^d.$ For instance, in the case of the line the invariant set of a
shift invariant
space could be $\mathbb{Z}, \frac{1}{n}\mathbb{Z} $ for some $n\in\mathbb{N}$ or $\mathbb{R}.$
\begin{definition} Let $\Phi \subset L^2(\mathbb{R}^d)$. We will say that $V=S(\Phi)$ is {\it $M$ extra-invariant} if $T_m f\in V$ for all $m\in M$ and for all $f\in V.$
If $M=\mathbb{R}^d$ then the space $V$ is translation invariant but generated by the integer translates of the set $\Phi$.
\end{definition}
One example of a translation invariant space in $\mathbb{R}$ is the Paley-Wiener space of functions that are bandlimited to $[-1/2, 1/2]$ defined by
\[
PW= \{f\in L^2(\mathbb{R})\colon \mbox{ supp }(\widehat{f})\subseteq [-1/2,1/2]\}.
\]
It is easy to prove that for a measurable set $\Omega\subset\mathbb{R}^d,$ the space
\begin{equation}\label{Wiener}
V_{\Omega}:=\{f\in L^2(\mathbb{R}^d)\colon \mbox{supp}(\widehat{f})\subset \Omega\}
\end{equation}
is translation invariant. Moreover, Wiener's theorem (see \cite{Hel64}) proves that any closed translation invariant subspace of $L^2(\mathbb{R}^d)$ is of the
form \eqref{Wiener}.
If $V$ is a shift invariant space of length $\ell$ and $M$ is an additive subgroup of $\mathbb{R}^d$ containing $\mathbb{Z}^d$, we will say that $V$ has {\it extra-invariance} $M$ if $V$ is $M-$invariant.
Note that in this case, if $\Phi$ is a set of generators of $V$, i.e. $V=S(\Phi)$, then
$$
S(\Phi)= \overline{\mbox{span}}\{T_k\phi\colon \phi\in\Phi, k\in\mathbb{Z}^d\}=
\overline{\mbox{span}}\{T_{\alpha}\phi\colon \phi\in\Phi, \alpha\in M\}.
$$
In \cite{ACHKM10} the authors characterize those shift invariant spaces $V\subset L^2(\mathbb{R})$ that have extra-invariance. They show that either $V$ is translation invariant, or there exists a maximum positive integer $n$
such that $V$ is $\frac{1}{n}\mathbb{Z}-$invariant.
The d-dimensional case is consider in \cite{ACP11}. There, a characterization of the extra invariance of $V$ when $M$ is not all $\mathbb{R}^d$ is obtained.
Given $M$ a closed subgroup of $\mathbb{R}^d$ containing $\mathbb{Z}^d$ and
$M^*=\{x\in\mathbb{R}^d\colon \langle x,m\rangle\in\mathbb{Z}\quad\forall m\in M\},$ the authors construct a special partition
$\{B_{\sigma}\}_{\sigma\in\mathcal{N}}$ of $\mathbb{R}^d,$
where each $B_{\sigma}$ is an $M^*-$periodic set and the index set $\mathcal{N}$ is a section of the quotient $\mathbb{Z}^d/M^*.$
More precisely, for each $\sigma\in\mathcal{N},$
\begin{equation}\label{def-Bsigma}
B_{\sigma}=\Omega+\sigma+M^*=\bigcup_{m^*\in M^*} (\Omega+\sigma)+m^*,
\end{equation}
where $\Omega$ is a section of the quotient $\mathbb{R}^d/\mathbb{Z}^d.$ We refer to \cite{ACP11} for more details.
Using this partition,
for each $\sigma\in\mathcal{N},$ they define the subspaces associated to a given SIS $V$
\begin{equation}\label{Usigma}
V_{\sigma}= \{f\in L^2(\mathbb{R}^d)\colon \widehat{f}= \chi_{B_{\sigma}}\widehat{g}, \text{ with } g\in V\}.
\end{equation}
Given $f\in L^2(\mathbb{R}^d)$ define for $\sigma \in \mathcal{N}$, the function $f^{\sigma}$ by $\widehat{f^{\sigma}}=\widehat{f}\chi_{B_{\sigma}}$.
The authors give a characterization of the $M-$invariance of $V$ in terms of the subspaces
$V_{\sigma}.$ More specifically they prove that
\begin{theorem}\label{extra-inv}
If $V\subset L^2(\mathbb{R}^d)$ is a SIS and $M$ is a closed subgroup of $\mathbb{R}^d$ containing $\mathbb{Z}^d,$ then the following are equivalent.
\begin{enumerate}
\item[(i)] $V$ is $M-$invariant,
\item[(ii)] $V_{\sigma}\subset V$ for all $\sigma\in\mathcal{N},$
\item[(iii)] $J_{V_{\sigma}}(\omega)\subset J_{V}(\omega)$ for almost every $\omega$ and each $\sigma\in\mathcal{N},$
\item[(iv)] if $V=S(\Phi)$ then $\tau\varphi^{\sigma}(\omega) \in J_V(\omega)$ a.e. $\omega\in\mathcal{U}$
for all $\varphi\in\Phi$ and all $\sigma\in\mathcal{N}$.
\end{enumerate}
\end{theorem}
\section{Optimality for the class of SIS with extra-invariance}\label{section1}
Here we consider the approximation problem for the class of finitely generated SIS with extra invariance under a given {\it proper} subgroup $M$ of $\mathbb{R}^d.$
Let us start introducing some notation. Let $m, \ell\in\mathbb{N},$ $M$ be a closed proper subgroup of $\mathbb{R}^d$ containing $\mathbb{Z}^d$ and $\mathcal{F}=\{f_1,\dots, f_m\}\subset L^2(\mathbb{R}^d).$
Define
\begin{equation}\label{classV}
\mathcal{V}_M^{\ell}=\{ V: V \;{\text {is a SIS of length at most}}\;\ell \;{\text {and}} \; V {\text{ is }} M{\text {-invariant}}\}.
\end{equation}
Let $\mathcal{N}=\{\sigma_1,\dots,\sigma_{\kappa}\}$ be a section of the quotient $\mathbb{Z}^d/M^*$ and $\{B_{\sigma}\::\sigma\in \mathcal{N}\}$ the partition defined in \eqref{def-Bsigma}.
For each ${\sigma}\in\mathcal{N},$
we consider $\mathcal{F}^{\sigma}=\{f_1^{\sigma},\dots, f_m^{\sigma}\}\subset L^2(\mathbb{R}^d)$ where, $f_j^{\sigma}$ is such that
$\widehat{f_j^{\sigma}}= \widehat{f_j}{\chi_{B_{\sigma}}}$ for $j=1, \dots, m.$
Also, let \mbox{$\widetilde{\mathcal{F}}=\{f_1^{\sigma_1},\dots, f_m^{\sigma_1},\dots\dots,f_1^{\sigma_{\kappa}},\dots, f_m^{\sigma_{\kappa}}\}.$}
For each $\omega\in\mathcal{U}$ let $G_{\widetilde{\mathcal{F}}}(\omega)$ be the associated Gramian matrix of the vectors in $\widetilde{\mathcal{F}}$ with eigenvalues
\[
\lambda_1(\omega)\ge\cdots\ge \lambda_{m\kappa}(\omega)\ge0.
\]
Using Lemma \ref{medibilidad}, we have that these eigenvalues are measurable functions.
Since $f_i^{\sigma_s}$ is orthogonal to $f_i^{\sigma_t}$ if $s \neq t$, the Gramian $G_{\widetilde{\mathcal{F}}}(\omega)$
is a diagonal block matrix with blocks $G_{\sigma}(\omega), \;\sigma\in\mathcal{N}.$ Here $G_{\sigma}(\omega)$ is the $m\times m$ Gramian associated to the data $\mathcal{F}^{\sigma}.$
On the other hand, using Lemma \ref{medibilidad} we have that
\[
G_{\sigma}(\omega)= U_{\sigma}(\omega) \Lambda_{\sigma}(\omega) U_{\sigma}^*(\omega) \quad a.e. \;\;\omega \in \mathcal{U}
\]
where
$U_{\sigma}$ are unitary and $\Lambda_{\sigma}(\omega):=\text{diag}(\lambda_1^{\sigma}(\omega),\dots, \lambda_m^{\sigma}(\omega))\in\mathbb{C}^{m\times m}$
and they are also measurable matrices as in Lemma \ref{medibilidad}. We also have $\lambda_1^{\sigma}(\omega)\ge \dots \ge \lambda_m^{\sigma}(\omega)$ for each $\sigma\in\mathcal{N}.$
Using the decompositions of the blocks $G_{\sigma}$ we have that
\begin{equation}\label{P1}
G_{\widetilde{\mathcal{F}}}(\omega) = U(\omega) \Lambda(\omega) U^*(\omega)
\end{equation}
where $U$ has blocks $U_{\sigma}$ in the diagonal, and $\Lambda$ is diagonal with blocks $\Lambda_{\sigma}$.
We want to recall here that for almost each $\omega$ the matrix $\Lambda(\omega)$ collects all the eigenvalues of the Gramian $G_{\widetilde{\mathcal{F}}}(\omega)$ and
the columns of the matrix $U(\omega)$ are the associated left eigenvectors. Note that an eigenvector associated to the
eigenvalue $\lambda_j^{\sigma}(\omega)$ has all the components not corresponding to the block $\sigma$ equal to zero.
Now for each fixed $\omega\in\mathcal{U},$ we consider $\{(i_1(\omega),j_1(\omega)), \dots, (i_{n}(\omega),j_{n}(\omega))\}$ with $i_s(\omega)\in\mathcal{N}$ and $j_s(\omega)\in\{1,\dots, m\}$ and $n=m\kappa$ such that
\[
\lambda_{j_1(\omega)}^{i_1(\omega)}\ge \dots \ge \lambda_{j_{n}(\omega)}^{i_{n}(\omega)}\ge 0
\]
are the ordered eigenvalues of $G_{\widetilde{\mathcal{F}}}(\omega),$ with corresponding left eigenvectors $Y^{(i_s(\omega), j_s(\omega))}\in\mathbb{C}^n,$ for $s=1, \cdots, n.$
Here $i_s(\omega)$ indicates the block of the matrix $G_{\widetilde{\mathcal{F}}}(\omega)$ in which the eigenvalue $\lambda_{j_s(\omega)}^{i_s(\omega)}(\omega)$ is found
and $j_s(\omega)$ indicates the displacement in this block of the matrix $G_{\widetilde{\mathcal{F}}}(\omega)$. More precisely, we have that $\lambda_{j_s(\omega)}^{i_s(\omega)}(\omega)$ coincides with $\lambda_{(i_s(\omega)-1) m +j_s(\omega)}(\omega),$ the $((i_s(\omega)-1) m +j_s(\omega))-$th eigenvalue of $G_{\widetilde{\mathcal{F}}}(\omega).$
When $\omega\in \mathcal{U}$ is fixed, we will write $i_s$ instead of $i_s(\omega)$ and $j_s$ instead of $j_s(\omega).$
We will prove now that $\gamma_s(\omega):=\lambda_{j_{s(\omega)}}^{i_s(\omega)}(\omega)$ is measurable as a function on $\omega$ for each $s=1, \cdots, n.$
Let $s\in\{1, \dots, n\}$ fixed. Let $i_s(\omega)\in\mathcal{N}$ and $j_s(\omega)\in\{1,\dots, m\}.$ We have that $\gamma_s(\omega)= \lambda_j^{\sigma}(\omega)$ for all $\omega\in E_{{\sigma}j}:= \{\omega\in\mathcal{U}\colon i_s(\omega)= \sigma, j_s(\omega)=j \}.$
We observe that
\[
E_{\sigma j}= \{\omega\in\mathcal{U}\colon i_s(\omega)=\sigma, j_s(\omega)=j\}= \{\omega\in \mathcal{U}\colon \lambda_s(\omega)= \lambda_j^{\sigma}(\omega)\}.
\]
Using Lemma \ref{medibilidad} applied to $G_{\widetilde{\mathcal{F}}}(\omega)$ and $G_{\sigma}(\omega),$ we have that $\lambda_s$ and $\lambda_j^{\sigma}$ are
measurable functions of $\omega.$ Therefore $E_{\sigma j}$ are measurable sets.
We further observe that $\gamma_s(\omega)= \lambda_j^{\sigma} (\omega),$ for $\omega\in E_{\sigma j}.$ So $\gamma_s(\omega)$ is a measurable function.
A similar argument shows that the eigenvectors are measurable.
Finally we define $h_s\colon \mathbb{R}^d\to \mathbb{C},$ for $s=1,\dots, \ell$
\begin{equation}\label{def-h}
h_s(\omega):= \theta_{j_s}^{i_s}(\omega) \sum_{k=1}^m y_{(i_s-1)m+k}^{(i_s, j_s)}(\omega) \widehat{f}_k^{i_s}(\omega),
\end{equation}
where $\theta_{j_s}^{i_s}(\omega)= (\lambda_{j_s}^{i_s}(\omega))^{-1/2}$ if $\lambda_{j_s}^{i_s}(\omega)\neq 0$ and $\theta_{j_s}^{i_s}(\omega)=0$ otherwise.
Now we are ready to state the main result of this section.
\begin{theorem}\label{solucion-PB}
Let $m, \ell\in\mathbb{N},$ and $M$ be a closed proper subgroup of $\mathbb{R}^d$ containing $\mathbb{Z}^d.$ Assume that $\mathcal{F}=\{f_1,\dots, f_m\}\subset L^2(\mathbb{R}^d)$ is given data and let $\mathcal{V}_M^{\ell}$ be the class defined in \eqref{classV}. Then, there exists a shift invariant space $V^*\in\mathcal{V}_M^{\ell}$ such that
\begin{equation}\label{solution}
V^*= \mathop{argmin}_{V\in\mathcal{V}_M^{\ell}} \sum_{j=1}^{m} \|f_j-P_{V} f_j\|^2.
\end{equation}
Furthermore, with the above notation,
\begin{enumerate}
\item[(1)] The eigenvalues $\{\lambda_j^{\sigma}(\omega): \sigma \in \mathcal{N} , j=1,\dots ,m\},$ are $\mathbb{Z}^d-$periodic, measurable functions in $L^2(\mathcal{U})$ and the error of approximation is
\[
\mathcal{E}(\mathcal{F}, M, \ell) := \sum_{j=1}^{m} \|f_j-P_{V^*} f_j\|^2= \int_{\mathcal{U}}\sum_{s= \ell +1}^{m\kappa} \lambda_{j_s}^{i_s}(\omega) \;d\omega.
\]
\item[(2)] The functions $\{h_1,\dots, h_{\ell}\}$ defined in \eqref{def-h} are in $L^2(\mathbb{R}^d)$ and if $\varphi_1,\dots,\varphi_{\ell}$
are defined by $\widehat{\varphi_j} = h_j$, then $\Phi=\{\varphi_1, \cdots, \varphi_{\ell}\}$ is a generator
set for the optimal subspace $V^*$ and the set $\{\varphi_i(\cdot -k), k\in \mathbb{Z}^d, i=1, \cdots, \ell\}$ is a Parseval frame for $V^*.$
\end{enumerate}
\end{theorem}
\color{black}
\begin{proof}
Let $\mathcal{V}^{\ell}$ be the class defined in Theorem \ref{Teorema-original}, that is $\mathcal{V}^{\ell}$ is the set of all shift invariant spaces $V$ that can be generated by $\ell$ or less generators. (Note that we do not ask the elements of the class $\mathcal{V}^{\ell}$ to have extra invariance.)
Define $V^* \in \mathcal{V}^{\ell}$ to be the optimal space given by Theorem \ref{Teorema-original} for the data $\widetilde{\mathcal{F}}$.
That is,
\begin{equation}\label{eq-optimal}
\sum_{\sigma \in \mathcal{N}} \sum_{j=1}^m \|f_j^{\sigma} - P_{V^*}f_j^{\sigma}\|^2 \le \sum_{\sigma \in \mathcal{N}} \sum_{j=1}^m \|f_j^{\sigma} - P_{V}f_j^{\sigma}\|^2 \qquad \forall \;\;V \in \mathcal{V}^{\ell}.
\end{equation}
We claim that $V^* \in \mathcal{V}_M^{\ell}$ (in particular is $M$ extra-invariant) and it is optimal in this class for the data $\mathcal{F},$ i.e.
\begin{equation}\label{optimality}
\sum_{j=1}^m \|f_j - P_{V^*}f_j\|^2 \le \sum_{j=1}^m \|f_j - P_{V}f_j\|^2 \qquad \forall \;\;V \in \mathcal{V}_M^{\ell}.
\end{equation}
Let us prove first that $V^*$ is $M$ extra-invariant. For this we will check that the generators of $V^*$ satisfy
condition (iv) in Theorem \ref{extra-inv}.
We have from \eqref{P1} that the Gramian $G_{\widetilde{\mathcal{F}}}(\omega)$ can be decomposed as $G_{\widetilde{\mathcal{F}}}(\omega) = U(\omega)\Lambda(\omega)U^*(\omega)$ with
eigenvalues $\{\lambda_j^{\sigma}(\omega) : \sigma\in\mathcal{N}, j=1,\dots,m\}.$
By Theorem \ref{Teorema-original}, the $\ell$ generators of $V^*$ have the form defined in \eqref{def-h},
\begin{equation}\label{P2}
\widehat{\varphi}_{s}(\omega)=\theta_{j_s}^{i_s}(\omega) \sum_{k=1}^m y_{(i_s-1)m+k}^{(i_s, j_s)}(\omega) \widehat{f}_k^{i_s}(\omega), \qquad {\text{ for }}
s=1,\dots,\ell.
\end{equation}
From \eqref{P2} it is clear that $\widehat{\varphi}_{s}$ is supported in $B_{i_s}$, since each $\widehat{f}_k^{i_s}$ is supported in $B_{i_s}$.
Then if we apply the cut off operator to these generators we obtain $\widehat{\varphi}_{s}^{\sigma}(\omega) = \widehat{\varphi}_{s}(\omega) {\text{ if }} \sigma = i_s(\omega) {\text{ and }} \widehat{\varphi}_{s}^{\sigma} = 0 \;{\text {otherwise}}.$
So, in any case $\varphi_{s}^{\sigma} \in V^*$ for all $\sigma \in \mathcal{N},\;\; s=1,\dots, \ell$ which proves the $M$-invariance of $V^*.$
What is left now is to prove that $V^*$ is optimal over the class $\mathcal{V}_M^{\ell}$, that is $V^*$ satisfies equation \eqref{optimality}.
For this note that if $V\in \mathcal{V}_M^{\ell}$ then $V = \bigoplus_{\sigma \in \mathcal{N}} V_{\sigma}.$ So we have for any $f\in L^2(\mathbb{R}^d)$,
\begin{equation*}
\|P_{V} f\|^2 = \|P_{V} \sum_{\sigma \in \mathcal{N}} f^{\sigma}\|^2 = \| \sum_{\sigma \in \mathcal{N}} P_{V}f^{\sigma}\|^2 =
\|\sum_{\sigma \in \mathcal{N}} P_{V^{\sigma}} f^{\sigma}\|^2=
\sum_{\sigma \in \mathcal{N}}\|P_{V^{\sigma}} f^{\sigma}\|^2=
\sum_{\sigma \in \mathcal{N}}\|P_{V} f^{\sigma}\|^2,
\end{equation*}
which implies together with \eqref{eq-optimal} that
\begin{equation*}
\sum_{j=1}^m \| P_{V^*}f_j\|^2 \ge \sum_{j=1}^m \| P_{V}f_j\|^2, \qquad \forall \;\;V \in \mathcal{V}_M^{\ell}.
\end{equation*}
The others claims of the theorem are a direct consequence of Theorem \ref{Teorema-original}.
\end{proof}
\bigskip
\section{Approximation with Paley-Wiener spaces}\label{section2}
\subsection{Preliminaries}
In this section the class of approximation subspaces will be finitely generated SIS that are translation invariant.
That is translation invariant spaces that are generated by the integer translates of a finite number of functions.
More precisely, given $\ell\in\mathbb{N}$ define $\T^{\ell}$ to be the set of all shift invariant spaces $V=S(\varphi_1,\dots, \varphi_{\ell})$ for some functions $\varphi_1,\dots, \varphi_{\ell}$ in $L^2(\mathbb{R}^d)$, and such that $V$ is translation invariant and
the integer translates of $\{\varphi_1,\dots, \varphi_{\ell}\}$ form a Riesz basis of $V$.
Given a set $\mathcal{F}=\{f_1,\dots, f_m\}\subset L^2(\mathbb{R}^d),$ we want to find $V^{*}\in \T^{\ell}$ such that
\begin{equation}\label{solucion-Tl}
V^*= \mathop{argmin}_{V\in\T^{\ell}} \sum_{j=1}^{m} \|f_j-P_{V} f_j\|^2.
\end{equation}
Here $P_{V}$ denotes the orthogonal projection on $V.$
Before going to the approximation problem, we will obtain a characterization of the class $\T^{\ell}.$
Using Wiener's theorem, we have that $V$ is a translation invariant space in $L^2(\mathbb{R}^d)$ if and only if there exists a measurable set $\Omega
\subset\mathbb{R}^d$ such that
$$V= \{f \in L^2(\mathbb{R}^d): \hat{f}(\omega)=0 {\text{ a.e. }} \omega \in \mathbb{R}^d \setminus \Omega \}.$$
Since $\Omega$ is unique up to measure zero, we will write $V = V_{\Omega}.$
\begin{definition}\label{multitile}
Let $\Omega\subset\mathbb{R}^d$ be measurable and $L\subset \mathbb{R}^d$ be a countable set. We say that $\Omega$ {\it tiles $\mathbb{R}^d$ when translated by $L$ at
level $\ell\in\mathbb{N}$}
if
\[
\sum_{t\in L} \chi_{\Omega}(\omega-t)=\ell, \quad \mbox{ for a.e. }\omega\in\mathbb{R}^d.
\]
In case of $L=\mathbb{Z}^d$ we will say that $\Omega$ is an {\it $\ell$ multi-tile}.
\end{definition}
It is known (see for example \cite{Ko13},) that $\Omega$ is an $\ell$ multi-tile of $\mathbb{R}^d$, if and only if, up to measure zero, $\Omega$ is the union of $\ell$ measurable and disjoint
$1$ tile sets. i.e. $\Omega$ is a quasi-disjoint union of $\ell$ sets of representatives of $\mathbb{R}^d/\mathbb{Z}^d.$
\begin{lemma}\label{multi}
A measurable set $\Omega\subset \mathbb{R}^d, \;\ell$ multi-tiles $\mathbb{R}^d$
if and only if $$\Omega = \Omega_1\cup\dots\cup \Omega_{\ell } \cup N,$$ where $N$ is a zero measure set,
and the sets $\Omega_j$, $ 1\leq j\leq \ell$ are measurable, disjoint and each of them tiles $\mathbb{R}^d$ by translations on $\mathbb{Z}^d$.
\end{lemma}
The following proposition characterizes the set $\Omega$ for the elements in $\T^{\ell}.$
\begin{proposition}\label{propPW}
A subspace
$V$ is in $\T^{\ell}$ if and only if
$V=V_{\Omega}$ with $\Omega$ a measurable $\ell$ multi-tile of $\mathbb{R}^d.$
\end{proposition}
\begin{proof}
Assume first that $V \in \T^{\ell}$, so $V=V_{\Omega}$ for some measurable $\Omega \subset \mathbb{R}^d$.
Also, as a consequence of Wiener's theorem, for almost all $\omega \in \mathcal{U}$ we have $J_V(\omega) \cong \ell^2(O_{\omega})$ with
$O_{\omega}=\{ k\in \mathbb{Z}^d: w+k \in \Omega\}.$ To see this, we note that $J_V(\omega) \subset \ell^2(O_{\omega})$. For the other inclusion,
fix $\omega\in \mathcal{U}.$ Using that $\Omega = \bigcup_{k\in\mathbb{Z}^d}E_k$ where $E_k = (\mathcal{U}+k) \cap \Omega$, we have that $k\in O_{\omega}$,
if and only if $\omega + k \in E_k$. Hence,
if $ a \in \ell^2(O_{\omega})$ consider the function $G_{\omega} (\xi) = \sum_{k \in O_{\omega}} a_k \chi_{E_k}(\xi)$.
Since $G_{\omega}$ is in $L^2(\Omega),$
the function $g$ defined by $\widehat{g}=G_{\omega}$ is in $V$, and $\widehat{g}(\omega+k) = a_k$
if $k \in O_{\omega}.$ Therefore, $g \in V$ and $a = \tau g(\omega) \in J_V(\omega).$
Now, since $V=S(\varphi_1,\dots, \varphi_{\ell}),$ and the integer translates of $\varphi_1,\dots, \varphi_{\ell}$ form a Riesz basis
of $V,$ using Lemma \ref{Lema 3.2} we obtain that $\{\tau\varphi_1(\omega), \dots,\tau\varphi_{\ell}(\omega)\}$ form a Riesz basis of $J_V(\omega)$ with the same Riesz bounds for a.e. $\omega\in \mathcal{U}.$
We conclude that dim($J_V(\omega))= \ell$ a. e. $\omega \in \mathcal{U}.$
Since $V$ is translation invariant, by the observation above dim$(J_V(\omega))= \#O_w$. Then $\#O_{\omega} = \ell$ for almost all $\omega \in \mathcal{U},$
which implies that $\Omega$ is an $\ell$ multi-tile. (Here $\#A$ denote the cardinal of the set $A).$
For the converse, assume that $\Omega$ is a measurable $\ell$ multi-tile of $\mathbb{R}^d$. Define $V = V_{\Omega}$.
So, $V$ is translation invariant.
By Lemma \ref{multi} we have that $\Omega = \Omega_1\cup\dots\cup \Omega_{\ell }$ up to a measure zero set,
where each $\Omega_j$ is a set of representatives or $\mathbb{R}^d/\mathbb{Z}^d.$
We define $\varphi_j$ by its Fourier transform: $\widehat\varphi_j = \chi_{\Omega_j},\quad j=1,\dots,\ell.$
Since $\{ e^{2\pi i \omega k} \widehat\varphi_j: k \in \mathbb{Z}^d\}$ is an orthonormal basis of $L^2(\Omega_j),$ we have that
$\{e^{2\pi i \omega k} \widehat\varphi_j : k \in \mathbb{Z}^d, j=1,\dots, \ell \}$ is an orthonormal basis of $L^2(\Omega),$ and so,
$\{t_k\varphi_j:k \in \mathbb{Z}^d, j=1,\dots, \ell \}$ is an orthonormal basis of $V$, in particular a Riesz basis.
\end{proof}
\subsection{The approximation problem for Paley-Wiener Spaces}
Now we come back to our approximation problem.
In order to find an optimal subspace in the class $\T^{\ell}$ for a set of data $\mathcal{F}=\{f_1,\dots, f_m\},$
it is enough to find the associated $\ell$ multi-tile $\Omega$ in $\mathbb{R}^d.$
It is not difficult to see that if we allow $\Omega$ to be {\it {any}} $\ell$ multi-tile the minimum in \eqref{solucion-Tl} may not exist.
So we will restrict $\Omega$ to be inside a cube that could be arbitrarily large.
Let us fix $N\in\mathbb{N}$.
Define
\begin{align*}
C_N &:= [-(N+1/2),N+1/2]^d,\\
M_N^{\ell} &:=\{\Omega\subset C_N: \, \Omega \mbox{ is measurable and $\ell$ multi-tiles } \mathbb{R}^d\} {\text{ and }}\\
\T_N^{\ell} &:= \{V\in \T^{\ell}: V= V_{\Omega} {\text{ with }} \Omega \in M_N^{\ell} \}.
\end{align*}
With this notation we can state the main result of this section.
\begin{theorem}
Assume that $m, \ell \in\mathbb{N}$ and a set $\mathcal{F}=\{f_1,\dots, f_m\}\subset L^2(\mathbb{R}^d)$, are given. Then for each $N \geq \ell$
there exists a Paley-Wiener space $V^*\in \T^{\ell}_N$ that satisfies
\begin{equation}
V^*= \mathop{argmin}_{V\in\\ \T^{\ell}_N} \sum_{j=1}^{m} \|f_j-P_{V} f_j\|^2,
\end{equation}
where $\T^{\ell}_N$ is the class defined above.
\end{theorem}
\begin{proof}
First we observe that if a solution space $V^*$ exists then
\begin{equation}\label{Problem2-particular}
V^*= \mathop{argmin}_{V\in\T^{\ell}_N} \sum_{j=1}^{m} \|f_j-P_{V} f_j\|^2 \;
= \mathop{argmax}_{V\in\T^{\ell}_N} \sum_{j=1}^{m} \|P_{V} f_j\|^2,
\end{equation}
and using the definition of $\T^{\ell}_N,$ we have that
\begin{equation}\label{maximun}
\mathop{max}_{V\in\T^{\ell}_N} \sum_{j=1}^{m} \|P_{V} f_j\|^2= \mathop{max}_{\Omega\in
M^{\ell}_N} \sum_{j=1}
^{m} \|P_{V_{\Omega}} f_j\|^2.
\end{equation}
So, we need to find $\Omega \in M^{\ell}_N$ that yields the maximum in \eqref{maximun}.
Using Lemma \ref{Lema 3.1} we see that for each $\Omega \in M^{\ell}_N$,
\begin{align}\label{xx}
\sum_{j=1}^{m} \|P_{V_{\Omega}} f_j\|^2&= \sum_{j=1}^{m} \|P_{\widehat{V_{\Omega}}} \widehat{f_j}\|^2\\
\nonumber&= \sum_{j=1}^{m} \int_{\mathcal{U}} \|P_{J_{V_{\Omega}}}(\omega)( \tau f_j(\omega))\|^2_{\ell^2(\mathbb{Z}^d)}\, d\omega\\
\nonumber&= \int_{\mathcal{U}} \sum_{j=1}^{m} \|P_{J_{V_{\Omega}}}(\omega) (\tau f_j(\omega))\|^2_{\ell^2(\mathbb{Z}^d)}\, d\omega.
\end{align}
Recall that $P_{J_{V_{\Omega}}}(\omega) $ denotes the orthogonal projection onto the closed subspace ${J_{V_{\Omega}}}(\omega)$ of $\ell^2(\mathbb{Z}^d).$
Furthermore, if $\Omega \in M^{\ell}_N,$ we know from the proof of Proposition \ref{propPW} that $\dim(J_{V_{\Omega}}(\omega))=
\ell$ for a.e. $\omega\in \mathcal{U} .$ Note that $J_{V_{\Omega}}(\omega)$ agrees with the subspace of $\ell^2(\mathbb{Z}^d)$ of the sequences supported
in $O_{\omega}.$
Then there exists a unique set of $\ell$ integer vectors ${\bf{k}}^{\Omega}(\omega)=\{k_1^{\Omega}(\omega),\dots, k_{\ell}^{\Omega}(\omega)\}\subset \mathbb{Z}^d$ such that $\mbox{span}\{\delta_{k_j^{\Omega}(\omega)}: j=1,\dots,\ell\}= J_{V_{\Omega}}(\omega),$
for a.e. $\omega\in \mathcal{U}$. Here $\delta_j$ denotes the canonical vector in $\ell^2(\mathbb{Z}^d).$
i.e. $\delta_j(s) = 0$ if $s \neq j$ and $1$ otherwise.
Note that , since $\Omega \subset C_N$ necessarely $\|k_j^{\Omega}(\omega)\|_{\infty} \leq N,$ for each $j$ and $\omega.$
Combining this observation with \eqref{xx} we obtain,
\begin{equation}\label{xxx}
\sum_{j=1}^{m} \|P_{V_{\Omega}} f_j\|^2= \int_{\mathcal{U}} \sum_{j=1}^{m} \sum_{s=1}^{\ell} |\widehat{f_j}(\omega+k_s^{\Omega}(\omega))|^2\, d\omega.
\end{equation}
So, now we need to maximize the left hand side in \eqref{xxx} over all the sets $\Omega \in M^{\ell}_N.$
Note that given $\Omega \in M^{\ell}_N,$ for almost each $\omega \in \mathcal{U},$ the set $\Omega$ contains exactly $\ell$ elements
from the sequence $\{\omega + k, k \in \mathbb{Z}^d\}.$
Then we can pick for each $\omega \in \mathcal{U}$ (up to a set of zero measure) $\ell$ translations $k_s^*(\omega)$
such that $\sum_{j=1}^{m} \sum_{s=1}^{\ell} |\widehat{f_j}(\omega+k_s^*(\omega))|^2$ is maximum over all
sets of $\ell$ translations ${\bf{k}}=\{k_1,\dots, k_{\ell}\}\subset \mathbb{Z}^d,$ with $\|k_j\|_{\infty} \leq N.$
The maximum exists since the fibers of $f_j$ are $\ell^2(\mathbb{Z}^d)$-sequences and the number of translations considered
is finite.
Call $\mathcal{K}$ the set of admisibles translations i.e. $\mathcal{K} = \{{\bf{k}}=\{k_1,\dots, k_{\ell}\}\subset \mathbb{Z}^d :\|k_j\|_{\infty} \leq N\}$ and for ${\bf{k}} \in \mathcal{K}$ set $ H_{\bf{k}} (\omega) = \sum_{j=1}^{m} \sum_{s=1}^{\ell} |\widehat{f_j}(\omega+k_s(\omega))|^2.$
Our goal is to construct a set $\Omega$ such that the associated space $V_{\Omega}$ is optimal. So the idea is to construct the optimal set $\Omega^*$
considering for each $\omega \in \mathcal{U}$ the optimal translations $\{\omega + k_s^*(\omega) : s=1,\dots,\ell\},$ and then taking the union over almost all $\omega \in \mathcal{U}.$
For this we define for each ${\bf{k}} =\{k_1,\dots, k_{\ell}\} \in \mathcal{K}$ the following subset of $\mathcal{U},$
\[
E_{\bf{k}}=\left\{\omega\in \mathcal{U}\colon H_{\bf{k}} (\omega)\geq H_{\bf{r}} (\omega),\;
\forall \;{\bf{r}}=\{{r_1},\dots,{r_{\ell}}\} \in \mathcal{K}\right\},
\]
i.e., $E_{\bf{k}}$ is the set of $\omega\in \mathcal{U}$ for which the maximum is attained for ${\bf{k}}=\{k_1,\dots, k_{\ell}\}.$
Note that $E_{\bf{k}}$ could be the empty set for some ${\bf{k}}=\{k_1,\dots, k_{\ell}\}$ and the sets
$E_{\bf{k}}$ may not be disjoint.
Finally we define our optimal set as,
\[
\displaystyle \Omega^*= \bigcup_{\begin{smallmatrix}
{\bf{k}} \in \mathcal{K}
\end{smallmatrix}} \bigcup_{j=1}^{\ell} E_{\bf{k}} +k_j.
\]
We will now prove that $\Omega^*$ is measurable. First we note that $E_{\bf{k}}$ is a measurable set for each ${\bf{k}} \in \mathcal{K}$ since,
\[
\displaystyle E_{\bf{k}}= \bigcap_{{\bf{r}}\in \mathcal{K}} F_{{\bf{r}}}^{\bf{k}},
\]
where,
\[
F_{{\bf{r}}}^{\bf{k}}=\left\{\omega\in \mathcal{U}\colon H_{\bf{k}} (\omega)\geq H_{\bf{r}} (\omega)
\right\}.
\]
Now, since $F_{\bf{r}}^{\bf{k}}$ is measurable for all ${\bf{r}}\in \mathcal{K},$ we obtain that $E_{\bf{k}}$ is measurable and so is $\Omega^*$.
Furthermore, by construction, $\Omega^*$ is in $M_N^{\ell}.$ Since for all $\Omega \in M_N^{\ell}$ we have that,
$$
\sum_{j=1}^{m} \sum_{s=1}^{\ell} |\widehat{f_j}(\omega+{k_s^{\Omega}}(\omega))|^2
\leq \sum_{j=1}^{m} \sum_{s=1}^{\ell} |\widehat{f_j}(\omega+k_s^*(\omega))|^2 \;\;{\text{for almost all }} \omega \in \mathcal{U},
$$
taking the integral over $\mathcal{U}$ we get
\begin{align*}
\sum_{j=1}^{m} \|P_{V_{\Omega}} f_j\|^2&= \int_{\mathcal{U}} \sum_{j=1}^{m} \sum_{s=1}^{\ell} |\widehat{f_j}(\omega+k_s^{\Omega}(\omega))|^2\, d\omega \\
&\leq \int_{\mathcal{U}} \sum_{j=1}^{m} \sum_{s=1}^{\ell} |\widehat{f_j}(\omega+k_s^*(\omega))|^2\, d\omega=\sum_{j=1}^{m} \|P_{V_{\Omega^*}} f_j\|^2 .
\end{align*}
This shows that $\Omega^*\in M_N^{\ell}$ is optimal over all $\Omega\in M_N^{\ell}.$
We conclude
that $V_{\Omega^*}\in\T^{\ell}_N$ is a solution for the data $\mathcal{F}$.
\end{proof}
\begin{remark}
Notice that if $\Omega_N^*$ is the optimal multi-tile set for the class $\T^{\ell}_N$ for some data $\mathcal{F}$,
then the approximation error is given by
\[
\mathcal{E}_N(\mathcal{F},\ell) = \int_{\mathbb{R}^d \setminus \Omega_N^*} \sum_{j=1}^{m} |\widehat{f_j}(\omega)|^2 d\omega =
\int_{C_N\setminus \Omega_N^*} \sum_{j=1}^{m} |\widehat{f_j}(\omega)|^2 d\omega +
\int_{\mathbb{R}^d\setminus C_N} \sum_{j=1}^{m} |\widehat{f_j}(\omega)|^2 d\omega .
\]
Clearly $\mathcal{E}_N(\mathcal{F},\ell)\geq \mathcal{E}_{N+1}(\mathcal{F},\ell)$.
So $\mathcal{E}(\mathcal{F},\ell):=\lim_{N\rightarrow \infty} \mathcal{E}_N(\mathcal{F},\ell)$ is somehow the optimal error.
Since $\mathcal{F}\subset L^2(\mathbb{R}^d)$ then the second integral goes to zero when $N$ goes to infinite,
for functions with good decay at infinite we will be close to the optimal error for conveniently large $N.$
\end{remark}
\begin{remark}
In Proposition \ref{propPW} we show, for an element of $\T^{\ell}$, how to construct a set of generators that gives a Riesz basis of translates in $\mathbb{Z}^d.$ There are many ways to construct other sets of generators that gives Riesz basis of translates.
Recently Grepstad-Lev in \cite{GL14} constructed a basis of exponentials for $L^2(\Omega)$ when $\Omega \subset \mathbb{R}^d$ is a multi-tile.
Later on, Kolountzakis \cite{Ko13} gave a simpler proof of this result in a slightly more general form.
Precisely they prove the following result.
\begin{theorem}[Theorem 1,\cite{Ko13}]\label{exponenciales}
Suppose $\Omega\subset\mathbb{R}^d$ is bounded, measurable and multi-tiles $\mathbb{R}^d$ when translated by $\mathbb{Z}^d$ at level $\ell.$ Then there exist vectors
$a_1,\dots, a_{\ell}\in\mathbb{R}^d$ such that the exponentials
\[
e^{-2\pi i(a_j+k)\omega} \quad j=1, \dots, \ell, \quad k\in\mathbb{Z}^d
\]
form a Riesz basis for $L^2(\Omega).$
\end{theorem}
From Theorem \ref{exponenciales}, we can obtain immediately a set of generators for $V_{\Omega}$.
Let $\varphi$ be such that $\widehat{\varphi} = \chi_\Omega.$
If $a_1,\dots, a_{\ell}\in\mathbb{R}^d$ are as in Theorem \ref{exponenciales}, then $V_{\Omega}= S(\varphi_1,\dots,\varphi_{\ell})$ with
$\varphi_j = t_{a_j}\varphi,\;j=1,\dots,\ell,$ and the translates of $\varphi_1,\dots,\varphi_{\ell}$ form a Riesz basis of $V_{\Omega}.$
In general, all the Riesz basis for $V_{\Omega}$ can be described in the following way:
Let $A = \{a_{js}\}\in [L^2(\mathcal{U})]^{\ell\times\ell}$ be a measurable matrix, such that $0 < c_1 \leq \lambda(\omega) \leq c_2$ for every eigenvalue $\lambda(\omega)$
and for almost each $\omega \in \mathcal{U}$. Set $k(\omega) = (k_1(\omega),\dots, k_{\ell}(\omega))$ such that $w+k_s(\omega) \in \Omega.$
Define $\varphi_j$ such that $\widehat\varphi_j(\omega+k_s(w)) = a_{js}(\omega)$. Using the results stated in subsection \ref{sis} it is not difficult to see that
$\varphi_1,\dots,\varphi_{\ell}$ are measurable, $V_{\Omega}= S(\varphi_1,\dots,\varphi_{\ell})$ and the translates of $\varphi_1,\dots,\varphi_{\ell}$ form a Riesz basis of $V_{\Omega}.$
\end{remark}
\section{The discrete case}\label{section0}
The optimal subspace $V^*$ in
Theorem \ref{solucion-PB} is the closest to the data $\mathcal{F}$
over all subspaces $V$ in the class $\mathcal{V}_M^{\ell}.$
It is not difficult to see that almost each fiber space $J_{V^*}(\omega) \subset \ell^2(\mathbb{Z}^d)$ of $V^*$ is the closest
to the fibers of our data, $\tau(\mathcal{F})(\omega)=\{\tau f_1(\omega),\dots,\tau f_m(\omega)\}$ over a certain class of closed subspaces of
$\ell^2(\mathbb{Z}^d)$ that we will call $\mathcal{D}_{\mathcal{N}}^{\ell}.$
Clearly this class of subspaces is determine by the class $\mathcal{V}_M^{\ell}$. So, Theorem \ref{solucion-PB}, implies an approximation result in $\ell^2(\mathbb{Z}^d)$,
for a very particular class determined by the extra invariance. Therefore it is interesting to see if this approximation result in $\ell^2(\mathbb{Z}^d)$, extends to more
general classes.
We will define in what follows a very general class $\mathcal{D}_{\mathcal{N}}^{\ell}.$ The cases coming from the continuous
case will be particular cases of our general definition. The proof of the optimality that we obtain is more general and can not follows from Theorem \ref{solucion-PB}.
In what follows, the Hilbert space we consider is $\ell^2(\mathbb{Z}^d).$
We define the class of approximating subspaces in the following way:
Let $\mathcal{N}$ be an arbitrary finite set and $\{D_{\sigma}\colon \sigma\in\mathcal{N}\}$ a partition of $\mathbb{Z}^d,$ that is, $\mathbb{Z}^d= \bigcup_{\sigma\in\mathcal{N}} D_{\sigma},$ where the union is disjoint.
For $a \in \ell^2(\mathbb{Z}^d),$ we denote $a^{\sigma} = {\bf{1}_{D_{\sigma}}} a,$
where ${\bf{1_{D_{\sigma}}}}$ denotes the indicator of $D_{\sigma}.$
Given $S\subset \ell^2(\mathbb{Z}^d)$ a closed subspace we define
\[
S_{\sigma}=\{a^{\sigma}: a\in S\}, \quad \mbox{ for each } \sigma\in\mathcal{N}.
\]
We define the class of approximating subspaces by
\begin{equation}\label{classD}
\mathcal{D}_{\mathcal{N}}^{\ell}=\{S\subset\ell^2(\mathbb{Z}^d)\colon S {\text{ is a subspace, }} \dim(S)\le\ell \mbox{ and } S_{\sigma}\subseteq S,\;\; \forall \sigma\in\mathcal{N}\}.
\end{equation}
Note that $S \in \mathcal{D}_{\mathcal{N}}^{\ell}$ if and only if $ \dim(S)\le\ell$ and $S$ is the orthogonal sum of the subspaces $S_{\sigma}$ i. e.,
\[
S=\oplus_{\sigma\in \mathcal{N}}S_{\sigma}.
\]
For a given set $\mathcal{A}= \{a_1,\dots, a_m\}\subset \ell^2(\mathbb{Z}^d)$ consider for each, $\sigma\in\mathcal{N},$ the Gramian matrix $G_{\sigma}\in \mathbb{C}^{m\times m}$ of the data
$\mathcal{A}_{\sigma} = \{a_1^{\sigma},\dots, a_m^{\sigma}\},$ that is
$(G_{\sigma})_{k,l}= \langle a_k^{\sigma}, a_l^{\sigma},\rangle,\; k,l=1,\dots,m,$
with eigenvalues $\lambda_1^{\sigma}\geq \dots,\geq\lambda_m^{\sigma},$ and orthonormal corresponding left eigenvectors $y_1^\sigma,\dots, y_m^\sigma.$
Now set $\Lambda=\{\lambda_j^\sigma: j=1,\dots,m,\;\sigma\in\mathcal{N}\}$ and collect in $\Lambda_{\ell}$ the $\ell$ first biggest eigenvalues of
$\Lambda$ that is if $\lambda \in \Lambda_{\ell}$ then $\lambda \geq \mu$ for all $\mu \in \Lambda\setminus \Lambda_{\ell}.$
Write $\Lambda_{\ell}=\{\lambda_1,\dots,\lambda_{\ell}\}$.
For each $s=1,\dots, \ell$ we define the sequence $q_s \in \ell^2(\mathbb{Z}^d)$ in the following way:
Since $\lambda_s= \lambda^{\sigma_s}_{j_s}$ for some $\sigma_s \in \mathcal{N}$ and some $j_s=1,\dots,m$, then
$\lambda_s$ is an eigenvalue of $G_{\sigma_s}$. Let $y_{j_s}^{\sigma_s}$ be the corresponding left eigenvector
$y_{j_s}^{\sigma_s}=(y_{j_s}^{\sigma_s}(1),\dots, y_{j_s}^{\sigma_s}(m)).$
Then define, if $\lambda_s \in \Lambda_{\ell}, \lambda_s \neq 0$
\begin{equation}\label{def-q}
q_{s} := (\lambda_s)^{-1/2} \sum_{k=1}^{m} y^{\sigma_s}_{j_s}(k) a_k^{\sigma_s}.
\end{equation}
If $\lambda_s=0$ we define $q_s$ to be the zero sequence.
With this notation we can state the main theorem of this section:
\begin{theorem}\label{PA}
Let $m, \ell\in\mathbb{N}$ and $\mathcal{N}$ a finite set. Assume that a set $\mathcal{A}= \{a_1,\dots, a_m\}\subset \ell^2(\mathbb{Z}^d)$ is given.
Then there exists $S^*\in\mathcal{D}_{\mathcal{N}}^{\ell}$ that satisfies
\begin{equation}\label{eqPA}
\sum_{j=1}^{m} \|a_j-P_{S^*} a_j\|^2\le \sum_{j=1}^{m} \|a_j-P_{S} a_j\|^2, \quad \forall \,S\in\mathcal{D}_{\mathcal{N}}^{\ell}.
\end{equation}
Moreover, we have that
\begin{enumerate}
\item[(1)] $S^*=\mbox{span}\{q_1, \cdots, q_{\ell}\}$ where $q_1, \cdots, q_{\ell}$ are defined in \eqref{def-q}.
Also, the vectors $\{q_1, \cdots, q_{\ell}\}$ form a Parseval frame for $S^*.$
\item[(2)] The error in the approximation is
\[
\mathcal{E}(\mathcal{A}, \mathcal{N}, \ell)= \sum_{\lambda \in \Lambda\setminus\Lambda_{\ell}}\lambda.
\]
\end{enumerate}
\end{theorem}
\begin{proof}
First, we observe that \eqref{eqPA} is equivalently to,
\[
\sum_{j=1}^{m} \|P_{S^*} a_j\|^2\ge \sum_{j=1}^{m} \|P_{S} a_j\|^2, \quad \forall\, S\in\mathcal{D}_{\mathcal{N}}^{\ell}.
\]
Furthermore, if $S\in\mathcal{D}_{\mathcal{N}}^{\ell}$ then
\begin{equation*}
\sum_{j=1}^{m} \|P_{S} a_j\|^2= \sum_{j=1}^{m} \|\sum_{\sigma\in\mathcal{N}}P_{S_{\sigma}} a_j\|^2
= \sum_{j=1}^{m} \|\sum_{\sigma\in\mathcal{N}}P_{S_{\sigma}} a_j^{\sigma}\|^2
= \sum_{\sigma\in\mathcal{N}} \sum_{j=1}^{m}\|P_{S_{\sigma}} a_j^{\sigma}\|^2,
\end{equation*}
where $a_j= \sum_{\sigma\in\mathcal{N}} a_j^{\sigma}.$
In order to construct an optimal subspace $S^*\in\mathcal{D}_{\mathcal{N}}^{\ell}$ for the data $\mathcal{A}$, we will find an optimal subspace $S_{\sigma}$ for each $\sigma\in\mathcal{N}$, of dimension at most $\alpha_{\sigma}\in \{1,\dots,\ell\}$
for the data $\mathcal{A}_{\sigma} = \{a_1^{\sigma},\dots, a_m^{\sigma}\}$.
The existence of the optimal subspaces are provided by Theorem 4.1 of \cite{ACHM07}.
We need $\sum_{\sigma\in\mathcal{N}} {\text{dim}}(S_{\sigma}) = \sum_{\sigma\in\mathcal{N}} \alpha_{\sigma} = $ dim$(S^*) \leq \ell$.
Thus if $\mathcal{Q}=\{\alpha=\{\alpha_{\sigma}\} : 0\le \alpha_{\sigma}\le \ell {\text{ and }} \sum_{\sigma\in\mathcal{N}} \alpha_{\sigma}\le \ell \},$ then for each choice of $\alpha \in\mathcal{Q}$
we will find optimal subspaces $\{S_{\sigma}^{\alpha}: \sigma\in\mathcal{N}\}$ and define
$S^{\alpha}=\oplus_{\sigma\in\mathcal{N}} S_{\sigma}^{\alpha}.$
The candidate for $S^*$ is the space $S^{\alpha}$ which minimize the expression \eqref{eqPA} over all $\alpha \in \mathcal{Q}.$
Let $\beta \in \mathcal{Q}$ be the minimizer. Hence $\beta$ satisfies,
\begin{equation}\label{max}
\sum_{\sigma\in\mathcal{N}} \sum_{j=1}^{m}\|P_{S_{\sigma}^{\alpha}} a_j^{\sigma}\|^2 \leq
\sum_{\sigma\in\mathcal{N}} \sum_{j=1}^{m}\|P_{S_{\sigma}^{\beta}} a_j^{\sigma}\|^2 \quad\forall \; \alpha \in \mathcal{Q}.
\end{equation}
Therefore, the subspace $S^*:= S^{\beta}=\oplus_{\sigma\in\mathcal{N}} S_{\sigma}^{\beta}$ is the optimal subspace
we need.
It is straightforward to see that $S^*\in\mathcal{D}_{\mathcal{N}}^{\ell}$ and that $S^*$ is optimal.
Using Theorem 4.1 of \cite{ACHM07} we obtain that for each $\alpha \in \mathcal{Q}$, the error of approximation for the data
$\mathcal{A}_{\sigma}$ and the class of subspaces of dimension at most $\alpha_{\sigma}$ is given by
$$
\mathcal{E}(\mathcal{A}_{\sigma},\alpha_{\sigma})=\sum_{s=\alpha_{\sigma}+1}^m \lambda_s^{\sigma}.
$$
So the distance between the $\alpha$-optimal subspace $S^{\alpha}$ and the data $\mathcal{A}$ is,
\begin{equation}\label{err}
E(\alpha) = \sum_{\sigma\in\mathcal{N}} \mathcal{E}(\mathcal{A}_{\sigma},\alpha_{\sigma}) =\sum_{\sigma\in\mathcal{N}} \sum_{s=\alpha_{\sigma}+1}^{m} \lambda_s^{\sigma}.
\end{equation}
Let $\kappa$ be the number of elements in $\mathcal{N}$. We see that $E(\alpha)$ is minimum when the $m\kappa-\ell$ eigenvalues used in \eqref{err} are the smallest
from the set $\Lambda=\{\lambda_j^\sigma: j=1,\dots,m,\;\sigma\in\mathcal{N}\}.$
Therefore if we set $\Lambda_{\ell}\subset \Lambda$ the set of the $\ell$ biggest eigenvalues from $\Lambda$,
the optimal $\beta=\{\beta_{\sigma}\} \in \mathcal{Q}$ satisfies that
$$
\bigcup_{\sigma\in\mathcal{N}}\{\lambda^{\sigma}_1,\dots,\lambda^{\sigma}_{\beta_{\sigma}}\}=\Lambda_{\ell}.
$$
Therefore,
\[
\mathcal{E}(\mathcal{A},\mathcal{N}, \ell) = \sum_{\sigma\in\mathcal{N}}\mathcal{E}(\mathcal{A}_{\sigma}, \beta_{\sigma})
= \sum_{\sigma\in\mathcal{N}} \sum_{j=1}^{m}\|a_j^{\sigma} - P_{S_{\sigma}^{\beta}} a_j^{\sigma}\|^2
= \sum_{\sigma\in\mathcal{N}} \sum_{s=\beta_{\sigma}+1}^{m} \lambda_s^{\sigma}= \sum_{\lambda \in \Lambda\setminus\Lambda_{\ell}}\lambda.
\]
\medskip
In order to construct the generators of $S^{\beta}$ it is enough to construct the generators of each $S^{\beta}_{\sigma}$.
Since $S^{\beta}_{\sigma}$ are optimal subspaces for the data $\mathcal{A}_{\sigma} $
according with Theorem 4.1 of \cite{ACHM07} the generators of $S^{\beta}_{\sigma}$ are given by \eqref{def-q}.
That is the set $\{q_s: \sigma_s=\sigma\}$ is a Parseval frame of $S^{\beta}_{\sigma}$.
Since the subspaces $S^{\beta}_{\sigma}$ are mutually orthogonal,
$\{q_1,\dots,q_{\ell}\}$ is a set of Parseval frame generators for the optimal space $S^*=S^{\beta}.$
\end{proof}
\begin{remark}
As explained at the beginning of this section there is a reason to consider this particular class of subspaces for the the discrete case.
If a SIS V is $M$ extra-invariant for some proper subgroup $M$ of $\mathbb{R}^d$ containing $\mathbb{Z}^d$,
then its fiber spaces $J_V(\omega)$ satisfies exactly the conditions that we imposed on the class
$\mathcal{D}_{\mathcal{N}}^{\ell}$ where the partition of $\mathbb{Z}^d$ is $\{B_{\sigma}\cap \mathbb{Z}^d: \sigma \in \mathcal{N}\} $ and $B_{\sigma}$ and $\mathcal{N}$ are as in \eqref{def-Bsigma}.
So, the discrete result (Theorem \ref{PA}) provides a different proof of Theorem \ref{solucion-PB}
using properties of range functions (\ref{U}),
without the need of Theorem \ref{Teorema-original}. Actually this proof includes Theorem \ref{Teorema-original}.
\end{remark}
\medskip
{\bf Acknowledgements.} We thank Ursula Molter and Victoria Paternostro for carefully reading the manuscript.
We also want to thank the anonymous referee for her/his comments that helped to improve the manuscript.
|
3,212,635,537,469 | arxiv | \section{The LOFT mission and the on-board instruments}
\label{sec:intro}
The LOFT mission\cite{feroci11}, selected by ESA in the context of the cosmic vision program in February 2011, is designed to exploit the diagnostics of rapid X-ray flux
and spectral variability in order to probe directly the motion of accreting matter down to near vicinity (several Schwarzschild radii) of black holes (BHs) and neutron stars (NSs). Moreover it will determine the equation of state of
matter at supranuclear density, through the measurement of the mass and radius of neutron star.
The LOFT payload comprises two instruments: the Large Area Detector (LAD\cite{zane12}) and the Wide Field Monitor (WFM\cite{brandt12}).
The former is a collimated experiment (field of view, FOV, is $\sim$1~deg) with a total effective area for X-ray photons detection of about
10~m$^2$ at 8~keV (the operating energy range is 2-80~keV). The time resolution is of $\sim$10~$\mu$s, and the
spectral resolution $<$260~eV (FWHM EoL at 6~keV). Such a large area instrument could be designed to fit into the envelope of a medium-size mission
thanks to the low power consumption and mass per unit area of the Silicon Drift Detectors (SDDs) and the Micro-Channel Plate collimators\cite{feroci11}.
The WFM is a coded mask imager, observing more than 1/3 of the sky at once. It makes use of the similar SDDs to the LAD's
(but optimized for imaging purpose), and has thus a similar time and spectral resolution. The main goal of the WFM is to detect
source state transitions as well as
galactic and cosmological bright events suitable for observations with the LAD. The positional accuracy
of the WFM is of 1~arcmin. The LOFT Burst Alert System (LBAS) will broadcast the position and trigger time of bright events
to the ground within a delay of $\sim$30~s (see also http://www.isdc.unige.ch/loft).
Among the main science drivers of the missions is the measurement of several General Relativistic (GR)
effects in the strong field regime (so far GR
has been probed only through measurements in the weak field domain, i.e. $r$$\sim$10$^5$--10$^6$$r_{\rm g}$).
As an example, the quasi-periodic oscillations (QPOs) in the X-ray flux of accreting NSs and BHs, are produced
in the innermost regions of the accretion flow towards these objects and have been associated to
relativistic frequencies of motion, which are much different in this regime from their Newtonian
equivalents. By studying in detail these QPOs with the LAD it will be possible to understand
their origin, thus getting direct access to so far untested GR effects (such as strong-field
frame-dragging, and periastron
precession, and the existence of an innermost stable orbit around BHs and, possibly, NSs).
By making use of the large area and spectral capabilities of the LAD, the variable very broad
profile of the Fe K line in galactic binaries
and Active Galactic Nuclei will be measured at high signal-to-noise ratio, thus allowing the detection of
Lense-Thirring nodal precession of the inner accretion disk ($\sim$$10 r_{\rm g}$) and the measurement of the mass and spin of BHs.
Apart from detecting suitable sources and events to be re-pointed with the LAD, the WFM will also produce very valuable science by itself.
In its large FOV, the detections of about 150 GRBs and thousands type-I X-ray bursts is expected every year
(together with any other relatively bright transient event), and for a number of them the WFM will provide high time and
spectral resolution data in a wide energy range (from 2 up to 80~keV).
\begin{figure}
\begin{center}
\includegraphics[width=1.8in]{LOFT.ps}
\includegraphics[width=1.8in]{effarea.ps}
\caption{{\it Left}: Artist's impression of the LOFT satellite. The SDD of the LAD are placed over the 6 panels extending from
the spacecraft optical bench. These panels are stowed during launch and deployed once in space. The WFM is located at the top of the optical bench.
{\it Right}: LAD effective area as a function of the energy (other X-ray astronomy satellites are shown for comparison).}
\label{fig:loft}
\end{center}
\end{figure}
\vspace{-0.5truecm}
|
3,212,635,537,470 | arxiv | \section{Introduction}
\label{sec:intro}
Giant resonances, excited
by various probes, show, at an initial stage of the excitation process,
a regular motion with a definite vibrational frequency \cite{speth,harakeh}.
These regular motions are then damped due to the coupling with
a huge number of background states, and finally the so called compound states are realized.
We now have understood the both ends of these processes: The frequency of the giant resonance,
for instance, can be well evaluated by the random phase approximation (RPA).
Compound states, on the other hand, are also well described by the random matrix
theory with the Gaussian orthogonal ensemble (GOE) \cite{dyson,mehta},
which characterizes a classical chaotic motion.
It is still not well understood, however, how the dynamics changes from regular
to chaotic \cite{mottelson}.
In order to answer this question, it is very useful to study the fluctuation
properties of the strength functions: The structure at the large energy scale of
the strength function corresponds to the behavior of the initial stage, while
the fluctuation properties at small energy scale correspond to the long time behavior.
We proposed and have used a novel fluctuation analysis based on the quantity we call the local
scaling dimension to study
the fluctuation properties of the strength functions \cite{aiba}.
This method is devised to quantitatively characterize the fluctuation at each energy scale,
and is suitable for the investigation of the fine structure of the strength function.
The strength distribution of giant resonances and its fluctuation have also been studied
experimentally. Recently, the fine structure of the strength distribution of the
giant quadrupole resonance (GQR) in
$^{208}$Pb \cite{shevchenko,shevchenko2,lacroix} or
the Gamow-Teller resonance (GTR) in $^{90}$Zr \cite{kalmykov} were measured and theoretical
analysis has also been done.
In the previous paper \cite{aiba2}, we investigated the GQR in $^{40}$Ca,
where the strength function was calculated by means of the second Tamm-Dancoff
approximation (TDA), namely, the 1p1h
and 2p2h model space is included.
The results of the local scaling dimension analysis were as follows:
At small energy scale, the behavior of the local scaling dimension is almost the same
as that of the GOE, which exhibits the complexity of 2p2h background states.
On the other hand, a clear deviation from the GOE was found at the intermediate energy
scale and it was found that this energy corresponds to the spreading
width of 1p1h states.
Hence, we can say that the spreading width of 1p1h states is detected
as deviation from the GOE limit in $^{40}$Ca.
For $^{40}$Ca the Landau damping is important for the damping
process of the giant resonance. Namely, the strength is first fragmented over
a wide range of 1p1h states, and this fragmentation characterizes a global
profile of the total strength function.
However, as the mass of nuclei increases, the relative importance of the Landau damping
may change.
Accordingly, 2p2h states may also contribute to the global profile of the
strength function.
Therefore, it is very important to investigate how the difference between the damping process
of light nuclei and that of heavy nuclei does affect the properties of the fluctuation
of the strength function.
In this paper, we study the isoscalar (IS) GQR of $^{208}$Pb,
where the strength function is calculated
with the second TDA in the same manner as in $^{40}$Ca,
and study the fluctuation of the strength function by means of the local scaling dimension.
Comparing results with those of $^{40}$Ca we would like to clarify which
properties of the damping process
are reflected in the fluctuation of the strength function
and make clear the physical origin of the difference.
This paper is organized as follows:
In Sec.\ \ref{sec:lsd}, we briefly explain the local scaling dimension.
The strength function for IS GQR in $^{208}$Pb is calculated in Sec.\ \ref{sec:numerical}, where
the adopted Hamiltonian and the model space are shown.
In Sec.\ \ref{sec:measures}, we discuss the nearest-neighbor level spacing distribution,
$\Delta_3$ statistics as well as a histogram of the strength distribution.
In Sec.\ \ref{sec:results}, we apply the local scaling dimension to the IS GQR strength
function in $^{208}$Pb. Detail of damping process is studied in Sec.\ \ref{sec:damping},
where the physical origin for the difference of the fluctuation property of the
strength function between $^{40}$Ca and $^{208}$Pb is also discussed.
Finally, Sec.\ \ref{sec:conclusion} is devoted to conclusion.
\section{Local Scaling Dimension}
\label{sec:lsd}
We briefly explain the local scaling dimension.
See Refs.\ \cite{aiba,aiba2} for details.
The strength function is expressed as \cite{Bohr-Mottelson2}
\begin{equation}
S(E)=\sum_i S_i\delta(E-E_i+E_0).
\label{defstr}
\end{equation}
Here $E_i$ and $S_i$ denote the energy and the strength of
exciting the $i$th energy level, respectively.
Strengths are normalized as $\sum_i S_i=1$.
To study the fluctuation at each energy scale,
we consider binned distribution of the
strength by dividing whole energy interval under
consideration into $N$ bins with length $\epsilon$. Strength
contained in $n$th bin is denoted by $p_n$,
\begin{equation}
p_n\equiv\sum_{i\in n{\rm th~ bin}}S_i.
\label{defp}
\end{equation}
To characterize the distribution of the binned strengths,
we introduce the moments of $p_n$, which are called in literature
the partition function $\chi_m(\epsilon)$ defined by
\begin{equation}
\chi_m(\epsilon)\equiv \sum_{n=1}^N p_n^m \\
=N\langle p_n^m\rangle.
\label{partition}
\end{equation}
Finally, by extending the idea of the generalized fractal dimensions \cite{hentschel,halsey}
to non-scaling
cases in a straightforward way, we can define the local scaling dimension as,
\begin{equation}
D_m(\epsilon)\equiv \frac{1}{m-1}
\frac{\partial\log\chi_m(\epsilon)}
{\partial\log\epsilon}.
\label{scaledim}
\end{equation}
Since the local scaling dimension has a definite physical
meaning similar to that of the generalized fractal dimension,
the value of $D_m(\epsilon)$ can quantitatively characterize the fluctuation
of the strength function at each energy scale $\epsilon$.
In the actual calculation of the local scaling dimension, we
define it by means of the finite difference under the change
of a factor 2,
\begin{equation}
D_m(\sqrt{2}\epsilon) = \frac{1}{m-1}\frac{
\log\chi_m(2\epsilon)-
\log\chi_m(\epsilon)}
{\log 2},
\label{approscaledim}
\end{equation}
rather than the derivative in Eq.\ (\ref{scaledim}).
\section{Numerical Calculation of Strength Function}
\label{sec:numerical}
We calculated the strength function of the IS GQR in $^{208}$Pb
within the second TDA
including the 1p1h and 2p2h excitations. Single-particle
wave-functions and energies were obtained for a Woods-Saxon potential
including the Coulomb interaction.
The effective mass parameter $m^*/m$, which scales the Woods-Saxon single-particle
energies $\varepsilon_{\rm WS}$
as $\varepsilon_{\rm HF}=\varepsilon_{\rm WS}/(m^*/m)$ to simulate the bare (Hartree-Fock)
single-particle energies $\varepsilon_{\rm HF}$, is set to be 1 in this calculation.
As the residual interaction, the Landau-Migdal-type interaction \cite{Schwe}
including the density-dependence was adopted.
The model space was constructed in terms of single-particle states
within the four major shells, two below and two above the Fermi surface,
and included all 1p1h states and 2p2h states whose unperturbed energies are
less than 15MeV. Resultant number of 1p1h states and 2p2h states
are 39 and 8032, respectively. We diagonalized the Hamiltonian
within this model space and obtained the strength function for
the isoscalar quadrupole operator.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=6cm]{fig1.eps}
\end{center}
\caption{
Calculated strength function of the IS GQR in$^{208}$Pb.
Dotted curve shows the smooth strength function by means of the
Strutinsky method with the smoothing width 0.2 MeV.
}
\label{fig_strfun}
\end{figure}
Figure \ \ref{fig_strfun} shows the calculated strength function.
The average of the excitation energy weighted by the strength
is about 10.5 MeV,
and the standard deviation around the average is about 2.6 MeV,
where all levels are considered.
The peak position lies at the same value as the average.
These values are consistent with the (p,p')
experimental data \cite{shevchenko}.
Moreover, the agreement of the global shape with the experimental data is
also good.
The dotted curve in Fig.\ \ref{fig_strfun} represents the smooth strength
function by means of the Strutinsky method\ \cite{ring-schuck} with the smoothing width 0.2 MeV.
The value of the FWHM of this smooth strength function is 0.63 MeV.
In order to quantitatively characterize the spreading of the strength function
around the largest peak, the FWHM is more appropriate than the standard deviation\
\cite{bertsch2}. Thus, we use the FWHM as a measure of the total width $\Gamma$ of
the strength function, which gives $\Gamma=0.63$ MeV.
Hereafter, when we estimate the value of the FWHM, the same procedure as above is adopted,
namely, we calculate the FWHM for the smooth strength function by means of the Strutinsky
method with the smoothing width 0.2 MeV.
\section{Fluctuation at small scale}
\label{sec:measures}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=9.2cm]{fig2.eps}
\end{center}
\caption{
The nearest-neighbor level spacing distribution for (a) $^{40}$Ca and
(b) $^{208}$Pb. For $^{208}$Pb 3321 levels between 9.9 MeV and 13.1 MeV,
while for $^{40}$Ca 804 levels between 20 MeV and 30 MeV are considered.
Level spacings were unfolded by the Strutinsky method with a smoothing width
0.5 MeV for $^{208}$Pb and 5.0 MeV for $^{40}$Ca, respectively.
The solid curve represents the Wigner distribution.
}
\label{fig_nns}
\end{figure}
Before going to the detailed discussion of the local scaling
dimension, we briefly show the results for other fluctuation measures: the
nearest-neighbor level spacing distribution (NND), the strength
distribution, and $\Delta_3$ statistics.
Here, the NND and the strength distribution are measures characterizing
the fluctuation at small energy scale limit.
We present the results of $^{40}$Ca as well as those of $^{208}$Pb
for the sake of comparison.
Figure\ \ref{fig_nns} shows the NND. For both nuclei the NND follows the
Wigner distribution well.
We present the strength distribution in Fig.\ \ref{fig_strdis} where
a histogram of the square-root of normalized strengths is plotted.
We also find that for both $^{208}$Pb and $^{40}$Ca the
distribution follows the Porter-Thomas
one rather well.
These two figures indicate that for both nuclei
the fluctuation of the strength as well as that of the energy level spacing
is governed by the GOE at least at small energy scale limit as expected.
Figure\ \ref{fig_delta3} shows the $\Delta_3$ statistics.
We again find that at small energy range the $\Delta_3$ follows the GOE
line for both $^{208}$Pb and $^{40}$Ca,
although at intermediate energy scales, $L_{\rm max}\simeq 20$ or 15 for
$^{208}$Pb or $^{40}$Ca, respectively, the $\Delta_3$ starts to deviate from the GOE
line to upward.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=9.2cm]{fig3.eps}
\end{center}
\caption{
The histogram of the square-root of normalized strengths $\bar{S}_i^{1/2}$
associated with IS GQR in (a) $^{40}$Ca and (b) $^{208}$Pb.
The solid curve represents the Porter-Thomas
distribution which becomes a Gaussian when plotted as a function
of $\bar{S}_i^{1/2}$. See the caption of Fig.\ \ref{fig_nns} for
the number of considered levels and also see Sec.\ \ref{sec:cal_lsd}
for the normalization
of the strengths.
}
\label{fig_strdis}
\end{figure}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=9.2cm]{fig4.eps}
\end{center}
\caption{
The $\Delta_3$ statistics for (a) $^{40}$Ca and (b) $^{208}$Pb.
The horizontal axis $L$ shows the value of the energy interval
for the unfolded spectrum.
The solid curve represents the $\Delta_3$ for the GOE level fluctuation.
See Fig.\ \protect\ref{fig_nns} for other parameters.
}
\label{fig_delta3}
\end{figure}
\section{Results of local scaling dimension}
\label{sec:results}
\subsection{Calculation of the local scaling dimension}
\label{sec:cal_lsd}
Since we are not interested in the global shape of the strength function,
we actually adopt the normalized strength function $\bar{S}(E)$ for the
fluctuation analysis as in the case of $^{40}$Ca \cite{aiba2}. The normalized strength function
$\bar{S}(E)$ is given by
\begin{equation}
{\bar S}(E)=\sum_i{\bar S}_i\delta(E-{\bar E}_i+{\bar E}_0),
\label{eq_norstrfun}
\end{equation}
where the normalized strength $\bar{S}_i $ of the $i$th level is defined by
\begin{equation}
\bar{S}_i \equiv{\cal N}\frac{S_i\tilde{\rho}(E_i)}{
\tilde{S}(E_i)}.
\label{eq_norstr}
\end{equation}
Here, $\tilde{\rho}(E)$ and $\tilde{S}(E)$ denote the level density and
the strength function, respectively, smoothed by the Strutinsky method \cite{ring-schuck}.
${\cal N}$ is a normalization factor to guarantee $\sum_i\bar{S}_i=1$.
We determine the width parameter $\omega$ of the Strutinsky smoothing function as
follows:
We note that the smoothed strength function $\tilde{S}(E)$
should represent the global profile of the original strength function $S(E)$ at large
energy scale, but
at the same time, we would like to choose $\omega$ as large as possible since
we do not want to wash out the fluctuations at smaller energy scales.
Figure\ \ref{fig_fwhm} shows the FWHM of the smoothed strength
function $\tilde{S}(E)/\tilde{\rho}(E)$
as the function of the smoothing width $\omega$.
The linear increase of the FWHM
at large values of $\omega \agt 0.6$ MeV indicates that the value of $\omega$ is
too large, while with smaller values $\omega \alt 0.5$ MeV
the FWHM stays at an approximately
constant value, reflecting the total width.
We therefore adopt 0.5 MeV as the value of
the smoothing width $\omega$ in order to satisfy
the above requirements.
We use the equidistant energy level ${\bar E}_i$ in Eq.\ (\ref{eq_norstrfun}),
namely,
${\bar E}_i=id$, where $d$ denotes the average level spacing.
Finally, we adopted the energy range from 9.9 MeV to 13.1 MeV, where
3321 levels are included.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=6cm]{fig5.eps}
\end{center}
\caption{
FWHM of the smoothed strength function $\tilde{S}(E)/\tilde{\rho}(E)$ of IS GQR in $^{208}$Pb
as a function of
smoothing width $\omega$ used in the Strutinsky method.
The dotted line is fitted to data and gives $\sim 0.98\omega+0.2$.
}
\label{fig_fwhm}
\end{figure}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=6cm]{fig6.eps}
\end{center}
\caption{
Normalized strength function Eq.\ (\ref{eq_norstrfun}) of IS GQR in$^{208}$Pb.
Smoothing width $\omega=0.5$ MeV was used.
}
\label{fig_norstrfun}
\end{figure}
The normalized strength function is plotted in Fig.\ \ref{fig_norstrfun}.
The local scaling dimension is derived from this normalized strength function.
\subsection{Behavior of the local scaling dimension}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=9.2cm]{fig7.eps}
\end{center}
\caption{
Partition function (a) and local scaling dimension (b) for the
IS GQR in $^{208}$Pb,
and those in $^{40}$Ca are also shown at (c) and (d).
Curves in each figure correspond to $m=2$ - 5 from upper to lower.
Dotted curves in (b) and (d) represent $D_2(\epsilon)$ for the GOE.
}
\label{fig_pat_ldim}
\end{figure}
Figure\ \ref{fig_pat_ldim} (a) and (b) represent the partition function
and the local scaling dimension, respectively, of IS GQR in $^{208}$Pb.
The horizontal axes in both figures represent the bin width $\epsilon$
of energy in unit of $d$, where $d$ represents the average level
spacing over the energy range 9.9 - 13.1 MeV ($d=0.96$ keV).
The partition function clearly deviates from the linear
relation in the log-log plot.
This means that for the GQR strength function the self-similar property
does not hold.
We can also see a more detailed structure in the figure of the local scaling
dimension.
At the smallest energy scale $\epsilon\simeq d$, the value of the local scaling
dimension is small, $D_2\simeq 0.35$, which means that the fluctuation
is very large at small energy scales. As the energy scale or the bin width increases,
the values of $D_m(\epsilon)$ monotonically increase.
Finally, at about $\epsilon\simeq 100d$ the values of $D_m(\epsilon)$
converges to unity, which indicates that at large energy scales, the strength
function appears smooth.
The most important feature in Fig.\ \ref{fig_pat_ldim} (b)
is that the local scaling dimension for $^{208}$Pb almost follows
the GOE line
at almost all the energy scales.
This should be contrasted with the case of $^{40}$Ca \cite{aiba2}:
The partition function and the local scaling dimension for $^{40}$Ca are
shown in Fig.\ \ref{fig_pat_ldim} (c) and (d), respectively,
for a comparison.
When the energy scale is small, the local scaling dimension almost follows
the GOE line.
As the energy scale increases, however, we can find a dip and a deviation from the
GOE line at about 1.7 MeV (Note that $d=12$ keV for $^{40}$Ca).
We verified that an occurrence of the dip is not due to a statistical
error. Moreover, further studies indicate that the energy where the minimum is
located is approximately related to the value of the spreading width of 1p1h states.
Note that if we look only at the small energy scale limit or large energy scale limit,
we can not find the difference between $^{208}$Pb and $^{40}$Ca. Studies of fluctuation at
intermediate energy scales lead to the finding of the difference.
In the following we shall investigate the mechanism which
brings about the difference in fluctuations at intermediate energy scales.
\section{Studies of damping process}
\label{sec:damping}
Let us now investigate origins of
the difference between the cases of $^{40}$Ca and
$^{208}$Pb.
In our previous study of the GQR in $^{40}$Ca, we have shown that the
behavior of the local scaling dimension, shown in Fig.\ \ref{fig_pat_ldim} (d),
can be
interpreted in terms of the doorway damping mechanism. We here
employ the same picture in order to clarify the damping mechanism
of the GQR in $^{208}$Pb.
The doorway damping mechanism consists of a two-step process which
is illustrated in Fig.\ \ref{fig_pic_Ca}.
The giant resonance is spread over the 1p1h states due to the Landau damping, the width
of which is denoted by $\Gamma_{\rm L}$.
The average spacing of 1p1h states is denoted by $D_{\rm 1p1h}$.
The 1p1h states are considered here as the ``doorway" states
of the damping process.
The 1p1h states then couple to more complicated
background states (2p2h states) through the residual two-body interaction.
The coupling causes the spreading width of 1p1h states, which we denote
$\gamma_{12}$.
We define the GQR TD state as the Tamm-Dancoff (TD) state with the largest
quadrupole strength among all TD states, where the TD states mean the states
obtained in the TDA, i.e., by the diagonalization within the model space limited
to the 1p1h configurations.
The GQR TD state also couples to 2p2h states,
and hence it should have the spreading width due to the coupling.
This is similar to $\gamma_{12}$, but we introduce a separate
symbol $\Gamma_2$ since the
GQR TD state is a special state consisting of a coherent superposition of
many unperturbed 1p1h excitations.
$d_{\rm 2p2h}$ is the
average spacing of background 2p2h states.
The residual interaction also acts
among the 2p2h states, and the mixing among the 2p2h states causes
a spreading width of the 2p2h states, which we denote $\gamma_{22}$.
In the following we shall evaluate all these quantities in order to
clarify the damping mechanism of the GQR in $^{208}$Pb
(Sec.\ \ref{sec:mechanism} and Sec.\ \ref{sec:spreading_width} ).
We also study whether there are specific states
among 2p2h states which strongly couple with the GQR mode (Sec.\ \ref{sec:surfacce_vib})
and then discuss
the difference of the nature associated with
the fluctuation of strength function between $^{40}$Ca and $^{208}$Pb
(Sec\ \ref{sec:comparison}).
\begin{figure}[tb]
\begin{center}
\includegraphics[width=7cm]{fig8.eps}
\end{center}
\caption{
Schematic drawing of the doorway damping mechanism of the giant resonance,
and related quantities.
}
\label{fig_pic_Ca}
\end{figure}
\subsection{Mechanism producing the total width}
\label{sec:mechanism}
\subsubsection{Landau damping}
For $^{40}$Ca, the Landau damping is important, so that the strengths are
already fragmented in the 1p1h levels.
Therefore we first would like to investigate in $^{208}$Pb,
how the strength is distributed in the TDA where only the 1p1h states are included.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=6cm]{fig9.eps}
\end{center}
\caption{
TDA strength function for the IS quadrupole operator in $^{208}$Pb.
See Fig.\ \ref{fig_strfun} for the dotted curve.
}
\label{fig_tdastr}
\end{figure}
Figure\ \ref{fig_tdastr} shows the TDA strength function,
which is obtained by means of the TDA, namely by neglecting 2p2h
states, of the IS quadrupole operator.
Different from the case of $^{40}$Ca, strengths in the GQR region
is considerably concentrated on the single peak located at about 10.7 MeV.
Because of this,
the TDA strength function is very different from the
full strength function in Fig.\ \ref{fig_strfun}.
At the same time, we also see only a small effect of the Landau damping.
In fact, the strength concentration on the single peak at $E=10.7$ MeV is 59\%
of the strengths
in the energy interval 9 - 13 MeV.
The Landau damping width $\Gamma_{\rm L}$ may be evaluated in terms of
a smoothed profile of the strength function plotted with the dotted curve
in Fig.\ \ref{fig_tdastr}.
Its FWHM reads 0.21 MeV.
On the other hand, if we closely look at Fig.\ \ref{fig_tdastr}, we find that there is
the second largest peak just below the largest one and that these two levels
dominate the whole structure.
The level spacing between these two levels can be considered as
a typical spreading of strength and may be a more direct
quantitative measure of the Landau damping
width $\Gamma_{\rm L}$: The level spacing 0.18 MeV gives $\Gamma_{\rm L}=0.18$ MeV.
\subsubsection{damping due to 2p2h states}
The Landau damping width $\Gamma_{\rm L}=0.18$ MeV is not enough to explain
the total width $\Gamma=0.63$ MeV of Sec.\ \ref{sec:numerical}.
Then, we would like to study a role of 2p2h states in the damping process,
namely, the fragmentation of the GQR TD state located at $E=10.7$ MeV
in Fig.\ \ref{fig_tdastr} over 2p2h states.
We shall investigate the damping width $\Gamma_2$ caused by the coupling to
2p2h states.
To estimate this width, we perform a calculation where we include only the GQR TD state
and 2p2h states, where
the coupling between the GQR TD state and 2p2h states
as well as the interaction among 2p2h states are taken into account.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=6cm]{fig10.eps}
\end{center}
\caption{
Strength function by neglecting all TD states except the GQR TD state.
3342 2p2h states lying in 9 MeV - 13 MeV are
considered.
See Fig.\ \ref{fig_strfun} for the dotted curve.
}
\label{fig_GQRTDstr}
\end{figure}
Figure\ \ref{fig_GQRTDstr} shows the resulting strength function.
The estimated FWHM is 0.41 MeV, i.e., $\Gamma_2=0.41$ MeV.
If the Landau damping and the 2p2h damping are independent of each other,
and neighboring TD states around the GQR TD states also have the
same spreading width as $\Gamma_2$,
the following approximate relation holds:
\begin{equation}
\Gamma\simeq\Gamma_{\rm L}+
\Gamma_2.
\label{Gamma}
\end{equation}
The values, $\Gamma_{\rm L}=0.18$ MeV and $\Gamma_2=0.41$ MeV, estimated
above indeed satisfy this relation. Consequently,
the total width $\Gamma=0.63$ MeV is approximately explained as a sum of
the Landau damping width $\Gamma_{\rm L} $ and the 2p2h damping
width $\Gamma_2$.
The importance of the 2p2h damping is contrasted with the case of $^{40}$Ca,
where the total width can be explained essentially by the Landau damping width,
i.e., $\Gamma\simeq \Gamma_\text{L}$.
\subsection{Spreading width of 1p1h states and 2p2h states}
\label{sec:spreading_width}
For the case of $^{40}$Ca, the strength is fragmented over many 1p1h states by the Landau
damping, and strength in each 1p1h state is further spread due to the coupling with 2p2h
states.
Let us evaluate the spreading width $\gamma_{12}$ of the 1p1h states due to this coupling.
We shall also evaluate the spreading width $\gamma_{22}$ of 2p2h states, which
is caused by the residual coupling among 2p2h states.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=9.2cm]{fig11.eps}
\end{center}
\caption{
Averaged strength function of (a) TD states and (b) 2p2h states.
Average was performed over levels lying in 9 MeV - 13 MeV.
The number of levels is 12 and 3342 for TD states and 2p2h states,
respectively.
}
\label{fig_avedoorstr}
\end{figure}
We evaluate $\gamma_{12}$ by using the strength
functions of TD states as in Ref.\ \cite{aiba2}.
Namely, we calculate the strength function of each
TD state.
Averaging the strength functions over whole TD states,
we obtain Fig.\ \ref{fig_avedoorstr} (a).
The FWHM of this averaged strength function gives an evaluation of the spreading width
$\gamma_{12}$.
We read $\gamma_{12 }=0.38$ MeV.
(Note that we define $\gamma_{12 }$ as the spreading width of TD states instead
of that of unperturbed 1p1h states.)
The value of spreading width of 2p2h states $\gamma_{22}$ is also evaluated in the
same manner.
From Fig.\ \ref{fig_avedoorstr} (b) we also obtain $\gamma_{22}=0.75$ MeV as the estimate
of the spreading width of 2p2h states.
These results will be used in Sec.\ \ref{sec:comparison}
For the sake of comparison, let us estimate the spreading width by assuming the Fermi golden rule.
The root mean square of matrix elements between 1p1h states and 2p2h states is
calculated as
$(\overline{ \langle {\rm 1p1h} |V_{\rm 12}|{\rm 2p2h}\rangle^2})^{1/2}=9.3\times10^{-3}$ MeV.
Similarly, we calculate
$(\overline{ \langle {\rm 2p2h} |V_{\rm 22}|{\rm 2p'2h'}\rangle^2})^{1/2}=1.0\times10^{-2}$ MeV.
Since the level spacing of 2p2h states is $d_{\rm 2p2h}=1.2$ keV, the spreading
widths $\gamma_{12}$ and $\gamma_{22}$ are approximately estimated in the Fermi golden rule as
$\gamma^{\rm FG}_{12}=2\pi \overline{ \langle {\rm 1p1h} |V_{\rm 12}|{\rm 2p2h}\rangle^2}/d_{\rm 2p2h}=0.46$
MeV and
$\gamma^{\rm FG}_{22}=2\pi \overline{ \langle {\rm 2p2h} |V_{\rm 22}|{\rm 2p'2h'}\rangle^2}/d_{\rm 2p2h}=0.53$
MeV, respectively, which are
in approximate agreement with the direct evaluation within 30\%.
\subsection{Search for strongly coupled states in 2p2h states}
\label{sec:surfacce_vib}
In the picture of Fig.\ \ref{fig_pic_Ca} 2p2h states are assumed to play a role as
the chaotic background and provide the GOE
fluctuation to the strength function.
However, if the GQR TD state couples with not all 2p2h states equally
but specific states in 2p2h states strongly, there is a possibility
for this hierarchical structure in 2p2h states to give rise to a deviation from
the GOE fluctuation.
We, here, would like to investigate whether
whole 2p2h states are rather equally coupled with the GQR TD state or whether
there are specific states in 2p2h states
which strongly couple with that state.
As a candidate of such specific states, we can consider
the low-energy surface vibration plus
1p1h states:
In Refs. \cite{bertsch2,bertsch,broglia,bortignon,lacroix2}, the importance of the coupling
to the surface vibration in the wide range of damping phenomena
including the damping of a single particle motion as well as that
of giant resonances
was discussed.
As for the giant resonance, which is composed of a
coherent superposition of 1p1h states, this means that
the damping occurs via the
coupling with the specific 2p2h states, namely,
the surface vibration plus 1p1h (s.v.+1p1h) states.
Since our model does not assume the particle-vibration coupling a priori,
it is not trivial whether our model also has a mechanism that enhances the coupling
with the low-energy surface vibration.
Therefore, we would like to study whether the s.v.+1p1h states are particularly strongly
coupled with the GQR TD state within our model.
To do so, we calculate the FWHM of the following approximate strength function:
\begin{equation}
S(E)=-\frac{1}{\pi}{\rm Im}\left(
E-E_c-\sum_\alpha \frac{V_{c\alpha}^2}{E-\omega_\alpha+i\gamma_{22}/2}\right)^{-1},
\label{doorwaystr}
\end{equation}
where, $E_c$ and $\omega_\alpha$ denote the energy of the GQR TD state and the energy
of the $\alpha$th s.v.+1p1h state, respectively. $V_{c\alpha}$ represents the coupling
matrix element between the GQR TD state and the s.v.+1p1h state $\alpha$.
Only $J^\pi=2^+$, $3^-$ modes are included as surface vibrations:
We took only the lowest TD state as $J^\pi=2^+$ surface vibrational mode.
On the other hand, we must pay attention to the collectivity of the octupole mode.
Figure\ \ref{fig_octstr} shows the TDA strength function for the IS octupole operator.
Compared with the experimental data \cite{spear}, the energy of the lowest state is too high,
and strengths are fragmented over several states.
Thus, we took into account the lowest nine states for the octupole mode.
Note that s.v.+1p1h states thus defined are not orthogonal.
In this sense Eq.\ (\ref{doorwaystr}) is an approximation which neglects
the non-orthogonality.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=6cm]{fig12.eps}
\end{center}
\caption{
TDA strength function for the IS octupole operator in $^{208}$Pb.
}
\label{fig_octstr}
\end{figure}
The strength function based on Eq.\ (\ref{doorwaystr}) is presented in
Fig.\ \ref{fig_doorwaystr}.
The width $\Gamma_2^{\rm (s.v.)}$ estimated by the FWHM is 0.074 MeV.
This value is significantly smaller than the width $\Gamma_2=0.41$ MeV
of the GQR TD state caused by the coupling to the whole 2p2h states.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=6cm]{fig13.eps}
\end{center}
\caption{
Strength function of the GQR TD state evaluated by considering only surface
vibration plus 1p1h (s.v.+1p1h) states based on Eq.\ (\ref{doorwaystr}).
$\gamma_{22}=0.75$ MeV is used.
}
\label{fig_doorwaystr}
\end{figure}
From the estimate by the Fermi golden rule, we can give more detailed comparison
between the width for the case of s.v.+1p1h states and that for the whole 2p2h states.
It is noted in Table\ \ref{table1} that the spreading width $\Gamma_2^{\rm (s.v.)}=0.074$
MeV and $\Gamma_2=0.41$ MeV are well accounted for by the estimate.
In the Fermi golden rule the spreading width is governed by two factors;
1) the average value of squared coupling matrix elements $\overline{ V_{c\alpha}^2}$
between the GQR TD state and the states that couple to it, and 2) the
level density of the coupling states.
From Table\ \ref{table1}, we see that the large difference between the two
widths simply reflects the difference between the number of s.v.+1p1h states 909
and 2p2h states 3142 whereas the coupling strength of s.v.+1p1h states
$\overline{ V_{c\alpha}^2}=0.65\times10^{-4}$ MeV$^2$ is comparable to
the coupling strength
$\overline{V_{c\alpha}^2}=0.72\times10^{-4}$ MeV$^2$ for the whole 2p2h states.
Table\ \ref{table1} and Fig.\ \ref{fig_doorwaystr} suggest that our model does not
contain the enhancement of the coupling with the surface vibrations in the damping
of the GQR.
Therefore we consider in the following the 2p2h states as background states
which do not have specific structures.
\begin{table}[t]
\caption{
Averaged value of squared coupling matrix elements
$\overline{V_{c\alpha}^2}$
between the GQR TD state and surface vibration plus 1p1h
states or the whole 2p2h states(third column), the associated spreading
width $\Gamma_2^{\rm FG}$ of the GQR TD state
evaluated by the Fermi golden rule (fourth column), and
the spreading width
$\Gamma_2$ estimated by
the FWHM of the strength function based on Eq.\ (\ref{doorwaystr}) (fifth column).
Second column shows the number of states
considered.
The second row shows the results obtained by including only
the s.v.+1p1h states while the third row shows those
for the case of the whole 2p2h states.
}
\label{table1}
\begin{ruledtabular}
\begin{tabular}{lcccr}
&\#&$\overline{V_{c\alpha}^2}$ (MeV$^2$)&$\Gamma_2^{\rm FG}$ (MeV)&$\Gamma_2$ (MeV)\\
\hline
s.v.+1p1h&909&$0.65\times 10^{-4}$&0.092&0.074\\
2p2h&3342&$0.72\times 10^{-4}$&0.38&0.41\\
\end{tabular}
\end{ruledtabular}
\end{table}
\subsection{Physical origin of the difference between $^{40}$Ca and $^{208}$Pb}
\label{sec:comparison}
In the above subsections, we have
evaluated the physical quantities such as
the various spreading widths, with which we have discussed
the damping process of $^{40}$Ca and $^{208}$Pb,
especially the mechanism of producing the total width of the strength function.
Here, using these quantities we would like to discuss the
physical origin of the difference between the fluctuation of the strength fluctuation of $^{40}$Ca
and that of $^{208}$Pb.
Table\ \ref{table2} summarizes the values of the above physical quantities related to the initial
stage of the damping process for both $^{40}$Ca and $^{208}$Pb.
We have shown in our previous study \cite{aiba} that the damping process through
the doorway states causes large fluctuations which have characteristic
energy scales, and that the fluctuations emerge in
the local scaling dimension. For instance, the energy scale of the
spreading width $\gamma_{12}$ of the doorway states is the quantity
which shows up prior to the other quantities. It is noted, however,
the size of the fluctuations depends on the mutual relations among the
quantities mentioned above, and indeed we have examined in \cite{aiba}
the relations
which are needed to detect the effect of the spreading width $\gamma_{12}$.
\begin{table}[tb]
\caption{
Values of physical quantities related to the damping of the GQR for
$^{40}$Ca and $^{208}$Pb.
Unit of the energy is keV for all cases.
}
\label{table2}
\begin{ruledtabular}
\begin{tabular}{lccccccr}
&$\Gamma$&$\Gamma_{\rm L}$&$\Gamma_2$&$\gamma_{12}$&$D_{\rm 1p1h}$&$\gamma_{22}$&$d_{\rm 2p2h}$\\
\hline
$^{40}$Ca&4000&4000&1500&1500&500&5200&11\\
$^{208}$Pb&630&180&410&380&230&750&1.2\\
\end{tabular}
\end{ruledtabular}
\end{table}
It is trivial that the local scaling dimension can detect the spreading width when the
spreading of 1p1h states does not cause the overlap of these states, namely when
$\gamma_{12} < D_{\rm 1p1h}$.
In addition to this case, the local scaling dimension still keeps the information of the
spreading width even if the 1p1h states start to overlap with each other,
i.e. $\gamma_{12} \simeq D_{\rm 1p1h}$.
Studying more quantitatively with the use of the doorway damping model of Ref.\ \cite{aiba},
we found the condition to detect the
effect of the spreading width as
\begin{enumerate}
\item[(A)] $\gamma_{12} \le 4D_{\rm 1p1h}$.
\end{enumerate}
Furthermore, we need the second condition:
\begin{enumerate}
\item[(B)] $\gamma_{12} < \Gamma_{\rm L}$.
\end{enumerate}
This simply means that the spreading width $\gamma_{12}$ of the doorway states
(1p1h states) need to be smaller than the total width $\Gamma$.
Since $\Gamma\simeq\Gamma_\text{L}+\Gamma_2$ and $\gamma_{12}\simeq\Gamma_2$,
the requirement $\gamma_{12} < \Gamma$ can be written as (B).
In addition to (A) and (B), we need the third condition:
\begin{enumerate}
\item[(C)] $D_{\rm 1p1h} < \Gamma_{\rm L}$.
\end{enumerate}
This is because we need more than one doorway states within the
the energy interval $\Gamma_{\rm L}$ in order to have fluctuating behavior
in the strength function.
Let us first look at the case of $^{40}$Ca.
From Table\ \ref{table2},
the relation $\gamma_{12} =3.0 D_{\rm 1p1h}$ is derived, and this relation
fulfills the condition (A).
On the other hand, relations $\gamma_{12} =0.38 \Gamma_{\rm L}$
and $D_{\rm 1p1h} =0.13 \Gamma_{\rm L}$
are also derived from Table \ \ref{table2}, and
these relations satisfy both conditions (B) and (C).
As a result, in the case of $^{40}$Ca, we can see a deviation from the GOE fluctuation
in the local scaling dimension and indeed the energy scale where the deviation
is seen is related to the value of $\gamma_{12}$.
For $^{208}$Pb, on the other hand, we find in Table\ \ref{table2} that
$\gamma_{12}=1.7D_{\rm 1p1h}$, while
$\Gamma_L$ is smaller than
$\gamma_{12}$ and $D_{\rm 1p1h}$, i.e.,
$\gamma_{12} =2.1 \Gamma_{\rm L}$ and $D_{\rm 1p1h} =1.3 \Gamma_{\rm L}$.
The first relation satisfies the condition (A).
The latter two relations, however, break the condition (B) and (C).
Accordingly, for the case of $^{208}$Pb, the deviation from the GOE due to the effect of
$\gamma_{12}$ can not be seen.
The situation in $^{208}$Pb is illustrated in
Fig.\ref{fig_pic_Pb}.
Essential physical origin of this difference is that for $^{208}$Pb the Landau damping
width is small compared with that of $^{40}$Ca.
The smallness or largeness of the value of the Landau damping width affects the fluctuation
property of the strength function.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=8cm]{fig14.eps}
\end{center}
\vspace{-0.5cm}
\caption{
Schematic picture of the initial stage of the damping process for
GQR in $^{208}$Pb.
}
\label{fig_pic_Pb}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
We studied the fluctuation properties of the strength function of
IS GQR for $^{208}$Pb by means of the local scaling dimension,
and compared the results with those of $^{40}$Ca. The strength function
was obtained by the second TDA including 2p2h states as well as 1p1h states.
For $^{40}$Ca, we find a fluctuation different from GOE around the energy scale which is
approximately related to the spreading width of the 1p1h states.
On the other hand, for $^{208}$Pb we can not find the fluctuation different from the GOE at
almost all the energy scales.
The different behavior of the fluctuation detected by the local scaling dimension analysis
is due to the difference of
the ratio of the Landau damping width $\Gamma_{\rm L}$ to the spreading width of the
1p1h states $\gamma_{12}$.
Recently, the analysis of the strength function of the IS GQR in $^{208}$Pb obtained
by (p,p') inelastic scattering experiment was performed by means of the wavelet
transform\ \cite{shevchenko}.
The authors suggest from the positions of the local maxima in the wavelet power
that there exist three energy scales in the fluctuation of the strength function:
I. 120 keV, II. 440, 850 keV, III. 1500 keV.
Existence of higher two energy scales is not inconsistent with our results,
since our analysis says nothing about the fluctuation at about energy scale II, which
may correspond to the total width $\Gamma$ in our model, or higher energy scales.
However, the existence of the smallest energy scale $\sim 120$ keV may conflict
with our results: If there is such an energy scale in our strength function, our analysis
must detect it as a deviation from the GOE fluctuation.
Therefore, it is very important to study the origin of this discrepancy.
In particular, it is interesting to clarify the relation between two method, namely,
the local scaling dimension and the wavelet power.
Studies in this direction are now in progress.
\begin{acknowledgments}
The authors acknowledge helpful discussion with
K. Matsuyanagi. We are also indebted to A. Richter, and P.
von Neumann-Cosel for many fruitful discussion.
The numerical calculations were performed at the
Yukawa Institute Computer Facility as well as at the RCNP Computer
Facility.
\end{acknowledgments}
|
3,212,635,537,471 | arxiv | \section{Introduction}
The main idea of this paper comes from the observation that the double ramification (DR) cycle, i.e. the class in the cohomology of the moduli space of stable curves representing the most natural compactification of the locus of smooth curves whose marked points support a principal divisor \cite{Hai13}, can be seen as a partial cohomological field theory (CohFT) \cite{LRZ15,KM94} with an infinite dimensional phase space.\\
In \cite{Bur15,BR16a,BDGR18} it was shown how to associate to any partial CohFT an integrable hierarchy of Hamiltonian systems of evolutionary PDEs in one space and one time dimensions. The number of dependent variables in these system of PDEs equals the dimension of the phase space of the partial CohFT. This integrable system is called the DR hierarchy and its properties and generalizations (including a quantization which exists in the case of actual CohFTs) where studied in \cite{BR16b,BDGR16,BGR17,BR18}.\\
This paper wants to answer the question: ``what is the DR hierarchy associated to the infinite rank partial CohFT given by the DR cycle?''. As the general construction of the DR hierarchy already involves intersection numbers of a partial CohFT with the top Hodge class and a DR cycle, choosing as CohFT a second DR cycle leads us directly to having to compute intersection numbers of two different DR cycles and the top Hodge class. This is, of course, a question of its own geometric interest and, as we show in Chapter~\ref{section:quadratic} of this paper, it has a very explicit answer.\\
In fact there is a natural deformation of the DR cycle which gives a one-parameter family of partial CohFTs. It comes from Hain's formula \cite{Hai13}, expressing the DR cycle (restricted to the moduli space of stable curves of compact type) as the $g$-th power of (the pullback to the moduli space of stable curves of compact type) of the class of the theta divisor on the universal Jacobian. If we consider instead the exponential of such theta class, putting into play all of its powers, we get a more general but still very explicit infinite rank partial CohFT.\\
After showing in Chapter \ref{section:DR} how to trade the resulting infinite rank DR hierarchy in one space and one time dimensions as a rank $1$ hierarchy in two space and one time dimensions, we set out to compute it explicitly. The main result of this paper is in Chapter \ref{section:ncKdV}, where we show that the DR hierarchy of our infinite rank partial CohFT coincides with the noncommutative KdV hierarchy, the natural generalization of the ordinary KdV hierarchy on a circle to a torus with a noncommutative Moyal structure.\\
One of the applications is that, through our mixed intersection theory and integrable systems techniques, we are able to compute the intersection numbers of the top Hodge class, one DR cycle, any power of the theta class and any power of the psi class at one marked point. Note also that noncommutative integrable systems have never appeared before in the study of the intersection theory on the moduli spaces of curves. Therefore, we expect that our result will provide a new tool for understanding the structure of the cohomology ring of the moduli spaces and, in particular, the structure of the DR cycle, which became the object of an intensive research in recent years (see e.g.~\cite{JPPZ17,HPS19,Pix18}).\\
\noindent{\bf Acknowledgements}. We would like to thank Johannes Schmitt for providing us with an early version of the SageMath package \texttt{admcycles}, presented in \cite{DSZ20}. The program was used for preliminary computational experiments in the early phases of this paper. The work is supported by the Mathematical Center in Akademgorodok, the agreement with the Ministry of Science and Higher Education of the Russian Federation no.~075-15-2019-1675. This project has also received funding from the European Union's Horizon~2020 research and innovation programme under the Marie Sk\l odowska-Curie grant agreement no.~797635.
\section{Quadratic double ramification integrals}\label{section:quadratic}
For a pair of nonnegative integers $(g,n)$ in the stable range, i.e. satisfying $2g+2-n>0$, let ${\overline{\mathcal{M}}}_{g,n}$ be the moduli space of stable curves with genus $g$ and $n$ marked points labeled by the set $[n]:=\{1,\ldots,n\}$. For integers $a_1,\ldots,a_n$, such that $\sum a_i=0$, the double ramification cycle $\mathrm{DR}_g(a_1,\ldots,a_n) \in H^*({\overline{\mathcal{M}}}_{g,n},\mathbb C)$ is the Poincar\'e dual to the pushforward to ${\overline{\mathcal{M}}}_{g,n}$ of the virtual fundamental class of the moduli space of rubber stable maps from curves of genus $g$ with $n$ marked points to $\mathbb P^1$ relative to $0$ and $\infty$ with ramification profile given by $(a_1,\dots,a_n)$. Here ``rubber'' means that we consider maps up to the $\mathbb C^*$-action in the target $\mathbb P^1$ and a~positive (negative) coefficient~$a_i$ indicates a pole (zero) at the $i$-th marked point of order $a_i$ ($-a_i$), while $a_i=0$ just indicates that the $i$-th marked point is unconstrained. For future convenience we will also define the class $\mathrm{DR}_g(a_1,\ldots,a_n)$ to vanish in case $\sum a_i\neq0$.\\
Let us introduce the class $\Theta(a_1,\ldots,a_n)\in H^2({\overline{\mathcal{M}}}_{g,n},\mathbb C)$ defined by
\begin{gather}\label{eq:Theta-class}
\Theta(a_1,\ldots,a_n):=\sum_{j=1}^n\frac{a_j^2\psi_j}{2}-\frac{1}{4}\sum_{h=0}^g\sum_{J\subset [n]}a_J^2\delta_h^J,
\end{gather}
if $\sum a_i=0$ and zero otherwise, where $\psi_i$, $1,\leq i\leq n$, is the first Chern class of the $i$-th tautological line bundle, and, for $J\subset [n]$ and $0\leq h\leq g$ in the stable range $2h-1+|J|>0$ and $2(g-h)-1+(n-|J|)>0$, $a_J:=\sum_{j\in J}a_j$ and $\delta^J_h \in H^2({\overline{\mathcal{M}}}_{g,n},\mathbb C)$ is the class of the irreducible boundary divisor of ${\overline{\mathcal{M}}}_{g,n}$ formed by stable curves with a separating node at which two stable components meet, one of genus $h$ and marked points labeled by $|J|$ and the other of genus~$g-h$ and marked points labeled by the complement $J^c$ (naturally, $\delta^J_h=0$ if at least one of the stability conditions $2h-1+|J|>0$ and $2(g-h)-1+(n-|J|)>0$ is not satisfied).\\
By a result of Hain \cite{Hai13}, we know that
\begin{gather}\label{eq:Hain}
\left.\mathrm{DR}_g(a_1,\ldots,a_n)\right|_{{\mathcal{M}}_{g,n}^\mathrm{ct}}=\frac{1}{g!}\Theta(a_1,\ldots,a_n)^g|_{{\mathcal{M}}_{g,n}^\mathrm{ct}}.
\end{gather}
where ${\mathcal{M}}_{g,n}^\mathrm{ct}\in {\overline{\mathcal{M}}}_{g,n}$ is the locus of stable curves with no non-separating nodes. Moreover, always by \cite{Hai13}, the class $\Theta(a_1,\ldots,a_n)|_{{\mathcal{M}}_{g,n}^\mathrm{ct}}$ represents the pullback to ${\mathcal{M}}_{g,n}^\mathrm{ct}$ of the theta divisor on the universal Jacobian over ${\mathcal{M}}_{g,n}^\mathrm{ct}$, which implies the following relation in~$H^*({\mathcal{M}}_{g,n}^\mathrm{ct},\mathbb C)$,
\begin{gather}\label{eq:Theta-relation}
\left.\Theta(a_1,\ldots,a_n)^{g+1}\right|_{{\mathcal{M}}^\mathrm{ct}_{g,n}}=0.
\end{gather}
Let $\lambda_i\in H^{2i}({\overline{\mathcal{M}}}_{g,n},\mathbb C)$, $1\leq i \leq g$, be the $i$-th Chern class of the Hodge bundle on ${\overline{\mathcal{M}}}_{g,n}$. We have \cite{FP00}
\begin{gather}\label{eq:lambdag-relation}
\lambda_g|_{{\overline{\mathcal{M}}}_{g,n}\setminus {\mathcal{M}}_{g,n}^\mathrm{ct}} = 0.
\end{gather}
The classes $\lambda_i$, $\mathrm{DR}_g(a_1,\ldots,a_n)$ and $\Theta(a_1,\ldots,a_n)$ are algebraic, i.e. they belong in fact to the Chow ring $A^*({\overline{\mathcal{M}}}_{g,n})$ . By the localization exact sequence, see e.g. \cite[Section~1.8]{Ful98}, for all~$k$,
$$A_k({\overline{\mathcal{M}}}_{g,n} \setminus {\mathcal{M}}_{g,n}^\mathrm{ct}) \xrightarrow{i_*} A_k({\overline{\mathcal{M}}}_{g,n}) \xrightarrow{j^*} A_k ({\mathcal{M}}_{g,n}^\mathrm{ct}) \to 0,$$
where $i$ and $j$ are the inclusion maps of ${\overline{\mathcal{M}}}_{g,n} \setminus {\mathcal{M}}_{g,n}^\mathrm{ct}$ and ${\mathcal{M}}_{g,n}^\mathrm{ct}$ into ${\overline{\mathcal{M}}}_{g,n}$, and by equation (\ref{eq:lambdag-relation}), we deduce that, if $\alpha \in A_k({\overline{\mathcal{M}}}_{g,n})$ is such that $\alpha|_{{\mathcal{M}}_{g,n}^\mathrm{ct}}:= j^*\alpha=0$, then $\lambda_g \cdot \alpha = 0 \in A^*({\overline{\mathcal{M}}}_{g,n})$. This allows to deduce from identities (\ref{eq:Hain}) and (\ref{eq:Theta-relation}) the following relations in $H^*({\overline{\mathcal{M}}}_{g,n},\mathbb C)$:
\begin{align}
&\lambda_g \mathrm{DR}_g(a_1,\ldots,a_n)=\frac{1}{g!}\lambda_g \Theta(a_1,\ldots,a_n)^g, \label{eq:lambdag-Hain} \\
&\lambda_g \Theta(a_1,\ldots,a_n)^{g+1}=0. \label{eq:lambdag-Theta-relation}
\end{align}\\
For two pairs of nonnegative integers $(g_1,n_1+1)$ and $(g_2,n_2+1)$ in the stable range, let $\mathrm{gl}\colon{\overline{\mathcal{M}}}_{g_1,n_1+1}\times {\overline{\mathcal{M}}}_{g_2,n_2+1} \to {\overline{\mathcal{M}}}_{g,n}$ be the map that glues two stable curves of genus $g_1$ and $g_2$ with marked points labeled by the sets $I=\{i_1,\ldots,i_{n_1},{n+1}\}$ and $J=\{j_1,\ldots,j_{n_2},n+2\}$, respectively, with $I\sqcup J=[n+2]$, at their last marked points to form stable curve with a separating node with genus $g=g_1+g_2$, and marked points labeled by $[n]$. It is easy to see from the definitions that we have
\begin{gather}\label{eq:gluing-Theta}
\mathrm{gl}^*\Theta(a_1,\ldots,a_n) = \Theta\left(a_{i_1},\ldots,a_{i_{n_1}},-k\right) + \Theta\left(a_{j_1},\ldots,a_{j_{n_2}},k\right),
\end{gather}
where $k=\sum_{k=1}^{n_1}a_{i_k}=-\sum_{k=1}^{n_2}a_{j_k}$ and the classes on the right-hand side are pulled back from each of the two factors in the product ${\overline{\mathcal{M}}}_{g_1,n_1+1}\times {\overline{\mathcal{M}}}_{g_2,n_2+1} $. By~\cite{BSSZ15}, we also have
\begin{gather*}
\mathrm{gl}^*\mathrm{DR}_g(a_1,\ldots,a_n) = \mathrm{DR}_{g_1}\left(a_{i_1},\ldots,a_{i_{n_1}},-k\right) \mathrm{DR}_{g_2}\left(a_{j_1},\ldots,a_{j_{n_2}},k\right).
\end{gather*}
In the following we will denote by $\mathrm{DR}_{g_1}\left(a_{i_1},\ldots,a_{i_{n_1}},-k\right)\boxtimes \mathrm{DR}_{g_2}\left(a_{j_1},\ldots,a_{j_{n_2}},k\right)$ the pushforward $\mathrm{gl}_*\mathrm{gl}^*\mathrm{DR}_g(a_1,\ldots,a_n) \in H^*({\overline{\mathcal{M}}}_{g,n},\mathbb C)$.\\
We have the following result on the intersection number of two double ramification cycles and the top Chern class of the Hodge bundle.
\begin{theorem}\label{eq:quadratic DR integral} Let $a_1,a_2,b_1,b_2 \in \mathbb Z$ and $g\in\mathbb Z_{\geq 0}$, then
\begin{equation}
\int_{{\overline{\mathcal{M}}}_{g,3}}\lambda_g\mathrm{DR}_g(a_1,a_2,-a_1-a_2)\mathrm{DR}_g(b_1,b_2,-b_1-b_2)=\frac{(a_1b_2-a_2b_1)^{2g}}{2^{3g}g!(2g+1)!!}.
\end{equation}
\end{theorem}
\begin{proof}
By equation (\ref{eq:lambdag-Hain}), Theorem \ref{eq:quadratic DR integral} is equivalent to the formula
\begin{gather}\label{eq:DRTheta integral}
\int_{{\overline{\mathcal{M}}}_{g,3}}\lambda_g\Theta(a_1,a_2,-a_1-a_2)^g\mathrm{DR}_g(b_1,b_2,-b_1-b_2)=\frac{(a_1b_2-a_2b_1)^{2g}}{2^{3g}(2g+1)!!},\quad g\ge 0.
\end{gather}
Denote the left-hand side of equation~\eqref{eq:DRTheta integral} by $f_g(\overline{a},\overline{b})$, where $\overline{a}=(a_1,a_2,a_3)$, $\overline{b}=(b_1,b_2,b_3)$ and $a_3=-a_1-a_2$, $b_3=-b_1-b_2$. Clearly, $f_0(\overline{a},\overline{b})=1$, so suppose that $g\ge 1$. In order to compute the integral, let us express one of the classes $\Theta(\overline{a})$ using formula~\eqref{eq:Theta-class}. Using relation~\eqref{eq:Theta-relation} and the formula for the intersection of a psi class with a double ramification cycle~\cite{BSSZ15}, for any $1\le i\le 3$ we compute
\begin{align*}
\int_{{\overline{\mathcal{M}}}_{g,3}}\lambda_g\psi_i\Theta(\overline{a})^{g-1}\mathrm{DR}_g(\overline{b})=&\frac{2g-1}{2g+1}\int_{{\overline{\mathcal{M}}}_{g,3}}\lambda_g\Theta(\overline{a})^{g-1}\mathrm{DR}_1(b_i,-b_i)\boxtimes\mathrm{DR}_{g-1}(b_j,b_k,b_i)+\\
&-\frac{2b_j}{(2g+1)b_i}\int_{{\overline{\mathcal{M}}}_{g,3}}\lambda_g\Theta(\overline{a})^{g-1}\mathrm{DR}_{g-1}(b_i,b_k,b_j)\boxtimes\mathrm{DR}_1(b_j,-b_j)+\\
&-\frac{2b_k}{(2g+1)b_i}\int_{{\overline{\mathcal{M}}}_{g,3}}\lambda_g\Theta(\overline{a})^{g-1}\mathrm{DR}_{g-1}(b_i,b_j,b_k)\boxtimes\mathrm{DR}_1(b_k,-b_k)=\\
=&\left[\frac{2g-1}{2g+1}\frac{b_i^2}{24}+\frac{2b_j}{(2g+1)(b_j+b_k)}\frac{b_j^2}{24}+\frac{2b_k}{(2g+1)(b_j+b_k)}\frac{b_k^2}{24}\right]f_{g-1}(\overline{a},\overline{b})=\\
=&\frac{(2g+1)b_i^2-6b_jb_k}{24(2g+1)}f_{g-1}(\overline{a},\overline{b}),
\end{align*}
where $\{i,j,k\}=\{1,2,3\}$. As a result,
\begin{gather*}
\int_{{\overline{\mathcal{M}}}_{g,3}}\lambda_g\left(\sum_{i=1}^3a_i^2\psi_i\right)\Theta(\overline{a})^{g-1}\mathrm{DR}_g(\overline{b})=f_{g-1}(\overline{a},\overline{b})\sum_{\substack{i;\,j<k\\\{i,j,k\}=\{1,2,3\}}}a_i^2\frac{(2g+1)b_i^2-6b_jb_k}{24(2g+1)}.
\end{gather*}
Next we compute
\begin{align*}
&\int_{{\overline{\mathcal{M}}}_{g,3}}\lambda_g\left(\frac{1}{2}\sum_{h=0}^g\sum_{J\subset [3]}a_J^2\delta_h^J\right)\Theta(\overline{a})^{g-1}\mathrm{DR}_g(\overline{b})=\\
&\hspace{2cm}=\sum_{\substack{i;\,j<k\\\{i,j,k\}=\{1,2,3\}}}a_i^2\int_{{\overline{\mathcal{M}}}_{g,3}}\lambda_g\Theta(\overline{a})^{g-1}\mathrm{DR}_1(b_i,-b_i)\boxtimes\mathrm{DR}_{g-1}(b_j,b_k,b_i)=\\
&\hspace{2cm}=f_{g-1}(\overline{a},\overline{b})\sum_{i=1}^3\frac{a_i^2b_i^2}{24}.
\end{align*}
Summarizing the above computations we get
\begin{align*}
f_g(\overline{a},\overline{b})=&\frac{1}{2}\int_{{\overline{\mathcal{M}}}_{g,3}}\lambda_g\left(\sum_{i=1}^3a_i^2\psi_i-\frac{1}{2}\sum_{h=0}^g\sum_{J\subset [3]}a_J^2\delta_h^J\right)\Theta(\overline{a})^{g-1}\mathrm{DR}_g(\overline{b})=\\
=&\frac{1}{2}f_{g-1}(\overline{a},\overline{b})\left(\sum_{\substack{i;\,j<k\\\{i,j,k\}=\{1,2,3\}}}a_i^2\frac{(2g+1)b_i^2-6b_jb_k}{24(2g+1)}-\sum_{i=1}^3\frac{a_i^2b_i^2}{24}\right)=\\
=&-\frac{f_{g-1}(\overline{a},\overline{b})}{8(2g+1)}\sum_{\substack{i;\,j<k\\\{i,j,k\}=\{1,2,3\}}}a_i^2 b_jb_k=\\
=&\frac{(a_1b_2-a_2b_1)^2}{8(2g+1)}f_{g-1}(\overline{a},\overline{b}),
\end{align*}
that proves formula~\eqref{eq:DRTheta integral} and completes the proof of the theorem.
\end{proof}
\section{An infinite rank partial CohFT and its DR hierarchy}\label{section:DR}
Recall the following definition, which is a generalization first considered in \cite{LRZ15} of the notion of cohomological field theory (CohFT) from \cite{KM94}.
\begin{definition}
For a pair of nonnegative integers $(g,n)$ in the stable range $2g-2+n>0$, a partial CohFT is a system of linear maps $c_{g,n}\colon V^{\otimes n} \to H^\mathrm{even}({\overline{\mathcal{M}}}_{g,n},\mathbb C)$, where $V$ is an arbitrary finite dimensional $\mathbb C$-vector space, called the phase space, together with a special element $e_1\in V$, called the unit, and a symmetric nondegenerate bilinear form $\eta\in (V^*)^{\otimes 2}$, called the metric, such that, chosen any basis $e_1,\ldots,e_{\dim V}$ of V, the following axioms are satisfied:
\begin{itemize}
\item[(i)] the maps $c_{g,n}$ are equivariant with respect to the $S_n$-action permuting the $n$ copies of~$V$ in $V^{\otimes n}$ and the $n$ marked points in ${\overline{\mathcal{M}}}_{g,n}$, respectively.
\item[(ii)] $\pi^* c_{g,n}( \otimes_{i=1}^n e_{\alpha_i}) = c_{g,n+1}(\otimes_{i=1}^n e_{\alpha_i}\otimes e_1)$ for $1 \leq\alpha_1,\ldots,\alpha_n\leq \dim V$, where $\pi\colon{\overline{\mathcal{M}}}_{g,n+1}\to{\overline{\mathcal{M}}}_{g,n}$ is the map that forgets the last marked point.\\
Moreover $c_{0,3}(e_{\alpha}\otimes e_\beta \otimes e_1) =\eta(e_\alpha\otimes e_\beta) =:\eta_{\alpha\beta}$ for $1\leq \alpha,\beta\leq \dim V$.
\item[(iii)] $\mathrm{gl}^* c_{g_1+g_2,n_1+n_2}( \otimes_{i=1}^n e_{\alpha_i}) = c_{g_1,n_1+1}(\otimes_{i\in I} e_{\alpha_i} \otimes e_\mu)\eta^{\mu \nu} c_{g_2,n_2+1}( \otimes_{j\in J} e_{\alpha_j}\otimes e_\nu)$ for $2g_1-1+n_1>0$, $2g_2-1+n_2>0$ and $1 \leq\alpha_1,\ldots,\alpha_n\leq \dim V$, where $I \sqcup J = \{1,\ldots,n\}$, $|I|=n_1$, $|J|=n_2$, and $\mathrm{gl}\colon{\overline{\mathcal{M}}}_{g_1,n_1+1}\times{\overline{\mathcal{M}}}_{g_2,n_2+1}\to {\overline{\mathcal{M}}}_{g_1+g_2,n_1+n_2}$ is the corresponding gluing map and where $\eta^{\alpha\beta}$ is defined by $\eta^{\alpha \mu}\eta_{\mu \beta} = \delta^\alpha_\beta$ for $1\leq \alpha,\beta\leq \dim V$.\\
\end{itemize}
\end{definition}
Remark that a notion of infinite rank partial CohFT, i.e. a partial CohFT with an infinite dimensional phase space $V$, requires some care. One needs to clarify what is meant by the matrix~$(\eta^{\alpha\beta})$ and to make sense of the, a priori infinite, sum over $\mu$ and $\nu$, both appearing in Axiom (iii). One possibility is demanding that the image of the linear map $V^{\otimes (n-1)}\to H^*({\overline{\mathcal{M}}}_{g,n},\mathbb C) \otimes V^*$ induced by $c_{g,n}\colon V^{\otimes n}\to H^*({\overline{\mathcal{M}}}_{g,n},\mathbb C)$ is contained in $H^*({\overline{\mathcal{M}}}_{g,n},\mathbb C) \otimes \eta^\sharp(V)$, where $\eta^\sharp\colon V\to V^*$ is the injective map induced by the bilinear form $\eta$. Then in Axiom~(iii), instead of using an undefined bilinear form $(\eta^{\alpha\beta})$ on $V^*$, one can use the bilinear form on $\eta^\sharp(V)$ induced by $\eta$. This solves the problem with convergence.\\
A useful special case is the following. Let the basis $\{e_\alpha\}_{\alpha \in I}$ of $V$ be countable and, for any~$(g,n)$ in the stable range and each $e_{\alpha_1},\ldots,e_{\alpha_{n-1}} \in V$, let the set $\{\beta \in I\, |\, c_{g,n}(\otimes_{i=1}^{n-1} e_{\alpha_i}\otimes e_\beta)\neq 0\}$ be finite. In particular this implies that the matrix $\eta_{\alpha\beta}$ is row- and column-finite (each row and each column have a finite number of nonzero entries), which is equivalent to $\eta^\sharp(V)\subseteq \mathrm{span}(\{e^\alpha\}_{\alpha \in I})$, where $\{e^\alpha\}_{\alpha \in I}$ is the dual ``basis''. Let us further demand that the injective map $\eta^\sharp\colon V \to \mathrm{span}(\{e^\alpha\}_{\alpha \in I})$ is surjective too, i.e. that a unique two-sided row- and column-finite matrix $\eta^{\alpha\beta}$, inverse to $\eta_{\alpha\beta}$, exists (it represents the inverse map $(\eta^\sharp)^{-1}\colon\mathrm{span}(\{e^\alpha\}_{\alpha \in I})\to V$). Then the equation appearing in Axiom (iii) is well defined with the double sum only having a finite number of nonzero terms for each boundary divisor.\\
We will now construct an example of such infinite rank partial CohFT.
\begin{proposition}\label{prop:CohFT}
Let $\mu$ be a formal parameter. The classes $c_{g,n}(\otimes_{i=1}^n e_{a_i}) := \exp(\mu^2\Theta(a_1,\ldots,a_n))$ form an infinite rank partial cohomological field theory with a phase space $V=\mathrm{span}(\{e_a\}_{a \in \mathbb Z})$, where the unit is~$e_0$ and the metric, written in the basis $\{e_a\}_{a \in \mathbb Z}$, is $\eta_{a b}=\delta_{a+b,0}$.
\end{proposition}
\begin{proof}
Axioms (i) and (ii) follow directly from the definition of the classes $\Theta(a_1,\ldots,a_n)$. For Axiom (iii) notice that, for fixed $e_{a_1},\ldots,e_{a_{n-1}}$, we have $c_{g,n}(\otimes_{i=1}^{n-1} e_{a_i} \otimes e_b) = 0$ unless $b= -\sum_{i=1}^{n-1} a_i$. Moreover we have $\eta^{a b}= \delta_{a+b,0}$ and equation (\ref{eq:gluing-Theta}) implies Axiom (iii) where the double sum consists of just one term.
\end{proof}
In \cite{Bur15} it was shown how to associate an integrable system of evolutionary Hamiltonian PDEs, called the double ramification (DR) hierarchy, to any CohFT and in \cite{BDGR18} it was remarked how a partial CohFT is sufficient for the construction to work. Let us see how such construction generalizes to the infinite rank partial CohFT introduced in Proposition \ref{prop:CohFT}. Recall from \cite{Bur15,BR16a} that the DR Hamiltonian densities are the generating series
\begin{align}\label{eq:DR Hamiltonians}
g_{a,d}:=\sum_{\substack{g\ge 0,\,n\ge 1\\2g-1+n>0}}\frac{(-\varepsilon^2)^g}{n!}\sum_{\substack{b,b_1,\ldots,b_n\in\mathbb Z\\ a_1,\ldots,a_n\in\mathbb Z}}\left(\int_{\mathrm{DR}_g\left(b,b_1,\ldots,b_n\right)}\lambda_g\psi_1^d c_{g,n+1}\left(e_a\otimes \otimes_{j=1}^n e_{a_j}\right)\right)\prod_{j=1}^n p_{b_j}^{a_j} e^{-ibx},
\end{align}
for $a\in \mathbb Z$ and $d\in \mathbb Z_{\geq 0}$, seen as formal power series in the formal variables $\varepsilon,\mu,e^{ix},p^a_b$, $a,b \in \mathbb Z$.\\
Thanks to the fact that the intersection numbers appearing in equation (\ref{eq:DR Hamiltonians}) vanish unless $\sum b_j=-b$ and that, by formula (\ref{eq:lambdag-Hain}), the class $\lambda_g \mathrm{DR}_g(b,b_1,\ldots,b_n)$ is a polynomial in $b_1,\ldots,b_n$ homogeneous of degree $2g$, the above generating functions can be expressed uniquely (see e.g.~\cite{BR16a}) as a degree $0$ differential polynomial, i.e. a formal power series in $\varepsilon,\mu$ and the new formal variables $u^a_k$, $a\in \mathbb Z$, $k\in \mathbb Z_{\geq 0}$, of degree $0$ with respect to the grading $\deg u^a_k=k$, $\deg\varepsilon = -1$. The relation between the new variables $u^*_*$ and the old ones $p^*_*,e^{ix}$ is given by the formula $u^a_k = {\partial}_x^k \left(\sum_{b\in \mathbb Z}p^a_b e^{ibx}\right)$. The expression of the operator ${\partial}_x$ in the new variables $u^*_*$ is given by
$$
{\partial}_x=\sum_{\substack{a\in \mathbb Z\\ j\geq 0}} u^a_{j+1} \frac{{\partial}}{{\partial} u^a_j}.
$$
Specifically, thanks to the fact that the intersection numbers appearing in equation (\ref{eq:DR Hamiltonians}) vanish unless $\sum a_j=-a$, we obtain $g_{a,d}\in \mathbb C[u^{<0}_*][[u^{\geq 0}_*,\varepsilon,\mu]]^{[0]}$, where we put the superscript $[0]$ to denote the space of differential polynomials of degree $0$.\\
The DR Hamiltonians are defined as the local functionals $\overline g_{a,d}:=\int g_{a,d} dx$, which denote the equivalence classes of $g_{a,d}$ in the quotient vector space $\left(\mathbb C[u^{<0}_*][[u^{\geq 0}_*,\varepsilon,\mu]]/(\Im({\partial}_x)\oplus \mathbb C)\right)^{[0]}$. Notice that, with respect to the formal variables $p^*_*,e^{ix}$, the symbol $\int g_{a,d} dx$ represents the coefficient of $e^{i 0 x}$ in the formal power series $g_{a,d}$.\\
A result of \cite{Bur15} says that the DR Hamiltonians mutually commute,
\begin{gather}\label{eq:DRcommute}
\{\overline g_{a_1,d_1},\overline g_{a_2,d_2}\} = 0, \qquad a_1,a_2\in \mathbb Z,\quad d_1,d_2 \in \mathbb Z_{\geq 0},
\end{gather}
with respect to the Poisson brackets of two local functionals ${\overline f}_1 = \int f_1 dx, {\overline f}_2 = \int f_2 dx$, with $f_1,f_2 \in \mathbb C[u^{<0}_*][[u^{\geq 0}_*,\varepsilon,\mu]]^{[0]}$, given by
\begin{gather}
\{{\overline f}_1,{\overline f}_2\} = \int \left(\sum_{a_1,a_2\in \mathbb Z}\frac{\delta {\overline f}_1}{\delta u^{a_1}} \delta_{a_1+a_2,0} \ {\partial}_x\left(\frac{\delta {\overline f}_2}{\delta u^{a_2}}\right)\right) dx,
\end{gather}
where $\frac{\delta {\overline f}}{\delta u^a} = \sum_{k\geq 0}(-{\partial}_x)^k \frac{{\partial} f}{{\partial} u^a_k}$ for ${\overline f} = \int f dx$ and $f\in \mathbb C[u^{<0}_*][[u^{\geq 0}_*,\varepsilon,\mu]]^{[0]}$.\\
Now we make the following observation, specific for the infinite rank partial CohFT we are dealing with. For fixed $d\in\mathbb Z_{\geq0}$, let us collect the DR Hamiltonian densities $g_{a,d}$, for all $a\in\mathbb Z$, into a single generating function $g_d:=\sum_{a\in \mathbb Z} g_{a,d} e^{-iay}$ by use of the extra formal variable $e^{iy}$. Because the classes $c_{g,n+1}\left(e_a\otimes \otimes_{j=1}^n e_{a_j}\right)$ are polynomials in $a_1,\ldots,a_n$ of top degree $2g$, where in particular the coefficient of $\mu^{2j}$ is a polynomial of degree $2j$, and $\sum a_j=-a$, we can consider the formal change of variables
$$
u_{k_1,k_2} = {\partial}_y^{k_2}\left( \sum_{a\in \mathbb Z} u^a_{k_1} e^{i a y} \right)= {\partial}_x^{k_1} {\partial}_y^{k_2} \left(\sum_{a,b\in \mathbb Z} p^a_b e^{iay+ibx}\right)
$$
and express $g_d$ uniquely as a differential polynomial in these new variables, specifically $g_d \in \mathbb C[[u_{*,*},\varepsilon,\mu]]^{[(0,0)]}$, where $\deg u_{k_1,k_2} = (k_1,k_2)$, $\deg \varepsilon = (-1,0)$, $\deg \mu = (0,-1)$. Naturally, we have
\begin{align*}
{\partial}_x=&\sum_{k_1,k_2\geq 0} u_{k_1+1,k_2} \frac{{\partial}}{{\partial} u_{k_1,k_2}},\\
{\partial}_y=&\sum_{k_1,k_2\geq 0} u_{k_1,k_2+1} \frac{{\partial}}{{\partial} u_{k_1,k_2}}.
\end{align*}
We will denote $u_{0,0}$ simply by $u$.\\
The DR Hamiltonian densities $g_{a,d}$ can be recovered from $g_d$ by the formula $g_{a,d}=\int \left(g_d e^{iay}\right)dy$, which extracts the coefficient of $e^{-iay}$ from $g_d$. Hence $\overline g_{a,d} = \iint\left(g_d e^{iay}\right)dxdy$. This suggest to restrict our attention to the Hamiltonians $\overline g_{0,d}$, whose densities depend on $e^{iy}$ through $u_{*,*}$ only. These are the simplest and most commonly considered kind of local functionals.\\
Let $\overline g_d = \overline g_{0,d} = \iint g_d\ dx dy$ be the equivalence class of $g_d$ in the quotient vector space $\left(\mathbb C[[u_{*,*},\varepsilon,\mu]]/(\Im({\partial}_x)\oplus\Im({\partial}_y)\oplus\mathbb C)\right)^{[(0,0)]}$. Then, from equation (\ref{eq:DRcommute}), we deduce
\begin{gather}
\{\overline g_{d_1},\overline g_{d_2}\} = 0, \qquad d_1,d_2 \in \mathbb Z_{\geq 0},
\end{gather}
where the Poisson bracket of two local functionals ${\overline f}_1 = \iint f_1\ dxdy, {\overline f}_2 = \iint f_2\ dxdy$, with $f_1,f_2 \in \mathbb C[[u_{*,*},\varepsilon,\mu]]^{[(0,0)]}$ is given by
\begin{gather}
\{{\overline f}_1,{\overline f}_2\} = \iint \left(\frac{\delta {\overline f}_1}{\delta u} {\partial}_x\left(\frac{\delta {\overline f}_2}{\delta u}\right)\right) dx dy,
\end{gather}
where $\frac{\delta {\overline f}}{\delta u} = \sum_{k_1,k_2\geq 0}(-{\partial}_x)^{k_1}(-{\partial}_y)^{k_2} \frac{{\partial} f}{{\partial} u_{k_1,k_2}}$ for ${\overline f} = \iint f\ dxdy$ and $f\in \mathbb C[[u_{*,*},\varepsilon,\mu]]^{[(0,0)]}$.\\
The evolutionary PDEs generated via the above Poisson structure by the DR Hamiltonians~$\overline g_d$ are all compatible and have the form
\begin{gather}\label{eq:DRhierarchy}
\frac{{\partial} u}{{\partial} t_d} = {\partial}_x \frac{\delta \overline g_d}{\delta u},\qquad d\in \mathbb Z_{\geq 0}.
\end{gather}
\section{The noncommutative KdV hierarchy and the main theorem}\label{section:ncKdV}
The classical construction of the KdV hierarchy as a system of Lax equations admits a generalization, where one doesn't require the pairwise commutativity of the $x$-derivatives of the dependent variable. The formal algebraic construction is the following. Let $(\widetilde{\mathcal{A}},(u_k)_{k\ge 0},{\partial}_x)$ be a triple, where $\widetilde{\mathcal{A}}$ is an associative, not necessarily commutative, algebra with a multiplication denoted by $*$, $u_k\in\widetilde{\mathcal{A}}, k\ge 0$, is a sequence of elements generating $\widetilde{\mathcal{A}}$ and ${\partial}_x\colon\widetilde{\mathcal{A}}\to\widetilde{\mathcal{A}}$ is a linear operator satisfying the properties
$$
{\partial}_x u_k=u_{k+1},\qquad {\partial}_x(f*g)={\partial}_x f*g+f*{\partial}_x g,\quad f,g\in\widetilde{\mathcal{A}}.
$$
Let us consider the algebra of pseudo-differential operators on $\widetilde{\mathcal{A}}$ of the form
$$
\sum_{i\le n}a_i*{\partial}_x^i,\quad n\in\mathbb Z,\quad a_i\in\widetilde{\mathcal{A}}[\varepsilon,\varepsilon^{-1}].
$$
Consider an operator $L:={\partial}_x^2+2\varepsilon^{-2}u$. The noncommutative KdV hierarchy on $\widetilde{\mathcal{A}}$ is defined by (see e.g.~\cite{Ham05,DM00})
\begin{gather}\label{eq:ncKdV hierarchy}
\frac{{\partial} L}{{\partial} t_n}=\frac{\varepsilon^{2n}}{(2n+1)!!}\left[\left(L^{n+1/2}\right)_+,L\right]_*,\quad n\ge 1,
\end{gather}
where we put the subscript $*$ in the notation for a commutator in order to emphasize that it is taken with respect to the noncommutative product $*$. The first equation of the hierarchy is
$$
\frac{{\partial} u}{{\partial} t_1}=\frac{1}{2}{\partial}_x(u*u)+\frac{\varepsilon^2}{12}u_{xxx}.
$$
In what follows we will work with a specific example from the class of noncommutative KdV hierarchies.\\
The graded algebra of differential polynomials in two space dimensions introduced in Section~\ref{section:DR}, $\mathbb C[[u_{*,*},\varepsilon,\mu]]$, where $\deg u_{k_1,k_2} = (k_1,k_2)$, $\deg \varepsilon = (-1,0)$, $\deg \mu = (0,-1)$, can be endowed with the following graded associative Moyal star-product. Let $f,g \in \mathbb C[[u_{*,*},\varepsilon,\mu]]$, with $\deg f = (i_1,i_2)$, $\deg g = (j_1,j_2)$, then define
\begin{gather}\label{eq:Moyal}
f* g:=f\ \exp\left(\frac{i \varepsilon\mu}{2}(\overleftarrow{{\partial}_x} \overrightarrow{{\partial}_y}-\overleftarrow{{\partial}_y} \overrightarrow{{\partial}_x})\right)\ g =\sum_{n\ge 0}\sum_{k_1+k_2=n}\frac{(-1)^{k_2}(i\varepsilon\mu)^n}{2^n k_1!k_2!}({\partial}_x^{k_1} {\partial}_y^{k_2} f) ({\partial}_x^{k_2}{\partial}_y^{k_1} g),
\end{gather}
with $\deg{(f * g)} = (i_1+j_1,i_2+j_2)$.\\
The following result identifies the equations of the DR hierarchy (\ref{eq:DRhierarchy}) with the noncommutative KdV hierarchy with respect to the Moyal star-product (\ref{eq:Moyal}).
\begin{theorem}\label{theorem:main}
The flows $\frac{{\partial}}{{\partial} t_d}$, $d\ge 1$, of the DR hierarchy are given by the noncommutative KdV hierarchy
\begin{gather}\label{eq:ncKdV}
\frac{{\partial} L}{{\partial} t_d}=\frac{\varepsilon^{2n}}{(2n+1)!!}\left[\left(L^{d+1/2}\right)_+,L\right]_*,\quad d\ge 1,
\end{gather}
where the product $*$ is the Moyal star-product (\ref{eq:Moyal}).
\end{theorem}
\begin{proof}
We prove the theorem in two steps.\\
{\it Step 1}. Let us prove that the flow $\frac{{\partial}}{{\partial} t_1}$ of the DR hierarchy (\ref{eq:DRhierarchy}) is given by
\begin{gather}\label{eq:first flow of DR hierarchy}
\frac{{\partial} u}{{\partial} t_1}=\frac{1}{2}{\partial}_x(u*u)+\frac{\varepsilon^2}{12}u_{xxx}.
\end{gather}
For this we have to compute the integrals
\begin{multline}\label{Theta-integrals}
\int_{{\overline{\mathcal{M}}}_{g,n+1}}\lambda_g\psi_1\Theta(0,a_1,\ldots,a_n)^k\mathrm{DR}_g(0,b_1,\ldots,b_n)=\\
=(2g-2+n)\int_{{\overline{\mathcal{M}}}_{g,n}}\lambda_g\Theta(a_1,\ldots,a_n)^k\mathrm{DR}_g(b_1,\ldots,b_n).
\end{multline}
Relation (\ref{eq:lambdag-Theta-relation}) implies that integral~\eqref{Theta-integrals} is nonzero only if $n=3$ and $k=g$ or if $n=2$, $k=0$ and $g=1$. In the second case integral~\eqref{Theta-integrals} is equal to $\frac{b_1^2}{12}$, which gives the second term on the right-hand side of equation~\eqref{eq:first flow of DR hierarchy}.\\
Regarding integral~\eqref{Theta-integrals} for $n=3$ and $k=g$, by Theorem~\ref{eq:quadratic DR integral}, we have
\begin{align*}
&\sum_{g\ge 0}\sum_{ a_1, a_2, b_1, b_2\in\mathbb Z}\frac{(-\varepsilon^2\mu^2)^g}{g!}\int_{{\overline{\mathcal{M}}}_{g,4}}\lambda_g\psi_1\Theta(0,-a_1-a_2,a_1,a_2)^g\mathrm{DR}_g(0,-b_1-b_2,b_1,b_2)p^{ a_1}_{ b_1}p^{ a_2}_{ b_2}=\\
&\hspace{2cm}=\sum_{g\ge 0}\sum_{a_1,a_2,b_1,b_2\in\mathbb Z}(-\varepsilon^2\mu^2)^g\frac{(a_2 b_1-a_1 b_2)^{2g}}{2^{2g}(2g)!}p^{ a_1}_{ b_1}p^{ a_2}_{b_2}=\\
&\hspace{2cm}=\sum_{g\ge 0}\sum_{k_1+k_2=2g}\sum_{ a_1, a_2, b_1, b_2\in\mathbb Z}\frac{(-1)^{k_2} (-\varepsilon^2\mu^2)^g}{2^{2g} k_1!k_2!}( a_2 b_1)^{k_1}( a_1 b_2)^{k_2} p^{ a_1}_{ b_1}p^{ a_2}_{ b_2}=\\
&\hspace{2cm}=\sum_{g\ge 0}\sum_{k_1+k_2=2g}\frac{(-1)^{k_2}(-\varepsilon^2\mu^2)^g}{2^{2g} k_1!k_2!}u_{k_1,k_2}u_{k_2,k_1}=\\
&\hspace{2cm}=u*u.
\end{align*}
Equation~\eqref{eq:first flow of DR hierarchy} is then proved.\\
{\it Step 2}. Let us now check that all other flows $\frac{{\partial}}{{\partial} t_d}$ of the DR hierarchy, for $d\geq 2$, are described by the noncommutative KdV hierarchy.\\
Let $f \in \mathbb C[[u_{*,*},\varepsilon,\mu]]$ and let $\deg f =:(\deg_x f,\deg_y f)$ so that, in particular,
$$
\deg_x u_{a,b}=a,\quad\deg_y u_{a,b}=b,\quad \deg_x\varepsilon=\deg_y\mu=-1.
$$
We see that both the flows of the DR hierarchy (\ref{eq:DRhierarchy}) and the flows of the noncommutative KdV hierarchy (\ref{eq:ncKdV}) have the form
\begin{gather}\label{eq:general hierarchy}
\frac{{\partial} u}{{\partial} t_d}=P_d(u_{*,*},\mu,\varepsilon)=\sum_{i\ge 0}P_{d,i}(u_{*,*},\mu)\varepsilon^i,\quad d\ge 1,
\end{gather}
where $P_{d,i}(u_{*,*},\mu)$ are polynomials in the variables $u_{*,*}$ and $\mu$ satisfying
\begin{align}
&P_1=\frac{1}{2}{\partial}_x(u*u)+\frac{\varepsilon^2}{12}u_{xxx},\label{eq:properties of hierarchies1}\\
&P_{d,0}={\partial}_x\left(\frac{u^{d+1}}{(d+1)!}\right),\label{eq:properties of hierarchies2}\\
&\deg_x P_{d,i}=i+1,\qquad\deg_y P_{d,i}=0\label{eq:properties of hierarchies3}.
\end{align}
It remains to check that a hierarchy of commuting flows of the form~\eqref{eq:general hierarchy}, satisfying properties~\eqref{eq:properties of hierarchies1}--\eqref{eq:properties of hierarchies3}, is unique. This is guaranteed by the following lemma.
\begin{lemma}
Suppose that $P(u_{*,*})$ is a polynomial in the variables $u_{*,*}$ of degrees $\deg_x P=d\ge 2$, $\deg_y P=q\ge 0$, and such that the flows
\begin{align*}
\frac{{\partial} u}{{\partial} t}=&uu_x,\\
\frac{{\partial} u}{{\partial}\tau}=&P(u_{*,*}),
\end{align*}
commute. Then $P=0$.
\end{lemma}
\begin{proof}
Without loss of generality we can assume that $P$ is homogeneous with respect to an auxiliary gradation given by $\tdeg u_{a,b}:=1$. So we assume that $\tdeg P=k\ge 1$.\\
For a vector $\overline{a}=(a_1,\ldots,a_k)\in\mathbb Z^k$ denote $|\overline{a}|:=\sum a_i$. Let
\begin{align*}
\mathcal{P}_k:=&\{\lambda=(\lambda_1,\ldots,\lambda_k)\in\mathbb Z^k_{\ge 0}|\lambda_1\ge\ldots\ge\lambda_k\},\\
\mathcal{P}_{k,q}:=&\{\lambda\in\mathcal{P}_k||\lambda|=q\}.
\end{align*}
The set $\mathcal{P}_{k,q}$ is endowed with the lexicographical order. We will use the standard notation
$$
m_p(\lambda):=\sharp\{1\le i\le k|\lambda_i=p\},\quad\lambda\in\mathcal{P}_k,\quad p\ge 0.
$$
The sequence $(m_0(\lambda),m_1(\lambda),\ldots)$ uniquely determines $\lambda$ that justifies the notation
$$
\lambda=(0^{m_0(\lambda)}1^{m_1(\lambda)}\ldots).
$$\\
Introduce a basis $f_{\lambda}$, $\lambda\in\mathcal{P}_k$, in the space of symmetric polynomials $\mathbb C[x_1,\ldots,x_k]^{S_k}$ by
$$
f_\lambda(x_1,\ldots,x_k):=\frac{1}{k!}\sum_{\sigma\in S_k}x_{\sigma(1)}^{\lambda_1}\ldots x_{\sigma(k)}^{\lambda_k},\quad\lambda\in\mathcal{P}_k.
$$
For $\lambda\in\mathcal{P}_k$ we call a polynomial $f\in\mathbb C[x_1,\ldots,x_k]$ $\lambda$-symmetric, if it is invariant with respect to the permutation of any pair of variables $x_i$ and $x_j$, $i\ne j$, such that $\lambda_i=\lambda_j$. We also introduce a notation for the symmetrization of a polynomial $f\in\mathbb C[x_1,\ldots,x_k]$ with respect to the variables $x_1,\ldots,x_a$, $1\le a\le k$:
$$
\mathrm{Sym}_{x_1,\ldots,x_a}f:=\frac{1}{a!}\sum_{\sigma\in S_a}f(x_{\sigma(1)},\ldots,x_{\sigma(a)},x_{a+1},\ldots,x_k).
$$\\
Making the substitutions $u(x,y)=\sum_{\alpha\in\mathbb Z}u^\alpha(y)e^{i\alpha x}$ and $t\mapsto\frac{t}{i}$, the flows $\frac{{\partial}}{{\partial} t}$ and $\frac{{\partial}}{{\partial}\tau}$ can be written in the form
\begin{align*}
\frac{{\partial} u^\alpha}{{\partial} t}=&\sum_{\substack{\alpha_1,\alpha_2\in\mathbb Z\\\alpha_1+\alpha_2=\alpha}}\alpha_1 u^{\alpha_1} u^{\alpha_2},&&\alpha\in\mathbb Z,\\
\frac{{\partial} u^\alpha}{{\partial}\tau}=&\sum_{\lambda\in\mathcal{P}_{k,q}}\sum_{\substack{\alpha_1,\ldots,\alpha_k\in\mathbb Z\\\sum\alpha_i=\alpha}}P_{\lambda}(\alpha_1,\ldots,\alpha_k)u^{\alpha_1}_{\lambda_1}\ldots u^{\alpha_k}_{\lambda_k},&&\alpha\in\mathbb Z,
\end{align*}
where $P_\lambda$ is a $\lambda$-symmetric polynomial in $\alpha_1,\ldots,\alpha_k$ of degree $d$. Here $u^\alpha_a:={\partial}_y^a u^\alpha$.\\
The commutator of the flows $\frac{{\partial}}{{\partial} t}$ and $\frac{{\partial}}{{\partial}\tau}$ is given by
\begin{align}
\frac{{\partial}}{{\partial} t}\frac{{\partial} u^\alpha}{{\partial}\tau}-\frac{{\partial}}{{\partial}\tau}\frac{{\partial} u^\alpha}{{\partial} t}=&\sum_{\beta\in\mathbb Z,\,b\ge 0}{\partial}_y^b\left(\sum_{\substack{\beta_1,\beta_2\in\mathbb Z\\\beta_1+\beta_2=\beta}}\beta_1 u^{\beta_1} u^{\beta_2}\right)\frac{{\partial}}{{\partial} u^\beta_b}\sum_{\lambda\in\mathcal{P}_{k,q}}\sum_{\substack{\alpha_1,\ldots,\alpha_k\in\mathbb Z\\\sum\alpha_i=\alpha}}P_{\lambda}(\alpha_1,\ldots,\alpha_k)u^{\alpha_1}_{\lambda_1}\ldots u^{\alpha_k}_{\lambda_k}\notag\\
&-\sum_{\substack{\alpha_1,\alpha_2\in\mathbb Z\\\alpha_1+\alpha_2=\alpha}}\alpha_1\left(\sum_{\lambda\in\mathcal{P}_{k,q}}\sum_{\substack{\beta_1,\ldots,\beta_k\in\mathbb Z\\\sum\beta_i=\alpha_1}}P_{\lambda}(\beta_1,\ldots,\beta_k)u^{\beta_1}_{\lambda_1}\ldots u^{\beta_k}_{\lambda_k}\right)u^{\alpha_2}\notag\\
&-\sum_{\substack{\alpha_1,\alpha_2\in\mathbb Z\\\alpha_1+\alpha_2=\alpha}}\alpha_1 u^{\alpha_1}\sum_{\lambda\in\mathcal{P}_{k,q}}\sum_{\substack{\beta_1,\ldots,\beta_k\in\mathbb Z\\\sum\beta_i=\alpha_2}}P_{\lambda}(\beta_1,\ldots,\beta_k)u^{\beta_1}_{\lambda_1}\ldots u^{\beta_k}_{\lambda_k}=\notag\\
=&\sum_{\lambda\in\mathcal{P}_{k+1,q}}\sum_{\substack{\alpha_1,\ldots,\alpha_{k+1}\in\mathbb Z\\\sum\alpha_i=\alpha}}Q_{\lambda}(\alpha_1,\ldots,\alpha_{k+1})u^{\alpha_1}_{\lambda_1}\ldots u^{\alpha_{k+1}}_{\lambda_{k+1}},\label{sum}
\end{align}
where a polynomial $Q_\lambda$ is $\lambda$-symmetric.\\
Let ${\widetilde{\lambda}}:=\max\{\lambda\in\mathcal{P}_{k,q}|P_\lambda\ne 0\}$ and ${\widehat{\lambda}}:=({\widetilde{\lambda}}_1,\ldots,{\widetilde{\lambda}}_k,0)\in\mathcal{P}_{k+1,q}$. We see that the sum in line~\eqref{sum} has the form
$$
\sum_{\substack{\alpha_1,\ldots,\alpha_{k+1}\in\mathbb Z\\\sum\alpha_i=\alpha}}Q_{{\widehat{\lambda}}}(\alpha_1,\ldots,\alpha_{k+1})u^{\alpha_1}_{{\widehat{\lambda}}_1}\ldots u^{\alpha_{k+1}}_{{\widehat{\lambda}}_{k+1}}+\sum_{\substack{\lambda\in\mathcal{P}_{k+1,q}\\\lambda<{\widehat{\lambda}}}}\sum_{\substack{\alpha_1,\ldots,\alpha_{k+1}\in\mathbb Z\\\sum\alpha_i=\alpha}}Q_{\lambda}(\alpha_1,\ldots,\alpha_{k+1})u^{\alpha_1}_{\lambda_1}\ldots u^{\alpha_{k+1}}_{\lambda_{k+1}},
$$
and
\begin{align*}
&Q_{{\widehat{\lambda}}}(\alpha_1,\ldots,\alpha_{k+1})=\\
=&\mathrm{Sym}_{\alpha_{s+1},\ldots,\alpha_{k+1}}\Bigg[\sum_{i=1}^k(\alpha_iP_{{\widetilde{\lambda}}}(\alpha_1,\ldots,\alpha_i+\alpha_{k+1},\ldots,\alpha_k)-\alpha_iP_{\widetilde{\lambda}}(\alpha_1,\ldots,\alpha_k))\\
&\hspace{2.7cm}+\sum_{i=1}^s\alpha_{k+1}P_{\widetilde{\lambda}}(\alpha_1,\ldots,\alpha_i+\alpha_{k+1},\ldots,\alpha_k)-\alpha_{k+1}P_{{\widetilde{\lambda}}}(\alpha_1,\ldots,\alpha_k)\Bigg],
\end{align*}
where $s:=\sharp\{1\le i\le k|{\widetilde{\lambda}}_i\ge 1\}$.\\
We decompose the polynomial $P_{\widetilde{\lambda}}$ in the following way:
$$
P_{\widetilde{\lambda}}(\alpha_1,\ldots,\alpha_k)=\sum_{\substack{\overline{a}\in\mathbb Z^s_{\ge 0},\,\mu\in\mathcal{P}_{k-s}\\|\overline{a}|+|\mu|=d}}C_{\overline{a},\mu}\alpha_1^{a_1}\ldots\alpha_s^{a_s}f_\mu(\alpha_{s+1},\ldots,\alpha_k),\quad C_{\overline{a},\mu}\in\mathbb C.
$$
Let
\begin{align*}
P^{(p)}_{\widetilde{\lambda}}(\alpha_1,\ldots,\alpha_k):=&\sum_{\substack{\overline{a}\in\mathbb Z^s_{\ge 0},\,|\overline{a}|=p\\\mu\in\mathcal{P}_{k-s,d-p}}}C_{\overline{a},\mu}\alpha_1^{a_1}\ldots\alpha_s^{a_s}f_\mu(\alpha_{s+1},\ldots,\alpha_k),&& p\ge 0,&&\\
P^{(p,\mu)}_{\widetilde{\lambda}}(\alpha_1,\ldots,\alpha_k):=&\left(\sum_{\overline{a}\in\mathbb Z^s_{\ge 0},\,|\overline{a}|=p}C_{\overline{a},\mu}\alpha_1^{a_1}\ldots\alpha_s^{a_s}\right)f_\mu(\alpha_{s+1},\ldots,\alpha_k),&& p\ge 0,&& \mu\in\mathcal{P}_{k-s,d-p}.
\end{align*}
Let ${\widetilde{d}}:=\max\left\{p\ge 0\left|P_{\widetilde{\lambda}}^{(p)}\ne 0\right.\right\}$ and ${\widetilde{\mu}}=(0^{m_0}1^{m_1}\ldots):=\max\left\{\mu\in\mathcal{P}_{k-s,d-{\widetilde{d}}}\left|P_{\widetilde{\lambda}}^{({\widetilde{d}},\mu)}\ne 0\right.\right\}$. Note that
$$
\mathrm{Sym}_{\alpha_{s+1},\ldots,\alpha_{k+1}}(f_\mu(\alpha_{s+1},\ldots,\alpha_k)\alpha_{k+1})=f_{\widehat{\mu}}(\alpha_{s+1},\ldots,\alpha_{k+1}),
$$
where ${\widehat{\mu}}:=(0^{m_0}1^{m_1+1}2^{m_2}\ldots)$. It is now easy to see that
$$
Q_{\widehat{\lambda}}^{{\widetilde{d}},{\widehat{\mu}}}(\alpha_1,\ldots,\alpha_{k+1})=\left(d+s-1+\sum_{i\ge 2}m_i\right)\left(\sum_{\overline{a}\in\mathbb Z^s_{\ge 0},\,|\overline{a}|={\widetilde{d}}}C_{\overline{a},{\widetilde{\mu}}}\alpha_1^{a_1}\ldots\alpha_s^{a_s}\right)f_{\widehat{\mu}}(\alpha_{s+1},\ldots,\alpha_{k+1}),
$$
which is nonzero, because $d\ge 2$ and $P^{({\widetilde{d}},{\widetilde{\mu}})}_{\widetilde{\lambda}}\ne 0$. This contradicts the assumption that the flows~$\frac{{\partial}}{{\partial} t}$ and~$\frac{{\partial}}{{\partial}\tau}$ commute. The lemma is proved.
\end{proof}
This completes the proof of the theorem.
\end{proof}
|
3,212,635,537,472 | arxiv | \section{Analytic method for Fig~\ref{2-loop-UV-diagrams}(c)}
\noindent
This Appendix derives the analytic result
for the 2-loop \emph{half-boiled egg} diagram of Fig~\ref{2-loop-UV-diagrams}(c).
By the variable transformation
\begin{center}
$x_1= \tau\,(1-\xi)$,~
$x_2= \tau\,\xi$,~
$x_3= (1-\tau)\,\tau'\,(1-\xi')$,~
$x_4= (1-\tau)\,\tau'\,\xi'$,~
$x_5= (1-\tau)\,(1-\tau'),$
\end{center}
the functions $U, V$ are given by
\begin{equation}
U=\tau \,F,\quad F = 1-\tau+\tau\,\xi\,(1-\xi) \nonumber
\end{equation}
\begin{equation}
W/s=\tau \,G, \quad G=(1-\tau)\,(1-\tau')\,((1-\tau)\,\tau'+\tau\,\xi\,(1-\xi)) \nonumber
\end{equation}
and
\begin{equation}
J_c^{S2}=\int_0^1 d\tau ~\tau(1-\tau)^2
\int_0^1 d\xi \int_0^1 d\tau' \,\tau'
\frac{1}{U^{1-3\varepsilon}\,(UV)^{1+2\varepsilon}}
\label{Jc}
\end{equation}
where we assume $m_3=m_4$ to perform the $\xi'$-integration.
The integral is divergent at $\tau=0$ to produce $\displaystyle{1/\varepsilon}$
singularity, and the separation of the singularity is done as follows.
Let us denote $H=M^2 F- s\,G,$ and let us use the suffix 0 for the function
defined at $\tau=0$, i.e., $F_0=F\,(\tau=0)=1$ and $H_0=H\,(\tau=0)
=[\tau'm_3^2+(1-\tau')\,m_5^2]-s\,(1-\tau')\,\tau'$.
The integral is separated into two terms as
\begin{equation}
J_c^{S2}=I_A+I_B \nonumber
\end{equation}
where the first term $I_A$ has a UV singularity. The integrands of
$I_A$ and $I_B$ are given according to Eq~\eqref{Jc} and
\begin{equation}
\frac{1}{U^{1-3\varepsilon}(UV)^{1+2\varepsilon}}
=\frac{1}{\tau^{2-\varepsilon}}
\frac{1}{F^{1-3\varepsilon}H^{1+2\varepsilon}}
=\frac{1}{\tau^{2-\varepsilon}}\left[
\frac{1}{F_0^{1-3\varepsilon}H_0^{1+2\varepsilon}}
+\left( \frac{1}{F^{1-3\varepsilon}H^{1+2\varepsilon}}
-\frac{1}{F_0^{1-3\varepsilon}H_0^{1+2\varepsilon}} \right)
\right] \, . \nonumber
\end{equation}
The divergent term $I_A$ is trivial in the $\xi$-integral and is calculated as
\[
I_A=
\int_0^1 d\tau \,\frac{(1-\tau)^2}{\tau^{1-\varepsilon}}
\int_0^1 d\tau' \frac{\tau'}{H_0^{1+2\varepsilon}}
\]
\begin{equation}
=\left(\frac{1}{\varepsilon}-\frac{3}{2}+\frac{7}{4}\varepsilon-\frac{15}{8}\varepsilon^2\right)
\times \left( I_A^{(0)}+I_A^{(1)}\varepsilon+I_A^{(2)}\varepsilon^2+I_A^{(3)}\varepsilon^3\right)
\label{expIA}
\end{equation}
and the non-divergent term $I_B$ is
\[
I_B=
\int_0^1d\tau
\int_0^1 d\xi \int_0^1 d\tau' \,\frac{(1-\tau)^2 \tau'}{\tau^{1-\varepsilon}}
\left(\frac{1}{F^{1-3\varepsilon}H^{1+2\varepsilon}}-\frac{1}{H_0^{1+2\varepsilon}}\right)
\]
\begin{equation}
=I_B^{(1)}+I_B^{(2)}\varepsilon+I_B^{(3)}\varepsilon^2
\label{expIB}
\end{equation}
The terms $ I_A^{(0)}, I_A^{(1)}, I_A^{(2)}, I_A^{(3)}$ and
$I_B^{(1)}, I_B^{(2)}, I_B^{(3)}$ in Eqs~\eqref{expIA} and~\eqref{expIB} can be obtained by
expanding the integrands in powers of $\varepsilon$.
We evaluated these terms numerically by {\sc Dqage} and {\sc Dqags} from {\sc Quadpack} and the DE formula, for the values $s=M^2=1,$ yielding
the expression in Eq~(\ref{Ihexp}).
\section{Introduction}
\label{intro}
High energy physics collider experiments target the precise measurement of parameters in the standard model and beyond, and detection of any deviations of the experimental data from the theoretical predictions, leading to the study of new phenomena. In modern physics, there are three basic interactions acting on particles: weak, electromagnetic and strong interactions. When we consider a scattering process of elementary particles, the cross section reflects the dynamics that govern the motion of the particles, caused by the interaction.
All information on a particle interaction is contained in the amplitude according to the (Feynman) rules of Quantum Field Theory. Generally, with a given particle interaction, a large number of configurations (represented by Feynman diagrams) is associated. Each diagram represents one of the possible configurations of virtual processes, and it describes a part of the total amplitude.
The square sum of the amplitudes delivers the probability or cross section of the process. Based on the Feynman rules, the amplitude can be obtained in an automatic manner:
\emph{(i)} {determine the physics process (external momenta and order of perturbation);}
\emph{(ii)} {draw all Feynman diagrams relevant to the process;}
\emph{(iii)} {describe the contributions to the amplitude.}
Feynman diagrams are constructed in such a way that the initial state particles are connected to the final state particles by propagators and vertices. Particles meet at vertices according to a coupling constant $g,$ which indicates the strength of the interaction. The amplitude is expanded as a perturbation series in $g,$ where the leading (lowest) order of approximation corresponds to the tree level of the Feynman diagrams. Higher orders require the evaluation of loop diagrams,
so that the computation of loop integrals is very important for the present and future high-energy experiments.
When few masses occur in the computation of loop integrals, analytic approaches are generally feasible.
However, in the presence of a wide range of masses, analytic evaluation becomes very complicated or impossible.
For one-loop integrals, explicit analytic methods have been established by many authors,
but alternative approaches are compulsory
for multi-loop integrals with a variety of masses and momenta.
We propose a fully numerical approach based on multi-dimensional integration and extrapolation,
and demonstrate results of the technique for multi-loop integrals with and without masses.
In the computation of loop integrals we have to handle singularities.
Depending on the value of internal masses and external momenta, the integrand denominator
may vanish in the interior of the integration domain.
The term $i\rho$ (subtracted from $V$) in the denominator of the loop integral
representation of Eq~\eqref{Lloop} is intended to prevent the integral from diverging
if $V$ vanishes in the domain.
The idea of our numerical extrapolation approach is to consider $\rho$ not as an
infinitesimal small number for the analytic continuation but
as a finite number, to make the integral non-singular.
We choose a sequence of $\rho$ values, $\rho_\ell\rightarrow 0$
(e.g., a geometric sequence), so that multi-dimensional integration yields consecutive
$I(\rho_{\ell})$ corresponding to $\rho_\ell.$
The sequence of $I(\rho_{\ell})$ is extrapolated numerically to approximate the value
of the loop integral in the limit as $\rho_\ell\rightarrow 0.$
For physical kinematics where an imaginary part is present, it can be treated numerically
as well as the real part,
since the integrand is not singular for finite $\rho_\ell.$
In previous work we have demonstrated various loop integral computations
using this type of method not only in the Euclidean but also in the
physical region~\cite{edcpp03,eddacat03,acat07,acat08,ddacat10,cpc11y,jocs11}.
For the infrared divergent case we have two prescriptions. One is to introduce a
small fictitious mass for the massless particles and the other is to use
dimensional regularization.
We have shown results for several problem classes in~\cite{iccs05b,acat07,cpp10,acat11,jocs11}.
In this paper we concentrate on loop integrals with UV singularities, which satisfy asymptotic
expansions in the dimensional regularization parameter $\varepsilon$
(see Eq~\eqref{Lloop}, where the space-time dimension $n$ will be set to $n = 4-2\varepsilon$
to account for UV singularity).
Based on multi-dimensional integration and numerical extrapolation,
we present a novel numerical regularization method for integrals with UV singularities,
applied to 1-, 2-, 3- and 4-loop diagrams.
We compare with results in the literature, including those of Laporta~\cite{laporta01} $-$ whose method is based on
the numerical solution of systems of difference equations,
the sector decomposition approach by Smirnov and Tentyukov~\cite{smirnov10},
and the analytic results by Baikov and Chetyrkin~\cite{baikov10}.
The integration strategies in this paper adhere to \emph{automatic integration}, which is
a black-box approach for generating an approximation ${\mathcal Q}f$ to an integral
\begin{equation}\label{blackbox}
{\mathcal I\hspace*{-0.4mm}}f = \int_{\mathcal D} f(\vec{x}) ~d\vec{x},
\end{equation}
as well as an absolute error estimate $E_af,$
in order to satisfy a specified accuracy requirement of the form
\begin{equation}
\label{accuracy}
|\,{\mathcal Q}f-{\mathcal I\hspace*{-0.4mm}}f\,| ~\le ~ E_af ~\le~ \max\,\{\,t_a\,,\,t_r\,|\,{\mathcal I\hspace*{-0.4mm}}f\,|~\}
\end{equation}
for a given integrand function
$f: {\mathcal D}\subset {\mathbb R}^d \rightarrow {\mathbb R},$
a $d$-dimensional domain $\mathcal D,$ and (absolute/relative)
error tolerances $t_a$ and $t_r.$
If it is found that Eq~\eqref{accuracy} cannot be achieved, an error indicator
should be returned.
In order to achieve the accuracy requirement, the actual error should not exceed the error estimate
$E_af,$ and the error estimate should not exceed the weaker of the
absolute and relative error tolerances (indicated by the maximum taken on the right
of`\eqref{accuracy}).
When a relative or an absolute accuracy (only) needs to be satisfied we set $t_a = 0$ or $t_r = 0,$ respectively.
If both $t_a \ne 0$ and $t_r \ne 0,$ the weaker of the two error tolerances is imposed; if
$t_a = t_r = 0$ then the program will reach an abnormal termination.
This type of accuracy requirement is based on~\cite{deboor71}
and used extensively in {\sc Quadpack}~\cite{pi83}.
Known methods for parallelization of these procedures include:\\
\emph{(i)} Parallelization on the {\color{black}rule} or {\color{black}points} level: typically in \emph{non-adaptive}
algorithms, e.g., for \emph{Monte-Carlo (MC)} algorithms and composite rules
using \emph{grid} or \emph{lattice} points. Then in $If = \int_{\mathcal D}f
\approx \sum_k w_k {\color{black}f(\vec{x}_k)},$ the function evaluations
{\color{black}$f(\vec{x}_k)$} are performed in parallel.\\
\noindent
\emph{(ii)} Parallelization on the {\color{black}region} level: in \emph{adaptive}
(region-partitioning) methods. These lead to {\color{black}task pool strategies},
which may benefit from load balancing on distributed memory systems;
or maintain a shared priority queue on shared memory systems.\\
\noindent
\emph{(iii)} We added multi-threading to {\color{black}iterated integration}~\cite{icmsq12,ccp12,ddacat13}:
the {\color{black}inner integrals} are independent and computed in parallel.
For example, over a subregion $\mathcal S = {\mathcal D}_1\times {\mathcal D}_2$ ~(with inner region ${\mathcal D}_2$) consider
$\int_{\mathcal S} F({\vec x}) \,d{\vec x} \approx \sum_k w_k {\color{black}\vec F({\vec x_k})},$ ~with~
${\color{black}\vec F({\vec x_k})} = \int_{{\mathcal D}_2}f(\vec{x_k},{\vec y}) \,d{\vec y}.$
The integrations in the different coordinate directions can be performed {\color{black}adaptively}, which we
achieved with iterated versions of the 1D programs {\sc Dqags} or {\sc Dqage} from
{\sc Quadpack}~\cite{pi83,jocs11}.
We further apply numerical extrapolation techniques for convergence acceleration
of a sequence of integrals with respect to a parameter $\gamma.$ For linear
extrapolation, an asymptotic expansion of the form
\begin{equation}
{\color{black}{\mathcal I}(\gamma) \sim \sum_{k\ge \kappa} C_k \,\varphi_k(\gamma), ~~~~~~\mbox{as~~} \gamma\rightarrow 0}
\label{asymp}
\end{equation}
is assumed, where ${\cal I}(\gamma)$ represents the integral and the
sequence of $\varphi_k(\gamma)$ is known.
If the structure of the expansion is unknown we resort to a non-linear extrapolation with the
$\epsilon$-algorithm~\cite{shanks55,wynn56,sidi96,sidi03,sidi11}.
This paper gives an overview of our recent work.
Section~\ref{Feynman} provides background and notations for multi-loop Feynman integrals
and diagrams, and discusses the use of extrapolation or convergence acceleration.
Section~\ref{integration} describes iterated integration,
the {\sc ParInt} adaptive strategies, and the double exponential transformation method.
Numerical results obtained for a set of
2-loop self-energy, vertex and box diagrams are discussed in Section~\ref{2-loop-integral}; 3-loop massless and massive self-energy
diagrams are covered in Section~\ref{3-loop-self}, and 4-loop massless self-energy diagrams in Section ~\ref{4-loop-self}.
Results from parallel distributed computations were obtained on the \emph{thor} cluster of the
Center for High Performance Computing and Big Data at WMU, where we used
16-core cluster nodes with Intel(R) Xeon(R) E5-2670, 2.6\,GHz dual processors and 128\,GB of memory,
and the cluster's Infiniband interconnect for message passing via MPI.
Some sample sequential and parallel results were collected from runs on
Intel(R) Xeon(R) CPU E5-1660 3.30GHz, E5-2687W v3 3.10\,GHz,
and on a 2.6\,GHz Intel(R) Core i7 Mac-Pro with 4 cores and 16\,GB memory under OS X.
For the inclusion of OpenMP~\cite{openmp} multi-threading compiler directives in the iterated
integration code (based on the Fortran version of {\sc Quadpack}), we used the (GNU) \emph{gfortran} compiler
and the Intel Fortran compiler, with the flags \emph{-fopenmp} and \emph{-openmp}, respectively.
{\sc ParInt} and its integrand functions were compiled with \emph{gcc (mpicc)}.
Besides Intel processors, we used POWER7(R) 3.83\,GHz on the KEKSC system A of the Computing Research Center at KEK (SR16000 model M1), with
the HITACHI Fortran90 compiler that enables automatic parallelization with the flag ~{\it -parallel}.
\section{Feynman loop integrals and extrapolation}
\label{Feynman}
\subsection{General form of Feynman loop integrals}
\label{Feynman-gen}
{\color{black}Higher-order corrections}
are required for accurate theoretical predictions of the {\color{black}cross section} for particle interactions.
Loop diagrams are taken into account, leading to the evaluation of {\color{black}loop integrals}.
The derivation of a closed analytic form is generally hard or impossible for
higher-order loop integrals with arbitrary internal masses and external momenta.
Thus we resort to numerical calculations.
A scalar $L$-loop integral with $N$ internal lines can be represented in Feynman parameter
space by
\begin{equation}
{\color{black}\mathcal I\hspace*{-0.4mm}
= {\mathcal I\hspace*{-0.4mm}}_N = (-1)^N \frac{\Gamma\left(N-\frac{{\color{black}n}L}{2}\right)}{(4\pi)^{{\color{black}n}L/2}}
\int_{0}^{1}\prod_{r=1}^{N}dx_{r}\, \delta(1-\sum x_{r})\,
\frac{1}{U^{n/2}(V-i{\color{black}\varrho})^{N-{\color{black}n}\,L/2}}},
\label{Lloop}
\end{equation}
where
\begin{equation}
V=M^2-\frac{1}{U}W,\qquad M^2=\sum m_r^2 x_{r} \nonumber
\label{LloopB}
\end{equation}
and $m_r$ is the mass for the propagator associated with $x_r.$
Here
$U$ and $W$ are polynomials determined by the topology of the corresponding
diagram and physical parameters ($U = 1$ for 1-loop ($L = 1$) integrals), and
$n$ is the space-time dimension.
We further denote
\begin{equation}
{\mathcal I\hspace*{-0.4mm}}_N ~=~
\frac{1}{(4\pi)^{\,n\,L/2}} \,I
~~=~~ (-1)^N \frac{\Gamma\left(N-\frac{n\,L}{2}\right)}{(4\pi)^{\,n\,L/2}} \,J,
\label{LloopIJ}
\end{equation}
defining $I$ and $J$ as integrals with a factor different from that of ${\mathcal I\hspace*{-0.4mm}}_N,$
in order to draw comparisons with results in the literature.
We sometimes also use the following notation for Feynman parameters,
\begin{equation}
x_{j\,k\cdots}=x_j+x_k+\cdots\,. \nonumber
\label{LloopC}
\end{equation}
The integration in Eq~\eqref{Lloop} is taken over the $N$-dimensional unit cube.
However, as a result of the $\delta$-function one of the $x_r$ can be expressed in terms
of the other ones in view of $\sum_{j=1}^N x_j = 1,$
which reduces the integral dimension to $N-1$ and the domain to
the $d = (N-1)$-dimensional unit simplex
\begin{equation}
{\mathcal S}_d = \{\,(x_1,x_2,\ldots,x_d)~\in~{\mathbb R}^d ~~|~
\sum_{j=1}^d x_j \le 1 \mbox{ and } x_j \ge 0\,\}.
\label{simplex}
\end{equation}
When the behavior of a singularity of the integrand is moderate, we can carry out the
integration within the unit simplex domain without variable transformation.
For the numerical integration where a steeper singularity appears,
the unit simplex domain of Eq~\eqref{Lloop} can be transformed
to the $(N-1)$-dimensional unit cube, using
\begin{small}
\begin{align}
& x_1 = \tilde{x}_1 \nonumber\\
& x_2 = (1-x_1)\,\tilde{x}_2 \nonumber\\
& x_3 = (1-x_1-x_2)\,\tilde{x}_3 \label{cubetrans} \\
& \ldots \nonumber\\
& x_{N-1} = (1-x_1-x_2-\ldots-x_{N-2})\,\tilde{x}_{N-1} \nonumber
\end{align}
\end{small}
with Jacobian ~$(1-x_1)\,(1-x_1-x_2)\ldots(1-x_1-x_2-\ldots-x_{N-2}),$ ~i.e.,
\begin{footnotesize}
\begin{align}
& \int_0^1 dx_1 \int_0^{1-x_1} dx_2 \int_0^{1-x_1-x_2-\ldots-x_{N-2}} \,dx_{N-1} \,f\,(x_1,x_2,\ldots,x_{N-1}) \label{transint} \\
& = \int_0^1 d\tilde{x}_1 ~(1-x_1) \int_0^1 d\tilde{x}_2 ~(1-x_1-x_2) \ldots \int_0^1 d\tilde{x}_{N-1} ~(1-x_1-x_2-\ldots-x_{N-2}) ~~f\,(x_1,(1-x_1)\,\tilde{x}_2,\ldots,(1-x_1-x_2-\ldots-x_{N-2})\,\tilde{x}_{N-1}) \nonumber.
\end{align}
\end{footnotesize}
We find that the approximations thus obtained are often more accurate than those generated with the
multivariate simplex rules in {\sc ParInt} (see Section~\ref{parint}), without the transformation.
Further, for some integrands with severe boundary singularities,
we use the \emph{double-exponential} transformation by Takahasi and
Mori~\cite{takahasi74,davis84,sugihara97}, which is
given in Section~\ref{DE}. We show examples of its application in
Sections~\ref{3-loop-massive-UV} and~\ref{uv-4ls}.
Furthermore, we introduce another type of variable transformations related to the topology of Feynman diagrams
to increase the accuracy of the results for some integrals in Sections~\ref{2ls},~\ref{uv-vertex} and~\ref{3-loop-massive-UV}. The integration domain is mapped to the
unit cube. Unlike the first two transformations, we determine
the latter using a heuristic approach.
Loop integrals are notorious for singularities due to vanishing
denominators, which may lead to divergence (e.g., IR or UV divergence) of the integral.
In the absence of IR and UV singularities, we have ${\color{black}n = 4}.$
For {\color{black}dimensional regularization} in case of IR singularities we set
${\color{black}n = 4+2\varepsilon}$ ~(cf.,~\cite{acat11}), and for UV singularities,
${\color{black}n = 4-2\varepsilon}.$ We apply the regularization by a
numerical extrapolation as ${\color{black}\varepsilon \rightarrow 0}$
(cf., Section~\ref{extrap-exp}).
The term \,$i\,{\color{black}\hspace*{-0.3mm}\varrho}$\, prevents
the integrand denominator in Eq~\eqref{Lloop} from vanishing in the interior of the domain, and can be used for regularization.
A regularization to keep the integral from diverging was achieved by extrapolation as
$\varrho \rightarrow 0$ in~\cite{edcpp03,eddacat03,acat07,acat08,ddacat10,cpc11y,jocs11}.
Results given in~\cite{cpp10} applied iterated
integration with {\sc Quadpack} programs and a double extrapolation with respect to
$\varrho$ and $\varepsilon$ to deal with interior as well as IR singularities.
However, even for finite integrals,
setting $\varepsilon = 0$ or $\varrho = 0$ in the integrand of Eq~\eqref{Lloop} may not yield the
desired accuracy and it may be advantageous to extrapolate as $\varepsilon \rightarrow 0$
or $\varrho \rightarrow 0$
(cf., Section~\ref{3ls-massless-finite-integrals}).
\subsection{Numerical extrapolation}
\label{extrap-exp}
For an extrapolation with respect to the dimensional regularization parameter $\varepsilon,$
the integral of Eq~\eqref{Lloop} is evaluated as a sequence of
${\cal I}(\varepsilon)$ for decreasing $\varepsilon = \varepsilon_\ell,$ which assumes
an asymptotic expansion of the form of Eq~\eqref{asymp} for $\gamma = \varepsilon.$
For example, the $\varphi_k(\varepsilon)$ functions in Eq~\eqref{asymp} may be integer powers of $\varepsilon,$
$\varphi_k(\varepsilon) = \varepsilon^k.$
Then for finite integrals, $\kappa = 0$ in Eq~\eqref{asymp} and the integral is represented by $C_0.$
{\color{black}Linear extrapolation} can be applied when the $\varphi_k(\varepsilon)$ functions are known.
In that case, ${\cal I}(\varepsilon)$ is approximated for decreasing values of $\varepsilon = \varepsilon_\ell,$
and Eq~\eqref{asymp} is truncated after $2, 3, \ldots, \nu$ terms to form linear systems
of increasing size in the $C_k$ variables. This is a generalized form of Richardson
extrapolation~\cite{brezinski80,sidi03}.
If the integral approximation becomes harder with smaller $\varepsilon,$ we can use
slowly decreasing sequences $\{\varepsilon_\ell\},$ such as a geometric sequence with base $1/1.15.$
Another sequence of interest is based on the Bulirsch sequence $\{b_\ell\}: 1,2,3,4,6,8,12,16,24,\ldots$ (see~\cite{bulirsch64}); we employ
$\{1/b_j\}_{j\ge j_0},$ from a starting index $j_0$ in the Bulirsch sequence.
The stability of linear extrapolation using geometric, harmonic and Bulirsch type sequences
was studied by Lyness~\cite{lyness76} with respect to the mesh ratio
of composite rules.
The condition of the system was found best for geometric and worse for the harmonic sequences,
with the Bulirsch sequence behavior in between.
We resort to non-linear extrapolation when the structure of the asymptotic expansion is not
known. In previous work we have made ample use of the $\epsilon$-algorithm~\cite{shanks55,wynn56,sidi96,sidi03,sidi11},
which can be applied with geometric sequences of $\varepsilon.$
The extrapolation results given in this paper are
achieved with a version of the $\epsilon$-algorithm code from {\sc Quadpack}~\cite{pi83}.
In between calls, the implementation retains the last two lower diagonals of the triangular
extrapolation table. When a new element $\mathcal I(\varepsilon_\ell)$ of the input sequence is supplied,
the algorithm calculates a new lower diagonal, together with an estimate or measure of the \emph{distance}
of each newly computed element from preceding neighboring elements.
With the location of the ``\emph{new}" element in the table relative to $e_0, ~e_1,
~e_2, ~e_3,$ pictured as:~~
\begin{tabular}{ccc}
& $e_0$ & \\
$e_3$ & $e_1$ & \mbox{\emph{new}}\\
& $e_2$ &
\end{tabular}
~~~we have that $\mbox{\emph{new}} = e_1+1/(1/(e_1-e_3)+1/(e_2-e_1)-1/(e_1-e_0)),$
and the distance measure for the \emph{new}
element is set to $|e_2-e_1| + |e_1-e_0| + |e_2-\mbox{\emph{new}}|.$
The new lower diagonal element with the smallest value of the distance measure is then returned
as the result for this call to the extrapolation code.
Note that the accuracy of the extrapolated result is generally limited by the accuracy of the
input sequence.
For an extrapolation as $\varrho \rightarrow 0$ in Eq~\eqref{Lloop},
the integral is approximated by a sequence of numerical results for
${\cal I}(\varrho_\ell)$ with decreasing $\varrho_\ell.$
An asymptotic expansion of the form of Eq~\eqref{asymp},
${\cal I}(\varrho) \sim \sum_{k\,\ge \,\kappa} C_k\,\varphi_k(\varrho)$
for $\varrho = \varrho_\ell$
is assumed, where the $\varphi_k(\varrho)$ functions are generally unknown, and
we perform non-linear extrapolation with the $\epsilon$-algorithm.
\section{Numerical Integration Methods}\label{integration}
Though various integration methods may be applicable in our approach,
we currently use three types of integration methods as presented in subsequent sections.
These are: numerical iterated integration, parallel adaptive integration and double-exponential
transformation methods.
\subsection{Numerical iterated integration}\label{multiter}
\input{cpc15paper-sec3}
\subsection{{\sc ParInt} package}\label{parint}
\input{cpc15paper-sec4}
\subsection{Double-exponential transformation}\label{DE}
\input{cpc15paper-sec5}
\section{Parallel numerical iterated integration}\label{multiter}
For iterated integration over a $d$-dimensional
product region we express Eq~\eqref{blackbox} as
\begin{equation}
{\mathcal I\hspace*{-0.4mm}}f = \int_{\alpha_1}^{\beta_1} dx_1 \int_{\alpha_2}^{\beta_2} dx_2 \ldots
\int_{\alpha_d}^{\beta_d} dx_d ~f(x_1,x_2,\ldots,x_d),
\label{iterint}
\end{equation}
where the limits of integration are given by functions $\alpha_j =
\alpha_j\,(x_1,x_2,\ldots,x_{j-1})$ and $\beta_j = \beta_j\,(x_1,x_2,\ldots,x_{j-1}).$
In particular, the boundaries of the $d$-dimensional unit simplex ${\mathcal S}_d$
given by Eq~\eqref{simplex}
are $\alpha_j = 0$ and $\beta_j = 1-\sum_{k=1}^{j-1} x_k.$
For the numerical integration over the interval $[\alpha_j,\beta_j], ~1 \le j \le d$ in Eq~\eqref{iterint} we can apply, e.g., the
1D adaptive integration code {\sc Dqage} from
the {\sc Quadpack} package~\cite{pi83} in each coordinate direction,
and select the $(K = 15)$-point Gauss-Kronrod rule pair via an input parameter,
for the integral (and error) approximation on each subinterval.
If an interval $[a,b]$ arises in the partitioning of \,$[\alpha_j,\beta_j],$
then the local integral approximation over $[a,b]$ is of the form
\begin{equation}
\int_a^b dx_j ~{\mathcal F}(c_1,\ldots,c_{j-1},x_j) ~\approx~ \sum_{k=1}^K w_k \,{\mathcal F}(c_1,\ldots,c_{j-1},x^{(k)}),
\label{localab}
\end{equation}
where the $w_k$ and $x^{(k)}, 1 \le k \le K,$ are the weights and abscissae of the local rule
scaled to the interval $[a,b]$ and applied in the $x_j$-direction. For $j = 1$ this is the
outer integration direction.
The function evaluation
\begin{equation}
{\mathcal F}(c_1,\ldots,c_{j-1},x^{(k)}) = \int_{\alpha_{j+1}}^{\beta_{j+1}} dx_{j+1} \ldots
\int_{\alpha_d}^{\beta_d} dx_d ~f\,(c_1,\ldots,c_{j-1},x^{(k)},x_{j+1},\ldots,x_d) , ~~~~~1 \le k \le K,
\label{innerintegral}
\end{equation}
is itself an integral in the $x_{j+1},\ldots,x_d$\,-\,directions for $1 \le j < d,$ and is computed by the
method(s) for the inner integrations.
For $j = d,$ Eq~\eqref{innerintegral} is the evaluation of the integrand function
\begin{equation}
{\mathcal F}(c_1,\ldots,c_{d-1},x^{(k)}) = f\,(c_1,\ldots,c_{d-1},x^{(k)}). \nonumber
\end{equation}
Note that successive coordinate directions may be combined into layers in the iterated
integration scheme.
Furthermore, the error incurred in any inner integration
will contribute to the integration error in all of its subsequent
outer integrations~\cite{fritsch81,kahaner88,iccs10}.
Since the ${\mathcal F}(\,)$ evaluations on the right of Eq~\eqref{localab} are independent of one another
they can be evaluated in parallel.
Important benefits of this approach include that:\\
\emph{(i)} the granularity of the parallel integration is large, especially when the inner
integrals ${\mathcal F(\,)}$ are of dimension $\ge 2;$\\
\emph{(ii)} the points where the function ${\mathcal F}$ is evaluated in parallel
are the same as those of the sequential evaluation; i.e.,
apart from the order of the summation in Eq~\eqref{localab},
the parallel calculation is essentially the same as the sequential one.
This important property facilitates the debugging of parallel code.
As another characteristic, the parallelization does not increase the total amount of computational work.
In addition, the memory required for the procedure is determined by (the sum of)
the amounts of memory needed for the data pertaining to the subintervals incurred in each
coordinate direction (corresponding to the length of the recursion stack for a recursive implementation).
Consequently the total memory increases linearly as a function of the dimension $d.$
Note that successive coordinate directions may be combined into layers in the iterated
integration scheme.
To achieve the multi-threading, OpenMP~\cite{openmp} compiler directives were inserted in the iterated
integration code. For the Fortran version of {\sc Quadpack} we used the (GNU) \emph{gfortran} compiler
and the Intel Fortran compiler, with the flags -\emph{fopenmp} and -\emph{openmp}, respectively.
\section{{\sc ParInt} package}\label{parint}
Written in C and layered over MPI~\cite{openmpi}, the {\sc ParInt} methods
(parallel adaptive, quasi-Monte Carlo and Monte Carlo)
are implemented as tools for \emph{automatic}
integration, where the user defines the integrand function and the domain, and specifies
a relative and absolute error tolerance for the computation ($t_r$ and $t_a,$ respectively).
For {\sc Parint} the integrand is generally defined as a vector function
with $m$ components,
\begin{equation}
\vec{f}: {\mathcal D}\subset {\mathbb R}^d \rightarrow {\mathbb R}^m,
\label{function}
\end{equation}
over a (finite) $d$-dimensional (hyper-rectangular or simplex) domain $\mathcal D.$
Denoting the exact integral by
\begin{equation}
{\mathcal I\hspace*{-0.4mm}}\vec{f} = \int_{\mathcal D} \vec{f}\,(\vec{x})~d\vec{x},
\label{general}
\end{equation}
then the objective of Eq~\eqref{accuracy} is generalized to
returning an approximation ${\mathcal Q}\vec{f}$ and absolute error estimate $E_a\vec{f}$ such that
\begin{equation}
||\, {\mathcal Q}\vec{f}-{\mathcal I\hspace*{-0.4mm}}\vec{f} \,|| \le ||\, E_a\vec{f} \,|| \le \max\{\, t_a, \,t_r\, ||\, {\mathcal I\hspace*{-0.4mm}}\vec{f} \,|| \,\}
\label{acc}
\end{equation}
(in infinity norm).
In order to satisfy the error criterion of Eq~\eqref{acc} the program tests throughout whether
$$||\, E_a\vec{f} \,|| \le \max\,\{\, t_a, \,t_r\, ||\, {\mathcal Q}\vec{f} \,|| \,\}$$
is achieved.
We used the vector function integration capability in~\cite{ccp14} for a simultaneous
computation of the entire entry sequence for extrapolation, obtained as the $m$ components
of the integral ${\mathcal I\hspace*{-0.4mm}}\vec{f}.$
The available
cubature rules in {\sc ParInt} (to compute the integral approximation over the domain or
its subregions) include a set of rules for the $d$-dimensional cube~\cite{ge80,ge83,gnzbe91a},
the 1D (Gauss-Kronrod) rules used in {\sc Quadpack} and a set of rules for the $d$-dimensional simplex~\cite{gnz90,gm78,ddmathcomp79}.
Some results in this paper are computed over the $d$-dimensional simplex using
iterated integration with Gauss-Kronrod rules.
In other cases, multivariate rules of polynomial degree 7 or 9 are used
over the $d$-dimensional unit cube.
A formula is said to be of a particular polynomial degree $k$
if it renders the exact value of the integral for integrands
that are polynomials of degree $\le k,$ and there are polynomials of degree $k+1$ for which the
formula is not exact.
The number of function evaluations per (sub)region is constant, and the total number of
subregions generated, or the number of function evaluations in the course of the integration,
is considered a measure of the computational effort.
\subsubsection{{\sc ParInt} adaptive methods}
\label{sect:parint-adaptive}
In the adaptive approach, the integration domain is divided initially among the workers. Each on its own part of the
domain, the workers engage in an adaptive partitioning strategy similar to that of {\sc Dqage}
from {\sc Quadpack}~\cite{pi83} and of {\sc Dcuhre}~\cite{gnzbe91bn} by successive bisections.
The workers then each generate a local priority queue as a task pool of subregions.
\begin{wrapfigure}[9]{r}{2.1in}
\begin{small}
\vspace*{-7mm}
\begin{tabbing}
88\=This\= is the adaptive meta-algorit\=hm. \kill
\> Evaluate initial region and update results \\
\> Initialize priority queue with initial region \\
\> {\bf while} (eval. limit not reached and estim. err. $>$ tolerance) \\
\> \> Retrieve region from priority queue \\
\> \> Split region \\
\> \> Evaluate new subregions and update results~~~~ \\
\> \> Insert new subregions into priority queue
\end{tabbing}
\end{small}
\vspace*{-4mm}
\caption{Adaptive integration meta-algorithm}
\label{meta-alg}
\end{wrapfigure}
\indent
The priority queue is implemented as a max-heap
keyed with the estimated integration errors over the subregions,
so that the subregion with the largest estimated error is stored in the root of the heap.
If the user specifies a maximum size for the heap structure on the worker,
the task pool is stored as a \emph{deap} or \emph{double-ended heap}, which allows
deleting of the maximum as well as the minimum element efficiently, in order to maintain a constant size
of the data structure once it reaches its maximum.
A task consists of the selection of the associated subregion and its subdivision
(generating two children regions), integration over the children, deletion of the parent region
(root of the heap) and insertion of the children into the heap (see Figure~\ref{meta-alg}).
The bisection of a region is performed perpendicularly to the coordinate direction in which the
integrand is found to vary the most, according to $4^{th}$-order differences computed in each
direction~\cite{gnzbe91bn}.
The subdivision procedure continues until
the global error estimate falls below the tolerated error, or
the total number of function evaluations exceeds the user-specified maximum.
\subsubsection{Load balancing}
\label{sect:loadbalancing}
For a regular integrand behavior and $p$ MPI processes distributed evenly over homogeneous
processors, the computational load would ideally decrease by a factor of about $p.$
Otherwise it may be possible to improve the parallel time (and space) usage by load balancing,
to attempt keeping the loads on the worker task pools balanced.
The receiver-initiated, scheduler based load balancing strategy in {\sc ParInt} is
an important mechanism of the distributed integration algorithm~\cite{DGE96,cpp01,akdpdcs03,adkvpdcs04}.
The message passing is performed in a non-blocking and asynchronous manner, and permits overlapping of
computation and communication,
which benefits {\sc ParInt}'s efficiency
on a hybrid platform (multi-core and distributed) where multiple processes are assigned to each node.
As a result of the asynchronous processing and message passing on MPI, {\sc ParInt}
executes on a hybrid platform by assigning multiple processes across the nodes.
The user has the option of turning load balancing on or off,
as well as allowing or dis-allowing the controller to also act as a worker.
\subsubsection{Use of {\sc ParInt}}
\label{use}
{\sc ParInt} can be invoked from the command line, or by calling the {\verb pi_integrate() } function
in a program for computing an integral of the form of Eq~\eqref{general}. A user guide is provided in~\cite{parintweb}.
The call sequence passes a pointer to the integrand function, typed as a pointer to
a function that returns an integer, and where the parameters of the integrand function
correspond to the integral dimension, argument vector $\vec{x},$ number of component functions
{\verb nfuncs } (corresponding to $m$ in Eq~\eqref{function}) and the resulting
component values of the function $\vec{f}\,(x).$
Apart from {\verb nfuncs }, further input parameters of {\verb pi_integrate() } are: an integer
identifying the cubature/quadrature rule to use, the
maximum number of function evaluations allowed, the region type (hyper-rectangle
or simplex) and specification. The output parameters are: the integral and error component approximations
{\verb result[] } and {\verb error[] }, and a user-declared pointer to a status structure.
The execution time is returned as part of the output
printed by the {\sc ParInt} \emph{pi\_print\_results()} function.
When {\sc ParInt} is used as a stand-alone executable, it uses the {\sc ParInt} Plug-in Library
(PPL) mechanism to specify integrand functions. The functions are written by the user, added to the
library (along with related attributes), and then compiled using a {\sc ParInt}-supplied compiler into \emph{plug-in
modules} (.ppl files). A single PPL file is loaded at runtime by the {\sc ParInt} executable.
Using a function library enables quick access to a predefined set of functions and lets {\sc ParInt}
users add and remove integrand functions dynamically without re-compiling the {\sc ParInt} binary.
Once these functions are stored in the library, they can be selected by name for integration.
For an execution on MPI, the MPI host file ({\verb myhostfile }) contains lines of the
form: {\verb"node_name slots=ppn"} where {\verb"ppn"} is the number of processes to be used on each
participating node.
A typical MPI run from the command line may be of the form\\
\begin{small}
{\verb"mpirun -np 64 --hostfile myhostfile ./parint -f fcn -lf 10000000 "
{\verb"-ea 0.0 -er 5.0e-10"}\\
\end{small}
For example, with four nodes listed in ~{\verb myhostfile }~ and ~{\verb"ppn = 16"},~
a total of 64 processes is requested on the specified nodes.
The integrand function of this run is named {\verb fcn } in the user's library; the maximum number of function
evaluations is 10000000, and the absolute and relative error tolerances are 0 and 5.0e-10, respectively.
Optionally the {\sc ParInt} installation can be configured to use long doubles instead of doubles.
\section{Double exponential transformation}\label{DE}
The \emph{Double Exponential (tanh-sinh) formula}, referred to here as \emph{DE formula} in short, was proposed by Takahasi and Mori in 1974~\cite{takahasi74,davis84,sugihara97}.
It is an efficient method for the numerical approximation of an integral whose integrand is a holomorphic function
with end-point singularities. This formula transforms the integration variable in $\int_{0}^{1} f(x)\,dx$ to~
$x=\phi\,(t)=\frac{1}{2}(\rm{tanh\,(\frac{\pi}{2}\rm{sinh\,(t)})+1}).$
Then ~${\mathcal I\hspace*{-0.4mm}}=\int_{0}^{1} f(x) \,dx =\int_{-\infty}^{\infty} f\,(\phi(t))\,\phi'(t)\,dt$ ~with~
$\phi'(t)=\frac{\pi\,\rm{cosh\,(t)}}{4\,\rm{cosh^2(\frac{\pi}{2}\rm{sinh\,(t))}}}.$
After the transformation, the trapezoidal rule is applied leading to
\begin{equation}
I_{h}^{N_{eval}}=\sum_{k=-N_{-}}^{k=N_{+}}f\,(\phi\,(kh))\,\phi'(kh),
\label{DEsum}
\end{equation}
with mesh size $h$ and $N_{eval}=N_{-}+N_{+}+1$ function evaluations.
A major issue in numerical integration with the DE formula is the treatment of overflows
at large $|t|$ and for
large $|N_{eval}|,$ and it is sometimes necessary to evaluate the integrand using multi-precision arithmetic even though it takes more CPU time than double precision.
This helps alleviating the loss of trailing digits in the evaluation of the integrand near the end-points.
For multi-dimensional loop integrals, we use the DE formula in a repeated integration scheme.
Apart from our sequential implementation we also developed code for multi-core systems using a parallel
library such as OpenMP or a compiler with auto-parallelization capabilities.
For an execution in a multi-precision environment, a dedicated accelerator system
consisting of multiple FPGA (Field Programmable Gate Array) boards was
developed and its performance results were presented by Daisaka et al.~\cite{daisaka15}.
\section{2-loop integrals with massive internal lines}
\label{2-loop-integral}
In this section we calculate the integral $I$ of Eq~\eqref{LloopIJ} for $L=2$ and $n=4-2\varepsilon$
according to
\begin{eqnarray}
\label{UV}
I &=& (-1)^N {\Gamma\left(N-4+2\varepsilon \right)}
\int_{0}^{1}\prod_{r=1}^{N}dx_{r}\, \delta(1-\sum x_{r})\frac{1}{U^{2-\varepsilon}(V-i\varrho)^{N-4+2\varepsilon}}\\
&=& (-1)^N {\Gamma\left(N-4+2\varepsilon \right)}
\int_{{\mathcal S}_{N-1}} \frac{1}{U^{2-\varepsilon}(V-i\varrho)^{N-4+2\varepsilon}}.\nonumber
\end{eqnarray}
IR divergence occurs through a singularity arising when $V$
vanishes at the boundaries of the domain. This problem can be addressed by dimensional regularization
with ${\color{black}n = 4+2\varepsilon},$ which we implemented numerically in~\cite{cpp10,acat11,jocs11}
using an extrapolation as $\varepsilon\rightarrow 0$ ($\varepsilon > 0$).
It is assumed that the denominator does not vanish in the interior of the integration domain,
so we can set $\varrho = 0.$
In this paper, we concentrate on UV divergence, which occurs when $U$ vanishes at the boundaries. The $\Gamma$-function in Eq~\eqref{UV} contributes to UV divergence when $N\le 4$.
We treat UV divergence by
a dimensional regularization with ${\color{black}n = 4-2\varepsilon},$ implemented by a numerical
extrapolation as $\varepsilon\rightarrow 0$ after either iterated integration
with {\sc Dqags} from {\sc Quadpack},
multivariate adaptive integration with {\sc ParInt} or the DE formula.
\subsection{2-loop self-energy integrals}\label{2ls}
Fig~\ref{2-loop-UV-diagrams} depicts 2-loop self-energy diagrams with $N = 3, \,4$ and 5
internal lines. We refer to the 2-loop self-energy diagrams (a-d) as the
\emph{sunrise-sunset}, \emph{lemon}, \emph{half-boiled egg} and \emph{Magdeburg} diagrams, and to
the corresponding integrals as $I_a^{S2}$, $I_b^{S2}$, $I_c^{S2}$ and $I_d^{S2}$, respectively.
As show in in Fig~\ref{2-loop-UV-diagrams}, the entering momentum is $p,$ and we
denote $s=p^2$.
\begin{figure}
\begin{center}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.22\linewidth]{./figures/2ls-sunset-20161029.eps}
\end{subfigure}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.22\linewidth]{./figures/2ls-lemon-20161029.eps}
\end{subfigure}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.22\linewidth]{./figures/2ls-halfboiled-20161029.eps}
\end{subfigure}
\vspace*{0.2in}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.22\linewidth]{./figures/2ls-magdeburg-20161029.eps}
\end{subfigure}
\caption{2-loop self-energy diagrams with massive internal lines:
(a)~2-loop \emph{sunrise-sunset} $N = 3$ (Laporta~\cite{laporta01}\,, Fig 2(b)),
(b)~2-loop \emph{lemon} $N = 4$ (Laporta~\cite{laporta01}\,, Fig 2(c)),
(c)~2-loop \emph{half-boiled egg} $N = 5$,
(d)~2-loop \emph{Magdeburg} $N = 5$ ((Laporta~\cite{laporta01}\,, Fig 2(d))}.
\label{2-loop-UV-diagrams}
\end{center}
\end{figure}
Analytic results for the
integrals have been derived by many authors. We use the following formulas
for the functions $U$ and $V$ in Eq~\eqref{UV}:\\
$-$ for the \emph{\color{black}sunrise-sunset diagram}~(Fig~\ref{2-loop-UV-diagrams}(a)),
$U = x_1 x_2 + x_2 x_3 + x_3 x_1, ~W/s = x_1 x_2 x_3;$ \\
$-$ for the \emph{\color{black}lemon diagram}~(Fig~\ref{2-loop-UV-diagrams}(b)), we have
$U = x_{12} x_{34} + x_1 x_2, ~W/s = x_4\,(x_1 x_2 + x_2 x_3 + x_3 x_1);$ \\
$-$ for the \emph{\color{black}half-boiled egg diagram}~(Fig~\ref{2-loop-UV-diagrams}(c)),
$U = x_{12} x_{345} + x_1 x_2, ~W/s = x_5\,(x_{12} x_{34} + x_1 x_2);$\\
$-$ for the \emph{Magdeburg diagram}~(Fig~\ref{2-loop-UV-diagrams}(d)),
$U = x_{14} x_{25} + x_3 x_{1245}, ~W/s = x_{1}x_{4}x_{235} + x_{2}x_{5}x_{134} + x_3\,(x_1x_5 + x_2x_4).$\\
In the numerical evaluation,
we take particular values for energy and masses, i.e., $s = p^2 = 1,$ and all masses $m_r=1$
in order to make comparisons with results in the references.
The integrals are expanded with respect to the dimensional regularization parameter $\varepsilon.$
The integrals $I_a^{S2}, I_b^{S2}$ are divergent as $\displaystyle{1/\varepsilon^2},$ which is
the product of $\displaystyle{1/\varepsilon}$ from the $\Gamma$-function part and
$\displaystyle{1/\varepsilon}$ from the integral part.
The integral $I_c^{S2}$ is divergent as $\displaystyle{1/\varepsilon},$ which
comes from the integral part. The integral $I_d^{S2}$ is finite.
The expansions are of the form of Eq~\eqref{asymp},
\begin{equation}
{\color{black}S(\varepsilon) \sim \sum_{k\ge \kappa} C_k\,\varepsilon^k~~~~~~\mbox{as~~} \varepsilon\rightarrow 0},
\label{expand}
\end{equation}
and we use linear extrapolation to approximate the coefficients of the leading terms.
We multiply the integrals $I_{a}^{S2},$ $I_{b}^{S2}$ and $I_{d}^{S2}$ of Eq~\eqref{UV}
with the factor $ \Gamma(1+\varepsilon)^{-2}$ for comparison with the results
in Laporta~\cite{laporta01}, since the latter are computed with this factor.
The \emph{\color{black}half-boiled egg diagram} is not covered in~\cite{laporta01}.
We give the analytic formula for $J_c^{S2}$ in Appendix A of this paper.
\begin{eqnarray}
I_{a}^{S2}({\varepsilon}) ~\Gamma(1+{\varepsilon})^{-2} = \sum_{k\ge -2} C_k \,{\varepsilon^k} &=& -1.5\,{\varepsilon^{-2}}-4.25\,{\varepsilon^{-1}}-7.375-17.22197253479\,{\varepsilon}\ldots \label{Isexp}\\
\nonumber
I_{b}^{S2}({\varepsilon}) ~\Gamma(1+{\varepsilon})^{-2} = \sum_{k\ge -2} C_k \,{\varepsilon^k} &=& 0.5\,{\varepsilon^{-2}}+0.6862006357658\,{\varepsilon^{-1}}-0.6868398873414\\
&+&1.486398391913\,{\varepsilon}\ldots \label{Ilexp}\\
\nonumber
\\
\nonumber
J_{c}^{S2}({\varepsilon}) = \sum_{k\ge -1} C_k \,{\varepsilon^k} &=& 0.6045997880781\,{\varepsilon^{-1}}-0.1756970002260\,-0.2977242542666\,{\varepsilon}\\
&+&0.4140155361099\,{\varepsilon^2}\ldots \label{Ihexp}\\
\nonumber \\
\nonumber
I_{d}^{S2}({\varepsilon}) ~\Gamma(1+{\varepsilon})^{-2} = \sum_{k\ge 0} C_k \,{\varepsilon^k} &=& 0.9236318265199-1.284921671848\,{\varepsilon}+2.689507626490\,{\varepsilon^2}\\
&-&5.338399227511\,{\varepsilon^3}\ldots \label{Imexp}
\end{eqnarray}
Note that the value of $\kappa$ in Eq~\eqref{expand} corresponds with the index of the first coefficient
$C_\kappa$ in the expansion. In that case we find that, if $\kappa$ is replaced by
$\kappa-1$ for the extrapolation, then the first coefficient converges to $C_{\kappa-1} = 0.$
\begin{table}
\caption{\footnotesize{ Results UV \emph{sunrise-sunset} integral, $I_{a}^{S2} \,\Gamma(1+\varepsilon)^{-2}$ \,(on Mac Pro),~
rel. err. tol. $t_r = 10^{-9}$ (outer), $10^{-9}$ (inner), $T[s]$ = Time (elapsed user time in $s$),~ $\varepsilon = 1.2^{-10-\ell}$ (starting at $1.2^{-10}$),~~$E_r = $ outer integration estim. rel. error}}
\begin{scriptsize}
\begin{center}
\begin{tabular}{ccccccc}\hline
& \multicolumn{6}{l}{{\sc Integral ~$I_{a}^{S2} \,\Gamma(1+\varepsilon)^{-2}$ ~Extrapolation}} \\
\hspace*{-1mm}$\ell$\hspace*{-1mm} & \hspace{-1mm}{ $E_r$ }\hspace{-1mm} & \hspace{-1mm}{\sc T[s]}
\hspace{-1mm} & {\sc Res.} ~$C_{-2}$ & \hspace{-1mm}{\sc Res.} ~$C_{-1}$\hspace{-1mm} & \hspace*{-0mm}{\sc Res.} ~$C_0$ & \hspace{-1mm}{\sc Res.} ~$C_1$ \\% & & & & & & & \\
\hlin
\hspace*{-1mm}$0$\hspace*{-1mm}& \hspace{-1mm}3.2e-10\hspace{-1mm} & \hspace{-1mm}0.015\hspace{-1mm} & \hspace{-1mm}\hspace{-1mm} & \hspace{-1mm}\hspace{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$1$\hspace*{-1mm}& \hspace{-1mm}4.7e-10\hspace{-1mm} & \hspace{-1mm}0.013\hspace{-1mm} & \hspace{-1mm}-1.156740414\hspace{-1mm} & \hspace{-1mm}-8.2070492\hspace{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$2$\hspace*{-1mm}& \hspace{-1mm}6.6e-10\hspace{-1mm} & \hspace{-1mm}0.013\hspace{-1mm} & \hspace{-1mm} -1.603088981\hspace{-1mm} & \hspace{-1mm}-2.1269693\hspace{-1mm} & \hspace*{-1.5mm}-20.5343\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$3$\hspace*{-1mm}& \hspace{-1mm}9.2e-10\hspace{-1mm} & \hspace{-1mm}0.013\hspace{-1mm} & \hspace{-1mm}-1.476861131\hspace{-1mm} & \hspace{-1mm}-4.9718825\hspace{-1mm} & \hspace*{-1.5mm}0.60362\hspace*{-1mm} & \hspace{-3mm}-51.7769\hspace{-1mm} \\
\hspace*{-1mm}$4$\hspace*{-1mm}& \hspace{-1mm}4.0e-11\hspace{-1mm} & \hspace{-1mm}0.028\hspace{-1mm} & \hspace{-1mm}-1.504342324\hspace{-1mm} & \hspace{-1mm}-4.0584835\hspace{-1mm} & \hspace*{-1.5mm}-10.6252\hspace*{-1mm} & \hspace{-3mm}\,8.73336\hspace{-1mm} \\
\hspace*{-1mm}$5$\hspace*{-1mm}& \hspace{-1mm}6.9e-11\hspace{-1mm} & \hspace{-1mm}0.029\hspace{-1mm} & \hspace{-1mm}-1.499360223\hspace{-1mm} & \hspace{-1mm}-4.2880409\hspace{-1mm} & \hspace*{-1.5mm}-6.46343\hspace*{-1mm} & \hspace{-3mm}-28.3731\hspace{-1mm} \\
\hspace*{-1mm}$6$\hspace*{-1mm}& \hspace{-1mm}1.3e-10\hspace{-1mm} & \hspace{-1mm}0.029\hspace{-1mm} & \hspace{-1mm}-1.500078550\hspace{-1mm} & \hspace{-1mm}-4.2438757\hspace{-1mm} & \hspace*{-1.5mm}-7.57341\hspace*{-1mm} & \hspace{-3mm}-13.7781\hspace{-1mm} \\
\hspace*{-1mm}$7$\hspace*{-1mm}& \hspace{-1mm}2.2e-10\hspace{-1mm} & \hspace{-1mm}0.028\hspace{-1mm} & \hspace{-1mm}-1.499992106\hspace{-1mm} & \hspace{-1mm}-4.2507887\hspace{-1mm} & \hspace*{-1.5mm}-7.34157\hspace*{-1mm} & \hspace{-3mm}-18.0041\hspace{-1mm} \\
\hspace*{-1mm}$8$\hspace*{-1mm}& \hspace{-1mm}3.6e-10\hspace{-1mm} & \hspace{-1mm}0.028\hspace{-1mm} & \hspace{-1mm}-1.500000763\hspace{-1mm} & \hspace{-1mm}-4.2499043\hspace{-1mm} & \hspace*{-1.5mm}-7.38015\hspace*{-1mm} & \hspace{-3mm}-17.0657\hspace{-1mm} \\
\hspace*{-1mm}$9$\hspace*{-1mm}& \hspace{-1mm}5.4e-10\hspace{-1mm} & \hspace{-1mm}0.029\hspace{-1mm} & \hspace{-1mm} -1.499999886\hspace{-1mm} & \hspace{-1mm}-4.2500174\hspace{-1mm} & \hspace*{-1.5mm}-7.37374\hspace*{-1mm} & \hspace{-3mm}-17.2650\hspace{-1mm} \\
\hspace*{-1mm}$10$\hspace*{-1mm}& \hspace{-1mm}7.8e-10\hspace{-1mm} & \hspace{-1mm}0.029\hspace{-1mm} & \hspace{-1mm}-1.500000026\hspace{-1mm} & \hspace{-1mm}-4.2499948\hspace{-1mm} & \hspace*{-1.5mm}-7.37544\hspace*{-1mm} & \hspace{-3mm}-17.2010\hspace{-1mm} \\
\hline
\multicolumn{3}{r}{Eq~\eqref{Isexp}:} & -1.5 & -4.25 &\hspace*{-1.0mm} -7.375 & \hspace{-3mm}~-17.2220\\ \hline
\end{tabular}
\end{center}
\end{scriptsize}
\label{table1}
\end{table}
\begin{table}
\caption{\footnotesize{ {\color{black}Results UV \emph{lemon} integral}, $I_{b}^{S2} \,\Gamma(1+\varepsilon)^{-2}$ \,(on Mac Pro),~
rel. err. tol. $t_r = 10^{-10}$ (outer), $5\times 10^{-11}$ (inner two), $T[s]$ = Time (elapsed user time in $s$),~ $\varepsilon = 1/b_\ell$ (starting at 1/4),~~$E_r = $ outer integration estim. rel. error}}
\begin{scriptsize}
\begin{center}
\begin{tabular}{ccccccc}\hline
& \multicolumn{6}{l}{{\sc Integral ~$I_{b}^{S2} \,\Gamma(1+\varepsilon)^{-2}$~ Extrapolation}} \\
\hspace*{-1mm}$b_\ell$\hspace*{-1mm} & \hspace{-1mm}{ $E_r$ }\hspace{-1mm} & \hspace{-1mm}{\sc T[s]}
\hspace{-1mm} & {\sc Res.} ~$C_{-2}$ & \hspace{-1mm}{\sc Res.} ~$C_{-1}$\hspace{-1mm} & \hspace*{-0mm}{\sc Res.} ~$C_0$ & \hspace{-1mm}{\sc Res.} ~$C_1$\hspace{-1mm} \\% & & & & & & & \\
\hlin
\hspace*{-1mm}$4$\hspace*{-1mm}& \hspace{-1mm}3.5e-11\hspace{-1mm} & \hspace{-1mm}0.36\hspace{-1mm} & \hspace{-1mm}\hspace{-1mm} & \hspace{-1mm}\hspace{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$6$\hspace*{-1mm}& \hspace{-1mm}8.8e-11\hspace{-1mm} & \hspace{-1mm}0.34\hspace{-1mm} & \hspace{-1mm}0.5130221162587\hspace{-1mm} & \hspace{-1mm}0.52467607220\hspace{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$8$\hspace*{-1mm}& \hspace{-1mm}2.9e-12\hspace{-1mm} & \hspace{-1mm}0.40\hspace{-1mm} & \hspace{-1mm}0.5031467341833\hspace{-1mm} & \hspace{-1mm}0.62342989295\hspace{-1mm} & \hspace*{-1.5mm}-0.237009170\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$12$\hspace*{-1mm}& \hspace{-1mm}3.4e-12\hspace{-1mm} & \hspace{-1mm}0.41\hspace{-1mm} & \hspace{-1mm}0.5004379328119\hspace{-1mm} & \hspace{-1mm}0.67218831764\hspace{-1mm} & \hspace*{-1.5mm}-0.518724512\hspace*{-1mm} & \hspace{-1mm}0.52008986\hspace{-1mm} \\
\hspace*{-1mm}$16$\hspace*{-1mm}& \hspace{-1mm}1.5e-11\hspace{-1mm} & \hspace{-1mm}0.39\hspace{-1mm} & \hspace{-1mm}0.5000485801347\hspace{-1mm} & \hspace{-1mm}0.68386889795\hspace{-1mm} & \hspace*{-1.5mm}-0.643317369\hspace*{-1mm} & \hspace{-1mm}1.08075772\hspace{-1mm} \\
\hspace*{-1mm}$24$\hspace*{-1mm}& \hspace{-1mm}4.7e-11\hspace{-1mm} & \hspace{-1mm}0.38\hspace{-1mm} & \hspace{-1mm}0.5000037328535\hspace{-1mm} & \hspace{-1mm}0.68593187289\hspace{-1mm} & \hspace*{-1.5mm}-0.679195194\hspace*{-1mm} & \hspace{-1mm}1.37495588\hspace{-1mm} \\
\hspace*{-1mm}$32$\hspace*{-1mm}& \hspace{-1mm}4.1e-11\hspace{-1mm} & \hspace{-1mm}0.43\hspace{-1mm} & \hspace{-1mm}0.5000002195177\hspace{-1mm} & \hspace{-1mm}0.68617780639\hspace{-1mm} & \hspace*{-1.5mm}-0.685884585\hspace*{-1mm} & \hspace{-1mm}1.46545941\hspace{-1mm} \\
\hspace*{-1mm}$48$\hspace*{-1mm}& \hspace{-1mm}1.4e-11\hspace{-1mm} & \hspace{-1mm}0.44\hspace{-1mm} & \hspace{-1mm}0.5000000087538\hspace{-1mm} & \hspace{-1mm}0.68619930431\hspace{-1mm} & \hspace*{-1.5mm}-0.686757991\hspace*{-1mm} & \hspace{-1mm}1.48373011\hspace{-1mm} \\
\hspace*{-1mm}$64$\hspace*{-1mm}& \hspace{-1mm}1.3e-11\hspace{-1mm} & \hspace{-1mm}0.31\hspace{-1mm} & \hspace{-1mm}0.5000000002937\hspace{-1mm} & \hspace{-1mm}0.68620057333\hspace{-1mm} & \hspace*{-1.5mm}-0.686834471\hspace*{-1mm} & \hspace{-1mm}1.48614633\hspace{-1mm} \\
\hspace*{-1mm}$96$\hspace*{-1mm}& \hspace{-1mm}3.2e-11\hspace{-1mm} & \hspace{-1mm}0.31\hspace{-1mm} & \hspace{-1mm}0.5000000000039\hspace{-1mm} & \hspace{-1mm}0.68620063534\hspace{-1mm} & \hspace*{-1.5mm}-0.686839872\hspace*{-1mm} & \hspace{-1mm}1.48639673\hspace{-1mm} \\
\hline
\multicolumn{3}{r} {{Eq}~\eqref{Ilexp}:} & 0.5 & 0.68620063577 &\hspace*{-1.0mm} -0.686839887 & 1.48639839\\ \hline
\end{tabular}
\end{center}
\end{scriptsize}
\label{table2}
\end{table}
\begin{table}
\caption{\footnotesize
Results UV \emph{half-boiled egg} integral, $J_{c}^{S2}$ \,(on Mac Pro),~
rel. err. tol. $t_r = 10^{-12}$ (outer), $5\times 10^{-13}$ (inner three), $T[s]$ = Time (elapsed user time in $s$),~
$\varepsilon = 1/b_\ell$ (starting at 1.0),~~$E_r = $ outer integration estim. rel. error}
\begin{scriptsize}
\begin{center}
\begin{tabular}{ccccccc}\hline
& \multicolumn{6}{l}{{\sc Integral ~$J_{c}^{S2}$~ Extrapolation}} \\
\hspace*{-1mm}$b_\ell$\hspace*{-1mm} & \hspace{-1mm}{ $E_r$ }\hspace{-1mm} & \hspace{-1mm}{\sc T[s]}
\hspace{-1mm} & {\sc Res.} ~${C}_{-1}$ & \hspace{-1mm}{\sc Res.} ~${C}_{0}$\hspace{-1mm} & \hspace*{-0mm}{\sc Res.} ~${C}_1$ & \hspace{-1mm}{\sc Res.} ~${C}_2$\hspace{-1mm} \\% & & & & & & & \\
\hlin
\hspace*{-1mm}$1$\hspace*{-1mm}& \hspace{-1mm}4.2e-13\hspace{-1mm} & \hspace{-1mm}7.3\hspace{-1mm} & \hspace{-1mm}\hspace{-1mm} & \hspace{-1mm}\hspace{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$2$\hspace*{-1mm}& \hspace{-1mm}3.9e-14\hspace{-1mm} & \hspace{-1mm}11.4\hspace{-1mm} & \hspace{-1mm}0.6121795323700\hspace{-1mm} & \hspace{-1mm}-0.26893337928\hspace{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$3$\hspace*{-1mm}& \hspace{-1mm}2.6e-13\hspace{-1mm} & \hspace{-1mm}10.8\hspace{-1mm} & \hspace{-1mm}0.6219220162954\hspace{-1mm} & \hspace{-1mm}-0.29816083105\hspace{-1mm} & \hspace*{-1.5mm}-0.019484967\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$4$\hspace*{-1mm}& \hspace{-1mm}5.2e-13\hspace{-1mm} & \hspace{-1mm}10.3\hspace{-1mm} & \hspace{-1mm}0.6084345649676\hspace{-1mm} & \hspace{-1mm}-0.21723612309\hspace{-1mm} & \hspace*{-1.5mm}-0.128876997\hspace*{-1mm} & \hspace{-1mm}0.08092471\hspace{-1mm} \\
\hspace*{-1mm}$6$\hspace*{-1mm}& \hspace{-1mm}1.1e-13\hspace{-1mm} & \hspace{-1mm}14.9\hspace{-1mm} & \hspace{-1mm}0.6050661595977\hspace{-1mm} & \hspace{-1mm}-0.18355206939\hspace{-1mm} & \hspace*{-1.5mm}-0.246771185\hspace*{-1mm} & \hspace{-1mm}0.24934498\hspace{-1mm} \\
\hspace*{-1mm}$8$\hspace*{-1mm}& \hspace{-1mm}7.9e-13\hspace{-1mm} & \hspace{-1mm}9.2\hspace{-1mm} & \hspace{-1mm}0.6046439346384\hspace{-1mm} & \hspace{-1mm}-0.17679647004\hspace{-1mm} & \hspace*{-1.5mm}-0.286882556\hspace*{-1mm} & \hspace{-1mm}0.35912347\hspace{-1mm} \\
\hspace*{-1mm}$12$\hspace*{-1mm}& \hspace{-1mm}9.6e-13\hspace{-1mm} & \hspace{-1mm}12.5\hspace{-1mm} & \hspace{-1mm}0.6046028724387\hspace{-1mm} & \hspace{-1mm}-0.17581097725\hspace{-1mm} & \hspace*{-1.5mm}-0.296039426\hspace*{-1mm} & \hspace{-1mm}0.40100691\hspace{-1mm} \\
\hspace*{-1mm}$16$\hspace*{-1mm}& \hspace{-1mm}2.3e-13\hspace{-1mm} & \hspace{-1mm}19.4\hspace{-1mm} & \hspace{-1mm}0.6045999612478\hspace{-1mm} & \hspace{-1mm}-0.17570617438\hspace{-1mm} & \hspace*{-1.5mm}-0.297527045\hspace*{-1mm} & \hspace{-1mm}0.41176667\hspace{-1mm} \\
\hspace*{-1mm}$24$\hspace*{-1mm}& \hspace{-1mm}8.7e-13\hspace{-1mm} & \hspace{-1mm}19.1\hspace{-1mm} & \hspace{-1mm}0.6045997948488\hspace{-1mm} & \hspace{-1mm}-0.17569752162\hspace{-1mm} & \hspace*{-1.5mm}-0.297707921\hspace*{-1mm} & \hspace{-1mm}0.41374216\hspace{-1mm} \\
\hspace*{-1mm}$32$\hspace*{-1mm}& \hspace{-1mm}7.7e-13\hspace{-1mm} & \hspace{-1mm}24.9\hspace{-1mm} & \hspace{-1mm}0.6045997882885\hspace{-1mm} & \hspace{-1mm}-0.17569702304\hspace{-1mm} & \hspace*{-1.5mm}-0.297723238\hspace*{-1mm} & \hspace{-1mm}0.41399119\hspace{-1mm} \\
\hspace*{-1mm}$48$\hspace*{-1mm}& \hspace{-1mm}4.1e-13\hspace{-1mm} & \hspace{-1mm}33.2\hspace{-1mm} & \hspace{-1mm}0.6045997880782\hspace{-1mm} & \hspace{-1mm}-0.17569700033\hspace{-1mm} & \hspace*{-1.5mm}-0.297724241\hspace*{-1mm} & \hspace{-1mm}0.41401488\hspace{-1mm} \\
\hline
\multicolumn{3}{r} {{Eq}~\eqref{Ihexp}:} & 0.6045997880781 & -0.17569700023 &-0.297724254 &0.41401554 \\ \hline
\end{tabular}
\end{center}
\end{scriptsize}
\label{table3}
\end{table}
\begin{table}
\caption{\footnotesize
Results UV \emph{Magdeburg} integral, $I_{d}^{S2} \,\Gamma(1+\varepsilon)^{-2}$ \,(by {\sc ParInt} on \emph{thor}
in \emph{long double} precision),~
rel. err. tol. $t_r = 10^{-13},$ max. \# evals = 1B, $T[s]$ = Time (elapsed user time in $s$),~
$\varepsilon = 2^{-\ell}$ (starting at 1.0),~~$E_r = $ estim. rel. error}
\begin{scriptsize}
\begin{center}
\begin{tabular}{ccccccc}\hline
& \multicolumn{6}{l}{{\sc Integral ~$I_{d}^{S2} \,\Gamma(1+\varepsilon)^{-2}$~ Extrapolation}} \\
\hspace*{-1mm}$\ell$\hspace*{-1mm} & \hspace{-1mm}{ $E_r$ }\hspace{-1mm} & \hspace{-1mm}{\sc T[s]}
\hspace{-1mm} & {\sc Res.} ~$C_0$ & \hspace{-3mm}{\sc Res.} ~$C_1$\hspace{-1mm} & \hspace*{-0mm}{\sc Res.} ~$C_2$ & \hspace{-1mm}{\sc Res.} ~$C_3$\hspace{-1mm} \\% & & & & & & & \\
\hlin
\hspace*{-1mm}$0$\hspace*{-1mm}& \hspace{-1mm}8.5e-14\hspace{-1mm} & \hspace{-3mm}0.8\hspace{-4mm} & \hspace{-1mm}\hspace{-1mm} & \hspace{-1mm}\hspace{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$1$\hspace*{-1mm}& \hspace{-1mm}1.0e-13\hspace{-1mm} & \hspace{-3mm}19.7\hspace{-4mm} & \hspace{-1mm}0.69130084611470\hspace{-1mm} & \hspace{-2mm}-0.142989490499\hspace{-1mm} & \hspace*{-1mm}\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$2$\hspace*{-1mm}& \hspace{-1mm}1.6e-13\hspace{-1mm} & \hspace{-3mm}7.5\hspace{-4mm} & \hspace{-1mm}0.84949643770104\hspace{-1mm} & \hspace{-2mm}-0.617576265258\hspace{-1mm} & \hspace*{-1mm}0.3163911832\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$3$\hspace*{-1mm}& \hspace{-1mm}4.8e-13\hspace{-1mm} & \hspace{-3mm}6.9\hspace{-4mm} & \hspace{-1mm}0.90878784906010\hspace{-1mm} & \hspace{-2mm}-1.032616144771\hspace{-1mm} & \hspace*{-1mm}1.1464709422\hspace*{-1mm} & \hspace{-2mm}-0.47433129\hspace{-1mm} \\
\hspace*{-1mm}$4$\hspace*{-1mm}& \hspace{-1mm}8.5e-13\hspace{-1mm} & \hspace{-3mm}6.6\hspace{-4mm} & \hspace{-1mm}0.92198476262012\hspace{-1mm} & \hspace{-2mm}-1.230569848171\hspace{-1mm} & \hspace*{-1mm}2.0702548914\hspace*{-1mm} & \hspace{-2mm}-2.05796092\hspace{-1mm} \\
\hspace*{-1mm}$5$\hspace*{-1mm}& \hspace{-1mm}1.2e-12\hspace{-1mm} & \hspace{-3mm}6.5\hspace{-4mm} & \hspace{-1mm}0.92353497740505\hspace{-1mm} & \hspace{-2mm}-1.278626506507\hspace{-1mm} & \hspace*{-1mm}2.5508214747\hspace*{-1mm} & \hspace{-2mm}-3.98022725\hspace{-1mm} \\
\hspace*{-1mm}$6$\hspace*{-1mm}& \hspace{-1mm}3.0e-12\hspace{-1mm} & \hspace{-3mm}6.6\hspace{-4mm} & \hspace{-1mm}0.92362889210499\hspace{-1mm} & \hspace{-2mm}-1.284543132600\hspace{-1mm} & \hspace*{-1mm}2.6730984140\hspace*{-1mm} & \hspace{-2mm}-5.02831530\hspace{-1mm} \\
\hspace*{-1mm}$7$\hspace*{-1mm}& \hspace{-1mm}2.4e-12\hspace{-1mm} & \hspace{-3mm}6.6\hspace{-4mm} & \hspace{-1mm}0.92363178137723\hspace{-1mm} & \hspace{-2mm}-1.284910070175\hspace{-1mm} & \hspace*{-1mm}2.6885097922\hspace*{-1mm} & \hspace{-2mm}-5.30131686\hspace{-1mm} \\
\hspace*{-1mm}$8$\hspace*{-1mm}& \hspace{-1mm}2.8e-12\hspace{-1mm} & \hspace{-3mm}6.6\hspace{-4mm} & \hspace{-1mm}0.92363182617006\hspace{-1mm} & \hspace{-2mm}-1.284921492347\hspace{-1mm} & \hspace*{-1mm}2.6894768694\hspace*{-1mm} & \hspace{-2mm}-5.33613164\hspace{-1mm} \\
\hspace*{-1mm}$9$\hspace*{-1mm}& \hspace{-1mm}2.8e-12\hspace{-1mm} & \hspace{-3mm}6.6\hspace{-4mm} & \hspace{-1mm}0.92363182651847\hspace{-1mm} & \hspace{-2mm}-1.284921670382\hspace{-1mm} & \hspace*{-1mm}2.6895071354\hspace*{-1mm} & \hspace{-2mm}-5.33832809\hspace{-1mm} \\
\hspace*{-1mm}$10$\hspace*{-1mm}& \hspace{-1mm}2.9e-12\hspace{-1mm} & \hspace{-3mm}6.6\hspace{-4mm} & \hspace{-1mm}0.92363182651995\hspace{-1mm} & \hspace{-2mm}-1.284921671903\hspace{-1mm} & \hspace*{-1mm}2.6895076534\hspace*{-1mm} & \hspace{-2mm}-5.33840357\hspace{-1mm} \\
\hspace*{-1mm}$11$\hspace*{-1mm}& \hspace{-1mm}2.9e-12\hspace{-1mm} & \hspace{-3mm}6.6\hspace{-4mm} & \hspace{-1mm}0.92363182651990\hspace{-1mm} & \hspace{-2mm}-1.284921671790\hspace{-1mm} & \hspace*{-1mm}2.6895075765\hspace*{-1mm} & \hspace{-2mm}-5.33838110\hspace{-1mm} \\
\hspace*{-1mm}$12$\hspace*{-1mm}& \hspace{-1mm}2.9e-12\hspace{-1mm} & \hspace{-3mm}6.6\hspace{-4mm} & \hspace{-1mm}0.92363182651992\hspace{-1mm} & \hspace{-2mm}-1.284921671898\hspace{-1mm} & \hspace*{-1mm}2.6895077252\hspace*{-1mm} & \hspace{-2mm}-5.33846839\hspace{-1mm} \\
\hspace*{-1mm}$13$\hspace*{-1mm}& \hspace{-1mm}3.7e-12\hspace{-1mm} & \hspace{-3mm}6.6\hspace{-4mm} & \hspace{-1mm}0.92363182651991\hspace{-1mm} & \hspace{-2mm}-1.284921671798\hspace{-1mm} & \hspace*{-1mm}2.6895075774\hspace*{-1mm} & \hspace{-2mm}-5.33835823\hspace{-1mm} \\
\hspace*{-1mm}$14$\hspace*{-1mm}& \hspace{-1mm}2.9e-12\hspace{-1mm} & \hspace{-3mm}6.6\hspace{-4mm} & \hspace{-1mm}0.92363182651991\hspace{-1mm} & \hspace{-2mm}-1.284921671840\hspace{-1mm} & \hspace*{-1mm}2.6895076182\hspace*{-1mm} & \hspace{-2mm}-5.33839951\hspace{-1mm} \\
\hline
\multicolumn{3}{r} {{Eq}~\eqref{Imexp}:} & 0.9236318265199~ & \hspace*{-1mm}-1.284921671848 & 2.6895076265 & \hspace{-1mm}-5.33839923\\ \hline
\end{tabular}
\end{center}
\end{scriptsize}
\label{table-Magdeburg}
\end{table}
Tables~\ref{table1}, ~\ref{table2}, ~\ref{table3} and ~\ref{table-Magdeburg} show the convergence of the extrapolation method
for the integrals of Eqs~\eqref{Isexp}-\eqref{Imexp}.
While the integral $I_{d}^{S2}$ has no UV-divergent terms and starts from a finite term ($\kappa=0$),
the coefficients ${C}_0$, ${C}_1$, ${C}_2$ and ${C}_3$ can be obtained using extrapolation.
To evaluate $I_{d}^{S2}$, we transform the variables as:
\begin{eqnarray}
\nonumber
x_1&=&y_{1m} y_{3m} y_4, ~~~~~~x_2~=~y_{1m} y_{3m} y_{4m},\\
x_3&=&y_1 y_{2m}, ~~~~~~x_4~=~y_{1m} y_3,\\
\nonumber
x_5&=&y_1 y_2
\end{eqnarray}
with $y_{im}=1-y_{i}$ and Jacobian $y_1 y_{1m}^2 y_{3m}$.
The accuracy and time of the calculation
of the integral sequence for the extrapolation in Table~\ref{table-Magdeburg}
are improved considerably by the transformation.
For the first three integrals,
an iterated integration is applied with {\sc Dqags} from {\sc Quadpack},
on a Mac Pro, 2.6\,GHz Intel Core i7, with 16\,GB memory, under OS X.
The value of $E_r$ is (the absolute value of)
the estimated relative error returned by the outer integration
(not accounting for the inner integration error). It is listed for each integration,
as well as the elapsed user time $T[s]$ (in seconds). The time for
the extrapolation is negligible compared to that of the integration. We use a standard linear
system solver to solve very small systems (of sizes $2\times 2$ up to around $15\times 15$
for the cases in this paper).
Table~\ref{table-Magdeburg} illustrates an application of {\sc ParInt} for the Magdeburg integral,
on the \emph{thor} system of the High Performance Computing and Big Data Center at WMU,
with dual Intel Xeon E5-2670, 2.6\,GHz processors, 16 cores and 128\,GB of memory per node.
For the distributed computation with {\sc ParInt}, using 16 processes per node and 64 processes total,
the MPI host file has four lines of the form \emph{nx slots=16}
where \emph{nx} represents a selected node name.
The running time is reported (in seconds) from {\sc ParInt}, and comprises all computation
not inclusive of the process spawning and {\sc ParInt} initialization at the start of the program.
The cubature rule of degree 9 is used for integration over the subregions (see Section~\ref{parint}),
to an allowed maximum number of one billion (1B) integrand evaluations over all processes,
and a requested accuracy of $t_r = 10^{-13}$ in \emph{long double} precision.
The total estimated relative error is denoted by $E_r$ in Table~\ref{table-Magdeburg}.
The extrapolation parameter for Tables~\ref{table2} and~\ref{table3} adheres to
$\{1/b_\ell\}$ where $\{b_\ell\}$ is the Bulirsch sequence~\cite{bulirsch64} started
at an early index.
Tables~\ref{table1} and~\ref{table-Magdeburg} give results for a geometric sequence
of the extrapolation parameter $\varepsilon = \varepsilon_\ell.$
The convergence results in Tables~\ref{table1}-\ref{table2} and~\ref{table-Magdeburg} are compared with
with the expansion coefficients available from~\cite{laporta01} (see Eqs~\eqref{Isexp}-\eqref{Ilexp},~\eqref{Imexp}.
Table~\ref{table3} shows excellent approximations to the analytic result of Eq~\eqref{Ihexp}
derived in the Appendix.
Throughout the extrapolation we keep track of the difference with the previous result as
a measure of convergence. Increases of the distance between successive extrapolation results
are an indicator that the convergence is no longer improving and the procedure can be terminated.
\subsection{2-loop vertex integrals}\label{uv-vertex}
\begin{figure}
\begin{center}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.22\linewidth]{./figures/2lv-laporta3b-20161029.epsi}
\end{subfigure}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.22\linewidth]{./figures/2lv-laporta3c-20161029.epsi}
\end{subfigure}
\caption{\footnotesize 2-loop vertex (UV-divergent) diagram with massive internal lines:
(a) $N = 4$ (Laporta~\cite{laporta01}\,, Fig 3(b) ), (b) $N = 5$ (Laporta~\cite{laporta01}\,, Fig 3(c) )}
\label{laporta-3ab}
\end{center}
\end{figure}
Fig~\ref{laporta-3ab}(a) and (b) depict 2-loop vertex diagrams with $N = 4$ and $5$ internal lines,
and the integrals of Eq~\eqref{UV} are denoted by $I_a^{V2}$ and $I_b^{V2}$, respectively.
The former is divergent as $\displaystyle{1/\varepsilon^2},$ which is
the product of $\displaystyle{1/\varepsilon}$ arising from the $\Gamma$-function factor and
$\displaystyle{1/\varepsilon}$ from the integral.
The latter is divergent as $\displaystyle{1/\varepsilon}$ arising from the integral.\\
$-$ For $I_a^{V2}$, we have
$U = x_{12}x_{34} + x_3 x_4$ and
$W = p_{1}^2 x_1 x_3 x_4 + p_{2}^2 x_2 x_3 x_4 + p_{3}^2 x_1 x_2 x_{34}.$ \\
$-$ For $I_b^{V2}$, we have
$U=x_{124}x_3+x_{1234}x_5 $ and
$W=(p_1^2 x_1+p_2^2 x_2)\,(x_3x_{45}+x_4x_5)+p_3^2x_1x_2x_{35}$. \\
\begin{table}
\caption{\footnotesize
{\color{black}Results UV \emph{vertex} integral}, $I_{a}^{V2}$ \,(on Mac Pro),~
rel. err. tol. $t_r = 10^{-10}$ (outer), $5\times 10^{-11}$ (inner three), $T[s]$ = Time (elapsed user time in $s$),~
$\varepsilon = 1/b_\ell$ (starting at 1/3),~~$E_r = $ outer integration estim. rel. error}
\begin{scriptsize}
\begin{center}
\begin{tabular}{ccccccc}\hline
& \multicolumn{6}{l}{{\sc Integral ${I}_{a}^{V2} ~\Gamma(1+\varepsilon)^{-2}$~ Extrapolation}} \\
\hspace*{-1mm}$b_\ell$\hspace*{-1mm} & \hspace{-1mm}{ $E_r$ }\hspace{-1mm} & \hspace{-1mm}{\sc T[s]}
\hspace{-1mm} & {\sc Res.} ~${C}_{-2}$ & \hspace{-1mm}{\sc Res.} ~${C}_{-1}$\hspace{-1mm} & \hspace*{-0mm}{\sc Res.} ~${C}_0$ & \hspace{-1mm}{\sc Res.} ~${C}_1$\hspace{-1mm} \\% & & & & & & & \\
\hlin
\hspace*{-1mm}$3$\hspace*{-1mm}&\hspace{-1mm} 8.7e-12 \hspace{-1mm} & \hspace{-1mm}$1.4$\hspace{-1mm} & \hspace{-1mm}\hspace{-1mm} & \hspace{-1mm}\hspace{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$4$\hspace*{-1mm}&\hspace{-1mm} 2.3e-11 \hspace{-1mm} & \hspace{-1mm}$1.5$\hspace{-1mm} & \hspace{-1mm}0.51489021736\hspace{-1mm} & \hspace{-1mm}0.535680679\hspace{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$6$\hspace*{-1mm}&\hspace{-1mm} 3.1e-10 \hspace{-1mm} & \hspace{-1mm}$1.8$\hspace{-1mm} & \hspace{-1mm}0.50586162735\hspace{-1mm} & \hspace{-1mm}0.598880809\hspace{-1mm} & \hspace*{-1.5mm}-0.1083431\hspace*{-1mm} &\hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$8$\hspace*{-1mm}&\hspace{-1mm} 8.2e-11 \hspace{-1mm} & \hspace{-1mm}$5.2$\hspace{-1mm} & \hspace{-1mm}0.50111160609\hspace{-1mm} & \hspace{-1mm}0.660631086\hspace{-1mm} & \hspace*{-1.5mm}-0.3648442\hspace*{-1mm} &\hspace{-2mm}0.342002\hspace{-1mm} \\
\hspace*{-1mm}$12$\hspace*{-1mm}&\hspace{-1mm} 1.2e-09 \hspace{-1mm} & \hspace{-1mm}$2.5$\hspace{-1mm} & \hspace{-1mm}0.50015901670\hspace{-1mm} & \hspace{-1mm}0.680635463\hspace{-1mm} & \hspace*{-1.5mm}-0.5153534\hspace*{-1mm} & \hspace{-2mm}0.822107\hspace{-1mm} \\
\hspace*{-1mm}$16$\hspace*{-1mm}&\hspace{-1mm} 1.7e-09 \hspace{-1mm} & \hspace{-1mm}$4.2$\hspace{-1mm} & \hspace{-1mm}0.50001764652\hspace{-1mm} & \hspace{-1mm}0.685300679\hspace{-1mm} & \hspace*{-1.5mm}-0.5733151\hspace*{-1mm} & \hspace{-2mm}1.161395\hspace{-1mm} \\
\hspace*{-1mm}$24$\hspace*{-1mm}&\hspace{-1mm} 4.4e-09 \hspace{-1mm} & \hspace{-1mm}$20.5$\hspace{-1mm} & \hspace{-1mm}0.50000135665\hspace{-1mm} & \hspace{-1mm}0.686098882\hspace{-1mm} & \hspace*{-1.5mm}-0.5885950\hspace*{-1mm} & \hspace{-2mm}1.307352\hspace{-1mm} \\
\hspace*{-1mm}$32$\hspace*{-1mm}&\hspace{-1mm} 7.7e-09 \hspace{-1mm} & \hspace{-1mm}$19.1$\hspace{-1mm} & \hspace{-1mm}0.50000007798\hspace{-1mm} & \hspace{-1mm}0.686192225\hspace{-1mm} & \hspace*{-1.5mm}-0.5912981\hspace*{-1mm} & \hspace{-2mm}1.347595\hspace{-1mm} \\
\hspace*{-1mm}$48$\hspace*{-1mm}&\hspace{-1mm} 3.8e-10 \hspace{-1mm} & \hspace{-1mm}$8.6$\hspace{-1mm} & \hspace{-1mm}0.50000000083\hspace{-1mm} & \hspace{-1mm}0.686200327\hspace{-1mm} & \hspace*{-1.5mm}-0.5916415\hspace*{-1mm} & \hspace{-2mm}1.355242\hspace{-1mm} \\
\hline
\multicolumn{3}{r} {Eq}~\eqref{eq-3a}: & 0.5 & 0.686200636 & -0.5916667 & \hspace{-1mm}1.356197\\ \hline
\end{tabular}
\end{center}
\end{scriptsize}
\label{tab:vertex3a}
\end{table}
\begin{table}[h]
\caption{\footnotesize
Results UV \emph{vertex} integral, $I_{b}^{V2} \,\Gamma(1+\varepsilon)^{-2}$ \,(by {\sc ParInt} on \emph{thor} in \emph{long double} precision),~
rel. err. tol. $t_r = 10^{-13},$ max. \# evals = 10B, $T[s]$ = Time (elapsed user time in $s$),~
$\varepsilon = 1.2^{-\ell}$ (starting at $1.2^{-8}$),~~$E_r = $ estim. rel. error}
\begin{scriptsize}
\begin{center}
\begin{tabular}{ccccccc}\hline
& \multicolumn{6}{l}{{\sc Integral ~$I_{b}^{V2} \,\Gamma(1+\varepsilon)^{-2}$~ Extrapolation}} \\
\hspace*{-1mm}$\ell$\hspace*{-1mm} & \hspace{-1mm}{ $E_r$ }\hspace{-1mm} & \hspace{-1mm}{\sc T[s]}
\hspace{-1mm} & {\sc Res.} ~$C_{-1}$ & \hspace{-3mm}{\sc Res.} ~$C_0$\hspace{-1mm} & \hspace*{-0mm}{\sc Res.} ~$C_1$ & \hspace{-1mm}{\sc Res.} ~$C_2$\hspace{-1mm} \\% & & & & & & & \\
\hlin
\hspace*{-1mm}$8$\hspace*{-1mm}& \hspace{-1mm}9.9e-14\hspace{-1mm} & \hspace{-3mm}0.7\hspace{-4mm} & \hspace{-1mm}\hspace{-1mm} & \hspace{-1mm}\hspace{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$9$\hspace*{-1mm}& \hspace{-1mm}5.7e-14\hspace{-1mm} & \hspace{-3mm}0.9\hspace{-4mm} & \hspace{-1mm}0.653547537693\hspace{-1mm} & \hspace{-2mm}0.10873442398\hspace{-1mm} & \hspace*{-1mm}\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$10$\hspace*{-1mm}& \hspace{-1mm}8.7e-14\hspace{-1mm} & \hspace{-3mm}1.3\hspace{-4mm} & \hspace{-1mm}0.667737620294\hspace{-1mm} & \hspace{-2mm}-0.02549804326\hspace{-1mm} & \hspace*{-1mm}0.148227487\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$11$\hspace*{-1mm}& \hspace{-1mm}5.7e-14\hspace{-1mm} & \hspace{-3mm}1.7\hspace{-4mm} & \hspace{-1mm}0.670486635108\hspace{-1mm} & \hspace{-2mm}-0.06852379156\hspace{-1mm} & \hspace*{-1mm}0.536826159\hspace*{-1mm} & \hspace{-2mm}-0.37763368\hspace{-1mm} \\
\hspace*{-1mm}$12$\hspace*{-1mm}& \hspace{-1mm}9.8e-14\hspace{-1mm} & \hspace{-3mm}6.9\hspace{-4mm} & \hspace{-1mm}0.671112937626\hspace{-1mm} & \hspace{-2mm}-0.08297974141\hspace{-1mm} & \hspace*{-1mm}0.660237921\hspace*{-1mm} & \hspace{-2mm}-0.83947237\hspace{-1mm} \\
\hspace*{-1mm}$13$\hspace*{-1mm}& \hspace{-1mm}9.8e-14\hspace{-1mm} & \hspace{-3mm}8.4\hspace{-4mm} & \hspace{-1mm}0.671231024200\hspace{-1mm} & \hspace{-2mm}-0.08675821868\hspace{-1mm} & \hspace*{-1mm}0.707808436\hspace*{-1mm} & \hspace{-2mm}-1.13401646\hspace{-1mm} \\
\hspace*{-1mm}$14$\hspace*{-1mm}& \hspace{-1mm}9.6e-14\hspace{-1mm} & \hspace{-3mm}9.3\hspace{-4mm} & \hspace{-1mm}0.671250129509\hspace{-1mm} & \hspace{-2mm}-0.08757395495\hspace{-1mm} & \hspace*{-1mm}0.722045637\hspace*{-1mm} & \hspace{-2mm}-1.26401791\hspace{-1mm} \\
\hspace*{-1mm}$15$\hspace*{-1mm}& \hspace{-1mm}9.4e-14\hspace{-1mm} & \hspace{-3mm}10.3\hspace{-4mm} & \hspace{-1mm}0.671252763777\hspace{-1mm} & \hspace{-2mm}-0.08772025175\hspace{-1mm} & \hspace*{-1mm}0.725452770\hspace*{-1mm} & \hspace{-2mm}-1.30714663\hspace{-1mm} \\
\hspace*{-1mm}$16$\hspace*{-1mm}& \hspace{-1mm}1.0e-13\hspace{-1mm} & \hspace{-3mm}65.7\hspace{-4mm} & \hspace{-1mm}0.671253072371\hspace{-1mm} & \hspace{-2mm}-0.08774214433\hspace{-1mm} & \hspace*{-1mm}0.726115949\hspace*{-1mm} & \hspace{-2mm}-1.31834841\hspace{-1mm} \\
\hspace*{-1mm}$17$\hspace*{-1mm}& \hspace{-1mm}9.4e-14\hspace{-1mm} & \hspace{-3mm}13.8\hspace{-4mm} & \hspace{-1mm}0.671253102986\hspace{-1mm} & \hspace{-2mm}-0.08774488225\hspace{-1mm} & \hspace*{-1mm}0.726221896\hspace*{-1mm} & \hspace{-2mm}-1.32067610\hspace{-1mm} \\
\hspace*{-1mm}$18$\hspace*{-1mm}& \hspace{-1mm}9.9e-14\hspace{-1mm} & \hspace{-3mm}18.1\hspace{-4mm} & \hspace{-1mm}0.671253105554\hspace{-1mm} & \hspace{-2mm}-0.08774516889\hspace{-1mm} & \hspace*{-1mm}0.726235878\hspace*{-1mm} & \hspace{-2mm}-1.32106853\hspace{-1mm} \\
\hspace*{-1mm}$19$\hspace*{-1mm}& \hspace{-1mm}9.3e-14\hspace{-1mm} & \hspace{-3mm}39.5\hspace{-4mm} & \hspace{-1mm}0.671253105741\hspace{-1mm} & \hspace{-2mm}-0.08774519476\hspace{-1mm} & \hspace*{-1mm}0.726237453\hspace*{-1mm} & \hspace{-2mm}-1.32112425\hspace{-1mm} \\
\hspace*{-1mm}$20$\hspace*{-1mm}& \hspace{-1mm}1.0e-13\hspace{-1mm} & \hspace{-3mm}76.0\hspace{-4mm} & \hspace{-1mm}0.671253105743\hspace{-1mm} & \hspace{-2mm}-0.08774519512\hspace{-1mm} & \hspace*{-1mm}0.726237480\hspace*{-1mm} & \hspace{-2mm}-1.32112544\hspace{-1mm} \\
\hspace*{-1mm}$21$\hspace*{-1mm}& \hspace{-1mm}1.0e-13\hspace{-1mm} & \hspace{-3mm}77.4\hspace{-4mm} & \hspace{-1mm}0.671253105751\hspace{-1mm} & \hspace{-2mm}-0.08774519670\hspace{-1mm} & \hspace*{-1mm}0.726237628\hspace*{-1mm} & \hspace{-2mm}-1.32113350\hspace{-1mm} \\
\hspace*{-1mm}$22$\hspace*{-1mm}& \hspace{-1mm}1.1e-13\hspace{-1mm} & \hspace{-3mm}77.8\hspace{-4mm} & \hspace{-1mm}0.671253105748\hspace{-1mm} & \hspace{-2mm}-0.08774519610\hspace{-1mm} & \hspace*{-1mm}0.726237559\hspace*{-1mm} & \hspace{-2mm}-1.32112888\hspace{-1mm} \\
\hline
\multicolumn{3}{r} {{Eq}~\eqref{eq-3b}:} & 0.671253105748& \hspace*{-1mm}-0.08774519609 & 0.726237563 & \hspace{-1mm}-1.32112949\\ \hline
\end{tabular}
\end{center}
\end{scriptsize}
\label{tab:vertex3b1.2}
\end{table}
\begin{table}
\caption{\footnotesize
Results UV \emph{vertex} integral, $I_{b}^{V2} \,\Gamma(1+\varepsilon)^{-2}$ \,(by {\sc ParInt} on \emph{thor} in \emph{long double} precision),~
rel. err. tol. $t_r = 10^{-13},$ max. \# evals = 10B, $T[s]$ = Time (elapsed user time in $s$),~
$\varepsilon = 1.5^{-\ell}$ (starting at 1.0),~~$E_r = $ estim. rel. error}
\begin{scriptsize}
\begin{center}
\begin{tabular}{ccccccc}\hline
& \multicolumn{6}{l}{{\sc Integral ~$I_{b}^{V2} \,\Gamma(1+\varepsilon)^{-2}$~ Extrapolation}} \\
\hspace*{-1mm}$\ell$\hspace*{-1mm} & \hspace{-1mm}{ $E_r$ }\hspace{-1mm} & \hspace{-1mm}{\sc T[s]}
\hspace{-1mm} & {\sc Res.} ~$C_{-1}$ & \hspace{-3mm}{\sc Res.} ~$C_0$\hspace{-1mm} & \hspace*{-0mm}{\sc Res.} ~$C_1$ & \hspace{-1mm}{\sc Res.} ~$C_2$\hspace{-1mm} \\% & & & & & & & \\
\hlin
\hspace*{-1mm}$0$\hspace*{-1mm}& \hspace{-1mm}1.0e-13\hspace{-1mm} & \hspace{-3mm}0.31\hspace{-4mm} & \hspace{-1mm}\hspace{-1mm} & \hspace{-1mm}\hspace{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$1$\hspace*{-1mm}& \hspace{-1mm}1.0e-13\hspace{-1mm} & \hspace{-3mm}0.77\hspace{-4mm} & \hspace{-1mm}0.552646148416\hspace{-1mm} & \hspace{-2mm}-0.03261093175\hspace{-1mm} & \hspace*{-1mm}\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$2$\hspace*{-1mm}& \hspace{-1mm}2.7e-14\hspace{-1mm} & \hspace{-3mm}0.26\hspace{-4mm} & \hspace{-1mm}0.645884613605\hspace{-1mm} & \hspace{-2mm}-0.09301315450\hspace{-1mm} & \hspace*{-1mm}0.139857698\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$3$\hspace*{-1mm}& \hspace{-1mm}3.9e-14\hspace{-1mm} & \hspace{-3mm}0.44\hspace{-4mm} & \hspace{-1mm}0.659977890791\hspace{-1mm} & \hspace{-2mm}-0.02607008786\hspace{-1mm} & \hspace*{-1mm}0.240272298\hspace*{-1mm} & \hspace{-2mm}-0.04756481\hspace{-1mm} \\
\hspace*{-1mm}$4$\hspace*{-1mm}& \hspace{-1mm}4.9e-14\hspace{-1mm} & \hspace{-3mm}0.92\hspace{-4mm} & \hspace{-1mm}0.668110702931\hspace{-1mm} & \hspace{-2mm}-0.04000901078\hspace{-1mm} & \hspace*{-1mm}0.428597729\hspace*{-1mm} & \hspace{-2mm}-0.02705818\hspace{-1mm} \\
\hspace*{-1mm}$5$\hspace*{-1mm}& \hspace{-1mm}1.2e-14\hspace{-1mm} & \hspace{-3mm}1.32\hspace{-4mm} & \hspace{-1mm}0.670596883473\hspace{-1mm} & \hspace{-2mm}-0.07279551667\hspace{-1mm} & \hspace*{-1mm}0.588431945\hspace*{-1mm} & \hspace{-2mm}-0.63020875\hspace{-1mm} \\
\hspace*{-1mm}$6$\hspace*{-1mm}& \hspace{-1mm}3.8e-14\hspace{-1mm} & \hspace{-3mm}1.96\hspace{-4mm} & \hspace{-1mm}0.671155085836\hspace{-1mm} & \hspace{-2mm}-0.08439565952\hspace{-1mm} & \hspace*{-1mm}0.680218075\hspace*{-1mm} & \hspace{-2mm}-0.98346458\hspace{-1mm} \\
\hspace*{-1mm}$7$\hspace*{-1mm}& \hspace{-1mm}9.5e-14\hspace{-1mm} & \hspace{-3mm}13.9\hspace{-4mm} & \hspace{-1mm}0.671242833695\hspace{-1mm} & \hspace{-2mm}-0.08721867268\hspace{-1mm} & \hspace*{-1mm}0.715417521\hspace*{-1mm} & \hspace{-2mm}-1.20334533\hspace{-1mm} \\
\hspace*{-1mm}$8$\hspace*{-1mm}& \hspace{-1mm}1.0e-13\hspace{-1mm} & \hspace{-3mm}76.5\hspace{-4mm} & \hspace{-1mm}0.671252362159\hspace{-1mm} & \hspace{-2mm}-0.08768802398\hspace{-1mm} & \hspace*{-1mm}0.724477468\hspace*{-1mm} & \hspace{-2mm}-1.29252918\hspace{-1mm} \\
\hspace*{-1mm}$9$\hspace*{-1mm}& \hspace{-1mm}1.0e-13\hspace{-1mm} & \hspace{-3mm}77.7\hspace{-4mm} & \hspace{-1mm}0.671253068976\hspace{-1mm} & \hspace{-2mm}-0.08774095520\hspace{-1mm} & \hspace*{-1mm}0.726041833\hspace*{-1mm} & \hspace{-2mm}-1.31636903\hspace{-1mm} \\
\hspace*{-1mm}$10$\hspace*{-1mm}& \hspace{-1mm}9.3e-15\hspace{-1mm} & \hspace{-3mm}26.8\hspace{-4mm} & \hspace{-1mm}0.671253104515\hspace{-1mm} & \hspace{-2mm}-0.08774498285\hspace{-1mm} & \hspace*{-1mm}0.726222803\hspace*{-1mm} & \hspace{-2mm}-1.32059155\hspace{-1mm} \\
\hspace*{-1mm}$11$\hspace*{-1mm}& \hspace{-1mm}2.1e-13\hspace{-1mm} & \hspace{-3mm}78.3\hspace{-4mm} & \hspace{-1mm}0.671253105720\hspace{-1mm} & \hspace{-2mm}-0.08774518885\hspace{-1mm} & \hspace*{-1mm}0.726236811\hspace*{-1mm} & \hspace{-2mm}-1.32108845\hspace{-1mm} \\
\hspace*{-1mm}$12$\hspace*{-1mm}& \hspace{-1mm}4.9e-13\hspace{-1mm} & \hspace{-3mm}79.5\hspace{-4mm} & \hspace{-1mm}0.671253105748\hspace{-1mm} & \hspace{-2mm}-0.08774519595\hspace{-1mm} & \hspace*{-1mm}0.726237540\hspace*{-1mm} & \hspace{-2mm}-1.32112754\hspace{-1mm} \\
\hline
\multicolumn{3}{r} {{Eq}~\eqref{eq-3b}:} & 0.671253105748& \hspace*{-1mm}-0.08774519609 & 0.726237563 & \hspace{-1mm}-1.32112949\\ \hline
\end{tabular}
\end{center}
\end{scriptsize}
\label{tab:vertex3b}
\end{table}
In order to compare with Laporta's results~\cite{laporta01}, we put $m_r=1$ and $p_1^2=p_2^2=p_3^2=1,$
and multiply the integrals with a factor $\Gamma(1+{\varepsilon})^{-2}.$
The expansions from~\cite{laporta01} are:
\begin{eqnarray}
\nonumber
I_{a}^{V2}({\varepsilon}) ~\Gamma(1+{\varepsilon})^{-2} = \sum_{k\ge -2} C_k \,{\varepsilon^k} &=& 0.5\,{\varepsilon^{-2}}+0.6862006357658\,{\varepsilon^{-1}}-0.5916667014024\\
&+&1.356196533114\,{\varepsilon}\ldots \label{eq-3a}\\
\nonumber \\
\nonumber
I_{b}^{V2}({\varepsilon}) ~\Gamma(1+{\varepsilon})^{-2} = \sum_{k\ge -1} C_k \,{\varepsilon^k} &=& 0.671253105748\,{\varepsilon^{-1}}-0.08774519609257+0.7262375626947\,{\varepsilon}\\
&-&1.32112948587\,{\varepsilon^2}\ldots \label{eq-3b}
\end{eqnarray}
The numerical results for $I_a^{V2}$ obtained with iterated integration by {\sc Dqags},
and extrapolation using a Bulirsch sequence and
linear system solver are shown in Table~\ref{tab:vertex3a}.
Tables~\ref{tab:vertex3b1.2}-\ref{tab:vertex3b} show results for $I_{b}^{V2}$ with geometric sequences of
base 1/1.2 and 1/1.5, respectively,
achieved by {\sc ParInt} using 64 processes on four 16-core nodes of the \emph{thor} cluster.
Both deliver very accurate results, with
the final results in Table~\ref{tab:vertex3b1.2} slightly closer to the analytic values.
The extrapolation in Table~\ref{tab:vertex3b} converges somewhat faster.
For the computation of $I_{b}^{V2}$, we transform the variables as for $I_d^{S2}$
in Section~\ref{2ls}.
This transformation maps the integration domain to the 4-dimensional unit cube
and also guards against the loss of significant digits
near the boundaries.
\subsection{2-loop box integrals}\label{2lb}
The 2-loop box integrals according to Eq~\eqref{UV} for the diagrams in Fig~\ref{laporta-diagrams}
are all finite integrals and can be evaluated with $\varepsilon=0$.
Integral approximations obtained with {\sc ParInt} for the {\it double-triangle} ($N = 5$),
{\it tetragon-triangle} ($N = 6$), {\it pentagon-triangle} ($N = 7$), {\it ladder} and {\it crossed
ladder} ($N = 7$) diagrams were presented in~\cite{ddacat13}.
The $U$ and $W$ functions can also be found in the reference.
In the numerical evaluation, we set $s=t=1,\ p_j^2=1,\ m_r=1$ for simplicity
and for comparisons with other results in the literature.
\begin{figure}
\begin{center}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.25\linewidth]{./figures/2lb-laporta4c-20161029.epsi} \quad
\end{subfigure}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.25\linewidth]{./figures/2lb-laporta4d-20161029.epsi} \quad
\end{subfigure}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.25\linewidth]{./figures/2lb-laporta4g-20161029.epsi} \\
\end{subfigure}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.25\linewidth]{./figures/2lb-laporta4h-20161029.epsi} \quad
\end{subfigure}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.25\linewidth]{./figures/2lb-laporta4i-20161029.epsi}
\end{subfigure}
\caption{\footnotesize 2-loop box diagrams with massive internal lines (finite diagrams)
(a) {\it double-triangle} $N = 5$ (Laporta~\cite{laporta01},\, Fig 4(c)),
(b) {\it tetragon-triangle} $N = 6$ (Laporta~\cite{laporta01},\, Fig 4(d)),
(c) {\it pentagon-triangle} $N = 7$ (Laporta~\cite{laporta01},\, Fig 4(g)),
(d) {\it ladder} $N = 7$ (Laporta~\cite{laporta01},\, Fig 4(h)),
(e) {\it crossed ladder} $N = 7$ (Laporta~\cite{laporta01},\, Fig 4(i))}
\label{laporta-diagrams}
\end{center}
\end{figure}
This subsection provides timing results obtained with {\sc ParInt} on the \emph{thor} cluster,
corresponding to the five diagrams in Fig~\ref{laporta-diagrams}.
Table~\ref{parint-test-specs} gives a brief overview of pertinent test specifications, $T_1, ~T_p$
and the speedup $S_p = T_1/T_p$ for $p = 64.$
The times $T_p$ are expressed in seconds, as a function of the number of MPI processes $p, 1\le p\le 64.$
When referring to numbers of integrand evaluations, \emph{million} and
\emph{billion} are abbreviated by ``M" and ``B", respectively.
For instance, the times of the $N = 5$ double triangle diagram decrease
from 32.6 seconds at $p = 1,$ to \,0.74 seconds at $p = 64$ for $t_r = 10^{-10}$ (reaching a speedup of 44);
whereas the $N = 7$ crossed diagram times range between 27.6 seconds and 0.49 seconds for $t_r = 10^{-7}$
(with speedup exceeding 56).
The transformation of Eq~\eqref{cubetrans} was used.
For the two-loop crossed box problem as an example, we ran {\sc ParInt} in \emph{long double} precision. The results for
an increasing allowed maximum number of evaluations and increasingly strict (relative) error tolerance
$t_r$ (using 64 processes) are given in Table~\ref{parint-results-crossed}, as well as the corresponding double precision
results. Listed are: the integral approximation, relative error estimate $E_r,$ number of
function evaluations reached and time taken in \emph{long double} precision, followed by the
relative error estimate, number of function evaluations reached and running time in \emph{double} precision.
For a comparable number of function evaluations, the time using long doubles is slightly less than twice
the time taken using doubles. The \emph{iflag} parameter returns 0 when the requested accuracy is assumed
to be achieved, and 1 otherwise. Reaching the maximum number of evaluations results in abnormal termination
with \emph{iflag} = 1. The integral approximation for the \emph{double} computation is not listed
in Table~\ref{parint-results-crossed}; it
is consistent with the \emph{long double} result within the estimated error (which appears to be over-estimated).
Using doubles the program terminates abnormally for the requested relative accuracy of $t_r = 10^{-10}.$
Normal termination is achieved in this case around 239B evaluations with long doubles.
Fig~\ref{parint-timings} shows {\sc ParInt} timing plots for the diagrams of
Fig~\ref{laporta-diagrams}, depicting a considerable time decrease as a function of the number of processes $p.$
Plots with similar orders of the time are grouped in Fig~\ref{parint-timings}(a) and in
Fig~\ref{parint-timings}(b).
Timing results obtained with parallel (multi-threaded) iterated integration were given in~\cite{ddacat13}.
\begin{table}
\centering
\caption{\footnotesize Test specifications and range of times in Fig~\ref{parint-timings}(a)-(d)}
\begin{footnotesize}
\begin{tabular}{lccccccc}
& & & & & & & \\
\hline
{\sc Diagram} & {\sc Figure/Timing Plot} & $N$ & {\sc Rel Tol} & {\sc Max evals} & $T_1[s]$ & $T_{64}[s]$ & {\sc Speedup} \\
& & & $E_r$ & & & & $S_p$ for $p = 64$ \\
\hline
double triangle & Fig~\ref{laporta-diagrams}(a) / Fig~\ref{parint-timings}(a) & 5 & $10^{-10}$ & 400M & 32.6 & 0.74 & 44.1 \\
crossed ladder & Fig~\ref{laporta-diagrams}(e) / Fig~\ref{parint-timings}(a) & 7 & $10^{-7}$ & 300M & 27.6 & 0.49 & 56.3 \\
tetragon triangle & Fig~\ref{laporta-diagrams}(b) / Fig~\ref{parint-timings}(b) & 6 & $10^{-9}$ & 3B & 213.6 & 5.06 & 42.2 \\
ladder & Fig~\ref{laporta-diagrams}(d) / Fig~\ref{parint-timings}(b) & 7 & $10^{-8}$ & 2B & 189.9 & 4.33 & 43.9 \\
pentagon triangle & Fig~\ref{laporta-diagrams}(c) / Fig~\ref{parint-timings}(c) & 7 & $10^{-8}$ & 5B & 507.9 & 8.83 & 57.5 \\
crossed ladder & Fig~\ref{laporta-diagrams}(e) / Fig~\ref{parint-timings}(d) & 7 & $10^{-9}$ & 20B & 1892.5 & 34.6 & 54.7 \\ \hline
\end{tabular}
\end{footnotesize}
\label{parint-test-specs}
\end{table}
\begin{table}
\centering
\caption{\footnotesize {\sc ParInt} long double and double results for crossed diagram Fig~\ref{laporta-diagrams}(e)}
\begin{scriptsize}
\begin{tabular}{crcccrrccrr}
& & & & & & & & & & \\
\hline
$t_r$ & Max & \multicolumn{5}{c}{\emph{long double} precision} & \multicolumn{4}{c}{\emph{double} precision} \\
& Evals & {\sc Integral Approx} & $E_r$ & \emph{iflag} & \#{\sc Evals} & {\sc Time}[s] & $E_r$ &\emph{iflag} & \#{\sc Evals} & {\sc Time} \\
\hline
$10^{-08}$ & 600M & 0.085351397048123 & 3.5e-07 & 1 & 600000793 & 1.7
& 3.4E-07 & 1 & 600001115 & 0.95 \\
& 1B & 0.085351397753978 & 1.7e-07 & 1 & 1000000141 & 2.9
& 1.6e-07 & 1 & 1000000141 & 1.6 \\
& 2B & 0.085351398064779 & 2.9e-08 & 1 & 2000000443 & 6.0
& 2.9e-08 & 1 & 2000000765 & 3.3 \\
& 6B & 0.085351398130559 & 5.6e-09 & 0 & 4164032999 & 14.3
& 8.0e-09 & 0 & 4424455329 & 8.6 \\
$10^{-09}$ & 10B & 0.085351398143465 & 5.5e-09 & 1 & 10000001571 & 29.7
& 5.5e-09 & 1 & 10000000927 & 16.4 \\
& 50B & 0.085351398152623 & 9.9e-10 & 0 & 35701321579 & 124.4
& 4.5e-10 & 0 & 9799638359 & 16.0 \\
$10^{-10}$ & 80B & 0.085351398153315 & 5.5e-10 & 1 & 80000001137 & 240.2
& 5.5e-10 & 1 & 80000000171 & 133.5 \\
& 100B & 0.085351398153507 & 4.1e-10 & 1 & 100000000093 & 302.0
& 4.1e-10 & 1 & 100000000093 & 168.3 \\
& 300B & 0.085351398153798 & 9.1e-11 & 0 & 238854968513 & 642.3
& 1.3e-10 & 1 & 300000000279 & 587.2 \\ \hline
\end{tabular}
\end{scriptsize}
\label{parint-results-crossed}
\end{table}
\begin{figure}[t]
\begin{center}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.40\linewidth]{./figures/N=5er1e-10-crossN=7er1e-7s0.eps} \quad
\end{subfigure}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.40\linewidth]{./figures/ladderN=7er1e-8N=6er1e-8s0c.eps}
\end{subfigure}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.41\linewidth]{./figures/Fig1c-N=7-er1e-8s0.eps} \quad
\end{subfigure}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.41\linewidth]{./figures/crossedN=7er1e-9s0.eps}
\end{subfigure}
\caption{{\sc ParInt} timing plots for Fig~\ref{laporta-diagrams} diagrams
as a function of the number of procs. $p.$ (a) $N = 5$ [Fig~\ref{laporta-diagrams}(a)], $t_r = 10^{-10}$ and $N = 7$ [Fig~\ref{laporta-diagrams}(e)], $t_r = 10^{-7};$ (b) $N = 6$ [Fig~\ref{laporta-diagrams}(b)], $t_r = 10^{-8}$ and $N = 7$ [Fig~\ref{laporta-diagrams}(d)], $t_r = 10^{-8};$ (c) $N = 7$ [Fig~\ref{laporta-diagrams}(c)], $t_r = 10^{-8};$ (d) $N = 7$ [Fig~\ref{laporta-diagrams}(e)], $t_r = 10^{-9}$}
\label{parint-timings}
\end{center}
\end{figure}
\section{3-loop self-energy integrals} \label{3-loop-self}
In this section we deal with the integral determined by Eq~\eqref{LloopIJ} for
$L=3$ and $n=4-2\varepsilon,$
\begin{equation}
I = (-1)^N {\Gamma\left(N-6+3\varepsilon \right)}
\int_{0}^{1}\prod_{r=1}^{N}dx_{r}\,
\delta(1-\sum x_{r})\frac{1}{U^{2-\varepsilon}(V-i\varrho)^{N-6+3\varepsilon}}.
\label{threeLOOP}
\end{equation}
\noindent
UV divergence occurs when $U$ vanishes at the boundaries. The $\Gamma$-function in Eq~\eqref{threeLOOP} contributes to UV divergence when $N\le 6$.
As shown in the figures, the entering momentum is $p,$ and we denote $s=p^2$.
We calculate integrals adhering to Eq~\eqref{threeLOOP},
denoted by $I_a^{S3},\, I_b^{S3},\,\cdots, I_j^{S3},$
for the diagrams of Fig~\ref{3ls-self}(a)-(j), respectively.
In the following four subsections,
we consider massless/massive internal lines and
UV finite/divergent cases.
\subsection{3-loop finite integrals with massless internal lines}\label{3ls-massless-finite-integrals}
The integrals $I_a^{S3},\, I_b^{S3},\, I_c^{S3},\, I_d^{S3}$ are finite,
with the $U,\ W$ functions given in Eqs~\eqref{SEthrLa}-\eqref{SEthrLd} below.
\begin{equation}
\mathrm{(a)}\quad
\left\{
\begin{array}{l}
U=
x_4 x_7 x_{12356} + x_{12} x_{47} x_{56}
+x_3\, ( x_{12} x_4+ x_{56} x_7+ x_{12} x_{56} )
\\
W/s=
x_4 x_7 x_{15} x_{236}
+x_4 \,(x_1 x_2 x_{356} + x_{12} x_{36} x_5) \\
\quad +x_7 \,(x_5 x_6 x_{123} +x_1 x_{23} x_{56} )
+x_3 \,(x_1 x_2 x_{56} + x_{12} x_5 x_6)
\end{array}
\right.
\label{SEthrLa}
\end{equation}
\begin{equation}
\mathrm{(b)}\quad
\left\{
\begin{array}{l}
U=
x_7 \,( x_{12} x_{3456} + x_{34} x_{56} )
+ x_{13} x_{24} x_{56} + x_1 x_2 x_{34} + x_{12} x_3 x_4
\\
W/s=
x_7 \,( x_{15} x_{26} x_{34} +x_1 x_2 x_{56} + x_{12} x_5 x_6)
+(x_1 x_3+ x_{13} x_5) \,(x_2 x_4+x_2 x_6+x_4 x_6)
\end{array}
\right.
\label{SEthrLb}
\end{equation}
\begin{equation}
\mathrm{(c)}\quad
\left\{
\begin{array}{l}
U=
x_5 x_8 x_{123467}
+x_5 x_{12} x_{3467} + x_8 x_{1234} x_{67}
+ x_{12} x_{34} x_{67}
\\
W/s=
x_5 x_8 x_{136} x_{247}
+ x_5 \,(x_1 x_2 x_{3467} + x_{12} x_{36} x_{47} ) \\
\quad + x_8 \,(x_6 x_7 x_{1234} + x_{13} x_{24} x_{67} )
+ x_1 x_2 x_{34} x_{67} + x_{12} x_3 x_4 x_{67} + x_{12} x_{34} x_6 x_7
\end{array}
\right.
\label{SEthrLc}
\end{equation}
\begin{equation}
\mathrm{(d)}\quad
\left\{
\begin{array}{l}
U=
x_5 x_8 x_{123467}
+x_5 x_{124} x_{367}
+x_8 x_{123} x_{467}
+ x_{12} x_3 x_{467} + x_{123} x_4 x_{67}
\\
W/s=
x_5 x_8 x_{136} x_{247}
+x_5 \,( x_{17} x_{24} x_{36} +x_1 x_7 x_{2346} ) \\
\quad +x_8 \,( x_{26} x_{13} x_{47} +x_2 x_6 x_{1347} )
+ x_{34} \,(x_1 x_2 x_{67} + x_{12} x_6 x_7)
+x_3 x_4 x_{17} x_{26}
\end{array}
\right.
\label{SEthrLd}
\end{equation}
In this subsection we take $m_r=0$ for the internal lines.
In the absence of divergences we set $\varrho=\varepsilon=0$ in
Eq~\eqref{threeLOOP} for $I_a^{S3},\, I_b^{S3},\, I_c^{S3},$
and show the results in Table~\ref{table-rst-eps0}.
However, the integral $I_d^{S3}$ is problematic with $\varrho=\varepsilon=0$
in view of integrand singularities, and we use the extrapolation method
with either $\varepsilon=0$ and in the limit as $\rho \rightarrow 0,$
or with $\varrho=0$ and in the limit as $\varepsilon \rightarrow 0.$
The analytic result is the same for the four diagrams (see~\cite{baikov10}),
\begin{equation}\label{BC-3loop}
I_a^{S3}=I_b^{S3}=I_c^{S3}=I_d^{S3}=
20 \,\zeta_5 = 20.738555102867\ldots\,. \nonumber
\end{equation}
\begin{figure}[h]
\begin{center}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.22\linewidth]{./figures/3ls-fig6a-20161029.epsi}
\end{subfigure}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.22\linewidth]{./figures/3ls-fig6b-20161029.epsi}
\end{subfigure}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.22\linewidth]{./figures/3ls-fig6c-20161029.epsi}
\end{subfigure}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.22\linewidth]{./figures/3ls-fig6d-20161029.epsi}
\end{subfigure}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.22\linewidth]{./figures/3ls-fig6e-20161029.epsi}
\end{subfigure}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.22\linewidth]{./figures/3ls-fig6f-20161029.epsi}
\end{subfigure}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.22\linewidth]{./figures/3ls-fig6g-20161029.epsi}
\end{subfigure}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.22\linewidth]{./figures/3ls-fig6h-20161029.epsi}
\end{subfigure}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.22\linewidth]{./figures/3ls-fig6i-20161029.epsi}
\end{subfigure}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.22\linewidth]{./figures/3ls-fig6j-20161029.epsi}
\end{subfigure}
\caption{3-loop self-energy diagrams with massive and massless internal lines
(finite and UV-divergent diagrams), cf. Laporta~\cite{laporta01}, Baikov and Chetyrkin~\cite{baikov10}:
(a) $N=7$, (b) $N=7$, (c) $N=8$, (d) $N=8$, (e) $N=7$,
(f) $N=8$, (g) $N=4$, (h) $N=6$, (i) $N=7$, (j) $N=7$}
\label{3ls-self}
\end{center}
\end{figure}
{\sc ParInt} numerical results and timings were given in~\cite{iccs15} for the 3-loop diagrams of Fig~\ref{3ls-self}(a)-(d),
from runs on 16-core nodes
of the \emph{thor} cluster.
For the integration over each subregion, the rule of degree 9 is used (see Section~\ref{parint}),
which evaluates the 6D integrand at 453 points and the 7D integrand at 717 points.
These integrals are transformed to the unit cube according to Eq~\eqref{cubetrans}.
Table~\ref{table-rst-eps0} lists the results obtained directly with $\varrho = \varepsilon = 0$
for Fig~\ref{3ls-self}(a)-(c) by {\sc ParInt} using 48 processes: integral approximation,
relative error estimate $E_r,$ and time in seconds for various total numbers
of function evaluations in double and long double precision.
Fig~\ref{fig:performance} shows times and speedups
as a function of the number of processes $p,$ for a computation
of the integrals of Fig~\ref{3ls-self}(a)-(c) using 10B integrand evaluations
in double precision.
Denoting the time in seconds for $p$ processes by $T_p[s],$ the corresponding speedup
given by $S_p = T_1/T_p$ is nearly optimal --
which would coincide with the diagonal in the graph, or slightly
superlinear ($ > p$) over the given range of $p.$
\begin{table}
\caption{\footnotesize {\sc ParInt} accuracy and times with 48 procs.~for loop integrals of Fig~\ref{3ls-self}(a)-(c) with massless internal lines,
using \,$\varrho = 0,$\, and various numbers of function evaluations; $E_r = $
integration estim. rel. error}
\begin{scriptsize}
\begin{center}
\begin{tabular}{cccccccc}\hline
& & \multicolumn{3}{c}{\sc double precision}
& \multicolumn{3}{c}{\sc long double precision} \\
Diagram & \# {\sc Fcn.} \hspace*{-1mm} & {\sc Integral} \hspace{-1mm} & \hspace{-1mm}{\sc Rel. err. }\hspace{-1mm} & \hspace{-2mm}{\sc Time}
\hspace*{0mm} & {\sc Integral} \hspace{-1mm} & \hspace{-1mm}{\sc Rel. err. }\hspace{-1mm} & \hspace{-2mm}{\sc Time} \\
& {\sc Evals.} \hspace*{-1mm} & {\sc Result} \hspace{-1mm} & \hspace{-1mm}{{\sc Est.} $E_r$ }\hspace{-1mm} & \hspace{-2mm}{\sc T[s]}\hspace{-1mm} & \hspace{-1mm}{\sc Result} \hspace{-1mm} & \hspace{-1mm}{{\sc Est.} $E_r$ }\hspace{-1mm} & \hspace{-1mm}{\sc T[s]} \\
~~\\
\hline
~~\\
\emph{Exact:}& & 20.73855510 & & & 20.73855510 & & \\
\hline
~~\\
Fig~\ref{3ls-self} (a) & 5B & 20.73871652 & 2.21e-05 & 9.0 & 20.73871522 & 2.5e-05 & 16.0 \\
& 10B & 20.73856839 & 3.50e-06 & 17.9 & 20.73856878 & 3.42e-06 & 32.1 \\
& 25B & 20.73855535 & 3.79e-07 & 44.9 & 20.73855539 & 3.71e-07 & 80.5 \\
& 50B & 20.73855508 & 9.07e-08 & 90.3 & 20.73855508 & 8.94e-08 & 161.1 \\
& 75B & 20.73855507 & 4.26e-08 & 135.6 & 20.73855507 & 4.23e-08 & 242.1 \\
& 100B & 20.73855508 & 2.59e-08 & 180.8 & 20.73855508 & 2.56e-08 & 323.2 \\
\hline
~~\\
Fig~\ref{3ls-self} (b)& 5B & 20.73933800 & 3.69e-05 & 9.7 & 20.73933292 & 3.63e-05 & 17.5 \\
& 10B & 20.73872210 & 6.64e-06 & 19.4 & 20.73872078 & 6.61e-06 & 35.1 \\
& 25B & 20.73857098 & 8.32e-07 & 48.6 & 20.73857018 & 8.12e-07 & 87.9 \\
& 50B & 20.73855716 & 1.96e-07 & 98.2 & 20.73855718 & 1.95e-07 & 175.9 \\
& 75B & 20.73855576 & 9.28e-08 & 146.4 & 20.73855575 & 9.12e-08 & 264.4 \\
& 100B & 20.73855540 & 5.68e-08 & 196.7 & 20.73855540 & 5.43e-08 & 352.6 \\
\hline
~~\\
Fig~\ref{3ls-self} (c)& 5B & 20.74194961 & 1.19e-03 & 10.3 & 20.74196270 & 1.19e-03 & 19.6 \\
& 10B & 20.73886434 & 3.64e-04 & 20.7 & 20.73880437 & 3.51e-04 & 39.2 \\
& 25B & 20.73827908 & 5.04e-05 & 51.7 & 20.73827607 & 5.04e-05 & 98.2 \\
& 50B & 20.73841802 & 1.09e-05 & 103.4 & 20.73842495 & 9.79e-06 & 196.7 \\
& 75B & 20.73848662 & 3.66e-06 & 146.4 & 20.73848624 & 3.70e-06 & 295.0 \\
& 100B & 20.73851402 & 1.96e-06 & 207.5 & 20.73851338 & 1.98e-06 & 393.9 \\
\hline
\end{tabular}
\end{center}
\end{scriptsize}
\label{table-rst-eps0}
\end{table}
\begin{figure}[h]
\begin{center}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.32\linewidth]{./figures/Times-rst.epsi}
\end{subfigure}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.32\linewidth]{./figures/Speedup-rst.epsi} \\%
\end{subfigure}
\caption{{\sc ParInt} parallel performance for computation
of integrals (Fig~\ref{3ls-self}(a-c)) using 10B evaluations:
(a) Computation times (in seconds), (b) $S_{peedup}=T_1/T_p$}
\label{fig:performance}
\end{center}
\end{figure}
\begin{table}
\begin{center}
\caption{{\footnotesize Integration with {\sc ParInt} using 64 procs., max. \# evals = 150B, $\varrho=\varrho_\ell = 2^{-\ell}, \ell=20,21,\ldots$ and extrapolation with $\epsilon$-algorithm for Fig~\ref{3ls-self}(d) integral with massless internal lines}}
\label{table:extrap-v}
\begin{scriptsize}
~~\\
\begin{tabular}{cccccc}\hline
& \multicolumn{3}{c}{{\sc Integral Fig}~\ref{3ls-self}(d)}
& \multicolumn{2}{c}{\sc Extrapolation} \\
{\hspace*{-5mm}$\ell$}\hspace*{-5mm} & {\sc Integral} \hspace{-1mm} & \hspace{-1mm}{ $E_r$ }\hspace{-1mm} & \hspace{-1mm}{\sc T[s]}
\hspace{-1mm} & \hspace{-1mm}{\sc Last} & Selected \\
\hline
& & & & & \\
\hspace*{-1mm} 20\hspace*{-1mm}& \hspace{-1mm} 19.69036128576084 \hspace{-1mm} & \hspace{-1mm}1.44e-07\hspace{-1mm} & \hspace{-1mm}474.5 & & \\
\hspace*{-1mm} 21\hspace*{-1mm}& \hspace{-1mm} 19.91633676759658 \hspace{-1mm} & \hspace{-1mm}1.64e-07\hspace{-1mm} & \hspace{-1mm}474.6 & & \\
\hspace*{-1mm} 22\hspace*{-1mm}& \hspace{-1mm} 20.09256513888053 \hspace{-1mm} & \hspace{-1mm}1.84e-07\hspace{-1mm} & \hspace{-1mm}474.7 & 20.71685142 & 20.71685142 \\
\hspace*{-1mm} 23\hspace*{-1mm}& \hspace{-1mm} 20.23033834092921 \hspace{-1mm} & \hspace{-1mm}1.94e-07\hspace{-1mm} & \hspace{-1mm}474.7 & 20.72393791 & 20.72393792 \\
\hspace*{-1mm} 24\hspace*{-1mm}& \hspace{-1mm} 20.33827222472266 \hspace{-1mm} & \hspace{-1mm}2.16e-07\hspace{-1mm} & \hspace{-1mm}474.7 & 20.73801873 & 20.73801873 \\
\hspace*{-1mm} 25\hspace*{-1mm}& \hspace{-1mm} 20.42297783943613 \hspace{-1mm} & \hspace{-1mm}2.36e-07\hspace{-1mm} & \hspace{-1mm}474.7 & 20.73801873 & 20.73801873 \\
\hspace*{-1mm} 26\hspace*{-1mm}& \hspace{-1mm} 20.48955227010659 \hspace{-1mm} & \hspace{-1mm}2.53e-07\hspace{-1mm} & \hspace{-1mm}474.7 & 20.73815511 & 20.73815511 \\
\hspace*{-1mm} 27\hspace*{-1mm}& \hspace{-1mm} 20.54194208640818 \hspace{-1mm} & \hspace{-1mm}2.74e-07\hspace{-1mm} & \hspace{-1mm}474.7 & 20.73825979 & 20.73825979 \\
\hspace*{-1mm} 28\hspace*{-1mm}& \hspace{-1mm} 20.58321345028519 \hspace{-1mm} & \hspace{-1mm}2.95e-07\hspace{-1mm} & \hspace{-1mm}474.7 & 20.73811441 & 20.73811441 \\
\hspace*{-1mm} 29\hspace*{-1mm}& \hspace{-1mm} 20.61575568045697 \hspace{-1mm} & \hspace{-1mm}3.14e-07\hspace{-1mm} & \hspace{-1mm}474.8 & 20.74104946 & 20.73840576 \\
\hspace*{-1mm} 30\hspace*{-1mm}& \hspace{-1mm} 20.64143527978811 \hspace{-1mm} & \hspace{-1mm}3.34e-07\hspace{-1mm} & \hspace{-1mm}474.6 & 20.73854289 & 20.73833347 \\
\hspace*{-1mm} 31\hspace*{-1mm}& \hspace{-1mm} 20.66171281606668 \hspace{-1mm} & \hspace{-1mm}3.51e-07\hspace{-1mm} & \hspace{-1mm}474.7 & 20.73861767 & 20.73855952 \\
\hspace*{-1mm} 32\hspace*{-1mm}& \hspace{-1mm} 20.67774379215522 \hspace{-1mm} & \hspace{-1mm}4.16e-07\hspace{-1mm} & \hspace{-1mm}474.6 & 20.73855567 & 20.73847582 \\
\hline
& & & {Eq}~\eqref{BC-3loop}: & 20.73855510 & 20.73855510 \\
\hline
\end{tabular}
\end{scriptsize}
\end{center}
\end{table}
With $\varrho = 0$ the integrand has boundary singularities.
For example, the integrals of Fig~\ref{3ls-self}(a)-(d) have a zero denominator
with $U = 0$ \,at\, $x_2 = 1$ and the other variables 0
(on the boundary of the unit simplex).
Thus the integrand program codes test for zero denominators.
However some of the computations overflow by integrand evaluations in the vicinity of the singularities, which is found to occur for Fig~\ref{3ls-self} (d) in double precision at 1B\,=\,$10^9$ evaluations or higher.
\begin{table}
\caption{{\footnotesize Integration with {\sc ParInt} using 64 procs., max. \# evals = 100B, $\varepsilon=\varepsilon_\ell = 2^{-\ell}, \ell=8,9,\ldots$ and extrapolation with $\epsilon$-algorithm for Fig~\ref{3ls-self}(d) integral with massless internal lines}}
\label{table:dimreg-r}
\begin{scriptsize}
\begin{center}
\begin{tabular}{cccccc}\hline
& \multicolumn{3}{c}{{\sc Integral Fig}~\ref{3ls-self}(d)}
& \multicolumn{2}{c}{\sc Extrapolation} \\
{\hspace*{-5mm}$\ell$}\hspace*{-5mm} & {\sc Integral} \hspace{-1mm} & \hspace{-1mm}{ $E_r$ }\hspace{-1mm} & \hspace{-1mm}{\sc T[s]}
\hspace{-1mm} & \hspace{-1mm}{\sc Last} & Selected \\
\hline
& & & & & \\
\hspace*{-1mm} 8\hspace*{-1mm}& \hspace{-1mm} 21.21987706233486 \hspace{-1mm} & \hspace{-1mm}8.84e-08 \hspace{-1mm} & \hspace{-1mm}648.5 & & \\
\hspace*{-1mm} 9\hspace*{-1mm}& \hspace{-1mm} 20.97727482739239 \hspace{-1mm} & \hspace{-1mm}8.69e-08 \hspace{-1mm} & \hspace{-1mm}648.0 & & \\
\hspace*{-1mm} 10\hspace*{-1mm}& \hspace{-1mm} 20.85743468356065 \hspace{-1mm} & \hspace{-1mm}8.61e-08\hspace{-1mm} & \hspace{-1mm}649.0 & 20.74044694 & 20.74044693 \\
\hspace*{-1mm} 11\hspace*{-1mm}& \hspace{-1mm} 20.79787566374119 \hspace{-1mm} & \hspace{-1mm}8.56e-08\hspace{-1mm} & \hspace{-1mm}647.7 & 20.73903010 & 20.73903010 \\
\hspace*{-1mm} 12\hspace*{-1mm}& \hspace{-1mm} 20.76818590083646 \hspace{-1mm} & \hspace{-1mm}8.53e-08\hspace{-1mm} & \hspace{-1mm}648.1 & 20.73855734 & 20.73855734 \\
\hspace*{-1mm} 13\hspace*{-1mm}& \hspace{-1mm} 20.75336343266232 \hspace{-1mm} & \hspace{-1mm}8.39e-08\hspace{-1mm} & \hspace{-1mm}648.3 & 20.73855626 & 20.73855626 \\
\hspace*{-1mm} 14\hspace*{-1mm}& \hspace{-1mm} 20.74595780282081 \hspace{-1mm} & \hspace{-1mm}8.29e-08\hspace{-1mm} & \hspace{-1mm}647.6 & 20.73855580 & 20.73855580 \\
\hspace*{-1mm} 15\hspace*{-1mm}& \hspace{-1mm} 20.74225639032920 \hspace{-1mm} & \hspace{-1mm}8.21e-08\hspace{-1mm} & \hspace{-1mm}647.6 & 20.73855592 & 20.73855592 \\
\hline
& & & {Eq}~\eqref{BC-3loop}: & 20.73855510 & 20.73855510 \\
\hline
\end{tabular}
\end{center}
\end{scriptsize}
\end{table}
For the integral $I_d^{S3},$ we first take $\varepsilon=0$ and use the following form with non-zero
$\varrho:$
\begin{equation}
\Re{e}\,I_d^{S3} = {\Gamma\left(2 \right)}
\int_{0}^{1}\prod_{r=1}^{8}dx_{r}\, \delta(1-\sum x_{r})
\frac{V^2-\varrho^2}{U^2(V^2+\varrho^2)^2}.
\label{ReN=8integral}
\end{equation}
Table~\ref{table:extrap-v}
shows an extrapolation as $\varrho\rightarrow 0$ using the $\epsilon$-algorithm of
Wynn~\cite{shanks55,wynn56,sidi96,sidi03,sidi11} (see Section~\ref{extrap-exp}).
The $\varrho_\ell$ geometric sequence is computed with base 2, \,$\varrho_\ell = 2^{-\ell}$
and the integration is performed in long double precision using 150B evaluation points.
The \emph{Selected} column lists the element along the new lower diagonal that is presumed
the best, based on its distance from the neighboring elements as computed by the $\epsilon$-algorithm
function from {\sc Quadpack}.
The \emph{Last} column lists the final (utmost right) element computed in the lower
diagonal. Overall the $\epsilon$-algorithm function from {\sc Quadpack} appears successful at
selecting a competitive element as its result for the iteration.
For an extrapolation as $\varepsilon \rightarrow 0$ we set $\varrho=0,$ so that
\begin{equation}
I_d^{S3} = {\Gamma\left(2+3\varepsilon \right)}
\int_{0}^{1}\prod_{r=1}^{8}dx_{r}\,
\delta(1-\sum x_{r})
\frac{1}{U^{2-\varepsilon}V^{2+3\varepsilon}}.
\label{IDS3}
\end{equation}
Baikov and Chetyrkin~\cite{baikov10} derive asymptotic expansions in integer powers of
\,$\varepsilon$\, for 3- and 4-loop integrals arising from diagrams with massless propagators.
Table~\ref{table:dimreg-r} gives an extrapolation in $\varepsilon$
for the integral of the Fig~\ref{3ls-self}(d) diagram,
using 100B evaluations for the integrations in long double precision.
The results show good agreement with the literature~\cite{baikov10,smirnov10}.
The integrand with $\varrho = 0$ in Eq~\eqref{IDS3} has a singular behavior
with $UV = 0$ at the boundaries of the domain.
The extrapolation converges faster than that with respect to $\varrho$ in
Table~\ref{table:extrap-v}.
The times are larger compared to those of
Table~\ref{table:extrap-v},
likely by calling the \emph{pow} function (in the C programming language) for each integrand evaluation, whereas the integrand
of Eq~\eqref{ReN=8integral}
for the $\varrho$ extrapolation can be calculated using only multiplications, divisions,
additions and subtractions.
\subsection{3-loop finite integrals with massive internal lines}
\begin{table}
\caption{\footnotesize Parallel performance of {\sc ParInt} for 3-loop diagrams of Fig~\ref{3ls-self} (a)-(f) with massive internal lines, abs. tolerance
$t_a = 5\times 10^{-10},$ and max. number of evaluations = 10B}
\centering
\begin{scriptsize}
\begin{tabular}{cccllccc}
& & & & & & & \\
\hline
3-loop & $N$ & Result & Result & Result & $T_1[s]$ & $T_{64}[s]$ & $S_{64}$ \\
diag. & & Laporta~\cite{laporta01} & $p = 1$ & $p = 64$ & & & \\
\hline
& & & & & & & \\
{Fig~\ref{3ls-self}~(a)} & 7 & 2.00250004111 & {\color{black} 2.0025000411}3 & {\color{black} 2.002500041}2 & 879.3 & 13.4 & 65.6 \\
{Fig~\ref{3ls-self}~(b)} & 7 & 1.34139924145 & {\color{black} 1.3413992414}7 & {\color{black} 1.341399241}6 & 1026.2 & 14.4 & 71.3 \\
{Fig~\ref{3ls-self}~(c)} & 8 & 0.27960892328 & {\color{black} 0.27960892}27 & {\color{black} 0.27960892}0 & 1019.7 & 15.9 & 64.1 \\
{Fig~\ref{3ls-self}~(d)} & 8 & 0.14801330396 & {\color{black} 0.148013303}6 & {\color{black} 0.14801330}26 & 976.6 & 16.4 & 59.5 \\
{Fig~\ref{3ls-self}~(e)} & 7 & 1.32644820827 & {\color{black} 1.32644820}6 & {\color{black} 1.326448}19 & 902.7 & 15.8 & 57.1 \\
{Fig~\ref{3ls-self}~(f)} & 8 & 0.18262723754 & {\color{black} 0.182627237}2 & {\color{black} 0.18262723}68 & 1018.3 & 15.8 & 64.4 \\ \hline
\end{tabular}
\end{scriptsize}
\label{parint-test3l-specs}
\end{table}
For a set of 3-loop self-energy diagrams with massive internal lines given in Fig~\ref{3ls-self} (a)-(f),
corresponding numerical results and {\sc ParInt} performance results are shown in Table~\ref{parint-test3l-specs}.
The $U,\ W$ functions for (a)-(d) are given in the previous subsection and
those for (e) and (f) are listed in Eqs~\eqref{SEthrLe}-\eqref{SEthrLf} below.
\begin{equation}
\mathrm{(e)}\quad
\left\{
\begin{array}{l}
U=
x_6 x_7 x_{1245} + x_7 x_{12} x_{345}
+(x_1 x_2+x_3 x_7) \,x_{45} + x_{14} x_{25} x_{36}
+ x_{12} x_4 x_5
\\
W/s=
x_6 \,(x_7 x_{12} x_{345} +(x_1 x_2+x_3 x_7) \,x_{45}
+ x_{12} x_4 x_5 +x_3 x_{14} x_{25} )
\end{array}
\right.
\label{SEthrLe}
\end{equation}
\begin{equation}
\mathrm{(f)}\quad
\left\{
\begin{array}{l}
U=
(x_3 x_5+x_5 x_8+x_3 x_8) \,x_{12467}
+x_5 x_{12} x_{467} +x_3 x_{124} x_{67}
+x_4 x_{12} x_{67} + x_4 x_8 x_{1267}
\\
W/s=
(x_3 x_5+x_5 x_8+x_3 x_8) \,x_{16} x_{247}
+x_5 \,(x_1 x_2 x_{467} + x_{12} x_{47} x_6) \\
\quad +x_3 \,(x_1 x_6 x_{247} + x_{16} x_{24} x_7)
+x_4 \,(x_8 x_{16} x_{27} + x_1 x_2 x_{67} + x_{12}x_6 x_7 )
\end{array}
\right.
\label{SEthrLf}
\end{equation}
In order to compare our integral approximations with Laporta's~\cite{laporta01},
we set all masses $m_r = 1$ and $s = 1,$ and furthermore divide the integral by $~\Gamma(1+\varepsilon)^3.$
The integrals are transformed from the (unit) simplex to the (unit) cube
according to the transformation of Eqs~\eqref{cubetrans} and~\eqref{transint}
and the integration is taken over the cube, using a basic integration rule of polynomial
degree 9 (see Section~\ref{parint}) and a maximum total of 10B evaluations.
The function evaluations are distributed over all the processes.
The absolute tolerance is $t_a = 5\times10^{-10}$ and the maximum number of
integrand evaluations is $10B = 10^{10}$ (which is reached in producing
the results of Table~\ref{parint-test3l-specs}).
The results in Table~\ref{parint-test3l-specs} are given for $p = 1$ and for $p = 64$ {\sc MPI} processes.
$T_1$ is the time with one process and $T_{64}$ is the parallel time on the \emph{thor} cluster with $p = 64$ processes, distributed over four 16-core, 2.6\,GHz compute nodes
and using the Infiniband interconnect for message passing via MPI.
The speedup $S_{64} = T_1/T_{64}$
indicates good scalability of the parallel implementation (see also~\cite{acat14,ccp14}).
Note that superlinear speedups ($S_{64}$) are obtained in some cases, where the
speedup exceeds the number of processes. This is partially due to the fact that the
timing is done within {\sc ParInt} after the processes are started.
It may also be noted that the adaptive partitioning reaches somewhat more accuracy sequentially.
Each process has its own priority queue, keyed with the absolute error estimates over their
region. This may lead to unnecessary work by the processes locally,
which increases with the number of processes.
Tables~\ref{table1-3ls} and~\ref{table4-3ls} are computed with consecutive calls to {\verb pi_integrate() } in
a loop and linear extrapolation, for the functions depicted in Fig~\ref{3ls-self}(b) and (f), respectively.
The values of $C_0, C_1$ and $C_2$ are listed ($\kappa = 0$ in Eq~\eqref{asymp}).
Results from extrapolation with the $\epsilon$-algorithm are shown in Table~\ref{table2-3ls}
for the diagram in Fig~\ref{3ls-self}(b). More extrapolations
are needed with the $\epsilon$-algorithm than with linear extrapolation. In this case
the latter is more accurate and efficient but utilizes knowledge of the structure of the
asymptotic expansion, i.e., that $\varphi_k(\varepsilon) = \varepsilon^k;$
this is not assumed for the non-linear extrapolation with the $\epsilon$-algorithm.
Tables~\ref{table3-3ls} and~\ref{table5-3ls} illustrate the vector function integration capability
of {\sc ParInt} to calculate the entry sequence for the extrapolation (as a vector integral result - see~\eqref{general}) with one call to
{\verb pi_integrate()}. This procedure delivers excellent accuracy and efficiency.
Note that the integration of the vector function in Table~\ref{table3-3ls} took 2403.4 seconds,
compared to the total time of 3766.6 seconds for the integrations listed in Table~\ref{table1-3ls}.
With regard to Table~\ref{table5-3ls}, the time for integrating the vector function was
2180.2 seconds, vs. the total (sum) of 3426.3 seconds for the iterations in Table~\ref{table4-3ls}.
\input{results.txt}
\subsection{3-loop UV-divergent integrals with massless internal lines}\label{3-loop-massless-UV}
This section handles the integral associated with the massless diagram of
Fig~\ref{3ls-self}(g) (the \emph{3-loop sunrise-sunset} diagram named $P_{3}$ in~\cite{baikov10}),
which has a $1/\varepsilon$ singularity in the dimensional regularization parameter,
arising from the $\Gamma$-function factor \,($\Gamma(N-6+3\varepsilon)$)\,
in Eq~\eqref{threeLOOP}.
The polynomials $U$ and $W$ for \,$I_g^{S3}$\, are given by
\begin{equation}
\mathrm{(g)}\quad
\left\{
\begin{array}{l}
U= x_1 x_2 x_3 + x_1 x_2 x_4 + x_1 x_3 x_4 + x_2 x_3 x_4
\\
W/s=x_1 x_2 x_3 x_4
\end{array}
\right.
\label{SEthrLg}
\end{equation}
We take $\varrho = 0$ in Eq~\eqref{threeLOOP}, and the numerical evaluation is done with $s=1$.
In order to compare the result to that of Baikov and Chetyrkin~\cite{baikov10}, we
multiply with the factor $n(\varepsilon)^L$ where $n(\varepsilon)$ is defined
(in their footnote 11, p. 193) as
\begin{equation}
n(\varepsilon) = \frac{\Gamma(2-2\varepsilon)}{\Gamma(1+\varepsilon)~\Gamma(1-\varepsilon)^2}\,,
\label{factorn}
\end{equation}
leading to the expansion
\begin{align}
&
~n(\varepsilon)^3 \,I_g^{S3} = \frac{1}{36}\,\frac{1}{\varepsilon}+\frac{35}{216}+\frac{991}{1296}\,\varepsilon
+\,\ldots \nonumber \\
& = \,0.027777777777\,\frac{1}{\varepsilon}+0.162037037037+0.764660493827\,\varepsilon
+\,\ldots \label{p3-act}
\end{align}
\begin{table}
\caption{\footnotesize
{\color{black}Results UV \emph{3-loop} integral}, $n(\varepsilon)^3 \,I_g^{S3}$ (on 4 nodes/64 procs thor cluster),
abs. err. tol. $t_a = 10^{-12},$ $T[s]$ = Time (elapsed user time in s);
$\varepsilon = \varepsilon_\ell = 2^{-\ell},~ \ell = 8,9,\ldots,$ ~~$E_a = $ integration estim. abs. error}
\label{p3ext}
\begin{scriptsize}
\begin{center}
\begin{tabular}{cccccc}\hline
& \multicolumn{2}{c}{{\sc Integral Fig}~\ref{3ls-self}(g) }
& \multicolumn{3}{c}{{\sc Extrapolation}} \\
\raisebox{1mm}{\hspace*{-5mm}$\ell$}\hspace*{-5mm} & \hspace{-1mm}{ $E_a$ }\hspace{-1mm} & \hspace{-1mm}{\sc T[s]}
\hspace{-1mm} & {\sc Result} ~${C}_{-1}$ & \hspace{-1mm}{\sc Result} ~${C}_{0}$\hspace{-1mm} & \hspace*{-0mm}{\sc Result} ~${C}_1$ \\% & & & & & & \\
\hlin
\hspace*{-1mm}$8$\hspace*{-1mm}& \hspace{-1mm}2.8e-14\hspace{-1mm} & \hspace{-1mm}0.37\hspace{-1mm} & \hspace{-1mm}\hspace{-1mm} & \hspace{-1mm}\hspace{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm} \\
\hspace*{-1mm}$9$\hspace*{-1mm}& \hspace{-1mm}1.4e-13\hspace{-1mm} & \hspace{-1mm}0.65\hspace{-1mm} & \hspace{-1mm}0.0277718241800302\hspace{-1mm} & \hspace{-1mm}0.166588879051\hspace{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm} \\
\hspace*{-1mm}$10$\hspace*{-1mm}& \hspace{-1mm}1.7e-13\hspace{-1mm} & \hspace{-1mm}0.96\hspace{-1mm} & \hspace{-1mm}0.0277777978854937\hspace{-1mm} & \hspace{-1mm}0.162001073255\hspace{-1mm} & \hspace*{-1.5mm}0.78298552 \hspace*{-1mm} \\
\hspace*{-1mm}$11$\hspace*{-1mm}& \hspace{-1mm}1.6e-13\hspace{-1mm} & \hspace{-1mm}0.56\hspace{-1mm} & \hspace{-1mm}0.0277777777439553\hspace{-1mm} & \hspace{-1mm}0.162037166892\hspace{-1mm} & \hspace*{-1.5mm}0.76450558 \hspace*{-1mm} \\
\hspace*{-1mm}$12$\hspace*{-1mm}& \hspace{-1mm}1.7e-13\hspace{-1mm} & \hspace{-1mm}1.01\hspace{-1mm} & \hspace{-1mm}0.0277777777777756\hspace{-1mm} & \hspace{-1mm}0.162037037022\hspace{-1mm} & \hspace*{-1.5mm}0.76466073 \hspace*{-1mm} \\
\hline
\multicolumn{3}{r} \emph{Eq}~\eqref{p3-act}: & 0.0277777777777777 & 0.162037037037 &\hspace*{-1.0mm} \hspace*{-1.5mm}0.76466049\hspace*{-1mm} \\
\hline
\end{tabular}
\end{center}
\end{scriptsize}
\end{table}
Based on integrations with {\sc ParInt}, a maximum of 10B function evaluations
and an absolute error tolerance of $10^{-12}$
(on the computation of the integral
${\mathcal I}_g^{S3}/\Gamma(-2+3\varepsilon)$),
the results in
Table~\ref{p3ext} are produced using linear extrapolation.
{\sc ParInt} returns a 0 error flag for the integrals in the input sequence to the
extrapolation, indicating that a successful termination
is assumed according to Eqs~\eqref{accuracy} or~\eqref{acc} for the requested accuracy.
\subsection{3-loop UV-divergent integrals with massive internal lines}
\label{3-loop-massive-UV}
In this subsection we calculate the integrals corresponding to massive diagrams of
Fig~\ref{3ls-self}(h)-(j).
The integral $I_{h}^{S3}$ is divergent as $\displaystyle{1/\varepsilon},$ resulting from
the $\Gamma$-function factor.
On the other hand, the integrals $I_{i}^{S3}$ and $I_{j}^{S3}$ are divergent as $\displaystyle{1/\varepsilon}$ due to the integral part.
The $U, W$ functions for $I_h^{S3},\, I_i^{S3},\, and\, I_j^{S3}$ are listed in Eqs~\eqref{SEthrLh}-\eqref{SEthrLj} below.
\begin{equation}
\mathrm{(h)}\quad
\left\{
\begin{array}{l}
U=
x_5 \,( x_{12} x_{346} +x_3 x_{46} )
+x_3 x_{14} x_{26} +x_1 x_2 x_{46} + x_{12} x_4 x_6
\\
W/s=
x_3 \,( x_5 x_{12} x_{46}
+x_1 x_2 x_{46} + x_{12} x_4 x_6 )
\end{array}
\right.
\label{SEthrLh}
\end{equation}
\begin{equation}
\mathrm{(i)}\quad
\left\{
\begin{array}{l}
U=x_{56}\,(x_{12}x_{34} + x_{1234}x_7) + x_5x_6 x_{1234}
\\
W/s=x_{56}\,(x_1 x_2 x_3 + x_1 x_2 x_4 + x_1 x_3 x_4 + x_2 x_3 x_4 + x_{13} x_{24} x_7 )
+x_5x_6 x_{13} x_{24}
\end{array}
\right.
\label{SEthrLi}
\end{equation}
\begin{equation}
\mathrm{(j)}\quad
\left\{
\begin{array}{l}
U=x_{37}\,(x_{12}x_{45} + x_{1245}x_6) + x_3x_7 x_{126}
\\
W/s=x_{37}\,(x_1 x_2 x_4 + x_1 x_2 x_5 + x_1 x_4 x_5 + x_2 x_4 x_5 + x_{14} x_{25} x_6 )\\
\quad +x_3x_7 \,( x_1 x_{25} +x_2x_5 + x_{25} x_6 )
\end{array}
\right.
\label{SEthrLj}
\end{equation}
\noindent
Expansions for these integrals from \cite{laporta01} are:
\begin{align}
&I_{h}^{S3}({\varepsilon}) ~\Gamma(1+{\varepsilon})^{-3} = \sum_{k\ge -1} C_k {\varepsilon^k} = 2.404113806319\,{\varepsilon^{-1}}-9.7634244476+34.99888166\,{\varepsilon}-116.0420478\,\varepsilon^2\ldots \label{I6hexp}
\end{align}
\begin{align}
&I_{i}^{S3}({\varepsilon}) ~\Gamma(1+{\varepsilon})^{-3} = \sum_{k\ge -1} C_k {\varepsilon^k} = 0.923631826520\,{\varepsilon^{-1}}-2.4234916344+8.3813497101\,{\varepsilon}-26.99362122\,\varepsilon^2\ldots \label{Ioexp}
\end{align}
\begin{align}
&I_{j}^{S3}({\varepsilon}) ~\Gamma(1+{\varepsilon})^{-3} = \sum_{k\ge -1} C_k {\varepsilon^k} = 0.923631826520\,{\varepsilon^{-1}}-2.1161697185+6.9295446853\,{\varepsilon}-21.50327838\,\varepsilon^2\ldots \label{Ipexp}
\end{align}
The evaluation is performed with $s=1,\, m_r=1$.
Numerical results by {\sc ParInt} on the \emph{thor} cluster,
for the asymptotic expansion coefficients of $I_{h}^{S3}\,\Gamma(1+{\varepsilon})^{-3}$
in Eq~\eqref{I6hexp}, are listed in Table~\ref{table5}.
A geometric sequence in base $2^{-1}$ is used for $\varepsilon.$
For the computation of $I_{i}^{S3}$ and $I_{j}^{S3}$, the variables are transformed as:
\begin{eqnarray}\label{KK-trans}
\nonumber
x_1&=&y_{1m} y_2 y_4 y_5, ~~~~x_2~=~y_{1m} y_2 y_4 y_{5m},\\
\nonumber
x_3&=&y_{1m} y_2 y_{4m} y_6, ~~~x_4~=~y_{1m} y_2 y_{4m} y_{6m},\\
x_5&=&y_{1} y_3, ~~~~x_6~=~y_{1} y_{3m},\\
\nonumber
x_7&=&y_{1m} y_{2m}
\end{eqnarray}
with $y_{im}=1-y_{i}$ and Jacobian $y_1 y_{1m}^4 y_2^3 y_4 y_{4m}$ for the former, and
\begin{eqnarray}\label{TI-trans}
\nonumber
x_1&=&y_1 y_2 y_4, ~~~~x_2=y_1 y_2 y_{4m}, \\
\nonumber
x_3&=&y_{1m} y_3 y_6, ~~~x_4=y_1 y_{2m} y_5, \\
x_5&=&y_1 y_{2m} y_{5m}, ~~~x_6=y_{1m} y_{3m}, \\
\nonumber
x_7&=&y_{1m} y_3 y_{6m}
\end{eqnarray}
with $y_{im}=1-y_{i}$ and Jacobian $y_1^{3} y_{1m}^2 y_{2m} y_2 y_3$ for the latter.
These variable transformations are beneficial to smoothen the integrand
boundary singularities.
The above two variable transformations can be adopted for both $I_{i}^{S3}$ and $I_{j}^{S3}$.
However, the transformation \eqref{KK-trans} works better for $I_{i}^{S3}$ and \eqref{TI-trans} works better for $I_{j}^{S3}.$
\begin{table}
\caption{\footnotesize
{\color{black}Results UV \emph{3-loop} integral} of Fig~\ref{3ls-self}(h) (on 4 nodes with 16 procs per node of \emph{thor} cluster),
err. tol. $t_a = 10^{-12},$ $T[s]$ = Time (elapsed user time in s);
$\varepsilon = \varepsilon_\ell = 2^{-\ell},~ \ell = 2,3,\ldots,$ ~~$E_r = $ integration estim. abs. error}
\begin{scriptsize}
\begin{center}
\begin{tabular}{ccccccc}\hline
& \multicolumn{2}{c}{{\sc Integral Fig~\ref{3ls-self}}(h)}
& \multicolumn{4}{c}{{\sc Extrapolation}} \\
\raisebox{1mm}{\hspace*{-5mm}$\ell$}\hspace*{-5mm} & \hspace{-1mm}{ $E_r$ }\hspace{-1mm} & \hspace{-1mm}{\sc T[s]}
\hspace{-1mm} & {\sc Result} ~${C}_{-1}$ & \hspace{-1mm}{\sc Result} ~${C}_{0}$\hspace{-1mm} & \hspace*{-0mm}{\sc Result} ~${C}_1$ & \hspace{-1mm}{\sc Result} ~${C}_2$\hspace{-1mm} \\% & & & & & & & \\
\hlin
\hspace*{-1mm}$2$\hspace*{-1mm}& \hspace{-1mm}5.7e-10\hspace{-1mm} & \hspace{-1mm}17.6\hspace{-1mm} & \hspace{-1mm}\hspace{-1mm} & \hspace{-1mm}\hspace{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$3$\hspace*{-1mm}& \hspace{-1mm}1.8e-09\hspace{-1mm} & \hspace{-1mm}28.3\hspace{-1mm} & \hspace{-1mm}1.990363419875\hspace{-1mm} & \hspace{-1mm}-3.3722810037\hspace{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$4$\hspace*{-1mm}& \hspace{-1mm}3.2e-09\hspace{-1mm} & \hspace{-1mm}28.3\hspace{-1mm} & \hspace{-1mm}2.330127359814\hspace{-1mm} & \hspace{-1mm}-7.4494482830\hspace{-1mm} & \hspace*{-1.5mm}10.87244608\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} \\
\hspace*{-1mm}$5$\hspace*{-1mm}& \hspace{-1mm}4.9e-09\hspace{-1mm} & \hspace{-1mm}28.3\hspace{-1mm} & \hspace{-1mm}2.397223358974\hspace{-1mm} & \hspace{-1mm}-9.3281362595\hspace{-1mm} & \hspace*{-1.5mm}25.90194989\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} -34.353152 \\
\hspace*{-1mm}$6$\hspace*{-1mm}& \hspace{-1mm}6.3e-09\hspace{-1mm} & \hspace{-1mm}28.3\hspace{-1mm} & \hspace{-1mm}2.403788052525\hspace{-1mm} & \hspace{-1mm}-9.7220178725\hspace{-1mm} & \hspace*{-1.5mm}33.25440667\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} -84.769999 \\
\hspace*{-1mm}$7$\hspace*{-1mm}& \hspace{-1mm}7.0e-09\hspace{-1mm} & \hspace{-1mm}28.3\hspace{-1mm} & \hspace{-1mm}2.404106076736\hspace{-1mm} & \hspace{-1mm}-9.7634244078\hspace{-1mm} & \hspace*{-1.5mm}34.83180676\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} -110.00840 \\
\hspace*{-1mm}$8$\hspace*{-1mm}& \hspace{-1mm}7.4e-09\hspace{-1mm} & \hspace{-1mm}28.3\hspace{-1mm} & \hspace{-1mm}2.404113714590\hspace{-1mm} & \hspace{-1mm}-9.7633776138\hspace{-1mm} & \hspace*{-1.5mm}34.99091852\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} -115.46366 \\
\hspace*{-1mm}$9$\hspace*{-1mm}& \hspace{-1mm}7.7e-09\hspace{-1mm} & \hspace{-1mm}28.3\hspace{-1mm} & \hspace{-1mm}2.404113805535\hspace{-1mm} & \hspace{-1mm}-9.7634238138\hspace{-1mm} & \hspace*{-1.5mm}34.99868012\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} -116.01362 \\
\hspace*{-1mm}$10$\hspace*{-1mm}& \hspace{-1mm}8.5e-09\hspace{-1mm} & \hspace{-1mm}28.3\hspace{-1mm} & \hspace{-1mm}2.404113806117\hspace{-1mm} & \hspace{-1mm}-9.7634244078\hspace{-1mm} & \hspace*{-1.5mm}34.99888129\hspace*{-1mm} & \hspace{-1mm}\hspace{-1mm} -116.04259 \\
\hline
\multicolumn{3}{r} \emph{Eq}~\eqref{I6hexp}: & 2.404113806319 & -9.7634244476 &\hspace*{-1.0mm} \hspace*{-1.5mm}34.99888166\hspace*{-1mm} &\hspace{-1mm}\hspace{-1mm} -116.04205 \\ \hline
\end{tabular}
\end{center}
\end{scriptsize}
\label{table5}
\end{table}
\begin{table}
\caption{\footnotesize
{\color{black}Results UV \emph{3-loop} diagram of Fig~\ref{3ls-self}(i) with massive internal lines, using 36 threads on Intel(R) Xeon(R) E5-2687W v3 3.10\,GHz. DE is applied with mesh size $h=0.1265988$ and number of evaluations $N_{eval}=104$ (cf., Eq~\eqref{DEsum}). For extrapolation, $\varepsilon = \varepsilon_{\ell} = 1.15^{-\ell}, \ell=17,18,\ldots$}}
\begin{scriptsize}
\begin{center}
\begin{tabular}{ccccl}\hline
\multicolumn{2}{l}{{\sc Integral Fig~\ref{3ls-self}}(i)}& \multicolumn{3}{c}{{\sc Extrapolation}} \\
{\hspace*{-5mm}$\ell$}\hspace*{-5mm} & {\sc Result} ~${C}_{-1}$ & \hspace{-1mm}{\sc Result} ~${C}_{0}$\hspace{-1mm} & \hspace*{-0mm}{\sc Result} ~${C}_1$ & \hspace*{-1.5mm}{\sc Result} ~${C}_2$\\ \hline
\hspace*{-1mm}$17$\hspace*{-1mm}& & \hspace{-1mm}\hspace{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm} \\
\hspace*{-1mm}$18$\hspace*{-1mm}& \hspace{-1mm}0.89291935327\hspace{-1mm} & \hspace{-1mm}-1.506522015\hspace{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm} & \\
\hspace*{-1mm}$19$\hspace*{-1mm}& \hspace{-1mm}0.91852761612\hspace{-1mm} & \hspace{-1mm}-2.187886833\hspace{-1mm} & \hspace*{-1.5mm}0.45102458\hspace*{-1mm} & \\
\hspace*{-1mm}$20$\hspace*{-1mm}& \hspace{-1mm}0.92289114325\hspace{-1mm} & \hspace{-1mm}-2.375404012\hspace{-1mm} & \hspace*{-1.5mm}7.17894748\hspace*{-1mm} & \hspace*{-1.5mm}-12.57809\hspace*{-1mm} \\
\hspace*{-1mm}$21$\hspace*{-1mm}& \hspace{-1mm}0.92353815728\hspace{-1mm} & \hspace{-1mm}-2.415386426\hspace{-1mm} & \hspace*{-1.5mm}8.09798189\hspace*{-1mm} & \hspace*{-1.5mm}-21.89098\hspace*{-1mm} \\
\hspace*{-1mm}$22$\hspace*{-1mm}& \hspace{-1mm}0.92362147849\hspace{-1mm} & \hspace{-1mm}-2.422338750\hspace{-1mm} & \hspace*{-1.5mm}8.32777907\hspace*{-1mm} & \hspace*{-1.5mm}-25.65197\hspace*{-1mm} \\
\hspace*{-1mm}$23$\hspace*{-1mm}& \hspace{-1mm}0.92363082823\hspace{-1mm} & \hspace{-1mm}-2.423351621\hspace{-1mm} & \hspace*{-1.5mm}8.37298426\hspace*{-1mm} & \hspace*{-1.5mm}-26.71586\hspace*{-1mm} \\
\hspace*{-1mm}$24$\hspace*{-1mm}& \hspace{-1mm}0.92363174411\hspace{-1mm} & \hspace{-1mm}-2.423477056\hspace{-1mm} & \hspace*{-1.5mm}8.38025256\hspace*{-1mm} & \hspace*{-1.5mm}-26.94683\hspace*{-1mm} \\
\hspace*{-1mm}$25$\hspace*{-1mm}& \hspace{-1mm}0.92363182249\hspace{-1mm} & \hspace{-1mm}-2.423490372\hspace{-1mm} & \hspace*{-1.5mm}8.38122798\hspace*{-1mm} & \hspace*{-1.5mm}-26.98707\hspace*{-1mm} \\
\hspace*{-1mm}$26$\hspace*{-1mm}& \hspace{-1mm}0.92363182726\hspace{-1mm} & \hspace{-1mm}-2.423491361\hspace{-1mm} & \hspace*{-1.5mm}8.38131792\hspace*{-1mm} & \hspace*{-1.5mm}-26.99177\hspace*{-1mm} \\
\hline
\hspace*{-1mm} {Eq}~\eqref{Ioexp}:\hspace*{-1mm} & \hspace*{-1mm}0.92363182652\hspace*{-1mm}&\hspace{-1mm} -2.423491634\hspace{-1mm}&\hspace*{-1.5mm}8.38134971\hspace*{-1mm} & \hspace*{-1.5mm}-26.99362\hspace*{-1mm} \\ \hline
\end{tabular}
\end{center}
\end{scriptsize}
\label{table5i}
\end{table}
Numerical results achieved with DE on Intel(R) Xeon(R) E5-2687W v3 3.10\,GHz are shown in Tables~\ref{table5i} and ~\ref{table5j}. Using IEEE 754-2008 binary128, extensive computation times are incurred (28 hours per iteration for Table~\ref{table5i} and 10 hours for Table~\ref{table5j}), as a trade-off for high accuracy. Similar or slightly less accuracy but far shorter computation times (between 670 and 2540 seconds per iteration) are reported in~\cite{cpp16}, using {\sc ParInt} in \emph{long double} precision on 4 nodes and 16 MPI processes per node of the \emph{thor} cluster.
\section{4-loop self-energy integrals with massless internal lines}\label{4-loop-self}
In this section we calculate the integral in Eq~\eqref{LloopIJ} for
$L=4$ and $n=4-2\varepsilon,$
\begin{equation}
I = (-1)^N \,{\Gamma\left(N-8+4\varepsilon \right)}
\int_{0}^{1}\prod_{r=1}^{N}dx_{r}\,
\delta\,(1-\sum x_{r})\,\frac{1}{U^{2-\varepsilon}\,(V-i\varrho)^{N-8+4\varepsilon}}.
\label{fourLOOP}
\end{equation}
\begin{table}
\caption{\footnotesize
{\color{black}Results UV \emph{3-loop} diagram of Fig~\ref{3ls-self}(j) with massive internal lines, using 36 threads on Intel(R) Xeon(R) E5-2687W v3 3.10\,GHz. DE is applied with mesh size $h=0.1253191$ and number of evaluations $N_{eval}=94$ (cf., Eq~\eqref{DEsum}). For extrapolation, $\varepsilon = \varepsilon_{\ell} = 1.15^{-\ell}, \ell=10,11,\ldots$}}
\begin{scriptsize}
\begin{center}
\begin{tabular}{ccccl}\hline
\multicolumn{2}{l}{{\sc Integral Fig~\ref{3ls-self}}(j)}& \multicolumn{3}{c}{{\sc Extrapolation}} \\
{\hspace*{-5mm}$\ell$}\hspace*{-5mm} & {\sc Result} ~${C}_{-1}$ & \hspace{-1mm}{\sc Result} ~${C}_{0}$\hspace{-1mm} & \hspace*{-0mm}{\sc Result} ~${C}_1$ & \hspace*{-1.5mm}{\sc Result} ~${C}_2$\hspace*{-1mm}\\ \hline
\hspace*{-1mm}$10$\hspace*{-1mm}& & \hspace{-1mm}\hspace{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm} \\
\hspace*{-1mm}$11$\hspace*{-1mm}& \hspace{-1mm}0.79879040550\hspace{-1mm} & \hspace{-1mm}-0.6424371659\hspace{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm} \\
\hspace*{-1mm}$12$\hspace*{-1mm}& \hspace{-1mm}0.87785768590\hspace{-1mm} & \hspace{-1mm}-1.3301603488\hspace{-1mm} & \hspace*{-1.5mm}1.48816624\hspace*{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm}\\
\hspace*{-1mm}$13$\hspace*{-1mm}& \hspace{-1mm}0.90817425166\hspace{-1mm} & \hspace{-1mm}-1.7560535047\hspace{-1mm} & \hspace*{-1.5mm}3.46958790\hspace*{-1mm} & \hspace*{-1.5mm}-3.0528730\hspace*{-1mm}\\
\hspace*{-1mm}$14$\hspace*{-1mm}& \hspace{-1mm}0.91891701861\hspace{-1mm} & \hspace{-1mm}-1.9730679987\hspace{-1mm} & \hspace*{-1.5mm}5.10026802\hspace*{-1mm} & \hspace*{-1.5mm}-8.4546800\hspace*{-1mm}\\
\hspace*{-1mm}$15$\hspace*{-1mm}& \hspace{-1mm}0.92233670919\hspace{-1mm} & \hspace{-1mm}-2.0663458474\hspace{-1mm} & \hspace*{-1.5mm}6.10815188\hspace*{-1mm} & \hspace*{-1.5mm}-13.847111\hspace*{-1mm}\\
\hspace*{-1mm}$16$\hspace*{-1mm}& \hspace{-1mm}0.92331256399\hspace{-1mm} & \hspace{-1mm}-2.1009045295\hspace{-1mm} & \hspace*{-1.5mm}6.61235817\hspace*{-1mm} & \hspace*{-1.5mm}-17.726240\hspace*{-1mm}\\
\hspace*{-1mm}$17$\hspace*{-1mm}& \hspace{-1mm}0.92356140680\hspace{-1mm} & \hspace{-1mm}-2.1120455643\hspace{-1mm} & \hspace*{-1.5mm}6.82339433\hspace*{-1mm} & \hspace*{-1.5mm}-19.918551\hspace*{-1mm}\\
\hspace*{-1mm}$18$\hspace*{-1mm}& \hspace{-1mm}0.92361796674\hspace{-1mm} &\hspace{-1mm}-2.1151864872\hspace{-1mm} & \hspace*{-1.5mm}6.89861152\hspace*{-1mm} & \hspace*{-1.5mm}-20.933028\hspace*{-1mm}\\
\hspace*{-1mm}$19$\hspace*{-1mm}& \hspace{-1mm}0.92362939967\hspace{-1mm} &\hspace{-1mm}-2.1159628755\hspace{-1mm} & \hspace*{-1.5mm}6.92167297\hspace*{-1mm} & \hspace*{-1.5mm}-21.326253\hspace*{-1mm}\\
\hspace*{-1mm}$20$\hspace*{-1mm}& \hspace{-1mm}0.92363145072\hspace{-1mm} &\hspace{-1mm}-2.1161313491\hspace{-1mm} & \hspace*{-1.5mm}6.92779243\hspace*{-1mm} & \hspace*{-1.5mm}-21.455677\hspace*{-1mm}\\
\hspace*{-1mm}$21$\hspace*{-1mm}& \hspace{-1mm}0.92363177660\hspace{-1mm} &\hspace{-1mm}-2.1161634494\hspace{-1mm} & \hspace*{-1.5mm}6.92920277\hspace*{-1mm} & \hspace*{-1.5mm}-21.492152\hspace*{-1mm}\\
\hspace*{-1mm}$22$\hspace*{-1mm}& \hspace{-1mm}0.92363182245\hspace{-1mm} &\hspace{-1mm}-2.1161688298\hspace{-1mm} & \hspace*{-1.5mm}6.92948625\hspace*{-1mm} & \hspace*{-1.5mm}-21.501020\hspace*{-1mm}\\
\hspace*{-1mm}$23$\hspace*{-1mm}& \hspace{-1mm}0.92363182806\hspace{-1mm} &\hspace{-1mm}-2.1161696094\hspace{-1mm} & \hspace*{-1.5mm}6.92953519\hspace*{-1mm} & \hspace*{-1.5mm}-21.502856\hspace*{-1mm}\\
\hspace*{-1mm}$24$\hspace*{-1mm}& \hspace{-1mm}0.92363182875\hspace{-1mm} &\hspace{-1mm}-2.1161697224\hspace{-1mm} & \hspace*{-1.5mm}6.92954359\hspace*{-1mm} & \hspace*{-1.5mm}-21.503232\hspace*{-1mm}\\
\hline
\hspace*{-1mm} {Eq}~\eqref{Ipexp}:\hspace*{-1mm} & \hspace*{-1mm}0.92363182652 \hspace*{-1mm}&\hspace{-1mm} -2.1161697185\hspace{-1mm}&\hspace*{-1.5mm}6.92954468\hspace*{-1mm} & \hspace*{-1.5mm}-21.503278\hspace*{-1mm}\\ \hline
\end{tabular}
\end{center}
\end{scriptsize}
\label{table5j}
\end{table}
\section{4-loop self-energy integrals with massless internal lines}\label{4-loop-self}
\noindent
UV divergence occurs when $U$ vanishes at the boundaries.
The $\Gamma$-function in Eq~\eqref{fourLOOP} contributes to UV divergence when $N\le 8$.
As show in in figures, the entering momentum is $p,$ and we
denote $s=p^2$.
We address the integrals adhering to Eq~\eqref{fourLOOP}
for the diagrams of Fig~\ref{4ls-massless-diagrams},
which are denoted by $I_a^{S4},\, I_b^{S4},\,I_c^{S4},\, I_d^{S4}.$
We only consider the massless case, i.e., $m_r=0$.
In the numerical evaluation, we set the value $s=1$.
\subsection{4-loop finite integrals}
Let us consider the finite integrals $I_a^{S4},\, I_b^{S4}$
corresponding to the diagrams
of Fig~\ref{4ls-massless-diagrams}(a) and (b)
(named $M_{44}$ and $M_{45}$ in Baikov and Chetyrkin~\cite{baikov10}).
The $U, W$ functions are given in Eqs~\eqref{SEforLa}-\eqref{SEforLb}.
\begin{equation}
\mathrm{(a)}\quad
\left\{
\begin{array}{l}
U=
x_7 x_8 x_9 x_{123456}
+x_7 x_8 x_{1256} x_{34}
+x_7 x_9 x_{126} x_{345}
+x_8 x_9 x_{16} x_{2345}
+x_7 x_5 x_{126} x_{34} \\
\quad +x_8 x_{16} x_{25} x_{34}
+x_9 x_2 x_{16} x_{345}
+x_2 x_5 x_{16} x_{34}
\\
W/s=
x_7 x_8 x_9 x_{123} x_{456}
+x_7 x_8 \,( x_{12} x_3 x_{456} + x_{123} x_4 x_{56} )
+x_7 x_9 \,( x_{12} x_3 x_{456} + x_{123} x_{45} x_6 ) \\
\quad +x_8 x_9 \,( x_1 x_{23} x_{456} + x_{123} x_{45} x_6 )
+x_7 x_5 \,( x_{12} x_3 x_{46} + x_{123} x_4 x_6 ) \\
\quad +x_8 \,( x_{12} x_{34} x_5 x_6+x_1 x_2 x_{34} x_{56} + x_{16} x_3 x_4 x_{25} )
+x_9 x_2 \,( x_{13} x_{45} x_6+x_1 x_3 x_{456} ) \\
\quad +x_2 x_5 \,( x_1 x_3 x_{46} + x_{13} x_4 x_6 )
\end{array}
\right.
\label{SEforLa}
\end{equation}
\begin{equation}
\mathrm{(b)}\quad
\left\{
\begin{array}{l}
U=
x_7 x_8 x_9 x_{123456}
+x_7 x_8 \,( x_{12} x_{34} + x_{34} x_{56} )
+x_7 x_9 x_{126} x_{345}
+x_8 x_9 x_{156} x_{234} \\
\quad +x_7 x_5 x_{126} x_{34}
+x_8 x_2 x_{156} x_{34}
+x_9 \,( x_{16} x_2 x_{345} + x_{126} x_{34} x_5 )
+x_2 x_5 x_{16} x_{34}
\\
W/s=
x_7 x_8 x_9 x_{123} x_{456}
+x_7 x_8 \,( x_{12} x_4 x_{356} + x_{124} x_3 x_{56} )
+x_7 x_9 \,( x_{12} x_3 x_{456} + x_{123} x_{45} x_6 ) \\
\quad +x_8 x_9 \,( x_1 x_{23} x_{456} + x_{123} x_4 x_{56} )
+x_7 x_5 \,( x_{12} x_3 x_{46} + x_{123} x_4 x_6 )
+x_8 x_2 \,( x_{13} x_4 x_{56} +x_1 x_3 x_{456} ) \\
\quad +x_9 \,( x_1 x_2 x_{36} x_{45} + x_{12} x_{36} x_4 x_5+ x_{14} x_{25} x_3 x_6 )
+x_2 x_5 \,( x_1 x_3 x_{46} + x_{13} x_4 x_6 )
\end{array}
\right.
\label{SEforLb}
\end{equation}
Since the corresponding integrals are finite, Eq~\eqref{fourLOOP} is evaluated
with $ \varrho = \varepsilon = 0$.
These diagrams have $N=9$ internal lines, leading to an 8-dimensional integral in the numerical evaluation.
Table~\ref{tab:4ls-eps0} lists the results for the integrals,
obtained with {\sc ParInt} executed on \emph{thor} in double precision, using the cubature rule
of degree 9 in 8 dimensions (see Section~\ref{parint}), which evaluates the function at 1105 points per subregion.
The analytic values given in~\cite{baikov10} are:
\begin{equation}\label{BC-M44}
I_a^{S4}=\frac{441\,\zeta_7}{8} = 55.5852539156784\,,
\end{equation}
\begin{equation}\label{BC-M45}
I_b^{S4}= 36\,\zeta_3^2 = 52.017868743610 \,.
\end{equation}
$I_a^{S4}$ and $I_b^{S4}$ were evaluated numerically by Smirnov and Tentyukov using FIESTA~\cite{smirnov10}.
The finite terms are given as $55.58537 \pm 0.00031$ and $52.0181 \pm 0.0003,$
respectively, with $1.5$ M samples.
\begin{figure}
\begin{center}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.22\linewidth]{./figures/4ls-fig8a-20161029.epsi}
\end{subfigure}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.22\linewidth]{./figures/4ls-fig8b-20161029.epsi}
\end{subfigure}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.22\linewidth]{./figures/4ls-fig8c-20161029.epsi}
\end{subfigure}
\begin{subfigure}[ ]
\centering
\includegraphics[width=0.22\linewidth]{./figures/4ls-fig8d-20161029.epsi}
\end{subfigure}
\caption{4-loop self-energy diagrams with massless internal lines,
cf., Baikov and Chetyrkin~\cite{baikov10}:
(a) $N=9$, (b) $N=9$, (c) $N=5$, (d) $N=8$}
\label{4ls-massless-diagrams}
\end{center}
\end{figure}
\begin{table}
\caption{\footnotesize {\sc ParInt} accuracy and times (on 4 nodes/64 procs. \emph{thor} cluster) for the loop integrals of the
diagrams of Fig~\ref{4ls-massless-diagrams}(a) and (b) ($M_{44}$ and $M_{45}$ in Baikov and Chetyrkin~\cite{baikov10})
with $\varrho = \varepsilon = 0,$ using various numbers of function evaluations.
}
\begin{footnotesize}
\begin{center}
\begin{tabular}{ccccc}\hline
Diagram & \# {\sc Fcn.} \hspace*{-1mm} & {\sc Integral} \hspace{-1mm} & \hspace{-1mm}{\sc Rel. err. }\hspace{-1mm} & \hspace{-2mm}{\sc Time} \\
& {\sc Evals.} \hspace*{-1mm} & {\sc Result} \hspace{-1mm} & \hspace{-1mm}{{\sc Est.} $E_r$ }\hspace{-1mm} & \hspace{-2mm}{\sc T[s]}\hspace{-1mm} \\
~~\\
\hline
~~\\
Fig~\ref{4ls-massless-diagrams} (a)
& 100B & 55.594725 & 9.87e-04 & 185.1 \\
& 200B & 55.586822 & 3.53e-04 & 370.0 \\
& 300B & 55.585150 & 1.80e-04 & 554.3 \\
\hline
& {Eq}~\eqref{BC-M44}: & 55.585254 & & \\
\hline
~~\\
Fig~\ref{4ls-massless-diagrams} (b)
& 100B & 52.026428 & 9.84e-04 & 239.5 \\
& 200B & 52.019118 & 3.63e-04 & 479.9 \\
& 275B & 52.017714 & 2.20e-04 & 658.8 \\
\hline
& {Eq}~\eqref{BC-M45}: & 52.017869 & & \\
\hline
\end{tabular}
\end{center}
\end{footnotesize}
\label{tab:4ls-eps0}
\end{table}
\subsection{4-loop UV-divergent integrals}\label{uv-4ls}
This section handles the integrals associated with the massless diagrams of
Fig~\ref{4ls-massless-diagrams}(c) and (d) (the 4-loop \emph{sunrise-sunset} and
\emph{Shimadzu} diagrams
named $M_{01}$ and $M_{36},$ respectively, in~\cite{baikov10}),
which have a UV singularity from the $\Gamma$-function factor in Eq~\eqref{fourLOOP}.
We put $\varrho = 0$ in
the integrals for Fig~\ref{4ls-massless-diagrams} (c) and (d), and consider the
expansions in $\varepsilon.$
For the integral $I_c^{S4}$ in Fig~\ref{4ls-massless-diagrams},
the $U,\, W$ functions are
\begin{equation}
\mathrm{(c)}\quad
\left\{
\begin{array}{l}
U=x_1 x_2 x_3 x_4 + x_1 x_2 x_3 x_5 + x_1 x_2 x_4 x_5 + x_1 x_3 x_4 x_5 + x_2 x_3 x_4 x_5
\\
W/s=x_1 x_2 x_3 x_4 x_5
\end{array}
\right.\,.
\label{SEforLc}
\end{equation}
\noindent
The numerical results are compared with the expansion in Baikov and Chetyrkin~\cite{baikov10},
\begin{align}
&
n(\varepsilon)^4 I_c^{S4}
= -\frac{1}{576}\,\frac{1}{\varepsilon}-\frac{13}{768}-\frac{9823}{82944}\,\varepsilon
+\,\ldots \nonumber \\
& = -0.001736111111\,\frac{1}{\varepsilon}-0.016927083333-0.118429301698\,\varepsilon
+\,\ldots \label{BandC-m01}
\end{align}
Note that $I = I_c^{S4}$ of Eq~\eqref{fourLOOP} is multiplied with $n(\varepsilon)^4,$
where $n(\varepsilon)$ is defined in Eq~\eqref{factorn}.
Based on integrations with {\sc ParInt}, using a maximum of 10B function evaluations
and an absolute error tolerance of $10^{-12}$ (on the computation of the integral
$I_{c}^{S4}/\Gamma(-3+4\varepsilon)$), the results in Table~\ref{tab:M01} are produced using linear extrapolation.
{\sc ParInt} returns a 0 error flag for the integrals constituting the input sequence to the
extrapolation, indicating that a successful termination
is assumed according to Eqs~\eqref{accuracy} or~\eqref{acc} for the requested accuracy.
\begin{table}
\caption{\footnotesize
{\color{black}Results UV \emph{4-loop sunrise-sunset} integral}, Fig~\ref{4ls-massless-diagrams} (c) (on 4 nodes/64 procs. \emph{thor} cluster),
err. tol. $t_a = 10^{-12},$ $T[s]$ = Time (elapsed user time in seconds);
$\varepsilon = \varepsilon_\ell = 2^{-\ell},~ \ell = 8,9,\ldots,$ ~~$E_a = $ integration estim. abs. error}
\begin{scriptsize}
\begin{center}
\begin{tabular}{cccccc}\hline
& \multicolumn{2}{c}{{\sc Integral $I_{c}^{S4}$}}
& \multicolumn{3}{c}{{\sc Extrapolation}} \\
\raisebox{1mm}{\hspace*{-5mm}$\ell$}\hspace*{-5mm} & \hspace{-1mm}{ $E_a$ }\hspace{-1mm} & \hspace{-1mm}{\sc T[s]}
\hspace{-1mm} & {\sc Result} ~${C}_{-1}$ & \hspace{-1mm}{\sc Result} ~${C}_{0}$\hspace{-1mm} & \hspace*{-0mm}{\sc Result} ~${C}_1$ \\% & & & & & & \\
\hlin
\hspace*{-1mm}$8$\hspace*{-1mm}& \hspace{-1mm}3.9e-14\hspace{-1mm} & \hspace{-1mm}30.2\hspace{-1mm} & \hspace{-1mm}\hspace{-1mm} & \hspace{-1mm}\hspace{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm} \\
\hspace*{-1mm}$9$\hspace*{-1mm}& \hspace{-1mm}3.8e-14\hspace{-1mm} & \hspace{-1mm}34.3\hspace{-1mm} & \hspace{-1mm} -0.001735179254977\hspace{-1mm} & \hspace{-1mm}\hspace{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm} \\
\hspace*{-1mm}$10$\hspace*{-1mm}& \hspace{-1mm}3.6e-14\hspace{-1mm} & \hspace{-1mm}34.1\hspace{-1mm} & \hspace{-1mm}-0.001736115881839\hspace{-1mm} & \hspace{-1mm}-0.016918557081\hspace{-1mm} & \hspace*{-1.5mm}-0.012276556 \hspace*{-1mm} \\
\hspace*{-1mm}$11$\hspace*{-1mm}& \hspace{-1mm}4.1e-14\hspace{-1mm} & \hspace{-1mm}50.8\hspace{-1mm} & \hspace{-1mm}-0.001736111099910\hspace{-1mm} & \hspace{-1mm}-0.016927126297\hspace{-1mm} & \hspace*{-1.5mm}-0.011837812 \hspace*{-1mm} \\
\hspace*{-1mm}$12$\hspace*{-1mm}& \hspace{-1mm}4.2e-14\hspace{-1mm} & \hspace{-1mm}58.5\hspace{-1mm} & \hspace{-1mm}-0.001736111111130\hspace{-1mm} & \hspace{-1mm}-0.016927083216\hspace{-1mm} & \hspace*{-1.5mm}-0.011842959 \hspace*{-1mm} \\
\hspace*{-1mm}$13$\hspace*{-1mm}& \hspace{-1mm}1.5e-14\hspace{-1mm} & \hspace{-1mm}48.1\hspace{-1mm} & \hspace{-1mm}-0.001736111111109\hspace{-1mm} & \hspace{-1mm}-0.016927083381\hspace{-1mm} & \hspace*{-1.5mm}-0.011842916 \hspace*{-1mm} \\
\hline
\multicolumn{3}{r} \emph{Eq}~\eqref{BandC-m01}: & -0.001736111111111 & -0.016927083333 &\hspace*{-1.0mm} \hspace*{-1.5mm}-0.011842930\hspace*{-1mm} \\
\hline
\end{tabular}
\end{center}
\end{scriptsize}
\label{tab:M01}
\end{table}
For $I_d^{S4}$ of the $N=8$ diagram shown in Fig~\ref{4ls-massless-diagrams} (d)
(\emph{Shimadzu}, named $M_{36}$ in~\cite{baikov10,smirnov10}), the
$U, W$ functions are:
\begin{equation}
\mathrm{(d)}\quad
\left\{
\begin{array}{l}
U=
x_7 x_8 \,( x_{12} x_{3456} + x_{34} x_{56} )
+ x_{78} x_{1234} x_5 x_6 \\
\quad +x_7 \,( x_5 x_3 x_{124} +x_6 x_4 x_{123} +x_3 x_4 x_{12} )
+x_8 \,( x_5 x_2 x_{134} +x_6 x_1 x_{234} +x_1 x_2 x_{34} ) \\
\quad +x_5 x_6 x_{14} x_{23} + x_5 x_2 x_3 x_{14} + x_6 x_1 x_4 x_{23}
+ x_1 x_2 x_3 x_4
\\
W/s=
x_7 x_8 x_{12} x_{34} x_{56}
+ x_{78} x_{12} x_{34} x_5 x_6
+x_7 x_{12} x_3 x_4 x_{56}
+x_8 x_1 x_2 x_{34} x_{56} \\
\quad +x_5 x_6 \,( x_1 x_2 x_{34} + x_{12} x_3 x_4 )
+ x_{56} x_1 x_2 x_3 x_4
\end{array}
\right.
\label{SEforLd}
\end{equation}
\noindent
The expansion given in~\cite{baikov10} is
\begin{align}
&
n(\varepsilon)^4 I_{d}^{S4} =
\frac{5\zeta_{5}}{\varepsilon}-5\zeta_{5}-7\zeta_{3}^2 +\frac{25}{2}\zeta_{6}
+ \,(35\zeta_{5}+7\zeta_{3}^2-\frac{25}{2}\zeta_{6}-21\zeta_{3}\zeta_{4}+\frac{127}{2}\zeta_{7})\,\varepsilon +\, \ldots,\nonumber \\
& = \frac{5.184638776}{\varepsilon}- 2.582436090 +70.39915145\,\varepsilon
+\,\ldots \label{BandC-m36}
\end{align}
This is $I = I_{d}^{S4}$ of Eq~\eqref{fourLOOP} multiplied with $n(\varepsilon)^4,$
where $n(\varepsilon)$ is defined in Eq~\eqref{factorn}.
The numerical result by FIESTA is shown
in~\cite{smirnov10}
and it is $5.184645 \pm 0.000042$.
The results by the DE formula~\eqref{DEsum} with $N_{eval}=49$ and mesh size $h=0.125$ in all dimensions,
and linear extrapolation, are shown in Table~\ref{tableM36}.
The starting $\varepsilon$ is $2^{-10}.$
The time required for each iteration using 64 threads on KEKSC System A, SR16000 Model M1 (POWER7(R) processor) is below 20 minutes.
\begin{table}
\caption{\footnotesize{ Results UV \emph{4-loop Shimadzu} integral, $I_{d}^{S4}$, Fig~\ref{4ls-massless-diagrams} (d), (on KEKSC 64 threads);
$\varepsilon = \varepsilon_\ell = 2^{-\ell},~ \ell = 10,11,\ldots,$.}}
\begin{scriptsize}
\begin{center}
\begin{tabular}{cccc}\hline
\multicolumn{1}{c}{{\sc Integral $I_d^{S4}$}}
& \multicolumn{3}{c}{{\sc Extrapolation}} \\
\raisebox{1mm}{\hspace*{-5mm}$\ell$}\hspace*{-5mm} & {\sc Result} ~${C}_{-1}$ & \hspace{-1mm}{\sc Result} ~${C}_{0}$\hspace{-1mm} & \hspace*{-0mm}{\sc Result} ~${C}_1$ \\% & & & & & & \\
\hlin
\hspace*{-1mm}$10$\hspace*{-1mm}& \hspace{-1mm}\hspace{-1mm} & \hspace{-1mm}\hspace{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm} \\
\hspace*{-1mm}$11$\hspace*{-1mm}& \hspace{-1mm}5.18460577\hspace{-1mm} & \hspace{-1mm}-2.47956688\hspace{-1mm} & \hspace*{-1.5mm}\hspace*{-1mm} \\
\hspace*{-1mm}$12$\hspace*{-1mm}& \hspace{-1mm}5.18463921\hspace{-1mm} & \hspace{-1mm}-2.58230627\hspace{-1mm} & \hspace*{-1.5mm} 70.1367604\hspace*{-1mm} \\
\hspace*{-1mm}$13$\hspace*{-1mm}& \hspace{-1mm}5.18463922\hspace{-1mm} & \hspace{-1mm}-2.58243393\hspace{-1mm} & \hspace*{-1.5mm} 70.3982056\hspace*{-1mm} \\
\hspace*{-1mm}$14$\hspace*{-1mm}& \hspace{-1mm}5.18463923\hspace{-1mm} & \hspace{-1mm}-2.58243413\hspace{-1mm} & \hspace*{-1.5mm} 70.3991764\hspace*{-1mm} \\
\hline
\multicolumn{1}{r} {Eq}\hspace*{-0.6mm}\eqref{BandC-m36}: &5.18463878 & -2.58243609 &70.3991515 \\
\hline
\end{tabular}
\end{center}
\end{scriptsize}
\label{tableM36}
\end{table}
\section{Conclusions} \label{conc}
In this paper we describe a fully numerical method for Feynman loop integrals, based on numerical multi-dimensional integration and linear or non-linear extrapolation.
We use three categories of numerical integration methods, iterated integration with {\sc Dqage (Dqags)} from {\sc Quadpack}, multivariate adaptive integration with {\sc ParInt}, and the DE formula.
For the numerical extrapolation, we employ nonlinear extrapolation with geometric sequences of the
extrapolation parameter, and linear extrapolation with Bulirsch or geometric sequences.
The main advantage of the method is its general applicability to multi-loop integrals with arbitrary physical masses and external momenta, without resorting to special problem formulations.
Using dimensional regularization, both {\it UV-divergent} and {\it finite} terms are estimated.
We have shown that the technique works well for sets of diagrams with up to four loops and up to four external lines, with or without UV-divergence,
and the numerical results
reveal excellent agreement with expansions
in the literature~\cite{laporta01,baikov10,smirnov10}.
We also demonstrate the effectiveness of variable transformations for some integrals.
Regardless of whether or not variables are transformed, the formulation of the numerical method
is not affected and it is only necessary to replace the integrand and Jacobian.
The way to transform variables for a loop integral is not unique, and a general technique to find
the most effective transformation is not currently known.
However, the effectiveness of transformations can be assessed by examining the
behavior of the integrations and the ensuing convergence of
the extrapolation.
The experience gained with the transformations in this paper will yield guidelines
to construct a more general procedure, which will be studied in future work.
The computation time of the numerical multivariate integration increases
with the number of internal lines, i.e., with the dimension of integration.
For example, though we understand the importance of scattering processes
with more external legs than five or six in two-loop order,
this is beyond the scope of the current paper, and we plan on
addressing these and related types of problems in future work.
Furthermore, while the examples shown here are limited to scalar loop integrals,
they can easily be extended to more general cases with physical masses and external momenta,
by including the associated numerator in the integrand (see~\cite{eddacat03}).
The present work includes integrals with massive and massless internal particles. For massive particles, a mass value of one is assigned in order to compare results with the literature~\cite{laporta01}.
Massless cases are compared, e.g., with results in~\cite{baikov10}.
In view of the numerical nature of the methods there is in principle no limitation for the general mass values,
even though challenges may arise with respect to computing time vs. computational precision.
We are helped in dealing with this trade-off by the evolution in computer architecture and
the computational techniques.
\section*{Acknowledgments}
We acknowledge the support from the National Science Foundation under Award Number 1126438,
and the Center for High Performance Computing and Big Data at Western Michigan University.
This work is further supported by Grant-in-Aid for Scientific
Research (15H03668) of JSPS, and the Large Scale Simulation Program Nos. 15/16-06 and 16/17-21 of KEK.
\input{cpc15paper-appendix}
\section*{References}
\bibliographystyle{iopart-num}
|
3,212,635,537,473 | arxiv |
\section{The Distance Oracle}
In this section we give a detailed presentation of both our algorithm
for answering distance queries and our distance oracle data structure.
Combining
Lemmas~\ref{lem:spaceboundDS},~\ref{lem:querytime}
and~\ref{lem:correctness} with $r = n^{2/3}$
directly implies Theorem~\ref{Thm:main}.
\subsection{The Data Structure}
We present the algorithm for building our data structure.
\medskip
\textsc{Preprocessing} $G$
\begin{enumerate}
\itemsep0pt
\item Compute an $r$-division $\mathcal{R}$ of $G$. Let $\delta$ be the
set of all boundary vertices.
\item Store for each internal vertex the region to which it belongs.
\item\label{step:allboundaryvert}
Compute and store the distances from each vertex to each
boundary vertex.
\item\label{step:allinternalpairs} For each region $R \in \mathcal{R}$,
compute and store the distances between any pair
of internal vertices of $R$.
\item\label{step:recursivedecomp} For each region $R$, for each
vertex $u \notin R$, for each hole
$H$, compute $\text{Vor}_{H}(R,u)$ and store a
separator decomposition as described in
Section~\ref{sec:recursivedecomp}.
\item\label{step:querysep}
For each region $R$, for each edge $(x,y) \in R$, for each hole
$H$,
compute and store the data structure described in
Section~\ref{sec:PreprocRegion}.
\end{enumerate}
\begin{lemma}
\label{lem:spaceboundDS}
The total size of the data structure computed by \textsc{Preprocessing}
is $O(n^2/\sqrt{r} + n \cdot r)$.
\end{lemma}
\begin{proof}
Recall that by definition of the $r$-division, there are $O(n/\sqrt{r})$
boundary vertices and $O(n/r)$ regions.
Thus, the number of distances
stored at step~\ref{step:allboundaryvert} of the
algorithm is at most $O(n^2/\sqrt{r})$.
For a given region, storing the pairwise distances between all
its internal vertices takes $O(r^2)$ space.
Since there are $O(n/r)$ regions in total,
Step~\ref{step:allinternalpairs}
takes memory
$O(n \cdot r)$.
We now bound the space taken by Step~\ref{step:recursivedecomp}.
There are $n/r$ choices for $R$ and $n$ choices for $u$.
By Lemma~\ref{Lem:sepspace}, each decomposition can be stored using
$O(\sqrt{r})$ space. Thus, this step takes
$O(n^2/\sqrt{r})$ total space.
We finally bound the space taken by Step~\ref{step:querysep}.
There $n/r$ choices for $R$ and $r$ choices for an edge $(x,y)$
By Lemma~\ref{Lem:RegionDS}, for a given edge $(x,y)$,
the data structure takes $O(r)$ space.
Hence, the total space taken by this step is $O(n \cdot r)$
and the lemma follows.
\end{proof}
\begin{theorem}
\label{thm:preprocesstime}
The execution of \textsc{Preprocessing} takes
$O(n^2)$
time and $O(n \cdot r + n^2/\sqrt{r} )$ space.
\end{theorem}
\begin{proof}
We analyze the procedure step by step. Computing an $r$-division
with $O(1)$ holes
can be done in linear time and space using
the algorithm of Klein et al., see~\cite{KleinMS13}.
We now analyze Step~\ref{step:allboundaryvert}.
There are $O(n/\sqrt{r})$ boundary vertices. Computing single-source
shortest paths can be done in linear time using the algorithm
of Henzinger et al.~\cite{Henzinger97}. Hence,
Step~\ref{step:allboundaryvert}
takes at most $O(n^2/\sqrt{r})$ time and space.
Step~\ref{step:allinternalpairs} takes
$O(n \cdot r)$ time and space
using the following algorithm. The algorithm proceed region by region
and hole by hole.
For a given region and hole, the algorithm adds an edge
between each pair
of boundary vertices that are on the hole
of length equal to the distance between these
vertices in the whole graph. Note that this is already in memory
and was computed at Step~\ref{step:allboundaryvert}.
Now, for each vertex of the region, the algorithm runs a
shortest path algorithm. Since there are $O(\sqrt{r})$ boundary
vertices, the number of edges added is $O(r)$.
Thus, the algorithm is run on a graph that has at most
$O(r)$ edges and vertices.
The algorithm spends at most $O(r)$
time per vertex of the region.
Since there are $O(n/r)$ regions and $O(r)$ vertices
per region, both the running time and the space
are $O(n \cdot r)$.
Step~\ref{step:recursivedecomp} takes $O(n^2)$ using the following
algorithm.
The algorithm proceeds vertex by vertex, region by region, hole
by hole.
For a given vertex $u$, a given region $R$, and a given hole $H$
the algorithm computes $\text{Vor}_{H}(R,u)$. This can be done
by adding a ``dummy'' vertex reprensenting $u$ and connecting it to
each boundary vertex $x$ of the hole by an edge of length $\text{dist}(u,x)$.
Thus, this takes time $O(r)$ using the single-source shortest path
algorithm of Henzinger et al~\cite{Henzinger97}. Furthermore,
by Lemma~\ref{Lem:sepspace} and Corollary~\ref{Cor:Separatortime},
computing the separator decomposition of
Section~\ref{sec:recursivedecomp} given $\text{Vor}_{H}(R,u)$
takes $\tilde{O}(\sqrt{r})$ time and $O(\sqrt r)$ space.
Thus, over all vertices, regions and holes, this step takes
$O(n^2)$ time and $O(n^2/\sqrt{r})$ memory.
Finally, we show that
Step~\ref{step:querysep} takes $\tilde{O}(n \cdot r)$ time and
$O(n \cdot r)$ space. The algorithm proceeds region by region,
hole by hole, and edge by edge. By Lemma~\ref{Lem:RegionDS},
for a given edge of the region,
computing the
data structure of Section~\ref{sec:PreprocRegion} takes
$O(r \cdot \log r)$
time and space. Since the total number of region is $O(n/r)$
and the total number of edges per region is $O(r)$, the proof
is complete.
\end{proof}
\begin{corollary}\label{cor:cabellotime}
There exists a distance oracle with total space $O(n^{11/6})$
and expected preprocessing time $O(n^{11/6})$.
\end{corollary}
\begin{proof}
We apply Procedure \textsc{Preprocessing} with $r= n^{1/3}$.
By Lemma~\ref{lem:spaceboundDS}, the total size of the
data structure output is $O(n^{11/6})$.
We now analyse the total preprocessing time.
For Steps~\ref{step:allboundaryvert},~\ref{step:allinternalpairs},
and~\ref{step:querysep}, we mimicate the analysis of the proof
of Theorem~\ref{thm:preprocesstime} and obtain
a total preprocessing time of $O(n \cdot r + n^2/\sqrt{r})$.
We now explain how to speed-up Step~\ref{step:recursivedecomp}.
We show that Step~\ref{step:recursivedecomp} can be done in time
$O(n \cdot r^{5/2})$ using Cabello's data structure~\cite{Cabello17}
for computing weighted Voronoi diagrams of a given region.
More formally, Cabello introduces a data structure that allows to
compute weighted Voronoi diagrams of a given region in expected
time $\tilde{O}(\sqrt{r})$. This data structure
has preprocessing time $O(r^{7/2})$.
Hence the total preprocessing time for computing the
data structure for all the regions is $\tilde{O}(n \cdot r^{5/2})$.
Then, for each vertex $u$, each region $R$, each hole $H$,
the algorithm
\begin{enumerate}
\item uses the data structure to compute the weighted Voronoi
diagram $\text{Vor}_{H}(R,u)$ in expected time
$\tilde{O}(\sqrt{r})$ and
\item computes the separator decomposition of
Section~\ref{sec:recursivedecomp} in time
$\tilde{O}(\sqrt{r})$ (by Lemma~\ref{Lem:sepspace}
and Corollary~\ref{Cor:Separatortime}).
\end{enumerate}
This results in an expected preprocessing time of $O(n \cdot r^{5/2} +
n^2/\sqrt{r})$. Choosing $r= n^{1/3}$ yields a bound of
$O(n^{11/6})$.
\end{proof}
\subsection{Algorithm for Distance Queries}\label{sec:query}
This section is devoted to the presentation of our
algorithms for answering distance queries between pairs of vertices.
We show that any distance query between two vertices $u$, $v$ can be
performed in $O(\log r)$ time. In the following, let $u,v$ be two
vertices of the graph. The algorithm is the following.\\
\smallskip
\textsc{Distance Query} $u,v$
\begin{enumerate}
\item If $u,v$ belong to the same region or if either $u$ or $v$ is
a boundary vertex, the query can be answered in $O(1)$ time since
the distances between vertices of the same region and between boundary
vertices and the other vertices of the graph are stored explicitly.
\item If $u$ and $v$ are internal vertices that
belong to two different regions we proceed as
follows. Let $R$ be the region containing $v$ and $\delta R$ be the
set of boundary vertices of region $R$. The boundary vertices are
partitioned into holes
$\mathcal{H} = \{H_0, \ldots, H_k\}$,
such that $\bigcup_{H \in \mathcal{H}} H = \delta R$.
For each $H \in \mathcal{H}$,
we apply the following procedure.
Let $\mathcal{V}$ be the weighted Voronoi
diagram where the sites are the vertices of $H$ and the
weight of $x \in H$ is the distance from $u$ to $x$.
We now aim at determining to which cell of $\mathcal{V}$, $v$
belongs. We use the binary search procedure of
Lemma~\ref{lem:mainseparator} on the
decomposition of $R$ induced by the separators of the weighted
Voronoi diagram.
More precisely, we use the algorithm described in
Section~\ref{sec:recursivedecomp}, Lemma~\ref{lem:mainseparator}, and
the query algorithm described in Section~\ref{sec:PreprocRegion},
Lemma~\ref{Lem:RegionDS} to identify a set of
at most six Voronoi cells so that one of them contains
$v$. This induces a set of at most six boundary vertices
$X = \{x_0,\ldots,x_k\}$ that represent the centers of the cells.
Finally, we have the distances from both $u$ and $v$ to
all the boundary vertices in $X$.
Let $v(H) = \min_{x \in X} \text{dist}(u,x) + \text{dist}(x,v)$.
The algorithm returns $\min_{H \in \mathcal{H}} v(H)$.
\end{enumerate}
\begin{lemma}[Running time]
\label{lem:querytime}
The \textsc{Distance Query} takes $O(\log r)$ time.
\end{lemma}
\begin{proof}
Consider a distance query from a vertex $u$ to a vertex $v$ and assume
that those vertices are internal vertices of two different regions
as otherwise the query takes $O(1)$ time.
Observe that we can determine in $O(1)$ time to which region $v$
belongs. Fix a hole $H$.
Let $\mathcal{V}$ be the weighted Voronoi
diagram where the sites are the vertices of $H$ and the
weight of $x \in \delta R$ is the distance from $u$ to $x$.
We consider the decomposition of the region of $v$ of
$\Vor{R}{u}$.
Lemma~\ref{Lem:RegionDS} shows that the query time for the data
structure defined in Section~\ref{sec:PreprocRegion} is $t = O(1)$.
Applying Lemma~\ref{lem:mainseparator} with $t=O(1)$ implies
that the total time to determine
in which Voronoi cell $v$ belongs is at most $O(\log r)$.
Finally, computing $\min_{x \in X} \text{dist}(u,x) + \text{dist}(x,v)$ takes $O(1)$
time. By definition of the $r$-division
there are $O(1)$ holes.
Therefore,
we conclude that the running time of the \textsc{Distance Query}
algorithm is $O(\log r)$.
\end{proof}
We now prove that the algorithm indeed returns the correct distance
between $u$ and $v$.
\begin{lemma}[Correctness]
\label{lem:correctness}
The \textsc{Distance Query} on input $u,v$ returns the length
of the shortest path between vertices $u$ and $v$ in the graph.
\end{lemma}
\begin{proof}
We remark that the distance from any vertex to a boundary vertex is
stored explicitly and thus correct.
Hence, we consider the case where $u$ and $v$ are internal
vertices of different regions.
Let $P$ be the shortest path from $u$ to $v$ in $G$.
Let $x \in P$ be the last boundary vertex of $R$ on the path from $u$
to $v$ and let $H_x$ be the hole containing $x$.
Let $\mathcal{V}$ be
the weighted Voronoi
diagram
where the sites are the boundary vertices of $H_x$ and the
weight of $y \in H_x$ is the distance from $u$ to $y$.
We need to argue that the data structure of
Section~\ref{sec:PreprocRegion}, Lemma~\ref{Lem:RegionDS} satisfies
the conditions of Lemma~\ref{lem:mainseparator}.
Observe that the separators defined in
Section~\ref{sec:recursivedecomp} consist of two shortest
paths $P_R(b_1,x)$ and $P_R(b_2,y)$ where $b_1,b_2 \in \delta R$ and
$(y,z)
is an edge of $R$. Hence, the set of vertices of the subgraph
$\Box(b_1,b_2,y,z)$ correspond to the set of
all the vertices that are one of the two sides of the separator.
Thus, by Lemma~\ref{Lem:RegionDS}
the data structure described in Section~\ref{sec:PreprocRegion}
satisfies the condition of Lemma~\ref{lem:mainseparator},
with query time $t = O(1)$.
We argue that $v$ belongs to the Voronoi cell of $x$.
Assume towards contradiction that $v$ is in the Voronoi cell
of $y \neq x$, we would have
$\text{dist}(y, v) + w(y) \le \text{dist}(x,v) + w(x)$,
where $w$ is the weight function associated with the Voronoi diagram.
Thus, this implies that
$\text{dist}(y, v) + \text{dist}(y,u) \le \text{dist}(x,v) + \text{dist}(x,u)$.
Therefore, there exists a shortest path from $u$ to $v$ that
goes through $y$.
Now observe that if $v$ belongs to the Voronoi cell of $y$, the
shortest path from $v$ to $y$ does not go through $x$.
Hence, assuming unique shortest paths between pairs of vertices,
we conclude that the last boundary vertex on the path
from $u$ to $v$ is $y$ and not $x$, a contradiction. Thus,
$v$ belongs to the Voronoi cell of $x$.
Combining with Lemma~\ref{Lem:RegionDS}, it follows that
the Voronoi cell of $x$ is in the set of Voronoi cells $X$
obtained at the end of the recursive procedure.
Observe that for any $x' \in X$ there exists a path
(possibly with repetition of vertices) of length
$\text{dist}(x',u) + \text{dist}(x',v)$.
Therefore, since we assume unique shortest paths between pairs of
vertices, we conclude that
$\text{dist}(u,v) =
\text{dist}(x,u) + \text{dist}(x,v) = \min_{x' \in X} \text{dist}(x',v) +
\text{dist}(x',u) = v(H_x) = \min_{H} v(H)$.
\end{proof}
\section{High-level description}\label{sec:HighLevel}
We now give a high-level description of our distance oracle where we omit the details needed to get our preprocessing time bounds. Our data structure is constructed on top of an $r$-division of
the graph. For each region $R$ of the $r$-division we store a look-up
table of the distance in $G$ between each ordered pair of vertices $u,v\in V(R)$. We also store a
look-up table of distances in $G$ from each vertex $u\in V$ to the boundary vertices of
$R$. In total this part requires $O(nr + n^2/\sqrt{r})$ space.
The difficult case is when two vertices $u$ and $v$
from different regions are queried. To do this we will use weighted Voronoi
diagrams. More specifically, for every vertex
$u$, every region $R$, and every hole $H$ of $R$, we construct a recursive separator decomposition
of the weighted Voronoi diagram of $u$ w.r.t.~$R$ and $H$. The goal is to determine the boundary vertex $w$ such that $v$ is contained in the Voronoi cell of $u$. If we can do this,
we know that $d_G(u,v) = d_G(u,w) + d_{R_H}(w,v)$ for one of the holes $H$ of $R$. To determine this we use a
carefully selected recursive decomposition. This decomposition is stored in a compact way and we show how it enables binary search to find $w$ in $O(\log r)$ time
In order to store all of the above mentioned parts efficiently we will employ
the compact representation of the abstract Voronoi diagram (namely $\Vor{R}{u}$ as defined in the previous section). This requires only $O(\sqrt r)$ space for each choice of $u$, $R$, and $H$ for a total of $O(n^2/\sqrt r)$ space. This also dominates the space for storing the recursive decompositions.
Finally, we store for each graph $R_H$ and each possible separator of $R_H$ the set of vertices on one side of the separator, since this is needed to perform the binary search. This is done in a compact way requiring only $O(r^2)$ space per region. Thus,
the total space requirement of our distance oracle is $O(nr + n^2/\sqrt{r})$. Picking $r=n^{2/3}$ gives the desired $O(n^{5/3})$ space bound.
\section{Introduction}
Efficiently storing distances between the pairs of vertices of a graph is a fundamental
problem that has receive a lot of attention over the years.
Many graph algorithms and real-world problems require that the distances between pairs of
vertices of a graph can be accessed efficiently.
Given an edge-weighted digraph $G = (V,E)$ with $n$ vertices,
a \emph{distance oracle} is a data structure that can efficiently
answer distance queries between pairs of vertices $u,v\in V$.
A naive approach consists in storing an $n\times n$ distance matrix,
giving a distance query time of $O(1)$ by a simple table lookup.
The obvious downside is the huge $\Theta(n^2)$ space requirement which is in
many cases impractical.
For example, several popular routing heuristics
(e.g.: for the travelling salesman problem) require fast access to distances
between pairs of vertices. Unfortunately the inputs are usually too big to allow
to store an $n \times n$ distance matrix (see e.g.:~\cite{TSPlib})\footnote{In these cases,
the inputs are then embedded into the 2-dimensional plane so that the distances
can be computed in $O(1)$ time at the expense of working with incorrect distances.}.
Fast and compact data structures for distances are also critical in many routing problems.
One important challenge in these applications is to process a large number of online queries
while keeping the space usage low, which is important for systems with limited
memory or memory hierarchies.
Therefore, the alternative naive approach consisting in simply storing the graph $G$ and
answering a query by running a shortest path algorithm on the entire graph
is also prohibitive for many applications.
Since road networks and planar graphs share many properties,
planar graphs are often used for modeling various
transportation networks (see e.g.:~\cite{MozesS12}).
Therefore obtaining good space/query-time trade-offs for planar distance oracles has been
studied thoroughly over the past
decades~\cite{Djidjev96,ArikatiCCDSZ96,ChenX00,FakR06,WNthesis,Cabello12,MozesS12}.
If $S$ represents the space usage and $Q$ represents the query time, the
trivial solutions described above would suggest a trade-off of
$Q = n^2/S$\footnote{Using the $O(n)$ shortest path algorithm for planar graphs of
Henzinger et al.~\cite{Henzinger97}.}.
Up to logarithmic factors, this trade-off is achieved by
the oracles of Djidjev~\cite{Djidjev96} and Arikati, et
al.~\cite{ArikatiCCDSZ96}. The oracle of Djidjev further improves on this
trade-off obtaining an oracle with $Q = n/\sqrt{S}$ for the range $S\in
[n^{4/3}, n^{3/2}]$ suggesting that this trade-off might instead be the correct
one. Extending this trade-off to the full range of $S$ was the subject of
several subsequent papers by Chen and Xu~\cite{ChenX00},
Cabello~\cite{Cabello12}, Fakcharoenphol and Rao~\cite{FakR06}, and finally
Mozes and Sommer~\cite{MozesS12} (see also the result of
Nussbaum~\cite{Nussbaum11}) obtaining a query time of $Q = n/\sqrt{S}$ for
the entire range of $S\in[n,n^2]$ (again ignoring constant and logarithmic
factors)
It is worth noting that the above mentioned trade-off between space usage and
query time is no better than the trivial solution of simply storing the
$n\times n$ distance matrix when constant (or even polylogarithmic) query time
is needed. In fact the best known result in this case due to
Wulff-Nilsen~\cite{WNthesis} who manages to obtain very slightly subquadratic
space of $O(n^2\operatorname{polyloglog}(n)/\log(n))$ and constant query time. It has been a major open question
whether an exact oracle with truly subquadratic (that is, $O(n^{2-\varepsilon})$
for any constant $\varepsilon > 0$) space usage and constant or even polylogarithmic query time
exists. Furthermore, the trade-offs obtained in the literature
suggest that this might not be the case.
In this paper we break this quadratic barrier:
\begin{theorem}
\label{Thm:main}
Let $G = (V,E)$ be a weighted planar digraph with $n$
vertices. Then there exists a data structure with $O(n^2)$ preprocessing time and $O(n^{5/3})$ space and a data structure with $O(n^{11/6})$ space and $O(n^{11/6})$ expected preprocessing time. Given any two query vertices $u,v\in V$, both oracles report the shortest path distance from $u$ to $v$ in $G$ in $O(\log n)$ time.
\end{theorem}
In addition to \Cref{Thm:main} we also obtain a distance oracle with a
trade-off between space
and query time.
\begin{theorem}\label{thm:tradeoff}
Let $G = (V,E)$ be a weighted planar digraph with $n$ vertices. Let $S$
denote the space, $P$ denote the preprocessing time, and $Q$ denote the
query time. Then there exists planar distance oracles with the following
properties:
\begin{itemize}
\item $P = O(n^2)$, $S\ge n^{3/2}$, and $Q =
O(\frac{n^{5/2}}{S^{3/2}}\log n)$.
\item $P = S$, $S\ge n^{16/11}$, and $Q =
O(\frac{n^{11/5}}{S^{6/5}}\log n)$.
\end{itemize}
\end{theorem}
In particular,
this result improves on the current state-of-the-art~\cite{MozesS12}
trade-off between space and query time for $S \ge n^{3/2}$. The main idea
is to use two $r$-divisions, where we apply our structure from \Cref{Thm:main}
to one and do a brute-force search over the boundary nodes of the other.
\paragraph{Recent developments}
We note that the main focus of this paper is on space usage and query time,
and the the preprocessing time follows directly from our proofs (and by
applying the result of \cite{Cabello17} for subquadratic time).
After posting a preliminary version of this paper on
arXiv~\cite{Cohen-AddadDW17},
the algorithm of Cabello~\cite{Cabello17} was
improved by Gawrychowski et al.~\cite{GawrychowskiKMSW17} to run in
$\tilde{O}(n^{5/3})$ deterministically. As noted in \cite{GawrychowskiKMSW17}
this also improves the preprocessing time of our \Cref{Thm:main} to
$\tilde{O}(n^{5/3})$ while keeping the space usage at $O(n^{5/3})$.
It is also possible to use Gawrychowski et al. to speed-up the pre-processing time
of our distance oracle described in \Cref{thm:tradeoff}.
This yields a distance oracle (in the notation of
\Cref{thm:tradeoff}) with $P = \tilde{O}(S)$, $S\ge
n^{3/2}$, and $Q = O(\frac{n^{5/2}}{S^{3/2}}\log n)$, thus eliminating entirely
the need of the second bullet point of \Cref{thm:tradeoff}.
\subsection*{Techniques}
We derive structural results on Voronoi diagrams for planar graphs when the
centers of the Voronoi cells lie on the same face.
The key ingredients
in our algorithm are a novel and technical separator decomposition and point
location structure for the regions in an $r$-division allowing us to perform
binary search to find a boundary vertex $w$ lying on a shortest-path between a
query pair $u,v$. These structures are applied on top of weighted Voronoi
diagrams, and our point location structure relies heavily on partitioning
each region into small ``easy-to-handle'' wedges which are shared by many such
Voronoi diagrams.
More high-level ideas are given in
Section~\ref{sec:HighLevel}.
Our approach bears some similarities with the recent breakthrough of Cabello~\cite{Cabello17}.
Cabello showed that abstract Voronoi diagrams~\cite{Klein89,KleinLN09} studied in computational
geometry combined with planar $r$-division can be used to obtain fast planar graphs algorithms
for computing the diameter and wiener index.
We start from Cabello's approach of using abstract Voronoi diagrams.
While Cabello focuses on developing fast algorithms for computing abstract Voronoi diagrams
of a planar graph, we introduce a decomposition theorem for abstract Voronoi diagrams of planar graphs
and a new data structure for point location in planar graphs.
\subsection{Related work
In this paper, we focus on distance oracles that report shortest path distances exactly. A closely related area is approximate
distance oracles. In this case, one can obtain near-linear space and constant or near-constant query time at the cost of a small $(1+\epsilon)$-approximation factor in the distances reported~\cite{Thorup04, Klein02, kawarabayashi2011linear, kawarabayashi2013more, WN16}.
One can also study the problem in a dynamic setting, where the graph undergoes
edge insertions and deletions. Here the goal is to obtain the best trade-off between update and
query time. Fakcharoenphol and Rao~\cite{FakR06} showed how to obtain
$\tilde{O}(n^{2/3})$ for both updates and queries and a trade-off of $O(r)$ and
$O(n/\sqrt{r})$ in general. Several follow up works have improved this result
to negative edges and shaving further logarithmic
factors~\cite{Klein05,ItalianoNSW11,KaplanMNS12,GawrK16}. Furthermore, Abboud
and Dahlgaard~\cite{AbboudD16} have showed that improving this bound to
$O(n^{1/2-\varepsilon})$ for any constant $\varepsilon > 0$ would imply a truly subcubic algorithm for
the All Pairs Shortest Paths (APSP) problem in general graphs.
In the seminal paper of Thorup and
Zwick~\cite{ThorupZ05}, a $(2k-1)$-approximate distance
oracle is presented for undirected edge-weighted $n$-vertex general graphs using $O(kn^{1+1/k})$ space and $O(k)$ query time for any integer $k\ge 1$. Both query time and space has subsequently been improved to $O(n^{1+1/k})$ space and $O(1)$ query time while keeping an approximation factor of $2k-1$~\cite{wulff2012approximate, Chechik14, Chechik15}. This is near-optimal, assuming the widely believed and partially proven girth conjecture of Erd{\H{o}}s~\cite{erdHos1964extremal}.
\section{Preliminaries and Notations}
Throughout this paper we denote the input graph by $G$ and we assume that it is a directed planar graph with a fixed embedding.
We assume that $G$ is connected (when ignoring edge orientations) as otherwise each connected component can be treated separately.
Section~\ref{sec:recursivedecomp} will make use of the geometry of the
plane and associate Jordan curves to cycle separators.
Let $H$ be a planar embedded edge-weighted digraph.
We use $V(H)$ to denote the set of
vertices of $H$ and we denote by $H^*$
the dual of $H$ (with parallel edges and loops) and view it as an undirected graph. We assume a natural embedding of
$H^*$ {\it i.e.,}~ each dual vertex is in the interior of its corresponding primal face and each dual edge crosses its corresponding primal edge
of $H$ exactly once and intersects no other edges of $G$. We let $d_H(u,v)$ denote the shortest path distance from vertex $u$ to vertex $v$ in $H$.
\paragraph{$r$-division}
We will rely on the notion of $r$-division introduced by
Frederickson~\cite{Frederickson87} and further developed by
Klein et al.~\cite{KleinMS13}.
For a subgraph $H$ of $G$, a vertex $v$ of $H$ is a {\em boundary
vertex} if $G$ contains an edge not in $H$ that is incident to $v$. We let $\delta H$ denote the set of boundary vertices of $H$. Vertices of $V(H)\setminus\delta H$ are called {\em internal vertices} of $H$.
A {\em hole} of a subgraph $H$ of $G$ is a face of $H$ that is not
a face of $G$.
Let $c_1$ and $c_2$ be constants. For a number $r$, an {\em
$r$-division with few holes}
of (connected) graph $G$ (with respect to $c_1, c_2$)
is a collection $\mathcal R$ of
subgraphs of $G$, called {\em regions}, with the following properties.
\begin{enumerate}
\itemsep0pt
\item Each edge of $G$ is in exactly one region. \label{def:rdivisiona}
\item The number of regions is at most $c_1 |V(G)|/r$.\label{def:rdivisionb}
\item Each region contains at most $r$ vertices.\label{def:rdivisionc}
\item Each region has at most $c_2\sqrt r$ boundary vertices.\label{def:rdivisiond}
\item Each region contains only $O(1)$ holes.
\end{enumerate}
We make the simplifying assumption that each hole $H$ of each region $R$ is a simple cycle and that all its vertices belong to $\delta R$. We can always reduce to this case as follows. First, turn $H$ into a simple cycle by duplicating vertices that are visited more than once in a walk of the hole. Then for each pair of consecutive boundary vertices in this walk, add a bidirected edge between them unless they are already connected by an edge of $R$; the new edges are embedded such that they respect the given embedding of $R$. We refer to the new simple cycle obtained as a hole and it replaces the old hole $H$.
We also make the simplifying assumption that each face of a region $R$ is either a hole or a triangle and that each edge of $R$ is bidirected. This can always be achieved by adding suitable infinite-weight edges that respect the current embedding of $R$.
\paragraph{Non-negative weights and unique shortest paths}
As mentioned earlier, we may assume w.l.o.g.~that $G$ has non-negative edge weights. Furthermore, we assume uniqueness of shortest paths {\it i.e.,}~
for any two vertices $x,y$ of a graph $G$, there is a unique path from
$x$ to $y$ that minimizes the sum of the weights of its edges. This can be achieved either with random perturbations of edge weights or deterministically with a slight overhead as described in~\cite{Cabello17}; we need the shortest paths uniqueness assumption only for the preprocessing step and thus the overhead only affects the preprocessing time and not the query time of our distance oracle.
\paragraph{Voronoi diagrams}
We now define the key notion of Voronoi diagrams.
Let $G$ be a graph and $r>0$.
Consider an $r$-division with few holes $\mathcal{R}$ of $G$ and a region
$R \in \mathcal{R}$ and let $H$
be a hole of $R$. Let $u$ be a vertex of $G$ not in $R$.
Let $R_H$ be the graph obtained from $R$ by adding inside each hole $H'\ne H$ of $R$, a new vertex in its interior and infinite-weight bidirected edges between this vertex and the vertices of $H'$ (which by the above simplifying assumption all belong to $\delta R$), embedding the edges such that they are pairwise non-crossing and contained in $H'$.
Some of the following definitions are illustrated in Figure~\ref{fig:wt_vor}. Consider the shortest path tree $T_u$ in $G$ rooted at
$u$. For any vertex $x\in V(T_u)$, define $T_u(x)$ to be the subtree of $T_u$
rooted at $x$.
For each vertex $x \in V(H)$ we define the \emph{Voronoi cell}
of $x$ (w.r.t.~$u$, $R$, and $H$)
as the set of vertices of $R_H$ that belong to the subtree $T_u(x)$
and not to any subtree $T_u(y)$ for $y \in (V(H) - \{x\})\cap V(T_u(x))$.
The \emph{weighted Voronoi diagram} of $u$ w.r.t.~$R$ and $H$ is the collection of all the Voronoi cells of the vertices in $H$.
For each vertex $x \in H$ we say that its \emph{weight}
is the shortest path distance from $u$ to $x$ in $G$.
Note that since we assume unique shortest paths and
bidirectional edges, the weighted Voronoi diagram of $u$ w.r.t.~$R$ and $H$
is a partition of the vertices of $R_H$. Furthermore, each Voronoi cell contains exactly one vertex of $H$.
For any Voronoi cell $C$, we define its boundary edges to be the edges of $R_H$ that have exactly one endpoint in $C$. Let $B_H^*$ be the subgraph of $R_H^*$ consisting of the (dual) boundary edges over all Voronoi cells w.r.t.~$u$ and $R_H$; we ignore edge orientations and weights so that $B_H^*$ is an unweighted undirected graph. We define $\text{Vor}_{H}(R,u)$ to be the multigraph
obtained from $B_H^*$ by replacing each maximal path whose interior vertices have degree two by a single edge whose embedding coincides with the path it replaces. When $H$ is clear from context, we simply write $\Vor{R}{u}$.
\begin{figure}[ht]
\centering
\includegraphics[width=.5\textwidth]{wt_vor.pdf}
\caption{Illustration of the weighted Voronoi diagram of a vertex $u$ (not shown) w.r.t.~a region $R$ (grey) and a hole $H$ (dotted edges and seven grey vertices). Edges of $R$ are bidirected and have weight $1$. The
number next to a vertex of $H$ is the weight of that vertex. Vertices of the
Voronoi diagram (which are dual vertices and hence faces of the primal) and its boundary are shown in black except the one embedded inside $H$. The graph $\text{Vor}_{H}(R,u)$ has seven edges.
Note that this illustration does \emph{not} have unique shortest paths as
assumed in this paper. Note also that the shortest path from $u$ to the
node with weight $5$ goes through the nearby node with weight $2$. Thus the
rest of the shortest-path tree from $u$ has been assigned to the node with
weight $5$. Note also that the illustration is not triangulated.}
\label{fig:wt_vor}
\end{figure}
\section{Preprocessing a Region}\label{sec:PreprocRegion}
Given a query separator $S$ in a graph $R_H = R$ and given a query vertex $w$ in $R$, our data structure needs to determine in $O(1)$ time the side of $S$ that $w$ belongs to. In this section, we describe the preprocessing needed for this.
In the following, fix $R$ as well as an ordered pair $(u,v)$ of vertices of $R$ such that either $(u,v)$ or $(v,u)$ is an edge of $R$. The preprocessing described in the following is done over all such choices of $R$ and $(u,v)$ (and all holes $H$).
The vertices of $\delta R$ are on a simple cycle and we identify $\delta R$ with this cycle which we orient clockwise (ignoring the edge orientations of $R$). We let $b_0,\ldots,b_k$ denote this clockwise ordering where $b_k = b_0$. It will be convenient to calculate indices modulo $k$ so that, e.g., $b_{k+1} = b_1$
Given vertices $w$ and $w'$
in $R$, let $P(w,w')$ denote the shortest path in $R$ from $w$ to $w'$. Given two vertices $b_i,b_j\in\delta R$, we let $\delta(b_i,b_j)$ denote the subpath of cycle $\delta R$ consisting of the vertices from $b_i$ to $b_j$ in clockwise order, where $\delta(b_i,b_j)$ is the single vertex $b_i$ if $i = j$ and $\delta(b_i,b_j) = \delta R$ if $j = i + k$. We let $\Delta(w,b_i,b_j)$ denote the subgraph of $R$ contained in the closed and bounded region of the plane with boundary defined by $P(b_i,w)$, $P(b_j,w)$, and $\delta(b_i,b_j)$. We refer to $\Delta(w,b_i,b_j)$ as a \emph{wedge} and call it a \emph{basic wedge} if $b_i$ and $b_j$ are consecutive in the clockwise order, i.e., if $j = i+1\pmod k$. We need the following lemma.
\begin{lemma}\label{Lem:PieQuery}
Let $w$ be a given vertex of $R$. Then there is a data structure with $O(r)$ preprocessing time and size which answers in $O(1)$ time queries of the following form: given a vertex $x\in V(R)$ and two distinct vertices $b_{i_1},b_{i_2}\in\delta R$, does $x$ belong to $\Delta(w,b_{i_1},b_{i_2})$?
\end{lemma}
\begin{proof}
Below we present a data structure with the bounds in the lemma which only answers restricted queries of the form ``does $x$ belong to $\Delta(w,b_0,b_i)$?'' for query vertices $x\in V(R)$ and $b_i\in\delta R$. In a completely symmetric manner, we obtain a data structure for restricted queries of the form ``does $x$ belong to $\Delta(w,b_i,b_k)$?'' for query vertices $x\in V(R)$ and $b_i\in\delta R$. We claim that this suffices to show the lemma. For consider a query consisting of $x\in V(R)$ and $b_i,b_j\in\delta R$. If $b_0\in\delta(b_i,b_j)$ then $\Delta(w,b_i,b_j) = \Delta(w,b_i,b_k)\cup\Delta(w,b_0,b_j)$ and otherwise, $\delta(w,b_i,b_j) = \Delta(w,b_0,b_j)\cap\Delta(w,b_i,b_k)$. Hence, answering a general query can be done using two restricted queries and checking if $b_0\in\delta(b_1,b_2)$ can be done in constant time by comparing indices of the query vertices.
It remains to present the data structure for restricted queries of the form ``does $x$ belong to $\Delta(w,b_0,b_i)$?''. In the preprocessing step, each $v\in V(R)$ is assigned the smallest index $i_v\in\{0,\ldots,k\}$ for which $v\in\Delta(w,b_0,b_{i_v})$. Clearly, this requires only $O(r)$ space and below we show how to compute these indices in $O(r)$ time.
Consider a restricted query specified by a vertex $x$ of $R$ and a boundary vertex $b_i\in\delta R$ where $0\le i\le k$. Since $x\in\Delta(w,b_0,b_i)$ iff $i_x\le i$, this query can clearly be answered in $O(1)$ time.
It remains to show how the indices $i_v$ can be computed in a total of $O(r)$ time. Let $R'$ be $R$ with all its edge directions reversed. In $O(r)$ time, a SSSP tree $T'$ from $w$ in $R'$ is computed. Let $T$ be the tree in $R$ obtained from $T'$ by reversing all its edge directions; note that all edges of $T$ are directed towards $w$ and for each $v\in V(R)$, the path from $v$ to $w$ in $T$ is a shortest path from $v$ to $w$ in $R$.
Next, $\Delta(w,b_0,b_0) = P(b_0,w)$ is computed and for each vertex $v\in\Delta(w,b_0,b_0)$, set $i_v = 0$. The rest of the preprocessing algorithm consists of iterations $i = 1,\ldots,k$ where iteration $i$ assigns each vertex $v\in V(\Delta(w,b_0,b_i))\setminus V(\Delta(w,b_0,b_{i-1}))$ the index $i_v = i$. This correctly computes indices for all vertices of $R$. In the following, we describe how iteration $i$ is implemented.
First, the path $P(b_i,w)$ is traversed in $T$ until a vertex $v_i$ is encountered which previously received an index. In other words, $v_i$ is the first vertex on $P(b_i,w)$ belonging to $\Delta(w,b_0,b_{i-1})$. Note that $v_i$ is well-defined since $w\in\Delta(w,b_0,b_{i-1})$. Vertices that are in $V(P(b_i,v_i))\setminus\{v_i\}$ or in a subtree of $T$ rooted in a vertex of $V(P(b_i,v_i))\setminus\{v_i\}$ and extending to the right of this path are assigned the index value $i$. Furthermore, vertices belonging to a subtree of $T$ rooted in a vertex of $V(P(b_{i-1},v_i))\setminus\{v_i\}$ and extending to the left of this path are assigned the index value $i$, except those on $P(b_{i-1},v_i)$ (as they belong to $\Delta(w,b_0,b_{i-1})$).
Since $R$ is connected, it follows that the vertices assigned an index of $i$ are exactly those belonging to $V(\Delta(w,b_0,b_i))\setminus V(\Delta(w,b_0,b_{i-1}))$ and that the running time for making these assignments is $O(|V(\Delta(w,b_0,b_i))\setminus V(\Delta(w,b_0,b_{i-1}))| + |P(b_{i-1},v_i) - v_i| + 1)$. Over all $i$, total running time is $O(r)$; this follows by a telescoping sums argument and by observing that vertex sets $V(P(b_{i-1},v_i))\setminus\{v_i\}$ are pairwise disjoint.
\end{proof}
Given distinct vertices $b_{i_1},b_{i_2}\in\delta R$, if $P(b_{i_1},u)$ and $P(b_{i_2},v)$ do
not cross (but may touch and then split), let $\Box(b_{i_1},b_{i_2},u,v)$ denote the subgraph of
$R$ contained in the closed and bounded region of the plane with boundary
defined by $P(b_{i_1},u)$, $P(b_{i_2},v)$, $\delta(b_{i_1},b_{i_2})$, and an edge of $R$
between vertex pair $(u,v)$. In order to simplify notation, we shall omit $u$ and $v$ and simply write $\Box(b_{i_1},b_{i_2})$.
It follows from planarity that there is at most one $b_i\in\delta R$ such that $(u,v)$ belongs to $E(\Delta(u,b_i,b_{i+1}))\setminus E(P(b_{i+1},u))$ when ignoring edge orientations. If $b_i$ exists, we refer to it as
$b_{uv}$; otherwise $b_{uv}$ denotes some dummy vertex not belonging to $R$.
The goal in this section is to determine whether a given query vertex belongs to a given query subgraph $\Box(b_{i_1},b_{i_2})$. The following lemma allows us to decompose this subgraph into three simpler parts as illustrated in
Figure~\ref{fig:canonical}. We will show how to answer containment queries for
each of these simple parts.
\begin{figure}[htbp]
\centering
\includegraphics[width=.5\textwidth]{canonical.pdf}
\caption{Example of decomposing the region into three parts using $b_i$:
The dashed wedges represent shortest paths to $u$, the dotted edges
represent shortest paths to $v$, and the dashed-dotted box
$\Box(b_i,b_{i+1})$.}
\label{fig:canonical}
\end{figure}
\begin{lemma}\label{Lem:SepDecomposition}
Let $b_{i_1}$ and $b_{i_2}$ be distinct vertices of $\delta R$ and assume that
$P(b_{i_1},u)$ and $P(b_{i_2},v)$ are vertex-disjoint. Then $\Box(b_{i_1},b_{i_2}) = \Delta(u,b_{i_1},b_i)\cup\Box(b_i,b_{i+1})\cup\Delta(v,b_{i+1},b_{i_2})$ where $b_i = b_{uv}$ if $b_{uv}\in\delta(b_{i_1},b_{i_2-1})$ and $b_i = b_{i_2-1}$ otherwise.
\end{lemma}
\begin{proof}
Figure~\ref{fig:sepProof} gives an illustration of the proof. We first show the following result: given a vertex $b_j\in\delta(b_{i_1+1},b_{i_2})$ such that $b_{uv}\notin\delta(b_{i_1},b_{j-1})$, $P(b_{j'},u)$ is contained in $\Box(b_{i_1},b_{i_2})$ for each $b_{j'}\in\delta(b_{i_1},b_j)$. The proof is by induction on the number $i$ of edges in $\delta(b_{i_1},b_{j'})$. The base case $i = 0$ is trivial since then $b_{j'} = b_{i_1}$ so assume that $i > 0$ and that the claim holds for $i - 1$. If $(v,u)$ is the last edge on $P(b_{j'},u)$, the induction step follows from uniqueness of shortest paths. Otherwise, neither $(u,v)$ nor $(v,u)$ belong to $\Delta(u,b_{j'-1},b_{j'})$ (since $b_{j'-1}\ne b_{uv}$). By the induction hypothesis, $P(b_{j'-1},u)$ is contained in $\Box(b_{i_1},b_{i_2})$ and since $P(b_{j'},u)$ cannot cross $P(b_{j'-1},u)$, $P(b_{j'},u)$ cannot cross $P(b_{i_1},u)$. Also, $P(b_{j'},u)$ cannot cross $P(b_{i_2},v)$ since then either $(u,v)$ or $(v,u)$ would belong to $\Delta(u,b_{j'-1},b_{j'})$. Since $b_{j'}\notin\{b_{i_1+1},b_{i_2-1}\}$, it follows that $P(b_{j'},u)$ is contained in $\Box(b_{i_1},b_{i_2})$ which completes the proof by induction.
Next, assume that $b_{uv}\notin\delta(b_{i_1},b_{i_2-1})$ so that $b_i = b_{i_2-1}$. Note that $\Delta(v,b_{i+1},b_{i_2}) = P(b_{i_2},v)$. Picking $b_j = b_{i_2}$ above implies that $\Delta(u,b_{i_1},b_i)$ is contained in $\Box(b_{i_1},b_{i_2})$ and hence $\Box(b_{i_1},b_{i_2}) = \Delta(u,b_{i_1},b_i)\cup\Box(b_i,b_{i+1})\cup\Delta(v,b_{i+1},b_{i_2})$.
Now consider the other case of the lemma where $b_i = b_{uv}\in\delta(b_{i_1},b_{i_2-1})$. Picking $b_j = b_i$ above, it follows that $\Delta(u,b_{i_1},b_i)$ is contained in $\Box(b_{i_1},b_{i_2})$. It suffices to show that $P(b_{i+1},v)$ is contained in $\Box(b_{i_1},b_{i_2})$ since this will imply that $\Box(b_i,b_{i+1})$ is well-defined and that $\Box(b_i,b_{i+1})\cup\Delta(v,b_{i+1},b_{i_2})$ is contained in $\Box(b_{i_1},b_{i_2})$ and hence that $\Delta(u,b_{i_1},b_i)\cup\Box(b_i,b_{i+1})\cup\Delta(v,b_{i+1},b_{i_2}) = \Box(b_{i_1},b_{i_2})$.
Assume for contradiction that $P(b_{i+1},v)$ is not contained in $\Box(b_{i_1},b_{i_2})$.
By uniqueness of shortest paths, $P(b_{i+1},u)$ does not cross $P(b_i,u)$. Since $b_i = b_{uv}$, we have that when ignoring edge orientations, $(u,v)$ belongs to $E(\Delta(u,b_i,b_{i+1}))\setminus E(P(b_{i+1},u))\subseteq E(\Delta(u,b_{i_1},b_{i+1}))\setminus E(P(b_{i+1},u))$. Hence $P(b_{i+1},u)$ is not contained in $\Box(b_{i_1},b_{i_2})$ so it crosses $P(b_{i_2},v)$. Let $x$ be a vertex on $P(b_{i+1},u)\cap P(b_{i_2},v)$ such that the successor of $x$ on $P(b{i+1},u)$ is not contained in $\Box(b_{i_1},b_{i_2})$. By our assumption above that $P(b_{i+1},v)$ is not contained in $\Box(b_{i_1},b_{i_2})$, there is a first vertex $y$ on $P(b_{i+1},v)$ such that its successor $y'$ does not belong to $\Box(b_{i_1},b_{i_2})$. By uniqueness of shortest paths, $y$ cannot belong to $P(b_{i_2},v)$ so it must belong to $P(b_{i_1},u)$. This also implies that $y\neq x$ since $x\in P(b_{i_2},v)$ and $P(b_{i_1},u)$ and $P(b_{i_2},v)$ are vertex-disjoint. Since $P(y,v)$ is a subpath of $P(b_{i+1},v)$ and $y\neq x$, shortest path uniqueness implies that $P(y,v)$ and $P(b_{i+1},u)$ are vertex-disjoint.
Since $P(b_{i+1},y)$ is contained in $\Box(b_{i_1},b_{i_2})$, $v$ belongs to the subgraph of $\Delta(u,b_{i_1},b_{i+1})$ contained in the closed region of the plane bounded by $P(b_{i+1},y)$, $P(y,u)$, and $P(b_{i+1},u)$. Since $P(y,v)$ does not intersect $P(b_{i+1},u)$, $P(y',v)$ thus intersects either $P(b_{i+1},y)$ or $P(y,u)$. However, it cannot intersect $P(b_{i+1},y)$ since then $P(b_{i+1},v)$ would be non-simple. By uniqueness of shortest paths, $P(y',v)$ also cannot intersect $P(y,u)$ since $P(y',v)$ is a subpath of $P(y,v)$ and $y'\notin P(y,u)$. This gives the desired contradiction, concluding the proof.
\end{proof}
\begin{figure}[htbp]
\centering
\includegraphics[width=.5\textwidth]{3_split_proof.pdf}
\caption{Illustration of the proof of Lemma~\ref{Lem:SepDecomposition}. The
figure shows how unique shortest paths imply a contradiction (highlighted
with grey) if the path
from $b_{i+1}$ to $v$ is not contained in $\Box(b_{i_1},b_{i_2})$.}
\label{fig:sepProof}
\end{figure}
Let $\mathcal P$ be a collection of subpaths such that for each path $\delta(b_{i_1},b_{i_2})$ in $\mathcal P$, $b_{uv}\notin\delta(b_{i_1},\ldots,b_{i_2-1})$ and $b_{vu}\notin\delta(b_{i_1+1},\ldots,b_{i_2-1})$. We may choose the paths such that $|\mathcal P| = O(1)$ and such that all edges of $\delta R$ except $(b_{uv},b_{vu})$ (if it exists) belongs to a path of $\mathcal P$. It is easy to see that this is possible by considering a greedy algorithm which in each step picks a maximum-length path which is edge-disjoint from previously picked paths and which satisfies the two stated requirements.
The next lemma allows us to obtain a compact data structure to answer queries of the form ``does face $f$ belong to $\Box(b_i,b_{i+1})$'' for given query face $f$ and query index $i$.
\begin{lemma}\label{Lem:CompressedStrips}
Let $P = \delta(b_{i_1},b_{i_2})\in\mathcal P$ be given. Then
\begin{enumerate}
\item an index $j(P)$ exists with $i_1\le j(P)\le i_2$ such that $\Box(b_i,b_{i+1})$ is undefined for $i_1\le i < j(P)$ and well-defined for $j(P)\le i < i_2$,
\item for each face $f\ne\delta R$ of $R$, there is at most one index $j_f(P)$ with $j(P)\le j_f(P)\le i_2 - 2$ such that $f\subseteq\Box(b_{j_f(P)},b_{j_f(P)+1})$ and $f\nsubseteq\Box(b_{j_f(P)+1},b_{j_f(P)+2})$, and
\item for each face $f\ne\delta R$ of $R$, there is at most one index $j_f'(P)$ with $j(P)\le j_f'(P)\le i_2 - 2$ such that $f\nsubseteq\Box(b_{j_f'(P)},b_{j_f'(P)+1})$ and $f\subseteq\Box(b_{j_f'(P)+1},b_{j_f'(P)+2})$.
\end{enumerate}
Furthermore, there is an algorithm which computes the index $j(P)$ and for each face $f\ne\delta R$ of $R$ the indices $j_f(P)$ and $j_f'(P)$ if they exist. The total running time of this algorithm is $O(r\log r)$ and its space requirement is $O(r)$.
\end{lemma}
\begin{proof}
Let path $P = \delta(b_{i_1},b_{i_2})\in\mathcal P$ and face $f\ne\delta R$ of $R$ be given. To simplify notation in the proof, we shall omit reference to $P$ and write, e.g., $j$ instead of $j(P)$.
Let $j$ be the smallest index such that $\Box(b_j,b_{j+1})$ is well-defined; if $j$ does not exist, pick instead $j = i_2$. We will show that $j$ satisfies the first part of the lemma. This is clear if $j = i_2$ so assume therefore in the following that $j < i_2$.
We prove by induction on $i$ that $\Box(b_i,b_{i+1})$ is well-defined for $j\le i < i_2$. By definition of $j$, this holds when $i = j$. Now, consider a well-defined subgraph $\Box(b_i,b_{i+1})$ where $j\le i \le i_2 - 2$. We need to show that $\Box(b_{i+1},b_{i+2})$ is well-defined. Since $b_i\ne b_{uv}$, $P(b_{i+1},u)$ is contained in $\Box(b_i,b_{i+1})$ and since $b_{i+1}\ne b_{vu}$, $P(b_{i+2},v)$ is contained in the closed region of the plane bounded by the boundary of $\Box(b_i,b_{i+1})$ and not containing $\Box(b_i,b_{i+1})$. In particular, $\Box(b_{i+1},b_{i+2})$ is well-defined. This shows the first part of the lemma.
Next, we show that $j$ can be computed in $O(r\log r)$ time. Checking that $\Box(b_i,b_{i+1})$ is well-defined (i.e., that paths $P(b_i,u)$ and $P(b_{i+1},v)$ do not cross) for a given index $i$ can be done in $O(r)$ time. Because of the first part of the lemma, a binary search algorithm can be applied to identify $j$ in $O(\log r)$ steps where each step checks if $\Box(b_i,b_{i+1})$ is well-defined for some index $i$. This gives a total running time of $O(r\log r)$, as desired. Space is clearly $O(r)$.
To show the second part of the lemma, assume that there is an index $j_f$ with $i_1\le j_f\le i_2 - 2$ such that $f\subseteq\Box(b_{j_f},b_{j_f+1})$ and $f\nsubseteq\Box(b_{j_f+1},b_{j_f+2})$. It follows from the observations in the inductive step above that $\Delta(u,b_{j_f},b_{j_f+1})$ contains exactly the faces of $R$ contained in $\Box(b_{j_f},b_{j_f+1})$ and not in $\Box(b_{j_f+1},b_{j_f+2})$ which implies that $f\subseteq\Delta(u,b_{j_w},b_{j_w+1})$. Since no face of $R$ belongs to more than one graph of the form $\Delta(u,b_i,b_{i+1})$, $j_f$ must be unique, showing the second part of the lemma.
Next, we give an $O(r)$ time and space algorithm that computes indices $j_f$. Let $T$ be constructed as in the proof of Lemma~\ref{Lem:PieQuery}. Initially, vertices of $P(b_{i_2-1},u)$ are marked and all other vertices of $R$ are unmarked. The remaining part of the algorithm consists of iterations $i_2 - 2,\ldots,j$ in that order. In iteration $i$, $P(b_i,u)$ is traversed until a marked vertex $v_i$ is visited and then the vertices of $P(b_i,v_i)$ are marked. The faces of $R$ contained in the bounded region of the plane defined by $P(b_i,v_i)$, $P(b_{i+1},v_i)$, and $\delta(b_i,b_{i+1})$ are exactly those that should be given an index value of $i$. The algorithm performs this task by traversing each subtree of $T$ emanating to the right of $P(b_i,v_i)$ and each subtree of $T$ emanating to the left of $P(b_{i+1},v_i)$; for each vertex visited, the algorithm assigns the index value $i$ to its incident faces.
We now show that the algorithm for computing indices $j_f$ has $O(r)$ running time. Using the same arguments as in the proof of Lemma~\ref{Lem:PieQuery}, the total time to traverse and mark paths $P(b_i,v_i)$ is $O(r)$. The total time to assign indices to faces is $O(r)$; this follows by observing that the time spent on assigning indices to faces incident to a vertex of $T$ is bounded by its degree and this vertex is not visited in other iterations.
The third part of the lemma follows with essentially the same proof as for the second part.
\end{proof}
We can now combine the results of this section to obtain the data structure described in the following lemma.
\begin{lemma}\label{Lem:RegionDS}
Let $(u,v)$ be a vertex pair connected by an edge in $R$. Then there is a data structure with $O(r\log r)$ preprocessing time and $O(r)$ space which answers in $O(1)$ time queries of the
following form: given a vertex $w\in R$ and two distinct vertices $b_i,
b_j\in\delta R$ such that $P(b_i,u)$ and $P(b_j,v)$ are vertex-disjoint, does $w$
belong to $\Box(b_i,b_j,u,v)$?
\end{lemma}
\begin{proof}
We present a data structure $\mathcal D(u,v)$ satisfying the lemma. First we
focus on the preprocessing. Boundary vertices $b_{uv}$ and $b_{vu}$ and set
$\mathcal P$ as defined above are precomputed and stored. Vertices of $\delta R$ are labeled with indices $b_0,\ldots,b_{|V(\delta R)|-1}$ according to a clockwise walk of $\delta R$. Each path of the form $\delta(b_{i_1},b_{i_2})$ (including each path in $\mathcal P$) is represented by the ordered index pair $(i_1,i_2)$. Checking if a given boundary vertex belongs to such a given path can then be done in $O(1)$ time.
Next, two instances of the data structure in
Lemma~\ref{Lem:PieQuery} are set up, one for $u$ denoted $\mathcal D_u$,
and one for $v$ denoted $\mathcal D_v$. Then the following is done for each path $P = \delta(b_{i_1},b_{i_2})\in\mathcal P$. First, the algorithm in Lemma~\ref{Lem:CompressedStrips} is applied. Then if $\Box(b_{j(P)},b_{j(P)+1},u,v)$ is well-defined, its set of faces $F_{j(P)}$ is computed and stored; otherwise, $F_{j(P)} = \emptyset$. Similarly, if $\Box(b_{i_2-1},b_{i_2},u,v)$ is well-defined, its set of faces $F_{i_2-1}$ is computed and stored, and otherwise $F_{i_2-1} = \emptyset$. If $(b_{uv},b_{vu})\in\delta R$, $\mathcal D(u,v)$ computes and stores the set $F_{uv}$ of faces of $R$ contained in $\Box(b_{uv},b_{vu},u,v)$. This completes the
description of the preprocessing for $\mathcal D(u,v)$. It is clear that
preprocessing time is $O(r\log r)$ and that space is $O(r)$.
Now, consider a query specified by a vertex $w\in R$ and two distinct vertices
$b_{i_1},b_{i_2}\in\delta R$ such that $P(b_{i_1},u)$ and $P(b_{i_2},v)$ are pairwise vertex-disjoint. First, $\mathcal D(u,v)$ identifies a boundary vertex
$b_i$ such that $\Box(b_{i_1},b_{i_2},u,v) =
\Delta(u,b_{i_1},b_i)\cup\Box(b_i,b_{i+1})\cup\Delta(v,b_{i+1},b_{i_2})$; this is
possible by Lemma~\ref{Lem:SepDecomposition}. Then $\mathcal D_u$ and
$\mathcal D_v$ are queried to determine if
$w\in\Delta(u,b_{i_1},b_i)\cup\Delta(v,b_{i+1},b_{i_2})$; if this is the case then
$w\in\Box(b_{i_1},b_{i_2},u,v)$ and $\mathcal D(u,v)$ answers ``yes''. Otherwise,
$\mathcal D(u,v)$ identifies an arbitrary face $f\ne\delta R$ of $R$ incident to $w$. At this point, the only way that $w$ can belong to $\Box(b_{i_1},b_{i_2},u,v)$ is if $w$ belongs to the interior of $\Box(b_i,b_{i+1},u,v)$ which happens iff $f$ is contained in $\Box(b_i,b_{i+1},u,v)$. If $(b_i,b_{i+1}) = (b_{uv},b_{vu})$, $\mathcal D(u,v)$ checks if $f\in F_{uv}$ and if so outputs ``yes''. Otherwise, there exists a path $P\in\mathcal P$ containing $(b_i,b_{i+1})$ and $\mathcal D(u,v)$ identifies this path. It follows from Lemma~\ref{Lem:CompressedStrips} and from the definition of $\mathcal P$ that $f$ is contained in $\Box(b_i,b_{i+1},u,v)$ iff at least one of the following conditions hold:
\begin{enumerate}
\item $j_f(P)$ and $j_f'(P)$ are well-defined and $j_f'(P) < i \le j_f(P)$.
\item $f\in F_{i_2-1}$, $j_f'(P)$ is well-defined, and $i > j_f'(P)$,
\item $f\in F_{j(P)}$, $j_f(P)$ is well-defined, and $i\le j_f(P)$,
\item $f\in F_{j(P)}$ and $j_f(P)$ is undefined,
\end{enumerate}
Data structure $\mathcal D(u,v)$ checks if any one these conditions hold and if
so outputs ``yes''; otherwise it outputs ``no''. Two of the cases are
illustrated in Figure~\ref{fig:path_cases}.
\begin{figure}[htbp]
\centering
\includegraphics[width=.5\textwidth]{path_cases.pdf}
\caption{Illustration of how to determine if the face $f$ belongs to
$\Box(b_i,b_{i+1},u,v)$. The illustration includes cases 1 and 4 in
the proof of Lemma~\ref{Lem:RegionDS}.
The large gray subpath indicates the part, where $f$ is contained in
each box defined by consecutive boundary nodes.}
\label{fig:path_cases}
\end{figure}
It remains to show that $\mathcal D(u,v)$ has query time $O(1)$. By Lemma~\ref{Lem:SepDecomposition}, identifying $b_k$ can be done in $O(1)$ time. Querying $\mathcal D_u$ and $\mathcal D_v$ takes $O(1)$ time by Lemma~\ref{Lem:PieQuery}. Checking whether $f\in F_{uv}$ can clearly be done in $O(1)$ time since this set of faces is stored explicitly. With our representation of paths by the indices of their endpoints, identifying $P$ takes $O(1)$ time. Finally, since sets $F_{j(P)}$ and $F_{j(P)}$ are explicitly stored, the four conditions above can be checked in $O(1)$ time.
\end{proof}
\section{Recursive Decomposition of Regions}\label{sec:recursivedecomp
In this section, we consider a region $R$ in an $r$-division of a planar
embedded graph $G = (V,E)$, a vertex $u\in V - V(R)$, and a hole $H$ of $R$
which we may assume is the outer face of $R$. To simplify notation, we identify
$R$ with $R_H$ and let $\delta R$ denote the boundary vertices of $R$ belonging
to $H$, i.e., $\delta R = V(H)$ (by our simplifying assumption regarding holes
in the preliminaries). The dual vertex corresponding to $H$ is denoted
$v_{\infty}(R,u)$ or just $v_{\infty}$.
We assume that $R$ contains at least three boundary vertices.
Recall that each face of $R$ other than the outer face is a triangle and so,
each vertex of $\Vor{R}{u}$ other than $v_{\infty}$ has degree $3$.
Moreover, every cell
of $\Vor{R}{u}$ contains exactly one boundary vertex, therefore the
boundary of each cell of $\Vor{R}{u}$ contains exactly one occurrence of
$v_{\infty}$ and contains at least one other vertex.
Also note that the cyclic ordering of cells of
$\Vor{R}{u}$ around $v_{\infty}$ is the same as the cyclic ordering $\delta R$
of boundary vertices of $R$.
Construct a plane multigraph $\Trivor{R}{u}$ from $\Vor{R}{u}$ as follows. First, for every Voronoi cell $C$, add an edge from $v_{\infty}$ to each vertex of $C$ other than $v_{\infty}$
itself;
these edges are embedded such that they are fully contained in $C$
and such that they are pairwise non-crossing. For each such edge $e$, denote by $C(e)$ the
cell it is embedded in.
To complete the construction
of $\Trivor{R}{u}$, remove every edge incident to $v_{\infty}$
belonging to $\Vor{R}{u}$. The construction of $\Trivor{R}{u}$ is illustrated
in Figure~\ref{fig:multigraph}.
\begin{figure}[htbp]
\centering
\includegraphics[width=.4\textwidth]{sep1.pdf}
\caption{Illustration of $\Trivor{R}{u}$ highlighted in bold black edges.
To avoid clutter, most edges incident to $v_\infty$ are only sketched.}
\label{fig:multigraph}
\end{figure}
\paragraph{Recursive decomposition using a Voronoi diagram}
In this section, we show how to obtain a recursive decomposition of
$\Trivor{R}{u}$ into subgraphs called \emph{pieces}. A piece $Q$ is decomposed
into two smaller pieces by a cycle separator $S$ of size $2$ containing
$v_{\infty}$. Each of the two subgraphs of $Q$ is obtained by replacing the
faces of $Q$ on one side of $S$ by a single face bounded by $S$. The separator
$S$ is balanced w.r.t.~the number of faces of $Q$ on each side of $S$. The
recursion stops when a piece with at most six faces is obtained. It will be
clear from our construction below that the collection of cycle separators over
all recursive calls form a laminar family, i.e., they are pairwise
non-crossing.
We assume a linked list representation of each piece $Q$ where edges are ordered clockwise around each vertex.
Lemma~\ref{Lem:Separator} below shows how to find the cycle separators needed to obtain the recursive decomposition into pieces. Before we can prove it, we need the following result.
\begin{lemma}\label{Lem:PieceStructure}
Let $Q$ be one of the pieces obtained in the above recursive decomposition. Then for every vertex $v$ of $Q$ other than $v_{\infty}$,
\begin{enumerate}
\item $v$ has at least two edges incident to $v_{\infty}$
\item for each edge $(v,w)$ of $Q$ where $w\neq v_{\infty}$, the edge preceding and the edge following $(v,w)$ in the clockwise ordering around $v$ are both incident to $v_{\infty}$, and
\item for every pair of edges $e_1 = (v,v_{\infty})$ and $e_2 = (v,v_{\infty})$ where $e_2$ immediately follows $e_1$ in the clockwise ordering around $v$, if both edges are directed from $v$ to $v_{\infty}$ then the subset of the plane to the right of $e_1$ and to the left of $e_2$ is a single face of $Q$.
\end{enumerate}
\end{lemma}
\begin{proof}
The proof is by induction on the depth $i\ge 0$ in the recursion tree of the
node corresponding to piece $Q$. Assume that $i = 0$ and let $v\neq
v_{\infty}$ be given. The first and third part of the lemma follow
immediately from the construction of $\Trivor{R}{u}$ and the assumption that $|\delta R|\ge 3$. To show the second
part, it suffices by symmetry to consider the edge $e$ following $(v,w)$ in
the cyclic ordering of edges around $v$ in $Q = \Trivor{R}{u}$. Since $e$
and $(v,w)$ belong to the same face of $Q$ and since each face of $Q$ contains $v_{\infty}$ and at most three edges, the second part
follows
Now assume that $i > 0$ and that the claim holds for smaller values. Let $v\neq
v_{\infty}$ be given. Consider the parent piece $Q'$ of $Q$ in the
recursive decomposition tree and let $S$ be the cycle separator that was
used to decompose $Q'$. Then $S$ contains $v_{\infty}$ and one additional
vertex $v'$. To show the inductive step, we claim that we only need to consider the case
when $v' = v$. This is clear for the first and second part since if $v'\ne v$ then $v$ has the same set of incident edges in $Q'$ and in $Q$. It is also clear for the third part since $Q$ is a subgraph of $Q'$.
It remains to show the induction step when $v' = v$. The first part follows since the two edges of $S$ are incident to $v$ and to $v_{\infty}$ and belong to $Q$. The second part follows by observing that the clockwise ordering of edges around $v$ in $Q$ is obtained from the clockwise ordering around $v$ in $Q'$ by removing an interval of consecutive edges in this ordering; furthermore, the first and last edge in the remaining interval are both incident to $v_{\infty}$. Applying the induction hypothesis shows the second part.
For the third part, if $e_2$ immediately follows $e_1$ in the clockwise ordering around $v$ in $Q'$ then the induction hypothesis gives the desired. Otherwise, $e_1$ and $e_2$ must be the two edges of $S$ and $Q$ is obtained from $Q'$ by removing the faces to the right of $e_1$ and to the left of $e_2$ and replacing them by a single face bounded by $S$.
\end{proof}
In the following, let $Q$ be a piece with more than six faces. The following lemma shows that $Q$ has a balanced cycle separator of size $2$ which can be found in $O(|Q|)$ time.
\begin{lemma}\label{Lem:Separator}
$Q$ as defined above contains a $2$-cycle $S$ containing $v_{\infty}$ such that the number of faces of $Q$ on each side of $S$ is a fraction between $1/3$ and $2/3$ of the total number of faces of $Q$. Furthermore, $S$ can be found in $O(|Q|)$ time.
\end{lemma}
\begin{proof}
We construct $S$ iteratively. In the first iteration, pick an arbitrary vertex $v_1\neq v_{\infty}$ of $Q$ and let $S_1$ consist of two distinct arbitrary edges, both incident to $v_1$ and $v_{\infty}$. This is possible by the first part of Lemma~\ref{Lem:PieceStructure}.
Now, consider the $i$th iteration for $i > 1$ and let $v_{i-1}$ and $v_{\infty}$ be the two vertices of the $2$-cycle $S_{i-1}$ obtained in the previous iteration. If $S_{i-1}$ satisfies the condition of the lemma, we let $S = S_{i-1}$ and the iterative procedure terminates.
Otherwise, one side of $S_{i-1}$ contains more than $2/3$ of the faces of $Q$. Denote this set of faces by $\mathcal F_{i-1}$ and let $E_{i-1}$ be the set of edges of $Q$ incident to $v_{i-1}$, contained in faces of $\mathcal F_{i-1}$, and not belonging to $S_{i-1}$. We must have $E_{i-1}\neq\emptyset$; otherwise, it follows from the third part of Lemma~\ref{Lem:PieceStructure} that $\mathcal F_{i-1}$ contains only a single face of $Q$ (bounded by $S_{i-1}$), contradicting our assumption that $Q$ contains more than six faces and that $\mathcal F_{i-1}$ contains more than $2/3$ of the faces of $Q$.
If $E_{i-1}$ contains an edge incident to $v_{\infty}$, pick an arbitrary such edge $e_{i-1}$. This edge partitions $\mathcal F_{i-1}$ into two non-empty subsets; let $\mathcal F_{i-1}'$ be the larger subset. We let $v_i = v_{i-1}$ and let $S_i$ be the $2$-cycle consisting of $e_{i-1}$ and the edge of $S_{i-1}$ such that one side of $S_i$ contains exactly the faces of $\mathcal F_{i-1}'$.
Now, assume that none of the edges of $E_{i-1}$ are incident to $v_{\infty}$. Then by the second part of Lemma~\ref{Lem:PieceStructure}, $E_{i-1}$ contains exactly one edge $e_{i-1}$. We let $v_i\neq v_{i-1}$ be the other endpoint of $e_{i-1}$ and we let $S_i$ consist of the two edges incident to $v_i$ which belong to the two faces of $\mathcal F_{i-1}$ incident to $e_{i-1}$.
To show the first part of the lemma, it suffices to prove that the above iterative procedure terminates. Consider two consecutive iterations $i > 1$ and $i+1$ and assume that the procedure does not terminate in either of these. We claim that then $\mathcal F_i\subset \mathcal F_{i-1}$. If we can show this, it follows that $|\mathcal F_1| > |\mathcal F_2| >\ldots$ which implies termination.
If $E_{i-1}$ contains an edge incident to $v_{\infty}$ then $\mathcal F_{i-1}'$ contains more than $1/3$ of the faces of $Q$. Since the procedure does not terminate in iteration $i+1$, $\mathcal F_{i-1}'$ must in fact contain more than $2/3$ of the faces so $\mathcal F_i = \mathcal F_{i-1}'\subset \mathcal F_{i-1}$, as desired.
Now, assume that none of the edges of $E_{i-1}$ are incident to $v_{\infty}$. Then one side of $S_i$ contains exactly the faces of $\mathcal F_{i-1}$ excluding two. Since we assumed that $Q$ contains more than six faces, this side of $S_i$ contains more than $2/3$ of these faces. Hence, $\mathcal F_i\subset\mathcal F_{i-1}$, again showing the desired.
For the second part of the lemma, note that counting the number of faces of $Q$ contained in one side of a $2$-cycle containing $v_{\infty}$ can be done in the same amount of time as counting the number of edges incident to $v_{\infty}$ from one edge of the cycle to the other in either clockwise or counter-clockwise order around $v_{\infty}$. This holds since every face of $Q$ contains $v_{\infty}$. It now follows easily from our linked list representation of $Q$ with clockwise orderings of edges around vertices that the $i$th iteration can be executed in $O(|\mathcal F_{i-1}| - |\mathcal F_i|)$ time for each $i > 1$. This shows the second part of the lemma.
\end{proof}
\begin{corollary}\label{Cor:Separatortime}
Given $\Vor{R}{u}$ and $\Trivor{R}{u}$, its recursive decomposition can
be computed in $O(\sqrt r\log r)$ time.
\end{corollary}
\begin{proof}
$\Vor{R}{u}$ has complexity $|\Vor{R}{u}| = O(\sqrt r)$ and $\Trivor{R}{u}$ can be found in time linear in this complexity. Since the recursive decomposition of $\Trivor{R}{u}$ has $O(\log r)$ levels and since the total size of pieces on any single level is $O(|\Trivor{R}{u}|) = O(|\Vor{R}{u}|) = O(\sqrt r)$, the corollary follows from Lemma~\ref{Lem:Separator}.
\end{proof}
\begin{lemma}
\label{Lem:sepspace}
The recursive decomposition of $\Trivor{R}{u}$ can be stored using $O(\sqrt r)$ space.
\end{lemma}
\begin{proof}
Observe that the number of nodes of the tree decomposition is
$O(\sqrt{r})$ and each separator consists of two edges and so takes
$O(1)$ space.
\end{proof}
\paragraph{Embedding of $\Trivor{R}{u}$:}
We now provide a more precise definition of the embedding of $\Trivor{R}{u}$.
Let $f_{\infty}$ be the face of $R$ corresponding to $v_{\infty}$ in $R^*$, i.e., $f_{\infty}$ is the hole $H$.
Consider the graph $\tilde{R}$ that consists of $R$ plus
a vertex $\tilde{v}_{\infty}$ located in $f_{\infty}$ and an edge between
each vertex of $f_{\infty}$ and $\tilde{v}_{\infty}$. The rest of $\tilde{R}$ is
embedded consistently with respect to the embedding of
$R$.
Now, consider the following embedding of $\Trivor{R}{u}$.
First, embed $v_{\infty}$ to $\tilde{v}_{\infty}$. We now specify the embedding of each edge
adjacent to $v_{\infty}$.
Recall that each edge $e$ that is adjacent to $v_{\infty}$ lies in
a single cell $C(e)$ of $\Vor{R}{u}$.
For each such edge $e$ going from $v_{\infty}$ to a vertex $w^*$ of $\Trivor{R}{u}$, we
embed it so that it follows the edge from $\tilde{v}_{\infty}$ to the boundary
vertex $b_e$ of $C(e)$,
then the shortest path in $\tilde{R}$ from $b_e$ to the vertex of $C(e)$ on the face
corresponding to $w^*$ in $\tilde{R}$. Note that by definition of $\Trivor{R}{u}$ such
a vertex exists. We also remark that since the edges follow shortest
paths and because of the uniqueness of the shortest paths
they may intersect but not cross (and hence do not contradict
the definition of $\Trivor{R}{u}$).
It follows that there exists a 1-to-1 correspondence between 2-cycle separators going through
$v_{\infty}$ of $\Trivor{R}{u}$ and cycle separators of $\tilde{R}$ consisting of
an edge $(u,v)$, the shortest paths between $u$ and a boundary vertex $b_1$ and $v$
and a boundary vertex $b_2$ and $(b_1,\tilde{v}_\infty)$ and
$(b_2,\tilde{v}_\infty)$. We call the set $\{b_1,u,v,b_2\}$ the \emph{representation}
of this separator.
This is illustrated in
Figure~\ref{fig:sep2}.
Thus, for any 2-cycle separator $S$ going through $v_{\infty}$,
we say that the set of vertices of $R$ that is in the interior (resp. exterior) of $S$
is the set of vertices of $R$ that lie in the bounded region of the
place defined by the Jordan curve corresponding to
the cycle separator in $\tilde{R}$ that is in 1-1 correspondence
with $S$.
\begin{figure}[htbp]
\centering
\includegraphics[width=.5\textwidth]{sep2.pdf}
\caption{Example of a $2$-cycle separator of $\Trivor{R}{u}$ and its
corresponding embedding into the actual graph.}
\label{fig:sep2}
\end{figure}
We can now state the main lemma of this section.
\begin{lemma}
\label{lem:mainseparator}
Let $w$ be a vertex of $R$. Assume there exists a data structure that
takes as input a the representation $\{b_1,x,y,b_2\}$
of a 2-cycle separator $S$ of
$\Vor{R}{u}$ going through $v_{\infty}$
and answers in $t$ time queries of
the following form: Is $w$ in the bounded closed subset of the plane
with boundary $S$?
Then there exists an algorithm running in time $O(t \log r)$
that returns
a set of at most $6$ Voronoi cells of $\Vor{R}{u}$ such that
one of them contains $w$.
\end{lemma}
\begin{proof}
The algorithm uses the recursive decomposition of $\Trivor{R}{u}$
described in this section. Note
that the decomposition consists of 2-cycle separators
going through $v_{\infty}$. Thus, using the above embedding, each of the 2-cycle
of the decomposition corresponds to a separator
consisting of an edge $(x,y)$ and the shortest paths $P_R(x,b_1)$ and
$P_R(y,b_2)$ where $b_1,b_2$ are boundary vertices of $R$.
Additionally, $y$ belongs to the Voronoi cell of $b_2$ in $\Vor{R}{u}$ and
$x$ belongs to the Voronoi cell of $b_1$ in $\Vor{R}{u}$. Therefore,
$P_R(x,b_1)$ and $P_R(y,b_2)$ are vertex disjoint and so the data structure can
be used to decide on which side of such a separator $w$ is.
The algorithm is the following: proceed recursively along the recursive
decomposition of $\Trivor{R}{u}$ and for each $2$-cycle separator of the
decomposition use the data structure to decide in $t$ time in which side
of the 2-cycle $w$ is located and then recurse on this side. If $w$ belongs to both sides, i.e., if $w$ is on the $2$-cycle separator, recurse on an arbitrary side. The algorithm stops when there are at most $6$ faces
of $\Trivor{R}{u}$ and then it returns the Voronoi cells of $\Vor{R}{u}$
intersecting those $6$ faces.
Observe that the separators do not cross.
Thus, when the algorithm obtains
at a given recursive call that $w$ is in the interior (resp. exterior)
of a 2-cycle $S$ and in the exterior (resp. interior) of the 2-cycle separator
$S'$ corresponding to the next recursive call, we can deduce that $w$
lies in the intersection of the interior of $S$ and the exterior of $S'$ and
hence deduce that it belongs to a Voronoi cell that lies in this area of the
plane.
Note that by Lemma~\ref{Lem:Separator} the number of faces of $\Trivor{R}{u}$
in a piece decreases by a constant factor at each step. Thus, since the number
of boundary vertices is $O(\sqrt{r})$, the procedure takes
at most $O(t \log r)$ time.
Finally, observe that each face of $\Trivor{R}{u}$
that is adjacent to $v_{\infty}$ lies in a single Voronoi cell of $\Vor{R}{u}$.
Thus, since at the end of the recursion there are at most $6$ faces in the piece,
they correspond to at most $6$ different Voronoi cells of $\Vor{R}{u}$. Hence the
algorithm returns at most $6$ different Voronoi cells of $\Vor{R}{u}$.
\end{proof}
\section{Trade-off}\label{sec:tradeoff}
We now prove \Cref{thm:tradeoff}. We first consider the case with $P = O(n^2)$
and then extend it to the case with efficient preprocessing time.
Let $r_1\le r_2$ be positive integers to be defined later. The case $r_1\le
r_2$ will correspond exactly to $S\ge n\sqrt{n}$. The data structure works as
follows.
\begin{enumerate}
\item We compute an $r_1$-division and an $r_2$-division of $G$ named
$\mathcal{R}_1$ and $\mathcal{R}_2$ respectively. Let $\delta_1$
respectively $\delta_2$ denote all the boundary vertices of
$\mathcal{R}_1$ resp. $\mathcal{R}_2$.
\item For each region $R$ of $\mathcal{R}_1$ and $\mathcal{R}_2$, compute
and store the pairwise distances of the nodes of $R$.
\item For each $u\in \delta_1$ and $v\in \delta_2$, compute and store the
distance between $u$ and $v$ in $G$.
\item For each $u\in \delta_1$, region $R\in \mathcal{R}_2$ and hole $H\in
R$, compute $\text{Vor}_{H}(R,u)$ and store a separator decomposition
as described in Section~\ref{sec:recursivedecomp}.
\item For each Region $R\in\mathcal{R}_2$, for each edge $(x,y)\in R$, for
each hole $H$, compute and store the data structure described in
Section~\ref{sec:PreprocRegion}.
\end{enumerate}
We start by bounding the space.
\begin{lemma}\label{lem:tradeoff_size}
The total size of the data structure described above is
\[
O\!\left(nr_2 + \frac{n^2}{\sqrt{r_1r_2}}\right)\ .
\]
\end{lemma}
\begin{proof}
Consider the steps above. By definition of $\mathcal{R}_1$ and
$\mathcal{R}_2$ step 2 uses $O(nr_2)$ space (since we assumed $r_1\le r_2$)
and step 4 uses $\frac{n}{\sqrt{r_1}} \cdot \frac{n}{\sqrt{r_2}}$ space. By
Lemma~\ref{Lem:sepspace} step 5 takes $O(n/\sqrt{r_2})$ for each node of
$\delta_1$ giving $O(n^2/\sqrt{r_1r_2})$ in total. By
Lemma~\ref{Lem:RegionDS} the total space of step 6 is $O(nr_2)$.
\end{proof}
Now consider a query pair $u,v$. If $u$ and $v$ belong to the same region in
$\mathcal{R}_1$ or $\mathcal{R}_2$ we return the stored distance.
Otherwise we iterate over each boundary node $w$ in the region of $u$ in
$\mathcal{R}_1$. For each such boundary node we compute the distance to $v$
using the data structures of steps 5 and 6 above similar to the
query algorithm from Section~\ref{sec:query}. This is possible since we have
stored the distances between all the needed boundary nodes in step 4. The
minimum distance returned over all such $w$ is the answer to the query.
From the description above it is clear that we get a query time of
$O(\sqrt{r_1}\log(r_2))$. The correctness follows immediately from the
discussion in the proof of Theorem~\ref{Thm:main}. What is left now is to
balance the space to obtain Theorem~\ref{thm:tradeoff}.
The expression of Lemma~\ref{lem:tradeoff_size} is balanced when
\[
r_2 = \frac{n^{2/3}}{r_1^{1/3}}\ .
\]
Now, since we assumed that $S \ge n\sqrt{n}$ we can focus on the case when
$r_2\ge \sqrt{n}$ and thus we get $r_1\le r_2$ as we required. Plugging into
the definition of $Q$ we get exactly
\[
Q = O(\sqrt{r_1}\log n) = \frac{n^{5/2}}{S^{3/2}}\log n\ ,
\]
which gives us Theorem~\ref{thm:tradeoff}.
For pre-processing time, we consider two cases similar to
\Cref{thm:preprocesstime} and \Cref{cor:cabellotime}. It follows directly from
the discussion above and \Cref{thm:preprocesstime} that the preprocessing can
be performed in $O(n^2)$. We may, however, also consider pre-processing time as
a parameter similar to space and query time. This gives a 3-way trade-off. In
\Cref{cor:cabellotime} we showed how to lower pre-processing time by increasing
the space. Here we discuss the case of lowering pre-processing time further by
increasing the query time.
It follows from the discussion above and \Cref{cor:cabellotime} that we can
perform pre-processing of the above structure in $O(nr_1 + nr_2^{5/2} +
n^2/\sqrt{r_1r_2})$ time and get the same space bound. If we assume that $r_1
\le r_2^{5/2}$ we get a data structure with query time $Q = O(\sqrt{r_1}\log
n)$ and space and pre-processing time $S = O(nr_2^{5/2} + n^2/\sqrt{r_1r_2})$.
Up to logarithimic and constant factors,
this gives us $Q = n^{11/5}/S^{6/5}$. For any $S \ge n^{16/11}$ this satisfies
the requirement that $r_1\le r_2^{5/2}$.
As an example, we get a data structure with space and pre-processing
time $O(n^{16/11})$ and a query time of $O(n^{5/11}\log n)$.
|
3,212,635,537,474 | arxiv | \section*{Results}
\subsection*{Degenerate vs Non-Degenerate cores}
A dislocation is characterised by its Burgers vector \(\mathbf{b}\) and line direction \(\boldsymbol{\xi}\)~\cite{anderson_2017_disl}.
On a continuum level, \(\mathbf{b}\) is defined as \(\oint_C \left(\partial \mathbf{u}/\partial l \right) \text{d} l\), where \(\mathbf{u}\) is the displacement vector and the integral \(\text{d}l\) is along an arbitrary closed path \(C\) enclosing the dislocation core.
On a discrete level, the contour integral is replaced by a summation of discrete segments \(k\) along a closed path,
\begin{equation}
\label{eqn:dd_dfn}
\mathbf{b} = \sum_{k=1}^N \left( \mathbf{u}^{(k)}_{ij} - \mathbf{U}^{(k)}_{ij} \right)=\sum_{k=1}^N \mathbf{d}_k,
\end{equation}
where \(\mathbf{u}_{ij}\) and \(\mathbf{U}_{ij}\) are displacement vectors between atom \(i\) and \(j\) before and after the introduction of the dislocation and \(\mathbf{d}_k\) is the differential displacement (DD) between atoms \(i\) and \(j\).
Each \(\mathbf{d}_k\) contributes a fraction of the total Burgers vector \(\mathbf{b}\)~\cite{anderson_2017_disl}.
In the BCC structure, the screw dislocation Burgers vector \(\mathbf{b}\) can split into three \(\mathbf{d}_k\) enclosing the core.
Figure~\ref{fig:gamma_disl_Li_Ta_W} shows the core structures and \(\gamma\)-surfaces (energy associated with translating one part of a crystal with respect to another along a crystal plane) of BCC Li, Ta and W calculated using DFT~\cite{supp_mat}.
Lithium exhibits a D-core while TMs Ta and W show ND-cores.
The core structures are dictated by a core energy \(E_\text{c}\), which is primarily associated with the shear displacements \(\mathbf{d}_1\), \(\mathbf{d}_2\), \(\mathbf{d}_3\) in the \(\langle 111\rangle\) direction between neighbouring atoms enclosing the core centre (orange arrows in Fig.~\ref{fig:gamma_disl_Li_Ta_W}a-c).
For Li, Ta and W, the current DFT calculations show equal splitting of \(\mathbf{d}_1 \cong \mathbf{d}_2 \cong \mathbf{d}_3 \cong \mathbf{b}/3\) on three equivalent \(\{110\}\) planes at the core centre, i.e., all core centres are identical (Fig.~\ref{fig:gamma_disl_Li_Ta_W}).
This is consistent with nearly all previous BCC TMs core structure simulation results, irrespective of the description of atomic interactions~\cite{rodney_2017_am} (DFT or interatomic potentials).
This core-splitting implies that the core centre (screw components) fulfils the \(\langle 110 \rangle\)-dyad symmetry of the BCC lattice and is thus ND.
Nevertheless, examination of the \(\gamma\)-surface shows that between the two ground states, the minimum energy path (MEP, black line in Fig.~\ref{fig:gamma_disl_Li_Ta_W}d-f) deviates from \(\langle 111 \rangle\) (white line).
The core energy can thus be further reduced by core splitting along the MEP with some edge components in \(\mathbf{d}_k\).
The principal differences between the D-/ND-cores lie in the details of the \(\mathbf{d}_k\) outside the core centre.
Applying the topological requirement of Eq.~\eqref{eqn:dd_dfn}, each \(\mathbf{d}_k\) must be compensated by two additional DD vectors in the triangular loop formed by one orange and two blue arrows (i.e., \(\sum \mathbf{d}_k=0\)).
The D-core (Fig.~\ref{fig:gamma_disl_Li_Ta_W}a) resolves each \(\mathbf{d}_k\) by a DD vector \(\mathbf{d}^\prime_k\) (blue arrow) of \textasciitilde{}\(\mathbf{b}/3\) along a \(\{110\}\) plane and a small DD (not visible in Fig.~\ref{fig:gamma_disl_Li_Ta_W}a).
For the ND-cores (Fig.~\ref{fig:gamma_disl_Li_Ta_W}b,c), each \(\mathbf{d}_k\) is compensated by two identical \(\mathbf{d}^{\prime\prime}_k\) (blue arrows) of \textasciitilde{}\(\mathbf{b}/6\) on two equivalent \(\{110\}\) planes. The D-/ND-core preference thus appears to be determined by the relative energy costs of the structures outside the core centre: one with a single \(\mathbf{d}^\prime_k\approx\mathbf{b}/3\) and the other with a pair of \(\mathbf{d}^{\prime\prime}_k\approx\mathbf{b}/6\). These energy costs can be approximated by the stacking fault energy \(\gamma\) at \(\mathbf{b}/3\) and \(\mathbf{b}/6\); the ND core structure is preferred when \(2\gamma(\mathbf{b}/6) <\gamma(\mathbf{b}/3)\), as a well-known \(\gamma\)-line criterion~\cite{duesbery_1998_am}.
\begin{figure*}[!htbp]
\centering
\includegraphics[width=0.7\textwidth]{fig_2.pdf}
\caption{\label{fig:prop_core_type} \textbf{Dislocation core structure versus proposed governing criteria}. \textbf{a} \(\gamma\)-line criterion. \textbf{b} Material index \(\chi\)-criterion.
Dislocation cores transform from non-degenerate (filled symbols) to degenerate (empty symbols) as \(\chi\) drops below \textasciitilde{}0.65. }
\end{figure*}
The above \(\gamma\)-line based criterion provides some rationalisation in pure BCC TMs~\cite{duesbery_1998_am,rodney_2017_am}. However, the DFT-based result of Li violates this criterion (Supplementary Fig.~\ref{fig:gamma_line_li_ta_w}), so do other cases calculated by interatomic potentials~\cite{fellinger_2010_prb} or DFT~\cite{romaner_2010_prl}. To determine the true mechanism governing the cores, we calculate the core structures of 7 BCC TMs (V, Nb, Ta, Cr, Mo, W and Fe) based upon 72 different interatomic potentials (Supplementary Table~\ref{tab:potential}). While the accuracies of the potentials vary, we may view them as 72 different types of atoms/average alloys with properties perturbed from respective pure BCC TM elements. These potentials yield both D- and ND-cores, thus providing a broad examination over the proposed \(\gamma\)-line and \(\Delta E\) criteria. To consolidate data across all elements, we use the material index \(\chi\) by normalising \(\Delta E\) of interatomic potentials with the intrinsic \(\Delta E^\text{P}\) of the pure element calculated by DFT.
Figure~\ref{fig:prop_core_type}a shows that the $\gamma$-line criterion does not distinguish different core structures. No correlation is seen between the core structure and the ratio \(2\gamma(\mathbf{b}/6)/\gamma(\mathbf{b}/3)\), or element type, or interatomic potential formalism. Rather, Fig.~\ref{fig:prop_core_type}b shows that the material index \(\chi\) provides a prediction of core structure.
More specifically, D-cores are seen when \(\chi < 0.45\) and ND-cores are observed for \(\chi>0.65\). When \(\chi\) approaches 1, the intrinsic value of pure TMs, all potentials yield the ND core in agreement with DFT calculations~\cite{rodney_2017_am}.
This simple \(\chi\)-criterion is highly predictive, failing in only 3 of 72 cases examined~\cite{supp_mat}.
Near the threshold \( 0.45 < \chi < 0.65\), both D- and ND-cores are seen, indicating that other subtle factors (e.g., elastic constants or the shapes of the \(\gamma\)-surfaces) may play a role.
The \(\chi\)-criterion thus appears to be broadly applicable (DFT validations are provided below).
While empirical, it is rooted in core structure geometry~\cite{kroupa_1964_cjpb}, as illustrated below.
\begin{figure*}[!htbp]
\centering
\includegraphics[width=0.8\textwidth]{fig_3.pdf}
\caption{\label{fig:bcc_fcc} \textbf{Atomic plane stacking sequences in BCC, HCP and FCC structures and EADs of screw dislocations.}
\textbf{a-c} Atomic stacking of BCC \(\{110\}\), HCP \(\{0001\}\) and FCC \(\{111\}\) planes.
\textbf{d-e} Atomic structure of the screw dislocation core viewed from two directions.
Displacements of atoms on the second A-layer in the \([110]\) direction on the \((1\bar{1}0)\) plane (orange arrows) creates a quasi-ABC stacking, similar to that for \(\{111\}\) planes in FCC.
\textbf{f-i} EADs (arrows) in cores of Li-MEAM and W-EAM. Atoms are coloured based on their positions in the \(\langle 111 \rangle\) direction.}
\end{figure*}
\subsection*{Screw Dislocation Core Geometry}
On BCC \(\{110\}\) planes, the atomic stacking follows an ABAB stacking sequence, similar to the ABAB close-packing of \(\{0001\}\) planes in hexagonal close-packed (HCP) structures (Fig.~\ref{fig:bcc_fcc}a-c).
This stacking can be converted to quasi close-packing if atoms in the B-layer are displaced to reside above the centroids of the triangles formed by the A-layer atom triplets (blue dashed triangle in Fig.~\ref{fig:bcc_fcc}a).
The required displacements are along the \(\langle 110 \rangle\)-direction on \(\{110\}\) planes (orange arrows in Fig.~\ref{fig:bcc_fcc}e) and can be provided by non-screw components of the dislocation displacement field.
Each fractional Burgers vector \(\mathbf{d}_k\) has an excess displacement relative to the elastic field~\cite{supp_mat}; the details vary between the D- and ND-cores.
We illustrate these excessive atomic displacements (EADs) (Fig.~\ref{fig:bcc_fcc}, f-i) in the D-core of Li computed by an MEAM potential~\cite{supp_mat} and in the ND-core of W described by an EAM potential~\cite{setyawan_2018_jap}.
In the D-core of Li (Fig.~\ref{fig:bcc_fcc}f), the EADs mainly shift 6 atoms (3 at the core centre and 3 neighbouring them) in \(\langle 110 \rangle\) directions on three \(\{110\}\) planes.
Taking any pair of atoms in the pink shaded box as an example (Fig.~\ref{fig:bcc_fcc}f, others have similar EADs by symmetry), both atoms have EADs along \(\langle 110 \rangle\). These displacements (arrows in Fig.~\ref{fig:bcc_fcc}f,g) move the two atoms on the A-layer (yellow atoms in Fig.~\ref{fig:bcc_fcc}e) towards the centroids of triangles formed by the B-layer atom triplets above them (black triangle), thus forming a local close-packed structure similar to that in FCC/HCP (\emph{cf.} Fig.~\ref{fig:bcc_fcc}b,c,e).
The displacements also put these two atoms in the C-layer position with respect to the A and B layers beneath them, forming a local quasi-ABC stacking sequence.
Kinematically, the EADs lead to similar differential displacements (\(\mathbf{d}_k \approx \mathbf{d}^{\prime}_k \approx \mathbf{b}/3\)) for the two atoms on the same \(\{110\}\) plane and hence the D-core structure.
Energetically, the resulting close packed, quasi-ABC stacking is favourable at the core of Li since \(\Delta E = - 0.8\) meV/atom.
In contrast, the EADs of W-EAM-ND (Fig.~\ref{fig:bcc_fcc}h,i) are relatively small and not aligned on \(\{110\}\) planes.
In this case, displacements in the \(\langle 110 \rangle \) directions on \(\{110\}\) planes incur a large energy penalty since the ABC close-packed structure is highly unfavourable (\(\Delta E = 384\) meV/atom).
Therefore, the D-/ND-core competition is determined by the energy cost associated with displacements along \(\langle 110 \rangle\) on \(\{110\}\) planes that lead to locally close packed structures.
Li and W represent extreme scenarios of the D- and ND-cores; similar examination of the Fe-GAP-ND~\cite{dragoni_2018_prm} core (\(\Delta E= 159\) meV/atom) and Fe-MEAM-D~\cite{asadi_2015_prb} core (\(\Delta E= 44\) meV/atom) shows that their EADs fall between these two cases (Supplementary Fig.~\ref{fig:fe_core_eecd}).
The structure formed at the D-core is not precisely FCC, but only similar in close-packing sequence; the geometric description here should be viewed as suggestive.
Nevertheless, the model explains the trend in Fig.~\ref{fig:prop_core_type}, where the transition from ND- to D-core starts as \(\chi\) drops below a critical value \textasciitilde{}0.65 (towards smaller \(\Delta E\)).
Smaller \(\Delta E\) implies that FCC structure becomes less unfavourable (relative to BCC) and forming the quasi-ABC structure is increasingly favourable.
Our extensive study of core structures using different interatomic potentials indicates that relating the energy of the quasi-ABC structure to that of the perfect FCC structure is of sufficient accuracy to predict D- and ND-cores.
The current model, based on \(\Delta E\), is also consistent with DFT calculations for all BCC TMs performed to date~\cite{rodney_2017_am}.
For all BCC TMs, \(\Delta E\) is large (from 138 to 483 meV/atom, Supplementary Fig.~\ref{fig:deltaE}) and FCC structures are highly unfavourable, suggesting limited EADs in \(\langle 110 \rangle\) directions on \(\{110\}\) planes.
Hence all screw dislocations prefer the ND-core structure in pure BCC TMs (Fig.~\ref{fig:gamma_disl_Li_Ta_W}b,c and other DFT calculations~\cite{dezerald_2014_prb}).
On the other hand, alkaline metals have nearly zero \(\Delta E\) and show D-cores (Fig.~\ref{fig:gamma_disl_Li_Ta_W}a for Li-DFT and Supplementary Fig.~\ref{fig:alkali_core} for Li, Na and K-interatomic potentials).
\begin{figure*}[!htbp]
\centering
\includegraphics[width=0.75\textwidth]{fig_4.pdf}
\caption{\label{fig:vca} \textbf{Prediction of \(\chi\) as a function of solute concentrations for binary W-TM alloys from VCA DFT calculations.}
The solubility of each element (El) in W is denoted by \(\widehat{\text{El}}\) at the top. The elements are colour-coded by their group numbers in the periodic table.
}
\end{figure*}
\begin{figure*}[!htbp]
\centering
\includegraphics[width=0.90\textwidth]{fig_5.pdf}
\caption{\label{fig:chi_prop} \textbf{Prediction of key material properties as a function of \(\chi\) for binary W-TM alloys. } \textbf{a} ND and D cores. \textbf{b} Unstable stacking fault energy \(\gamma_{\text{us}}\). \textbf{c} Peierls barrier \(\Delta E_{\text{PB}}\) of the screw dislocation. All properties are calculated by VCA DFT. Lower \(\chi\) favours the degenerate core, reduces \(\gamma_{\text{us}}\) and \(\Delta E_{\text{PB}}\), all independent of solute types.
}
\end{figure*}
\subsection*{Validation and Prediction in Transition Metal Alloys}
The above analysis provides a geometric link between the material index \(\chi\) and the screw dislocation core structure; \(\chi\) can be thought of as a surrogate for local bonding in the core.
It serves as an indicator of core structure and provides a physical/thermodynamic basis for understanding core structure in all BCC metals. Since slip occurs along the Burgers vector direction on \(\{110\}\) planes, the slip process can be related to a transformation between the BCC structure and the quasi-FCC structure (Fig.~\ref{fig:bcc_fcc}). Therefore, the energy barriers associated with dislocation nucleation \(\gamma_\text{us}\) and glide (Peierls barrier) \(\Delta E_\text{PB}\) should also be related to the transformation energy barrier or \(\Delta E\)/\(\chi\). Solid solution alloying can effectively modulate \(\Delta E\) and thus be used as a means of controlling the screw dislocation core structure and slip behaviour. The material index \(\chi\) thus has practical, engineering application for BCC TM alloy design. Since \(\chi\) depends on the energy difference of two elementary structures, DFT calculations can be used to predict \(\chi\) as a function of solute concentration \(c\) and identify favourable solute types/concentrations. We demonstrate this strategy in binary BCC TM alloys, and extensively in W-TMs, a well-studied system in experiments~\cite{stephens_1970_mmtb,li_2012_am} and simulations~\cite{romaner_2010_prl}.
We employ the DFT-based virtual crystal approximation (VCA~\cite{nordheim_1931_ap}) to predict \(\chi\) and core properties for alloys.
VCA replaces all atoms in an alloy by virtual atoms with a composition-weighted pseudopotential and valence electron number based upon the alloy constituent elements~\cite{bellaiche_2000_prb}.
While VCA accuracy is limited to average, homogeneous alloy properties, it retains a general first-principles DFT framework~\cite{supp_mat}.
Figure~\ref{fig:vca} shows the predicted \(\chi\) calculated for 28 binary W-TM alloys (Supplementary Fig.~\ref{fig:vca_mo_vasp} for Mo-TM alloys).
At low solute concentrations ($<$10 \%), \(\chi\) increases for elements to the left of the solvent TM in the periodic table and decreases sharply (favourably) for elements to its right.
For W/Mo, increasing solute valence electron concentration (VEC~\cite{stephens_1972_jlcm,klopp_1975_jlcm,guo_2011_jap}) decreases \(\chi\) and its slope.
Figure~\ref{fig:chi_prop} shows the core structure, \(\gamma_\text{us}\) and \(\Delta E_\text{PB}\) as a function of \(\chi\) for binary W-TM alloys.
For all alloys, ND- and D-cores are seen when \(\chi\) is above and below $0.8$, similar to predictions by interatomic potentials (Fig.~\ref{fig:prop_core_type}b).
Both \(\gamma_\text{us}\) and \(\Delta E_\text{PB}\) scale linearly with \(\chi\); i.e.,
\begin{align}
\label{eqn:chi_scaling}
\gamma_\text{us} (\chi) &= \gamma_\text{us}^\text{P}\left[1+ k_\text{us} (\chi - 1)\right] \\
\Delta E_\text{PB} (\chi) &= \Delta E_\text{PB}^\text{P} \left[1 + k_\text{PB} (\chi - 1)\right] .
\end{align}
where \(k_\text{us} = 0.66\) and \(k_\text{PB} = 2.75\) as fit from the data.
The above expressions can be extrapolated to Group VI-based TM alloys.
Solutes to the left/right of the solvent increase/decrease \(\gamma_\text{us}\) and \(\Delta E_\text{PB}\), which is broadly consistent with solute hardening/softening effects measured in Group VI binary BCC TM alloys (Supplementary Fig.~\ref{fig:hs_gp_vi_exp}) and DFT-based calculations in Mo/W-TMs~\cite{trinkle_2005_science,hu_2017_am}.
The VCA calculations for W-Re show that \(\chi\) decreases with increasing Re for all concentrations.
At 20\% Re (\(\chi=0.73\)), the ND-core transforms to the D-core (Supplementary Fig.~\ref{fig:wre_core}), in agreement with the transformation seen at 25\% Re in explicit solute-based DFT calculations~\cite{romaner_2010_prl} and the observation of a change to alternating \(\{110\}\) slips
in experiments~\cite{li_2012_am}.
The core transformation is also seen in Mo-20\% Re (Supplementary Fig.~\ref{fig:more_core}).
In addition, reducing \(\chi\) can favourably decrease the barrier for dislocation nucleation (\(\gamma_\text{us}\)) at crack tips and lattice friction (\(\Delta E_\text{PB}\)) of the screw dislocation, the two limiting factors governing DBT in BCC TMs~\cite{gumbsch_1998_science}. At sufficiently low \(\chi\) (e.g., 0.66, or 25 at.\% Re), the gap between the stress intensity factor \(K_\text{Ie}\) for dislocation emission and \(K_\text{Ic}\) for cleavage may be reduced to enable ductile behaviour in W at higher temperatures (Supplementary Fig.~\ref{fig:kie_kic_wre_calc})~\cite{supp_mat}. In fact, increasing the Re concentration to $\geq25$\% is the basis of many commercial W-Re alloys which exhibit high ductility and a DBT below room temperature.
For W-Ta alloys, \(\chi\) is always greater than 1 and no core transformation is predicted, in agreement with DFT calculations~\cite{li_2012_am} and experiments where a switch in slip behaviour is not observed~\cite{li_2012_am}.
Ta also increases \(\gamma_\text{us}\) and \(\Delta E_\text{PB}\) (Fig.~\ref{fig:chi_prop}b,c), and thus has deleterious effects on W alloy ductility, again consistent with experiments~\cite{wurster_2011_jnm}. While Fig.~\ref{fig:vca} suggests many solutes (Tc, Fe, Os, Ru, Rh, Co, Ir etc.) possessing the ductilizing/softening effects of Re~\cite{klopp_1975_jlcm,caillard_2020_am}, their practical application is limited by their low solubility~\cite{thaddeus_1986_bapd} and/or cost.
Re is thus one of the most practical solutes for ductilizing W.
The present predictions provide explicit recommendations of solutes and concentrations that may be explored through non-equilibrium techniques (e.g., mechanical alloying or plasma sintering) to overcome solubility limits.
Overall, the \(\chi\)-based approach identifies solutes capable of ductilizing/softening Group VI TMs that are broadly consistent with nearly all experiments performed to date~\cite{ren_2018_ijrmhm} (Supplementary Fig.~\ref{fig:hs_gp_vi_exp}).
The applicability of the \(\chi\)-approach is further examined in binary TM alloys with solvents from Group V.
For V-TM alloys, solutes to the left of V in the periodic table (decreasing VEC) reduce \(\chi\) (Supplementary Fig.~\ref{fig:vca_v_vasp}) and are expected to have softening effects, while solutes to its right increase \(\chi\) and lead to hardening.
The V-results are opposite to that in Group VI but fully consistent with hardening/softening measurements of Group V TMs (Supplementary Fig.~\ref{fig:hs_gp_v_exp}), as well as DFT calculations based upon inter-string potential~\cite{medvedeva_2007_prb} and a double-kink nucleation~\cite{zhang_2015_sm} models.
As another example, Ti was observed to ductilize V at 77 K~\cite{fraser_1962_cmq}, in agreement with \(\chi\)-model predictions.
While solute/VEC effects vary among elements, the proposed material index \(\chi\) succinctly and quantitatively captures such effects of quantum-mechanics origin via inexpensive DFT calculations.
\section*{Discussion}
The importance of \(\chi\) for \(\gamma_\text{us}\) and \(\Delta E_\text{PB}\) and thus overall material plasticity and toughness is indisputable~\cite{gumbsch_1998_science}.
Effects of \(\chi\) on the screw dislocation core structures are more subtle but significant.
A change from the ND-core to D-core can have multiple effects on dislocation behaviour; the most important of which is the activation of additional slip systems.
The D-core slips on alternating \(\{110\}\) planes with net slip on multiple \(\{112\}\) planes, while the ND-core is highly restricted to a single \(\{110\}\) plane~\cite{vitek_2008_dis}.
A complete switching of slip plane only occurs at high solute concentrations (corresponding to \(\chi\)<0.8) and may be influenced by additional factors such as stress state and temperature~\cite{weinberger_2013_imr,caillard_2020_am}.
At moderate solute concentrations, both D- and ND-core dislocations can co-exist, depending on local solute concentrations.
Their co-existence activates 24 slip systems; i.e., twice the number of ND core-favoured \{110\} slip systems.
The importance of an increase in the number of active slip systems should not be underestimated. The ductility of W depends on texture and grain size~\cite{zhang_2009_msea}; single crystals are generally ductile and polycrystals brittle.
Activation of additional slip systems can be critical; providing the flexibility required for strain compatibility at grain boundaries/junctions and reducing the texture-sensitivity of ductility in structural alloys.
At low temperature, pure BCC TMs tend to be brittle as slip is inhibited by high lattice friction and limited to \(\{110\}\) systems.
With increasing temperatures, the ductility of these pure TMs increases dramatically as \(\Delta E_\text{PB}\)/\(\gamma_\text{us}\) is reduced and additional slip planes (\(\{112\}\) and \(\{123\}\)) become thermally activated.
Such effects of entropy can be equivalently achieved by solute-addition changes to the core structure, \(\gamma_\text{us}\) and \(\Delta E_\text{PB}\), as captured by \(\chi\).
In other words, plastic deformation of pure TMs observed at high temperature can be achieved by alloying with ductilizing solutes (e.g., Re in W and Mo) at low temperature.
This implies that appropriate choice of solutes can increase ductility and lower the DBT temperature, as seen in W/Mo-Re alloys.
The \(\chi\)-based predictions are also fully consistent with general ductile/brittle behaviour in emerging compositionally complex alloys. Refractory high entropy alloy NbTaMoW can be considered as adding group V BCC TM to group VI TM, which raises \(\chi\). The ND core is expected to be prevalent, consistent with recent DFT calculations~\cite{yin_2020_npjcm}. NbTaMoW is also generally brittle at low temperatures. In contrast, HfTiVNb can be considered as adding group IV elements to the group V TM, which favorably reduces \(\chi\). Preliminary DFT calculations show that HfTiVNb adopts both the ND and D core, depending on the local solute environment. HfTiVNb possesses tensile ductility above 25\% at room temperature in experiments~\cite{an_2021_mh}.
Solutes also have local effects, which influence \(\chi\) and all core-related properties locally; e.g., dislocation lattice friction by modifying double-kink nucleation/propagation, dislocation cross-slip and interactions~\cite{trinkle_2005_science}.
These local effects vary between BCC TMs and should not be overlooked. The \(\chi\)-approach does not replace established temperature-dependent solid-solution softening/hardening models~\cite{trinkle_2005_science,zhao_2018_msmse,ghafarollahi_2021_am}.
Instead, \(\chi\) offers a fundamental understanding of the physical basis for controlling dislocation behaviour and quantitatively predicts solute effects at low temperatures where softening/toughening is most needed.
Since \(\chi\) is built upon crystal geometry and bonding characteristics (captured through energy differences of two simple crystal structures), and is validated in extensive DFT calculations and across nearly all interatomic potentials available, it is expected to be general and can be used to rapidly identify favourable solutes in the entire family of BCC TM alloys (e.g., Nb/Fe-based alloys~\cite{romaner_2014_pml}).
In summary, we revealed the mechanistic origin of the critical screw dislocation properties and presented a general model with a fundamental material index \(\chi\) for predicting and controlling screw dislocation average properties in all BCC materials. The \(\chi\)-model rationalises core structures in pure alkaline metals, BCC TMs and their alloys, and is quantitatively related to lattice friction and barrier for dislocation nucleation. More importantly, \(\chi\) can be computed rapidly for any alloy composition using DFT, providing a practical computational approach for ductile and tough BCC TM alloy design. This approach was tested in several binary BCC TM systems and, for example, correctly identifies Re as the most practical solute for ductilizing W and the appropriate Re concentration range. The material index \(\chi\) does not predict ductility or toughness directly, but does, nevertheless, provide a fundamental constraint on the nature of plasticity in these materials. The proposed \(\chi\) approach is first-principles-based; hence it is applicable to a wide-range of complex alloys where ductility is controlled/limited by lattice friction, dislocation nucleation barrier or nature's choice of slip systems.
\newpage
\section*{Methods}
An extensive range of calculations are performed using various methods and models in the current work. We also used two density functional theory (DFT) packages to overcome some limitations in determining some specific properties and to cross-validate the generality of the \(\chi\)-model. We therefore first describe the general methods, followed by details of individual models. The parameters in the general methods apply to all the calculations unless mentioned otherwise in the respective models.
\subsection*{Density-functional theory calculations using VASP}
\label{sec:dft_method}
First-principles calculations are performed within the DFT framework using the Vienna Ab initio Simulation Package (VASP~\cite{kresse_1996_prb,kresse_1999_prb}) and Quantum ESPRESSO (QE~\cite{giannozzi_2009_jpcm}).
In VASP, the exchange-correlation functional is described by the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE~\cite{perdew_1996_prl}) parameterization. In the standard DFT calculations, the core electrons are replaced by the projector augmented wave (PAW~\cite{blochl_1994_prb}) pseudopotentials with the valence states shown in Supplementary Table~\ref{tab:dft_details}. A first-order Methfessel-Paxton method~\cite{methfessel_1989_prb} is used to smooth eigenstate occupancy. The plane-wave cutoff energy, \(\Gamma\)-centered Monkhorst-Pack k-point sampling mesh~\cite{monkhorst_1976_prb} and the smearing parameters sigma are established through convergence tests of 2-atom BCC unit cells with a threshold energy variation \(\Delta E_\text{c} < 1\) meV/atom. The final converged parameters are shown in Supplementary Table~\ref{tab:dft_details}.
In the virtual crystal approximation (VCA~\cite{bellaiche_2000_prb}) DFT calculations, virtual atoms are created with effective cores and valence electrons constructed based on the alloy constituents. In particular, the virtual atoms have their core electrons replaced by pseudopotentials \(\phi_\text{virtual}\), which are obtained via mixing that of the solvent and solute atoms and weighted by their atomic fraction \(X\). For example, for a binary \(\text{A}_{X} \text{B}_{1-X}\) alloy, a pseudopotential is created for the core electrons of the virtual atom representing the average alloy property, i.e.,
\begin{equation}\label{eqn:vca_weighted_core_ve}
\phi_\text{virtual} = X \phi_\text{A} + (1-X) \phi_\text{B}
\end{equation}
where \(\phi_{i}\) are the pseudopotentials of the respective atoms. In addition, the valence electron numbers of virtual atoms are calculated according to Eqn.~\ref{eqn:vca_weighted_core_ve}. The plane-wave kinetic cutoff energy is set at 450 eV and a fine \(\Gamma\)-centered Monkhorst-Pack k-point mesh~\cite{monkhorst_1976_prb} (k-spacing \(2\pi \times 0.02\)\AA\(^{-1}\)) is used for Brillouin zone integration. A first order Methfessel-Paxton smearing with a width of 0.2 eV is used to smooth the partial electron occupancies. The valence states used for different elements are listed in Supplementary Table~\ref{tab:vs_state_vca_vasp_qe}.
\subsection*{Molecular dynamics/statics simulations using LAMMPS}
\label{sec:md_method}
Molecular dynamics and static relaxation simulations are performed using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS~\cite{plimpton_1995_jcp}). Structure optimization is performed using the conjugate gradient method. Convergence is assumed when forces on all atoms drop below \(10^{-12}\) eV/\AA{} in calculations of \(\Delta E\) and \(\gamma\) surfaces, and \(10^{-10}\) eV/\AA{} in calculations of dislocation core structures. For the GAP potential for Fe~\cite{dragoni_2018_prm,bartok_2010_prl,bartok_2013_prb,bartok_2015_ijqc,gap_www}, convergence tolerance is relaxed to \(10^{-12}\) eV for energy variation and \(10^{-6}\) eV/\AA{} for atomic forces, due to its slow convergence and high computational cost.
\subsection*{Interatomic potential models for BCC metals}
The screw dislocation has been extensively studied in elementary BCC transition metals using DFT calculations. A rich set of interatomic potentials have also been developed using various formalisms over the past several decades. In this work, we examined nearly all interatomic potentials (publically accessible) for BCC metals developed up to date. These potentials are available from the National Institute of Standards and Technology (NIST) potential repository~\cite{nist_2013}, the Open Knowledgebase of Interatomic Models (OpenKIM) project~\cite{openkim_2011} or literature. In total, we examined 72 potentials, as shown in Supplementary Table~\ref{tab:potential}. These potentials use different functional forms in their descriptions of interatomic interactions. The choice made here is not biased toward any particular form, element or fitting procedure.
In addition, we developed two new MEAM potentials for BCC V (V1 and V2). The potential parameters are fitted with target properties shown in Supplementary Fig.~\ref{fig:meam_prop_v1_v2} using the particle swarm optimization algorithm. The two fit-for-purpose potentials have nearly identical properties, except for \(\Delta E\), which is intentionally controlled at 91 meV/atom and 124 meV/atom, respectively. With \(\Delta E\) from DFT at 243 meV/atom, V1 and V2 have \(\chi = 0.37 \) and 0.51, respectively. V1 adopts a D-core and V2 adopts a ND-core for the screw dislocation, consistent with the proposed \(\chi\)-criterion. A new MEAM interatomic potential for Li is also developed with \(\Delta E\) = -0.8 meV/atom close to that from DFT. All potential parameters for Li are shown in Supplementary Table~\ref{tab:meam_pot_param} and can be used in LAMMPS directly. The MEAM potential for Li exhibits a D-core, in agreement with DFT and consistent with the geometric model where D-core is favored if the FCC structure is not hugely energetically unfavourable.
\subsection*{Calculation of energy difference \(\Delta E\) between FCC and BCC structures}
\label{sec:calc_chi_pure_ele}
The energy differences \(\Delta E\) per atom between the FCC and BCC structures are calculated using two-atom BCC and 4-atom FCC unit cells by DFT in VASP. Convergence is assumed when energy variation drops below \(10^{-5}\) eV per electronic step and \(10^{-4}\) eV per ionic step. The final \(\Delta E\) results are shown in Supplementary Table~\ref{tab:delta_E_bcc_fcc}. Similarly, the energy difference \(\Delta E\) is also calculated using all the interatomic potentials in Supplementary Table~\ref{tab:potential} and convergence criterion described earlier. In all cases, the values of \(\Delta E\) do not depend on the employed supercell size within convergence tolerances.
\subsection*{Generalized stacking fault energy surface}
\label{sec:gamma_surf_calc_vasp}
The generalized stacking fault energy surfaces (\(\gamma\)-surface) of Li, Ta and W are calculated by DFT using VASP. A slab-supercell model is used, with its crystal orientation given in Supplementary Table~\ref{tab:supercell_dft}. The slab contains 12 \(\{110\}\)-plane atom layers and a 20 \AA{} vacuum layer. \(\gamma\)-surfaces are calculated using the classical method by Vitek. For each specific stacking fault position, a homogeneous slip displacement \(\mathbf{s}\) is applied to atoms in the upper half block of the supercell to create the stacking fault between the sheared and un-sheared atom block. Stacking fault energies are calculated with all atoms allowed to move in the direction perpendicular to the slip, \(\{110\}\) plane. The same convergence criteria as that for \(\Delta E\) are used for the \(\gamma\)-surface calculations.
In the calculations of \(\gamma\)-surface using interatomic potentials in LAMMPS, tilted supercells of dimensions \textasciitilde{}\(10 \times 10 \times 40\) (\AA{}\textsuperscript{3}) are used. The stacking fault energies are calculated with atoms constrained to move in the direction perpendicular to the slip plane only (same as in DFT, see Ref.\cite{yin_2017_am} for details).
\subsection*{Dislocation core structure}
\label{sec:disl_core}
Dislocation core structures are calculated using both a dipole configuration (dipole method) and a single dislocation with fixed boundary conditions (cluster method). For the dipole method, a fully periodic prism supercell is chosen with the dislocations arranged in quadrupolar positions, as shown in Supplementary Fig.~\ref{fig:dis_sche}b. The supercell vectors are
\begin{align}
\mathbf{c}_1 &= \mathbf{e}_{1}, \\
\mathbf{c}_2 &= 5\mathbf{e}_{2}, \\
\mathbf{c}_3 &= \dfrac{1}{2}\mathbf{e}_{1} + \dfrac{5}{2}\mathbf{e}_{2}+9\mathbf{e}_{3},
\end{align}
where (\(\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}) = (a/2[111],a[11\bar{2}],a[\bar{1}10])\) and \(a\) is the lattice parameter of the BCC structure determined in unit cell calculations. In total, the supercell contains 135 atoms.
A dislocation dipole, i.e., a pair of dislocations with opposite Burgers vectors \(\pm \mathbf{b}\), is introduced at easy-core/hard-core positions (Supplementary Fig.~\ref{fig:dis_sche}c) using the Babel package developed by E. Clouet~\cite{clouet_2021_babel}. In particular, the displacement field of arrays of dipoles (repeated 16 times in the \(\mathbf{c}_2\) and \(\mathbf{c}_3\) directions) is firstly applied to atoms in the supercell, followed by applying a homogeneous strain to accommodate the plastic strain introduced by the dipole (see Ref.~\cite{rodney_2017_am} for details). During structure optimization, the cell vectors are fixed and all atoms are allowed to move until the ionic force is below \(5\times 10^{-3}\) eV/\AA. DFT calculations show that the easy-core is the minimum energy configuration and the hard-core is a local maximum. The DFT-computed core structures (ND vs D) are shown in Fig.~\ref{fig:gamma_disl_Li_Ta_W} for BCC Li, Ta and W.
The dipole approach allows modelling of dislocation cores with a fully periodic supercell of relatively small sizes. It avoids the complexity of free surfaces, but introduces dislocation interactions and a non-negligible homogeneous strain scaling with supercell sizes. Therefore, we also used a hexagonal prism supercell (Supplementary Fig.~\ref{fig:dis_sche}a) with a single dislocation and performed dislocation core structure calculations with interatomic potentials. In this approach, supercells of radius \(R\) \textasciitilde{}92 \AA{} and length of one Burgers vector are first created with perfect BCC structures. The crystallographic \([11\bar{2}]\) direction is aligned with the \(x\)-axis, \([\bar{1}10]\) aligned with the \(y\)-axis and the screw dislocation line direction \([111]\) aligned with the \(z\)-axis, i.e., the hexagonal prism axis (Supplementary Fig.~\ref{fig:dis_sche}d). The \(z\)-direction has periodic boundary conditions while the \(x\) and \(y\) directions are treated as surfaces. This geometry keeps the \(\langle 111 \rangle\)-axis threefold symmetry of the BCC lattice. Each supercell contains \textasciitilde{}3600 atoms (depending on element types). A single dislocation is then introduced by displacing atoms according to the anisotropic elastic displacement field of the corresponding Volterra dislocation at the supercell center (easy-core position). Atoms within \textasciitilde{}12 \AA{} from the outer surface (grey area in Supplementary Fig \ref{fig:dis_sche}a) are always fixed to their elastic displacements. The initial dislocation structure is first optimized using conjugate gradient algorithm at 0 K, then equilibrated for 100,000 time steps (100,000 femtoseconds) at finite temperatures (\textasciitilde{}50 K) and optimized again at 0 K. This procedure is not always needed, but helps to obtain well-equilibrated core structures in some interatomic potentials. Only 0 K structure optimization is performed for the GAP potential for Fe~\cite{dragoni_2018_prm}. The obtained core structures (ND vs D) are summarized in Fig.~\ref{fig:prop_core_type}.
\subsection*{Differential displacement plot and atomistic visualisation}
Dislocation core structures are visualised using the differential displacement (DD) map between neighboring atoms~\cite{vitek_1970_pma}. For screw components (Fig.~\ref{fig:gamma_disl_Li_Ta_W}, Supplementary Figs.~\ref{fig:prop_core_type_v} and~\ref{fig:alkali_core}), the arrows are scaled so that the largest components (i.e., \textasciitilde{}\(\mathbf{b}/3\)) touch the neighboring atoms.
Excessive atomic displacements (EADs) (Fig.~\ref{fig:bcc_fcc}f-i and Supplementary Fig.~\ref{fig:fe_core_eecd}) are obtained by using the corresponding anisotropic Volterra dislocation field as references. The EADs of ND-/D-cores possess different symmetry; in the D-core, they have a \(\langle{111} \rangle\)-threefold screw axis symmetry while that in the ND-core have both the \(\langle{111} \rangle\)-threefold screw axis symmetry and the \(\langle {110} \rangle\)-diad axis symmetry. The EADs illustrate the key, subtle differences among different materials/models. They are generally very small, and are thus magnified by 6 times.
Atomic configurations (Fig.~\ref{fig:bcc_fcc}a-e) are visualised using the Open Visualisation Tool (OVITO~\cite{stukowski_2009_msmse}).
\subsection*{Calculation of \(\chi\) in binary alloys with DFT VCA in VASP }
\label{sec:chi_vca_vasp}
The values of \(\chi\) are calculated at a 2\%-increment for binary W-TM, Mo-TM and V-TM alloys using the VCA method in VASP. The supercells and convergence criteria are the same as that used for calculating \(\Delta E\) in pure elements. The DFT settings are described in the Methods: Density-functional theory calculations using VASP, except that the plane-wave kinetic cutoff energy is increased to 550 eV for all considered elements. We used the same cutoff energy for 84 alloys here to demonstrate the method of \(\chi\) to identify favourable solutes, as commonly employed in high throughput DFT calculations~\cite{lejaeghere_2016_science}. Refined calculations can be performed once favourable solutes are identified. The final results are shown in Fig.~\ref{fig:vca}, Supplementary Figs.~\ref{fig:vca_mo_vasp} and ~\ref{fig:vca_v_vasp}.
\subsection*{Peierls barrier of binary W-TM alloys with DFT VCA in VASP}
Peierls barrier can be defined as the energy barrier per unit length for a straight dislocation to move from one energy valley (equilibrium position) to the next one at zero stress and zero temperature. For the \(1/2\langle 111 \rangle\) screw dislocation in BCC structures, the two equilibrium positions correspond to the easy-core configuration in adjacent lattice positions (Supplementary Fig.~\ref{fig:dis_sche}c). In the current work, we use the nudged elastic band (NEB~\cite{henkelman_2000_11_jcp}) method to calculate the dislocation migration path between these two adjacent easy-core configurations and thus obtain the height of the energy barrier, i.e., the Peierls barrier. The transition path is calculated for pure W by DFT in VASP and for W-5\%Re, W-10\%Re and W-10\%Ta by DFT VCA in VASP. The dipole supercell with 7 linearly interpolated replicas are used for all calculations (see Refs.~\cite{samolyuk_2013_jpcm,dezerald_2014_prb} for details).
Supplementary Fig.~\ref{fig:vca_wtms_pe} shows the energy variation along the transition path. For all cases, the Peierls potential profiles have a near-sinusoidal profile with a peak in the middle replica corresponding to the saddle-core configuration. Re and Ta respectively decreases and increases the Peierls barrier relative to that of pure W, in agreement with previous DFT calculations~\cite{samolyuk_2013_jpcm,dezerald_2014_prb}.
The NEB method requires simultaneous structure optimizations in all replicas and is computational expensive. For calculations in a wide range of W-TM alloys, we employ the reaction coordinate method~\cite{xu_1998_cms} (i.e., the drag method) to compute their Peierls barriers as a function of solute concentrations and \(\chi\). In the drag method, the 5 intermediate replicas are linearly interpolated between the initial and final configurations (easy-1 and easy-2 in Supplementary Fig.~\ref{fig:dis_sche}c). In the middle replica, a single atom near the core (the light grey atom in Supplementary Fig.~\ref{fig:dis_sche}c) is fixed along the dislocation line direction while all other degrees of freedom are fully optimized. Under this constraint, the middle replica will evolve to the saddle-core configuration (peak energy in the transition path) found in the NEB method. Supplementary Table~\ref{tab:drag_vs_neb} shows the Peierls barriers computed by the NEB and the drag methods, respectively. Their agreement shows that the drag method, as an alternative to NEB, can be used to compute the Peierls barrier in a reliable and computationally efficient manner. The calculations for alloys are performed with the VCA method in VASP and convergence is assumed when the ionic forces fall below \(0.02\) eV/\AA . The obtained Peierls barriers of W-TM alloys are shown in Fig.~\ref{fig:chi_prop}c.
\subsection*{Unstable stacking fault energy of binary W-TM alloys with DFT VCA in VASP}
\label{sec:usf_wre_vca_vasp}
The unstable stacking fault energy \(\gamma_\text{us}\) dictates the energy barrier to nucleate a dislocation. In the current work, we calculate \(\gamma_\text{us}\) in the \(\langle 111 \rangle\) direction on the \(\{110\}\) plane as a function of solute concentration in binary BCC W-TM alloys by DFT VCA in VASP. In particular, \(\gamma_\text{us}\) of \{110\} and \{112\} planes are calculated for W-Re alloys and are used as material property inputs for analysis of crack tip behaviour using linear elastic fracture mechanics (LEFM). The DFT settings, supercell and convergence criterion are described in the Methods section: density-functional theory calculations using VASP, Generalized stacking fault energy surface and Supplementary Table~\ref{tab:supercell_dft}. The \(\gamma_{\text{us}}\) of \{112\} plane converges when the slab has 16 atomic layers or more. The converged \(\gamma_\text{us}\) values are shown in Fig.~\ref{fig:chi_prop}b and Supplementary Fig.~\ref{fig:usf_sf_wre_calc}a.
\subsection*{Surface energy of W-Re alloys with DFT VCA in VASP and QE}
\label{sec:wre_surf_e_vca_vasp}
The surface energies of binary W-Re alloys are calculated using VCA as implemented in both VASP and QE (see Supplementary Table~\ref{tab:supercell_dft} for supercells employed). The pseudopotentials and parameters are described in the Methods section: Density-functional theory calculations using VASP and Cross-validation of \(\chi\) in binary W-TM alloys with DFT VCA in QE. In addition, the kinetic cutoff energies for wavefunctions and charge densities are chosen as 952 eV (70 Ry) and 11424 eV (840 Ry) respectively in QE. For the ionic minimization, the convergence thresholds for total energy and forces are 1.36 $\times 10^{-3}$ eV (1 $\times 10^{-4}$ Ry) and 2.57 $\times 10^{-2}$ eV/\AA\ (1 $\times 10^{-3}$ Ry/bohr). The convergence criterion for electronic self-consistency is 1.36 $\times 10^{-5}$ eV (1 $\times 10^{-6}$ Ry).
For each surface, a fully periodic supercell is created with supercell vectors \((\mathbf{c}_1, \mathbf{c}_2, \mathbf{c}_3)\), where \(\mathbf{c}_1\) and \(\mathbf{c}_2\) are the in-plane supercell repeating vectors and \(\mathbf{c}_3\) is the out-of-plane vector. We first obtain the energy per atom \(E_\text{c}\) in the bulk by optimizing the supercell and ionic positions. The slab-vacuum supercell is then created by adding a vacuum layer in the out-of-plane direction. The slab-vacuum structure is optimized through ionic relaxation only in the out-of-plane direction while keeping all supercell vectors fixed. In both calculations of the bulk supercell and slab-vacuum supercell, a consistent k-point sampling is used in the in-plane directions, leading to better convergence with respect to \(\mathbf{c}_3\), i.e., slab layers~\cite{sun_2013_ss}. The surface energy \(\gamma\) is calculated as
\begin{align}
\gamma = \dfrac{E_\text{sv} - n_\text{sv}E_\text{c}}{2A},
\end{align}
where \(E_\text{sv}\) and \(n_\text{sv}\) are the total energy and number of atoms of the slab-vacuum system and \(A\) is the area of the surface, i.e., \(A = \lvert \mathbf{c}_1 \times \mathbf{c}_2 \rvert\). The supercell vectors are shown in Supplementary Table~\ref{tab:supercell_dft} and a slab with 12 atomic layers gives the converged surface energies, as shown in Supplementary Fig.~\ref{fig:usf_sf_wre_calc}c,d.
For pure W in stardard DFT calculations, both VASP and QE give similar surface energies; \(\gamma_{\text{s} \{100\}}\) and \(\gamma_{\text{s}\{110\}}\) are 4.03 J/m\(^{2}\) and 3.27 J/m\(^{2}\) in VASP, and 4.15 J/m\(^{2}\) and 3.28 J/m\(^{2}\) in QE respectively. In VCA QE, the surface energies of the \(\{100\}\) and \(\{110\}\) planes are predicted to gradually decrease with Re additions, consistent with previous calculations on W-Re alloys~\cite{yang_2018_prb,hu_2021_am}. However, in VCA VASP, the surface energies are unusual; they increase substantially with Re additions. For example, the surface energy is 11.12 J/m\(^{2}\) for the \(\{110\}\) plane of W-50\%Re, nearly 3.4 times of pure W (3.27 J/m\(^{2}\)). The VCA VASP-based results are also substantially different from that based on special quasi-random structure (SQS) supercell in standard VASP calculations~\cite{hu_2021_am}. In contrast, the results of VCA QE are close to those in SQS supercell calculations. Therefore, we use the surface energies based on VCA QE as inputs in subsequent LEFM analysis.
\subsection*{Elastic constants of W-Re alloys with DFT VCA in VASP}
\label{sec:wre_elas_vca_vasp}
The elastic constants of binary W-Re alloys are calculated using VCA method with VASP. The elastic contants are determined by fitting to the total energies versus strains within \(\pm\)1\%. Supplementary Fig.~\ref{fig:usf_sf_wre_calc}b shows the elastic constants as a function of Re concentrations. Overall, Re has weak effects on the elastic constants of W.
\subsection*{Critical stress intensity factors for cleavage and dislocation emission}
\label{sec:lefm_wre}
Pure W exhibits brittle cleavage behaviour on \(\{100\}\) and \(\{110\}\) planes at low temperature~\cite{gumbsch_1998_science}. The critical stress intensity factor \(K_\text{Ic}\) for Griffith cleavage is lower than \(K_\text{Ie}\) for dislocation emission~\cite{rice_1992_jmps} for sharp cracks on these two planes under mode-I loading. In binary W-Re alloys, \(K_\text{Ic}\) and \(K_\text{Ie}\) may change due to Re-effect. Here, we compute the respective \(K_\text{Ic}\) and \(K_\text{Ie}\) as a function of Re concentration for mode-I loaded sharp cracks.
Since W is nearly isotropic, we use isotropic LEFM theory. In particular, \(K_\text{Ic}\) for Griffith cleavage under plane strain Mode-I loading is computed as
\begin{equation}
K_\text{Ic}=\sqrt{\frac{G}{D}},
\end{equation}
where
\begin{equation}
G = 2 \gamma_\text{s}, \gamma_\text{s} \text{ is the surface energy}
\end{equation}
and
\begin{equation}
D=\frac{1-{\nu}^{2}}{E}, \nu \text{ is the Poisson's ratio and } E \text{ is the Young's modulus}.
\end{equation}
The critical stress intensity for dislocation emission \(K_\text{Ie}\) is calculated according to the Rice criterion~\cite{rice_1992_jmps}
\begin{equation}
K_\text{Ie}=\frac{1}{f(\theta)} \sqrt{\frac{2\mu}{1-\nu} [1+(1-\nu)\tan^{2}\phi]\gamma_\text{us}}
\end{equation}
where \(f(\theta)=\cos^{2}(\theta/2)\sin(\theta/2)\), \(\theta\) is the angle between the crack plane and slip plane, \(\phi\) is the angle between the dislocation Burgers vector and crack front direction in the slip plane, and \(\mu\) is shear modulus, \(\nu\) is Possion's ratio, \(\gamma_\text{us}\) is the unstable stacking fault energy of the slip plane.
All material properties (\(\gamma_\text{s}, \gamma_\text{us}, E, \nu, \mu,\)) depend on Re concentrations in W and are calculated based on first-principle DFT VCA calculations. Supplementary Fig.~\ref{fig:kie_kic_wre_calc} shows the results for 4 crack systems (\(\{100\}\langle 001 \rangle\), \(\{100\}\langle 011 \rangle\), \(\{110\}\langle 001 \rangle\), \(\{110\}\langle 011 \rangle\)) susceptible to cleavage fracture in BCC W. In pure W and for a \(\{100\}\) sharp crack (Supplementary Fig.~\ref{fig:kie_kic_wre_calc} c), \(K_\text{Ic} = 1.90 \text{ MPa}\cdot\text{m}^{1/2}\), which is \(\sim\)40\% and \(\sim\)42\% lower than \(K_\text{Ie}\) for dislocation emission in the \(\{100\}\langle 001 \rangle\) and \(\{100\}\langle 011 \rangle\) crack systems. Similar behaviours are seen in cracks on \(\{110\}\) planes (Supplementary Fig.~\ref{fig:kie_kic_wre_calc}g). In pure W, \(K_{\text{Ic}} = 1.72 \text{ Mpa}\cdot\text{m}^{1/2}\), which is \(\sim\)41\% and \(\sim\)31\% lower than \(K_\text{Ie}\) for dislocation emission in the \(\{110\}\langle 001 \rangle\) and \(\{110\}\langle 011 \rangle\) crack systems. The isotropic solution is nearly identical to that of the anisotropic solution~\cite{mak_2021_jmps} and both results suggest that \(\{100\}\) and \(\{110\}\) planes are intrinsically brittle in W. With increasing Re concentrations, all \(K_{\text{Ie}}\) decrease sharply. However, \(K_{\text{Ic}}\) is insensitive to Re additions. As a result, the gaps between \(K_{\text{Ie}}\) and \(K_{\text{Ic}}\) decrease sharply in all the four crack systems, which is potentially beneficial to make W ductile. Nevertheless, no cross-over between \(K_{\text{Ie}}\) and \(K_{\text{Ic}}\) is predicted in the analysis based on material properties at 0 K.
To extend the prediction to room temperatures, we employ a finite-temperature ductility criterion \(D=K_{\text{Ie}}/K_{\text{Ic}}\) proposed recently~\cite{mak_2021_jmps} based on experimental observations. In BCC W, ductile behaviour can be achieved at room temperatures if \(D < D_\text{c}\); \(D_\text{c} \approx 1.56/1.26\) for \(\{100\}/\{110\}\) planes respectively. In W-20\%Re, \(D_{\{100\}\langle 001 \rangle} = 1.53\) and \(D_{\{110\}\langle 011 \rangle} = 1.30\), close to the respective threshold values \(D_\text{c}\) (Supplementary Fig.~\ref{fig:kie_kic_wre_calc}d,h). With higher Re concentrations, \(D\) decreases further and satisfies the ductile criterion of both planes. At 20\% Re concentration (\(\chi=0.73\)), the ND core also transforms to the D core. This concentration is also close to many commercial W-Re alloys exhibiting high ductility and a DBT below room temperatures. The finite-temperature \(D_\text{c}\) is established empirically based on DBT experiments, it may include the effects of core transformation and associated change in plastic slip behaviour. Since \(D > 1\) at 0 K, the mechanism of Re-ductilization effects is thus more likely related to core transformation. However, we can not rule out thermal activation in dislocation nucleation at crack tips. Nevertheless, the above analysis is based on well-established linear elastic fracture mechanics and DFT-computed material properties. The primary assumption is linear elasticity. Therefore, the results are robust and consistent with ductile behaviour in BCC W-Re alloys at room temperatures. More importantly, the critical material properties (\(\gamma_\text{us}\) and ND/D core structures) and associated brittle vs ductile behaviour can be related to the material index \(\chi\).
\def\section*{}{\section*{}}
\section*{References}
\textbf{References 1-70 are for the main text.}
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3,212,635,537,475 | arxiv | \section{Introduction}
In a reductive algebraic group over $\mathbb{C}$ split over $\mathbb R$ with a fixed
choice of Chevalley generators in the Lie algebra, there is a well-defined
notion of positive, or $\mathbb R_{>0}$-valued points due to Lusztig
\cite{Lus:TotPos94}. In the case of $GL_n$ with the standard choices,
the resulting ``$GL_n(\mathbb R_{>0})$'' recovers the classical notion of
totally positive matrices, that is, matrices such that all minors
are positive. For general $G$ the set $G_{>0}$ is therefore
called the totally positive part of $G$. The closure $G_{\geq 0}$
of $G_{>0}$ (in the real topology) is called the totally nonnegative
part of $G$.
These notions have a natural extension to flag varieties $G/P$. That is,
there is a notion of $(G/P)_{>0}$, and of $(G/P)_{\geq 0}$, the
closure of
$(G/P)_{>0}$, which Lusztig has described as a
``remarkable polyhedral subspace" \cite{Lus:TotPos94}.
Lusztig has proved that $(G/P)_{\geq 0}$ is contractible
\cite{Lus:IntroTotPos} and the first author
has shown that it is a union of semi-algebraic cells \cite{Rie:CelDec,Rie:PhD}.
Moreover,
in \cite{Rie:TotPosPoset} the first author showed that the
closure of a totally
nonnegative cell in $G/P$ is a union of totally nonnegative cells
and described the closure relations in terms of the
Weyl group. The combinatorics of these closure
relations was then studied by the second author in \cite{Williams:poset},
where it was shown that the partially ordered set (poset) of cells of
$(G/P)_{\geq 0}$ is in fact the poset of cells of a
regular CW complex. Recall that a CW complex is a union of cells with additional
requirements on how cells are {\it glued}; a
{\it regular} CW complex is one
where the closure of each cell is homeomorphic to a closed ball
and the closure minus the interior of each cell is homeomorphic to a sphere.
The combinatorial results of \cite{Williams:poset}
prompted the second author to conjecture
that $(G/P)_{\ge 0}$ is a regular CW complex, which in particular
would imply that $(G/P)_{\geq 0}$ is homeomorphic to a closed ball.
In \cite{PSW}, Postnikov, Speyer, and the second author proved that
the non-negative part of the Grassmannian is a CW complex, by introducing
an auxiliary toric variety to each parameterization of a cell, and
constructing a glueing map from the non-negative part of that toric
variety to the closure of the corresponding cell. The construction
of the toric variety and glueing map relied on
explicit positivity properties
of the parameterizations of the cells, which had been described in terms
of certain graphs in
\cite{Postnikov}.
In this paper we generalize the previous result and show
that the non-negative part of any flag variety
$(G/P)_{\ge 0}$ is a CW complex. As in \cite{PSW}, we again
construct a toric variety for each parameterization of a cell.
However, in our proof we use the parameterizations of the cells due to Marsh
and the first author \cite{MarRie:ansatz}, and use Lusztig's canonical basis \cite{Lus:CanonBasis}
in order to prove that they have the desired positivity properties.
Once we have proved that
$(G/P)_{\ge 0}$ is a CW complex,
the combinatorics from \cite{Williams:poset}
implies that the closures of the individual cells have Euler
characteristic one.
The following result is our main theorem.
\begin{thm}\label{th:main}
$(G/P)_{\ge 0}$ is a CW complex.
\end{thm}
In \cite{Williams:poset}, the second author proved the following result.
\begin{thm}\cite{Williams:poset} \label{th:Eulerian}
The poset of cells of $(G/P)_{\ge 0}$ is the poset of cells of some
regular CW complex; therefore the poset
of cells of $(G/P)_{\ge 0}$ is Eulerian.
\end{thm}
In other words, the alternating sum of cells in the closure of a cell
of $(G/P)_{\ge 0}$ is $1$.
This result combined with Theorem \ref{th:main} implies the following.
\begin{cor}
The Euler characteristic of the closure of a cell of
$(G/P)_{\ge 0}$ is $1$.
\end{cor}
The structure of this paper is as follows. In Section \ref{s:prelim}, we review
basic results on algebraic groups and flag varieties. In Sections \ref{s:TotPosGB}
and \ref{s:Toric} we introduce the notion of total positivity for real reductive groups
and flag varieties, and toric varieties, respectively. In Section \ref{s:ConstructToric}
we construct a toric variety associated to a parameterization of a cell. In Section
\ref{s:proof} we prove a key proposition,
and
in Section \ref{s:Partial}, we prove the main result.
\section{Preliminaries}\label{s:prelim}
\subsection{}
We recall some basic notation and results from algebraic groups,
see e.g. \cite{Springer:AlgGroupBook}.
Let $G$ be a simply connected semisimple linear algebraic group over
$\mathbb{C}$ split over $\mathbb R$. We identify $G$ and any related spaces with their
$\mathbb R$-valued points in their real topology (as real manifolds or subsets thereof).
We write $\mathbb R^*$ for $\mathbb R\setminus\{0\}$.
Let $T$ be a split torus and $B^+$ and $B^-$ opposite Borel
subgroups containing $T$. We denote the character and the cocharacter
groups of $T$ by $X^*(T)$ and $X_*(T)$, respectively. Let $<\ ,\ >$
denote the dual pairing between $X^*(T)$ and $X_*(T)$. The unipotent
radicals of $B^+$ and
$B^-$ are denoted $U^+$ and $U^-$, respectively.
Let $\{\alpha_i\ | \ i\in I\}\subset X^*(T)$ be the set of simple roots
associated to $B^+$ and $\{\alpha_i^\vee\ |\ i\in I\}\subset X_*(T)$ the
corresponding coroots. Then we have the simple root subgroups
$U^+_{\alpha_i}\subseteq U^+$ and $U^-_{\alpha_i}\subseteq U^-$.
Furthermore assume we are given homomorphisms
\begin{equation*}
\phi_i: SL_2(\mathbb \mathbb R)\to G, \qquad i\in I,
\end{equation*}
such that
\begin{equation*}
\phi_i\left(\begin{pmatrix}t &0\\ 0& t^{-1}
\end{pmatrix}\right)=\alpha_i^\vee(t), \qquad t\in \mathbb R^*,
\end{equation*}
and such that
\begin{equation*}
\phi_i\left(\begin{pmatrix}1 & m\\ 0&1\end{pmatrix}\right):=x_i(m),
\quad
\phi_i\left(\begin{pmatrix}1 & 0\\ m&1\end{pmatrix}\right):=y_i(m),
\end{equation*}
define isomorphisms $x_i:\mathbb R\to U^+_{\alpha_i}$ and $y_i:
\mathbb R\to U^-_{\alpha_i}$. Following \cite{Lus:TotPos94}, the datum
$(T,B^+,B^-, x_i,y_i, i\in I)$ is called a {\it pinning} for $G$.
\subsection{}\label{s:automorphism} If $G$ is not simply laced, then one can
construct a simply laced group $\dot G$ and an automorphism
$\tau$ of $\dot G$ defined over $\mathbb R$, such that there is
an isomorphism, also defined over $\mathbb R$, between $G$ and
the fixed point subset $\dot G^\tau$ of $\dot G$. Moreover
the groups $G$ and $\dot G$ have compatible pinnings. Explicitly
we have the following.
Let $\dot G$ be simply connected and simply laced.
We apply the same notations as above for $G$, but with an added dot,
to our simply laced group $\dot G$. So we have a pinning
$(\dot T,\dot B^+,\dot B^-, \dot x_i,\dot y_i, i\in \dot I)$ of $\dot G$,
and $\dot I$ may be identified with the vertex set of the Dynkin
diagram of $\dot G$.
Let $\sigma$ be a permutation of $\dot I$ preserving
connected components of the Dynkin diagram, such that,
if $j$ and $j'$ lie in the same orbit under $\sigma$
then they are {\it not} connected by an edge.
Then $\sigma$ determines an automorphism $\tau$ of $\dot G$
such that
\begin{enumerate}
\item
$\tau(\dot T)=\dot T$,
\item
$\tau(x_i(m))=x_{\sigma(i)}(m)$ and $\tau(y_i(m))=y_{\sigma(i)}(m)$ for all
$i\in \dot I$ and $m\in \mathbb R$.
\end{enumerate}
In particular $\tau$ also preserves $\dot B^+,\dot B^-$. Let $\bar I$ denote
the set of $\sigma$-orbits in $\dot I$, and for $\bar i\in \bar I$, let
\begin{eqnarray*}
x_{\bar i}(m)&:=&\prod_{i\in \bar i}\ x_i(m),\\
y_{\bar i}(m)&:=&\prod_{i\in \bar i}\ y_i(m).
\end{eqnarray*}
The fixed
point group $\dot G^{\tau}$ is a simply connected algebraic group with pinning
$({\dot T}^\tau,\dot B^{+ \tau},\dot B^{-\, \tau}, x_{\bar i}, y_{\bar i}, \bar i\in \bar I)$.
There exists, and we choose, $\dot G$ and $\tau$ such that $\dot G^{\tau}$
is isomorphic to our group $G$ via an isomorphism compatible with the pinnings.
\subsection{}
Let $W=N_G(T)/T$ be the Weyl group of $G$. For $i\in I$ the elements
\begin{equation*}
\dot s_i=x_i(-1)y_i(1)x_i(-1)
\end{equation*}
represent the simple reflections $s_i\in W$. If $w=s_{i_1}\dotsc s_{i_m}$
is a reduced expression for $w$ then we write $\ell(w)=m$ for the length
of $w$. We note also that the representative
\begin{equation*}
\dot w=\dot s_{i_1}\dotsc \dot s_{i_m}
\end{equation*}
of $w$ is well-defined, independent of the reduced expression.
Inside $W$ there is a longest element which is denoted by $w_0$.
\subsection{}
Let $J$ be a subset of $I$. The parabolic subgroup
$W_J\subseteq W$ is the subgroup generated by all of the
$s_j$ with $j\in J$. Let $w_J$ denote the longest element in
$W_J$. We also consider the set $W^J$
of minimal-length coset representatives for $W/W_J$, and the
set $W^J_{max}=W^Jw_J$ of maximal-length coset representatives.
The parabolic subgroup $W_J$ of $W$
corresponds to a parabolic subgroup $P_J$
in $G$ containing $B^+$. Namely,
$P_J$ is the subgroup of $G$ generated by $B^+$ and the
elements $\dot w$ for $w\in W_J$.
Let $\mathcal P^J$ be the set of parabolic subgroups $P$
conjugate to $P_J$. This is a homogeneous space for the conjugation
action of $G$ and can be identified with the
partial flag variety $G/P_J$ via
$$
G/P_J\overset\sim\longrightarrow \mathcal P^J\ :\ gP_J\mapsto gP_J g^{-1}.
$$
In the case $J=\emptyset$ we are identifying the full flag variety $G/B^+$
with the variety $\mathcal B$ of Borel subgroups in $G$. We have the
usual projection from the full flag variety to any
partial flag variety which takes the form
$\pi=\pi^J:\mathcal B\to\mathcal P^{J}$, where $\pi(B)$ is the
unique parabolic subgroup of type $J$ containing $B$.
The conjugate of a parabolic subgroup $P$ by
an element $g\in G$ will be denoted by
$g\cdot P:=gPg^{-1}$.
\subsection{}
Recall the Bruhat decomposition for the full flag variety,
$$
\mathcal B=\bigsqcup_{w\in W} B^+\dot w\cdot B^+,
$$
and the Bruhat order $\le$ on $W$.
The Bruhat cell $B^+\dot w\cdot B^+$ is isomorphic
to $\mathbb \mathbb R^{\ell(w)}$. And the Bruhat order has the property
$$
v\le w\ \ \iff \ \
B^+\dot v\cdot B^+\subseteq\overline {B^+\dot w\cdot B^+},
$$
for $v,w\in W$.
It is a well-known consequence of Bruhat decomposition
that $\mathcal B\times\mathcal B$ is the union of the
$G$-orbits $\mathcal O(w)=G\cdot (B^+,\dot w\cdot B^+)$,
with $G$ acting
diagonally. Therefore to any pair $(B_1,B_2)$ of Borel
subgroups one can associate a unique $w\in W$ such that
\begin{equation*}
(B_1,B_2)=(g\cdot B^+,g\dot w \cdot B^+)
\end{equation*}
for some $g\in G$. We write
$$
B_1\overset w\to B_2
$$
in this case and call $w$ the relative position of $B_1$ and $B_2$.
\subsection{}
Finally, let us consider the two opposite Bruhat
decompositions
\begin{equation*}
\mathcal B=\bigsqcup_{w\in W} B^+\dot w\cdot B^+=\bigsqcup_{v\in W}
B^-\dot v\cdot B^+.
\end{equation*}
Note that $B^-\dot v\cdot B^+\cong\mathbb R^{\ell(w_0)-\ell(v)}$. The
closure relations for these opposite Bruhat cells are
given by $B^-\dot v'\cdot B^+\subset
\overline{B^-\dot v\cdot B^+}$ if and only if $v\le v'$.
We define
\begin{equation*}
\mathcal R_{v,w}:=B^+\dot w\cdot B^+\cap B^-\dot v\cdot B^+,
\end{equation*}
the intersection of opposed Bruhat cells. This intersection is empty
unless $v\le w$, in which case it is smooth of dimension
$\ell(w)-\ell(v)$, see \cite{KaLus:Hecke2,Lus:PartFlag}.
\section{Total Positivity for $G$ and $\mathcal B$}\label{s:TotPosGB}
Real projective space $\mathbb{P}^n$ has a natural open subset: the set of
lines spanned by vectors with all coordinates positive. This subset is
called the totally positive part of $\mathbb{P}^n$, and its closure, the set
of lines spanned by vectors with all coordinates non-negative, is called
the totally non-negative part of $\mathbb{P}^n$. These subsets can be defined
more generally \cite{Lus:TotPos94}
for any split semisimple real algebraic group and any
partial flag manifold of such a group.
\subsection{Total positivity in $G$}\label{s:totpos}The totally nonnegative part $G_{\ge 0}$ of $G$ is defined
by Lusztig \cite{Lus:TotPos94} to be the semigroup
inside $G$ generated by the sets
\begin{align*}
&\{x_i(t)\ |\ t\in \mathbb R_{>0}, i\in I\},\\
&\{y_i(t)\ |\ t\in \mathbb R_{>0}, i\in I\}, \text{ and}\\
&T_{>0}:=\{ t\in T\ |\ \text{$\chi(t)>0$ all $\chi\in X^*(T)$}\}.
\end{align*}
When $G=SL_n(\mathbb R)$ then by a theorem of A.~Whitney
this definition agrees with the classical notion
of totally nonnegative matrices inside $SL_n(\mathbb R)$,
that is those matrices all of whose minors are
nonnegative.
We recall some basic facts about
total positivity for $G$ from \cite{Lus:TotPos94}.
Let
$U^+_{\ge 0}:=G_{\ge 0}\cap U^+$ and $U^-_{\ge 0}:=G_{\ge 0}\cap U^-$.
For $w\in W$ and $s_{i_1}\dotsc s_{i_m}=w$ a reduced expression define
\begin{eqnarray*}
U^+(w)&:=&\{x_{i_1}(t_1)x_{i_2}(t_2)\dotsc x_{i_m}(t_m)\ |\ t_i\in\mathbb R_{>0}\},\\
U^-(w)&:=&\{y_{i_1}(t_1)y_{i_2}(t_2)\dotsc y_{i_m}(t_m)\ |\ t_i\in\mathbb R_{>0}\}.
\end{eqnarray*}
These sets are independent of the chosen
reduced expression and give
\begin{eqnarray*}
U^+(w)&=&U^+_{\ge 0}\cap B^-\dot w B^-,\\
U^-(w)&=& U^-_{\ge 0}\cap B^+\dot w B^+.
\end{eqnarray*}
In particular $U^+_{\ge 0}=\bigsqcup_{w\in W} U^+(w)$ and $U^-_{\ge 0}=
\bigsqcup_{w\in W}U^-(w)$. Moreover $U^+(w)$ and $U^-(w)$ are isomorphic to
$\mathbb R_{>0}^{\ell(w)}$ using the $t_i$ as coordinates.
The totally positive parts for $U^+$ and $U^-$
are defined by
\begin{equation*}
U^+_{>0}:=U^+(w_0),\qquad U^-_{>0}:=U^-(w_0).
\end{equation*}
\subsection{Total positivity in $\mathcal B$}\label{s:totposflag}
The totally positive and totally nonnegative parts of the
flag variety $\mathcal B$ are defined by
\begin{eqnarray*}
\mathcal B_{>0}&:=&\{y\cdot B^+\ |\ y\in U^-_{>0}\},\\
\mathcal B_{\ge 0}&:=&\overline{\mathcal B_{>0}}.
\end{eqnarray*}
The set $\mathcal B_{\ge 0}$ has a
decomposition into strata,
\begin{equation*}
\mathcal R^{>0}_{v,w}:=\mathcal R_{v,w}\cap \mathcal B_{\ge 0},
\end{equation*}
where $v\le w$.
These strata were defined and conjectured
to be semi-algebraic cells by Lusztig \cite{Lus:TotPos94},
a result which was later proved in \cite{Rie:CelDec}.
The conjecture was proved again in a different way in \cite{MarRie:ansatz},
this time with explicit parametrizations of the cells
given. We recall these parametrizations
now.
Let $v\le w$ and let $\mathbf w=(i_1,\dotsc, i_m)$ encode a
reduced expression $s_{i_1}\dotsc s_{i_m}$ for $w$. Then there
exists a unique subexpression $s_{i_{j_1}}\dotsc s_{i_{j_k}}$
for $v$ in $\mathbf w$ with the
property that, for $l=1,\dotsc,k$,
\begin{equation*}
s_{i_{j_1}}\dotsc s_{i_{j_{l}}}s_{i_r}>
s_{i_{j_1}}\dotsc s_{i_{j_{l}}} \quad \text{whenever $j_l< r\le j_{l+1}$,}
\end{equation*}
where $j_{k+1}:=m$. This is loosely speaking the rightmost reduced subexpression for $v$ in
$\mathbf w$. It is called the `positive subexpression' in \cite{MarRie:ansatz}, and
we use the notation
\begin{eqnarray*}
\mathbf v_+&:=&\{j_1,\dotsc, j_k\},\\
\mathbf v_+^c&:=&\{1,\dotsc, m\}\setminus \{j_1,j_2,\dotsc, j_k\},
\end{eqnarray*}
when referring to this special subexpression for $v$ in $\mathbf w$.
Now we can define a map
\begin{eqnarray*}
\phi_{\mathbf v_+,\mathbf w}:(\mathbb{C}^*)^{\mathbf v^c_+} &\to& \mathcal R_{v,w},\\
( t_r)_{r\in\mathbf v^c_+}&\mapsto &g_1\dotsc g_m\cdot B^+,
\end{eqnarray*}
where
\begin{equation*}
g_r=\begin{cases}
\dot s_{i_r}, &\text{ if $r\in\mathbf v_+$,}\\
y_{i_r}(t_r) & \text{ if $r\in \mathbf v^c_+$.}
\end{cases}
\end{equation*}
It is shown in \cite{MarRie:ansatz} that the map $\phi_{\mathbf v_+,\mathbf w}$
is an embedding with image
the open `Deodhar stratum' of $\mathcal R_{v,w}$ associated to $\mathbf w$,
see \cite{Deo:Decomp}.
\begin{thm}\cite[Theorem~11.3]{MarRie:ansatz}\label{t:param}
The restriction of $\phi_{\mathbf v_+,\mathbf w}$ to $(\mathbb R_{>0})^{\mathbf v_+^c}$
defines an isomorphism of semi-algebraic sets,
$$
\phi^{>0}_{\mathbf v_+,\mathbf w}: (\mathbb R_{>0})^{\mathbf v_+^c} \to \mathcal R_{v,w}^{>0}.
$$
\end{thm}
\subsection{Changes of coordinates under braid relations} \label{s:BraidRels}
\bigskip
In the simply laced case
there is a simple change of coordinates \cite{Lus:TotPos94, Rie:MSgen}
which describes how
two parameterizations of the same cell are related when considering
two reduced expressions which differ by a commuting relation or
a braid relation.
If $s_i s_j = s_j s_i$ then $y_i(a) y_j(b)=y_j(b) y_i(a)$
and $y_i(a) \dot s_j = \dot s_j y_i(a)$.
If $s_i s_j s_i = s_j s_i s_j$ then
\begin{enumerate}
\item
$y_i(a) y_j(b) y_i(c) = y_j(\frac{bc}{a+c}) y_i(a+c) y_j(\frac{ab}{a+c})$.\\
\item $y_i(a) \dot s_j y_i(b) = y_j(\frac{b}{a}) y_i(a) \dot s_j$\\
\item
$\dot s_j \dot s_i y_j(a)=
y_i(a) \dot s_j \dot s_i$.
\end{enumerate}
The changes of coordinates have also
been computed for more general braid relations
and have been observed to be subtraction-free \cite{BZ,Rie:MSgen}.
\subsection{Closure relations}
We have the following closure relations.
\begin{thm}{\cite{Rie:TotPosPoset}}\label{t:B} Let $v,w\in W$. Then
$$
\overline{\mathcal R_{v,w}^{>0}}=\bigsqcup_{
v\le v'\le w'\le w}
\mathcal R^{>0}_{v',w'}.
$$
\end{thm}
\subsection{Total positivity and canonical bases, for simply laced $G$}\label{s:CanonBasis}
\bigskip
We now assume that $G$ is simply laced. Let $\bf U$ be the enveloping
algebra of the Lie algebra of $G$; this can be defined by generators
$e_i, h_i, f_i$ ($i\in I$) and the Serre relations. For any
dominant character
$\lambda$
there is a finite-dimensional simple $\bf U$-module $V(\lambda)$
with a non-zero vector $\eta$ such that $e_i \eta = 0$
and $h_i \eta = <\lambda,\alpha_i^\vee> \eta$ for all $i \in I$. Moreover, the
pair $(V(\lambda), \eta)$ is determined up to unique isomorphism.
There is a unique $G$-module structure on $V(\lambda)$ such that
for any $i\in I, a\in \mathbb R$ we have that $x_i(a)$ acts by
\begin{equation*}
\exp(a e_i):V(\lambda) \to V(\lambda),
\end{equation*}
and $y_i(a)$ acts by
\begin{equation*}
\exp(a f_i): V(\lambda) \to V(\lambda).
\end{equation*}
Then
$x_i(a)\eta = \eta$ for all $i\in I$, $a\in \mathbb R$, and
$t \eta = \lambda(t) \eta$ for all $t \in T$. Let ${\mathcal B}(\lambda)$
be the canonical basis of $V(\lambda)$ that contains $\eta$.
See \cite{Lus:CanonBasis} for details on the canonical basis.
In relation to $G_{\geq 0}$, the basis
${\mathcal B}(\lambda)$ has the following positivity property
\cite[Prop. 3.2]{Lus:TotPos94}.
\begin{thm} \cite{Lus:TotPos94}
Let $g\in G_{\geq 0}$. Then the
matrix entries of $g: V \to V$ with respect to ${\mathcal B}(\lambda)$ are non-negative
real numbers.
\end{thm}
\begin{rem}\label{rem:positive1}
We remark that it is easy to see that a little more is true.
The
matrix entries of any $x_i(a)$ or
$y_i(a): V \to V$ with respect to ${\mathcal B}(\lambda)$
are given by positive polynomials in $a$.
\end{rem}
The following lemma comes from \cite{Lus:PartFlag}.
\begin{lem}\label{lem:positive2} \cite[1.7(a)]{Lus:PartFlag}.
For any $w\in W$, the vector $\dot w \eta$ is the unique element of
${\mathcal B}(\lambda)$ which lies in the extremal weight space
$V(\lambda)^{w(\lambda)}$. In particular, $\dot w \eta \in {\mathcal B}(\lambda)$.
\end{lem}
\section{Positivity for toric varieties}\label{s:Toric}
In this section we define projective toric varieties and recall
some basic results.
We may define a (generalized) projective toric variety
as follows \cite{Cox, Sottile}.
Let $S=\{\mathbf{m}_i \ \vert \ i=1, \dots, \ell\}$ be any finite subset
of $\mathbb{Z}^n$, where $\mathbb{Z}^n$ can be thought of as the character group
of the torus $(\mathbb{C}^*)^n$.
Here
$\mathbf{m}_i=(m_{i1}, m_{i2},\dots ,m_{in})$.
Then consider
the map $\phi: (\mathbb{C}^*)^n \to \mathbb{P}^{\ell-1}$ such that
$\mathbf{x}=(x_1, \dots , x_n) \mapsto [\mathbf{x^{m_1}}, \dots , \mathbf{x^{m_\ell}}]$.
Here $\mathbf{x^{m_i}}$ denotes $x_1^{m_{i1}} x_2^{m_{i2}} \dots x_n^{m_{in}}$.
We then define the toric variety $X_S$
to be the Zariski closure of the image of this map.
The {\it real part} $X_S(\mathbb R)$ of $X_S$ is defined to be the
intersection of $X_S$ with $\mathbb R\mathbb{P}^{\ell-1}$; the
{\it positive part} $X_S^{>0}$ is defined to be the image of
$(\mathbb R_{>0})^n$ under $\phi$; and the {\it non-negative part}
$X_S^{\geq 0}$ is defined to be the closure (in $X_S(\mathbb R)$) of
$X_S^{>0}$.
Note that $X_S$ is not necessarily a toric variety in the sense of \cite{Fulton}, as
it may not be normal;
however, its normalization is a toric variety in this sense. See \cite{Cox} for more details.
Observe that if $S$ is the set of lattice points in the standard simplex, then
$X_S^{>0}$ and
$X_S^{\geq 0}$ are the totally positive and totally non-negative
parts of real projective space.
Let $P$ be the convex hull of $S$.
The restriction of the moment map is a homeomorphism from $X_S^{\geq 0}$ to $P$
(see \cite[Section 4.2, page 81]{Fulton}
and \cite[Theorem 8.4]{Sottile}).
In particular,
$X_S^{\geq 0}$ is homeomorphic to a closed ball.
The following lemma, proved in \cite{PSW}, will be an important tool in the proof of our
main result.
\begin{lem}\label{important} \cite{PSW}
Suppose we have a map $\Phi: (\mathbb R_{>0})^n \to \mathbb{P}^{N-1}$ given by
\begin{equation*}
(t_1, \dots , t_n) \mapsto [h_1(t_1,\dots,t_n), \dots , h_N(t_1,\dots,t_n)],
\end{equation*}
where the $h_i$'s are Laurent polynomials with positive coefficients. Let $S$ be the
set of all exponent vectors in $\mathbb{Z}^n$ which occur among the (Laurent) monomials
of the $h_i$'s, and let $P$ be the convex hull of the points of $S$.
Then the map $\Phi$ factors through the totally positive part
$X_S^{>0}$, giving a map
$\tau_{>0}: X_S^{>0} \to \mathbb{P}^{N-1}$. Moreover $\tau_{>0}$ extends continuously to the
closure to give a well-defined map
$\tau_{\ge 0}:X_S^{\ge 0} \to \overline{\tau_{>0}(X_{S}^{>0})}$.
\end{lem}
\begin{proof}
Let $S = \{\mathbf{m_1},\dots,\mathbf{m_{\ell}}\}$.
Clearly the map $\Phi$ factors
as the composite map $t=(t_1,\dots,t_n) \mapsto
[\mathbf{t^{m_1}}, \dots , \mathbf{t^{m_\ell}}] \mapsto [h_1(t_1,\dots,t_n),\dots,
h_N(t_1,\dots,t_n)]$,
and the image of $(\mathbb R_{>0})^n$ under the first map is precisely
$X_S^{>0}$.
The second map, which we will call $\tau_{>0}$,
takes a point $[x_1,\dots, x_{\ell}]$ of $X_S^{>0}$ to
$[g_1(x_1,\dots,x_{\ell}), \dots, g_N(x_1,\dots,x_{\ell})]$,
where the $g_i$'s are homogeneous polynomials of degree $1$ with positive coefficients.
By construction, each $x_i$ occurs in at least one of the $g_i$'s.
Since $X_S^{\geq 0}$ is the closure inside $X_S$ of $X_S^{>0}$,
any point $[x_1,\dots,x_{\ell}]$ of $X_S^{\geq 0}$ has all $x_i$'s non-negative;
furthermore, not all of the $x_i$'s are equal to $0$. And now since the $g_i$'s
have positive coefficients and they involve {\it all} of the $x_i$'s, the image of
any point $[x_1,\dots,x_{\ell}]$ of $X_S^{\geq 0}$ under $\tau_{>0}$ is well-defined.
Therefore $\tau_{>0}$ extends continuously to the closure to give a well-defined map
$\tau_{\ge 0}:X_S^{\ge 0} \to \overline{\tau_{>0}(X_{S}^{>0})}$.
\end{proof}
\bigskip
\section{Construction of a toric variety associated to a parametrization of a cell}\label{s:ConstructToric}
We begin by stating a key proposition.
\begin{prop}\label{p:main} Given $G$ we can construct a positivity preserving embedding
$i:G/B\to \mathbb P^N$, for some $N$ with the following property.
For any totally nonnegative cell $\mathcal R_{v,w}^{>0}$ and parameterization
$\phi^{>0}_{\mathbf v_+, \mathbf w}$ as in Theorem~\ref{t:param}, the composition
$$
i\circ\phi^{>0}_{\mathbf v_+, \mathbf w}:(\mathbb R_{>0})^{\mathbf v^c_+}\overset{\sim}\longrightarrow\mathcal R_{v,w}^{>0}\hookrightarrow
\mathbb P^N
$$
takes the form
$$
i \circ\phi^{>0}_{\mathbf v_+, \mathbf w} : \mathbf t= ( t_r)_{r\in\mathbf v^c_+}\mapsto [p_1(\mathbf t),\dotsc,p_{N+1}(\mathbf t)],
$$
where the $p_j$'s are polynomials with nonnegative coefficients.
\end{prop}
This proposition will be proved in the next section.
Now assuming Proposition \ref{p:main} is true, we can prove the following.
\begin{cor} \label{c:glue}
There is a map $\tau_{>0}:X_{\mathbf v_+,\mathbf w}^{>0}\to \mathbb{P}^N$ which extends continuously to
the closure to give a well-defined map
$$
\tau_{\ge 0}:X_{\mathbf v_+,\mathbf w}^{\ge 0}\to \overline{\tau_{>0}(X_{\mathbf v_+,\mathbf w}^{>0})}.
$$
Moreover we have
$$\overline{\tau_{>0}(X_{\mathbf v_+,\mathbf w}^{>0})}=i(\overline{\mathcal R_{v,w}^{>0}})
\overset\sim\longrightarrow \overline{\mathcal R_{v,w}^{>0}}.
$$
The resulting map $X_{\mathbf v_+,\mathbf w}^{\ge 0}\to \overline{\mathcal R_{v,w}^{>0}}$
is surjective, an isomorphism on the strictly positive parts, and takes the boundary of $X_{\mathbf v_+,\mathbf w}^{\ge 0}$ to the boundary of $\overline{\mathcal R_{v,w}^{>0}}$.
\end{cor}
\begin{proof}
Let
$S_{\mathbf v_+, \mathbf w}$ be the set of all exponent vectors in $\mathbb{Z}^{n+1}$
which occur among the monomials of the $p_i$'s, and let
$P_{\mathbf v_+,\mathbf w}$ be the convex hull of the points of $S_{\mathbf v_+, \mathbf w}$.
Let
$X_{\mathbf v_+,\mathbf w}$ be the toric variety associated with
$P_{\mathbf v_+,\mathbf w}$. By Lemma \ref{important},
the map
$i \circ\phi^{>0}_{\mathbf v_+, \mathbf w}$ factors through
$X_{\mathbf v_+,\mathbf w}^{> 0}$,
\begin{equation*}
(\mathbb R_{>0})^{\mathbf v^c_+}\overset{\sim}\longrightarrow X_{\mathbf v_+,\mathbf w}^{> 0}\longrightarrow\mathcal R_{v,w}^{>0}\hookrightarrow
\mathbb P^N,
\end{equation*}
and
we get a
map
$\tau_{>0}$ from
$X_{\mathbf v_+,\mathbf w}^{> 0}$ to $\mathbb{P}^N$. Moreover, this map extends
continuously to a map $\tau_{\geq 0}$ from
$X_{\mathbf v_+,\mathbf w}^{\geq 0}$ to
$\overline{\tau_{>0}(X_{\mathbf v_+,\mathbf w}^{>0})}$.
Since the map
${i \circ\phi^{>0}_{\mathbf v_+, \mathbf w}}$ is a homeomorphism onto its image
and it factors through
$X_{\mathbf v_+,\mathbf w}^{> 0}$ as above,
the map $\tau_{\geq 0}$ restricts to a homeomorphism from
$X_{\mathbf v_+,\mathbf w}^{> 0}$ to
$i({\mathcal R_{v,w}^{>0}})$.
We now claim that that $\tau_{\geq 0}$
takes the boundary of $X_{\mathbf v_+,\mathbf w}^{\ge 0}$
to the boundary of $\overline{i(\mathcal R_{v,w}^{>0})}$.
To prove the claim, suppose that there is a point $x \in {\operatorname{bd}}(X_{\mathbf v_+,\mathbf w}^{\ge 0})$ such that
$\tau_{\geq 0}(x)=y$ is in the interior of $\overline{i(\mathcal R_{v,w}^{>0})}$. Since $x$ is in the boundary of
$X_{\mathbf v_+,\mathbf w}^{\ge 0}$ we can find
a sequence of points $\{x_i\}$ in $X_{\mathbf v_+,\mathbf w}^{> 0}$ which converge to $x$. Let
$y_i=\tau_{\geq 0}(x_i)$.
Since $\tau_{\geq 0}$ is a homeomorphism on the interior, each $y_i$ is in $i({\mathcal R_{v,w}^{>0}})$,
and since
$\tau_{\geq 0}$ is continuous, the sequence of points $y_i$ converges to $y$.
Let $g:i({\mathcal R_{v,w}^{>0}}) \overset \sim\to X_{\mathbf v_+,\mathbf w}^{> 0}$ denote
the inverse of the restriction of
$\tau_{\geq 0}$ to the interiors. We now have that
$g$ maps the
convergent sequence $\{y_i\}$ in $i({\mathcal R_{v,w}^{>0}})$ to the
divergent sequence $\{x_i\}$ in $X_{\mathbf v_+,\mathbf w}^{> 0}$, which
contradicts the continuity
of $g$.
\end{proof}
Since $X_{\mathbf v_+,\mathbf w}^{\ge 0}$ is homeomorphic to a closed ball the above corollary
provides us with a glueing map for $\mathcal R_{v,w}^{>0}$.
\bigskip
\section{proof of Proposition \ref{p:main}}\label{s:proof}
In this section we will prove Proposition \ref{p:main}.
\subsection{Simply laced case}\label{s:simplylaced}
We suppose $G$ is simply laced. Consider the
representation
$V=V(\rho)$ of $G$ with
a fixed highest weight vector $\eta$ and
corresponding canonical basis $\mathcal B(\rho)$.
Let $i:\mathcal B\to \mathbb P(V)$ denote the embedding
which takes $g\cdot B_+\in \mathcal B$ to the line
$\left <g\cdot \eta\right >$.
This is the unique $g \cdot B_+$-stable line in $V$.
We specify points in the projective space $\mathbb P(V)$
using homogeneous coordinates corresponding to
$\mathcal B(\rho)$.
Now let $\mathbf w_0=(i_1,\dotsc, i_N)$ be a fixed reduced expression of $w_0$.
\begin{lem}\label{l:w0} Let $v\in W$, and let $\mathbf v_+$ be the positive subexpression for
$v$ in $\mathbf w_0$. The composition
$$
i\circ \phi_{\mathbf v_+,\mathbf w_0}^{>0}: (\mathbb R_{>0})^{\mathbf v_+^c}\to \mathcal R_{v,w_0}^{>0} \to
\mathbb P(V)
$$
is given by polynomials with positive coefficients.
\end{lem}
\begin{proof}
Consider first a reduced expression of $w_0$ which ends in $v$, i.e.\
consider a reduced expression $s_{i_1} \dots s_{i_k} s_{i_{k+1}} \dots
s_{i_m}$ of $w_0$, where $s_{i_{k+1}} \dots s_{i_m}$
is a reduced expression of $v$ (which is clearly positive distinguished).
Let $\dot v = \dot s_{i_{k+1}} \dots \dot s_{i_m}$. Then
$i\circ \phi_{\mathbf v_+,\mathbf w_0}^{>0}$
maps $(a_1, \dots , a_k)$ to
$\left <y_{i_1}(a_1) \dots y_{i_k}(a_k) \dot v \cdot \eta\right >$.
By Lemma \ref{lem:positive2}, $\dot v \cdot \eta$ is a canonical basis vector,
and so by Remark \ref{rem:positive1}, the coefficients which express
$\left <y_{i_1}(a_1) \dots y_{i_k}(a_k) \dot v \cdot \eta\right >$
in terms of the canonical basis are positive polynomials in the $a_i$'s.
Now let the reduced expression for $w_0$ be arbitrary.
This reduced expression can be obtained from our previous
reduced expression $s_{i_1} \dots s_{i_k} s_{i_{k+1}} \dots
s_{i_m}$ using braid relations,
so the results in Section~\ref{s:BraidRels} imply that
the parameterization in question will change by a sequence of
substitutions which involve only positive subtraction-free
rational expressions. It follows that the
image of the map $i\circ \phi_{\mathbf v_+,\mathbf w_0}^{>0}$
is given by rational expressions with positive coefficients.
Since we are in projective space, we can clear denominators to
obtain polynomials with positive coefficients.
\end{proof}
This proves Proposition~\ref{p:main} for simply laced $G$
in the case where $w=w_0$.
We now turn to the case of arbitrary $w$.
\begin{prop}\label{p:mainsimplylaced}
Proposition~\ref{p:main} holds when $G$ is simply laced.
\end{prop}
\begin{proof}
Choose $v,w\in W$ with $v \leq w$.
Choose a reduced expression $\mathbf w_0 = (i_1, \dots , i_N)$
for $w_0$ such that $(i_1, \dots , i_r)$ is a reduced expression
for $w_0 w^{-1}$. Then as in the proof of Lemma 4.3 in
\cite{Rie:TotPosPoset}, we have
\begin{equation*}
\mathcal R^{>0}_{v,w_0} =
\{y_{i_1}(t_1) \dots y_{i_r}(t_r) g_{r+1} \dots g_N \cdot B^+ \ \vert \
g_{r+1} \dots g_N \cdot B^+ \in
\mathcal R^{>0}_{v,w} \}.
\end{equation*}
Here each $g_{r+j}$ is either
$\dot s_{i_{r+j}}$ or $y_{i_{r+j}}(t_{r+j})$ and the $g$'s give
a parameterization of
$\mathcal R^{>0}_{v,w} $.
It's clear that our parameterization
of
$\mathcal R^{>0}_{v,w}$ is obtained from the above parameterization of
$\mathcal R^{>0}_{v,w_0}$ by setting $t_1, \dots t_r$ to $0$.
Since setting certain variables to zero in a positive polynomial
results in another positive polynomial,
Proposition~\ref{p:main} in the simply laced case now follows from Lemma \ref{l:w0}.
\end{proof}
\subsection{General type case}
\bigskip
Let $\dot G$ be the simply laced group with automorphism
corresponding to $G$, as introduced in Section~\ref{s:automorphism}.
Note that we have already proved Proposition~\ref{p:main} for $\dot G$.
Explicitly, we considered in Section~\ref{s:simplylaced} the projective
space determined by
the $\dot\rho$-representation of $\dot G$, that is, $\mathbb P(V(\dot \rho))$,
with homogeneous coordinates determined by the canonical basis $\mathcal B(\dot \rho )$.
We have seen that for any
totally nonnegative cell $\dot {\mathcal R}^{>0}_{\dot v,\dot w}$ with parameterization
$\phi_{\dot{\mathbf v}_+,\dot{\mathbf w}}$, the composition $i\circ\phi_{\mathbf v_+,\mathbf w}$
of this parameterization with the usual embedding,
$i:\dot G/\dot B^+\to\mathbb P(V(\dot \rho))$, is given by polynomials
with positive coefficients.
To treat the general case, we now identify $G$ with $\dot G^\tau$ and use
all of the notation from Section~\ref{s:automorphism}. For
any $\bar i\in \bar I$
there is a simple reflection $s_{\bar i}$ in $W$, which is
represented in $\dot G$ by
$$
s_{\bar i}:=\prod_{i\in\bar i}\dot s_i.
$$
In this way any reduced expression $\mathbf w=(\bar i_1,\bar i_2,\dotsc, \bar i_m)$ in $W$ gives rise to a reduced
expression $\dot{\mathbf w}$ in $\dot W$ of length
$\sum_{k=1}^m |\bar i_k|$, which is determined
uniquely up to commuting elements \cite[Prop. 3.3]{Nanba}.
To a subexpression $\mathbf v$ of $\mathbf w$
we can then associate a unique subexpression
$\dot {\mathbf v}$ of $\dot{\mathbf w}$ in
the obvious way.
\begin{lem}\label{l:folding} Let $\mathbf w$ be a reduced expression for $w$
in $W$ and $v\le w$.
If $\mathbf v_+$ is the positive subexpression for $v$
in $\mathbf w$, then $\mathbf{\dot{v}_+}$ is the positive subexpression for $v$ (now viewed as element of $\dot W$) in $\dot{\mathbf w}$.
\end{lem}
\begin{proof}
Let $\mathbf w =
(\bar i_1,\bar i_2,\dotsc, \bar i_m)$ be a reduced expression
for $w$ in $W$ and let
$v_{(j)}$ denote the product (in order) of all simple generators
of $\mathbf v_+$ which come from $(\bar i_1, \bar i_2,\dotsc,\bar i_j)$.
The fact that $\mathbf v_+$ is positive in $W$ means that
$v_{(j-1)} < v_{(j-1)} s_{\bar i_j}$ for all $j=1, \dots,n$.
In particular,
$\ell (v_{(j-1)} s_{\bar i_j}) = \ell(v_{(j-1)})+1$ in $W$.
For the remainder of this proof let $\dot{v}_{(j-1)}$ denote the element
$v_{(j-1)}$ viewed as an element of $\dot W$.
Then by the relationship of lengths in $W$ versus $\dot W$,
$\ell(\dot{v}_{(j-1)} \prod_{i_j \in \bar{i_j}} \dot{s}_{i_j})
= \ell(\dot{v}_{(j-1)}) + |\bar{i_j}|$ in $\dot W$.
This fact together with the fact that the $\dot{s}_{i_j}$'s
for $i_j \in \bar{i_j}$ commute with each other implies
that for any $i_j \in \bar{i_j}$,
$\ell(\dot{v}_{(j-1)} \dot{s}_{i_j}) = \ell(\dot{v}_{(j-1)})+1$
in $\dot W$.
Therefore for any $i_j \in \bar{i_j}$,
$\dot{v}_{(j-1)} \dot{s}_{i_j} > \dot{v}_{(j-1)}$.
Letting $\bar{i_j}=\{i_{j_1},\dots,i_{j_r}\}$, this now shows that
$\dot{v}_{(j-1)} \dot{s}_{i_{j_1}} \dots \dot{s}_{i_{j_{k+1}}} >
\dot{v}_{(j-1)} \dot{s}_{i_{j_1}} \dots \dot{s}_{i_{j_k}}$ for
all $0 \leq k \leq r-1$ (note that we are again using
the commutativity of the simple generators coming from $\bar{i_j}$). A
little thought now
shows that
$\mathbf{\dot{v}_+}$ is positive in $\mathbf{ \dot{w}}$.
\end{proof}
\begin{lem}\label{l:folding2} Let $v,w$ be in $W$ with $v\le w$.
\begin{enumerate}
\item
We have
$$
\mathcal R_{v,w}^{>0}=\dot{\mathcal R}_{v,w}^{>0}\cap \mathcal B^{\tau}.
$$
In particular the composition $i':\mathcal R_{v,w}\hookrightarrow \dot{\mathcal R}_{v,w}
\to \mathbb P(V(\dot\rho))$ is positivity preserving.
\item
Suppose $\mathbf w=(\bar i_1,\dotsc, \bar i_m)$ is a reduced expression for $w$ in $W$, and $\mathbf v_+=(j_1,\dotsc, j_k)$ is the positive subexpression for $v$. Then we have a commutative diagram,
\begin{equation*}
\begin{CD}
\mathcal R_{v,w}^{>0}& @>\iota>> &\dot{\mathcal R}_{v,w}^{>0}\\
@A\phi^{>0}_{\mathbf v_+,\mathbf w}AA & & @AA\phi^{>0}_{\dot{\mathbf v}_+,\dot{\mathbf w}}A\\
\mathbb R_{>0}^{\mathbf v_+^c} &@>\bar \iota>> & \mathbb R_{>0}^{\dot {\mathbf v}_+^c},
\end{CD}
\end{equation*}
where the top arrow is the usual inclusion, the vertical arrows are both isomorphisms,
and the map $\iota$ has the form
$$(t_1,\dotsc, t_k)\mapsto (t_1,\dotsc, t_1, t_2,\dotsc, t_2,\dotsc, t_k),$$
where each $t_l$ is repeated $|\bar i_{j_l}|$ times on the right hand side.
\end{enumerate}
\end{lem}
\begin{proof}
(1) We have $\mathcal B_{\ge 0}=\dot{\mathcal B}_{\ge 0}\cap \dot{\mathcal B}^{\tau}$ by \cite{Lus:TotPos94}.
Clearly $\mathcal R_{v,w}^{>0}\subset \dot{\mathcal R}_{v,w}^{>0}$. However since
$\mathcal B_{\ge 0}= \bigsqcup_{v,w\in W} \mathcal R_{v,w}^{>0}=\dot{\mathcal B}_{\ge 0}\cap \dot{\mathcal B}^{\tau}$ it follows that
$\mathcal R_{v,w}^{>0}= \dot{\mathcal R}_{v,w}^{>0}\cap\dot{\mathcal B}^{\tau}$.
(2) This is a consequence of Lemma~\ref{l:folding}.
\end{proof}
We can now combine the two parts of Lemma~\ref{l:folding2} with
Proposition~\ref{p:mainsimplylaced}
to prove Proposition~\ref{p:main} for general type $G$.
\begin{proof}[Proof of Proposition~\ref{p:main}]
Firstly, Proposition~\ref{p:mainsimplylaced} gives the map
$$
i\circ\phi^{>0}_{\dot{\mathbf v}_+,\dot{\mathbf w}}: \mathbb R_{>0}^{\dot {\mathbf v}_+^c}\to
\dot{\mathcal R}^{>0}_{v,w}\overset i\hookrightarrow\mathbb P(V(\dot\rho)).
$$
Secondly, for non simply laced $G$ we will use
the inclusion to projective space
given by
$i':{\mathcal R}_{v,w}\hookrightarrow\mathbb P(V(\dot\rho))$, which is
positivity preserving by Lemma~\ref{l:folding2} (1).
And thirdly, by Lemma~\ref{l:folding2} (2), we have that
$$
i'\circ\phi^{>0}_{{\mathbf v}_+,{\mathbf w}}: \mathbb R_{>0}^{ {\mathbf v}_+^c}\to
{\mathcal R}^{>0}_{v,w}\overset {i'}\hookrightarrow\mathbb P(V(\dot\rho))
$$
can be rewritten as
$$i'\circ\phi^{>0}_{{\mathbf v}_+,{\mathbf w}}= i\circ\iota\circ\phi^{>0}_{{\mathbf v}_+,{\mathbf w}}
=(i\circ\phi^{>0}_{\dot{\mathbf v}_+,\dot{\mathbf w}})\circ\bar\iota.
$$
By Proposition~\ref{p:mainsimplylaced} for $\dot G$
and the description of $\bar\iota$ in Lemma~\ref{l:folding2} (2)
we see that $i'\circ\phi^{>0}_{{\mathbf v}_+,{\mathbf w}}$ is given by positive
polynomials. This proves Proposition~\ref{p:main} for $G$.
\end{proof}
\bigskip
\section{Generalization to partial flag varieties}\label{s:Partial}
In this section we generalize the previous results
to partial flag varieties.
\subsection{Lusztig's decomposition of $\mathcal P^J$}\label{s:strata}
The stratification of $\mathcal B$ into smooth pieces $\mathcal R_{v,w}$ has
an analogue for partial flag varieties introduced by Lusztig in
\cite{Lus:PartFlag}.
Consider a triple of Weyl group elements $x,u,w\in W$ with $x\in W^J_{max}$,
$w\in W^J$ and $u\in W_J$. Then
$\mathcal P^J_{x,u,w}\subset \mathcal P^J$ is defined as the
set of all $P\in \mathcal P^J$ such that there exist Borel subgroups
$B_L$ and $B_R$ inside $P$ satisfying
\begin{equation*}
B^+\overset w\longrightarrow B_L\overset u\longrightarrow B_R\overset {x^{-1} w_0}\longrightarrow B^-.
\end{equation*}
An equivalent characterization of $\mathcal P^J_{x,u,w}$ is
\begin{equation*}
\mathcal P^J_{x,u,w}=\pi^J(\mathcal R_{x,w u})=\pi^J(\mathcal R_{x u^{-1}, w}).
\end{equation*}
It is not hard to see that $B_L$ and $B_R$ are uniquely determined
as the Borel subgroups in $P$ `closest to' $B^+$ respectively $B^-$
with regard to their relative position, and the projection maps
$\mathcal R_{x,wu}\to \mathcal P^J_{x,u,v}$ and $\mathcal R_{x u^{-1},w}\to
\mathcal P^J_{x,u,v}$ are isomorphisms.
In particular $\mathcal P^J_{x,u,w}$ is nonempty if and only
if $x\le w u$, in which case it is smooth of dimension
$\ell(w)+\ell(u)-\ell(x)$.
Let us denote the indexing set for this decomposition of $\mathcal P^J$
by $Q^J$. So
\begin{equation*}
Q^J:=\{(x,u,w)\in W^J_{max}\times W_J\times W^J\ |\ x\le wu \}.
\end{equation*}
\subsection{Totally nonnegative cells in
$\mathcal P^J$}\label{s:P}
The totally positive and nonnegative parts of
$\mathcal P^J$ are defined in
\cite{Lus:IntroTotPos} by
\begin{eqnarray*}
\mathcal P^J_{>0}&=&\pi^J(\mathcal B_{>0}),\\
\mathcal P^J_{\ge 0}&=&\pi^J(\mathcal B_{\ge 0}).
\end{eqnarray*}
Since $\pi^J$ is closed it follows that $\mathcal P^J_{\ge 0}=
\overline{\mathcal P^J_{>0}}$.
We decompose $\mathcal P^J_{\ge 0}$ by intersecting it with the
strata $\mathcal P^J_{x,u,w}$ from Section~\ref{s:strata}.
From the definitions and the fact that reduction preserves total
positivity it follows that (\cite[Lemma 3.2]{Rie:PhD})
\begin{equation*}
\mathcal P^J_{x,u,w;>0}:=\mathcal P^J_{x,u,w}\cap\mathcal P^J_{\ge 0}
=\pi^J(\mathcal R_{x,wu}^{>0})=\pi^J(\mathcal R_{x u^{-1}, w}^{>0}).
\end{equation*}
Keeping in mind that
$\pi^J:\mathcal R_{x,wu}\to\mathcal P^J_{x,u,w}$, say,
is an isomorphism, we see that
\begin{equation*}
\mathcal P^{J}_{x,u,w;>0}\cong \mathcal R^{>0}_{x,wu}\cong
\mathcal \mathbb R_{>0}^{\ell(w)+\ell(u)-\ell(x)},
\end{equation*}
for any triple $(x,u,w)\in Q^J$.
\begin{thm}\label{t:P}\cite{Rie:TotPosPoset} Let $(x,u,w)\in Q^J$.
Then
$\overline{\mathcal P^J_{x,u,w;>0}}$ is the disjoint union of
${\mathcal P^J_{x,u,w;>0}}$ and some lower-dimensional cells.
\end{thm}
\subsection{Proof of Theorem \ref{th:main}}
Finally we are ready to prove Theorem \ref{th:main}.
Let us recall the notion of a CW complex.
In a CW complex $X$, a
cell is {\it attached} by
glueing a closed $i$-dimensional ball $D^i$ to the $(i-1)$-skeleton
$X_{i-1}$, i.e.\ the union of all lower dimensional cells.
The glueing is specified by a continuous function $f$ from
$\partial D^i = S^{i-1}$ to $X_{i-1}$. CW complexes are defined
inductively as follows. Given $X_0$ a discrete space
(a discrete union of $0$-cells), we inductively construct
$X_i$ from $X_{i-1}$ by attaching some collection of $i$-cells. The
resulting colimit space $X$ is called a {\it CW complex} provided
it is given the weak topology and the {\it closure-finite} condition
is satisfied for its closed cells. Recall that the closure-finite
condition requires that every closed cell is covered by a finite
union of open cells.
\begin{proof}
The cell decomposition of
$\mathcal P^J_{\ge 0}$ has
only finitely many cells; therefore
the closure-finite condition in the definition of a CW complex
is automatically satisfied.
What we need to do is define the attaching maps for the cells.
Recall from Corollary \ref{c:glue} that for each parameterization of
a cell
${\mathcal R_{v,w}^{>0}}$ of $\mathcal B_{\geq 0}$
we have a toric variety
$X_{\mathbf v_+,\mathbf w}$ and
a map $\tau_{\geq 0}: X_{\mathbf v_+,\mathbf w}^{\ge 0}\to \overline{\mathcal R_{v,w}^{>0}}$
which is surjective,
an isomorphism on the strictly positive parts,
and which takes the boundary of $X_{\mathbf v,\mathbf w}^{\ge 0}$ to the boundary of $\overline{\mathcal R_{v,w}^{>0}}$. Since
$X_{\mathbf v_+,\mathbf w}^{\ge 0}$
is homeomorphic to a closed ball, this provides a glueing map for the cell
${\mathcal R_{v,w}^{>0}}$
of $\mathcal B_{\geq 0}$.
To construct the glueing map for
the cell $P^J_{x,u,w;>0}$ of
$\mathcal P^J_{\ge 0}$, we compose the map $\pi^J$ from
$\mathcal B_{\ge 0}$ to
$\mathcal P^J_{\ge 0}$ with $\tau_{\geq 0}$.
Since
$\pi^J:\mathcal R_{x,wu}^{>0}\to\mathcal P^J_{x,u,w;>0}$ is a homeomorphism,
we obtain a map
$\pi^J \circ \tau_{\geq 0} $ from
$X_{\mathbf x_+,\mathbf {wu}}^{\ge 0}\to \overline{\mathcal P^J_{x,u,w;>0}}$
which again is surjective and
an isomorphism on the strictly positive parts.
Now as in the proof of Corollary \ref{c:glue}, it must take
the boundary of
$X_{\mathbf x_+,\mathbf wu}^{\ge 0}$ to the boundary of
$\overline{\mathcal P^J_{x,u,w;>0}}$.
Therefore
the composition $\tau_{\geq 0} \circ \pi^J$ provides a glueing
map for cells of
$\mathcal P^J_{\ge 0}$.
The only thing that remains to check is that the boundary of each
$i$-cell is mapped to the $i-1$-skeleton. But this follows from
Theorem \ref{t:P}.
\end{proof}
As explained in the Introduction, Theorem \ref{th:main} together with
Theorem \ref{th:Eulerian} from \cite{Williams:poset} implies the following
result.
\begin{cor}
The Euler characteristic of the closure of a cell of
$(G/P)_{\ge 0}$ is $1$.
\end{cor}
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3,212,635,537,476 | arxiv |
\section*{Acknowledgements}
This research was supported by the NSF (CRII-1656998), Schmidt Sciences, and cloud computing credits from Amazon.
We thank John Langford and Dipendra Misra for helpful and insightful discussions with regards to our learning algorithm. We also thank the anonymous reviewers for their helpful comments.
\section{Analysis}
\label{sec:analysis}
\begin{table}
\begin{footnotesize}
\begin{center}
\begin{tabular}{|p{2cm}|c|c|c|} \hline
Class & \textsc{Alchemy} & \textsc{Scene} & \textsc{Tangrams} \\ \hline
multi-turn ref. & & 5 & 13 \\ \hline
impossible multi-turn ref. & & 5 & 13 \\ \hline
unclear label & & 19 & 12 \\ \hline
in-turn ref. & & 13 & 7 \\ \hline
misunderstands task & & 7 & 3 \\ \hline
\end{tabular}
\end{center}
\end{footnotesize}
\caption{Counts of common errors in the three domains. }
\label{tab:analysis}
\end{table}
\paragraph{Error Analysis}
We analyze the performance of our best models with respect to errors that occur on an instruction-level, i.e. when the agent is provided with the correct initial state.
We observe \as{fifty} examples in each domain.
Table~\ref{tab:analysis} shows the counts of major error categories across the domains.
We consider errors with resolving references both from previous utterances (multi-turn) and in the environment (in-turn).
Some multi-turn reference resolution errors are due to the model not having access to previous world states.
For example, in \textsc{Tangrams}, the instruction \nlstring{put it back in the same place} refers to a previously-removed item.
Because the agent only has access to the world state after following this instruction, it does not observe what kind of item was previously removed, and therefore cannot correctly identify which item it should add back.
Additionally, we find that a significant number of errors are due to ambiguous or incorrect instructions.
For example, the \textsc{Scene} instruction \nlstring{person in green appears on the right end} is ambiguous.
In the data, it is interpreted as \nlstring{person in green} referring to a person already in the environment, who moves to the 10th position.
However, it can also be interpreted as a new person in green appearing in the 10th position.
\begin{figure*}[t]
\fbox{
\centering
\begin{minipage}{0.45\linewidth}
\includegraphics[width=\linewidth,clip,trim=65 75 65 20,page=1, right]{figs/attention} \\
\end{minipage}
\begin{minipage}{0.45\linewidth}\vspace{9mm}
\includegraphics[width=0.6\linewidth,clip,trim=70 125 550 125,page=2, left]{figs/attention} \\
\end{minipage}}
\caption{Attention}
\label{fig:attention}
\end{figure*}
\paragraph{Attention Analysis}
Figure~\ref{fig:attention} shows attention distributions for an example in \textsc{Alchemy}.
The supplementary material contains more examples of attention.
\section{Discussion}
\label{sec:discuss}
We propose a model to reason about context-dependent instructional language that display strong dependencies both on the history of the interaction and the state of the world.
Future modeling work may include using intermediate world states from previous turns in the interaction, which is required for some of the most complex references in the data.
We propose to train our model using SESTRA, a learning algorithm that takes advantage of single-step reward observations to overcome learned biases in on-policy learning.
Our learning approach requires additional reward observations in comparison to conventional reinforcement learning.
However, it is particularly suitable to recovering from biases acquired early during learning, for example due to biased action spaces, which is likely to lead to incorrect blame assignment in neural network policies.
When the domain and model are less susceptible to such biases, the benefit of the additional reward observations is less pronounced.
One possible direction for future work is to use an estimator to predict rewards for all actions, rather than observing them.
\section{Experimental Setup}
\label{sec:exp}
\paragraph{Evaluation}
Following \citet{Long:16context}, we evaluate task completion accuracy using exact match between the final state and the annotated goal state.
We report accuracy for complete interactions (5utts), the first three utterances of each interaction (3utts), and single instructions (Inst). For single instructions, execution starts from the annotated start state of the instruction.
\paragraph{Systems}
We report performance of ablations and two baseline systems: \textsc{PolicyGradient}: policy gradient with cumulative episodic reward without a baseline, and \textsc{ContextualBandit}: the contextual bandit approach of \citet{Misra:17instructions}. Both systems use the reward with the shaping term and our model.
We also report supervised learning results (\textsc{Supervised}) by heuristically generating correct executions and computing maximum-likelihood estimate using context-action demonstration pairs. Only the supervised approach uses the heuristically generated labels.
Although the results are not comparable, we also report the performance of previous approaches to SCONE.
All three approaches generate logical representations based on lambda calculus.
In contrast to our approach, this requires an ontology of hand built symbols and rules to evaluate the logical forms.
\citet{Fried:17} uses supervised learning with annotated logical forms.
\paragraph{Training Details}
For test results, we run each experiment five times and report results for the model with best validation interaction accuracy.
For ablations, we do the same with three experiments.
We use a batch size of $20$.
We stop training using a validation set sampled from the training data.
We hold the validation set constant for each domain for all experiments.
We use patience over the average reward, and select the best model using interaction-level (5utts) validation accuracy.
We tune $\lambda$, $\delta$, and $M$ on the development set.
The selected values and other implementation details are described in the Supplementary Material.
\section{Introduction}
\label{sec:intro}
An agent executing a sequence of instructions must address multiple challenges, including grounding the language to its observed environment, reasoning about discourse dependencies, and generating actions to complete high-level goals.
For example, consider the environment and instructions in Figure~\ref{fig:example}, in which a user describes moving chemicals between beakers and mixing chemicals together.
To execute the second instruction, the agent needs to resolve \nlstring{sixth beaker} and \nlstring{last one} to objects in the environment.
The third instruction requires resolving \nlstring{it} to the rightmost beaker mentioned in the second instruction, and reasoning about the set of actions required to mix the colors in the beaker to brown.
In this paper, we describe a model and learning approach to map sequences of instructions to actions.
Our model considers previous utterances and the world state to select actions, learns to combine simple actions to achieve complex goals, and can be trained using goal states without access to demonstrations.
\begin{figure}[t]
\fbox{
\centering
\begin{minipage}{0.95\linewidth}
\begin{center}
\includegraphics[width=0.7\linewidth,clip,trim=112 425 567 167]{figs/start-goal} \\
\end{center}
\vspace{-8pt}
\footnotesize
\nlstring{throw out first beaker} \\[1pt]
$\act{pop~1}, \act{stop}$\\[3pt]
\nlstring{pour sixth beaker into last one} \\[1pt]
$\act{pop~6}, \act{pop~6}, \act{push~7~o}, \act{push~7~o}, \act{stop}$ \\[3pt]
\nlstring{it turns brown} \\[1pt]
$\act{pop~7}, \act{pop~7}, \act{pop~7}, \act{push~7~b}, \act{push~7~b}, \act{push~7~b}, \act{stop}$ \\[3pt]
\nlstring{pour purple beaker into yellow one} \\[1pt]
$\act{pop~3}, \act{push~5~p}, \act{stop}$\\[3pt]
\nlstring{throw out two units of brown one} \\[1pt]
$\act{pop~7}, \act{pop~7}, \act{stop}$
\begin{center}
\vspace{-6pt}
\includegraphics[width=0.7\linewidth,clip,trim=112 371 567 211]{figs/start-goal} \\
\end{center}
\end{minipage}}
\vspace{-5pt}
\caption{Example from the SCONE~\cite{Long:16context} \textsc{Alchemy} domain, including a start state (top), sequence of instructions, and a goal state (bottom). Each instruction is annotated with a sequence of actions from the set of actions we define for \textsc{Alchemy}.}
\label{fig:example}
\vspace{-10pt}
\end{figure}
The majority of work on executing sequences of instructions focuses on mapping instructions to high-level formal representations, which are then evaluated to generate actions~\cite[e.g.,][]{Chen:11,Long:16context}.
For example, the third instruction in Figure~\ref{fig:example} will be mapped to $\const{mix}(\const{prev\_arg1})$, indicating that the mix action should be applied to first argument of the previous action~\cite{Long:16context,Guu:17rl-mml}.
In contrast, we focus on directly generating the sequence of actions.
This requires resolving references without explicitly modeling them, and learning the sequences of actions required to complete high-level actions; for example, that mixing requires removing everything in the beaker and replacing with the same number of brown items.
A key challenge in executing sequences of instructions is considering contextual cues from both the history of the interaction and the state of the world.
Instructions often refer to previously mentioned objects (e.g., \nlstring{it} in Figure~\ref{fig:example}) or actions (e.g., \nlstring{do it again}).
The world state provides the set of objects the instruction may refer to, and implicitly determines the available actions. For example, liquid can not be removed from an empty beaker.
Both types of contexts continuously change during an interaction. As new instructions are given, the instruction history expands, and as the agent acts the world state changes.
We propose an attention-based model that takes as input the current instruction, previous instructions, the initial world state, and the current state.
At each step, the model computes attention encodings of the different inputs, and predicts the next action to execute.
We train the model given instructions paired with start and goal states without access to the correct sequence of actions.
During training, the agent learns from rewards received through exploring the environment with the learned policy by mapping instructions to sequences of actions.
In practice, the agent learns to execute instructions gradually, slowly correctly predicting prefixes of the correct sequences of increasing length as learning progress.
A key challenge is learning to correctly select actions that are only required later in execution sequences.
Early during learning, these actions receive negative updates, and the agent learns to assign them low probabilities.
This results in an exploration problem in later stages, where actions that are only required later are not sampled during exploration.
For example, in the \textsc{Alchemy} domain shown in Figure~\ref{fig:example}, the agent behavior early during execution of instructions can be accomplished by only using $\act{pop}$ actions.
As a result, the agent quickly learns a strong bias against $\act{push}$ actions, which in practice prevents the policy from exploring them again.
We address this with a learning algorithm that observes the reward for all possible actions for each visited state, and maximizes the immediate expected reward.
We evaluate our approach on SCONE~\cite{Long:16context}, which includes three domains, and is used to study recovering predicate logic meaning representations for sequential instructions.
We study the problem of generating a sequence of low-level actions, and re-define the set of actions for each domain.
For example, we treat the beakers in the \textsc{Alchemy} domain as stacks and use only $\act{pop}$ and $\act{push}$ actions.
Our approach robustly learns to execute sequential instructions with up to $89.1\%$ task-completion accuracy for single instruction, and $62.7\%$ for complete sequences.
Our code is available at \mbox{\url{https://github.com/clic-lab/scone}}.
\section{Learning}
\label{sec:learning}
We estimate the policy parameters $\theta$ using an exploration-based learning algorithm that maximizes the immediate expected reward.
Broadly speaking, during learning, we observe the agent behavior given the current policy, and for each visited state compute the expected immediate reward by observing rewards for all actions.
We assume access to a set of training examples $\{(\bar{\inputtoken}^{(j)}_i, s_{i, 1}^{(j)}, \langle \bar{\inputtoken}^{(j)}_1, \dots, \bar{\inputtoken}^{(j)}_{i-1} \rangle, g^{(j)}_i) \}_{j=1,i=1}^{N,n^{(j)}}$, where each instruction $\bar{\inputtoken}^{(j)}_i$ is paired with a start state $s_{i, 1}^{(j)}$, the previous instructions in the sequence $\langle \bar{\inputtoken}^{(j)}_1, \dots, \bar{\inputtoken}^{(j)}_{i-1} \rangle$, and a goal state $g^{(j)}_i$.
\begin{figure}[t!]
\begin{tabular}{@{}p{7cm}}
\vspace{-0.275in}
\begin{algorithm}[H]
\caption{SESTRA: \textbf{S}ingl\textbf{e}-\textbf{st}ep \textbf{R}eward Observ\textbf{a}tion.}\label{alg:learning}
\begin{algorithmic}[1]
\footnotesize
\Require Training data $\{(\bar{\inputtoken}^{(j)}_i, s_{i, 1}^{(j)}, \langle \bar{\inputtoken}^{(j)}_1, \dots, \bar{\inputtoken}^{(j)}_{i-1} \rangle, $ $g^{(j)}_i) \}_{j=1,i=1}^{N,n^{(j)}}$, learning rate $\mu$, entropy regularization coefficient $\lambda$, episode limit horizon $M$.
\Definitions $\pi_\theta$ is a policy parameterized by $\theta$, $\act{BEG}$ is a special action to use for the first decoder step, and $\act{STOP}$ indicates end of an execution. $T(s, a)$ is the state transition function, $H$ is an entropy function, $R^{(j)}_i(s, a,s')$ is the reward function for example $j$ and instruction $i$, and $\textsc{RMSProp}$ divides each weight by a running average of its squared gradient~\cite{Tieleman:12}.
\Ensure Parameters $\theta$ defining a learned policy $\pi_\theta$.
\For{$t = 1, \dots, T, j = 1, \dots, N$}\label{algline:epoch}
\For{$i = 1, \dots, n^{(j)}$}
\State $\bar{e} \leftarrow \langle ~ \rangle, k \gets 0, a_0 \gets \mathtt{BEG}$
\State \Comment{Rollout up to $\act{STOP}$ or episode limit.}
\While{$a_{k} \neq \mathtt{STOP} \land k < M$}\label{algline:rollout}
\State $k \leftarrow k + 1$
\State $\tilde{s}_k \gets (\bar{\inputtoken}_i, \langle \bar{\inputtoken}_1, \dots, \bar{\inputtoken}_{i-1}\rangle, s_k, \bar{e}[:k])$
\State \Comment{Sample an action from policy.}
\State $a_k \sim \pi_\theta (\tilde{s}_k, \cdot)$
\State $s_{k+1} \leftarrow T(s_k, a_k)$
\State $\bar{e} \leftarrow [ \bar{e} ; \langle (s_k, a_k)\rangle ]$\label{algline:rolloutend}
\EndWhile
\State $\Delta \gets \bar{0}$
\For{$k' = 1,\dots,k$}
\State \Comment{Compute the entropy of $\pi_\theta(\tilde{s}_{k'}, \cdot)$.}
\State $\Delta \gets \Delta + \lambda \nabla_\theta H(\pi_\theta(\tilde{s}_{k'}, \cdot))$ \label{algline:entropy}
\For{$a \in \mathcal{A}$}
\State $s' \gets T(s_{k'}, a)$
\State \Comment{Compute gradient for action $a$.}
\State $\Delta \leftarrow \Delta + R^{(j)}_i(s_{k'}, a, s')\nabla_\theta \pi_\theta(\tilde{s}_{k'}, a)$ \label{algline:action}
\EndFor
\EndFor
\State $\theta \leftarrow \theta + \mu \textsc{RMSProp}\left(\dfrac{\Delta}{k}\right)$ \label{algline:update}
\EndFor
\EndFor
\State \Return $\theta$
\end{algorithmic}
\label{alg:learn}
\end{algorithm}
\end{tabular}
\vspace{-15pt}
\end{figure}
\paragraph{Reward}
The reward $R^{(j)}_i : \mathcal{S} \times \mathcal{S} \times \mathcal{A} \rightarrow \mathbb{R}$ is defined for each example $j$ and instruction $i$:
\begin{small}
\begin{equation*}
R^{(j)}_i(s, a, s') = P^{(j)}_i(s, a, s') + \phi^{(j)}_i(s') - \phi^{(j)}_i(s)\;\;,
\end{equation*}
\end{small}
\noindent
where $s$ is a source state, $a$ is an action, and $s'$ is a target state.\footnote{While the reward function is defined for any state-action-state tuple, in practice, it is used during learning with tuples that follow the system dynamics, $s' = T(s, a)$.} $P^{(j)}_i(s, a, s')$ is a problem reward and \mbox{$\phi^{(j)}_i(s') - \phi^{(j)}_i(s)$} is a shaping term.
The problem reward $P^{(j)}_i(s, a, s')$ is positive for stopping at the goal $g^{(j)}_i$ and negative for stopping in an incorrect state or taking an invalid action:
\begin{small}
\begin{equation}
\nonumber P^{(j)}_i(s, a, s') =
\begin{cases}
1.0 & a = \mathtt{STOP} \land s' = g^{(j)}_i \\
-1.0 & a = \mathtt{STOP} \land s' \neq g^{(j)}_i \\
-1.0 - \delta & s = s' \\
-\delta & \text{otherwise}
\end{cases}\;\;,
\end{equation}
\end{small}
\noindent
where $\delta$ is a verbosity penalty.
The case $s = s'$ indicates that $a$ was invalid in state $s$, as in this domain, all valid actions except $\mathtt{STOP}$ modify the state.
We use a potential-based shaping term \mbox{$\phi^{(j)}_i(s') - \phi^{(j)}_i(s)$}~\cite{Ng:99rewardshaping}, where $\phi^{(j)}_i(s) = - || s - g^{(j)}_i||$ computes the edit distance between the state $s$ and the goal, measured over the objects in each state.
The shaping term densifies the reward, providing a meaningful signal for learning in nonterminal states.
\paragraph{Objective}
We maximize the immediate expected reward over all actions and use entropy regularization.
The gradient is approximated by sampling an execution $\bar{e} = \langle (s_1, a_1), \dots, (s_k, a_k) \rangle$ using our current policy:
\begin{small}
\begin{eqnarray}
\nonumber \nabla_\theta \mathcal{J} &=& \dfrac{1}{k}\sum_{k'=1}^{k}\biggl( \sum_{a \in \mathcal{A}}R\left(s_k, a, T(s_k, a)\right)\nabla_\theta\pi(\tilde{s}_k, a ) \\
\nonumber && + \lambda \nabla_\theta H(\pi(\tilde{s}_k, \cdot))\biggr)\;\;,
\end{eqnarray}
\end{small}
\noindent
where $H(\pi(\tilde{s}_k, \cdot)$ is the entropy term.
\paragraph{Algorithm}
Algorithm~\ref{alg:learning} shows the Single-step Reward Observation (\textsc{SESTRA}) learning algorithm.
We iterate over the training data $T$ times (line~\ref{algline:epoch}).
For each example $j$ and turn $i$, we first perform a rollout by sampling an execution $\bar{e}$ from $\pi_\theta$ with at most $M$ actions (lines~\ref{algline:rollout}-\ref{algline:rolloutend}).
If the rollout reaches the horizon without predicting $\act{stop}$, we set the problem reward $P^{{(j)}}_{i}$ to $-1.0$ for the last step.
Given the sampled states visited, we compute the entropy (line~\ref{algline:entropy}) and observe the immediate reward for all actions (line~\ref{algline:action}) for each step.
Entropy and rewards are used to accumulate the gradient, which is applied to the parameters using \textsc{RMSProp}~\cite{Dauphin:15} (line~\ref{algline:update}).
\paragraph{Discussion}
Observing the rewards for all actions for each visited state addresses an on-policy learning exploration problem. Actions that consistently receive negative reward early during learning will be visited with very low probability later on, and in practice, often not explored at all.
Because the network is randomly initialized, these early negative rewards are translated into strong general biases that are not grounded well in the observed context.
Our algorithm exposes the agent to such actions later on when they receive positive rewards even though the agent does not explore them during rollout.
For example, in \textsc{Alchemy}, $\act{pop}$ actions are sufficient to complete the first steps of good executions. As a result, early during learning, the agent learns a strong bias against $\act{PUSH}$ actions.
In practice, the agent then will not explore $\act{push}$ actions again.
In our algorithm, as the agent learns to roll out the correct $\act{pop}$ prefix, it is then exposed to the reward for the first $\act{push}$ even though it likely sampled another $\act{pop}$.
It then unlearns its bias towards predicting $\act{pop}$.%
\begin{figure}[t]
\fbox{
\centering
\begin{minipage}{0.95\linewidth}
\begin{center}
\includegraphics[width=\linewidth,clip,trim=197 319 376 242]{figs/lols}
\end{center}
\end{minipage}}
\vspace{-5pt}
\caption{Illustration of LOLS~\cite[left; ][]{Chang:15LearningTSbetter} and our learning algorithm (SESTRA, right). LOLS branches a single time, and samples complete rollout for each branch to obtain the trajectory loss. SESTRA uses a complete on-policy rollout and single-step branching for all actions in each sample state.}
\label{fig:learning}
\vspace{-5pt}
\end{figure}
Our learning algorithm can be viewed as a cost-sensitive variant of the oracle in \textsc{Dagger}~\cite{Ross:11dagger}, where it provides the rewards for all actions instead of an oracle action.
It is also related to Locally Optimal Learning to Search~\cite[LOLS;][]{Chang:15LearningTSbetter} with two key distinctions: (a) instead of using different roll-in and roll-out policies, we use the model policy; and (b) we branch at each step, instead of once, but do not rollout from branched actions since we only optimize the immediate reward. Figure~\ref{fig:learning} illustrates the comparison.
Our summation over immediate rewards for all actions is related the summation of estimated Q-values for all actions in the Mean Actor-Critic algorithm~\cite{Asadi:17mac}.
Finally, our approach is related to \citet{Misra:17instructions}, who also maximize the immediate reward, but do not observe rewards for all actions for each state.
\section{Model}
\label{sec:model}
\begin{figure*}
\centering
\includegraphics[width=\textwidth,clip,trim=70 311 273 163]{figs/arch}
\vspace{-15pt}
\caption{Illustration of the model architecture while generating the third action $a_3$ in the third utterance $\bar{\inputtoken}_3$ from Figure~\ref{fig:example}. Context vectors computed using attention are highlighted in blue. The model takes as input vector encodings from the current and previous instructions $\bar{\inputtoken}_1$, $\bar{\inputtoken}_2$, and $\bar{\inputtoken}_3$, the initial state $s_1$, the current state $s_3$, and the previous action $a_2$. Instruction encodings are computed with a bidirectional RNN. We attend over the previous and current instructions and the initial and current states. We use an MLP to select the next action.}
\label{fig:arch}
\vspace{-15pt}
\end{figure*}
\newcommand{X}{X}
\newcommand{S}{S}
\newcommand{\mathbf{z}}{\mathbf{z}}
We map sequences of instructions $\langle \bar{\inputtoken}_1, \dots, \bar{\inputtoken}_n \rangle$ to actions by executing the instructions in order.
The model generates an execution $\bar{e} = \langle (s_1, a_1), \dots, (s_{m_i}, a_{m_i})\rangle$ for each instruction $\bar{\inputtoken}_i$.
The agent context, the information available to the agent at step $k$, is $\tilde{s}_k = (\bar{\inputtoken}_i, \langle \bar{\inputtoken}_1, \dots, \bar{\inputtoken}_{i-1}\rangle, s_k, \bar{e}[:k])$, where $\bar{e}[:k]$ is the execution up until but not including step $k$.
In contrast to the world state, the agent context also includes instructions and the execution so far.
The agent policy $\pi_\theta(\tilde{s}_k, a)$ is modeled as a probabilistic neural network parametrized by $\theta$, where $\tilde{s}_k$ is the agent context at step $k$ and $a$ is an action.
To generate executions, we generate one action at a time, execute the action, and observe the new world state.
In step $k$ of executing the $i$-th instruction, the network inputs are the current utterance $\bar{\inputtoken}_i$, the previous instructions $\langle \bar{\inputtoken}_1, \dots, \bar{\inputtoken}_{i-1}\rangle$, the initial state $s_1$ at beginning of executing $\bar{\inputtoken}_i$, and the current state $s_k$.
When executing a sequence of instructions, the initial state $s_1$ is either the state at the beginning of executing the sequence or the final state of the execution of the previous instruction.
Figure~\ref{fig:arch} illustrates our architecture.
We generate continuous vector representations for all inputs.
Each input is represented as a set of vectors that are then processed with an attention function to generate a single vector representation~\cite{Luong:15nmtattention}.
We assume access to a domain-specific encoding function $\func{enc}(s)$ that, given a state $s$, generates a set of vectors $S$ representing the objects in the state.
For example, in the \textsc{Alchemy} domain, a vector is generated for each beaker using an RNN.
Section~\ref{sec:data} describes the different domains and their encoding functions.
We use a single bidirectional RNN with a long short-term memory~\cite[LSTM;][]{Hochreiter:97lstm} recurrence to encode the instructions.
All instructions $\bar{\inputtoken}_1$,\dots,$\bar{\inputtoken}_i$ are encoded with a single RNN by concatenating them to $\bar{\inputtoken}'$.
We use two delimiter tokens: one separates previous instructions, and the other separates the previous instructions from the current one.
The forward LSTM RNN hidden states are computed as:\footnote{To simplify the notation, we omit the memory cell (often denoted as $\mathbf{c}_j$) from all LSTM descriptions. We use only the hidden state $\mathbf{h}_j$ to compute the intended representations (e.g., for the input text tokens). All LSTMs in this paper use zero vectors as initial hidden state $\mathbf{h}_0$ and initial cell memory $\mathbf{c}_0$.}
\begin{small}
\begin{equation}
\nonumber \overrightarrow{\mathbf{h}_{j+1}} = \overrightarrow{\rnn^E}\left(\embedding^I(x'_{j+1}); \overrightarrow{\mathbf{h}_j}\right)\;\;,
\end{equation}
\end{small}
\noindent
where $\embedding^I$ is a learned word embedding function and $\overrightarrow{\rnn^E}$ is the forward LSTM recurrence function.
We use a similar computation to compute the backward hidden states $\overleftarrow{\mathbf{h}_j}$.
For each token $x'_j$ in $\bar{\inputtoken}'$, a vector representation $\mathbf{h}_j' = \left[ \overrightarrow{\mathbf{h}_j}; \overleftarrow{\mathbf{h}_j}\right]$ is computed.
We then create two sets of vectors, one for all the vectors of the current instruction and one for the previous instructions:
\begin{small}
\begin{eqnarray}
\nonumber X^c &=& \{ \mathbf{h}'_j \}_{j = \MakeUppercase{j}}^{\MakeUppercase{j} + \length{\bar{\inputtoken}_i} } \\
\nonumber X^p &=& \{ \mathbf{h}'_j \}_{j = 0}^{j < \MakeUppercase{j}}
\end{eqnarray}
\end{small}
\noindent
where $\MakeUppercase{j}$ is the index in $\bar{\inputtoken}'$ where the current instruction $\bar{\inputtoken}_i$ begins.
Separating the vectors to two sets will allows computing separate attention on the current instruction and previous ones.
To compute each input representation during decoding, we use a bi-linear attention function~\cite{Luong:15nmtattention}.
Given a set of vectors $H$, a query vector $\mathbf{h}^q$, and a weight matrix $\mathbf{W}$, the attention function $\func{attend}(H, \mathbf{h}^q, \mathbf{W})$ computes a context vector $\mathbf{z}$:
\begin{small}
\begin{eqnarray}
\nonumber \alpha_i &\propto& \exp(\mathbf{h}_i^\intercal \mathbf{W} \mathbf{h}^q ): i = 0, \dots, \length{H} \\
\nonumber \mathbf{z} &=& \sum_{i=1}^{\length{H}}\alpha_i \mathbf{h}_i\;\;.
\end{eqnarray}
\end{small}
We use a decoder to generate actions.
At each time step $k$, we compute an input representation using the attention function, update the decoder state, and compute the next action to execute.
Attention is first computed over the vectors of the current instruction, which is then used to attend over the other inputs.
We compute the context vectors $\mathbf{z}_k^c$ and $\mathbf{z}_k^p$ for the current instruction and previous instructions:
\begin{small}
\begin{eqnarray}
\nonumber \mathbf{z}^c_k &=& \func{Attend}(X^c, \mathbf{h}^d_{k-1}, \mathbf{W}^c) \\
\nonumber \mathbf{z}^p_k &=& \func{Attend}(X^p, [\mathbf{h}^d_{k-1}, \mathbf{z}^c_k], \mathbf{W}^p)\;\;,
\end{eqnarray}
\end{small}
\noindent
where $\mathbf{h}^d_{k-1}$ is the decoder hidden state for step $k-1$, and $X^c$ and $X^p$ are the sets of vector representations for the current instruction and previous instructions.
Two attention heads are used over both the initial and current states.
This allows the model to attend to more than one location in a state at once, for example when transferring items from one beaker to another in \textsc{Alchemy}.
The current state is computed by the transition function $s_k = T(s_{k-1}, a_{k-1})$, where $s_{k-1}$ and $a_{k-1}$ are the state and action at step $k-1$.
The context vectors for the initial state $s_1$ and the current state $s_k$ are:
\begin{small}
\begin{eqnarray}
\nonumber \mathbf{z}^s_{1,k} &=& [\func{Attend}( \func{enc}(s_1), [\mathbf{h}^d_{k-1}, \mathbf{z}^c_k], \mathbf{W}^{s_b, 1}) ; \\ \nonumber &&~ \func{Attend}(\func{enc}(s_1), [\mathbf{h}^d_{k-1}, \mathbf{z}^c_k], \mathbf{W}^{s_b, 2})] \\
\nonumber \mathbf{z}^{s}_{k, k} &=& [\func{Attend}( \func{enc}(s_k), [\mathbf{h}^d_{k-1}, \mathbf{z}^c_k], \mathbf{W}^{s_c, 1}) ; \\ \nonumber &&~ \func{Attend}(\func{enc}(s_k), [\mathbf{h}^d_{k-1}, \mathbf{z}^c_k], \mathbf{W}^{s_c, 2})]\;\;,
\end{eqnarray}
\end{small}
\noindent
where all $\mathbf{W}^{*,*}$ are learned weight matrices.
We concatenate all computed context vectors with an embedding of the previous action $a_{k-1}$ to create the input for the decoder:
\begin{small}
\begin{eqnarray}
\nonumber \mathbf{h}_k &=& \tanh([\mathbf{z}^c_k ; \mathbf{z}^p_k; \mathbf{z}^{s}_{1,k}; \mathbf{z}^{s}_{k,k}; \embedding^O(a_{k-1})]\mathbf{W}^d + \mathbf{b}^d) \\
\nonumber \mathbf{h}^d_k &=& \rnn^D\left(\mathbf{h}_k; \mathbf{h}^d_{k-1}\right)\;\;,
\end{eqnarray}
\end{small}
\noindent
where $\embedding^O$ is a learned action embedding function and $\rnn^D$ is the LSTM decoder recurrence.
Given the decoder state $\mathbf{h}^d_k$, the next action $a_k$ is predicted with a multi-layer perceptron (MLP).
The actions in our domains decompose to an action type and at most two arguments.\footnote{We use a $\act{NULL}$ argument for unused arguments.} For example, the action $\act{push~1~B}$ in \textsc{Alchemy} has the type $\act{push}$ and two arguments: a beaker number and a color.
Section~\ref{sec:data} describes the actions of each domain.
The probability of an action is:
\begin{small}
\begin{eqnarray}
\nonumber \mathbf{h}^a_k &=& \tanh(\mathbf{h}^d_k \mathbf{W}^a) \\
\nonumber s_{k,a_T} &=& \mathbf{h}^a_k \mathbf{b}_{a_T} \\
\nonumber s_{k,a_1} &=& \mathbf{h}^a_k \mathbf{b}_{a_1} \\
\nonumber s_{k,a_2} &=& \mathbf{h}^a_k \mathbf{b}_{a_2} \\
\nonumber p(a_k = a_T(a_1, a_2) \mid \tilde{s}_k ; \theta) &\propto & \\ \nonumber && \hspace{-2.5em} \exp(s_{k, a_T} + s_{k,a_1} + s_{k,a_2})\;\;,
\end{eqnarray}
\end{small}
\noindent
where $a_T$, $a_1$, and $a_2$ are an action type, first argument, and second argument.
If the predicted action is $\act{stop}$, the execution is complete.
Otherwise, we execute the action $a_k$ to generate the next state $s_{k+1}$, and update the agent context $\tilde{s}_{k}$ to $\tilde{s}_{k+1}$ by appending the pair $(s_k,a_k)$ to the execution $\bar{e}$ and replacing the current state with $s_{k+1}$.
The model parameters $\theta$ include: the embedding functions $\embedding^I$ and $\embedding^O$; the recurrence parameters for $\overrightarrow\rnn^E$, $\overleftarrow\rnn^E$, and $\rnn^D$; $\mathbf{W}^C$, $\mathbf{W}^P$, $\mathbf{W}^{s_b, 1}$, $\mathbf{W}^{s_b, 2}$, $\mathbf{W}^{s_c, 1}$, $\mathbf{W}^{s_c, 2}$, $\mathbf{W}^d$, $\mathbf{W}^a$, and $\mathbf{b}^d$; and the domain dependent parameters, including the parameters of the encoding function $\func{ENC}$ and the action type, first argument, and second argument weights $\mathbf{b}_{a_T}$, $\mathbf{b}_{a_1}$, and $\mathbf{b}_{a_2}$.
\section{Technical Overview}
\label{sec:overview}
\paragraph{Task and Notation}
Let $\mathcal{S}$ be the set of all possible world states, $\mathcal{X}$ be the set of all natural language instructions, and $\mathcal{A}$ be the set of all actions.
An instruction $\bar{\inputtoken} \in \mathcal{X}$ of length $\length{\bar{\inputtoken}}$ is a sequence of tokens $\langle x_1, ... x_{\length{\bar{\inputtoken}}} \rangle$.
Executing an action modifies the world state following a transition function $T : \mathcal{S} \times \mathcal{A} \rightarrow \mathcal{S}$. For example, the \textsc{Alchemy} domain includes seven beakers that contain colored liquids. The world state defines the content of each beaker. We treat each beaker as a stack. The actions are $\act{pop~N}$ and $\act{push~N~C}$, where $ 1 \leq \act{N} \leq 7$ is the beaker number and $\act{C}$ is one of six colors. There are a total of $50$ actions, including the $\act{stop}$ action. Section~\ref{sec:data} describes the domains in detail.
Given a start state $s_1$ and a sequence of instructions $\langle \bar{\inputtoken}_1, \dots, \bar{\inputtoken}_n \rangle$, our goal is to generate the sequence of actions specified by the instructions starting from $s_1$.
We treat the execution of a sequence of instructions as executing each instruction in turn.
The execution $\bar{e}$ of an instruction $\bar{\inputtoken}_i$ starting at a state $s_1$ and given the history of the instruction sequence $\langle \bar{\inputtoken}_1, \dots, \bar{\inputtoken}_{i-1}\rangle$ is a sequence of state-action pairs $\bar{e} = \langle (s_1, a_1), ..., (s_m, a_m) \rangle$, where $a_k \in \mathcal{A}$, $s_{k+1} = T (s_k, a_k)$.
The final action $a_m$ is the special action $\act{STOP}$, which indicates the execution has terminated.
The final state is then $s_m$, as $T(s_k, \mathtt{STOP}) = s_k$.
Executing a sequence of instructions in order generates a sequence $\langle \bar{e}_1, ..., \bar{e}_n \rangle$, where $\bar{e}_i$ is the execution of instruction $\bar{\inputtoken}_i$.
When referring to states and actions in an indexed execution $\bar{e}_i$, the $k$-th state and action are $s_{i, k}$ and $a_{i, k}$.
We execute instructions one after the other: $\bar{e}_1$ starts at the interaction initial state $s_1$ and $s_{i+ 1,1} = s_{i,\length{\bar{e}_i}}$, where $s_{i + 1,1}$ is the start state of $\bar{e}_{i+1}$ and $s_{i, \length{\bar{e}_i}}$ is the final state of $\bar{e}_i$.
\paragraph{Model}
We model the agent with a neural network policy (Section~\ref{sec:model}).
At step $k$ of executing the $i$-th instruction, the model input is the current instruction $\bar{\inputtoken}_i$, the previous instructions $\langle \bar{\inputtoken}_1,\dots, \bar{\inputtoken}_{i- 1}\rangle$, the world state $s_1$ at the beginning of executing $\bar{\inputtoken}_i$, and the current state $s_{k}$.
The model predicts the next action $a_{k}$ to execute.
If $a_{k} = \act{stop}$, we switch to the next instruction, or if at the end of the instruction sequence, terminate.
Otherwise, we update the state to $s_{k+1} = T(s_{ k}, a_{k} )$.
The model uses attention to process the different inputs and a recurrent neural network (RNN) decoder to generate actions~\cite{Bahdanau:14neuralmt}.
\paragraph{Learning}
We assume access to a set of $N$ instruction sequences, where each instruction in each sequence is paired with its start and goal states.
During training, we create an example for each instruction.
Formally, the training set is $\{(\bar{\inputtoken}^{(j)}_i, s_{i, 1}^{(j)}, \langle \bar{\inputtoken}^{(j)}_1, \dots, \bar{\inputtoken}^{(j)}_{i-1} \rangle, g^{(j)}_i) \}_{j=1,i=1}^{N,n^{(j)}}$, where $\bar{\inputtoken}^{(j)}_i$ is an instruction, $s_{i, 1}^{(j)}$ is a start state, $\langle \bar{\inputtoken}^{(j)}_1, \dots, \bar{\inputtoken}^{(j)}_{i-1} \rangle$ is the instruction history, $g^{(j)}_i$ is the goal state, and $n^{(j)}$ is the length of the $j$-th instruction sequence.
This training data contains no evidence about the actions and intermediate states required to execute each instruction.\footnote{This training set is a subset of the data used in previous work~\cite[Section 6,][]{Guu:15:traverse}, in which training uses all instruction sequences of length $1$ and $2$.}
We use a learning method that maximizes the expected immediate reward for a given state (Section~\ref{sec:learning}).
The reward accounts for task-completion and distance to the goal via potential-based reward shaping.
\paragraph{Evaluation}
We evaluate exact task completion for sequences of instructions on a test set $\{(s^{(j)}_1, \langle \bar{\inputtoken}_1^{(j)}, \dots,\bar{\inputtoken}_{n_j}^{(j)}\rangle, g^{(j)})\}_{j=1}^N$, where $g^{(j)}$ is the oracle goal state of executing instructions \mbox{$\bar{\inputtoken}_1^{(j)}$, \dots,$\bar{\inputtoken}_{n_j}^{(j)}$} in order starting from $s^{(j)}_1$.
We also evaluate single-instruction task completion using per-instruction annotated start and goal states.
\section{Related Work}
\label{sec:related}
Executing instructions has been studied using the SAIL corpus~\cite{MacMahon:06} with focus on navigation using high-level logical representations~\cite{Chen:11,Chen:12,Artzi:13,Artzi:14} and low-level actions~\cite{Mei:16neuralnavi}.
While SAIL includes sequences of instructions, the data demonstrates limited discourse phenomena, and instructions are often processed in isolation.
Approaches that consider as input the entire sequence focused on segmentation~\cite{Andreas:15navi}.
Recently, other navigation tasks were proposed with focus on single instructions~\cite{Anderson:17,Janner:17spatialrep}.
We focus on sequences of environment manipulation instructions and modeling contextual cues from both the changing environment and instruction history.
Manipulation using single-sentence instructions has been studied using the Blocks domain~\cite{Bisk:16nl-robots,Bisk:17,Misra:17instructions,Tan:17blocks}.
Our work is related to the work of \citet{Branavan:09} and \citet{Vogel:10}.
While both study executing sequences of instructions, similar to SAIL, the data includes limited discourse dependencies.
In addition, both learn with rewards computed from surface-form similarity between text in the environment and the instruction.
We do not rely on such similarities, but instead use a state distance metric.
Language understanding in interactive scenarios that include multiple turns has been studied with focus on dialogue for querying database systems using the ATIS corpus~\cite{Hemphill:90atis,Dahl:94}.
\citet{Tur:10atis} surveys work on ATIS.
\citet{Miller:96}, \citet{Zettlemoyer:09}, and \citet{Suhr:18atis} modeled context dependence in ATIS for generating formal representations.
In contrast, we focus on environments that change during execution and directly generating environment actions, a scenario that is more related to robotic agents than database query.
The SCONE corpus~\cite{Long:16context} was designed to reflect a broad set of discourse context-dependence phenomena.
It was studied extensively using logical meaning representations~\cite{Long:16context,Guu:17rl-mml,Fried:17}.
In contrast, we are interested in directly generating actions that modify the environment. This requires generating lower-level actions and learning procedures that are otherwise hardcoded in the logic (e.g., mixing action in Figure~\ref{fig:example}).
Except for \citet{Fried:17}, previous work on SCONE assumes access only to the initial and final states during training.
This form of supervision does not require operating the agent manually to acquire the correct sequence of actions, a difficult task in robotic agents with complex control.
Goal state supervision has been studied for instructional language~\cite[e.g.,][]{Branavan:09,Artzi:13,Bisk:16nl-robots}, and more extensively in question answering when learning with answer annotations only~\cite[e.g.,][]{Clarke:10,Liang:11,Kwiatkowski:13,Berant:13,Berant:14paraphrasing,Berant:15imitation,Liang:17freebase}.
\section{Results}
\label{sec:results}
\begin{table*}[t]
\begin{footnotesize}
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|} \hline
& \multicolumn{3}{c|}{\textsc{Alchemy}} & \multicolumn{3}{c|}{\textsc{Scene}} & \multicolumn{3}{c|}{\textsc{Tangrams}} \\ \cline{2-10}
System & Inst & 3utts & 5utts & Inst & 3utts & 5utts & Inst & 3utts & 5utts \\ \hline\hline
\citet{Long:16context} & -- & $56.8$ & $52.3$ & --& $23.2$ & $14.7$ & -- & $64.9$ & $27.6$ \\ \hdashline[0.5pt/1pt]
\citet{Guu:17rl-mml} & -- & $66.9$ & $52.9$ & -- & $64.8$ & $46.2$ & -- & $65.8$ & $37.1$\\ \hline\hline
\citet{Fried:17} & -- & -- & $72.0$ & -- & -- & $72.7$ & -- & -- & $69.6$ \\ \hline\hline
\textsc{Supervised} & $89.4$ & $73.3$ & $62.3$ & $88.8$ & $78.9$ & $66.4$ & $86.6$ & $81.4$ & $60.1$ \\ \hline\hline
\textsc{PolicyGradient} & $0.0$ & $0.0$ & $0.0$ & $0.0$ & $1.3$ & $0.2$ & $84.1$ & $77.4$ & $54.9$\\ \hdashline[0.5pt/1pt]
\textsc{ContextualBandit} & $73.8$ & $36.0$ & $25.7$ & $15.1$ & $2.9$ & $4.4$ & $84.8$ & $76.9$ & $57.9$ \\ \hdashline[0.5pt/1pt]
Our approach & $89.1$ & $74.2$ & $62.7$ & $87.1$ & $73.9$ & $62.0$ & $86.6$ & $80.8$ & $62.4$ \\ \hline
\end{tabular}
\end{center}
\end{footnotesize}
\vspace{-5pt}
\caption{Test accuracies for single instructions (Inst), first-three instructions (3utts), and full interactions (5utts).}
\vspace{-3pt}
\label{tab:test}
\end{table*}
\newcommand{\nstdev}[2]{${#1}$ \vspace{-3pt}\newline \hspace*{8pt}\begin{tiny}$\pm {#2}$ \end{tiny}}
\newcommand{0.8cm}{0.8cm}
\begin{table*}[t]
\begin{footnotesize}
\begin{center}
\begin{tabular}{|p{3.2cm}|>{\centering\arraybackslash}p{0.8cm}|>{\centering\arraybackslash}p{0.8cm}|>{\centering\arraybackslash}p{0.8cm}|>{\centering\arraybackslash}p{0.8cm}|>{\centering\arraybackslash}p{0.8cm}|>{\centering\arraybackslash}p{0.8cm}|>{\centering\arraybackslash}p{0.8cm}|>{\centering\arraybackslash}p{0.8cm}|>{\centering\arraybackslash}p{0.8cm}|} \hline
& \multicolumn{3}{c|}{\textsc{Alchemy}} & \multicolumn{3}{c|}{\textsc{Scene}} & \multicolumn{3}{c|}{\textsc{Tangrams}} \\ \cline{2-10}
System & Inst & 3utts & 5utts & Inst & 3utts & 5utts & Inst & 3utts & 5utts \\ \hline\hline
\textsc{Supervised} & $92.0$ & $83.3$ & $71.4$ & $85.3$ & $72.7$ & $60.6$ & $86.1$ & $81.9$ & $58.3$ \\ \hline \hline
\textsc{PolicyGradient} & $0.0$ & $0.0$ & $0.0$ & $0.9$ & $1.0$ & $0.5$ & $85.2$ & $74.9$ & $52.3$\\ \hdashline[0.5pt/1pt]
\textsc{ContextualBandit} & $58.8$ & $6.9$ & $5.7$ & $12.0$ & $0.5$ & $1.5$ & $85.6$ & $78.4$ & $52.6$ \\ \hline\hline
Our approach & $\mathbf{92.1}$ & $\mathbf{82.9}$ & $\mathbf{71.8}$ & $\mathbf{83.9}$ & $\mathbf{68.7}$ & $56.1$ & $88.5$ & $82.4$ & $60.3$ \\ \hdashline[0.5pt/1pt]
-- previous instructions & $90.1$ & $77.1$ & $66.1$ & $79.3$ & $60.6$ & $45.5$ & $76.4$ & $55.8$ & $27.6$\\ \hdashline[0.5pt/1pt]
-- current and initial state & $25.7$ &$4.5$ &$3.3$ & $17.5$ & $0.0$ & $0.0$ & $45.4$ & $15.1$ & $3.5$ \\ \hdashline[0.5pt/1pt]
-- current state & $89.8$ & $78.0$ & $62.9$ & $83.0$ & $\mathbf{68.7}$ & $54.0$ & $87.6$ & $78.4$ & $60.8$ \\ \hdashline[0.5pt/1pt]
-- initial state & $81.1$ & $68.6$ & $42.9$ & $82.7$ & $67.7$ & $\mathbf{57.1}$ & $\mathbf{88.6}$ & $\mathbf{82.9}$ & $\mathbf{63.3}$ \\ \hline\hline
Our approach ($\mu \pm \sigma$) & \nstdev{91.5}{1.4} & \nstdev{80.4}{2.6} & \nstdev{69.5}{5.0} & \nstdev{62.9}{17.7} & \nstdev{37.8}{23.5} & \nstdev{29.0}{21.1} &\nstdev{88.2}{0.6} & \nstdev{80.8}{2.8} & \nstdev{59.2}{2.3}\\ \hline
\end{tabular}
\end{center}
\end{footnotesize}
\vspace{-5pt}
\caption{Development results, including model ablations. We also report mean $\mu$ and standard deviation $\sigma$ for all metrics for our approach across five experiments. We bold the best performing variations of our model.}
\label{tab:ablations}
\vspace{-6pt}
\end{table*}
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};
\end{axis}
\end{tikzpicture}
\end{center}
\vspace{-10pt}
\caption{Instruction-level training accuracy per epoch when training five models on \textsc{Scene}, demonstrating the effect of randomization in the learning method. Three of five experiments fail to learn effective models. The red and blue learning trajectories are overlapping.}
\label{fig:scenelearning}
\vspace{-15pt}
\end{figure}
Table~\ref{tab:test} shows test results.
Our approach significantly outperforms \textsc{PolicyGradient} and \textsc{ContextualBandit}, both of which suffer due to biases learned early during learning, hindering later exploration.
This problem does not appear in \textsc{Tangrams}, where no action type is dominant at the beginning of executions, and all methods perform well.
\textsc{PolicyGradient} completely fails to learn \textsc{Alchemy} and \textsc{Scene} due to observing only negative total rewards early during learning. Using a baseline, for example with an actor-critic method, will potentially close the gap to \textsc{ContextualBandit}. However, it is unlikely to address the on-policy exploration problem.
Table~\ref{tab:ablations} shows development results, including model ablation studies.
Removing previous instructions (-- previous instructions) or both states (-- current and initial state) reduces performance across all domains.
Removing only the initial state (-- initial state) or the current state (-- current state) shows mixed results across the domains.
Providing access to both initial and current states increases performance for \textsc{Alchemy}, but reduces performance on the other domains. We hypothesize that this is due to the increase in the number of parameters outweighing what is relatively marginal information for these domains.
In our development and test results we use a single architecture across the three domains, the full approach, which has the highest interactive-level accuracy when averaged across the three domains ($62.7$ 5utts).
We also report mean and standard deviation for our approach over five trials.
We observe exceptionally high variance in performance on \textsc{Scene}, where some experiments fail to learn and training performance remains exceptionally low (Figure~\ref{fig:scenelearning}). This highlights the sensitivity of the model to the random effects of initialization, dropout, and ordering of training examples.
\begin{table}
\begin{footnotesize}
\begin{center}
\begin{tabular}{|p{4cm}|c|c|c|} \hline
Class & \textsc{Alc} & \textsc{Sce} & \textsc{Tan} \\ \hline
State reference & $23$ & $13$ & $7$ \\ \hdashline[0.5pt/1pt] %
Multi-turn reference & $12$ & $5$ & $13$ \\ \hdashline[0.5pt/1pt] %
Impossible multi-turn reference & $2$ & $5$ & $13$ \\ \hdashline[0.5pt/1pt] %
Ambiguous or incorrect label & $2$ & $19$ & $12$ \\ \hline %
\end{tabular}
\end{center}
\end{footnotesize}
\vspace{-5pt}
\caption{Common error counts in the three domains.}
\label{tab:error_analysis}
\vspace{-5pt}
\end{table}
We analyze the instruction-level errors made by our best models when the agent is provided the correct initial state for the instruction.
We study fifty examples in each domain to identify the type of failures.
Table~\ref{tab:error_analysis} shows the counts of major error categories.
We consider multiple reference resolution errors.
State reference errors indicate a failure to resolve a reference to the world state.
For example, in \textsc{Alchemy}, the phrase \nlstring{leftmost red beaker} specifies a beaker in the environment. If the model picked the correct action, but the wrong beaker, we count it as a state reference.
We distinguish between multi-turn reference errors that should be feasible, and these that that are impossible to solve without access to states before executing previous utterances, which are not provided to our model.
For example, in \textsc{Tangrams}, the instruction \nlstring{put it back in the same place} refers to a previously-removed item.
Because the agent only has access to the world state after following this instruction, it does not observe what kind of item was previously removed, and cannot identify the item to add.
We also find a significant number of errors due to ambiguous or incorrect instructions.
For example, the \textsc{Scene} instruction \nlstring{person in green appears on the right end} is ambiguous.
In the annotated goal, it is interpreted as referring to a person already in the environment, who moves to the 10th position.
However, it can also be interpreted as a new person in green appearing in the 10th position.
We also study performance with respect to multi-turn coreference by observing whether the model was able to identify the correct referent for each occurrence included in the analysis in Table~\ref{tab:analysis}.
The models were able to correctly resolve $92.3\%$, $88.7\%$, and $76.0\%$ of references in \textsc{Alchemy}, \textsc{Scene}, and \textsc{Tangrams} respectively.
Finally, we include attention visualization for examples from the three domains in the Supplementary Material.
\section{Architecture and Training Details}
\label{sec:sup:arch}
\paragraph{Model Architecture}
We use an embedding size of $50$ for words and action types and arguments.
Action embedding is a concatenation of the embeddings of each part, including the action type and the two arguments; an embedded action is a vector of size $150$.
Embeddings of colors in \textsc{Alchemy} and \textsc{Scene}, and shapes in \textsc{Tangrams}, are of size $10$.
Positional embeddings are of size $10$.
$\mathbf{W}^d$ and $\mathbf{W}^a$ are square matrices.
All matrices are initialized by sampling from the uniform distribution $U\left(\left[-\sqrt{\frac{6}{M+N}}, \sqrt{\frac{6}{M+N}}\right]\right)$~\cite{Glorot:2010understanding}, where $M$ and $N$ are the matrix dimensionality.
All RNNs are single-layer LSTMs.
For the main model, both the instruction encoder and action sequence decoder use a hidden size of $100$ in each direction.
The action sequence decoder is initialized by first setting the hidden state and cell memory to zero-vectors, and passing in a zero-vector to update the states, after which attention is computed for the first time.
For \textsc{Alchemy}, the world state encoder has a hidden size of $20$.
For \textsc{Scene}, the world state encoder has a hidden size of $5$.
\paragraph{Training}
We apply dropout in three places: (a) in each attention computation after multiplying by $\mathbf{W}$; (b) after computing $\mathbf{h}_k$, the input to each decoder step; and (c) for all attention keys except for the current utterance.
For \textsc{PolicyGradient}, \textsc{ContextualBandit}, and our approach, we optimize parameters using \textsc{RMSProp}~\cite{Tieleman:12}.
For supervised learning, we use \textsc{Adam}~\cite{Kingma:14adam} for optimization.
We use a learning rate of $0.001$ for all experiments.
Our validation set is a held-out subset containing $7.0\%$ of the training data.
We stop training by observing the instruction-level reward on the validation set.
We use patience for early stopping. We reset patience to $50 \cdot 1.005^x$ the $x$-th time the reward has improved on the validation set, decrease by one each epoch reward does not improve, and stop when patience runs out.
Regardless of patience, we terminate training after $200$ epochs.
We tune $\lambda$, $\delta$, and $M$ on the development set.
In \textsc{Alchemy}, $\lambda = 0.1$, $\delta=0.15$, and $M = 7$.
In \textsc{Scene}, $\lambda = 0.07$, $\delta = 0.2$, and $M = 5$.
In \textsc{Tangrams}, $\lambda=0.1$, $\delta=0.0$, and $M=5$.
\section{Attention Analysis}
\label{sec:sup:attn}
\begin{figure*}[t]
\fbox{
\centering
\includegraphics[width=\linewidth,clip,trim=10 60 50 35,right]{figs/alchemy_attention.pdf}}
\caption{Example of the attention distributions for executing the instruction \nlstring{It turns completely brown} in \textsc{Alchemy}.
This is the fifth instruction in the interaction.
The correct action sequence mixes the chemicals in the sixth beaker by removing the three units and re-adding three brown units. Our model correctly predicts this sequence.
We show the different attention distributions when generating this sequence of actions.
Clockwise starting from the top left: (a) attention over the current instruction; (b) two attention heads over the initial state; (c) two attention heads over the current world state, which changes following each action; and (d) the attention over the previous instructions in the interaction.
}
\label{fig:attention}
\end{figure*}
\begin{figure*}[t]
\centering
\fbox{
\centering
\centering
\includegraphics[width=\linewidth,clip,trim=0 200 350 70,right]{figs/scene_attention.pdf} }
\caption{Example of attention for a randomly selected instruction from the development set for \textsc{Scene}.
The instruction \nlstring{A person with a blue shirt appears to the left of him} is the second in the interaction, following the instruction \nlstring{The person with a red shirt and a blue hat moves to the right end}.
The correct action sequence consists of a single action, $\act{ADD\_PERSON\ 9\ B}$, where a person wearing a blue shirt appears in position $9$, to the left of the person in the red shirt. Our model predicts this action correctly.
We show the different attention distributions when generating this sequence of a single action.
From top to bottom: (a) attention over the current instruction; (b) attention over the previous instruction; and (c) attention over the world state.
As the sequence contains a single action only, the current and initial world states are the same, and their distributions are shown together.
There are two attention heads over both the initial (top two rows) and current (bottom two rows) world states.}
\label{fig:scene}
\end{figure*}
\begin{figure*}[t]
\fbox{
\centering
\includegraphics[width=\linewidth,clip,trim=20 175 30 100, right]{figs/alchemy2_attention.pdf}}
\caption{Example of attention for a randomly selected instruction from the development set for \textsc{Alchemy}. The instruction executed is \nlstring{Pour green beaker into orange one}, the fifth instruction in the sequence. We show the different attention distributions when generating the correct action sequence, which removes green items from the sixth beaker and adds the same number of green items to the beaker containing orange. Clockwise starting from the top left: (a) attention on the current instruction; (b) the two attention heads over the initial state; (c) the two attention heads over the current state as it changes during execution; and (d) attention over previous instructions.}
\label{fig:alchemy}
\end{figure*}
\begin{figure*}[t]
\fbox{
\centering
\includegraphics[width=\linewidth,clip,trim=290 415 170 40, right]{figs/tangrams_attention.pdf} }
\caption{Example of attention for a randomly selected instruction from the development set for \textsc{Tangrams}. The instruction executed is \nlstring{Switch the first and second figure}, the fourth instruction in the sequence.
We show the different attention distributions when generating the correct action sequence, which removes the figure in position two and adds it in position one, thereby swapping the first two items. Clockwise starting from the top left: (a) attention on the current instruction; (b) the two attention heads over the initial state; (c) the two attention heads over the current state as it changes during execution; and (d) attention over previous instructions.}
\label{fig:tangrams}
\end{figure*}
Figure~\ref{fig:attention} shows attention distributions for a hand-picked example in \textsc{Alchemy}.
We show the attention probabilities ($\alpha$ in Section~\ref{sec:model}) for the current and previous utterances, initial state, and current state throughout execution.
In this example, the previous-instruction attention puts most of the weight on \nlstring{brown one} during generation, which is the referent of \nlstring{it} in the current instruction.
The initial and current state attentions are placed heavily on the beaker being manipulated.
However, for randomly selected examples, we observe that the attention distribution does not always correspond to intuitions about what should be attended on.
Figures~~\ref{fig:scene}, \ref{fig:alchemy}, and ~\ref{fig:tangrams} show examples of attention distributions for three random instructions in the development sets of the three domains where the action sequence was predicted correctly.
\section{Data Analysis}
\label{sec:sup:data}
We analyze SCONE to identify the frequency of various discourse phenomena in the three domains, including explicit coreference and ellipsis, which is implicit reference to previous entities.
We observe references to previous objects (e.g., beakers in \textsc{Alchemy}), actions, locations (e.g., positions in \textsc{Scene}), and world states.
We analyze thirty development set interactions for each domain for presence of these references. We define the age of each referent as the number of turns since it was last explicitly mentioned.
This illustrates the extent to which this dataset challenges models for context-dependent reasoning.
\begin{table}
\begin{footnotesize}
\begin{center}
\begin{tabular}{|l|l|c|c|c|c|c} \hline
& & $1$ & $2$ & $3$ & $4$ \\ \hline
\multirow{3}{*}{Coref.} & Beaker & $24$ & $7$ & $2$ & $0$ \\ \ddline{2-6}
& Action & $3$ & $0$ & $0$ & $0$\\ \ddline{2-6}
& Action + Arguments & $1$ & $0$ & $0$ & $0$ \\ \hline
\multirow{1}{*}{Ellipsis} & Beaker & $0$ & $0$ & $3$ & $1$ \\ \hline
\end{tabular}
\end{center}
\end{footnotesize}
\caption{Count of phenomena in \textsc{Alchemy}.}
\label{tab:alchemy}
\end{table}
\paragraph{\textsc{Alchemy}} Table~\ref{tab:alchemy} shows phenomena counts in \textsc{Alchemy}.
Each interaction contains on average $1.4$ references dependent on the interaction history.
Each non-first utterance contains on average $0.3$ references.
The most common form of reference is explicit coreference (Coref.) to previously-mentioned beakers, for example \nlstring{mix it}.
Other references are to previous actions, referring to the action only (e.g., \nlstring{same with the last beaker}) or the action as well as the arguments (e.g., \nlstring{same for one more unit}, referring to draining one unit from a previously-used beaker).
Ellipsis occurred four times in the thirty evaluated interactions, for example \nlstring{then, drain 1 unit}, implicitly referring to a specific beaker to drain from.
\begin{table}
\begin{footnotesize}
\begin{center}
\begin{tabular}{|l|l|c|c|c|c|c} \hline
& & $1$ & $2$ & $3$ & $4$ \\ \hline
\multirow{3}{*}{Coref.} & Person & $42$ & $16$ & $5$ & $3$ \\ \ddline{2-6}
& Hat & $2$ & $0$ & $0$ & $0$ \\ \ddline{2-6}
& Action + Arguments & $3$ & $0$ & $0$ & $0$ \\ \ddline{2-6}
& Position & $2$ & $0$ & $0$ & $0$ \\ \hline
\end{tabular}
\end{center}
\end{footnotesize}
\caption{Count of phenomena in \textsc{Scene}.}
\label{tab:scene}
\end{table}
\paragraph{\textsc{Scene}} Table~\ref{tab:scene} shows phenomena counts in \textsc{Scene}.
Each interaction contains on average $2.4$ references dependent on the interaction history.
Each non-first utterance contains on average $0.6$ references.
The most common form of reference is explicit coreference (Coref.) to previously-mentioned people, for example \nlstring{he moves to the left end}.
Coreference also occurs on hat colors (e.g., \nlstring{he gives it back}), actions along with their arguments (e.g., \nlstring{they did it again} referring to trading specific hats), and positions (e.g., \nlstring{he moves back}).
\begin{table}
\begin{footnotesize}
\begin{center}
\begin{tabular}{|l|l|c|c|c|c|c} \hline
& & $1$ & $2$ & $3$ & $4$ \\ \hline
\multirow{3}{*}{Coref.} & Object & $13$ & $0$ & $0$ & $0$ \\ \ddline{2-6}
& Object via Arguments & $6$ & $10$ & $2$ & $1$ \\ \ddline{2-6}
& Position & $0$ & $2$ & $0$ & $0$ \\ \ddline{2-6}
& Action & $5$ & $2$ & $0$ & $0$ \\ \ddline{2-6}
& Action + Arguments & $1$ & $0$ & $0$ & $0$ \\ \hline
\multirow{2}{*}{Ellipsis} & Position & $3$ & $0$ & $0$ & $0$ \\ \ddline{2-6}
& Action + Arguments & $1$ & $0$ & $0$ & $0$ \\ \hline
\end{tabular}
\end{center}
\end{footnotesize}
\caption{Count of phenomena in \textsc{Tangrams}.}
\label{tab:tangrams}
\end{table}
\paragraph{\textsc{Tangrams}} Table~\ref{tab:tangrams} shows phenomena counts in \textsc{Tangrams}.
Each interaction contains on average $1.7$ references dependent on the interaction history.
Each non-first utterance contains on average $0.4$ references.
The most common form of reference is on objects via reference to a previous step, for example \nlstring{put the item you just removed in the second spot}.
This requires recalling actions taken in previous turns, including the actions' arguments and the previous world state.
Coreference (Coref.) also occurs for positions (e.g., \nlstring{...where the last deleted figure was}), actions (e.g., \nlstring{do the same with the second to last figure and one before it}), and actions along with the previously-used arguments (e.g., \nlstring{repeat the first step}).
Ellipsis occurs for positions (e.g., \nlstring{add it again}, implicitly referring to the item's previous location) and actions along with their arguments (e.g., \nlstring{undo the last step}).
\section{Domain-Specific Implementation Details}
\label{sec:sup:domains}
For each domain \textsc{Alchemy}, \textsc{Scene}, and \textsc{Tangrams}, we describe the world state representation, state distance function, transition function, and the state encoder.
For all states $s$, $s = T(s, \mathtt{STOP})$.
\paragraph{\textsc{Alchemy}}
The world state in \textsc{Alchemy} is a sequence of beakers $\langle \bar{b}_1, \bar{b}_2, ..., \bar{b}_N \rangle$ of fixed length $N = 7$.
Each beaker $\bar{b}_i = \langle c_{i, 1}, c_{i, 2}, ... c_{i, \length{\bar{b}_i}} \rangle$ is a variable length sequence containing chemical units $c$, each one of six possible colors.
The distance between two world states is the sum over distances for each corresponding beaker pair. The distance between two beakers is the edit distance of the list of chemical units in each.
The action space of \textsc{Alchemy} includes two action types, $\act{pop}$ and $\act{push}$.
The $\act{pop}$ action takes one argument: $\act{n} \in \{1, \dots, N\}$ denoting the beaker to pop a chemical unit from.
The $\act{push}$ action takes two arguments: $\act{n}$ and $\act{c}$, one of six colors.
The transition function $T$ is defined by two cases: (a) $T(s, a = \act{push~ n~ c})$ will return a state where $\act{c}$ is added to the beaker with index $\act{n}$; and (b) $T(s, a = \act{pop~ n~})$ will remove the top element from the beaker with index $\act{n}$, or if the beaker with index $\act{n}$ is empty, the input state $s$ is returned.
The state encoding function $\func{Enc}$ is parameterized by (a) $\embedding^c$, an embedding function for each color; (b) $\embedding^p$, a positional embedding function for each beaker position; and (c) $\rnn^B$, a forward RNN used to encode each beaker.
We encoder each beaker $\bar{b}_i$ with an RNN:
\begin{small}
\begin{equation*}
\mathbf{h}^b_{i, j} = \rnn^B\left(\embedding^c(c_{i, j}); \mathbf{h}^b_{i, j-1}\right)\;\;.
\end{equation*}
\end{small}
\noindent $\func{ENC}$ returns a set of $N$ vectors $\{ \mathbf{h}_i \}_{i=1}^N$ , where each $\mathbf{h}_i = [\mathbf{h}^b_{i, \length{\bar{b}_i}}; \embedding^p(i)]$ represents a beaker.
\paragraph{\textsc{Scene}}
The world state in \textsc{Scene} is a sequence of positions $S = \langle p_1, p_2, ..., p_N \rangle$ of fixed length $N=10$.
Each position is a tuple $p_i = \langle s_i, h_i \rangle$, where $s_i$ is a shirt color $h_i$ is a hat color. There are six colors, and a special $\mathtt{NULL}$ marker indicating no shirt or hat is present.
The distance between two world states is the sum over positions of the number of steps required to modify two corresponding positions to be the same given the domain actions space.
The action space of \textsc{Scene} includes four action types: $\act{appear\_person}$, $\act{appear\_hat}$, $\act{remove\_person}$, and $\act{remove\_hat}$.
$\act{appear\_person}$ and $\act{appear\_hat}$ take two arguments: a position index $\act{N}$ and a color $\act{C}$.
$\act{remove\_person}$ and $\act{remove\_hat}$ take one argument: a position index $\act{N}$.
The transition function $T$ is defined by four cases: (a) $T(s, a = \act{appear\_person~ N~ C})$ returns a state where position $\act{N}$ contains shirt color $\act{C}$ if the shirt color in position $\act{N}$ is $\mathtt{NULL}$, otherwise the action is invalid and the input state $s$ is returned; (b) $T(s, a = \act{appear\_hat~ N~ C})$ is defined analogously to $\act{appear\_person}$; (c) $T(s, a = \act{remove\_person~ N})$ returns a state where the shirt color at position $\act{N}$ is set to $\mathtt{NULL}$ if there is a color at position $\act{N}$, otherwise the action is invalid and the input state $s$ is returned; and (d) $T(s, a = \act{remove\_hat~ N})$ is defined analogously to $\act{remove\_person}$.
The state encoding function $\func{Enc}$ is parameterized by (a) $\embedding^c$, an embedding function for shirt and hat colors; (b) $\embedding^p$, a positional embedding for each position in the scene; and (c) $\rnn^S$, a bidirectional RNN over all positions in order.
Each position is embedded using a function $\phi'(p_i) = [\embedding^c(s_i); \embedding^c(h_i); \embedding^p(i)]$.
We compute a sequence of forward hidden states:
\begin{small}
\begin{equation*}
\overrightarrow{\mathbf{h}}^s_{i} = \overrightarrow{\rnn^S}\left(\phi'(p_i); \overrightarrow{\mathbf{h}}^s_{i -1}\right)\;\;.
\end{equation*}
\end{small}
\noindent The backward RNN is equivalent. $\func{ENC}$ returns the set $\{ \mathbf{h}_i\}_{i = 1}^N$, where $\mathbf{h}_i = [ \overrightarrow{\mathbf{h}}^s_{i} ; \overleftarrow{\mathbf{h}}^s_{i}; \phi'(p_i)]$ represents a position.
\paragraph{\textsc{Tangrams}}
The world state in \textsc{Tangrams} is a list of positions $T = \langle p_1, p_2, ..., p_n \rangle$ of a variable length $n$.
Each position contains one of five unique shapes.
The distance function between states is the edit distance between the lists, with a cost of two for substitutions.
The action space of \textsc{Tangrams} includes two action types, $\act{INSERT}$ and $\act{REMOVE}$.
The $\act{INSERT}$ action takes two arguments: a position $\act{N} \in \{ 1, \cdots, M\}$, where $M$ is the maximum length of a state in the \textsc{Tangrams} dataset, and a shape type $\act{T}$, which is one the five possible shapes.
The $\act{REMOVE}$ action takes a single argument: a position $\act{N}$.
The transition function $T$ is defined by two cases: (a) $T(s, a = \act{INSERT~ N~ T})$ returns a state where the shape $\act{T}$ is in position $\act{N}$ and all objects to its right shifted by one position if $\act{T}$ is not already in the state, otherwise the action is invalid and $s$ is returned; and (b) $T(s, a = \act{REMOVE~ N})$ returns a state where the object in position $\act{N}$ was removed if $\act{N} \leq n$, otherwise the action is invalid and $s$ is returned.
The state encoding function $\func{ENC}$ is parameterized by (a) $\mathbf{h}_{NULL}$, a vector used when $n=0$; (b) $\phi^s$, an embedding function for the shapes; and (c) $\embedding^p$, a positional embedding of the position $i$.
$\func{ENC}$ returns a set $\{ \mathbf{h}_i\}_{i=1}^n$, where $\mathbf{h}_i = [\embedding^p(i); \phi^s(p_i)]$ is the position encoding, or it returns $\{\mathbf{h}_{NULL}\}$ if the state contains no objects.
\section{SCONE Domains and Data}
\label{sec:data}
\begin{table}[t!]
\centering
\begin{footnotesize}
\begin{tabular}{|l|c|c|c|} \hline
& \textsc{Alc} & \textsc{Sce} & \textsc{Tan} \\
\hline
\# Sequences (train) & $3657$ & $3352$ & $4189$\\
\# Sequences (dev) & $245$ & $198$ & $199$ \\
\# Sequences (test) & $899$ & $1035$ & $800$ \\
\hdashline[0.5pt/1pt]
Mean instruction & \multirow{ 2}{*}{\stdev{8.0}{3.2}} & \multirow{ 2}{*}{\stdev{10.5}{5.5}} & \multirow{ 2}{*}{\stdev{5.4}{2.4}} \\
~~~length & && \\
Vocabulary size & $695$ & $816$ & $475$\\
\hline
\end{tabular}
\end{footnotesize}
\vspace{-5pt}
\caption{Data statistics for \textsc{Alchemy} (\textsc{Alc}), \textsc{Scene} (\textsc{Sce}), and \textsc{Tangrams} (\textsc{Tan}).}
\label{tab:stats}
\vspace{-5pt}
\end{table}
\begin{table}[t!]
\begin{footnotesize}
\begin{center}
\begin{tabular}{|l|c|l|c|c|c|c|c} \hline
& Refs/Ex & & $1$ & $2$ & $3$ & $4$ \\ \hline
\multirow{2}{*}{\textsc{Alchemy}} & \multirow{2}{*}{1.4} & Coref. & $28$ & $7$ & $2$ & $0$ \\ \ddline{3-8}
& & Ellipsis & $0$ & $0$ & $3$ & $1$ \\ \hline
\multirow{2}{*}{\textsc{Scene}} & \multirow{2}{*}{2.4} & Coref. & $49$ & $16$ & $5$ & $3$ \\ \ddline{3-8}
& & Ellipsis & $0$ & $0$ & $0$ & $0$ \\ \hline
\multirow{2}{*}{\textsc{Tangrams}} & \multirow{2}{*}{1.7} & Coref. & $25$ & $14$ & $2$ & $1$ \\ \ddline{3-8}
& & Ellipsis & $4$ & $0$ & $0$ & $0$ \\ \hline
\end{tabular}
\end{center}
\end{footnotesize}
\vspace{-5pt}
\caption{Counts of discourse phenomena in SCONE from $30$ randomly selected development interactions for each domain. We count occurrences of coreference between instructions (e.g., \nlstring{he leaves} in \textsc{Scene}) and ellipsis (e.g., \nlstring{then, drain 2 units} in \textsc{Alchemy}), when the last explicit mention of the referent was $1$, $2$, $3$, or $4$ turns in the past. We also report the average number of multi-turn references per interaction (Refs/Ex).}
\label{tab:analysis}
\vspace{-10pt}
\end{table}
SCONE has three domains: \textsc{Alchemy}, \textsc{Scene}, and \textsc{Tangrams}.
Each interaction contains five instructions.
Table~\ref{tab:stats} shows data statistics.
Table~\ref{tab:analysis} shows discourse reference analysis.
State encodings are detailed in the Supplementary Material.
\paragraph{\textsc{Alchemy}}
Each environment in \textsc{Alchemy} contains seven numbered beakers, each containing up to four colored chemicals in order.
Figure~\ref{fig:example} shows an example.
Instructions describe pouring chemicals between and out of beakers, and mixing beakers.
We treat all beakers as stacks. There are two action types: \act{push} and \act{pop}.
\act{pop} takes a beaker index, and removes the top color.
\act{push} takes a beaker index and a color, and adds the color at the top of the beaker.
To encode a state, we encode each beaker with an RNN, and concatenate the last output with the beaker index embedding. The set of vectors is the state embedding.
\paragraph{\textsc{Scene}}
Each environment in \textsc{Scene} contains ten positions, each containing at most one person defined by a shirt color and an optional hat color.
Instructions describe adding or removing people, moving a person to another position, and moving a person's hat to another person.
There are four action types: \act{add\_person}, \act{add\_hat}, \act{remove\_person}, and \act{remove\_hat}.
\act{add\_person} and \act{add\_hat} take a position to place the person or hat and the color of the person's shirt or hat.
\act{remove\_person} and \act{remove\_hat} take the position to remove a person or hat from.
To encode a state, we use a bidirectional RNN over the ordered positions. The input for each position is a concatenation of the color embeddings for the person and hat.
The set of RNN hidden states is the state embedding.
\paragraph{\textsc{Tangrams}}
Each environment in \textsc{Tangrams} is a list containing at most five unique objects.
Instructions describe removing or inserting an object into a position in the list, or swapping the positions of two items.
There are two action types: \act{insert} and \act{remove}.
\act{insert} takes the position to insert an object, and the object identifier.
\act{remove} takes an object position.
We embed each object by concatenating embeddings for its type and position. The resulting set is the state embedding.
|
3,212,635,537,477 | arxiv | \section{Introduction}
Detection of a pure spin current, i.e., the flow of spin angular momentum without a charge current, with spin-precession signals in semiconductors (SCs) has been reported by measuring four-terminal nonlocal voltages \cite{Johnson,Jedema_Nature} in lateral spin-valve (LSV) devices with SC layers such as GaAs \cite{Lou_NatPhys,Ciorga_PRB,Peterson_PRB}, GaN \cite{Bhattacharya_APL,Park_NC}, Si \cite{Jonker_APL,Shiraishi_PRB,Suzuki_APEX,Saito_IEEE,Ishikawa_PRB,Jansen_PRAP}, and Ge \cite{Zhou_PRB,Fujita1,Fujita2,Yamada1,Yamada2}.
The four-terminal nonlocal measurements \cite{Johnson,Jedema_Nature} are surely important to demonstrate reliable spin transport and to investigate spin relaxation phenomena in SCs \cite{Peterson_PRB,Ishikawa_PRB,Fujita1,Fujita2,Yamada1}.
On the other hand, the transport of spin-polarized charge currents flowing between two ferromagnets (FMs) through SC should also be understood for semiconductor spintronic applications \cite{AwschalomFlatte, Dery_Nature, Zutic_RMP}.
To date, there have been lots of reports on the electrical detection of the transport of spin-polarized charge carriers by using local two-terminal spin-transport measurements in FM-SC-FM structures \cite{Mattana_GaAs, Appelbaum_Si,Hamaya_QD1,Sasaki_APL,Ciorga_AAD,Bruski_APL,PLi_PRL,Sasaki_PRAP,Saito_JAP,Kawano_PRmat,Oltscher_2DEG}.
However, because only a few local spin signals have been discussed by simultaneously showing a comparison with the nonlocal spin transport signals in SC-based LSVs \cite{Ciorga_AAD,Saito_JAP,Sasaki_APL, Bruski_APL,Oltscher_2DEG},
there are some unclear physics relevant to the magnitude of the local two-terminal spin signals.
According to the one-dimensional spin diffusion models \cite{TakahashiMaekawa,Jedema_PRB, Kimura_Metal}, the magnitude of the local spin signal is twice as large as that of the nonlocal spin signal.
For all metallic LSVs, most local spin signals are able to be explained by the conventional models \cite{Jedema_PRB, Kimura_Metal, KimuraHamaya}.
On the other hand, the correlation between local and nonlocal spin signals was not straightforward in SC-based LSVs \cite{Saito_JAP,Sasaki_APL, Bruski_APL}; Sasaki {\it et al.} and Bruski {\it et al.} showed the relatively large (4 $\sim$ 10 times) magnitude of local spin signals compared to the theoretical ones in Si- and GaAs-based LSVs, respectively.
They have so far regarded one of the origins as an enhancement in the spin diffusion length of the SC spin-transport layers at finite bias voltages \cite{Sasaki_APL, Bruski_APL}, where Yu {\it et al}. theoretically suggested the presence of the spin-drift effect in the nondegenerate SC layers in FM-SC hybrid systems \cite{Yu}.
However, because the previous study on Si \cite{Sasaki_APL} used strongly degenerate SC layers and the FM/MgO/SC tunnel contacts with non-Ohmic electrical properties, the effect of the bias voltage on the local spin signals remains an open question. At least, the influence of the FM/SC interfaces on detecting local spin signals should be discussed in FM-SC hybrid systems.
Here we experimentally study the magnitude of the local spin accumulation signals as a function of bias voltages applied between the two ferromagnetic contacts in FM-SC hybrid systems.
In this study, we use LSVs consisting of the spin injector and detector with relatively low resistance area product ($RA$) and degenerate Ge as a spin transport layer, as shown in our previous works \cite{Fujita1,Fujita2,Yamada1,Yamada2}.
We experimentally find extraordinarily nonmonotonic responses including the sign change of the two-terminal spin-accumulation signals with respect to the bias voltage applied between the two FM/SC contacts.
These features cannot be explained by the spin-drift effect in the SC layer, discussed in previous works for Si \cite{Sasaki_APL} and GaAs \cite{Bruski_APL}.
A possible mechanism is discussed by considering the asymmetric bias dependence of the spin polarization at the FM/SC interfaces.
These behavior can be observed up to room temperature, meaning that it is pretty important for spintronic applications to simultaneously use a low $V_{\rm bias}$ condition and highly efficient spin injector and detector in FM-SC-FM hybrid systems.
\section{Experimental}
To explore the two-terminal local spin signals in FM-SC hybrid systems, we have prepared LSVs with an $n$-Ge spin-transport channel and two ferromagnetic contacts, as shown in Fig. 1(a).
First, an undoped Ge(111) layer ($\sim$28 nm) at 350 $^\circ$C (LT-Ge) was grown on commercial undoped Si(111) substrates ($\rho$ $\sim$ 1000 $\Omega$cm), followed by an undoped Ge(111) layer ($\sim$70 nm) grown at 700 $^\circ$C (HT-Ge), where we utilized the two-step growth technique by molecular beam epitaxy (MBE) \cite{Sawano_TSF}.
Next, a 70-nm-thick phosphorous (P)-doped $n^{+}$-Ge(111) layer (doping concentration $\sim$ 10$^{19}$ cm$^{-3}$) was grown by MBE at 350 $^\circ$C on top of it as the spin transport layer.
To promote the tunneling conduction at FM/Ge interfaces, a P $\delta$-doped Ge layer with an ultra-thin Si layer was grown on top of the $n$$^{+}$-Ge layer \cite{MYamada_APL}.
We have so far developed Schottky-tunnel contacts with a $\delta$-doping layer near the interface to obtain the tunneling conduction of electrons via FM/SC interfaces \cite{Ando_APL,Kasahara_JAP}.
As a spin injector and detector, we grew Co$_{2}$FeAl$_{x}$Si$_{1-x}$ (CFAS) layers \cite{Fujita2} on top of it by nonstoichiometric growth techniques with Knudsen cells by MBE \cite{Hamaya_PRL,Fujita2}.
We note that a highly spin-polarized material, Co$_{2}$FeAl$_{x}$Si$_{1-x}$ (CFAS), is utilized as a spin injector and detector for Ge, where CFAS is one of the Heusler alloys \cite{Hono,Inomata}.
High quality heterointerfaces between CFAS and Ge were guaranteed by direct structural observations \cite{Lazarov}.
Just like our previous works \cite{Fujita1,Fujita2,Yamada1,Yamada2}, the FM/$n$$^{+}$-Ge contacts enabled Schottky tunnel conduction of electrons for electrical spin injection and detection.
Finally, the grown layers were patterned into the contacts with a width of 0.4 $\mu$m (FM1) or 1.0 $\mu$m (FM2).
The detailed fabrication processes were presented in Fig. S1 in Supplemental material \cite{Supplemental}.
Here we first fabricated two different devices named device A and device B.
Device A has a channel width ($w$) of 5.0 $\mu$m and the edge-to-edge distance ($d$) between CFAS contacts is 2.0 $\mu$m, as schematically shown in Fig. 1(a).
The top view of the actual device is shown in Fig. 1(b).
On the other hand, device B has a $w$ of 7.0 $\mu$m and a $d$ of 0.4 $\mu$m (not shown here).
As reference devices, we also fabricated device C and device D, as shown in Supplemental material \cite{Supplemental}.
To observe room-temperature signals, we further fabricated device E with the same CFAS contacts and $d =$ 0.35 $\mu$m.
As depicted in Fig. 1(a), local and nonlocal voltage measurements were carried out in the two- and four-terminal schemes in the same device \cite{Johnson,Jedema_Nature,TakahashiMaekawa,Jedema_PRB, Kimura_Metal,KimuraHamaya}. In the two-terminal scheme, the spin polarized electrons are injected and extracted beneath the FM/SC contacts, leading to the nonequilibrium spin accumulation in the SC layer.
The magnitude and sign of the created spin accumulation depend on the polarity of the spin polarization.
\begin{figure}
\begin{center}
\includegraphics[width=7.5cm]{Fig1r.eps}
\caption{(Color online) (a) Schematic illustration of a FM-SC-FM LSV, in which measurement schemes for nonlocal and local voltage detections are drawn. (b) Optical micrograph of a FM-SC-FM LSV (device A). (c) Nonlocal magnetoresistance curve measured at $I = -0.5$ mA at 8 K in device A. (d) Local magnetoresistance curve in the same conditions ($I =$ $-$0.5 mA at 8 K). The blue dotted curve is a minor-loop meaning that the anti-parallel magnetization state between FM1 and FM2 is stable. }
\end{center}
\end{figure}
\section{Results}
\subsection{Spin accumulation signals}
Figure 1(c) shows a representative nonlocal spin signal ($\Delta R_{\rm NL} = \Delta V_{\rm NL}/I$) of device A by applying in-plane magnetic fields ($B_{\rm y}$) at $I =$ $-$0.5 mA at 8 K.
Here the negative sign of $I$ ($I < 0$) indicates that the spin polarized electrons are injected into the SC layer from FM, i.e., spin injection condition via the Schottky-tunnel barrier. The value of $RA$ for the contacts in device A is $\sim$200 $\Omega$$\mu$m$^{2}$, which is the same order as those in our previous works \cite{Fujita2,Yamada1}.
The observed hysteretic nature depends clearly on the parallel and anti-parallel magnetization states between FM1 and FM2, as depicted in the arrows in Fig. 1(c).
In the nonlocal measurements by applying out-of-plane magnetic fields ($B_{\rm z}$), we have also observed spin-precession signals (Hanle-effect curves), indicating reliable spin transport in the SC layer [see Fig. S2(a) in the Supplemental material \cite{Supplemental}].
Using the same device (device A), we measured a local spin signal ($\Delta R_{\rm L} = \Delta V_{\rm L}/I$) by applying $B_{\rm y}$ in the same conditions ($I =$ $-$0.5 mA at 8 K), as shown in Fig. 1(d), where the small negative ones due to the anisotropic magnetoresistance (AMR) effect in the larger FM electrode (FM2) can be seen within $\pm$16 mT.
Although this feature cannot be observed in some cases, these AMR signals are rather proof of the formation of the antiparallel states after $B_{\rm y}$ exceeds $\pm$16 mT.
Clear positive $\Delta R_{\rm L}$ changes ($|{\Delta}R_{\rm L}|$) with hysteretic behavior are successfully observed when $B_{\rm y}$ exceeds $\pm$16 mT, meaning that the positive $|{\Delta}R_{\rm L}|$ implies a conventional spin-dependent transport of electrons through the SC layer.
To verify the reliability, we also plotted minor-loop data, measured in the same condition, as a blue dashed curve.
The evident minor-loop means that the observed positive $\Delta R_{\rm L}$ changes in Fig. 1(d) are surely attributed to the spin-dependent transport of electrons through the SC layer.
This is a proof of the presence of the nonequilibrium spin accumulation in the SC layer in an FM-SC-FM LSV.
In addition, we obtained a Hanle-effect curve even in the local measurements by applying $B_{\rm z}$, as shown in Fig. S2(b) in the Supplemental material \cite{Supplemental}.
As we focus on the magnitude of the local spin signal $|{\Delta}R_{\rm L}|$, the ratio of $|{\Delta}R_{\rm L}|$/$|{\Delta}R_{\rm NL}|$ is $\sim$2.7, which is slightly deviated from the value interpreted in terms of the one-dimensional spin diffusion models \cite{Jedema_PRB, Kimura_Metal}.
It should be noted that the $|{\Delta}R_{\rm L}|$/$|{\Delta}R_{\rm NL}|$ value is relatively small compared to those in LSVs with Si \cite{Saito_JAP,Sasaki_APL} and GaAs \cite{Bruski_APL}.
\subsection{Bias voltage effect on spin accumulation}
To investigate the correlation between the magnitudes of local and nonlocal spin signals, we firstly explore the bias-current ($I$) dependence of the local spin signals as voltage changes ($\Delta V_{\rm L}$).
Figure 2(a) shows $\Delta V_{\rm L}$ as a function of $B_{\rm y}$ for device A for various $I$ values applied between the two FM contacts at 8 K.
Interestingly, we can clearly see the sign changes in $\Delta V_{\rm L}$ even in the same $I$ polarity, indicating that the created spin accumulation does not depend linearly on $I$.
To verify the extraordinary behavior in detail, we summarize the detected $\Delta V_{\rm L}$ values as a function of $I$ in Fig. 2(b).
For devices A and B, there are sign changes in the same bias polarity, leading to the {\it wave-like} nonmonotonic variation in $\Delta V_{\rm L}$.
These {\it wave-like} behavior have never been observed in local spin transport measurements of FM-SC-FM LSVs.
\begin{figure}
\begin{center}
\includegraphics[width=7.5cm]{Fig2r.eps}
\caption{(Color online) (a) Local spin accumulation signals at 8 K at $I = -4.5, -1.5, +0.3,$ and $+4.0$ mA for device A.
The sign changes in $\Delta V_{\rm L}$ even in the same $I$ polarity are observed and the magnitude of $\Delta V_{\rm L}$ ($|\Delta V_{\rm L}|$) in the high $I$ region becomes smaller than that in the low $I$ region.
(b) $I$ dependence of $\Delta V_{\rm L}$ at 8 K for devices A (open circle) and B (open square). The amplitude of the wave for device B is larger than that for device A because the $d$ value in device B is smaller than that in device A. (c) $V_{\rm bias}$ dependence of $\Delta V_{\rm L}$ at 8 K for devices A (closed circle) and B (closed square). }
\end{center}
\end{figure}
In a standard theory based on the one-dimensional spin drift-diffusion in FM1-SC-FM2 systems including tunnel barriers \cite{Fert_PRB},
$\Delta V_{\rm L}$ is increased with increasing $I$ and the sign of $\Delta V_{\rm L}$ is associated with the polarity of $I$ as follows.
\begin{equation}
\label{eq: Fertmodel}
\Delta V_{\rm L} = \frac{2I(\beta r_{\rm F} + \gamma_{1} r_{\rm b}^*)(\beta r_{\rm F} + \gamma_{2} r_{\rm b}^*)}{(r_{\rm b}^* + r_{\rm F}){\rm cosh}(\frac{d}{\lambda_{\rm N}}) + (\frac{r_{\rm N}}{2})\{1 + (\frac{r_{\rm b}^*}{r_{\rm N}})^2\}{\rm sinh}(\frac{d}{\lambda_{\rm N}})},
\end{equation}
where $\gamma_{\rm 1}$ and $\gamma_{\rm 2}$ are the spin polarizations of the FM1/SC and FM2/SC interfaces, $r_{\rm b}^*$ indicates the $RA$ value for the FM/SC interfaces. $\beta$ and $r_{\rm F}$ are the spin polarization and the spin resistance of the FM bulk. $\lambda_{\rm N}$ and $r_{\rm N}$ are the spin diffusion length and the spin resistance of the SC layer, respectively.
If the $\gamma_{\rm 1}$ and $\gamma_{\rm 2}$ values are constant and the spin-dependent transport of electrons through the FM1/SC/FM2 structure stems from the spin accumulation in the SC layer including FM/SC interfaces, the tendency observed in Fig. 2(b) can not be explained by the standard theory in Eq. (\ref{eq: Fertmodel}).
To understand the behavior in Fig. 2(b), we summarize $\Delta V_{\rm L}$ as a function of bias voltage ($V_{\rm bias}$) applied between the two FM contacts.
Figure 2(c) displays the $\Delta V_{\rm L}$ values as a function of $V_{\rm bias}$, also including the {\it wave-like} behavior.
Considering the correlation between $\Delta V_{\rm L}$ and $I$ in Eq. (\ref{eq: Fertmodel}), we can recognize that the important parameter for understanding the {\it wave-like} behavior is not $I$ but $V_{\rm bias}$.
Thus, we should take the change in the $\gamma_{\rm 1}$ and $\gamma_{\rm 2}$ values with varying $V_{\rm bias}$ into account in the following sections.
\subsection{Asymmetric bias-voltage dependence}
To understand the {\it wave-like} nonlinear behavior in Figs. 2(b) and 2(c), we focus again on the nonlocal spin accumulation signals ($\Delta V_{\rm NL}$) as a function of the interface voltage ($V_{\rm int}$) at the FM/SC contact used in the same devices.
Figure 3(a) shows the plot of $\Delta V_{\rm NL}$ versus $V_{\rm int}$ for devices A and B, where two kinds of $\Delta V_{\rm NL}$ can be obtained by exchanging between the spin injector and detector for each device and $V_{\rm int}$ stands for the bias voltage applied to the interface of the spin injector (FM1 or FM2) detected by the three-terminal current-voltage measurements, as shown in the inset figures.
As references, we have also given the plot of $\Delta V_{\rm NL}$ versus spin injection current ($I_{\rm inj}$) for devices A and B in Fig. S3 in the Supplemental material \cite{Supplemental}.
For $V_{\rm int} < 0$, i.e., spin-injection conditions of electrons from FM to SC, almost all the positive $\Delta V_{\rm NL}$ increases with increasing $|V_{\rm int}|$.
On the other hand, for $V_{\rm int} > 0$ (spin extraction condition), the enhancement in the negative $\Delta V_{\rm NL}$ values is markedly suppressed, and then, the $\Delta V_{\rm NL}$ values approach to zero at around $V_{\rm int} =$ + 0.3 V.
These asymmetric features with respect to $V_{\rm int} =$ 0 V lead to the strong nonlinearities in the $V_{\rm int}$ dependence of $\Delta V_{\rm NL}$.
Although similar nonlinearities in the nonlocal spin accumulation signals were already observed in CoFe/GaAs-LSVs \cite{Salis_PRBR2}, the origin of the asymmetry in $\Delta V_{\rm NL}$ versus the bias voltage ($V$) applied to the FM/SC interface in Ref.\cite{Salis_PRBR2} was discussed based on the change in the spin-injection efficiency.
However, the feature in Fig. 3(a) implies that the spin accumulation created in the spin-extraction condition ($V_{\rm int} > 0$) strongly affects the asymmetry in $\Delta V_{\rm NL}$ versus $V_{\rm int}$.
Thus, we should reconsider the spin extraction efficiency (or the sensitivity of the spin detection) at the FM/SC interfaces, which has already been discussed \cite{Chantis2,Ando_PRB,Crooker_PRB,Shiogai_PRB}.
In Fig. S4 in the Supplemental material \cite{Supplemental}, we showed the $V_{\rm int}$ dependence of $\Delta V_{\rm NL}$ for device C and device D, where device C was an LSV with CFAS contacts annealed at 300 $^{\circ}$C and device D was an LSV with as-prepared CoFe contacts.
Unlike Fig. 3(a), the asymmetry with respect to $V_{\rm int} =$ 0 V was small for each FM/SC contacts.
Also, in Fig. S4 in the Supplemental material \cite{Supplemental}, we could see no {\it wave-like} behavior in the $V_{\rm bias}$ dependence of $\Delta V_{\rm L}$ for both devices C and D [see Supplemental material \cite{Supplemental}].
Here there was no influence of both the post annealing at 300 $^{\circ}$C and the change in the FM contact from CFAS to CoFe on the extracted parameters such as $\lambda_{\rm N}$ and the spin lifetime of the used SC layer ($n$-Ge) \cite{Vlado2}.
From these investigations, we can conclude that the $V_{\rm bias}$ dependence of $\Delta V_{\rm L}$ is affected by the asymmetry of the spin injection/detection efficiencies at the FM/SC contacts in FM-SC hybrid systems.
This interpretation is completely different from the previous ones on the basis of spin-drift effect induced by the electric-field to the SC channel layers \cite{Sasaki_APL, Bruski_APL}.
\begin{figure*}
\begin{center}
\includegraphics[width=14cm]{Fig3r.eps}
\caption{(Color online) (a) $V_{\rm int}$ dependence of $\Delta V_{\rm NL}$ at 8 K for devices A (circle) and B (square).
The open and closed symbols denote the data for FM1 and FM2, respectively, as a spin injector and the insets show the measurement schemes.
(b) $V_{\rm int}$ dependence of spin polarization ($P_{\rm inj}$) created by FM1 or FM2 in devices A and B.
(c) Normalized $\gamma_{\rm 1}$$\gamma_{\rm 2}$$I$ as a function of $V_{\rm bias}$ for devices A and B. The dashed curves indicate the experimental data shown in Fig. 2(c). }
\end{center}
\end{figure*}
\subsection{Estimation of the interface spin polarization}
When two-terminal local measurements in FM-SC-FM LSVs are considered, the value of $V_{\rm bias}$ shown in Fig. 2(c) is related to the both $V_{\rm int}$ values at the spin injector and the spin detector in addition to the bias voltage applied to the SC channel layer ($V_{\rm SC}$).
Because the $V_{\rm SC}$ value is sufficiently small compared to $V_{\rm int}$ values in our devices, we can roughly regard $V_{\rm bias}$ as the sum of the $V_{\rm int}$ values at the spin injector and the spin detector; if $V_{\rm bias} >$ 0 in a two-terminal local measurement in the FM1-SC-FM2 device, we should take the spin polarizations of the FM1/SC and FM2/SC interfaces at $V_{\rm int}$ ($>$ 0) in a spin extraction condition and at $V_{\rm int}$ ($<$ 0) in a spin injection condition, respectively.
Because the Schottky-tunnel contacts in our LSVs have almost no rectifying characteristics, as shown in our previous works \cite{Fujita1,Fujita2}, we can tentatively ignore the influence of the large rectification between the spin injection and spin extraction conditions.
In this situation, we can roughly evaluate the spin polarization at the FM/SC contacts from $\Delta V_{\rm NL}$ in Fig. 3(a).
When we regard the spin polarizations created from spin injector and spin detector contacts as $P_{\rm inj}$ and $P_{\rm det}$, the correlation among $\Delta V_{\rm NL}$, $P_{\rm inj}$, and $P_{\rm det}$ is expressed as follows \cite{Lou_NatPhys, Johnson, Jedema_Nature}.
\begin{equation}
\label{eq: NL}
\Delta V_{\rm NL} ={\frac{{P_{\rm inj}}{P_{\rm det}}{I_{\rm inj}}{\rho_{\rm N}}{\lambda_{\rm N}}}{S}}{\exp\left(-\frac{d}{\lambda_{\rm N}}\right)},
\end{equation}
where $\rho_{\rm N}$ and $S$ are the resistivity (17.4 $\Omega$$\mu$m) and the cross section area (device A : 0.35 $\mu$m$^{2}$, device B : 0.49 $\mu$m$^{2}$) of the SC layer.
The value of $\lambda_{\rm N}$ has already been clarified to be 0.56 $\mu$m at 8 K \cite{Fujita1}.
When the FM1/SC contact was used as a spin injector in the nonlocal voltage measurements, the value of $P_{\rm inj}$ can change with increasing $V_{\rm int}$.
On the other hand, the spin polarization of the non-biased contact (FM2/SC), $P_{\rm det}$, can be regarded as a constant value.
Only at low $V_{\rm int}$ conditions, we can roughly consider that the assumption of $|P_{\rm inj}| = |P_{\rm det}|$ is valid, leading to the values of $P_{\rm det} =$ 0.25 for device A and 0.11 for device B, respectively.
Using these $P_{\rm det}$ values and the above parameters, we can estimate the value of $P_{\rm inj}$ for various $I_{\rm inj}$, which can be converted to $V_{\rm int}$.
As consequences, the plots of the estimated $P_{\rm inj}$ versus $V_{\rm int}$ for the FM1/SC and FM2/SC contacts in devices A and B are presented in Fig. 3(b).
With increasing $|V_{\rm int}|$, the $P_{\rm inj}$ value is decreased, and the decrease in $P_{\rm inj}$ at $V_{\rm int}$ $>$ 0 (spin extraction condition) is slightly larger than that at $V_{\rm int}$ $<$ 0 (spin injection condition).
Note that the negative spin polarization created by the FM1/SC and FM2/SC contacts can be seen at $V_{\rm int} \sim$ $+$ 0.3 V.
Regarding the above $P_{\rm inj}$ values separately estimated for FM1 and FM2 as $\gamma_{\rm 1}$ and $\gamma_{\rm 2}$ of the FM1/SC and FM2/SC interfaces, we can also discuss the local spin accumulation voltage $\Delta V_{\rm L}$ in the FM1-SC-FM2 LSVs by using Eq. (\ref{eq: Fertmodel}).
Here $\beta = 1$ was used for FM1 and FM2 because the CFAS Heusler alloy was expected to be a half-metal \cite{Hono,Inomata}.
Even if we use $\beta = 1$, $\beta r_{\rm F}$ can be sufficiently smaller than $\gamma_{1} r_{\rm b}^*$ and $\gamma_{2} r_{\rm b}^*$ in this study.
Also, $r_{\rm N} (=\rho_{\rm N} \times \lambda_{\rm N}) =$ 9.74 $\Omega$$\mu$m$^{2}$.
In the local measurement conditions, the $r_{\rm b}^*$ values are varied within the range of 70 $\Omega$$\mu$m$^{2}$ $\le r_{\rm b}^*$ $\le$ 470 $\Omega$$\mu$m$^{2}$.
In this condition, we can roughly consider the relation, $\Delta V_{\rm L} \propto$ $\gamma_{1}$$\gamma_{\rm 2}$$I$.
As mentioned before, when we assume that $V_{\rm bias}$ $>$ 0 ($<$ 0) in the local measurement consists of $V_{\rm int}$ in a spin extraction (in a spin injection) for the FM1/SC contact and $V_{\rm int}$ in a spin injection (in a spin extraction) for the FM2/SC contact, we can calculate $\gamma_{1}$$\gamma_{\rm 2}$$I$ as a function of $V_{\rm bias}$ from Eq. (\ref{eq: Fertmodel}), as shown in Fig. 3(c), for devices A and B.
For example, as $V_{\rm bias} =$ $+$0.16 V in device B, the bias current of $I =$ $+$0.5 mA flows in the FM1-SC-FM2 structure.
In this condition, we can determine that the values of $V_{\rm int}$ (extraction) for the FM1/SC contact and of $V_{\rm int}$ (injection) for the FM2/SC contact are $+$0.11 V and $-$0.033 V, respectively.
At these $V_{\rm int}$ values, we can assign $\gamma_{\rm 1}$ and $\gamma_{\rm 2}$ to 0.028 and 0.042, respectively, resulting in $\gamma_{1}$$\gamma_{\rm 2}$$I \sim$ 0.00115.
The maximum value of $|\gamma_{1}$$\gamma_{\rm 2}$$I|$ was 0.00249 at $V_{\rm bias} =$ $-$0.17 V for device B.
In Fig. 3(c), the normalized values of $\gamma_{1}$$\gamma_{\rm 2}$$I$ are plotted, together with the data in Fig. 2(c).
As a result, the {\it wave-like} nonmonotonic behavior can qualitatively be reproduced only by considering the change in the $\gamma_{\rm 1}$ and $\gamma_{\rm 2}$ values with increasing $V_{\rm int}$ for both devices A and B. Thus, a predominant origin of the {\it wave-like} nonmonotonic behavior in Fig. 2(c) is the presence of the asymmetric $V_{\rm int}$ dependence of $\gamma_{\rm 1}$ and $\gamma_{\rm 2}$ in the FM1-SC-FM2 LSVs.
\section{Discussion}
Figure 4 shows schematic diagrams of the possible spatial distribution of the spin-dependent chemical potentials ($\mu_{\uparrow}$, $\mu_{\downarrow}$) in the SC layer, together with that in the FM/SC interface in the parallel and anti-parallel states under two-terminal local measurements.
The following is one of the possible situations of the spin accumulation in the SC layer.
In Fig. 4(a), as the magnetization state is switched from the parallel state to the anti-parallel one, the sign of the created spin accumulation near the spin detector (FM1/SC) is inverted, resulting in the change in the spatial distribution of the spin-dependent chemical potential in the SC layer.
The spatial distribution illustrated in Fig. 4(a) is dominant in low $V_{\rm bias}$ region for our LSVs.
With increasing $V_{\rm bias}$ [Fig. 4(b)], the spin extraction efficiency becomes markedly small due to the reduction in $\gamma_{1}$ at the FM1/SC detector interface.
Thus, the spin accumulation created near the spin injector (FM2/SC) interface decays exponentially from the spin injector side toward the detector one as shown in Fig. 4(b).
Because the spin extraction efficiency is negligibly small, the spin accumulation in the SC layer cannot be detected, leading to $\Delta V_{\rm L} \sim$ 0 even at a finite $V_{\rm bias}$.
In high $V_{\rm bias}$ region [Fig. 4(c)], the spin extraction efficiency at the FM1/SC detector interface recovers with a sign reversal of $\gamma_{1}$.
As a result, even in the parallel state, the spatial distribution of the spin-dependent chemical potential created in the SC layer is similar to that in the anti-parallel state shown in Fig. 4(a).
After the switching of the magnetization state from the parallel to the anti-parallel, that becomes like the parallel state shown in Fig. 4(a).
These phenomena mean that it is very important for the two-terminal spin-accumulation signals in FM-SC-FM LSVs to understand the spin extraction efficiency (or the sensitivity of the spin detection) at the FM/SC detector interface \cite{Chantis2,Ando_PRB,Crooker_PRB,Shiogai_PRB} and to utilize a low $V_{\rm bias}$ region.
\begin{figure}
\begin{center}
\includegraphics[width=7.5cm]{Fig4.eps}
\caption{(Color online) Schematic diagrams of possible spatial distribution of the spin-dependent chemical potentials ($\mu_{\uparrow}$, $\mu_{\downarrow}$) in the FM/SC interfaces and the SC layer in our LSVs for (a) $\gamma_{1}, \gamma_{2} > 0$, (b) $\gamma_{1} \approx 0, \gamma_{2} > 0$, and (c) $\gamma_{1} < 0, \gamma_{2} > 0$, with increasing $V_{\rm bias}$.
Here this figure is based on a condition of $V_{\rm bias} >$ 0 for the present device shown in Fig. 1, and the FM1/SC and FM2/SC contacts are the spin detector and spin injector, respectively.
The left and right figures are parallel and anti-parallel magnetization states between FM1 and FM2. }
\end{center}
\end{figure}
By comparing device A (B) in Fig. 3(a) and device C in Fig. S4 in the Supplemental material \cite{Supplemental}, there is a large difference in the $V_{\rm int}$ dependence of $\Delta V_{\rm NL}$.
When we checked the interface quality of the FM/SC contacts by using high angle annular dark field scanning transmission electron microscopy imaging, the degradation of the contact quality was clarified after the annealing \cite{Vlado2}. This implies that the sign inversion of $\gamma$ in our LSVs depends on the FM/SC interface quality at least.
However, it should be noted that the $V_{\rm bias}$ dependence of $\Delta V_{\rm NL}$ in device C is almost the same as that in device D with CoFe contacts, as shown in Fig. S5 in the Supplemental material \cite{Supplemental}.
This means that the sign inversion of $\gamma$ depends not only on the interface quality but also on the spin-injector/detector FM material.
Since the tunnel probability of electrons through the FM/SC interface depends on the density of states of the FM material \cite{Valenzuela_PRL}, we should generally consider the FM material used as the spin-injector and detector.
\begin{figure}[t]
\begin{center}
\includegraphics[width=7.5cm]{Fig5.eps}
\caption{(Color online) $V_{\rm bias}$ dependence of $\Delta V_{\rm L}$ from 150 to 296 K for device E.
The amplitude of the wave is reduced with increasing external temperatures. The inset shows a two-terminal magnetoresistance curve at 296 K. }
\end{center}
\end{figure}
Because we could not observe room-temperature local spin signals for devices A and B, we fabricated device E with smaller $d$ ($d =$ 0.35 $\mu$m, $RA \sim$ 100 $\Omega$$\mu$m$^{2}$).
Figure 5 shows $\Delta V_{\rm L}$ as a function of $V_{\rm bias}$ up to room temperature for device E.
As shown in the inset of Fig. 5, we can evidently see the two-terminal local magnetoresistance in a FM-SC-FM hybrid system with FM/SC Schottky tunnel contacts even at room temperature (296 K).
Note that similar behavior shown in Fig. 2(b) can be observed from 150 to 296 K, indicating the reproducibility of the nonmonotonic bias dependence of spin accumulation up to room temperature.
In high $V_{\rm bias}$ regions ($V_{\rm bias} >$ 0.25 V, $V_{\rm bias} <$ $-$0.45 V), $|\Delta V_{\rm L}|$ becomes smaller than those in low $V_{\rm bias}$ regions.
From the behavior shown in Fig. 5, together with Figs. 2(c), 3(a), and 3(b), the influence of the high $V_{\rm bias}$ on the $\Delta V_{\rm L}$ is extremely large.
For obtaining large $|\Delta V_{\rm L}|$ in FM-SC hybrid systems at room temperature, it is pretty important to simultaneously use a low $V_{\rm bias}$ condition and highly efficient spin injector and detector.
\section{Conclusion}
We experimentally found extraordinarily nonmonotonic response of the two-terminal spin signals, i.e., local spin-accumulation signals, in FM-SC-FM LSVs.
With respect to the bias voltage applied between the two FM/SC contacts, the local spin accumulation signal showed the {\it wave-like} variation including the sign changes.
These features were not reproduced for devices with the degraded contact interface quality.
A possible mechanism was discussed by considering the asymmetric bias dependence of the spin polarization at the FM/SC interfaces.
These behavior can be observed up to room temperature, meaning that it is pretty important for spintronic applications to simultaneously use a low $V_{\rm bias}$ condition and highly efficient spin injector and detector in FM-SC-FM hybrid systems.
\section{Acknowledgements}
Y.F. and K.H. acknowledge Dr. Ron Jansen of AIST in Japan for useful discussion to interpret the experimental data.
This work was partly supported by Grants-in-Aid for Scientific Research (A) (No. 16H02333) and (S) (No. 17H06120) from the Japan Society for the Promotion of Science (JSPS), and a Grant-in-Aid for Scientific Research on Innovative Areas `Nano Spin Conversion Science' (No. 26103003) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT). Y.F. acknowledges JSPS Research Fellowships for Young Scientists.
M.Y. acknowledges scholarships from the Toyota Physical and Chemical Research Institute Foundation and Grants-in-Aid for Scientific Activity Start-up (No. 17H06832) from JSPS.
|
3,212,635,537,478 | arxiv | \section{Introduction}
\label{sec:intro}
Modern NLP is dominated by large pre-trained models, systems which are large, complex, and costly to train.
As a result, much research effort is put into questions of tuning and configuring the various layers and training regimes for improving prediction quality on a growing number of tasks~\cite{rogers2020primer}.
Unfortunately, not as much research asks questions about the decisions made at the most upstream parts of the models, those that deal with input tokenization and subword vocabulary creation.
In this exploratory work, we isolate a single decision point which appears to be resolved arbitrarily by existing model developers, with no consensus but also no underlying theory:
\textbf{should subword tokenizers mark word boundaries at the beginning or the end?}
The immediate effects of such a decision are easy to describe: an out-of-vocabulary word\footnote{We use the term \textit{word} to refer to a space-delimited string of characters, post-pre-tokenization if such a process exists. Consequentially, \textit{subword} refers to tokens which are either word-length or form a proper substring of a word.} which is represented by multiple tokens, or \say{pieces}, e.g.~the plural noun \textit{polynomials}, shares the token \textit{polynomial} with its singular form in initial-boundary-marking tokenizers, but not in final-boundary-marking ones.
Conversely, a word prepended by a prefix like \textit{neoconservative} shares the stem token in final-boundary-marking tokenizers and not in the initial variants.
In lexical compounds, the boundary marking decision can affect whether \textit{thunderstorm} shares a token with its internal head, \textit{storm}, or its modifier \textit{thunder}.
The consequence for the unshared stem, in such cases, is often further segmentation since the suffix \textit{storm} (or prefix \textit{thunder}) rarely appears in the training corpus in a similar boundary-relative position.
It might appear at first that in English, a heavily-suffixing language \cite{wals-26}, a tokenizer that marks word-initial pieces is more appropriate.
Such a claim requires empirical support, but consideration of common practice can also be offered to challenge it:
for one, pre-tokenization such as punctuation separation and accent normalization is not always applied consistently when moving on to a downstream text.
A model that was trained on untreated text may find it difficult to process an NER dataset (for example) where punctuation is separated from preceding words, rendering a word-final-marking tokenizer more robust to change;
some tokenizers like BERT's Wordpiece~\cite{devlin-etal-2019-bert} \say{mark} a class of tokens by omission, i.e.~marking the non-initial pieces rather than initial ones.
This discrepancy surfaces edge case effects when compared with a seemingly-equivalent tokenizer like GPT-2's~\cite{radford2019language}, which marks initial pieces but only if they are prepended by a space (i.e.~not sentence-initial pieces).
We survey models' decisions for this question in \autoref{tab:systems}.
\input{tab_systems}
We offer three courses of analysis for the English Unigram LM tokenizer~\cite{kudo-2018-subword}, recently found to outperform BPE~\cite{sennrich-etal-2016-neural} on both statistical measures and downstream performance~\cite{bostrom-durrett-2020-byte}.
We conduct type-level, token-level and corpus-level statistical analysis on the initial-vs.-final variable~(\S\ref{sec:prop},\ref{sec:stat}), and add a morphological coverage test (\S\ref{sec:morph}) motivated by the question of subword models' ability to pick up meaningful linguistic generalizations using corpus statistics only.
We find that: (a) pre-tokenizing input text affects the boundary marking decision: unprocessed corpora produce better final-marking vocabularies, whereas pretokenized ones produce better initial-marking vocabularies;
(b) corpus compression efficiency trades off against information-theoretic measures;
(c) more morphemes are detected by final-marking vocabularies, but the techniques are complementary to each other and finding a combination of both can prove useful.
\section{Model Properties}
\label{sec:prop}
The unigram language model~\cite[Unigram LM,][]{kudo-2018-subword} is a method for creating subword vocabularies by optimizing a frequency objective over a large corpus:
a vocabulary
is initialized to include all subword sequences in the corpus above a frequency threshold, and then portions of the vocabulary are removed iteratively such that the likelihood scores for corpus-level word segmentations are minimally affected, until a pre-defined vocabulary size has been reached.
In the final vocabulary, each piece is assigned a score which is simply the log-probability of its appearance in the training corpus given the vocabulary, allowing for a decoding algorithm to select the most probable segmentation for a space-delimited word.
We train four Unigram LM models on a pre-specified corpus of 1 million sentences randomly sampled from the March 2019 dump of English Wikipedia.
We use the Sentencepiece package\footnote{Version 0.1.91, \url{https://github.com/google/sentencepiece}} with full character coverage, trimming the piece vocabulary by a factor of 75\% iteratively until reaching a size of 32,000.
Our evaluated conditions are whether or not the text is pretokenized (using version 3.7 of the StanfordNLP tokenizer~\cite{qi2018universal}; no other normalization, such as lowercasing, is performed), and whether the word-initial or word-final piece is marked:\footnote{We accomplish this by simply reversing all training and inference texts, and applying the model as-is.}
\textsc{RawInit}: no pretokenization, word-initial marked;
\textsc{RawFin}: no pretokenization, word-final marked;
\textsc{StanInit}: Stanford pretokenization, word-initial marked;
\textsc{StanFin}: Stanford pretokenization, word-final marked.
At the type level, we note that normalized unigram frequencies are encoded as piece scores in the model artifact, so we can compute the entropy of the piece distributions.
A better-balanced model should present higher entropy, although this may also be the artifact of excessive splitting of frequent character sequences into several \say{cases}.
We find that under both pretokenization conditions, the \textsc{Fin} condition produces higher-entropy distributions: 7.549 vs. 7.509 on \textsc{Raw}, 7.400 vs. 7.285 on \textsc{Stan}.
\begin{figure*}
\centering
\begin{tabular}{cc}
\includegraphics[width=7.5cm,height=4cm]{figures/rawtok_unilm_dists.png}
&
\includegraphics[width=7.5cm,height=4cm]{figures/stantok_unilm_dists.png}
\\
\end{tabular}
\caption{Token length distributions for models trained on (left) raw, (right) Stanford-pretokenized samples.}
\label{fig:lengths}
\end{figure*}
Next, we consider the distributions of piece character lengths.
Intuitively, a model which finds more long pieces can more efficiently compress the information encoded in the corpus; \citet{bostrom-durrett-2020-byte} argued that a distribution with mean centered around 6 resembles that of English morphemes better.
As demonstrated in \autoref{fig:lengths}, these two arguments are at odds in our experiment: in the \textsc{Raw} condition, the final-marking model produces shorter pieces, but centered at 6.
In the \textsc{Stan} condition, the situation is reversed.
\section{Corpus analysis}
\label{sec:stat}
Our next evaluation considers the utility of each marking schema in the context of a large corpus.
We report several surface-level statistics which are not immediately translatable to downstream model efficacy, but are \say{clean} in the sense that they introduce no stochastic noise other than corpus selection, and may still indicate a desired preference for downstream model inputs.
We measure each statistic on the same \textbf{training} corpus as initially passed to the tokenizer; on a different sample of \textbf{in-domain text} from the same Wikipedia dump; and on an \textbf{out of domain} sample from Reddit's \texttt{r/politics} feed extracted in the first half of 2015.
\paragraph{Token ratio.}
We count the total pieces tokenized by the model, as a measure for efficient text encoding, and divide by the number of space-delimited words in the corpus (lower = better).
\paragraph{Corpus-level entropy.}
We distill the piece-type distribution across each corpus into a single entropy number, under the assumption that a good encoding balances token distribution out rather than relying on a \say{heavy head} of certain types (higher = better).
\paragraph{Bigram LM perplexity.}
We train a basic bigram language model with add-$\epsilon$ smoothing on each tokenizer's original training corpus and apply it to each inference corpus, measuring overall base-2 perplexity.
We expect differences between \textsc{Init} and \textsc{Fin} to be mainly the result of in-word prediction ability.
We use $\epsilon\leftarrow 0.005$ (lower = better).
\input{tab_corpus}
Our results, presented in \autoref{tab:corpus_stats}, show a number trends, all of which remain consistent across corpora and domains.
While type-level entropy is always lower in the \textsc{Init} marking scheme, other findings suggest a trade-offs between compression ability (better for \textsc{Init} on raw text but for \textsc{Fin} when pre-tokenized) and prediction regularity (vice versa).
It may be the case that final-marked tokens are easier to predict after non-final tokens in a left-to-right bigram language model, suggesting these results may be particularly important for downstream regressive models like GPT-$n$, as opposed to masked language models.
\section{Morphological recovery}
\label{sec:morph}
Given the difference in segmentation strategy, and ultimately in the difference in the quality of the representations,
we might ask how well both the initial- and final-marking implementations of Unigram LM recover morphologically well-formed subword sequences.
To that end, we consulted the morphological section of the English Lexicon Project \newcite{balota2007english} available from the \texttt{citylex} repository.\footnote{\url{https://github.com/kylebgorman/citylex}}
We extracted morphological information from all words in the database, identifying bound (e.g., \say{-ing}) and unbound morphemes (e.g., \say{cat}), resulting in a \say{gold} set of 18,794 strings corresponding to linguistically-motivated morphological forms.
\input{fig_morph}
\paragraph{Coverage.}
Tokenization from sentences has a sizeable effect on the performance of subword vocabularies on perplexity.
It is feasible that pre-tokenization will likewise influence morphological recovery.
The \textsc{RawInit} model recovered 5,170 (27\%) morphemes, while \textsc{RawFin} models recovered 6,049 (32\%), suggesting that initial-marking can be detrimental to morphological recovery.
Assessing the union of the subword vocabularies learned by both models reveals 7,898 (42\%) morphological subwords.
Some morphemes that were identified only by \textsc{RawInit} included 109 rarer initial morphemes (e.g., \textit{co-}), while \textsc{RawFin} identified some 482 non-initial morphemes that likely came from compounds (e.g., \textit{-office}, \textit{-league}).
We find similar but less stark findings for the pre-tokenized text, where the \textsc{StanInit} model recovered 5,391 morphemes (28\%), against 5,537 (29\%) for \textsc{StanFin}, with a union of 7,568 covered morphemes (40\%).
\paragraph{Likelihood.}
We compared the log probabilities that the \textsc{Init} and \textsc{Fin} models assign to each subword.
We sought to test whether higher language model scores are allocated to subword strings that are probable morphemes than to non-morphemes.
First, we characterized morphological agreement between the models as a single number: 0 (neither model identifies a subword as a morpheme); 1 (recognized as a morpheme in exactly one model); and 2 (both models identify the morpheme) as possible values.
We used a linear regression to predict subword tokens' log probabilities as a function of (a) morphological agreement score (0-2); and (b) the identity of the tokenization model from which the log probability was taken.
The model presented significantly higher log probabilities at higher agreement levels: a mean value of -10.7 for the 2 class, vs. -11.3 and -12.7 for (1) and (0) respectively ($\beta$ = 0.95, $t$ = 43.5, $p$ $<$ .001), but this was true to similar degrees for both the \textsc{Init} or \textsc{Fin} model.
The results are qualitatively similar for both the pretokenized and raw corpora.
We visualize the data used in these analyses in \autoref{fig:scores}, which shows the probabilities assigned by each model depending on a subword's morphological status.
Altogether, the results of these experiments suggest that the greatest power is in assessing the overlap between the \textsc{Init} and \textsc{Fin} models --- significant morphological faithfulness can be achieved by integrating both vocabularies.
\section{Morphological recovery}
\label{sec:morph}
Given the difference in segmentation strategy, and ultimately in the difference in the quality of the representations,
we might ask how well both the initial- and final-marking implementations of Unigram LM recover morphologically well-formed subword sequences.
To that end, we consulted the morphological section of the English Lexicon Project \newcite{balota2007english} available from the \texttt{citylex} repository.\footnote{\url{https://github.com/kylebgorman/citylex}}
We extracted morphological information from all words in the database, identifying bound (e.g., \say{-ing}) and unbound morphemes (e.g., \say{cat}), resulting in a \say{gold} set of 18,794 strings corresponding to linguistically-motivated morphological forms.
\input{fig_morph}
\paragraph{Coverage.}
Tokenization from sentences has a sizeable effect on the performance of subword vocabularies on perplexity.
It is feasible that pre-tokenization will likewise influence morphological recovery.
The \textsc{RawInit} model recovered 5,170 (27\%) morphemes, while \textsc{RawFin} models recovered 6,049 (32\%), suggesting that initial-marking can be detrimental to morphological recovery.
Assessing the union of the subword vocabularies learned by both models reveals 7,898 (42\%) morphological subwords.
Some morphemes that were identified only by \textsc{RawInit} included 109 rarer initial morphemes (e.g., \textit{co-}), while \textsc{RawFin} identified some 482 non-initial morphemes that likely came from compounds (e.g., \textit{-office}, \textit{-league}).
We find similar but less stark findings for the pre-tokenized text, where the \textsc{StanInit} model recovered 5,391 morphemes (28\%), against 5,537 (29\%) for \textsc{StanFin}, with a union of 7,568 covered morphemes (40\%).
\paragraph{Likelihood.}
We compared the log probabilities that the \textsc{Init} and \textsc{Fin} models assign to each subword.
We sought to test whether higher language model scores are allocated to subword strings that are probable morphemes than to non-morphemes.
First, we characterized morphological agreement between the models as a single number: 0 (neither model identifies a subword as a morpheme); 1 (recognized as a morpheme in exactly one model); and 2 (both models identify the morpheme) as possible values.
We used a linear regression to predict subword tokens' log probabilities as a function of (a) morphological agreement score (0-2); and (b) the identity of the tokenization model from which the log probability was taken.
The model presented significantly higher log probabilities at higher agreement levels: a mean value of -10.7 for the 2 class, vs. -11.3 and -12.7 for (1) and (0) respectively ($\beta$ = 0.95, $t$ = 43.5, $p$ $<$ .001), but this was true to similar degrees for both the \textsc{Init} or \textsc{Fin} model.
The results are qualitatively similar for both the pretokenized and raw corpora.
We visualize the data used in these analyses in \autoref{fig:scores}, which shows the probabilities assigned by each model depending on a subword's morphological status.
Altogether, the results of these experiments suggest that the greatest power is in assessing the overlap between the \textsc{Init} and \textsc{Fin} models --- significant morphological faithfulness can be achieved by integrating both vocabularies.
\section{Model Properties}
\label{sec:prop}
The unigram language model~\cite[Unigram LM,][]{kudo-2018-subword} is a method for creating subword vocabularies by optimizing a frequency objective over a large corpus:
a vocabulary
is initialized to include all subword sequences in the corpus above a frequency threshold, and then portions of the vocabulary are removed iteratively such that the likelihood scores for corpus-level word segmentations are minimally affected, until a pre-defined vocabulary size has been reached.
In the final vocabulary, each piece is assigned a score which is simply the log-probability of its appearance in the training corpus given the vocabulary, allowing for a decoding algorithm to select the most probable segmentation for a space-delimited word.
We train four Unigram LM models on a pre-specified corpus of 1 million sentences randomly sampled from the March 2019 dump of English Wikipedia.
We use the Sentencepiece package\footnote{Version 0.1.91, \url{https://github.com/google/sentencepiece}} with full character coverage, trimming the piece vocabulary by a factor of 75\% iteratively until reaching a size of 32,000.
Our evaluated conditions are whether or not the text is pretokenized (using version 3.7 of the StanfordNLP tokenizer~\cite{qi2018universal}; no other normalization, such as lowercasing, is performed), and whether the word-initial or word-final piece is marked:\footnote{We accomplish this by simply reversing all training and inference texts, and applying the model as-is.}
\textsc{RawInit}: no pretokenization, word-initial marked;
\textsc{RawFin}: no pretokenization, word-final marked;
\textsc{StanInit}: Stanford pretokenization, word-initial marked;
\textsc{StanFin}: Stanford pretokenization, word-final marked.
At the type level, we note that normalized unigram frequencies are encoded as piece scores in the model artifact, so we can compute the entropy of the piece distributions.
A better-balanced model should present higher entropy, although this may also be the artifact of excessive splitting of frequent character sequences into several \say{cases}.
We find that under both pretokenization conditions, the \textsc{Fin} condition produces higher-entropy distributions: 7.549 vs. 7.509 on \textsc{Raw}, 7.400 vs. 7.285 on \textsc{Stan}.
\begin{figure*}
\centering
\begin{tabular}{cc}
\includegraphics[width=7.5cm,height=4cm]{figures/rawtok_unilm_dists.png}
&
\includegraphics[width=7.5cm,height=4cm]{figures/stantok_unilm_dists.png}
\\
\end{tabular}
\caption{Token length distributions for models trained on (left) raw, (right) Stanford-pretokenized samples.}
\label{fig:lengths}
\end{figure*}
Next, we consider the distributions of piece character lengths.
Intuitively, a model which finds more long pieces can more efficiently compress the information encoded in the corpus; \citet{bostrom-durrett-2020-byte} argued that a distribution with mean centered around 6 resembles that of English morphemes better.
As demonstrated in \autoref{fig:lengths}, these two arguments are at odds in our experiment: in the \textsc{Raw} condition, the final-marking model produces shorter pieces, but centered at 6.
In the \textsc{Stan} condition, the situation is reversed.
\section{Introduction}
\label{sec:intro}
Modern NLP is dominated by large pre-trained models, systems which are large, complex, and costly to train.
As a result, much research effort is put into questions of tuning and configuring the various layers and training regimes for improving prediction quality on a growing number of tasks~\cite{rogers2020primer}.
Unfortunately, not as much research asks questions about the decisions made at the most upstream parts of the models, those that deal with input tokenization and subword vocabulary creation.
In this exploratory work, we isolate a single decision point which appears to be resolved arbitrarily by existing model developers, with no consensus but also no underlying theory:
\textbf{should subword tokenizers mark word boundaries at the beginning or the end?}
The immediate effects of such a decision are easy to describe: an out-of-vocabulary word\footnote{We use the term \textit{word} to refer to a space-delimited string of characters, post-pre-tokenization if such a process exists. Consequentially, \textit{subword} refers to tokens which are either word-length or form a proper substring of a word.} which is represented by multiple tokens, or \say{pieces}, e.g.~the plural noun \textit{polynomials}, shares the token \textit{polynomial} with its singular form in initial-boundary-marking tokenizers, but not in final-boundary-marking ones.
Conversely, a word prepended by a prefix like \textit{neoconservative} shares the stem token in final-boundary-marking tokenizers and not in the initial variants.
In lexical compounds, the boundary marking decision can affect whether \textit{thunderstorm} shares a token with its internal head, \textit{storm}, or its modifier \textit{thunder}.
The consequence for the unshared stem, in such cases, is often further segmentation since the suffix \textit{storm} (or prefix \textit{thunder}) rarely appears in the training corpus in a similar boundary-relative position.
It might appear at first that in English, a heavily-suffixing language \cite{wals-26}, a tokenizer that marks word-initial pieces is more appropriate.
Such a claim requires empirical support, but consideration of common practice can also be offered to challenge it:
for one, pre-tokenization such as punctuation separation and accent normalization is not always applied consistently when moving on to a downstream text.
A model that was trained on untreated text may find it difficult to process an NER dataset (for example) where punctuation is separated from preceding words, rendering a word-final-marking tokenizer more robust to change;
some tokenizers like BERT's Wordpiece~\cite{devlin-etal-2019-bert} \say{mark} a class of tokens by omission, i.e.~marking the non-initial pieces rather than initial ones.
This discrepancy surfaces edge case effects when compared with a seemingly-equivalent tokenizer like GPT-2's~\cite{radford2019language}, which marks initial pieces but only if they are prepended by a space (i.e.~not sentence-initial pieces).
We survey models' decisions for this question in \autoref{tab:systems}.
\input{tab_systems}
We offer three courses of analysis for the English Unigram LM tokenizer~\cite{kudo-2018-subword}, recently found to outperform BPE~\cite{sennrich-etal-2016-neural} on both statistical measures and downstream performance~\cite{bostrom-durrett-2020-byte}.
We conduct type-level, token-level and corpus-level statistical analysis on the initial-vs.-final variable~(\S\ref{sec:prop},\ref{sec:stat}), and add a morphological coverage test (\S\ref{sec:morph}) motivated by the question of subword models' ability to pick up meaningful linguistic generalizations using corpus statistics only.
We find that: (a) pre-tokenizing input text affects the boundary marking decision: unprocessed corpora produce better final-marking vocabularies, whereas pretokenized ones produce better initial-marking vocabularies;
(b) corpus compression efficiency trades off against information-theoretic measures;
(c) more morphemes are detected by final-marking vocabularies, but the techniques are complementary to each other and finding a combination of both can prove useful.
\section{Related Work}
Little prior work has compared different tokenization schemes.
\citet{bostrom-durrett-2020-byte} compared the standard BPE implementation in BERT to a subword vocabulary learned using the Unigram LM algorithm, finding that Unigram LM was better able to identify morphological segments.
In a study comparing BPE to Morfessor, \citet{banerjee2018meaningless} found that combining the two segmentation algorithms produced a superior subword vocabulary for machine translation.
In a study of the usefulness of subword segmentation in Hebrew, \newcite{klein2020getting} found that forcing a morphologically-informed parsing scheme, rather than relying on word pieces from multilingual BERT \cite{devlin-etal-2019-bert}, improved performance on tagging accuracy.
However, while the morphological typology of a language can play a significant role in the success of a language model, recent work suggests that morphological variability is less important than other types of variability \cite{mielke-etal-2019-kind}.
\section{Conclusion}
We find that the oft-ignored decision of how to mark subword tokens for the sake of word recoverability carries noticeable implications to the efficacy of the resulting model's representation of vocabularies and corpora, as well as the subwords' correspondence to morphemes.
This variable interacts in a nontrivial manner with the degree to which the training corpus has been pre-processed.
We urge researchers to pay heed to these preliminary decisions, and make their reasoning or supporting experimental evidence explicit when reporting model performance.
One promising avenue for future work may be to try and synthesize initial- and final-word marking into one tokenization model which can recover a large amount of morphemes, while still benefiting from the compression abilities of both techniques.
\bibliographystyle{acl_natbib}
\section{Corpus analysis}
\label{sec:stat}
Our next evaluation considers the utility of each marking schema in the context of a large corpus.
We report several surface-level statistics which are not immediately translatable to downstream model efficacy, but are \say{clean} in the sense that they introduce no stochastic noise other than corpus selection, and may still indicate a desired preference for downstream model inputs.
We measure each statistic on the same \textbf{training} corpus as initially passed to the tokenizer; on a different sample of \textbf{in-domain text} from the same Wikipedia dump; and on an \textbf{out of domain} sample from Reddit's \texttt{r/politics} feed extracted in the first half of 2015.
\paragraph{Token ratio.}
We count the total pieces tokenized by the model, as a measure for efficient text encoding, and divide by the number of space-delimited words in the corpus (lower = better).
\paragraph{Corpus-level entropy.}
We distill the piece-type distribution across each corpus into a single entropy number, under the assumption that a good encoding balances token distribution out rather than relying on a \say{heavy head} of certain types (higher = better).
\paragraph{Bigram LM perplexity.}
We train a basic bigram language model with add-$\epsilon$ smoothing on each tokenizer's original training corpus and apply it to each inference corpus, measuring overall base-2 perplexity.
We expect differences between \textsc{Init} and \textsc{Fin} to be mainly the result of in-word prediction ability.
We use $\epsilon\leftarrow 0.005$ (lower = better).
\input{tab_corpus}
Our results, presented in \autoref{tab:corpus_stats}, show a number trends, all of which remain consistent across corpora and domains.
While type-level entropy is always lower in the \textsc{Init} marking scheme, other findings suggest a trade-offs between compression ability (better for \textsc{Init} on raw text but for \textsc{Fin} when pre-tokenized) and prediction regularity (vice versa).
It may be the case that final-marked tokens are easier to predict after non-final tokens in a left-to-right bigram language model, suggesting these results may be particularly important for downstream regressive models like GPT-$n$, as opposed to masked language models.
\section{Related Work}
Little prior work has compared different tokenization schemes.
\citet{bostrom-durrett-2020-byte} compared the standard BPE implementation in BERT to a subword vocabulary learned using the Unigram LM algorithm, finding that Unigram LM was better able to identify morphological segments.
In a study comparing BPE to Morfessor, \citet{banerjee2018meaningless} found that combining the two segmentation algorithms produced a superior subword vocabulary for machine translation.
In a study of the usefulness of subword segmentation in Hebrew, \newcite{klein2020getting} found that forcing a morphologically-informed parsing scheme, rather than relying on word pieces from multilingual BERT \cite{devlin-etal-2019-bert}, improved performance on tagging accuracy.
However, while the morphological typology of a language can play a significant role in the success of a language model, recent work suggests that morphological variability is less important than other types of variability \cite{mielke-etal-2019-kind}.
\section{Conclusion}
We find that the oft-ignored decision of how to mark subword tokens for the sake of word recoverability carries noticeable implications to the efficacy of the resulting model's representation of vocabularies and corpora, as well as the subwords' correspondence to morphemes.
This variable interacts in a nontrivial manner with the degree to which the training corpus has been pre-processed.
We urge researchers to pay heed to these preliminary decisions, and make their reasoning or supporting experimental evidence explicit when reporting model performance.
One promising avenue for future work may be to try and synthesize initial- and final-word marking into one tokenization model which can recover a large amount of morphemes, while still benefiting from the compression abilities of both techniques.
\bibliographystyle{acl_natbib}
|
3,212,635,537,479 | arxiv | \section{Introduction}
The \emph{Ramsey number} $r(m,n)$ is the minimum integer $N$ such that every red/blue-coloring of the edges of the complete graph $K_N$ on $N$ vertices contains either a red $K_m$ or a blue $K_n$. Ramsey's theorem guarantees the existence of $r(m,n)$ and determining or estimating Ramsey numbers is a central problem in combinatorics. Classical results of Erd\H{o}s--Szekeres and Erd\H{o}s imply that $2^{n/2} \leq r(n,n) \leq 2^{2n}$ for $n \geq 2$. The only improvements to these bounds over the last seventy years have been to lower order terms (see~\cite{Conlon0, Sp}), with the best known lower bound coming from an application of the Lov\'asz local lemma~\cite{ErLo}.
Off-diagonal Ramsey numbers, where $m$ is fixed and $n$ tends to infinity, have also received considerable attention. In progress that has closely mirrored and often instigated advances on the probabilistic method, we now know that
\[
r(3,n)=\Theta(n^2/\log n).
\]
The lower bound here is due to Kim \cite{Ki} and the upper bound to Ajtai, Koml\'os and Szemer\'edi \cite{AjKoSz}. Recently, Bohman and Keevash \cite{BoKe1} and, independently, Fiz Pontiveros, Griffiths and Morris \cite{FiGrMo} improved the constant in Kim's lower bound via careful analysis of the triangle-free process, determining $r(3,n)$ up to a factor of $4+o(1)$.
More generally, for $m\ge 4$ fixed and $n$ growing, the best known lower bound is
\[
r(m,n) = \Omega_m(n^{\frac{m+1}{2}}/(\log{n})^{\frac{m+1}{2}-\frac{1}{m-2}}),
\]
proved by Bohman and Keevash \cite{BoKe} using the $H$-free process, while the best upper bound in this setting is
\[
r(m,n) = O_m(n^{m-1}/(\log n)^{m-2}),
\]
again due to Ajtai, Koml\'os and Szemer\'edi \cite{AjKoSz}. Here the subscripts denote the variable(s) that the implicit constant is allowed to depend on.
There are many interesting variants of the classical Ramsey problem. One such variant is the \emph{size Ramsey number} $\hat{r}(m, n)$, defined as the smallest $N$ for which there exists a graph $G$ with $N$ edges such that every red/blue-coloring of the edges of $G$ contains either a red $K_m$ or a blue $K_n$. It was shown by Chv\'atal (see Theorem 1 in the foundational paper of Erd\H os, Faudree, Rousseau and Schelp \cite{ErFaRoSc}) that $\hat{r}(m, n)$ is just the number of edges in the complete graph on $r(m,n)$ vertices, that is,
\[
\hat{r}(m, n) = \binom{r(m,n)}{2}.
\]
We will be concerned with a much-studied game-theoretic variant of the size Ramsey number, introduced independently by Beck~\cite{Beck93} and by Kurek and Ruci\'nski~\cite{KuRu}. The $(m,n)$-online Ramsey game is a game between two players, Builder and Painter, on an infinite set of initially isolated vertices. Each turn, Builder places an edge between two nonadjacent vertices and Painter immediately paints it either red or blue. The \emph{online Ramsey number} $\tilde{r}(m,n)$ is then the smallest number of turns $N$ that Builder needs to guarantee the existence of either a red $K_m$ or a blue $K_n$.
It is a simple exercise to show that $\tilde{r}(m,n)$ is related to the usual Ramsey number $r(m,n)$ by
\begin{equation} \label{eq:trivial}
\frac{1}{2} r(m,n) \le \tilde{r}(m,n) \le \binom{r(m,n)}{2}.
\end{equation}
In the diagonal case, the upper bound in (\ref{eq:trivial}) has been improved by Conlon \cite{Conlon}, who showed that for infinitely many $n$,
\[
\tilde{r}(n,n) \le 1.001^{-n} \binom{r(n,n)}{2}.
\]
The main result of this paper is a new lower bound for online Ramsey numbers.
\begin{thm}\label{thm:general-lower-bound} If, for some $m,n,N\ge1$, there exist $p\in(0,1)$, $c\le\frac{1}{2}m$,
and $d\le\frac{1}{2}n$ for which
\[
p^{\binom{m}{2}-c(c-1)}(2N)^{m-c}+(1-p)^{\binom{n}{2}-d(d-1)}(2N)^{n-d} \le \frac{1}{2},
\]
then $\tilde{r}(m,n)>N$.
\end{thm}
In particular, if $\tilde{r}(n) :=\tilde{r}(n,n)$ is the diagonal online Ramsey number, Theorem~\ref{thm:general-lower-bound} can be used to improve the classical bound $\tilde{r}(n) \ge 2^{n/2 - 1}$ by an exponential factor. Indeed, taking $p=\frac{1}{2}$ and $c = d \approx(1-\frac{1}{\sqrt{2}})n$ in
Theorem~\ref{thm:general-lower-bound}, we get the following immediate corollary.
\begin{cor} \label{cor:diagonalonline}
For the diagonal online Ramsey numbers $\tilde{r}(n)$,
\[
\tilde{r}(n)\ge 2^{(2-\sqrt{2})n-O(1)}.
\]
\end{cor}
As for the off-diagonal case, when $m$ is fixed and $n\rightarrow \infty$, Theorem~\ref{thm:general-lower-bound} can be also used to substantially improve the best-known lower bound. In this case, we take $c\approx(1-\frac{1}{\sqrt{2}})m$, $d=0$, and $p=C\frac{m\log n}{n}$
for a sufficiently large $C>0$ to obtain the following corollary.
\begin{cor} \label{cor:offdiagonalonline}
For fixed $m\ge3$ and $n$ sufficiently large in terms of $m$,
\[
\tilde{r}(m,n)\ge n^{(2-\sqrt{2})m-O(1)}.
\]
\end{cor}
For general $m$, Corollary~\ref{cor:offdiagonalonline} gives the best known lower bounds for the off-diagonal online Ramsey number. However, it is possible to do better for $m=3$ by using a smarter Painter strategy which deliberately avoids building red triangles.
\begin{thm} \label{thm:alterations} For $n\rightarrow \infty$,
\[
\tilde{r}(3,n) = \Omega \left(\frac{n^3}{\log^2 n}\right).
\]
\end{thm}
Roughly speaking, Painter's strategy is to paint every edge blue initially, but to switch to painting randomly if both endpoints of a freshly built edge have high degree. Also, when presented with an edge that would complete a red triangle, Painter always paints it blue. The bound given in Theorem~\ref{thm:alterations} is $n$ times the bound on the usual Ramsey number that comes from applying the Lov\'asz Local Lemma~\cite{ErLo}. However, our argument is closer in spirit to an earlier proof of the same bound given by Erd\H{o}s~\cite{Er} using alterations. This method for lower bounding $r(3,n)$ was later generalized to all $r(m,n)$ by Krivelevich~\cite{Kr1} and we suspect that Theorem~\ref{thm:alterations} can be generalized to $\tilde{r}(m,n)$ in the same way.
In the other direction, we prove a new upper bound on the off-diagonal online Ramsey number.
\begin{thm} \label{thm:online-upper-bound}
For any fixed $m\ge3$,
\[
\tilde{r}(m,n)= O_m\left(\frac{n^{m}}{\left(\log n\right)^{\lfloor m/2\rfloor-1}}\right).
\]
\end{thm}
In particular, note that Theorems~\ref{thm:alterations} and~\ref{thm:online-upper-bound} determine the asymptotic growth rate of $\tilde{r}(3,n)$ up to a polylogarithmic factor, namely,
\[\Omega\left(\frac{n^3}{\log^2 n}\right) \leq \tilde{r}(3,n) \leq O\left(n^3\right).\]
Theorem~\ref{thm:online-upper-bound} has a similar flavor to the improvement on diagonal online Ramsey numbers made by the first author~\cite{Conlon} and work on the so-called vertex online Ramsey numbers due to Conlon, Fox and Sudakov~\cite{CoFoSu}. It is obtained by adapting the standard Erd\H{o}s--Szekeres proof of Ramsey's theorem to the online setting and applying a classical result of Ajtai, Koml\'os and Szemer\'edi~\cite{AjKoSz} bounding $r(m,n)$.
In order to prove Theorem~\ref{thm:general-lower-bound}, we specialize to the case where Painter plays randomly. This is sufficient because Builder, who we may assume has unlimited computational resources, will always respond in the best possible manner to Painter's moves. Therefore, if a random Painter can stop this perfect Builder from winning within a certain number of moves with positive probability, an explicit strategy exists by which Painter can delay the game up to this point. This motivates the following key definition.
\begin{defn}
For $m,n \ge 3$ and $p\in(0,1)$, define $\tilde{r}(m,n;p)$ to be the number of turns Builder needs to win the $(m,n)$-online Ramsey game with probability at least $\frac{1}{2}$ against a Painter who independently paints each edge red with probability $p$ and blue with probability $1-p$. The \emph{online random Ramsey number} $\tilde{r}_{\text{rand}}(m,n)$ is the maximum value of $\tilde{r}(m,n;p)$ over $p\in(0,1)$.
\end{defn}
We note that there is a rich literature on simplifying the study of various combinatorial games by specializing to the case where one or both players play randomly (see~\cite{Beck, HKSS, Kr2}). For example, a variant of the online Ramsey game with random Builder instead of random Painter was studied by Friedgut et al. \cite{FKRRT}.
We make the following conjectures about the growth rate of $\tilde{r}_{\text{rand}}(m,n)$.
\begin{conj}\label{conj:2/3}
\begin{enumerate}[label=(\alph*)]
\item The diagonal online random Ramsey numbers satisfy
\[
\tilde{r}_{\text{\normalfont rand}}(n,n) = 2^{(1+o(1))\frac{2}{3}n}.
\]
\item The off-diagonal online random Ramsey numbers ($m\ge 3$ fixed and $n\rightarrow\infty$) satisfy
\[
\tilde{r}_{\text{\normalfont rand}}(m,n) = n^{(1+o(1))\frac{2}{3}m}.
\]
\end{enumerate}
\end{conj}
These conjectures are motivated by a connection with another problem, which we now describe.
Let $p\in(0, 1)$ be a fixed probability and suppose Builder plays the following one-player game, which we call the {\it Subgraph Query Game}, on the random graph $G(\mathbb{Z},p)$ with infinitely many vertices. The edges of the graph are initially hidden. At each step, Builder queries a single pair of vertices and is told whether the pair is an edge of the graph or not. Equivalently, the graph starts out empty and each edge is successfully built by Builder with probability $p$ (each edge may be queried at most once). In what follows, we use the terms ``query'' and ``build'' interchangeably.
Builder's goal is to find a copy of a given graph $H$ in the ambient random graph as quickly as possible. We call this problem of minimizing the number of steps in the Subgraph Query Game the \emph{Subgraph Query Problem}. When $H = K_m$, this may be seen as a variant of the online random Ramsey game, but where Builder is only interested in finding a red copy of $K_m$.
A version of this problem was studied independently by Ferber, Krivelevich, Sudakov and Vieira \cite{FKSV,FKSV2}, although they were interested in querying for long paths and cycles in $G(n,p)$. For instance, they showed that if $p \ge \frac{\log n + \log \log n + \omega(1)}{n}$, then it is possible to find a Hamiltonian cycle with high probability in $G(n,p)$ after $(1+o(1))n$ positive answers. In contrast, we are mainly interested in the setting where $H$ is a fixed graph to be found in a much larger random graph.
\begin{defn} If $p\in(0,1)$, define $f(H,p)$ to be the minimum (over all Builder strategies) number of turns Builder needs to be able to build a copy of $H$ with probability at least $1/2$ in the Subgraph Query Game, if each edge is built successfully with probability $p$.
\end{defn}
It might appear equally reasonable to study the minimum number of turns in which one can build at least one copy of $H$ {\it in expectation}. However, for certain $H$, such as a clique $K_m$ together with many leaves off a single vertex, it is possible to describe a strategy which has a tiny probability of successfully constructing copies of $H$, but upon success immediately builds a large number of copies, attaining low success probability but high expectation. Such a strategy is undesirable for application to online random Ramsey numbers, so we use the first definition instead.
Conjecture~\ref{conj:2/3} is motivated by the following conjecture regarding $f(K_m, p)$. The upper bound in this conjecture is proved in Section~\ref{sec:upper-bound-strat}.
\begin{conj} \label{conj:query-2/3} For any $m \ge 4$,
\begin{equation*}
f(K_m, p) = 2^{o(m)}p^{-\frac{2}{3}m + c_m},
\end{equation*}
where
\[
c_m = \begin{cases}
\frac{m}{2m-3} & m\equiv0\pmod3\\
\frac{2}{3} & m\equiv1\pmod3\\
\frac{2m+8}{6m-3} & m\equiv2\pmod3.
\end{cases}
\]
\end{conj}
The following result shows that the Subgraph Query Problem and the online random Ramsey game are closely related.
\begin{thm} \label{thm:query-online-connection}
For any $m,n\ge 3$ and $p\in(0,1)$,
\begin{equation*} \label{eq:query-online-connection}
\tilde{r}(m,n;p) \le \min\{f(K_m, p), f(K_n,1-p)\} \le 3\tilde{r}(m,n;p).
\end{equation*}
\end{thm}
Using Theorem~\ref{thm:query-online-connection}, we can show that Conjecture~\ref{conj:query-2/3} implies both cases of Conjecture~\ref{conj:2/3}.
We can also determine an approximately optimal value for the probability parameter $p$ in the online Ramsey game with random Painter.
\begin{thm}\label{thm:opt-p}
For $m\ge 3$ fixed and $n\rightarrow\infty$, there exists a $p = \Theta(m/n\log(n/m))$ for which
\[
\tilde{r}_{\text{\normalfont rand}}(m,n) \le 3 \tilde{r}(m,n; p).
\]
\end{thm}
We say that a graph has a \emph{$k$-matching} if it contains $k$ vertex-disjoint edges.
Our main result on the Subgraph Query Problem shows that graphs with large matchings are hard to build in few steps. We write $V(H)$ and $E(H)$ for the vertices and edges of $H$ and let $v(H) = |V(H)|$ and $e(H)=|E(H)|$.
\begin{thm}\label{thm:f-bound}
If $H$ is a graph that contains a $k$-matching, then
\[
f(H,p) = \Omega_H( p^{-(e(H)-k(k-1))/(v(H)-k)}).
\]
\end{thm}
Together with the upper bound construction described in Section~\ref{sec:upper-bound-strat}, this is enough to settle the growth rate of $f(K_m, p)$ for $m\le 5$. In particular, it proves Conjecture~\ref{conj:query-2/3} for $m=4, 5$.
\begin{thm} \label{thm:nle5}
The asymptotic growth rates of $f(K_m, p)$ for $m=3,4,5$ are
\begin{align*}
f(K_{3},p) & = \Theta(p^{-3/2})\\
f(K_{4},p) & = \Theta(p^{-2})\\
f(K_{5},p) & = \Theta(p^{-8/3}).
\end{align*}
\end{thm}
Asymptotically, the optimal $k$ to pick in Theorem~\ref{thm:f-bound} for $G = K_m$ is $k = (1-1/\sqrt{2})m$. With this value, we get the following bound on $f(K_m, p)$ which corresponds to Corollaries~\ref{cor:diagonalonline} and~\ref{cor:offdiagonalonline} in the online Ramsey number setting.
\begin{cor} \label{cor:f-bound-asymp}
For all $m\ge 3$,
\[
f(K_m,p) = \Omega_m(p^{-(2-\sqrt{2})m+O(1)}).
\]
\end{cor}
In studying the function $f(H,p)$, we were naturally led to consider the following function. When $H$ is a graph with no isolated vertices, define $t(H,p,N)$ to be the maximum expected number of copies of $H$ that can be built in $N$ moves in the Subgraph Query Game with parameter $p$, the maximum taken over all possible Builder strategies.
However, if $H$ has isolated vertices, the expected value is zero or infinite. Instead, if $H$ has exactly $k$ isolated vertices $v_1,\ldots,v_k$, we define
\[
t(H,p,N) \coloneqq (2N)^{k}t(H\backslash \{v_1,\ldots,v_k\},p,N)
\]
to capture the fact that the game with $N$ turns involves at most $2N$ vertices and therefore might as well be played on $2N$ fixed vertices.
Studying the threshold value of $N$ for which $t(H,p,N) \ge 1$ leads to Theorem~\ref{thm:f-bound} above. Intuitively, we expect the best strategy for building a copy of $H$ to be the same as the one which expects to build a single copy of $H$ in as few turns as possible.
Another natural question about the function $t(H,p,N)$ is: if $N$ is very large, what is the maximum number of copies Builder can expect to build in the Subgraph Query Game? Here we show that for $N$ sufficiently large the strategy of taking $O(\sqrt{2N})$ vertices and building all pairs of edges between them is asymptotically optimal for maximising $t(K_m,p,N)$, even though it is decidedly suboptimal for trying to build a single copy of $K_m$.
\begin{thm}\label{thm:t-large-N}
For all $m\ge 2, p \in (0,1), \varepsilon>0$, there exists $C > 0$ such that if $N\ge Cp^{-(2m-1)}(\log (p^{-1}))^2$, then
\[
t(K_{m},p,N) = (1\pm \varepsilon)p^{\binom{m}{2}}(2N)^{\frac{m}{2}}.
\]
\end{thm}
The rest of the paper is organized as follows.
In Section~\ref{sec:generallowerbounds}, we motivate and prove Theorem~\ref{thm:general-lower-bound}, our lower bound on the online Ramsey number, via the method of conditional expectations. In Section~\ref{sec:alterations}, we prove the lower bound Theorem~\ref{thm:alterations} for $\tilde{r}(3,n)$ using a Painter strategy designed to avoid red triangles.
We prove the upper bound Theorem~\ref{thm:online-upper-bound} in Section~\ref{sec:off-diagonal-upper}. Then, in Section~\ref{sec:subgraph-query}, we study the Subgraph Query Problem for its own sake, proving the upper bound in Conjecture~\ref{conj:query-2/3} as well as Theorems~\ref{thm:query-online-connection},~\ref{thm:opt-p},~\ref{thm:f-bound},~\ref{thm:nle5} and~\ref{thm:t-large-N}. We include a handful of open problems raised by our research in the closing remarks.
Unless otherwise indicated, all logarithms are base $e$. For clarity of presentation, we omit floor and ceiling signs when they are not crucial. We also do not attempt to optimize constant factors in the proofs.
\section{General lower bounds}\label{sec:generallowerbounds}
\subsection{Motivation}
In this section, we prove Theorem~\ref{thm:general-lower-bound}
via a weighting argument, motivated by the method of conditional expectations
and a result of Alon \cite{Alon} on the maximum number of copies of a
given graph $H$ in a graph with a fixed number of edges.
The first idea, the derandomization technique known as the method
of conditional expectations (see Alon and Spencer \cite{AlSp}), can be
used to give the following ``deterministic'' proof of the classical lower
bound on diagonal Ramsey numbers. We will show that
\[
\binom{r(n, n)}{n}2^{-\binom{n}{2}+1}\ge1.
\]
Suppose that for some $N$,
\begin{equation}
\binom{N}{n}2^{-\binom{n}{2}+1}<1.\label{eq:totalweight}
\end{equation}
Paint the edges of $K_{N}$ one at a time as follows. To each vertex subset
$U$ of order $n$, assign a weight $w(U)$ which is the
probability that $U$ becomes a monochromatic clique if the edges which remain uncolored at that time are colored uniformly at randomly. That is, writing $e(U)$ for the number of edges already colored in $U$,
\[
w(U)=\begin{cases}
2^{-\binom{n}{2}+1} & e(U)=0\\
2^{-\binom{n}{2}+e(U)} & \text{$e(U)>0$ and all already colored edges in $U$ are the same color}\\
0 & \text{otherwise}.
\end{cases}
\]
At every step, the total weight $\sum_{U}w(U)$
is equal to the expected number of monochromatic cliques if the remaining edges are painted uniformly at random. It is therefore possible to
paint each edge so as not to increase the total weight. Since the condition
$\sum_{U}w(U)<1$ is initially guaranteed by (\ref{eq:totalweight}), we can
maintain this condition throughout the course of the game, ending with a coloring where there is no
monochromatic clique of order $n$.
We now wish to apply such a weighting argument to the online
Ramsey game. The key observation is that if $\tilde{r}(n, n)$
is close to $r(n, n)$, then, since the graph built by Builder has at least
$r(n, n)$ vertices, it must be extremely
sparse. In particular, most of the weight should be concentrated on sets $U$
almost none of whose edges are ever built.
This is where the idea behind Alon's result \cite{Alon} comes in. For any fixed graph $H$, that paper solves the problem of determining the maximum possible number of copies of $H$ in a graph with a prescribed number of edges. Roughly speaking, Alon showed that the maximum number of copies of $H$ can be controlled by the size of the maximum matching in $H$. We show that this heuristic also applies to the online Ramsey game, though it will be more convenient for our calculations to work with minimum vertex covers instead of maximum matchings.
To make this idea work, instead of controlling the total weight function $\sum_{U}w(U)$, we restrict the sum to
subsets $U$ with a large minimum vertex cover, which
are comparatively few in number. Even if the total weight $\sum_{U}w(U)$
becomes large, the amount of weight supported on sets $U$ with a
large vertex cover is much smaller, and this is the only weight
that stands a chance to make it to the finish line and complete a
monochromatic clique.
\subsection{The proof}
Using the weighting argument described informally above, we now prove a lower bound on the value of $\tilde{r}(m,n;p)$, where Painter plays randomly, independently coloring each edge red with probability $p$ and blue with probability $1-p$.
\begin{thm}\label{thm:random-lower-bound}
If, for some $m,n,N\ge1$ and $p\in(0,1)$, there exist $c\le\frac{1}{2}m$
and $d\le\frac{1}{2}n$ for which
\[
p^{\binom{m}{2}-c(c-1)}(2N)^{m-c}+(1-p)^{\binom{n}{2}-d(d-1)}(2N)^{n-d} \le \frac{1}{2},
\]
then $\tilde{r}(m,n;p)>N$.
\end{thm}
We would like to show that regardless of Builder's strategy, the online random Ramsey game lasts for more than $N$ steps with probability at least $1/2$.
Suppose the game ends in at most $N$ turns and, without loss of generality, is played on $2N$ vertices. Let $G_{t}$, for $0\le t\le N$, be the state
of the graph after $t$ turns. Assign to each subset $U\subset V(G)$
an evolving weight function
\begin{align*}
w(U,t) & = \begin{cases}
p^{\binom{|U|}{2}-e(G_{t}[U])} & G_{t}[U]\text{ is monochromatic red}\\
0 & \text{otherwise.}
\end{cases}
\end{align*}
The value of $w(U,t)$ is the probability that $U$ becomes a red
clique if the remaining edges are built.
We say that $C \subset V(G)$ is a {\it vertex cover} of $G$ if every edge is incident to some vertex $v \in C$. If $U\subset V(G)$, let $c(U,t)$ be the size of the minimum vertex cover of $G_{t}[U]$. Note that $c(U,t)$ is a nondecreasing function of $t$. For each pair $(k, c)$ with $k \geq 2c$, we will be interested in the total weight supported on sets of order $k$ with $c(U,t) \geq c$,
\begin{align*}
w_{k,c}(t) & = \sum_{|U|=k,c(U,t)\ge c}w(U,t).
\end{align*}
Since $w(U,N)$ is nonnegative and $w(U,N)=1$ if and only if $U$ is a red clique, we see that for all $c \le m/2$, $w_{m,c}(N)$ is an upper bound for
the number of red copies of $K_m$ built after $N$ turns. We
would like to upper bound the expected value of $w_{m,c}(N)$.
\begin{lem}\label{lem:weight}
With $w_{m,c}(t)$ as above, regardless of Builder's strategy,
\[
\mathbb{E}w_{m,c}(N)\le p^{\binom{m}{2}-c(c-1)}(2N)^{m-c}.
\]
\end{lem}
\begin{proof}
Each $U$ with the property $c(U,N)\ge c$ first achieves this property
at a time $t_{c}(U)$. We say that $U$ is $c$-critical at this time.
Write
\[
w_{k,c}^{*}(t)=\sum_{|U|=k,t_{c}(U)=t}w(U,t)
\]
to be the contribution of the $c$-critical sets $U$ to $w_{k,c}(t)$.
Crucially, if we focus on the family of $U$ for which $t_{c}(U)=t$,
their expected total weight will remain $w_{k,c}^{*}(t)$ indefinitely.
Thus,
\[
\mathbb{E}w_{k,c}(N)=\sum_{t\le N}\mathbb{E}w_{k,c}^{*}(t).
\]
Now, a set $U$ which is $c$-critical at time $t$ must be
the vertex-disjoint union of the edge $e_{t}$ that Builder builds
at time $t$ and a set $U'$ of size $k-2$ with a vertex cover of order $c-1$. Also, because $U$ has a vertex cover of order $c-1$ before adding
this edge $e_{t}$, the edges incident to $e_{t}$ must also be incident
to one of the $c-1$ vertices in the vertex cover of $U'$, so $e_{t}$
is incident to a total of at most $2c-2$ edges in $U$. It follows
that after turn $t=t_{c}(U),$
\[
w(U,t)\le p^{2k-2c-2}w(U',t),
\]
where in particular if $U'$ is already not monochromatic then neither
is $U$. The exponent comes from the fact that among the total $2(k-2)$
edges between $e_{t}$ and $U'$ at least $2(k-2)-2(c-1)=2k-2c-2$
are thus far unbuilt and still contribute factors of $p$ to the weight of
$w(U,t)$. Thus, since each $U'$ completes at most one set $U$ which is $c$-critical at time $t$,
\[w_{k,c}^{*}(t) \le p^{2k-2c-2}w_{k-2,c-1}(t).\]
Further, note that there can only be $c$-critical sets at time $t$ if $e_t$ is colored red, which occurs with probability $p$. Otherwise, $w_{k,c}^{*}(t) = 0$. Taking expectations and using the fact that $\mathbb{E}w_{k,m}(t)$
is nondecreasing in $t$ gives
\begin{align*}
\mathbb{E}w_{k,c}^{*}(t) & \le p \cdot \mathbb{E}[p^{2k-2c-2}w_{k-2,c-1}(t)] \\
& \le p^{2k-2c-1}\mathbb{E}w_{k-2,c-1}(N).
\end{align*}
Summing over all $t$,
\[\mathbb{E}w_{k,c}(N) \le N\cdot p^{2k-2c-1}\mathbb{E}w_{k-2,c-1}(N).\]
Iterating this last inequality, we conclude that
\begin{align*}
\mathbb{E}w_{m,c}(N) & \le N^{c}\cdot p^{2mc-3c^{2}}\mathbb{E}w_{m-2c,0}(N)\\
& \le N^{c}\cdot p^{2mc-3c^{2}}\cdot (2N)^{m-2c}p^{\binom{m-2c}{2}}\\
& \le p^{\binom{m}{2}-c(c-1)}(2N)^{m-c},
\end{align*}
as desired.
\end{proof}
The same analysis with the blue weight function
\begin{align*}
w'(U,t) & = \begin{cases}
(1-p)^{\binom{|U|}{2}-e(G_{t}[U])} & G_{t}[U]\text{ is monochromatic blue}\\
0 & \text{otherwise}
\end{cases}
\end{align*}
leads to the conclusion that $\mathbb{E}w'_{n,d}(N)\le (1-p)^{\binom{n}{2}-d(d-1)}(2N)^{n-d}$ for all $n \geq 2d$. The assumption of Theorem~\ref{thm:random-lower-bound} then implies that the expected number of red $K_m$ plus the expected number of blue $K_n$ is at most $1/2$. This implies that the probability of containing either is at most $1/2$, completing the proof of Theorem~\ref{thm:random-lower-bound}. Theorem~\ref{thm:general-lower-bound} follows as an immediate corollary.
\section{Lower bound via alterations}
\label{sec:alterations}
In this section, we improve the lower bound for the off-diagonal online Ramsey numbers $\tilde{r}(3,n)$ using a different Painter strategy. Our proof extends an alteration argument of Erd\H os \cite{Er} which shows that
\[
r(3,n) \ge \frac{cn^2}{\log^2 n},
\]
for some constant $c>0$. The main idea of Erd\H os' proof was to show that in a random graph $G(r,p)$ with $p\approx r^{-1/2}$, only a small fraction of the edges need to be removed to destroy all triangles. Moreover, with high probability, removing these edges doesn't significantly affect the graph's independence number.
Our proof involves a randomized strategy which pays particular attention to avoiding red triangles. Instead of painting entirely randomly, Painter's strategy is modified in two ways to avoid creating red triangles. First, if an edge is built incident to a vertex of degree less than $(n-1)/4$, Painter always paints it blue. Second, if painting an edge red would create a red triangle, Painter again always paints it blue. In all other cases, Painter paints edges red with probability $p$ and blue with probability $1-p$.
In order to show that this Painter strategy works, we first prove a structural result about Erd\H{o}s--R\'enyi random graphs. Roughly speaking, this lemma implies that if an edge is removed from each triangle in $G(r,p)$, the remaining graph still has small independence number.
\begin{lem} \label{lem:rand-struct}
Suppose $n$ is sufficiently large, $p = 20 \log n / n$, $r = 10^{-6} n^2 /(\log n)^2$
and $G \sim G(r,p)$ is an Erd\H{o}s--R\'{e}nyi random graph. Then, with high probability, there does not exist a set $S\subset V(G)$ of order $n$ such that more than $\frac{n^2}{10}$ pairs of vertices in $S$ have a common neighbor outside $S$.
\begin{proof}
Let $E_1$ be the event that the maximum degree of $G$ is at most $2rp$. For a given vertex subset $S$ of order $n$, let $E_1(S)$ be the event that every vertex outside $S$ has at most $2rp$ neighbors in $S$. Thus, $E_1$ implies $E_1(S)$ for all $S$.
For a set $S$ of size $n$, let $E_2(S)$ be the event that at most $\frac{n^2}{10}$ pairs of vertices in $S$ have a common neighbor outside $S$ and let $E_2$ be the event that $E_2(S)$ holds for all $S$. We will show $E_1 \wedge E_2$ occurs w.h.p. which in turn implies that $E_2$ itself occurs w.h.p.
The distribution of $\deg(v)$ for a single vertex $v\in G$ is the binomial distribution $B(r-1,p)$. Using the Chernoff bound (see, e.g., Appendix A in \cite{AlSp}), we find that
\[
\Pr[\deg(v) > 2rp] < \left(\frac{e}{4}\right)^{rp} < \exp\left(-\frac{n}{5\cdot 10^5\log n}\right).
\]
Taking the union over all vertices of $G$, it follows that
\[
\Pr[\overline{E_1}] < r\exp\left(-\frac{n}{5\cdot 10^5\log n}\right),
\]
so $E_1$ occurs w.h.p.
Fix a set $S$ of $n$ vertices. For $v \in V(G) \backslash S$, define $\deg_S(v)$ to be the number of neighbors of $v$ in $S$. Since $E_1$ implies $E_1(S)$, we have
\[
\Pr[E_1 \wedge \overline{E_2(S)}] \le \Pr[E_1(S) \wedge \overline{E_2(S)}].
\]
We will show that this last probability is so small that we may union bound over all $S$.
For $E_1(S)$ to occur, the possible values of $\deg_S(v)$ range through $[0, 2rp]$. We will cut off the bottom of this range and divide the rest into dyadic intervals. Let $D_0 = -1, D_1 = 4enp, D_2 = 8enp, D_3 = 16enp, \ldots, D_k = 2rp$ so that $D_i = 2D_{i-1}$ for each $2\le i \le k-1$ and $D_k \le 2D_{k-1}$. The number of intervals $k$ satisfies $k\le \log_2(r/n) \le 2\log n$.
Define $d_i$ to be the number of $v \in V(G) \backslash S$ satisfying $D_{i-1} < \deg_{S}(v) \le D_i$. For $\overline{E_2 (S)}$ to occur, it must be the case that
\[
\sum_{v\not\in S} \binom{\deg_S(v)}{2} \ge \frac{n^2}{10},
\]
as the left hand side counts each pair in $S$ with a common neighbor outside $S$ at least once. In particular,
\begin{equation}\label{eq:degree-prof}
\sum_{i=1}^{k} d_i \binom{D_i}{2} \ge \frac{n^2}{10}.
\end{equation}
Notice that since $D_1 = 4enp = 80e\log n$ and $d_1 \le r$,
\[
d_1 \binom{D_1}{2} \le r \cdot D_1 ^2 = \frac{64e^2}{10^4} n^2 < \frac{n^2}{20},
\]
so at least half the contribution of (\ref{eq:degree-prof}) must come from $i\ge 2$. Thus,
\begin{equation}\label{eq:degree-prof-2}
\sum_{i=2}^{k} d_i \binom{D_i}{2} \ge \frac{n^2}{20}.
\end{equation}
We would like to bound the probability that $E_1(S)$ and (\ref{eq:degree-prof-2}) occur simultaneously. Let $T$ be the family of all sequences $(d_i)_{i=1}^k$ which sum to $r-n$ and satisfy (\ref{eq:degree-prof-2}). Given the choice of $(d_i)_{i=1}^k$, the number of ways to assign vertices to dyadic intervals $(D_{i-1},D_i]$ is at most $\binom{r-n}{d_1,d_2,\ldots, d_k}$.
If $i\ge 2$ and a vertex $v$ is assigned to $(D_{i-1},D_i]$, the probability that $\deg_S(v)$ lies in that interval is at most
\[
\Pr[\deg_S(v) > D_{i-1}]\le \binom{n}{D_{i-1}} p^{D_{i-1}} \le \left(\frac{enp}{D_{i-1}}\right)^{D_{i-1}}.
\]
If $i = 1$, then we simply use the trivial bound $\Pr[\deg_S(v) \in (D_0, D_1]] \le 1$. Thus,
\begin{align*}
\Pr[E_1(S) \wedge \overline{E_2(S)}] &\le \sum_{(d_i)\in T} \binom{r-n}{d_1,d_2,\ldots, d_k} \prod_{i=1}^{k} \Pr[\deg_S(v) \in (D_{i-1}, D_i]] \\
& \le \sum_{(d_i)\in T} \binom{r-n}{d_1,d_2,\ldots, d_k} \prod_{i=2}^{k} \left( \left(\frac{enp}{D_{i-1}}\right)^{D_{i-1}}\right)^{d_i} \\
& \le \sum_{(d_i)\in T} \prod_{i=2}^{k} \left(r \cdot \left(\frac{enp}{D_{i-1}}\right)^{D_{i-1}}\right)^{d_i},
\end{align*}
where we used $\binom{r-n}{d_1, d_2,\ldots, d_k} < r^{d_2 + \cdots + d_k}$. Next, the number of compositions of $r-n$ into $k$ parts is at most $r^k$, so $|T| \le r^k$ and we have
\begin{align}
\Pr[E_1(S) \wedge \overline{E_2(S)}] & \le r^k \max_{(d_i)\in T} \prod_{i=2}^{k} \left(r \cdot \left(\frac{enp}{D_{i-1}}\right)^{D_{i-1}}\right)^{d_i} \nonumber \\
& \le r^k \max_{(d_i)\in T} \exp\left(\sum_{i=2}^k d_i \log A_i\right), \label{eq:deg-opt}
\end{align}
where
\[
A_i = r \cdot \left(\frac{enp}{D_{i-1}}\right)^{D_{i-1}}.
\]
It remains to maximize the exponent in (\ref{eq:deg-opt}) subject to (\ref{eq:degree-prof-2}). Consider the function
\[
f(D) = \frac{1}{D^2}\log \left(r \cdot \left(\frac{enp}{D}\right)^{D}\right) = \frac{\log r}{D^2} + \frac{\log(enp)}{D} - \frac{\log D}{D}.
\]
Notice that $D_1 = 4enp = 80e\log n$ so that for $D\ge D_1$,
\[
r \cdot \left(\frac{enp}{D}\right)^{D} \le
r \cdot \left(\frac{enp}{D_1}\right)^{D_1}\le r\cdot 2^{-80e\log n} < 1.
\]
Thus, $f(D)$ takes negative values on $[D_1, D_k]$. Its derivative is
\[
f'(D) = - \frac{2\log r}{D^3} - \frac{\log{enp}}{D^2} + \frac{\log D}{D^2} - \frac{1}{D^2} = \frac{D(\log D - \log(e^2np)) - 2\log r}{D^3}.
\]
Since $r \le n^2$, we find that whenever $D \ge D_1 = 4enp = 80e\log n$,
\[
f'(D) \ge \frac{D\log(4/e) - 2\log r}{D^3} \ge \frac{80e\log(4/e)\cdot \log n - 4\log n}{D^3} > 0,
\]
and so $f(D)$ is monotonically increasing on $[D_1, D_k]$ and attains its maximum value at $D_k = 2rp$. With $2rp = 4\cdot 10^{-5} n/\log n$ and $n$ sufficiently large, observe that
\[
\left(\frac{enp}{2rp}\right)^{2rp} = \left(\frac{10^6e(\log n)^2}{2n}\right)^{4\cdot 10^{-5} n/\log n} \le \exp(-2\cdot 10^{-5} n),
\]
so that this maximum value is
\[
f(2rp) \le \frac{10^{10}(\log n)^2}{16 n^2} \cdot \log (n^2 \cdot \exp(-2\cdot 10^{-5} n)) \le - \frac{10^5(\log n)^2}{16 n}.
\]
In particular, because $\binom{D}{2} \ge D^2/3$ for $D \ge 3$ and $f(D)$ is always negative,
\begin{align*}
\sum_{i=2}^k d_i \log A_i & = \sum_{i=2}^k d_i \binom{D_i}{2}\cdot \frac{\log A_i}{\binom{D_i}{2}} \\
& \le 3 \sum_{i=2}^k d_i \binom{D_i}{2}\cdot f(D_i) \\
& \le 3 f(D_k) \sum_{i=2}^k d_i \binom{D_i}{2} \\
& \le 3 f(2rp) \cdot \frac{n^2}{20} \\
& \le - n (\log n)^2
\end{align*}
for any $(d_i)\in T$.
Returning to (\ref{eq:deg-opt}), it follows that
\[
\Pr[E_1(S) \wedge \overline{E_2(S)}] \le r^k \max_{(d_i)\in T} \exp\left(\sum_{i=2}^k d_i \log A_i\right) \le r^k \exp(-n (\log n)^2).
\]
There are at most $\binom{r}{n} \le e^{2n\log n}$ subsets $S$ of size $n$ to consider and $r^k \le r^n \le e^{2n \log n}$ as well, so
\begin{align*}
\Pr[\overline{E_1 \wedge E_2}] & = \Pr[\overline{E_1} \vee \bigvee_S E_2(S)] \\
& \le \Pr[\overline{E_1}] + \sum_S \Pr[E_1 \wedge \overline{E_2(S)}] \\
& \le \Pr[\overline{E_1}] + \sum_S \Pr[E_1(S) \wedge \overline{E_2(S)}] \\
& \le \Pr[\overline{E_1}] + \exp(4n\log n) \cdot \exp\left(-n(\log n)^2\right). \\
\end{align*}
Both summands on the right vanish rapidly, so $E_2$ holds w.h.p., as desired.
\end{proof}
\end{lem}
With this lemma in hand, we are now ready to prove Theorem~\ref{thm:alterations}.
\vspace{3mm}
\noindent
{\it Proof of Theorem~\ref{thm:alterations}.}
Let $p = 20 \log n / n$, $r = 10^{-6} n^2/(\log n)^2$ and $N=\frac{(n-1)r}{8}$.
We will give a randomized strategy for Painter such that, regardless of Builder's strategy, after $N$ edges are colored there is neither a red $K_3$ nor a blue $K_n$ w.h.p. Thus, there exists a strategy for Painter which makes the game last more than $N$ steps and the desired bound $\tilde{r}(3,n)>N$ follows. Note that proving the result with positive probability suffices, but our argument shows it w.h.p.~for no additional cost.
We now describe Painter's strategy. Initially, all vertices are considered inactive; a vertex is activated when its degree reaches at least $(n-1)/4$. The active vertices are labeled with the natural numbers in $[r]$ when they reach degree at least $(n-1)/4$, using an arbitrary underlying order on the vertices to break ties. Since $N = (n-1)r/8$, there will never be more than $r$ active vertices.
When Builder builds an edge $(u,v)$, this edge is considered inactive if either $u$ or $v$ is inactive immediately after $(u,v)$ is built and active otherwise. The status of an edge remains fixed once it is built, so that inactive edges remain inactive even if both of its incident vertices are active at a later turn. Painter automatically colors inactive edges blue.
If Builder builds an active edge $(u,v)$, Painter first checks if $u$ and $v$ have a common neighbor $w$ such that $(u,w)$ and $(v,w)$ are both red. For brevity's sake, we call such a vertex $w$ a {\it red common neighbor} of $u$ and $v$. If so, Painter paints $(u,v)$ blue so as to not build a red triangle and we call such an edge {\it altered}. Otherwise, Painter paints it red with probability $p$ and blue with probability $1-p$. Following this strategy, Painter guarantees that no red triangles are built. It suffices to show that w.h.p.~no blue $K_n$ is built either.
Here is an equivalent formulation of Painter's strategy. At the start of the game, Painter samples an Erd\H{o}s--R\'enyi graph $G = G([r],p)$ on the labels which he keeps hidden from Builder. Inactive edges are painted blue. When an active edge between vertices labelled $i$ and $j$ is built, it is painted red if and only if $i\sim j$ in $G$ and these two vertices currently have no red common neighbor.
Now, we apply Lemma~\ref{lem:rand-struct} to the graph $G$. Letting $E_2(S)$ be the event that an $n$-set $S$ has at most $n^2/10$ pairs with outside common neighbors and $E_2 = \bigwedge_S E_2(S)$, we see that $\Pr[\overline{E_2}] \rightarrow 0$ as $n\rightarrow \infty$.
For a set $S\subseteq [r]$ of labels, write $T(S)$ for the set of active vertices with labels in $S$. We seek to bound the probability of the event $B(T(S))$ that $T(S)$ is a blue $n$-clique at the end of the game. Because any blue $n$-clique would have all of its vertices active (as each vertex of the $n$-clique would have degree at least $n-1\ge (n-1)/4$), if none of the events $B(T(S))$ occurs, then no blue $K_n$ is ever built. Once we show that the probability of a single $B(T(S))$ is sufficiently small, we will apply the union bound over all $S$ to show that w.h.p.~no blue $K_n$ is built.
First, note that if any edge $(u,v)$ in $T(S)$ is altered (and hence blue), we may assume that their red common neighbors lie outside $T(S)$. Otherwise, there must be two red edges inside $T(S)$ already and $T(S)$ can never become a blue $n$-clique.
With this in mind, conditioning on the event $E_2(S)$, at most $n^2/10$ altered blue edges are built in $T(S)$. Within $T(S)$ there can be at most $n^2/4$ inactive edges. Assuming $B(T(S))$ occurs, there are at least
\[
\binom{n}{2} - \frac{n^2}{4} - \frac{n^2}{10} \ge \frac{n^2}{8}
\]
edges between vertices of $T(S)$ that are both active and unaltered. For $B(T(S))$ to occur, each of these active and unaltered edges would have to be colored blue on its turn. On the other hand, each of these edges has a chance $p$ of being colored red on that turn.
Thus, we find that
\[
\Pr[B(T(S))|E_2(S)] \le (1-p)^{\frac{n^2}{8}},
\]
with one factor of $1-p$ for each unaltered active edge built in $T(S)$. Thus,
\[
\Pr[\bigvee_S B(T(S))] \le \Pr[E_2 \wedge \bigvee_S B(T(S))] + \Pr[\overline{E_2}].
\]
The second summand goes to zero, so it suffices to show the first does as well. We have
\begin{align*}
\Pr[E_2 \wedge \bigvee_S B(T(S))] & \le \sum_S \Pr[E_2 \wedge B(T(S))] \\
& \le \sum_S \Pr[E_2(S) \wedge B(T(S))] \\
& \le \sum_S \Pr[B(T(S))|E_2(S)] \\
& \le \binom{r}{n} (1-p)^{\frac{n^2}{8}}.
\end{align*}
Using $1-p \le e^{-p}$, the right-hand side is at most
\[
r^n e^{-pn^2/8} \le e^{n\log r -pn^2/8} = e^{-(\frac{1}{2} + o(1))n\log n},
\]
also tending to zero as $n\rightarrow \infty$. Thus, the probability that either $\overline{E_2}$ or some $B(T(S))$ occurs tends to zero. Therefore, with high probability no blue $K_n$ is built.
\hfill \qed
\section{Off-diagonal upper bounds} \label{sec:off-diagonal-upper}
In Section~\ref{sec:generallowerbounds}, we proved lower bounds of the form $\tilde{r}(m,n)\ge\Omega(n^{(2-\sqrt{2})m+o(m)})$ on the off-diagonal online Ramsey numbers through an analysis of the online random
Ramsey number. It is easy to give an upper
bound of the form $\tilde{r}(m,n) \leq O(n^{2m-2})$ simply by applying
the Erd\H os--Szekeres bound for classical Ramsey numbers and the
trivial observation that $\tilde{r}(m,n)\le\binom{r(m,n)}{2}$.
However, the simple inductive proof of the Erd\H os--Szekeres bound suggests a
Builder strategy that does considerably better. Namely, build many edges from one
vertex until it has a large number of edges of one color, then proceed
inductively in that neighborhood.
This strategy is particularly well suited to the online Ramsey game because
the number of edges built is only slightly more than linear in the
number of vertices used, allowing us to derive a bound of the form $\tilde{r}(m,n)\le O(n^{m})$.
A slight variation on this argument allows us to bound the online Ramsey number in terms of the bounds for classical Ramsey numbers.
\begin{lem} \label{lem:classical-to-online-upper-bound}
Let $m \leq n$ be positive integers with $m$ fixed. Let $m_0=\lfloor m/2 \rfloor + 1$ and $n_0=\lfloor \sqrt{n} \rfloor$. Suppose $\mathcal{L}$ is a positive real such that for all $m_0 \le m' \le m$ and $n_0 \le n' \le n$,
\begin{align*}
r(m_0, n') & \le \frac{1}{\mathcal{L}}\binom{m_{0}+n'-2}{m_{0}-1}, \\
r(m',n_0) & \le \frac{1}{\mathcal{L}}\binom{m'+n_{0}-2}{m'-1}.
\end{align*}
Then
\[
\tilde{r}(m,n)\le \frac{C_m n}{\mathcal{L}}\binom{m+n-2}{m-1}
\]
for a constant $C_m$ depending only on $m$.
\end{lem}
\begin{proof}
We describe a general Builder strategy for the online Ramsey game
with parameters $m$ and $n$ and some savings parameter $\mathcal{L}$. Let $f(m,n)=\frac{1}{\mathcal{L}}\binom{m+n-2}{m-1}$,
so we have $f(m-1,n)+f(m,n-1)=f(m,n)$ by Pascal's identity.
Begin by building $f(m,n) - 1$ edges out of a given initial vertex $v_{1}$.
If $f(m-1,n)$ of these edges are colored red, we proceed to
the red neighborhood of $v_1$; otherwise, we proceed to
the at least $f(m,n-1)$ vertices in the blue neighborhood of $v_1$. If at some step we reach a neighborhood with $f(m - i, n - j)$ vertices, we build $f(m - i, n -j) -
1$ edges inside this neighborhood from one of the vertices, which we label
$v_{i+j+1}$. If $f(m - i - 1, n - j)$ of these edges are colored red, we proceed
to the red neighborhood of $v_{i+j+1}$; otherwise, we proceed to the at least
$f(m-i,n-j-1)$ vertices in the blue neighborhood of $v_{i+j+1}$.
We stop once $m$ reaches $m_0$ or $n$ reaches $n_0$, ending up with either
$f(m_{0},n')$ vertices for some $n_{0}\le n'\le n$ or $f(m',n_{0})$ vertices for some $m_{0}\le m'\le m$. Once we reach this stage, we build all edges in the remaining set.
Suppose now that we arrive at a set $S$ of order $f(m_0, n')$. By construction, there are $\ell = m + n - m_0 - n'$ vertices $v_1, \dots, v_\ell$ such that $m-m_0$ of the vertices $v_i$ are joined in red to every $v_j$ with $j > i$ and every $w \in S$. The remaining $n - n'$ vertices $v_i$ are joined in blue to every $v_j$ with $j > i$ and every $w \in S$. But since
\[
r(m_0, n') \le \frac{1}{\mathcal{L}}\binom{m_{0}+n'-2}{m_{0}-1} = f(m_0,n'),
\]
the complete graph on $S$ contains either a red $K_{m_0}$ or a blue $K_{n'}$, either of which can be completed to a red $K_m$ or a blue $K_n$ by using the appropriate subset of $v_1, \dots, v_\ell$. If we had instead arrived at a set of order $f(m',n_0)$, a similar analysis would have applied.
Note that the total number of edges built in the branching phase is at most
$(m+n)f(m,n)$, while the number built by filling in the final clique is at most $
\max(f(m_{0},n)^{2}, f(m,n_{0})^{2})$. Using the choice of $m_0$ and $n_0$, the total number of edges built is easily seen to be at most a constant in $m$ times the previous expression.
\end{proof}
From here we derive Theorem~\ref{thm:online-upper-bound}.
\vspace{3mm}
\noindent
{\it Proof of Theorem~\ref{thm:online-upper-bound}.}
We apply the bound
\[
r(m,n) = O_m(n^{m-1}/\log ^{m-2} n),
\]
due to Ajtai, Koml\'os and Szemer\'edi~\cite{AjKoSz}. In particular, suppose $m_0=\lfloor m/2 \rfloor + 1$, $n_0=\lfloor \sqrt{n} \rfloor$ and $m', n'$ satisfy $m_0 \le m' \le m$ and $n_0 \le n' \le n$. Then, for some constants $C, C' >0$ depending only on $m$, we have
\[
r(m_0, n') \le \frac{C}{\log ^{m_0-2} n'} (n')^{m_0-1} \le \frac{C'}{\log^{\lfloor m/2 \rfloor - 1} n}\binom{m_{0}+n'-2}{m_{0}-1}
\]
and
\[
r(m', n_0) \le \frac{C}{\log ^{m'-2} n_0} n_0^{m'-1} \le \frac{C'}{\log^{\lfloor m/2 \rfloor - 1} n}\binom{m'+n_{0}-2}{m'-1},
\]
verifying the conditions of Lemma~\ref{lem:classical-to-online-upper-bound} with $\mathcal{L} = \Omega_m(\log^{\lfloor m/2 \rfloor - 1} n)$. It follows by that lemma that there exists another constant $C''>0$ depending only on $m$ for which
\[
\tilde{r}(m,n) \le \frac{C'' n}{\log^{\lfloor m/2 \rfloor - 1} n} \binom{m+n-2}{m-1}.
\]
Fixing $m\ge 3$ and taking $n\rightarrow\infty$, this implies
\[
\tilde{r}(m,n)= O_m\left(\frac{n^{m}}{\left(\log n\right)^{\lfloor m/2\rfloor-1}}\right),
\]
as desired.
\hfill \qedsymbol
\vspace{3mm}
We remark that while the statement and proof of Lemma~\ref{lem:classical-to-online-upper-bound} are designed for the case where $m$ is a constant, they can be easily modified to make them meaningful for all $m$ and $n$.
\section{The Subgraph Query Problem} \label{sec:subgraph-query}
The vertex cover argument in Section~\ref{sec:generallowerbounds} was motivated by our study of the closely-related Subgraph Query Problem. Indeed, one can view this problem as an instance of the online Ramsey game with a random Painter where Builder single-mindedly tries to build a clique in one color, ignoring the other color entirely.
Let $p\in(0,1)$ be the probability that Builder successfully builds any given edge in the Subgraph Query Problem. We are primarily interested in the quantity $f(H,p)$, which we defined as the minimum $N$ for which there exists a Builder strategy which builds a copy of $H$ with probability at least $\frac{1}{2}$ in $N$ turns. Of secondary interest is the quantity $t(H,p,N)$, which we define as the maximum, over all Builder strategies, of the expected number of copies of $H$ that can be built in $N$ turns. It is easy to see that
\begin{equation*}
t(H,p,N) < \frac{1}{2} \implies f(H,p) > N.
\end{equation*}
Thus, upper bounds on $t(H,p,N)$ yield lower bounds on $f(H,p)$.
\subsection{Connection with online Ramsey numbers}\label{sec:query-online-connection}
We first check that the Subgraph Query Problem gets easier when edges are built with higher probability.
\begin{lem}\label{lem:f-monotone}
For any $m\ge 3$, $f(K_m,p)$ is a nonincreasing function of $p\in (0,1)$.
\end{lem}
\begin{proof}
Suppose $p < q$ and $f(K_m, p) = N$. This means that in the Subgraph Query Problem with parameter $p$, Builder has an $N$-move strategy $S$ to win with probability at least half. Strategy $S$ is defined by Builder's choice of edge to build at each step, given the data of which edges were successfully built in previous steps.
Builder's strategy for the Subgraph Query Problem with parameter $q$ is as follows. For each edge that Builder successfully builds, Builder then flips a biased coin that comes up heads $\frac{p}{q}$ of the time. If the coin comes up tails, Builder pretends the edge actually failed to build, and acts according to strategy $S$ with respect to only the edges for which the coin came up heads.
Just looking at the edges which come up heads, Builder is exactly following strategy $S$, and so builds a $K_m$ with probability at least $1/2$ in $N$ steps.
\end{proof}
We now prove Theorem~\ref{thm:query-online-connection}, which connects the Subgraph Query Problem to the online Ramsey game. Recall the statement:
\[
\tilde{r}(m,n;p) \le \min\{f(K_m, p), f(K_n,1-p)\} \le 3\tilde{r}(m,n;p).
\]
\vspace{3mm}
\noindent
{\it Proof of Theorem \ref{thm:query-online-connection}.}
We first show the left side of the inequality. Let $N=\min\{f(K_m, p), f(K_n,1-p)\}$ and suppose that $f(K_m,p)$ is the smaller of the two. Then there exists an $N$-move Builder strategy which builds a $K_m$ with probability at least half. Now, let Builder play the online Ramsey game against a random Painter with the same probability parameter $p$. Builder's strategy will be to treat red edges as successfully built and blue edges as failed. In this way, Builder wins the online Ramsey game in $N$ moves with probability at least half, by constructing a red $K_m$. Similarly, if $f(K_n, 1-p)$ were smaller, Builder would instead treat blue edges as successfully built and red edges as failed. This would then guarantee the construction of a blue $K_n$ with probability at least half.
Now we show the right side of the inequality. Suppose $N=\tilde{r}(m,n;p)$, so in the online Ramsey game against random Painter with parameter $p$, there exists an $N$-move Builder strategy which builds a red $K_m$ or blue $K_n$ with probability at least half. In particular, this same strategy guarantees either a red $K_m$ with probability at least $\frac{1}{4}$ or a blue $K_n$ with probability at least $\frac{1}{4}$.
Suppose the first is true. Then Builder plays the Subgraph Query Game using this same strategy, treating red edges as successfully built and blue as failed. In $N$ moves, he has at least a $\frac{1}{4}$ chance of successfully building a $K_m$. Repeating this strategy three independent times on three different vertex sets, Builder uses $3N$ moves to build a $K_m$ with probability at least
\[
1-\Big(1-\frac{1}{4}\Big)^3 = \frac{37}{64} > \frac{1}{2},
\]
showing that $f(K_m,p) \le 3\tilde{r}(m,n;p)$ in this case. Similarly, if the second case occurs, $f(K_n,1-p) \le 3\tilde{r}(m,n;p)$. Either way, the smaller of $f(K_m,p)$ and $f(K_n, 1-p)$ is bounded above by $3\tilde{r}(m,n;p)$.
\hfill \qed
\vspace{3mm}
Now we show that Conjecture~\ref{conj:query-2/3} about the Subgraph Query Problem directly implies Conjecture~\ref{conj:2/3} about online random Ramsey numbers.
\vspace{3mm}
\noindent
{\it Proof that Conjecture~\ref{conj:query-2/3} implies Conjecture~\ref{conj:2/3}.}
Assume Conjecture~\ref{conj:query-2/3}, i.e., $f(K_m, p) = 2^{o(m)}p^{-\frac{2}{3} m + c_m}$ for all $m\ge 3$, $p\in(0,1)$.
By Theorem~\ref{thm:query-online-connection}, we have
\begin{equation}\label{eq:online-to-f}
\tilde{r}(m,n;p) = \Theta(\min\{f(K_m,p), f(K_n,1-p)\}).
\end{equation}
In the diagonal case of the online Ramsey game, (\ref{eq:online-to-f}) together with Lemma \ref{lem:f-monotone} implies that $p=\frac{1}{2}$ gives the online random Ramsey number to within a constant factor. Thus,
\[
\tilde{r}_{\text{\normalfont rand}}(n,n) = 2^{\frac{2}{3} n + o(n)}.
\]
This proves part (a).
In the off-diagonal case, a value of $p$ nearly optimizing the right hand side of (\ref{eq:online-to-f}) satisfies $p = \Theta(\frac{m}{n}\log{\frac{n}{m}})$ by Theorem~\ref{thm:opt-p}, which is proved in Subsection \ref{sec:recursive-graph-building}. Plugging in this value of $p$, we get
\[
\tilde{r}_{\text{\normalfont rand}}(m,n)=2^{o(m)}\Big(\Theta\left(\frac{m}{n}\log{\frac{n}{m}}\right)\Big)^{-\frac{2}{3}m+c_m},
\]
which implies case (b) of Conjecture~\ref{conj:2/3}.
\hfill \qed
\vspace{3mm}
\subsection{The Branch and Fill Strategy} \label{sec:upper-bound-strat}
We now prove the upper bound in Conjecture~\ref{conj:query-2/3}.
We will say it is possible to build a graph $H$ in $O(T)$ turns, where $T=T(p)$ is a function of $p$, if for any $p\in(0,1)$ it is possible, in the Subgraph Query Game played with probability $p$, to build a copy of $H$ in $O(T)$ time with probability at least $\frac{1}{2}$. It is a simple fact about randomized algorithms that if one can achieve any constant success probability in $O(T)$ time then one can iterate the algorithm to succeed with probability $1-\varepsilon$ in $O(T \log{\varepsilon^{-1}})$ time.
We describe a Builder strategy to prove the upper bound in Conjecture~\ref{conj:query-2/3} and conjecture that this is essentially the optimal strategy for the Subgraph Query Problem for cliques.
\begin{lem} \label{lem:branch-and-fill}
Let $a\ge 1$, $b \ge 2$ and $n=a+b+1$ satisfy $2a+3-b\ge 0$. Then
\[
f(K_{n},p) = O_n(p^{-\frac{2a+b+1}{2}+\frac{\alpha}{b}}),
\]
where $\alpha = \min(1, \frac{b(2a+3-b)}{2(b-1)})$.
\end{lem}
\begin{proof}
To build a clique $K_{n}$ in $O(T)$ turns, where $T = p^{-\frac{2a+b+1}{2}+\frac{\alpha}{b}}$, we follow a strategy with three phases:
\begin{enumerate}
\item Build a clique $U$ on $a$ vertices. By induction, the number of
turns needed will be negligible.
\item Find $p^{a}T$ common neighbors of $U$ in $O_n(T)$
time with high probability. This is done by repeatedly picking a new
vertex $v$ and trying to build each of the edges between $v$ and the vertices in
$U$ until one fails. Let $W$ be the set of common neighbors found in
this way.
\item Among the vertices of $W$, pick a vertex $w_{1}$
and try to build all edges incident to $w_{1}$ within $W$. Let $W_{1}=N(w_{1})\cap W$
be the neighborhood determined. Try to build all $\binom{|W_{1}|}{2}$
edges within $W_{1}$. Remove $\{w_{1}\}\cup W_{1}$ from $W$ and
repeat a total of $p^{-\alpha}$ times, picking $w_{2},\ldots,w_{p^{-\alpha}}$,
finding their neighborhoods, and filling them in. Here, $\alpha\in[0,1]$ is a parameter which we have not yet specified.
\end{enumerate}
After the process is complete, if any one of the $W_{i}$ contains
a $b$-clique $W_{i}'$, then we are done, since $U\cup\{w_{i}\}\cup W_{i}'$ forms an $n$-clique.
It remains
to determine the success probability and the number of steps taken
in the above process. By the standard Chernoff bounds, the sizes of all
the sets $W_{i}$ concentrate around their means with high
probability. Hence, with high probability,
\[|W_{i}| = (1+o(1))p^{a+1}(1-p)^{i-1}T.\]
A standard application of Janson's inequality (see Chapters 8 and 10 of~\cite{AlSp}) then implies
\[
\Pr[W_{i}\text{ contains a \ensuremath{b}-clique}] = \Omega_b\left(\min(p^{\binom{b}{2}} |W_i|^b, 1)\right) = \Omega_{b}\left(\min(p^{(a+1)b+\binom{b}{2}}(1-p)^{(i-1)b}T^{b}, 1)\right).
\]
If $i$ ranges up to $p^{-\alpha}$ and $\alpha\le1$, then the decay
factor $(1-p)^{(i-1)b}$ is $\Theta_{b}(1)$ and can be safely ignored.
Since the event that each $W_{i}$ contains a $b$-clique is independent
of all the others, we need only pick $p,T,\alpha$ for which the expression
$p^{-\alpha}p^{(a+1)b+\binom{b}{2}}T^{b}$ is a positive
constant. If this is the case, then with at least constant probability
our strategy constructs an $n$-clique.
We also need to know that the total number of turns taken is $O_n(T)$.
This is true in Phases 1 and 2 by design. With high probability, the number
of turns taken in filling out each $W_i$ is $O_a(p^{a}T+p^{2(a+1)}T^{2})$. Since this is repeated $p^{-\alpha}$ times, it suffices to have
\[
T = O_a(p^{\alpha-2(a+1)})
\]
for the number of turns to be $O(T)$.
It remains to optimize the value of $T$ subject to
the constraints
$T = O_a(p^{\alpha-2(a+1)})$ and
$p^{-\alpha}p^{(a+1)b+\binom{b}{2}}T^{b} = \Omega_b(1)$.
As long as $2a + 3 - b \ge 0$, this system has solutions. Solving for $\alpha$ which minimizes $T$, we find that any
\[
\alpha\le \frac{b(2a+3-b)}{2(b-1)}
\]
works, as long as the decay condition $\alpha \le 1$ was also satisfied.
\end{proof}
Lemma~\ref{lem:branch-and-fill} provides upper bounds for $f(K_m,p)$ for all $m\ge 4$, where the shape of the power of $p$ depends on the residue class of $m$ modulo $3$.
\begin{thm} \label{thm:detailed-upper-bound}
If $p\in(0,1)$, then $f(K_3,p) = O(p^{-3/2})$ and, for $m\ge4$,
\[
f(K_m, p) = O_m(p^{-\frac{2}{3}m + c_m}),
\]
where
\[
c_m = \begin{cases}
\frac{m}{2m-3} & m\equiv0\pmod3\\
\frac{2}{3} & m\equiv1\pmod3\\
\frac{2m+8}{6m-3} & m\equiv2\pmod3.
\end{cases}
\]
\end{thm}
\begin{proof}
For $m = 3$, the bound is simple. Query $\Theta(p^{-3/2})$ pairs containing a given vertex $v_1$ and then, among the $\Theta(p^{-1/2})$ neighbors successfully found, query all pairs. For sufficiently large implied constants, the probability that we build a triangle containing $v_1$ is at least $1/2$.
When $m\ge 4$, we use Lemma~\ref{lem:branch-and-fill}, taking
\[
(a,b)=\begin{cases}
\Big(\frac{m-3}{3},\frac{2m}{3}\Big) & m\equiv0\pmod3\\
\Big(\frac{m-4}{3},\frac{2m+1}{3}\Big) & m\equiv1\pmod3\\
\Big(\frac{m-2}{3},\frac{2m-1}{3}\Big) & m\equiv2\pmod3.
\end{cases}
\]
This gives the required result.
\end{proof}
We conjecture that the bounds in Theorem~\ref{thm:detailed-upper-bound} are best possible up to the constant factor. In the next two subsections, we prove this is the case for $m\le 5$.
\subsection{Recursive graph building} \label{sec:recursive-graph-building}
Recall that $f(H,p)$ is the number of queries
needed in the Subgraph Query Problem to build a copy of $H$ with probability at least $\frac{1}{2}$. When $H=K_m$, we can prove a lower bound on $f(H,p)$ by combining Theorem~\ref{thm:query-online-connection} with Theorem~\ref{thm:random-lower-bound}.
\begin{prop}\label{prop:f-bound-clique}
If $m\ge 3$ and $c\le\frac{1}{2}m$, then
\[
f(K_m,p) \ge \frac{1}{4} p^{-(\binom{m}{2} - c(c-1))/(m-c)}.
\]
\end{prop}
\begin{proof}
Take $N = \frac{1}{4} p^{-(\binom{m}{2} - c(c-1))/(m-c)}$, which is chosen so that
\[
p^{\binom{m}{2} - c(c-1)}(2N)^{m-c} \le \frac{1}{4}.
\]
Since $(1-p)^{\binom{n}{2}}(2N)^n\rightarrow 0$ as $n\rightarrow\infty$, there is some $n$ sufficiently large for which
\[
p^{\binom{m}{2} - c(c-1)}(2N)^{m-c}+ (1-p)^{\binom{n}{2}}(2N)^n \le \frac{1}{2}.
\]
With $d=0$, this choice of $m,n,N,p,c,d$ satisfies the conditions of Theorem~\ref{thm:random-lower-bound}, so $\tilde{r}(m,n;p) > N$. By Theorem~\ref{thm:query-online-connection}, $f(K_m,p) \ge \tilde{r}(m,n;p)$, giving the required result.
\end{proof}
We now describe a general
method for obtaining a similar lower bound on $f(H,p)$ when $H$ is not a clique. As
before, define $t(H,p,N)$ to be the maximum expected number of copies of
$H$ that can be constructed in $N$ queries. The main result of this section bounds $t(H,p,N)$ when $H$ contains a large matching. To this end, recall that a graph has a $k$-matching if it contains $k$ disjoint edges.
\begin{thm} \label{thm:t-bound}
Let $H$ be a graph containing a $k$-matching. Then there exists an absolute constant $A>1$ for which
\[
t(H,p,N)\le (Ae(H))^{e(H)} p^{e(H)-k(k-1)}(2N)^{v(H)-k},
\]
whenever $pN \ge 1$.
\end{thm}
For any edge $e\in H$,
write $H\backslash e$ for the graph formed by removing the edge
$e$ from $H$. If $U$ is a subset of the vertices of $H$, write $H\backslash U$
for the induced subgraph of $H$ on the complement of $U$. We begin by proving
the following pair of recursive bounds on $t(H,p,N)$.
\begin{lem}\label{lem:recursive-H}
If $H$ is a simple labeled graph, then
\begin{equation}
t(H,p,N)\le p\sum_{e\in E(H)}t(H\backslash e,p,N)\label{eq:recursion1}
\end{equation}
and
\begin{equation}
t(H,p,N)\le (1+o(1))pN \min_{(u,v)\in E(H)}t(H\backslash\{u,v\},p,N),\label{eq:recursion2}
\end{equation}
where the $o(1)$ term tends to $0$ as $pN \rightarrow \infty$.
\end{lem}
\begin{proof}
Suppose Builder follows an optimal strategy which achieves $t(H,p,N)$
expected copies of $H$ in $N$ turns. For each copy $H_{i}$ of $H$
that appears during the game, distinguish the edge $e_{i}$ which
is built last in $H_{i}$. For each $e\in E(H)$, let $t_{e}(H,p,N)$
be the maximum expected number of copies of $H$ that Builder can build, only counting those copies of $H$ in which $e$ is the last edge built. Then, clearly,
\[
t(H,p,N)\le\sum_{e\in E(H)}t_{e}(H,p,N).
\]
Furthermore, $t_{e}(H,p,N)\le p t(H\backslash e,p,N)$, since each
copy of $H\backslash e$ can become exactly one copy of $H$ with
success rate $p$ if $e$ is built. Inequality \eqref{eq:recursion1}
follows.
As for recursion \eqref{eq:recursion2}, note simply that the number
of copies of $H$ is bounded by the number of choices for the images
of the vertices $u,v$ which are connected by an edge times the number of copies of
$H\backslash\{u,v\}$. By the Chernoff bound, the number of choices of an edge is tightly
concentrated around $pN$, so the inequality follows.
\end{proof}
It remains to apply these inequalities recursively.
\vspace{3mm}
\noindent
{\it Proof of Theorem~\ref{thm:t-bound}. } By (\ref{eq:recursion2}), there is an absolute constant $A > 1$ for which
\begin{equation}\label{eq:recursion2-explicit}
t(H,p,N)\le ApN \min_{(u,v)\in E(H)} t(H\backslash\{u,v\},p,N)
\end{equation}
whenever $pN \ge 1$.
We proceed by induction on the number of edges in $H$. When $H$ is
an empty graph on $m$ vertices, the result is trivial with $k=0$.
Let $H$ be a labeled graph for which the induction hypothesis is
true for every graph with fewer edges than $H$. Let $e\in E(H)$ run over all edges of $H$. We break into two cases:
\vspace{2mm}
{\it Case 1.} Every $H\backslash e$ contains a $k$-matching. Then, by induction
and \eqref{eq:recursion1}, it follows that
\begin{align*}
t(H,p,N) & \le p\sum_{e\in E(H)}t(H\backslash e,p,N)\\
& \le pe(H) (A(e(H)-1))^{e(H)-1} \cdot p^{e(H)-1-k(k-1)}(2N)^{v(H)-k}\\
& \le (Ae(H))^{e(H)} p^{e(H)-k(k-1)}(2N)^{v(H)-k},
\end{align*}
as desired.
\vspace{2mm}
{\it Case 2.} There exists $e\in E(H)$ for which $H\backslash e$ contains no $k$-matching.
Then, let $e_{2},\ldots,e_{k}$ be $k-1$ edges which complete a $k$-matching
of $H$ containing $e$. The edges incident
to $e$ must all be incident to one of the $e_{i}$ or else $H\backslash e$
would contain a $k$-matching. Also, $e$ cannot form a $4$-cycle
with any $e_{i}$ for the same reason. From these two facts one finds
that $e$ can be incident to at most $2(k-1)$ other edges in total.
Let $H'$ be the graph obtained from $H$ by removing the two vertices
of $e$ from $H$. Applying the induction hypothesis on $H'$,
which is a graph on $v(H)-2$ vertices with at least $e(H)-(2k-1)$
edges and a $(k-1)$-matching, we find that
\[
t(H',p,N)\le (Ae(H'))^{e(H')} p^{e(H)-(2k-1)-(k-1)(k-2)}(2N)^{v(H)-2-(k-1)}.
\]
Combining this with inequality \eqref{eq:recursion2-explicit}, we have
\begin{align*}
t(H,p,N) & \le A pN\cdot t(H',p,N)\\
& \le (Ae(H))^{e(H)} p^{e(H)-k(k-1)}(2N)^{v(H)-k},
\end{align*}
as desired.\hfill\qedsymbol
\vspace{3mm}
For our purposes, we will always assume $pN \ge 1$. Otherwise, with high probability at most a constant number of edges are built successfully in the Subgraph Query Game, so $t(H,p,N)$ will be negligibly small.
Since $t(H,p,N) < 1/2$ implies $f(H,p) > N$, Theorem~\ref{thm:t-bound} immediately implies Theorem~\ref{thm:f-bound}. Comparing this with Proposition~\ref{prop:f-bound-clique}, we note that while Theorem~\ref{thm:f-bound} gives a bound for all graphs $H$, it gives an inferior quantitative dependence on $e(H)$. While this stronger quantitative dependence in Proposition~\ref{prop:f-bound-clique} seems to be only a minor benefit, it was needed in the proof of Theorem~\ref{thm:opt-p}, which is why we retained the proof.
For large $m$, this bound only gives Corollary~\ref{cor:f-bound-asymp}, that $f(K_m, p) = \Omega_m( p^{-(2-\sqrt{2})m +O(1)})$, which is still far from the conjectured growth rate $p^{-\frac{2}{3}m+O(1)}$. However, for $m \le 5$, Theorem~\ref{thm:f-bound} can be used to pin down the asymptotic growth rate of $f(K_m, p)$, proving Theorem~\ref{thm:nle5}.
\vspace{3mm}
\noindent
{\it Proof of Theorem~\ref{thm:nle5}. }
The upper bounds for these cases are proved in Section~\ref{sec:upper-bound-strat}. Apply Theorem~\ref{thm:f-bound} by taking $k=1$ for $m=3$ and $k=2$ for $m=4, 5$ to get the desired lower bounds. \hfill \qedsymbol
\vspace{3mm}
\noindent
When $m\ge 6$, the matching argument of Theorem~\ref{thm:t-bound} does not seem sufficient for determining the exact growth rate of $f(K_m, p)$. Indeed, we will now exhibit an infinite family of graphs for which Theorem~\ref{thm:t-bound} is tight.
For $k\ge 1$, let $H_k$ be the graph on $2k$ vertices $a_i, b_i$, $1\le i \le k$, such that $a_i \sim a_j$ for all $i\ne j$, $b_i \not\sim b_j$ for all $i \ne j$, and $a_i \sim b_j$ if and only if $i\le j$. Thus $H_k$ is a split graph consisting of a $k$-clique, a $k$-independent set, and a half graph between them. We show that Theorem~\ref{thm:t-bound} is tight for $H_k$ up to a constant factor.
Note that the construction below requires $N$ to grow like a tower of $p^{-1}$'s of height $k$. It is possible that the same lower bound is false in the regime $N \le p^{-C}$ for any $C = C(k) > 0$.
\begin{thm}\label{thm:t-bound-tight}
For every $k\ge 1$, the graph $H_k$ defined above contains a $k$-matching and, for any $p\in(0,1)$,
\[
t(H_k, p, N) = \Omega_k( p^{e(H_k)-k(k-1)}N^{v(H_k)-k}),
\]
provided $N$ is sufficiently large in terms of $p$.
\end{thm}
\begin{proof}
In fact, $H_k$ has $k^2$ edges, $2k$ vertices, and contains a unique $k$-matching ${(a_i, b_i)}_{i\le k}$. It will suffice to show that for all $p\in (0,1)$ and $N$ sufficiently large in terms of $p$,
\[
t(H_k, p, N) = \Omega_k( p^{k}N^{k}).
\]
Builder's strategy will involve constructing a nested sequence of vertex sets $U_{1},U_{2},\ldots,U_{k}$.
The first set $U_{1}$ is just an arbitrary set of $N/k$ vertices.
In each successive $U_{i}$, assuming $|U_i|
\ge \sqrt{N}$ we can pick $N_{i}=N/(k|U_{i}|)$ vertices
$a_{i}^{(1)},a_{i}^{(2)},\ldots,a_{i}^{(N_{i})}\in U_{i}$ and try to build
all edges from each $a_{i}^{(j)}$ to every other vertex in $U_{i}$. This step takes at most $N/k$ turns.
The set $U_{i+1}$ is then defined to be the common neighborhood of $a_{i}^{(1)},\ldots,a_{i}^{(N_{i})}$ within $U_i$.
Repeating this process $k$ times, we use at most $N$ turns. For $N$ sufficiently large, with high probability the edge density from $a_{i}^{(1)},a_{i}^{(2)},\ldots,a_{i}^{(N_{i})}$ to the rest of $|U_i|$ is $(1+o(1))p$.
Thus, the number of copies of $H_{k}$ built in this way is bounded below by
\[
\prod_{i}(N_{i}\cdot p|U_{i}|)\ge (1+o(1))(pN)^{k}/k^{k},
\]
since we can choose $a_{i}$ out of any of the $N_{i}$ vertices $a_{i}^{(1)},\ldots,a_{i}^{(N_{i})}$
and $b_{i}$ out of any of its $(1+o(1))p|U_{i}|$ neighbors.
As long as $N$ is large enough that $|U_{k}|\ge\sqrt{N}$ with high probability, there will be enough vertices in the last set $U_k$ to perform the strategy. This argument
successfully constructs $\Theta_k(p^kN^k)$ copies of $H_k$. Taking $N$ to be a tower of $(2+2p^{-1})$'s of height $k$ is sufficient.
\end{proof}
We finish the subsection with an application of the preceding results and prove Theorem~\ref{thm:opt-p}.
Recall that this theorem states that for $m$ fixed and $n\rightarrow\infty$, a value of $p$ for which $\tilde{r}_{\text{\normalfont rand}}(m,n) \le 3\tilde{r}(m,n;p)$ satisfies $p = \Theta(\frac{m}{n}\log{\frac{n}{m}})$.
\vspace{3mm}
\noindent
{\it Proof of Theorem \ref{thm:opt-p}.}
By Theorem~\ref{thm:detailed-upper-bound},
\begin{equation*}\label{eq:f-known}
f(K_m,p) = O_m(p^{-2m/3}),
\end{equation*}
and in fact it can be checked from the proof that the explicit dependence on $m$ is polynomial. Moreover, using Proposition~\ref{prop:f-bound-clique} with $c=0$, we have that
\[
f(K_m,p) \ge \frac{1}{4} p^{-\frac{m-1}{2}} \ge \frac{1}{4}p^{-m/3},
\]
since $\frac{m-1}{2}\ge \frac{m}{3}$ for $m\ge 3$. Putting all this together, there exists an absolute constant $A>0$ for which
\begin{equation}\label{eq:f-known-explicit}
\frac{1}{4} p^{-m/3} \le f(K_m,p) \le m^A p^{-2m/3}
\end{equation}
for all $m\ge 3$, $p\in (0,1)$.
By Theorem~\ref{thm:query-online-connection}, we have
\[
\tilde{r}(m,n;p) \le \min\{f(K_m, p), f(K_n, 1-p)\} \le 3\tilde{r}(m,n;p).
\]
Pick some $p_0 \in (0,1)$ which maximizes the function $\min\{f(K_m, p), f(K_n, 1-p)\}$. Such a $p_0$ exists because $f(K_m,p)$ is nonincreasing in $p$, $f(K_n,1-p)$ is nondecreasing, and both are integer-valued. Then, $\tilde{r}_{\text{\normalfont rand}}(m,n)\le 3\cdot \tilde{r}(m,n;p_0)$. It remains to check that we could have chosen $p_0 = \Theta(\frac{m}{n}\log{\frac{n}{m}})$. By (\ref{eq:f-known-explicit}) and the fact that the bounds are continuous, we have
\[
\frac{1}{4}p_0^{-\frac{1}{3}m} \le n^A (1-p_0)^{-\frac{2}{3}n}
\]
and
\[
\frac{1}{4}(1-p_0)^{-\frac{1}{3}n} \le m^A p_0^{-\frac{2}{3}m}.
\]
Since $m\ge 3$ is fixed and $n\rightarrow \infty$, the first inequality implies $p_0\rightarrow 0$. In particular, $\log(1-p_0)=-p_0 +O(p_0^2)$. Taking the logarithm of both sides in the inequalities above, we have
\[
-\frac{1}{3} m \log p_0 -\log 4 \le A \log n + \frac{2}{3} n (p_0 +O(p_0 ^2))
\]
and
\[
\frac{1}{3}n(p_0 +O(p_0 ^2)) -\log 4 \le A \log m - \frac{2}{3}m\log p_0.
\]
Taking $n\rightarrow \infty$ and dividing through by $mp_0$, these inequalities combine to show
\[
\frac{\log (1/p_0)}{p_0} = \Theta\left(\frac{n}{m}\right)
\]
and it follows that $p_0 = \Theta(\frac{m}{n}\log{\frac{n}{m}})$, as desired. \hfill\qed
\vspace{3mm}
\subsection{The value of $t(K_m,p,N)$ for large $N$}\label{sec:t-large-N}
In this section, we investigate the behavior of the function $t(K_m,p,N)$ as $N\rightarrow \infty$. We find that when $N$ is very large, the essentially optimal strategy for building as many copies of $K_m$ as possible is to fill in the edges of a clique on $\sqrt{2N}$ vertices. This is in stark contrast with the rather delicate procedure described in Section~\ref{sec:upper-bound-strat} to build a single copy of $K_m$.
\subsubsection{Chernoff bounds and subjumbledness}
We will need a standard lemma (see, for example,~\cite[Theorem 2.1]{KrSu}) saying that with high probability all moderately large induced subgraphs of a random graph $G(N,p)$ have the expected number of edges. Recall that if $U\subset V(G)$ is a vertex subset of $G$, we write $G[U]$ for the induced subgraph on $U$.
\begin{lem}
\label{lem:chernoff}If $G=G(N,p)$ and $\varepsilon>0$, then, with high probability,
\[
e(G[U])=(1\pm\varepsilon)p\binom{|U|}{2}
\]
for all $|U|=\Omega_{\varepsilon}(p^{-1}\log N)$.
\end{lem}
In the literature (see~\cite{KrSu} and its references), this pseudorandomness property is usually called jumbledness. We also use this term, though in a slightly different way to how it is usually used.
\begin{defn}
A graph $G$ is {\it $(p,M,\varepsilon)$-jumbled} if, for every $U\subseteq V(G)$
with $|U|\ge M$,
\[
e(G[U])=(1\pm\varepsilon)p\binom{|U|}{2}.
\]
A graph $G$ is {\it $(p,M,\varepsilon)$-subjumbled} if it is a subgraph of some $(p,M,\varepsilon)$-jumbled graph.
\end{defn}
In what follows, we will show that subjumbled graphs
cannot have too many cliques. For the graph-building problem, the
heuristic is that it's not possible to build more copies of $H$ in
a known jumbled graph $G$ with $pN$ queries than it is
with $N$ queries in $G(N,p)$.
\subsubsection{Degeneracy}
Define a graph to be $d$-degenerate if there exists an ordering of
the vertices $v_{1},\ldots,v_{n}$ such that $|N(v_{i})\cap\{v_{1},\ldots,v_{i-1}\}|\le d$ for all $i$. The following simple lemma is well known.
\begin{lem}
\label{lem:degeneracy}Every graph with $E$ edges is $\sqrt{2E}$-degenerate.
\end{lem}
\begin{proof}
We exhibit the degenerate ordering by picking the vertices backwards
from $v_{n}$ to $v_{1}$. At each step, pick $v_{i}$ to be the minimal
degree vertex in the current graph and delete it. Note that $d(v_{i})\le i-1$
because there are only $i$ points left and also $d(v_{i})\le\frac{2E}{i}$
because the sum of the degrees is at most $2E$ and $v_{i}$ has minimal
degree. It follows that at every step $d(v_{i})\le\min(i,\frac{2E}{i})\le\sqrt{2E}$, as desired.
\end{proof}
This is not quite sufficient for our purposes, but it gives the main idea. What we really need is a better understanding of degeneracy in jumbled graphs.
In what follows, given a candidate ordering $v_{1},\ldots,v_{n}$ of the vertices,
we write $N^{-}(v_{i})=N(v_{i})\cap\{v_{1},\ldots,v_{i-1}\}$ and $d^{-}(v_{i})=|N^{-}(v_{i})|$.
\begin{lem} \label{lem:main-degeneracy}
Any $(p,M,\varepsilon)$-subjumbled graph on $N$ edges is $\max((1+\varepsilon)M\sqrt{p},(1+\varepsilon)\sqrt{2pN})$-degenerate.
\end{lem}
\begin{proof}
Let $H$ be a graph on $N$ edges that is a subgraph of some $(p,M,\varepsilon)$-jumbled graph $G$. We again pick vertices
of the graph $H$ in order of increasing degree among the remaining
vertices. Let the resulting order be $v_{1},\ldots,v_{n}$ and write
$U_{i}=\{v_{1},\ldots,v_{i}\}$. The construction guarantees that $v_{i}$ is
of minimal degree in $G[U_{i}]$.
If $i\le M$, then the subgraph $H[v_{1},\ldots,v_{i}]$ has at most
as many edges as $H[v_{1},\ldots,v_{M}]$, which has at most $(1+\varepsilon)pM^{2}/2$
edges. Thus,
\begin{align*}
d^{-}(v_{i}) & \le \min\Big(i,\frac{(1+\varepsilon)pM^{2}}{i}\Big)\\
& \le (1+\varepsilon)M\sqrt{p}.
\end{align*}
Otherwise, if $i>M$, the induced subgraph $H[v_{1},\ldots,v_{i}]$
has at most $(1+\varepsilon)pi^{2}/2$ edges and it clearly cannot have more
than $e(H)=N$ edges. Because $v_{i}$ is of minimal degree
in this induced subgraph,
\begin{align*}
d^{-}(v_{i}) & \le \frac{2}{i}\min\Big(\frac{(1+\varepsilon)pi^{2}}{2},N\Big)\\
& = \min((1+\varepsilon)pi,2Ni^{-1})\\
& \le (1+\varepsilon)\sqrt{2pN}
\end{align*}
and so every vertex has $d^{-}(v_{i})\le\max((1+\varepsilon)M\sqrt{p},(1+\varepsilon)\sqrt{2pN})$,
as desired.
\end{proof}
\subsubsection{Counting cliques}
We are ready to prove the following lemma. Recall the standard notation that $t(K,H)$ is the number of labeled graph homomorphisms from $K$ to $H$. Up to a lower order term, this is the same as counting labeled copies of $K$ in $H$. In fact, the equality is exact in the case we care about, where $K$ is a clique and $H$ is a simple graph without self-loops.
\begin{lem} \label{lem:t-bound}
For all $p\in(0,1)$, $m,M\ge2$ and $0<\varepsilon<1$, if $H$ is
a $(p,M,\varepsilon)$-subjumbled graph with $N$ edges, then
\begin{equation*}
t(K_{m},H)\le(1+O_{m}(\varepsilon+p^{1/2}N^{-1/2}))p^{\binom{m}{2}}(2p^{-1}N)^{\frac{m}{2}}+O_{m}\Big(\sum_{k=2}^{m-1}p^{\frac{m+k(k-3)}{2}}\cdot M^{m-k}\cdot N^{\frac{k}{2}}\Big).\label{eq:lem:t-bound}
\end{equation*}
\end{lem}
\begin{proof}
Take a degenerate ordering $v_{1},\ldots,v_{n}$ of $H$ such that
$v_{i}$ is of minimum degree in $H[v_{1},\ldots,v_{i}]$. By Lemma~\ref{lem:main-degeneracy},
\[
d^{-}(v_{i})\le\max((1+\varepsilon)M\sqrt{p},(1+\varepsilon)\sqrt{2pN}),
\]
where the second term dominates as soon as $N\ge M^{2}/2$.
Conditioning on whether or not $v_{n}$ is in the copy of $K_{m+1}$
we are counting, we see that
\[
t(K_{m+1},H)-t(K_{m+1},H\backslash v_{n})=(m+1)t(K_{m},H[N^{-}(v_{n})]).
\]
In particular, writing $t(m,N)=\max_{e(H)=N}^{*}t(K_{m},H)$, where
the maximum is taken over all graphs $H$ with $N$ edges that are subgraphs of
some $(p,M,\varepsilon)$-jumbled graph, we find that
\begin{equation}
t(m+1,N)\le\max_{d\le U(N)}\Big[t(m+1,N-d)+(m+1)t(m,e^{+}(d))\Big],\label{eq:t-system}
\end{equation}
where $U(N)=\max((1+\varepsilon)M\sqrt{p},(1+\varepsilon)\sqrt{2pN})$
and $e^{+}(d)$ is any upper bound on the number of edges in a graph on $d$
vertices that is a subgraph of a $(p,M,\varepsilon)$-jumbled graph.
The function $e^{+}$ we take is
\[
e^+(d) = \begin{cases}
\frac{d^{2}}{2} & d<M\sqrt{p}\\
(1+\varepsilon)\frac{pM^{2}}{2} & M\sqrt{p}\le d<M\\
(1+\varepsilon)\frac{pd^{2}}{2} & d\ge M.
\end{cases}
\]
To see that $e^+(d)$ is indeed an upper bound on the number of edges in a graph on $d$ vertices that is a subgraph of a $(p,M,\varepsilon)$-jumpled graph, we use the trivial bound when $d$ is small, extend to a
size $M$ set to use jumbledness when $d$ is somewhat close to $M$, and use
jumbledness directly for $d$ larger than $M$.
We are left to bound $t$ using the system of inequalities (\ref{eq:t-system}).
Write
\[
t^*(m,N) = p^{\binom{m}{2}}(2p^{-1}N)^{\frac{m}{2}}
\]
for the approximate optimum value of $t(m,N)$. We induct on $m$. The base case is $t(2,N)=2N$. Assume, by induction, that for some $m\ge 2$,
\[
t(m,N)\le(1+O_{m}(\varepsilon+p^{1/2}N^{-1/2}))t^*(m,N)+O_{m}\Big(\sum_{k=2}^{m-1}p^{\frac{m+k(k-3)}{2}}\cdot M^{m-k}\cdot N^{\frac{k}{2}}\Big).
\]
We would like to show that the same inequality holds for $m+1$. Iterating (\ref{eq:t-system}), there exists a sequence $(d_i)_{i\ge 1}$ of positive integers summing to $N$ for which
\[
d_i \le U\Big(N-\sum_{1\le j < i} d_j\Big)
\]
and
\[
t(m+1,N) \le (m+1)\sum_{i\ge 1} t(m,e^+(d_i)),
\]
which implies, by the induction hypothesis, that
\begin{align}
t(m+1,N) & \le (1+O_m(\varepsilon + p^{1/2}N^{-1/2}))(m+1)\sum_{i\ge 1}t^*(m,e^+(d_i)) \nonumber \\
& + O_{m+1}\Big(\sum_{k=2}^{m-1}p^\frac{m+k(k-3)}{2} \cdot M^{m-k} \cdot \Big[\sum_{i\ge 1}e^+(d_i)^{\frac{k}{2}}\Big]\Big).\label{eq:t-recursive}
\end{align}
Since $e^+(d)$ is constant on the range $M\sqrt{p}\le d < M$, the optimal choice of $d_i$ will never have any points in this range. The main term of (\ref{eq:t-recursive}) can thus be separated into the sum over $d_i < M\sqrt{p}$ and the sum over $d_i \ge M$:
\begin{equation} \label{eq:ranges}
\sum_{i\ge 1}t^*(m,e^+(d_i)) \le \sum_{d_i \ge M} t^*\Big(m, (1+\varepsilon)pd_i^2/2\Big) + \sum_{d_i < M\sqrt{p}} t^*\Big(m, (1+\varepsilon)d_i^2/2\Big).
\end{equation}
Note that $m\ge 2$, so $t^*(m,N)$ is a convex nondecreasing function in $N$. Also, the function $e^{+}(d)$ is nondecreasing and convex in $d$ except for the jump discontinuity at $d=M\sqrt{p}$. Therefore, in each of the ranges above, $t^*(m, \cdot)$ and $e^+(\cdot)$ are both convex nondecreasing functions.
To bound the first sum in (\ref{eq:ranges}), we pass to an integral. Write
\[
N_i = N - \sum_{j\le i} d_i.
\]
Then
\begin{align*}
\sum_{d_i \geq M} t^*(m,e^+(d_i)) & \le \sum_{i\ge 1} t^*\Big(m, (1+\varepsilon)pd_i^2/2\Big) \\
& = \sum_{i\ge 1} \int_{N_i}^{N_{i-1}}\frac{t^*(m, (1+\varepsilon)pd_i^2/2)}{d_i} dx.
\end{align*}
Because $t^*(m,(1+\varepsilon)pd^2/2)/d$ is an increasing function of $d$ and $d_i \le U(N_{i-1}) = (1+\varepsilon)\sqrt{2pN_{i-1}}$, we have
\begin{align*}
\sum_{i\ge 1} \int_{N_i}^{N_{i-1}}\frac{t^*(m, (1+\varepsilon)pd_i^2/2)}{d_i} dx
& \le \sum_{i\ge 1} \int_{N_i}^{N_{i-1}}\frac{t^*(m, (1+\varepsilon)p((1+\varepsilon)\sqrt{2pN_{i-1}})^2/2)}{(1+\varepsilon)\sqrt{2pN_{i-1}}} dx \\
& \le \sum_{i\ge 1} \int_{N_i+(1+\varepsilon)\sqrt{2pN}}^{N_{i-1}+(1+\varepsilon)\sqrt{2pN}}\frac{t^*(m, (1+\varepsilon)p((1+\varepsilon)\sqrt{2px})^2/2)}{(1+\varepsilon)\sqrt{2px}} dx \\
& \le \int_{0}^{N+(1+\varepsilon)\sqrt{2pN}}\frac{t^*(m,(1+\varepsilon)^3p^{2}x)}{\sqrt{2px}}dx.
\end{align*}
We had to shift integrals in the second step to guarantee that every value of $x$ in the range of integration is at least $N_{i-1}$.
Next, $t^*(m,N)$ is a polynomial in $N$, so we can absorb the $(1+\varepsilon)$ into the error term. Similarly, we can pull out an error term of $(1+(1+\varepsilon)\sqrt{2p/N})$ from the bounds of the integral to simplify. Reorganizing various error terms, we get
\[
\int_{0}^{N+(1+\varepsilon)\sqrt{2pN}}\frac{t^*(m,(1+\varepsilon)p^{2}x)}{\sqrt{2px}}dx
\le (1+O_{m+1}(\varepsilon + p^{1/2}N^{-1/2}))\int_{0}^{N}\frac{t^*(m,p^{2}x)}{\sqrt{2px}}dx.
\]
Finally, explicitly evaluating the integral, we have
\begin{align*}
\int_{0}^{N}\frac{t^*(m,p^{2}x)}{\sqrt{2px}}dx & = \int_{0}^{N}p^{\binom{m}{2}}(2p^{-1}p^{2}x)^{\frac{m}{2}}\frac{dx}{\sqrt{2px}}\\
& = 2^{\frac{m-1}{2}}p^{\frac{m^{2}-1}{2}}\int_{0}^{N}x^{\frac{m-1}{2}}dx\\
& = \frac{1}{m+1}2^{\frac{m+1}{2}}p^{\frac{m^{2}-1}{2}}x^{\frac{m+1}{2}}\Big|_{0}^{N}\\
& = \frac{1}{m+1}p^{\binom{m+1}{2}}(2p^{-1}N)^{\frac{m+1}{2}} \\
& = \frac{1}{m+1}t^*(m+1, N).
\end{align*}
Estimating the second sum in (\ref{eq:ranges}) trivially, we get
\begin{equation*}
\sum_{i\ge 1}t^*(m,e^+(d_i)) \le (1+O_{m+1}(\varepsilon+p^{1/2}N^{-1/2}))t^*(m+1,N)+O_{m+1}\Big(\frac{N}{M\sqrt{p}}t^*\Big(m,\frac{pM^{2}}{2}\Big)\Big).
\end{equation*}
To check the error terms in (\ref{eq:t-recursive}) match up
is similar: break up each sum into the sums over $d_i \ge M$ and $d_i < M\sqrt{p}$. The first sum is estimated by an integral and the second trivially. The result is
\[
O_{m+1}\Big(\sum_{k=2}^{m-1}p^\frac{m+k(k-3)}{2} \cdot M^{m-k} \cdot \Big[\sum_{i\ge 1}e^+(d_i)^{\frac{k}{2}}\Big]\Big) \le O_{m+1}\Big(\sum_{k=2}^{m}p^{\frac{m+1+k(k-3)}{2}}\cdot M^{m+1-k}N^{\frac{k}{2}}\Big),
\]
which is the right error term for $t(m+1,N)$, completing the induction.
\end{proof}
In particular, and this is essential, the implicit constants in this lemma do not
depend on $M$. As an immediate corollary, we now prove Theorem~\ref{thm:t-large-N}. Note that the $N$ above is the number of edges in $H$, which will correspond to $(1+o(1))pN$ below if $N$ is the number of queries made in the Subgraph Query Game.
\vspace{3mm}
\noindent
{\it Proof of Theorem~\ref{thm:t-large-N}.}
Applying the Chernoff bound from Lemma~\ref{lem:chernoff}, we see
that for any $\varepsilon>0$ we can take some $M=Cp^{-1}\log N$ so that the random graph $G(2N,p)$ is $(p,M,\varepsilon)$-jumbled with high probability. Also with high probability, the number of edges built in $N$ queries is $(1+o(1))pN$. It is easy to check that the exponentially small probabilities with which either of these are false have negligible impact on the value of $t(K_m,p,N)$. The subgraph $H$ built by Builder
must therefore satisfy the hypotheses of Lemma~\ref{lem:t-bound}
with $(1+o(1))pN$ edges.
The main term dominates the error terms for $N$ sufficiently large, giving the expected answer which is just $p^{\binom{m}{2}}(2N)^{\frac{m}{2}}$,
the number of $m$-cliques in $G(\sqrt{2N},p)$. This happens once the main term outgrows the largest error term, the term with $k=m-1$. This happens at $N = \Omega(p^{-(2m-3)}M^2)$, so it suffices to have $N \ge \omega(p^{-(2m-1)}\log^2{(p^{-1})})$. This proves the upper bound in Theorem~\ref{thm:t-large-N}. Of course, the lower bound is proved by the strategy of building all edges among $\sqrt{2N}$ vertices. \hfill \qedsymbol
\section{Concluding remarks} \label{sec:concluding-remarks}
It is an interesting problem to close the gap in the bounds for the online Ramsey number $\tilde{r}(m,n)$. In particular, we know that there are positive constants $c,c'$ for which $cn^3/(\log n)^2 \leq \tilde{r}(3,n) \leq c' n^3$ and it seems plausible that these bounds could be brought closer together. Indeed, we conjecture that the lower bound can be improved to $cn^3/\log n$ by considering the following Painter strategy motivated by the triangle-free process~\cite{B09}. Painter applies the triangle-free process to obtain an auxiliary triangle-free graph $G$ on vertex set $\{1, 2, \dots, r\}$ with $r=c_0 n^2/\log n$. Painter does not reveal this auxiliary graph. As before, we label vertices that reach degree $n/4$ with $1,\ldots,r$ as they arrive at degree $n/4$. When Builder adds an edge between two vertices in which both vertices have degree at least $n/4$, then these vertices have labels, say $i$ and $j$, and Painter paints the edge with the color of the edge $ij$ in $G$. Otherwise, they color the edge blue. This coloring clearly contains no red triangles, but it remains to show that it contains no blue $K_n$.
In studying the online Ramsey number, we were usually led by the idea that Builder's optimal strategy is to fill out an extremely sparse graph on the vertex set they touch. However, if Builder is restricted to play on a small vertex set, this intuition seems to go awry. If we define $\tilde{r}(m,n;N)$ in the same manner as the online Ramsey number but with the additional restriction that only $N$ vertices are allowed, then we conjecture that the function $\tilde{r}(m,n;N)$ increases substantially as $N$ decreases from $2 \tilde{r}(m,n)$, the maximum number of vertices in a graph with $\tilde{r}(m,n)$ edges, down to its minimal meaningful value $r(m,n)$.
The order of growth of $f(K_m, p)$ is still open for $m\ge 6$. In particular, Theorems~\ref{thm:f-bound} and~\ref{thm:detailed-upper-bound} show that $f(K_6, p) = \Omega(p^{-13/4})$ and $f(K_6,p) = O(p^{-10/3})$ and we conjecture that the upper bound is correct. This belief is rooted in our conviction that the upper bound for $t(H,p,N)$ given by Theorem~\ref{thm:t-bound}, upon which Theorem~\ref{thm:f-bound} relies, is not tight when $N$ is on the order of $f(H,p)$. Because of the examples in Theorem~\ref{thm:t-bound-tight}, these upper bounds can be tight when $N$ is very large, so further progress on this problem would need to be more sensitive to the size of $N$. It is plausible that any advance on this question and its generalizations could also impinge on our estimates for online Ramsey numbers.
\medskip
\noindent {\bf Acknowledgements.} We are extremely grateful to Joel Spencer for pointing out a serious flaw in our previous proof of Theorem~\ref{thm:alterations} which had been based on a generalization of the Lov\'asz Local Lemma~\cite{ErSp}. In the current version, we have a correct proof using a different approach. We would also like to thank the anonymous referee for some helpful remarks and Benny Sudakov for bringing the paper of Krivelevich~\cite{Kr1} to our attention.
|
3,212,635,537,480 | arxiv | \section{Introduction}
Models with a joint understanding of language and vision such as CLIP~\cite{radford2021learning}, ALIGN~\cite{jia2021scaling}, and UNITER~\cite{chen2019uniter} have found their use in a number of applications ranging from guiding image generation and editing using language~\cite{ramesh2022hierarchical}, to open vocabulary object detection~\cite{gu2021open}, image segmentation~\cite{Luddecke_2022_CVPR,zhou2021denseclip}, and retrieval. These models are trained on massive collections of image and text pairs, and their remarkably good visual and language representations is reflected by their strong performance on many standard image understanding tasks.
We focus on zero-shot learning capabilities of CLIP on the texture domains. Our motivation is two-fold. First, texture can be used to describe the appearance of a wide range of objects categories, especially in fine-grained domains.
Second, there is a rich vocabulary to describe textures corresponding to color, pattern, structure, periodicity, stochasticity, and other properties. While prior work has evaluated zero-shot learning capabilities of CLIP on some texture datasets such as DTD~\cite{cimpoi14describing}, in this work we conduct a more detailed study in the context of texture understanding expanding in three directions. First, we incorporate a wider variety of datasets including FMD~\cite{Sharan-JoV-14}, KTH-TIPS~\cite{fritz2004kth} and KTH-TIPS2~\cite{mallikarjuna2006kth}. Second, we investigate CLIP's ability to handle compositional attributes of texture on \dtdd~\cite{wu2020dtd2}, which contains descriptions of textures in the DTD dataset. Third, we analyze how well CLIP recognizes texture attributes on real world images of birds described by the color and texture of body parts.
The study reveals that CLIP performs remarkably well on these tasks. Larger image encoders (e.g., ViT-L/14~\cite{dosovitskiy2020image}) are consistently better for zero-shot classification. Prompt tuning has a smaller impact on larger image encoders. CLIP also handles compositions well, and outperforms custom models trained on specific domains like DTD, especially in their ability to handle rare color and pattern combinations.
On the CUB dataset~\cite{WahCUB_200_2011}, the top 10 classification accuracy on zero-shot learning improves from $24.9\%$ to $56.6\%$ when texture attributes are added to the categories described by their scientific names. However, we also find that image encoders of CLIP have a significant foreground bias which can be problematic when referring to background regions and non-central objects.
\section{Background}
\paragraph{Zero-shot learning with CLIP.} Models such as CLIP, ALIGN, UNITER jointly learn an image encoder $\mathbf{\Theta}$ and a text encoder $\mathbf{\Phi}$ such as $\mathbf{\Theta(x)} \approx \mathbf{\Phi(y)}$ for image-text pairs ${(\mathbf{x},\mathbf{y})}$. CLIP uses bidirectional encoder representations using transformers (BERT~\cite{vaswani2017attention}) for text, and convolutional networks (e.g., ResNet~\cite{he2016resnet}) or transformer (ViT~\cite{dosovitskiy2020image}) encoders are used for images.
CLIP was trained on a massive, curated dataset of 400 million image-text pairs, resulting in encoders that have good generalization abilities across visual recognition tasks.
To use CLIP for zero-shot learning, the description of a category $\textbf{y}$, often referred to as a prompt $\mathbf{p(y)}$, is encoded using the language model to obtain class prototypes $\mathbf{\Phi(p(y))}$.
The encoded images are then classified based on the similarity to the class prototypes.
The authors of CLIP showed that this strategy works remarkably well for a wide variety of datasets in computer vision.
While the class label can be directly used as the prompt, i.e., $\mathbf{p(y) = y}$, better prototypes can be obtained by using a structure phrase involving the category as a prompt, e.g., ``a photo of a cat'' instead of ``cat". Prompts reflect the style of text accompanying images on the web and can significantly impact performance.
While a large literature exists on designing prompts~\cite{liu2021pre,li2021prefix,lester2021power}, we explore a small space of hand-designed prompts on zero-shot texture recognition.
\paragraph{Texture datasets and tasks.} While many texture datasets exist in the literature, we focus on those that reflect describable properties of textures to benchmark CLIP's ability for zero-shot learning. This includes the Flickr material dataset (FMD) containing 10 material categories (e.g., wood, paper), Describable texture dataset (DTD) containing 47 describable attributes of texture (e.g., swirly, banded, zigzagged), KTH-TIPS and KTH-TIPS2a containing materials 10 and 11 materials taken under different lighting conditions. We also use the \dtdd dataset where images of DTD are annotated with descriptions of each image. \dtdd, unlike DTD contains multiple attributes that describe texture patterns in a compositional manner (e.g., red polka-dots or multicolored banded). We also explore the use of these attributes to describe species of birds in CUB Dataset~\cite{WahCUB_200_2011}. The dataset contains color and texture attributes of various body parts of the bird for each individual image. We collect these statistics at the entire set of images within a category to construct texture and color based descriptions of each category, which serve as a basis for zero-shot recognition. We compare the performance of the model against simply using the category name. CLIP has already seen many examples of each bird category associated with the images (possibly the entire CUB dataset is part of its training set). To simulate zero-shot learning on novel categories we compare against a baseline where we use the scientific names of the birds instead of their common ones. CLIP performs poorly in this setting, but by incorporating color and texture attributes the results improve significantly.
\section{Experiments}
\subsection{Zero-shot texture classification}
\paragraph{Datasets and evaluation metrics.} We report the average per-image accuracy on all four datasets. For DTD we evaluate on the first split (``test1.txt") with 1180 images across 47 classes. For FMD we evaluated on the entire dataset (10 material classes with 100 images per class). For KTH-TIPS we evaluate on 810 images equally split across 10 classes with categories such as ``brown bread", ``sandpaper", ``cotton", etc. While for KTH-TIPS2a we use 4608 images across 11 classes, with roughly the same number of images per class. The datasets are publicly available and linked via the project's github repository \url{https://github.com/ChenyunWu/CLIP_Texture}.
\paragraph{Results.} Table~\ref{tab:clip-texture-perf} shows the zero-shot classification accuracy of CLIP on DTD, FMD and KTH datasets. For this task we use the prompt ``a photo of a [c] pattern" for each category ``c" in the dataset. We observe that transformer variants (e.g., ViT-B/32 and ViT-L/14) are generally better than the ResNet (e.g., RN50 and RN101) counterparts.
The ViT-L/14@336 transformer trained on larger images (336$\times$336 vs. 224$\times$224) performs the best.
Table~\ref{tab:clip-prompt-perf} shows how the accuracy varies across prompts for two different image encoders. The best prompt varies across datasets but on average ``a photo of a [c] object" and ``a photo of a [c] pattern" performs best. Prompts have a larger impact on the performance of the smaller ViT-B/32 model compared to ViT-L/14 model indicated by the larger variance in performance across prompts. Table~\ref{tab:clip-texture-more} shows some additional results using the ViT-L/14@336 image encoder. The accuracy on FMD reaches 93.6\% using this encoder for the prompt ``a photo of [c] object", surprisingly outperforming current state-of-the-art ($\approx$ 85\%) based on bilinear representations and their variants~\cite{cimpoi2015deep,lin2015bilinear,lin2018second,gao2019global}. This could indicate potential overlap between FMD and the training set of CLIP. We are unable to verify this as the training dataset of CLIP is not publicly available, nor described in detail in the original paper.
\begin{table}[t]
\setlength{\tabcolsep}{5pt}
\small
\centering
\caption{\textbf{Performance of CLIP on zero-shot texture recognition.} Zero-shot accuracy for various image encoders using ``a photo of a [c] pattern" as the prompt.}
\begin{tabular}{l|cccc|c}
Model & DTD & FMD & KTH & KTH2a & Average \\
\hline
RN50 & 40.7 & 83.4 & 49.1 & 62.8 & 59.0\\
RN101 & 42.0 & 79.0 & 48.5 & 51.3 & 55.2 \\
ViT-B/32 & 41.1 & 83.8 & 58.4 & 59.5 & 60.7 \\
ViT-B/16 & 44.7 & 87.9 & 57.4 & 61.1 & 62.8\\
ViT-L/14 & 50.4 & 89.5 & 63.5 & 64.5 & 67.0 \\
ViT-L/14@336 & 50.7 & 90.5 & 63.9 & 66.0 & 67.8
\end{tabular}
\label{tab:clip-texture-perf}
\vspace{-2em}
\end{table}
\begin{table}
\setlength{\tabcolsep}{1pt}
\small
\centering
\caption{\textbf{Effect of prompt tuning.} Zero-shot accuracy for different prompts and image encoders. Prompt tuning has a larger impact on some datasets (e.g., KTH) and encoders (e.g., ViT-B/32). The best prompt varies across the datasets and encoders.}
\label{tab:clip-prompt-perf}
\begin{tabular}{l|cccc|cccc}
& \multicolumn{4}{c|}{ViT-B/32} & \multicolumn{4}{c}{ViT-L/14} \\
Prompt & DTD & FMD & KTH & KTH2a & DTD & FMD & KTH & KTH2a\\
\hline
[c] & 41.1 & 80.0 & 48.6 & 46.7 & 50.4 & 88.7 & 58.3 & 68.0 \\
a photo of a [c] & 43.1 & 79.9 & 50.4 & 49.9 & 52.3 & 89.0 & 61.0 & 69.4 \\
a photo of a [c] background & 43.1 & 79.9 & 50.4 & 49.9 & 50.4 & 89.3 & 59.3 & 69.8 \\
a photo of a [c] object & 42.3 & 83.2 & 56.3 & 59.7 & 53.0 & 92.3 & 59.6 & 70.0 \\
a photo of a [c] pattern & 41.1 & 83.8 & 58.4 & 59.5 & 50.4 & 89.5 & 63.5 & 64.5 \\
\hline
std. dev. & $\pm$1.0 & $\pm$2.0 & $\pm$4.3 & $\pm$6.0 & $\pm$1.3 & $\pm$1.5 & $\pm$2.0 & $\pm$2.3
\end{tabular}
\vspace{-2em}
\end{table}
\begin{table}
\setlength{\tabcolsep}{4pt}
\centering
\small
\caption{\textbf{Additional results using ViT-L/14@336 image encoder.} Zero-shot accuracy is shown on four texture datasets.}
\begin{tabular}{l|c|cccc|c}
Prompt & Model & DTD & FMD & KTH & KTH2a & Average \\
\hline
a photo of a [c] object & ViT-L/14@336 & 53.3 & 93.6 & 59.4 & 69.5 & 69.0\\
a photo of a [c] pattern & ViT-L/14@336 & 50.7 & 90.5 & 63.9 & 66.0 & 67.8
\end{tabular}
\label{tab:clip-texture-more}
\end{table}
\subsection{Performance on Describable Texture in Detail Dataset}
\begin{table*}[t]
\setlength{\tabcolsep}{8pt}
\centering
\caption{\textbf{Retrieval performance of DTML and CLIP on \dtdd.} Various performance metrics on image and phrase retrieval are shown for CLIP and DTML.}
\begin{tabular}{c|c|*{6}{c}}
Task & \multicolumn{1}{c|}{Model} & \multicolumn{1}{c}{MAP} & \multicolumn{1}{c}{MRR} & P@5 & \multicolumn{1}{c}{P@20} & \multicolumn{1}{c}{R@5} & \multicolumn{1}{c}{R@20} \\
\hline
\multirow{2}[2]{*}{Phrase retrieval}
& {DTML} & \textbf{31.6} & \textbf{72.5} & \textbf{40.6} & \textbf{22.9} & \textbf{20.2} & \textbf{44.5} \\
& CLIP & 12.2 & 40.0 & 17.6 & 11.4 & 8.4 & 21.5\\
\hline
\multirow{2}[2]{*}{Image retrieval}
& DTML & \textbf{13.5} & 31.1 & 16.5 & \textbf{14.5} & 5.2 & \textbf{17.3} \\
& CLIP & 12.7 & \textbf{32.1} & \textbf{16.9} & 13.2 & \textbf{6.1} & 17.3\\
\end{tabular}
\label{tab:dtd2}
\end{table*}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figs/clip-vs-dtml.pdf}
\vspace{-1em}
\caption{\textbf{Best and worst performing attributes on \dtdd for CLIP and DTML.} Each cloud represents top and bottom 80 attributes based on average precision on the retrieval task for CLIP (left) and DTML (right). On the right 80 attributes with the highest difference in performance between the two models. Font sizes of attributes are proportional to their frequency in \dtdd.}
\label{fig:dtd2_cloud}
\end{figure}
We first compare CLIP with DTML on phrase and image retrieval on \dtdd. DTML is the metric-learning baseline presented in~\cite{wu2020dtd2}, which trains an off-the-shelf BERT text encoder and a ResNet101 image encoder using a triplet-based metric learning loss on the \dtdd dataset. A linear layer on top of BERT and all the layers of ResNet101 are trained (or fine-tuned). For phrase retrieval we rank the 655 frequent phrases in \dtdd according to their distances to the query image, while for image retrieval we rank the images in the test set based on distances to the query text. For CLIP we use the prompt ``an image of [c] texture" where ``c" is the category. All results for CLIP are using the ViT-B/32 image encoder.
Table~\ref{tab:dtd2} shows the retrieval performance of CLIP and DTML on \dtdd.
CLIP obtains similar performance on image retrieval but is worse on phrase retrieval compared to DTML.
Figure~\ref{fig:dtd2_cloud} shows the best and worst attributes for each model.
We calculate the image retrieval average precision (AP) for each phrase and plot top and bottom 80 phrases.
We also visualize phrases with the largest difference of AP between the two models.
The two models are both good at phrases that describe common colors and patterns, but their worst performing phrases are different.
CLIP is better than DTML on rare colors such as ``orange", ``pink", ``purple". CLIP is also better on attributes related to materials or certain types of objects (e.g., ``wood", ``marble", ``glass") with are relatively rare. However, CLIP performs worse used to describe patterns and textures frequent in \dtd (e.g., ``rough", ``lined", ``grooved").
\begin{table*}[t]
\setlength{\tabcolsep}{4pt}
\centering
\caption{\textbf{R-precision of image retrieval on \dtdd.} CLIP understands compositional attributes despite not trained on this dataset. For example, on the ``two-colors" the performance is significantly better. It also exhibits a significant foreground bias as indicated by the lower performance on the ``background" task.}
\begin{tabular}{c|cccc}
\textbf{Model} & \textbf{~Foreground~} & \textbf{Background~} & \textbf{Color+Pattern~} & \textbf{Two-colors}\\
\hline
DTML & \textbf{46.5$\pm$20.6} & 52.0$\pm$6.3 & 41.7$\pm$22.8 & 27.4$\pm$15.1\\
CLIP & 38.0$\pm$14.9 & \textbf{60.2$\pm$5.5} & \textbf{45.2$\pm$23.5} & \textbf{55.2$\pm$16.2} \\
\hline
Chance & 50.0 & 50.0 & 7.4 & 5.5\\
\end{tabular}
\label{tab:clip:comp}
\end{table*}
\paragraph{Attributes as prompts.}
\dtdd contains multiple attributes for each image which could be incorporated into the prompt design for each category. We include the 20 most frequent attributes for each category in the prompt as ``an image of [$p_1$, $p_2$,\ldots, $p_{20}$] texture", where $p_i$ is the $i^{th}$ most-frequent phrase.
For example, the ``gauzy" category is described as ``an image of gauzy, sheer, transparent, light, thin, white, translucent, soft, see through, delicate, netted, meshy, airy, silky, fabric, see-through, folded, wavy, curtains, cloth texture."
This improves the accuracy to {54.8\%} from 41.1\% when only including the category as prompts with the ViT-B/32 encoder on DTD.
\paragraph{Synthetic Textures.} We conduct the compositionality modeling analysis on synthetic texture images the same as described in Section 5.3 of~\cite{wu2020dtd2}.
Given a query phrase such as ``blue and red", the task is rank the positive and hard negative images (which are ``blue" or ``red" but not both). The R-precision for the retrieval task is listed in Table~\ref{tab:clip:comp}. CLIP achieves a significant improvement on ``Two-colors" and a slight improvement on ``Color+Pattern" over the DTML model.
The larger training set of CLIP allows better generalization to rare or novel combinations in \dtdd.
We also see a slight improvement for CLIP on ``Background" compared against DTML but it performs lower than random guesses on ``Foreground". This suggests that CLIP
likely has a foreground bias. We investigate this aspect further on CUB Dataset.
\subsection{Performance on Caltech-UCSD Birds Dataset}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{figs/cub_examples.pdf}
\vspace{-2em}
\caption{\textbf{Examples of birds and their attributes.}
On the left are some bird species with the CUB annotations indicating color and texture of body parts. On the right are automatically generated prompts based on these attributes for zero-shot learning.
}
\label{fig:classify_eg}
\end{figure*}
\begin{table*}[t]
\centering
\setlength{\tabcolsep}{4pt}
\caption{\textbf{Phrase and image retrieval on CUB.} We experiment with 17 attributes that are included in both CUB and \dtdd.}
\begin{tabular}{c|c|*{6}{c}}
Task & \multicolumn{1}{c|}{Model} & \multicolumn{1}{c}{MAP} & \multicolumn{1}{c}{MRR} & P@5 & \multicolumn{1}{c}{P@20} & \multicolumn{1}{c}{R@5} & \multicolumn{1}{c}{R@20} \\
\hline
\multirow{2}[2]{*}{Phrase retrieval}
& DTML & 52.6 & 68.6 & \textbf{46.4} & - & \textbf{45.8} & - \\
& CLIP & \textbf{54.1} & \textbf{75.9} & 43.2 & - & 43.4 & - \\
\hline
\multirow{2}[2]{*}{Image retrieval}
& DTML & 35.3 & 53.7 & 44.7 & 43.8 & 0.2 & 0.7 \\
& {CLIP} & \textbf{50.1} & \textbf{91.7} & \textbf{72.9} & \textbf{71.8} & \textbf{0.5} & \textbf{1.6}\\
\end{tabular}
\label{tab:clip_cub}
\end{table*}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{figs/clip-vs-dtml-retrieval.pdf}
\vspace{-2em}
\caption{\textbf{Image retrieval on CUB.} For each query we show 5 ground-truth and top images retrieved using CLIP and DTML. CLIP primarily associates attributes with the foreground --- e.g., the query ``blue" is only associated with blue birds for CLIP, while the DTML retrieves both blue birds and the background water.}
\label{fig:cub_eg}
\end{figure*}
Once again, we compare the two models, DTML and CLIP, on the CUB dataset \cite{WahCUB_200_2011}.
Images in this dataset differ from the types of images in DTD which poses a significant domain shift for DTML.
For evaluation we select 17 attributes that both occur in \dtdd and CUB dataset.
Images with an attribute on any part, i.e., ``striped" on wing, upper-parts, or back, are counted as positives for the attribute, i.e., ``striped".
Table~\ref{tab:clip_cub} shows the retrieval performance of DTML and CLIP.
CLIP performs better than DTML on image retrieval and they perform similarly on phrase retrieval.
Figure~\ref{fig:cub_eg} shows example retrieved images. CLIP focuses on the foreground while DTML recognizes attributes from the background as well. For example, CLIP retrieves ``blue" birds, while DTML retrieves images with a ``blue" background such as water.
DTML retrievals are from different categories, but CLIP tends to return images of the same category, which implies that CLIP image features are highly related to categories such that images of the same category are close to each other in the embedding space.
\begin{table*}
\centering
\caption{\textbf{Zero-shot classification top-k accuracy on CUB test set using CLIP with different levels of category names and attributes.} In the brackets we show the number of categories for each level, .e.g., there are 115 different genus names.}
\begin{tabular}{c|cc|cc|cc|cc|cc|c}
\textbf{category} & \multicolumn{2}{c|}{\textbf{name}(200)} & \multicolumn{2}{c|}{\textbf{species}(200)} & \multicolumn{2}{c|}{\textbf{genus}(115)} & \multicolumn{2}{c|}{\textbf{family}(39)} & \multicolumn{2}{c|}{\textbf{order}(12)} & \textbf{``bird''}(1)\\
\hline
\textbf{\#attribute} & 0 & 15 & 0 & 15& 0 & 15& 0 & 15& 0 & 15&15\\
\hline
{top-1} & 51.8 & 50.2 & 6.6 & 14.3 & 15.4 & 14.4 & 10.8 & 12.9 & 6.6 & 12.5 & 12.1 \\
{top-5} & 82.7 & 81.6 & 19.2 & 41.0 & 18.8 & 40.5 & 11.5 & 36.0 & 6.7 & 35.4 & 35.8 \\
{top-10} & 91.0 & 91.3 & 24.9 & 56.6 & 20.8 & 53.3 & 17.3 & 51.9 & 11.3 & 51.5 & 52.3\\
\end{tabular}
\label{tab:cub_classify}
\end{table*}
\paragraph{Zero-Shot Classification with Attributes.} To construct a list of attributes for each category, for each attribute we estimate the ratio of images that contain the attribute within a category to the images that contain that attribute across all categories.
Top $k$ such attributes are added to prompts for the category in the format ``an image of a [$P$] [$C$] with [$P_1$] [$N_1$], [$P_2$] [$N_2$], ..."
where $C$ is the category label, $P$ are attributes of the entire bird, $N_i$ are body parts (such as ``belly", ``tail") and $P_i$ are attributes for the part $N_i$.
Figure~\ref{fig:classify_eg} shows three examples of constructed descriptions.
The attributes reflect subtle differences among the three similar species, e.g., ``Rhinoceros Auklet" is more ``duck-like" with ``buff leg", ``Parakeet Auklet" has ``white eye" and ``white belly", ``Crested Auklet" has ``crested head" and ``black nape".
We also use different choices for the category label, varying them from the common name to scientific names at different levels of the biological taxonomy. The common name is often used to describe the category and be associated with the corresponding images on the Internet. Scientific names at various levels in the taxonomy are less likely to have been observed by the language models.
Classification results are shown in Table~\ref{tab:cub_classify}.
We report the top-k accuracy for the 200-way classification.
When no attribute is added, we construct the description simply as “an image of a [$C$]” where [$C$] can be the common name, species, genus, family, or order.
In such cases species from the same genus, family or order have the same category embedding resulting in ties.
There is a significant performance drop when using scientific names compared to the common names. CLIP has learned plenty about the common names during training as it exploits images available on the web but fails when using species names which are relatively rare. Adding attributes to the descriptions improves performance by a large margin in this case, especially on the top 5/10 accuracy. With the help of attributes, we achieve similar performance when we replace the scientific names with the generic category description ``bird”.
\section{Conclusion and Limitations}\label{sec:conclusion}
We analyze how well CLIP recognizes describable properties of texture in natural images.
Remarkably, CLIP achieves strong zero-shot performance for texture classification, outperforming strong baselines on texture image and phrase retrieval on \dtdd.
Texture understanding is also effective on natural images of birds for attributes that describe the color and texture of body parts. This brings up the exciting possibility of applying similar models on other fine-grained domains such as fashion, Fungi, and Butterflies.
At the same time, we observe a foreground bias in the model, which might not be desirable when referring to attributes of the background or non-central object in the image.
We hope this contributes to a better understanding of performance and biases of CLIP for various applications.
\clearpage
{
\bibliographystyle{ieee_fullname}
|
3,212,635,537,481 | arxiv | \section{Introduction}
Ever since the inception of the Einstein's relativistic field equations, mathematical models for Lagrangian and Hamiltonian field theories have been at the forefront of research in mathematical physics. In this paper $k$ stands for the number of parameters on which the fields depend (e.g. $k=4$ in the case of spacetime), and we will always assume that the fields take values in a configuration manifold $Q$. In a few words, the polysymplectic model we will use in this paper, first introduced by G\"{u}nther \cite{Gunter}, makes use of a family of $k$ closed two-forms and it characterizes the field theory in terms of $k$ vector fields on a suitable vector bundle over $Q$. Polysymplectic geometry has proven to be appropriately applicable to many different contexts (see e.g.\ the references in the recent monograph \cite{bookpoly} on this topic). Recent contributions, both on the side of the mathematical fundamentals \cite{McClain1, Blacker1,Regularizedpolysymplectic} and on the side of the physics applications \cite{McClain2,Blacker2,PolysymplecticBF}, testify that polysymplectic geometry is today an active field of research.
The recent literature on the geometry of field theories has seen an ever increasing interest in symmetries and in symmetry reduction of the field equations (see e.g.\ \cite{Castrillon0,Castrillon1,Castrillon2,GbRat,LagPoincare_JGP,Ellis2,Routhfields,blacker2020reduction} for a non-exhaustive list, in different geometric frameworks, and for both Lagrangian and Hamiltonian field theories). In this context, symmetry is understood as the invariance of the Lagrangian or Hamiltonian function under the appropriate lift of the action of a Lie group $G$ on the configuration manifold $Q$. Specifically for polysymplectic structures, we will build in this paper on earlier results of e.g.\ \cite{Polyreduction,MunSal,LTM_LP,Symmetriesk}.
Since field theories are a generalization of classical mechanical systems, our methodology is mostly inspired by the many contributions on this topic in geometric mechanics. Symmetry reduction of Hamiltonian systems can best be understood in terms of either Poisson reduction or symplectic (and cotangent bundle) reduction (see e.g. \cite{stagesbook,MarsdenRatiuIntroduction}). In the last decades it has become clear that there exist two analoguous types of symmetry reduction for Lagrangian systems: Lagrange-Poincar\'e reduction \cite{LagRedbyStag} and Routh reduction \cite{RouthMarsden,Routh_CrampinTom,quasi}. The extension of the Lagrange-Poincar\'e reduction to polysymplectic Lagrangian field theories has been investigated in \cite{LTM_LP}. In this paper we will explore both cotangent bundle reduction and Routh reduction for polysymplectic manifolds. As far as we are aware, these two specific types of reduction have not been studied elsewhere. There exist, however, results on Routh reduction for field theories in other frameworks. For example, the paper \cite{Routhfields} investigates aspects of the field theory version of Routh reduction in a variational framework.
We start Section~\ref{sec:Preliminaries} by recalling both the standard polysymplectic structure on the cotangent bundle of $k^1$-covelocities $(T^1_k)^*Q$ for a Hamiltonian field theory, and the Poincar\'e-Cartan polysymplectic structure that leads to the $k$-symplectic Lagrangian field equations. We then state, for a general polysymplectic structure with momentum map $J$, a version of the so-called polysymplectic reduction theorem of \cite{Polyreduction}: we mention both the aspects concerning the reduced polysymplectic form (Theorem~\ref{thm:polyred}) and those concerning the reduced dynamics (Theorem~\ref{thm:polyred2}). We would like to point out that this type of reduction only applies under rather restrictive conditions, which are very specific to the field theory case and which do not show up in the case of a classical mechanical system. In classical mechanics, we know via Noether's theorem that there is (roughly speaking) a correspondence between the invariance of the system under symmetry groups and conservation laws, and that the symplectic reduction theorem heavily relies on this property. This correspondence is no longer valid in the case of a Lagrangian field theory: there the invariance under a symmetry group only results in the vanishing of a divergence, and in general no first integrals can be derived from this property. The polysymplectic reduction theorem of \cite{Polyreduction} circumvents this problem: one needs to ask for the dynamics to be tangent to a level set of momentum. This is the reason why this property also appears in our Theorems~\ref{thm:polyred2} and \ref{thm:red1}. We will see in the final section, however, that this does not prevent us from discussing some explicit solutions of interesting examples and applications.
In \cite{Polyreduction} there is no description of the reconstruction procedure after reduction. We discuss in Section~\ref{sec:polyred} first the notion of a $k$-vector field and its integrability and we give a new perspective on how and when the integrability of an invariant $k$-vector field can be inherited by its reduced $k$-vector field (Proposition~\ref{pro:reconstruction}). Based on results of \cite{LTM_LP}, we state in Theorem~\ref{thm:reconstruction} how the integral sections of the original field theory can be reconstructed from those of the reduced field theory.
As stated before, one of the main goals of this paper is to extend Routh reduction to polysymplectic field theories. We will base our methodology on geometric constructions that are similar to those in \cite{quasi} for the mechanical case. That is: we will explain Routh reduction via two distinct steps. First, in Section~\ref{sec:cotangentreduction} we refine the polysymplectic reduction theorem to the specific case of the $k$-cotangent bundle $(T^1_k)^*Q$ and its canonical momentum map $J$. This results in Proposition~\ref{thm:identification}, which is our analogue of the cotangent bundle reduction theorem (see e.g.\ \cite{stagesbook}). This Theorem has an interest on its own because, together with Theorem~\ref{thm:polyred2}, we obtain in this way the complete picture of the polysymplectic reduction of a Hamiltonian system on the standard polysymplectic manifold $(T^1_k)^*Q$.
In the second step we introduce the momentum map $J_L$ of a Lagrangian field theory and we use Lemma~\ref{lem:lema1} to identify the reduced manifold $J_L^{-1}(\mu)/G_\mu$ of the Lagrangian polysymplectic structure in Proposition~\ref{prop:identificationJL}. With the help of the so-called Routhian function, we show how its Legendre-type transformation relates the two reduced polysymplectic structures of $J_L^{-1}(\mu)/G_\mu$ and $J^{-1}(\mu)/G_\mu$. By pulling back the geometric structures from the Hamiltonian side, we can summarize in Theorem~\ref{thm:reducedform} the discussion with an explicit expression of the reduced polysymplectic structure on $J_L^{-1}(\mu)/G_\mu$ for the Lagrangian side. Finally, in Theorem~\ref{thm:red1} we relate the reduced Lagrangian field equations to the reduced polysymplectic form on $J_L^{-1}(\mu)/G_\mu$. The picture is then completed with a version of the reconstruction problem in Theorem~\ref{thm:reconstruction2}. The paper ends with some examples and applications: we discuss the case where the configuration manifold is a Lie group, the case where the Lagrangian is derived from a single invariant Riemannian metric (cfr.\ harmonic maps), and we end with the explicit coordinate calculations for a concrete example.
\section{Preliminaries}\label{sec:Preliminaries}
\subsection{Polysymplectic manifolds}
We will use Einstein's conventions for sums over $i=1,\dots,{\rm dim(Q)}$, but no sum signs for sums over $a=1,\dots,k$.
Let $N$ be a manifold of dimension $n(k+1)$. A \emph{$k$-polysymplectic structure} in $N$ is a closed non-degenerate $\R^k$-valued two-form $\Omega$, say
\begin{equation*}
\Omega= \sum_a \omega^a \otimes e_a,
\end{equation*}
with $1\leq a\leq k$, where $\omega^a$ are two-forms on $N$ and $\{e_1,\dots,e_k\}$ is a basis of $\R^k$. Clearly, having a $k$-polysymplectic structure is equivalent to the existence of a family of $k$ closed two-forms $(\omega^1,\dots,\omega^k)$ such that $\ker\omega^1\cap\dots\cap\ker \omega^k=\{0\}$. The case $k=1$ corresponds to a symplectic manifold; however, for $k>1$, we do not have in general a family of $k$ symplectic forms, since each of the two-forms $\omega^1,\dots,\omega^k$ might be degenerate. Introduced by G\"{u}nther in~\cite{Gunter}, polysymplectic structures provide a simple formalism to study a broad class of field theories, specifically those for which the Lagrangian or Hamiltonian is explicitly independent of the space-time coordinates.
A first and important example of polysymplectic manifold is the cotangent bundle of $k^1$-covelocities
\[
(T^1_k)^*Q =T^*Q\oplus\stackrel{k}{\dots}\oplus T^*Q
\]
equipped with the family of two-forms $(\omega_Q^1,\dots,\omega_Q^k)$ defined by $\omega_Q^a=(\pi_Q^a)^*\omega_Q$, where $\omega_Q$ the canonical symplectic form on $T^*Q$ and $\pi_Q^a\colon (T^1_k)^*Q\to T^*Q$ is the projection defined by
\[
\pi_Q^a(\alpha_q^1,\dots, \alpha_q^k)= \alpha_q^a.
\]
Elements in $(T^1_k)^*Q$ will be denoted using bold face characters, such as $\bm{\alpha}_q=(\alpha_q^1,\dots, \alpha_q^k)$ and, when the basepoint $q$ is understood, we will drop it and write $\bm{\alpha}=(\alpha^1,\dots,\alpha^k)$. If $q^i$ are coordinates on an open set $U\subset Q$, with $i=1,\dots,\dim Q$, then there are induced coordinates $(q^i,p_i^a)$ on $(\pi_Q^a)^{-1}(U)$ with $p_i^a$ the components of $\alpha_q^a$ in the usual basis of $T_q^*Q$. Our convention is to choose $\omega_Q=-d\theta_Q$ with $\theta_Q= p_i dq^i$ the tautological one-form on $T^*Q$, and therefore in the coordinates $(q^i,p_i^a)$ we have
\begin{equation}\label{eq:omega-a}
\omega_Q^a= dq^i\wedge dp_i^a=-d\theta^a_Q,
\end{equation}
where $\theta^a_Q=(\pi_Q^a)^*\theta_Q= p_i^a dq^i$.
The cotangent bundle of $k^1$-covelocities plays the role of the phase space for the Hamiltonian field theory in the $k$-symplectic framework. The dual of this space, which plays a similar role in the Lagrangian picture, is the tangent bundle of $k^1$-covelocities. It is denoted
\[
T^1_kQ=TQ\oplus\stackrel{k}{\dots} \oplus TQ,
\]
and can be identified with $J^1_0(\R^k,Q)$, the manifold of 1-jets of maps from $\R^k$ to $Q$ with source at $0\in\R^k$. An element in $T^1_kQ$ is denoted $\pmb{v}_q=(v_{1q},\dots,v_{kq})$ or, more often, simply $\pmb{v}=(v_{1},\dots,v_{k})$. Given coordinates $q^i$ on $Q$, the induced coordinates on $T^1_kQ$ are $(q^i,v^i_a)$, with $v^i_a$ the components of $v_{aq}$ in the natural basis of $T_qQ$. The projection of $(T^1_k)Q$ onto $Q$ will be denoted by $\tau_Q^k\colon T^1_kQ\to Q$. The natural pairing between elements of $(T^1_k)^*Q$ and $T^1_kQ$ is written as
\[
\langle\bm{\alpha}_q,\pmb{v}_q\rangle=\langle\alpha_q^1,v_{1q}\rangle +\dots+\langle \alpha_q^k,v_{kq}\rangle.
\]
A section $\pmb{X}$ of $\tau_Q^k\colon T^1_kQ\to Q$ is called a $k$-vector field on $Q$. If $\tau_Q^{k,a}\colon T^1_kQ\to TQ$ denotes the projection on the $a$-th component, we will denote
\[
X_a= \tau_Q^{k,a}\circ \pmb{X}\colon Q\to TQ,
\]
and write
\[
\pmb{X}=(X_1,\dots,X_k).
\]
Therefore, we might think of a $k$-vector field $\pmb{X}$ on $Q$ as a family of $k$ (standard) vector fields $X_a$ on $Q$.
We will use $t=(t^1,\dots,t^k)$ to denote points in $\R^k$. For any map $f\colon N\to M$ between manifolds, we write $T_nf$ or simply $Tf$ for the tangent map of $f$ at $n\in N$.
\begin{definition}\label{def:integrable} An \emph{integral section} of $\pmb{X}$ through $p\in Q$ is a map $\varphi\colon U\subset\R^k\to Q$, defined on some neighbourhood $U\ni 0$, such that
\[
\varphi(0)=p,\qquad T_t\varphi\circ \left(\left.\fpd{}{t^a}\right|_t\right)=X_a(\varphi(t)),\quad \text{for every $t\in U$}.
\]
We say that $\pmb{X}$ is \emph{integrable} if there is an integrable section through every point of $Q$.
\end{definition}
We will usually take $U=\R^k$ in the definition of integral section above. For any map
\[
\phi=(\phi^1,\dots,\phi^{\dim(Q)})\colon U\subset \R^k\to Q,
\]
the first prolongation is the map $\phi^{(1)}\colon U\to T^1_kQ$ given by
\[
\phi^{(1)}(t)=\left(\phi(t),T_t\phi\,(\partial/\partial t^1),\dots,T_t\phi\,(\partial/\partial t^k)\right).
\]
An integral section of $\pmb{X}$ is then a map which satisfies $\varphi^{(1)}(t)=\pmb{X}\circ \varphi(t)$, or
\begin{equation}\label{eq:integralsection}
\fpd{\varphi^i}{t^a}=X^i_a\circ\varphi,
\end{equation}
where $X_a= X_a^i\,\partial/\partial q^i$. From Definition~\ref{def:integrable}, one can show that in order for $\pmb{X}$ be integrable one needs to rectify the family of vector fields $X_1,\dots,X_k$. It is well-known that this is possible iff the condition $[X_a,X_b]=0$ holds for each $a,b$. This condition can be expressed as
\[
X_a^i\fpd{X_b^j}{q^i}- X_b^i\fpd{X_a^j}{q^i}=0,
\]
which is precisely the integrability condition of the PDE~\eqref{eq:integralsection} when one applies the chain rule.
\subsection{Hamiltonian and Lagrangian \texorpdfstring{$k$}{k}-symplectic field theory}\label{sec:HLfieldtheory}
Given a polysymplectic manifold $(N,\omega^a)$, the vector bundle morphism $\flat_\omega\colon T^1_k N\to T^*N$ is defined as
\[
\flat_\omega(v_1,\dots,v_k)= \sum_a v_a\lrcorner \omega^a.
\]
It can be easily shown that $\flat_\omega$ is surjective: indeed, it is the dual -up to a sign- of the injective map over the identity $TN\to (T^1_k)^*N$ given by $v\mapsto (v\lrcorner\omega^1,\dots,v\lrcorner \omega^k)$.
Consider a Hamiltonian $H\colon (T_k^1)^*Q\to \R$. The $k$-symplectic Hamiltonian eqs. for a $k$-vector field $\pmb{X}$ on $(T_k^1)^*Q$ are
\begin{equation}\label{eq:k-HAM}
\flat_{\omega_Q} (\pmb{X})=dH, \tag{k-HAM}
\end{equation}
i.e.\ $\sum_a X_a\lrcorner \omega_Q^a=dH$ ($\omega_Q^a$ have been defined in~\eqref{eq:omega-a}). Since $ \flat_{\omega_Q}$ is surjective, the equations~\eqref{eq:k-HAM} always admit a solution which, in general, will not be unique. The main interest of these equations is that the integrable sections of $\pmb{X}$ give solutions of the familiar Hamilton-De\,~Donder-Weyl equations in Hamiltonian field theory (cf. the book \cite{bookpoly} for a full discussion). One finds in this way the so-called admissible solutions~\cite{RomanRey}.
The equations~\eqref{eq:k-HAM} are a particular case of a more general class of $k$-symplectic Hamiltonian equations: if $(N,\omega^a)$ is a polysymplectic manifold and $H\colon N\to\R$ is the Hamiltonian, the $k$-symplectic Hamiltonian eqs. for a $k$-vector field $\pmb{X}$ on $N$ are
\begin{equation}\label{eq:k-Sym}
\flat_{\omega} (\pmb{X})=dH. \tag{k-Sym}
\end{equation}
Of course, one recovers~\eqref{eq:k-HAM} in the case $N=(T_k^1)^*Q$, and we speak of $k$-symplectic Hamiltonian equations for both~\eqref{eq:k-HAM} and ~\eqref{eq:k-Sym}.
We now recall how the Euler-Lagrange equations in Lagrangian field theory can be obtained in the $k$-symplectic setting; a detailed discussion, including a comprehensive list of references, can be found in~\cite{bookpoly}. The Euler-Lagrange field equations for a Lagrangian $L\colon T^1_k Q\to\R$ of the form $L(q^i,v^i_a)$ are the following set of second-order PDEs:
\begin{equation}\label{eq:EL}
\sum_a \left.\fpd{}{t^a}\right|_t \left(\left.\fpd{L}{v^i_a}\right|_{\psi(t)}\right)=\left.\fpd{L}{q^i}\right|_{\psi(t)},\quad \left.v^i_a(\psi(t))\right|_{\psi(t)} =\left.\fpd{\psi^i}{t^a}\right|_t, \tag{EL}
\end{equation}
where $\psi=(\psi^1,\dots,\psi^k)\colon \R^k\to T^1_kQ$. We are interested in describing solutions of~\eqref{eq:EL} for a regular Lagrangian using a $k$-symplectic approach. This can be done in a way that closely resembles the case of a regular Lagrangian in classical mechanics, where the Legendre transformation is used to construct a symplectic form and an energy function that govern the dynamics.
Given a Lagrangian $L\colon T^1_k Q\to\R$ , the Legendre transformation is the map
\[
\pmb{F} L\colon T^1_kQ\to (T^1_k)^*Q
\]
with components $(FL)^a\colon T^1_kQ\to T^*Q $ given by
\begin{equation}\label{eq:Legendre}
\langle(FL)^a(\pmb{v}),w_q\rangle = \left.\frac{d}{ds}\right|_{s=0}L(v_{1q},\dots,v_{aq}+sw_q,\dots,v_{kq}),\qquad w_q\in TQ.
\end{equation}
In coordinates, it reads
\[
\pmb{F} L(q^i,v^i_a)=\left(q^i,\fpd{L}{v^i_a}\right).
\]
The energy function associated to the Lagrangian $L$, $E_L\colon T^1_kQ\to\R$, is defined as
\begin{equation}\label{eq:energy}
E_L(\pmb{v})=\langle \F L (\pmb{v}),\pmb{v}\rangle -L(\pmb{v}),
\end{equation}
or in coordinates:
\[
E_L(q^i,v^i_a)=\sum_{a} v^i_a\fpd{L}{v^i_a}-L.
\]
\begin{definition} A Lagrangian $L\colon T^1_k Q\to\R$ is \emph{regular} if $\pmb{F} L$ is a local diffeomorphism. If, additionally, $\pmb{F} L$ is a global diffeomorphism then $L$ is \emph{hyperregular}.
\end{definition}
Locally, the Lagrangian $L$ is regular if the matrix
\begin{equation*}
\left(\fpd{^2 L}{ v^i_a\partial v^j_b}\right)\qquad 1\leq i,j\leq n,\;\; 1\leq a,b\leq k,
\end{equation*}
has maximal rank everywhere on $T^1_kQ$ (this maximal rank equals $kn$). When the Lagrangian is regular, the family of 2-forms
\[
\omega^a_{Q,L}=(\F L)^*\omega^a_Q= dq^i\wedge d\left(\fpd{L}{v^i_a}\right),\qquad a=1,\dots,k,
\]
are a polysymplectic structure on $T^1_kQ$. In this case, we will consider the following set of equations for a $k$-vector field $\pmb{\Gamma}$ on $T_k^1Q$:
\begin{equation}\label{eq:k-EL}
\sum_a \Gamma_a\lrcorner \omega_{Q,L}^a=dE_L, \tag{k-EL}
\end{equation}
which will be referred to as \emph{(k-symplectic) Euler-Lagrange equations} (or k-EL for short). They are of course a particular case of~\eqref{eq:k-Sym}. It can be shown that, defining the Hamiltonian as $H=E_L\circ (\F L)^{-1}$, the Legendre transformation gives a bijection between the solutions of \eqref{eq:k-HAM} and \eqref{eq:k-EL} and their integral sections; for a proof, see \S 6.4 in~\cite{bookpoly}.
The importance of~\eqref{eq:k-EL} lies in the following result:
\begin{theorem}\label{thm:regularsolutions} Let $L$ be a regular Lagrangian and $\pmb{\Gamma}$ be an integrable solution of \eqref{eq:k-EL}. Then the integral sections $\varphi\colon Q\to T^1_kQ$ of $\pmb{\Gamma}$ are prolongations of maps $\phi\colon \R^k\to Q$ and are solutions of \eqref{eq:EL}.
\end{theorem}
An integrable solution $\pmb{\Gamma}$ of \eqref{eq:k-EL} will be called an integrable Lagrangian SOPDE. In other words, Theorem~\ref{thm:regularsolutions} states that, if an integrable Lagrangian SOPDE for a given regular Lagrangian $L$ is known, we can find solutions of the Euler-Lagrange equations of $L$ by finding integral sections $\varphi=\phi^{(1)}\colon Q\to T^1_kQ$ of $\Gamma$. By definition, the projection $\phi=\tau_Q^k\circ\phi^{(1)}\colon \R^k\to Q$ satisfies the Euler-Lagrange equations
\begin{equation}\label{eq:ELphi}
\sum_a \left.\fpd{}{t^a}\right|_t \left(\left.\fpd{L}{v^i_a}\right|_{\phi^{(1)}(t)}\right)=\left.\fpd{L}{q^i}\right|_{\phi^{(1)}(t)}.
\end{equation}
We will also say that a map $\phi\colon \R^k\to Q$ is a solution of~\eqref{eq:EL} if it satisfies the equations~\eqref{eq:ELphi} above. The situation is similar to the Hamiltonian case: here one finds the admissible solutions of the Euler-Lagrange equations, namely those solutions that can be retrieved as an integral section of a some integral solution $\pmb{\Gamma}$, see~\cite{RomanRey}.
The term SOPDE stands for ``second order partial-differential equation'', which means that $\pmb{\Gamma}=(\Gamma_1,\dots, \Gamma_k)$ is of the form
\begin{equation}\label{eq:SOPDE}
\Gamma_a=v^i_a \fpd{}{q^i}+ \sum_b (\Gamma_a)^i_b \fpd{}{v^i_b},
\end{equation}
for some functions on $(\Gamma_a)^i_b$ on $T^1_kQ$. This is equivalent with the fact that all the integral curves of $\Gamma$ are prolongations of maps $\phi\colon \R^k\to Q$ (see e.g.~\cite{MunSal} for a proof). The solutions of~\eqref{eq:k-EL} for regular Lagrangians are always of this type. By definition, an integral section $\varphi=(\varphi^i,\varphi^i_a)\colon \R^k\to T^1_kQ$ of the SOPDE~\eqref{eq:SOPDE} satisfies
\[
\fpd{\varphi^i}{t^a}= \varphi^i_a,\qquad \fpd{\varphi^i_a}{t^b}= (\Gamma_a)^i_b\circ \varphi,
\]
which may as well be written as a system of second order partial differential equations:
\[
\fpd{^2\varphi^i}{t^a\partial t^b}(t)= (\Gamma_a)^i_b\left(\varphi^i(t),\fpd{\varphi^i(t)}{t^c}\right).
\]
In particular, the equality of the mixed partial derivatives gives the integrability condition $(\Gamma_a)^i_b=(\Gamma_b)^i_a$, valid for any integrable SOPDE $\pmb{\Gamma}$.
Later, when we study the reduced Lagrangian field theories we will need extensions of some of the notions above. In particular the reduced space will be a pullback bundle $\Pi^* T^1_kQ$ for some bundle $\Pi\colon P\to Q$.
In this case:
\begin{enumerate}[label={(\roman*})]
\item An element of $\Pi^* T^1_kQ$ will be denoted by $\pmb{v}=(p,v_{1q},\dots,v_{kq})$, with $\Pi(p)=q$.
\item A Lagrangian on the reduced space is a function $L\colon \Pi^* T^1_kQ\to \R$.
\item The Legendre transformation is a map $\pmb{F} L\colon \Pi^* T^1_kQ\to \Pi^*(T^1_k)^*Q$. If $q^i$ are coordinates on $Q$ and $(q^i,y^\alpha)$ are bundle coordinates on $P$, the Legendre transformation is simply
\[
\pmb{F} L(q^i,y^\alpha,v^i_a)=\left(q^i,y^\alpha,\fpd{L}{v^i_a}\right).
\]
\item The energy $E_L\colon \Pi^* T^1_kQ\to\R$ is the function
\[
E_L(q^i,y^\alpha, v^i_a)=\sum_{a} v^i_a\fpd{L}{v^i_a}-L.
\]
\end{enumerate}
The intrinsic definition of $\pmb{F} L$ and $E_L$ is obtained from the natural pairing between $\Pi^*(T^1_k)^*Q$ and $\Pi^* T^1_kQ$ as in the standard case.
\subsection{Actions and connections}\label{subsec:connections}
If $G$ is a Lie group, we will denote by $\lag$ its Lie algebra and by ${\rm Ad}\colon G\times \lag\to\lag$ the adjoint action. The coadjoint action will be denoted ${\rm Coad}\colon G\times \lag\to\lag$, and is defined as usual by:
\[
{\rm Coad}_g(\mu) = {\rm Ad}_{g^{-1}}^*\mu.
\]
We consider the spaces $\lagk=\lag\times \stackrel{k}{\dots} \times \lag$ and $\lagdk = \lag^*\times \stackrel{k}{\dots} \times \lag^*$. They are equipped with the $k$-Adjoint and $k$-Coadjoint actions, defined as:
\begin{align*}
{\rm Ad}^k\colon G\times \lagk &\to\lagk,\\
(g,\xi_1,\dots,\xi_k)&\mapsto ({\rm Ad}_g \xi_1,\dots,{\rm Ad}_g \xi_k),
\end{align*}
and
\begin{align*}
{\rm Coad}^k\colon G\times \lagdk &\to\lagdk,\\
(g,\mu_1,\dots,\mu_k)&\mapsto ({\rm Ad}_{g^{-1}}^*\mu_1,\dots,{\rm Ad}_{g^{-1}}^*\mu_k),
\end{align*}
respectively.
In this paper we will only consider free and proper actions of a Lie group $G$ on a manifold $Q$. This guarantees that $\pi \colon Q\to Q/G$ is a principal bundle. If $\Phi\colon G\times Q\to Q$ is an action, we will denote by $\xi_Q$ the infinitesimal generator of the action corresponding to $\xi\in\lag$. For concreteness, we will work with left actions, and given $q\in Q$ we will write $\Phi_g(q)=g\cdot q$ when there is no risk of confussion.
Given an action $\Phi$ on $Q$, there are induced $G$-actions $\Phi^{TQ}$ on $TQ$ and $\Phi^{T^1_kQ}$ on $T^1_kQ$. They are defined as follows:
\begin{align*}
\Phi^{TQ}_g(v_q)&=T_q\Phi_g(v_q), \\
\Phi^{T^1_kQ}_g(v_{1q},\dots,v_{kq})&=(T_q\Phi_g(v_{1q}),\dots,T_q\Phi_g(v_{kq})).
\end{align*}
We will write the actions above as $g\cdot v_q$ and $g\cdot \pmb{v}_q$ respectively. There are also induced actions on $T^*Q$ and $(T^1_k)^*Q$ defined similarly as follows:
\begin{align*}
\Phi^{T^*Q}_g(\alpha_q)&=T^*_{g\cdot q}\Phi_{g^{-1}}(\alpha_q), \\
\Phi^{(T^1_k)^*Q}_g(\alpha^1_{q},\dots,\alpha^k_{q})&=(T^*_{g\cdot q}\Phi_{g^{-1}}(\alpha^1_q),\dots,T^*_{g\cdot q}\Phi_{g^{-1}}(\alpha^k_q)).
\end{align*}
We will also write $g\cdot \alpha_q$ and $g\cdot \bm{\alpha}_q$ for the actions above. Note, in particular, that the pairings satisfy $\langle \alpha_q,v_q\rangle=\langle g\cdot \alpha_q, g\cdot v_q\rangle$ and $\langle \pmb{\alpha}_q,\pmb{v}_q\rangle=\langle g\cdot \pmb{\alpha}_q, g\cdot \pmb{v}_q\rangle$.
Recall that a principal connection on the principal bundle $Q\to Q/G$ can be given in terms of a connection 1-form $\Ac\colon TQ\to \lag$ which satisfies:
\begin{enumerate}[label={(\roman*})]
\item $\Ac(\xi_Q)=\xi$, for each $\xi\in\lag$,
\item $\Ac(g\cdot v)={\rm Ad}_{g} (\Ac(v))$, for each $g\in G$ and $v\in TQ$.
\end{enumerate}
Alternatively, a principal connection defines an (invariant) horizontal subbundle $\mathcal{H}\subset TQ$:
\[
\mathcal{H}=\ker(\Ac).
\]
We will use the following definition for the curvature $K$ of $\Ac$: for two vector fields $X,Y$ on $Q$
\[
K(X,Y)=-{\rm Ver}\big([{\rm Hor}(X),{\rm Hor}(Y)]\big),
\]
where ${\rm Hor}(\cdot)$ and ${\rm Ver}(\cdot)$ denote the horizontal and vertical parts w.r.t. $\Ac$. By definition, the curvature of two vector fields is again a vector field.
Given a principal connection $\Ac$, we will sometimes denote by $\pmb{\Ac}\colon T^1_kQ\to \lagk$ the map:
\[
\pmb{\Ac}(v_1,\dots,v_k)=\left(\Ac(v_1),\dots,\Ac(v_k)\right).
\]
The map $\pmb{\Ac}$ is, in the terminology of \cite{LTM_LP}, a \emph{simple principal $k$-connection}.
\section{Polysymplectic reduction and reconstruction}\label{sec:polyred}
The aim of this section is to recall the polysymplectic reduction theorem in~\cite{Polyreduction} and to discuss, when possible, the reconstruction of solutions. We will also study in detail the case of the cotangent bundle of $k^1$-velocities, which will be used later in Section~\ref{sec:Routh} when we examine the reduction of the tangent bundle of $k^1$-velocities.
\subsection{Polysymplectic reduction}
Assume that $(N,\omega^a)$ is a polysymplectic manifold, and that $\Phi_g\colon N\to N$ is a polysymplectic action of a Lie group $G$ with momentum map
\[
J=(J^1,\dots,J^k)\colon N\to \lag^*\times \stackrel{k}{\dots} \times \lag^*=\lagdk,
\]
where $\lag$ is the Lie algebra of $G$. This means that $\Phi_g^*\omega^a=\omega^a$ (for each $a=1,\dots,k$) and that ${\xi_N}\lrcorner \omega^a=d J^a_\xi$, with $J^a_\xi\colon N\to\R$ defined as $J^a_\xi(x)=\langle J^a(x),\xi\rangle$ for each $x\in N$. We also require equivariance of the momentum map w.r.t.\ the $k$-Coadjoint action (see Section~\ref{subsec:connections}), namely
\[
J\circ\Phi_g={\rm Coad}^k_g\circ J.
\]
Let us denote by $G_\mu\subset G$ the isotropy group of $\mu=(\mu_1,\dots,\mu_k)$ under the ${\rm Coad}^k$ action and by $\lag_\mu$ its Lie algebra. One checks easily that the following holds:
\[
G_\mu= G_{\mu_1}\cap\dots\cap G_{\mu_k},\qquad \lag_\mu=\lag_{\mu_1}\cap\dots\cap \lag_{\mu_k},
\]
where $G_{\mu_a}$ is the isotropy group of $\mu_a$ under the usual coadjoint action ${\rm Coad}$ of $G$, and $\lag_{\mu_a}$ is the Lie algebra of $G_{\mu_a}$. In particular, $G_\mu$ is a subgroup of $G_{\mu_a}$ for each $a$.
For future reference, we need to specify two technical conditions for the momentum map:
\begin{enumerate}[label={(\roman*})]
\item[(C1)] $\ker (T_xJ^a)=T_x(J^{-1}(\mu))+\ker\omega^a\mid_x+T_x(G_{\mu_a}\cdot x)$,\; for all $x\in J^{-1}(\mu)$, and for each $a$.
\item[(C2)] $T_x(G_{\mu}\cdot x)=\cap_a \big[T_x(G_{\mu_a}\cdot x)+\ker\omega^a\mid_x\big]\cap T_x(J^{-1}(\mu))$,\; for all $x\in J^{-1}(\mu)$.
\end{enumerate}
\vspace{-.7pc}The notation $G_{\mu_a}\cdot x=\{g\cdot x\st g\in G_{\mu_a}\}$ stands for the orbit of $x$ under the $G_{\mu_a}$ action.
The polysymplectic reduction theorem (Theorem 3.17 in~\cite{Polyreduction}) is as follows:
\begin{theorem}\label{thm:polyred} Under the same conditions (C1) and (C2) above, let $\mu=(\mu_1,\dots,\mu_k)$ be a regular value of $J$ and assume that $G_\mu$ acts freely and properly on $J^{-1}(\mu)$. Then the reduced space $ J^{-1}(\mu)/G_\mu$ admits a unique polysymplectic structure $(\omega^1_\mu,\dots,\omega^k_\mu)$ satisfying $\pi_\mu^*\omega^a_\mu=i_\mu^*\omega^a$, where $\pi_\mu\colon J^{-1}(\mu)\to J^{-1}(\mu)/G_\mu$ is the canonical projection and $i_\mu\colon J^{-1}(\mu)\to N$ is the canonical inclusion.
\end{theorem}
There is also a direct dynamical consequence of Theorem~\ref{thm:polyred} if we are given an invariant Hamiltonian and consider invariant solutions.
\begin{definition}\label{def:invariantvf} A $k$-vector field is \emph{$G$-invariant} if satisfies:
\[
\Phi_g^{T^1_kP}\circ \pmb{X}=\pmb{X}\circ\Phi_g.
\]
\end{definition}
In other words, each of the components $X_a$ are $G$-invariant vector fields in the usual sense, namely $T\Phi_g\circ X_a=X_a\circ\Phi_g$. When the group is clear from the context, we will simply say ``invariant'' $k$-vector field. The following is proved in~\cite{Polyreduction} (note that the result is stated here in a slightly different form):
\begin{theorem}\label{thm:polyred2} Under the same conditions of Theorem~\ref{thm:polyred}, let $H\colon N\to\R$ be a $G$-invariant Hamiltonian and denote by $H_\mu$ its reduction to $N_\mu\equiv J^{-1}(\mu)/G_\mu$. Let $\pmb{X}=(X_1,\dots,X_k)$ be a solution of~\eqref{eq:k-Sym} with Hamiltonian $H$ and assume that:
\begin{enumerate}[label={(\roman*})]
\item $\pmb{X}$ is $G_\mu$-invariant.
\item The restriction $X_a\mid_{J^{-1}(\mu)}$ is tangent to $J^{-1}(\mu)$.
\end{enumerate}
Then the projection $\overline{\pmb{X}}_{\mu}$ of $\pmb{X}\mid_{J^{-1}(\mu)}$ on $N_\mu$ is a solution of~\eqref{eq:k-Sym} with Hamiltonian $H_\mu$.
\end{theorem}
We will write
\[
\pmb{X}_{\mu}\equiv \pmb{X}\mid_{J^{-1}(\mu)}.
\]
A $k$-vector field $\pmb{X}$ such that $X_a\mid_{J^{-1}(\mu)}$ is tangent to $J^{-1}(\mu)$ (as in condition $(ii)$ of Theorem~\ref{thm:polyred2}) is said to be tangent to $J^{-1}(\mu)$. This means that $\pmb{X}_{\mu}$ can be considered as a $k$-vector field on $J^{-1}(\mu)$. For such a $k$-vector field, one identifies integral sections of $\pmb{X}$ through points in $J^{-1}(\mu)$ with integral sections of $\pmb{X}\mid_{J^{-1}(\mu)}$.
\begin{remark} Theorem 4.4 in \cite{Polyreduction} (which corresponds to Theorem~\ref{thm:polyred2} above) requires the stronger condition that $\pmb{X}$ is $G$-invariant (rather than only $G_\mu$-invariant). But, reading through the proof of Theorem 4.4, it is clear that $G_\mu$-invariance suffices to reduce the $k$-vector field $\pmb{X}$.
\end{remark}
We draw the attention of the reader to the fact that, in the present context, Noether's Theorem reads \cite{Symmetriesk}
\begin{equation}\label{eq:noetherfields}
\sum_a X_a (J_a^\xi)=0
\end{equation}
for each $\xi\in\lag$, where $J_a^\xi \colon N\to\R$ is defined as the contraction $\langle J_a,\xi\rangle$. This is the divergence property that we referred to in the introduction. Condition $(ii)$ in Theorem~\ref{thm:polyred2} is much more restrictive than~\eqref{eq:noetherfields}: it requires that each of the terms $X_a(J_a^\xi)$ vanish \emph{separately}. This means, in particular, that only very specific solutions $\pmb{X}$ can be reduced to solutions of the reduced problems. However, it has been show in~\cite{Polyreduction} that a large class of examples (constructed from an invariant metric on $\lag$) contain solutions which fit within this category.
\subsection{Reconstruction}
We will now discuss a method to lift an integral section of the reduced $k$-vector field $\overline{\pmb{X}}_{\mu}$ to an integral section of the original vector field $\pmb{X}$, when possible.
To simplify the notation, we will describe the situation as follows. We assume that a manifold $P$ has an action $\Phi_g\colon P\to P$ of a Lie group $G$, and we write $\pi_P\colon P\to P/G$ for the corresponding principal bundle. At the end of this section we will take $P=J^{-1}(\mu)$ and $G=G_\mu$ to relate the results to the polysymplectic reduction case. Since the basic setting here is a principal bundle $P\to P/G$, which is precisely that of the Lagrange-Poincaré reduction, many of our results can be found independently in~\cite{LTM_LP}.
We consider an invariant $k$-vector field $\pmb{X}$ on $P$. It defines, uniquely, a reduced $k$-vector field $\overline{\pmb{X}}$ on $P/G$ by the following relation:
\[
T\pi_P\circ X_a=\overline{X}_a\circ\pi_P.
\]
We first address the relation between the integrability of the (invariant) $k$-vector field $\pmb{X}$ and of the corresponding reduced vector field $\overline{\pmb{X}}$. It is clear that the integrability of $\pmb{X}$ implies that of $\overline{\pmb{X}}$: indeed, it suffices to observe that, in view of the integrability of $\pmb{X}$, we have
\[
[\overline{X}_a,\overline{X}_b]=[T\pi_P(X_a),T\pi_P(X_b)]=T\pi_P\left([X_a,X_b]\right)=0.
\]
We now seek to find conditions for the converse. For simplicity and concreteness in the proofs, one may assume that a principal connection $\Ac$ on the bundle $\pi_P\colon P\to P/G$ has been chosen. While this is not necessary yet (see~\cite{LTM_LP}), we will anyhow have to pick a principal connection on $P\to P/G$ later for the effective reconstruction of solutions. Each of the vector fields $X_a$ decomposes then as
\[
X_a={\rm Hor}(X_a)+{\rm Ver}(X_a) = \overline{X}_a^h+{\rm Ver}(X_a),
\]
where $(\cdot)^h\colon \pi_P^*T(P/G)\to TP$ is the horizontal lift (we will omit the point where the lift occurs when is clear from the context). Let us assume that $\overline{\pmb{X}}$ is integrable. The bracket $[X_a,X_b]$ is decomposed as follows:
\begin{align}\label{eq:bracket}
[X_a,X_b]&={\rm Hor}([X_a,X_b])+ {\rm Ver}([X_a,X_b])= [\overline{X}_a,\overline{X}_b]^h+{\rm Ver}([X_a,X_b])\nonumber \\
&={\rm Ver}([X_a,X_b]).
\end{align}
We observe that $[X_a,X_b]$ is vertical (this condition does not depend on the chosen connection). In other words, we have the following:
\begin{lemma}\label{lem:integrability}
Assume that $\overline{\pmb{X}}$ is integrable. Then $\pmb{X}$ is integrable if and only if the vertical part of the Lie brackets $[X_a,X_b]$ vanishes.
\end{lemma}
Given an integral section $\overline{\phi}\colon\R^k\to P/G$ of $\overline{\pmb{X}}$ at $[p]=\pi_P(p)$, we can construct the pull-back bundle $\overline{\phi}^*P$:
\begin{equation}\label{dia:pullbackconnection}
\begin{tikzcd}
\R^k\times P\arrow[ddr,"\tilde p_1"',bend right=30]\arrow[drr,"\tilde p_2",bend left=30]&&[3em] \\
&\overline{\phi}^*P\arrow[r,"p_2"]\arrow[d,"p_1"']\arrow[ul]& P\arrow[d,"\pi_P"] \\[2em]
&\R^k \arrow[r, "\overline{\phi}"' ]& P/G
\end{tikzcd}\hspace{2em}
\begin{tikzcd}
&(t,p)\arrow[r,"p_2"]\arrow[d,"p_1"']&[2em] p\arrow[d,"\pi_P"] \\[2em]
&t \arrow[r, "\overline{\phi}"' ]& {[p]}
\end{tikzcd}
\end{equation}
Recall that $\overline{\phi}^*P\subset \R^k\times P$ is the submanifold
\[
\overline{\phi}^*P=\{(t,p)\in \R^k\times P\st \overline{\phi}(t)=\pi_P(p)\}.
\]
It is a $G$-bundle over $\R^k$ with action $g\cdot (t,p)=(t,g\cdot p)$. A tangent vector in $T(\overline{\phi}^*P)$ at $(t,p)$ is a pair $(v_t,v_p)$ with $T\overline{\phi}(v_t)=T\pi_P(v_p)$. The $k$-vector field $\pmb{X}$ defines a distribution $\mathcal{D}_{\pmb{X}}\subset TP$ of dimension $k$:
\[
\mathcal{D}_{\pmb{X}}=\langle X_1,\dots,X_k\rangle.
\]
We consider the distribution $\mathcal{H}(\pmb{X},\overline{\phi})$ in $\overline{\phi}^*P$ obtained as the inverse image via $p_2$ of $\mathcal{D}_{\pmb{X}}$:
\[
\mathcal{H}(\pmb{X},\overline{\phi})=(Tp_2)^{-1}(\mathcal{D}_{\pmb{X}}),
\]
i.e. at a point $(t,p)\in \overline{\phi}^*P$ we have
\[
\left.\mathcal{H}(\pmb{X},\overline{\phi})\right|_{(t,p)}=(T_{(t,p)}p_2)^{-1}\big(\left.\mathcal{D}_{\pmb{X}}\right|_{p}\big).
\]
Let $Z_1=(V_1,X_1)$ be a vector field on $\overline{\phi}^*P\subset \R^k\times P$ such that $Tp_2(Z_1)=X_1$. Then $T\overline{\phi}(V_1)=T\pi_P(X_1)=\overline{X}_1$, and therefore $V_1=\partial/\partial t^1$. Similarly one finds $Z_2,\dots,Z_k$. Thus:
\begin{equation}\label{eq:distribution}
\mathcal{H}(\pmb{X},\overline{\phi}) =\langle Z_1,\dots,Z_k\rangle
=\left\langle \fpd{}{t^1}+X_1,\dots,\fpd{}{t^k}+X_k\right\rangle.
\end{equation}
Note that each of the vector fields
\[
\left.Z_a\right|_{(t,p)}=\left.\fpd{}{t^a}\right|_t+\left.X_a\right|_p
\]
at a point $(t,p)\in \overline{\phi}^*P$ is interpreted as a vector field on $\overline{\phi}^*P$ as follows: $Z_a$ defines a vector field on $\R^k\times P$ tangent to $\overline{\phi}^*P$, and therefore its restriction to $\overline{\phi}^*P$ defines a vector field on $\overline{\phi}^*P$. We will not make a notational distinction between $Z_a\in \mathfrak{X}(\R^k\times P)$ and its restriction $Z_a\in \mathfrak{X}(\overline{\phi}^*P)$.
\begin{lemma} The distribution $\mathcal{H}(\pmb{X},\overline{\phi})$ defines a principal connection on $p_1\colon \overline{\phi}^*P\to\R^k$.
\end{lemma}
\begin{proof} Clearly, $\mathcal{H}(\pmb{X},\overline{\phi})$ defines a distribution of dimension $k$ in $\overline{\phi}^*P$ which is complementary to the vertical distribution of the bundle $p_1$. It is moreover an invariant distribution since each of the $X_a$ is.
\end{proof}
The horizontal and vertical parts of a vector field $Y\in\mathfrak{X}(\overline{\phi}^*P)$ w.r.t.\ the previous connection will be denoted by
\[
Y= {\rm Ver}_{\mathcal{H}(\pmb{X},\overline{\phi})}(Y)+{\rm Hor}_{\mathcal{H}(\pmb{X},\overline{\phi})}(Y)
\]
to distinguish them from those associated to $\Ac$. Take $0\in\mathfrak{X}(\R^k)$ and $X\in\mathfrak{X}(P)$ vector fields with $0+X\in T(\overline{\phi}^*P)$, then $T\pi_P(X)=0$ or, in other words, $X={\rm Ver}(X)$. Since vertical vector fields (w.r.t.\ $p_1$) on $T(\overline{\phi}^*P)$ are precisely of the form $0+X$, a vertical vector in $\overline{\phi}^*P$ can be though of as a vertical vector on $P$. We then have:
\begin{proposition}\label{pro:reconstruction} Let $\pmb{X}$ be an invariant $k$-vector field on $P$. Then $\pmb{X}$ is integrable if, and only if the following two conditions are satisfied:
\begin{enumerate}[label={(\roman*})]
\item The reduced vector field $\overline{\pmb{X}}$ is integrable.
\item For each integral section $\overline{\phi}\colon\R^k\to P/G$ of $\overline{\pmb{X}}$ the connection $\mathcal{H}(\pmb{X},\overline{\phi})$ is flat.
\end{enumerate}
\end{proposition}
\begin{proof} If $Z_a=\partial/\partial t^a+ X_a$ and $Z_b=\partial/\partial t^b+ X_b$ are two horizontal vector fields on $\overline{\phi}^*P$,
their bracket reads
\[
[Z_a,Z_b] =0+ [X_a,X_b]
\]
and the curvature is
\[
K_{\pmb{X},\overline{\phi}}(Z_a,Z_b)=- {\rm Ver}_{\mathcal{H}(\pmb{X},\overline{\phi})}(0+ [X_a,X_b])=-{\rm Ver}([X_a,X_b]).
\]
Note that in the preceding expression we have abused slightly the notation to identify the vertical subbundles of $T(\overline{\phi}^*P)$ and $TP$. Thus, if the curvature vanishes for each pair $Z_a,Z_b$, one obtains precisely the condition for the integrability of $\pmb{X}$ under the assumption that $\overline{\pmb{X}}$ is integrable, see~\eqref{eq:bracket}. We conclude that if $\pmb{X}$ is integrable then both (i) and (ii) hold, and conversely.
\end{proof}
The same result can be found in \S 3.2 of~\cite{LTM_LP}, where the connection on $\overline{\phi}^*P$ is defined in an alternative way using the connection associated to $\pmb{X}$. Note that one can also obtain this connection as the restriction of
the distribution $\mathcal{H}$ defined by
\begin{equation*}
\mathcal{H}(\pmb{X})=\left\langle \fpd{}{t^1}+ X_1^i\fpd{}{q^i},\dots, \fpd{}{t^k}+ X_k^i\fpd{}{q^i}\right\rangle\subset T(\R^k\times Q)
\end{equation*}
to the submanifold $\overline{\phi}^*P\subset \R^k\times P$.
The proof of Propositon~\ref{pro:reconstruction} shows the following: if $[p]\in P/G$ is a point and we take an integral section $\overline{\phi}$ of $\overline{\pmb{X}}$ through $[p]$, then given any point $p\in P$ with $\pi_P(p)=[p]$ there exists an integral section $\phi$ of $\pmb{X}$ through $p\in P$. It also clear that $\pi_P\circ\phi=\overline{\phi}$ since
\[
T(\pi_P\circ\phi)(\partial/ \partial t^a)=T\pi\circ X_a=\overline{X}_a=T\overline{\phi} (\partial/ \partial t^a),
\]
and both $\pi_P\circ\phi$ and $\overline{\phi}$ agree on $0\in\R^k$ (with value $[p]$). The proof of Propositon~\ref{pro:reconstruction}, however, does not give a constructive procedure to find such $\phi$. This procedure, namely the effective reconstruction of such a $\phi$ starting from a reduced integral section $\overline{\phi}$, has been described already in~\cite{LTM_LP} in the polysymplectic formalism, and we will only recollect here the main results without proofs.
We will denote by $\overline{\pmb{X}}^h$ the horizontal lift of $\pmb{X}$ w.r.t.\ $\pmb{\Ac}$, which is by definition the $k$-vector field on $P$ with components
\[
\overline{\pmb{X}}^h=(\overline{X}_1^h,\dots,\overline{X}_k^h),
\]
where the notation $(\cdot)^h$ for the vector fields in the brackets is the usual horizontal lift w.r.t. $\Ac$. By construction, $\overline{\pmb{X}}^h$ projects onto $\overline{\pmb{X}}$ and we can write just like in~\eqref{eq:bracket}
\[
[\overline{X}_a^h,\overline{X}_b^h]=[\overline{X}_a,\overline{X}_b]^h-K(X_a,X_b),
\]
where we have used that ${\rm hor}(X_a)=\overline{X}_a^h$. Clearly, $\overline{\pmb{X}}^h$ is integrable iff both of the following conditions are satisfied: (i) $\overline{\pmb{X}}$ is integrable, and (ii) the curvature $K(X_a,X_b)$ vanishes for each $a,b=1,\dots k$.
\begin{definition} An integral section
\[
\overline{\phi}_h\colon \R^k\to P
\]
of $\overline{\pmb{X}}^h$ is a \emph{horizontal lift of $\overline{\phi}$} if $\pi_P\circ \overline{\phi}_h=\overline{\phi}$.
\end{definition}
When $\overline{\pmb{X}}^h$ is integrable such an horizontal lift always exists for the same reasons that $\phi$ does (see the argument after the proof of Propositon~\ref{pro:reconstruction}). To determine the desired integral section $\phi\colon \R^k\to P$ of $\pmb{X}$ one looks for the map $g\colon \R^k\to G$ such that
\begin{equation}\label{eq:phase}
\phi(t)=g(t)\cdot \overline{\phi}_h(t).
\end{equation}
If one uses~\eqref{eq:phase} in the defining relation for the integral section $\phi$, i.e.
\[
\phi^{(1)}=\pmb{X}\circ\phi,
\]
one arrives at the following \emph{reconstruction equation}~\cite{LTM_LP}:
\begin{equation}\label{eq:reconstruction}
g^{-1}\cdot g^{(1)}=\pmb{\Ac}(\pmb{X}\circ \overline{\phi}_h).
\end{equation}
The previous equation compares directly to the well-known case of mechanics. Indeed, when $k=1$, the expression~\eqref{eq:reconstruction} reduces to
\begin{equation}\label{eq:reconstruction2}
g^{-1}\cdot \dot g=\Ac(X_H\circ \overline{\gamma}_h),
\end{equation}
which is the reconstruction equation in the case of reduction in Hamiltonian mechanics (see for example~\cite{phases} for details in the context of symplectic reduction). Here $X_H$ the Hamiltonian vector field and $\overline{\gamma}_h$ is the horizontal lift of the reduced solution $\overline{\gamma}$.
When Proposition~\ref{pro:reconstruction} is applied to the case $P=J^{-1}(\mu)$ of interest, we have the following:
\begin{theorem}\label{thm:reconstruction} Under the same conditions as Theorem~\ref{thm:polyred2}, let $\overline{\phi}_\mu\colon \R^k\to N_\mu$ be an integral section of $\overline{\pmb{X}}_{\mu}$ such that the connection $\mathcal{H}(\pmb{X}_\mu,\overline{\phi}_\mu)$ is flat. Then there exists an integral section $\phi_\mu\colon \R^k\to J^{-1}(\mu)$ of $\overline{\pmb{X}}_{\mu}$ with $\pi_\mu\circ\phi_\mu=\overline{\phi}_\mu$.
Moreover, if for a given principal connection $\Ac$ the horizontal lift $\overline{\pmb{X}}_{\mu}^h$ is integrable, then such an integral section $\phi_\mu$ can be computed as
\[
\phi_{\mu}(t)=g(t)\cdot(\overline{\phi}_\mu)_h(t),
\]
where $g(t)$ satisfies the reconstruction equation~\eqref{eq:reconstruction}.
\end{theorem}
We point out that in Theorem~\ref{thm:reconstruction} the connection $\mathcal{H}(\pmb{X}_\mu,\overline{\phi}_\mu)$ is a $G_\mu$ principal connection on the bundle $\overline{\phi}_\mu^*(J^{-1}(\mu))$.
\begin{remark} We will not obtain the general coordinate expressions for the reconstruction equations. In the Lagrangian case (Section \ref{sec:Routh}) such expressions can be adapted from those in~\cite{LTM_LP}.
\end{remark}
\section{The reduction of the cotangent bundle of \texorpdfstring{$k^1$}{k1}-covelocities}\label{sec:cotangentreduction}
We will now discuss in detail how the polysymplectic reduction theorem applies to the particular case of the cotangent bundle of $k^1$-covelocities $(T^1_k)^*Q$. The field equations in the $k$-symplectic Hamiltonian field theory look for integral sections of some Hamiltonian $k$-vector field defined on $(T^1_k)^*Q$, so this reduction is important in its own. Besides, we will build on these results to discuss Routh reduction later in Section~\ref{sec:Routh}.
Our starting point is a free and proper action $\Phi_g\colon Q\to Q$ of $G$ on $Q$ and a Hamiltonian $H\colon (T^1_k)^*Q\to\R$ which is invariant under the canonical prolongation of the action to $(T^1_k)^*Q$. It is well-known~\cite{Gunter} that in this case an equivariant momentum map is given by the following family of maps $J^a\colon (T_k^1)^*Q\to \lag^*$:
\begin{equation}\label{eq:liftedmmap}
\langle J^a(\bm{\alpha}_q),\xi\rangle=\langle\alpha^a_q, \xi_{Q}(q)\rangle ,\quad \bm{\alpha}_q=(\alpha_q^1,\dots, \alpha_q^k),\;\text{for all } \xi\in\lag.
\end{equation}
It is shown in~\cite{Polyreduction} that this example is amenable to polysymplectic reduction, i.e. that both (C1) and (C2) in Theorem~\ref{thm:polyred} hold under the assumption of a free action (actually under the weaker condition of an infinitesimally free action).
We need some notations before moving on. Given a principal connection $\Ac$ on $Q\to Q/G$ and a value $\nu\in\lag^*$, it is possible to define 1-form $\Ac_\nu$ on $Q$ as follows:
\begin{equation}\label{eq:1form}
\Ac_\nu(v_q)=\langle\nu,\Ac(v_q)\rangle.
\end{equation}
The 2-form $d\Ac_\nu$ reduces to a 2-form on $Q/G_\nu$, with $G_\nu$ the isotropy group of $\nu\in\lag$ under the coadjoint action. This follows easily from the $G_\nu$-equivariance of $\Ac_\nu(v_q)$ (cf. \cite{RouthMarsden} for further details). We will denote this reduced form on $Q/G_\nu$ by $B_\nu$; in reduction terminology, this reduced form is often called the ``magnetic term''.
For a given regular value $\mu=(\mu_1,\dots,\mu_k)\in\lagdk$, we consider the family of 2-forms on $Q$ given by $d\Ac_{\mu_a}$, whose expression is defined in~\eqref{eq:1form}. Since $G_\mu\subset G_{\mu_a}$, each of them drops to a 2-form on $Q/G_\mu$ that we will denote $\Bc_{\mu_a}$.
\begin{theorem}\label{thm:identification} Any choice of a principal connection $\Ac$ on $Q\to Q/G$ gives a polysymplectomorphism
\begin{equation}\label{eq:identification}
\left((T^1_k)^*Q\right)_\mu \simeq Q/G_\mu \times_{Q/G} \big(\smallunderbrace{T^*(Q/G)\oplus\;\dots\oplus T^*(Q/G)\;}_{k\; {\rm copies}}\big),
\end{equation}
where the space on the right-hand side (RHS) is endowed with the polysyplectic structure
\begin{equation*}
\omega_\mu^a= ({\rm pr}^a_2)^*\omega_{Q/G}-({\rm pr}_1)^*\Bc_{\mu_a}.
\end{equation*}
\end{theorem}
The Whitney sum in~\eqref{eq:identification} is over $Q/G$. The notation ${\rm pr}^a_2$ stands for the projection of the $a$-th component of the RHS of~\eqref{eq:identification} onto $T^*(Q/G)$, and similarly ${\rm pr}_1$ is the projection onto $Q/G_\mu$. If we write $p_\mu\colon Q/G_\mu\to Q/G$ for the projection $[q]_\mu\mapsto [q]$, the space on the RHS of~\eqref{eq:identification} is a pullback bundle:
\begin{equation*}\label{dia:pullback}
\begin{tikzcd}
p_\mu^*\left((T^1_k)^*(Q/G)\right)\arrow[r,dashed]\arrow[d,dashed]& (T^1_k)^*(Q/G)\arrow[d] \\Q/G_\mu\arrow[r, "p_\mu"' ]& Q/G
\end{tikzcd}
\end{equation*}
\begin{proof} We extend and adapt the proof of the cotangent bundle reduction theorem (see~\cite{stagesbook} and references therein) to the polysymplectic setting.
\textsc{Step 1 (The level set of zero):} Asume for a moment that $\mu=\pmb{0}\equiv (0,\dots,0)$. Then from the definition of the momentum map~\eqref{eq:liftedmmap}, we have an identification
\[
(J^a)^{-1}(0)=T^*Q\oplus\dots\oplus \underbrace{(V\pi)^\circ}_{\text{$a$-th comp.}} \oplus\dots\oplus T^*Q \subset (T^1_k)^*Q
\]
where $V\pi\subset TQ$ is the vertical subbundle w.r.t.\ the projection $\pi\colon Q\to Q/G$, and $(V\pi)^\circ\subset T^*Q$ denotes its annihilator. Thus:
\[
J^{-1}(\pmb{0})=\cap_a \left[(J^a)^{-1}(0)\right] = (V\pi)^\circ\oplus\dots\oplus (V\pi)^\circ\subset (T^1_k)^*Q.
\]
There is a canonical identification between $(V\pi)^\circ$ and $T^*(Q/G)\times_{Q/G}Q$
\begin{align*}
(V\pi)^\circ &\to Q\times_{Q/G}T^*(Q/G), \\
\alpha_q &\mapsto (q,\tilde\alpha_{[q]}),
\end{align*}
where $\tilde\alpha_{[q]}$ is uniquely defined by the relation $\langle\tilde\alpha_{[q]},\tilde v_{[q]}\rangle = \langle\alpha_q,v_q\rangle$ for any $v_q$ with $T\pi(v_q)=\tilde v_{[q]}$ (this is well-defined). Therefore, we also have an identification
\[
\mathcal{T}\colon J^{-1}(\pmb{0}) \to Q\times_{Q/G}\left(T^*(Q/G)\oplus\dots\oplus T^*(Q/G)\right) =\pi^*\left((T^1_k)^*(Q/G)\right).
\]
Moreover, under the previous identification, $G$ acts on $\pi^*\left((T^1_k)^*(Q/G)\right)$ as:
\[
g\cdot (q,\alpha^1_{[q]},\dots,\alpha^k_{[q]})= (g\cdot q,\alpha^1_{[q]},\dots,\alpha^k_{[q]}).
\]
\textsc{Step 2 (The ``momentum shift''):} Now let $\mu$ be arbitrary. For each $a=1,\dots,k$, we consider the 1-form $\Ac_{\mu_a}$ on $Q$ . Define the map (which is the generalization of the ``momentum shift'' to the polysymplectic setting):
\begin{align*}
\mathcal{S}\colon (T^1_k)^*Q&\to (T^1_k)^*Q, \\
(\alpha^1_q,\dots,\alpha^k_q)&\mapsto (\alpha^1_q-(\Ac_{\mu_1})_q,\dots,\alpha^k_q-(\Ac_{\mu_k})_q).
\end{align*}
Note that $\mathcal{S}$ depends on both $\mu$ and the chosen principal connection $\Ac$. The map $\mathcal{S}$ is such that $\mathcal{S}^*\theta^a_Q=\theta^a_Q-(\pi^a_Q)^*\Ac_{\mu_a}$. To see this, note that if we write $(v_1,\dots,v_k)\in T\big((T^1_k)^*Q\big)$ we have
\begin{align*}
\mathcal{S}^*(\theta^a_Q)_{(\alpha^1_q,\dots,\alpha^n_q)}(v_1,\dots,v_k)&=(\theta_Q)_{\alpha^a_q-\Ac_{\mu_a}}(T\pi^a_Q(v_a))\\
&=\big( (\theta^a_Q)_{(\alpha^1_q,\dots,\alpha^n_q)}-(\pi^a_Q)^*\Ac_{\mu_a}\big)(v_1,\dots,v_k),
\end{align*}
where we have used the definition of $\theta^a_Q$. In particular, taking the exterior derivative and using $\omega^a_Q=-d\theta^a_Q$ we find
\begin{equation}\label{eq:shift}
\mathcal{S}^* \omega^a_Q= \omega^a_Q+(\pi^a_Q)^*d\Ac_{\mu_a}.
\end{equation}
We consider now the restriction of $\mathcal{S}$ to $J^{-1}(\mu)$, which we denote by the same symbol. It maps, diffeomorphically, $J^{-1}(\mu)$ onto $J^{-1}(\pmb{0})$ (``shifts the momentum''):
\[
\mathcal{S}\colon J^{-1}(\mu)\to J^{-1}(\pmb{0}).
\]
Indeed, for each $a$ we have:
\[
(J^a\circ \mathcal{S})(\alpha^1_q,\dots,\alpha^n_q)=J^a(\alpha^1_q-\Ac_{\mu_1},\dots,\alpha^k_q-\Ac_{\mu_k})=\mu_a-\mu_a=0.
\]
Composing with $\mathcal{T}$ we get a diffeomorphism
\begin{align*}
\tau=\mathcal{T}\circ\mathcal{S}\colon J^{-1}(\mu)&\to \pi^*\left((T^1_k)^*(Q/G)\right)
\end{align*}
which is $G_\mu$-equivariant (this follows from the fact that each $\Ac_{\mu_a}$ is $G_\mu$-equivariant), and hence reduces to a diffeomorphism:
\[
\tau_\mu\colon J^{-1}(\mu)/G_\mu\to p_\mu^*\left((T^1_k)^*(Q/G)\right).
\]
\textsc{Step 3 (The symplectic form):} Since $\tau_\mu$ is a diffeomorphism, it only remains to check that it relates both polysymplectic structures. If we let $\tilde{\omega}^a_\mu$ be the reduced polysymplectic structure on $J^{-1}(\mu)/G_\mu$ given by Theorem~\ref{thm:polyred}, then we must check that
\[
\tilde{\omega}^a_\mu=\tau_\mu^*\omega^a_\mu=\tau_\mu^* \left[({\rm pr}^a_2)^*\omega_{Q/G}-({\rm pr}_1)^*\Bc_{\mu_a}\right].
\]
The situation is summarized in the following commutative diagram:
\begin{equation*}
\begin{tikzcd}
(T^1_k)^*Q\arrow[rr,"\mathcal{S}"] & & (T^1_k)^*Q &\\
J^{-1}(\mu)\arrow[rr,"\tau"]\arrow[u,"i_\mu"]\arrow[dd,"\pi_\mu"']&& \pi^*\left((T^1_k)^*(Q/G)\right)\arrow[dd,"\pi^0_\mu"]\arrow[u,"j_0=(\mathcal{T}^{-1}\circ i_0)"'] \arrow[dr] &\\
&& & T^*(Q/G)\\
J^{-1}(\mu)/G_\mu \arrow[rr,"\tau_\mu"']&& p_\mu^*\left((T^1_k)^*(Q/G)\right) \arrow[ur,"{\rm pr}^a_2"']\arrow[r,"{\rm pr}_1"'] & Q/G_\mu
\end{tikzcd}
\end{equation*}
We will denote by $i_0\colon J^{-1}(0)\to (T^1_k)^*Q$ the inclusion at $\mu=\pmb{0}$ and write
\[
j_0=(\mathcal{T}^{-1}\circ i_0)\colon \pi^*\left((T^1_k)^*(Q/G)\right) \to (T^1_k)^*Q.
\]
Also, we will write $\pi^0_\mu\colon \pi^*\left((T^1_k)^*(Q/G)\right)\to p_\mu^*\left((T^1_k)^*(Q/G)\right)$ for the quotient projection
\[
\pi^0_\mu([q],\bm{\alpha}_{[q]})= ([q]_\mu,\bm{\alpha}_{[q]}).
\]
Since $\tilde{\omega}^a_\mu$ is uniquely determined from the relation
$\pi_\mu^*\tilde{\omega^a_\mu}=i_\mu^*\omega_Q^a$, we only need to show that
\[
\pi_\mu^* \tau_\mu^* \big(({\rm pr}^a_2)^*\omega_{Q/G}-({\rm pr}_1)^*\Bc_{\mu_a}\big)=i_\mu^*\omega_Q^a,
\]
or
\begin{equation}\label{eq:equalityforms}
\tau^*(\pi^0_{\mu})^* \big(({\rm pr}^a_2)^*\omega_{Q/G}-({\rm pr}_1)^*\Bc_{\mu_a}\big)=i_\mu^*\omega_Q^a.
\end{equation}
From the definitions of the maps involved, it is not hard to check that the following relations hold:
\[
(\pi^0_{\mu})^* ({\rm pr}^a_2)^*\omega_{Q/G}=j_0^*\omega^a_Q,\qquad (\pi^0_{\mu})^* ({\rm pr}_1)^*\Bc_\mu =j_0^*(\pi^a_Q)^*d\Ac_{\mu_a}.
\]
Therefore, the equality~\eqref{eq:equalityforms} will hold provided
\[
\tau^*j_0^*(\omega^a_Q-(\pi^a_Q)^*d\Ac_{\mu_a})= i_\mu^*\omega_Q^a.
\]
But this follows if we rewrite~\eqref{eq:shift} in the form
\[
\omega^a_Q=\mathcal{S}^* \omega^a_Q(\pi^a_Q)^*d\Ac_{\mu_a}=\mathcal{S}^*(\omega^a_Q(\pi^a_Q)^*d\Ac_{\mu_a},
\]
and now:
\[
i_\mu^*\omega_Q^a=i_\mu^*\mathcal{S}^*(\omega^a_Q-(\pi^a_Q)^*d\Ac_{\mu_a})=\tau^*j_0^*(\omega^a_Q-(\pi^a_Q)^*d\Ac_{\mu_a})=\tau^*(\pi^0_{\mu})^* \big(({\rm pr}^a_2)^*\omega_{Q/G}-({\rm pr}_1)^*\Bc_{\mu_a}\big).
\]
\end{proof}
\section{Routh reduction in the polysymplectic formalism}\label{sec:Routh}
In this section we obtain a polysymplectic version of Routh reduction. To do that, we will follow~\cite{quasi} and adapt the techniques to the polysymplectic setting. In a nutshell, the methodology consists on applying the polysymplectic reduction Theorem~\ref{thm:polyred} to the given regular Lagrangian system on $T^1_kQ$.
\subsection{The reduction of the tangent bundle of \texorpdfstring{$k^1$}{k1}-velocities}
Building on the results of the previous section, we now discuss the reduction of a Lagrangian field theory defined on $T^1_kQ$. The assumption is that we have a Lagrangian $L\colon T^1_kQ\to \R$ which is hyperregular and $G$-regular (to be defined later, see Definition~\ref{def:Gregularity}), and a free and proper action $\Phi_g$ on $Q$ such that $L$ is invariant w.r.t. its canonical prolongation to
$T^1_kQ$. The requirement of hyperregularity can be replaced for that of regularity with small changes. We will use the following result:
\begin{lemma}\label{lem:lema1} Let $(N,\omega^a)$ and $(N',\omega'^a)$ be polysymplectic manifolds and $f\colon N\to N'$ a polysymplectomorphism. Assume that $G$ acts canonically on both $N$ and $N'$ with momentum maps $J\colon N\to \lagdk$ and $J'\colon N'\to \lagdk$, and that $f$ is equivariant w.r.t. the $G$-actions and satisfies $f^*J'=J$.
\begin{equation*}
\begin{tikzcd}
(N,\omega^a)\arrow[rr,"f"]\arrow[dr,"J"']&& (N',\omega'^a)\arrow[dl,"J'"] \\
& \lagdk &
\end{tikzcd}
\end{equation*}
\noindent Then, if $f_\mu\colon N_\mu\to N'_\mu$ denotes the map between the reduced polysymplectic spaces induced by $f$, $f_\mu$ is a polysymplectomorphism, i.e.
\[
f_\mu^*\omega'^a_\mu=\omega^a_\mu.
\]
\end{lemma}
\begin{proof} The proof follows from the characterization of the reduced polysymplectic forms and is similar to that of Theorem~5 in~\cite{quasi} in the symplectic case.
First we observe that, since $f^*J'=J$, the map $f$ restricts to a diffeomorphism $f_r\colon J^{-1}(\mu)\to J'^{-1}(\mu)$ which is $G_\mu$-equivariant, and thus $f_\mu\colon N_\mu\to N'_\mu$ is a diffeomorphism:
\begin{equation*}
\begin{tikzcd}[column sep=huge,row sep= large]
(N,\omega^a)\arrow[r,"f"]& (N',\omega'^a) \\
J^{-1}(\mu)\arrow[d,"\pi_\mu"']\arrow[u,"i_\mu"]\arrow[r,"f_r"]& J'^{-1}(\mu)\arrow[d,"\pi'_\mu"]\arrow[u,"i'_\mu "'] \\
(N_\mu,\omega^a_\mu) \arrow[r,"f_\mu"] & (N'_\mu,\omega'^a_\mu)
\end{tikzcd}
\end{equation*}
We need to show that $f_\mu^*\omega'^a_\mu=\omega^a_\mu$. Since $\pi_\mu$ and $\pi'_\mu$ are submersions, it suffices to check that:
\[
f_r^*(\pi'_\mu)^* \omega'^a_\mu=(\pi_\mu)^*\omega^a_\mu.
\]
But this is easily shown using the characterization of the reduced polysymplectic forms:
\begin{align*}
f_r^*(\pi'_\mu)^* \omega'^a_\mu&=f_r^*(i'_\mu)^* \omega'^a=(i_\mu)^*f^*\omega'^a\\
&=(i_\mu)^*\omega^a=(\pi_\mu)^*\omega^a_\mu.
\end{align*}
\end{proof}
Recall that a regular Lagrangian $L$ defines the diffeomorphism $\pmb{F} L\colon T^1_kQ\to (T^1_k)^*Q$, see~\eqref{eq:Legendre}. We first want to observe that, since $L$ is invariant, the Legendre transformation is equivariant:
\[
\F L(g\cdot \pmb{v}_q)=g\cdot \F L(\pmb{v}_q).
\]
The proof, componentwise, is similar to the mechanical case which can be found for example in~\cite{Foundations}, Corollary 4.2.14. With this in mind, it is clear that the map
\[
J_L=J\circ \F L\colon T^1_kQ\to\lagdk
\]
is an equivariant momentum map for the polysymplectic action of $G$ on $T^1_k Q$. Its components are the maps
\[
J_L^a=J^a\circ \F L\colon T^1_kQ\to \lag^*,
\]
i.e.
\begin{equation*}
\langle J_L^a(\pmb{v}_q),\xi\rangle=\langle(F L)^a(\pmb{v}_q), \xi_{Q}(q)\rangle ,\quad \pmb{v}_q=(v_{1q},\dots,v_{kq}),\;\text{for all } \xi\in\lag.
\end{equation*}
We will need the following map:
\begin{align*}
\mathfrak{J}_L\colon T^1_kQ\times \lagk&\to \lagdk,\\
(\pmb{v}_q,\bm{\xi})&\mapsto \mathfrak{J}_L(\pmb{v}_q,\bm{\xi})=J_L(\pmb{v}_q+\bm{\xi}_Q(q)),
\end{align*}
where $\bm{\xi}=(\xi_1,\dots,\xi_k)\in\lagk$ and $\bm{\xi}_Q(q)\in T^1_kQ$ is given by
\[
\bm{\xi}_Q(q)=\left((\xi_1)_Q(q),\dots,(\xi_k)_Q(q)\right).
\]
\begin{definition}\label{def:Gregularity} Let $L\colon T^1_kQ\to \R$ be a Lagrangian. We say that $L$ is $G$-regular if, for each $\pmb{v}_q\in T^1_kQ$, the map $\mathfrak{J}_L(\pmb{v}_q,\cdot)\colon \lagk\to \lagdk$ is a diffeomorphism.
\end{definition}
This definition is the natural extension of the notion of a $G$-regular Lagrangian to the polysymplectic setting~\cite{Routhstages}. There are other equivalent definitions of $G$-regularity~\cite{quasi}, and one might as well work with the extension of those to the $k$-symplectic framework, but we find Definition~\ref{def:Gregularity} to be most practical. The importance of $G$-regularity comes from the following fact:
\begin{proposition}\label{prop:identificationJL} If $L$ is $G$-regular, then there is a diffeomorphism
\[
J_L^{-1}(\mu)/G_\mu \simeq Q/G_\mu \times_{Q/G} \big(\smallunderbrace{T(Q/G)\oplus\;\dots\oplus T(Q/G)\;}_{k\; {\rm copies}}\big)
\]
over the identity on $Q/G_\mu$.
\end{proposition}
\begin{proof} We first show that there is a diffeomorphism $\tilde\tau$ between $J_L^{-1}(\mu)$ and $\pi^*\left(T^1_k(Q/G)\right)$. We observe that $J_L^{-1}(\mu)=\cap_a \left[(J^a_L)^{-1}(\mu)\right]$. We define $\tilde\tau$ as follows:
\begin{align*}
\tilde\tau \colon \cap_a \left[(J^a_L)^{-1}(\mu)\right] &\to \pi^*\left(T^1_k(Q/G)\right),\\
(v_{1q}\dots,v_{kq})&\mapsto (q,T\pi(v_{1q}),\dots, T\pi(v_{kq})).
\end{align*}
We will now find explicitly the inverse $\tilde\tau$. An element in $\pi^*\left(T^1_k(Q/G)\right)$ is of the form
\[
(q,v_{1[q]},\dots,v_{k[q]})=\left(q,T\pi(v_{1q}),\dots,T\pi(v_{kq})\right)\in \pi^*\left(T^1_k(Q/G)\right)
\]
for some $v_{1q},\dots,v_{kq}\in TQ$. Then we define the inverse of $\tilde\tau$ as:
\begin{align*}
(\tilde\tau)^{-1} \colon \pi^*\left((T^1_k)(Q/G)\right) &\to \cap_a \left[(J^a_L)^{-1}(\mu)\right],\\
\left(q,T\pi(v_{1q}),\dots,T\pi(v_{kq})\right)&\mapsto \left(v_{1q}+(\xi_1)_Q(q),\dots,v_{kq}+(\xi_k)_Q(q)\right)=\pmb{v}_q+\bm{\xi}_Q(q),
\end{align*}
where $\xi_a\in\lag$ can be chosen so that $J_L(\pmb{v}_q+\bm{\xi}_Q(q))=\mu$ because of the assumed $G$-regularity of $L$. We note that, if we had made a different choice for the vectors $v_{1q},\dots,v_{kq}$, say $w_{1q},\dots,w_{kq}$, then we would have $\pmb{v}_q=\pmb{w}_q+\bm{\eta}_Q(q)$ for some $\bm{\eta}\in\lagk$ and the result would be the same due to $G$-regularity. This means that $(\tilde\tau)^{-1}$ is well-defined.
The $G_\mu$-action on space $J_L^{-1}(\mu)$ is pushed forward to the space $\pi^*\left(T^1_k(Q/G)\right)$, where it reads:
\[
g\cdot(q,v_{1[q]},\dots,v_{k[q]})= (g\cdot q,v_{1[q]},\dots,v_{k[q]}).
\]
Therefore, the map $\tilde\tau$ drops to a diffeomorphism
\[
\tilde\tau_\mu\colon J_L^{-1}(\mu)/G_\mu\to \pi^*\left(T^1_k(Q/G)\right)/G_\mu= p_\mu^*\left(T^1_k(Q/G)\right),
\]
and this gives the desired identification.
\end{proof}
The conclusion is that we have the following diagram:
\begin{equation}\label{dia:red}
\begin{tikzcd}
\pi^*\left(T^1_k(Q/G)\right)\arrow[d,"\tilde\pi^0_\mu"']&J_L^{-1}(\mu)\arrow[l,"\tilde\tau"']\arrow[r,"\F L"]\arrow[d,"\tilde\pi_\mu"']& J^{-1}(\mu)\arrow[d,"\pi_\mu"]\arrow[r,"\tau"] & \pi^*\left((T^1_k)^*(Q/G)\right)\arrow[d,"\pi^0_\mu"]\\
p_\mu^*\left(T^1_k(Q/G)\right)&J_L^{-1}(\mu)/G_\mu \arrow[l,"\tilde\tau_\mu"]\arrow[r,"(\F L)_\mu"']& J^{-1}(\mu)/G_\mu \arrow[r,"\tau_\mu"'] & p_\mu^*\left((T^1_k)^*(Q/G)\right)
\end{tikzcd}
\end{equation}
where $\tilde\pi^0_\mu$ and $\tilde\pi_\mu$ are quotient maps. The map $(\F L)_\mu$, obtained from the equivariant diffeomorphism $\F L$, is then a polysymplectomorphism (Lemma~\ref{lem:lema1}). We point out that, as mentioned earlier, we also denote by $\F L$ the restriction of $\F L$ to $J^{-1}(\mu)$.
\subsection{The Routhian}
Using the momentum shift $\mathcal{S}$ we write the first row of Diagram~\eqref{dia:red} as:
\[
\begin{tikzcd}
\pi^*\left(T^1_k(Q/G)\right)&J_L^{-1}(\mu)\arrow[l,"\tilde\tau"']\arrow[r,"\F L"]& J^{-1}(\mu)\arrow[r,"\mathcal{S}"] & J^{-1}(\pmb{0})\arrow[r,"\mathcal{T}"] & \pi^*\left((T^1_k)^*(Q/G)\right).
\end{tikzcd}
\]
The composition $\mathcal{S}\circ \F L\colon J_L^{-1}(\mu)\to J^{-1}(0)$ can be described as follows:
\begin{align*}
\langle\mathcal{S}\circ (F L)^a(\pmb{v}),w_q\rangle &= \left.\frac{d}{ds}\right|_{s=0}L(v_{1q},\dots,v_{aq}+sw_q,\dots,v_{kq})-\langle\Ac_{\mu_a},w_q \rangle\\
&=\left.\frac{d}{ds}\right|_{s=0}(L-(\pi_Q^a)^*\Ac_{\mu_a})(v_{1q},\dots,v_{aq}+sw_q,\dots,v_{kq}).
\end{align*}
This means that if we define the function $R\colon T^1_kQ\to\R$
\begin{align}
R(\pmb{v})&=L(\pmb{v})-(\pi_Q^1)^*\Ac_{\mu_1}(\pmb{v})-\dots- (\pi_Q^k)^*\Ac_{\mu_k}(\pmb{v})\nonumber \\
&=L(\pmb{v})-\Ac_{\mu_1}(v_{1q})-\dots- \Ac_{\mu_k}(v_{kq}),
\end{align}
then its restriction to $J_L^{-1}(\mu)$ satisfies $\F R=\mathcal{S}\circ \F L$. We will now denote by
\[
\Ro\colon \pi^*\left(T^1_k(Q/G)\right)\to\R
\]
the function induced by $R$ on $\pi^*\left(T^1_k(Q/G)\right)$, and by
\[
\Ro_\mu\colon \pi_\mu^*\left(T^1_k(Q/G)\right)\to\R
\]
its reduction (the function induced by $\Ro$ on the quotient by $G_\mu$).
\begin{definition}
The function $\Ro_\mu$ (but also of $R$ and $\Ro$) is called the \emph{Routhian}.
\end{definition}
We recall that the meaning of the fiber derivatives $\F\Ro$ and $\F\Ro_\mu$ has been described at the end of Section~\ref{sec:HLfieldtheory}.
\begin{lemma} The following holds:
\begin{enumerate}[label={(\roman*})]
\item $\Ro$ is $G_\mu$-invariant.
\item $\F\Ro=\tau\circ \F L\circ(\tilde\tau)^{-1}$.
\item $\F\Ro_\mu=\tau_\mu \circ (\F L)_\mu\circ (\tilde\tau_\mu)^{-1}$.
\end{enumerate}
\end{lemma}
\begin{proof} For each $a$, the term $\Ac_{\mu_a}$ in $\Ro$ is $G_\mu$-invariant (see Section~\ref{subsec:connections}). Since by assumption $L$ is $G_\mu$-invariant, $R$ is $G_\mu$-invariant and so is $\Ro$. This proves $(i)$. To prove $(ii)$, we first observe that if $\bm{\alpha}_q\in J^{-1}(\bm{0})$ and $\pmb{v}_q\in J_L^{-1}(\bm{\mu})$ then
\[
\langle\alpha^a_{q},v_{aq}\rangle = \langle\mathcal{T}^a(\bm{\alpha}_q),\tilde\tau^a(\pmb{v}_q)\rangle,
\]
where
\[
\mathcal{T}^a\colon J^{-1}(\pmb{0}) \to T^*(Q/G), \quad \tilde\tau^a \colon J_L^{-1}(\mu) \to T(Q/G),
\]
are the $a$-th components of the maps $\mathcal{T}$ and $\tilde\tau$, respectively. This is a direct consequence of the definitions of $\mathcal{T}$ and $\tilde\tau$. It follows that, if $\pmb{v}_q$ and $\pmb{w}_q$ are elements of $T^1_kQ$, then
\[
\langle\mathcal{S}^a\circ (F L)^a(\pmb{v}),w_{aq}\rangle = \langle\mathcal{T}^a\circ \mathcal{S}\circ (\F L)(\pmb{v}),\tilde\tau^a(\pmb{w})\rangle ,
\]
where $\mathcal{S}^a\colon T^*Q\to T^*Q$ is the map $\mathcal{S}^a(\beta_{q})=\beta_q-\Ac_{\mu_a}$.
If $\tilde\tau(\pmb{v})$ and $\tilde\tau(\pmb{w})$ are arbitrary elements in $\pi^*\left(T^1_k(Q/G)\right)$, then using the fact that $\tilde\tau$ is linear on the fibers:
\begin{align*}
\langle(\F \Ro)^a(\tilde\tau(\pmb{v})),\tilde\tau^a(\pmb{w})\rangle&=\left.\frac{d}{ds}\right|_{s=0}\Ro(\tilde\tau^1(\pmb{v}),\dots,\tilde\tau^a(\pmb{v}+s\pmb{w}),\dots,\tilde\tau^k(\pmb{v}))\\
&=\left(\left.\frac{d}{ds}\right|_{s=0}L(v_{1q},\dots,v_{aq}+sw_{aq},\dots,v_{kq})\right)-\Ac_{\mu_a}(w_{aq})\\
&=\langle(F L)^a(\pmb{v})),w_{aq}\rangle-\Ac_{\mu_a}(w_{aq})
=\langle\mathcal{S}^a\circ (F L)^a(\pmb{v}),w_{aq}\rangle \\
&=\langle\mathcal{T}^a\circ \mathcal{S}\circ (\F L)(\pmb{v}),\tilde\tau^a(\pmb{w})\rangle=\langle\tau^a\circ \F L\circ(\tilde\tau)^{-1}(\tilde\tau(\pmb{v})),\tilde\tau^a(\pmb{w})\rangle,
\end{align*}
with $\tau^a\colon J^{-1}(\mu)\to T^*(Q/G)$ the $a$-th component of $\tau$. Hence $\F\Ro=\tau\circ \F L\circ(\tilde\tau)^{-1}$ as desired. The proof of $(iii)$ is similar.
\end{proof}
We have now the following diagram (see~Diagram~\eqref{dia:red})
\begin{equation}\label{dia:red2}
\begin{tikzcd}[column sep=large]
T_k^1Q \arrow[r,"\F L"]\arrow[d,"{\rm Poly-Red}"'] & (T_k^1)^*Q\arrow[d,"{\rm Poly-Red}"]\\
p_\mu^*\left(T^1_k(Q/G)\right)\arrow[r,"\F \Ro_\mu"]& p_\mu^*\left((T^1_k)^*(Q/G)\right)
\end{tikzcd}
\end{equation}
where both $\F L$ and $\F \Ro_\mu$ are polysymplectomophisms (we are using again Lemma~\ref{lem:lema1}). The arrows ``Poly-Red'' above account for polysymplectic reduction followed by an identification for each of the reduced spaces $J^{-1}(\mu)/G_\mu$ and $J_L^{-1}(\mu)/G_\mu$.
We see in Diagram~\eqref{dia:red2} that the function $\Ro_\mu$ plays the role of a reduced Lagrangian in the space $p_\mu^*\left(T^1_k(Q/G)\right)$; we will give precise meaning to this analogy soon. So far, we have proved the following result:
\begin{theorem}\label{thm:reducedform} Let $L\colon T^1_k Q\to\R$ be a regular, $G$-invariant and $G$-regular Lagrangian and consider the polysymplectic manifold $(T^1_k Q,\omega^a_{Q,L})$. Let $\mu\in\lagdk$ be a regular value of the momentum map and fix a principal connection $\Ac$ on $\pi\colon Q\to Q/G$.
Then the reduced polysymplectic space can be identified with
\[
p_\mu^*\left(T^1_k(Q/G)\right).
\]
The reduced polysymplectic forms are given, for each $a=1,\dots,k$, by
\begin{equation}\label{eq:reducedform}
\overline{\omega}^a_\mu= (\F \Ro_\mu)^*\left(({\rm pr}^a_2)^*\omega_{Q/G}-({\rm pr}_1)^*\Bc_{\mu_a}\right).
\end{equation}
\end{theorem}
\subsection{The reduced Lagrangian field theory}
The goal now is to relate solutions of the original Lagrangian field theory with solutions of a reduced Lagrangian field theory.
\begin{lemma}\label{lem:energy} The energy $E_L\colon T^1_kQ\to\R$ is $G_\mu$-invariant. Its reduction is the function $E_{\Ro_\mu}$ defined as follows:
\[
E_{\Ro_\mu}(\pmb{v})\equiv \langle \F \Ro_\mu(\pmb{v}),\pmb{v} \rangle -\Ro_\mu (\pmb{v}),\qquad \pmb{v}\in p_\mu^*\left(T^1_k(Q/G)\right).
\]
\end{lemma}
\begin{proof} The fact that $E_L\colon T^1_kQ\to\R$ is invariant follows directly from the equivariance of the Legendre transformation $\F L$. For the second part, note that $E_L$ might be as written in the form
\[
E_L(\pmb{w})=\langle \F L (\pmb{w}),\pmb{w}\rangle -L(\pmb{w})=\langle \F R (\pmb{w}),\pmb{w}\rangle -R(\pmb{w}),\qquad \pmb{w}\in T^1_kQ.
\]
and therefore its pullback to $\pi^*(T^1_k(Q/G))$ is
\[
\big((\tilde\tau^{-1})^*E_L\big)(\pmb{v})=\langle \F \Ro (\pmb{v}),\pmb{v}\rangle -\Ro(\pmb{v}),\qquad \pmb{v}\in \pi^*(T^1_k(Q/G)).
\]
From here the claim follows directly.
\end{proof}
Theorem~\ref{thm:reducedform}, together with Lemma~\ref{lem:energy}, show that the Routhian $\Ro_\mu$ plays the role of the Lagrangian function in the reduced Lagrangian field theory: it encodes both the polysymplectic form and the energy function. It should be noted, however, that the reduced polysymplectic form~\eqref{eq:reducedform} has an additional term, so the reduced equations of motion are not the usual EL equations. This additional term does not depend on the Routhian, but on the chosen principal connection $\Ac$.
We can finally state the main reduction theorem:
\begin{theorem}\label{thm:red1} Let $L\colon T^1_kQ\to\R$ be an hyperregular, $G$-invariant and $G$-regular Lagrangian, and fix a principal connection $\Ac$ on $\pi\colon Q\to Q/G$. Then, if $\pmb{\Gamma}$ is a $G_\mu$-invariant solution of the $k$-symplectic Euler-Lagrange equations
\[
\sum_a \Gamma_a\lrcorner \omega_{Q,L}^a=dE_L,
\]
which is tangent to $J_L^{-1}(\mu)$, the reduced $k$-vector field $\overline{\pmb{\Gamma}}_\mu$ on $p_\mu^*(T^1_k(Q/G))$ satisfies the following $k$-symplectic system:
\[
\sum_a \left(\overline{\Gamma}_\mu\right)_a\lrcorner \overline{\omega}_\mu^a=dE_{\Ro_\mu},
\]
where $\overline{\omega}_\mu^a$ are given by~\eqref{eq:reducedform}.
\end{theorem}
\begin{proof} This follows from Theorem~\ref{thm:polyred} taking into account the observations above, Theorem \ref{thm:reducedform} and Lemma \ref{lem:energy}.
\end{proof}
We recall that $\pmb{\Gamma}_\mu$ denotes the restriction of $\pmb{\Gamma}$ to $J_L^{-1}(\mu)$. With regards to reconstruction, our situation is a particular case of Theorem~\ref{thm:reconstruction}:
\begin{theorem}\label{thm:reconstruction2} Under the same conditions of Theorem~\ref{thm:red1}, let $\overline{\phi}_\mu\colon \R^k\to p_\mu^*(T^1_k(Q/G))$ be an integral section of $\overline{\Gamma}_\mu$ such that the connection $\mathcal{H}(\pmb{\Gamma}_\mu,\overline{\phi}_\mu)$ is flat. Then there exists an integral section $\phi_\mu\colon \R^k\to J_L^{-1}(\mu)$ of $\pmb{\Gamma}_\mu$ with $\pi_\mu\circ \phi_\mu=\overline{\phi}_\mu$.
\end{theorem}
\section{Examples}
\subsection{The case of a Lie group}
Given a Lie group $G$, we consider a (regular and $G$-regular) left invariant Lagrangian $L\colon T^1_kG\to \R$. This general situation appears, for example, when one studies Lagrangians given by invariant metrics on $G$ (which we will discuss later as a separate example). In the mechanical case ($k$=1), this example is discussed in detail in \cite{LGC_class}.
We first recall some well-known facts. When $G$ acts on itself by left translation, the principal bundle of interest is simply the projection of $G$ onto the neutral element $\{e\}$ (or any other point). The infinitesimal generators are the right-invariant vector fields, and every tangent vector $v_g\in G$ is vertical. This bundle admits a canonical flat principal connection given by $\Ac(v_g)=v_g\cdot g^{-1}$. If $\nu\in\lag^*$, the contraction $\Ac_\nu=\langle\nu,\Ac\rangle$ satisfies
\begin{equation}\label{eq:ex0}
d\Ac_\nu (g\cdot\xi,g\cdot\eta)=\langle\nu,[{\rm Ad}_g\xi,{\rm Ad}_g\eta]\rangle=\langle {\rm Ad}_g^*\nu,[\xi,\eta]\rangle,
\end{equation}
where we have used Cartan's structure equations.
We identify on the left $TG\to G\times\lag$, $v_g\mapsto g^{-1}\cdot v_g$, so that $L$ may be thought of as a function of $G\times \lagk$. Invariance then implies that $L$ is of the form
\[
L(g,g\cdot\xi_1,\dots,g\cdot\xi_k)=\ell(\xi_1,\dots,\xi_k)
\]
where $\ell\colon\lagk\to \R$. If we denote by $\pmb{F} \ell\colon\lagk\to\lagdk$ the fiber derivative of $\ell$, with components $(F\ell)^a\colon \lagk\to \lag^*$ given by
\[
\langle (F\ell)^a(\xi_1,\dots,\xi_k),\eta\rangle=\left.\frac{d}{ds}\right|_{s=0}\ell(\xi_1,\dots,\xi_a+s\eta,\dots,\xi_k),
\]
then one finds
\begin{equation}\label{eq:ex1}
J_L(g,g\cdot\xi_1,\dots,g\cdot\xi_k)=\big({\rm Ad}^*_{g^{-1}}\big)^k\pmb{F} \ell(\xi_1,\dots,\xi_k).
\end{equation}
One can prove~\eqref{eq:ex1} directly applying the reasoning in the mechanical case to the components $J_L^a$ of the momentum map $J_L$, see \cite{LGC_class}.
In view of \eqref{eq:ex1}, fixing a momentum $\mu=(\mu_1,\dots,\mu_k)$ we have:
\begin{equation}\label{eq:ex3}
J_L(g,g\cdot\xi_1,\dots,g\cdot\xi_k)=\mu\iff \pmb{F} \ell(\xi_1,\dots,\xi_k)=\big({\rm Ad}^*_{g}\big)^k\mu=({\rm Ad}^*_{g}\mu_1,\dots,{\rm Ad}^*_{g}\mu_k).
\end{equation}
We will use the following short notation. We let $\nu\in\lagdk$ be
\[
\nu\equiv (\nu_1,\dots,\nu_k)= ({\rm Ad}^*_{g}\mu_1,\dots,{\rm Ad}^*_{g}\mu_k)\equiv \big({\rm Ad}^*_{g}\big)^k\mu.
\]
Since every element on $TG$ is vertical w.r.t. the projection $G\to\{e\}$, from the last relation one checks easily that $G$-regularity implies that $\pmb{F} \ell$ is a diffeomorphism. We shall write
\[
\tau\equiv (\pmb{F} \ell)^{-1}\colon \lagdk\to\lagk
\]
to denote its inverse. The relation \eqref{eq:ex3} of the momentum constraint is then
\[
\pmb{F} \ell(\xi_1,\dots,\xi_k)=\big({\rm Ad}^*_{g}\big)^k\mu=\nu\iff (\xi_1,\dots,\xi_k)=\tau(\nu).
\]
We need to recall the definition of the $k$-coadjoint orbit through $\mu\in\lagdk$, which is a polysymplectic manifold. We will denote it by $\mathcal{O}_\mu =\mathcal{O}_{(\mu_1,\dots,\mu_k)}$ and it is defined as the orbit of the ${\rm Coad}^k$-action, i.e.
\[
\mathcal{O}_\mu=\{g\cdot \mu\st g\in G\}\subseteq\lagdk.
\]
It has been shown that $G/G_\mu\simeq \mathcal{O}_\mu$, see~\cite{Polyreduction}.
From the general considerations in this paper (in particular, Proposition~\ref{prop:identificationJL}), applied to the present case $Q=G$, we have $J_L^{-1}(\mu)/G_\mu\simeq G/G_\mu\simeq \mathcal{O}_\mu$. We will now describe how the Routhian $\Ro_\mu\colon \mathcal{O}_\mu\to \R$ is obtained. We will use the $\Ac$ introduced before. The Routhian $R\colon G\times \lagk\to \R$ is then:
\begin{align*}
R(g,g\cdot\xi_1,\dots,g\cdot\xi_k)&=L(g,g\cdot\xi_1,\dots,g\cdot\xi_k)- \langle \mu_1,\Ac(g\cdot\xi_1)\rangle - \dots - \langle \mu_k,\Ac(g\cdot\xi_k)\rangle\\
&=\ell(\xi_1,\dots,\xi_k)-\langle {\rm Ad}_g^*\mu_1,\xi_1\rangle - \dots - \langle {\rm Ad}_g^*\mu_k,\xi_k\rangle\\
&=\ell(\xi_1,\dots,\xi_k)-\langle \nu_1,\xi_1\rangle - \dots - \langle \nu_k,\xi_k\rangle.
\end{align*}
The restriction of $R$ to $J_L^{-1}(\mu)$, denoted $\Ro\colon G\to \R$ as in Section \ref{sec:Routh}, is then
\begin{equation}\label{eq:ex2}
\Ro(g)= \left.\left(\ell(\xi_1,\dots,\xi_k)-\langle \nu_1,\xi_1\rangle - \dots - \langle \nu_k,\xi_k\rangle\right)\right|_{(\xi_1,\dots,\xi_k)=\tau(\nu)}.
\end{equation}
We make an observation before moving on. The notation in \eqref{eq:ex2} is meant to imply that the occurrences of $\xi_1,\dots,\xi_k$ are replaced in terms of $\nu_1,\dots,\nu_k$ using the map $\tau$. Thus, the momentum map fixes the ``$\lagk$-component'' of $L$, and as a result $\Ro$ is a function of $G$ alone. In other words, the level set of $\mu$ is identified with $G$.
Proceeding with the reduction by $G_\mu$, we finally find that the (reduced Routhian) $\Ro_\mu\colon \mathcal{O}_\mu\to \R$ is
\[
\Ro_\mu(\nu_1,\dots,\nu_k)= \left.\left(\ell(\xi_1,\dots,\xi_k)-\langle \nu_1,\xi_1\rangle - \dots - \langle \nu_k,\xi_k\rangle\right)\right|_{(\xi_1,\dots,\xi_k)=\tau_\mu(\nu)},
\]
with $\tau_\mu$ the restriction of $\tau$ to $\mathcal{O}_\mu$. We have used that the diffeomorphism $G/G_\mu\to \mathcal{O}_\mu$ is given by $g\mapsto g\cdot \mu$.
\begin{remark} If one wants to compute the equations of motion one needs to take into account the force terms $\Bc_{\mu_a}$. Following \cite{LGC_class}, one can relate these terms to the the Kirillov–Kostant–Souriau symplectic forms on $\mathcal{O}_{\mu_a}\subseteq \lag^*$.
\end{remark}
\subsection{Invariant metrics on a Lie group}
A particular case of the construction in the previous example is obtained when the Lagrangian $L$ is constructed from a left-invariant metric $\mathcal{G}$ on $G$. Let us write
\begin{equation}\label{eq:ex4}
L(g,g\cdot\xi_1,\dots,g\cdot\xi_k)=\frac{1}{2}\mathcal{G}(\xi_1,\xi_1)+\dots+\frac{1}{2}\mathcal{G}(\xi_k,\xi_k)=\ell(\xi_1,\dots,\xi_k).
\end{equation}
The expression of $L$ makes it simple to compute the maps $\pmb{F}\ell$ and $\tau$. First, the fiber derivative $\pmb{F}\ell$ has components
\[
(F\ell)^1(\xi_1,\dots,\xi_k)=\flat(\xi_1),\; (F\ell)^2(\xi_1,\dots,\xi_k)=\flat(\xi_2),\; \dots,
\]
where $\flat\colon\lag\to\lag^*$ is the isomorphism induced by the metric $\langle \flat(\xi),\eta\rangle=\mathcal{G}(\xi,\eta)$. Therefore the inverse $\tau\colon \lagdk\to \lagk$ has components:
\[
(\tau)^1(\nu_1,\dots,\nu_k)=\flat^{-1}(\nu_1),\; (\tau)^2(\nu_1,\dots,\nu_k)=\flat^{-1}(\nu_2),\; \dots.
\]
Let us denote by $\widetilde{\mathcal{G}}=(\flat^{-1})^*(\mathcal{G})$ the metric induced on $\lag^*$. The Routhian is then
\begin{align*}
\Ro_\mu(\nu)&= \left.\left(\frac{1}{2}\mathcal{G}(\xi_1,\xi_1)+\dots+\frac{1}{2}\mathcal{G}(\xi_k,\xi_k)-\langle \nu_1,\xi_1\rangle - \dots - \langle \nu_k,\xi_k\rangle\right)\right|_{\xi_a=\flat^{-1}(\nu_a)}\\
&=\frac{1}{2}\widetilde{\mathcal{G}}(\nu_1,\nu_1)+\dots+\frac{1}{2}\widetilde{\mathcal{G}}(\nu_k,\nu_k)-\langle \nu_1,\flat^{-1}(\nu_1)\rangle - \dots - \langle \nu_k,\flat^{-1}(\nu_k)\rangle\\
&=\frac{1}{2}\widetilde{\mathcal{G}}(\nu_1,\nu_1)+\dots+\frac{1}{2}\widetilde{\mathcal{G}}(\nu_k,\nu_k)-\widetilde{\mathcal{G}}(\nu_1,\nu_1)-\dots-\widetilde{\mathcal{G}}(\nu_k,\nu_k)\\
&=-\left(\frac{1}{2}\widetilde{\mathcal{G}}(\nu_1,\nu_1)+\dots+\frac{1}{2}\widetilde{\mathcal{G}}(\nu_k,\nu_k)\right).
\end{align*}
This coincides, up to a sign, with the reduced Hamiltonian $H_\mu$ in the Example 4.3.1 of \cite{Polyreduction}. This is to be expected: since there are no fiber coordinates -``velocities''- in $\Ro_\mu$, the usual ansatz to compute the Hamiltonian gives
\[
H_\mu(q,p)=\left(\langle\F \Ro_\mu(v),v\rangle-\Ro_\mu(q,v)\right)_{v=(\F \Ro_\mu)^{-1}(p)}=-\Ro_\mu(q,v).
\]
The case of an invariant metric on Lie group described in this example, where the Lagrangian is given by \eqref{eq:ex4}, is of interest because it admits a solution $\pmb{\Gamma}$ which can be reduced to a solution of the Routhian $\Ro_\mu$ (in the sense of Theorem~\ref{thm:red1}). The explicit construction of such a solution $\pmb{\Gamma}$ can be done adapting the Hamiltonian version in the Example 4.3.1 of \cite{Polyreduction} mentioned above. From a Lagrangian perspective, we will see in the next example that such a solution $\pmb{\Gamma}$ appears very naturally when dealing with invariant metrics.
\subsection{Invariant metrics: a concrete example}
We will now discuss a concrete example in dimension 4 that has been studied in the context of Lagrange-Poincar\'e reduction in $k$-symplectic geometry in~\cite{LTM_LP}.
Before presenting the actual example, let us recall some facts. Consider the natural Lagrangian $L\colon TQ\to \R$ that one may associate to a Riemannian metric $\mathcal{G}$ on $Q$:
\begin{equation}\label{eq:ex7}
\tilde {L}(q,v)=\frac{1}{2}\mathcal{G}(v,v)=\frac{1}{2}g_{ij}(q)v^iv^j.
\end{equation}
This Lagrangian is regular, and its -unique- solution is the geodesic spray
\[
\Gamma_{\tilde L}=v^j\fpd{}{q^j}-\Gamma^i_{jk}v^jv^k\fpd{}{v^i},
\]
where $\Gamma^i_{jk}$ are the Christoffel symbols of $\mathcal{G}$. If $\mathcal{G}$ is invariant under a $G$-action, then the Lagrangian is also $G$-invariant and $\Gamma_{\tilde L}$ is $G$-invariant. The second-order vector field $\Gamma_{\tilde L}$ is completely determined from the relation $i_{\Gamma_{\tilde{L}}}\omega_{Q,\tilde{L}}=dE_{\tilde L}$. For this invariant Lagrangian, Noether's theorem can be written in the form:
\[
0=\Gamma_{\tilde L}\left(J_{\tilde L}(q,v),\xi\right)= \Gamma_{\tilde L}\left(\mathcal{G}(v,\xi_Q(q))\right).
\]
We now consider a Lagrangian $L\colon T^1_kQ\to\R$ of the form
\begin{equation}\label{eq:ex5}
L(v_1,\dots,v_k)=\frac{1}{2}\mathcal{G}(v_1,v_1)+\dots+\frac{1}{2}\mathcal{G}(v_k,v_k),
\end{equation}
where $\mathcal{G}$ is a $G$-invariant metric (the previous example falls in this category when $Q=G$). The form of the Lagrangian~\eqref{eq:ex5} is motivated by the following well-known observation: the defining relation for a harmonic map between an Euclidean space and a Riemannian manifold $(Q,{\mathcal G})$ can be thought of as a Lagrangian field equation, when the Lagrangian of the field theory is exactly of the form (30). We will now show that a Lagrangian of that form admits an invariant harmonic as solution, which can further be reduced by means of the Routh procedure we have described in the previous sections.
\begin{proposition} Consider the $k$-vector field $\pmb{\Gamma}$ on $T^1_kQ$ with components
\[
\Gamma_a=v^i_a\fpd{}{q^i}- \Gamma^i_{jk}v^j_av^k_a\fpd{}{v^i_a}.
\]
Then:
\begin{enumerate}[label={(\roman*})]
\item $\pmb{\Gamma}$ is a solution of \eqref{eq:k-EL} for the Lagrangian \eqref{eq:ex5};
\item $\pmb{\Gamma}$ is tangent to $J_L^{-1}(\mu)$ (see the paragraph after Theorem~\ref{thm:polyred2});
\item $\pmb{\Gamma}$ is $G$-invariant (in particular, $G_\mu$ invariant for any $\mu$).
\end{enumerate}
\end{proposition}
\begin{proof} For simplicity, lets us denote $\tau^1\equiv \tau_Q^{k,1}\colon T^1_kQ\to TQ$ the projection onto the first factor, in coordinates
\[
\tau^1(v_1^i,v_2^i,\dots,v_k^i)\to (v_1^i).
\]
We see that $\Gamma_1$ is $\tau^1$-projectable and satisfies (note that all of the terms in $\partial/\partial{v^i_b}$ will vanish unless $b=1$):
\[
T\tau^1(\Gamma_1)=v^i_1\fpd{}{q^i}- \Gamma^i_{jk}v^j_1v^k_1\fpd{}{v^i_1}.
\]
If we define $\tilde L\colon TQ\to\R$ as before~\eqref{eq:ex7}, we recognize in $T\tau^1(\Gamma_1)$ the geodesic spray of $\mathcal{G}$, hence
\[
(\Gamma_1\lrcorner \omega^1_{Q,L})(\cdot)=(\Gamma_1\lrcorner (\tau^1)^*(\omega_{Q,\tilde L}))(\cdot)=(\tau^1)^*dE_{\tilde L}(\cdot).
\]
Proceeding similarly with the remaining components we have
\[
(\Gamma_1\lrcorner \omega^1_{Q,L})+(\Gamma_2\lrcorner \omega^2_{Q,L})+\dots+(\Gamma_k\lrcorner \omega^k_{Q,L})= (\tau^1)^*dE_{\tilde L}+(\tau^2)^*dE_{\tilde L}+\dots+ (\tau^k)^*dE_{\tilde L}.
\]
But the right-hand side equals $dE_L$, which shows that $\pmb{\Gamma}$ satisfies~\eqref{eq:k-EL}. This proves $(i)$.
To show $(ii)$, we observe that
\begin{equation}\label{eq:momentumlocked}
\langle J_L^a(q,v_1,\dots,v_k),\xi \rangle=\langle (FL)^a(q,v_1,\dots,v_k),\xi_Q(q) \rangle=\mathcal{G}(v_a,\xi_Q(q)).
\end{equation}
Let us denote by $\mathcal{G}_\xi\colon TQ\to \R$ the function $\mathcal{G}_\xi(v)=\mathcal{G}(v,\xi_Q)$. Then if we fix $a$, we have that:
\begin{align*}
\Gamma_a (\langle J_L^a(q,v_1,\dots,v_k),\xi \rangle)&=\Gamma_a\left[\mathcal{G}_\xi(v_a)\right]=\Gamma_a\left[(\tau^a)^*\mathcal{G}_\xi(v_1,\dots,v_k)\right]\\
&=(\tau^a)^*\left(\Gamma_{\tilde L}\mathcal{G}_\xi(v_a)\right)\\
&=0.
\end{align*}
This means that, for each $a$, $\Gamma_a$ is tangent to $(J_L^a)^{-1}(\mu)$. Therefore each $\Gamma_a$ is also tangent to $(J_L)^{-1}(\mu)$, which is the intersection of all the sets $(J_L^a)^{-1}(\mu)$. This proves $(ii)$. Finally, from the definition of the action on $T^1_kQ$ it is clear that $\Gamma_a$ is invariant if $\Gamma_{\tilde L}$ is. This proves $(iii)$ and completes the proof.
\end{proof}
Let us choose the mechanical connection $\Ac$ on $Q\to Q/G$. It is defined by declaring the horizontal subspace to be the orthogonal (w.r.t. the metric $\mathcal{G}$) of the vertical subbundle $V\pi$. In other words, a vector $v_q\in TQ$ is horizontal if the following condition holds:
\[
v_q\; \text{is horizontal} \iff \mathcal{G}(v_q,\xi_Q(q))=0,\; \text{for all}\; \xi\in\lag.
\]
The use of this principal connection greatly simplifies the computation of the Routhian. Our approach here follows \cite{RouthMarsden}.
It is customary to call the map $I_q\colon \lag\to\lag^*$ defined by
\[
\langle I_q(\xi),\eta\rangle=\mathcal{G}(\xi_Q(q),\eta_Q(q))
\]
the locked inertia tensor; we remark that, in general, it depends on the chosen point $q\in Q$. Let us denote $\zeta_a=\Ac(v_a)\in\lag$. When $\mu=J_{L}(\pmb{v})$, from \eqref{eq:momentumlocked} we have
\[
\mathcal{G}({\rm Ver}(v_a),{\rm Ver}(v_a))= \mathcal{G}\big((\zeta_a)_Q(q),(\zeta_a)_Q(q)\big)=\langle I_q(\zeta_a),\zeta_a\rangle=\langle \mu_a,\zeta_a\rangle,
\]
and we can write $\mathcal{G}({\rm Ver}(v_a),{\rm Ver}(v_a))=\langle\mu_a,I_q^{-1}(\mu_a)\rangle$. Then, if for each $v_a$ we decompose it in its horizontal and vertical parts, we have for the Routhian $\Ro\colon J_L^{-1}(\mu)\to\R$:
\begin{align}\label{eq:ex6}
\Ro(q,v_1,\dots,v_k)&=\sum_a \left[\frac{1}{2}\mathcal{G}({\rm Hor}(v_a),{\rm Hor}(v_a)+ \frac{1}{2}\mathcal{G}({\rm Ver}(v_a),{\rm Ver}(v_a)-\langle\mu_a,\zeta_a\rangle\right]\nonumber \\
&=\sum_a\left[\frac{1}{2}\mathcal{G}({\rm Hor}(v_a),{\rm Hor}(v_a)- \frac{1}{2}\langle\mu_a,I_q^{-1}(\mu_a)\rangle\right].
\end{align}
We remark that the terms $\langle\mu_a,I_q^{-1}(\mu_a)\rangle$ are $G_\mu$-invariant, so $\Ro$ can be easily reduced to define the function $\Ro_\mu$ on the quotient. Note that \eqref{eq:ex6} is the field-theoretic analogue of the expression in Proposition 3.5 in \cite{RouthMarsden}. Of course, such a simple expression can only be obtained because the Lagrangian \eqref{eq:ex5} is a sum of Lagrangians on $TQ$, pulled-back to $T^1_kQ$.
We will now discuss a concrete example which has also appeared in the context of Lagrange-Poincar\'e reduction in the $k$-symplectic formalism \cite{LTM_LP}. Consider the Lie group $G$ of matrices of the form:
\[
\begin{pmatrix}
1 & y\cos\theta+x\sin\theta & -y\sin\theta+x\cos\theta & z\\
0 & \cos\theta & -\sin\theta & x\\
0 & \sin\theta & \cos\theta & -y\\
0 & 0 &0 & 1
\end{pmatrix}.
\]
For details on $G$, we refer the reader to \cite{Thompson} (case ``$A_{4,10}$'' on page 423) or \cite{LTM_LP}. Take $Q=\R\times G$ and consider the Lagrangian $L\colon T^1_kQ\to\R$ given by \eqref{eq:ex7} where the metric in $Q$ is:
\[
\mathcal{G}=dq^2+\gamma\,dq\,d\theta+dx^2+dy^2-y\,dx\,d\theta+x\,dy\,d\theta+dz\,d\theta,
\]
with $dx\,d\theta=(dx\otimes d\theta+d\theta\otimes dx)/2$ the usual symmetric product notation (and similarly for the other terms) . The natural left action of $G$ on $Q$ gives isometries.
We take the following square matrices in dimension 4 as the basis of the Lie algebra $\lag$ of $G$:
\[
e_x=e_{13}-e_{24},\quad e_y=e_{12}-e_{34},\quad e_z=e_{14},\quad e_{\theta}=-e_{23}+e_{32}.
\]
The notation $e_{ij}$ denotes the matrix with one entry equal to one at $(i,j)$ (and zero otherwise). The Lie algebra structure of $\lag$ is given by the following nonvanishing brackets:
\begin{equation}\label{eq:brackets}
[e_x,e_y]=-2e_z,\qquad [e_x,e_\theta]=e_y,\qquad [e_y,e_\theta]=-e_x.
\end{equation}
With this choice of basis, the infinitesimal generators are:
\[
E_x= \fpd{}{x}-y\fpd{}{z},\quad E_y= \fpd{}{y}+x\fpd{}{z},\quad E_z=\fpd{}{z},\quad E_\theta= \fpd{}{\theta}-x\fpd{}{y}+y\fpd{}{x}.
\]
These can be easily obtained since the infinitesimal generators for the left translation on a Lie group are precisely the right-invariant vector fields. Their brackets satisfy their lowercase analogs except for a sign; for instance, we have $[E_x,E_y]=2E_z$.
A set of left-invariant vector fields has been obtained in~\cite{LTM_LP}:
\begin{align*}
F_x&= \cos\theta\left(\fpd{}{x}+y\fpd{}{z}\right)-\sin\theta\left(\fpd{}{y}-x\fpd{}{z}\right),\quad F_z=\fpd{}{z},\\
F_y&= \sin\theta\left(\fpd{}{x}+y\fpd{}{z}\right)+\cos\theta\left(\fpd{}{y}-x\fpd{}{z}\right),\quad F_\theta=\fpd{}{\theta}.
\end{align*}
We consider the following vector field on $Q$:
\[
X=\fpd{}{x}-\gamma\fpd{}{q}.
\]
The definition of $X$ is such that $\mathcal{G}(X,E_a)=0$ for all $a=x,y,z,\theta$, so that $X$ represents a horizontal vector field projecting onto $\partial/\partial q$ orthogonal to the vertical subspace, or in other words $X$ encodes the mechanical connection for the metric $\mathcal{G}$. This is a flat connection.
It is clear that $\mathcal{G}(X,X)=1$. With regards to the vertical part of the metric, if we compute it in the basis of infinitesimal generators we find:
\[
\mathcal{F}_{ab}\equiv\mathcal{G}(E_a,E_b)=
\begin{pmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 0 & 1/2\\
0 & 0 & 1/2 & 0
\end{pmatrix}.
\]
One shows in a similar way that $\mathcal{G}(F_a,F_b)=\mathcal{F}_{ab}$. In other words, the vertical part of the metric is a bi-invariant metric on $G$ (one can show as well that $\mathcal{G}$ is a bi-invariant metric on $Q$ when the natural action of $G$ on the right is considered). But then the locked inertia tensor $I_q$ does not depend on $q$; the matrix $\mathcal{F}$, which gives the metric at the Lie algebra level, is such that
\[
\langle I_q(\xi),\eta\rangle= \sum_{a,b}\xi^a\mathcal{F}_{ab}\eta^b
\]
when we use the basis $\{e_x,e_y,e_z,e_\theta\}$ of $\lag$. Looking at \eqref{eq:ex7} we see that all of the terms
\[
-\frac{1}{2}\langle\mu_a,I_q^{-1}(\mu_a) \rangle
\]
are constant and do not contribute to the Euler-Lagrange equations of $\Ro$, so we may ignore them (but keep the name for $\Ro$, for simplicity). One concludes that the reduced Routhian $\Ro_\mu\colon Q/G_\mu\times T^1_k\R\to\R$ is given by the following expression:
\[
\Ro_\mu([q,g],v_1,\dots,v_k)= \left[\frac{1}{2}\mathcal{G}({\rm Hor}(v_1),{\rm Hor}(v_1)\right]+\dots+\left[\frac{1}{2}\mathcal{G}({\rm Hor}(v_k),{\rm Hor}(v_k)\right].
\]
The coordinate formula for $\Ro_\mu$ is then
\[
\Ro_\mu(v_a)= \frac{1}{2}\Big((v^1_1)^2+\dots+(v^1_k)^2\Big).
\]
It is independent of the choice of $\mu\in\lagdk$ or the base point $(q,[g])\in Q/G_\mu$. As a matter of fact, one could still use the residual symmetry in $q$ to further reduce: the reason, as mentioned above, is that the metric $\mathcal{G}$ is bi-invariant for the Lie group $Q=\R\times G$ with product $(q_1,g_2)\cdot (q_2,g_2)=(q_1+q_2,g_1g_2)$. Reducing by the whole of $Q$ directly would take us to the results in the previous example.
Let us choose $\nu=(0,0,1,0)=e^z$, where $e^z$ is the dual of $e_z$). A computations shows that $G_\nu$ is the following Abelian subgroup of $G$:
\[
G_\nu=\left\{
\begin{pmatrix}
1 & 0 & 0 & z\\
0 & \cos\theta & -\sin\theta & 0\\
0 & \sin\theta & \phantom{-}\cos\theta & 0\\
0 & 0 &0 & 1
\end{pmatrix}, \; x\in\R,\,z\in S^1
\right\} \simeq \R\times S^1.
\]
One can check this directly using the infinitesimal condition for $\xi\in G_\nu$: an element $\xi\in\lag_\nu$ satisfies ${\rm ad}_\xi^*\nu=0$ or $\langle \nu,[\xi,\eta]\rangle=0$ for $\eta\in\lag$ arbitrary. From the adjoint relations in~\eqref{eq:brackets} we have
\[
2(\xi^y\eta^x-\xi^x\eta^y)=0
\]
where we have written $\xi=\xi^x e_x+\xi^y e_y+\xi^z e_z+\xi^\theta e_\theta\in\lag$ (and similar for $\eta$). This means that $\xi^x=\xi^y=0$, and therefore $\lag_\nu={\rm span}\{e_z,e_\theta\}$, which corresponds to the subgroup $G_\nu$ above.
If, for the sake of the argument, we carry out the reduction at
\[
\mu= (\nu,\dots,\nu)\in\lagdk,
\]
then $G_\mu=G_\nu\cap \dots\cap G_\nu=G_\nu=\R\times S^1$ and $\Ro_\mu$ is defined on $ Q/(\R\times S^1)\times T^1_k\R$.
The reduced $k$-vector field
\[
\overline{\Gamma}_a=v^1_a\fpd{}{q}
\]
is a solution of~\eqref{eq:k-EL} for $\Ro_\mu$. There is no force term because the the base of $Q\to Q/G=\R$ is one-dimensional, and this implies that all of the $B_\mu$ (which are 2-forms on the base) must vanish. It is one of the benefits of our approach that finding solutions of the reduced Routhian equations is very simple in this case. The reconstruction of the integral sections of the original problem, however, involves solving a PDE problem, which is not a trivial matter.
\paragraph{Acknowledgments.} We would like to thank J.C.\ Marrero for some clarifications concerning reference~\cite{Polyreduction}. S.\ Capriotti, V. D\'iaz and E.\ Garc\'{\i}a-Tora\~{n}o Andr\'{e}s are thankful to FONCYT for funding through project PICT 2019-00196. T.\ Mestdag thanks the Research Foundation -- Flanders (FWO) for its support through Research Grant 1510818N.
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3,212,635,537,482 | arxiv | \section{Introduction}
One of the most important problems of combinatorial abelian group theory is to determine
the number of subgroups of a finite abelian group. This topic has enjoyed a constant evolution
starting with the first half of the $20^{\mathrm{th}}$ century. Since a finite
abelian group is a direct product of abelian $p$-groups, the above
counting problem can be reduced to $p$-groups. Formulas which give the
number of subgroups of type $\mu$ of a finite $p$-group of type
$\lambda$ were established by S.~Delsarte \cite{Del1948}, P.~E.~Djubjuk \cite{Dju1948} and
Y.~Yeh \cite{Yeh1948}. An excellent survey on this subject together with connections to symmetric
functions was written by M.~L.~Butler \cite{But1994} in 1994.
Another way to find the total number of subgroups of finite abelian
$p$-groups was described by G.~Bhowmik \cite{Bho1996} by using
divisor functions of matrices. By invoking different arguments,
formulas in the case of rank two $p$-groups were obtained by G.~C\u
alug\u areanu \cite{Cal2004}, M.~T\u arn\u auceanu
\cite{Tar2007,Tar2010}, M.~Hampejs, N.~Holighaus, L.~T\'oth,
C.~Wiesmeyr \cite{HHTW2014}, L.~T\'oth \cite{Tot2014} and for rank
three $p$-groups by M.~Hampejs, L.~T\'oth \cite{HamTot2013},
J.-M.~Oh \cite{Oh2013}. Note that the papers
\cite{HHTW2014,HamTot2013,Tot2014} include also direct formulas for
the groups $\Z_m\times \Z_n$ and $\Z_m\times \Z_n\times \Z_r$,
respectively, where $m,n,r\in \N^*:= \{1,2,\ldots\}$ are arbitrary.
The purpose of the current paper is to count the number of subgroups
of a given exponent in a finite abelian $p$-group. Explicit formulas
are obtained for rank two and rank three $p$-groups. The numbers of
subgroups of exponent $p$, respectively $p^2$ in an arbitrary
$p$-group are also considered. We prove that if two finite abelian groups have the same number of
subgroups of any exponent, then they are isomorphic. We also deduce compact formulas for
the number of subgroups of a given exponent of the group $\Z_m\times
\Z_n$, where $m,n\in \N^*$ are arbitrary. Furthermore, we obtain an
exact formula for the sum of exponents of the subgroups of
$\Z_m\times \Z_n$, and an asymptotic formula for the arithmetic
means of exponents of the subgroups of $\Z_n\times \Z_n$.
For the proofs we use two different approaches. The first one is
based on the known formula for the number of subgroups of a given
type in an abelian $p$-group, given in terms of gaussian
coefficients (Theorem \ref{Th_subgroups_type}). The second method,
applicable only for rank two groups, uses the representation of the
subgroups of $\Z_m\times \Z_n$ obtained by the second author in
\cite{Tot2014} (Theorem \ref{Th_repr}).
Most of our notation is standard and will usually not be repeated here.
For basic notions and results on group theory we refer the reader to \cite{Suz}.
\section{First approach}
Let $G$ be a finite abelian group of order $n$ and $G=G_1\times G_2\times\cdots\times G_m$ be
the primary decomposition of $G$, where $G_i$ is a $p_i$-group ($i=1,2,\ldots,m$). For every divisor
$d=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_m^{\alpha_m}$ of $n$ we denote
\begin{equation*}
\cale_d(G)=\{H \leq G : \exp(H)=d\}.
\end{equation*}
Since the subgroups $H$ of $G$ are of type $H= H_1\times H_2\times\cdots\times H_m$ with $H_i\leq G_i$ ($i=1,2,\ldots,m$), we infer that
\begin{equation} \label{eq_prod}
|{\cale}_d(G)|= \prod_{i=1}^m |\cale_{p_i^{\alpha_i}}(G_i)|.
\end{equation}
Equality \eqref{eq_prod} shows that the problem of counting the number of
subgroups of exponent $d$ in $G$ is reduced to $p$-groups. So, in this section we will assume that $G$ is a finite abelian
$p$-group, that is a group of type $\mathbb{Z}_{p^{\lambda_1}} \times \mathbb{Z}_{p^{\lambda_2}}\times \cdots \times
\mathbb{Z}_{p^{\lambda_k}}$ with $\lambda_1\geq\lambda_2\geq \ldots \geq\lambda_k\ge 1$.
In this case we will say that $G$ is of type $\lambda$, where $\lambda$ is the partition $(\lambda_1,\lambda_2,\ldots,\lambda_k,0,\ldots)$,
and we will denote it by $G_{\lambda}$.
We recall the following well--known result, which gives the number of subgroups of type $\mu$ of $G_{\lambda}$ (see \cite{Del1948,Dju1948,Yeh1948}).
\begin{thm} \label{Th_subgroups_type} For every partition $\mu \preceq\lambda$ {\rm (}i.e., $\mu_i\leq\lambda_i$ for every $i\in \N^*${\rm )} the
number of subgroups of type $\mu$ in $G_{\lambda}$ is
\begin{equation*}
\alpha_{\lambda}(\mu;p) =\prod_{i\geq 1} p^{(a_i-b_i)b_{i+1}}\binom{a_i-b_{i+1}}{b_i-b_{i+1}}_{\hspace{-1mm}p}\,,
\end{equation*}
where $\lambda' =(a_1,a_2,\ldots,a_{\lambda_1},0,\ldots)$, $\mu'=(b_1,b_2,\ldots,b_{\mu_1},0,\ldots)$ are the partitions conjugate
to $\lambda$ and $\mu$, respectively, and
\begin{equation*}
\binom{n}{k}_{\hspace{-1mm}p} = \displaystyle\frac{\prod_{i=1}^n(p^i-1)}{\prod_{i=1}^k(p^i-1) \prod_{i=1}^{n-k}(p^i-1)}
\end{equation*}
is the gaussian binomial coefficient {\rm (}it is understood that $\prod_{i=1}^m (p^i-1)=1$ for $m=0${\rm )}.
\end{thm}
By using Theorem \ref{Th_subgroups_type} a way to compute the number of subgroups of exponent $p^i$ in $G_{\lambda}$ can be inferred, namely
\begin{equation*}
|\cale_{p^i}(G_{\lambda})| = \hspace{-3mm}\displaystyle\sum_{\mu\preceq\lambda,\, \mu_1=i}\hspace{-3mm}
\alpha_{\lambda}(\mu;p) \quad (i=0,1,\ldots,\lambda_1),
\end{equation*}
which is a polynomial in $p$ with integer coefficients. If $i\geq 1$ and $k\geq 2$ are arbitrary, then the polynomial
$|\cale_{p^i}(G_{\lambda})|$ can not be given explicitly, but we will do this in some
particular cases. Namely, we will consider the following cases: $k\in \{2,3\}$ and $i\geq 1$ arbitrary, $i\in \{1,2\}$ and $k\geq 2$ arbitrary.
We remark first that since $\sum_{H\in\cale_{p^i}(G_{\lambda})}H\leq G_{\lambda}$ and $\sum_{H\in\cale_{p^i}(G_{\lambda})}H$ is
of type $\mathbb{Z}_{p^i}\times \cdots\times \mathbb{Z}_{p^i}\times\mathbb{Z}_{p^{\lambda_r}}\times \cdots
\times\mathbb{Z}_{p^{\lambda_k}}$, where $r = \min\{j: \lambda_j <i\}$, we have
\begin{equation*}
|\cale_{p^i}(G_{\lambda})|= |\cale_{p^i}(\mathbb{Z}_{p^i}\times \cdots \times\mathbb{Z}_{p^i}
\times\mathbb{Z}_{p^{\lambda_r}}\times \cdots \times\mathbb{Z}_{p^{\lambda_k}})|
\end{equation*}
with the convention that the direct product $\mathbb{Z}_{p^{\lambda_r}}\times\cdots\times\mathbb{Z}_{p^{\lambda_k}}$ is trivial for
$i\leq\lambda_k$.
Our first result gives a precise expression for $|\cale_{p^i}(G_{\lambda})|$ in the case $k=2$.
\begin{prop} \label{Prop_exp_p} Let $G_{\lambda} =\mathbb{Z}_{p^{\lambda_1}}\times\mathbb{Z}_{p^{\lambda_2}}$ with $\lambda_1\geq\lambda_2\geq 1$. Then
\begin{align*}
|\cale_{p^i}(G_{\lambda})| = \begin{cases}
\dd \frac{p^{i+1}+p^i-2}{p-1}\,, &1\leq i\leq\lambda_2\,,\\ & \\
\dd \frac{p^{\lambda_2+1}-1}{p-1}\,, &\lambda_2<i\leq\lambda_1\,.
\end{cases}
\end{align*}
\end{prop}
\begin{proof} We have $\lambda=(\lambda_1,\lambda_2,0,\ldots)$ and consequently
\begin{align*}
\lambda'=(2,2,\ldots,2,1,1,\ldots,1,0,\ldots),
\end{align*}
where the number of $2$'s is $\lambda_2$ and the number of $1$'s is $\lambda_1-\lambda_2$. Then
$$
|\cale_{p^i}(G_{\lambda})|=\dd\sum_{j=0}^i N_{i,j},
$$
where
$N_{i,j}$ denotes the number of subgroups of type $(i,j)$ in $G_{\lambda}$, or equivalently in $\mathbb{Z}_{p^i}\times\mathbb{Z}_{p^i}$
if $i\leq\lambda_2$ or in $\mathbb{Z}_{p^i}\times \mathbb{Z}_{p^{\lambda_2}}$ if $\lambda_2<i\leq\lambda_1$.
In the first case we obtain $N_{i,j}=(p+1)p^{i-j-1}$ for $j=0,1,\ldots,i-1$, and $N_{i,i}=1$. Therefore
$$
|\cale_{p^i}(\mathbb{Z}_{p^i}\times\mathbb{Z}_{p^i})|=1+\dd\sum_{j=0}^{i-1}(p+1)p^{i-j-1} = \dd\frac{p^{i+1}+p^i-2}{p-1}\,.
$$
In the second case we obtain
$N_{i,j}=p^{\lambda_2-j}$ for $j=0,1,\ldots,\lambda_2$. Therefore
$$
|\cale_{p^i}(\mathbb{Z}_{p^i}\times\mathbb{Z}_{p^{\lambda_2}})|= \dd\sum_{j=0}^{\lambda_2}p^{\lambda_2-j}=\dd\frac{p^{\lambda_2+1}-1}{p-1}\,.
$$
This completes the proof.
\end{proof}
\begin{exm} We have
\begin{align*}
|\cale_{p^i}(\mathbb{Z}_{p^4}\times\mathbb{Z}_{p^2})|=\begin{cases}
1,&i=0,\\
p+2,&i=1,\\
p^2+2p+2,&i=2,\\
p^2+p+1,&i=3 \text{\rm{ or }} i=4. \end{cases}
\end{align*}
\end{exm}
In particular, by summing all quantities $|\cale_{p^i}(G_{\lambda})|$ ($i=0,1,\ldots,\lambda_1$) we obtain the total number of subgroups of
$G_{\lambda}$ (see also \cite[Prop.\ 2.9]{Tar2007}, \cite[Th.\ 3.3]{Tar2010}).
\begin{cor} \label{Cor_total_nr_subgroups_2} The total number of subgroups of $G_{\lambda}=\mathbb{Z}_{p^{\lambda_1}}\times\mathbb{Z}_{p^{\lambda_2}}$,
where $\lambda_1\geq\lambda_2\geq 1$, is
\begin{equation} \label{s_p}
\frac{1}{(p{-}1)^2}\left[(\lambda_1{-}\lambda_2{+}1)p^{\lambda_2{+}2}{-}(\lambda_1{-}\lambda_2{-}1) p^{\lambda_2{+}1}{-}(\lambda_1{+}\lambda_2{+}3)p{+}(\lambda_1{+}\lambda_2+1)\right].
\end{equation}
\end{cor}
\begin{exm} The total number of subgroups of $\mathbb{Z}_{p^4} \times \mathbb{Z}_{p^2}$ is
$3p^2+5p+7$.
\end{exm}
Now consider the case of rank three $p$-groups. We need the following lemma.
\begin{lem} \label{Lemma} Let $N_{i,j,\ell}$ denote the number of subgroups of type $(i,j,\ell)$ {\rm (}$i\geq j\geq \ell\geq 0${\rm )}
in $G_{\lambda} =\mathbb{Z}_{p^{\lambda_1}}\times \mathbb{Z}_{p^{\lambda_2}}\times \mathbb{Z}_{p^{\lambda_3}}$
with $\lambda_1\geq\lambda_2\geq \lambda_3\geq 1$. Then
\begin{align*}
N_{i,j,\ell}= \begin{cases}
p^{2i-2\ell-3}(p+1)(p^2+p+1), & \ell<j<i\leq \lambda_3,\\
p^{2(i-\ell-1)}(p^2+p+1), & \ell<j=i\leq \lambda_3 \text{ or } \ell=j<i\leq \lambda_3, \\
1, & \ell=j=i\leq \lambda_3, \\
p^{\lambda_3+i-2\ell-2}(p+1)^2, & \ell<j\leq \lambda_3 < i\leq \lambda_2, \\
p^{\lambda_3+i-2j-1}(p+1), & \ell=j\leq \lambda_3 < i\leq \lambda_2, \\
p^{2\lambda_3+i-j-2\ell-1}(p+1), & \ell \leq \lambda_3 < j<i\leq \lambda_2, \\
p^{2(\lambda_3-\ell)}, & \ell \leq \lambda_3 < j=i\leq \lambda_2, \\
p^{\lambda_2+2\lambda_3-j-2\ell}, & \ell \leq \lambda_3 < j\leq \lambda_2<i \leq \lambda_1, \\
p^{\lambda_2+\lambda_3-2\ell-1}(p+1), & \ell <j \leq \lambda_3 \leq \lambda_2 <i \leq \lambda_1, \\
p^{\lambda_2+\lambda_3-2\ell}, & \ell = j \leq \lambda_3 \leq \lambda_2 <i \leq \lambda_1.
\end{cases}
\end{align*}
\end{lem}
\begin{proof} We use Theorem \ref{Th_subgroups_type}. We distinguish the following three cases:
I. If $i\leq \lambda_3$, then $N_{i,j,\ell}$ is the number of subgroups of type $(i,j,\ell)$ in $\mathbb{Z}_{p^i}\times
\mathbb{Z}_{p^i} \times\mathbb{Z}_{p^i}$. Here we need to consider $\lambda=(i,i,i,0,\ldots)$ with $\lambda'=(3,3,\ldots,3,0\ldots)$, where
the number of $3$'s is $i$, and $\mu=(i,j,\ell,0,\ldots)$ with
$$\mu'=(3,3,\ldots,3,2,2,\ldots,2,1,1,\ldots,1,0,\ldots),$$ where the number of $3$'s is $\ell$, the number
of $2$'s is $j-\ell$, the number of $1$'s is $i-j$. In the subcase $\ell<j<i\leq \lambda_3$ we deduce
\begin{equation*}
N_{i,j,\ell}=(p^2)^{j-\ell-1}p(p+1)(p^2)^{i-j-1}(p^2+p+1)= p^{2i-2\ell-3}(p+1)(p^2+p+1).
\end{equation*}
The subcases $\ell<j=i\leq \lambda_3$, $\ell=j<i\leq \lambda_3$ and $\ell=j=i\leq \lambda_3$ are treated similar.
II. If $\lambda_3<i\leq \lambda_2$, then $N_{i,j,\ell}$ is the number of subgroups of type $(i,j,\ell)$ in $\mathbb{Z}_{p^i}\times
\mathbb{Z}_{p^i} \times\mathbb{Z}_{p^{\lambda_3}}$. We consider $\lambda=(i,i,\lambda_3,0,\ldots)$ with
$\lambda'=(3,3,\ldots,3,2,2,\ldots,2,0\ldots)$, where the number of $3$'s
is $\lambda_3$, the number of $2$'s is $i-\lambda_3$, and $\mu$, $\mu'$ like in the case I.
For example, in the subcase $\ell=j\leq \lambda_3<i\leq \lambda_2$
we obtain
\begin{equation*}
N_{i,j,\ell}=(p^2)^{\lambda_3-j}p^{i-\lambda_3-1}(p+1)= p^{\lambda_3+i-2j-1}(p+1).
\end{equation*}
III. If $\lambda_2<i\leq \lambda_1$, then $N_{i,j,\ell}$ is the number of subgroups of type $(i,j,\ell)$ in $\mathbb{Z}_{p^i}\times
\mathbb{Z}_{p^{\lambda_2}} \times\mathbb{Z}_{p^{\lambda_3}}$. We consider $\lambda=(i,\lambda_2,\lambda_3,0,\ldots)$
with $$\lambda'=(3,3,\ldots,3,2,2,\ldots,2,1,1,\ldots,1,0\ldots),$$
where the number of $3$'s is $\lambda_3$, the number of $2$'s is $\lambda_2-\lambda_3$, the number of $1$'s is $i-\lambda_2$,
and $\mu$, $\mu'$ like in the case I.
\end{proof}
\begin{prop} \label{Prop_exp_p_rank_3} Let $G_{\lambda} =\mathbb{Z}_{p^{\lambda_1}}\times\mathbb{Z}_{p^{\lambda_2}}\times \mathbb{Z}_{p^{\lambda_3}}$
with $\lambda_1\geq\lambda_2\geq \lambda_3\geq 1$. Then
\begin{align*}
|\cale_{p^i}(G_{\lambda})| = \begin{cases}
\dd \frac{p^{2i-1}\left((i+1)p^4+(i-1)p^3-p^2-(i+2)p-i\right)+3}{(p^2-1)(p-1)}\,, &1\leq i\leq\lambda_3,\\
&\\
\dd \frac{(\lambda_3+1)p^{\lambda_3+i}(p^2-1)(p+1)-2p^{2\lambda_3+2}+2}{(p^2-1)(p-1)}\,, &\lambda_3<i\leq\lambda_2,\\
&\\
\dd \frac{(\lambda_3+1)p^{\lambda_2+\lambda_3+1}(p^2-1)-p^{2\lambda_3+2}+1}{(p^2-1)(p-1)}\,, &\lambda_2<i\leq\lambda_1\,.
\end{cases}
\end{align*}
\end{prop}
\begin{proof} We have
\begin{equation*}
|\cale_{p^i}(G_{\lambda})|=\sum_{0\leq \ell \leq j\leq i} N_{i,j,\ell}
\end{equation*}
and use Lemma \ref{Lemma}.
Case I. If $i\leq \lambda_3$, then
\begin{align*}
|\cale_{p^i}(G_{\lambda})| = & N_{i,i,i} + \sum_{\ell =0}^{i-1} \sum_{j=\ell+1}^{i-1} N_{i,j,\ell} + \sum_{\ell=0}^{i-1} N_{i,i,\ell} +
\sum_{\ell=0}^{i-1} N_{i,\ell,\ell} \\
= & 1+ \sum_{\ell =0}^{i-1} \sum_{j=\ell+1}^{i-1} p^{2i-2\ell-3}(p+1)(p^2+p+1) + 2 \sum_{\ell=0}^{i-1} p^{2(i-\ell-1)}(p^2+p+1) \\
= & 1+ \frac{p^2+p+1}{(p^2-1)(p-1)}\left((i-1)p^{2i+1}-ip^{2i-1}+p \right) + 2(p^2+p+1) \frac{p^{2i}-1}{p^2-1} \\
= & \frac{p^{2i-1}((i+1)p^4+(i-1)p^3-p^2-(i+2)p-i)+3}{(p^2-1)(p-1)}\,,
\end{align*}
by direct computations.
II. If $\lambda_3<i\leq \lambda_2$, then we have
\begin{align*}
|\cale_{p^i}(G_{\lambda})| = & \sum_{\ell =0}^{\lambda_3} \sum_{j=\ell+1}^{\lambda_3} N_{i,j,\ell} + \sum_{\ell=0}^{\lambda_3}
\sum_{j=\lambda_3+1}^{i-1} N_{i,j,\ell} + \sum_{\ell=0}^{\lambda_3} N_{i,i,\ell} + \sum_{\ell=0}^{\lambda_3} N_{i,\ell,\ell} \\
= & \sum_{\ell =0}^{\lambda_3} \sum_{j=\ell+1}^{\lambda_3} p^{\lambda_3+i-2\ell-2} (p+1)^2 + \sum_{\ell=0}^{\lambda_3}
\sum_{j=\lambda_3+1}^{i-1} p^{2\lambda_3+i-j-2\ell-1}(p+1) \\
+ & \sum_{\ell=0}^{\lambda_3} p^{2(\lambda_3-\ell)} + \sum_{\ell=0}^{\lambda_3} p^{\lambda_3+i-2\ell -1}(p+1) \\
= & \frac{\lambda_3 p^{i-2}(p+1)^2+p^{\lambda_3}+p^i+p^{i-1}}{p^{\lambda_3}(p^2-1)} \left(p^{2\lambda_3+2}-1 \right) \\
- & \frac{p^{i-2}\left(p^{2\lambda_3+2}-(\lambda_3+1)p^2+ \lambda_3
\right)}{p^{\lambda_3}(p-1)^2} +
\frac{(p^{2\lambda_3+2}-1)(p^{i-\lambda_3-1}-1)}{(p-1)^2},
\end{align*}
which gives the above formula.
III. Finally, if $\lambda_2<i\leq \lambda_1$, then
\begin{gather*}
|\cale_{p^i}(G_{\lambda})| = \sum_{\ell =0}^{\lambda_3} \sum_{j=\lambda_3+1}^{\lambda_2} N_{i,j,\ell} + \sum_{\ell=0}^{\lambda_3-1}
\sum_{j=\ell+1}^{\lambda_3} N_{i,j,\ell} + \sum_{\ell=0}^{\lambda_3} N_{i,\ell,\ell} \\
= \sum_{\ell =0}^{\lambda_3} \sum_{j=\lambda_3+1}^{\lambda_2} p^{\lambda_2+2\lambda_3-j-2\ell} + \sum_{\ell=0}^{\lambda_3-1}
\sum_{j=\ell+1}^{\lambda_3} p^{\lambda_2+\lambda_3-2\ell-1} (p+1) + \sum_{\ell=0}^{\lambda_3} p^{\lambda_2+\lambda_3-2\ell} \\
= \frac{(p^{2\lambda_3+2}-1)(p^{\lambda_2-\lambda_3}-1)}{(p^2-1)(p-1)} + \frac{p^{\lambda_2-\lambda_3+1}}{(p^2-1)(p-1)}
\left(\lambda_3p^{2\lambda_3+2}-(\lambda_3+1)p^{2\lambda_3}+1\right)
\\ + p^{\lambda_2-\lambda_3}\frac{p^{2\lambda_3+2}-1}{p^2-1},
\end{gather*}
leading to the given result.
\end{proof}
Note that Proposition \ref{Prop_exp_p_rank_3} is valid also in the case $\lambda_3=0$, when it reduces to Proposition \ref{Prop_exp_p}.
\begin{exm} We have
\begin{align*}
|\cale_{p^i}(\mathbb{Z}_{p^4}\times\mathbb{Z}_{p^2}\times\mathbb{Z}_{p^2})| = \begin{cases}
1,&i=0,\\
2p^2+2p+3,&i=1,\\
3p^4+4p^3+6p^2+3p+3,&i=2,\\
3p^4+2p^3+2p^2+p+1,&i=3 \text{\rm{ or }} i=4. \end{cases}
\end{align*}
\end{exm}
\begin{cor} \label{Cor_total_nr_subgroups_3} The total number of subgroups of $G_{\lambda}=\mathbb{Z}_{p^{\lambda_1}} \times
\mathbb{Z}_{p^{\lambda_2}} \times \mathbb{Z}_{p^{\lambda_3}}$, where $\lambda_1 \geq \lambda_2 \geq \lambda_3\geq 1$, is
\begin{equation*}
\frac{A}{(p^2-1)^2(p-1)},
\end{equation*}
where
\begin{align*}
A = & (\lambda_3+1)(\lambda_1-\lambda_2+1)p^{\lambda_2+\lambda_3+5} +
2(\lambda_3+1)p^{\lambda_2+\lambda_3+4} \\
& - 2(\lambda_3+1)(\lambda_1-\lambda_2)p^{\lambda_2+\lambda_3+3} -
2(\lambda_3+1)p^{\lambda_2+\lambda_3+2} \\
&+ (\lambda_3+1)(\lambda_1-\lambda_2-1)p^{\lambda_2+\lambda_3+1} -
(\lambda_1+\lambda_2-\lambda_3+3)p^{2\lambda_3+4} \\
& -2 p^{2\lambda_3+3} + (\lambda_1 + \lambda_2 - \lambda_3-1) p^{2\lambda_3+2}\\
& + (\lambda_1 +\lambda_2 + \lambda_3+5) p^2 + 2p -(\lambda_1 + \lambda_2 + \lambda_3 +1).
\end{align*}
\end{cor}
\begin{proof} The total number of subgroups of $G_{\lambda}$ is
\begin{equation*}
\sum_{i=0}^{\lambda_1} |\cale_{p^i}(G_{\lambda})|= \sum_{0\leq i\leq \lambda_3} |\cale_{p^i}(G_{\lambda})| +
\sum_{\lambda_3<i\leq \lambda_2} |\cale_{p^i}(G_{\lambda})| + \sum_{\lambda_2 < i\leq \lambda_1} |\cale_{p^i}(G_{\lambda})|
\end{equation*}
and summing the quantities given in Proposition \ref{Prop_exp_p_rank_3} we deduce the result.
\end{proof}
We remark that an equivalent formula to that given in Corollary \ref{Cor_total_nr_subgroups_3} was obtained in \cite[Cor.\ 2.2]{Oh2013}
by using different arguments. Corollary \ref{Cor_total_nr_subgroups_3} is valid also in the case $\lambda_3=0$, when it reduces to Corollary
\ref{Cor_total_nr_subgroups_2}.
\begin{exm} The total number of subgroups of $\mathbb{Z}_{p^4} \times \mathbb{Z}_{p^2}\times \mathbb{Z}_{p^2}$ is
$9p^4+8p^3+12p^2+7p+9$.
\end{exm}
Next consider the number of subgroups of exponent $p$ (that is, the
number of elementary abelian subgroups) in $G_{\lambda}$, which
equals the total number of nontrivial subgroups of $(\Z_p)^k$. Since
$(\Z_p)^k$ is a $k$--dimensional linear space over $\Z_p$, the
number in question is exactly the total number of nonzero subspaces,
which is, as well--known, $\sum_{i=1}^k \binom{k}{i}_p$ (Galois
number). So we have the next result.
\begin{prop} \label{Prop_elem} Let $G_{\lambda}=\mathbb{Z}_{p^{\lambda_1}}
\times\mathbb{Z}_{p^{\lambda_2}}\times\cdots\times\mathbb{Z}_{p^{\lambda_k}}$
with $\lambda_1\geq\lambda_2\geq \ldots \geq\lambda_k\geq 1$. Then
\begin{equation*}
|\cale_p(G_{\lambda})|= \dd\sum_{r=1}^k \binom{k}{r}_{\hspace{-1mm}p}.
\end{equation*}
\end{prop}
We give a direct proof of this formula based on Theorem \ref{Th_subgroups_type}.
\begin{proof} We use Theorem \ref{Th_subgroups_type} in the case $\lambda=(1,1,\ldots,1,0,\ldots)$,
where the number of $1$'s is $k$, and $\mu=(1,1,\ldots,1,0,\ldots)$,
where the number of $1$'s is $r$ with $1\leq r\leq k$. Here
$\lambda'=(k,0,0,\ldots)$, $\mu'=(r,0,0,\ldots)$ and obtain that the
number of subgroups of type $\mu$ is $\binom{k}{r}_p$, while the
number of subgroups of exponent $1$ is exactly $\sum_{r=1}^k
\binom{k}{r}_p$.
\end{proof}
\begin{exm} We have for any $\lambda_1\geq\lambda_2\geq \lambda_3\geq \lambda_4\geq
1$,
\begin{equation*}
|\cale_p(\mathbb{Z}_{p^{\lambda_1}}\times \mathbb{Z}_{p^{\lambda_2}}
\times\mathbb{Z}_{p^{\lambda_3}} \times \mathbb{Z}_{p^{\lambda_4}})|
= p^4+3p^3+4p^2+ 3p+4.
\end{equation*}
\end{exm}
Concerning the number of subgroups of exponent $p^2$ in
$G_{\lambda}$ we have the next formula.
\begin{prop} \label{Prop_exp_p^2} Let $G_{\lambda}=\mathbb{Z}_{p^{\lambda_1}}\times\mathbb{Z}_{p^{\lambda_2}}
\times\cdots\times\mathbb{Z}_{p^{\lambda_k}}$ with
$\lambda_1\geq\lambda_2\geq \ldots \geq
\lambda_t>\lambda_{t+1}=\ldots = \lambda_k=1$, where $0\leq t\leq k$
is fixed {\rm (}$t=0$ if each $\lambda_j$ is $1$ and $t=k$ if each
$\lambda_j$ is $\geq 2${\rm )}. Then
\begin{equation*}
|\cale_{p^2}(G_{\lambda})|= \dd \sum_{\substack{1\leq r\leq t\\
0\leq s\leq k-r}} p^{r(k-r-s)} \binom{k-r}{s}_{\hspace{-1mm}p}
\binom{t}{r}_{\hspace{-1mm}p},
\end{equation*}
which is zero {\rm (}empty sum{\rm )} for $t=0$.
\end{prop}
\begin{proof} Let $t\geq 1$. Use Theorem \ref{Th_subgroups_type} for
$$
\lambda=(2,2,\ldots,2,1,1,\ldots,1,0,\ldots),
$$
where the number of
$2$'s is $t$ and the number of $1$'s is $k-t$ and
$$
\mu=(2,2,\ldots,2,1,1,\ldots,1,0,\ldots),
$$
where the number of
$2$'s is $r$ and the number of $1$'s is $s$ with $1\leq r\leq t$,
$0\leq s\leq k-r$. Now $\lambda'=(k,t,0,\ldots)$,
$\mu'=(r+s,r,0,\ldots)$ and obtain that the number of subgroups of
type $\mu$ is
\begin{equation*}
p^{r(k-r-s)} \binom{k-r}{s}_{\hspace{-1mm}p}
\binom{t}{r}_{\hspace{-1mm}p}
\end{equation*}
and the number of subgroups of exponent $p^2$ is deduced by summing
over $r$ and $s$.
\end{proof}
\begin{exm} {\rm ($k=4$, $t=2$)} We have
\begin{equation*}
|\cale_{p^2}(\mathbb{Z}_{p^4} \times \mathbb{Z}_{p^2} \times
\mathbb{Z}_{p} \times \mathbb{Z}_{p})| = p^5+5p^4+6p^3+4p^2+2p+2.
\end{equation*}
\end{exm}
In what follows let
$G_{\lambda}=\mathbb{Z}_{p^{\lambda_1}}\times\mathbb{Z}_{p^{\lambda_2}}
\times\cdots\times\mathbb{Z}_{p^{\lambda_k}}$ and $G_{\kappa}=
\mathbb{Z}_{p^{\kappa_1}}\times\mathbb{Z}_{p^{\kappa_2}}\times
\cdots\times\mathbb{Z}_{p^{\kappa_{\ell}}}$ be two finite abelian
$p$-groups, where $\lambda_1\geq\lambda_2\geq \ldots
\geq\lambda_k\geq 1$ and $\kappa_1\geq\kappa_2\geq \ldots
\geq\kappa_{\ell}\geq 1$. Assume that $G_{\lambda}$ and $G_{\kappa}$
have the same number of subgroups of exponent $p^i$ for every $i$,
i.e.
\begin{equation*}
|\cale_{p^i}(G_{\lambda})|=|\cale_{p^i}(G_{\kappa})| \quad (i\geq 0).
\end{equation*}
Then $\lambda_1=\kappa_1$. On the other hand, since the function
\begin{equation*}
f:\N^* \to \N^*, \quad f(k)=\sum_{i=1}^k \binom{k}{i}_p
\end{equation*}
is one--to--one, by Proposition \ref{Prop_elem} we infer that
$k=\ell$. Clearly, if $k=1$ one obtains $G_{\lambda}\cong
G_{\kappa}$. The same thing can be also said for $k=2$ and $k=3$ by
Propositions \ref{Prop_exp_p} and \ref{Prop_exp_p_rank_3}. Inspired
by these remarks, we state and prove the following result.
\begin{prop} \label{Prop_Isom} Two finite abelian $p$-groups $G_{\lambda}$ and $G_{\kappa}$ are isomorphic
if and only if they have the same number of subgroups of exponent
$p^i$, for every $i\geq 0$.
\end{prop}
\begin{proof} Let $\lambda = ( \lambda_1, \lambda_2,\ldots,\lambda_k,0,\ldots)$ and $\kappa=(\kappa_1,\kappa_2,\ldots,\kappa_{\ell},0,\ldots)$,
where $\lambda_k,\kappa_{\ell}>0$, be partitions such that for
$p$-groups $G_{\lambda}$ and $G_{\kappa}$ one has $|\cale_{p^i}(G_{\lambda})| = |\cale_{p^i}(G_{\kappa})|$, for every $i\geq 0$.
As noted before, $\lambda_1=\kappa_1$ and $k=\ell$ hold true.
Let us define $\lambda_{k+1} = \kappa_{k+1} = 0$. We prove, by reverse induction on $t\le k+ 1$ that
$\lambda_t = \kappa_t$. So, let us assume that $\lambda_i = \kappa_i$ for all $k + 1\geq i\geq t+1$, and prove that $\lambda_t = \kappa_t$.
Suppose to the contrary that w.l.o.g. $\lambda_t > \kappa_t$. Since $\lambda_1 = \kappa_1$, one has $t>1$. Consider the partitions
$\lambda^{\lambda_t}$ and $\kappa^{\lambda_t}$ defined with
\begin{align*}
\lambda^{\lambda_t} & = (\underbrace{\lambda_t,\ldots,\lambda_t}_t, \gamma_{t+1},\ldots,\gamma_k,0,\ldots), \\
\kappa^{\lambda_t} & = (\underbrace{\lambda_t,\ldots,\lambda_t}_s, \kappa_{s+1},\ldots,\kappa_t, \gamma_{t+1},\ldots,\gamma_k,0,\ldots),
\end{align*}
where $\gamma_i = \lambda_i = \kappa_i$, for $i\geq t+1$, and $s = \max \{j: \kappa_j \geq \lambda_t\}$. Note that by our assumption $s < t$, and
since $\kappa_1=\lambda_1\geq \lambda_t$ also $s\geq 1$.
From the definition of numbers $\alpha_{\omega}(\mu;p)$, it is clear that for any three partitions $\mu,\sigma,\tau$, which satisfy
$\mu \preceq\sigma \preceq \tau$, it holds
\begin{equation*}
\alpha_{\sigma}(\mu;p) \leq \alpha_{\tau}(\mu;p).
\end{equation*}
Using this remark, the fact that $\kappa^{\lambda_t} \precneqq \lambda^{\lambda_t}$ and $\alpha_{\lambda^{\lambda_t}}(\lambda^{\lambda_t};p)=1$,
one has
\begin{gather*}
1+ |\cale_{p^{\lambda_t}}(G_{\kappa})|= 1+ |\cale_{p^{\lambda_t}}(G_{\kappa^{\lambda_t}})|
= 1+ \sum_{\mu \preceq \kappa^{\lambda_t},\ \mu_1=\lambda_t} \alpha_{\kappa^{\lambda_t}}(\mu;p) \\
\leq \alpha_{\lambda^{\lambda_t}}(\lambda^{\lambda_t};p) + \sum_{\mu \preceq \kappa^{\lambda_t},\ \mu_1=\lambda_t}
\alpha_{\lambda^{\lambda_t}}(\mu;p)
\leq \sum_{\mu \preceq \lambda^{\lambda_t},\ \mu_1=\lambda_t} \alpha_{\lambda^{\lambda_t}}(\mu;p)\\
= |\cale_{p^{\lambda_t}}(G_{\lambda^{\lambda_t}})|= |\cale_{p^{\lambda_t}}(G_{\lambda})|,
\end{gather*}
a contradiction.
\end{proof}
\begin{cor} Two arbitrary finite abelian groups are isomorphic if and only if they have the same
number of subgroups of any exponent.
\end{cor}
Finally, we note that another interesting problem is to find the polynomial
$|\cale_{p^i}(G_{\lambda})|$ in the case $k=4$.
\section{Second approach}
We need the next result giving the representation of subgroups of
the group $\Z_m\times \Z_n$. For every $m,n\in \N^*$ let
\begin{gather*}
J_{m,n}:=\left\{(a,b,c,d,\ell)\in (\N^*)^5: a\mid m, b\mid a, c\mid
n, d\mid c, \frac{a}{b}=\frac{c}{d}, \right. \\ \left. \ell \le \frac{a}{b}, \,
\gcd\left(\ell,\frac{a}{b} \right)=1\right\}.
\end{gather*}
Note that here $\gcd(b,d)\cdot \lcm(a,c)=ad$ and $\gcd(b,d)\mid \lcm(a,c)$.
For $(a,b,c,d,\ell)\in J_{m,n}$ define
\begin{equation*}
K_{a,b,c,d,\ell}:= \left\{\left(i\frac{m}{a}, i\ell \frac{n}{c}+j\frac{n}{d}\right): 0\le i\le a-1, 0\le
j\le d-1\right\}.
\end{equation*}
\begin{thm} \label{Th_repr} {\rm (\cite[Th.\ 3.1]{Tot2014})} Let $m,n\in \N^*$.
i) The map $(a,b,c,d,\ell)\mapsto K_{a,b,c,d,\ell}$ is a bijection
between the set $J_{m,n}$ and the set of subgroups of $(\Z_m \times
\Z_n,+)$.
ii) The invariant factor decomposition of the subgroup
$K_{a,b,c,d,\ell}$ is
\begin{equation*}
K_{a,b,c,d,\ell} \simeq \Z_{\gcd(b,d)} \times \Z_{\lcm(a,c)}.
\end{equation*}
iii) The order of the subgroup $K_{a,b,c,d,\ell}$ is $ad$ and its exponent is $\lcm(a,c)$.
\end{thm}
Let $s_E(m,n)$ stand for the number of subgroups of exponent $E$ of the group $\Z_m\times \Z_n$.
\begin{prop} \label{Prop_m_n} For every $m,n\in \N^*$, $E\mid \lcm(m,n)$ we have
\begin{align}
s_E(m,n) & = \sum_{\substack{i\mid m, j\mid n\\ \lcm(i,j)=E}} \gcd(i,j) \label{1_1} \\
& = \frac1{E} \sum_{\substack{i\mid m, j\mid n\\ \lcm(i,j)=E}} ij \label{1_2}.
\end{align}
\end{prop}
\begin{proof} According to Theorem \ref{Th_repr},
\begin{align*}
s_E(m,n)= \sum_{\substack{a\mid m\\ b\mid a}} \sum_{\substack{c\mid n\\ d\mid c}} \sum_{\substack{a/b=c/d=e \\ \lcm(a,c)=E}} \phi(e),
\end{align*}
where $\phi$ is Euler's totient function. This can be written (with $m=ax$, $a=by$, $n=cz$, $c=dt$) as
\begin{gather*}
s_E(m,n) = \sum_{\substack{bxe=m \\ dze=n \\ e \lcm(b,d)=E}} \phi(e)
= \sum_{\substack{ix=m \\ jz=n}} \sum_{\substack{be=i\\ de=j\\ e \lcm(b,d)=E}} \phi(e) \\
= \sum_{\substack{i\mid m \\ j\mid n \\ \lcm(i,j)=E}} \sum_{e\mid \gcd(i,j)} \phi(e)
= \sum_{\substack{i\mid m \\ j\mid n \\ \lcm(i,j)=E}} \gcd(i,j),
\end{gather*}
which is \eqref{1_1}. Formula \eqref{1_2} is its immediate consequence.
\end{proof}
\begin{rem} {\rm Proposition \ref{Prop_exp_p} is a direct consequence of the above result. The total number $s(m,n)$ of subgroups of the group $\Z_m\times \Z_n$ is (see \cite[Th.\ 3]{HHTW2014}, \cite[Th.\ 4.1]{Tot2014})
\begin{align} \label{s_m_n}
s(m,n) = \sum_{i\mid m, j\mid n} \gcd(i,j)
\end{align}
and \eqref{1_1} shows the distribution of the number of subgroups according to their exponents. Formula \eqref{s_p} can be obtained also
by using \eqref{s_m_n}.}
\end{rem}
\begin{exm} The total number of subgroups of $\Z_{12}\times \Z_{18}$ is $s(12,18)=80$ and we have $s_1(12,18)=1$, $s_2(12,18)=4$,
$s_3(12,18)=5$, $s_4(12,18)=3$, $s_6(12,18)=20$, $s_9(12,18)=4$, $s_{12}(12,18)=15$, $s_{18}(12,18)=16$, $s_{36}(12,18)=12$.
\end{exm}
\begin{cor} \label{Th_n_n} {\rm ($m=n$)} For every $n\in \N^*$ and $E\mid n$,
\begin{align} \label{1_3}
s_E(n,n) & = \sum_{\substack{i\mid E, j\mid E \\ \gcd(E/i,E/j)=1}} \gcd(i,j),
\end{align}
which equals the number of cyclic subgroups of the group $\Z_E\times \Z_E$.
\end{cor}
The fact that $s_E(n,n)$ equals the number of cyclic subgroups of the group $\Z_E\times \Z_E$, but without
deriving formula \eqref{1_3} is \cite[Th.\ 8]{HHTW2014}, proved by different arguments.
\begin{proof} In the case $m=n$, for every $E\mid n$ we have by \eqref{1_1},
\begin{gather*}
s_E(n,n) = \sum_{\substack{i\mid n, j\mid n \\ \lcm(i,j)=E}} \gcd(i,j)
= \sum_{\substack{ia=E, jb=E \\ \lcm(E/a,E/b)=E}} \gcd(i,j) \\
= \sum_{\substack{ia=E, jb=E \\ \gcd(a,b)=1}} \gcd(i,j),
\end{gather*}
giving \eqref{1_3}, which equals the number of cyclic subgroups of the group $\Z_E\times \Z_E$ by
\cite[eq.\ (16)]{HHTW2014}.
\end{proof}
\begin{prop} \label{Prop_sum_exp} For every $m,n\in \N^*$ the sum of exponents of the subgroups of $\Z_m\times \Z_n$
is $\sigma(m)\sigma(n)$, where $\sigma(k)=\sum_{d\mid k} d$.
\end{prop}
\begin{proof} By \eqref{1_2} the sum of exponents of the subgroups of $\Z_m\times \Z_n$ is
\begin{gather*}
\sum_{E\mid \lcm(m,n)} E s_E(m,n) = \sum_{E\mid \lcm(m,n)} E \cdot \frac1{E} \sum_{\substack{i\mid m, j\mid n\\ \lcm(i,j)=E}} ij
= \sum_{\substack{i\mid m, j\mid n\\ \lcm(i,j)\mid \lcm(m,n)}} ij \\
= \sum_{i\mid m} i \sum_{j\mid n} j = \sigma(m)\sigma(n).
\end{gather*}
\end{proof}
By Proposition \ref{Prop_sum_exp} the arithmetic mean of exponents of the subgroups of $\Z_m\times \Z_n$ is
$\sigma(m)\sigma(n)/s(m,n)$, where $s(m,n)$ is given by \eqref{s_m_n}. Now consider the case $m=n$.
Let $A(n)$ stand for the arithmetic mean of exponents of the subgroups of $\Z_n\times \Z_n$. We have
\begin{align} \label{A}
A(n)= \frac{\sigma(n)^2}{s(n)},
\end{align}
where $s(n):=s(n,n)$.
Recall that a function $f:\N^* \to \C$ is said to be multiplicative if $f(nn')=f(n)f(n')$ whenever $\gcd(n,n')=1$. It
is well known that the sum-of-divisors function $\sigma$ is multiplicative. The function $s(n)=\sum_{i,j\mid n} \gcd(i,j)$ is
also multiplicative, as shown by the following direct proof: Let $\gcd(n,n')=1$. Then
\begin{gather*}
s(nn') = \sum_{i,j\mid nn'} \gcd(i,j) = \sum_{\substack{a,b\mid n\\ a',b'\mid n'}} \gcd(aa',bb') \\
= \sum_{a,b\mid n} \gcd(a,b) \sum_{a',b'\mid n'} \gcd(a',b') =s(n)s(n').
\end{gather*}
We conclude that the function $A$ given by \eqref{A} is
multiplicative and and for every prime power $p^{\nu}$,
\begin{align*}
A(p^{\nu})= \frac{(p^{\nu+1}-1)^2}{p^{\nu+2}+p^{\nu+1}-(2\nu+3)p+2\nu+1},
\end{align*}
cf. Corollary \ref{Cor_total_nr_subgroups_2}.
Since the exponent of every subgroup of $\Z_n\times \Z_n$ is a
divisor of $n$, whence $\leq n$, we deduce that $A(n)\leq n$ ($n\in
\N^*$). For the function $n\mapsto f(n):=A(n)/n \in (0,1]$ the
series taken over the primes
\begin{equation*}
\sum_p \frac{1-f(p)}{p}= \sum_p \frac{p-1}{p^2(p+3)}
\end{equation*}
is convergent, and it follows from a theorem of H.~Delange (see, e.g., \cite{Pos1988})
that the function $f$ has a non-zero mean value $M$ given by
\begin{align*}
M:= & \lim_{x\to \infty} \frac1{x} \sum_{n\le x} f(n) \\
= & \prod_p \left(1-\frac1{p}\right) \sum_{\nu=0}^{\infty} \frac{f(p^\nu)}{p^\nu}\\
= & \prod_p \left(1-\frac1{p}\right) \sum_{\nu=0}^{\infty} \frac{(p^{\nu+1}-1)^2}{p^{2\nu}(p^{\nu+2}+p^{\nu+1}-(2\nu+3)p+2\nu+1)}.
\end{align*}
the products being over the primes.
We prove the following more exact result.
\begin{prop} We have
\begin{align} \label{asympt}
\sum_{n\le x} A(n) = \frac{M}{2} x^2 + O\left(x \log^3 x \right).
\end{align}
\end{prop}
\begin{proof} Let $f(n)=\sum_{d\mid n} g(d)$ ($n\in \N^*$), that is $g=\mu*f$ in terms of the Dirichlet convolution, where $\mu$ is the
M\"{o}bius function. Here $g(p^\nu)=f(p^\nu)-f(p^{\nu-1})$ for every prime power $p^{\nu}$ ($\nu \in
\N^*$). Note that
\begin{equation*}
A(p^k)= \frac{(\sum_{i=0}^k p^i)^2}{\sum_{i=0}^k (2i+1)p^{k-i}}
\quad (k\geq 0),
\end{equation*}
so if we denote $S=\sum_{i=0}^{\nu-1} p^i$ and $T=\sum_{i=0}^{\nu-1}
(2i+1) p^{\nu-1-i}$, we have
\begin{equation*}
g(p^{\nu})= \frac{(Sp+1)^2}{p^{\nu}(Tp+2\nu +1)}-
\frac{S^2}{p^{\nu-1}T}=
\frac{2STp+T-pS^2(2\nu+1)}{p^{\nu}T(Tp+2\nu+1)}.
\end{equation*}
Since $S\leq T$ and $T\leq (2\nu-1)S$, we have
\begin{align*}
|g(p^{\nu})| & < \frac{\max \{2STp+T-pS^2(2\nu-1), pS^2(2\nu+1)-2STp\}}{p^{\nu}T(Tp+2\nu+1)} \\
& \leq \frac1{p^{\nu}} \max \left\{\frac{STp+T}{T(Tp+1)},\frac{pS^2(2\nu-1)}{T^2p} \right\} \\
& \leq \frac1{p^{\nu}} \max \{1,2\nu-1 \} = \frac{2\nu-1}{p^{\nu}},
\end{align*}
valid for every prime power $p^\nu$ ($\nu \in
\N^*$). Hence, $|g(n)|\le \tau(n^2)/n$ for every $n\geq 1$, where
$\tau(k)$ stands for the number of positive divisors of $k$.
We deduce that
\begin{equation*}
\sum_{n\le x} f(n)= \sum_{de\le x} g(d)= \sum_{d\le x} g(d) \sum_{e\le x/d} 1= \sum_{d\le x} g(d) \left( x/d +O(1)\right)
\end{equation*}
\begin{equation*}
=x \sum_{d=1}^{\infty} \frac{g(d)}{d} + O\left(x\sum_{d>x} \frac{|g(d)|}{d}\right) + O \left(\sum_{d\le x} |g(d)|\right)
\end{equation*}
\begin{equation*}
=M x + O\left(x\sum_{d>x} \frac{\tau(d^2)}{d^2}\right) + O\left(\sum_{d\le x} \frac{\tau(d^2)}{d}\right),
\end{equation*}
where in the main term the coefficient of $x$ is $M$ by Euler's product formula. It is known that $\sum_{n\le x}
\tau(n^2)= cx\log^2 x+O(x\log x)$ with a certain constant $c$ and partial summation shows that
$\sum_{n>x} \tau(n^2)/n^2= O((\log^2 x)/x)$, $\sum_{n\le x} \tau(n^2)/n= O(\log^3 x)$.
Therefore,
\begin{align} \label{f}
\sum_{n\le x} f(n) = M x + O \left(\log^3 x \right).
\end{align}
Now \eqref{asympt} follows from \eqref{f} by partial summation.
\end{proof}
Formula \eqref{asympt} and its proof are similar to those of \cite[Th.\ 3.1.3]{TT2015}.
\medskip
{\bf Acknowledgement:} The authors thank the referee for very careful reading of the manuscript, many useful comments and for the proof of
Proposition \ref{Prop_Isom}.
|
3,212,635,537,483 | arxiv | \section{Introduction}
The realistic quantum systems\cite{Divin98,Nielsen00} are usually
open to environments which result in the
uncontrollable decoherence and dissipation\cite{Breuer,Zoller}.
Therefore, it is of great importance to study the impacts of
environments on the feasible quantum information processing such as
quantum teleportation \cite{Bennett,Lee00,Bowen,Albeverio,Yeo06}.
Quantum teleportation is the protocol which enables the recreation
of an arbitrary unknown state at a remote place via the local
measurements and necessary classical communications. The key to the
standard teleportation is entangled states acting as the quantum
channel. Recently, some schemes via the noisy channel of the mixed states have been extensively studied in
\cite{Yeo02,Yeo03,Yeo031,Yeo05,Guo06,Zhang07,Zhu05,Oh02,Jung08,Bhaktavatsala}.
These investigations reveal an interesting point that the
entanglement of the channel mainly influences the quality of the
teleportation. The effects of the environments have
also been taken into account. The previous results of the above
works manifest that the type of noises acting on the quantum
channel can determine the efficiency of the teleportation. If the
fidelity is smaller than the critical value of $2/3$, the
entanglement of the channel will be decreased to zero. These
noises are considered in the Markovian limit with the assumption of
an infinitely short correlation time of the environment. However,
the conventionally employed Markovian approximation has faced
more and more challenges because of the rapid advance of
experimental techniques\cite{Breuer}. Recent studies have shown that
non-Markovian quantum processes play an increasingly important role
in many fields of physics \cite{Piilo08,
Piilo09,Yi,Fanchini,Bellomo,Li,Chen}. The apparent feature of the
non-Markovian dynamics is the memory effects from the practical
environment with a certain correlation time. To quantitatively
characterize the memory effect, some measures for the degrees of the
non-Markovianity have been introduced by \cite{Wolf,Piilo091,Xu10}. These
reasons motivate us to investigate how non-Markovian
environments influence a protocol of quantum teleportation.
In this paper, we consider a general mixed two-qubit states as the
noisy channel coupled to two local non-Markovian
reservoirs. The non-Markovian dynamics of the average fidelity of
the teleportaiton is derived. For the long time limit, we use the
minimal fidelity to quantify the success of the teleportation
protocol. Finally, a simple conclusion is given.
\section{The formalism}
In the following discussion, we investigate the quantum teleportation
in the noisy channel which is composed of the two noninteracting
parts. For each part, a qubit $S=A,B$ is locally coupled to a
reservoir $R_{S}=R_{A}, R_{B}$ with the memory effects. The
Hamiltonian of the quantum channel is described by
\begin{equation}
\label{eq:1} H^{c}=\sum_{S=A,B}H_{S}^{0}+H_{R_S}^0+H_{S, R_S}^{I}.
\end{equation}
The inherent Hamiltonian of each qubit $S$ is formed by
$H_{S}^{0}=\omega_0\sigma_{+}^{S}\sigma_{-}^{S}$ where $\omega_0$ is
the transition frequency of the effective two-level system with the
excited state $|e\rangle_{S}$ and ground state $|g\rangle_{S}$.
$\sigma_{\pm}^{S}$ denotes the raising and lowering operators respectively. The
local reservoir is given by the
$H_{R_S}^{0}=\sum_{j}\omega_{j}b_{R_S,j}^{+}b_{R_S,j}$ where the
index $j$ labels the field mode of the reservoir with the
corresponding frequency $\omega_{j}$. $b_{R_S,j}^{+},b_{R_S,j}$ are
the creation and annihilation operators of the mode for the reservoir $R_S$. The
qubit-reservoir interaction Hamiltonian is also obtained by $H_{S,
R_S}^{I}=\sigma_{+}^{S}B_{R_S}+\sigma_{-}^{S}B_{R_S}^{+}$ where
$B_{R_S}=\sum_{j}g_{R_S,j}b_{R_S,j}$ and $g_{R_S,j}$ represents the
coupling between the qubit and the reservoir.
We suppose that the qubits of the channel are initially prepared in
a general mixed states $\rho^{c}(0)$ which can be expressed in the
standard product basis $\{ |1\rangle=|ee\rangle,
|2\rangle=|eg\rangle, |3\rangle=|ge\rangle, |4\rangle=|gg\rangle\}$.
The elements of the density matrix satisfy that $\sum_{n=1}^{4}
\rho_{nn}^{c}(0)=1$ and
$\rho_{mn}^{c}(0)=[\rho_{nm}^{c}(0)]^{\ast}$. If the dynamical map
$\varepsilon_{S}$ for each qubit is known, the evolution of the
states for the quantum channel can be obtained by
\begin{equation}
\label{eq:2}\rho^{c}(t)=\varepsilon_{A}\otimes
\varepsilon_{B}[\rho^{c}(0)].
\end{equation}
In accordance with the result of \cite{Breuer}, the non-Markovian
dynamics $\varepsilon_{S}$ can be written as
\begin{align}
\label{eq:3}\varepsilon_{S}(|e\rangle\langle
e|)&=|G_{S}(t)|^2|e\rangle\langle
e|+[1-|G_{S}(t)|^2]|g\rangle\langle g|& \\ \nonumber
\varepsilon_{S}(|e\rangle\langle g|)&=G_{S}(t)|e\rangle\langle g|&\\
\nonumber \varepsilon_{S}(|g\rangle\langle e|)&=G_{S}^{\ast}(t)|g\rangle\langle e|&\\
\nonumber \varepsilon_{S}(|g\rangle\langle g|)&=|g\rangle\langle
g|,&
\end{align}
where each local reservoir is initially in the vacuum state and the
function $G_{S}(t)=G(t)$ is defined as the solution of the
integrodifferential equation
\begin{equation}
\label{eq:4} \dot{G}(t)=-\int_{0}^{t}f(t-\tau)G(\tau)d\tau.
\end{equation}
The two-point reservoir correlation function $f(t-\tau)$ is closely
related to the spectral density of the reservoir $J(\omega)$,
$f(t-\tau)=\int J(\omega)\exp [i(\omega_0-\omega)(t-\tau)]d\omega$.
For example, we take the detuning case of the Lorentzian spectral
density which describes the vacuum radiation field inside an
imperfect cavity \cite{Breuer},
\begin{equation}
\label{eq:5}J(\omega)=\frac {1}{\pi} \frac {W^2
\lambda}{(\omega_0-\omega-\delta)^2+\lambda^{2}},
\end{equation}
where $\delta=\omega_0-\omega_{c}$ is the detuning between the
center frequency of the cavity $\omega_{c}$ and the resonance
frequency $\omega_0$. $W$ represents the interaction strength
between the qubit and its local reservoir. The parameter $\lambda$
defines the spectral width of the reservoir. It is found out that the effective coupling between the qubit and local reservoir is dependent on the detuning \cite{Breuer}. The large values of $\delta$ represent the weak effective couplings \cite{Breuer}. This also means that the effects of the reservoirs on the protocol are small when the detuning $\delta$ is increased. We can
verify that the correlation function decays exponentially
$f(t-\tau)=W^2\exp[-(\lambda-i\delta)(t-\tau)]$. The correlation
time scale of the reservoir is approximately estimated by
$\tau_{R}\sim \lambda^{-1}$. As another necessary time parameter,
the relaxation time for each qubit is also obtained by $\tau_{S}\sim
\gamma_{0}^{-1}$ where $\gamma_0=2W^2/\lambda$ is regarded as the
decaying rate of the qubit in the Markovian limit of flat spectrum.
Applying the correlation function $f(t-\tau)$, we can obtain the
expression of $G(t)$,
\begin{equation}
\label{eq:6}G(t)=\exp[-\frac 12(\lambda-i\delta)t](\cosh\frac
{dt}2+\frac {\lambda-i\delta}{d}\sinh\frac {dt}2),
\end{equation}
with $d=\sqrt{(\lambda-i\delta)^2-2\gamma_0\lambda}$. If the time
scale $\tau_{R}$ is comparable with the relaxation time scale
$\tau_{S}$, i.e., $\tau_{R}>\tau_{S}/2$, the memory effects of the
reservoir should be taken into account. When $\lambda \ll \gamma_0$ and $\delta=0$, the function is simplified as $G(t)=e^{-\frac {\lambda t}{2}}(\cos \frac{|d|t}{2}+\frac {\lambda}{d}\sin \frac {|d|t}{2})$. It is clearly seen that the revivals of $G(t)$ occur in this case. While $\tau_{R}\ll \tau_{S}$ or $\lambda \gg \gamma_0$, the dynamical map reduces
to the Markovian decoherence where $|G(t)|=e^{-\frac {\gamma_{0}t}{2}}$ is monotonically
decreasing with time.
\section{Teleportation in the environments}
Without the loss of the generality, an arbitrary unknown single
qubit state $|\varphi_{in}\rangle=\cos\frac
{\theta}2|e\rangle+\sin\frac {\theta}2e^{i\phi}|g\rangle$ is
recognized as one input state for the standard teleportation. The
parameter $\theta\in[0,\pi]$ and $\phi\in[0,2\pi]$ represent the
polar and azimuthal angles respectively. The protocol uses a shared
quantum state as a quantum channel to transfer a third quantum state
between two distant parties. For a stationary quantum channel,
quantum teleportation can be performed when the parties $A$ and $B$
can determine which Bell state has the largest overlap with the
quantum channel. When the quantum channel varies with time, the
optimal Bell state will possibly change after the time $T$. In that
time, the quantum channel has evolved to $\rho_c(T)$. If the parties
$A$ and $B$ have complete knowledge of the evolution process, they
will be able to correctly select the optimal Bell state. According
to the work of \cite{Yeo05}, the definite expression of the output
state $\rho_{out}^{(m)}(T)$ can be written as
\begin{equation}
\label{eq:7}
\rho_{out}^{(m)}(T)=\sum_{j=0}^{3}p_{j}^{(m)}(T)\sigma_{j}\rho_{in}\sigma_{j},
\end{equation}
where the local operations $\sigma_{j}(j=1,2,3)$ are the three
components of the Pauli rotation, $\sigma_0=I$ denotes the unity
matrix and $\rho_{in}=|\varphi_{in}\rangle \langle \varphi_{in}|$.
The probabilities $p_{j}^{(m)}= \langle \Psi^{(j\oplus
m)}|\rho^{c}(t) |\Psi^{(j\oplus
m)}\rangle=\mathrm{Tr}[\rho_{Bell}^{(j\oplus m)}\rho^{c}(t)]$
satisfying that $\sum_{j=0}^{3}p_{j}^{(m)}=1$. Although the form of
Eq. (7) is different from the result of \cite{Bowen,Albeverio}, all
of the expressions discover the fact that the teleportation via the
channel of mixed states is equivalent to a generalized depolarizing
channel with the probabilities obtained by the maximally entangled
components of the channel. It is noted that the values of $m$ are
time dependent. Here $j\oplus m (m=0,1,2,3)$ represents summation
modulus $4$ and $\rho_{Bell}^{(j\oplus m)}=|\Psi^{(j\oplus
m)}\rangle \langle \Psi^{(j\oplus m)}|$ describes the density matrix
of the four Bell entangled states $\{ |\Psi^{(0,3)}\rangle=\frac
1{\sqrt 2}(|ee\rangle \pm |gg\rangle) , |\Psi^{(1,2)}\rangle=\frac
1{\sqrt 2}(|eg\rangle \pm |ge\rangle) \}$. The above equation
describes the output state through the channel with the
maximally entangled fraction $|\Psi^{(m)}\rangle $ after the
implementation of Bell measurements and corresponding pauli
operations.
In regard to the environments, the maximally entangled component of
the noisy channel can be varied with time. Therefore, the best
quality of the teleportation can be obtained by the optimal
estimation of the four possible output states. To efficiently
measure the quality of the protocol, the average fidelity between
the input state and output one can be written as $ F^{(m)}=\frac
1{4\pi}\int_{0}^{2\pi}\int_{0}^{\pi}\langle
\varphi_{in}|\rho_{out}^{(m)}|\varphi_{in}\rangle \sin\theta d\theta
d\phi.$ For a given noisy channel, we can optimize the standard
teleportation by choosing certain values of $m$ to maximize the
average fidelity. The achievable maximum of the four average
fidelity can be given by
\begin{equation}
\label{eq:8} F(t)=\frac 13+\frac 23\max_{m}\{ p_0^{(m)}(t)\},
\end{equation}
where the probabilities $p_0^{(m)}(t)=\langle
\Psi^{(m)}|\rho^{c}(t)|\Psi^{(m)}\rangle$ are only connected with
the diagonal and anti-diagonal elements of $\rho^{c}(t)$. The
maximum of $\{ p_0^{(m)}(t),(m=0,1,2,3) \}$ can be obtained as
\begin{equation}
\label{eq:9} \max_{m} \{ p_0^{(m)}(t) \}=\frac 12 \max \{ \mu_1(t),
\mu_2(t) \},
\end{equation}
where
$\mu_1(t)=\rho_{11}^{c}(t)+\rho_{44}^{c}(t)+2|\mathrm{Re}[\rho_{14}^{c}(t)]|$,
$\mu_2(t)=1-\rho_{44}^{c}(t)+2|\mathrm{Re}[\rho_{23}^{c}(t)]|$ and
$\mathrm{Re}[a]$ denotes the real part of $a$. In this case, the
diagonal elements of $\rho^{c}(t)$ are
\begin{align}\label{eq:10}
\rho_{11}^{c}(t)&=\rho_{11}^{c}(0)|G(t)|^4& \\ \nonumber
\rho_{22}^{c}(t)&=\rho_{11}^{c}(0)|G(t)|^2[1-|G(t)|^2]+\rho_{22}^{c}(0)|G(t)|^2&
\\ \nonumber \rho_{33}^{c}(t)&=\rho_{11}^{c}(0)|G(t)|^2[1-|G(t)|^2]+\rho_{33}^{c}(0)|G(t)|^2&
\\ \nonumber
\rho_{44}^{c}(t)&=1-\rho_{11}^{c}(t)-\rho_{22}^{c}(t)-\rho_{33}^{c}(t),&
\end{align}
and the anti-diagonal elements are also given by
\begin{align}
\label{eq:11} \rho_{14}^{c}(t)&=\rho_{14}^{c}(0)G^2(t)& \\ \nonumber
\rho_{23}^{c}(t)&=\rho_{23}^{c}(0)|G(t)|^2.&
\end{align}
For simplicity, we can take the maximally entangled states to be the
initial ones for the channel. When $\rho^{c}(0)=|\Psi^{(0,3)}\rangle
\langle \Psi^{(0,3)}|$, the optimized average fidelity is obtained
by
\begin{equation}\label{eq:12}F(t)=\frac 13[2+|G(t)|^4].
\end{equation}
During the decoherence, the probabilities $|\Psi^{(0,3)}\rangle$ of
the channel are always larger than those of $|\Psi^{(1,2)}\rangle$.
Figures 1-4 show that the dynamics of the optimized average fidelity
is dependent on the scaled time $\gamma_0t$ in the Markovian regime
of $\lambda=5\gamma_0$ and non-Markovian regime of
$\lambda=0.01\gamma_0$. In Fig. 1, the values of $F(t)$
are rapidly decreased to the critical value $2/3$ below which the
quality of the teleportation is worse than that of the classical
communication. With the increasing of the detuning $\delta$, the
decaying rate of the optimized average fidelity is smaller than that
of the resonance case $\delta=0$. In Fig. 2, the revivals of the average fidelity occur after a finite period
time of $F(t)=2/3$ when $\delta=0$. The large detuning of the
reservoir can ensure the preservation of high values of the
optimized average fidelity. This is the reason that the effective interactions between the qubits and reservoirs are weak when the detuning values $\delta$ are large. For $\rho^{c}(0)=|\Psi^{(1,2)}\rangle
\langle \Psi^{(1,2)}|$, the optimized average fidelity is expressed
by
\begin{equation}
\label{eq:17}F(t)=\frac 13+\frac 13\max\{1-|G(t)|^2,2|G(t)|^2 \}.
\end{equation}
From Fig. 3, we can clearly see that the values of the optimized
average fidelity are rapidly decreased to a certain value and then
gradually increased to the critical value $2/3$ in the Markovian
case. Meanwhile, the similar revivals of the average fidelity also
occur in Fig. 4 when the effects of the non-Markovian
environment are considered. When the detuning is large, the
high values of average fidelity can remain during the evolution. In this condition, it is noted that if
the probability $|\Psi^{(1,2)}\rangle$ of the channel is larger than
$|\Psi^{(0,3)}\rangle$, the value of average fidelity $F(t)>2/3$.
This means that the purity of the channel state can affect the
quality of the teleportation.
\section{The minimal fidelity for the protocol}
For the long time, the quantum channel is in the $|gg\rangle $
state. The average fidelity in this case is $2/3$, even though the
$|gg\rangle $ state is nearly the worst quantum channel for
teleportation. From this point of view, we may consider the minimal
fidelity as a better measure to quantify the success of the
teleportation protocol. The minimization can be performed over all
possible values of $\theta$ and $\phi$, i.e.,
$f(t)=\min_{\theta,\phi} \{\langle
\varphi_{in}|\rho_{out}^{(m)}(t)|\varphi_{in}\rangle \}$. Figure 5
shows the minimal fidelity of the teleportation protocol when the
initial entangled resource is chosen to be $|\Psi^{(0,3)}\rangle$.
The minimal fidelity for $\rho^{c}(0)=|\Psi^{(0,3)}\rangle \langle
\Psi^{(0,3)}|$ can be obtained by,
\begin{equation}
f(t)=\frac 12+\frac {\mathrm{Re}[G^2(t)]}2.
\end{equation}
It is clearly seen that the revival of the fidelity is mainly
dependent on the memory effect from the non-Markovian environment.
For the time limit $t\gg \frac {1}{\lambda}$, the minimal fidelity
for the initial channel of $|\Psi^{(0,3)}\rangle$ is $\frac 12$.
With increasing the detuning value $\delta$, the minimal fidelity
can be greatly improved at the large time scale. This point
demonstrates the memory effects from the non-Markovian environments
can enhance the success of the teleportation protocol. When the
initial channel state is $|\Psi^{(1,2)}\rangle$, the
minimal fidelity is plotted in Figure 6. The analytical expression
of $f(t)$ in this case is given by,
\begin{equation}
f(t)=|G(t)|^2.
\end{equation}
The revival of the fidelity for $\delta=0$ is clearly shown in
Figure 6(a). If the detuning value is increased, the minimal
fidelity can keep the high values for a long time. The large
detuning value helps for the efficiency of the teleportation. In
this case, the values of the minimal fidelity are decreased and
infinitely close to zero with time. This means that the protocol for
the entanglement channel of $|\Psi^{0,3}\rangle$ is much better in
the non-Markovian environment.
\section{Conclusion}
The dynamics of the average fidelity for the standard teleportation
is studied when the quantum channel is subject to the decoherence
from the non-Markovian reservoirs. For the long time limit, the
minimal fidelity is used to quantify the efficiency of the
teleportation protocol. We also investigate the effects of the
non-Markovianity of the channel on the teleportation.
It is also found out that the memory effects can cause the revivals
of the average fidelity and minimal fidelity. Compared with the
Markovian environment, the scheme of the teleportation in the
non-Markovian environment is much better because of the flowing of
information from the environment back to the communication channel.
\vskip 0.5cm
{\large \bf Acknowledgement}
It is a pleasure to thank Yinsheng Ling, Jianxing Fang for their
many fruitful discussions about the topic. The work was supported by
the Natural Science Foundation of China Grant No. 10904104 and No.
11074184.
|
3,212,635,537,484 | arxiv | \section{\textbf{Introduction}}
\label{Section: Introduction}
The wide penetration and overwhelming acceptance of internet in Nigeria and across the globe has facilitated learning, business and other activities that are internet enabled. The introduction of internet technology has impacted positively on everyday activities in the various sectors of human endeavours. However, this technological advancement has equally resulted to increased cyber threats, vulnerabilities and risks. In Nigeria, for instance, internet has made it possible the perpetration of different forms of cyber-crime on daily basis ranging from fraudulent electronic mails, pornography, identity theft, hacking, cyber harassment, spamming, Automated Teller Machine spoofing, piracy and phishing~\cite{omodunbi2016cybercrimes}. \\
According to~\cite{adesina2017cybercrime}, in the year 2016, the personal information of millions of people were stolen through cyber-crime, which comprises of 40 million people in United States of America (USA), 54 million in Turkey, 20 million in Korea, 16 million in Germany and over 20 million in China.\\
Moreover, these risks associated with this ugly phenomenon are likely to be more in developing countries such as Nigeria. In Nigeria, cyber-attacks are committed by people of different age ranges, both old and young are involved in this act though majority are young ones~\cite{hassan2012cybercrime}. The cyber space is the main channel through which financial fraud is being perpetrated in the Nigerian banking industry~\cite{ibrahimimpact}. This therefore calls for concerted effort to curb the menace of cyber-crime activities on individuals, organisations and governments. \\
One of the ways through which cyber-crime can be mitigated is by improving the cyber-hygiene culture of the internet users. Cyber-hygiene here refers to those cyber-security attitudes and behaviours which internet users are expected to adopt to ensure the safety and integrity of their data and also their devices in the case of cyber-attacks by the internet fraudsters~\cite{vishwanath2020cyber}. \\
This research through a pilot study seeks to find out if there is relationship between demographic factors (age and level of education) and cyber-hygiene among students and employees of University of Nigeria, Nsukka. An ethical approval was obtained from the university for the purpose of this research. This paper presents the result of the survey for determining the relationship between age and level of education of students and staff, and cyber-hygiene. The wide adoption of internet in tertiary institutions, coupled with quest for migration to online learning as a result o f the outbreak of Coronavirus-2019 (COVID-19) necessitated this research.\\
The remaining parts of this paper are organised as follows:
Section~\ref{Section: Review} presents the literature review, followed by methodology as Section~\ref{Section: Method}. Section~\ref{Section: Discussion} presents the Result and Discussion of the research analysis. The Conclusion forms the final section of the the paper. \\
\section\textbf{{Review of Related Literature}}
\label{Section: Review}
While there have been several studies involving various users and aspects of cyber-hygiene, there is currently is no substantial survey, which explores cyber-hygiene by considering individual differences and the users’ level of knowledge especially amongst students and employees in the universities in Nigeria. The main reasons for this research are to find out the effect of age and educational level on cyber-hygiene and also to ascertain the level of cyber-hygiene knowledge and behaviour among students and employees of University of Nigeria, Nsukka.\\
Previously many researchers have made findings on cyber-hygiene cultures of internet users. Talib~\textit{et al.}~\cite{talib2010analysis} found that 97\% of users did have antivirus software at home. Authors in~\cite{talib2010analysis} also reported that 72\% of people who are not trained on the topic did use firewall protection. There are also discrepancies in the data that describes the use of Spam protection. \\
The study conducted in~\cite{neigel2020holistic} focused on understanding the human factors and individual differences that influence cyber-hygiene. The results demonstrated in the work indicate that cyber-hygiene education need not target a particular sex or age group in terms of content or delivery method, which contradicts previous findings from~\cite{whitty2015individual}.
The research carried out by authors in~\cite{whitty2015individual} highlighted the importance of understanding the types of people who are more likely to engage in the risky behavior of sharing passwords. They found a number of variables that can be used to predict users that engage in the risky practice of sharing passwords, which are age, perseverance, and self-monitoring.\\
Research conducted in~\cite{barrerainfluence} seeks to examine relationships, if any, between cyber-security awareness level and the background of participants pursuing careers in the area of Information Systems (IS) and/or Information Technology (IT) at the bachelor’s level in three different geographic locations; Germany, United Kingdom, and the United States of America, focusing specifically on four demographic variables: gender, age, education Level Completed and Current Employment Status. They arrived at a conclusion that Awareness is frequently associated to operational situations, where specific reasons require individuals to have an identifiable awareness level for a specific context. Therefore, individuals and business organisations benefit from higher levels of security awareness, which ultimately reflects higher literacy levels and learning. Lastly, business continuity depends on how individuals respond to various situations, exercise caution in their decisions, and ultimately, how aware they are about current and future security risks in their doings.\\
A study to create awareness of security threat and avoidance was carried out in \cite{arachchilage2014security}. The study was carried out on anti-phishing education to guard against identity theft and related issues. It examined whether the knowledge of cyber-hygiene concepts has effect on users’ ability to avoid phishing attack. An online questionnaire were distributed and collected from 161 computer users from Brunel University and the University of Bedfordshire and their responses were used for analysis. The finding revealed that the knowledge of concepts enables internet users to avoid phishing attacks. It was concluded that educating users on the knowledge of concepts has significant positive effects on enabling users to avert phishing threats and attacks.\\
\section{Methodology}
\label{Section: Method}
\subsection{\textbf{Research Goal}}
The primary goal of this research is to determine the significance effect of certain demographic factors/variables such as age and level of education on cyber-hygiene culture among students and employees of University of Nigeria, Nsukka. The reasons for choosing the university are for convenience, cost reduction and availability of participants to researchers. In respect to this research, an ethical approval was sought from the university authority and it was given. Because of prolonged Academic Staff Union of Universities in Nigeria and lockdown necessitated by COVID-19 pandemic, the response rate was very low. Based on this reason, the researchers decided to take a pilot study from the received responses to test the respondent’s understanding of the research questionnaire. An updated version of the survey will be conducted as soon as the university reopens.\\
\subsection{\textbf{Instrument and Method of Data Collection}}
Questionnaire was the basic instrument used to gather required data from the chosen institution. A well-structured questionnaire was designed with Google form for the survey and distributed via online approach using the institution’s online mailing system and group WhatsApps to the chosen participants. Responses from the respondents were collated in a datasheet and were subjected for analysis.\\
\subsection{\textbf{Sampling Technique}}
Convenience sampling, a common non-probability sampling was applied in selecting the sample used for the survey. This sampling method is fast, cost-effective and it makes sample easily available.\\
\subsection{\textbf{Sample Size}}
A total of 145 responses were received and used for this pilot study. The respondents cut across different age ranges and educational levels to accommodate all relevant demographics.\\
\subsection{\textbf{Research Model}}
Here in our study, the analysis of the effect of some selected demographic factors on cyber-hygiene among students and employees of University of Nigeria, Nsukka was carried out. The demographic factors represent the independent variables, which include age and educational level whereas cyber-hygiene represents the dependent variable in this study. The percentage of internet usage for educational purposes increases with the student’s age, which shows the increasing e-learning prospect with the level of education as observed in \cite{tirumala2016survey}. Some common aspects of cyber-hygiene culture were chosen for this study namely, storage and virus attack hygiene, social network hygiene, authentication hygiene and social engineering hygiene. The first three which include virus attack hygiene, social network hygiene and authentication hygiene were adopted from~\cite{vishwanath2020cyber} with little modification in nomenclature. Social engineering hygiene was introduced by the research team as users’ cyber-hygiene dimension required to overcome the cyber-security threat considered in~\cite{fatokun2019impact}. Questions in section B and section C of the questionnaire were spread across these four aspects of cyber-hygiene. The association of the independent variables and the dependent variable is shown in Figure~\ref{fig:my_label} below.\\
\begin{figure} [h]
\centering
\includegraphics[width=8cm,height=4cm]{model2.jpg}
\caption{\textbf{Research Model}}
\label{fig:my_label}
\end{figure}
\subsection{\textbf{Research Questions}}
In order to ascertain if there is any significance relationship between demographic factors and cyber-hygiene culture, the following research questions were raised:
\begin{enumerate}
\item Does age of internet users have any significant relationship with cyber-hygiene culture?
\item Does level of education of internet users have any significant relationship with cyber-hygiene culture?
\end{enumerate}
\subsection{\textbf{Research Hypotheses}}
Two null hypotheses were formulated as follows by the researchers to answer the above research questions:
\begin{itemize}
\item H1: Age of internet user does not have any significant effect on cyber-hygiene
\item H2: Level of education of internet user does not have any significant effect on cyber-hygiene
\end{itemize}
\subsection{\textbf{Data Analysis}}
Our research questionnaire in this study consist of three sections, section A which was used to gather information on demographic factors, type of activities they perform with internet and the type of devices used for these activities. Questions on section B were asked to determine participants’ knowledge on threat and concept. This section has total of 12 questions, which comprises of multiple choice questions and five points Likert scale statements. Options for the five points Likert scale statements ranging from \textbf{Strongly Agree}, \textbf{Agree}, \textbf{Don’t Know}, \textbf{Disagree} and \textbf{Strongly Disagree}. In each of the statement, either Strongly Agree and Agree or Disagree and Strongly Disagree are considered as the correct option. Section C seeks to verify the level of cyber-hygiene culture exhibited by participants or their cyber-hygiene behaviour. On section C, respondents were asked to report the degree to which they engage on certain cyber-hygiene behaviour on a four points Likert scale with option ranging from Every time, Often, Rarely, and Never. In this case, either Every time and Often or Never is considered as the correct option. Questions in section B and C concentrates within the four chosen aspects of cyber-hygiene (\textbf{storage and virus, social network, authentication and social engineering}) used for this study. The respondents were asked to go through the definition of some technical terms such as cyber-hygiene, virus, authentication, social network and social engineering used in the study for clarification, easy understanding and better response. \\
According to~\cite{akman2010gender}, the test of statistical hypotheses is one of the main areas of statistical inference. Version 20 of the Statistical Packages for Social Sciences (SPSS) software was used for the analysis. Chi-square test, an analytical tool embedded in Statistical Packages for Social Sciences (SPSS) software was used to test the significance of these demographic variables. Also, multiple regression technique was subsequently used to determine the direction of the main impacts of the independent variables on the dependent one. Out of these tests, the following inferences made on this paper were drawn.\\
\section{\textbf{Result and Discussion}}
\label{Section: Discussion}
This research survey sought to find out from the participants the type of devices they use for internet, the activities they use internet for, their knowledge of cyber-hygiene concepts and threats, and their cyber-hygiene behaviour. Our findings are shown as tables and discussed in the following sections below:\\
\subsection{\textbf{Devices Used for the Internet}}
As shown in Table~\ref{tab: table1}, it was found that majority of the respondents, 135(93.1\%) indicated mobile phone as the most commonly used device, followed by laptop 103(71\%). The least used is other devices 2(1.4\%), followed by desktop 17(11.7\%) and finally, Tablet with 21(14.5\%). This varies from the result in~\cite{cain2018exploratory}, where the number of laptop users are more than the number of smart phone users. This may not be far from the low cost of mobile phones and its proliferation across the country compared to laptops.\\
\begin {table}[h!]
\begin {center}
\caption{\textbf{Devices used for internet}}
\label{tab: table1}
\begin{tabular}{l|c|r}
\textbf{Variable} & \textbf{Frequency} & \textbf{Percentage}\\
\hline
Laptop & 103 & 71\\
Desktop & 17 & 11.7\\
Mobile phone & 135 & 93.1\\
Tablet & 21 & 14.5\\
Other devices & 2 & 1.4\\
\end{tabular}
\end {center}
\end {table}
\subsection{\textbf{Uses of Internet}}
Table~\ref{tab: table2} shows various purposes for which the participants use internet for. It can be drawn from the table that apart from playing games with only 23.4\% (34, n=145), other purposes in which the participants use the internet for attracts high percentage of users ranging from 50.3 to 94.5. The highest is internet browsing with 94.5\% (137, n=145), followed by learning with 86.9\% (126, n=145). This findings show high level of adoption and usage of internet among students and employees of higher institutions, which calls for survey of their cyber-hygiene knowledge and behaviour.\\
\begin {table}[h!]
\begin {center}
\caption{\textbf{Uses of internet}}
\label{tab: table2}
\begin{tabular}{l|c|r}
\textbf{Variable} & \textbf{Frequency} & \textbf{Percentage}\\
\hline
Browsing & 137 & 94.5\\
Email & 117 & 80.7\\
Downloading music/video & 96 & 66.2\\
Social Networking & 118 & 81.4\\
Playing games & 34 & 23.4\\
Learning & 126 & 86.9\\
Business & 73 & 50.3\\
Banking & 91 & 62.8\\
\end{tabular}
\end {center}
\end {table}
\subsection\textbf{{Demographics of Respondents}}
From the demographic information presented in Table~\ref{tab: table3}, the age of the respondents were coded into five groups, respondents between 15-24years, 25-34year, 35-44years, 45-54years and above 54years. The distributions of respondents within these groups are as follows: 15-24years 40.7\% (59, n=145), 25-34year 25.5\% (37, n=145), 35-44years 20.0\% (29, n=145), 45-54years 11.0\% (16, n=145) and above 54years 2.8\% (4, n=145). As regards their educational level of the respondents, 0.7\% (1) has diploma, 44.1\% (64) are undergraduates, 21.4\% (31) are graduates, 20.0\% (29) are Masters degree holders and 13.8\% (20) have Doctorate degree.\\
\begin {table}[h!]
\begin {center}
\caption{\textbf{Socio-demographics of the respondents}}
\label{tab: table3}
\begin{tabular}{l|c|r}
\textbf{Variable} & \textbf{Frequency} & \textbf{Percentage}\\
\hline
\multirow {2} {*}
Age\\
15-24yrs & 59 & 40.7\\
25-34yrs & 37 & 25.5\\
34-44yrs & 29 & 20.0\\
45-54yrs & 16 & 11.0\\
Above 54yrs & 4 & 2.8\\
\hline
Educational Level\\
Diploma & 1 & 0.7\\
Undergraduate & 64 & 44.1\\
Graduate & 31 & 21.4\\
Masters & 29 & 20.0\\
Doctorate & 20 & 13.8\\
\end{tabular}
\end {center}
\end {table}
\subsection{\textbf{Age and Cyber hygiene}}
In order to find out the effect of the age of the participants on cyber-hygiene, chi-square tests were carried out to check the relationship between age and cyber-hygiene knowledge and also between age and cyber-hygiene behaviour. From the result of the chi-square test to determine the relationship between age and cyber-hygiene as shown in Table~\ref{tab: table4}, the p-value of (0.455), which is greater than 0.05 indicates no statistical significance. This means that there is no relationship between age and cyber-hygiene knowledge.
On the same hand, Table~\ref{tab: table5} presents a p-value of 0.551 for age and cyber-hygiene behaviour, meaning that age of the participants has no relationship with the cyber-hygiene behaviour. The result here conforms to what was obtained in-\cite{cain2018exploratory}.\\
\begin {table}[h!]
\begin {center}
\caption{\textbf{Chi-Square Test for Age and Cyber hygiene knowledge}}
\label{tab: table4}
\begin{tabular}{l|c|c|r}
\textbf{Quantity} & \textbf{Value} & \textbf{df} & \textbf{ Asym. Sig (2-sided)}\\
\hline
Pearson Chi-Square & 3.691 & 4 & 0.455\\
Likelihood Ratio & 3.175 & 4 & 0.270\\
Linear-by-Linear Association & 0.896 & 1 & 0.344\\
N of Valid Cases & 145 \\
\end{tabular}
\end {center}
\end {table}
\begin {table}[h!]
\begin {center}
\caption{\textbf{Chi-Square Test for Age and Cyber hygiene behaviour}}
\label{tab: table5}
\begin{tabular}{l|c|c|r}
\textbf{Quantity} & \textbf{Value} & \textbf{df} & \textbf{ Asym. Sig (2-sided)}\\
\hline
Pearson Chi-Square & 3.043 & 4 & 0.551\\
Likelihood Ratio & 3.093 & 4 & 0.542\\
Linear-by-Linear Association & 0.301 & 1 & 0.583\\
N of Valid Cases & 145 \\
\end{tabular}
\end {center}
\end {table}
\subsection{\textbf{Level of Education and Cyber hygiene}}
Chi-square tests were also conducted in two categories to determine the relationship between level of education as an independent variable and the dependent variable, cyber-hygiene. In the first category, the p-value of the chi-square test for level of education and cyber-hygiene knowledge as shown in Table \ref{tab: table6} is 0.628. The interpretation is that the level of education of the respondent does not have relationship with cyber-hygiene knowledge. Secondly, the chi-square test for level of education and cyber-hygiene behaviour has p-value of 0.285 as shown in Table \ref{tab: table7}. This also means that there is no relationship between the level of education of the respondents and cyber-hygiene behaviour.\\
\begin {table}[h!]
\begin {center}
\caption{\textbf{Chi-Square Test for Level of Education and Cyber-hygiene knowledge}}
\label{tab: table6}
\begin{tabular}{l|c|c|r}
\textbf{Quantity} & \textbf{Value} & \textbf{df} & \textbf{ Asym. Sig (2-sided)}\\
\hline
Pearson Chi-Square & 2.596 & 4 & 0.628\\
Likelihood Ratio & 2.985 & 4 & 0.560\\
Linear-by-Linear Association & 0.655 & 1 & 0.418\\
N of Valid Cases & 145 \\
\end{tabular}
\end {center}
\end {table}
\begin {table}[h!]
\begin {center}
\caption{\textbf{Chi-Square Test for Level of Education and Cyber-hygiene behaviour}}
\label{tab: table7}
\begin{tabular}{l|c|c|r}
\textbf{Quantity} & \textbf{Value} & \textbf{df} & \textbf{ Asym. Sig (2-sided)}\\
\hline
Pearson Chi-Square & 5.012 & 4 & 0.285\\
Likelihood Ratio & 5.488& 4 & 0.241\\
Linear-by-Linear Association & 0.135 & 1 & 0.714\\
N of Valid Cases & 145 \\
\end{tabular}
\end {center}
\end {table}
\subsection{\textbf{Percentage Representation of Good Knowledge and Good Behaviour}}
Tables \ref{tab: table8} and \ref{tab: table9} show the descriptive statistics of good knowledge scores and good behaviour score respectively. This was done to find out if the data is normally distributed, that is, if the median is the same as the mean. There are other tests used for normality but are not included here. Both tables show that data are normally distributed. That is it failed the tests. So, the median was used as the cut off point for the categorisation of the respondents into those with poor and good knowledge, and poor and good behaviour. Those who had a score of 10 and above had good knowledge, while those with lower than 10 score were grouped as having poor knowledge. The median for cyber-hygiene behaviour score is 6, therefore, those who had a score of 6 and above had good behaviour, while those with lower than 6 score were grouped as having poor behaviour.
As shown in Table \ref{tab: table10}, the number of the respondents with good knowledge is 78, representing 53.79\%, while 75 have good behaviour, that is 71.72\%. This result shows that a reasonable number of the respondents, almost half, do neither have good cyber-hygiene knowledge nor good cyber-hygiene behaviour. \\
\begin {table}[h!]
\begin {center}
\caption{\textbf{Descriptive for cyber-hygiene knowledge score}}
\label{tab: table8}
\begin{tabular}{l|c|r}
\textbf{Total cyber-hygiene knowledge score } & \textbf{ Statistic } & \textbf{Std. Error} \\
\hline
Mean & 9.39 & 0.123\\
95\% confidence upper range & 9.15\\
Interval of mean lower range & 9.64\\
5\% Trimmed Mean & 9.49\\
Median & 10.00\\
Variance & 2.20\\ Std. Deviation & 1.48\\
Minimum & 4.00\\
Maximum & 12.00\\
Range & 8.00\\
Interquartile Range & 2.00\\
Skewness & -0 .87 & 0.201\\
Kurtosis & 0.81 & 0.400\\
\end{tabular}
\end {center}
\end {table}
\begin {table}[h!]
\begin {center}
\caption{\textbf{Descriptive for Cyber-hygiene behaviour score}}
\label{tab: table9}
\begin{tabular}{l|c|r}
\textbf{ Total Cyber-hygiene behaviour score } & \textbf{ Statistic } & \textbf{Std. Error} \\
\hline
Mean & 5.84 & 0.248\\
95\% confidence upper range & 5.35\\
Interval of mean lower range & 6.33\\
5\% Trimmed Mean & 5.78\\
Median & 6.00\\
Variance & 8.93\\
Std. Deviation & 2.99\\
Minimum & 0.00\\
Maximum & 13.00\\
Range & 13.00\\
Interquartile Range & 5.00\\
Skewness & 0 .27 & 0.201\\
Kurtosis & -0.78 & 0.400\\
\end{tabular}
\end {center}
\end {table}
\begin {table}[h!]
\begin {center}
\caption{\textbf{Percentage Representation of Good Knowledge and Good Behaviour }}
\label{tab: table10}
\begin{tabular}{l|c|c|r}
\textbf{Descriptive variable} & \textbf{Total Sample} & \textbf{Good Knowledge} & \textbf{Good Behaviour}\\
$\hspace{1cm}$ & $N=145(100)\%$ & $N=75(51.72)\%$ & $N=78(53.79)\%$\\
\hline
\multirow {2} {*}
\underline{Age}\\
15-24yrs & 59(40.7) & 31(39.7) & 27(36.0)\\
25-34yrs & 37(25.5) & 19(24.4) & 21(28.0)\\
34-44yrs & 29(20.0) & 15(19.2) &17(22.7)\\
45-54yrs & 16(11.0) & 9(11.5) & 9(12.0)\\
Above 54yrs & 4(2.8) & 4(5.1) & 1(1.3)\\
\hline
Educational Level\\
Diploma & 1(0.7) & 0(0.0) & 0(0.0) \\
Undergraduate & 64(44.1) & 32(44.1) & 30(40.0)\\
Graduate & 31(21.4) & 19(24.4) & 21(28.0)\\
Masters & 29(20.0) & 15(19.2) & 14(18.7)\\
Doctorate & 20(13.8) & 12(15.4) & 10(13.3)\\
\end{tabular}
\end {center}
\end {table}
\section{\textbf{Conclusion and Recommendation}}
\label{Section: Conclusion}
The primary target of our study was to determine the effect of age and educational level on the cyber-hygiene knowledge of employees and students of University of Nigeria, Nsukka, through an online pilot study. Thus, the detailed findings of the research were presented in the result and discussion section of our paper. From the results we have, it was found that the two variables: age and level of education do not have significant effect on the cyber-hygiene knowledge and behaviour of students and employees of University of Nigeria, Nsukka. It was also discovered that a reasonable proportion of the participants of this survey have poor cyber-hygiene knowledge and behaviour, which suggests the need for proper awareness and training. \\
The result of our findings as regards the effect of age on cyber-hygiene correlates with the result in obtained in \cite{cain2018exploratory}, which also found that age has no statistical significance with cyber-hygiene.
Based on the findings from this research, it can be recommended that well focused training on cyber-hygiene best practices should be organised for both students and employees of the tertiary institutions. Also, the internet security units of tertiary institutions should put in place adequate security infrastructures to safeguard the institution’s information and devices against cyber-attacks.\\
\section{\textbf{Future Work}}
\label{Section: Future Work}
In future study, the effect of other factors such as gender, level of exposure, area of discipline that might impact on the cyber-hygiene will be investigated. One major limitation of this study was poor responses occasioned by closed down of institutions because of Coronavirus disease 2019 (COVID-19) pandemic. \\
\bibliographystyle{IEEEtran}
|
3,212,635,537,485 | arxiv | \section{Introduction}
\label{SEC_INTRO}
In recent years a great deal of attention has been paid to the study of the spin structure of the nucleon using deep inelastic, semi-inclusive hadron production (SIDIS) with two measured, unpolarized final state hadrons~\cite{Bacchetta:2011ip, Bacchetta:2012ty, Pisano:2015wnq}. It was shown in Refs.~\cite{Bianconi:1999cd, Bianconi:1999uc, Radici:2001na,Bacchetta:2003vn} that this process allows one to directly extract the collinear transversity parton distribution functions (PDF) from a specific azimuthal single spin asymmetry (SSA), where it is multiplied by the so-called interference DiFF (IFF) $H^\sphericalangle_1$. At the same time, the two-hadron SIDIS cross section on a longitudinally polarized nucleon also gives access to the helicity PDF, which in this case is convoluted with the helicity dependent DiFF $G_1^\perp$. The latter is interesting both because it has no analogue in the single unpolarized hadron production case and because it is related to the long-predicted quantity of longitudinal jet handedness~\cite{Efremov:1992pe}. Both $H^\sphericalangle_1$ and $G_1^\perp$ are T-odd and can be extracted from the azimuthal modulations in two hadron pair production process from back to back jets in $e^+e^-$ annihilation~\cite{Boer:2003ya}. The measurements from the BELLE experiment~\cite{Vossen:2011fk} yielded significant results for the asymmetry involving IFF $H_1^\sphericalangle$, that were used in Refs.~\cite{Courtoy:2012ry,Radici:2015mwa} to fit parametrizations of $H_1^\sphericalangle$. Recently the BELLE experiment produced preliminary results on SSA sensitive to $G_1^\perp$, showing no signal within the experimental uncertainties~\cite{Abdesselam:2015nxn, Vossen:2015znm}. Similarly, the recent results from the COMPASS experiment~\cite{Sirtl:2017rhi} also showed no significant signal for SIDIS asymmetry that involve helicity dependent DiFF.
The DiFFs have been already studied in the quark-jet model, both for the case of an unpolarized~\cite{Matevosyan:2013aka, Matevosyan:2013nla} and transversely polarized quark~\cite{Matevosyan:2013eia}. In the first studies the simplistic treatment of the polarization transfer during the quark hadronization nevertheless created unphysical modulations that had to be circumvented using additional assumptions. However, more recently the quark-jet hadronization framework has been developed to self-consistently describe the hadronization of a quark~\cite{Bentz:2016rav} with any polarization, where the polarization transfer from the fragmenting to the remnant quark in each hadron emission step has been calculated using the complete set of twist-two transverse momentum dependent (TMD) quark-to-quark splitting functions. Moreover, MC implementation of the polarized quark hadronization based on this framework was developed and implemented in Ref.~\cite{Matevosyan:2016fwi}, and both the unpolarized and the Collins fragmentation functions of pions produced in the hadronization of a light quark were computed. The input elementary quark-to-quark splitting functions (SF) used in that work were derived using Nambu--Jona-Lasinio (NJL) effective theory of quark interactions~\cite{Nambu:1961tp,Nambu:1961fr}.
In this work, we use this MC implementation of the quark-jet model to study the dihadron correlations in the hadronization of a quark with nonzero longitudinal polarization. We aim to calculate the relative size of the helicity dependent DiFF compared to the unpolarized DiFF for pion pairs. Further, we derive integral expressions for both these DiFFs in the case where only two hadrons are emitted by the quark in order to validate our MC results using an alternate method. Finally, these derived integral expressions will elucidate the mechanism for generating the dihadron correlations encoded by $G_1^\perp$. These results are independent of the transverse polarization component of the quark.
This paper is organized in the following way. In the next section we briefly review the formalism for DiFFs. In Sec.~\ref{SEC_MC} we briefly describe the details of the MC simulations and the newly developed method for extracting $G_1^\perp$. We also present the results for the simulations with different numbers of hadrons produced by the fragmenting quark. In Sec.~\ref{SEC_VALIDATION} we validate the MC results against integral expressions for the unpolarized and helicity dependent DiFFs derived for two-step hadronization process. In that section we also discuss the dynamical mechanism for generating helicity dependent two-hadron correlations. We present our conclusions in Sec.~\ref{SEC_CONCLUSIONS}.
\section{Formalism of the DiFFs}
\label{SEC_DIFF_FORM}
In this section we briefly review the kinematics and the formal definitions of the DiFFs, following Refs.~\cite{Bianconi:1999cd, Bianconi:1999uc, Radici:2001na,Bacchetta:2003vn, Boer:2003ya}. The kinematics of dihadron fragmentation is described by the momentum $k$ of the initial quark $q$ with mass $m$ fragmenting into two hadrons $h_1,h_2$ with momenta $P_1, P_2$ and masses $M_1,M_2$. We define the light-cone components of a 4-vector $a$ as $a^\pm = \frac{1}{\sqrt{2}}(a^0 \pm a^3)$. The light-cone momentum fractions of the two hadrons are then defined as $z_i = {P_i^+}/{k^+}$. The total $P$ and the relative $R$ momenta are defined as
\al
{
&
P \equiv P_h = P_1 + P_2,
\\
&
R = \frac{1}{2}( P_1 - P_2),
}
%
and the relevant light-cone momentum fractions are
\al
{
z &= z_1 + z_2,
\\
\xi &= \frac{z_1}{z} = 1- \frac{z_2}{z} \, .
}
The two most common coordinate systems used in describing DiFFs are the so-called $\perp$ and $T$ systems depicted in Figs.~\ref{PLOT_PERP_T_SYS} (a) and (b), respectively. We denote the components of a 3-vector $\vect{a}$, perpendicular to the $\hat{z}$ and $\hat{z}'$ axes in these two systems, as $\vect{a}_\perp$ and $\vect{a}_T$.
\begin{figure}[t]
\centering
\subfigure[] {
\includegraphics[width=0.8\columnwidth]{Plots/DiFF-Perp_System.pdf}
}
\\
\subfigure[] {
\includegraphics[width=0.8\columnwidth]{Plots/DiFF-T_System.pdf}
}
\vspace{-8pt}
\caption{Common frames used in defining DiFFs: (a) the $\perp$ system where the fragmenting quark's 3-momentum $\vect{k}$ is along the $\hat{z}$ axis (b) the $T$ system where the total 3-momentum of hadrons $\vect{P}$ is along the $\hat{z}'$ axis.}
\label{PLOT_PERP_T_SYS}
\vspace{1cm}
\end{figure}
We can relate the $\perp$ and $T$ components of vectors in the two systems by considering a Lorentz transformation, which preserves the light-cone momentum fractions and results in quark acquiring transverse momentum $\vect{k}_T$ in the $T$ system. The corresponding transformation matrix reads~\cite{Diehl:2000xz}
\al
{
\Lambda^\mu_\nu=
\left(
\begin{array}{cccc}
1 & \frac{\vect{k}_T^2}{(k^+)^2} & \frac{k^1}{k^+} & \frac{k^2}{k^+} \\
0 & 1 & 0 & 0 \\
0 & \frac{k^1}{k^+} & 1 & 0 \\
0 & \frac{k^2}{k^+} & 0 & 1
\end{array}
\right),
\label{EQ_LORENTZ}
}
where $\mu,\nu \in \{+, -, 1, 2\}$. Then one easily finds
\al
{
\vect{P}_{1T} &= \PPn{1} + z_1 \vect{k}_T,
\\
\vect{P}_{2T} &= \PPn{2} + z_2 \vect{k}_T \, .
}
%
The relations for the total and relative transverse momenta of the hadron pair imply that:
%
\al{
&
\vect{k}_T = -\frac{\vect{P}_\perp}{z},
%
\\&
\vect{R}_T = \frac{z_2 \PPn{1} - z_1 \PPn{2}}{z} = (1-\xi) \PPn{1} - \xi \PPn{2}.
}
The formal definition of DiFFS is given using the so-called quark-quark correlator in the $T$ system~\cite{Bianconi:1999cd, Radici:2001na, Boer:2003ya}
\al
{
\Delta_{ij}(k; P_1, P_2)&
\\ \nonumber
= \sum_X \int d^4 \zeta &e^{i k\cdot \zeta} \langle 0 |\psi_i(\zeta) | P_1 P_2, X \rangle \langle P_1 P_2, X | \bar{\psi}_j(0) | 0 \rangle.
}
In particular, the DiFFs are defined via projections of the quark-quark correlator, defined for a Dirac operator $\Gamma$ as
\al
{
\Delta^{\Gamma}&(z, \xi, \vect{k}_T^2, \vect{R}_T^2, \vect{k}_T \cdot \vect{R}_T)
\\ \nonumber
&= \frac{1}{4z} \int d k^+ \mathrm{Tr}[\Gamma \Delta(k, P_1, P_2)]|_{k^+ = P_h^+/z}.
}
The expressions for all four leading order DiFFs of a polarized quark into an unpolarized hadron pair are
\al
{
\label{EQ_DELTA_UNP}
\Delta^{[\gamma^+]} =& D_1(z, \xi, \vect{k}_T^2, \vect{R}_T^2, \vect{k}_T \cdot \vect{R}_T),
\\ \label{EQ_DELTA_LIN}
%
\Delta^{[\gamma^+\gamma_5]}
=& \frac{\epsilon_T^{ij} R_{Ti} k_{Tj}}{M_1 M_2} G_1^\perp(z, \xi, \vect{k}_T^2, \vect{R}_T^2, \vect{k}_T \cdot \vect{R}_T),
%
\\ \label{EQ_DELTA_TRANSV}
%
\Delta^{[i \sigma^{i+} \gamma_5]}
=& \frac{\epsilon_T^{ij} R_{Tj}}{M_1 + M_2} H_1^\sphericalangle(z, \xi, \vect{k}_T^2, \vect{R}_T^2, \vect{k}_T \cdot \vect{R}_T)
\\ \nonumber
& + \frac{\epsilon_T^{ij} k_{Tj}}{M_1 + M_2} H_1^\perp(z, \xi, \vect{k}_T^2, \vect{R}_T^2, \vect{k}_T \cdot \vect{R}_T),
}
where $D_1$ is the unpolarized DiFF and $G_1^\perp$ the helicity dependent DiFF describing the correlations between the total and relative transverse momenta of the pair and the longitudinal polarization of the quark. The two remaining DiFFs describe the correlations between the transverse polarization of the quark and the transverse momenta: the analogue of the Collins function $H_1^\perp$ encodes the correlations with the total transverse momentum, while the IFF $H_1^\sphericalangle$ encodes correlations with the relative transverse momentum.
The relevant parts of the integrated cross section for back-to-back creation of two dihadron pairs in $e^+e^-$ annihilation involve only the following integrals of the DiFFs~\cite{Boer:2003ya}
%
\al
{
\label{EQ_D1_MH}
D_1(z, M_h^2)
= & \int d \xi \int d \varphi_R \int d^2 \vect{k}_T
\\ \nonumber
& \times \ D_1(z, \xi, \vect{k}_T^2, \vect{R}_T^2, \vect{k}_T \cdot \vect{R}_T),
}
\al
{
\label{EQ_G1_MH}
G_1^\perp(z, M_h^2)
=& \int d \xi \int d \varphi_R \int d^2 \vect{k}_T
\\ \nonumber
& \times (\vect{k}_T\cdot \vect{R}_T) \ G_1^\perp(z, \xi, \vect{k}_T^2, \vect{R}_T^2, \vect{k}_T \cdot \vect{R}_T) \,
}
while for the transverse polarization-dependent DiFFs
\al
{
\label{EQ_HAng_MH}
H_1^\sphericalangle(z, M_h^2)
= & \int d \xi \int d \varphi_R \int d^2 \vect{k}_T
\\ \nonumber
& \times |\vect{R}_T| H_1^\sphericalangle(z, \xi, \vect{k}_T^2, \vect{R}_T^2, \vect{k}_T \cdot \vect{R}_T),
}
\al
{
\label{EQ_HPerp_MH}
H_1^\perp(z, M_h^2)
= & \int d \xi \int d \varphi_R \int d^2 \vect{k}_T
\\ \nonumber
& \times |\vect{k}_T| \ H_1^\perp(z, \xi, \vect{k}_T^2, \vect{R}_T^2, \vect{k}_T \cdot \vect{R}_T),
}
where $\varphi_R$ is the azimuthal angle of vector $\vect{R}_T$ and the invariant mass of the hadron pair $M_h$ is employed to replace $|\vect{R}_T|$ in the integrand using the relation
\al
{
R_T^2 &= \xi (1-\xi) M_h^2 - M_1^2 (1-\xi) - M_2^2 \xi \, ,
\\
M_h^2 &= P^2.
}
The unintegrated DiFFs in Eqs.~(\ref{EQ_DELTA_UNP})-(\ref{EQ_DELTA_TRANSV}) are even functions of the relative azimuthal angle $\varphi_{RK} \equiv \varphi_R-\varphi_{k}$ between the vectors $\vect{k}_T$ and $\vect{R}_T$, since they only depend on the scalar product $(\vect{k}_T \cdot \vect{R}_T)$. Thus, in general their Fourier decomposition in angle $\varphi_{RK}$ would involve only cosine moments. We define the $n$-th Fourier cosine moment of $G_1^\perp$ in expansion withe respect to $\varphi_{RK}$ as
\al
{
\label{EQ_G1_MOM_MH}
&G_1^{\perp,[n]}(z, M_h^2)
= \int d \xi \int d \varphi_R \int d^2 \vect{k}_T
\\ \nonumber
& \times |\vect{k}_T| |\vect{R}_T| \cos(n \cdot \varphi_{RK}) \ G_1^\perp(z, \xi, \vect{k}_T^2, \vect{R}_T^2, \vect{k}_T \cdot \vect{R}_T).
}
We note that in $e^+e^-$ annihilation cross section contains only the first moment of this decomposition, shown in \Eq{EQ_G1_MH}. On the other hand, the $F_{UL}^{\sin(\varphi_h-\varphi_R)}$ structure function (see, e.g. Eq.~(B3) in Ref.~\cite{Bianconi:1999cd}) in SIDIS cross section can in general be decomposed into infinite Fourier series with respect to $\cos(\varphi_h-\varphi_R)$. In turn, these moments can be expressed as convolutions of helicity PDF and combinations of Fourier cosine moments of $G_1^{\perp,[n]}$. A similar approach using decomposition in terms of spherical harmonic with azimuthal angle $\varphi_{RK}$ has been presented in Ref.~\cite{Gliske:2014wba}. Naturally, the Fourier cosine moments we use here can be related to these spherical harmonics, as they both encode the same initial functions.
\section{Monte Carlo simulations}
\label{SEC_MC}
\subsection{Extracting DiFFs from polarization-dependent number densities}
\label{SUBSEC_DIFF_EXTRACT}
In this subsection we describe the method developed for extracting the DiFFs using MC approach. Within this approach we calculate various number densities by averaging over a large number of MC simulations of the quark hadronization process. These number densities can be expressed in terms of the corresponding fragmentation functions, allowing us to extract them from these number densities using the corresponding azimuthal modulations. In the case of production of two unpolarized hadrons by a longitudinally polarized quark, the relevant number density, according to Eqs.~(\ref{EQ_DELTA_UNP}) and (\ref{EQ_DELTA_LIN}), can be expressed as
%
\al
{
\label{EQ_F_VEC}
F&(z, \xi , \vect{k}_T, \vect{R}_T; s_L) = D_1(z, \xi, \vect{k}_T^2, \vect{R}_T^2, \vect{k}_T \cdot \vect{R}_T)
\\ \nonumber
&+ s_L \frac{ (\vect{R}_T \times \vect{k}_T)\cdot \vect{\hat{z}} }{M_1 M_2} G_1^\perp(z, \xi, \vect{k}_T^2, \vect{R}_T^2, \vect{k}_T \cdot \vect{R}_T) \, .
}
This can also be expressed in terms of the azimuthal angles $\varphi_{k}$ and $\varphi_R$ of the vectors $\vect{k}_T$ and $\vect{R}_T$
\al
{
\label{EQ_F_ANG}
F&(z, \xi , \vect{k}_T, \vect{R}_T; s_L) = D_1(z, \xi, \vect{k}_T^2, \vect{R}_T^2, \cos(\varphi_{RK}))
\\ \nonumber
&
- s_L \frac{ R_T k_T \sin(\varphi_{RK}) }{M_1 M_2} G_1^\perp(z, \xi, \vect{k}_T^2, \vect{R}_T^2, \cos(\varphi_{RK})).
}
The unpolarized and helicity dependent DiFFs of Eqs.~(\ref{EQ_D1_MH},\ref{EQ_G1_MH}) then can be extracted from the number density
\al
{
D_1(z, M_h^2)=& \int d \xi \int d \varphi_R \int d^2 \vect{k}_T
\\ \nonumber
&\times \ F(z, \xi , \vect{k}_T, \vect{R}_T; s_L),
}
\al
{
G_1^\perp(z, M_h^2) = -&\frac{M_1 M_2}{s_L} \int d \xi \int d \varphi_R \int d^2 \vect{k}_T
\\ \nonumber
&\times \ \cot(\varphi_{RK}) F(z, \xi , \vect{k}_T, \vect{R}_T; s_L).
}
In this work we concentrate on calculating $M_h^2$ integrated DiFFs, which are defined as
\al
{
\label{EQ_D1_Z}
%
D_1(z)
= & \int d \xi \int d^2 \vect{R}_T \int d^2 \vect{k}_T
\\ \nonumber
& \times \ D_1(z, \xi, \vect{k}_T^2, \vect{R}_T^2, \vect{k}_T \cdot \vect{R}_T),
}
\al
{
\label{EQ_G1_Z}
%
G_1^\perp(z)
=& \int d \xi \int d^2 \vect{R}_T \int d^2 \vect{k}_T
\\ \nonumber
& \times (\vect{k}_T\cdot \vect{R}_T) \ G_1^\perp(z, \xi, \vect{k}_T^2, \vect{R}_T^2, \vect{k}_T \cdot \vect{R}_T) \, .
}
The corresponding expressions in terms of the number density are
\al
{
\label{EQ_D1_Z_EXTR}
D_1(z)=& \int d \xi \int d^2 \vect{R}_T \int d^2 \vect{k}_T
\\ \nonumber
&\times \ F(z, \xi , \vect{k}_T, \vect{R}_T; s_L),
}
\al
{
\label{EQ_G1_Z_EXTR}
G_1^\perp(z) = -& \frac{M_1 M_2}{s_L} \int d \xi \int d^2 \vect{R}_T \int d^2 \vect{k}_T
\\ \nonumber
&\times \ \cot(\varphi_{RK}) F(z, \xi , \vect{k}_T, \vect{R}_T; s_L).
}
Finally, throughout this work we will use the definition
\al
{
\tilde{G}_1^\perp(z) \equiv \frac{1}{M_1 M_2} G_1^\perp(z) ,
}
to simplify the notation.
\subsection{The quark-jet model simulations}
\label{SUBSEC_NJL_JET}
\begin{figure}[!b]
\centering
\includegraphics[width=0.8\columnwidth]{Plots/Q-Q-H-SPIN.pdf}
\vspace{-8pt}
\caption{The extended quark-jet framework.}
\label{PLOT_QUARK_JET}
\end{figure}
In this subsection we describe the quark-jet MC simulation framework, used in the current study of the DiFFs, which has been developed over recent years~\cite{Matevosyan:2011ey,Matevosyan:2011vj, Matevosyan:2013aka, Matevosyan:2013nla, Matevosyan:2013eia, Matevosyan:2016fwi}. We use the most recent evolution of the quark-jet framework, as detailed in Refs.~\cite{Bentz:2016rav, Matevosyan:2016fwi} to model the hadronization of a quark with a longitudinal polarization. The schematic depiction of the framework in Fig.~\ref{PLOT_QUARK_JET} displays the sequential emission of hadrons $h_1$, $h_2$, etc., where the polarization of the remnant quark after each emission is determined by the corresponding spin density matrix. We choose a pre-determined number of hadron emissions, $N_L$, for terminating each hadronization chain and use MC to calculate the number densities for different hadron pairs. The input quark-to-quark SFs are calculated using the Nambu--Jona-Lasinio (NJL) effective quark interaction model~\cite{Nambu:1961tp,Nambu:1961fr}. The details of the model calculations and parameters are described in detail in Ref.~\cite{Matevosyan:2016fwi}. In this work, we forgo the QCD evolution of the DiFFs~\cite{Ceccopieri:2007ip} we computed using the low-energy NJL effective model input. The evolution would be necessary for accurate direct comparisons with the experimental results obtained at various large energy scales. We performed such studies in our previous work for the unpolarized DiFFs calculated within the same model in Ref.~\cite{Matevosyan:2013aka}, showing that the QCD evolution shifts the shape of the DiFFs towards lower $z$ region. To mimic such effects for the qualitative comparisons made in this article, we use the ansatz with all the input SFs multiplied by a factor of $(1-z)^4$. It is worth noting, that the particular choice of the input SFs does not affect the qualitative features of the computed DiFFs. Rather, the dominant aspects are the transverse momentum and spin transfer mechanisms within the quark-jet framework. All the results shown in this work are obtained by setting $s_L=1$, and $\vect{s}_T{}=0$.
\begin{figure}[!b]
\centering
\includegraphics[width=0.8\columnwidth]{Plots/PhiRT_DEP_KTRT_NL2.pdf}
\vspace{-8pt}
\caption{The MC results for the longitudinal spin correlations in $\pi^+\pi^-$ pairs for a hadronization of a $u$ quark with $N_L=2$. The horizontal axis represents the azimuthal angle $\varphi_{RK}$, while the vertical axis represents the magnitudes of the $z$-integrated number densities and their modulations. The black solid curve depicts the total number density $F$, the red dashed and orange dash-dotted curves represent the even ($F_E$) and odd ($F_O$) parts of $F$ with respect to $\varphi_{RK}$, and the blue dotted curve is the $F_O$ multiplied by $\cot(\varphi_{RK})$.}
\label{PLOT_PHIRT_MOD}
\end{figure}
Our first task is to test the method of extracting the DiFFs described in Sec.~\ref{SUBSEC_DIFF_EXTRACT}, considering $\pi^+\pi^-$ pairs produced with the smallest possible value of $N_L=2$. For this purpose, in Fig.~\ref{PLOT_PHIRT_MOD} we show,using a black solid line, the $\varphi_{RK}$ dependence of the function $F$ from~\Eq{EQ_F_ANG}, integrating over all the other variables. It obviously is not an even function of $\varphi_{RK}$, indicating a nonzero term associated with $G_1^\perp$ in \Eq{EQ_F_ANG}. The dashed and dash-dotted lines represent the even and the odd parts of this function, corresponding to the unpolarized and the helicity dependent DiFF terms, respectively. The blue dotted line is the odd part of $F$, multiplied by $\cot(\varphi_{RK})$. It is clear, that the integral of the red dashed and the blue dotted lines over $\varphi_{RK}$ correspond to the $z$-integrated values of $D_1(z)$ and $\tilde{G}_1^\perp(z)$. Thus, we can ensure that the multiplication by the $\cot(\varphi_{RK})$ does not produce discontinuities for $\varphi_{RK} = \{0, \pi\}$, yielding reliable estimates for $\tilde{G}_1^\perp$. We have to note though, that such extraction requires a large enough number of MC simulations to sufficiently suppress the statistical fluctuations. In calculating the $z$-dependence of $D_1$ and $\tilde{G}_1^\perp$, we use a similar procedure involving the $z$-unintegrated form of $F$. In this work we used $10^{12}$ MC events to calculate the polarized number densities, allowing us to reliably extract the DiFFs for $100$ discretization points of $z\in[0,1]$ and $200$ discretization points for $\varphi_{RK}\in[0,2\pi)$.
\begin{figure}[!t]
\centering
\subfigure[]
{
\includegraphics[width=0.8\columnwidth]{Plots/D1_NL-X.pdf}
}
\\ \vspace{-8pt}
\subfigure[]
{
\includegraphics[width=0.8\columnwidth]{Plots/GPerp_NL-X.pdf}
}
\\ \vspace{-8pt}
\subfigure[]
{
\includegraphics[width=0.8\columnwidth]{Plots/Rat_GPerp_NL-X.pdf}
}
\vspace{-8pt}
\caption{Comparison of MC results for $D_1(z)$ (a), $\tilde{G}_1^{\perp}(z)$ (b), and their ratios (c) for various values of $N_L$.}
\label{PLOT_D1_GP_PIPL_PIMI}
\end{figure}
\begin{figure}[!t]
\centering
\subfigure[]
{
\includegraphics[width=0.8\columnwidth]{Plots/D1_ZCut_NL-X.pdf}
}
\\ \vspace{-8pt}
\subfigure[]
{
\includegraphics[width=0.8\columnwidth]{Plots/GPerp_ZCut_NL-X.pdf}
}
\\ \vspace{-8pt}
\subfigure[]
{
\includegraphics[width=0.8\columnwidth]{Plots/Rat_GPerp_ZCut_NL-X.pdf}
}
\vspace{-8pt}
\caption{Comparison of MC results for $D_1(z)$ (a), $\tilde{G}_1^{\perp}(z)$ (b), and their ratios (c) for for various values of $N_L$. The cuts $z_{1,2} \geq 0.1$ are imposed in these results.}
\label{PLOT_D1_GP_PIPL_PIMI_ZCUT}
\end{figure}
The results for the $z$-dependent extractions of $D_1$, $\tilde{G}_1^\perp$, and their ratios of the $\pi^+\pi^-$ pairs are depicted in Fig.~\ref{PLOT_D1_GP_PIPL_PIMI}. Here the black solid , the red dotted , the blue dashed and the orange dash-dotted lines depict the results for $N_L=2,3,4,$ and $6$, respectively. It is clear that the function $D_1(z)$ increases rapidly in the small-$z$ region with increasing number of produced hadrons $N_L$, due to the large combinatorial factors in selecting pairs from an expanding set of final particles. It is also remarkable to see a nonvanishing signal for $\tilde{G}_1^\perp(z)$, which gets smaller with an increasing number of produced hadrons because of the destructive interference of oppositely signed signals for the pairs produced at different ranks. Nevertheless, the results for the "analyzing power" of the $\tilde{G}_1^\perp(z)$ modulations of the number density $F$, depicted in Fig.~\ref{PLOT_D1_GP_PIPL_PIMI}(c), show that only in the large-$z$ region is there a significant signal that can be possibly measured.
The plots in Fig.~\ref{PLOT_D1_GP_PIPL_PIMI_ZCUT} are analogous to those in Fig.~\ref{PLOT_D1_GP_PIPL_PIMI}, except that here an additional cut is placed upon the minimum values of $z$ for each member of the pair, $z_{1,2}\geq 0.1$. Such cuts are common in experimental settings, and for example in SIDIS are mainly aimed at separating the current and target fragmentation regions. Thus it is important to evaluate the impact on our results. The plots show the expected suppression of the unpolarized DiFF with respect to the previous case, as the hadrons with increasing rank on average carry decreasing values of light-cone momentum $z$, thus the high-ranked hadrons are often excluded from the pairs due to the cut criterion. On the other hand, the impact on the helicity dependent DiFF is less significant, since the bulk of the effect is generated by the first two produced hadrons. This results in a slight increase of the analyzing power in the mid-$z$ range, making it less peaked towards large $z$.
Next, we plot the results for all possible types of pion pairs in Fig.~\ref{PLOT_GP_RAT_PAIRS} for $N_L=6$, both without and with the $z$ cut. It is interesting to see, that $\pi^+\pi^+$ pairs have an opposite sign and similar magnitude to the $\pi^+\pi^-$ case. Here, in the same-signed pairs we assign the hadron with the larger $z$ as the first member of the pair. Without such a choice for $z$-ordering, all the results for the same-signed pairs vanish, as expected from the symmetry considerations.
\begin{figure}[!b]
\centering
\subfigure[]
{
\includegraphics[width=0.8\columnwidth]{Plots/Rat_GPerp_NL-6_Pairs.pdf}
}
\\ \vspace{-8pt}
\subfigure[]
{
\includegraphics[width=0.8\columnwidth]{Plots/Rat_GPerp_ZCut_NL-6_Pairs.pdf}
}
\vspace{-8pt}
\caption{The ratio $\tilde{G}_1^{\perp}(z)/D_1(z)$ for different pion pairs (a), and the cuts of $z_{1,2} \geq 0.1$ (b), for simulations with $N_L=6$.}
\label{PLOT_GP_RAT_PAIRS}
\end{figure}
The SIDIS cross section for two hadron production contains all the cosine moments of $G_1^{\perp}$. In Fig.~\ref{PLOT_GP_MOM_PIPL_PIMI_ZCUT} we we present the first five moments for $\pi^+\pi^-$ and $N_L=6$, extracted from the polarized number density in a analogous manner to the first moment. Here only the results with cuts $z_{1,2} \geq 0.1$ are presented.
\begin{figure}[!t]
\centering
\includegraphics[width=0.8\columnwidth]{Plots/GPerp_ZCut_NL-X_CosMoms.pdf}
\vspace{-8pt}
\caption{The first five Fourier cosine moments of $\tilde{G}_1^{\perp}(z)$ for $\pi^+\pi^-$ pairs and $N_L=6$, imposing the cuts of $z_{1,2} \geq 0.1$.}
\label{PLOT_GP_MOM_PIPL_PIMI_ZCUT}
\end{figure}
\section{MC Validation}
\label{SEC_VALIDATION}
In this section we validate our MC simulations in two ways. First, we use combinatorial arguments to calculate the total number of all $\pi^+\pi^-$ pairs for a given $N_L$ and compare these calculations to the results from MC simulations. Second, we derive explicit integral relations for the $z$ dependence of the DiFFs for the case $N_L=2$, and compare them with the MC results.
\subsection{Unpolarized DiFFs - Total number of dihadron pairs}
It is useful to calculate the number of the dihadron pairs of a given type ($\pi^+ \pi^-$, $\pi^0\pi^0$, etc) for a fixed number of produced hadrons, where we integrate over all variables. Here we will limit ourselves to consider only pions produced by the $u,d$ quarks. Let us look at the initial $u$ quark producing $N_L$ pions, and at each step the total probability of producing a charged pion has an isospin factor of $1$, while the neutral pion has a isospin factor of $\frac{1}{2}$. In the quark-jet framework, the total probability of producing a hadron at each step is $1$, then the probability of producing a charged pion is $\frac{2}{3}$, while for a neutral is $\frac{1}{3}$.
It is important to note that, because of flavor conservation, the first produced charged pion should be a $\pi^+$, and each subsequent charged pion should have an alternate charge. Neutral pions can be produced at any stage without such limitations. First, we count the number of different cases when producing $n_0$ neutral pions, where $0\leq n_0 \leq N_L$. Let us start with one of the possible scenarios for $n_0$ produced hadrons
\al
{
(\overbrace{\pi^+, \pi^-, \pi^+, ...,}^{N_L- n_0} \overbrace{\pi^0, \pi^0, \pi^0}^{n_0}).
}
The number of all possible permutations of this set is $N_L!$. Thus number will be over-counting the cases with all permutations of just $\pi^0$s, which is $n_0!$. Also, for a given $n_0$, there is only a single ordering of the charged pion possible, as a permutation of any two same charged hadrons would yield an identical set, while a permutation of opposite signed hadrons would yield an invalid set that violates flavor conservation. Thus, we should also divide the total number of sets by all possible permutations of the charged pions, that is $(N_L - n_0)!$. Then, the number of such combinations is
%
\al
{
N_{n_0} = \frac{N_L!}{n_0 ! (N_L -n_0)!} \equiv C^{n_0}_{N_L}.
}
and the probability of producing any such combination is
\al
{
P(n_0) = \Big(\frac{2}{3} \Big)^{N_L - n_0} \Big(\frac{1}{3} \Big)^{ n_0}.
}
Just a quick verification of our formula can be obtained by calculating the total probability of producing $N_L$ hadrons (all possible combinations) in $N_L$ steps, given by
\al
{
P &= \sum_{n_0 =0}^{n_0 = N_L} N_{n_0} P(n_0) = 1,
}
as expected.
The number of various $\pi \pi$ pairs in each combination is
\al
{
&
N^ {(\pi^+ \pi^-)}(n_0) = U\Big( \frac{N_L - n_0}{2} \Big) \ D\Big( \frac{N_L - n_0}{2} \Big),
\\
&
N^ {(\pi^0 \pi^+)}(n_0) = n_0\ U\Big( \frac{N_L - n_0}{2} \Big) ,
\\
&
N^ {(\pi^- \pi^0)}(n_0) = D\Big( \frac{N_L - n_0}{2} \Big) \ n_0,
\\
&
N^ {(\pi^+ \pi^+)}(n_0) = C_{U\Big( \frac{N_L - n_0}{2} \Big)}^2,
\\
&
N^ {(\pi^- \pi^-)}(n_0) = C_{D\Big( \frac{N_L - n_0}{2} \Big)}^2,
\\
&
N^ {(\pi^0 \pi^0)}(n_0) = C_{n_0}^2,
}
where the $U(n), D(n)$ functions round up and down to the nearest integer.
Finally, the mean number of producing a $\pi^+\pi^-$ pair after $N_L$ emissions is
\al
{
\label{EQ_N_PIPI}
&\mathcal{N}^ {(\pi^+ \pi^-)}(N_L) = \sum_{n_0 =0}^{n_0 = N_L} N_{n_0} P(n_0) N^ {(\pi^+ \pi^-)}(n_0)
\\ \nonumber
&= \sum_{n_0 =0}^{n_0 = N_L} C^{n_0}_{N_L} \Big(\frac{2}{3} \Big)^{N_L - n_0} \Big(\frac{1}{3} \Big)^{ n_0} U\Big( \frac{N_L - n_0}{2} \Big) \ D\Big( \frac{N_L - n_0}{2} \Big).
}
It is also clear that this number is simply the integral over $z$ of the unpolarized DiFF extracted from MC simulations with $N_L$ produced hadrons.
%
\al
{
\label{EQ_D1_INT}
\mathcal{N}^ {(\pi^+ \pi^-)}_{MC}(N_L) = \int_0^1 dz \ D_{1, [N_L]}^{u\to\pi^+\pi^-}(z)
}
The results of the calculations both using \Eq{EQ_N_PIPI} and \Eq{EQ_D1_INT} for a range of values of $N_L$ are presented in Table~\ref{TABLE_N_PIPI}. We see very good agreement between the two methods, given the discretization errors of the MC simulations. The last row in Table~\ref{TABLE_N_PIPI} represents the results of MC simulations with cuts on minimum value of $z$ for each hadron in the pair, $z\geq z_{min}=0.1$.
\begin{table}[tb]
\centering
\begin{center}
\begin{tabular}
{ | C{0.5cm} | C{1.5cm} | C{1.5cm} | C{1.5cm} | C{1.5cm} m{-30pt} |}
\hline
$N_L$ & $\mathcal{N}^ {(\pi^+ \pi^-)}$ & $\mathcal{N}^ {(\pi^+ \pi^-)}_{N} $ & $\mathcal{N}^ {(\pi^+ \pi^-)}_{MC}$ & $\mathcal{N}^ {(\pi^+ \pi^-)}_{MC, z_{min}}$ & \\ [2.5ex] \hline \hline
2 & $\dfrac{4}{9}$ & 0.44444 & 0.4444 & 0.350175 & \\ [3.5ex] \hline
3 & $\dfrac{28}{27}$ & 1.03704 & 1.03694 & 0.683999 & \\ [3.5ex]\hline
4 & $\dfrac{152}{81}$ & 1.87654 & 1.87641 & 0.959588 & \\ [3.5ex]\hline
5 & $\dfrac{712}{243}$ & 2.93004 & 2.92992 & 1.11531 & \\ [3.5ex]\hline
6 & $\dfrac{3068}{729}$ & 4.2085 & 4.20882 & 1.18162 & \\ [3.5ex]\hline
7 & $\dfrac{12484}{2187}$ & 5.70828 & 5.70867 & 1.20282 & \\ [3.5ex]\hline
8 & $\dfrac{48752}{6561}$ & 7.43057 & 7.43047 & 1.20809 & \\ [3.5ex]\hline
\end{tabular}
\caption{The number of $\pi^+\pi^-$ pairs $\mathcal{N}^ {(\pi^+ \pi^-)}$ for a given number of produced hadrons $N_L$. The numbers in the third row are the approximate numerical values obtained via the \Eq{EQ_N_PIPI}, while those in the fourth row are the results of the numerical simulations and \Eq{EQ_D1_INT}. The last row shows the results form the same MC simulations with cuts on minimum value of $z$ for each hadron in the pair: $z_{1,2}\geq z_{min}=0.1$.}
\label{TABLE_N_PIPI}
\end{center}
\end{table}
\subsection{Two-step process and validation}
\label{SUBSEC_VALID_TWO}
Here we aim to validate our MC results for both unpolarized and helicity dependent DiFFs by deriving explicit integral relations when an initial $u$ quark produces only a single $\pi^+\pi^-$ pair. We used a similar approach in sections IIC and IVA of~\cite{Matevosyan:2016fwi} to derive similar expressions for the unpolarized and unfavored Collins function for a $u\to\pi^-$ fragmentation for the case of two-hadron emission. We briefly review the kinematics setup here, and a more detailed description can be found in Ref.~\cite{Matevosyan:2016fwi}.
In the quark-jet framework, the fragmenting quark $q$ (the initial $u$ quark), emits a hadron $h_1$ (a $\pi^+$ in our calculations), leaving a remnant quark $q_1$ (a $d$ quark) carrying light-cone momentum fraction $\eta_1$ and transverse momentum component $\pe{1}$, where the transverse direction is defined with respect to the three-momentum vector of $q$. The initial and remnant quark's spin 3-vectors are denoted as $\vect{s}_q = (\vect{0},s_L )$ and $\vect{s}_{q_1} = (\vect{s}_{T_1}, s_{L_1})$. In the second hadronization step, the quark $q_1$ emits a hadron $h_2$ (a $\pi^-$) with light-cone momentum fraction $\eta_2$ and transverse momentum $\pe{2}$ with respect to the 3-momentum direction of $q_1$. We can then easily calculate the momentum of $h_2$ in the initial frame using the Lorentz transformation in \Eq{EQ_LORENTZ}.The number density for the process of $u\to \pi^+\pi^-$ is simply the product of the corresponding number densities for each of these two steps
%
\al
{
\label{EQ_Q_to_2H}
F^{(2)}_{q\to h_1 h_2}(&\eta_1, \pe{1}, \eta_2, \pe{2}; \sq{q} )
\\ \nonumber
=& \ \hat{f}^{q\to q_1 }(\eta_1, \pe{1}; \sq{q} ) \cdot \hat{f}^{q_1\to h_2}(\eta_2, \pe{2}; \sq{q_1} ) \, ,
}
where the elementary probability densities can be expressed in terms of the elementary TMD SFs
\al
{
\label{EQ_FHAT_Q_Q1}
\hat{f}^{q\to q_1 }(z, \pe{}; & \vect{s} )
\\ \nonumber
= \hat{D}^{(q\to q_1)}&(z, p_\perp^2{}) + \frac{(\pe{}\times \vect{s}_T) \cdot \hat{\vect{z}} }{z M_{q_1}} \ \hat{H}^{\perp({q\to q_1})}(z,p_\perp^2{}),
}
\al
{
\label{EQ_FHAT_Q_H}
\hat{f}^{q \to h}(z, \pe{};& \vect{s} )
\\ \nonumber
= \hat{D}^{(q\to h)}&(z, p_\perp^2{}) + \frac{(\pe{}\times \vect{s}_T) \cdot \hat{\vect{z}}}{z m_h}\ \hat{H}^{\perp({q\to h})}(z,p_\perp^2{}),
}
The remnant quark's polarization is determined using the quark spin density matrix formalism, which for an initial longitudinally polarized quark reads
\al
{
\vect{s}_{T_1} =& \frac{1}{ \hat{f}^{q\to q_1}(\eta_1, \pe{1}; \sq{q}) }
\\ \nonumber
& \times \Bigg( \frac{ \pe{1}' }{\eta_1 M_{q_1}} \hat{D}_T^\perp(\eta_1,\psqn{1}) - \frac{\pe{1} }{\eta_1 M_{q_1}} s_L \hat{G}_T(\eta_1,\psqn{1}) \Bigg),
\\
s_{L_1} = & \frac{1 }{ \hat{f}^{q\to q_1}(\eta_1, \pe{1}; \sq{q}) } s_L \ \hat{G}_L(\eta_1,\psqn{1}),
}
and $\pe{1}' \equiv (-p_{1y},p_{1x})$. Here $\hat{D}$, $\hat{D}_T^\perp$,$\hat{G}_L$, $\hat{G}_T$, and $\hat{H}^{\perp}$ are the TMD elementary SFs. A quark model calculation of all the eight quark-to-quark and the two quark-to-hadron TMD SFs has been done in Ref.~\cite{Matevosyan:2016fwi} using the spectator approach. Note, that only $\vect{s}_{T_1} $ contributes to $F^{(2)}_{q\to h_1 h_2}$, as can be seen from \Eq{EQ_FHAT_Q_H}.
The momenta of $h_1$ and $h_2$ are obtained using the momentum conservations and the Lorentz transformations similar to that in \Eq{EQ_LORENTZ} to calculate the transverse momenta of $h_2$ in the initial quark's system
\al
{
& z_1 = 1-\eta_1,
&\Pe{1}= - \pe{1},
\\
&z_2 = \eta_1 \eta_2,
&\Pe{2} = \pe{2} + \eta_1\pe{1}.
}
The final results for the $z$-dependence of the unpolarized and helicity dependent DiFFs are
%
\al
{
\label{EQ_D1_NL2}
D_1^{(2)}(z)
= \int_{0}^{1} d \eta_1 \int_{0}^{1} & d \eta_2
\ \delta \Big(z - 1 +\eta_1 (1-\eta_2) \Big)
\\ \nonumber
& \times \ \hat{D}^{q\to q_1}(\eta_1) \ \hat{D}^{q_1\to h_2} ( \eta_2 ),
}
and
\al
{
\label{EQ_GP_NL2}
\nonumber
&\tilde{G}_1^{\perp (2)}(z)
= -\pi^2 \int_{0}^{1} d \eta_1 \int_{0}^{1} { d \eta_2} \ \delta \Big(z - 1 +\eta_1 (1-\eta_2) \Big)
\\
& \ \times
\ \int d p_{1\perp}^2 \int d p_{2\perp}^2
\frac{ \eta_2 (1- \eta_2)\psqn{1} - (1-\eta_1)\psqn{2} }{z}
\\ \nonumber
& \ \times
\ \frac{1 }{\eta_1M_{q_1} } \hat{G}_T^{q\to q_1}(\eta_1,\psqn{1})
\ \frac{1 }{\eta_2 M_2} \ \hat{H}^{\perp (q_1\to h_2)}(\eta_{2},\psqn{2}).
}
It is important to note, that the above results are obtained only using the quark-jet formalism and the spin transfer mechanism, and do not depend on the underlying quark models used to calculate the particular forms of the SFs.
The plots in Fig.~\ref{PLOT_D1_GP_NL2} depict the results for the unpolarized DiFF (a) and the helicity dependent DiFF (b), calculated both using the MC method (plotted with red points) and the explicit integral relations in Eqs.~(\ref{EQ_D1_NL2},\ref{EQ_GP_NL2}) (plotted with black lines) for $N_L=2$. We observe excellent agreement between the two methods, both for the unpolarized and the helicity dependent DiFFs, validating our calculations.
\begin{figure}[t]
\centering
\subfigure[]
{
\includegraphics[width=0.8\columnwidth]{Plots/D1_NL2.pdf}
}
\\ \vspace{-8pt}
\subfigure[]
{
\includegraphics[width=0.8\columnwidth]{Plots/GP_NL2.pdf}
}
\vspace{-8pt}
\caption{Comparison of results of $D_1(z)$ (a), and $\tilde{G}_1^{\perp}(z)$ (b) of integral expressions (IE) and MC results for $\pi^+\pi^-$ pair produced for $N_L=2$ in the quark-jet picture.}
\label{PLOT_D1_GP_NL2}
\end{figure}
It is also worth examining the structure of \Eq{EQ_GP_NL2}, which elucidates the microscopic mechanism for generating the helicity dependent DiFFs in the quark-jet framework. Here the "worm-gear" type elementary splitting function $G_T$ for the first quark-to-quark process, describing the correlation of the transverse polarization of the remnant quark with the longitudinal polarization of the fragmenting quark, is convoluted with the elementary Collins function $H^\perp$ for the second hadron emission. The latter describes the correlation of the emitted hadron's transverse momentum with the quark's transverse polarization. Thus, even in the case of longitudinal polarization, the Collins effect in the single hadron emission process, together with the momentum recoil mechanism of the quark-jet framework, is responsible for generating two-hadron correlations with the initial longitudinal polarization. This is a fascinating result, as naively the Collins effect is associated with the correlations involving transverse polarization. Finally, we have to note that it is possible for a helicity dependent two hadron correlation to be generated by the so-called interference mechanism of the hadron pair produced in different channels (let us say as decay products of resonances emitted by the quark), as detailed in Ref.~\cite{Bianconi:1999cd}. Nevertheless, such calculations are beyond the scope of this work.
\section{Conclusions}
\label{SEC_CONCLUSIONS}
Spin-dependent correlations in two-hadron fragmentation functions provide a wealth of information about the hadronization process. Moreover, they provide an additional method for exploring the momentum and spin structure of the nucleon via two-hadron SIDIS measurements. Nonetheless, these DiFFs are still not very well known. This is especially true for the helicity dependent DiFF $G_1^\perp$, as the most recent efforts to measure the asymmetries involving this function in back-to-back two hadron production in $e^+e^-$ annihilation produced no significant signal.
In this work we have described the helicity dependent two-hadron correlations in hadronization of a longitudinally polarized quark using the quark-jet hadronization framework. We derived the method for extracting the $G_1^\perp$ function from the number density of hadron pairs produced by a polarized quark in Sec.~\ref{SUBSEC_DIFF_EXTRACT}. Then, we used the NJL model input quark TMD splitting functions to perform MC simulations of the linearly polarized quark and calculated the corresponding number densities in Sec~\ref{SUBSEC_NJL_JET}. The results showed nonzero, but small signal for the helicity dependent DiFF, as depicted in Figs.~\ref{PLOT_D1_GP_PIPL_PIMI}(b) and \ref{PLOT_D1_GP_PIPL_PIMI_ZCUT}(b). The corresponding analyzing power is small and sharply peaked towards large values of $z$, and the cuts in $z$ often used in experimental setup only moderately enhance the signal in the medium-$z$ region. For comparison, the analyzing powers of pion Collins fragmentation functions, computed within the same model and plotted in Fig.~8(c) of Ref.~\cite{Matevosyan:2016fwi}, raise quickly and plateaux starting in the small-$z$ region at a value with equal or greater magnitude to the maximum of the analyzing power for the helicity dependent DiFF. These can help to explain the nonobservation of the signal in the existing experimental measurements~\cite{Abdesselam:2015nxn, Vossen:2015znm}. We also explored all the different pion pairs, showing that pairs of positively charged pions have a similar magnitude and opposite sign to the $G_1^\perp$ results expected for $\pi^+\pi^-$ pairs, making them good candidates for future studies.
Thus far, we discussed the first Fourier cosine moment of $G_1^\perp$ entering into $e^+e^-$ annihilation cross section, defined in \Eq{EQ_G1_MH}. The COMPASS Collaboration~\cite{Sirtl:2017rhi} has measured the SIDIS two hadron production asymmetries $A_{UL}^{\sin(\phi_h-\phi_R)}$ and $A_{UL}^{\sin(2\phi_h-2\phi_R)}$, which contain several of the cosine moments of $G_1^\perp$. The results of our calculations of the first five of these moments for $\pi^+\pi^-$ pairs for $N_L=6$, depicted in Fig.~\ref{PLOT_GP_MOM_PIPL_PIMI_ZCUT}, showed a significant decrease in the size of the signal with the increase of the moment $n$. These results supported the nonobservation of significant asymmetries in COMPASS measurements.
Finally, we validated our MC method in Sec.~\ref{SEC_VALIDATION}. Here we first derived expressions for the integrals of $D_1(z)$ over $z$ for different values of $N_L$, using combinatoric arguments. We showed, that these analytic results match extremely well with those obtained using the MC method. Further, we derived explicit relations for both $D_1(z)$ and $G_1^\perp(z)$ for the case of only two hadron production. Again, we found a perfect match to the results obtained using MC simulations. These steps validate our method completely. Moreover, there are two important insights obtained in that section. First, the results in Table~\ref{TABLE_N_PIPI} indicate quite rapid convergence of the results with a $z$-cut, as also seen seen from plots in Fig.~\ref{PLOT_D1_GP_PIPL_PIMI_ZCUT} for the $z$-unintegrated case. Second, \Eq{EQ_GP_NL2} provides an interesting insight into the microscopic mechanism for creating such two-hadron correlations with the longitudinal polarization of the fragmenting quark. Here the "worm-gear" type splitting function $G_T$ creates a correlation between the transverse spin of the intermediate quark in the hadronization process, which in turn is correlated with the transverse momentum of the second emitted hadron via Collins effect. Thus, it is safe to note, that the Collins effect has a crucial role also in creating this asymmetry.
Future work includes analogous studies of the DiFFs responsible for the two-hadron momentum correlations with the transverse spin of the quark. We performed a detailed study of the dependence of the unpolarized DiFF on both $z$ and $M_h^2$ within the quark-jet model in Ref.~\cite{Matevosyan:2013aka}, where the inclusion of the vector mesons and their strong decays proved crucial to describe the invariant mass spectrum. We also studied what are the relevant contributions of the "primary" emitted pions and koans versus those produced by the vector meson decays to the unpolarized DiFF. The recent experimental and MC studies of the unpolarized DiFFs by BELLE Collaboration in Ref.~\cite{Seidl:2015lla} strongly support these findings. The correlations between the $z$ and $M_h^2$ dependencies were also explored, and it has been shown that different $z$ regions emphasize the contributions of the different resonances in $M_h^2$ spectrum. The main motivation for exploring the $M_h^2$ dependence is to find the signatures of the interference effects between the two hadrons produced in the decays of the resonances, that would generate the polarization-dependent DiFFs. In our work we showed that the interference effects involving just the single hadron production (Collins effect) generates the helicity-dependent DiFF. In this work, we omitted the vector mesons altogether, as the first step in describing the polarized DiFFs within the self-consistent description of the polarized quark hadronization. Thus, we chose not to discuss the $M_h^2$ dependence knowing the obvious lack of the resonant structures there. We leave such detailed analysis to future work, when we consider the vector meson production and their strong decays. Finally, we very roughly mimicked the effects of QCD evolution on DiFFs in this work by using he $(1-z)^4$ ansatz for the input TMD splitting functions. In future work with a more complete model, we will use QCD evolution for critical comparisons of our results with the experiment.
\section*{ACKNOWLEDGEMENTS}
The work of H.H.M. and A.W.T. was supported by the Australian Research Council through the ARC Centre of Excellence for Particle Physics at the Terascale (CE110001104), and by an ARC Australian Laureate Fellowship FL0992247 and Discovery Project No. DP151103101, as well as by the University of Adelaide. A.K. was supported by A.I. Alikhanyan National Science Laboratory (YerPhI) Foundation, Yerevan, Armenia.
\bibliographystyle{apsrev4-1}
|
3,212,635,537,486 | arxiv | \section{Introduction}
Now generally accepted that about half of abundances of chemical elements heavier than iron are produced
by slow neutron captures (s--process) in the deep layers of asymptotic giant branch (AGB) stars.
Mixing brings freshly synthesized heavy elements to the stellar surface (the third dredge-up process)
and then stellar wind carries them away in the stellar enviroments. This is a simplified description of
the processes, the sequence of which enriches both circumstellar and interstellar medium with heavy metals.
For more detail see, e.g., \cite{Herwig,Kappeler} and references there. AGB stars are therefore
the principal suppliers of heavy metals and important suppliers of carbon and nitrogen to the
interstellar medium, thereby participating in the chemical evolution of galaxies. In this article,
we shall focus on the first observational evidences on the presence of circumstellar spectral features
of heavy metals in the optical spectra of nearest descendants of AGB-stars---protoplanetary nebulae (PPNe).
PPNe are objects at the post--asymptotic branch (post--AGB) stage of evolution.
These descendants of AGB stars are low--mass cores with typical masses of 0.6\,M$_{\odot}$ surrounded
by an extended and often structured gaseous--dusty envelope, which formed as a result of substantial mass
loss by the star during the preceding evolutionary stages.
Central stars are usually surrounded by envelopes in the form of extended haloes, arcs, lobes, and tori.
Some stars exhibit various combinations of the above features as well as bipolar and quadrupolar nebulae
with dust bars. Examples of the latter two types include the Egg nebula\,=\,RAFGL~2688 and {\it IRAS\/}\,19475+3119
nebulae, for which high-resolution images were taken by the Hubble Space Telescope~\cite{Siodmiak}.
All PPNe and about 80\% of planetary nebulae are asymmetric~\cite{Lagadec}. Much remains to be understood
about condensation of dust particles and formation of the dust fraction in AGB star envelopes
(see~\cite{Bieging} and references therein).
A circumstellar gas and dust envelope shows up in the form of peculiarities in the IR, radio, and optical
spectra of post-AGB supergiants. The optical spectra of PPNe differ from those of classical massive supergiants
by the presence of molecular bands superimposed onto the spectrum of an F--G supergiant and by the anomalous
behavior of the profiles of selected spectral features. These may include complex emission and absorption profiles
of H\,I, Na\,I, and He\,I lines, and metal emission features (see for detail~\cite{Envelope} and references
therein).
Furthermore, all these peculiarities are variable. In whole we see that the known types of spectral features
in the optical spectra of post--AGB stars are:
1) low-- or moderate--intensity symmetric metal absorptions without apparent distortions;
2) complex profiles of neutral hydrogen lines, which vary with time and include absorption and emission components;
3) absorption or emission bands of mostly carbon containing molecules; 4) envelope components of the Na\,I
and K\,I resonance lines, and also 5) narrow permitted or forbidden metal emission lines that form in envelopes.
The presence of type 2--5 features is the key difference of the spectra of PPNe from those of massive supergiants.
Here we analyze the manifestations of circumstellar envelopes in the optical spectra of PPNe focusing on the
homogeneous subsample of stars whose atmospheres, according to previous studies, underwent evolutionary
variations of the chemical composition. Section~2 briefly describes the employed observational
data and lists the studied stars and their basic parameters. In Section\,3
we analyze the available data on the peculiarities of the profiles of metal lines, found in high-resolution spectra,
as well as data on the presence of molecular bands and outflow velocities for objects with different envelope structures.
In Sections~4 and~5 we discuss the obtained results and summarize the main conclusions.
\section{OBSERVATIONAL DATA}
Over the past two decades more than 40 post-AGB candidates -- supergiants with IR excesses and several related
luminous stars with unclear evolutionary status have been spectroscopically monitored with the 6--m telescope
of the Special Astrophysical Observatory. As a result, a collection of high-quality spectra
has been acquired with the primary purpose of searching for anomalies of stellar chemical composition due
to the nucleosynthesis of chemical elements in the interiors of low- and intermediate-mass stars and the
subsequent dredge-up of the synthesis products to the surface layers of stellar atmospheres. These observational
data are also used to search for peculiarities in the PPNe spectra, to analyze the velocity fields in the atmospheres
and envelopes of these stars with mass loss, and to search for the likely long-term spectral and radial velocity pattern
variations.
Here we use the data acquired in the Nasmyth focus with the NES~\cite{nes} echelle spectrograph.
The NES spectrograph, equipped with a 2048$\times$2048 CCD and an image slicer,
produces a spectroscopic resolution of R$\approx60\,000$. Since 2011 the NES spectrograph has been equipped
with a 2048$\times$4096 CCD which made it possible to significantly extend the wavelength coverage.
The spectra of the faintest program objects (the optical component of the IR sources {\it IRAS}\,04296+3429 and 20000+3239)
were acquired with the PFES echelle spectrograph mounted in the
primary focus of the 6-m telescope~\cite{pfes}. This spectrograph, equipped with a 1k$\times$1k CCD,
produces a spectroscopic resolution of R$\approx$15\,000. We described the details of spectrophotometric
and position measurements of the spectra in our earlier papers, the corresponding references can be found
in the original papers listed in the paper by~\cite{Envelope}.
\begin{table}
\caption{Basic data for C--rich circumstellar envelopes of post-AGB stars. Details concerning the C$_2$ bands
in the third column see in the text. The last column gives the expansion velocity of the envelope
as determined from the position of Ba\,II circumstellar components.}
\medskip
\begin{tabular}{> \small c|> \small l|> \small l| > \small l|> \small l|> \small l}
\hline
Object & Morphology & Type of & \multicolumn{3}{c}{\small $V_{\rm exp}$, km\,s$^{-1}$} \\
\cline{4-6}
& of the & the C$_2$& & & \\[-10pt]
& envelope$^a$ & bands & CO & C$_2$ & BaII \\
\hline
04296+3429& bipolar\,+ & abs & 10.8$^b$ & 7.7$^g$ & \\ [-10pt]
& halo\,+ bar & emis & & 12$^h$ & \\
\hline
07134+1005& elongated & abs & 10.2$^b$ & 8.3$^g$ & \\ [-10pt]
& halo & abs & & 11$^i$ & \\
\hline
08005$-$2356& bipolar & uncertain& 100:$^c$ & 43.7$^g$ & \\ [-10pt]
& & abs & & 42$j$ & \\
\hline
19500$-$1709& bipolar & no &17.2, 29.5$^b$& &20 \& 30$^k$ \\ [-10pt]
& & &10, 30--40$^d$& & \\
\hline
20000+3239& elongated & abs & 12.0$^e$ & 12.8$^g$ & \\ [-10pt]
& halo & abs & & 11.1$^l$& \\
\hline
RAFGL\,2688 &multipolar\,+ & abs &17.9, 19.7$^f$& 17.3$^g$&\\ [-10pt]
& halo\,+\,arcs & emis & & 60$^m$ & \\
\hline
22223+4327& halo\,+ & abs. & 14--15$^f$& 15.0$^g$ & \\ [-10pt]
& small lobes & emis & & 15.2$^n$ & \\
\hline
22272+5435&elongated\,+ & abs &9.1--9.2$^b$& 9.1$^g$ & \\ [-10pt]
& halo\,+\,arcs & abs & & 10.8$^o$&10$^o$ \\
\hline
23304+6147& quadrupole\,+ & abs &9.2--10.3$^b$& 13.9$^g$& \\ [-10pt]
& halo\,+\,arcs & emis & & 15.5$^p$&15.1$^p$\\
\hline
\multicolumn{6}{l}{\small \it Notes: a -- morphology type of envelopes is taken from papers by \cite{Ueta2000,Sahai,Siodmiak,Lagadec};} \\ [-10pt]
\multicolumn{6}{l}{\small \it b -- \cite{Hrivnak}, c -- \cite{Hu}, d -- \cite{Bujar}, e -- \cite{Omont}, f -- \cite{Loup}, g -- \cite{Bakk97},}\\ [-10pt]
\multicolumn{6}{l}{\small \it h -- \cite{04296}, i --\cite{atlas}, j -- \cite{08005}, k -- \cite{19500}, l -- \cite{20000}, m -- \cite{Egg1},} \\ [-10pt]
\multicolumn{6}{l}{\small \it n -- \cite{22223}, o -- \cite{V354Lac}, p -- \cite{23304b}.} \\
\end{tabular}
\label{PPN}
\end{table}
\section{Main peculiarities in the optical spectra of post--AGB stars }
Our comprehensive study of the program stars allowed us to determine (or refine) their evolutionary status.
One of the results of our analysis is that the studied sample of luminous stars with IR excesses is not
homogeneous~\cite{VAK}. In this paper we consider the peculiarities of the optical spectra of
post--AGB stars paying special attention to the subsample of objects listed in Table\,\ref{PPN}.
It contains {\it IRAS} objects with central stars whose atmospheres are overabundant in carbon and
heavy metals. Their circumstellar envelopes have a complex morphology and are usually rich in carbon,
as evidenced by the presence of C$_2$, C$_3$, CN, CO, etc. molecular bands in their IR, radio, and
optical spectra. Presence and type of C$_2$ bands for stars in Table\,\ref{PPN} are published by
\cite{Bakk97,19500,08005,20000}; \cite{04296,Egg1,atlas,V354Lac,22223,23304b}.
Furthermore, the objects from Table\,\ref{PPN} are among those few PPNe whose IR--spectra exhibit the so far
unidentified emission band at 21\,$\mu$m~\cite{Kwok-21,Hrivnak2009}. Despite an extensive search for appropriate
chemical agents, so far no conclusive identification has been proposed for this rarely observed feature.
However, its very presence in the spectra of PPNe with carbon enriched envelopes suggests that this emission
may be due to the presence of a complex carbon-containing molecule in the envelope (see~\cite{Hrivnak2009,Li}
for details and references).
Overabundances of carbon and heavy-metal [s$/$Fe] overabundance (or lack thereof) in the atmosphere of the
central star were published earlier by \cite{04296,07134,08005,23304} for {\it IRAS}\,04296+3429,
07134+1005, 08005$-$2356, and 23304+6147 respectively; \cite{20000} for {\it IRAS}\,20000+3239; \cite{Egg2} and
\cite{Ishigaki} for RAFGL~2688; \cite{22223} for V448~Lac and \cite{V354Lac} for V354~Lac. Chemical abundances
for five stars from Table\,\ref{PPN} were also published by~\cite{Reyn}. As follows from Table\,\ref{Split},
all stars with excess carbon and heavy metals are objects with moderate iron deficiency and may belong to
the thick disk of the Galaxy.
The spectra of protoplanetary nebulae with F--K-type supergiant as central stars that have carbon-enriched atmospheres
show features of carbon-containing molecules C$_2$, C$_3$, CN, and CH$^+$. Position measurements of molecular
features in the spectra indicate that they form in expanding circumstellar envelopes. Authors~\cite{Bakk97} and
\cite{04296, atlas, 08005, 20000, Egg1, 22223, V354Lac, 23304b} used high-resolution optical spectra to analyze
molecular bands for several post--AGB stars including some objects from Table\,\ref{PPN}.
It appears that the emission in the Swan bands or Na\,I D--lines is observed in the spectra of PPNe with bright
and conspicuously asymmetric circumstellar nebulae. The results of spectroscopic observations of several PPNe
confirm this hypothesis. An analysis of the spectra taken with the 6--m telescope revealed C$_2$ Swan
emission bands of different intensities (relative to the continuum) in the spectra of the central stars of the
following sources: {\it IRAS}~04296+3429~\cite{04296}, 08005$-$2356~\cite{08005},
$RAFGL$~2688~\cite{Egg1}, {\it IRAS}~22223+4327~\cite{22223}, and {\it IRAS}~23304+6147~\cite{23304,23304b}.
According to HST images~\cite{Ueta2000,Siodmiak}, these objects have structured (and often bipolar) envelopes.
In addition to a sample of related C--rich objects with the above features, Table\,\ref{PPN} also includes the infrared
source {\it IRAS}~08005$-$2356. This object has so far been poorly studied, and no data are available either
on the peculiarities of the chemical composition of its atmosphere or on the presence of the 21-$\mu$m band.
However, {\it IRAS}~08005$-$2356 can be viewed as related to the objects of this sample
because its optical spectrum exhibits C$_2$~Swan bands, the hydrogen and metal lines in its spectrum have
emission-absorption profiles~\cite{08005}, and the circumstellar envelope is observed in CO emission~\cite{Hu}.
Note that our subsample of PPNe, thoroughly studied by high-resolution spectra (Table\,\ref{PPN}), practically
coincides with the list of C--rich and 21\,$\mu$m protoplanetary nebulae the photometric and spectral properties
of which were extensively studied in papers~\cite{Hrivnak2009,Hrivnak2010}. Of fundamental importance to us is the
conclusion in~\cite{Hrivnak2011} about the rare occurrence of binaries among post--AGB stars of the type considered,
which is based on the long-term investigation of the velocity field in PPN atmospheres conducted by the above authors.
Thus, the results of \cite{Hrivnak2009,Hrivnak2010,Hrivnak2011} provide further evidence for the homogeneity of
the PPNe subsample considered here.
\subsection{Peculiar metal absorptions in the optical spectra of selected post--AGB stars}
The systematic monitoring of PPN candidates, which we performed with a high spectral resolution, has
released a new result---an unknown earlier peculiarity: the splitting (or asymmetry) of strongest metal
absorptions due to distortion by envelope features. Now such peculiar profiles of the strongest absorptions
were revealed in the spectra of five stars listed in Table\,\ref{Split}: CY~CMi~\cite{atlas},
V354~Lac~\cite{V354Lac,V354LacK}, V448~Lac~\cite{22223}, V5112~Sgr~\cite{19500} and CGCS~6918
({\it IRAS}~23304+6147)~\cite{23304b}. Let us illustrate this effect using the example of the spectrum of the
high-latitude supergiant V5112~Sgr, where it is most pronounced. Figure\,\ref{V5112Sgr-Ba4934fr} shows
a fragment of the spectrum of V5112~Sgr taken on July~7, 2001 with a split Ba\,II~4934~\AA{} line.
At the same time, the profiles of strong absorptions of iron-group metals in the spectrum of this star
are neither asymmetric nor split, as is immediately apparent from the same Fig.\,\ref{V5112Sgr-Ba4934fr},
where we see a very strong but unsplit Fe\,II~4924~\AA{} line.
\begin{figure*}
\includegraphics[angle=-90,width=114mm,bb=30 80 550 790,clip]{fig1.ps}
\caption{Fragment of the spectrum for V5112~Sgr containing split Ba\,II\,4934\,\AA{} line in the July 7, 2001 spectrum.
The identification of main absorptions is indicated.}
\label{V5112Sgr-Ba4934fr}
\end{figure*}
Besides, a comparison of the line profiles in the spectra of V5112~Sgr taken during different nights
reveals substantial variability of the profile shapes and of the positions of the components of the split lines.
To illustrate the variability effect, we show in Fig.\,\ref{V5112Sgr-Ba4934} the Ba\,II~4934~\AA{} line profile,
which is most asymmetric and most variable. The different widths of the components are immediately apparent in both
Figs.\,\ref{V5112Sgr-Ba4934fr} and \ref{V5112Sgr-Ba4934}: the red component is about twice broader than the blue
components, which are offset substantially relative to the systemic velocity. This difference between the component
widths indicates that the red and blue components form under different physical conditions. It follows also from
Fig.\,\ref{V5112Sgr-Ba4934fr} that the position of the photospheric (red) component of the complex profile is variable,
whereas the blue components, which, as shown in~\cite{19500}, form in the envelope, are stable.
\begin{figure}
\includegraphics[width=74mm,bb=40 70 550 790,clip]{fig2.ps}
\caption{Variability of the Ba\,II~4934~\AA{} line profile in the spectra of V5112~Sgr taken in different years:
August 2, 2012 (the thin solid line); June 13, 2011 (the solid bold line); August 14, 2006 (the dotted line);
July 7, 2001 (the dashed line)~\cite{19500}.}
\label{V5112Sgr-Ba4934}
\end{figure}
\begin{figure}
\includegraphics[width=74mm,bb=30 70 550 780,clip]{fig3.ps}
\caption{Profile variations of the BaII\,6141\,\AA{} line in the spectra of V448~Lac at various epochs:
JD\,2454760.17 (bold), JD\,2454721.15 (thin), JD\,2453694.36 (dashed), and JD\,2452131.53 (dotted)}
\label{V448Lac-Ba6141}
\end{figure}
\begin{figure}
\includegraphics[width=64mm,bb=30 70 550 780,clip]{fig4.ps}
\caption{Profiles of the split lines in the October 12, 2013 spectrum of CGCS~6918: D$_2$\,NaI -- the bold line
and Ba\,II\,6141 -- the the thin line}
\label{Profiles-23304}
\end{figure}
\begin{table}
\caption{Main atmospheric parameters of post--AGB stars with peculiar profiles of strongest absorptions: effective temperature, Teff,
metallicity, [Fe/H]$_{\odot}$, and the mean excess of heavy metals synthesized in s--process, [s/Fe]$_{\odot}$}
\medskip
\begin{tabular}{> \small r > \small c > \small c > \small c> \small c > \small c}
\hline
Star & {\it IRAS} & Teff, {\it K} & [Fe/H]$_{\odot}$ & [s/Fe]$_{\odot}$ & Ref \\
\hline
CY~CMi & 07134+1005 & 7000 &$-1.0$ & +1.4 & a \\ [-10pt]
V5112~Sgr & 19500$-$1709& 8000 &$-0.6$ & +1.1 & b\\ [-10pt]
V448~Lac & 22223+4327 & 6500 &$-0.3$ & +0.9 & b \\ [-10pt]
V354~Lac & 22272+5435 & 5650 &$-0.8$ & +1.2 & c \\ [-10pt]
CGCS~6918 & 23304+6147 & 5900 &$-0.6$ & +1.2 & d \\
\hline
\multicolumn{6}{l}{\small \it a -- \cite{07134}, b -- \cite{Reyn}, c -- \cite{V354Lac}, d -- \cite{23304}.} \\
\end{tabular}
\label{Split}
\end{table}
The semiregular variable V354~Lac is the closest analog to V5112~Sgr among the objects listed in Tables\,\ref{PPN}
and \ref{Split} in terms of the structured envelope and chemical abundances. The spectroscopic monitoring of
this star~\cite{V354Lac} carried out at the SAO 6--m telescope with a resolution of $R=60\,000$ also revealed
the splitting of the strongest absorptions with the low-level excitation potential of $\chi_{\rm low}\le 1$~eV.
An analysis of the kinematic pattern showed that the blue component of the split line forms in the powerful gas
and dust envelope of V354~Lac. This splitting shows up most conspicuously in the profile of the strong
Ba\,II~6141~\AA{} line. The shift of the blue component of the Ba\,II line coincides with that of the circumstellar
component of Na\,I D lines, which forms in the same layers as the circumstellar C$_2$ Swan bands. This coincidence
indicates that the complex profile of the Ba\,II~6141~\AA{} line contains, in addition to the atmospheric component,
a component that forms in the circumstellar envelope.
Such splitting (or the profile asymmetry due to the more shallow slope of the blue wing) is also observed
for other Ba\,II ($\lambda\lambda$\,4554, 5853, 6496~\AA{}) lines as well as for the strong Y\,II~5402~\AA{},
La\,II~6390~\AA{}, and Nd\,II~5234, 5293~\AA{} lines. The lines of these ions in the spectrum of V354\,Lac are
enhanced to the extent that their intensities are comparable to those of neutral-hydrogen lines. For example,
the equivalent width of the Ba\,II~6141~\AA{} line reaches $W_{\lambda}\approx1$~\AA{}, and that of H$\beta$
reaches $W_{\lambda}\approx2.5$~\AA{}.
As follows from Figs.\,\ref{V448Lac-Ba6141} and \ref{Profiles-23304}, in the spectra of two stars---V448~Lac and
CGCS~6918 -- a central star of {\it IRAS}\,23304, the Ba\,II~6141~\AA{} line profile is or asymmetric or split for
different moments of observations. Note that the variations are displayed only by the short-wavelength profile
wings and the position of the core, whereas the positions and intensities of the long-wavelength wings of these
strong Ba\,II, La\,II, and Y\,II absorption lines do not vary in time.
Asymmetric and variable profiles of strongest lines (Y\,II, Ba\,II, Fe\,II and other strong absorption lines)
formed in the expanding stellar atmosphere were also found in~\cite{56126} in the spectra of CY~CMi.
However, as follows from Fig.\,5 in the paper by these authors, in the spectra of CY~CMi profiles are just
only asymmetric, but not split. Thus, we see a difference between the peculiarity types of the profiles of
three stars---V5112~Sgr, V354~Lac, and CGCS~6918 with split profiles of the strongest
absorptions of selected elements, and two stars---CY~CMi and V448~Lac with asymmetric but unsplit profiles.
Such a difference permit us to suggest that the morphology of the circumstellar envelope may be the main
factor that causes the peculiarity and variability of the profiles of the strongest lines. As is evident
from Table\,\ref{PPN}, the two stars V5112~Sgr, V354~Lac and CGCS~6918 with split absorptions have bipolar
(or quadrupolar) envelopes, whereas the absorptions are unsplit in the spectra of CY~CMi and V448~Lac
the envelope of these stars have a less structured environment. This hypothesis is further corroborated
by the three-component structure of the strong absorption profiles observed in the spectrum of V5112~Sgr,
where CO--observations show both the slow (V$_{\rm exp}$\,=\,10\,km\,s$^{-1}$) and the fast
($30$--$40$\,km\,s$^{-1}$) expansion~\cite{Bujar}. As follows from data in Table\,\ref{PPN},
the profiles of the split lines include a photospheric component and two envelope components, one of which,
like in the case of the CO--profile, arises in the envelope that formed at the AGB--stage and expands at
a velocity of V$_{\rm exp}(2)\approx 20$~km\,s$^{-1}$, and the other one arises in the
envelope that moves at a velocity of V$_{\rm exp}(1)\approx 30$~km\,s$^{-1}$ and formed later.
Worth mentioning here is another interesting phenomenon which was revealed in the spectrum of V5112~Sgr.
Klochkova~\cite{19500} showed that the optical spectrum of this high latitude post--AGB supergiant contains
weak absorptions whose positions are indicative of their formation in the circumstellar envelope. The velocities
V$_r$(DB) were measured in the 5780--6379~\AA{} wavelength interval from the positions of reliably identified 5780,
5797, 6196, 6234, and 6379~\AA{} features. The mean V$_r$(DIB) averaged over several spectra and determined with
an accuracy better than $\pm 0.5$~km\,s$^{-1}$ agrees excellently with the velocity determined from the blue
component of the Na\,I D--lines. The resulting agreement leads us to conclude that the weak bands found in the
spectrum of V5112\,Sgr origin in the circumstellar envelope. Kipper~\cite{Kipper2013} independently came to
similar conclusions for the same object based on the spectra of V5112~Sgr taken with a different instrument.
This DIBs identification at the present time is a unique result in the task of searching for these spectral
features in the circumstellar environment.
\section{Discussion of the results}
Note that the splitting of strong absorptions were observed in the spectra only those post--AGB stars that
are listed in the Table\,\ref{Split}, in atmospheres which were found significant excess carbon and heavy metals.
The second precondition for the splitting of the strong absorptions in the spectrum of these post--AGB stars
is presence of a complex structured envelope. We can suggest that the process of the formation
of a circumstellar envelope may contribute to the enrichment of this envelope by products of stellar nucleosynthesis.
The profiles of the split lines contain a photospheric and one--two envelope components, one of which,
like in the case of the CO--profile, arises in the envelope that formed at the AGB--stage, and the
other one---in the envelope that formed later. The circumstellar components of strong heavy-metal absorptions
have been conclusively identified in the spectra of V5112~Sgr~\cite{19500}, V354~Lac~\cite{V354Lac} and
CGCS~6918~\cite{23304b}.
The structure of the circumstellar nebulae of V354\,Lac and CGCS\,6918 may be more complex than it appears
in the HST observations (see Table\,\ref{PPN}). Polarimetric observations of V354\,Lac~\cite{Gledhill}
indicate the presence of a ring structure embedded in an extended nebula. Nakashima et al.~\cite{Nakashima2012} point out
that the axes of the optical and infrared images of the nebula are almost perpendicular to each other.
Based on the kinematic pattern of the nebula as determined from the mapping of the CO--emission,
authors~\cite{Nakashima2012} concluded that the structure of the nebula includes not only a torus and a spherical
component but also another element (possibly a jet).
Currently, no consensus has been reached concerning the development of deviations from spherical symmetry
in PPNe. Authors~\cite{Ueta2000, Siodmiak} analyzed high spatial resolution optical images of a sample
of PPNe taken by the Hubble Space Telescope and concluded that the optical depth of the circumstellar matter
is the crucial factor that determines the formation of a particular morphology of stellar envelopes.
The dense and often spherical envelope that formed during the AGB--stage is believed
to expand slowly, whereas the rapidly expanding feature is the axisymmetric part of the envelope that
formed later at the post--AGB stage~\cite{Castro2010}.
The sequence of these processes results in the development of an optical depth gradient in the direction
from the equator to the polar axis of the system. The presence of a companion and/or a magnetic field in
the system may also prove to be the physical factor that causes the loss of the spherical symmetry of
the stellar envelope during the short evolutionary interval between the AGB and post--AGB stages
(see~\cite{Huggins,Ferreira} and references therein).
In their recent paper~\cite{Koning} proposed a simple PPN model based on a pair of evacuated cavities
inside a dense spherical halo. The above authors demonstrated that all the morphological features observed
in real bipolar PPNe can be reproduced by varying the available parameters (mass density inside the cavity,
its size and orientation) of this model.
So far, the discovery of the heavy-metal enrichment of the circumstellar envelopes of the post--AGB supergiants
V5112~Sgr, V354~Lac and CGCS~6918 remain the only results. Here it should be recalled that Mauron \& Huggins~\cite{Mauron}
detected atomic metals (Ca, Fe and upper limits for Al, Ti, Mn and Sr) in the C--rich circumstellar envelope
of the AGB--star CW~Leo, which is a central star of the IR source {\it IRC}\,+10216.
The effect of splitting of metallic lines found requires to continue the high resolution spectroscopy of
very related post--AGB stars. The most promising objects could be {\it IRAS}~04296+3429 and {\it RAFGL}~2688.
It follows from Table\,\ref{PPN} that the totality of properties of their envelopes coincide with
those of the objects listed in Table\,\ref{Split}: they are C--enriched and have a very complex structure.
Besides, the atmospheres of the weak central stars of both sources are enriched in heavy metals~\cite{04296,Egg1}.
Evidently, the next step in study of the discovered splitting of the profiles should be theoretical modeling
of the spectra envelopes and calculation chemical abundances, which could be consider with regard to selective
depletion of chemical elements in a dusty environment.
\section{Main coclusions}
We used the results of high spectral resolution observations made with the 6--m telescope to analyze
the peculiarities of the optical spectra of a sample of post--AGB stars with atmospheres enriched in carbon and heavy
s--process metals and with carbon-enriched circumstellar envelopes.
We showed that presence of the peculiarities of the line profiles (the asymmetry and splitting of the profiles of strong absorptions)
is associated with the kinematic and chemical properties of the circumstellar envelope and the type of its morphology.
The splitting of the profiles of the strongest heavy-metal absorptions in the spectra of the V5112~Sgr, V354~Lac
and CGCS~6918 supergiants found as a result of our observations suggests that the formation of a structured circumstellar
envelope is accompanied by the enrichment of this envelope with the products of stellar nucleosynthesis.
Attempts to find a definite link between the peculiarities of the optical spectrum and the morphology of the circumstellar
environment are complicated by the fact that the observed structure of the envelope depends strongly on the inclination
of the symmetry axis to the line of sight and on the angular resolution of the spectroscopic and direct-imaging instruments.
In the spectrum of the high latitude post--AGB supergiant V5112\,Sgr were revealed weak absorptional DIBs 5780,
5797, 6196, 6234, and 6379~\AA{}. Their mean radial velocity V$_r$(DIBs) determined with an accuracy better than
$\pm 0.5$~km\,s$^{-1}$ agrees excellently with the velocity determined from the blue circumstellar component of
the Na\,I D--lines.
\section*{Acknowledgments}
This work was supported by the Russian Foundation for Basic Research (project No.\,14--02--00291\,a).
This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France, and NASA's Astrophysics Data System.
|
3,212,635,537,487 | arxiv | \section{Introduction}
\label{sec-intro}
Although class II methanol masers are now generally accepted to be
exclusively associated with massive star forming regions \citep[see
eg.][]{ellingsen2006, xu2008}, it is not yet clear how much can be
learned about the star formation environment from the masers. Underlying
this uncertainty is the fact that it is difficult to determine where in
the circumstellar environment the masers operate based solely on the
spatial distribution and velocity structure of the masers
\citep{beuther2002}. In the past very little attention has been given to
the time domain aspect of the masers. Various authors \citep[see
eg.][]{macleod1993, caswell1995, moscadelli1996, goedhart2002,
niezurawska2002, goedhart2003, goedhart2004, goedhart2005a} have
studied the variability of the 6.7 and 12.2 GHz masers over various
periods of time and have found that variability is a common feature
amongst these masers. The most systematic study on the variability of
6.7 GHz masers to date was done by \citet{goedhart2004} using the
Hartebeesthoek Radio Astronomy Observatory (HartRAO) 26-m telescope,
revealing a wide variety of time-dependent behaviour in the masers. Six
out of their list of 54 sources were identified as periodic, with
periods ranging from 133 to 504 days \citep{goedhart2007}. Of particular
interest here are the periodic masers in the star forming region
G9.62+0.20E which show repeated flaring activity with a period of about
244 days \citep{goedhart2003, gaylard2007}
The star forming region G9.62+0.20 has been the object of study by a
number of authors. \citet{garay1993} mapped this region with the VLA at
1.5, 4.9 and 15.0 GHz and identified five extended, compact and
ultracompact (UC) H\,{\sevensize II} regions, labelled A - E. A striking feature in
this star forming complex is the near perfect alignment of various
tracers (masers, hot molecular clumps and UC H\,{\sevensize II} regions) of star
forming activity extending between components C and D, with component E
lying between C and D \citep[see][]{hofner1994}. \citet{hofner1994}
interpreted the overall morphology of this star forming region as
suggestive of induced star formation progressing from the most evolved
H\,{\sevensize II} region (component A) with the most recently formed stars located in
the linear structure. The methanol masers in the star forming
complex are associated with components D and E and are in the
literature also referred to as G9.619+0.193 and G9.621+0.196
respectively \citep{phillips1998}. For component E, the masers have a
maximum linear extent of about 566 AU (2.75 milliparsec) based on the
12.2 GHz observations of \citet{goedhart2005b} and using a distance of
5.15 kpc (A. Sanna et al., in preparation).
Component E shows all the signs of a very early phase of massive star
formation. According to \citet{hofner1996} the continuum emission
follows a power law with a spectral index of $1.1 \pm 0.3$ between 2 cm
and 2.7 mm, which can be interpreted to be due to either an ionized
spherical stellar wind or a spherical homogeneous UC H\,{\sevensize II}
region with an excess of dust emission at 2.7 mm. If interpreted as an
UC H\,{\sevensize II} region, the size of the H\,{\sevensize II} region is only 2.5 mpc and is
excited by a B1 ZAMS star \citep{hofner1996}. \citet{franco2000} quote a
slightly flatter spectral energy distribution between 8.4 and 110 GHz,
with a spectral index of $0.95 \pm 0.06$ which they attribute to a
density gradient proportional to $r^{-2.5}$. Using the 1.3 cm data of
\citet{testi2000} for G9.62+0.20E, \citet{franco2000} derives an
electron density of about $6 \times 10^5~\mathrm{cm^{-3}}$ which could
rise to above $10^6~\mathrm{cm^{-3}}$ in a uniform core region. Taken
together with the size of the H\,{\sevensize II} region, G9.62+0.20E qualifies as a
hypercompact H\,{\sevensize II} region, implying a very early phase of massive star
formation.
Component E is also well defined in a number of sulphur and nitrogen
bearing molecules as well as in some organic molecules
\citep{su2005}. Near-infrared imaging of the star forming region shows
emission at components B, C and E \citep{testi1998}. \citet{persi2003}
subsequently found that the NIR source near component E has the colours
of a foreground star. \citet{debuizer2005} find weak emission at 11.7
$\mu$m towards the radio component E.
In this paper we present further monitoring data on G9.62+0.20E for
methanol masers at 6.7, 12.2, and 107 GHz. We present various analyses
of the data and consider a possible explanation for the variability
of the masers.
\section[]{Observations and data reduction}
\label{sec-obs}
{\it HartRAO observations:} Observations at 6.7 GHz and 12.2 GHz were
made with the 26-m Hartebeesthoek telescope. Both receivers provide
dual circular polarization. The system temperatures at zenith are
typically 60 K at 6.7 GHz and 100 K at 12.2 GHz. The spectrometer
provides 1024 channels per polarization, and the bandwidths used for
spectroscopy of 1 MHz at 6.7 GHz and 2 MHz at 12 GHz provide spectral
resolutions of 0.044 and 0.048 km s$^{-1}$ respectively.
For spectroscopic observing, the telescope pointing error was determined
through five short integrations at the cardinal half power points of the
beam and on source. The on-source spectra were scaled up by determining
the amplitude correction from the pointing error using a Gaussian model
for the main beam.
Calibration was based on monitoring of 3C123, 3C218 and Virgo A (which
is bright but partly resolved), using the flux scale of \citet{ott1994}.
The antenna temperature from these continuum calibrators was measured by
drift scans. Pointing errors in the north-south direction were measured
via drift scans at the beam half power points, and the on-source
amplitude was corrected using the Gaussian beam model. The bright
methanol maser source G351.42+0.64, which previous monitoring has shown
to exhibit only small variations in the strongest features
\citep{goedhart2004}, was observed in the same way as G9.62+0.20 to
provide a consistency check on the spectroscopy. These sources were
observed frequently, usually on the same days as G9.62+0.20.
{\it ARO 12m observations:} The observations with the ARO 12m
telescope\footnote{The Kitt Peak 12 Meter telescope is operated by the
Arizona Radio Observatory (ARO), Steward Observatory, University of
Arizona. } were done from December 3, 2003 to June 25, 2004 to observe
cycle 8 and from January 14, 2005 to April 22, 2005 to observe cycle
9. The peak for cycle 8 was expected in early July 2004. Poor
atmospheric conditions forced any further observations to be abandoned
after June 2004. The onset of flare 9 was expected to occur around
February 7, 2005 with a peak between March 7 - 13 and to decay to its
low state by middle April 2005. The receivers used were dual-channel SIS
mixers operated in single-sideband mode. The backends were filter banks
with 100kHz and 250 KHz resolution. The observing frequency was
107.01385 GHz. Each observing run lasted about three hours, during
which regular positional checking on planets were done.
DR-21 was used as calibrator as well as a reference source and was
observed in spectral line mode. Data reduction was done with the CLASS
package. Fig.\ref{fig:dr21_ts} shows the time series for DR-21.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[width=84mm,clip,angle=-90]{fig1.eps}}
\caption{Time series for the continuum emission of DR-21 at 107 GHz}
\label{fig:dr21_ts}
\end{figure}
The results are for the 250 KHz resolution and each point is the average
over all channels. We estimated the flux density
of DR-21 at 107 GHz using the continuum spectrum given by
\citet{righini1976}, which gives a flux density of 16.7 Jy. For each
observing run an appropriate conversion factor was calculated and
applied to the corrected antenna temperature of G9.62+0.20E.
\section{Results and analysis}
The average or representative spectra for the three maser transitions
are shown in Fig. \ref{fig:avgspec}. It is seen that, whereas for the
6.7 and 12.2 GHz spectra the strongest feature is located at about 1.3
$\mathrm{km~s^{-1}}$, the corresponding feature in the 107 GHz spectrum
is the weaker maser feature. The 107 GHz spectrum also shows a broad
thermal component. G9.62+0.20E has in the past been observed at 107 GHz
by \citet{valtts1995, valtts1999} and \citet{caswell2000}. The average
107 GHz spectrum in Fig. \ref{fig:avgspec} resembles the result of
\citet{caswell2000}. The observations of \citet{valtts1995, valtts1999}
show only one maser feature at -0.57 $\mathrm{km~s^{-1}}$ and also no
broad thermal component.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[width=84mm,clip,angle=0]{fig2.eps}}
\caption{Representative spectra for the masers at 6.7, 12.2 GHz. The 107
GHz spectrum is an average taken over all observations. The features
at 5.3 and 6.4 $\mathrm{km~s^{-1}}$ in the 6.7 GHz spectrum are
associated with G9.62+0.20D.}
\label{fig:avgspec}
\end{figure}
The full time series for all the identifiable features in the 6.7 and
12.2 GHz spectra are shown in Figs. \ref{fig:67ts} and
\ref{fig:122ts}. For the 6.7 GHz maser, eleven different features could be
identified in the single dish spectrum and ten for the 12.2 GHz
maser. The time series covers about 2670 days and 13 flares. Inspection
of Fig. \ref{fig:67ts} shows that while there are some similarities in
the time series of some of the 6.7 GHz maser features, there also are
some significant differences.
The strongest varying 12.2 GHz masers are significantly more
variable than the strongest varying 6.7 GHz masers. To quantify the
variability we calculated the relative amplitude, defined as
\begin{equation}
R = \frac{S_{max} - S_{min}}{S_{min}} = \frac{S_{max}}{S_{min}} - 1
\label{eq:r}
\end{equation}
for a number of features in the three transitions. The results are
given in Table \ref{tab:amplitudes}. The errors indicate the spread in
the relative amplitudes for the last six flares in the time series. No
error could be given for the single 107 GHz flare that has been
monitored. The 12.2 GHz
masers are indeed significantly more variable than the 6.7 and 107 GHz
masers while the relative amplitudes of the 6.7 and 107 GHz masers are
approximately the same.
\begin{table*}
\begin{minipage}{120mm}
\caption{Relative amplitudes (eq. \ref{eq:r}) for selected masers at 6.7, 12.2 and 107 GHz}
\label{tab:amplitudes}
\begin{tabular}{cccccc}
\hline
\multicolumn{2}{c}{6.7 GHz} & \multicolumn{2}{c}{12.2 GHz} & \multicolumn{2}{c}{107 GHz} \\
\hline
Velocity & Relative & Velocity & Relative & Velocity & Relative \\
$\mathrm{km\, s^{-1}}$ & amplitude & $\mathrm{km\, s^{-1}}$ & amplitude& $\mathrm{km\,
s^{-1}}$ & amplitude\\
1.18 & 0.26 $\pm$ 0.04 & 1.25 & 2.0 $\pm$ 0.4 & 1.14 & 0.31 \\
1.84 & 0.32 $\pm$ 0.05 & 1.63 & 2.4 $\pm$ 0.3 & & \\
2.24 & 0.33 $\pm$ 0.04 & & & & \\
\hline
\end{tabular}
\label{table:r}
\end{minipage}
\end{table*}
Figures \ref{fig:67ts} and \ref{fig:122ts} also show that the regular
flaring does not occur in all of the 6.7 and 12.2 GHz maser
features. For the 12.2 GHz masers it is seen that the flaring behaviour
is limited to three features namely $+1.25 ~\mathrm{km\:s^{-1}}$, $+1.63
~\mathrm{km\:s^{-1}}$ and $+2.13~ \mathrm{km\:s^{-1}}$. The strongest
flaring occurs at +1.25 and +1.63 $\mathrm{km\:s^{-1}}$. These are also
the strongest two features in the maser spectrum. The absence of flaring
behaviour outside the above velocity range is quite obvious. Because of
the smaller variability of the 6.7 GHz masers the question of which
features are flaring is not as simple as for the 12.2 GHz masers. Simple
visual inspection suggests that flaring occurs for the features at -0.2,
+1.2, +1.8, and +2.2 $\mathrm{km\:s^{-1}}$.
It is noteworthy that the most variable and brightest 12.2 GHz masers, at
+1.63 and +1.25 $\mathrm{km\:s^{-1}}$, lie close to each other at the
northern end of a north-south chain of maser features as seen in the
high resolution maps of \citet{goedhart2005b}. As already pointed out,
these are also the two brightest masers in the 12.2 GHz spectrum. The
maser features that show no periodic flaring lie significantly offset to
the south from the above two features.
We also checked for periodic variations in the light curves using the
Lomb-Scargle periodogram \citep{scargle1982} and Davies' L-statistic
\citep{davies1990}. For the 12.2 GHz masers both methods confirm the
existence of a periodic signal with period 244 days in the light curves
of the $+1.25~\mathrm{km\:s^{-1}}$, $+1.63 \mathrm{km\:s^{-1}}$ and
$+2.13~\mathrm{km\:s^{-1}}$ features at a very high level of
significance. For the 6.7 GHz masers the periodograms were significantly
more complicated than that of the 12.2 GHz masers. A full analysis and
discussion of the periodograms falls outside the scope of the present
paper and will not be discussed here. Within the context of the present
paper it is necessary to note that for the 12.2 GHz masers,
periodic (regular) flaring is observed only in three features. For the
6.7 GHz masers, flaring behaviour is strongest also for only a subset of
the maser features, some of which coincide in velocity with the 12.2 GHz
masers that show flaring behaviour.
\begin{figure*}
\centering
\includegraphics[width=200mm,clip,angle=270]{fig3.eps}
\caption{6.7 GHz time series}
\label{fig:67ts}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=200mm,clip,angle=270]{fig4.eps}
\caption{12.2 GHz time series}
\label{fig:122ts}
\end{figure*}
In Fig. \ref{fig:ts} we show the time series for cycles 9 and 10 for
which the 107 GHz masers have also been monitored. The vertical dashed
lines give the position of the peak of the flare at 12.2 GHz for the
1.25 $\mathrm{km\:s^{-1}}$ feature. For cycle 9 the scatter in the 107
GHz data is quite large and, although there might be a slight hint of a
rise in the flux density towards the expected time when the flare was at
its peak, no clear sign of a flare can be identified. The reason for the
large scatter in the data is most probably the poor atmospheric
conditions that prevailed at that time and which led to the termination
of the monitoring for cycle 9.
Closer inspection of the time series in Figs. \ref{fig:67ts},
\ref{fig:122ts}, and \ref{fig:ts} suggests that the flare profiles for
the 6.7, 12.2 and 107 GHz masers might be different. To investigate to
what extent the flare profiles are similar we first considered the
average flare profiles by calculating the normalized profile
\begin{equation}
J(t) = \frac{S(t) - S_{min}}{S_{max} - S_{min}}
\label{eq:jt}
\end{equation}
for the +1.25 $\mathrm{km\:s^{-1}}$ feature of the 12.2 GHz spectrum and
for the corresponding feature at 6.7 GHz. For both these features a 30
point running average was used after detrending the time series and
folding of the data modulo the period of 244 days. The 107 GHz data for
the one flare were scaled accordingly. The results are shown in
Fig. \ref{fig:flareprofile}. In spite of the fact that the average
profiles do not coincide exactly in time, it is remarkable to note that
the profiles seem to be very similar for the three maser transitions.
\begin{figure}
\centering
\includegraphics[width=84mm,clip,angle=0]{fig5.eps}
\caption{Time series for masers at 6.7, 12.2 and 107 GHz. For the 6.7
and 12.2 GHz masers only one velocity component has been
considered. The solid dashed vertical lines indicate the position of
the peak of the 12.2 GHz maser.}
\label{fig:ts}
\end{figure}
\begin{figure}
\includegraphics[width=85mm,clip,angle=0]{fig6.eps}
\caption{Comparison of the average flare profile for the 1.24
$\mathrm{km~s^{-1}}$ feature for the 6.7, 12.2, and 107 GHz
masers. For the 6.7 and 12.2 GHz masers the flare profile was obtained
from a 30 point running average.}
\label{fig:flareprofile}
\end{figure}
The same analysis as above was also carried out on individual 6.7 and
12.2 GHz flares to examine to what extent individual flares have the
same profile. The result is shown in Fig. \ref{fig:jt}. For clarity only
the flares that occured after MJD 52750 are shown and only for the
features at 1.25 $\mathrm{km\:s^{-1}}$. In spite of the large errors on
the 6.7 GHz data, it is seen that the flare profiles are basically the
same, also for individual flares. The similarity of the individual
flare profiles suggests that the characteristics of the mechanism
underlying the flaring remain the same from flare to flare and that it
affects the 6.7 and 12.2 GHz masers in the same way.
\begin{figure}
\centering
\includegraphics[width=85mm,clip]{fig7.eps}
\caption{Comparison of the individual normalized flare profiles for the
1.24 $\mathrm{km~s^{-1}}$ feature for the 6.7 and 12.2 GHz masers for
MJD $>$ 52750}
\label{fig:jt}
\end{figure}
We finally note that within the velocity resolution of the observations,
no velocity shifts of the different feature in the maser spectrum could
be detected during the flares.
\section{Discussion}
\citet{goedhart2005a} briefly considered a number of possible mechanisms
that might underlie the flaring behaviour of the masers in
G9.62+0.20E. These included: disturbances like shock waves or clumps of
matter passing through the masing region; variations in the flux of seed
photons or of pump photons due to stellar pulsations; periodic
outbursts, or the effects of a binary system. While it is not possible
to come to a final conclusion about the origin of the periodicity of the
masers with the present data, it is possible to at least exclude certain
mechanisms and point to possible explanations for the periodicity.
It has already been noted that the return of the masers to basically the
same quiescent level between flares is a strong indication that the
masing regions remain unaffected by whatever mechanism underlies the
flaring. Mechanisms such as shock waves or clumps of matter passing
through the masing region are therefore excluded. This implies that
whatever the underlying mechanism for the periodicity, the coupling with
the masing regions must be radiative, ie. either through the seed
photons or through heating and cooling of the dust that is responsible
for the pumping radiation field.
At present the information available on the masers are the single dish
light curves (as presented above) and the high resolution
interferometric mapping of \citet{goedhart2005b}. We consider the light
curve (flare profile) only since in general the light curves of periodic
sources contain significant information about the physical processes
that are responsible for the observed variability. We also present an
analysis that strongly suggests that the decay part of the light curve
might be due to the background H\,{\sevensize II} region decaying from a higher to a
lower state of ionization. It is necessary to point out here that since
the relative positions of the masers with respect to the H\,{\sevensize II} region is
not known to milli-arcsecond accuracy, a more complete understanding of
all aspects of the variability of the masers in G9.62+0.20E cannot be
given here.
For the present discussion we will focus on the strongest maser features
and take the average flare profile of the 12.2 GHz masers
(Fig. \ref{fig:flareprofile}) as representative of all flares. The flare
profile can be described by a rather steep rise followed by a decaying
part lasting about 100 days to eventually reach a minimum (or possibly
quiescent) state after which it appears to slowly increase before the
next flare starts again. Since, as we have already argued that effects
that physically affect the masing region can be ruled out, we are
basically left with two possibilities to understand the light curve:
stellar pulsation and a binary system.
A comparison of the maser light curves as described above with the light
curves of all classes of pulsating stars clearly shows that the maser
light curve has a completely different behaviour from that of any class
of pulsating stars. In fact, considering the physics of stellar
pulsation it is hard to see how to produce a light curve for a pulsating
star such that it resembles that of the masers in G9.62+0.20E. For the
masers the light curve is strongly suggestive of a flaring state that is
superposed on top of a base level emission. Pulsating stars are not in an
equilibrium state and therefore do not have such a quiescent state on
top of which pulsations or flares are superposed. Explaining the
observed maser light curves as being due to pulsation of the central
star would therefore appear to be very difficult.
\begin{figure}
\centering
\includegraphics[width=80mm,clip,angle=0]{fig8.eps}
\caption{Time dependence of $r^{-1}$ for different eccentricities. }
\label{fig:profile}
\end{figure}
The only alternative is that of a binary system. The light curve does
not suggest that we are dealing with an eclipsing effect for the masers.
A class of binary systems which potentially can meet the requirement of
providing radiative coupling between the source of variability and the
masing regions is the colliding-wind binaries. In fact,
\citet{zhekov1994} argued that, given the high binary frequency found in
young stars and the observed mass loss rates and wind velocities,
supersonically colliding-wind systems, should also be found amongst
pre-main sequence binary systems. Although it is not possible with the
available data to come to a conclusion here whether or not G9.62+0.20E
indeed harbours a colliding-wind binary, it is worth exploring this
possibility somewhat more by considering some properties of
colliding-wind binaries.
The main ingredient of a colliding-wind binary is the two oppositely
facing shocks separated by a contact discontinuity. The postshock
temperature is given by $T_{sh} = 3mv_w^2/16k$ \citep{stevens1992,
zhekov1994}, where $m$ is the mean mass of the particles that
constitute the wind, and $v_w$ the wind speed. Given that the central
star for G9.62+0.20E is estimated to be of spectral type B1 and using
the results of \citet{bernabeu1989}, a wind speed of
$800~\mathrm{km~s^{-1}}$ does not seem to be an unrealistic
assumption. Using this value for $v_w$ and assuming for simplicity a
pure hydrogen wind, a postshock temperature of $\sim 1.5 \times 10^7$ K
is found. \citet{pittard2005} presented the emissitivity for hot thermal
plasmas of temperatures $10^6$, $10^7$, and $10^8$ K, from which an
extrapolation to energies below 100 eV suggests that the post-shock gas
will emit photons from the visible up to X-ray energies. Using the
CHIANTI code \citep{dere1997, landi2006}, we confirmed that this is
indeed the case. This range of photon energies obviously includes
photons that can heat the inner edge of the circumstellar dust as well
as ionizing photons that can cause additional ionization in the H\,{\sevensize II}
region, respectively affecting the pumping radiation field and/or the
background source of seed photons. Due to the lack of a numerical model
it was not possible to calculate expected absolute fluxes of ionizing
photons produced at the shocks as well as of photons that can heat the
dust. A quantitative evaluation of the effects these photons might
have on the H\,{\sevensize II} region and the dust was therefor not possible.
A second property of colliding-wind systems relevant to this discussion,
is that the total luminosity at the shock scales like $r^{-1}$, where
$r$ is the distance between the two stars \citep{stevens1992,
zhekov1994}. Obviously this scaling property implies a modulating
effect of the total luminosity at the shock during the orbital motion
for eccentric orbits. To investigate the time dependence of $r^{-1}$,
we used Kepler's third law to calculate the semi-major axis for the
binary system. For this we used a mass of 17 $\mathrm{M_\odot} $ for the ionizing
star (spectral type B1) and arbitrarily adopted a mass of 8 $\mathrm{M_\odot} $ for
the secondary star. Since in Kepler's third law the semi-major axis
depends on the inverse of the third root of the total mass, the final
answer is not very sensitive to the mass of the secondary. Using the
period of 244 days it then follows that the semi-major axis of the orbit
is 2.23 AU. For an elliptic orbit the radial distance between the two
stars is given by $r = a(1-e^2)/(1 + e\cos\theta)$ with $a$ the
semi-major axis and $e$ the eccentricity. The resulting orbital
modulation, for eccentricities of 0.5, 0.7, and 0.9, of the luminosity
is shown in Fig. \ref{fig:profile} with $r$ in AU. The position of
periastron passage has been set at 43.5 days in all three cases.
Combining the above two basic properties of colliding-wind binary
systems, it is clear that such systems can provide a periodic source of
photons that can heat the circumstellar dust and thereby possibly affect
the pumping radiation field for the masers, as
well as ionizing photons that can cause additional ionization in the
H\,{\sevensize II} region. Although the exact projection of the masers against the
H\,{\sevensize II} region in G9.62+0.20E is not known, the work of
\citet{phillips1998} suggests that the H\,{\sevensize II} region might indeed be the
source of seed photons for the masers.
Comparison of Fig. \ref{fig:profile} with the maser flare profile,
however, shows that, even in the case of an eccentricity of 0.5, the
observed maser flare profile has a decay time that is significantly
longer than what is expected simply from the radiation pulse produced at
periastron passage. Within the framework of the colliding-wind binary
scenario, the observed decay must then be due to either the cooling of
dust or the recombining of the H\,{\sevensize II} region (or parts thereof) from a
higher degree of ionization to its pre-flare state.
Using the fact that the dust cooling time is proportional to $T^{-6}$
\citep{kruegel2003} for the optical thin case and proportional to
$T^{-4}$ in the optically thick case, and that in the optical thin case
the typical cooling times for small grains with a temperature of 60 K is
about 10 seconds \citep{kruegel2003}, an upper limit of about 10 hours
for the cooling time is found for the optically thick case. Assuming now
a grain temperature of 100K and that the cooling time under optically thin
conditions is also 10 seconds, which, due to the $T^{-6}$ dependence
should actually be less, a cooling time of 1.2 days is found for the
optically thick case. Thus, even in the optically thick case the dust
cooling time is only a small fraction of the decay time of the maser
flare, suggesting that the decay of the maser flare is most probably not
due to the cooling of dust.
To investigate the possibility that the decaying part of the maser flare
is due to the recombination of the H\,{\sevensize II} region or parts thereof, we note
that the recombination time scale of a hydrogen plasma is given by
$(\alpha n_e)^{-1}$ \citep{osterbrock1989}, with $\alpha$ the
recombination coefficient and $n_e$ the electron density. Using $\alpha
= 2.95 \times 10^{-13}~\mathrm{cm^3s^{-1}}$, characteristic decay times
between 40 and 400 days are found for densities between $n_e =
10^6~\mathrm{cm^{-3}}$ and $n_e = 10^5 ~\mathrm{cm^{-3}}$. This is
significantly longer than the dust cooling time and seems to be able to
account for the observed decay time of about 100 days.
\begin{figure}
\centering
\includegraphics[width=80mm,clip,angle=0]{fig9a.eps}
\caption{Normalized average flare profile for the 12.2 GHz masers with
the solid and dashed lines the fits from eq. \ref{eq:decay}. The vertical
dashed lines indicate the time intervals which was used for the
fitting. }
\label{fig:hiidecay}
\end{figure}
In view of these numbers, the question arises whether the decay of
the maser flare can indeed be ascribed to a change in the seed photon
flux from a recombining thermal plasma. In Appendix A we derive an
expression for the time dependence of the electron density in a volume
of ionized hydrogen going from a higher level of ionization to a lower
equilibrium ionization state, due to the recombination of the plasma.
It is shown that in the optically thin case, the intensity of the
associated free-free emission decreases with time according to
\begin{equation}
I_\nu(t) \propto \left[\frac{1 + u_0\tanh(\alpha n_{e,\star}t)}{u_0 + \tanh(\alpha
n_{e,\star}t)}\right]^{-2} \label{eq:decay}
\end{equation}
with $u_0 = n_{e,0}/n_{e,\star} > 1$. $n_{e,0}$ is the electron density
from where the decay starts at time $t = 0$, and $n_{e,\star}$ the
equilibrium electron density determined by ionization balance for the
ionizing stellar radiation. Fitting eq. \ref{eq:decay} to the decay part
of the flare will give an indication to what extent the decay of the
maser flare can be interpreted as being due to the recombination of a
hydrogen plasma. An initial non-linear regression analysis using the
above time dependence suggested that the decay of the maser flare
actually consists of two components that have to be fitted
separately. The first component is from days 48 to 66, and the second
from days 67 to 135.
Following the discussion in Appendix A, we thus fitted the two
components with eq. \ref{eq:decay} by systematically varying
$n_{e,\star}$. This allowed us to estimate $u_0$ and therefore $n_{e,0}
= u_0n_{e,\star}$, ie. the electron density at the time when the
ionization pulse was switched off or when its effect was not significant
anymore. For the interval between days 48 and 66 it was found that
$n_{e,0}$ varies from $1.57 \times 10^6~\mathrm{cm^{-3}}$ to $2.00
\times 10^6~\mathrm{cm^{-3}}$ for $n_{e,\star}$ ranging from
$1.0\times10^5~\mathrm{cm^{-3}}$ to $7.0 \times 10^5~\mathrm{cm^{-3}}$.
Similarly, for the interval between days 67 and 135, $n_{e,0}$ varies
between $6.0 \times 10^5~\mathrm{cm^{-3}}$ and $7.3 \times
10^5~\mathrm{cm^{-3}}$ for $n_{e,\star}$ ranging from $1.0 \times
10^4~\mathrm{cm^{-3}}$ to $2.0 \times 10^5~\mathrm{cm^{-3}}$. For each
interval the lower value for the range of $n_{e,\star}$ was chosen such
that $n_{e,0}$ is also the solution of eq. \ref{special}, ie. the
solution for the limiting case when $n_{e,\star} \ll n_{e,0}$. The upper
value for the range of $n_{e,\star}$ was chosen to be about one third or
half of the value of $n_{e,0}$ obtained for the lower value of the range
of $n_{e,\star}$.
In Fig. \ref{fig:hiidecay} we show the fit of eq. \ref{eq:decay} to
each of the two time intervals of the decay of the maser flare for
specific values of $n_{e,\star}$ from the abovementioned ranges of
values. For days 48 to 66 we used $n_{e,\star} = 3 \times
10^5~\mathrm{cm^{-3}}$ and for days 67 to 135 we used $n_{e,\star} = 1
\times 10^5~\mathrm{cm^{-3}}$. These two values correspond to $n_{e,0} =
1.66 \times 10^6~\mathrm{cm^{-3}}$ and $n_{e,0} = 6.36 \times
10^5~\mathrm{cm^{-3}}$, respectively. The quality of the fits is
quite remarkable, especially given that a priori there is no
information suggesting that the decay of the maser flare might
follow the recombination of a thermal hydrogen plasma. The values for
$n_{e,0}$ as estimated from the non-linear regression are also what is
expected for hypercompact H\,{\sevensize II} regions. In fact, the agreement with the
densities quoted by \citet{franco2000} for G9.62+0.20E is also
remarkable. At the least these results are strongly suggestive that
the decay of the maser flare might be due to the recombination of an
H\,{\sevensize II} region, or parts thereof, from a higher to a lower ionization
state. The entire maser flare might therefore be due to changes in the
background source.
We also note that within the framework of the hypothesis that the maser
flaring is due to changes in the background H\,{\sevensize II} region, day 48 gives an
upper limit of the time when the radiative pulse causing the increase in
the ionization level, has ``switched'' off and the H\,{\sevensize II} region is left
to decay to its equilibrium level. Referring to
Fig. \ref{fig:flareprofile} the start of the flare is at about day 10,
implying that the full width of the radiative pulse is at most about 38
days. This suggests a rather sharply peaked pulse which is what is
qualitatively expected from the colliding-wind binary scenario discussed
above.
To what extent the colliding-wind binary scenario is the only one that
can explain the maser flaring within the broader framework of binary
systems, is not clear yet. Our discussion above does not include the
presence of a possible accretion disk or even two accretion disks, as
well as outflows. Obviously such scenarios are much more complex, but
should nevertheless be able to explain the periodic flaring as well as
the shape of the flare profiles. Given the short cooling time of the
dust, it follows, if the entire flare profile is due to changes in the
pumping radiation field, that the primary driving source of the
variable infrared radiation field should basically have the same time
dependence as the maser flare profile. Furthermore, the physical process
should be rather stable from orbit to orbit to explain the similarity of
the flare profiles in the time series. Obviously, monitoring of
G9.62+0.20E in the mid-infrared might help resolve this problem.
\section{Summary and Conclusion}
We presented the results of the monitoring of methanol masers in
G9.62+0.20 at 6.7, 12.2, and 107 GHz. Like the 6.7 and 12.2 GHz masers,
the 107 GHz masers also show flaring behaviour, with a relative amplitude
that is the same as that of the 6.7 GHz masers. It was also found that
the flare profile is the same for the 6.7, 12.2 and 107 GHz masers, and
that for the 6.7 and 12.2 GHz masers the profiles of individual flares
also are the same. We have also shown that in the low phase of the
masers, i.e. between two flaring events, the ratios of maser flux
densities return to the same levels. From this behaviour we conclude
that the physical conditions in the masing region must be relatively
stable and that the source for the flaring behaviour lies outside the
masing region.
Comparison of the maser light curves with that of pulsating stars lead
us to conclude that stellar pulsation is not the underlying cause for
the observed maser flaring. This leaves a binary system as the only
other option. It was argued that, at least qualitatively, a
colliding-wind binary can provide the mechanism for a periodic
background source and/or a periodic pumping radiation field. Although
there might be heating of the dust due to radiation from the shocked
regions, and thus the continuum of G9.62+0.20E might then show
variability in the IR, it seems difficult to explain the decay time of
about 100 days as being due to the cooling of dust.
It was shown that the characteristic recombination time of an H\,{\sevensize II}
region with densities in the range $10^5~\mathrm{cm^{-3}}$ to
$10^6~\mathrm{cm^{-3}}$ can explain the observed decay time of the maser
flare. We also showed that during the intervals 48 to 66 days and 67 to
135 days, the decay of the maser flare is what can be expected for the
recombination of thermal plasmas with densities of approximately $1.6
\times 10^6~\mathrm{cm^{-3}}$ and $6.0 \times 10^5~\mathrm{cm^{-3}}$,
respectively. These values are in very good agreement with densities
derived independently from radio continuum measurement.
A number of questions, however, still need to be answered. Can we find
observational evidence for a colliding-wind binary? Is there a periodic
infrared, radio continuum, or X-ray signal associated with G9.62+0.20E?
Is the ionizing photon flux produced in the hot post-shock gas large
enough that it still can have an observable effect on H\,{\sevensize II} region in
terms of changes in the free-free emission? Are there other mechanisms
associated with young binary systems that can also account for the maser
flaring? It is also necessary to determine the exact position of the
masers relative to the H\,{\sevensize II} region. Obviously, significantly more work
still needs to be done before a final answer on the periodic flaring in
G9.62+0.20E can be given.
\section*{Acknowledgements}
DJvdW was supported by the National Research Foundation under Grant
number 2053475. We would like to thank Moshe Elitzur and Andrei
Ostrovskii for valuable discussions in the early part of this
project. DJvdW also would like to acknowledge discussions with Karl
Menten and Endrik Kr\"ugel.
\bibliographystyle{mn2e}
|
3,212,635,537,488 | arxiv |
\section{Introduction}
\label{sec:intro}
Technology in
smart wearables is advancing rapidly
with an increasing integration of
rich camera and sensor data
\cite{niknejad2020comprehensive,sun2017smart}.
At the same time, there has been remarkable
progress in machine vision technology for processing this visual information.
A key challenge of deploying advanced machine
vision algorithms in the wearable setting is that
state-of-the-art deep neural networks are
computationally demanding, particularly
for mobile devices that are limited in
power and processing resources
for high-resolution images \cite{wu2019machine}.
Mobile edge computing combined with the massive mobile broadband capabilities of \gls{5g}
cellular wireless systems offers the possibility of offloading these computationally intensive vision processing tasks to the network edge \cite{leung2021ieee,wang2019edge}.
Importantly, \gls{5g} systems can leverage the \gls{mmw} bands
which afford vastly greater spectrum for higher-rate and lower-latency connectivity compared to standard 4G ones \cite{rangan2014millimeter,shafi20175g,uwaechia2020comprehensive}.
With mmWave connectivity,
a mobile device or wearable
can upload high-resolution video data to edge servers, where
much greater computational processing can be performed while keeping resources closer to the user to reduce the overall latency.
Wireless offloading can thereby enable support for multiple cameras for an enlarged field-of-view. Edge connectivity may also provide real-time access to data from other users, converging to new cooperative service strategies.
In this work, we study the potential of
wireless offloading of machine vision processing
for a powerful, smart wearable for the Blind-and-Visually
Impaired (BVI). The system, called VIS$^4$ION\xspace (Visually Impaired Smart Service System for Spatial Intelligence and Navigation) \cite{intro-7, intro-8, intro-9, arc-1, arc-2, arc-7, arc-10, arc-11} is a human-in-the-loop, sensing-to-feedback advanced wearable that supports a host of microservices during BVI navigation, both outdoors and indoors. The current VIS$^4$ION\xspace system is implemented as
an instrumented backpack; more specifically, a series of miniaturized sensors are integrated into the support straps and connected to an embedded system for computational analysis; real-time feedback is provided through a binaural bone conduction headset and an optional reconfigured waist strap turned haptic interface.
\begin{figure*}
\centering
\includegraphics[width=15cm]{figures/wireless/concept.png}
\caption{\textbf{Wireless offloading study:} The VIS$^4$ION\xspace wearable jacket from \cite{intro-8, intro-9, arc-1, arc-2, arc-8, arc-11}
is outfitted with multiple cameras for
\SI{360}{\degree} view. Due to power limitations, local processing of the camera data may be limited to a single camera at low resolution. When high rate wireless
connectivity is available, multi-camera,
high-resolution
data can be sent to an edge server where greater processing capabilities is available. The
detection results are then returned in the downlink. The paper assesses the feasibility of
this approach in urban environments under
realistic wireless channel conditions and
deployment assumptions.}
\label{fig:overview}
\end{figure*}
A key limitation of the current VIS$^4$ION\xspace system
is that all the machine vision is performed
locally by an embedded processor in the backpack, which limits the image resolution and the frame rate at which visual computation (e.g., object detection) can be performed. Furthermore, the battery needed to enable prolonged operation adds considerably to
the backpack weight.
Here, we investigate the wireless offloading
of the vision computations to edge servers as shown in Fig.~\ref{fig:overview}.
In the system studied, the wearable is augmented
with multiple high-resolution cameras to increase the field of view (device-wise) and enhance functionality (the current system has a single stereo camera).
When wireless connectivity is available, the camera
data will be uploaded over a cellular network to
a mobile edge server. We analyze the system in the
case where the cellular wireless link
can include both traditional lower data rate
carriers (e.g.,\ sub-6-GHz carriers in 4G)
as well as higher data rate 5G
carriers in the mmWave band.
Since the data rate in the multi-carrier system
may be variable, we consider an adaptive video scheme where
the number of camera feeds and bit rate per
camera are adapted based on the estimated
uplink wireless rate and delay.
The requirements for such a wireless system
are considerable and well beyond those considered in prior
vision offloading studies. For example,
as we will see in Section~\ref{sec:video}, accurate object detection for pedestrian scenes
at reasonable distances
can require over \SI{100}{Mbps}
if four cameras are used. Moreover, based on
physiological markers (see Section~\ref{sec:delay_requirements}),
the total maximum end-to-end delay of the system
will likely need to be less than \SI{100}{ms}.
After removing the time for video acquisition,
compression, and inference, there is a limited
time for uplink and downlink transmission.
As described in the previous work section below, most prior applications
of edge computing for machine vision processing
with video rate adaptation
(e.g.,~\cite{jedari2020video,li2019edge,sun2019mvideo,liu2019edge,ran2018deepdecision})
have considered relatively low-resolution,
single-camera data where the requirements
are much less strict. In these cases,
sub-6-GHz carriers with relatively limited bandwidths
are generally sufficient.
In contrast, we will see that the
5G mmWave bands are uniquely capable
of meeting the peak requirements for the enhanced VIS$^4$ION\xspace wearable.
Yet, 5G mmWave connectivity presents considerable technical
challenges of its own when used for offloading.
Most importantly, data rates in mmWave
outdoor links are highly variable since the signals
have limited range and are strongly
susceptible to blockage
from buildings, pedestrians, and other objects
in the environment \cite{maccartney2017rapid,slezak2018empirical}.
In addition, mmWave links are highly directional
and require continuous beam tracking to maintain
connectivity \cite{rangan2014millimeter,heath2016overview}.
This beam management and rate prediction
can cause significant additional delays
\cite{giordani2018tutorial}.
The broad goal of this paper is to provide
a detailed assessment of the feasibility of 5G mmWave
machine vision edge processing in a high data rate low-latency application.
Our study does not include the important factor of power
consumption that arises from mmWave connectivity,
or the potential power savings from avoiding
local computation. Power analyses of mmWave
devices transceivers and beam tracking
can be found in
\cite{dutta2019case,skrimponis2020towards,shah2021power}, along with power measurements of commercial devices
in \cite{narayanan2021variegated}. Our focus here is on the \textit{functional} benefits
of 5G connectivity such as support for increased
number of cameras and higher resolution and
object detection range.
Our analysis follows four main steps, each
of which bears significant new
contributions to handle the unique nature
of the mmWave offloading system for the enhanced
VIS$^4$ION\xspace wearable:
\begin{itemize}
\item \textit{Creation of the NYU-NYC StreetScene dataset}:
First, to evaluate object detection, we curated a custom dataset,
\textit{NYU-NYC StreetScene}, of high-resolution
videos taken during the `Last Mile' pedestrian segment of commuting
in NYC. The videos were manually annotated
with objects specific to BVI navigation.
A depth estimation method was developed for selected objects (standing people) so that the
distance of the objects could also be estimated
-- key to assessing the detection range.
The dataset is made public and is itself
a contribution of the work \cite{rizzolab}.
\item \textit{Evaluation of the impact of video resolution and bit rate on object detection:}
Using the dataset,
we conducted an extensive study to evaluate the impact of
video resolution and bit rate on the object detection accuracy and the reliable detection range.
As discussed in the prior work section,
previous analyses such as \cite{liu2019edge,ran2018deepdecision}
considered only low-resolution images,
and did not explicitly study the detection range.
Moreover, \cite{huang2017speed} did not consider the
effect of compression.
\item \textit{Wireless network evaluation:}
We next conducted detailed, realistic wireless network simulations of end users engaged in `Last Mile' pedestrian commuting similar
to those from which the video was captured.
To assess the unique capabilities of 5G,
we simulated both a 5G mmWave carrier at \SI{28}{GHz},
and a traditional 4G \gls{lte} carrier at \SI{1.9}{GHz}.
To accurately predict propagation at both
frequencies, we used state-of-the-art ray tracing
\cite{remcom} combined with new
highly detailed 3D models
acquired from GeoPipe \cite{geopipe}.
The use of such detailed models in wireless simulation
is the first of its kind. The channel data
was integrated into a widely-used
end-to-end network simulator, ns-3 \cite{ns3}, that captures
blocking, beam tracking, 4G and 5G protocol functionalities as well as delays in the
core network and edge network.
\item \textit{Performance analysis and availability:} Combining the
video and wireless analysis with inference
times, we compared
the performance of three scenarios:
local processing only, offloading with LTE,
and offloading with 5G mmWave + LTE.
For each option, we were able to determine key performance numbers such as the object detection accuracy, range of detection, number of cameras that
can be supported, and end-to-end latency.
In addition, since the channel quality is variable,
we determined the percentage of time that the performance
values can be obtained for a given scenario. We further evaluated the performance achievable with an adaptive offloading scheme, which switches between edge and local computing and varies the video resolution based on the wireless throughput.
In summary, we completed a thorough assessment of combining state-of-the-art
machine vision and mobile edge computing for contemporary advanced wearables.
\end{itemize}
We point out that this present work focuses on cellular wide area technologies
such as 4G and 5G since our target application is outdoor mobility. Of course,
in indoor settings and hotspots, Wireless Local Area Networks (WLANs), including high data rate versions such
as \cite{anastasi2003ieee,khorov2018tutorial}, may be
available -- see also the prior work section below
for studies in mobile edge computing with WiFi.
The study of indoor navigation with wireless
offloading with high data rate WLAN
is an interesting topic of future research.
\subsection*{Related Prior Work}
With the growing use of computationally intensive
deep learning methods for vision tasks, there has been
significant work in studying offloading of
this computation via edge computing; see, e.g., \cite{jedari2020video} for an excellent recent survey.
However, very few consider the unique
challenges of high-resolution images transmitted over massive broadband 5G links as needed by the VIS$^4$ION\xspace
system. For example, some of the most recent
works are as follows:
\begin{itemize}
\item Edge-AI \cite{li2019edge} studies
edge computing for image classification.
Since the study is on relatively limited
data rate 4G links, the work studies
dynamically partitioning the layers in a CNN
for vision classification.
In this work, we assume the processing
is entirely done at the edge or local device.
\item mVideo \cite{sun2019mvideo}
considers offloading large batches of surveillance images to the cloud for face detection. This
work also only considers 4G.
\item Hochsteetler et al. \cite{hochstetler2018embedded} studies
inference time on a very low-power edge device
without access to an edge server.
\item Liu et al. \cite{liu2019edge} considered edge-assisted object detection in mobile Augmented Reality (AR) applications. To meet the stringent delay requirement, they combine edge computing for object detection and fast local object tracking, which we adopt in our work as well. They also propose slice-based processing so that video transmission and object detection can be run sequentially over successive slices to reduce the edge computing delay. Their simulations only consider indoor WiFi connections between the VR headset and the nearby server, while we consider users walking through urban streets using 4G and 5G cellular networks. They also investigated the effect of video resolution and bit rate on the object detection accuracy and processing delay. However, they only examine resolutions up to 720P and consider the faster R-CNN object detector, while we consider resolutions up to 2.2K and focus on the more popular YOLO detector.
\item Ran et al. \cite{ran2018deepdecision} also study edge computing for AR applications. They assume the local processor (smartphone) can run either a tiny-YOLO or a big-YOLO model, while the edge server only runs the big-YOLO model. They consider the trade-off among decision variables including spatial resolution, frame rate, mobile power consumption, edge vs. local processing, object detection model through measurement studies and propose a measurement-driven optimization framework to determine the optimal setting for these decision variables to optimize a weighted average of the detection accuracy and frame processing rate, under the delay, bandwidth, and power constraint. However, the resolution range considered in their study is very low (only up to 480$\times$480). They also do not simulate real wireless networks.
\item
Jiang et al.~\cite{jiang2018chameleon} propose methods to adapt frame size and frame rate as well as detection models to meet a target detection accuracy based on video content. They leverage the spatial (cross cameras) and temporal correlation of the optimal configuration to reduce the computation cost of profiling. However, this work does not consider the impact of video compression (and hence the bit rate).
\item Huang et al.~\cite{huang2017speed}
consider the impact of frame size and frame rate (as well as detection model configuration) on the object detection accuracy. This work is not evaluated in the context of edge computing and hence does not consider the effect of compression.
\end{itemize}
In addition to the above studies on
mobile edge computing,
there is also a large body of work on delivering
low-latency services in 5G. Indeed, one of
the core design requirements of 5G are so-called
Ultra-Reliable Low Latency Communications
(URLLCs) \cite{li20185g,popovski2019wireless}
targeting air-link latencies of 1 to \SI{10}{ms}.
For the application in this paper,
the URLLC features of 5G
are critical in meeting the overall delay requirements. However, we will also see that
several other operations contribute to the overall
delay including video framing, video encoding,
delays in the core network, and inference time.
One goal of this work can be seen as evaluating
end-to-end delays with a realistic assessment of the
major components of the overall application.
Finally, on a more general note, the work
\cite{maier2020internet} describes an Internet of
No Things and ``seeing the invisible". The focus
of this paper is more limited, specifically
increasing the range and field of view via
computational offloading.
\section{The VIS$^4$ION\xspace System and Offloading Architecture} \label{sec:architecture}
\subsection{Motivation}
Immobility is a fundamental challenge for persons with BVI \cite{intro-1}. Loss of sight leads to loss of spatial cognition \cite{intro-2}. Spatial cognition can be defined as the knowledge or cognitive representation of the structure, entities/objects, and relationships within space \cite{intro-3}. The overwhelming majority of physical spaces are not visually accessible or do not allow for safe and efficient travel \cite{intro-4, intro-5}. This limits spatial cognition for the blind and leads to inefficiencies and peril during navigation. This gap requires new tools to bridge such accessibility barriers and to promote independence in daily tasks. A host of assistive technologies for citizens with BVI have been proposed. The white cane \cite{intro-8} is arguably the most widely used \cite{intro-13, intro-14} and affordable tool, but its short perceptive range limits its function as a direct extension of physical touch \cite{intro-15,intro-16,intro-17,intro-18,intro-19,intro-20}. High-tech hardware-based wearable devices have been developed to provide assistive features such as outdoor navigation \cite{chanana2017assistive,bai2019wearable,wang2017enabling}. However, they are generally either high cost or overly cumbersome \cite{wahab2011smart,elmannai2017sensor,farcy2006electronic}. In contrast, software-based solutions that run on ordinary smartphones are more affordable and accessible for BVIs. For example, Microsoft Seeing AI \cite{intro-17} and Blind Square \cite{intro-18} are widespread sensory substitution and navigation apps. However, these applications are not capable of offering advanced computer-vision-based assistive services or features due to the smartphones’ limited on-board sensing capabilities and computing power.
\begin{figure}
\centering
\includegraphics[trim={0 6cm 0 0 },clip,width=8cm]{figures/backpack.pdf}
\caption{\textbf{VIS$^4$ION\xspace} \textbf{backpack:} The current version of the VIS$^4$ION\xspace wearable jacket from \cite{intro-8, intro-9, arc-1, arc-2, arc-8, arc-11}
is an instrumented book bag with
a single ZED camera, NVIDIA Jetson GPU for local
vision processing, and haptic and audio feedback.
In this work, we consider augmenting it
(proposed augmentation shown in blue)
with three additional cameras for rear, left
and right visibility and wireless connectivity
for edge processing. }
\label{fig:backpack}
\end{figure}
\subsection{The Current VIS$^4$ION\xspace System and Its Limitations}
To address these challenges, we recently developed VIS$^4$ION\xspace, a Visually Impaired Smart Service System for Spatial Intelligence and Navigation \cite{intro-7, intro-8, intro-9, arc-2, arc-7, intro-12}.
The system is implemented as
a mobile sensor-to-feedback wearable device in the form of an instrumented bookbag -- See Fig.~\ref{fig:backpack}.
This smart service system is capable of real-time scene understanding with human-in-the-loop navigation assistance, supporting both mobility and orientation \cite{arc-1, arc-8}. VIS$^4$ION\xspace has four components: (1) distance and ranging/image sensors scaffolded into the shoulder straps of a backpack; these sensors (including a stereo camera and an Inertial Measurement Unit) extract pertinent information about the environment; (2) an embedded system (micro-computer) with both computing and communication capability (inside backpack); (3) a haptic interface (waist strap) that communicates spatial information computed from the sensory data to the end-user in real time via an intuitive, torso-based, ergonomic, and personalized vibrotactile scheme; and (4) a headset that contains both binaural bone conduction speakers and a noise-cancelling microphone for oral communication \cite{intro-8, intro-9, arc-2, arc-10}. The system leverages stereo cameras as its primary sensory input and employs advanced computer vision algorithms on Nvidia Jetson processing boards.
The goal of the embedded system is to enable continuous mapping, localization, and surveillance within a dynamically changing environment \cite{intro-8, intro-9, arc-1, arc-2, arc-8, arc-10}.
The key limitation in the current system is that
the video processing is performed entirely
locally, which is computationally and power intensive and limits the performance of visual analytics.
Indeed, the wearable runs off a laptop battery with approximately \SI{66}{Wh} at \SI{0.5}{kg}
yielding 2-3 hours of function if continuously running the vision processing.
To stay within these power limits, the wearable
uses the Jetson Xavier NX to perform object detection. With a standard YOLO model
the system is able to process
only WVGA resolution video at a rate of 10-13 frames per second (FPS) --- See Table~\ref{tab:ComplexityVsResolution}; results are often even poorer in mobile phone applications employing similar approaches \cite{ignacio}.
As we discuss below, the low resolution results in poor
detection accuracy and limited range.
Another limitation of the current system is that it deploys only a single stereo camera providing a field of view of approximately 90 degrees horizontally and 60 degrees vertically. At about 3 meters of distance or range from the end user, a 90-degree field of view is very restrictive, leaving potentially pertinent spatial obstacles out of the perceptive capabilities of the system, ones that may be encountered with even slight orientation shifts in forward paths. While this may be circumvented by simply using ultra-wide angle cameras, the geometric distortion in such cameras can degrade performance of visual analytics \cite{playout2021adaptable}. In order to address these shortcomings, we propose to embed multiple cameras in the VIS$^4$ION\xspace backpack to provide omnidirectional coverage.
Although multiple cameras or 360 degrees of perception may seem superfluous for a human with no disability, a person with a disability
may benefit significantly from a system that can
provide advanced notice and anticipate danger omnidirectionally. These full-field approaches to environmental analysis are now a common practice in myriad autonomous systems, from robots to cars and drones \cite{darms2009obstacle,davidson2021fov}.
\subsection{Wireless Offloading System Studied}
Wireless offloading of vision processing
to a mobile edge server offers two key
potential benefits:
(1) greater processing capability at the edge can enable analysis of multiple high-resolution camera streams for fast and more accurate object detection, over a wider field of view and greater range (distance); and
(2) reducing processing on the wearable
can prolong the battery life and/or reduce the battery weight.
To assess these potential gains, we
consider two augmentations to the backpack,
depicted in blue in Fig.~\ref{fig:backpack}.
First, we consider a version of the
wearable with four stereo cameras,
for example, to cover four sectors of
90 degrees each.
Second, to process multiple cameras at higher
resolution, we consider adding cellular connectivity
to an edge server with higher process capabilities.
The system will adapt the number of camera streams to be uploaded to the edge server
and their target bit rates based on the estimated uplink network capacity. Furthermore, the compression configuration (e.g., frame rate and spatial resolution) of each video stream will also be adjusted based on the target rates (adaptive).
The mobile edge will analyze the videos from multiple cameras using a deep learning network for object detection (and other tasks such as environmental mapping) and send the results
back to the wearable over the downlink; see also
Fig.~\ref{fig:overview}.
When the wireless connection is temporarily down (e.g., due to blockage), the local processor can analyze one video stream at a lower resolution, while storing the captured high-resolution video within its on-board memory. The high-resolution video can then be opportunistically uploaded to the edge server when the user reestablishes a high-bandwidth wireless connection, to enable the mapping of various environments, perform post-hoc behavioral analysis, etc.
\subsection{Delay Requirements}
\label{sec:delay_requirements}
For real-time pedestrian navigation, there is no generally agreed upon requirement for the tolerable total delay between the time an object appears in the environment of a pedestrian and the time it should be detected and reported to the pedestrian. In our previous work, we suggested 100 ms \cite{leigh2015neurology}. This benchmark is predicated on a physiologic marker, the high-end of the duration range for a large-amplitude saccade (fast eye movement); such an eye movement would be used by a normal-sighted pedestrian to identify a potential hazard. This stringent delay requirement enables the detection of dynamic, high-velocity objects (e.g., a suddenly appearing scooter in a pedestrian walkway).
\section{Impact of Video Compression and Spatial Resolution on Object Detection Accuracy and Range} \label{sec:video}
\subsection{Overview}
Due to the high bit rate of raw high-resolution video, compression is needed to stream image frames to the edge server via a throughput-constrained link.
In this section, we analyze how video compression (including reducing video resolution)
impacts object detection performance and inference time.
The analysis will provide the bit rate and delay requirements for the wireless
uploading described in Section~\ref{sec:wireless}.
Although the augmented VIS$^4$ION\xspace system will use multiple stereo cameras to enable a wider field of view and distance estimation, the analysis in this section focuses on a single monocular video. How to combine detection results from multiple views or make use of depth information in object detection is an interesting subject of
future work.
Thus, in the wireless evaluation in the next section,
we will consider only uploading of
multiple monocular streams.
Given a target rate for a camera, the video can be compressed at different spatial resolutions (frame size in terms of pixels) and temporal resolutions (frame rate), as illustrated in Fig~\ref{fig:VideoSTAR}. With the chosen spatiotemporal resolution, the bit rate is controlled by the quantization stepsize, which controls the amplitude resolution and affects the pixel quality.
While there has been significant work in relating
spatial, temporal, and amplitude resolution (STAR)
to \textit{perceptual} video quality
\cite{QSTAR,RSTAR,STARoptimization},
the effect of STAR on object detection accuracy
is less understood. As mentioned in the Introduction,
most prior works have only studied relatively
low-resolution images.
Here,
we conduct a study to systematically evaluate the impact of spatial and amplitude resolution on the object detection accuracy using a popular object detection deep learning model (YOLO \cite{YOLO}).
We leave out the consideration of the temporal resolution at this time because the YOLO model works on video frames independently. This study enables us to determine the optimal spatial resolution for a given bit rate, and the achievable detection accuracy under the optimal resolution at this rate.
We will further characterize the effect of the object distance (from the camera) on the detection accuracy under different spatial and amplitude resolutions, to provide recommendations/guidelines on the necessary spatial resolution and bit rate to meet the desired detection range for wearables that support pedestrian navigation applications. Finally, we characterize the computational cost of YOLO (including inference time) at different spatial resolutions, which provides guidance on the tolerance for the roundtrip delay with wireless offloading.
\begin{figure}
\centering
\includegraphics[width=8cm]{figures/video/VideoSTAR.png}
\caption{Under the same bit rate constraint, one can represent a video using different combinations of spatial, temporal, and amplitude resolutions as shown here with an example video
compressed to \SI{1}{Mbps}. The bottom row shows a crop from each version of the video to better illustrate the differences in compression artifacts.}
\label{fig:VideoSTAR}
\end{figure}
\subsection{Creation of the NYU-NYC StreetScene Data Set}
\label{sec:DataPreparation}
Currently, there are no public datasets containing high-resolution videos captured from the perspective of a typical pedestrian. A significant effort in
this work is the creation of a new, manually-annotated
`StreetScene' video dataset for this purpose.
\paragraph*{Video collection}
To test the performance of the YOLO model for detecting objects of interest for pedestrian navigation, we recorded a set of videos
while wearing the current
VIS$^4$ION\xspace backpack which has a single
stereo camera on the front shoulder strap.
The camera model is the ZED camera from StereoLabs \cite{zed2} --
a lightweight, powerful
recording device,
ideal for wearables requiring spatial intelligence.
A total of 9 videos were captured with the ZED
at the 2.2K spatial resolution and \SI{15}{Hz} temporal resolution, with a total video length of 43 minutes.
\paragraph*{Object annotation}
We manually annotated the bounding boxes for 15 objects of interest, listed in Table~\ref{tab:object_type},
along with the number of occurrences for each object. We annotated every 30th frame of each video (only left view). This annotated dataset is publicly available at \cite{rizzolab}.
Because YOLO was trained where the `traffic light' includes both `vehicle traffic light' and `pedestrian signal', we grouped these two separately annotated objects into the same object type when applying the YOLO model. Furthermore, because the detection performance of YOLO on `bench', `stop sign' and `dining table' is very poor, we only report the detection performance for detecting the remaining 11 objects.
\begin{table*}[]
\resizebox{\textwidth}{!}{
\begin{tabular}{|c|c|c|cc|c|c|c|c|c|c|c|c|c|cc|}
\hline
\cellcolor{BlueGreen!50} & \cellcolor{BlueGreen!50} & \cellcolor{BlueGreen!50} & \multicolumn{2}{c|}{\tblhdr{traffic light}} & \cellcolor{BlueGreen!50} &
\cellcolor{BlueGreen!50} & \cellcolor{BlueGreen!50} & \cellcolor{BlueGreen!50} & \cellcolor{BlueGreen!50} &
\cellcolor{BlueGreen!50} & \cellcolor{BlueGreen!50} &
\cellcolor{BlueGreen!50} &
\multicolumn{3}{c|}{\tblhdr{Not reported}}
\\ \cline{4-5} \cline{14-16}
\multirow{-2}{*}{\tblhdr{Object}} & \multirow{-2}{*}{\tblhdr{person}} & \multirow{-2}{*}{\tblhdr{car}} & \multicolumn{1}{c|}{
\begin{tabular}[c]{@{}c@{}} \tblsubhdr{vehicle} \\
\tblsubhdr{traffic light}\end{tabular}} & \begin{tabular}[c]{@{}c@{}}\tblsubhdr{pedestrian}\\ \tblsubhdr{signal}\end{tabular} &\cellcolor{BlueGreen!50}
\multirow{-2}{*}{\begin{tabular}[c]{@{}c@{}}\tblhdr{potted} \\ \tblhdr{plant}\end{tabular}} & \multirow{-2}{*}{\tblhdr{bicycle}} & \multirow{-2}{*}{\tblhdr{truck}} &
\multirow{-2}{*}{\tblhdr{chair}} &
\cellcolor{BlueGreen!50}
\multirow{-2}{*}{\begin{tabular}[c]{@{}c@{}}\tblhdr{fire} \\ \tblhdr{hydrant}\end{tabular}} &
\multirow{-2}{*}{\tblhdr{bus}} & \multirow{-2}{*}{\tblhdr{umbrella}} & \multirow{-2}{*}{\begin{tabular}[c]{@{}c@{}} \tblhdr{motor} \\ \tblhdr{cycle}\end{tabular}} & \tblsubhdr{bench} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}\tblsubhdr{stop} \\ \tblsubhdr{sign}\end{tabular}} & \begin{tabular}[c]{@{}c@{}} \tblsubhdr{dining} \\ \tblsubhdr{table}\end{tabular} \\ \hline
Occurrences & 9783 & 5442 & \multicolumn{1}{c|}{1977} & 651 & 1122 & 704 & 459 & 450 & 370 & 349 & 335 & 162 & 247 & \multicolumn{1}{c|}{217} & 180 \\ \hline
\end{tabular}}
\caption{Object types annotated in the NYU-NYC StreetScene Dataset.}
\label{tab:object_type}
\end{table*}
\paragraph*{Video compression}
We compressed all videos (left view only) in the StreetScene dataset using the FFmpeg software with the x265 codec \cite{FFmpeg,X265}, which follows the latest international video coding standard H.265/HEVC \cite{H265}.
We kept the same temporal resolution, and compressed the video either at the original 2.2K spatial resolution or reduced spatial resolutions (See Table~\ref{tab:SpatialResolution}) under different quantization parameters (QPs). Default down-sampling filters (`bicubic') in FFmpeg were used for the spatial downsampling. Considering the low-delay requirement of the navigation application, we used a Group of Picture (GOP) length of 60 frames, without B-frames, i.e, each GOP starts with one I-frame, followed by 59 P-frames.
\begin{table}[]
\centering
\begin{tabular}{|c|c|c|c|} \hline
\tblhdr{Resolution} & \tblhdr{Width} & \tblhdr{Height} & \tblhdr{Reduction factor$^*$}\\ \hline
2.2K& 2208 & 1242 & 1x \\ \hline
1080P & 1920 & 1080 & 0.87x \\ \hline
720P & 1280 & 720 & 0.58x\\ \hline
WVGA & 672 & 378 & 0.30x\\ \hline
\end{tabular}
\vspace{2mm}
$^*$ Defined relative to the 2.2K resolution (same reduction in both width and height).
\caption{Spatial resolutions considered}
\label{tab:SpatialResolution}
\end{table}
\paragraph*{Distance estimation for standing people}
To examine how distance affects the detection accuracy in the StreetScene dataset, which does not have accurate distance measurements
\footnote{The depth estimation from the stereo disparity in the ZED camera SDK is not very accurate and only works when the distance is within 20 meters.},
we developed a method to estimate the distance of standing people in our `StreetScene' dataset.
We focus on distance estimation for
this object type since this is a relatively small object whose detection can be greatly affected by the object distance.
Given that the variation of the physical size of standing people is relatively small, the size of the box bounding a standing person is mainly determined by the distance of the person from the camera. Based on this observation, we trained a distance estimation model (containing a few fully connected layers) based on the bounding box width and height using the KITTI dataset \cite{geiger2012we}, which has annotated bounding boxes and distances for standing people. To account for the difference in the camera used for the KITTI data and our ZED camera, we used our ZED camera to capture a set of videos with standing people at multiple distances against a variety of backgrounds,
with the distance captured using the positional tracking system of the ZED camera. Using this dataset, we were able to learn the mapping from the distance estimated by the model trained on the KITTI data to the distance from the video captured by the ZED camera. To apply this model on the people detected in the StreetScence dataset using the YOLO model, which includes both standing and sitting people in the same object category, we looked at the distribution of the height over width ratio among the standing and sitting people in the KITTI data, and found that using a ratio threshold of 2.0 can fairly reliably separate standing people from sitting people. Therefore, we used this ratio threshold to detect standing people in the StreetScene dataset. By applying the distance estimation model to the detected bounding boxes for the standing people followed by the camera mapping, we generated the distance measurements of standing people in the StreetScene dataset.
\subsection{Effect of Spatial and Amplitude Resolution on Object Detection Accuracy}
We first examine the impact of spatial and amplitude resolutions (with corresponding bit rates) on the object detection accuracy on the \textit{StreetScene} dataset, in which objects appear at varying distances. Results show that the optimal spatial resolution varies with the target bit rate (which is constrained by the network throughput).
We applied a pretrained YOLO 5s model \cite{YOLO5s} on the decompressed videos in the StreetScence dataset to detect the 14 objects of interest.
Fig.~\ref{fig:wmAPvsRate} shows the weighted mean average precision (wmAP) over 11 objects (see Table~\ref{tab:object_type}) vs. bit rate.\footnote{Note that although the videos in the StreetScene dataset are captured and compressed at 15 Hz, we report the equivalent bit rates for videos at 30 Hz, which is necessary to meet the real-time navigation requirement as further detailed in Sec.~\ref{sec:delay_requirements}. This is accomplished by scaling the actual bit rates corresponding to different spatial resolutions and QPs with different scaling factors determined by a separate experiment where we compressed videos at 30 Hz and 15 Hz separately at multiple resolutions and QPs for several sample videos captured at 30 Hz.}
The weight of an object type is proportional to its occurrence frequency in the dataset.
The figure reveals that there is an optimal spatial resolution at each bit rate that will maximize the wmAP. Specifically, 720P is best for 0.35-6.0 Mbps, 1080P for 6.0-26.2 Mbps, 2.2K for higher bit rates. However, 2.2K provides only marginal improvement over 1080P above 26.2 Mbps. We note that this could be because the YOLO model was trained mainly on low-resolution images. Although WVGA is best at a very low rate (below 0.35 Mbps), the achievable AP is too low to be usable.
At 26.2 Mbps and using 1080P resolution, the weighted mean AP is about 54\%, which is still far from perfect. However, this relatively low detection accuracy is due to limitations of the YOLO 5s model, which was trained using uncompressed low-resolution images. Better detection models (e.g., models specifically trained for street scenes and/or models that are separately optimized for different resolutions) will likely further improve the detection accuracy. It is tenable that the trend of the detection accuracy vs. rate vs. spatial resolution would be preserved for future, more powerful models.
\begin{figure}
\centering
\includegraphics[width=9cm]{figures/video/weighted_mAP_vs_Bitrate_H265_0-25Mbps_all_2.2K.png}
\caption{Mean detection accuracy (weighted mean AP) for 11 object types vs. bit rate for different spatial resolutions. Different points in the same curve correspond to different QPs.}
\label{fig:wmAPvsRate}
\end{figure}
Fig.~\ref{fig:APpersonvsRate} presents the detection result for the person category. We see a similar trend as in Fig.~\ref{fig:wmAPvsRate}, although the specific rate points where higher resolutions take over the lower resolutions are slightly different. The AP for the person category is higher than the wmAP over 11 objects at similar bit rates, which shows that the YOLO model is more effective in detecting people than other object categories. This is consistent with the performance reported in \cite{YOLOv2}, likely because there are significantly more instances of people than other objects in the training set.
\begin{figure}
\centering
\includegraphics[width=9cm]{figures/video/person_AP_2.2K.png}
\caption{Detection accuracy (AP) for person vs. bit rate for different spatial resolutions. Different points in the same curve correspond to different QPs.}
\label{fig:APpersonvsRate}
\end{figure}
Sample frames from videos compressed to around 10 Mbps using different settings are shown in Fig.~\ref{fig:SampleImages}(a-b). From the outset, it is not clear which decompressed image will lead to improved object detection. However, from the detection results shown in Fig.~\ref{fig:SampleImages}(c-d), the YOLO model did better for image (b), which was represented with a high spatial resolution but low amplitude resolution.
\begin{figure*}
\begin{subfigure}{0.49\linewidth}
\centering
\includegraphics[width=0.95\linewidth]{figures/video/79_reswvga_qp10_3808.jpg}
\caption{\textbf{Low resolution:} WVGA, QP=10, PSNR=45dB}
\end{subfigure}
\begin{subfigure}{0.49\linewidth}
\centering
\includegraphics[width=0.95\linewidth]{figures/video/79_res1080_qp28_3808.jpg}
\caption{\textbf{High resolution:} 1080P, QP=28, PSNR=36dB}
\end{subfigure}
\qquad
\begin{subfigure}{0.49\linewidth}
\centering
\includegraphics[width=0.95\linewidth]{figures/video/79_reswvga_qp10_3808_detected_arrow_annotation.jpg}
\caption{YOLO detection results for (a)}
\end{subfigure}
\begin{subfigure}{0.49\linewidth}
\centering
\includegraphics[width=0.95\linewidth]{figures/video/79_res1080_qp28_3808_detected_arrow_annotation.jpg}
\caption{YOLO detection results for (b)}
\end{subfigure}
\caption{Sample detection results from videos both compressed at 10 Mbps but using different spatial and amplitude resolutions. Objects that are indicated by yellow arrows in (d) are missed in (c).
Note that although the image in (a) and (c) have lower spatial resolution than those in (b) and (d), we display them at the same size for easier visual comparison. Notice that image in (a) is blurred due to the spatial upsampling. }
\label{fig:SampleImages}
\end{figure*}
Table~\ref{tab:AccuracyVsResolution} summarizes, for each resolution, the bit rate at which the detection accuracy for multiple object detection (wmAP) plateaus, and the corresponding detection accuracy. We also list the AP for human detection, and the recall when the precision is 0.80. As we can see, using higher resolution video enables higher object detection accuracy, which translates to more correctly detected objects. For example, going from WVGA to 1080P, the recall for the "person" category increased from 48\% to 74\%, while keeping the false detection rate at 20\%. Therefore, by transmitting high resolution video to the edge server when the network throughput is sufficiently high, we are able to see ``more'' objects.
\begin{table}[]
\resizebox{8.5cm}{!}{
\centering
\begin{tabular}{|c|c|c|c|c|} \hline
\cellcolor{BlueGreen!50} Resolution
& \cellcolor{BlueGreen!50} \makecell[c]{Bitrate \\ (Mbps)}
& \cellcolor{BlueGreen!50} \makecell[c]{wmAP \\ (11 obj)}
& \cellcolor{BlueGreen!50} \makecell[c]{AP \\ (person)}
& \cellcolor{BlueGreen!50} \makecell[c]{Recall \\ (person)} \\ \hline
2.2K & 30.09 & 54.62 & 67.27 & 0.75\\ \hline
1080P & 26.00 & 54.00 & 66.11 & 0.74\\ \hline
720P & 18.01 & 50.27 & 60.31 & 0.68\\ \hline
WVGA & 9.29& 36.57 & 44.32 & 0.48 \\ \hline
\end{tabular}}
\caption{Impact of spatial resolution on the bit rate and the achievable detection accuracy. Recall for the person category is evaluated when the precision is 80\%. }
\label{tab:AccuracyVsResolution}
\end{table}
\subsection{Effect of Spatial Resolution on Detection Accuracy at Different Distances}
The performance measures reported so far are aggregated results for objects appearing at varying distances from the camera. Generally, detecting a faraway object is harder than a nearby object. On the other hand, being able to detect an object while it is still far away provides more time for navigation planning. Therefore, it is important to understand how the detection accuracy degrades as the distance increases and what is the maximum distance when an object can be detected reliably. How does the distance affect the detection also depends on the physical size of the objects. We show such results for the detection of standing pedestrians as a case study, wherein we use the algorithm described in Sec.~\ref{sec:DataPreparation} to detect standing pedestrians and furthermore estimate the distance of the detected person(s) from the bounding box size(s).
We quantized the distances to several bins and determined the AP within each bin.
Fig.~\ref{fig:standing_people_AP} illustrates how the detection accuracy drops as the object distance increases under different spatial resolutions. When the distance is very close, YOLO performs very well even at the WVGA resolution, but the accuracy drops quickly as the distance increases at this low resolution. As expected, higher spatial resolutions enjoy a slower decay rate. The 2.2K resolution leads to significantly better detection than 1080P only when the distance is greater than 21 m. Note that the 2.2K resolution did worse than other lower resolutions in the short distance range in this study. This is likely because the YOLO 5s model was trained on low-resolution video (close to WVGA). People within a short distance occupy a very large area in the 2.2K image, requiring bounding box sizes that rarely occur in the training data.
\begin{figure}
\centering
\includegraphics[width=0.95\linewidth]{figures/video/person_mAP_vs_Distance_mid_2.2K.png}
\caption{Detection accuracy (in AP) vs. distance for standing people under different resolutions. The results are obtained when videos at different resolutions are compressed to the bit rates listed in Table~\ref{tab:ComplexityVsResolution}. Note that the results would be similar even if the videos were uncompressed because the average detection accuracy for each particular resolution already plateaued at its corresponding bit rate, as shown in Fig.~\ref{fig:APpersonvsRate}.}
\label{fig:standing_people_AP}
\end{figure}
Fig.~\ref{fig:standing_people_range} shows the detection ranges for different spatial resolutions. Here, we see clearly that going from WVGA to 1080P, we are able to extend the detection range from about 6 m to 12 m. This study shows that we should use at least 1080P video to be able to reliably detect people at a distance important for navigation planning.
\begin{figure}
\centering
\includegraphics[width=0.95\linewidth]{figures/video/distance_range_for_person_2.2K.png}
\caption{Detection range vs. spatial resolution for standing people. The detection range is defined as the maximum distance at which recall $\geq 0.9$ and precision $\geq$ 0.83. Videos' bit rates are listed in Table III.}
\label{fig:standing_people_range}
\end{figure}
\subsection{Computational Complexity vs.\ Spatial Resolution}
Table~\ref{tab:ComplexityVsResolution} summarizes the computation complexity (measured by the FLOP count), the inference time per video frame, and corresponding speed (frame/sec or fps) on the embedded processor in our VIS$^4$ION\xspace backpack (Jetson Xavier NX running at 15 Watts, using a GPU at 1.1 GHz), the inference time per video frame and speed using an edge server equipped with an RTX 8000 GPU, of the YOLO 5s model, for videos at different spatial resolutions.
As will be explained in Sec.~\ref{sec:evaluation}, to meet the total delay requirement for real-time navigation, local processing should be completed within 67 ms, which is barely possible with the WVGA video, severely limiting the achievable object detection accuracy and detection range (cf. Table~\ref{tab:AccuracyVsResolution}, Fig.~\ref{fig:standing_people_range}). On the other hand, offloading the computation to the edge server allows us to process the 1080P video and consequently significantly increase the detection performance, while still meeting the delay constraint.
\begin{table}[]
\resizebox{8.5cm}{!}{
\centering
\begin{tabular}{|c|c|c|c|c|c|} \hline
\cellcolor{BlueGreen!50} Resolution & \cellcolor{BlueGreen!50} GFLOP & \cellcolor{BlueGreen!50} \makecell[c]{Local \\ Inference \\Time (ms)\\} &
\cellcolor{BlueGreen!50} \makecell[c]{Local \\ Inference \\Speed (fps)\\} &\cellcolor{BlueGreen!50} \makecell[c]{Server \\ Inference \\Time (ms)}
&\cellcolor{BlueGreen!50} \makecell[c]{Server \\ Inference \\Speed (fps)}
\\ \hline
2.2K & 57.22 & 232.02 & 4.31 & 23.4 & 42.7 \\ \hline
1080P & 43.38 & 178.25 & 5.6 &18.7 &53.5 \\ \hline
720P & 19.56 & 95.69 & 10.5 & 10.4 &96.2\\ \hline
WVGA & 5.36 & 75.02 & 13.3 & 5.1 & 196.1 \\ \hline
\end{tabular}}
\caption{Impact of spatial resolution on the detection model complexity, running time on local processor and edge server, respectively. The Jetson Xavier NX is used as the local processor, while the server uses an RTX 8000 GPU. }
\label{tab:ComplexityVsResolution}
\end{table}
\section{Wireless Evaluation} \label{sec:wireless}
Having analyzed the bit rate and latency
requirements for video processing, we now
simulate the wireless network to determine
what percentage of time these requirements can be met.
\subsection{User Route and Ray Tracing} \label{sec:raytracing}
\begin{figure}%
\centering
\includegraphics[width=0.95\linewidth]{figures/wireless/routes_map.png}%
\caption{Main walking route from which we extracted four specific sites represented by the red rectangles. Top to bottom, the sites are \textit{Lighthouse Guild}, \textit{Midtown}, \textit{Herald Square}, and \textit{NYU Langone}}
\label{fig:main_route}
\end{figure}
We simulate a hypothetical end-user commuting through the streets of Manhattan -- a challenging environment from a wireless perspective,
due to the tall building blockage.
In particular, we identified a walking route starting at Lighthouse Guild (a healthcare and research center that assists persons with BVI) and finishing at the NYU Tisch Hospital in Kips Bay, as depicted in Fig.~\ref{fig:main_route}. The environment of this route is similar to where the NYU-NYC StreetScene dataset described in Section~\ref{sec:video} was collected.
Along this route, we selected four sites (red rectangles of Fig.~\ref{fig:main_route}) to perform a realistic full stack \gls{5g} simulation. This simulation involved two main steps: 1) generation of ray-tracing data of the wireless environment from the realistic 3D layout of the city at these sites, and 2) end-to-end simulation using \gls{ns3} and the ray-tracing information obtained in the previous step. Here, we describe the generation of ray-tracing data with the figures for the top site in Fig.~\ref{fig:main_route} as an example.
In wireless communications, ray-tracing involves the calculation of the paths that electromagnetic waves, represented as rays, follow while propagating in a known 3D environment according to a set of transmitting and receiving positions. For each combination of transmitting and receiving locations, the output of a ray-tracing simulation consists of a set of propagation information for each ray: path loss, propagation time, phase offset, \gls{aod}, and \gls{aoa}. We used Remcom's \textit{Wireless InSite} \cite{remcom} software to generate ray-tracing data in our work.
This software has been successfully used in a number
of other studies \cite{XiaRanMez2020,Alkhateeb2019,khawaja2017uav}.
Accurate ray-tracing of a 3D scenario at \gls{mmw} and sub-6-GHz frequencies requires precise information of buildings materials, since each material affects the propagation of the electromagnetic wave differently at each frequency range. In addition, details of the vegetation in a certain area are essential for accurate ray-tracing at \gls{mmw} frequencies. To capture all this information, we imported in Remcom the 3D layout of the City at each of the 4 sites, as provided by Geopipe \cite{geopipe}.
Fig.~\ref{fig:3d_map} depicts the 3D layout with building materials and vegetation at the top site of Fig.~\ref{fig:main_route}.
This data from GeoPipe provides
one of the most accurate models for \gls{mmw}
ray tracing. In particular, the models contain
small building features which are known to influence
mmWave propagation significantly \cite{rappaport2015millimeter}.
For each material (i.e., concrete, wood, glass, etc.) and frequency range, Remcom uses different parameters for the reflection and diffraction coefficients, as well as different propagation properties.
We performed the simulation at two frequencies:
\SI{1.9}{GHz} for a typical sub-6 GHz 4G LTE carrier,
and \SI{28}{GHz} for a typical 5G \gls{mmw} carrier.
Moreover, we considered a real placement of the base stations in each of the four sites as provided by \cite{opendata}. This database contains information regarding the actual foreseen placement of \gls{5g} \gls{mmw} \glspl{bs} in New York City. We assumed that the LTE 4G and 5G mmWave base stations are
co-located, meaning that, at each
cell site, base station equipment is available for both frequencies. This co-location is common
since, once the operator has secured a site, it
generally utilizes it maximally. Note that
in 3GPP terminology,
a 4G base station is called eNB (evolved Node B),
and a 5G base station is called gNB (next generation Node B).
For ray tracing, each base station site represents a transmitting location,
whereas the receiver positions are placed one meter apart along the route taken by the hypothetical user at each specific site (more details in \ref{sec:wresults}). The ray tracing then
provides the channel from each \gls{bs} to each
position along the route at both frequencies.
Note that due to symmetry, the large scale
channel parameters are identical in the uplink
and downlink. Hence, we can use the estimated
channel in both directions.
\begin{figure}
\centering
\includegraphics[width=0.95\linewidth]{figures/wireless/3d_example.jpg}%
\caption{3D map of the top site, \textit{Lighthouse Guild}, in the route
in Fig.~\ref{fig:main_route}, with material information for each building and vegetation
provided courtesy of GeoPipe \cite{geopipe}.
}%
\label{fig:3d_map}%
\end{figure}
\subsection{Network Simulation}
The second step in the evaluation of wireless offloading
involves the end-to-end (full stack) network simulation using \gls{ns3}. \Gls{ns3} is an open-source, discrete event network simulator which affords end-to-end simulations (i.e., simulations that
model the entire network stack from the physical layer up to the application layer) with support for user mobility and traffic modeling, among other features. To compare the performance of \gls{mmw} connectivity with a standard sub-6 GHz system,
we run two simulations:
\begin{itemize}
\item An 4G LTE system at \SI{1.9}{GHz}
with \SI{40}{MHz} downlink and \SI{40}{MHz}
uplink total bandwidth; and
\item A 5G mmWave system at \SI{28}{GHz}
with \SI{400}{MHz} total bandwidth
that is Time-Division Duplexed (TDD) for the
uplink and downlink.
\end{itemize}
For both the 4G and 5G systems, the deployment
would likely be on multiple carriers,
as is common today. For example,
the LTE system could be two standard carriers
of \SI{20}{MHz} each and the 5G mmWave
system with four carriers of \SI{100}{GHz} each.
The parameter values for both systems are shown
in Table~\ref{tab:params} and are representative
of typical 4G and 5G simulation studies,
see, e.g., \cite{akdeniz2014millimeter,3gpp-channel,shafi20175g}.
In addition, details for the 4G-LTE \gls{ns3} module can be found at \cite{lte-ns3}.
To account for loading, we assume an individual UE
obtains a fraction $0.25$ of the total
bandwidth, which would represent a moderate loading
level relative to standard evaluation methodologies
\cite{3gpp-channel}.
Hence, we simulate the 4G user as operating
in a system with $10+10$\, \si{MHz} bandwidth
and the 5G user as operating in a system
with \SI{100}{MHz} total bandwidth. Morevoer, the \gls{5g} system operating at mmWave frequencies uses Numerology 2 and the TDD configuration allows symbols to be \textit{flexible}, meaning that each symbol can be used for either uplink or downlink traffic.
In both the 4G and 5G cases, we model the wearable as a \gls{ue} traversing the
path described above. The channels from the ray tracing in Section \ref{sec:raytracing} are imported into the \gls{ns3} simulator.
Since ray tracing captures only the
buildings and foliage, it does not capture blockage
from objects such as humans and vehicles in the
environment. As discussed in the Introduction,
modeling blockage is critical to accurately
assess mmWave coverage \cite{maccartney2017rapid,slezak2018empirical}.
To model this additional blockage,
we employ the \textit{Blockage Model A} in \cite{3gpp-channel} which is integrated into the \gls{ns3} simulator \cite{ns3}.
This model adopts a stochastic approach for capturing human and vehicular blocking. In particular, multiple 2D angular blocking regions, in terms of azimuth and elevation angular spreads, are generated around the \gls{ue}. One blocking region, denoted as self-blocking region, captures the effect of human body blocking, whereas $K_{\rm NSB}$ non-self-blocking regions with random sizes are used to model other sources of blockage ($K_{\rm NSB}$ can be changed to increase/decrease the density of blockers). Once the blocking regions are computed, each cluster (or ray, in the case of our ray-tracing data) is attenuated accordingly, based on the angular spreads and position of each blocking component. The parameter $T_{\rm NSB}$ denotes the time interval at which new blockers are randomly generated.
\begin{table}[t!]
\caption{Wireless network simulation parameters}
\label{tab:params}
\footnotesize
\centering
\begin{tabular}{|>{\RaggedRight}m{3.3cm}|>{\RaggedRight}m{1.6cm}|>{\RaggedRight}m{1.6cm}| }
\hline
\cellcolor{BlueGreen!50} & \multicolumn{2}{c|}{\tblhdr{Value}} \\ \cline{2-3}
\multirow{-2}{*}{\tblhdr{Parameter}} & \tblsubhdr{4G LTE} &
\tblsubhdr{5G mmWave} \\ \hline
Carrier frequency & \SI{1.9}{GHz}
& \SI{28}{GHz} \\ \hline
Total bandwidth (MHz) & 40+40 FDD & 400 TDD \\ \hline
Fraction of bandwidth available
to UE due to loading & 0.25 & 0.25 \\ \hline
Bandwidth to UE (MHz) & 10+10 FDD & 100 TDD \\ \hline
UE array & Single & $4 \times 4$ \\ \hline
BS array & Single & $8 \times 8$ \\ \hline
UE transmit power (dBm) & 25 & 25 \\ \hline
DL transmit power (dBm) & 30 & 30 \\ \hline
UL noise figure (dB) & 5 & 5 \\ \hline
DL noise figure (dB) & 5 & 5 \\ \hline
Number of HARQ processes, $N_{\rm HARQ}$ & 8 & 20 \\ \hline
Non-self-blocking components, $K_{\rm NSB}$ &
\multicolumn{2}{c|}{40} \\ \hline
Blockage update period, $T_{\rm BLK}$ &
\multicolumn{2}{c|}{\SI{100}{ms}} \\ \hline
TCP packet size & \multicolumn{2}{c|}{\SI{1024}{B}}
\\ \hline
Core network delay $D_{\rm core}$ &
\multicolumn{2}{c|}{\SI{5}{ms}} \\ \hline
\end{tabular}
\end{table}
Given the blocked channels, the \gls{ns3}
simulator then models the full stack communication.
At the Physical (PHY) and \gls{mac} layers, the modeling
includes beam tracking, \gls{cqi} reports, rate prediction, \gls{harq}, and scheduling. The total number of \gls{harq} processes for both systems is specified in Table~\ref{tab:params}.
At the higher layers, the simulator models all the \gls{rlc}
segmentation and buffering, \gls{rrc} signaling, and handovers.
In our simulations, we use the Acknowledged Mode (AM) for the \gls{rlc} layer for both 4G and 5G systems.
An important parameter in the network configuration
is the location of the edge server. In
cellular systems, data in the uplink
traverses a path: UE (wearable) $\rightarrow$
base station (4G eNB or 5G gNB) $\rightarrow$
core network $\rightarrow$ server in the public
Internet.
The downlink follows the reverse path.
In conventional deployments, operators have
relatively few gateway points from the core network
to servers in the public Internet.
The data may thus need to traverse a long path
in the core network to the closest gateway
resulting in high delay from
the base station to the server --- see, for example, measurements in commercial networks in \cite{narayanan2021variegated}.
Mobile edge computing reduces the core network
delay by placing the edge servers much closer
to the base station \cite{hu2015mobile}.
In this study, we will assume that the
one-way delay from the base station to the
edge server is $D_{\rm core} = 5$\, \si{ms}.
As we will see in the delay analysis below,
this lower delay will be critical to meet
the strict delay requirements for BVI navigation.
\subsection{Traffic Modeling}
Ideally, in the uplink,
we would model the adaptive
multi-camera
video encoding
application in the \gls{ns3} simulator. This analysis would
then be specific to the video adaption algorithm
used. To provide a more general and simpler
analysis, we instead model the uplink video
data as a single Transmission Control Protocol (TCP) stream with a full buffer up to
a maximum data rate of \SI{120}{Mbps}.
This maximum data rate is sufficient to support
four cameras at \SI{30}{Mbps} each.
Since TCP has congestion control, it will
automatically adjust the sender rate to
the available uplink link capacity.
As a simplification, we
assume that the video encoding
can be adapted to be exactly the same as the TCP rate.
Hence, the full buffer
TCP rate at any time can be regarded
as an approximation of the actual video
rate.
For the downlink, data from
the edge server to the wearable is used to carry
the object detection results.
We model this downlink traffic as a constant
bit rate application at 30 packets per seconds,
corresponding to the expected video frame rate.
We assume that the total rate is \SI{1}{Mbps},
which is ample to specify a large number of detected objects, including their bounding boxes and probabilities belonging to different object classes.
For low-latency streaming
applications, one should not use
TCP as a transport
protocol. For example, one can use User Datagram Protocol (UDP) or Real-time Transport Protocol (RTP) that are designed for real-time applications \cite{schulzrinne1996rtp}. Here, we use TCP only to simulate the available link rate, since TCP’s congestion control automatically adjusts to the link rate and therefore can be regarded as a proxy for the video rate adaptation that would need to be incorporated on top of any real-time transport protocol such as RTP.
\subsection{Simulation Results}
\label{sec:wresults}
\begin{figure}%
\centering
\includegraphics[width=0.9\linewidth]{figures/wireless/oneroute_heraldsquare_new.PNG}%
\caption{\Gls{ue} route for the \textit{Herald Square} displayed along with the 3D model used in Wireless InSite ray tracing tool
\cite{remcom}. The \gls{ue}'s RX positions are highlighted by the yellow line. \gls{bs} 1 and \gls{bs} 2 provide both 4G-\gls{lte} and \gls{5g} \gls{mmw} connectivity.}%
\label{fig:oneroute}%
\end{figure}
\begin{figure*}
\begin{subfigure}{0.5\linewidth}
\centering
\includegraphics[width=0.95\linewidth]{figures/wireless/sinr_time_heraldsquare.png}
\caption{\Gls{sinr} over time in dB.}
\end{subfigure}
\begin{subfigure}{0.5\linewidth}
\centering
\includegraphics[width=0.95\linewidth]{figures/wireless/thr_time_heraldsquare.png}
\caption{End-to-end throughput over time in Mbps.}
\end{subfigure}
\caption{UE \gls{sinr} and end-to-end throughput over time for the \textit{Herald Square} site.}
\label{fig:herald}
\end{figure*}
As mentioned above, we performed end-to-end wireless simulation of an end user walking through four sites in Manhattan, near \textit{Lighthouse Guild}, \textit{Midtown}, \textit{Herald Square}, and \textit{NYU Langone}
as shown in Fig.~\ref{fig:main_route}.
As one example,
Fig.~\ref{fig:oneroute} depicts the route of the user (\gls{ue}) for the \textit{Herald Square} site. The \gls{ue} starts walking from the north-west corner in the figure and moves south-east across 34$^{\rm th}$ Street (yellow line in the picture). In this site, two base stations are present, both
with 4G-\gls{lte} and 5G. Figure~\ref{fig:herald}(a) shows the \gls{sinr} over time for this scenario for the two \gls{mmw} base stations (gNB 1 and gNB 2), as well
as the maximum SINR for the two LTE cells.
We see that, in this case, the LTE SINR
is continuously high (mostly $>$\SI{40}{dB})
due to the favorable propagation of
the lower frequency (\SI{1.9}{GHz}) carrier.
In contrast, there is a period from approximately
80 to 170 seconds where the SINR from both
mmWave cells is low.
This time period corresponds to a
segment of the route where
the UE is in \gls{nlos} to both
mmWave cells.
The resulting TCP end-to-end throughput is shown in Fig.~\ref{fig:herald}(b). We see that the LTE rate is more continuously available. However,
the maximum rate is limited to $\sim$\SI{36}{Mbps}
corresponding to the maximum modulation
and coding scheme (MCS) with a \SI{10}{MHz}
bandwidth. In contrast, the mmWave system
can obtain the full \SI{120}{Mbps} rate,
but the rate falls below the LTE rate during some
parts of the NLOS segment.
\begin{figure*}
\begin{subfigure}{0.5\linewidth}
\centering
\includegraphics[width=0.95\linewidth]{figures/wireless/thr_cdf_mmWave_Only_LTE_Only_mmWave+LTE_all.png}
\caption{CDF of the delay-constrained end-to-end throughput. }
\end{subfigure}
\begin{subfigure}{0.5\linewidth}
\centering
\includegraphics[width=0.95\linewidth]{figures/wireless/rtdelay_cdf_mmWave_Only_LTE_Only_mmWave+LTE_all.png}
\caption{CDF of the unconstrained end-to-end roundtrip delay in milli-seconds.}
\end{subfigure}
\caption{Aggregated simulation results for a hypothetical user (\gls{ue}) walking across the four
sites in Manhattan. The delay-constrained
throughputs in the left plot
are plotted for different rountrip delay constraint values $D_{\rm max} = 30, 40, 50\, \si{ms}$.}
\label{fig:ns3_results}
\end{figure*}
The simulation results for each of the four
scenarios have been aggregated and depicted in Fig.~\ref{fig:ns3_results}. Fig.~\ref{fig:ns3_results}(a) plots what we will
call the \textit{delay-constrained throughput},
which is calculated as follows. We divide
the time into intervals of $T$ seconds,
where $1/T = 30$\,\si{Hz} corresponds to the expected
video frame rate ($T$ is the frame interval). For each TCP packet transmitted in the interval, we measure its uplink
delay from the UE to the edge server application.
We also measure the downlink delay for each feedback packet transmitted
in that interval as well. Note that these delays
contain all the air-link and core network delays.
For a given delay constraint, $D_{\rm max}$,
we define the delay-constrained throughput as $b/T$,
where $b$ is the number of uplink bits transmitted
in the interval $T$ for which the uplink +
downlink delay $\leq D_{\rm max}$.
The delay-constrained rate is computed separately
for the LTE and mmWave systems. Since mmWave systems
are always deployed with a sub-6 GHz fallback carrier,
we also estimate the rate of mmWave+LTE system
as the maximum of these two rates under the same delay constraint.
Fig.~\ref{fig:ns3_results}(a) plots the
delay constrained rates for the LTE and mmWave+LTE
systems under delay constraints of $D_{\rm max}=$\,
30, 40 and 50\,\si{ms}.
We see in Fig.~\ref{fig:ns3_results}(a)
that the LTE system achieves a peak uplink
rate of approximately \SI{36}{Mbps}, and
attains over \SI{20}{Mbps} more than 90\% of
the time. However, this rate is only achievable
with a delay constraint of $D_{\rm max}=50$\,\si{ms}.
At a tighter delay constraint of $D_{\rm max}=40$\,\si{ms}, the delay-constrained rate of approximately \SI{36}{Mbps} is supported less than 80\% of the time; at $D_{\rm max}=30$\,\si{ms},
there are virtually no LTE data within this delay
constraint. In contrast, the mmWave 5G
system coupled with \gls{lte} can obtain the peak rate of \SI{120}{Mbps}
at least 40\% of the time, even under a delay
constraint of $D_{\rm max}=30$\,\si{ms}.
However, for approximately 25\% of the time,
the delay-constrained rate of the mmWave+LTE
system is similar to that of the LTE system
owing to the fact that the mmWave only coverage
is not always available and falls back to the LTE
carrier.
To understand the delay differences,
Fig.~\ref{fig:ns3_results}(b) plots the CDF
of the delays of packets without any delay constraint. We see that for the
mmWave system, the minimum delay is $\approx$\,\SI{15}{ms} which includes two times
the core network delay of $D_{\rm core}=$\,\SI{5}{ms}
along with an addition \SI{5}{ms} for the transmission
of the uplink and downlink data.
The LTE packet delays are generally higher.
Although the core network delay is assumed to be
the same, the LTE frame structure as well as the lower throughput results
in a higher air-link delay.
Finally, it is useful
to compare these results with the
URLLC requirements
of 5G. As mentioned in the Introduction, the URLLC design goal is to achieve
air-link latencies of 1 to \SI{10}{ms} \cite{li20185g, popovski2019wireless}.
When the UE has 5G mmWave connectivity, we see a median
delay of \SI{15}{ms}, which is consistent
with an air-link round-trip
latency of \SI{5}{ms} along with our
assumed core network delay of \SI{5}{ms} each way.
However, our study also includes blockage
and environments where the 5G coverage is not uniformly
available. As a result, the UE must occasionally
fall back
to LTE links where the delay is higher
and the bandwidth cannot sustain the peak rates.
In these cases, the overall delay grows significantly,
beyond the 5G URLLC levels.
\newcommand{\badcell}[1]{\cellcolor{red!20}{#1}}
\newcommand{\medcell}[1]{\cellcolor{orange!20}{#1}}
\newcommand{\goodcell}[1]{\cellcolor{green!20}{#1}}
\begin{table*}[t]
\begin{center}
\begin{tabular}{|>{\raggedright}m{2.5cm}|>{\raggedright}m{1.3cm}|>{\raggedright}m{1.3cm}|
>{\raggedright}m{1.3cm}|>{\raggedright}m{1.3cm}|>{\raggedright}m{1.3cm}|>{\raggedright}m{1.3cm}|>{\raggedright}m{3.5cm}| }
\hline
\tblhdr{Item} &
\multicolumn{2}{c|}{\tblhdr{Local only}} &
\tblhdr{LTE only} &
\multicolumn{2}{c|}{\tblhdr{mmWave+LTE}} & \tblhdr{Adaptive}
& \tblhdr{Remarks}\tabularnewline \hline
\multicolumn{8}{|l|}{\tblsubhdr{Video configuration and object detection performance} }
\tabularnewline \hline
Number of monocular cameras & \badcell{1} & \badcell{1} & \badcell{1} & \badcell{1} & \goodcell{4} & \goodcell{Variable (1-4)$^*$} &
$^*$1 camera if throughput $\leq$ \SI{26}{Mbps}, more cameras if throughput >26 Mbps
\tabularnewline \hline
Camera resolution & WVGA & 720P & 1080P &
1080P & 1080P & mostly 1080P &
\tabularnewline \hline
wmAP (\%) for multiple objects & \badcell{36.6} & \medcell{50.3} & \goodcell{54.0} &
\goodcell{54.0} & \goodcell{54.0} & \goodcell{54.0} avg$^\dagger$ & See Table~\ref{tab:ComplexityVsResolution}
\tabularnewline \hline
AP (\%) for person & 44.3 & 60.3 & 66.1 & 66.1 & 66.1 & 65.8 avg$^\dagger$
& See Table~\ref{tab:ComplexityVsResolution}
\tabularnewline \hline
Detection range for person (meter) & 6 & 9 & 12 &12 &12 & mostly 12 & See Fig.~\ref{fig:standing_people_range}
\tabularnewline \hline
\multicolumn{8}{|l|}{\tblsubhdr{Bandwidth, Delay, and Availability} }
\tabularnewline \hline
Total uplink data rate (Mbps) & 0 & 0 & 26 & 26 & 104 & variable &
We assume each camera stream takes 26 Mbps except in the adaptive case.
\tabularnewline \hline
Video frame delay (ms) & 33 & 33 & 33 & 33 & 33 & 33 & We assume video is captured at 30 Hz.
\tabularnewline \hline
Video encoding delay (ms) & 0 & 0 & 17 & 17 & 17 & 17 & We assume that video encoding takes at most 1/60 (s) per frame, because today's smart phones can capture video at 60 Hz. Local processing does not need to compress video.
\tabularnewline \hline
Inference time (ms) & 75 & 96 & 19 & 19 & 19 & variable & See Table~\ref{tab:ComplexityVsResolution}. We assume multiple GPUs process separate frames simultaneously when multiple camera data are uploaded.
\tabularnewline \hline
Median round-trip time (RTT) (ms) & - & - & 37 & 15 & 15 & 15 &
See Fig.~\ref{fig:ns3_results}(b).
\tabularnewline \hline
Median total delay (ms) & \medcell{108} & \badcell{129} & \medcell{106} & \goodcell{84} &
\goodcell{84}& \goodcell{84} &
Frame delay+encoding delay+inference time+RTT
\tabularnewline \hline
\textcolor{black}{Availability with total delay $\leq 100$\,\si{ms} }& \badcell{0\%} & \badcell{0\%} & \badcell{0\%} & \medcell{75\%} & \medcell{65\%} & \medcell{78\%} & \multirow{2}{3.5cm}{Computed from CDF of delay constrained throughput. See text.}
\tabularnewline \cline{1-7}
\textcolor{black}{Availability with total delay $\leq 150$\,\si{ms}} & \goodcell{100\%} & \goodcell{100\%} & \goodcell{95\%} & \goodcell{97\%} & \medcell{67\%} & \goodcell{100\%} &
\tabularnewline \hline
\end{tabular}
\end{center}
\begin{tikzpicture}[xscale=1, yscale=1,every text node part/.style={align=left}]
\newcommand{5.3cm}{5.3cm}
\footnotesize
\node [minimum width=1cm] (space) {};
\node[draw, right of=space, fill=red!20,
minimum width=1.8cm, minimum height=0.5cm] (bad) {};
\node[right of=bad,xshift=3cm,align=left,text width=6cm] {Performance value is poor};
\node[draw, below of=bad, yshift=0.4cm, fill=orange!20,
minimum width=1.8cm, minimum height=0.5cm] (med) {};
\node[right of=med,xshift=3cm,align=left,text width=6cm]
{Performance value is medium};
\node[draw, below of=med, yshift=0.4cm, fill=green!20,
minimum width=1.8cm, minimum height=0.5cm] (good) {};
\node[right of=good,xshift=3cm,align=left,text width=6cm]
{Performance value is good};
\end{tikzpicture}
$^\dagger$The average wmAP and AP presented are for a total delay of $\leq 100$\,\si{ms}. These numbers are 53.9\% and 66.0\%, respectively for a total delay $\leq 150$\,\si{ms}.
There are many other feasible configurations, including \\
(1) Processing 1080P video locally for increased detection accuracy, at a total delay of \SI{211}{ms}.\\
(2) Sending both views of each stereo camera or one view plus depth map, with increased uplink rate, and reduced availability.
\caption{Comparison summary of example configurations in different connectivity scenarios.}
\label{tab:summary}
\end{table*}
\section{Performance Evaluation}
\label{sec:evaluation}
\subsection{Overview}
We now combine the video processing
requirements in Section~\ref{sec:video}
with the wireless simulation results in Section~\ref{sec:wireless} to assess the
potential benefits of wireless offloading.
We consider three scenarios for edge connectivity:
(1) \textbf{Local only},
where all the processing is performed on the
wearable; (2) \textbf{LTE only}, where edge processing
can be accessed by the LTE link; and (3)
\textbf{mmWave+LTE} where edge processing
can be accessed by LTE or mmWave, whichever
has the highest rate.
For each such scenario, we consider different
possible video options in terms of the
number of cameras, spatial resolution, and bit rate.
We can then use the wireless analysis
in Section~\ref{sec:wireless} to
assess the percent of time such video options
would be available based
within a delay budget close to the
target of \SI{100}{ms}.
Although there are a large number of possible video configurations, in the sequel,
we will focus on the options in Table~\ref{tab:summary} as these provide a good
demonstration of the capabilities of
the system. The table also highlights some of the key
values in red, orange, and green to draw attention
to the performance that are relative poor, medium,
or good.
We also examine an adaptive offloading strategy, which switches between edge and local computing and furthermore adapts the video resolution based on the wireless link throughput when edge computing is chosen. The remaining sub-sections will
describe the details of these options and their
analysis.
\subsection{Video Configurations}
In Table~\ref{tab:summary}, for edge
computing with mmWave or LTE
connectivity, we have considered the case where the
video from each monocular camera would be delivered at 1080P spatial resolution, \SI{30}{Hz} temporal resolution, at \SI{26}{Mbps}.
Based on the video analysis in Section~\ref{sec:video},
this configuration provides a high object detection
accuracy and good
detection range -- see the wmAP and detection range
rows in Table~\ref{tab:summary}. Going beyond 1080P resolution and 26 Mbps brings only very slight gains, and yet processing 2.2K video will consume substantially more computation time.
For the local processing scenario,
we have considered only WVGA and 720P.
With local processing, the inference time for higher
spatial resolution 1080P (see Table~\ref{tab:ComplexityVsResolution}) would substantially exceed the delay budget.
As shown in Table~\ref{tab:summary},
this lower resolution results in both a lower
object detection accuracy and reduced object
detection range.
\subsection{Delay Analysis}
The delay computations in Table~\ref{tab:summary}
consider four components:
\begin{itemize}
\item \textit{Video frame delay} which is the interval of one video frame (i.e., the inverse
of the frame rate). For edge
computing, the video frame interval needs to be considered because an object may appear any time between two adjacent frames.
\item \textit{Video encoding delay} which is the time to encode the video for edge computing.
\item \textit{Round-trip time (RTT)}
which is the total time to transmit the packets
from the wearable to the edge server and back.
\item \textit{Inference time} which is the
time for the object detection network (either
local or edge) to compute the detection results.
\end{itemize}
In our analysis, the video frame delay, encoding delay,
and inference time are fixed. The only variable
component is the RTT. Table~\ref{tab:summary}
shows the median RTT and the corresponding median
total time.
We see that mmWave+LTE offers dramatically lower median
RTT of \SI{15}{ms} relative to the median RTT in LTE
only of \SI{37}{ms}. Recall that the RTT includes
$2D_{\rm core}=10$\,\si{ms} of assumed delay
from the base station (gNB or eNB) through the core
network to the edge server and back.
Note that with direct local processing, only the
frame delay and inference time are required since there is no video encoding or communication.
As suggested in \cite{liu2019edge}, to avoid detection mismatch, we will assume that the local processor runs a simple object tracking algorithm that predicts the locations of objects detected for the last frame for which the edge server detection results were fed back. For example, if the total delay for edge processing is twice of the frame interval, at the time when frame $t$ is captured, detection results for frame $t-2$ will be used as the reference, the motion between frame $t$ and frame $t-2$ will be used to predict the locations of these detected objects in frame $t$. In the mean time, frame $t$ will be delivered to the edge server with its processing results to be fed back at time when frame $t+2$ will be captured. The motion vectors between frames generated for video compression can be leveraged for local object tracking. Such an approach should be able to track small movements of previously detected objects within a few frames, while also reporting any newly appearing objects within the 100 ms delay.
\subsection{Supportable Video Streams Under Different Delay Constraints}
\begin{figure*}
\begin{subfigure}{0.5\linewidth}
\centering
\includegraphics[width=0.95\linewidth]{figures/wireless/heatmap_mmwave_lte_26.png}
\caption{\Gls{mmw} + \gls{lte} case, all cameras with 1080P resolution @ 26 Mbps.}
\end{subfigure}
\begin{subfigure}{0.5\linewidth}
\centering
\includegraphics[width=0.95\linewidth]{figures/wireless/heatmap_lte_26.png}
\caption{\Gls{lte} only case, all cameras with 1080P resolution @ 26 Mbps.}
\end{subfigure}
\quad
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=0.475\linewidth]{figures/wireless/heatmap_mmwave_lte_26_lower.png}
\caption{\Gls{mmw} + \gls{lte} case, camera $1$ @ 26 Mbps, cameras 2, 3, 4 @ 10 Mbps.}
\end{subfigure}
\caption{Heat-map of probabilities of supporting different numbers of cameras under different roundtrip delay constraints.}
\label{fig:heatmap}
\end{figure*}
From the wireless simulation results in Figure~\ref{fig:ns3_results},
we derive the probability of supporting one or more camera streams under different round-trip delay constraints.
These are plotted as heat maps in Fig.~\ref{fig:heatmap}.
Fig.~\ref{fig:heatmap}(a) shows that, with multi-connectivity using both \gls{mmw} and \gls{lte} links, we can support one video stream over 75\% of the time, and support two streams over 73\% of the time, under the the roundtrip delay constraint of \SI{30}{ms}.
All four cameras can be supported 65\% of the time. Furthermore, we can support one and four cameras with high availability (91\% and 67\%, respectively) if the delay constraint is relaxed to \SI{40}{ms}.
On the other hand, with the \gls{lte} link only (Figure.~\ref{fig:heatmap}(b)) and a round-trip
delay of \SI{40}{ms},
the availability for supporting one and four cameras drops to 79\% and 0\%, respectively, since the peak \gls{lte} rate is approximately \SI{36}{Mbps}.
The \gls{lte} link can sustain one camera with high probability (93\%) only if the delay constraint is relaxed to \SI{50}{ms}.
For any configuration,
we can also compute the probability that the total
delay will meet a certain delay target. For example,
from Section~\ref{sec:delay_requirements}, the
estimated total delay requirement is \SI{100}{ms}.
This is not met by local processing. When using offloading,
the video frame delay, encoding delay, and inference
take a total of $33+17+19=69$\,\si{ms}, so there
would be $D_{\rm max}=100-69=31$\,\si{ms} for the RTT.
Similarly,
if we relax the total delay requirement to
\SI{150}{ms}, the communication delay constraint
would be $D_{\rm max}=150-69=81$\,\si{ms}.
The availability numbers listed in the final two
rows of Table~\ref{tab:summary} are the percentage of
time the delay-constrained throughput meets the minimum
uplink data rate requirement at the RTT of 31 and 81 \si{ms}, respectively.
The availability percentages for these two delay constraints are not shown in Fig.~\ref{fig:heatmap}, but we have extracted the numbers from the wireless simulation results in a similar manner.
Note that the availability for RTT of \SI{31}{ms} is the practically the same as for \SI{30}{ms}. Using Fig.~\ref{fig:heatmap}, one can also find the corresponding availability for total delays of \SI{110}{ms} (RTT of \si{40}{ms}) or \si{120}{ms} (RTT of \si{50}{ms}).
In practice, given the limited total throughput, it may be better to only upload the front facing camera stream at a high rate (for the highest detection accuracy), and use a lower rate for other cameras (side and back facing) to enhance situational awareness.
As an example, Fig.~\ref{fig:heatmap}(c) shows the probability of supporting one camera at the full rate
of \SI{26}{Mbps} and additional cameras at \SI{10}{Mbps} each. In this case, with mmWave+LTE
connectivity and a total delay 100 \si{ms}, we could increase the probability to support 4 cameras from 65\% to 72\% .
\subsection{Adaptive Offloading}
Given the variability in availability,
particularly in mmWave, it is natural
to consider a strategy
that adaptively selects the rate,
number of cameras,
and whether to use local or remote
processing based on the available
uplink bandwidth.
For example, as a simple
strategy, when the available throughput is $\geq$\SI{26}{Mbps}, we can transmit
one or more cameras.
When the throughput is lower than \SI{26}{Mbps}, we can still offload the video for edge computing, but at lower
rates, enabling adaptive switching between camera number and video quality based on bandwidth constraints and functional need (refer to
Fig.~\ref{fig:heatmap} as an example).
From Fig.~\ref{fig:wmAPvsRate}, when the throughput is between \SI{6}{Mbps} and \SI{26}{Mbps}, the system should deliver the video at 1080P but at a lower rate, leading to proportionally lower detection accuracy. When the rate is between \SI{1}{Mbps} and \SI{6}{Mbps}, the system should upload the video at 720P resolution.
In the ``adaptive'' column of Table~\ref{tab:summary}, the detection accuracy is derived by assuming an average detection accuracy of 51.5 and 63.2 for multi-object and person, respectively, when the bit rate is between 6 and 26 Mbps (which occurs 2\% of the time with a delay constraint of \SI{30}{ms}, from Fig.~\ref{fig:ns3_results}(a)); and an accuracy of 41.1 and 49.7, respectively, when the bit rate is between 1 Mbps and 6 Mbps (which occurs 1\% of the time).
The overall availability for adaptive offloading under total \SI{100} ms delay is the probability that the throughput is $\geq 1$ {Mbps} at a RTT constraint of \SI{30}{ms}.
Under the relaxed total delay constraint of \SI{150} ms, when the throughput is between 10 and 26 Mbps (with probability of 2.2\%), the system should still upload the 1080P video. When the throughput is below \SI{10} Mbps (with probability of 0.8\%), the wearable could locally process the uncompressed video at 720P video resolution for better detection performance (see Fig.~\ref{fig:wmAPvsRate}). Therefore, the availability is 100\%. The average accuracy would be 53.92\% and 66.00\%, for multi-object and person, respectively.
\section{Conclusions and Future Work} \label{sec:conclusion}
Mobile edge computing coupled with the high data
rate capabilities of mmWave holds significant promise for accessing powerful video analytics by wearable devices.
We have assessed the
feasibility of such capabilities for a
advanced smart wearable with multiple high-resolution cameras where the wireless and video requirements
are particularly demanding.
Several new elements were required in the analysis
including developing a large labeled video data set,
evaluation of object detection algorithms at variable resolutions and bit rates, and detailed and high accuracy wireless simulations with ray tracing.
Overall, wireless simulations provide a high level
of realism and can
identify the key limitations in
high data rate edge computing.
These tools can be applied in other applications
and may prove valuable
as video processing and
spatial intelligence becomes more widely-used
in mobile scenarios.
For the VIS$^4$ION\xspace application, our simulation
results suggest that at bandwidths and loading
similar to current deployments, systems in
traditional sub-6-GHz bands combined with low delay
mobile edge computing can provide gains by
offloading camera data
a large fraction of time, improving the accuracy
and detection range. However, meeting the
end-to-end delay requirements of \SI{100}{ms} is challenging.
The mmWave bands can reduce the delay to meet these requirements
and provide additional
capabilities including multiple
cameras at high resolution.
However, due to blockage and the
limited range
of mmWave signals, the peak performance is not uniformly available at typical
cell-site densities.
Thus, fall back to lower frequency carriers and local
processing combined with adaptation in the video resolution
and number of camera streams will be required.
In the current work, we have abstracted this adaptation
by assuming that the number of cameras and their bit rate can be adjusted to the available throughput.
An obvious line of future work is to actually simulate
a particular adaptive video application over wireless links and assess
its performance. Additionally, we have simply relied on the pre-trained YOLO network that
was trained on uncompressed low-resolution images.
A second line of work is to train multiple
detection networks for different resolutions and bit rates or a single network that can perform well across resolutions and bit rates.
More ambitiously,
one can also consider new compression
schemes that are trained end-to-end with
object detection accuracy as the goal, as opposed to the standard compression algorithms, which are optimized for image reconstruction.
\paragraph*{Acknowledgements}
This work was supported in part by the NSF grant 1952180 under the Smart and Connected Community program
as well as the industrial affiliates of
NYU WIRELESS.
In addition,
Azzino, Mezzavilla and Rangan were supported by
NSF grants 1925079, 1564142, and 1547332
and the Semiconductor Research Corporation (SRC).
Yuan and Wang were also supported by NSF grant 2003182. Yu Hao was also partially supported by NYUAD Institute (Research Enhancement Fund - RE132).
Special thanks are also given to Remcom that provided the Wireless Insite software and GeoPipe that
provided the 3D models.
\bibliographystyle{IEEEtran}
|
3,212,635,537,489 | arxiv | \section{Introduction}
The logistic equation describes the population growth. The model is initially published by Pierre Verhulst in 1838 \cite{Cu}. The continuous Logistic model is described by a first order ordinary differential equation. The model describes the population growth that may be limited by certain factors like population density \cite{AlSaYo, Au, Pa}.
The continuous form of the logistic equation is expressed in the form of a nonlinear ordinary differential equation,
\[\dot{x}(t)=rx(t)\left(1-\frac{x(t)}{K}\right).\]
In the above equation, $x(t)$ indicates population at time $t$, $r>0$ represents the Malthusian parameter expressing growth rate of species and $K$ denotes carrying capacity.
Motivated by its applications in different scientific areas (electricity, magnetism, mechanics, fluid dynamics, medicine, etc. \cite{Alm, BaRe, Hil, Kil}), fractional calculus is in development, which has led to great growth in its study in recent decades. The fractional derivative is a nonlocal operator \cite{Die, Pod}, making fractional differential equations good candidates for modeling situations in which is important to consider the history of the phenomenon studied \cite{FeSa}, unlike the models with classical derivative where this is not taken into account. There are several definitions of fractional derivatives. The most commonly used are the Riemann-Liouville fractional derivative and the Caputo fractional derivative. It is important to note that while the Riemann-Liouville fractional derivative \cite{Old}, is historically the most studied approach to fractional calculus, the Caputo fractional derivative is more popular among physicists and scientists due to the fact that the formulation of initial value problems with this type of derivative is more similar to the formulation with classical derivative.
The fractional order logistic equation has been discussed in the literature \cite{BhDaGe, MoQa}. A detailed study of existence, uniqueness, stability and approximate solutions of this equation can be found in \cite{AmNuAnSu, BaReTid, EsEmEs, KSQB, SwKhMa, SyMaSh}.
When the logistic equation is used to describe the natural evolution of a species, it is logical to think about the exploitation of this resource. For this reason we have decided to make a study on the maximization of the exploitation of a certain resource by using fractional derivatives.
The principal objective of this work is to study a fractional control problem that models the maximization of the profits obtained by exploiting a certain resource. The structure of this article is as follows: in section 2, the classical control problem and its solution are presented. In section 3, the fundamental concept of fractional derivatives, fractional control and variational problems, and the solution of a fractional control problem. The comparison between both problems and numerical approximations of the solutions are discussed in section 4. Finally, the section 5, is dedicated to the conclusions.
\section{A simple optimal control problem}\label{sec2}
A simple classical optimal control model will be presented, which consists of maximizing the extraction or harvest of a certain renewable resource, where the first-order integer derivative appears in the dynamic equation,
\begin{equation}
\left\{
\begin{array}{l}
\max \, \int \limits_{0}^{T} e^{-\delta t} h(x(t)) \, dt\\
\dot{x}(t)=rx(t)\left(1-\frac{x(t)}{K}\right)-h(x(t))\\
x(0)=x_0\\
x(T)=x_{T}\\
h_{min}\leq h(x(t)) \leq h_{max},\\
\end{array}\right.
\label{pclasico}
\end{equation}
where $T$ represents the final time, $x_0$ the initial condition, $x_T$ the final condition, $h_{min}$ and $h_{max}$ preset minimum and maximum harvest and $e^{-\delta t}$ represents a discount factor with $\delta\geq 0$ the instantaneous annual rate of discount which can be zero.
In this problem the following dynamic equation appears
\begin{equation}
\dot{x}(t)=rx(t)\left(1-\frac{x(t)}{K}\right)-h(t),
\label{eclogisticacosecha}
\end{equation}
which is the logistical growth of a certain resource, where $r>0$ is the intrinsic growth rate, $K>0$ the carrying capacity of the resource and $h(t)$ is the harvest.
\begin{remark}
In general, the function to be optimized is the following
$$ \left(p - \frac{c}{x} \right) h(x),$$
where $p$ is the price and $\frac{c}{x}$ the extraction cost, \cite{Cl}. In this work a simplification of this model will be done, since the resolution of this problem has a high level of complexity in its fractional version.
\end{remark}
Now we will proceed to solve the problem (\ref{pclasico}).
Clearing $h(x(t))$ from the dynamic equation (\ref{eclogisticacosecha}) and replacing it in the functional to maximize results,
$$\left\{
\begin{array}{l}
\max \, \int \limits_{0}^{T} e^{-\delta t} \left[rx(t)\left(1-\frac{x(t)}{K}\right)-\dot{x}(t)\right] \, dt\\
x(0)=x_0\\
x(T)=x_{T}\\
h_{min}\leq h(x(t)) \leq h_{max},\\
\end{array}\right.
$$
which means, it becomes a variational problem with Lagrangian
$$L(t,x,\dot{x})=e^{-\delta t} \left[rx(t)\left(1-\frac{x(t)}{K}\right)-\dot{x}(t)\right].$$
Then, its Euler-Lagrange equation results
$$
\begin{array}{l}
\dfrac{\partial L}{\partial x}-\dfrac{d}{dt} \dfrac{\partial L}{ \partial \dot{x}}=0,
\end{array}
$$
from which is obtained
$$
\begin{array}{l}
r\left(1-\frac{2}{K}x(t)\right)-\delta=0.
\end{array}
$$
That is, the optimal population is the constant
\begin{equation}
x^{*}(t)=\frac{K}{2}\left(1-\frac{\delta}{r}\right),
\label{condesc1}
\end{equation}
while the optimal harvest is
\begin{equation}
h^{*}(t)=rx^{*}(t)\left(1-\frac{x^{*}(t)}{K}\right),
\end{equation}
then,
\begin{equation}
h^{*}(t)=\frac{K}{4} \frac{r^2-\delta^2}{r}.
\label{condesch1}
\end{equation}
\begin{remark}
It can be seen that being the solution $x^{*}(t)$ a constant function, it will not be able to verify the imposed boundary conditions. For this reason, it can be said that the problem is singular.
For its resolution, the Nearest Feasible Paths theorem will be used, and its proof can be seen in \cite{Set}.
\end{remark}
\begin{theorem}\textbf{Nearest Feasible Paths}
Consider the following optimal control problem with one state and one control variable
$$\left\{
\begin{array}{l}
\max \limits_{u} \, \int \limits_{0}^{T} e^{-\delta t} F(x,u) \, dt\\
\dot{x}(t)=f(x,u)\\
x(0)=x_0\\
x(T)=x_T\\
u \in [0,Q]\subset \mathbb{R},
\end{array}\right.
$$
we define the Nearest Feasible Paths $x^{*}(t)$ to a given feasible steady state $(\overline{x},\overline{u})$ as the feasible path starting from $x_0$ in such a way that
\[\left|x^{*}-\overline{x}\right|\leq \left|x(t)-\overline{x}\right| \, \, \forall t>0,\]
for all feasible paths $x(t)$ starting from $x_0$. Further, we define the nearest approach segment $\hat{x}^{*}(t), \, t \in [0, t_{min}]$, as the feasible segment starting from $x_0$ in such a way that
\[\left|\hat{x}^{*}-\overline{x}\right|\leq \left|x(t)-\overline{x}\right| \, \, \forall t\in [0, t_{min}],\]
for all feasible segments $x(t)$ starting from $x_0$ and where $t_{min}$ is defined as the first time $\hat{x}^{*}(t)$ reaches $\overline{x}$. If $\overline{x}$ were the optimal stationary equilibrium and if the optimal path for the problem is the Nearest Feasible Paths, then the nearest approach segment from $x_0$ to $\overline{x}$, followed by a stay at $\overline{x}$, followed by an exit from $\overline{x}$ as late as possible ($t_{max}$) to attain $x_T$ at time $T$ is optimal.
\end{theorem}
With this result, the method consists in finding $x_{a}(t)$ the solution to the problem that begins in $x(0)=x_0$ and uses the minimum harvest $h_{min}$ if $x_0<x^{*}(t)$ or uses the maximum harvest $h_{max}$ if $x_0>x^{*}(t)$, until it reaches the value $x^{*}(t)$ in the time $t_{min}$ to determinate. We call its solution $x_a(t)$ and $t_{min}$ will be the first moment when $x_{a}(t)$ gets the value $x^{*}(t)$.
Then it must be found $x_{b}(t)$ solution to the problem with final condition $x (T)=x_T$, that uses the minimum harvest $h_{min}$ if $x_T>x^{*}(t)$ or uses the maximum harvest $h_{max}$ if $x_T<x^{*}(t)$. The time $t_{max}$ is determined so that it is the first time when $x_b(t)$ gets the value $x^{*}(t)$.
In summary, the following optimal solution is obtained
\begin{equation}
x^{*}(t)= \left\{ \begin{array}{lcc}
x_{a}(t) & \mathrm{if} & 0\leq t \leq t_{min} \\
\\ \frac{K}{2}\left(1-\frac{\delta}{r}\right) & \mathrm{if} & t_{min} \leq t\leq t_{max} \\
\\ x_{b}(t) & \mathrm{if} & t_{max} \leq t\leq T
\end{array}
\right.
\label{xclasic}
\end{equation}
and the optimal harvest obtained with this procedure is
\begin{equation}
h^{*}(t)= \left\{ \begin{array}{lcc}
h_{min} & \mathrm{if} & x(t)> x^{*}(t) \\
\\ \frac{K}{4} \frac{r^2-\delta^2}{r} & \mathrm{if} & x(t) = x^*(t)\\
\\ h_{max} & \mathrm{if} & x(t)< x^{*}(t),
\end{array}
\right.
\label{hclasic}
\end{equation}
taking into account that for it to be a feasible solution it must be satisfied
\[h_{min}\leq \frac{K}{4} \frac{r^2-\delta^2}{r} \leq h_{max}.\]
\begin{remark}
The problem without discount is obtained as a particular case of the given, writing $\delta=0$ in the equations (\ref{xclasic}) and (\ref{hclasic}).
\end{remark}
Following, this problem will be solved but in its fractional version, for this we will need to recognize tools of the fractional calculus, which allow us to change from fractional control problems to fractional variational problems and their corresponding fractional Euler-Lagrange equation.
\section{A fractional optimal control problem} \label{sec:frac}
\subsection{Introduction to fractional calculus}
In this section certain definitions and properties of the fractional calculus will be presented. For more details refer to \cite{Die, Old, Pod}.
\begin{definition}
The Mittag Leffler function with parameters $\alpha , \, \beta$, is defined by
\begin{equation}
E_{\alpha,\beta} (z) = \displaystyle \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(\alpha k + \beta)},
\label{mitag}
\end{equation}
for all $z\in \mathbb{C}$.
\end{definition}
\begin{definition}
The Gamma function, $\Gamma: (0, \infty)\rightarrow \mathbb{R}$, is defined by
\begin{equation}
\Gamma(t) = \int_{0}^{\infty} s^{t-1} e^{-s} \, ds.
\label{gamma}
\end{equation}
\end{definition}
\begin{definition}
The Riemann-Liouville fractional integral operator of order $\alpha \in \mathbb{R}^{+}_{0}$ is defined in $L^1[a,b]$ by
\begin{equation}
\,_{a}I_{t}^{\alpha} [f] (t) = \dfrac{1}{\Gamma(\alpha)} \int_{a}^{t} (t-s)^{\alpha -1} f(s) \, ds.
\label{frac1}
\end{equation}
\end{definition}
\begin{definition}(Left and Right Riemann-Liouville Fractional Derivatives) \label{defRL}
The left and right Riemann-Liouville fractional derivatives of order $\alpha \in \mathbb{R}^{+}_{0}$ are defined, respectively, by
\[ \,_{a}D_{t}^{\alpha}[f](t)= \dfrac{1}{\Gamma(n-\alpha)}\dfrac{d^{n}}{dt^{n}}\int_a^{t}(t-s)^{n-1-\alpha}f(s)ds\]
and
\[ \,_{t}D_{b}^{\alpha}[f](t)= \dfrac{(-1)^{n}}{\Gamma(n-\alpha)}\dfrac{d^{n}}{dt^{n}}\int_t^{b}(s-t)^{n-1-\alpha}f(s)ds,\]
with $n=\left\lceil \alpha \right\rceil$, that is if $f \in L^1[a,b]$.
\end{definition}
\begin{definition}(Left and Right Caputo Fractional Derivatives)
The left and right Caputo fractional derivatives of order $\alpha \in \mathbb{R}^{+}_{0}$ are defined, respectively, by
\[ \,_{a}^{C}D_{t}^{\alpha}[f](t)= \dfrac{1}{\Gamma(n-\alpha)}\int_a^{t}(t-s)^{n-1-\alpha}\dfrac{d^{n}}{ds^{n}}f(s)ds\]
and
\[ \,_{t}^{C}D_{b}^{\alpha}[f](t)= \dfrac{(-1)^{n}}{\Gamma(n-\alpha)}\int_t^{b}(s-t)^{n-1-\alpha}\dfrac{d^{n}}{ds^{n}}f(s)ds,\]
with $n=\left\lceil \alpha \right\rceil$, that is if $\dfrac{d^{n}f}{dt^{n}} \in L^1[a,b]$.
\end{definition}
\begin{remark}An important difference between Riemann-Liouville derivatives and Caputo derivatives is that, being K an arbitrary constant, is
\[ \,_{a}^{C}D_{t}^{\alpha} K= 0,\hspace{1cm} \,_{t}^{C}D_{b}^{\alpha}K=0, \]
however
\[ \,_{a}D_{t}^{\alpha}K=\dfrac{K}{\Gamma (1- \alpha)}(t-a)^{-\alpha}, \hspace{1cm} \,_{t}D_{b}^{\alpha}K=\dfrac{K}{\Gamma (1- \alpha)}(b-t)^{-\alpha}, \]
\[ \,_{a}D_{t}^{\alpha} (t-a)^{\alpha-1}=0, \hspace{1cm} \,_{t}D_{b}^{\alpha}(b-t)^{\alpha-1}=0.\]
In this sense, the Caputo fractional derivatives are similar to the classical derivatives.
\end{remark}
\subsection{Fractional control and variational problems}
To solve a fractional control problem, tools of fractional variational problems will be used. For this, a brief introduction to them is presented.
Consider the following problem of the fractional calculus of variations:
find a function $x \in \,_{a}^{\alpha}E$ that optimizes (minimizes or maximizes) the functional
\[J(x)= \int^{b}_{a} L(t,x,\,_{a}^{C}D_{t}^{\alpha}x) \, dt,\]
with a Lagrangian $L \in C^1([a,b]\times \mathbb{R}^2)$ and
\[\,_{a}^{\alpha}E=\{ x: [a,b] \rightarrow \mathbb{R}: x \in C^1([a,b]), \, \,_{a}^{C}D_{t}^{\alpha}x \in C([a,b]) \},\]
subject to the boundary conditions $x(a)=x_a \, ,\, \, x(b)=x_{b}$.
Now the Euler-Lagrange equation for this problem will be stated, its proof is in \cite{LaTo}.
\begin{theorem}
Let $x$ be an optimizer of $J$ in $\,_{a}^{\alpha}E$ with $L\in C^{2}\left([a,b]\times\mathbb{R}^{2}\right)$ subject to boundary conditions $x(a)=x_a \, ,\, \, x(b)=x_{b}$, then $x$ satisfies the fractional Euler-Lagrange differential equation
\begin{equation}
\dfrac{\partial{L}}{\partial x}+\,_{t}^{C}D_{b}^{\alpha}\dfrac{\partial{L}}{\partial \,_{a}^{C}D_{t}^{\alpha}x}=0.
\label{EulerLagrange}
\end{equation}
\label{teoEulerLagrange}
\end{theorem}
\subsection{Fractional model}
Below, the fractional version of the problem that has been proposed in the first section will be shown. This means, it is the same problem but where derivatives of a fractional order appear,
\begin{equation}
\left\{
\begin{array}{l}
\max \, \int \limits_{0}^{T} e^{-\delta t} h(x(t)) \, dt\\
\,^{C}_{0} D^{\alpha}_{t}\left[x\right](t)=rx(t)\left(1-\frac{x(t)}{K}\right)-h(x(t))\\
x(0)=x_0\\
x(T)=x_{T}\\
h_{min}\leq h(x(t)) \leq h_{max}.\\
\end{array}\right.
\label{pfraccionario}
\end{equation}
The only difference with the problem exposed above, is that in the dynamic equation
\begin{equation}
\,^{C}_{0} D^{\alpha}_{t}\left[x\right](t)=rx(t)\left(1-\frac{x(t)}{K}\right)-h(x(t)),\\
\label{dinamicafrac}
\end{equation}
the first-order derivative no longer appears, but now intervenes $\,^{C}_{0} D^{\alpha}_{t}\left[x\right](t)$ the left Caputo fractional derivative of order $0<\alpha\leq 1$.
On one hand, the derivatives with fractional order contains partially or totally the history, temporary future or the spatial behavior of the function, averaged in some way. This transforms the fractional differential equations on suitable candidates for the modeling of memory phenomenon or subsequent effects, those in which what happens at a point on the space or at an instant of time depends on an interval (spatial or temporal) that has the point or the instant.
The Riemann-Liouville fractional derivative had a determining role in the developing of the fractional calculus theory, and was used successfully in strictly mathematical applications. But when it was trying to carry out mathematical modeling of real physical phenomena using fractional differential equations, the problem of the initial conditions also of fractional order emerged. These types of conditions are not physically interpretable and presents a considerable obstacle when making practical use of fractional calculus. The Caputo differential operator, in contrast to the Riemann-Liouville operator, uses derivatives of integer order as initial conditions, that is, initial values that are physically interpretable as in the models with integer derivatives. The definition that follows represented a notable practical advance in the study of physical phenomena such as those of the viscoelastic type and others.
Finally, the fractional derivative at $t$ of a function $x$ is a non-local operator, depending on past values of $x$ (left derivatives) or future values of $x$ (right derivatives). In physics, the right fractional derivative of $x(t)$ is interpreted as a future state of the process $x(t)$. For this reason, the right derivative is usually neglected in applications, when the present state of the process does not depend on the results of the future development. However, the left fractional derivative of $x(t)$ is interpreted as a past state of the process $x(t)$, in which memory effects intervene.
Since the evolution of a certain resource depends on its past, we have decided to choose the left Caputo fractional derivative for modeling its evolution, see \cite{CaTo}.
Now the problem (\ref{pfraccionario}) will be solved.
Clearing $h(x(t))$ from the dynamic equation (\ref{dinamicafrac}) and replacing it in the functional to maximize, it results,
$$\left\{
\begin{array}{l}
\max \, \int \limits_{0}^{T} e^{-\delta t} \left[rx(t)\left(1-\frac{x(t)}{K}\right)-\,^{C}_{0} D^{\alpha}_{t}\left[x\right](t)\right] \, dt\\
x(0)=x_0\\
x(T)=x_{T}\\
h_{min}\leq h(x(t)) \leq h_{max}.\\
\end{array}\right.
$$
Again it results a variational problem, now fractional, with fractional Lagrangian
$$L(t,x,\,^{C}_{0} D^{\alpha}_{t}\left[x\right])=e^{-\delta t} \left[rx(t)\left(1-\frac{x(t)}{K}\right)-\,^{C}_{0} D^{\alpha}_{t}\left[x\right](t)\right],$$
belonging to $C^{2}\left([0,T]\times\mathbb{R}^{2}\right)$.
Using theorem \ref{teoEulerLagrange}, its fractional Euler-Lagrange equation (\ref{EulerLagrange}) results,
$$
\begin{array}{l}
\dfrac{\partial L}{\partial x}+ \,^{C}_{t} D^{\alpha}_{T}\left[ \dfrac{\partial L}{ \partial \,^{C}_{0} D^{\alpha}_{t}\left[x\right] } \right] =0,
\end{array}
$$
from which is obtained
$$
\begin{array}{l}
e^{-\delta t} r \left(1-\frac{2}{K}x(t)\right)-\delta (T-t)^{1-\alpha}e^{-\delta T} E_{1,2-\alpha}(\delta (T-t))=0,
\end{array}
$$
where $E_{1,2-\alpha}(\delta (T-t))$ is the Mittag-Leffler function of two parameters.
We can conclude that the optimal population is
\begin{equation}
\begin{array}{l}
x^{*}_{\alpha}(t)=\frac{K}{2}\left(1-\frac{\delta}{r} (T-t)^{1-\alpha} e^{-\delta (T-t)} E_{1,2-\alpha}(\delta (T-t))\right).
\end{array}
\label{condescalfa}
\end{equation}
And the optimal harvest is obtained from this expression
\begin{equation}
h^{*}_{\alpha}(t)=rx^{*}_{\alpha}(t)\left(1-\frac{x^{*}_{\alpha}(t)}{K}\right)-\,^{C}_{0} D^{\alpha}_{t}\left[x^{*}_{\alpha}\right](t).
\label{condeschalfa}
\end{equation}
Note that in the classical case the optimal population is constant, which is not happening in this case, therefore in the optimal harvest a term appears with the Caputo derivative of $x^{*}_{\alpha}(t)$, that due to its difficulty we must calculate it numerically.
\begin{remark}
We can see that, as in the classical case, the optimal solution $x^{*}_{\alpha}(t)$, although it is not a constant, it will also not verify the established boundary conditions. This means that we are once again faced with a singular problem.
For its resolution, it would be possible to resort again to a Nearest Feasible Paths theorem, but in a fractional version. However, it has not been carried out yet because the proof of the theorem requires the use of a fractional Green theorem, which at the moment is only available for rectangular regions\cite{OdMaTo}, and its version is necessary for all types of regions.
\end{remark}
\begin{remark}
Here is also the solution of the problem without discount ($\delta=0$) as a particular case of the solution of the problem (\ref{pfraccionario}).
The optimal population is
\begin{equation}
\begin{array}{l}
x^{*}_{\alpha}(t)=\frac{K}{2}\\
\end{array}
\label{sindescalfa}
\end{equation}
and the optimal harvest is obtained from this expression
\begin{equation}
h^{*}_{\alpha}(t)= \frac{rK}{4}.
\label{sindeschalfa}
\end{equation}
It can be observed that the solution is the same for the classical case, we have that for the case $\delta=0$ the solution and the optimal harvest is the same for all $0<\alpha\leq 1$. The reason of this is that there is no time-dependent factor that multiplies $\,^{C}_{0} D^{\alpha}_{t}\left[x^{*}_{\alpha}\right](t)$ in the fractional variational problem resulting from taking $\delta=0$,
$$\left\{
\begin{array}{l}
\max \, \int \limits_{0}^{T} \left[rx(t)\left(1-\frac{x(t)}{K}\right)-\,^{C}_{0} D^{\alpha}_{t}\left[x\right](t)\right] \, dt\\
x(0)=x_0\\
x(T)=x_{T}\\
h_{min}\leq h(x(t)) \leq h_{max},\\
\end{array}\right.
$$
with Lagrangian $L(t,x,\,^{C}_{0} D^{\alpha}_{t}\left[x\right])= rx(t)\left(1-\frac{x(t)}{K}\right)-\,^{C}_{0} D^{\alpha}_{t}\left[x\right](t)$.
Its fractional Euler-Lagrange equation is
$$
\begin{array}{l}
\dfrac{\partial L}{\partial x}+ \,^{C}_{t} D^{\alpha}_{T}\left[ \dfrac{\partial L}{ \partial \,^{C}_{0} D^{\alpha}_{t}\left[x\right] } \right] =0,\\
\end{array}
$$
which means that
$$
\begin{array}{l}
r \left(1-\frac{2}{K}x(t)\right)+\,^{C}_{t} D^{\alpha}_{T}\left[(-1)\right] =0,
\end{array}
$$
then,
$$
\begin{array}{l}
r \left(1-\frac{2}{K}x(t)\right)=0,
\end{array}
$$
since the Caputo derivative of a constant is also zero, as in the case of the classical derivative, then the same Euler-Lagrange equation is obtained for all $0<\alpha\leq 1$ and its solution is independent of that value.
\end{remark}
To perform a graphic analysis of the solutions to the discounted problem in both the fractional and classical cases, we must see a specific example.
\section*{Example and comparison}
Consider $r,\,K,\,\delta,\,x_0,\, x_T$, $T$, $h_{min}$ and $h_{max}$ in given values, as in the example \textit{The Pacific Halibut Fishery}, in \cite{Cl}.
As well, it is considered a fixed value of $\alpha$ to evade the problem of needing a fractional Nearest Feasible Paths theorem, being able to obtain the boundary conditions $x_0$ and $x_{10}$ of this fractional solution, in order to make a comparison.
The following problem will be considered,
$$\left\{
\begin{array}{l}
\max \, \int \limits_{0}^{10} e^{-0.01 t} h(x(t)) \, dt\\
\,^{C}_{0} D^{\alpha}_{t}\left[x\right](t)=0.71\,x(t)\left(1-\frac{x(t)}{80.5}\right)-h(x(t))\\
x(0)=38.6896\\
x(10)=40.25\\
10 \leq h(x(t)) \leq 15,\\
\end{array}\right.
$$
A comparison of the results will be made between $\alpha=1$ (classical version) and $\alpha=0.6$ (fractional version).
Solving the problem in its classical version as in the previous section, using (\ref{condesc1}) and (\ref{condesch1}) we have
\begin{equation}\label{xestrella}
x^{*}(t)=39.6831,
\end{equation}
\begin{equation}
h^{*}(t)=14.2859.
\end{equation}
It can be observed that $x^{*}(t)$ does not verify established boundary conditions, therefore, the Nearest Feasible Paths theorem will be used to solve the problem.
The method consists in finding $x_{a}(t)$ the solution to the problem that begins in $x(0)=38.6896$ and uses the minimum harvest $h_{min}=10$ until it reaches the value $x^{*}(t)=39.6831$ in the time $t_{min}$.
For this purpose,
$$\left\{
\begin{array}{l}
\dot{x}(t)=0.71 x(t)\left(1-\frac{x(t)}{80.5}\right)-10\\
x(0)=38.6896.\\
\end{array}\right.
$$
Its solution is
\begin{equation}
x_a(t)= \frac{20.9703+62.3013 \, e^{0.388979t}}{1.1523+e^{0.388979t}}.
\label{sola}
\end{equation}
And $t_{min}$ will be such that $x_a (t_{min})= x^{*}(t)=39.6831$. Results $t_{min}=0.232235$.
Following with the method, it must be found $x_{b}(t)$ solution to the problem that uses the minimum harvest $h_{min}=10$ until it reaches the value $x (T)=x(10)=40.25$.
For this purpose,
$$\left\{
\begin{array}{l}
\dot{x}(t)=0.71 x(t)\left(1-\frac{x(t)}{80.5}\right)-10\\
x(10)=40.25.\\
\end{array}\right.
$$
Its solution is
\begin{equation}
x_b(t)=\frac{889.931+62.3013 \, e^{0.388979t}}{48.9008+e^{0.388979t}}.
\label{solb}
\end{equation}
And $t_{max}$ will be such that $x_b (t_{max})= x^{*}(t)=39.6831$. Results $t_{max}=9.8678$.
Finally, the optimal solution will be obtained, using (\ref{xestrella}), (\ref{sola}) and (\ref{solb}), times $t_{min}$ and $t_{max}$ and the Nearest Feasible Paths theorem, it is obtained,
\begin{equation}
x^{*}(t)= \left\{ \begin{array}{lcc}
\displaystyle \frac{20.9703+62.3013 \, e^{0.388979t}}{1.1523+e^{0.388979t}} & \mathrm{if} & 0\leq t\leq 0.232235 \\
\\ 39.6831 & \mathrm{if} & 0.232235 \leq t\leq 9.8678 \\
\\ \displaystyle \frac{889.931+62.3013 \, e^{0.388979t}}{48.9008+e^{0.388979t}} & \mathrm{if} & 9.8678 \leq t\leq 10=T
\end{array}
\right.
\end{equation}
and the optimal harvest obtained with this procedure is
\begin{equation}
h^{*}(t)= \left\{ \begin{array}{lcc}
10 & \mathrm{if} & 0\leq t\leq 0.232235 \\
\\ 14.2859 & \mathrm{if} & 0.232235 \leq t\leq 9.8678 \\
\\ 10 & \mathrm{if} & 9.8678 \leq t\leq 10=T.
\end{array}
\right.
\end{equation}
Solving the problem in its fractional version as in the previous section, from (\ref{condescalfa}) and (\ref{condeschalfa}),
is obtained
\begin{equation}
\begin{array}{ll}
x^{*}_{0.6}(t)&= 40.25\left(1-\frac{0.01}{0.71} (10-t)^{0.4} e^{-0.01\, (10-t)} E_{1,1.4}(0.01\, (10-t))\right).
\end{array}
\end{equation}
While the optimal harvest is
\begin{equation}
h^{*}_{0.6}(t)=0.71 \,x^{*}_{0.6}(t)\left(1-\frac{x^{*}_{0.6}(t)}{80.5}\right)-\,^{C}_{0} D^{0.6}_{t}\left[x^{*}_{0.6}\right](t).
\end{equation}
In the graphic below, the optimal populations corresponding to both cases can be observed.
\begin{figure}[htb]
\centering
\includegraphics[width=85mm]{Graphics/OptimalPopulations.pdf}
\caption{Optimal populations of classic and fractional problems.}
\label{Figcosechadescx}
\end{figure}
It can be noticed in Figure \ref{Figcosechadescx} that the optimal solution given by the fractional problem, $x^{*}_{0.6}(t)$, is lower than the optimal population of the classical problem, $x^{*}(t)$, most of the time, it means that the version with $\alpha=0.6$
shows a deterioration of the state of the stock with respect to the case $\alpha=1$ which is only recovered at the end by the fact that it has to verify the final condition.
In the following graphics, the optimal harvests are considered. Since $\,^{C}_{0} D^{0.6}_{t}\left[x^{*}_{0.6}\right](t)$ cannot be obtained exactly, we will proceed to use a fractional numerical method of L1 type \cite{BaDiScTr, LiZe}.
A regular partition of $[0,t]$ is considered, as $0=t_0 \leq t_1 \leq$ ... $\leq t_m=t$, of size $\Delta t>0$, to approximate the Caputo derivative as follows
\[\,^{C}_{0} D^{\alpha}_{t}\left[f\right](t_{m}) =\displaystyle \sum_{k=0}^{m-1} b_{m-k-1}(f(t_{k+1}-f(t_k)),\]
where $b_k=\frac{\Delta t^{-\alpha}}{\Gamma(2-\alpha)}\left[(k+1)^{1-\alpha}-k^{1-\alpha}\right]$.
Using this method we have obtained.
\begin{figure}[htb]
\centering
\includegraphics[width=85mm]{Graphics/OptimalHarvest.pdf}
\caption{Optimal harvests of classical and fractional problems.}
\label{Figcosechadesch}
\end{figure}
It can be noticed in Figure \ref{Figcosechadesch} that although with the fractional problem the extraction of the resource is minor, which is logical because in the fractional model the resource grows more slowly, it only decreases at the end of the interval near $T$ and until the final extraction in $T$ turns out being larger than in the classical case.
This way, it is possible to make a comparison of the profit obtained in each case,
\begin{center}
Classical case profit: $ 134.411$.\\
Fractional case profit:$133.828$.\\
\end{center}
To make an analysis of this, consider the following graphic of the resources evolution without harvest with the given initial condition.
\begin{figure}[htb]
\centering
\includegraphics[width=85mm]{Graphics/Populations.pdf}
\caption{Populations of the classical and fractional problems without harvest.}
\label{Figlog}
\end{figure}
It can be noted that the use of a fractional dynamic equation, which makes the resource grow more slowly as in Figure \ref{Figlog}, does not vary considerably the profit compared to the classical case.
Lastly, it can be stated what happens to the population $x(t)$ if we take the optimal harvest of the classic problem $h_1^{*}$ and consider it in the fractional dynamic equation of the resource.
It must be solved
$$\left\{
\begin{array}{l}
\,^{C}_{0} D^{\alpha}_{t}\left[x\right](t)=0.71\,x(t)\left(1-\frac{x(t)}{80.5}\right)-14.2859\\
x(0)=38.6896.\\
\end{array}\right.
$$
Since this equation has no exact solution, it will be approximated using the Adams fractional method, which consists of using Euler's method to obtain $u_{n+1}^{P}$ (predictor), and the trapezoidal fraction rule to get $u_{n + 1}$ (corrector),
\[ \left\{ \begin{array}{ll}
u_{n+1}^{P}&=\displaystyle \sum_{j=0}^{m-1}\frac{t_{n+1}^{j}}{j!}u_{0}^{j}+\sum_{j=0}^{n}b_{j,n+1}f(t_j,u_j),\\
u_{n+1}&=\displaystyle \sum_{j=0}^{m-1}\frac{t_{n+1}^{j}}{j!}u_{0}^{j}+\sum_{j=0}^{n}a_{j,n+1}f(t_j,u_j)+a_{n+1,n+1}f(t_{n+1},u_{n+1}^{P}).\\
\end{array}\right.\]
For more details refer to \cite{BaDiScTr, LiZe}.
The following result was obtained
\begin{figure}[htb]
\centering
\includegraphics[width=85mm]{Graphics/DifferentHarvest.pdf}
\caption{Optimal populations of the classical and fractional with different harvest problems.}
\label{Figxconh1}
\end{figure}
In the Figure \ref{Figxconh1} it can be seen that the population obtained from the fractional dynamic equation corresponding to taking the optimal harvest of the classical control problem is lower than the obtained by taking the optimal harvest of the fractional problem, as we expected.
Furthermore, if we assume that the "true evolution" of the resource is considering $\alpha = 0.6$ in the dynamic equation and that the harvesting agency considers the dynamic equation of $\alpha = 1$ to be erroneous, the loss is not very great because the agency will use $h^{*}_1(t)$ and its profit will be $133.828$ and not $ 134.411$ as would have been estimated. Also note that the difference between the profits obtained could be more significant if the instantaneous profit function, which in this case is only the harvest, was a little more complex and depended on the stock too as in Remark 1.
\section{Conclusions}
\label{sec:conclusions}
In this article, we have studied a fractional control problem that models the maximization of the profits obtained by exploiting a certain resource. An explanation of the proposed model has been made. Due to the singularity of the problem, different resolution techniques have been developed, both for the classic case and the fractional case. Although we have seen the need of a non-existent fractional Nearest Feasible Paths theorem, we have been able to make a comparison between the classical and fractional results for a certain value of the fractional order. It is also observed that the order of time fractional derivative significantly affects the population growth. Hence, we conclude that fractional derivatives may be more suitable for modeling the evolution of natural resources that naturally have a resilience problem. As a future investigation, it is proposed that the extension of the fractional Nearest Feasible Paths theorem should be explored and the optimal control problem should be extended for more complex instantaneous profits functions.
\section*{Acknowledgments}
This work was partially supported by Universidad Nacional de Rosario through the projects ING568 ``Problemas de Control \'Optimo Fraccionario''. The first author was also supported by CONICET through a PhD fellowship.
|
3,212,635,537,490 | arxiv | \section{Introduction}
\label{sec:intro}
Internet-of-things (IoT), which aims at realizing ubiquitous connectivity for massive devices in a wireless manner \cite{Atzori2010Internet,stankovic2014research}, is one of the major applications of the forthcoming fifth generation (5G) and future wireless networks. Due to the exponential growth of the number of IoT devices, substantial amount of energy and radio spectrum resources is required to support such massive connections.
The lack of sufficient radio spectrum is one of the bottlenecks to the success of IoT. According to \cite{UniEurope2016Identification}, around 76 GHz spectrum resources are needed to accommodate the massive IoT connections if exclusive spectrum is allocated. While cognitive radio (CR) technology can be used to support the shared spectrum access for IoT \cite{Liang2011Cognitive,Khan2017Cognitive,Zhang2018Spectrum}, the required spectrum resource is still as large as 19 GHz \cite{UniEurope2016Identification}. On the other hand, energy constraint is another critical issue of IoT devices. Traditional transmitters in IoT devices use active radio frequency (RF) components such as converters and oscillators, which are costly and power-consuming, thus may not be suitable for low-power IoT devices.
Therefore, novel spectrum and energy efficient communication technologies need to be developed for the future IoT.
One solution to achieve energy efficient IoT is ambient backscatter communication (AmBC) \cite{Liu2013}, in which a passive backscatter device (BD) modulates its information over ambient RF signals (e.g., cellular and WiFi signals) without requiring active RF components. While suitable for passive IoT, due to the spectrum sharing nature, the backscatter transmission in AmBC may suffer from severe direct-link interference (DLI) \cite{Wang2016Ambient}, resulting in performance degradation for the BD transmission. To tackle the DLI problem, in \cite{Guo2018Exploiting}, multiple receive antennas are used to suppress the DLI, while in \cite{Yang2017-p-}, a novel BD waveform is designed with clever interference cancellation by exploiting the cyclic prefix of ambient orthogonal frequency division multiplexing (OFDM) signal. In \cite{Zhang2018Constellation}, an improved Gaussian mixture model (GMM) based algorithm is proposed to recover the BD symbols by exploiting the received signal constellation information. In \cite{Yang-2017-p1-6}, a cooperative receiver with multiple antennas is designed to decode both the primary signal and the backscattered signal. In \cite{Darsena-2017-p1-1}, the capacity of AmBC system over ambient OFDM signals with perfect DLI cancellation is analyzed. In \cite{Kang-2017-p1-6}, the AmBC system is designed to maximize the ergodic capacity of the BD by jointly optimizing the ambient RF source's transmit power and the BD's reflection coefficient. In \cite{Duan2017Achievable}, the sum rate of the multiple-input multiple-output (MIMO) primary system and the multi-antenna BD transmission is analyzed. In \cite{Liu2018Backscatter}, the achievable rate region of the primary and BD transmissions is studied based on a new multiplicative multiple-access channel model. The above studies all assume certain forms of cooperation at the receiver side to cancel out the DLI or suppress the DLI effect.
In this paper, we propose a novel passive IoT transmission scheme, namely symbiotic radio (SR), in which a BD is integrated with a primary transmission. In the proposed SR, the primary transmitter (PT) is designed to support both the primary and BD transmissions, and the primary receiver (PR) needs to decode the information from the PT as well as the BD. Based on different transmission rate of the BD, the proposed SR can be further divided into parasitic SR (PSR), for which the BD transmission may introduce interferences to the primary transmission, and commensal SR (CSR), for which the two transmissions benefit from each other. One of the typical applications for SR is smart home, in which a smartphone recovers the data from both its serving WiFi AP and a served IoT sensor in its vicinity.
Compared with the conventional AmBC systems, in the proposed SR, the BD transmission shares not only the radio spectrum and RF source but also the receiver with the primary system. In this paper, we consider a basic SR model consisting of a multiple-input single-output (MISO) primary system and a single-antenna BD. The PT jointly designs its transmit beamforming to assist in the primary and BD transmissions, while the PR cooperatively decodes the signals from both the PT and the BD. Thus, the BD can realize opportunistic transmission with the aid of the primary system; on the other hand the achievable rate of the primary system can be improved by properly exploiting the additional signal path from the BD.
The main contributions of this paper are summarized as follows:
\begin{itemize}
\item First, we establish a general system model for the proposed SR and further investigate two practical setups, PSR and CSR, for which the symbol period for BD transmission is assumed to be either same as or much longer than that of the primary system, respectively.
\item Second, we analyze the achievable rates of the primary and BD transmissions under the two SR setups. Specifically, the achievable rate of the primary transmission is derived by treating the BD signal as an interference for PSR and a multipath signal component for CSR. After the PR cancels out the primary signal, the achievable rate of the BD transmission is obtained.
\item Third, transmit beamforming problems are formulated and solved, which aim to maximize the weighted sum rate of the primary and BD transmissions or to minimize the PT's transmit power under rate constraints.
\item Fourth, we propose a novel optimal beamforming structure to reduce the computational complexity. Specifically, the optimal transmit beamforming vector is shown to be a linear combination of the primary direct-link channel vector and the backscatter-link channel vector.
\item At last, numerical examples are presented to show that for the CSR system, the BD can realize its own transmission, and meanwhile enhance the primary transmission rate by providing an additional signal path for the primary system.
\end{itemize}
\begin{table}[t]\label{table:abbr}
\caption{List of abbreviations}
\centering
\begin{spacing}{1.2}{
\begin{tabular}{|l|l|}
\hline
\textbf{Abbreviation} & \textbf{Description} \\
\hline
AmBC & Ambient Backscatter Communication \\
\hline
BD & Backscatter Device \\
\hline
CSCG & Circularly Symmetric Complex Gaussian\\
\hline
CSI & Channel State Information\\
\hline
CSR & Commensal Symbiotic Radio \\
\hline
DLI & Direct-link Interference \\
\hline
IoT & Internet of Things\\
\hline
MIMO & Multiple-input Multiple-output \\
\hline
MISO & Multiple-input Single-output \\
\hline
OFDM & Orthogonal Frequency Division \\
&Multiplexing\\
\hline
PDF & Probability Density Function\\
\hline
PR & Primary Receiver\\
\hline
PSD & Positive Semi-definite\\
\hline
PSR & Parasitic Symbiotic Radio\\
\hline
PT & Primary Transmitter\\
\hline
RF & Radio Frequency\\
\hline
SDR & Semi-definite Relaxation\\
\hline
SIC & Successive Interference Cancellation\\
\hline
SINR & Signal to Interference plus Noise Ratio\\
\hline
SNR & Signal to Noise Ratio\\
\hline
SR & Symbiotic Radio\\
\hline
TPM & Transmit Power Minimization\\
\hline
WSRM & Weighted Sum-rate Maximization\\
\hline
\end{tabular}
}
\end{spacing}
\end{table}
The rest of this paper is organized as follows. In Section \ref{sec:system-model}, we present the SR system model. In Section \ref{sec:ARA_analysis}, we derive the achievable rates of the primary and the BD transmissions for both PSR and CSR. In Section \ref{sec:ProblemF}, we formulate the weighted sum-rate maximization problem and the transmit power minimization problem. In Section \ref{sec:solution}, we present the SDR-based solutions to the formulated problems. In Section \ref{sec:LowC}, a more efficient algorithm with lower complexity is presented based on a novel beamforming structure. In Section \ref{sec:NumR}, numerical results are presented for performance evaluations. Finally, the paper is concluded in Section \ref{sec:Conclusion}.
The main notations in this paper are listed as follows: The lowercase, boldface lowercase, and boldface uppercase letters such as $t$, $\mathbf{t}$, and $\mathbf{T}$ denote the scalar, vector, and matrix, respectively. $|t|$ denotes the absolute value of $t$. $\|\mathbf{t}\|$ denotes the norm of vector $\mathbf{t}$. ${\cal{CN}}(\mu, \sigma^2)$ denotes the circularly symmetric complex Gaussian (CSCG) distribution with mean $\mu$ and variance $\sigma^2$. $\mathbb{E}[\cdot]$ denotes the statistical expectation. $t^{\ast}$ denotes the conjugate of $t$. $\mathbf{T}^{\mathrm{T}}$ and $\mathbf{T}^{\mathrm{H}}$ denotes the transpose and conjugate transpose of matrix $\mathbf{T}$, respectively. Finally, the list of abbreviations commonly appeared in this paper is given in Table \ref{table:abbr}.
\section{System Model}
\label{sec:system-model}
Fig.~\ref{fig:systemmodel} shows the system model of a symbiotic radio (SR) consisting of three nodes, namely the primary transmitter (PT) equipped with $M$ ($M > 1$) antennas, the single-antenna primary receiver (PR) and the single-antenna backscatter device (BD). The PT performs multi-antenna beamforming to transmit its primary information to the PR, and at the same time enables the BD to transmit information to the PR. Specifically, the BD modulates its own information over the incident (primary) signal from the PT by intelligently varying its reflection coefficient. The SR thus shares not only the same spectrum but also the same receiver with the primary system.
\begin{figure}[t]
\centering\includegraphics[width=.63\columnwidth]{systemmodel.eps}
\caption{System model of a symbiotic radio.}\label{fig:systemmodel}
\end{figure}
Block flat-fading channel models are considered in this paper. During each fading block, the direct-link channel from PT to PR is denoted by $\mathbf{h}_1 = [h_{1,1}, \ldots, h_{M,1}]^\mathrm{T} \in\mathbb C^{M\times 1}$, where $h_{m,1}, \forall m$, denotes the the channel coefficient between the PT's $m$-th antenna and the PR's antenna. Meanwhile, the backscatter-link channel, denoted by $g\mathbf{h}_{2}$, is the multiplication of the forward-link channel from PT to BD, denoted by $\mathbf{h}_{2}= [h_{1,2}, \ldots, h_{M,2}]^\mathrm{T} \in \mathbb{C}^{M\times 1}$, and the backward-link channel from BD to PR, denoted by $g \in \mathbb{C}$. We assume that the SR system operates in the time-division-multiplexing (TDD) mode, and the PT and the PR have perfect channel state information (CSI) of the direct link and the backscatter link. In practice, the CSI can be obtained by the training-based channel estimation scheme with two steps. First, the BD switches its impedance into the matched state, and the PT estimates the direct-link channel $\mathbf{h}_1$ via channel reciprocity. Second, the BD switches its impedance into a fixed and known backscatter state, and the PT estimates the backscatter-link channel $g \mathbf{h}_2$ by subtracting the estimated direct-link channel component ${\mathbf{h}}_1$ from the estimated composite channel $\mathbf{h}_1 + g \mathbf{h}_2$.
\section{Achievable Rate Analysis}\label{sec:ARA_analysis}
In this section, we analyze the achievable rate performance of the proposed SR. Let $s(n)$ be the signal transmitted by the PT with symbol period $T_s$, and $s(n)$ is assumed to follow the standard CSCG distribution, i.e., $s(n) \sim \mathcal{CN}(0,1)$. Denote the beamforming vector of the PT by $\mathbf{w}\in\mathbb{C}^{M\times 1}$. Let $c(n)$ be the BD signal to be transmitted, with symbol period $T_c$. The $c(n)$ varies with different reflection coefficients and is assumed to be distributed\footnote{We assume that the Gaussian codewords herein to derive the maximum achievable rate of the SR.} as $\mathcal{CN}(0,1)$. The backscattered signal from the BD is thus $\sqrt{\alpha} c(n)$, where the power reflection coefficient $\alpha \in[0,1]$ controls the power of the backscattered signal by the BD. It is noticed that there is no additive noise in the BD, since its integrated circuit only includes passive components \cite{Fuschini-2008-p33-35,Qian2017Noncoherent}.
In the following, we consider two setups based on different relationships between $T_s$ and $T_c$. One is PSR for which $T_s = T_c$, and the other is CSR, for which $T_c = N T_s$, where $N$ is an integer and $N \gg 1$.
\subsection{PSR Setup}\label{sec:setupESP}
Let $p$ be the transmit power of the PT. The PR receives the backscattered signal from the BD as well as the primary signal transmitted from the PT. In the $n$-th symbol period, the received signal at the PR, denoted by $y(n)$, is thus given by
\begin{equation}\label{eq:Tr1}
y(n) = \sqrt {p} \mathbf{h}_1^\mathrm{ H} \mathbf{w} s(n) + \sqrt{p} \sqrt{\alpha} c (n)g\mathbf{h}_2^\mathrm{H} \mathbf{w}s(n) + z(n),
\end{equation}
where $z(n)$ is the additive white Gaussian noise (AWGN) with zero mean and power $\sigma^2$, i.e., $z(n)\sim{~}\mathcal{CN}(0,\sigma^2)$.
In practice, the direct-link signal is typically stronger\footnote{Since the analog-to-digital convertor (ADC) in the receiver often has large dynamic range (e.g., 49.9 dB for an 8-bit ideal ADC\cite{Walden-1999-p539-550}.) and the line-of-sight (LoS) pathloss due to the transmission from the BD to PR is usually within this range (e.g., 28 dB for 5m distance), the two received signals generally will not exceed the dynamic range of ADC.} than the backscatter-link signal, due to the following two facts. First, the backscatter-link channel suffers from double attenuations, i.e., the forward-link channel $\mathbf{h}_2$ and the backward-link channel $g$. Second, compared to the incident primary signal, the backscattered signal from the BD further suffers from an obvious power loss due to the backscattering operation. As a result, the PR can first decode the primary signal $s(n)$, then cancels out the decoded signal $\hat{s}(n)$ from its received signal, and finally detects the BD signal $c(n)$. In the following, we analyze the achievable rate performance of such decoding scheme.
Since $s(n)$ and $c(n)$ have the same symbol rate, when the PR decodes the primary signal $s(n)$, it treats the BD signal as the background noise of which the average power is
$\mathbb{E}\left[\alpha p|g|^2 |c(n)|^2 |\mathbf{h}^\mathrm{ H}_2\mathbf{w}|^2\right]=\alpha p|g|^2 |\mathbf{h}^\mathrm{ H}_2\mathbf{w} |^2$. Thus the signal-to-interference-plus-noise ratio (SINR) for decoding $s(n)$ at the PR is given by
\begin{equation}\label{eq:SINR_PR}
\gamma_{s}^{(1)}=\frac{{p | {\mathbf{h}_1^\mathrm{H}\mathbf{w}}|^2}}{{\alpha p|g|^2 | {\mathbf{h}_2^\mathrm{H}\mathbf{w}} |^2 + {\sigma ^2}}}.
\end{equation}
The corresponding data rate of the primary system can be written as
\begin{equation}\label{eq:Rs1}
{R_{s}^{(1)}} = {{{\log }_2}(1 + \gamma_{s}^{(1)})}.
\end{equation}
After obtaining an estimation of the primary signal $\hat{s}(n)$, the PR utilizes the successive-interference-cancellation (SIC) technique to decode the BD signal $c(n)$. That is, the received primary signal component $\sqrt{p}\mathbf{h}_1^\mathrm{ H}\mathbf{w}\hat{{s}}(n)$ is subtracted from the received signal $y(n)$, yielding the following intermediate signal
\begin{equation}\label{eq:yc_est}
\hat{y}_c(n)=y(n)-\sqrt {p} \mathbf{h}_1^\mathrm{ H}\mathbf{w}\hat{s}(n).
\end{equation}
Assuming that the primary signal is removed perfectly, we have
\begin{equation}\label{eq:yc}
\hat{y}_c(n)=\sqrt{\alpha}\sqrt{p} s(n) g\mathbf{h}_2^\mathrm{H} \mathbf{w} c(n) + z(n).
\end{equation}
Given the primary signal $s(n)$, the signal-to-noise ratio (SNR) for decoding the BD signal is written as
\begin{equation}\label{eq:SNR_SR}
\gamma_{c}^{(1)} (s) = \frac{{\alpha p |s(n)|^2 |g|^2| { \mathbf{h}_2^\mathrm{ H}\mathbf{w}} |^2}}{{{\sigma ^2}}}.
\end{equation}
Thus the average data rate of the BD transmission is written by
\begin{equation}\label{eq:Rc1}
{ R_{c}^{(1)}} = \mathbb{E}_s\left[ {{{\log }_2} \left( 1 + \gamma_{c}^{(1)} (s) \right)} \right].
\end{equation}
When decoding $c(n)$, the primary signal $s(n)$ plays the role of fast-varying channel responses. The squared envelope $|s(n)|^2$ of $s(n)$ follows an exponential distribution, and its probability density function (PDF) is $f(x)=e^{-x},x>0$. Thus, the BD (i.e., backscatter-link) transmission rate $R_{c}^{(1)}$ can be derived as follows
\begin{align}\label{eq:Rc_pdf}
{ R_{c}^{(1)}} &= \int_{\rm{0}}^{{\rm{ + }}\infty } {{\mathrm{e}^{ - x}}{{\log }_2}} (1 + \beta x)\mathrm{d}x \nonumber\\ &={{-\mathrm{e}}^{\frac{1}{\beta }}}\rm{Ei}\left( - \frac{1}{\beta } \right)log_2\mathrm{e},
\end{align}
where $\beta = \frac{{{{\alpha p|g|^2 \left| {\mathbf{h}_2^\mathrm{H}\mathbf{w}} \right|}^2}}}{{{\sigma ^2}}}$ is the average received SNR of the backscatter link, and $\mathrm{Ei} {\left( x \right)} \triangleq \int_{-\infty}^{x} \frac{1}{u} \mathrm{e}^{u} \mathrm{d}u$ is defined for the exponential integral.
\remark
The ${{-\mathrm{e}}^{\frac{1}{x }}}\mathrm{Ei} \left( - \frac{1}{\it{x} }\right)$ is a monotonically increasing and concave function of $x$, for $x\geq0$. This can be easily verified by its first and second derivatives.
\subsection{CSR Setup}\label{sec:setupUSP}
In this subsection, we thus consider the CSR setup in which $T_c= NT_s$, where $N$ is an integer, and $N \gg 1$. Compared with the PSR setup, the BD transmission in CSR has much low rate than the primary transmission, thus it can provide an additional signal component by its scattering.
To differentiate CSR from the PSR, we let $c$ be the BD signal to be transmitted in one particular BD symbol period, which covers $N$ primary symbol periods. Thus, in the $n$-th primary symbol period within one BD symbol period, for $n=1, \ldots, N$, the received signal at the PR is given by
\begin{equation}\label{eq:Tr2}
y(n) = \sqrt {p} \mathbf{h}_1^\mathrm{ H} \mathbf{w} s(n) + \sqrt{p} \sqrt{\alpha} c g\mathbf{h}_2^\mathrm{H} \mathbf{w}s(n) + z(n).
\end{equation}
The second signal term in~\eqref{eq:Tr2} can be viewed as the output of the primary signal $s(n)$ passing through a slowly varying channel $\sqrt{\alpha} c g\mathbf{h}_2$. Thus the PR first decodes the primary signal $s(n)$ by treating the BD signal as a multipath component. The equivalent channel for decoding $s(n)$ is denoted by $\mathbf{h}_\mathrm{eq}=\mathbf{h}_1 +\sqrt{\alpha}{c} g\mathbf{h}_2$. Since the PR has no prior knowledge about the BD signal $c$, a training symbol from the PT is required to estimate the equivalent channel $\mathbf{h}_\mathrm{eq}$. Given $c$, the SNR for decoding $s(n)$ is written as
\begin{equation}\label{eq:SNR_PR2}
\gamma_{s}^{(2)}(c) = \frac{p \left| \mathbf{h}_\mathrm{eq}^\mathrm{H}(c) \mathbf{w} \right|^2}{\sigma ^2}.
\end{equation}
With a given $c$, the achievable rate of the direct link is thus given by
\begin{equation}\label{eq:NoncoherentDetection}
\tilde{R}_{s}^{(2)}(c)=\log_2\left(1+\gamma_{s}^{(2)}(c)\right),
\end{equation}
where we have ignored the training overhead in each BD symbol period due to large $N$.
Thus, for sufficiently large $N$, the average primary rate is
\begin{equation}\label{eq:Rs2}
{R_{s}^{(2)}} = \mathbb{E}_c\left[ {{{\log }_2}(1 + \gamma_{s}^{(2)}(c))} \right],
\end{equation}
where the expectation is taken over the random variable $c$.
\proposition
$\gamma_{s}^{(2)}$ is distributed as a noncentral chi-square distribution $\chi^2$ with the freedom of $2$, the non-centrality parameter $\lambda = \frac{p \left|\mathbf{h}^\mathrm{H}_1 \mathbf{w}\right|^2 }{\sigma ^2}$ and the Gaussian variance parameter $\Sigma= \frac{p \alpha |g|^2 \left|\mathbf{h}^\mathrm{H}_2 \mathbf{w}\right|^2 }{2\sigma ^2}$. Its PDF is given by
\begin{equation}\label{eq:NoncenChipdf}
f(x)= \frac{1}{2\Sigma}\mathrm{e}^{\left(-\frac{x+\lambda}{2\Sigma}\right)}I_0\left(\frac{\sqrt{x\lambda}}{\Sigma}\right),
\end{equation}
where $I_0\left(\cdot\right)$ is a modified Bessel function of the first kind given by
\begin{equation}\label{eq:Bessel}
I_0(x) = \sum_{m=0}^{\infty}\frac{1}{m!\Gamma(m+1)}\left(\frac{x}{2}\right)^{2m}.
\end{equation}
\begin{IEEEproof}
Please refer to Appendix \ref{proof:pro_Nonchi}.
\end{IEEEproof}
Notice that the non-centrality parameter $\lambda$ can be explained as the SNR of the direct link, while the Gaussian variance related parameter $2\Sigma$ can be interpreted as the SNR of the backscatter link. Let $x=\gamma_{s}^{(2)}$. From~\eqref{eq:NoncenChipdf}, the achievable rate $ R_{s}^{(2)}$ in~\eqref{eq:Rs2} can be expanded as follows,
\begin{equation}\label{eq:Rs2_integral}
{ R_{s}^{(2)}} = \int_{0}^{+\infty} \log_2(1+x) f(x) \mathrm{d} x.
\end{equation}
In order to obtain analytical insights, we consider the asymptotic case with high SNR $\gamma_{s}^{(2)}$.
\proposition For the case of SNR $\gamma_{s}^{(2)} \rightarrow + \infty$, the primary rate $R_{s}^{(2)}$ can be obtained with a closed-form as follows
\begin{equation}\label{eq:Rs2_closed}
R_{s}^{(2)} = \log_2 \lambda- \mathrm{Ei}\left(-\frac{\lambda}{2\Sigma}\right) \log_2 \mathrm{e}.
\end{equation}
\begin{IEEEproof}
Please refer to Appendix \ref{proof:pro_Rs}.
\end{IEEEproof}
Clearly, the first term in \eqref{eq:Rs2_closed} of Proposition 2 can be interpreted as the achievable rate for a traditional MISO system with transmit beamforming. Moreover, we have the following important observations for Proposition 2.
\remark First, compared to the traditional MISO system, the primary transmission in the SR achieves a rate gain of $\Delta R_{s}^{(2)} = - \mathrm{Ei}\left(-\frac{\lambda}{2\Sigma}\right) \log_2 \mathrm{e}$, since $\Delta R_{s}^{(2)}>0$. This implies that the existence of the backscattering BD can enhance the primary transmission rate by providing an additional scattered path for the primary system. Second, the rate gain of the primary system $\Delta R_{s}^{(2)}$ increases as the backscatter-link SNR $2\Sigma$ increases, for any given direct-link SNR $\lambda$, since the exponential integer function $\mathrm{Ei}(x)$ is monotonically decreasing for $x < 0$.
After decoding $s(n)$, the PR also applies the SIC technique to remove the direct-link interference. In a BD symbol duration, we denote the primary signal vector by $\mathbf{s}=\left[s(1),s(2),\dots,s(N)\right]^\mathrm{T}$, the noise vector by $\mathbf{z}=\left[z(1),z(2),\dots,z(N)\right]^\mathrm{T}$ and the received signal vector after the interference cancellation by $\mathbf{\hat{y}_c}=\left[\hat{y}_c(1),\hat{y}_c(2),\dots,\hat{y}_c(N)\right]^\mathrm{T}$. Assuming that the primary signal component is removed perfectly, we obtain the intermediate signal in a vector form as
\begin{equation}\label{eq:yc2}
\mathbf{\hat{y}_c}=\sqrt{\alpha} \sqrt{p}{g}\mathbf{h}_2^\mathrm{H} \mathbf{w} \mathbf{s} c + \mathbf{z}.
\end{equation}
Since $\mathbb{E} [|s(n)|^2]=1$, the SNR for decoding BD symbol $c$ via maximal ratio combining (MRC) can be approximated as (assuming $N \gg 1$)
\begin{equation}\label{eq:SNR_SR2}
\gamma_{c}^{(2)} = \frac{N \alpha p |g|^2\left| {\mathbf{h}_2^\mathrm{ H}\mathbf{w}} \right|^2}{\sigma^2}.
\end{equation}
In the CSR setup, since only one BD symbol is transmitted during $N$ successive primary-symbol periods, the primary signal $s(n)$ can be viewed as a spread-spectrum code with length $N$ for BD symbols. Accordingly, the SNR for decoding BD symbol $\gamma_{c}^{(2)}$ is increased by $N$ times, at the cost of symbol rate decreased by $\frac{1}{N}$. Hence, the BD achievable rate is given by
\begin{equation} \label{eq:Rc2}
{R_{c}^{(2)}} = \frac{1}{N} {{{\log }_2}(1 + \gamma_{c}^{(2)})}.
\end{equation}
\section{Transmit Beamforming Problem Formulation}\label{sec:ProblemF}
In this section, to further investigate the performance of the proposed SR, we consider two transmit beamforming optimization problems, i.e., weighted sum-rate maximization (WSRM) problem and transmit power minimization (TPM) problem
\subsection{Weighted Sum-Rate Maximization}
\label{sec:WSRMP}
In this subsection, we aim to maximize the weighted sum of the primary rate and the BD rate by optimizing the transmit beamforming vector $\mathbf{w}$. A general WSRM problem can be formulated as follows
\begin{subequations}
\begin{eqnarray}\label{eq:WSRMP_General}
\max_{\mathbf{w}}&& \rho R_s^{(i)} + (1-\rho) R_c^{(i)}, \\
\mathrm{s.t.} &&\|\mathbf{w}\|^2=1,\label{eq:PGen}
\end{eqnarray}
\end{subequations}
where the weight factor $\rho\in[0,1]$, the index $i\in\{1,2\}$ indicates the PSR setup and the CSR setup, respectively, and \eqref{eq:PGen} is the normalization constraint for the transmit beamforming design.
By following \cite{Shang2011Multiuser}, the achievable rate region can be adopted to characterize the optimal rate tradeoff between the primary and BD transmissions. Specifically, the rate region consists of all the achievable rate pairs that can be achieved by the PT and BD transmissions under the considered beamforming scheme. By varying the weight factor $\rho$ in \eqref{eq:WSRMP_General}, a sequence of WSRM problems can be solved to obtain the Pareto boundary for the rate region of the SR with transmit beamforming.
For the PSR setup, from \eqref{eq:Rs1} and \eqref{eq:Rc_pdf}, we have the following WSRM problem
\begin{subequations}
\textbf{\underline{P1:}}\label{eq:P1}
\begin{align}
\max_{\mathbf{w}}~~& R_1 (\mathbf{w}) \triangleq { \rho \log_2\left(1+\frac{{p | {\mathbf{h}_1^\mathrm{H}\mathbf{w}}|^2}}{{\alpha p|g|^2 | {\mathbf{h}_2^\mathrm{H}\mathbf{w}} |^2 + {\sigma ^2}}}\right)} - \nonumber\\
~&(1 \!-\! \rho){\mathrm{e}^{\frac{\sigma^2}{\alpha p|g|^2\left|\mathbf{h}_2^\mathrm{H} \mathbf{w}\right|^2 }}}{\mathrm{Ei}}\left( \! - \frac{\sigma^2}{\alpha p|g|^2\left|\mathbf{h}_2^\mathrm{H} \mathbf{w}\right|^2 } \! \right)\log_2\mathrm{e} \\
\mathrm{s.t.}~~
&\|\mathbf{w}\|^2=1.\label{eq:P1C1}
\end{align}
\end{subequations}
For the CSR setup, from \eqref{eq:Rs2} and \eqref{eq:Rc2}, we have the following WSRM problem
\textbf{\underline{P2:}}\label{eq:P2}
\begin{subequations}
\begin{align}
\max_{\mathbf{w}}~~& R_2 (\mathbf{w}) \triangleq \frac{1 \! -\! \rho}{N} {{{\log }_2}\left(1 \!+\! \frac{N \alpha p|g|^2 \left| {\mathbf{h}_2^\mathrm{ H}\mathbf{w}} \right|^2}{\sigma^2}\right)} \!+\! \nonumber\\
~&\rho \mathbb{E}_c\left[ {{{\log }_2}\left(1 + \frac{{p\left| {(\mathbf{h}_1 + \sqrt{\alpha}{c}g\mathbf{h}_2)^\mathrm{H}\mathbf{w}} \right|^2}}{{{\sigma ^2}}} \right)} \right] \\
\mathrm{s.t.}~~&\|\mathbf{w}\|^2=1. \label{eq:P2C1}
\end{align}
\end{subequations}
Both (P1) and (P2) are non-convex optimization problems, and thus it is difficult to obtain their optimal solutions in general.
\subsection{Transmit Power Minimization}
\label{sec:PMP}
Since energy consumption is another important performance metric, in this subsection, we aim to minimize the PT's transmit power under given primary and BD rate requirements by optimizing the transmit beamforming vector $\mathbf{w}$ and the transmit power $p$ jointly. A general optimization problem is given by
\begin{subequations}
\begin{align}
\min_{\mathbf{w},p} & \quad\quad p \\
\mathrm{s.t.} & \quad\quad R_s^{(i)} \geq \epsilon_s, \\
&\quad\quad R_c^{(i)} \geq \epsilon_c, \\
&\quad\quad \left\|\mathbf{w}\right\|^2=1 ,
\end{align}
\end{subequations}
where $\epsilon_s$ and $\epsilon_c$ are the rate requirements of the primary system and the BD, respectively.
For the PSR setup, the rate requirements can be equivalently converted into the SINR/SNR constraints. Then, the TPM problem can be rewritten as
\textbf{\underline{P3:}}
\begin{subequations}
\begin{align}
\min_{\mathbf{w},p} & \quad\quad p\\
\mathrm{s.t.} & \quad\quad \frac{{p | {\mathbf{h}_1^\mathrm{H}\mathbf{w}}|^2}}{{\alpha p|g|^2 | {\mathbf{h}_2^\mathrm{H}\mathbf{w}} |^2 + {\sigma ^2}}} \geq 2^{\epsilon_{s}}-1, \\
&\quad\quad \frac{{{{\alpha p|g|^2 \left| {\mathbf{h}_2^\mathrm{H}\mathbf{w}} \right|}^2}}}{{{\sigma ^2}}} \geq \gamma_\beta(\epsilon_c), \\
&\quad\quad \left\|\mathbf{w}\right\|^2=1,
\end{align}
\end{subequations}
where $\gamma_\beta(\epsilon_c)$ is the root of the equation $R_c^{(1)}=\epsilon_c$, that is
\begin{equation}\label{eq:Epsilonc2SNR}
{{-\mathrm{e}}^{\frac{1}{\beta }}}\rm{Ei}\left( - \frac{1}{\beta } \right)log_2\mathrm{e}=\epsilon_c.
\end{equation}
By converting the BD rate requirement into the SNR constraint, the TPM problem can be rewritten as follows
\textbf{\underline{P4:}}
\begin{subequations}
\begin{align}
\min_{\mathbf{w},p} & \quad\quad p \\
\mathrm{s.t.} & \quad \mathbb{E}_c\left[ {{{\log }_2}\left(1 + \frac{{p\left| {(\mathbf{h}_1 + \sqrt{\alpha} {c}g\mathbf{h}_2)^\mathrm{H}\mathbf{w}} \right|^2}}{{{\sigma ^2}}} \right)} \right] \geq \epsilon_{s},\label{eq:P4-C1} \\
&\quad \frac{{{{\alpha p|g|^2 \left| {\mathbf{h}_2^\mathrm{H}\mathbf{w}} \right|}^2}}}{{{\sigma ^2}}} \geq \frac{2^{N \epsilon_c}-1}{N}, \\
&\quad \left\|\mathbf{w}\right\|^2=1.
\end{align}
\end{subequations}
However, \eqref{eq:P4-C1} cannot be converted into a SNR constraint in (P4), since it is difficult to obtain a closed-form expression for the primary rate in terms of SNR.
It is easy to see that the TPM problems are always feasible, provided of course that none of the channel vectors is identically zero and the channel vectors $\mathbf{h}_1$ and $\mathbf{h}_2$ are not parallel to each other. However, it is also verified that both (P3) and (P4) are non-convex optimization problems which are difficult to solve optimally.
\section{Proposed Solutions}\label{sec:solution}
In this section, we propose algorithms to obtain generally suboptimal solutions to the problems formulated in the previous section
\subsection{Weighted Sum-Rate Maximization}
\subsubsection{PSR Setup} Denote $\mathbf{v}=\sqrt{p}\mathbf{w}$, $\mathbf{H}_{1}=\mathbf{h}_{1}\mathbf{h}_{1}^\mathrm{ H}$ and $\mathbf{H}_{2}=\alpha |g|^2 \mathbf{h}_{2}\mathbf{h}_{2}^\mathrm{ H}$ for convenience. By introducing the new variable $\mathbf{W}=\mathbf{vv}^{\mathrm{H}}$, (P1) is recast as the following equivalent optimization problem with a positive semi-definite (PSD) matrix variable $\mathbf{W}$.
\begin{algorithm}[t]
\caption{ for solving (P1-SDR)}\label{algorithm1}
\begin{algorithmic}[1]
\REQUIRE
\textcolor{black}{The power reflection coefficient $\alpha$; transmit power $p$; the CSI $\mathbf{h}_1$, $g \mathbf{h}_2$ and the noise power $\sigma^2$.}
\ENSURE
\textcolor{black}{The solution for (P1-SDR) $\mathbf{W}^{\star}$.}
\STATE Initialization: $\xi = \sigma^2$, the interval $\Delta\xi$, and iteration index $k=1$.
\WHILE {$\xi \leq \alpha p|g|^2 \left\|\mathbf{h}_2\right\|^2 + \sigma^2 $}
\STATE Given $\xi$, solve {\rm{(P1-SDR)}} by using CVX to obtain the optimal $\mathbf{W}_k^{\star} (\xi)$ and objective value $C_k (\xi)$.
\STATE $\xi \leftarrow \xi + \Delta\xi$.
\STATE $k \leftarrow k+1$.
\ENDWHILE
\STATE Obtain the optimal solution to (P1-SDR) as $\mathbf{W}^{\star} = \mathbf{W}_{k^{\star}}$, where $k^{\star} = \arg \underset{k}{\max} \;\; C_k(\xi)$.
\end{algorithmic}
\end{algorithm}
\textbf{\underline{P1-PSD:}}
\begin{subequations}
\begin{eqnarray}\label{eq:P1-PSD}
\max_{\mathbf{W}}&& \rho{{{\log }_2}\left(1 + \frac{\Tr(\mathbf{H}_{1}\mathbf{W})}{\Tr(\mathbf{H}_{2}\mathbf{W})+\sigma^2}\right)} - \hspace{-0.1cm}(1\hspace{-0.1cm}-\hspace{-0.1cm}\rho)\hspace{-0.3cm}\quad {\mathrm{e}^{\frac{\sigma^2}{\Tr(\mathbf{H}_2 \mathbf{W}) }}}\rm{Ei}\left( - \frac{\sigma^2}{\Tr(\mathbf{H}_2 \mathbf{W}) } \right)log_2\mathrm{e} \label{eq:P1-PSDObj1} \\
\mathrm{s.t.} && \Tr({\mathbf{W}})=p, \label{eq:P1-PSDC1} \\
&&\mathrm{Rank}(\mathbf{W}) = 1.\label{eq:P1-PSDC3}
\end{eqnarray}
\end{subequations}
To solve this problem, we replace the denominator in the objective function~\eqref{eq:P1-PSDObj1} with an auxiliary variable $\xi \triangleq \Tr(\mathbf{H}_2\mathbf{W})+\sigma^2$ and add an equality constraint $\Tr(\mathbf{H}_2\mathbf{W})+\sigma^2 = \xi$ accordingly. Moreover, by relaxing the nonconvex rank-one constraint \eqref{eq:P1-PSDC3}, (P1-PSD) can be recast to the following semi-definite relaxation (SDR) problem \cite{CVXBoyd04}
\textbf{\underline{P1-SDR:}}
\begin{subequations}
\begin{eqnarray}\label{P1-SDR}
\max_{\mathbf{W},\xi}&& {{{\rho \log }_2}\left(1+\frac{\Tr(\mathbf{H}_1\mathbf{W})}{\xi} \right)} - \ (1-\rho) {\mathrm{e}^{\frac{\sigma^2}{\xi-\sigma^2 }}}\rm{Ei}\left( - {\frac{\sigma^2}{\xi-\sigma^2 }} \right)log_2\mathrm{e}\\
\mathrm{s.t.}&&\Tr(\mathbf{W})= p,\\
&&\Tr(\mathbf{H}_2\mathbf{W})+\sigma^2= \xi.
\end{eqnarray}
\end{subequations}
Notice that for a given $\xi$, (P1-SDR) is a convex optimization problem which can be solved optimally by using software tools such as CVX \cite{grant2008cvx}. Then the optimal $\xi^{\star}$ can be obtained by one-dimensional exhaustive search over $\xi$. The details for solving (P1-SDR) are summarized in Algorithm 1.
If the SDR solution $\mathbf{W}^{\star}$ obtained by Algorithm 1 is of rank one, i.e., $\mathbf{W}^{\star}=\mathbf{w}^{\star}(\mathbf{w}^{\star})^{\mathrm{H}}$, then $\frac{\mathbf{w}^{\star}}{\sqrt{p}}$ is the optimal solution to (P1). Otherwise, we use the randomization-based method \cite{sidiropoulos2006transmit} to obtain an approximate (suboptimal) solution to (P1). Based on $\mathbf{W}^{\star}$, the steps to find the solution to (P1) are summarized in Algorithm~\ref{Algorithm_rand}.
\subsubsection{CSR Setup}
Let $\mathbf{W}=\mathbf{vv}^{\mathrm{H}}$, $\mathbf{H}_\mathrm{eq}=\mathbf{h}_\mathrm{eq}\mathbf{h}_\mathrm{eq}^\mathrm{H}$ and $\mathbf{H}_{2}=\alpha |g|^2 \mathbf{h}_{2}\mathbf{h}_{2}^\mathrm{ H}$ for convenience. Then (P2) is recast into the following equivalent problem
\begin{algorithm}[t!]
\caption{ for solving (P1)} \label{Algorithm_rand}
\begin{algorithmic}[1]
\REQUIRE \textcolor{black}{The solution to (P1-SDR) $\mathbf{W}^{\star}$.}
\ENSURE \textcolor{black}{The beamforming solution $\mathbf{w}^{\star}$.}
\STATE Initialization: the solution $\mathbf{W}^{\star}$ to (P1-SDR), a large positive integer $D$. \\
\STATE Compute the singular value decomposition (SVD) of $\mathbf{W}^{\star}$ as $\mathbf{W}^{\star}=\mathbf{U{\Sigma}U}^\mathrm{ H}$, with $\mathbf{U}=[\mathbf{u}_1 \cdots \mathbf{u}_M]$. \\
\IF {$\Rank ({\mathbf{W}^{\star}})=1$, }
\STATE $\mathbf{w}^{\star} = \mathbf{u}_1$.
\ELSE
\FOR{$d=1,\ldots,D$}
\STATE Generate a random vector $\mathbf{v}_d =\mathbf{U} \mathbf{\Sigma}^{\frac{1}{2}} \mathbf{e}_d$, where $\mathbf{e}_d=[\mathrm{e}^{j{\theta}_1},\mathrm{e}^{j{\theta}_2},...,\mathrm{e}^{j{\theta}_n}]^\mathrm{ H}$ and $\theta_i$ follows the uniform distribution $U(0,2\pi)$.
\ENDFOR
\RETURN $\mathbf{w}^{\star}=\frac{\mathbf{v}^{\star}}{\sqrt{p}}$, where $\mathbf{v}^{\star}= \arg \underset{d \in \calD} {\max} \ \ R_1 (\mathbf{v}_d)$.
\ENDIF
\end{algorithmic}
\end{algorithm}
\textbf{\underline{P2-PSD:}}
\begin{subequations}
\begin{eqnarray}\label{eq:P2-PSD}
\max_{\mathbf{W}}&& \rho\mathbb{E}_c\left[ {{{\log }_2}\left(1 + \frac{\Tr(\mathbf{H}_\mathrm{eq}(c)\mathbf{W})}{\sigma^2}\right)} \right]+ (1-\rho)\frac{1}{N} {{{\log }_2}\left(1 + \frac{N \Tr(\mathbf{H}_{2}\mathbf{W})}{\sigma^2}\right)} \label{P2Sobj}\\
\mathrm{s.t.}&&\Tr(\mathbf{W})=p,\label{P2SC1}\\
&&\mathrm{Rank}(\mathbf{W})=1.\label{P2SC2}
\end{eqnarray}
\end{subequations}
Similar to (P1-PSD), the SDR of (P2-PSD) is a convex optimization problem, which can be solved optimally and efficiently. Once obtaining the SDR solution $\mathbf{W}^{\star}$ to (P2-PSD), we can use an algorithm similar to Algorithm~\ref{Algorithm_rand} to find a generally approximate solution $\mathbf{w}^{\star}$ to (P2). The details are omitted here for brevity.
\subsection{Transmit Power Minimization}
A similar variable transformation as the WSRM problem can be applied to the TPM problem. The optimization problem (P3) is thus recast into the following equivalent problem
\textbf{\underline{P3-PSD:}}
\begin{subequations}
\begin{align}
\min_{\mathbf{W}} & \quad\quad \mathrm{Tr}(\mathbf{W}) \\
\mathrm{s.t.}&\quad\quad \frac{\Tr(\mathbf{H}_{1}\mathbf{W})}{\Tr(\mathbf{H}_{2}\mathbf{W})+\sigma^2} \geq 2^{\epsilon_s}-1, \\
&\quad\quad \frac{\Tr(\mathbf{H}_{2}\mathbf{W})}{\sigma^2} \geq \gamma_\beta(\epsilon_c),\\
&\quad\quad \mathrm{Rank}(\mathbf{W})=1.
\end{align}
\end{subequations}
Without the rank-one constraint, the SDR problem of (P3-PSD) can be solved by the CVX. Based on the SDR solution, Algorithm~\ref{Algorithm_TPM} is designed to find a rank-one solution $\mathbf{w}^\star$ together with a transmit power $p^\star$ to (P3).
\begin{algorithm}[t!]
\caption{ for solving (P3)} \label{Algorithm_TPM}
\begin{algorithmic}[1]
\REQUIRE \textcolor{black}{The SDR solution to (P3-PSD) $\mathbf{W}^{\star}$.}
\ENSURE \textcolor{black}{The transmit power $p^\star$ and the beamforming solution $\mathbf{w}^{\star}$.}
\STATE Initialization: the SDR solution $\mathbf{W}^{\star}$ to (P3-PSD), a large positive integer $D$. \\
\STATE Compute the singular value decomposition (SVD) of $\mathbf{W}^{\star}$ as $\mathbf{W}^{\star}=\mathbf{U{\Sigma}U}^\mathrm{ H}$, with $\mathbf{U}=[\mathbf{u}_1 \cdots \mathbf{u}_M]$.\\
\STATE $p = \Tr(\mathbf{\Sigma})$
\IF {$\Rank ({\mathbf{W}^{\star}})=1$, }
\RETURN $p^\star= p$ and $\mathbf{w}^{\star} = \mathbf{u}_1$.
\ELSE
\FOR{$d=1,\ldots,D$}
\STATE Generate a random vector $\mathbf{v}_d =\mathbf{U} \mathbf{\Sigma}^{\frac{1}{2}}\mathbf{e}_d$, where $\mathbf{e}_d=[\mathrm{e}^{j{\theta}_1},\mathrm{e}^{j{\theta}_2},...,\mathrm{e}^{j{\theta}_n}]^\mathrm{ H}$ and $\theta_i$ follows the uniform distribution $U(0,2\pi)$.
\IF{ (P3) is feasible with $p$ and $\mathbf{w}_d=\frac{\mathbf{v}_d}{\sqrt{p}}$. }
\RETURN $p^\star= p$ and $\mathbf{w}^{\star}=\mathbf{w}_d$.
\ENDIF
\ENDFOR
\ENDIF
\end{algorithmic}
\end{algorithm}
Different from Algorithm~\ref{Algorithm_rand}, once a feasible solution is obtained for the case $\Rank(\mathbf{W}^\star)\neq1$, Algorithm~\ref{Algorithm_TPM} will end early. The reason is that, in Algorithm~\ref{Algorithm_TPM}, the phase randomization does not affect the value of minimum transmit power $p$ and is used to find a feasible beamforming vector $\mathbf{w}^\star$ that satisfies the constraints of (P3) under a given transmit power.
The SDR technique can also be applied to solve the following equivalent problem of (P4).
\textbf{\underline{P4-PSD:}}
\begin{subequations}
\begin{align}
\min_{\mathbf{W}} & \quad\quad \mathrm{Tr}(\mathbf{W}) \\
\mathrm{s.t.}&\quad\quad \mathbb{E}_c\left[ \log_2\left(1+\frac{\mathbf{H}_\mathrm{eq}(c)\mathbf{W}}{\sigma^2}\right)\right] \geq {\epsilon_s}, \label{eq:P4-PSDC1}\\
&\quad\quad \frac{\Tr(\mathbf{H}_{2}\mathbf{W})}{\sigma^2} \geq \frac{2^{N \epsilon_c}-1}{N},\\
&\quad\quad \mathrm{Rank}(\mathbf{W})=1.
\end{align}
\end{subequations}
After solving the SDR of (P4-PSD), we can also obtain a generally approximate beamforming solution $\mathbf{w}^{\star}$ together with the transmit power $p^\star$ to (P4), by using an algorithm analogous to Algorithm~\ref{Algorithm_TPM}. The details are thus omitted for brevity.
\section{low-complexity beamforming optimization} \label{sec:LowC}
Notice that the complexity of solving the formulated problems increases exponentially as the dimension $M$ of the optimization matrix variable $\mathbf{W}$ increases, which may be unaffordable for the case of large-scale antenna array at the PT (i.e., $M\gg1$). In this section, we present a low-complexity beamforming optimization scheme for the considered SR system.
Denote the normalized channel vectors by $\mathbf{\tilde{h}}_1= \frac{\mathbf{h}_1}{{\left\| \mathbf{h}_1 \right\|}}$ and $\mathbf{\tilde{h}}_2= \frac{\mathbf{h}_2}{{\left\| \mathbf{h}_2 \right\|}}$. Then, we have the following proposition.
\begin{proposition}\label{sec:proposition1}
The optimal beamforming vector $\mathbf{w}^{\star}$ for each WSRM or TPM problem has the structure $\mathbf{w}^{\star} = \alpha_1\mathbf{\tilde{h}}_1 + \alpha_2\mathbf{\tilde{h}}_2$, where the complex weights $\alpha_1$ and $\alpha_2$ are subject to $|\alpha_1|^2+|\alpha_2|^2=1$.
\end{proposition}
\begin{IEEEproof}
Please refer to Appendix \ref{proof:pro1}.
\end{IEEEproof}
That is, the optimal beamforming vector $\mathbf{w}^{\star}$ lies in the space spanned by the normalized channel vectors $\mathbf{\tilde{h}}_1$ and $\mathbf{\tilde{h}}_2$.
To demonstrate the advantage of the above beamforming structure, we take (P2) as an example of the WSRM problems and (P3) as an example of the TPM problems.
According to Proposition 3, $\mathbf{w}^{\star}$ can be written as
\begin{equation}\label{eq.w_matrix}
\mathbf{w}^{\star} = \alpha_1\mathbf{\tilde{h}}_1 + \alpha_2\mathbf{\tilde{h}}_2 = \mathbf{B}\mathbf{a},
\end{equation}
where $\mathbf{B} =[\mathbf{\tilde{h}}_1,\mathbf{\tilde{h}}_2]\in\mathbb C^{M \times 2}$ and $\mathbf{a}=[\alpha_1,\alpha_2]^\mathrm{T} \in\mathbb C^{2\times 1}$.
From~\eqref{eq.w_matrix}, the problem (P2-PSD) can be rewritten as follows
\textbf{\underline{P2-PSD-L:}}
\begin{subequations}
\begin{eqnarray}\label{eq:P2-PSD-L}
\max_{\mathbf{A}}&&\rho\mathbb{E}_c\left[ {{{\log }_2}\left(1 + \frac{\Tr(\mathbf{G}_\mathrm{eq}\mathbf{A})}{\sigma^2}\right)} \right]+ (1-\rho)\frac{1}{N} {{{\log }_2}\left(1 + \frac{N \Tr(\mathbf{G}_{2}\mathbf{A})}{\sigma^2}\right)} \label{P2Lobj}\\
\mathrm{s.t.}&&\Tr(\mathbf{BAB}^\mathrm{H})= p,\label{P2LC1}\\
&&\mathrm{Rank}(\mathbf{A})=1,\label{P2LC2}
\end{eqnarray}
\end{subequations}
where $\mathbf{A}= p \mathbf{aa}^\mathrm{H}\in \mathbb C^{2\times 2}$,
$\mathbf{G}_\mathrm{eq}=\mathbf{B}^\mathrm{H}\mathbf{h}_\mathrm{eq}\mathbf{h}_\mathrm{eq}^\mathrm{H}\mathbf{B}\in\mathbb C^{2\times 2}$ and
$\mathbf{G}_{2}=\alpha |g|^2 \mathbf{B}^\mathrm{H}\mathbf{h}_2\mathbf{h}_2^\mathrm{H}\mathbf{B}\in\mathbb C^{2\times 2}$.
The problem (P2-PSD-L) can also be solved with the SDR technique.
As for the TPM problem, the similar variable transformation is applied. By introducing an additional variable $\mathbf{G}_\mathrm{1}=\mathbf{B}^\mathrm{H}\mathbf{h}_\mathrm{1}\mathbf{h}_\mathrm{1}^\mathrm{H}\mathbf{B}\in\mathbb C^{2\times 2}$, (P3-PSD) is rewritten as follows
\textbf{\underline{P3-PSD-L:}}
\begin{subequations}
\begin{align}
\min_{\mathbf{A}} & \quad\quad \Tr(\mathbf{BAB}^\mathrm{H}) \\
\mathrm{s.t.}&\quad\quad \frac{\Tr(\mathbf{G}_{1}\mathbf{A})}{\Tr(\mathbf{G}_{2}\mathbf{A})+\sigma^2} \geq 2^{\epsilon_s}-1, \\
&\quad\quad \frac{\Tr(\mathbf{G}_{2}\mathbf{A})}{\sigma^2} \geq \gamma_\beta(\epsilon_c),\\
&\quad\quad \mathrm{Rank}(\mathbf{A})=1.
\end{align}
\end{subequations}
Similar to (P3-PSD), the SDR problem of (P3-PSD-L) can be solved by the CVX.
The variable for both (P2-PSD) and (P3-PSD) is $\mathbf{W}\in \mathbb{C}^{M\times M}$, while the variable for both (P2-PSD-L) and (P3-PSD-L) is $\mathbf{A}\in \mathbb{C}^{2\times 2}$. Compared to (P2-PSD) and (P3-PSD) which optimize the $M$-by-$M$ matrix variable $\mathbf{W}$ directly, (P2-PSD-L) and (P3-PSD-L) only need to optimize a $2$-by-$2$ matrix variable $\mathbf{A}$, thus leading to significantly reduced computational complexity, especially when $M$ is practically large.
\section{Simulation Results} \label{sec:NumR}
In this section, simulation results are provided to evaluate the performance of the proposed SR. Independent and identically distributed (i.i.d.) Rayleigh fading is assumed for the direct-link channel $\mathbf{h}_1$ as well as the forward-link channel $\mathbf{h}_2$. The backward-link channel $g$ is assumed to be static, since the BD is typically close to the PR. We define the relative channel gain as $\Delta\Gamma \triangleq \frac{\alpha |g|^2 \mathbb{E}{\left[|{h}_{m,2}|^2\right]}}{\mathbb{E}{\left[|{h}_{m,1}|^2\right]}}$, which mainly depends on the large-scale path loss and the power reflection coefficient $\alpha$. In the simulations, without loss of generality, we set the PT-PR and the PT-BD path loss to be 0 dB, and thus we choose ${h}_{m,i} \sim {\cal{CN}} (0,1)$ for $i=1, 2$. In addition, we define the received SNR as the ratio of transmit power $p$ at the PT and noise power $\sigma^2$ at the PR. The noise power is assumed to be normalized to one. The numerical results are obtained by averaging over $10^4$ channel realizations.
\subsection{Weighted Sum-Rate Maximization}
In this subsection, we consider the WSRM and simulate the rate region performance with different SNR values. Both PSR and CSR setups are considered, and for CSR, we assume $N=128$.
\begin{figure}[t]
\centering\includegraphics[width=.63\columnwidth]{varSNRP1.eps}
\caption{Rate region of the PSR with transmit beamforming: $N=1, \Delta \Gamma = -10 ~\mathrm{dB}$ and $M=2$.}\label{fig:RateRegion}
\end{figure}
\begin{figure}
[t]
\centering
\subfigure[Primary and sum transmission rate.]{
\label{fig:PrimaryRate_20dB}
\includegraphics[width=.63\columnwidth]{SumRate20dB_Legacy.eps}}
\hspace{1in}
\subfigure[BD transmission rate.]{
\label{fig:BDRate_20dB}
\includegraphics[width=.63\columnwidth]{SumRate20dB_BD.eps}}
\caption{Rate performances versus the received SNR at PR: $\rho=0.5$, $\Delta \Gamma=-20~\mathrm{dB}$ and $M=4$.}
\label{fig:SumRate}
\end{figure}
Fig. \ref{fig:RateRegion} plots the achievable rate regions of the PSR by solving a sequence of WSRM problems with different $\rho$ varying from 0 to 1, for $N=1, \Delta\Gamma=-10 ~\mathrm{dB}$ and $M=2$. Notice that each point of the solid curve represents the pair of maximum primary rate and BD rate by solving (P1) with a given $\rho$, and the dash curve is the extension to the axes. It is observed that the achievable rate region enlarges with the increase of SNR. That is, the increase of SNR can improve both the primary and the BD transmission rates, as expected.
Fig. \ref{fig:SumRate} compares the rate performances versus different received SNRs, when $\rho=0.5, \Delta\Gamma=-20 ~\mathrm{dB}$ and $M=4$. In general, each rate curve increases as the SNR increases. Specifically, for the CSR setup with $N=128$, the system achieves a higher primary rate than that for the PSR setup with $N=1$. This is because that the decoding strategy for CSR exploits the BD signal as a multipath component rather than interference. On the other hand, for CSR, the BD rate is lower than that for the PSR case with $N=1$, due to the longer BD symbol period. In addition, in Fig. \ref{fig:SumRate} the low-complexity method and the conventional method are compared for problem (P2). We observe that by using the low-complexity (LC) beamforming structure, the WSRM problem has almost the same performance as that by using the conventional method.
By comparing Fig. \ref{fig:PrimaryRate_20dB} with Fig. \ref{fig:BDRate_20dB}, it is observed that the primary rate is much higher than the BD rate for each setup, due to the double attenuations in the backscatter link. It is also observed from Fig. \ref{fig:PrimaryRate_20dB} that the CSR system achieves a higher sum rate than the primary system without any BD. Although this sum-rate gain is only moderate, the practical significance of this result lies in that our proposed CSR system enables the backscatter communication concurrently with the primary transmission without any loss in spectral efficiency.
\begin{figure}
[t]
\centering
\subfigure[PSR: $N=1$.]{
\label{fig:Equal_Rc_cst_minP}
\includegraphics[width=.63\columnwidth]{Equal_Rc_cst_vs_minP.eps}}
\hspace{1in}
\subfigure[CSR: $N=128$.]{
\label{fig:Unequal_Rc_cst_minP}
\includegraphics[width=.63\columnwidth]{Unequal_Rc_cst_vs_minP.eps}}
\caption{Minimum transmit power $p$ versus BD rate requirement $\epsilon_c$.}
\label{fig:MPM_minP}
\end{figure}
\subsection{Transmit Power Minimization}
In this subsection, we investigate TPM problems under given rate requirements $\epsilon_s$ and $\epsilon_c$ for each setup. For ease of explanation, we generally investigate the TPM problem by varying the BD rate requirement $\epsilon_c$ with a fixed primary rate requirement $\epsilon_s$. We define our transmit power as
\begin{align}\label{eq:powerVsSNR}
P(\mathrm{dBm}) &= \mathrm{SNR}(\mathrm{dB}) + \mathrm{pathloss}(\mathrm{dB})+\sigma^2(\mathrm{dBm}) \nonumber\\
&= \mathrm{SNR}(\mathrm{dB}) + \sigma^2(\mathrm{dBm}).
\end{align}
Fig. \ref{fig:Equal_Rc_cst_minP} and Fig. \ref{fig:Unequal_Rc_cst_minP} plot the minimum transmit power versus the BD rate requirement $\epsilon_c$ for PSR ($N=1$) and CSR ($N=128$), respectively. In general, the minimum transmit power increases with the BD rate requirement $\epsilon_c$, but it increases more dramatically for CSR, due to the fact that the rate loss caused by the longer symbol period needs to be compensated with higher transmit power. Also, Fig. \ref{fig:Equal_Rc_cst_minP} shows that the low-complexity method has almost the same performance as the conventional method for TPM problem. Moreover, for both cases, it is observed that for lower $\epsilon_c$, the minimum transmit power increases as the primary rate requirement $\epsilon_s$ increases, but for higher $\epsilon_c$, the minimum transmit power remains the same with different $\epsilon_s$.
\begin{figure}[t]
\centering\includegraphics[width=.63\columnwidth]{Equal_Rc_cst_vs_Rs.eps}
\caption{Primary transmission rate $R^{(1)}_s$ versus BD rate requirement $\epsilon_c$: PSR case.}\label{fig:Equal_Rc_cst_Rs}
\end{figure}
\begin{figure}[t]
\centering\includegraphics[width=.63\columnwidth]{Equal_Rs_cst_vs_Rc.eps}
\caption{BD transmission rate $R^{(1)}_c$ versus primary rate requirement $\epsilon_s$: PSR case.}\label{fig:Equal_Rs_cst_Rc}
\end{figure}
\begin{figure}[t]
\centering\includegraphics[width=.63\columnwidth]{Unequal_Rc_cst_vs_Rs.eps}
\caption{Primary transmission rate $R^{(2)}_s$ versus BD rate requirement $\epsilon_c$: CSR case with $N=128$.}\label{fig:Unequal_Rc_cst_Rs}
\end{figure}
\begin{figure}[t]
\centering\includegraphics[width=.63\columnwidth]{Unequal_Rs_cst_vs_Rc.eps}
\caption{BD transmission rate $R^{(2)}_c$ versus primary rate requirement $\epsilon_s$: CSR case with $N=128$.}\label{fig:Unequal_Rs_cst_Rc}
\end{figure}
Fig. \ref{fig:Equal_Rc_cst_Rs} illustrates the effect of the BD rate requirement $\epsilon_c$ on the primary rate for PSR case. The primary rate $R_s^{(1)}$ increases slowly as the BD rate requirement increases, since higher BD rate requirement results in more transmit power. However, the curve with $\epsilon_s = 4~\mathrm{bps/Hz}$ is flat at first due to the tight primary rate constraint.
Furthermore, we investigate the BD rate performance $R_c^{(1)}$ versus the primary rate requirement $\epsilon_s$. A set of unimodal curves with different $\epsilon_c$ is shown in Fig. \ref{fig:Equal_Rs_cst_Rc}. The phenomenon can be explained as follows. For lower $\epsilon_s$, the BD rate constraint is the bottleneck that limits the transmit power. The BD rate constraint becomes slack as $\epsilon_s$ increases. Since more power is needed to fulfill the primary rate requirement, thus the BD rate increases. However, in high $\epsilon_s$ regime, the transmit power goes to infinity and the primary rate requirement $\epsilon_s$ is only related to the beamforming vector $\mathbf{w}$ as shown below
\begin{equation}\label{eq:MPM_high}
\frac{\left|\mathbf{h}_{1}^{\mathrm{H}}\mathbf{w}\right|^2}{\alpha |g|^2\left|\mathbf{h}_{2}^{\mathrm{H}}\mathbf{w}\right|^2}\geq 2^{\epsilon_s}-1.
\end{equation}
Once the beamforming vector solution is decided by \eqref{eq:MPM_high}, the minimum transmit power depends on the constraint $p\geq{R^{(1)}_{c}}^{-1}(\epsilon_c)/\alpha |g|^2\left|\mathbf{h}_{2}^{\mathrm{H}}\mathbf{w}\right|^2$ and reaches the optimal one when the equality holds. Thus, the BD rate constraint will be tight again.
Similar simulation results are observed for the CSR setup with $N = 128$. Fig. \ref{fig:Unequal_Rc_cst_Rs} shows the effect of the BD rate requirement $\epsilon_c$ on the primary rate $R^{(2)}_s$. As the BD rate requirement $\epsilon_c$ increases, the curves remain unchanged first, then increase gradually and finally coincide with each other. This is due to the fact that, when $\epsilon_c$ is low, the primary rate requirement is the bottleneck that limits the transmit power. As $\epsilon_c$ gradually increases, the BD rate requirement becomes the bottleneck.
In Fig. \ref{fig:Unequal_Rs_cst_Rc}, the BD rate constraint is tight first and then becomes slack as $\epsilon_s$ increases. Compared with Fig. \ref{fig:Equal_Rs_cst_Rc}, the BD rate constraint will not be tight again in Fig. \ref{fig:Unequal_Rs_cst_Rc}. This is due to the fact that the backscattered signal is treated as a multipath component, and there is no interference for the primary transmission in this CSR setup.
\section{Conclusions}\label{sec:Conclusion}
In this paper, a novel technique, called symbiotic radio (SR), has been proposed for passive IoT, in which a backscatter device (BD) is integrated with a primary communication system, and the primary transmitter and receiver are designed to optimize both the primary and BD transmissions. We first present the SIC-based decoding strategy and analyze the achievable rate performance under both PSR and CSR setups. Then, we formulate two problems to maximize the weighted sum rate and minimize transmit power for the considered system, respectively, by optimizing the beamforming vector at the PT. Both problems are recast into equivalent optimization problems with a PSD matrix variable and solved approximately via the technique of SDR. We also propose a novel transmit beamforming structure to reduce the computational complexity of the beamforming optimization. Simulation results show that not only the BD transmission is enabled, but also the primary system achievable rate is improved by exploiting the BD's scattering in the CSR setup.
\appendices
\section{Proof of Proposition 1}\label{proof:pro_Nonchi}
Given channels and transmit beamforming vector, the random variable $T= \frac{\sqrt{p}}{\sigma}\mathbf{h}_1^\mathrm{ H} \mathbf{w} + \frac{\sqrt{p}}{\sigma}g\mathbf{h}_2^\mathrm{H} \mathbf{w} \sqrt{\alpha}c$, is a linear transformation of a Gaussian random variable $c\sim \mathcal{CN} (0,1)$. Thus we have $T\sim \mathcal{CN}(\frac{\sqrt{p}}{\sigma}\mathbf{h}_1^\mathrm{ H} \mathbf{w},\frac{p \alpha |g|^2 \left|\mathbf{h}^\mathrm{H}_2 \mathbf{w}\right|^2 }{\sigma ^2})$ and its real part and imaginary part are distributed as
\begin{align}
\mathrm{Re} \left\{T\right\} \sim &\mathcal{N} \left(\mathrm{Re} \left\{\frac{\sqrt{p}}{\sigma}\mathbf{h}_1^\mathrm{ H} \mathbf{w}\right\},\frac{p \alpha |g|^2 \left|\mathbf{h}^\mathrm{H}_2 \mathbf{w}\right|^2 }{2 \sigma ^2} \right), \\
\mathrm{Im} \left\{T\right\} \sim &\mathcal{N} \left(\mathrm{Im} \left\{\frac{\sqrt{p}}{\sigma}\mathbf{h}_1^\mathrm{ H} \mathbf{w}\right\},\frac{p \alpha |g|^2 \left|\mathbf{h}^\mathrm{H}_2 \mathbf{w}\right|^2 }{2 \sigma ^2} \right).
\end{align}
Since $\mathrm{Re} \left\{T\right\}$ and $\mathrm{Im} \left\{T\right\}$ are independent Gaussian random variables with the same variance $\Sigma = \frac{p \alpha |g|^2 \left|\mathbf{h}^\mathrm{H}_2 \mathbf{w}\right|^2 }{2 \sigma ^2} $, the SNR $\gamma_{s}^{(2)}=\mathrm{Re} \left\{T\right\}^2 + \mathrm{Im} \left\{T\right\}^2$ is distributed as a noncentral chi-square distribution with the non-centrality parameter
\begin{align}
\lambda &= \mathrm{Re} \left\{\frac{\sqrt{p}}{\sigma}\mathbf{h}_1^\mathrm{ H} \mathbf{w}\right\}^2+ \mathrm{Im} \left\{\frac{\sqrt{p}}{\sigma}\mathbf{h}_1^\mathrm{ H} \mathbf{w}\right\}^2, \nonumber\\
&= \frac{p \left|\mathbf{h}^\mathrm{H}_1 \mathbf{w}\right|^2 }{\sigma ^2},
\end{align}
and its PDF is given by
\begin{equation}\label{eq:pdf_rs2Pro}
f(x)= \frac{1}{2\Sigma}\mathrm{e}^{\left(-\frac{x+\lambda}{2\Sigma}\right)}I_0\left(\frac{\sqrt{x\lambda}}{\Sigma}\right).
\end{equation}
The proof is thus completed.
\section{Proof of Proposition 2}\label{proof:pro_Rs}
As the SNR $\gamma_{s}^{(2)}$ is sufficient large, $\log_2(1+\gamma_{s}^{(2)})\simeq \log_2(\gamma_{s}^{(2)})$, thus we have
\begin{align}
{R_{s}^{(2)}} & = \mathbb{E}_c\left[ {{{\log }_2}( \gamma_{s}^{(2)}(c))} \right], \\
&=\log_2\int_{0}^{\infty}\ln x\frac{1}{2\Sigma}\mathrm{e}^{\left(-\frac{x+\lambda}{2\Sigma}\right)}I_0\left(\frac{\sqrt{x\lambda}}{\Sigma}\right)\mathrm{d}x. \label{eq:Rs_int_m=1}
\end{align}
From \cite{Moser2007Some}, the expected value of the logarithm of a non-central chi-square random variable $V$ with an even number $2m$ of degrees of freedom is given as
\begin{equation}\label{eq:expected_nonchi}
\mathbb{E}\left[\ln V \right] = q_\mathrm{m}(s^2),
\end{equation}
where $s^2$ is the non-centrality parameter, and the function $q_\mathrm{m}(\cdot)$ is defined as follows
\begin{align}\label{eq:qmDefinition}
q_\mathrm{m} \triangleq &\ln(x)- \mathrm{Ei}(-x)+\sum_{j=1}^{m-1}(-1)^j\left[e^{-x}(j-1)!-\frac{(m-1)!}{j(m-1-j)!}\right]\left(\frac{1}{x}\right)^j, x>0
\end{align}
That is
\begin{equation}\label{eq:intSolution}
\int_{0}^{\infty}\ln v\cdot\left(\frac{v}{s^2}\right)^{\frac{m-1}{2}}e^{-v-s^2}I_{m-1}\left(2s\sqrt{v}\right)\mathrm{d}v=q_m(s^2),
\end{equation}
for any $m\in\mathbb{N}$ and $s^2\geq0$. Applying the linear transformation $v=\frac{x}{2\Sigma},s^2 = \frac{\lambda}{2\Sigma}$ to \eqref{eq:Rs_int_m=1}, we have
\begin{subequations}
\begin{align}
{R_{s}^{(2)}}& = \log_2\mathrm{e} \int_{0}^{\infty} \ln v \mathrm{e}^{(-v-s^2)}I_0\left(2s\sqrt{v}\right) \mathrm{d}v \nonumber \\
&~~~+\log_2\left(2\Sigma\right)\int_{0}^{\infty}\mathrm{e}^{(-v-s^2)}I_0\left(2s\sqrt{v}\right) \mathrm{d}v, \\
&=\log_2\mathrm{e} \cdot q_1(s^2)+\log_2(2\Sigma),\label{eq:pdf_int}\\
&=\log_2\mathrm{e}\cdot q_1\left(\frac{\lambda}{2\Sigma}\right) + \log_2(2\Sigma), \\
&= \log_2 \lambda- \mathrm{Ei}\left(-\frac{\lambda}{2\Sigma}\right) \log_2 \mathrm{e}.
\end{align}
\end{subequations}
Equation \eqref{eq:pdf_int} is due to the fact that the second term is an integral over a noncentral chi-square distribution. Thus, the proof is completed.
\section{Proof of Proposition 3}\label{proof:pro1}
Let the beamforming vector be
\begin{equation}\label{eq:w}
\mathbf{w}= \sum\limits_{{i} = 1}^{ 2} {{\alpha _i}\tilde{\mathbf{h}}_i}+\sum\limits_{{j} = 1}^{{M - 2}} {{\eta _j}\tilde {\mathbf{t}}_j^{\perp}},
\end{equation}
where $\tilde {\mathbf{t}}_j^{\perp}$ is the basis of the null space of $\{ \tilde{\mathbf{h}}_i \}$, i.e., $\tilde{\mathbf{h}}_i^\mathrm{H}\tilde {\mathbf{t}}_j^{\perp}=0$. It can be verified that $\tilde {\mathbf{t}}_j^{\perp}$ cannot contribute to improve the SNR in the objective functions of WSRM problems (P1) and (P2), while $\tilde{\mathbf{h}}_i,i=1,2$ can help to improve the SNR in the objective function.
For the SINR expression in the objective function in (P1), the beamforming vector satisfies the condition that $\mathbf{w}=a\tilde{\mathbf{h}}_1 + b\tilde{\mathbf{h}}^\perp_2+\sum\limits_{{j} = 1}^{{M - 2}} {{\eta _j}\tilde {\mathbf{t}}_j^{\perp}}$, where $\tilde{\mathbf{h}}_2^\mathrm{H}\tilde {\mathbf{h}}_2^{\perp}=0$,
$\tilde{\mathbf{h}}_1^\mathrm{H}\tilde {\mathbf{h}}_2^{\perp} > 0 $,
and $\tilde {\mathbf{h}}_2^{\perp}=x\tilde{\mathbf{h}}_1+y\tilde{\mathbf{h}}_2$, thus $\mathbf{w}$ satisfies the structure~\eqref{eq:w}.
It is easy to verify that $\tilde{\mathbf{h}}_1$ and $\tilde{\mathbf{h}}_2^{\perp}$ help improve the SINR of the primary transmission while $\tilde {\mathbf{t}}_j^{\perp}$ does not.
Since the component $\tilde {\mathbf{t}}_j^{\perp}$ cannot contribute to improve the value of the objective function, we only need to optimize the coefficients of $\tilde{\mathbf{h}}_1$ and $\tilde{\mathbf{h}}_2$ to find the optimal beamforming vector $\mathbf{w}^{\star}$.
In addition, since the beamforming vector $\mathbf{w}^{\star}$ is a normalized one, the complex weights $\alpha_1$ and $\alpha_2$ are subject to $|\alpha_1|^2+|\alpha_2|^2=1$.
Since the TPM problems (P3) and (P4) have the same SNR or SINR expressions, the same results hold for the TPM problems. The proof is thus completed.
\renewcommand{\baselinestretch}{1}
\bibliographystyle{IEEEtran}
|
3,212,635,537,491 | arxiv | \section{Introduction}
The world astronomy community is about to embark on
wide-area
galaxy
surveys that
aim
to use large-scale structure probes
to study the origin of cosmic acceleration.
These range from
ground-based imaging and spectroscopic surveys such as the Subaru Hyper
Suprime-Cam (HSC) Survey
\footnote{\url{http://www.naoj.org/Projects/HSC/index.html}}\citep[see
also][]{Miyazakietal:12}, the Dark Energy Survey (DES)
\footnote{\url{http://www.darkenergysurvey.org}}, the Kilo-Degrees
Survey (KIDS) \footnote{\url{http://www.astro-wise.org/projects/KIDS/}},
the LSST \footnote{\url{http://www.lsst.org/lsst/}},
the Baryon
Oscillation Spectrograph Survey (BOSS)
\footnote{\url{http://cosmology.lbl.gov/BOSS/}}, the Extended BOSS survey
(eBOSS) \footnote{\url{http://www.sdss3.org/future/eboss.php}}, the
BigBOSS \footnote{\url{http://bigboss.lbl.gov}}, and the Subaru Prime
Focus Spectrograph (PFS) Survey
\footnote{\url{http://sumire.ipmu.jp/pfs/intro.html}}\cite[see
also][]{Takadaetal:12}
to space-based optical and near-infrared missions
such as the Euclid project
\footnote{\url{http://sci.esa.int/science-e/www/area/index.cfm?fareaid=102}}
and the WFIRST project
\footnote{\url{http://wfirst.gsfc.nasa.gov}}\citep[see also][]{WFIRST}.
Each of these surveys approaches the nature of cosmic acceleration using
multiple large-scale structure probes: weak gravitational lensing,
baryon acoustic oscillations, clustering statistics of large-scale
structure tracers such as galaxies and clusters, the redshift-space
distortion effects, and the abundance of massive clusters
\citep[see][for a recent review]{Weinbergetal:12}.
Among the cosmological probes,
weak lensing measurements directly trace the distribution
of matter in the universe without assumptions about galaxy biases
and redshift space distortions
\citep[see][for a
review]{BartelmannSchneider:01}. They are potentially the most powerful
cosmological probe in the coming decade
\citep{Hu:99,Huterer:02,TakadaJain:04}. Recent results such as the Planck
lensing measurement \citep{PlanckLens:13} and the CFHT Lens Survey
\citep{Heymansetal:13,Kilbingeretal:13} are demonstrating the growing
power of these measurements.
Most of the useful weak lensing signals
are in the
nonlinear clustering regime, over the range of multipoles around
$l\simeq $ a few thousands \citep{JainSeljak:97,HutererTakada:05}.
Due
to mode-coupling nature of the nonlinear structure formation, the weak
lensing field at angular scales of interest display large non-Gaussian
features. Thus, the two-point correlation function or the
Fourier-transformed counterpart, power spectrum, no longer fully
describes the statistical properties of the weak lensing field. Using ray-tracing
simulations and analytical methods such as the halo model approach,
previous works have shown that
the non-Gaussianity due to nonlinear structure formation
causes significant correlations between the power spectra at different multipoles
\citep{Jainetal:00,WhiteHu:00,CoorayHu:01,Sembolonietal:07,Satoetal:09,TakadaJain:09,Kiesslingetal:11,Kayoetal:13,KayoTakada:13}. In
particular, \citet{Satoetal:09} studied the power spectrum covariance
using 1000 ray-tracing simulation realizations, and showed that the non-Gaussian
error covariance degrades the information content by a factor of 2--3
for multipoles of a few thousands compared to the Gaussian information
of
the initial density field.
What is the source of this non-Gaussian covariance that ``loses'' so
much of the hard-earned information in both the lensing power spectrum and galaxy redshift surveys?
\citet{Satoetal:09} \citep[see
also][]{TakadaJain:09,Kayoetal:13,TakadaHu:13,KayoTakada:13,Lietal:14}
showed that
{\em super-sample variance} due to super-survey modes of length scales
comparable with or greater than a survey size is the leading source of
non-Gaussian covariance \citep[see also][for the pioneer
work]{Hamiltonetal:06}\footnote{Recently \citet{TakadaHu:13} developed
a unified theory of the power spectrum covariance including both the
weakly or deeply nonlinear versions of super-sample variance, which are
called beat coupling
\citep{Hamiltonetal:06} or halo sample variance
\citep{Satoetal:09}, derived
based on the perturbation theory
or the halo model approach, respectively. For angular scales of interest in this
paper, the halo sample variance gives a dominant contribution to the
sample variance \citep{TakadaJain:09}. For this reason,
we will often refer to the
halo sample variance as the super-sample variance in this paper.}.
The nonlinear version of the super-sample
variance can be physically interpreted as follows. If a survey region is
embedded in a coherent over- or under-density region, the abundance of
massive halos is up- or down-scattered from the ensemble-averaged
expectation as interpreted via halo bias theory
\citep{MoWhite:96,Moetal:97,ShethTormen:99} \citep[see
also][for the derivation of the super-sample variance of the halo number
counts]{HuKravtsov:03,HuCohn:06}. Then the modulation of halo abundance
in turn causes upward- and downward-fluctuations
in the amplitudes of weak lensing
power spectrum measured from the same survey region
\citep{TakadaBridle:07,Satoetal:09,Kayoetal:13}. For angular scales
ranging from $l\sim 100$ to a few thousands, massive halos with $M\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}
10^{14}M_\odot/h$ give a dominant contribution to the super-sample
variance of lensing power spectrum.
The information lost in
the power spectrum measurement can be recovered through measurements of
higher-order
correlation functions of the weak lensing field. They
add complementary information
that cannot be extracted by the power spectrum, even if measured from
the same survey region
\citep{TakadaJain:03,TakadaJain:03a,Sembolonietal:11,TakadaJain:04,Kayoetal:13,SatoNishimichi:13,KayoTakada:13}.
There have also been a number of different approaches suggested for extracting
this complementary information: (1) performing a
nonlinear transformation of the weak lensing
field and then studying the power spectrum of the transformed field
\citep{Neyrincketal:09,Seoetal:11,Zhangetal:11,Yuetal:12,Joachimietal:11,Seoetal:12}; or (2)
using the statistics of rare peaks in the weak lensing mass map
\citep{Miyazakietal:02,Hamanaetal:04,Kratochviletal:10,Munshietal:12,Shirasakietal:12,Hamanaetal:12}.
Inspired by these previous works, the purpose of this paper is to study
a method of combining
the abundance
of massive halos with the weak lensing power spectrum,
in order to reduce the super-sample variance
contamination. Massive halos of $M\lower.5ex\hbox{$\; \buildrel > \over \sim \;$} 10^{14}M_\odot/h$ are
relatively easy to identify through a number of techniques such as identifying a
concentration of member galaxies in multi-color data
\citep{Rykoffetal:13} or identifying peaks
in $X$-ray observations or in
high-angular-resolution microwave surveys.
By comparing the observed abundance of massive halos in the
survey region with the expectation for a fiducial cosmological model, we
can infer the effect of super-survey modes and therefore improve the
weak lensing power spectrum measurement. Based on this motivation, we
will first derive the covariance between the weak lensing
power spectrum and the number counts of massive halos for a given survey
region, using a method to model the likelihood function of halo number
counts \citep{HuCohn:06} and the halo model approach \citep[see
also][for the similar-idea study]{TakadaBridle:07}. Then, assuming that
the observed number counts of massive halos is available, we propose a
method of suppressing the 1-halo term contribution of the massive halos
to the weak lensing power spectrum measurement -- a Gaussianization
method. We will study how upcoming wide-area imaging surveys allow us to
implement the Gaussianization method in order to recover the information
content of the weak lensing power spectrum, compared to the maximum
information content of the initial Gaussian density field.
The structure of this paper is as follows. In Section~\ref{sec:idea}, we
motivate the method in this
paper. In Section~\ref{sec:formulation}, we describe a formulation to
model the joint likelihood function of the halo number counts and the
matter power spectrum when both the two observables are drawn from the
same survey volume. Then we discuss a ``Gaussianization'' method of
matter power spectrum estimation, which is feasible by combining with
the number counts of massive halos. In
Section~\ref{sec:wl}, we apply the formulas to the weak lensing power
spectrum measurement, assuming that massive halos in the surveyed
light-cone volume are identified. Assuming survey parameters for
upcoming wide-area galaxy surveys, we show how the Gaussianization
method of suppressing the 1-halo term contribution of massive halos can
recover the information content of the weak lensing power spectrum.
Section~\ref{sec:conclusion} is devoted to discussion and
conclusion. Unless explicitly denoted, we employ a $\Lambda$-dominated
cold dark matter ($\Lambda$CDM) model that is consistent with the WMAP
results.
\section{Basic idea}
\label{sec:idea}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth,angle=-90]{dN-dCl_different-ells.ps}
\caption{The cross-correlation between the halo number counts and the
amplitudes of lensing power spectrum at different multipole bins,
measured from 1000 ray-tracing realizations for a $\Lambda$CDM model,
each of which has an area of 25 square degrees and has contributions of
super-survey modes (the $N$-body simulations used have the projected
angular scale greater than the ray-tracing area, $5$ degrees on a side).
The halo
counts in the $x$-axis is for halos with masses greater than
$10^{14}M_\odot/h$. The different cross symbols are for different
realizations, and the red solid contours show 68 or 95 percentile
regions that are computed by smoothing the distribution with a
two-dimensional Gaussian kernel that has widths of 1/50th the plotted
ranges in the $x$- and $y$-axes. Note that the plotted ranges of $x$-
and $y$-axes are the same in all the panels.
For multipole bins
around $l\simeq 1000-3000$,
the power
spectrum amplitudes are highly correlated with the number counts of
halos with $M\ge 10^{14}M_\odot$.
\label{fig:dn-dcl}
}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.44\textwidth,angle=-90]{dN-dCl_2d_l1200.ps}
\includegraphics[width=0.44\textwidth,angle=-90]{dN-dCl_2d_l1200_wohsv.ps}
\caption{This plot compares the
halo model predictions (Eqs.~\ref{eq:joint_n-wl}--\ref{eq:cov_n-wl})
to
the results of the numerical simulations shown l in the previous figure for the
multipole bin centered at $l=1245$). {\em Left panel}:The
model includes the halo sampling variance (HSV) contribution, which
arises from the super-survey modes of scales comparable with or outside
the survey region. With the HSV term, the halo model predictions agrees with the simulation results. The
upper- or left-side panels
show the projected one-dimensional likelihood functions for each
observable, and again shows that the halo model predictions well
reproduce the simulation results. {\em Right}: The model predictions
without the HSV cannot reproduce the simulation results. The plot shows
a only weak correlation between the two observables.
These results suggest that the number counts of the massive halos in a
given survey region can be used to correct for the HSV contribution to
the power spectrum measurement. \label{fig:dn-dcl-model} }
\end{figure}
There are several sources of statistical fluctuations in measurement of the
weak lensing power spectrum and the number counts of halos.
The finite size of a survey implies that there will be Poisson noise in the halo
number counts and cosmic variance due to the finite number of Fourier modes available for the
power spectrum measurement.
{\em
Super-sample variance} \citep{Hamiltonetal:06,TakadaHu:13}
or the halo sample variance (HSV)
\citep{Satoetal:09,Kayoetal:13}
is an important additional source of statistical fluctuations.
\citet{Satoetal:09} \citep[see also][]{Kayoetal:13} showed that the HSV gives a significant contribution
to the power spectrum covariance at $l\lower.5ex\hbox{$\; \buildrel > \over \sim \;$} 1000$.
Numerical simulations clearly show the coherent fluctuation in the power
spectrum due to halo sample variance. Fig.~\ref{fig:dn-dcl} shows how
the number of massive halos in a light-cone volume is correlated with
the amplitude of lensing power spectrum measured from the same volume.
For this figure (and for other analyses in this paper), we used 1000
simulation realizations, generated in \cite{Satoetal:09} \citep[see
also][]{Satoetal:11}, each of which has an area of 25 square degrees and
contains both the distribution of massive halos and the lensing field
for source galaxies at redshift $z_s=1$. The ray-tracing simulations
are done in a light-cone volume, and have contributions from the
super-survey modes, because the $N$-body simulations used for modeling
the nonlinear large-scale structure contain the modes of projected
length scales greater than the light-cone size (5~degrees on a side) at
each lens redshift bin \citep[see Fig.~1 in][]{Satoetal:09}. Hence, with
the ray-tracing simulations, we can study the effect of super-survey
variance on the halo number counts and the power spectrum estimation.
For the number counts of halos, we included massive halos with masses
$M\ge 10^{14}M_\odot$. The cross symbols in each panel denote the
different realizations, and the solid contours show 68 or 95 percentile
regions of the distribution. Shown here is the fractional variations of
the two observables, where the quantities in the denominator,
$\bar{C}_l$ or $\bar{N}$, are their mean values among the 1000
realizations. The two observables are highly correlated with each other
at high multipole bins, $\ell \lower.5ex\hbox{$\; \buildrel > \over \sim \;$} 1000$. For massive halos with $M\ge
10^{14}M_\odot/h$, the spectrum amplitude of multipole bin centered at
$l=1245$ shows a strongest correlation with the number counts,
displaying an almost linear relation of $\Delta N/\bar{N}\propto \Delta
C_l/\bar{C}_l$.
In Fig.~\ref{fig:dn-dcl-model}, we demonstrate that the analytical
model developed in Sections~\ref{sec:formulation} and \ref{sec:wl}
reproduces the
simulation result in Fig.~\ref{fig:dn-dcl}. To compute the model
predictions, we assume a multivariate Gaussian distribution of the two
observables, assuming that their widths and the cross-correlation
strength are given by the covariances and the cross-covariance computed
based on the halo model, as we will develop in Section~\ref{sec:wl}.
The right panel explicitly shows that, if the
HSV effect is ignored, the model predicts a only weak correlation
between the two observables, which does not match the simulation result.
The results in Figs.~\ref{fig:dn-dcl} and \ref{fig:dn-dcl-model} imply
that, by using the observed number counts of massive halos in each
survey volume, one can calibrate or correct for the super-sample
variance effect
in the power spectrum estimation.
This is the question that we address in the
following sections.
\section{Formulation: Covariance of halo number counts and
matter power spectrum}
\label{sec:formulation}
\subsection{Likelihood function of halo number counts}
In this section, we briefly review the likelihood function of cluster
number counts taking into account the super-sample variance, based on
the method developed in \citet{HuKravtsov:03}, \citet{HuCohn:06} and
\cite{TakadaBridle:07}.
Consider a finite-volume survey of comoving volume $V_s$ that has
an over- or under-density given by
\begin{equation}
\bar\delta_m(V_s)\equiv \int\!\rm.^3\bm{x}~
\delta_m(\bm{x})W(\bm{x}; V_s),
\end{equation}
where $W(\bm{x}; V_s)$ is the survey window function;
$W(\bm{x})=1$ if $\bm{x}$ is inside a survey region, otherwise
$W(\bm{x})=1$, and is
defined so as to
satisfy the normalization condition $\int\!\!\ensuremath{\mathrm{d}}^3\bm{x}W(\bm{x})=1$.
We can use
halo bias theory \citep{MoWhite:96,ShethTormen:99} to estimate how this super-survey
mode modulates the predicted
number counts of halos in mass range $[M,M+dM]$ from its
ensemble average expectation:
\begin{equation}
\bar{N}(M)=V_s\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M}\ensuremath{\mathrm{d}} M \rightarrow
N(M)=V_s\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M}\ensuremath{\mathrm{d}} M\left[1+b(M)\bar{\delta}_m(V_s)\right],
\label{eq:n_mod}
\end{equation}
where $\ensuremath{\mathrm{d}} n/\ensuremath{\mathrm{d}} M$ is the halo mass function,
$V_s(\ensuremath{\mathrm{d}} n/\ensuremath{\mathrm{d}} M)\ensuremath{\mathrm{d}} M$ is
an ensemble average expectation
of the number counts,
and $b(M)$ is the halo bias. Note that we use the
same model ingredients in \cite{OguriTakada:11} to compute these
quantities for a given cosmological model.
For a sufficiently large volume in which we are most interested, the
density fluctuation $\bar\delta_m$ is considered to be well in the
linear regime, and the probability distribution of $\bar\delta_m$ is
approximated by a Gaussian distribution:
\begin{equation}
P(\bar\delta_m)=\frac{1}{\sqrt{2\pi}\sigma_m(V_s)}\exp\left[
-\frac{\bar\delta_m^2}{2\sigma_m^2(V_s)}\right].
\label{eq:gauss_dm}
\end{equation}
The variance $\sigma_m(V_s)$ is the rms mass density fluctuations
of the survey volume $V_s$, defined in terms of the linear mass
power spectrum as
\begin{equation}
\sigma^2_m(V_s)\equiv \int\!\!\frac{\ensuremath{\mathrm{d}}^3\bm{k}}{(2\pi)^3}~P^L_m(k)\left|
\tilde{W}(\bm{k}; V_s)
\right|^2,
\end{equation}
where $P^L_m(k)$ is the linear matter power spectrum, and
$\tilde{W}(\bm{k})$ is the Fourier transform of the survey window
function (the window function is generally
anisotropic in Fourier space, depending on the geometry of survey
region).
$\bar\delta_m$ and $\sigma_m$ have
contributions from Fourier modes of scales comparable with or outside
the survey volume, so are therefore {\em not} a direct observable.
We next construct an estimator of the number counts of halos in
different mass bins:
$\hat N_1, \hat N_2, ...$, and $\hat N_k$
in mass bins of
$M_1, M_2, ...$ and $M_k$, respectively. Assuming a joint Poisson distribution,
the joint probability
distribution
is given as
\begin{equation}
{\cal L}(\hat N_1, \hat N_2,..., \hat N_k; \bar\delta_m) =
\prod_{i=1}^{k}
\frac{N_i^{\hat N_i}}{\hat N_i!} \exp(-N_i),
\label{eq:p_nd}
\end{equation}
where
\begin{equation}
N_i\equiv \bar{N}_i[1+b_i \bar\delta_m(V_s)],
\end{equation}
$\bar{N}_i \equiv V_s(\ensuremath{\mathrm{d}} n/\ensuremath{\mathrm{d}} M_i)dM_i$ and $b_i\equiv
b(M_i)$. In the following,
quantities with hat symbol ``$\hat{~ }$'' denote
estimators or observables that can be estimated from a survey,
and the quantities with bar symbol ``$\bar{~ }$'', except for
$\bar\delta_m(V_s)$,
denote the
ensemble-average expectation values.
Since $\bar{\delta}_m(V_s)\ll 1$ for a case we are interested in,
expanding the likelihood function (Eq.~\ref{eq:p_nd}) to second order in
$\bar\delta_m$ yields
\begin{eqnarray}
{\cal L}(\hat N_1, \hat N_2,...,\hat N_k; \bar{\delta}_m) &\simeq &
\left[
\prod_{i=1}^k \frac{\bar{N}_i^{\hat N_i}}{\hat N_i!}\exp(-\bar{N}_i)
\right]
\left(1 + \sum_j b_j \hat N_j \bar\delta_m + \sum_{j,j'}b_j \hat N_j b_{j'}
\hat N_{j'}\bar{\delta}_m^2
+ \sum_j \frac{b_j\hat N_j (b_j\hat N_j-1)}{2} \bar\delta_m^2 \right) \nonumber \\
&& \times \left(1 -\sum_j \bar{N}_j b_j \bar\delta_m +
\sum_{j,j'}\bar{N}_j b_j \bar{N}_{j'}b_{j'} \bar{\delta}_m^2+
\frac{1}{2}\sum_j\bar{N}_j^2 b_j^2 \bar\delta_m^2\right)\nonumber \\
&&\hspace{-2em}\simeq
\left[
\prod_{i=1}^k \frac{\bar{N}_i^{\hat N_i}}{\hat N_i!}\exp(-\bar{N}_i)
\right]
\left[
1+\sum_{j}\left(b_j\hat N_j-\bar{N}_jb_j\right)\bar\delta_m \right.
\nonumber\\
&&\hspace{-0em}+
\left.
\left\{
\sum_{j,j'}\left(b_j\hat N_jb_{j'}\hat N_{j'}-b_j\hat N_jb_{j'}\bar{N}_{j'}+b_{j}\bar{N}_j
b_{j'}\bar{N}_{j'}
\right)
+\frac{1}{2}\sum_j\bar{N}_j^2 b_j^2
+\sum_{j}\frac{b_j\hat N_j(b_j\hat N_j-1)}{2}\right\}\bar\delta_m^2
\right].
\end{eqnarray}
By integrating over the density contrast $\bar\delta_m$ with its
probability distribution (Eq.~\ref{eq:gauss_dm}), we can derive the
joint probability distribution for the number counts of halos that include
marginalizing over the amplitude of the super-survey mode
$\bar\delta_m$:
\begin{eqnarray}
{\cal L}(\hat N_1,\hat N_2,\dots, \hat N_k)&=&
\left[
\prod_{i=1}^k \frac{\bar{N}_i^{\hat N_i}}{\hat N_i!}\exp(-\bar{N}_i)
\right]
\left[
1+\frac{1}{2}\left\{
\left(\sum_jb_j(\hat N_j-\bar{N}_j)\right)^2-\sum_jb_j^2\hat N_j
\right\}
\sigma_m^2
\right].
\label{eq:p_app}
\end{eqnarray}
This is a slight generalization of Eq.~(16) in \cite{HuCohn:06}. Since
the quantities $\bar{N}_i$, $b_i$ and $\sigma_m^2$ can be computed
once
a cosmological model, the survey window function and the halo mass
bins are specified,
we
can evaluate the joint probability distribution for the observed
number counts $\left\{\hat{N}_i\right\}$ for the assumed cosmological model.
In practice,
we need to also include observational effects such as detector noise
and halo mass proxy uncertainty, but we do not consider the effects in
this paper for simplicity.
By using the probability distribution function (Eq.~\ref{eq:p_app}), we
can find the following summation rules for the halo number counts of a single
mass bin:
\begin{eqnarray}
&&\sum_{\hat N=0}^{\infty} {\cal L}(\hat N)=1, \nonumber\\
&&\ave{\hat N}\equiv \sum_{\hat N=0}^{\infty} \hat N{\cal L}(\hat N)=\bar{N}, \nonumber\\
&&\ave{\hat N^2}\equiv \sum_{\hat N=0}^{\infty} \hat N^2{\cal L}(\hat N)=\bar{N}+\bar{N}^2+b^2\bar{N}^2\sigma_m^2,
\label{eq:p_app_form1}
\end{eqnarray}
Note again $\bar{N}=V_s(\ensuremath{\mathrm{d}} n/\ensuremath{\mathrm{d}} M)\Delta M$, the ensemble-average expectation
value of the
number counts corresponding to the counts for an infinite-volume survey.
Hence, the variance of the halo number counts
is found to be
\begin{equation}
\sigma^2(\hat N)\equiv \ave{\hat N^2}-\ave{\hat{N}}^2=\bar{N}+
b^2\bar{N}^2\sigma_m^2.
\end{equation}
The first term is a Poisson noise contribution arising due to a finite
number of sampled halos.
The second term is the halo sample variance (HSV) contribution arising
due to
super-survey modes.
\citet{Crocceetal:10} showed that,
using cosmological simulations of a sufficiently
large volume, the above formula can accurately describe
sample variances of
the halo number counts measured from subdivided volumes of $N$-body
simulation,
where the sub-volumes were considered in order to study the effect
of super-survey modes on the sample variance.
Next let us consider the joint probability distributions for the halo
number counts, $\hat{N}_i$ and $\hat{N}_j$, in two mass bins $M_i$ and
$M_j$, respectively ($i\ne j$). From Eq.~(\ref{eq:p_app}), we can find that the joint
distribution can be rewritten as
\begin{eqnarray}
{\cal L}(\hat N_i,\hat N_j)={\cal L}(\hat N_i){\cal L}(\hat N_j)+
\frac{e^{-\bar{N}_i}\bar{N}_i^{\hat N_i}}{\hat N_i !}
\frac{e^{-\bar{N}_j}\bar{N}_j^{\hat N_j}}{\hat N_j !}
b_ib_j(\hat N_i-\bar{N}_i)(\hat N_j-\bar{N}_j)\sigma_m^2.
\label{eq:n_cov1}
\end{eqnarray}
Then we can find the following summation rules:
\begin{eqnarray}
&&\sum_{\hat N_i=0}^{\infty}
\sum_{\hat N_j=0}^{\infty}{\cal L}(\hat N_i,\hat N_j)=1, \nonumber\\
&&\ave{\hat{N}_i\hat{N}_j}\equiv
\sum_{\hat N_i=0}^{\infty}
\sum_{\hat N_j=0}^{\infty}{\cal L}(\hat N_i,\hat N_j)\hat N_i\hat N_j=\bar{N}_i\bar{N}_j
\left(1+b_ib_j\sigma_m^2\right), \hspace{2em} (i\ne j).
\label{eq:n_cov2}
\end{eqnarray}
Similarly, the variance reads
\begin{equation}
\sigma^2(\hat N_i\hat{N}_j)\equiv \ave{\hat N_i \hat
N_j}-\ave{\hat N_i}\ave{\hat N_j} =b_ib_j \bar{N}_i\bar{N}_j
\sigma_m^2.
\end{equation}
Thus the number fluctuations in halos of two mass bins are positively
correlated with each other. The similar formula hold for more than three
bins.
Combining Eqs.~(\ref{eq:n_cov1}) and (\ref{eq:n_cov2}), we can re-write
the ensemble average of the halo number counts as
\begin{equation}
\ave{\hat N_i\hat N_j}=\bar N_i\delta^K_{ij}+\bar N_i \bar
N_j\left(1+b_ib_j
\sigma_m^2\right),
\label{eq:n_cov3}
\end{equation}
where $\delta^K_{ij}$ is the Kronecker delta function: $\delta_{ij}^K=1$
if $i=j$, otherwise $\delta^K_{ij}=0$.
Similarly, we can find
\begin{equation}
\ave{\hat N_i \hat N_j \hat N_k}=\bar{N}_i\bar{N}_j\bar{N}_k
\left[
1+\left(
b_ib_j+b_jb_k + b_k b_i
\right)\sigma_m^2
\right],
\end{equation}
for $i\ne j, j\ne k$ and $k\ne i$.
\subsection{Matter power spectrum and the covariance matrix}
In this section, using the halo model formulation
\citep{Seljak:00,PeacockSmith:00,MaFry:00,Scoccimarroetal:01,TakadaJain:03a}
\citep[see also][for a review]{CooraySheth:02} as well as the joint likelihood of
halo number counts (Eq.~\ref{eq:p_app}), we derive the covariance matrix
for the three-dimensional matter power spectrum including the
super-sample variance contribution.
In the halo model approach, the matter
power spectrum is given by a
sum of the 1- and 2-halo terms that arise
from correlations of matter within the same one halo and between
different halos, respectively:
\begin{eqnarray}
P^{1h}(k)&=&\int\!\ensuremath{\mathrm{d}} M\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M}
\left(
\frac{M}{\bar{\rho}_{m}}\right)^2
\left|\tilde{u}_M(k)\right|^2, \nonumber \\
P^{2h}(k)&=&
\int\!\ensuremath{\mathrm{d}} M\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M}\frac{M}{\bar{\rho}_m}
\int\!\ensuremath{\mathrm{d}} M'\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M'}\frac{M'}{\bar{\rho}_m} P_{hh}(k; M, M'),
\label{eq:ps_halomodel}
\end{eqnarray}
where $\tilde{u}_{M}(k)$ is the Fourier transform of the average mass
profile of halos with mass $M$, and $P_{hh}(k;M, M')$ is the power
spectrum between two halos of masses $M$ and $M'$. Throughout this paper
we assume an Navarro-Frenk-White
(NFW) halo mass profile \citep{Navarroetal:97}. The factor $(M/\bar\rho_m)$ in
the above equation accounts for the fact that more massive halos contain
more dark matter particles.
We assume that the ensemble average of
the halo power spectrum is given by
\begin{equation}
P_{hh}(k; M, M')\simeq
b(M)b(M')\tilde{u}_{M}(k)\tilde{u}_{M'}(k)P^{L}_m(k).
\label{eq:ens_2h}
\end{equation}
If we were working with real data, rather than assuming an NFW profile, we could measure the
halo profile by stacking clusters identified by other techniques.
If recalling that the halo number counts in a given mass range is given
as $N=V_s\left(\ensuremath{\mathrm{d}} n/\ensuremath{\mathrm{d}} M\right)\Delta M$ in an ensemble average
sense, Eq.~(\ref{eq:ps_halomodel}) leads us to define estimators of the
1- and 2-halo power spectra in terms of the observed halo number counts
as
\begin{eqnarray}
\hat{P}^{1h}(k)&=&\frac{1}{V_s}\sum_i \hat N_i
\hat{p}_i^{1h}(k),\nonumber \\
\hat{P}^{2h}(k)&=&\frac{1}{V_s^2}\sum_{i,j} \hat N_i\hat N_j
\hat{p}_{ij}^{2h}(k),
\label{eq:ps1h-2h_est}
\end{eqnarray}
where $\hat N_i\equiv \hat N(M_i)$ and we have approximated the
integration in Eq.~(\ref{eq:ps_halomodel}) by a discrete summation over
different halo mass bins. $\hat{p}^{1h}_i(k)$ and $\hat{p}^{2h}_{ij}(k)$
are estimators that are given in terms of the mass density field. More
specifically, $\hat{p}^{1h}_i(k)$ arises from the matter distribution
inside halos of the $i$-th mass bin, $M_i$, and the ensemble average
gives the average mass profile of the halos. $\hat{p}^{2h}_{j}(k)$ is
from the mass field that governs clustering of different halos in mass
bins $M_i$ and $M_j$,
and the
ensemble average
gives
the linear mass power spectrum,
weighted with the halo biases $b_i$ and $b_j$
(see
Eq.~\ref{eq:ens_2h}).
We assume that, since the super-survey mode,
$\bar{\delta}_m$, contributes only to the monopole of the Fourier modes
in a finite survey region (that is, $\bar{\delta}_m$ is a constant,
background mode across the survey volume),
$\bar{\delta}_m$ is not correlated with
$\hat{p}_i^{1h}(k)$ and $\hat{p}_{ij }^{2h}(k)$; $\ave{\bar\delta_m
\hat{p}^{1h}_i}= \ave{\bar\delta_m \hat{p}^{2h}_{ij}}=0$.
In other words,
we
assume that the
super-survey mode affects the matter power spectrum $\hat{P}(k)$ only
through its effect on the halo number counts $\hat{N}_i$.
As derived in
detail in Appendix~\ref{app:pscov}, by using the joint
probability distribution function for the halo number counts,
${\cal L}(\hat N_1, \hat{N}_2, \dots)$ (Eq.~\ref{eq:p_app}),
we can derive the power spectrum
covariance as
\begin{equation}
{\rm Cov}[\hat{P}(k),\hat{P}(k')]=\frac{2}{N_{\rm mode}(k)}P(k)^2\delta^K_{kk'}
+\frac{1}{V_s}\bar{T}(k,k')+\left[
\int\!\ensuremath{\mathrm{d}} M\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M}b(M)p^{1h}_M(k)
\right]
\left[
\int\!\ensuremath{\mathrm{d}} M'\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M'}b(M')p^{1h}_{M'}(k')
\right]\sigma_m^2(V_s).
\label{eq:pscov}
\end{equation}
Here $N_{\rm
mode}(k)$ is the number of independent Fourier modes centered at $k$,
where we mean by ``independent'' that the Fourier modes are
discriminated by the fundamental mode of a given survey, $k_f\simeq
2\pi/V_s^{1/3}$. For the case of $k\gg k_f$,
\begin{equation}
N_{\rm mode}(k)\simeq \frac{4\pi k^2\Delta k}{k_f^3}=\frac{k^2\Delta k V_s}{2\pi^2},
\label{eq:Nmode}
\end{equation}
where $\Delta k$ is the bin width.
Eq.~(\ref{eq:pscov}) reproduces Eq.~(11) in \citet{Kayoetal:13}.
$\bar{T}(k,k')$ is the
angle-averaged trispectrum \citep[see around Eq.~14 in][]{Satoetal:09}.
The first and second terms on the r.h.s. of Eq.~(\ref{eq:pscov}) are
standard terms of the power spectrum covariance that have been
considered in the literature \citep{Scoccimarroetal:99}. The terms both
scale with survey volume as $\propto 1/V_s$; a larger survey volume
reduces the amplitudes. The third term of Eq.~(\ref{eq:pscov}) is the
super-sample variance or the HSV contribution
\citep{Satoetal:09,Kayoetal:13,TakadaHu:13}. Very similarly to the
effect on the halo number counts, a coherent over- or under-density
mode
in a given survey region
causes an upward or
downward scatter in the power spectrum amplitudes, respectively.
At large $k$ limit,
where the 1-halo term is dominant, the HSV term
behaves
like
${\rm Cov}[P(k),P(k')]_{\rm HSV}\propto
P^{1h}(k)P^{1h}(k')\sigma_m^2$ or the correlation coefficient
matrix
$r(k,k')\equiv
{\rm Cov}[P(k),P(k')]_{\rm HSV}/[P(k)P(k')]\propto \sigma_m^2
$.
That is, the HSV adds powers to the diagonal and off-diagonal
components of the covariance matrix in the exactly same way.
The
dependence of the HSV term on survey volume differs from
other terms as it scales with survey volume via $\sigma_m$, which depends
on the linear mass power spectrum $P^L_m(k)$ convolved with
the survey window
function that has a width of $1/L~ (V_s\sim L^3)$ in Fourier space
\citep[see Section~3.1 in][for the details]{Kayoetal:13}.
\subsection{Cross-correlation between the halo number counts and the
matter power spectrum}
Eq.~(\ref{eq:ps1h-2h_est}) implies that the power spectrum estimators
are correlated with the halo number counts \citep{TakadaBridle:07}, if
the two observables are drawn from the same survey region. Similarly, as
shown in
Appendix~\ref{app:cross},
we derive the
cross-covariance between
the halo number counts of the
$i$-th mass bin, $M_i$ and the power spectrum amplitude
at the $k$-bin:
\begin{eqnarray}
{\rm Cov}[\hat{N}_i,\hat{P}(k)]&=& \frac{1}{V_s}\bar{N}_{i}p_{i}^{1h}(k)+
b_{i}\bar{N}_{i} \sigma_m^2\int\!\!\ensuremath{\mathrm{d}} M\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M} b(M)p_M^{1h}(k)
\nonumber\\
&&
+\bar{N}_i\sigma_m^2
\int\!\ensuremath{\mathrm{d}} M\ensuremath{\mathrm{d}} M'\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M}\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M'}\left[
2b(M_i)b(M)+b(M)b(M')
\right]p^{2h}_{MM'}(k).
\label{eq:crosscov}
\end{eqnarray}
The second and third terms with $\sigma_m^2$ explicitly show that the
halo number counts is correlated with the power spectrum amplitudes
through the super-survey modes. Note that the above formula has a
similar form to that in \cite{TakadaBridle:07}.
The first term is negligible
compared to other terms if a mass bin of halos is sufficiently narrow.
\subsection{A Gaussianized estimator of matter power spectrum:
suppressing the 1-halo term contribution of massive halos }
\label{sec:dp}
As shown in \citet{Satoetal:09} and \cite{Kayoetal:13}, the non-Gaussian
errors significantly degrade the cumulative signal-to-noise ratio or the
information content of power spectrum measurement compared to the
Gaussian expectation which is originally contained in
the initial density field of structure
formation. The degradation is significant in the
nonlinear regime,
and is mainly from the HSV
contribution.
In particular, for the nonlinear scales around
$k\simeq$ a few $h/{\rm Mpc}$ (corresponding to
angular scales of $l\simeq 10^3$ for the weak lensing power spectrum), the
HSV effect arises mainly from massive halos with masses $M\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}
10^{14}M_\odot/h$. Such massive halos are relatively easy to identify in
the survey region, e.g., from a concentration of member galaxies. These
suggest that, by combining the observed number counts of massive halos
with a measurement of power spectrum, we may be able to correct for the
HSV effect on the power spectrum -- a Gaussianization method of the
power spectrum measurement.
In this section, we study this method.
Note that this method is {\em not} feasible
if the two observables are drawn from {\em different} survey regions.
First, let us consider an ideal case: suppose that we have a measurement
of the halo number counts $\left\{\hat{N}_2,\hat N_2, \dots,
\hat{N}_k\right\}$ in mass bins of $M_1, M_2, \dots, M_k$ for a given
survey of comoving volume $V_s$. Also suppose that we have an estimator of the
mass density field, $\delta_m(\bm{x})$, in order to estimate the matter
power spectrum for the same survey region.
For these assumptions,
we can define an estimator of the matter
power spectrum with suppressing the 1-halo term contribution:
\begin{equation}
\widehat{\Delta P(k)}=\hat{P}(k)-\frac{1}{V_s}\sum_i \hat N_i p^{1h}_i(k)
=\hat{P}^{2h}(k)+\frac{1}{V_s}\sum_i \hat N_i
\left[\hat{p}^{1h}_i(k)-p^{1h}_i(k )\right],
\label{eq:est_dpk}
\end{equation}
where $p^{1h}_i(k)$ is a theory template for the 1-halo term power
spectrum of halos in the $i$-th mass bin $M_i$, for an assumed
cosmological model (e.g., an NFW profile for the assumed cosmology).
The ensemble average of
the power spectrum estimator (Eq.~\ref{eq:est_dpk}) reads
\begin{eqnarray}
\ave{\widehat{\Delta P}(k)}&=&\ave{\hat{P}^{2h}(k)}+\frac{1}{V_s}\sum_i \ave{
\hat N_i\left[\hat{p}^{1h}_i(k)-p^{1h}_i(k)\right]
}\nonumber\\
&=&
P^{2h}(k)+\frac{1}{V_s}\sum_i\ave{\hat N_i}\left[\ave{\hat{p}^{1h}(k)}-p^{1h}(k)\right]\nonumber\\
&=& P^{2h}(k),
\end{eqnarray}
where we have assumed that the 1-halo term power template
spectrum matches the underlying true spectrum after the ensemble
average. Thus the ensemble average of the estimator (\ref{eq:est_dpk})
leaves only the 2-halo term.
The covariance matrix for the estimator (\ref{eq:est_dpk})
is found from Eq.~(\ref{eq:pscov}) to be
\begin{eqnarray}
{\rm Cov}\left[\widehat{\Delta P}(k),\widehat{\Delta P}(k')
\right]\simeq \frac{2}{N_{\rm mode}(k)}\delta_{kk'}^K P^{2h}(k)^2.
\end{eqnarray}
The power spectrum estimator (Eq.~\ref{eq:est_dpk}) suppressing the
1-halo term contribution obeys a Gaussian error covariance. In other
words, it reduces the non-Gaussian errors including the HSV effect, and
can recover the Gaussian information content.
In reality, we can only identify halos with
masses greater than a certain mass threshold $M_{\rm th}$. Given this limitation,
the power spectrum estimator with suppressing the 1-halo term
contribution needs to be modified from Eq.~(\ref{eq:est_dpk}) as
\begin{equation}
\widehat{\Delta P}(k)\equiv \hat{P}(k)-\frac{1}{V_s}\sum_{i; M_i\ge M_{\rm th}}\hat N_i
p_i(k)=\hat{P}^{2h}(k)+\frac{1}{V_s}\sum_{i; M_i<M_{\rm
th}}\hat N_i\hat{p}^{1h}(k)
+\frac{1}{V_s}\sum_{i;M_i\ge M_{\rm
th}}\hat N_i\left[\hat{p}^{1h}_i(k)-p^{1h}_i(k)\right].
\label{eq:est_dpk2}
\end{equation}
The ensemble average of the estimator (\ref{eq:est_dpk2}) yields
\begin{eqnarray}
\ave{\widehat{\Delta P}(k)}&=&
P^{2h}(k)+\int_0^{M_{\rm th}}\!\!\ensuremath{\mathrm{d}} M~ \frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M}p_M^{1h}(k)\nonumber\\
&\equiv & P^{2h}(k)+P^{1h}_{M<M_{\rm th}}(k),
\end{eqnarray}
where we have introduced the notation defined as $P^{1h}_{M<M_{\rm
th}}\equiv \int_0^{M_{\rm th}}\ensuremath{\mathrm{d}} M~(\ensuremath{\mathrm{d}} n/\ensuremath{\mathrm{d}} M)p^{1h}_M(k)$, the
1-halo term contribution arising from halos with masses $M<M_{\rm
th}$. Thus the estimator reduces the 1-halo term arising form massive
halos with $M>M_{\rm th}$.
Likewise, we can compute the covariance matrix of the Gaussianized power
spectrum estimator:
\begin{eqnarray}
{\rm Cov}[\widehat{\Delta P}(k),\widehat{\Delta P}(k')]_{M_{\rm th}}
&=&\frac{2}{ N_{\rm mode}(k)}
\delta^K_{kk'}
\left[P^{2h}(k)+P^{1h}_{M<M_{\rm th}}(k)\right]^2
+\frac{1}{V_s}
\left[\bar{T}(k,k')
-\bar{T}^{1h}(k,k'; M>M_{\rm th})
\right]
\nonumber\\
&&\hspace{2em}
+\left[
\int^{M_{\rm th}}_0\!\ensuremath{\mathrm{d}} M\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M}b(M)p^{1h}_M(k)
\right]
\left[
\int^{M_{\rm th}}_0\!\ensuremath{\mathrm{d}} M'\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M'}b(M')p^{1h}_{M'}(k')
\right]\sigma_m^2(V_s),
\end{eqnarray}
where $T^{1h}(k,k; M>M_{\rm th})$ is the 1-halo term of matter
trispectrum containing only the contributions from massive halos with
$M>M_{\rm th}$. The estimator does suppress the non-Gaussian error
contributions arising from massive halos with masses $M>M_{\rm th}$.
Then the question we want to address is whether the modified power
spectrum estimator can recover the information content.
To be comprehensive,
the cross-covariance between the number counts and the power spectrum
is
\begin{eqnarray}
{\rm Cov}[\hat N_i, \widehat{\Delta P(k)}]&=&
b_{i}\bar{N}_{i} \sigma_m^2\int^{M_{\rm th}}_0\!\!\ensuremath{\mathrm{d}} M\frac{\ensuremath{\mathrm{d}}
n}{\ensuremath{\mathrm{d}} M} b(M)p_M^{1h}(k)
\nonumber\\
&&+\bar{N}_i\sigma_m^2
\int\!\ensuremath{\mathrm{d}} M\ensuremath{\mathrm{d}} M'\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M}\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M'}\left[
b(M_i)b(M)+b(M_i)b(M')+b(M)b(M')
\right]p^{2h}_{MM'}(k).
\end{eqnarray}
The power spectrum estimator (Eq.~\ref{eq:est_dpk}) suppresses the 1-halo
term contribution of the cross-covariance arising form massive halos
with $M>M_{\rm th}$.
\section{Application to weak lensing power spectrum}
\label{sec:wl}
Since the weak lensing power spectrum is a projection of the three dimensional
power spectrum, we can extend the
covariance
calculations and the Gaussianization
methodology of Section~\ref{sec:formulation} to the weak lensing observables.
\subsection{Angular number counts of halos, weak lensing power spectrum
and their covariance matrices}
There are a number of potential methods of obtaining a mass-limited halo
sample. Perhaps, the most attractive approach is to simultaneously
carry out a CMB survey and a weak lensing survey for the same
region of the sky. In the next few
years, the HSC and the new-generation high-angular resolution,
high-sensitivity CMB experiment, ACTPol \citep{ACTPol:10},
will survey overlapping regions of the sky as will
the DES and the SPTPol \citep{SPTPol:12}.
The weak lensing surveys will calibrate the SZ flux-mass
relation and the SZ surveys will provide a mass-selected sample of halos
for the joint analysis envisioned in this paper. Further, if the imaging
survey has overlapping footprints with a wide-area spectroscopic survey,
the spectroscopic data can determine redshifts of the identified
clusters from the spectroscopic redshifts of member galaxies, such as
BCGs,
and/or can
improve an identification of massive clusters from a concentration of
the spectroscopic member galaxies in the small spatial region
\citep{ReidSpergel:09,Hikageetal:12,Masakietal:13}. This is indeed the
case for combinations of the HSC survey with the BOSS or PFS surveys,
and the space-based Euclid and WFIRST projects. Soon
the all-sky eROSITA
survey, scheduled to be launched in
2015\footnote{\url{http://www.mpe.mpg.de/eROSITA}},
can be used to improve
the completeness/purity of the cluster catalog.
In a full analysis, we would need to include
the scatter in the mass observable relation; however, for this paper, we will simplify
the presentation by assuming that we can directly count
the number of halos in the
mass range of $[M_i, M_i+\Delta M]$ and over the entire redshift range
$0<z<z_{\rm max}$, in a light-cone volume with solid
angle $\Omega_s$:
\begin{equation}
\bar{N}^{2D}_{i}\equiv \Omega_s \int_0^{\chi_H}\!\!\ensuremath{\mathrm{d}}\chi~ \chi^2
\left.\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M}\right|_{M_i}\Delta M.
\label{eq:n_2D}
\end{equation}
The estimator for the angular number counts of
halos is given as
\begin{equation}
\hat{N}^{2D}_{i}=\sum_{\tilde{b}=1}\hat{N}_{i(\tilde{b})},
\end{equation}
where $\hat{N}_{i(b)}$ is the {\em observed}
number counts of halos in the mass
range $[M_i,M_i+\Delta M]$ and in the redshift range
$[\chi_b,\chi_b+\Delta \chi]$, which has the comoving volume of
$\Omega_s\chi^2\Delta
\chi$.
The ensemble average
$\ave{\hat{N}_{i(b)}}\equiv \bar{N}_{i(b)}=\Omega_s
\chi_b^2\Delta \chi (\ensuremath{\mathrm{d}} n/\ensuremath{\mathrm{d}} M_i) \Delta M$.
Hereafter we sometimes use notation ``$b$'' or ``$b'$'' to
denote the $b$- or $b'$-th redshift bin (do not confuse with the halo
bias $b(M_i)$ or $b_i$).
Similarly to the derivation used in Eqs.~(\ref{eq:n_cov1}) and
(\ref{eq:n_cov2}),
the covariance matrix of the number counts (Eq.~\ref{eq:n_2D}) are computed as
\begin{eqnarray}
{\rm Cov}\left[
\hat{N}^{2D}_{i},\hat{N}^{2D}_{j}
\right]&=& \bar{N}^{2D}_{i}\delta_{ij}^K+
\sum_{\tilde b}
\bar{N}_{i(\tilde b)}\bar{N}_{j(\tilde
b)}\sigma_m^2(\Omega_{s};\chi_{\tilde b})\nonumber\\
&&\hspace{-7em}=\bar{N}^{2D}_{i}\delta^K_{ij}+
\sum_{\tilde b=1}
\left[
\Omega_{\rm s}\chi_{\tilde b}^2\Delta \chi
b(M_i)\left.\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M}\right|_{M_i}\Delta M
\right]
\left[
\Omega_{\rm s}\chi_{\tilde b}^2\Delta \chi
b(M_j)\left.\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M}\right|_{M_j}\Delta M
\right]
\sigma_m^2(\Omega_{s}; \chi_{\tilde b})\nonumber\\
&&\hspace{-7em}\simeq
\bar{N}^{2D}_{i}\delta^K_{ij}+
\Omega_s^2\int^{\chi_b}_0\!\!\ensuremath{\mathrm{d}}\chi~
\left[
\chi^2
b(M_i)\left.\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M}\right|_{M_i}\Delta M
\right]
\left[
\chi^2
b(M_j)\left.\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M}\right|_{M_j}\Delta M
\right]
\Delta \chi \sigma_m^2(\Omega_{s}; \chi)\nonumber\\
&&\hspace{-7em}\simeq
\bar{N}^{2D}_{i}\delta^K_{ij}+
\Omega_s^2\int^{\chi_b}_0\!\!\ensuremath{\mathrm{d}}\chi~
\left[
\chi^2
b(M_i)\left.\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M}\right|_{M_i}\Delta M
\right]
\left[
\chi^2
b(M_j)\left.\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M}\right|_{M_j}\Delta M
\right]
\int\!\!\frac{\ensuremath{\mathrm{d}}^2\bm{k}_\perp}{(2\pi)^2}
P^L_m(k_\perp;\chi)\left|
\tilde{W}_\perp(\bm{k}_\perp;\Omega_s)
\right|^2,
\label{eq:cov_n2d}
\end{eqnarray}
where
$\sigma^2_m(\Omega_s;\chi_b)$ is the rms mass
fluctuations of the volume around the $b$-th redshift bin $\chi_b$,
$V(\chi_b)=\Omega_s\chi_b^2\Delta \chi$,
and $\tilde{W}_\perp(\bm{k}_\perp;\Omega_s)$ is the Fourier
transform of the angular survey window function assuming the flat-sky
approximation.
In the third line on the r.h.s., we backed the summation to the
integration.
We have also assumed that the halo number counts of different redshift
bins are uncorrelated with each other; to be more precise, we used the
following ensemble average:
\begin{equation}
\ave{N_{i(b)}N_{j(b')}}=\bar{N}_{i(b)}\bar{N}_{j(b')}
+\delta^K_{bb'}\delta^K_{ij}\bar{N}_{i(b)}+
\delta^K_{bb'}\bar{N}_{i(b)}\bar{N}_{j(b)}\left[
1+b_{i}b_{j}\sigma_m^2(\Omega_s; \chi_b)
\right],
\end{equation}
where $b_i\equiv b(M_i)$ and so on.
In the fourth line on the r.h.s., we used the
following calculation, based on the Limber's approximation \citep{Limber:54}:
\begin{eqnarray}
\Delta \chi \sigma^2_m(k;
\Omega_s,\chi)&=&\Delta\chi\int\!\frac{\ensuremath{\mathrm{d}}^2\bm{k}_\perp}{(2\pi)^2}
\int^{\infty}_{-\infty}\!\frac{\ensuremath{\mathrm{d}} k_\parallel}{2\pi}P^L_m(k;\chi)\left|
\tilde{W}_\perp(\bm{k}_\perp;\Omega_s)
\right|^2 |W_\parallel(k_\parallel;\chi)|^2\nonumber\\
&\simeq & \Delta\chi\int\!\frac{\ensuremath{\mathrm{d}}^2\bm{k}_\perp}{(2\pi)^2}
P^L_m(k_\perp;\chi)\left|
\tilde{W}_\perp(\bm{k}_\perp;\Omega_s)
\right|^2
\int^{\infty}_{-\infty}\frac{\ensuremath{\mathrm{d}} k_\parallel}{2\pi}
\left|
\tilde{W}_\parallel(k_\parallel)
\right|^2\nonumber\\
&=&\int\!\!\frac{\ensuremath{\mathrm{d}}^2\bm{k}_\perp}{(2\pi)^2}
P^L_m(k_\perp;\chi)\left|
\tilde{W}_\perp(\bm{k}_\perp;\Omega_s)
\right|^2,
\label{eq:sigm_dchi}
\end{eqnarray}
where $W_\parallel$ is the radial selection function, and we have
assumed in the second line on the r.h.s that only the density
fluctuations with Fourier modes perpendicular to the line-of-sight
direction contribute the rms of super-survey modes. For the integration
of the radial selection function, we have used the identity \citep[see
Eq.~8 in][]{TakadaHu:13}:
$\int\!d\chi W_\parallel(\chi)=
\Delta\chi\int\!d\chi
W_\parallel(\chi)^2=\Delta\chi\int\!\frac{dk_\parallel}{2\pi}
|\tilde{W}_\parallel(k_\parallel)|^2=1$.
We have assumed
that the radial bin width is sufficiently large compared to the
wavenumber $k$ relevant for the power spectrum of interest.
Eq.~(\ref{eq:cov_n2d}) matches
Eq.~(13) in \citet{TakadaBridle:07}.
The weak lensing angular power spectrum is a projection of the matter power spectrum.
Thus, we can use
the halo model approach and the Limber's approximation
to represent the weak lensing power spectrum as the sum of the 1- and 2-halo terms:
\begin{equation}
C_l=C_l^{1h}+C_l^{2h},
\end{equation}
where $C_l^{1h}$ and $C_{l}^{2h}$ are the 1- and 2-halo terms defined as
\begin{eqnarray}
C_l^{1h }&=&\int\!\ensuremath{\mathrm{d}}\chi\frac{\ensuremath{\mathrm{d}}^2V}{\ensuremath{\mathrm{d}}\chi \ensuremath{\mathrm{d}}\Omega}
\int\!\ensuremath{\mathrm{d}} M\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M} \left|\kappa_M(l; \chi)\right|^2
=\frac{1}{\Omega_s}\int\!\ensuremath{\mathrm{d}}\chi
\left[
\Omega_s \chi^2
\int\!\ensuremath{\mathrm{d}} M\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M}\right] \left|\kappa_M(l; \chi)\right|^2,\\
C_l^{2h}&=&\int\!\ensuremath{\mathrm{d}}\chi W^{\rm GL}(\chi)^2\chi^{-2}
P^{2h}\!\left(k=\frac{l}{\chi}; \chi\right)
\nonumber\\
&=&
\frac{1}{\Omega_{\rm s}^2}\int\!\ensuremath{\mathrm{d}}\chi~ \chi^{-6}\left[
\Omega_s \chi^2 \int\!\!\ensuremath{\mathrm{d}} M\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M}
\right]\left[\Omega_s \chi^2 \int\!\ensuremath{\mathrm{d}} M'\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M'}
\right]\frac{M}{\bar\rho_m}\frac{M'}{\bar\rho_m}
W_{\rm GL}^2 P_{hh}(k; M,M'),
\end{eqnarray}
where $k=l/\chi$,
$\tilde{\kappa}_M(l;\chi)$ is the two-dimensional Fourier
transform of the projected NFW profile \citep[see Eq.~28 in][for the
definition]{OguriTakada:11}, and $W_{\rm GL}$ is the lensing efficiency
function \citep[see Eq.~19 in][]{OguriTakada:11} that has a dimension
of $[{\rm Mpc}^{-1}]$. From the above equation,
we find that
the
estimators of 1- and 2-halo term lensing power spectra can be rewritten as functions of
observed halo number counts:
\begin{eqnarray}
\hat{C}^{1h}_l &\equiv& \frac{1}{\Omega_s}\sum_b\sum_{i}\hat{N}_{i(b)}
\hat{C}^{1h}_i(l;\chi_a),\nonumber\\
\hat{C}^{2h}_l &\equiv & \frac{1}{\Omega_s^2\Delta \chi}
\sum_b\sum_{i,j}\hat{N}_{i(b)} \hat{N}_{j(b)}\chi^{-6}_b
\hat{P}^{2h}_{ij}\!\!\left(k=\frac{l}{\chi_b}; \chi_b\right)
\end{eqnarray}
where $\hat{C}_i^{1h}(l)$ is the estimator arising from the mass density
field inside halos of mass $M_i$, and
$\hat{P}^{2h}_{ij}(k)$ arises from the mass density field at
large scales that governs the distribution between different halos of masses
$M_i $ and $M_j$.
Note that
$\hat{P}^{2h}_{ij}$ is defined as
$\hat{P}^{2h}_{ij}\equiv (M_i/\bar\rho_m)(M_j/\bar\rho_m)
W_{\rm GL}^2\hat{P}_{ij}^{2h}$, and has a dimension
of $[({\rm Mpc})^7]$.
Using the similar derivation to Eq.~(\ref{eq:pscov}), the covariance
matrix of the lensing power spectrum is found to be
\begin{eqnarray}
{\rm Cov}[\hat{C}_l,\hat{C}_{l'}]&=&\frac{2}{N_{\rm
mode}(l)}C_l^2\delta^K_{ll'}
+\frac{1}{\Omega_s}\bar{T}(l,l')\nonumber\\
&&\hspace{-2em}+\int\!\!\ensuremath{\mathrm{d}}\chi~ \left[
\chi^2\int\!\ensuremath{\mathrm{d}} M\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M}b(M)\left|\kappa_M(l;\chi)\right|^2
\right]
\left[
\chi^2\int\!\ensuremath{\mathrm{d}} M'\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M'}b(M')\left|\kappa_{M'}(l';\chi)\right|^2
\right]\int\!\!\frac{\ensuremath{\mathrm{d}}^2\bm{k}_\perp}{(2\pi)^2}P^L_m(k_\perp)
\left|\tilde{W}_\perp(k_\perp;\Omega_s)\right|^2,
\nonumber\\
\label{eq:clcov}
\end{eqnarray}
where $N_{\rm mode}(l)\equiv l\Delta l \Omega_s/(2\pi)$ and $\Delta l$ is the
bin width.
Again the above equation reproduces Eq.~(18) in \cite{Satoetal:09}
\citep[see also Eq. 14 in][]{Kayoetal:13}. In reality an accuracy of the
lensing power spectrum measurement is affected by intrinsic shape
noise. Assuming random shape orientations in between different galaxies,
the measured lensing power spectrum is contaminated by the shape noise
as
\begin{equation}
C_l^{\rm obs}=C_l + \frac{\sigma_\epsilon^2}{\bar{n}_g},
\end{equation}
where $\sigma_\epsilon$ is the rms of intrinsic ellipticity per
component and $\bar{n}_g$ is the mean number density of source
galaxies. By replacing $C_l$ with $C_l^{\rm obs}$ in
Eq.~(\ref{eq:clcov}), we can take into account the shape noise
contamination to the covariance matrix. We assume
$\sigma_{\epsilon}=0.22$ as for the fiducial value.
Similarly, the cross-covariance between the angular number counts and
the lensing power spectrum is given as
\begin{eqnarray}
{\rm Cov}\left[\hat{N}^{2D}_{i},\hat{C}_l\right]&=&
\frac{1}{\Omega_s}\int_0^{\chi_b}\!\!\ensuremath{\mathrm{d}}\chi~\Omega_s \chi^2\frac{\ensuremath{\mathrm{d}}
n}{\ensuremath{\mathrm{d}} M_i}\Delta M\left|
\kappa_{M_i}(l;\chi)
\right|^2\nonumber\\
&&\hspace{-6em}+\int_0^{\chi_b}\!\ensuremath{\mathrm{d}}\chi~
\Omega_s\chi^2b(M_i)\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M_i}\Delta
M\left[\chi^2\int\!\ensuremath{\mathrm{d}} M\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}}
M}b(M)\left|\kappa_M(l;\chi)\right|^2\right]
\int\!\!\frac{\ensuremath{\mathrm{d}}^2\bm{k}_\perp}{(2\pi)^2}P^L_m(k_\perp)
\left|\tilde{W}_\perp(k_\perp;\Omega_s)\right|^2
\nonumber\\
&&\hspace{-6em}
+\int_0^{\chi_b}\!\ensuremath{\mathrm{d}}\chi~ \Omega_s\chi^2\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M_i}\Delta
M
\chi^4 \int\!\ensuremath{\mathrm{d}} M\ensuremath{\mathrm{d}} M'~\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M}\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M'}
\left[2b(M_i)b(M)+b(M)b(M')\right]\nonumber\\
&&\times
\chi^{-6}W_{\rm GL}^2
\frac{M}{\bar\rho_0}
\frac{M'}{\bar\rho_0}
P_{hh}\!\!\left(k=\frac{l}{\chi};\chi, M,M'\right)
\int\!\!\frac{\ensuremath{\mathrm{d}}^2\bm{k}_\perp}{(2\pi)^2}P^L_m(k_\perp)
\left|\tilde{W}_\perp(k_\perp;\Omega_s)\right|^2.
\end{eqnarray}
Note that the shape noise does not contaminate to the cross-covariance.
\subsection{Joint likelihood function of the angular halo number counts
and the weak lensing power spectrum}
Having derived all the covariance matrices of the halo number counts
and the weak lensing power spectrum as well as their cross-covariance, we
can advocate the joint likelihood
function for the two
observables.
Assuming that the two observables obey a multivariate
Gaussian distribution, we can derive the joint likelihood function as
\begin{equation}
{\cal L}(\hat{N}_{i}^{2D},\hat{C}_l)\propto
\exp\left[-\frac{\chi^2}{2}\right]\equiv
\exp\left[-\frac{1}{2}\bm{D}^{T}\bm{C}^{-1}\bm{D}
\right],
\label{eq:joint_n-wl}
\end{equation}
where the data vector $\bm{D}$ and the covariance matrix $\bm{C}$ are
defined as
\begin{eqnarray}
\bm{D}&\equiv& \left(
\begin{array}{cc}
\hat{N}_{i}^{2D} - \bar{N}_{i}^{2D}
&
\hat{C}_l -\bar{C}_l
\end{array}
\right),
\nonumber\\
\bm{C}&\equiv&
\left(
\begin{array}{cc}
{\rm Cov}[\hat{N}_{i}^{2D},\hat{N}_{j}^{2D}] &
{\rm Cov}[\hat{N}_{i}^{2D},\hat{C}_l] \\
{\rm Cov}[\hat{C}_l,\hat{N}_{j}^{2D}] &
{\rm Cov}[\hat{C}_l,\hat{C}_{l'}]
\\
\end{array}
\right).
\label{eq:cov_n-wl}
\end{eqnarray}
Here $\bm{C}^{-1}$ is the inverse of
the covariance matrices. Note that the products of the data vector and
the inverse of the covariance matrix
run over
all the halo mass
bins
as well as the multipole bins.
We used the above equation to compute the joint likelihood function in
Fig.~\ref{fig:dn-dcl-model}. The 68 or 95\% percentile of the
distribution for the two observables is obtained from the range
satisfying $\chi^2\ge 2.3$ or $6.17$, respectively. We again
notice that the halo model (Eq.~\ref{eq:joint_n-wl}) well
reproduces the simulation results, giving a justification of our method
and the multivariate Gaussian assumption at the angular scales.
\subsection{Information content of
the Gaussianized weak lensing power spectrum}
According to the discussion in Section~\ref{sec:dp}, we can define an
estimator of the power spectrum suppressing the 1-halo term
contribution, assuming that all halos with masses greater than a certain
mass threshold $M_{\rm th}$ in the surveyed light-cone volume
are identified from a
given survey volume:
\begin{equation}
\left.\widehat{\Delta C_l}\right|_{M_{\rm th}}=\hat{C}_l -\frac{1}{\Omega_s}\sum_b \sum_{i;
M>M_{\rm th}}
\hat{N}_{i(b)}{C}^{1h}_i(l;\chi_b),
\end{equation}
where $\hat N_{i(b)}$ is the observed counts of halos with $M\ge M_{\rm
th}$ and in the redshift range of $[\chi_b,\chi_b+\Delta\chi]$, and
$C^{1h}_i(l; \chi_b)$ is the theory temperate of the 1-halo term power
spectrum for halos with $M_i$ and at redshift $\chi_b$.
The theoretical template can be estimated from stacked lensing
of the sampled halos, e.g., using the method in \cite{OguriTakada:11}.
The ensemble average of the estimator and the covariance are
\begin{eqnarray}
\ave{\left. \widehat{\Delta C_l}\right|_{M_{\rm th}}}&=&
C_l^{2h}+\left.C_l^{1h}\right|_{M<M_{\rm th}}\equiv
C_l^{2h}+\int\!\ensuremath{\mathrm{d}}\chi~\frac{\ensuremath{\mathrm{d}}^2V}{\ensuremath{\mathrm{d}}\chi \ensuremath{\mathrm{d}}\Omega}\int^{\rm M_{\rm
th}}_0\!\ensuremath{\mathrm{d}} M~\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M}\left|\kappa_M(l;\chi)\right|^2,
\label{eq:cl_r1h_exp}
\end{eqnarray}
and
\begin{eqnarray}
{\rm Cov}\left[
\left.\widehat{\Delta C_l}\right|_{M_{\rm th}},
\left.\widehat{\Delta C_{l'}}\right|_{M_{\rm th}}
\right]&=&
\frac{2}{N_{\rm mode}(l)}
\left[
C^{2h}_l + \left. C_l^{1h}\right|_{M<M_{\rm th}}
+\frac{\sigma_\epsilon^2}{\bar{n}_g}\right]^2\delta^K_{ll'}
+\frac{1}{\Omega_s}
\left[
\bar{T}_\kappa(l,l')
-\bar{T}_\kappa^{1h}(l,l'; M>M_{\rm th})
\right]
\nonumber\\
&&\hspace{-12em}+
\int\!\!\ensuremath{\mathrm{d}}\chi~ \left[
\chi^2\int_0^{M_{\rm th}}\!\ensuremath{\mathrm{d}} M\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M}b(M)
\left|\kappa_M(l;\chi)\right|^2
\right]
\left[
\chi^2\int_0^{M_{\rm th}}\!\ensuremath{\mathrm{d}} M'\frac{\ensuremath{\mathrm{d}} n}{\ensuremath{\mathrm{d}} M'}b(M')\left|\kappa_{M'}(l';\chi)\right|^2
\right]\int\!\!\frac{\ensuremath{\mathrm{d}}^2\bm{k}_\perp}{(2\pi)^2}P^L_m(k_\perp)
\left|\tilde{W}_\perp(k_\perp;\Omega_s)\right|^2,\nonumber\\
\label{eq:cl_r1h_cov}
\end{eqnarray}
where we have also included the intrinsic shape noise contribution.
In the following, we assume
a circular-shaped
survey geometry with area $\Omega_s=\pi\Theta_s^2$, yielding
$\tilde{W}_\perp(k;\Omega_{s})=2J_1(\chi k\Theta_s)/(\chi k\Theta_s)$.
The information content inherent in the power spectrum measurement for a
given survey is defined in \cite{Tegmarketal:97} \citep[see also][]{TakadaJain:09} as
\begin{equation}
\left(\frac{S}{N}\right)^2\equiv \sum_{l_i,l_j<l_{\rm max}}
\left.{\Delta C_{l_i}}\right|_{M_{\rm th}}
[{\bf{C}^{-1}}]_{l_il_j}
\left.{\Delta C_{l_j}}\right|_{M_{\rm th}},
\label{eq:sn}
\end{equation}
where the summation runs over multipole bins up to a given maximum
multipole $l_{\rm max}$, $\left.{\Delta C_l}\right|_{M_{\rm th}}$ is the
expectation value of the power spectrum (Eq.~\ref{eq:cl_r1h_exp}), and
$\bm{C}^{-1}$ denotes the inverse of the covariance matrix
(Eq.~\ref{eq:cl_r1h_cov}). The inverse of $S/N$ is equivalent to a
fractional error of measuring the power spectrum amplitude when using
the information up to the maximum multipole $l_{\rm max}$,
assuming that the shape of
the power spectrum is perfectly known. For a Gaussian field, Eq. (\ref{eq:sn})
predicts
$S/N\propto l_{\rm max}$.
\subsection{Results}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth,angle=-90]{cl.ps}
\caption{Weak lensing power spectrum for source galaxies whose mean
redshift $\langle z_s\rangle=1$. The dashed curve is the spectrum
obtained by projecting the linear matter power spectrum weighted with
the lensing efficiency function, while the top dotted curve is the
result for the nonlinear matter power spectrum. The solid curves are
the spectra where the 1-halo term contribution arising from massive
halos with $M>1, 2, 3 $ or $5~ [10^{14}M_\odot/h]$ is
subtracted, respectively. For comparison, the thin solid curves are the
relative shape noise contamination for the number densities of
$\bar{n}_g=10$, 30 and 100~arcmin$^{-2}$, respectively,
where we assumed
$\sigma_\epsilon=0.22$ for the rms intrinsic
ellipticities.
\label{fig:cl}}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth,angle=-90]{sn_wosn_v2.ps}
\caption{Cumulative signal-to-noise ($S/N$) ratio (Eq.~\ref{eq:sn}) for
the weak lensing power spectrum measurement as a function of maximum
multipole $l_{\rm max}$, expected for a wide-area survey that is
characterized by $\Omega_{\rm s}=1500$~sq. degrees and $\langle
z_s\rangle=1$ for survey area and the mean redshift of source galaxies,
respectively. Note that we did not include the shape noise
contamination. The top dashed line is the $S/N$ for a Gaussian field,
which has a scaling of $S/N|_{\rm Gaussian}\propto l_{\rm max}$. The
bottom dotted curve is the $S/N$ computed by using the full non-Gaussian
error covariance including the halo sampling variance contribution. The
solid curves are the $S/N$ values for a {\em Gaussianized} weak lensing
field, where the 1-halo term contribution for halos with masses greater
than a given mass threshold, as indicated by the legend, is subtracted
assuming that such massive halos are identified in the survey region.
The Gaussianization method using massive halos with $M\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}
10^{14}M_\odot/h$ recovers the information content by up to a factor of
a few, especially over $300 \lower.5ex\hbox{$\; \buildrel < \over \sim \;$} l_{\rm max}\lower.5ex\hbox{$\; \buildrel < \over \sim \;$} 3000$.
\label{fig:sn_wosn}}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth,angle=-90]{sn_ng100_v2.ps}
\includegraphics[width=0.4\textwidth,angle=-90]{sn_ng30_v2.ps}
\caption{Similar to the previous figure, but the shape noise
contamination is included in the power spectrum covariance
calculation. We here considered $\bar{n}_g=100$ or
30~arcmin$^{-2}$
for the number density of galaxies in the left or right panels,
respectively, which roughly correspond to the number densities for
the WFIRST and LSST-type surveys, respectively.
Other survey parameters are kept fixed
to the fiducial values, $\Omega_{s}=1500$
deg$^2$ and $\langle z_s\rangle=1$. Note that
the top curve for a Gaussian field includes
the shot noise contamination in the covariance calculation.
\label{fig:sn_ng100}}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth,angle=-90]{sn_ng20_v2.ps}
\includegraphics[width=0.4\textwidth,angle=-90]{sn_ng10_v2.ps}
\caption{Similar to the previous figure, but we assumed
$\bar{n}_g=20$ or 10~arcmin$^{-2}$ in the left or right panels,
which roughly correspond to the Subaru HSC- or DES/Euclid/KiDS-type
surveys, respectively.
\label{fig:sn_ng20}}
\end{figure}
To estimate an expected performance of the Gaussianized weak lensing field for
an upcoming imaging survey, we need to assume the fiducial cosmological
model and the survey parameters. For the fiducial cosmological model, we
assume a $\Lambda$CDM model that is consistent with the WMAP 7-year
result in \citet{Komatsuetal:10}. We use the same model ingredients in
\cite{OguriTakada:11} to compute the halo model predictions. As for
survey parameters, we employ the parameters that resemble the planned
Subaru Hyper Suprime-Cam (HSC) survey: $\langle z_s \rangle=1$,
$\bar{n}_g=20~$arcmin$^{-2}$, $\sigma_\epsilon=0.22$ and $\Omega_{\rm
s}=1500$~deg$^2$ for the mean redshift of galaxies, the mean number
density, the rms intrinsic ellipticities, and the survey area,
respectively. For the redshift distribution of imaging galaxies, we
employ Eq.~(17) in \citet{OguriTakada:11}, where we set the parameter
$z_0=1/3$ so as to have $\langle z_s\rangle =3z_0=1$. We will also study
how the results are changed by varying the shot noise contamination.
The solid curves in Fig.~\ref{fig:cl} show the power spectra when the
1-halo term contribution arising from massive halos with $M\ge 1, 2, 3$
or 5 $[10^{14}M_\odot/h]$ is subtracted, respectively.
The subtraction reduces the power spectrum amplitudes at high
multipole. The
lower
mass cuts produce the greatest suppression of power.
Fig.~\ref{fig:sn_wosn} shows the expected cumulative signal-to-noise
ratio ($S/N$) for a Subaru HSC-type survey when implementing the
Gaussianization method of weak lensing power spectrum, combined with the
number counts of massive halos. Here the ``cumulative'' $S/N$ is
obtained by integrating the signal-to-noise ratio of the weak lensing
power spectrum over angular scales from $l=10$ up to a given maximum
multipole $l_{\rm max}$ as denoted in the $x$-axis. Note that, for the
results in this plot, we did not include the shape noise contamination
in order to study a best-available improvement of the Gaussianization
method. The top dashed line is the $S/N$ for a two-dimensional Gaussian
field, which gives the maximum $S/N$ value as the weak lensing field or
the underlying matter distribution originates from the initial Gaussian
field as in the CMB field. The bottom dotted curve is the $S/N$ value
when using the full power spectrum covariance including the non-Gaussian
errors, where the non-Gaussian errors due to super-survey modes gives a
dominant contribution at $l_{\rm max}\lower.5ex\hbox{$\; \buildrel > \over \sim \;$} $a few 100. The dotted curve
is similar to Fig.~9 in \citet{Satoetal:09} \citep[see also Fig.~10
in][]{Kayoetal:13}.
The solid curves show the results when implementing the Gaussianization
method. More precisely, the curves are the results for the Gaussianized
power spectra, where the 1-halo term power spectrum arising from massive
halos with $M\ge 1, 2,3,$ or $5~ [10^{14}~M_\odot/h]$, respectively, is
subtracted from the total power, assuming that the massive halos of each
mass range are identified in the survey region. The 1-halo term
subtraction significantly increases the information content. With
decreasing the mass threshold, it recovers the information content,
almost the full information as does in a Gaussian field, up to higher
$l_{\rm max}$.
To be more precise, the Gaussianization method recovers about 80
or 60 per cent at $l_{\rm max}=1000$ for $M_{\rm th}=1$ or
$3\times 10^{14}~M_\odot/h$, while it recovers about 50 or 30 per cent at $l_{\rm
max}=2000$, respectively.
Compared to the
$S/N$ value without the Gaussianization method, the improvement is up to
a factor of 2 or 1.4 for $M_{\rm th}=1$ or $3\times 10^{14}~M_\odot/h$
for the range of $l_{\rm max}=1000$--2000,
which is equivalent to a factor 4 or 2 larger survey, respectively. The
improvement means that adding the abundance of massive halos to the
power spectrum measurement can correct for the super-sample covariance
contamination, because the super-sample covariance is a dominant source
to cause a saturation in the information content at $l_{\rm max}\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}
500$ \citep[Fig.~9 in][]{Satoetal:09}. Even for the higher $l_{\rm.
max}$ such as $l_{\rm max}\lower.5ex\hbox{$\; \buildrel > \over \sim \;$} 2000$, where less massive halos of
$M\lower.5ex\hbox{$\; \buildrel < \over \sim \;$} 10^{14}M_\odot/h$ becomes important in the 1-halo term
contribution, the figure still displays a significant improvement, by up
to a factor of 2. Upcoming imaging surveys are aimed at constraining
cosmological parameters from the lensing power spectrum information up
to $l_{\rm max}=1000$--2000, beyond which
complex baryonic physics in the nonlinear clustering
can be important. Our results are very promising in
a sense that the Gaussianization method allows for an efficient masking
of the mass distribution in such a highly-nonlinear region, the region inside
massive halos, when measuring the power spectrum.
Figs.~\ref{fig:sn_ng100} and \ref{fig:sn_ng20} show the results when
including the shape noise contamination to the covariance, for different
number densities of source galaxies; $\bar{n}_g=10, 20, 30, $ and
100~arcmin$^{-2}$, respectively. The range includes the number densities
expected for the planned weak lens surveys; $70$ for WFIRST, 30 for
LSST/Euclid, 20 for HSC and 10~arcmin$^{-2}$ for DES/KiDS, respectively.
Note that other survey parameters (area and the mean redshift) are kept
fixed to their fiducial values as in Fig.~\ref{fig:sn_wosn}. The
relative improvement in the $S/N$ values with and without the
Gaussianization method does not largely change for different survey
areas. The figures show that a survey having a higher number density
can have a greater benefit from the Gaussianization method as there is
more information on small scales that can be recovered. To be
more precise, the Gaussianization method for a Subaru HSC-type survey
recovers about 90 or 70 per cent information of the Gaussian plus shape
nose case at $l_{\rm max}=1000$ for $M_{\rm th}=1$ or $3\times
10^{14}~M_\odot/h$, while it recovers about 75 or 55 per cent
information at $l_{\rm max}=2000$, respectively. On the other hand, the
Gaussianization method gives about 1.6 improvement at $l_{\rm max}=1000$
-- 2000 compared to the results without the Gaussianization method for
$M_{\rm th}=10^{14}M_\odot/h$ (for a Subaru-type survey), while it gives
about 1.4 improvement for $M_{\rm th}=3\times 10^{14}M_\odot/h$. These
improvements are equivalent to a factor 2 -- 2.5 wider survey area. The
dependence of these results on survey area is very weak; therefore these
improvements hold for other surveys, as can also be found from Fig.~2 in
\citet{Kayoetal:13} or Fig.~1 in \citet{TakadaHu:13}.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth,angle=-90]{chi2_csys.ps}
\caption{Shown is how uncertainties in the 1-halo term power
spectrum template, which is used for the 1-halo subtraction from the
total power spectrum, affect the Gaussianization method. We model a
misestimation of the 1-halo term template by including a bias in the
normalization parameter of the halo concentration, $c(M,z)=c_0
(1+z)^{-0.71}[M/(2\times 10^{12}M_\odot/h)]^{-0.081}$, with the
fiducial value $c_0=7.85$ (see text for details). The plot shows the
cumulative deviation between the true and misestimated power spectra up
to a certain maximum multipole $l_{\rm max}$, when the average halo
profile of halos with $M\ge M_{\rm th}$ is misestimated by an amount of
$c_{0, {\rm sys}}=0.5c_{0, {\rm true}}$ (a factor 2 bias);
$\Delta\chi^2=\sum_{l\le l_{\rm max}}\left[C_{l_i}(c_{0, {\rm true}})
-C_{l_i}(c_{0,{\rm sys}})\right] {\bf{C}^{-1}}_{ij} \left[C_{l_j}(c_{0,
{\rm true}}) -C_{l_j}(c_{0,{\rm sys}})\right] $ (see also text for the
details). Note that $\sqrt{\Delta\chi^2}$ is plotted. The dotted and
solid curves are the results for $M_{\rm th}=1$ and $3\times
10^{14}~M_\odot/h$, respectively. The thick and thin respective curves
are the results with and without the shape noise contamination for a
Subaru HSC-type survey ($\bar{n}_g=20~$arcmin$^{-2}$ and $\Omega_{\rm
s}=1500$ deg$^2$). Although a factor 2 bias in the concentration
parameter seems the worst case scenario, the cumulative deviation for
$l_{\rm max}=10^3$ is only up to a few-$\sigma$ deviation for the
Subaru-type survey.
\label{fig:csys}}
\end{figure}
The Gaussianization method requires a knowledge of the average
mass profile for massive halos in order to subtract the inferred 1-halo
term from the total power spectrum. The NFW profile seen in N-body
simulations is characterized by two parameters, the halo concentration
and halo mass. For lensing perspective, the halo mass is sensitive to
the area-weighted, integrated lensing signal up to the virial radius,
while the halo concentration needs to be estimated from the scale radius
which is the radius to divide the inner and outer profiles in the NFW
model. We expect that stacked lensing of the massive halos can be used
to estimate the 1-halo profile
\citep{Okabeetal:10,OguriTakada:11,Okabeetal:13}, probably with the aid
of priors from N-body simulations. However, this estimate itself is
limited by statistical measurement uncertainties, suffers from
degeneracies with cosmological
parameters, and can be affected by uncertainties in the astrophysical
effects such as baryonic effects on halo formation/structure
\citep[e.g.][]{Gnedinetal:04}.
What is the required accuracy of the halo profile template? To
address this question, we study how variants in the halo profile affect
a performance of the Gaussianization method as follows. We model the
halo profile variants by allowing a possible misestimation in the
normalization parameter of the halo mass and concentration relation,
$c(M,z)=c_0 (1+z)^{-0.71}[M/(2\times 10^{12}M_\odot/h)]^{-0.081}$, where
we employ $c_0=7.85$ as for the fiducial value following the N-body
simulation results in \citet{Duffyetal:08}. This treatment is also
motivated by the study of \citet{Zentneretal:13}, where they showed that
the baryonic effects on the halo profile, which are indicated from
hydrodnynamical simulations, can be taken into account by including the
halo concentration parameters as nuisance parameters in weak lensing
cosmology. \citet{OguriTakada:11} showed that, if the stacked lensing
measurement for a Subaru HSC-type survey is used to estimate the halo
profile parameters (the normalization, the mass slope and the
redshift-dependence slope) simultaneously with cosmological parameters,
the marginalized, fractional accuracy for the normalization parameter
$c_0$ is about 50 percent, i.e. $|\sigma(c_0)|/c_0\simeq 0.5$, even
including possible miscentering effects of the halos. This accuracy is
considered as the worst case scenario, because in practice some priors
from N-body simulations can be used and/or the halo concentration
estimation can be improved by using detailed studies for representative
massive halo sample, e.g., based on the method combining strong and weak
lensing measurements \citep[e.g.][]{Broadhurstetal:05}.
Based on the above consideration, we assume that the halo mass
profile for massive halos, used for the 1-halo term subtraction, is
misestimated by an amount of factor 2, i.e. $c_{0,{\rm sys}}=0.5\times
c_{0,{\rm true}}$. Fig.~\ref{fig:csys} shows the cumulative deviation
between the true and model spectra for a Subaru HSC-type survey
($\bar{n}_g=20$~arcmin$^{-2}$ and $\Omega_s=1500$~deg$^2$):
\begin{equation}
\Delta\chi^2=\sum_{l\le l_{\rm max}}\left[C_{l_i}(c_{0, {\rm true}})
-C_{l_i}(c_{0,{\rm sys}})\right] {\bf{C}^{-1}}_{ij} \left[C_{l_j}(c_{0,
{\rm true}}) -C_{l_j}(c_{0,{\rm sys}})\right],
\label{eq:dchi2}
\end{equation}
where $C_l(c_{0,{\rm true}})$ is the true spectrum with the true 1-halo
term being subtracted, $C_l(c_{0,{\rm sys}})$ is the model spectrum with
the biased 1-halo term being subtracted, and $\bm{C}^{-1}$ is the
inverse of the covariance matrix. The figure plots
$\Delta\chi=\sqrt{\Delta\chi^2}$; $\Delta\chi=1$ means the $\pm 1\sigma$
deviation between the true and model spectra. The figure shows that,
even if we consider a factor 2 bias in the concentration parameter, the
cumulative deviations up to $l_{\rm max}=10^3$ are only up to a few
$\sigma$-deviation for $M_{\rm th}\ge 10^{14}M_\odot/h$ and a
Subaru-type survey. Recalling that the maximum multipole $l_{\rm
max}\sim 10^3$ corresponds to
$N_{\rm mod}\simeq \pi l_{\rm max}^2/[(2\pi)^2/\Omega_{s}]/2\sim 10^5$
for the total number of Fourier modes, where $(2\pi)/\sqrt{\Omega_s}$ is
the fundamental Fourier mode, only a few $\sigma$-deviation compared to
the huge data points is considered encouraging. This can be understood
as follows. The weak lensing signals up to $l_{\rm max}\sim 10^3$ are
not sensitive to the inner structure of halos or equivalently probe the
regime of $\tilde{u}_M(k)\simeq 1$ for the halo profile; the 1-halo term
up to $l_{\rm max}\sim 10^3$ is determined mainly by the abundance of
the massive halos (see Eq.~\ref{eq:ps_halomodel}). The deviations
become increasingly larger for the larger maximum multipoles. Thus we
conclude that the requirement on the halo profile template needed for
the Gaussianization method is not so stringent; stacking lensing can
probably achieve the desired accuracy.
\section{Conclusion and Discussion}
\label{sec:conclusion}
In this paper, using the halo model approach and the likelihood function
of halo number counts, we have derived the joint likelihood function of
the halo number counts and the weak lensing
power spectrum. The joint likelihood properly takes into account the
cross-correlation between the two observables when they are measured from the same
survey region. The cross-correlation in
the nonlinear regime is mainly due to the super-sample variance that
arises from the modes of length scales comparable with or greater than a
survey volume, which cannot therefore be directly observed. For instance, due to the super-survey sample variance,
the weak lensing power
spectrum amplitudes around $l$ of a few thousands have a significant
correlation with the number counts of massive halos with $M\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}
10^{14}~M_\odot/h$.
We showed that our analytical model of the joint likelihood
function can well reproduce the distributions of halo number counts and
weak lensing power spectra seen from 1000 ray-tracing simulations (see
Fig.~\ref{fig:dn-dcl-model}).
Given the strong correlation between the two observables, we have
proposed a method of combining the observed number counts of massive
halos with a measurement of matter or weak lensing power spectrum in the
same survey region, in order to suppress or correct for the super-sample
variance contamination -- the Gaussianization method of power spectrum
measurement (see Section~\ref{sec:wl}). Massive halos with $M\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}
10^{14}M_\odot$ are relatively easy to identify in a survey region,
e.g., from a concentration of member galaxies in the small spatial region
or $X$-ray and SZ observations if available. The Gaussianization can be
done by subtracting the 1-halo term power spectrum contribution,
weighted with the observed number counts of massive halos, from the
power spectrum. The method requires a theory template of the average
mass profile of the massive halos. In the paper, we can use the NFW profile
based on $N$-body simulations. If we had survey data to use with this
method,
we could
use the stacked lensing method
\citep{OguriTakada:11,Okabeetal:10,Okabeetal:13} to directly estimate
the average mass profile around such massive halos from the data.
This subtraction automatically corrects for the super-sample variance,
by using the observed number counts of massive halos that are affected
by super-survey modes. We showed that the weak lensing power spectrum
subtracting the 1-halo term can improve the information content, almost
recovering the full information content in a Gaussian field that should
have been in the initial density field as does the CMB field
(Fig.~\ref{fig:sn_wosn}). If we can measure the number of halo
with $M\ge 1$ or $3\times
10^{14}M_\odot/h$, then the increase in the information content
can be up to a factor of 2 or 1.4 if the angular power spectrum is used
up to $l_{\rm max}\simeq 2000$. This is
equivalent to a factor 2 or 4 increase is survey area. A survey having a larger
number density of galaxies, such as $\bar{n}_g=20$--100~arcmin$^{-2}$,
has a greater benefit from the Gaussianization method; the power
spectrum measurement is otherwise limited by the shape noise
contamination (Figs.~\ref{fig:sn_ng100} and \ref{fig:sn_ng20}).
The Gaussianization method suppressing the 1-halo term contamination in
the power spectrum measurement has an additional practical advantage.
Massive halos are a source of nonlinear clustering, and
the matter distribution inside massive halos is in the deeply
nonlinear regime and is affected by complex baryonic physics that is
difficult to accurately model from first principles
\citep{HutererTakada:05,Sembolonietal:12,Zentneretal:13}. Thus our
method can mask out the contribution arising from the highly-complex
nonlinear physics in a power spectrum measurement, and then allows for
the use of the cleaned lensing power spectrum to do cosmology
\citep[see also][for the similar-idea based
method]{Baldaufetal:10,Mandelbaumetal:12}.
The method can almost recover the Gaussian information, and we therefore
need not to further measure the higher-order functions of the weak
lensing field to extract the full information of weak lensing.
In this paper, we consider a method of subtracting a theory (or the
data-calibrated) template of the 1-halo term from the measured power
spectrum. An alternative approach would be to subtract the 1-halo
term contribution cluster by cluster in the two-dimensional shear map. For each halo
region, one can assume an expected shear field around the halo, subtract
the contribution from the measured shear field, and then measure the
power spectrum of the modified shear field. This method may have a
practical advantage in that it can properly take into account variations
in the expected shear field for each halo region. However, our method
in this paper almost recovers the Gaussian information content at
angular scales of interest, and therefore the 1D and 2D based methods
would be almost equivalent -- in other words, there is no significant
contributions arising from
the higher-order moments of the shear field around each halo.
However, the results we have shown in this paper are based on several
simplified assumptions. First, we assumed that we can select all massive
halos with masses greater than a sharp mass threshold in the survey
region. In reality, halo mass needs to be inferred from observables,
which therefore involves an unavoidable uncertainty in relating the
observables to halo masses -- scatters and bias in the halo-mass proxy
relation. An imperfect knowledge of the halo mass proxy causes an
uncertainty in the use of massive halos for cosmology.
The stacked lensing of sampled halos divided in halo observable bins can
also be used to calibrate the cluster-mass proxy relation, as studied in
\citet{OguriTakada:11}. In addition, we have ignored possible
systematic errors inherent in weak lensing measurements such as
photometric redshift errors and imperfect shape measurement
\citep[][]{Hutereretal:06,Nishizawaetal:10}. Hence we
need to further carefully study how the Gaussianization method in this
paper can be applied in the presence of the systematic errors. In this
paper, we ignored the super-sample variance in the weakly nonlinear
regime, which can be derived based on the perturbation theory
\citep{TakadaHu:13}. The perturbation theory version of the super-sample
variance is not significant compared to other non-Gaussian errors at
scales of interest, as studied in \citet{TakadaJain:09}, but this effect
also needs to be taken into account for an actual application.
The formulation developed in this paper would offer various
applications. Our method is based on the fact that all large-scale
structure probes, drawn from the same survey volume, arise from the same
underlying matter distribution and therefore are correlated with each
other through the super-sample variance effect. Ideally, we want to
develop a theory to describe the joint likelihood function of all the
observables in order to extract or reconstruct the full information of
the underlying matter field or equivalently the information of the
initial Gaussian field. Since the super-survey modes are not a direct
observable, we need to properly taken into account the super-sample
variance contribution. Our results suggest that the observed number
counts of massive halos in a given survey volume can be used to
``self-calibrate'' the super-sample variance effect on the power
spectrum measurement or other large-scale structure probes
in the nonlinear regime.
Our method can be easily extended to the higher-order
functions of matter or weak lensing field and also to weak lensing
tomography. These are our future work and will be presented elsewhere.
\bigskip
\section*{Acknowledgments}
We thank Wayne Hu, Bhuvnesh Jain, Issha Kayo, Elisabeth Krause, Tsz Yan
Lam, Roland de Putter, and Emmanuel Schaan for useful discussion and
valuable comments. We also thank Masanori Sato for providing us with
the ray-tracing simulation data and the halo catalogs used in this work.
MT greatly thanks Department of Astrophysical Sciences, Princeton
University for its warm hospitality during his visit, where this work
was initiated.
MT also thanks the Aspen Center for Physics and the NSF Grant \#1066293
and Institut f\"ur Theoretische Physik,
Universit\"at Heidelberg, for their warm hospitality during his visit,
where this work was partly done. DNS acknowledges support from the NASA AST theory program
and the US Euclid Science Team. This work is in part
supported in part by Grant-in-Aid for Scientific Research from the JSPS
Promotion of Science (No. 23340061),
by Grant-in-Aid for Scientific Research on
Priority Areas No. 467 ``Probing the Dark Energy through an Extremely
Wide \& Deep Survey with Subaru Telescope'', by World Premier
International Research Center Initiative (WPI Initiative), MEXT, Japan,
by the FIRST program ``Subaru Measurements of Images and Redshifts
(SuMIRe)'', CSTP, Japan, and by the exchange program between JSPS and
DFG.
\bibliographystyle{mn2e} |
3,212,635,537,492 | arxiv | \section{Introduction}
Air absorption is an important source of attenuation in room acoustics and its emulation is
critical to achieving realistic room acoustic simulations~\cite{vorlander2020auralization}. In
general, air attenuation is a frequency-dependent and distance-based effect caused by
viscothermal effects and relaxation processes~\cite{pierce2019acousticsChap9}. While these
physical processes may be modelled directly with wave-based
models~\cite{wochner2005atmospheric,jimenez2016time,hamilton2020viscothermal}, the effect of air
absorption is typically modelled with a set of digital
filters~\cite{moorer1979reverberation,jot1991digital,huopaniemi1997modeling,savioja1999creating,schroder2011physically,saarelma2018challenges,kates2020airabsorption}.
Filter approaches have the potential to accurately reproduce air attenuation, but a common
underlying problem is that filters must be tuned to accurately apply air attenuation. This
comes with a trade-off between simplicity and accuracy -- ranging from, e.g.,
octave-band approaches~\cite{scheibler2018pyroomacoustics} to optimised IIR
filters~\cite{kates2020airabsorption} and window-method FIR
filters~\cite{saarelma2018challenges}.
Recently, a more physically-motivated filtering approach was proposed~\cite{hamilton2021dafx},
which is based on an approximate Green's function to the viscothermal wave equation and
side-steps many of the issues of pre-existing filter approaches. However, that approximate
Green's function approach is best suited to controlled indoor air conditions where power-law
attenuation is
sufficient for audible frequencies. The aim of this paper is to present a more general method
that allows for post-processing a pre-computed room impulse response (RIR) with a complete air
attenuation profile (i.e., classical power-law absorption plus two relaxation
effects~\cite{pierce2019acousticsChap9,iso19939613,bass1995atmospheric}) that can be
used for all air conditions (temperature and humidity). This is particularly important when
simulating partly-outdoor spaces, such as, e.g., medieval cathedrals~\cite{postma2016acoustics}
or other heritage/historic
sites~\cite{iannace2014acoustics,murphy2017acoustic,katz2020exploring,dorazio2020understanding}.
The general aim of the proposed method is to use an input signal to drive a set of
exponentially-damped plane waves, which travel a given distance such that the resulting output
can be seen as frequency- and distance-based air attenuation having been applied to the input
signal. As will be seen, this can be achieved with a system of dissipative wave equations which
can be solved accurately and efficiently by a modal time-stepping scheme.
This paper is structured as follows. Section~\ref{secback} presents background theory for air
absorption and lossy wave propagation. Section~\ref{secnum1} gives numerical schemes for lossy
wave propagation, building towards a modal scheme of interest, along with numerical evaluations. Section~\ref{secmodal} presents
the frequency-dependent modal scheme and procedure for adding air absorption to RIRs, along with
numerical experiments testing its performance against a reference solution.
Conclusions and final remarks are given in Section~\ref{secconc}.
\begin{figure}[t]
\centering
\includegraphics[scale=0.48,clip,trim=1.8cm 7cm 2cm 7cm]{abs_i3da.pdf}
\caption{Distance attenuation in dB/km as a function of frequency, for varying air conditions
(temperature in degrees Celsius and relative humidity in percent).}
\label{figabs}
\end{figure}
\section{Background\label{secback}}
\subsection{Air absorption}
Air absorption can be seen as a frequency-dependent distance-based damping on plane waves, written as:
\begin{equation}
p(t,x) = \hat{p}e^{-\alpha(\omega)x}e^{i(k x -\omega t)}
\label{eqnplanewavespace}
\end{equation}
where $p$ is a pressure, with a complex amplitude $\hat{p}$, real wavenumber $k$, real angular frequency $\omega$,
time $t$, and position~$x$. $\alpha(\omega)$ is a
frequency-dependent attenuation coefficient in Nepers/m which represents air absorption.
Formulae for $\alpha(\omega)$ as a function of air temperature and humidity are left out for brevity but may be found
in~\cite{iso19939613,bass1995atmospheric}. See Fig.~\ref{figabs} for typical behaviour.
For the purpose of this study it is sufficient to characterise this absorption by the property that $\alpha(\omega)\ll \omega/c$,
such that $k\approx\omega/c$, where $c$ is the speed of sound in air, which also means that the
phase velocity can be assumed constant and equal to $c$.
\subsection{Dissipative wave equation}
In order to build up a physical system to achieve the desired result, we start by
considering the one-dimensional lossy wave equation:
\begin{equation}
\partial_t^2 p + 2\sigma \partial_t p - c^2 \partial_x^2 p = 0
\label{eqnlossywave}
\end{equation}
where $p=p(t,x)$ is our wave-variable of interest (e.g., a pressure quantity), $\partial_t^2$ and $\partial_x^2$ denote
second-order partial derivatives with respect to time and space, respectively, and $\sigma$ is a loss constant in s$^{-1}$.
A dispersion relation can be derived by considering a trial solution of the
form:
\begin{equation}
p(t,x) = \hat{p}e^{i(\hat{k} x -\omega t)} = \hat{p}e^{-\alpha x}e^{i(k x -\omega t)}
\label{eqnplanealphaconst}
\end{equation}
with complex wavenumber $\hat{k}=k+i \alpha$, where $\alpha$ is a constant (i.e., so far
dissipation is not frequency-dependent).
By insertion into~\eqref{eqnlossywave} we get:
\begin{equation}
\hat{k}^2 = \frac{\omega^2}{c^2} + \frac{2i\sigma\omega}{c^2} \quad \Rightarrow \quad \hat{k} = \frac{\omega}{c}\sqrt{\left(1+ \frac{2i\sigma}{\omega}\right)}
\label{eqnlossydisp}
\end{equation}
We assume $\sigma\ll \omega$, for which we get $k\approx \omega/c$ and the approximate square root:
\begin{equation}
\hat{k} \approx \frac{\omega}{c} + i\frac{\sigma}{c}
\label{eqnlossydispapprox}
\end{equation}
Equating this to $\hat{k}=k+i \alpha$ we recover $\alpha\approx \sigma/c$.
It will be useful to set a boundary condition for this dissipative wave system that induces
one-way travelling \mbox{(plane-)}wave solutions. One manner this can be achieved is through a forced
boundary condition~\cite{kinsler2000fundamentals}.\footnote{One-way wave
equations exist and could also be used (e.g., convection-diffusion
equations~\cite{strikwerda2004finite}). Second-order (two-way) wave
equations are preferred here for favourable properties of associated numerical schemes.}
Returning to the time-space
domain, let us consider this system on the domain $x\in[0,+\infty)$. At the left
boundary we impose the condition:
\begin{equation}
p(t,0) = u(t)
\label{eqnbc0}
\end{equation}
where $u(t)$ is some input signal used to drive the signal (with, otherwise, zero initial
conditions for $x>0$).
In the lossless case, $\sigma=0$, it is straightforward to check that this boundary
condition results in a rightward-travelling wave solution:
\begin{equation}
p(t,x) = u(t-x/c)
\label{eqntrav}
\end{equation}
When $\sigma>0$, the system is dispersive and travelling-wave solutions -- analogous
to~\eqref{eqntrav} -- are not available. However, in the case of a time-harmonic
$u(t)=\hat{u}e^{-j\omega t}$, it is straightforward to see that the solution on the domain
$x\in[0,\infty)$ takes the
form:
\begin{equation}
p(t,x) = \hat{u}e^{i(\hat{k} x -\omega t)}
\end{equation}
with $\hat{k}$ given by~\eqref{eqnlossydisp}.
Let us then assume $u(t)$ can be written
as the following sum of time-harmonic signals~$u_q(t)$:
\begin{equation}
u(t) = \sum_{q=0}^{Q-1} u_q(t)\,, \quad u_q(t) = \hat{u}_q e^{-i\omega_q t}
\label{eqnufourier}
\end{equation}
where $\omega_q$ are distinct (real) angular frequencies.
Clearly, each component $u_q(t)$ would have the associated pressure solution $p_q(t,x) =
\hat{p}_q e^{-\alpha x}e^{j(k_qx-\omega_q t)}$, and by
linearity we have the solution:
\begin{equation}
p(t,x) = \sum_{q=0}^{Q-1} p_q(t,x) = \sum_{q=0}^{Q-1} \hat{u}_q e^{-\alpha x}e^{i(k_qx-\omega_q t)}
\label{eqnsumpressures}
\end{equation}
Thus, the general solution to this system, for $x\in[0,\infty)$, may be seen as a Fourier decomposition of damped,
travelling plane waves, driven by the input signal $u(t)$.
\begin{comment}
Leading to the use of this system to add air absorption to a room impulse response (RIR),
consider a finite-duration signal $h(t)$ with duration $T'$, such that $h(t)=0$ for $t<0$ and $t\ge T_e$.
Now letting $u(t)=h(T_e-t)$, with $\sigma=0$ one finds:
\begin{equation}
p(T_e,x) = h(x/c)
\end{equation}
Thus, a finite-duration signal $h(t)$ can propagate through this system and be fully recovered
after time $T_e$, and the value of $h(t)$ can be read out at $p(T_e,x)$ for $x=ct$. In the
lossy case then we have an analogous result through the lens of a Fourier decomposition, where
the distance travelled by some value $h(t)$ will incur an attenuation based on the corresponding
distance $x=ct$.
\end{comment}
\section{Numerical schemes for lossy wave equation\label{secnum1}}
In this section we consider numerical schemes to simulate this lossy wave equation system.
Consider a grid function $p^n_l \approxeq p(nT,(l+1/2)X)$ with grid spacing $X$ and time-step
$T$, chosen to be
$T=X/c$. Furthermore let $u^{n}=u((n-1/2) T)$. In terms of these discrete functions, the lossless
travelling wave solution~\eqref{eqntrav} would translate to:
\begin{equation}
\label{eqnwavedisc}
p^{n}_l = u^{n-l}
\end{equation}
\subsection{FDTD schemes}
Although the intention is to derive and use a modal scheme, it will be informative to first discuss a
simple FDTD scheme. The simplest finite-difference time-domain (FDTD) scheme in this
setting~\cite{bilbao2009numerical} would be:
\begin{equation}
\label{eqnfdtd}
p^{n+1}_l = \frac{1}{1+\sigma T }\left(p^n_{l+1} + p^n_{l-1} +(\sigma T-1) p^{n-1}_{l}\right)
\end{equation}
This scheme is known to be stable for $\sigma\geq 0$~\cite{bilbao2009numerical}. To implement the boundary condition
applied at $l=0$, we can simply force the value at $l=0$:
\begin{equation}
\label{eqnbc1}
p^{n+1}_0 = u^{n+1}
\end{equation}
The above can be seen as a ``hard source'' -- as opposed to a ``soft source'', which would be
a forcing term that additive to a pressure
update~\cite{sheaffer2014sources,botts2014sources}.
In the lossless case ($\sigma=0$), this scheme permits the solution~\eqref{eqnwavedisc} and is thus
\emph{exact} (this can be verified by hand). This implies the scheme~\eqref{eqnfdtd} is stable
in the lossless case. It is expected that with the inclusion of the loss term $2\sigma \partial_t p$ --
discretised here using the trapezoidal rule -- stable is maintained under~\eqref{eqnbc1},
although a stability analysis with the ``hard source'' boundary is not immediately available.
A close alternative boundary condition featuring a ``soft source'' is also available as:
\begin{equation}
\label{eqnbc2}
p^{n+1}_0 = \frac{1}{1+\sigma T}\left(p^n_{1} + p^n_{0} +(\sigma T-1) p^{n-1}_{0}\right) + F^n
\end{equation}
where the ``soft source'' forcing term is denoted $F^n$:
\begin{equation}
\label{eqnforce}
F^n = \frac{1+\sigma T/2}{1+\sigma T} u^{n+1} - \frac{1-\sigma T/2}{1+\sigma T} u^n
\end{equation}
The derivation for this update appears in the Appendix.
It is straightforward to check that this scheme is also exact in the lossless case.
Furthermore, a discrete
energy balance maybe be obtained for this scheme (with $\sigma\geq 0$), from which stability can
be proven (this is left out for brevity).
In the presence of loss ($\sigma>0$) these schemes are no longer exact, but they remain
consistent with their underlying models.\footnote{This is primarily due to the
fact that the introduction of loss in~\eqref{eqnfdtd} upsets a delicate balance of approximation errors in space and
time from the lossless case~\cite{bilbao2009numerical}.}
\subsection{Semi-discrete modal schemes}
With the target of simulating the general case of frequency-dependent dissipation, it will be
convenient to consider modal schemes. To start we consider a semi-discrete modal scheme for
this dissipative wave problem. For this we take the view that the solution is composed of modes
(as per~\eqref{eqnsumpressures}) of the form:
\begin{equation}
p_q(t,x) = P_q(t)\cos(k_qx)
\end{equation}
with $P_q(t)$ unknown.
Inserting this trial solution into our PDE gives:
\begin{equation}
\ddot{P}_q(t) + 2\sigma \dot{P}_q(t) - c^2k_q^2 P_q(t) = 0
\end{equation}
This ODE system ($q=0,\dots,Q-1$) can be solved with leapfrog integration, on a temporal-grid function $P^n_q \approxeq
P_q(nT)$, or it can be solved with an
``exact'' recursion for the damped harmonic oscillator~\cite{cieslinski2006simulations,bilbao2009numerical,botts2015extension}:
\begin{equation}
P_q^{n+1} = 2e^{-\sigma T}\cos(\omega_q T)P_q^{n} - e^{-2\sigma T}P_q^{n-1}
\label{eqnmodalupdate}
\end{equation}
where $\omega_q = \sqrt{c^2 k_q^2 + \sigma^2}$. This recursion is ``exact'' for solutions of
the form $P^n_q = \hat{p}_q e^{-j\hat{\omega}_qnT}$, which can be verified by insertion into the
above. In this semi-discrete time-space domain, the numerical solution would then be:
\begin{equation}
p(nT,x) = \sum_{q=0}^{Q-1} P^n_q \cos(k_q x)
\end{equation}
\subsection{Fully-discrete modal schemes}
At this point we consider a fully-discrete modal scheme for our wave problem. For practical
purposes we must truncate the domain of interest to some length $L_x$, such that $x\in[0,L_x]$.
For our purposes, it is sufficient to assume $L_x = cT_d$ where $T_d$ is a simulation duration
of interest (this will be set to the duration of our pre-simulated input room impulse response).
We consider a grid function $p^n_l\approxeq p(nT,(l+1/2)X)$, with $l=0,\dots,N_x-1$, with $N_x =
\lceil L_x/X \rceil$, and $T=X/c$. For the modes, we choose the orthonormal cosine basis:
\begin{equation}
\Phi_{q,l} = a_q\cos\left( \frac{q\pi(l+1/2)}{N_x}\right)
\label{eqncosinePhi}
\end{equation}
where $a_0 = \sqrt{\frac{1}{N_x}}$ and $a_q = \sqrt{\frac{2}{N_x}}$ for $q>0$.
As such, the solution is now assumed to take the form:
\begin{equation}
p^n_l = \sum_{q=0}^{Q-1} P^n_q \Phi_{q,l}
\end{equation}
with $Q=N_x$.
It is insightful to write the full solution in matrix form. Let us define:
\begin{subequations}
\begin{align}
\mathbf{p}^n &= \left[p^n_0,p^n_1,\dots,p^n_{N_x-1}\right]^{\mathrm{T}}
\\
\mathbf{P}^n &= \left[P^n_0,P^n_1,\dots,P^n_{Q-1}\right]^{\mathrm{T}}
\\
\mathbf{\Phi}_{q} &= \left[\Phi_{q,0},\Phi_{q,1},\dots,\Phi_{q,N_x-1}\right]^{\mathrm{T}}
\\
\mathbf{V} &= \left[\mathbf{\Phi}_{0} | \mathbf{\Phi}_{1}| \dots |\mathbf{\Phi}_{Q-1}\right]
\end{align}
\end{subequations}
where $\mathbf{V}$ is a $N_x \times N_x$ matrix, and $\mathbf{p}^n$, $\mathbf{P}^n$ and
$\mathbf{\Phi}_{q}$ are $N_x\times 1$ vectors (and ``$^\mathrm{T}$'' denotes transposition),
with $Q=N_x$.
With these we have the ability to transform between time-space and modal domains via the matrix-vector products:
\begin{equation}
\mathbf{P}^n= \mathbf{V}^\mathrm{T} \mathbf{p}^n
\,,\qquad
\mathbf{p}^n= \mathbf{V} \mathbf{P}^n
\end{equation}
and we note that $\mathbf{V}\mathbf{V}^\mathrm{T} = \mathbf{I}$, where $\mathbf{I}$ is an
identity matrix. These matrix-vector products represent the Discrete Cosine Transform (DCT-II)
and its inverse~\cite{strang1999discrete}, respectively, which can be computed efficiently using
a FFT-based algorithm~\cite{vanloan1992computational}.
The fully-discrete analogue to the modal update~\eqref{eqnmodalupdate} is now:
\begin{equation}
\mathbf{P}^{n+1} = \mathbf{A}\circ\mathbf{P}^{n} - \mathbf{B}\circ\mathbf{P}^{n-1}
\end{equation}
where ``$\circ$'' denotes an element-wise product, and
\begin{subequations}
\begin{align}
\mathbf{A} &= 2\left[e^{-\sigma T}\cos(\omega_0 T), \dots,e^{-\sigma T}\cos(\omega_{Q-1}T)\right]^{\mathrm{T}}
\\
\mathbf{B} &= \left[e^{-2\sigma T}, \dots,e^{-2\sigma T}\right]^{\mathrm{T}}
\end{align}
\end{subequations}
Next we consider the boundary condition~\eqref{eqnbc0}, which we would like to
implement in a ``hard'' manner analogous to~\eqref{eqnbc1}. In order to derive this, let us
introduce an intermediate grid function $\tilde{p}_l^{n}$, for which
$\tilde{p}_l^{n}=p_l^{n}$ only when $l>0$.
Let us also define:
\begin{subequations}
\begin{align}
\tilde{\mathbf{p}}^n &= \left[\tilde{p}^n_0,\tilde{p}^n_1,\dots,\tilde{p}^n_{N_x-1}\right]^{\mathrm{T}}
\\
\tilde{\mathbf{P}}^n &= \mathbf{V}^\mathrm{T} \tilde{\mathbf{p}}^n
\\
\boldsymbol{\delta} &= \left[1,0,\dots,0\right]^{\mathrm{T}}
\end{align}
\end{subequations}
i.e., $\tilde{\mathbf{p}}^n$ is the vector form of $\tilde{p}_l^{n}$ and $\tilde{\mathbf{P}}^n$ its modal form, and
$\boldsymbol{\delta}$ is a Kronecker delta in vector form.
We would like to express $p_l^{n+1}$ as $\tilde{p}_l^{n+1}$ plus a correction, such that~\eqref{eqnbc1} is
satisfied. Clearly, this would be as simple as setting:
\begin{equation}
p_0^{n+1} = \tilde{p}_0^{n+1} + (u^{n+1}-\tilde{p}_0^{n+1})
\label{eqncorrectionFDTD}
\end{equation}
This could also be written as:
\begin{equation}
\mathbf{p}^{n+1} = \tilde{\mathbf{p}}^{n+1} + \boldsymbol{\delta}(u^{n+1}-\boldsymbol{\delta}^{\mathrm{T}}\tilde{\mathbf{p}}^{n+1})
\label{eqncorrect0}
\end{equation}
Left-multiplying by $\mathbf{V}^\mathrm{T}$, we have in the
modal domain:
\begin{equation}
\mathbf{P}^{n+1} = \tilde{\mathbf{P}}^{n+1} + \mathbf{V}^{\mathrm{T}}\boldsymbol{\delta}\left(u^{n+1} - \boldsymbol{\delta}^{\mathrm{T}}\mathbf{V}\tilde{\mathbf{P}}^{n+1}\right)
\end{equation}
Let us now define:
\begin{equation}
\boldsymbol{\phi} = \mathbf{V}^{\mathrm{T}}\boldsymbol{\delta}
= \left[\Phi_{0,0},\Phi_{1,0},\dots,\Phi_{Q-1,0} \right]^{\mathrm{T}}
\end{equation}
We can then propose the following scheme update with a ``hard source'' correction (forced-boundary):
\begin{subequations}
\label{eqnmodalupdatematrix}
\begin{align}
\tilde{\mathbf{P}}^{n+1} &= \mathbf{A}\circ\mathbf{P}^{n} - \mathbf{B}\circ\mathbf{P}^{n-1}
\\
\mathbf{P}^{n+1} &= \tilde{\mathbf{P}}^{n+1} + \boldsymbol{\phi}\left(u^{n+1} - \boldsymbol{\phi}^{\mathrm{T}}\tilde{\mathbf{P}}^{n+1}\right)
\end{align}
\end{subequations}
This update is exact for the lossless case since it is, in fact, the exact FDTD scheme
(\eqref{eqnfdtd} and~\eqref{eqnbc1} via~\eqref{eqncorrectionFDTD})
transformed to a modal update.\footnote{This will be not be shown, but the proof is
carried out using the known eigenvalues/eigenvectors of the corresponding Laplacian
matrix (see, e.g.,~\cite{strang1999discrete}) or by Taylor series expansion (see,
e.g.,~\cite{etgen1989accurate}).} It is not expected to be exact in the presence of loss since
the hard boundary is applied at $x=X/2$ and not $x=0$, which means there is a small distance
over which dissipation is not incurred (this error is negligible for practical purposes).
Alternatively, one can apply a simpler ``soft source'' update, for which the modal update becomes:
\begin{equation}
\label{eqnmodalsoft}
\mathbf{P}^{n+1} = \mathbf{A}\circ\mathbf{P}^{n} - \mathbf{B}\circ\mathbf{P}^{n-1}
+ F^{n}\boldsymbol{\phi}
\end{equation}
where $F^{n}$ is the forcing term~\eqref{eqnforce}.\footnote{It may be noted that~\eqref{eqnmodalsoft} is similar to the modal time-stepping update
presented in~\cite{botts2015extension} -- particular in the choices of time-stepping update and of a cosine
modal basis. The numerical modal method in~\cite{botts2015extension}, paired with domain decomposition, was
ultimately used for 3-D wave simulations with damping that similarly mimics air attenuation.
However, the scheme
here is limited to one spatial dimension, and the
``soft'' forcing term given here is of a different nature than that in~\cite{botts2015extension}
-- derived here specifically to satisfy the boundary condition~\eqref{eqnbc0}.}
This update is also exact in the lossless
case, but it is expected to be less accurate than~\eqref{eqnmodalupdatematrix} when $\sigma>0$.
However, this update has the advantage that stability is straightforward to establish through
Z-transform analysis.
\begin{figure}[t]
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[scale=0.53,clip,trim=1cm 0cm 1cm 0cm]{figs/i3da_fig2_1.000_0.010.png}
\caption{$t=10$\,ms}
\end{subfigure}
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[scale=0.53,clip,trim=1cm 0cm 1cm 0cm]{figs/i3da_fig2_1.000_0.040.png}
\caption{$t=40$\,ms}
\end{subfigure}
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[scale=0.53,clip,trim=1cm 0cm 1cm 0cm]{figs/i3da_fig2_1.000_0.060_insert.png}
\caption{$t=60$\,ms}
\label{figsim1c}
\end{subfigure}
\caption{Examples of simulated one-way dissipative wave propagation at different time
instances $t$, using raised cosine input $u(t)$ with 60dB decay time
$6\ln(10)\sigma^{-1}=1.0$\,s and speed of sound $c=343$\,m/s. All schemes using
time-step $T=(48\textrm{\,kHz})^{-1}$. The last image displays an overlay to show detail
in wake behaviour.}
\label{figsim1}
\end{figure}
\begin{figure*}[t]
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[scale=0.40,clip,trim=0cm 0cm 1cm 0cm]{figs/i3da_fig3_10.000_20.000_0.100.png}
\caption{$t=0.1$\,s}
\end{subfigure}
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[scale=0.40,clip,trim=0cm 0cm 1cm 0cm]{figs/i3da_fig3_10.000_20.000_0.200.png}
\caption{$t=0.2$\,s}
\end{subfigure}
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[scale=0.40,clip,trim=0cm 0cm 1cm 0cm]{figs/i3da_fig3_10.000_20.000_0.500.png}
\caption{$t=0.5$\,s}
\end{subfigure}
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[scale=0.40,clip,trim=0cm 0cm 1cm 0cm]{figs/i3da_fig3_10.000_20.000_1.000.png}
\caption{$t=1.0$\,s}
\end{subfigure}
\caption{Examples of simulated one-way dissipative wave propagation with modal schemes using
``soft'' and ``hard'' boundary conditions (and differences), at different time
instances $t$, using Kronecker delta input ($u^n=[1,0,\dots,0]$) with decay constants
$\sigma_q$ set from air
attenuation model for air temperature $10^\circ$C and 20\% relative humidity ($c=337$\,m/s).
Modal time-stepping using time-step $T=(48\textrm{\,kHz})^{-1}$.}
\label{figsim2}
\end{figure*}
\subsection{Validation of frequency-independent dissipative schemes}
A simple test is conducted to evaluate the numerical schemes presented thus far. The case of
zero loss will not be tested (for brevity) since exactness may be verified by hand. As for
the lossy case $\sigma>0$,
Fig.~\ref{figsim1} displays simulated one-way dissipative wave propagation with a raised-cosine
input signal $u(t)$ with dissipation set for a 60dB decay time of $6\ln(10)\sigma^{-1}=1.0$\,s.
Displayed in the figure are $p(t,x)$ simulated using the FDTD schemes with ``hard'' and
``soft'' boundary updates, as well as using the corresponding modal schemes. It can be seen
that the four simulations display similar behaviour, but differences are apparent in the wake of
the travelling wave (see zoom overlay in Fig.~\ref{figsim1c}). For example, the ``soft-FDTD''
does not satisfy the boundary condition with the chosen raised-cosine input
(one can note a slight DC offset at $x=0$), whereas the ``hard-FDTD'' output displays a slope in
the wake of the wave which is absent from modal outputs. In regards to the ``soft'' updates, it
should be noted that in this case the assumption $\sigma\ll \omega$ is not satisfied for low
frequencies.
For the remainder of this paper we will consider only modal methods, but it is worth mentioning
that what follows could be achieved with FDTD methods using a more general -- and
indeed more ``physically valid'' -- second-order wave equation with air absorption processes
included (after~\cite{hamilton2020viscothermal}). However, in that case we would be faced with
numerical dissipation errors -- even with a domain limited to one spatial
dimension\footnote{Numerical phase velocity errors will be minimal in such 1-D FDTD schemes (and
non-existent in the lossless case~\cite{hildebrand1968finitedifference,ames1977numerical}), but
numerical \emph{dissipation} errors may be significant in high frequencies when discretising loss terms using
conventional means (trapezoid rule) -- e.g., as reported
in~\cite{hamilton2014visc,hamilton2020viscothermal} with regards to the problem of air
absorption. Without resorting to higher-accurate schemes (with less favourable stability
properties), such dissipation errors can only be overcome with oversampling, which would
increase computational costs. An analogous issue has been reported in the context of FDTD
modelling of musical strings~\cite{desvages2018physical}, where a possible solution is to
optimise parameters for fictitious loss terms in the discrete domain to minimize numerical
dissipation errors~\cite{desvages2019optimised}.} -- and this would necessitate grid refinement;
whereas the ``exact'' modal schemes are essentially free from dissipation errors (and are,
accordingly, more efficient for this specific application).
\section{Modal scheme for simulating air absorption\label{secmodal}}
In this section we extend the modal scheme to the frequency-dependent case to incorporate air
attenuation. Since each mode in
the system is independent the modal damping coefficients can be set individually for each
mode (as in, e.g.,~\cite{botts2015extension,ducceschi2016plate}). It suffices then to replace
$\sigma$ by $\sigma_q=c\alpha(\omega_q)$. Note that with realistic air absorption (see, e.g.,
Fig.~\ref{figabs}), it can be assumed that $\alpha(\omega)\ll \omega/c$ for all frequencies of
interest (audible frequencies).
In terms of a PDE system, we have then $Q$ lossy wave equations ($q=0,\dots,Q-1$), each with its own damping
coefficient:
\begin{subequations}
\label{eqnsyseries}
\begin{align}
\partial_t^2 p_q + 2\sigma_q \partial_t p_q - c^2 \partial_x^2 p_q = 0
\\
p_q(t,0) = u_q(t)
\end{align}
\end{subequations}
We take the associated
solution:
\begin{equation}
p(t,x) = \sum_{q=0}^{Q-1} p_q(t,x) = \sum_{q=0}^{Q-1} \hat{u}_q e^{-\alpha_q x}e^{i(k_qx-\omega_q t)}
\end{equation}
where $ck_q= \sqrt{\omega_q^2 + \sigma_q^2}$, and
where $u(t)$ is related to $\hat{u}_q e^{-i\omega_q t}$ through~\eqref{eqnufourier}.
Our fully-discrete modal scheme with ``hard'' boundary keeps the form~\eqref{eqnmodalupdatematrix},
but now with
\begin{subequations}
\begin{align}
\mathbf{A} &= \left[2e^{-\sigma_0 T}\cos(\omega_0 T), \dots,2e^{-\sigma_{Q-1} T}\cos(\omega_{Q-1}T)\right]^{\mathrm{T}}
\\
\mathbf{B} &= \left[e^{-2\sigma_0 T}, \dots,e^{-2\sigma_{Q-1} T}\right]^{\mathrm{T}}
\end{align}
\end{subequations}
Meanwhile, the modal scheme with the ``soft source'' boundary now takes the form:
\begin{equation}
\label{eqnmodalsoftq}
\mathbf{P}^{n+1} = \mathbf{A}\circ\mathbf{P}^{n} - \mathbf{B}\circ\mathbf{P}^{n-1}
+\mathbf{F}^{n}\circ\boldsymbol{\phi}
\end{equation}
with $\mathbf{A}$ and $\mathbf{B}$ changed as noted previously, and the forcing term taking the
\emph{vector} form:
\begin{subequations}
\begin{align}
\mathbf{F}^n &= \left[F^n_0,F^n_1,\dots,F^n_{Q-1}\right]^{\mathrm{T}}
\\
F_q^n&=\frac{1+\sigma_q T/2}{1+\sigma_q T} u^{n+1} - \frac{1-\sigma_q T/2}{1+\sigma_q T} u^n
\end{align}
\end{subequations}
\subsection{Validation of modal schemes and comparisons with filter methods}
Numerical tests are presented in order to compare and validate the frequency-dependent modal
schemes in the context of air absorption. For the following tests, air conditions of
$10^\circ$C temperature and 20\% relative humidity are chosen so that air absorption
deviates significantly from classical power-law attenuation in the audible frequencies (see
Fig.~\ref{figabs}).
Fig.~\ref{figsim2} shows simulated one-way dissipative wave propagation with a
Kronecker delta input with dissipation set to mimic the desired air attenuation. Displayed in
the figure are $p(t,x)$ simulated using the modal schemes with ``hard'' and ``soft'' boundary
update, and differences between them. It can be seen that the schemes return similar results
with small differences.
\begin{figure*}[ht]
\centering
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[scale=0.34,clip,trim=0cm 0cm 1cm 0cm]{figs/i3da_fig4_10.000_20.000_1.000.png}
\caption{Modal approach}
\label{figsim3a}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[scale=0.34,clip,trim=0cm 0cm 1cm 0cm]{figs/i3da_fig5_butter-13.png}
\caption{Octave-band filter approach}
\label{figsim3b}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[scale=0.34,clip,trim=0cm 0cm 1cm 0cm]{figs/i3da_fig5_kates.png}
\caption{Time-varying optimised lowpass-filter approach}
\label{figsim3c}
\end{subfigure}
\caption{Simulated air attenuation using modal schemes and digital-filter approaches, and
target air attenuation for air temperature $10^\circ$C and 20\% relative humidity
($c=337$\,m/s). Modal time-stepping using time-step $T=(48\textrm{\,kHz})^{-1}$. Filter
methods use 48\,kHz sample rates. Octave-band filters are of third-order Butterworth type,
and optimised lowpass-filter approach is using the method presented in~\cite{kates2020airabsorption}.}
\end{figure*}
To validate the modal schemes in their reproduction of a target air attenuation, we can simply
measure the attenuation at a given time instant (corresponding to a distance travelled) and
compare to the ``analytic'' air absorption model (from~\cite{iso19939613}). For the same
conditions above, Fig.~\ref{figsim3a} shows the attenuation per distance (in km) simulated,
calculated at time instant $t=1.0$\,s, along with the target air attenuation curve. It can be
seen that agreement with the target air attenuation is excellent (as expected for these modal
schemes). Results are identical for other time instances and are thus left out for
brevity.\footnote{However, for very large distances dissipation can be sufficient to exceed the
relative accuracy of finite-precision, and -- only in those cases -- simulated measurements of
air attenuation will not return accurate results.}
In contrast, a more conventional filter approach is plotted in Fig.~\ref{figsim3b}, in which
third-order Butterworth octave-band filters are used as a analysis/synthesis filter bank used to
process a 48\,kHz-sample-rate Kronecker delta input with distance-based exponential damping applied to mimic air
absorption (sampled at 11 octave-band centre frequencies) -- as may be found in traditional
geometrical acoustics room simulations such as, e.g.,~\cite{scheibler2018pyroomacoustics}.
Ripples are apparent in the attenuation simulated by the filterbank approach, as one might
expect. This can be improved using a filterbank of 1/3-octave band filters (as in,
e.g.,~\cite{schroder2011physically}), which, if chosen well, leads to more tightly-fitting
ripples, as seen in the same figure (here, using 32 1/3-octave-band filters of third-order
Butterworth type). Another potential option would be the use of FIR partition-of-unity
filters~\cite{antoni2010orthogonal}.
Additionally, results from an alternative, recently-presented~\cite{kates2020airabsorption},
optimised IIR filter approach is plotted in Fig.~\ref{figsim3c}. This method is based on a set of triple-cascaded
first-order lowpass filters optimised for attenuation tuned to distances corresponding to each sampling
instant~\cite{kates2020airabsorption}. A Kronecker delta is filtered to measure
simulated attenuation tuned to different time instants (distances). In the figure it can be seen that this approach does well to reproduce
attenuation in high frequencies (where it is most significant), but errors appear in low
frequencies. On the other hand, one can note that simulated attenuation curves vary as a
function of time instant, and for longer times (higher attenuation) attenuation
filters deviate more from the target attenuation (in dB/km).\footnote{To produce data for the
optimised lowpass filter approach~~\cite{kates2020airabsorption}, Matlab code provided as
supplementary material to~\cite{kates2020airabsorption} was used, but reconfigured to optimise over
20\,Hz--20\,kHz with approximately 40k frequency samples.}
\subsection{Adding air absorption to simulated RIRs}
It remains to detail the procedure to add air absorption to a
pre-simulated RIR (simulated without air absorption) using the presented modal schemes. Consider then a finite-duration discrete
RIR signal $h[n]$ with sample rate $1/T$ (e.g., 48\,kHz), with $N_t$ samples, indexed by
$n'=0,\dots,N_t-1$. Of course, $h[n]$ can be seen as a linear combination of $N_t$ shifted and
scaled Kronecker deltas. We have shown that the distance-based attenuation is accurately
reproduced for a Kronecker delta input, so it follows that we can simply apply the same procedure to
the signal itself to process each individual sample. Since the system is linear and
time-invariant, all such samples can be processed simultaneously (i.e. in a single modal
simulation).
We must ensure that correct distance attenuation is applied to each sample
$n$, where that distance would be simply $d=c(n+1/2)T$ (in m).\footnote{Here we assume a constant
speed of sound $c$, but it is important in practice that any ``pre-delay'' in the signal be
removed (or compensated for in the application of attenuation).} For this we must \emph{time-reverse} the
input RIR as we feed it into the system (via the left-boundary condition);
i.e., we can set the forcing signal to our ODE system to be:
\begin{equation}
u^n = h[N_t-1-n]
\end{equation}
We can then run the forced modal schemes with $Q=N_t$ modes for $N_t$ time-steps, starting from zero initial conditions
($\mathbf{P}^{-1}=\mathbf{P}^{-2}=0$). The output RIR with air attenuation added, which we
will denote $h'[n]$, can then be read from the final state of our wave simulation as follows:
\begin{equation}
h'[n] = p^{N_t-1}_n\,, \quad n=0,\dots,N_t-1
\end{equation}
where $p^{N_t-1}_n$ are simply the elements of $\mathbf{p}^{N_t-1} = \mathbf{V}\mathbf{P}^{N_t-1}$.
Note that
the entire simulation can be run in the modal domain. Projection back to a spatial grid (with
$N_x$ points) is only needed at the end, and this can be accomplished using a fast inverse DCT-II
algorithm~\cite{vanloan1992computational}.
When losses are disabled ($\sigma_q=0$), the proposed procedure will simulate the solution~\eqref{eqnbc1} and
return $h'[n]=h[n]$, as desired. Also, it is recommended that the ``soft'' update be used for the
application of air absorption, since we have seen that ``hard'' and ``soft'' boundary
updates give similar results, and the soft update ultimately requires fewer operations.
\subsection{A reference solution}
At this point it is useful to introduce a ``reference solution'' for the problem of adding air
absorption to a RIR. Taking $h[n]$ and $h'[n]$ as a $N_t\times 1$ vectors:
\begin{subequations}
\begin{align}
\mathbf{h} &= \left[h[0],h[1],\dots,h[N_t-1]\right]^{\mathrm{T}}
\\
\mathbf{h}' &= \left[h'[0],h'[1],\dots,h'[N_t-1]\right]^{\mathrm{T}}
\end{align}
\end{subequations}
We can propose a reference solution:
\begin{equation}
\mathbf{h}' = \boldsymbol{\Omega}\mathbf{V}^\mathrm{T} \mathbf{h}
\label{eqnrefsol}
\end{equation}
where $\boldsymbol{\Omega}$ is a $N_t \times N_t$ matrix with elements at rows $q$ and columns
$l$ (with zero-based indexing) given by:
\begin{equation}
\Omega_{q,l} = e^{-\alpha_q x_l} \Phi_{q,l}
\end{equation}
where $x_l = (l+1/2)X$ and where $\Phi_{q,l}$ is given by~\eqref{eqncosinePhi}. This reference
solution is simply a decomposition of a signal onto an orthonormal cosine basis and a subsequent
projection back onto the same basis weighted by distance-based air attenuation, under the
assumption that the $l$th sample travels $x_l$ metres. This can ultimately be seen as a
limiting case of the frequency-band filterbank approach, where here $N_t$ frequency bands are
used. Note that when $\alpha_q=0$~\eqref{eqnrefsol}, $\boldsymbol{\Omega}=\mathbf{V}$,
and~\eqref{eqnrefsol} returns $\mathbf{h}'=\mathbf{h}$ (as desired). However, this is not
claimed to be an analytic solution to the dissipative wave problems seen before, nor to the more
general second-order wave systems with air absorption
processes~\cite{pierce2019acousticsChap9,hamilton2020viscothermal}; it should be seen as a
``brute-force'' application of air attenuation under the assumptions used in this study.
In terms of its compute cost, for $\alpha_q>0$ this reference solution is somewhat
impractical for the reason that $N_t$ should on the order of $10^4$--$10^5$, and only the
matrix-vector product $\mathbf{y}=\mathbf{V}^\mathrm{T}\mathbf{h}$ (DCT-II) can use a fast
FFT-based algorithm~\cite{vanloan1992computational}. The matrix-vector product
$\boldsymbol{\Omega}\mathbf{y}$ must be computed directly, and this would require $N_t^2$
evaluations of transcendental functions, and $N_t^2$ storage (if $\boldsymbol{\Omega}$ is
precomputed). More will be said about computational complexity later.
\subsection{Numerical examples}
In order to test the application of air absorption to a RIR using this modal scheme (with
``soft'' boundary update), we conduct a simple image source simulation~\cite{allen1979image}. A
room of size $7.2\times5.1\times4.3$ (in m) is chosen with a Sabine-absorption coefficient of
$0.045$, with a source and receiver at positions $(6.1,2.0,1.1)$ and $(3.3,3.1,1.3)$
respectively. Image source distances are randomly perturbed by a relative amount in
range $[-1\%,+1\%]$ in order to eliminate sweeping echoes due to perfect rectangular
symmetry~\cite{desena2015modeling} (and ultimately increasing diffuseness in the room). Air
conditions are as before (10$^\circ$C and 20\% relative humidity) with a corresponding speed of
sound $c=337$\,m/s. A RIR, $h[n]$, is simulated at 48kHz, and its spectrogram -- computed
with 1024-sample Hann-windowing and 75\% frame overlap -- is displayed in Fig.~\ref{figimage1}.
Using $h[n]$ as input to the modal scheme (using the time-reversed procedure described
previously) returns a modified RIR, $h'[n]$, whose spectrogram is displayed in
Fig.~\ref{figimage2}, where the effect of air absorption is clearly seen. The corresponding reference
solution is shown in Fig.~\ref{figimage3}. Additionally, air attenuation is added to $h[n]$ with the
overlap-add (OLA) approach (STFT processing) (after~\cite{southern2013room,saarelma2018challenges})
and shown in Fig.~\ref{figimage4}.\footnote{For OLA processing, a Hann window of length 1024
samples was used for analysis and synthesis windows, scaled for a constant unity overlap-add
envelope (with sufficient zero padding to provide perfect reconstruction in lossless case), with
air absorption applied to STFT frames by frequency-domain multiplication.} Agreements are
excellent between the three methods, at least from the point of view of spectrograms.
\begin{figure}[t]
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[scale=0.48,clip,trim=0.4cm 0cm 0.5cm 0cm]{figs/i3da_ism_1.png}
\caption{Spectrogram of $h[n]$, obtained with lossless-air image-source method}
\label{figimage1}
\end{subfigure}
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[scale=0.48,clip,trim=0.4cm 0cm 0.5cm 0cm]{figs/i3da_ism_3_soft.png}
\caption{Air attenuation added via modal scheme}
\label{figimage2}
\end{subfigure}
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[scale=0.48,clip,trim=0.4cm 0cm 0.5cm 0cm]{figs/i3da_ism_3_ref.png}
\caption{Air attenuation added with reference solution}
\label{figimage3}
\end{subfigure}
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[scale=0.48,clip,trim=0.4cm 0cm 0.5cm 0cm]{figs/i3da_ism_2_OLA.png}
\caption{Air attenuation added via STFT processing}
\label{figimage4}
\end{subfigure}
\caption{Spectrograms of room impulse response, simulated with image source method, with and
without air absorption added.}
\label{figISM}
\end{figure}
For a more objective comparison, it is useful to compute similarities between STFT frames
corresponding to the spectrograms in Fig.~\ref{figISM}. This can be accomplished with the
following geometric similarity index:
\begin{equation}
\chi[n] = \frac{\| Y_1(n,\omega)Y_2^*(n,\omega)\|_{\omega}}{\| Y_1(n,\omega)\|_{\omega} \| Y_2(n,\omega)\|_{\omega}}
\label{eqnsimilarity}
\end{equation}
Here $Y_1(n,\omega)$ and $Y_2(n,\omega)$ represent the STFTs of two discrete-time signals $y_1[n]$ and
$y_2[n]$, and the Euclidean norms are calculated across
the frequency-dimension, and $0\leq \chi[n]\leq 1$. Comparing to the reference solution, in Fig.~\ref{figismsim} we plot
$\log_{10}(1-\chi)$ for the hard-modal output, and the output from the OLA method, along with the output of the soft-modal
scheme (corresponding spectrogram not shown for brevity). It can be seen that the outputs generated are extremely
similar to the reference solution, with modal schemes performing best, generally by an order of
magnitude or more (in terms of frame similarity). Indeed, the OLA
method, at least as implemented here, would seem to perform more than well enough for practical use.
With that said, it should be remarked that this constitutes the first evaluation of the
accuracy of the OLA approach in the literature (to the author's knowledge).
\begin{figure}[t]
\centering
\includegraphics[scale=0.50,clip,trim=0.5cm 0cm 1cm 0cm]{figs/i3da_ism_4_soft_hard.png}
\caption{RIR STFT frame similarities~\eqref{eqnsimilarity} to reference solution (see Fig.~\ref{figISM}).}
\label{figismsim}
\end{figure}
\section{Conclusions and Final Remarks\label{secconc}}
A method to add general air absorption to simulated room impulse responses was provided in this
paper.
The method uses a fully-discrete modal scheme with boundary conditions chosen for one-way
dissipative wave propagation, such that accurate frequency-dependent, distance-based air
attenuation may be applied to an input RIR. FDTD schemes were used to help derive a ``soft
source'' boundary update, which was also compared to a ``hard'' forced-boundary update.
Frequency-dependent modal schemes were validated against target air attenuation curves, along
with existing filter-based approaches. An image source room simulation was used to test the
application of air attenuation to a RIR, and this was compared to air attenuation applied via
STFT-based time-frequency processing.
The method presented is advantageous for its generality (it can apply air attenuation for any air
conditions) and because it does not need tuning or optimisation as required in filter-based methods.
On the other hand, this method is more computationally demanding than filter-based methods,
which would be more applicable to real-time applications. This modal approach is better suited to
offline data generation (as needed for, e.g., auditory research~\cite{fogerty2020effect} or
speech dereverberation~\cite{naylor2010speech}). Additionally, this approach is well-suited to complement
(offline) RIR simulation with 3-D wave-based methods~\cite{hamilton2016gpuISMRA,lai2020ctk},
where modelling air absorption processes directly leads to significant increases in memory usage
(e.g., at minimum 50\% more memory than lossless wave-equation
schemes~\cite{hamilton2014visc,hamilton2020viscothermal}).
With regards to computational costs, the complexity of the presented modal is
$O(N_t^2)$ where $N_t$ is the number of samples to be computed, which is similar to a 1-D FDTD
scheme (in the lossy case~\cite{bilbao2009numerical}). On the other hand, time-frequency
processing via STFT would be $O(\frac{N_t}{H}M\log_2 M)$ where $M$ is the frame size (assumed
power of two) and $1\leq H\leq M$ is the hop size in samples. While it is possible for the OLA
method to have higher complexity than the modal approach (depending on choices of $M$ and $H$),
in practice it is expected that the complexity should be lower for STFT-based processing (usually $M\ll
N_t$) . Low-order IIR filter-based approaches would usually be $O(N_t)$, but frequency-band
approaches would reach $O(N_t^2)$ in the high-resolution limit of $N_t$ frequency bands. The
reference solution provided here can be seen as such a limiting filterbank -- and is accordingly also
$O(N_t^2)$ -- but should be less efficient than modal recursion because of the need for evaluation of
$N_t^2$ exponentially-damped-cosine matrix elements.
In regards to compute times, it must also be noted that modal updates are highly, and easily,
parallelisable (each mode can be updated in parallel), and processing a RIR with this modal
approach could be accomplished in only a few seconds on a modern desktop computer (with an
optimised multithreaded code -- see,
e.g.,~\cite{ducceschi2016plate}), even with $N_t$ on the order of $10^5$. To keep
the number of modes ($Q=N_t$) to a minimum the sampling rate of the input RIR should be not be
higher than necessary for audio purposes.
Finally, it should be mentioned that the presented approach is only applicable to static
scenes (precluding moving sources or receivers), but allows for source/receiver
directivity~\cite{bilbao2018directional,bilbao2019local}. It could also be used for outdoor
scenes provided that wind is not significant. On the other hand, for typical indoor conditions
(e.g., 20$^\circ$C, 50\% humidity), it may be sufficient to view air absorption as a simple
power-law attenuation, in which case the Green's function method recently presented
in~\cite{hamilton2021dafx} may be preferable with its $O(N_t)$ complexity and applicability
to real-time usage. In future work, the various methods of applying air attenuation to a
simulated RIR could be compared on a perceptual basis (through listening tests).
\section*{Acknowledgment}
Thanks to Michele Ducceschi, Charlotte Desvages, and James-Michael Leahy for fruitful
discussions on topics relating to numerical modelling of dissipative wave propagation.
|
3,212,635,537,493 | arxiv | \section{Introduction}
The extreme helium stars (EHes) are supergiants with peculiar chemical
composition. The atmospheres of these supergiants with effective
temperature in the range 8000 -- 35000 K are devoid of hydrogen, and are
enriched with helium, carbon, and nitrogen with respect to the
atmospheres of normal main-sequence stars. Helium is the
most abundant element in their atmospheres. There are about 21 known EHes.
There are 5 cool EHes, stars with effective temperatures 8000 to 13000 K, of
which 4 were analysed by Pandey et al. (2001). The remaining one cool EHe LSS\,3378
is analysed here.
Star No. 3378 (LSS\,3378) in the catalog of Stephenson and Sanduleak (1971)
was first identified as helium-rich B-type by Drilling (1973). Drilling
noticed the absence of Balmer absorption lines, presence of strong He\,{\sc i}
absorption lines, and the presence of C\,{\sc ii} line at 4267\AA\ in the
spectrum of LSS\,3378. Drilling also noted strong Mg\,{\sc ii} line at
4481\AA\ and Si\,{\sc ii} lines in absorption indicating a spectral class of
about B8 but, however, suggested somewhat earlier class than this because of the presence
of several weak O\,{\sc ii} lines.
Jeffery et al. (2001) reported the effective
temperature of LSS\,3378 to be about 10500 K by fixing $E_{B-V}$
using $IUE$ data and $UBV$ photometry. However, Jeffery et al. also reported that
the $IUE$ data for this star are very noisy.
Drilling et al. (1984) were the first to derive
an effective temperature for LSS\,3378, obtaining 9400$\pm$500K, using the same
procedure adopted by Jeffery et al.
R Coronae Borealis (RCB) stars, which are hydrogen-deficient F- and G-type supergiants,
overlap in their effective temperatures with cooler EHes. With this abundance
analysis of LSS\,3378, we have the chemical composition of all the 5 known
cool EHes which are closely related to RCB stars. Our abundance analysis is based on
the high resolution optical spectrum.
\section{Observations}
The spectrum of LSS\,3378 was obtained on 2002 June 19 with the 4-m Blanco telescope
and the Cassegrain echelle spectrograph at CTIO in Chile. Three exposures of
25 minutes each, spectra covering the wavelength interval 5000 -- 8200\AA\ without gaps,
were recorded at a resolving power of R = 30,000.
The Image Reduction and Analysis Facility (IRAF) software packages were used
to reduce the recorded spectra.
The Th-Ar hollow cathode lamp provided lines for wavelength calibration.
A final spectrum was obtained by co-adding these three individual wavelength calibrated
spectra from the three exposures. The maximum signal-to-noise (S/N) in the
continuum (per pixel) of the co-added spectrum is between 200 and 250 at the
middle of each echelle order.
The Na\,D lines in the spectrum of LSS 3378 are very strong and appear to
be saturated as seen in the spectra of other cool EHes: FQ\,Aqr, LS IV $-14^\circ109$,
BD $-1^\circ3438$, and LS IV $-1^\circ2$. These saturated Na\,D lines in the spectra of cool
EHes are certainly interstellar in origin. The Na\,D lines in the spectra of
cool EHes are stronger than in any other spectrum of normal star. Note that,
lines of ionized metals of the iron group are plentiful in the cool EHe's spectrum.
These lines are much stronger when compared with those observed in early A-type
and late B-type normal stars. This notable feature of the spectra of cool EHes
is attributable to the lower opacity in the atmosphere due to hydrogen deficiency.
\section{Abundance analysis - method}
\subsection{Procedure}
For the abundance analysis of LSS\,3378, same procedure is followed as
described in Pandey et al. (2001, 2004, 2006). The analysis
uses line-blanketed hydrogen-deficient model atmospheres computed
by the code STERNE (Jeffery, Woolf \& Pollacco 2001).
STERNE model atmosphere was combined with the Armagh LTE code
SPECTRUM (Jeffery, Woolf \& Pollacco 2001) to compute the equivalent width of a line
or a synthetic spectrum. In matching
a synthetic spectrum to an observed spectrum we include broadening
due to the instrumental profile, the microturbulent velocity $\xi$ and assign all
additional broadening, if any, to rotational broadening.
We use the standard rotational broadening
function $V(v\sin i,\beta)$ with
the limb darkening coefficient set at $\beta = 1.5$ (Jeffery, Woolf \& Pollacco 2001).
Observed unblended line profiles are used to obtain the projected rotational
velocity $v\sin i$. Synthetic line profile, including
the broadening due to instrumental profile, for
the adopted model atmosphere ($T_{\rm eff}$,$\log g, \xi$) and the abundance,
is found to be sharper than the observed. This extra broadening in the observed profile
is attributed to rotational broadening.
The adopted $gf$-values for C, N, O, are from Wiese, Fuhr \& Deters (1996), and
for rest of the elements are from
NIST database\footnote{http://physics.nist.gov/PhysRefData/ASD/lines\_form.html},
Kurucz's database\footnote{http://kurucz.harvard.edu}, Th\'{e}venin (1989, 1990),
and the compilations by R. E. Luck (private communication).
The Stark broadening and radiative broadening coefficients, if available, are mostly
taken from the Kurucz's database.
The data for
computing He\,{\sc i} profiles are the same as in Jeffery, Woolf \& Pollacco (2001).
Table A1 has the detailed line list used in our analysis.
\subsection{Atmospheric parameters}
The model atmospheres are defined by the effective temperature, the
surface gravity, and the chemical composition. The input composition of
He and C (the C/He ratio) is fully consistent with the C/He ratio derived from
the observed spectrum with that model; He and C abundances, particularly He,
dominate the continuous opacity. The input composition of rest of the elements
is solar with H/He fixed at 10$^{-4}$ by number.
The analysis involves the determination of effective temperature ($T_{\rm eff}$),
surface gravity ($\log g$), and microturbulent velocity ($\xi$) before estimating
the photospheric elemental abundances of the star. These parameters are determined
from the line spectrum. The microturbulent velocity $\xi$ (in km s$^{-1}$) is first
determined by the requirement that the abundance from
a set of lines of the same ion with similar excitation potentials be independent of
a line's equivalent width.
For an element represented in the
spectrum by two or more ions, imposition of ionization
equilibrium (i.e., the same abundance is required from lines of
different stages of ionization) defines a locus in the
($T_{\rm eff},\log g)$ plane.
Different pairs of ions of a common element provide
loci of very similar slope in the ($T_{\rm eff},\log g)$ plane.
An indicator yielding a locus with a contrasting slope
in the ($T_{\rm eff},\log g)$ plane is required to break the
degeneracy presented by ionization equilibria.
A potential indicator is a He\,{\sc i} line.
For stars hotter than about 10,000 K,
the He\,{\sc i} lines are less sensitive to $T_{\rm eff}$
than to $\log g$ on account of pressure broadening due to
the quadratic Stark effect.
The diffuse series lines are, in particular, useful because they are
less sensitive to the microturbulent velocity than the sharp lines.
A second indicator may be available: species represented
by lines spanning a range in excitation potential may serve as
a thermometer measuring $T_{\rm eff}$ with a weak dependence
on $\log g$.
\section{LSS\,3378 - abundance analysis results}
First to determine is the microturbulent velocity $\xi$. We adopt a model atmosphere
with $T_{\rm eff}$ = 10000 K, which is close to that found by Jeffery et al. (2001),
and adopt $\log g$ = 1.0
which is a fair assumption for these hydrogen-deficient supergiants. $\xi$ is found
to be 4.5 and 7.5 km s$^{-1}$ from Fe\,{\sc ii} and C\,{\sc i} lines, respectively.
We adopt $\xi$ = 6 km s$^{-1}$ for abundance determination, and Figure 1 illustrates
the method for obtaining $\xi$. Fe\,{\sc ii} lines used for determining $\xi$ were of
similar lower excitation potential and so was the case for C\,{\sc i} lines.
\begin{figure}
\epsfxsize=8truecm
\epsffile{lssfig1.ps}
\caption{Abundances from Fe\,{\sc ii} lines for LSS\,3378 versus their
reduced equivalent widths (log $W_{\lambda}/\lambda$).
A microturbulent velocity of $\xi \simeq 4.5$ km s$^{-1}$ is obtained from this
figure.}
\end{figure}
\begin{figure}
\epsfxsize=8truecm
\epsffile{lssfig2.ps}
\caption{Excitation equlilbrium for LSS\,3378 using Fe\,{\sc ii} lines, and
models with $\log g$ = 0.5.}
\end{figure}
\begin{figure}
\epsfxsize=8truecm
\epsffile{lssfig3.ps}
\caption{The $T_{\rm eff}$ vs $\log g$ plane for LSS\,3378. Loci satisfying
ionization equilibria are plotted -- see key on the figure.
The locus satisfying the excitation balance of Fe\,{\sc ii}
lines is shown by thick solid line.
The cross shows the adopted model atmosphere parameters.}
\end{figure}
\begin{figure}
\epsfxsize=8truecm
\epsffile{lssfig4.ps}
\caption{LSS\,3378's observed and synthesized He\,{\sc i} line profile
at 6678.15\AA. The He\,{\sc i} line profiles are synthesized using the model
$T_{\rm eff}$ = 10600 K and $\log g$ = 0.4, for different values of C/He - see key
on the figure.}
\end{figure}
The $T_{\rm eff}$ was estimated from Fe\,{\sc ii} lines spanning excitation
potentials from 3 eV to 11 eV. For $\xi$ = 4.5 km s$^{-1}$,
models ($T_{\rm eff}$, $\log g$) were found which gave the same abundance independent
of excitation potential. The $T_{\rm eff}$ determined from Fe\,{\sc ii} lines
somewhat depends on the adopted surface gravity.
Figure 2 illustrates the method for obtaining $T_{\rm eff}$.
Ionization equilibrium loci
for C\,{\sc i}/C\,{\sc ii} and N\,{\sc i}/N\,{\sc ii} are shown in Figure 3.
These with the $T_{\rm eff}$
determined from the excitation equilibrium of Fe\,{\sc ii} lines, which is a function
of surface gravity, gives the estimate of the
stellar parameters: $T_{\rm eff}$ = 10600$\pm$250 K, $\log g$ = 0.4$\pm$0.25 cgs, and
$\xi$ = 6$\pm$2 km s$^{-1}$ (Figure 3). Thus, the abundance analysis was conducted
for the model atmosphere (10600,0.4,6.0).
At this $T_{\rm eff}$ which is close to 11000 K, electron scattering (most of
the free electrons from photoionization of neutral helium), and photoionization of
neutral helium are the major sources of continuous opacity in the line forming
regions (Pandey et al. 2001). Hence, for LSS\,3378 it is evident that helium controls
the continuum opacity, and the C/He ratio is directly determined
from the measured equivalent widths of C\,{\sc i} and C\,{\sc ii}
lines observed in the spectrum of LSS\,3378.
The C\,{\sc i} lines give log $\epsilon$(C) = 9.53$\pm$0.19. Four C\,{\sc ii}
lines give 9.44$\pm$0.11. The abundances from C\,{\sc i} and C\,{\sc ii} lines
are in good agreement, and imply a C/He = 1\%.
At the $T_{\rm eff}$ of LSS\,3378, the He\,{\sc i} lines are mildly sensitive to the
C/He ratio. The C/He ratio may be derived by
fitting the He\,{\sc i} lines at 5048, 5876, and 6678\AA\ (Figure 4); the blending
Ne\,{\sc i} line at 6678\AA\ is taken into account.
These He\,{\sc i} lines give C/He = 0.8$\pm$0.1\%, where the uncertainty reflects only the scatter
of the three results.
LSS\,3378's final abundances, for key elements, derived for C/He = 1\% are
summarized in Table 1; also given are solar abundances from Table 2 of Lodders (2003)
for comparison.
The individual elemental abundances listed in Table 1 are given as
log $\epsilon(i)$, normalized such that log $\Sigma$$\mu_i \epsilon(i)$ = 12.15
where $\mu_i$ is the atomic weight of element $i$.
The lines used for abundance analysis including the mean abundance, and the
line-to-line scatter are given in Table A1. The abundance errors due to the
uncertainty in the adopted stellar parameters are given in Table 2.
The deduced $v\sin i$ is about 26 km s$^{-1}$.
\begin{table*}
\caption{Adopted key elemental abundances for LSS\,3378}
\begin{center}
\begin{tabular}{lrcccccccccccccccc} \hline
Star & H & He & C & N & O & Mg & Al & Si & P & S & Ti & Cr & Mn & Fe & Y & Zr & Ba \\
\hline
LSS\,3378 & 7.18 & 11.50 & 9.46 & 8.26 & 9.29 & 5.97 & 5.98 & 6.62 & 4.85 & 6.52 & 4.29 & 4.47 &
4.57 & 6.11 & 2.85 & 3.51 & 2.49 \\
Sun & 12.00& 10.98 & 8.46 & 7.90 & 8.76 & 7.62 & 6.54 & 7.61 & 5.54 & 7.26 & 5.00 & 5.72 &
5.58 & 7.54 & 2.28 & 2.67 & 2.25 \\
\hline
\end{tabular
\end{center}
\end{table*}
\section{Abundances}
The derived abundances of the EHe LSS\,3378 are compared with the measured
abundances, taken from the literature, of the other EHes (see Pandey et al. 2006).
The photospheric composition of LSS\,3378 reveals that the surface material
is contaminated by the products of H-burning, and He-burning reactions, as observed
for most of the EHes.
LSS\,3378's abundance ratios: Cr/Fe, Mn/Fe, S/Fe, Si/Fe, and Ti/Fe, are
as expected for that in metal-poor normal and unevolved stars except for the low
Mg/Fe ratio. The abundances
of Fe, Cr, and Mn$-$the iron peak elements, and S, Si, and Ti$-$the
$\alpha$-elements represent the initial metallicity of LSS\,3378 as these elements are
expected to be unaffected by H- and the He-burning, and
attendant nuclear reactions.
We choose Fe to be the indicator of the initial metallicty in
LSS\,3378 for spectroscopic convenience.
LSS\,3378's derived elemental abundances that are affected by evolution
are of H, C, N, O, Ne, Y, Zr, and Ba. Note that, the Ne abunadnaces from
Ne\,{\sc i} lines are affected by
departures from LTE, hence, not discussed here. However, the neon LTE abunadnces
from Ne\,{\sc i} lines are listed in Table A1. \\
{\it Hydrogen}$-$H abundance log $\epsilon$(H) is about 7.2 that fits the suggested trend
of increasing H with increasing $T_{\rm eff}$ for all the EHes with C/He ratio of
about 1\%, the exception being the hottest EHe
LS IV $+6^\circ2$ (see Pandey et al. 2006).\\
{\it Carbon}$-$The C/He ratio is 0.0083, the mean C/He ratio from 15 EHes, that excludes
the two EHes HD\,144941 and V652\,Her with much lower C/He ratio, is 0.0066.\\
{\it Nitrogen}$-$Nitrogen is enriched above its initial abundance expected based
on the Fe abundance. The observed N abundance is the result of complete
conversion of the initial C, N, and O to N via H-burning CN-cycle and the ON-cycles.\\
{\it Oxygen}$-$Oxygen abundance relative to Fe is overabundant by about 1.5 dex.
LSS\,3378 fits to the oxygen-rich group of EHes (see Pandey et al. 2006).\\
{\it Yttrium, Zr, {\rm and} Ba}$-$Relative to iron, Y, Zr, and Ba are overabundant with
respect to solar by about a factor of 80 (1.9 dex), 150 (2.2 dex), and 40 (1.6 dex),
respectively. Note that, the measured abundances are are based on one or two lines.
The observed Y, Zr, and Ba overabundances are attributed to contamination of the
atmosphere by $s$-process products.
\section{Conclusions}
The abundances of the cool EHe star LSS\,3378 are measured. The measured C/He ratio
is about 1\% similar to most of the EHes. LSS\,3378 fits the oxygen-rich group
of EHes discussed by Pandey et al. (2006). The measured abundances of Y, Zr, and Ba
suggest that in the EHe star LSS\,3378, $s$-process nucleosynthesis did occur in its
earlier evolution. With this analysis, a total of four EHes: LSS\,3378, PV\,Tel,
V1920\,Cyg, and LSE\,78, show a strong enhancement of Y and Zr attributable to an
$s$-process (Pandey et al. 2004, 2006).
An interesting similarity is suggested by the $s$-process abundances in LSS\,3378
and those of the RCB stars. The light $s$-process elements are more enhanced than the
heavy $s$-process elements in RCB stars (Asplund et al. 2000; Rao \& Lambert 2003;
Vanture, Zucker, \& Wallerstein 1999; Bond, Luck, \& Newman 1979).
Similar enhancement of light $s$-process elements (Y, Zr) over heavy $s$-process
element (Ba) is observed in LSS\,3378.
Enrichment of $s$-process elements is not expected for the EHes resulting from
a merger of a He with a C-O white dwarf as discussed by Pandey et al. (2004, 2006).
However, synthesis by neutrons via the $s$-process may occur during the merger and
needs to be explored.
To further improve the chemical analysis, non-LTE calculations should be performed
for key elements like N, O, Ne, Si, S, and Fe, and in particular neon.
\begin{table}
\caption{Abundance errors due to uncertainties in the stellar parameters: $\Delta$$T_{\rm eff}$,
$\Delta$$\log g$, and $\Delta$$\xi$. The abundance error due to $\Delta$$T_{\rm eff}$ is the difference
in abundances derived from the adopted model ($T_{\rm eff}$, $\log g$, $\xi$) and
a model ($T_{\rm eff}$+$\Delta$$T_{\rm eff}$, $\log g$, $\xi$). The abundance error due to
$\Delta$$\log g$ is the difference in abundances derived from the adopted
model ($T_{\rm eff}$, $\log g$, $\xi$) and a model ($T_{\rm eff}$, $\log g$+$\Delta$$\log g$, $\xi$).
The abundance error due to $\Delta$$\xi$ is the difference in abundances derived from the adopted
model ($T_{\rm eff}$, $\log g$, $\xi$) and a model ($T_{\rm eff}$, $\log g$, $\xi$+$\Delta$$\xi$).}
\begin{tabular}{lccc} \hline
\multicolumn{1}{l}{Species}&\multicolumn{1}{c}{$\Delta$$T_{\rm eff}$ = +300}&
\multicolumn{1}{c}{$\Delta$$\log g$ = +0.25}&
\multicolumn{1}{c}{$\Delta$$\xi$ = +2.0} \\
\multicolumn{1}{c}{}& \multicolumn{1}{c}{[K]} &
\multicolumn{1}{c}{[cgs]} &\multicolumn{1}{c}{[km s$^{-1}$]}\\
\hline
H\,{\sc i} & $-$0.19 & +0.22 & +0.09\\
C\,{\sc i} & $-$0.18 & +0.18 & +0.11\\
C\,{\sc ii} & +0.02 & +0.02 & +0.13\\
N\,{\sc i} & $-$0.18 & +0.18 & +0.12\\
N\,{\sc ii} & +0.04 & $-$0.02 & +0.09\\
O\,{\sc i} & $-$0.17 & +0.20 & +0.24\\
Ne\,{\sc i} & $-$0.04 & +0.11 & +0.41\\
Na\,{\sc i} & $-$0.17 & +0.17 & +0.09\\
Mg\,{\sc ii} & $-$0.22 & +0.18 & +0.28\\
Al\,{\sc ii} & $-$0.12 & +0.11 & +0.17\\
Al\,{\sc iii} & +0.04 & $-$0.05 & +0.09\\
Si\,{\sc ii} & $-$0.15 & +0.16 & +0.26\\
P\,{\sc ii} & $-$0.07 & +0.03 & +0.05\\
S\,{\sc ii} & 0.00 & 0.00 & +0.20\\
Ti\,{\sc ii} & $-$0.27 & +0.18 & +0.03\\
Cr\,{\sc ii} & $-$0.35 & +0.17 & $-$0.05\\
Mn\,{\sc ii} & $-$0.22 & +0.17 & +0.03\\
Fe\,{\sc ii} & $-$0.25 & +0.15 & +0.11\\
Y\,{\sc ii} & $-$0.27 & +0.19 & +0.01\\
Zr\,{\sc ii} & $-$0.28 & +0.19 & $-$0.01\\
Ba\,{\sc ii} & $-$0.25 & +0.19 & +0.01\\
\hline
\end{tabular}
\end{table}
The observations presented here were obtained at CTIO, National
Optical Astronomy Observatories (NOAO), which is operated by the Association of
Universities for Research in Astronomy Inc. (AURA) under a cooperative agreement
with the National Science Foundation, USA. We thank the referee Tony Lynas-Gray
for drawing our attention to the strong Na\,D profiles in LSS\,3378's spectrum.
|
3,212,635,537,494 | arxiv | \section{Introduction}
A {\em strong edge-coloring} of a graph $G$ is an edge-coloring in which every color class is an induced matching; that is, no edge can be incident to two edges with the same color. The {\em strong chromatic index $\chi_s'(G)$} is the minimum number of colors in a strong edge-coloring of $G$. This notion was introduced by Fouquet and Jolivet (1983, \cite{FJ83}) and one of the main open problems was proposed by Erd\H{o}s and Ne\v{s}et\v{r}il during a seminar in Prague in 1985, as follows,
\begin{conjecture}[Erd\H{o}s and Ne\v{s}et\v{r}il, 1985]
If $G$ is a simple graph with maximum degree $\Delta$, then $\chi_s'(G)\le 5\Delta^2/4$ if $\Delta$ is even, and $\chi_s'(G)\le (5\Delta^2-2\Delta+1)/4$ if $\Delta$ is odd.
\end{conjecture}
This conjecture is true for $\Delta\le 3$, see \cite{A92, HHT93}. For $\Delta=4$, Cranston \cite{C06} showed that $\chi_s'(G)\le 22$, two more than the conjectured upper bound $20$. Chung, Gy\'arf\'as, Trotter, and Tuza (1990, \cite{CGTT90}) confirmed the conjecture for $2K_2$-free graphs. Using probabilistic methods, Molloy and Reed \cite{MR97} proved that for sufficiently large $\Delta$, every graph with maximum degree $\Delta$ has strong chromatic index at most $1.998\Delta^2$. Readers are referred to Problem 17 in Chapter 6 of \cite{SSTM} for more information about strong edge colorings of graphs.
Sparse graphs have also attracted a lot of attention. Faudree, Gy\'arf\'as, Schelp, and Tuza (1989, \cite{FGST89}) showed that every planar graph with maximum degree $\Delta$ has strong chromatic index at most $4\Delta+4$. Hocquarda, Ochemb, and Valicov \cite{HOV12} showed a tight result which claims that every non-trivial outerplanar graph can be strongly edge-colored with at most $3\Delta-3$ colours. Hocquard, Montassier, Raspaud, and Valicov \cite{HMRV12} gave the optimal upper bound on $\chi_s'$ for subcubic graphs with certain maximum average degrees.
A graph is {\em $2$-degenerate} if every subgraph has minimum degree at most two. Outplanar graphs, non-regular subcubic graphs, and planar graphs with girth at least six are all $2$-degenerate graphs. Chang and Narayanan (2012, \cite{CN12}) recently proved that a $2$-degenerate graph with maximum degree $\Delta$ has strong chromatic index at most $10\Delta-10$. They actually proved the following stronger statement (see paragraphs 4 and 5 in \cite{CN12}).
\begin{theorem}(Chang and Narayanan, \cite{CN12})
Let $G$ be a $2$-degenerate graph with maximum degree $\Delta>0$. Let $B= \{1,2,\cdots, 5\Delta -5\}$ and $B'=\{1',2', \cdots, (5\Delta-5)'\}$. Then $G$ has a strong edge-coloring with colors from $B\cup B'$ so that \\
(a) Every pendant edge (if any) is colored with a color in $B$;\\
(b) If a pendant edge is colored with a color $c\in B$, then no edge within distance $1$ to the edge is colored with $c'$.
\end{theorem}
In this note, we prove a stronger result with a better upper bound. The induction in our proof is similar to their proof, except that we add an extra condition (3). This extra condition helps fix a gap in their proof (in the third paragraph on Page 5 in \cite{CN12}, if some edge incident to $v$ has color $c'$, then the priming process fails as one cannot color $ww'$ with $c'$). Also in our proof, instead of considering a subgraph by deleting one edge from $G$, we delete a bunch of edges and try to color those deleted edges one by one, and this helps us improve the upper bound.
\begin{theorem}\label{induction}
Let $G$ be a 2-degenerate graph with maximum degree at most $\Delta$. Let $B= \{1,2,\cdots, 4\Delta - 2\}$ and $B'=\{1',2', \cdots, (4\Delta-2)'\}$. Then $G$ has a strong edge-coloring with colors from $B\cup B'$ so that \\
(1) Every pendant edge is colored with a color in $B$;\\
(2) If a pendant edge is colored with a color $c \in B$, then no edge within distance 1 to the edge is colored with $c'$;\\
(3) No pair of colors $\{c,c'\}$ appears at the same vertex.
\end{theorem}
As a corollary, every $2$-degenerate graph with maximum degree $\Delta$ has strong chromatic index at most $8\Delta-4$.
\section{Proof of Theorem~\ref{induction}}
In this section, we prove Theorem~\ref{induction}.
We will use the following notion. A $k$-vertex ($k^-$-vertex, $k^+$-vertices) is a vertex of degree $k$ (at most $k$, at least $k$). If a $k$-vertex ($k^-$-vertex, $k^+$-vertex) $u$ is a neighbor of vertex $v$, then we also call $u$ to be a $k$-neighbor ($k^-$-neighbor, $k^+$-neighbor) of $v$. Let $B= \{1,2,\cdots, 4\Delta -2\}$ and $B'=\{1',2', \cdots, (4\Delta-2)'\}$ be the sets of colors. For a color $c \in B\cup B'$, denote $c'$ to be the corresponding color in the other color set (thus $(c')'=c$). For $A \subseteq B$ or $A \subseteq B'$, let $A' = \{c': c \in A\}$. If $f$ is a strong edge coloring of $G$ (or its subgraph), then we use $C_f(v)$ to denote the set of colors of the edges incident with $v\in V(G)$.
First of all, we quote the following lemma on $2$-degenerated graphs.
\begin{lemma}\label{2-degenerate}(Lemma 3, \cite{CN12})
Let $G$ be a 2-degenerate graph. Then $G$ has a vertex $v$ with degree at least $3$ that has at most two $3^+$-neighbors.
\end{lemma}
\begin{proposition}\label{recoloring}
Let $uv$ be a pendant edge in $G$ with $d(u) = 1$. If $f$ is an edge-coloring of $G$ with properties in Theorem~\ref{induction} so that $f(uv)=c$, then the coloring $g$, which differs from $f$ only on $uv$ with $g(uv)=c'$, is a strong edge-coloring which satisfies (2) and (3) but violates (1) (only) at the pendant edge $uv$.
\end{proposition}
\begin{proof}
It is clear that $uv$ is the only pendant edge colored with a color in $B'$ in $g$. Since $f(uv) = c$, by (2), $c,c'\not\in\cup_{u\in N(v)} C_f(u)=\cup_{u\in N(v)} C_g(u)$ and $c, c'$ do not appear on the edges incident with $v$ (except $uv$) in $f$ thus also in $g$. So $g$ satisfies (2) and (3).
\end{proof}
\medskip \noindent
{\bf Proof of Theorem~\ref{induction}.} Let $G$ be a smallest counterexample to Theorem~\ref{induction}. That is, $G$ is a 2-degenerate graph with maximum degree at most $\Delta$ and very proper subgraph of $G$ has a desired strong edge coloring.
By Lemma~\ref{2-degenerate}, let $v$ be a vertex adjacent to at most two $3^+$-vertices. Let $d=d(v) -2$, and $v_1,\cdots,v_d$ be $2^-$-neighbors of $v$, and $u$ and $w$ be the other two neighbors of $v$. We have the following claim.
\begin{claim}\label{1-vertex} $d(v_i) = 2$ for $i = 1,\cdots d$. \end{claim}
\begin{proof}
Without loss of generality, suppose that $v_1$ is a $1$-neighbor of $v$. Let $f$ be a strong edge-coloring of $G-v_1$ satisfying (1)-(3). Let
$A=\cup_{x\in N(v)} C_f(x)$. Then $(A\cap B')' \subseteq B$ and $(A\cap B) \cup (A\cap B')' \subseteq B$, and
$$|(A\cap B) \cup (A\cap B')' |\leq |A|\le \sum_{x\in N(v)} d(x) -1 \leq 2\Delta + 2(d(v) -3)-1\leq 4\Delta -7.$$
Therefore let $f(vv_1)=c$, where $c\in B-((A\cap B)\cup (A\cap B')')$. As $c, c'\not\in A$, $f$ is a strong edge-coloring of $G$.
To see a contradiction, we now show that $f$ satisfies (1)-(3). Since $f(vv_1)=c \in B$, Property (1) is met. Since $c \not \in A\cap B$ and $c \not \in (A\cap B')'$, $c\not\in A\cap B$ and $c'\not\in A\cap B'$, that is, $c\not\in A$. Therefore, Properties (2) and (3) are met.
\end{proof}
By Claim~\ref{1-vertex}, for each $i=1,2,\cdots, d$, let $N(v_i)=\{v, u_i\}$. Consider $G_1 = G-\{vv_i: 1 \leq i \leq d\}$. Then $G_1$ has a desired strong edge-coloring $f$. As $v_iu_i$ is a pendant edge in $G_1$, it is colored with a color in $B$.
We now consider a new edge-coloring $g$ of $G$ which may only differ from $f$ on edges $v_iu_i$. For each edge $v_iu_i$, if $(f(v_iu_i))'\not\in \{f(vu), f(vw)\}$, then let $g(v_iu_i)=(f(v_iu_i))'$, otherwise, $g(v_iu_i)=f(v_iu_i)$. By Proposition~\ref{recoloring}, $g$ is a strong edge-coloring of $G_1$ which satisfies (2) and (3). We are going to color the edge $vv_i$ one by one with available colors in $B$. As the edges $v_iu_i$ are not pendent edges in $G$, (1) is not violated in the new coloring. Note that the more edges $vv_i$ are colored, the less choices for the remaining uncolored edges, so we only need to consider when only one edge (say $vv_1$) is yet to color. We still use $g$ to denote the strong edge-coloring of $G-vv_1$ which is an extension of the coloring $g$ of $G_1$.
Let $S=\{g(vu), g(vw), g(v_1u_1)\}\cap B'$. Then $S' \subseteq B$. Note that by the construction of $g$, $g(vv_i) \in B$ for each $ i =2,3,\cdots, d$ and furthermore $g(v_iu_i)=c\in B$ if and only if $c'\in S$, thus if and only if $c\in S'$. Let $$\Omega =[\{g(vu), g(vw)\}\cap B]\cup S' \cup \{g(vv_i): 2\leq i \leq d\} \cup ([C_g(u_1)\cup C_g(u)\cup C_g(w)]\cap B).$$
Since $$|\Omega| \leq (d(u) + d(w) + (d(v) - 3) + d(u_1) \leq 4\Delta - 3,$$
pick one color $t \in B\setminus \Omega$ and let $g(vv_1)=t$, so we obtain an edge-coloring of $G$. By the choice of $t$, $g(vv_1) = t \not = g(v_1u_1)$ and no edge within distance $1$ to $vv_1$ is colored with $t$ except those on $v_iu_i$ with $i\ge 2$. But if $v_iu_i=t$ for some $i\ge 2$, then $t'\in \{f(vu), f(vw)\}$, so $t\in \Omega$, a contradiction. Therefore $g$ is a strong edge-coloring of $G$.
Lastly, we show that $g$ satisfies Properties (1)-(3). Since none of $v_1u_1$ and $vv_1$ are pendant edges in $G$, (1) and (2) are satisfied. So we just need to show that $g(v_1u_1)\not=t'$ and $t'\not\in C_g(v)$. If there is an edge incident with $v_1$ or $v$ is colored with $t'$, then $e \in \{v_1u_1, vw,vu\}$ since $g(vv_i) \in B$ for each $i = 2,3\cdots$. Hence $f(e) = t' \in S$ and thus $t = S'\subseteq \Omega$, a contradiction. Therefore (3) is met. This finishes the proof of Theorem~\ref{induction}.
\section{Final remarks}
In \cite{CN12}, they also studied the strong chromatic index of a chordless graph $G$ and showed that $\chi_s'(G)\le 8\Delta-8$. Their proof is very similar to the one for 2-degenerate graphs, thus also has a similar gap. We can improve their result to the following
\begin{theorem}
Let $G$ be a chordless graph with maximum degree at most $\Delta$. Let $B= \{1,2,\cdots, 3\Delta -1\}$ and $B'=\{1',2', \cdots, (3\Delta-1)'\}$. Then $G$ has a strong edge-coloring with colors from $B\cup B'$ so that \\
(1) Every pendant edge is colored with a color in $B$;\\
(2) If a pendant edge is colored with a color $c \in B$, then no edge within distance 1 to the edge is colored with $c'$;\\
(3) No pair of colors $\{c,c'\}$ appears at the same vertex.
\end{theorem}
So if a chordless $G$ has maximum degree $\Delta$, then $\chi_s'(G)\le 6\Delta-2$. The proof is very similar to that of Theorem~\ref{induction}, so we omit it here.
We do not believe that the bounds in this note are optimal. Let $G$ be the graph consisting of a triangle with $D-2$ leaves on each of the vertices on the triangle. Then $\Delta(G)=D$ and $\chi_s'(G)=3\Delta-3$, but $G$ is 2-degenerate and chordless. So $\chi_s'(G)\ge 3\Delta-3$ for any 2-degenerate or chordless graph $G$ with maximum degree $\Delta$.
|
3,212,635,537,495 | arxiv | \section{Introduction}
Software quality assurance forms an important part of the software development cycle.
Its importance is accepted throughout the software engineering community.
Verification and validation of an application are essential steps in software testing.
The increasing complexity of present-day software and the use of machine learning techniques presents new challenges to test software sufficiently.
However, machine learning techniques are used in research and in developing solutions for real-world applications \cite{Reichstein2019,EcoInf2017}.
Besides this, the nature of research software is exploratory.
The output for such software is usually unknown and cost-intensive to compute.
The unknown output of the application is known as the test oracle problem in software engineering.
There have been many approaches to test software that pose the test oracle problem.
Metamorphic testing (MT) is the most prominent approach among these, as seen by the increase in number of papers published and increased research efforts in recent times.
MT was first introduced by \cite{Chen1998}.
It has seen an increase adoption in research and real-world applications.
Comprehensive surveys of metamorphic testing may be found in~\cite{Zhang2020,Segura2020,Lin2020,Segura2019a,Segura2018,Segura2016}.
Metamorphic testing is based on the idea that most of the time it is easier to predict relations between outputs of a program, than understanding its input-output behavior \cite{Saha2020}.
The central element of metamorphic testing is the metamorphic relation (MR). An MR is a necessary property of the target application in relation to multiple inputs and their corresponding outputs \cite{Chen2018b}.
Identifying metamorphic relations is labor-, time-, and cost-intensive.
Experienced domain experts require a thorough understanding of the application and the desired behavior of MRs to identify metamorphic test scenarios.
Automating this process would reduce the cost and increase the probability of widespread adoption.
Using the available automated test case generation methods, we can develop test cases once the MRs are available as shown by Hui et al.~\cite{Hui2020}.
We can harness testing frameworks to automate application testing using the generated test cases and test suites.
MT is not without its limitations.
A major limit is that MT by itself cannot prove that the output of the application is correct~\cite{Segura2019a}.
It is also challenging to quantify the effectiveness of MT.
Since the basis of MT relies on comparing multiple successive outputs with respect to each other and the morphed inputs. If the initial outputs themselves were wrong and the successive outputs were within an expected variation satisfying the MR, MT would not detect the error in the software under test.
The other limitation of the MT approach is that for comprehensive testing the number of MRs required is large and thus leads to an extensive test suite.
MT finds faults in the application but does not point to the function with the bug, thus debugging the software under test is challenging.
One method is to follow the changes to input parameters for failed metamorphic tests, find the parameters with high covariance, and investigate their path for debugging.
Section~\ref{motivation} explains the motivation behind our research.
In Section~\ref{s-application} we introducing an example application, before formulating the problem to be solved in Section~\ref{s-problem}.
In Section~\ref{s-method}, we explain our proposed method and sketch the solution employing machine learning.
Section~\ref{s-future} of the paper highlights the challenges and an outlook to future work.
Section~\ref{data_availability} provides the details of the resources available for this paper.
\section{Motivation}\label{motivation}
Our application domain are ocean system models.
Currently, there are no papers based on MT focused on ocean modeling, to the best of our knowledge.
However, there are a few papers on applying MT to geographic information systems \cite{Lin2020,Hui2020}
Ocean system modeling software is often based on legacy code that has over the years undergone multiple development cycles leading to complex intertwined code.
They rarely employ a standard systematic testing approach to verify the software.
The challenges to this have been explained by \cite{Johanson2018}, \cite{Kanewala2014b}, and \cite{Kelly2008}.
The focus is on assimilating functions that work rather than developing the software adhering to the principles of software engineering.
Verification is based on the modeler's undocumented plausibility check to see if the output lies within the output range, the researcher is expecting.
Though researchers accept software testing as a value-adding step in developing software, they rarely identify or maintain separate test cases to verify their functions and software.
Some recent developments of ocean simulation software are collaborative, open-source, and in newer programming languages.
This has led to refactoring some existing components and modular addition of new functions to the existing codebase.
Testing efforts are focused on regression tests as part of the continuous integration framework in most collaborative open-source scientific software development~\cite{OSRS2020}. A goal should be to follow the FAIR principles not only for research data but also for research software~\cite{FAIR_Software_2020}.
The increasing use of machine learning techniques to compute intermediate input data points demands non-traditional software techniques to ensure the correctness of the software in use.
There is a substantial gap between adopting artificial intelligence in software development and testing capabilities for valid implementations of artificial intelligence methodologies.
Testing these systems has proven to be challenging.
Though there is an increase in the solution's effectiveness by adopting these techniques, the lack of methods to verify and the inability to explain the improvement in effectiveness has resulted in cautious adoption of these software development techniques.
With the increasing assimilation of artificial Intelligence methodologies as a part of software engineering, the need for better testing approaches for validating software posing test oracle problems has increased.
From our current knowledge, there does not exist an established standard testing procedure for verifying the developed scientific software.
Most of the verification efforts are based on unit tests written, managed, and executed by contributors at their discretion.
A structured quality assurance framework would help with the adoption of the testing capabilities across the research facility and lead to more reliable software development.
One of the major hurdles to adopting MT is the cost associated with identifying metamorphic relations.
The central idea of this short paper is to present an approach to automate identifying metamorphic tests.
This will increase the widespread application of MT as an efficient verification technique for the software that poses test oracle problems.
The future of Software Engineering will revolve around harnessing artificial intelligence techniques for solving real-world complex and interdependent problems.
Automated identification of metamorphic tests will lead to improved adoption of MT-based test suit development for automated program repair as shown by Jiang et al.~\cite{Jiang2020}.
\section{Example Application}\label{s-application}
We use a simple but realistic ocean-modeling application to show our approach~\cite{Rath2019}.
We can access the application on Binder ~\cite{Jupyter2018BinderScale.}.
The example application is to calculate a time series of Kinetic energy of the surface ocean for randomly generated data for a grid of ($10 \times 20$) for $30$ temporal resolutions
\begin{equation}
\label{eq:energy}
e(t, y, x) = \frac{1}{2}\frac{\int{\rm d}y{\rm d}x (u(t, y, x)^2 + v(t, y, x)^2)}{\int{\rm d}y{\rm d}x}
\end{equation}
where $e$ represents the time series of Kinetic energy, $t$ the time, $y$ and $x$ are spatial coordinates, and horizontal surface-velocities $u$ and $v$ are calculated from the sea-level $\eta$ using
\begin{equation}
\label{eq:uv}
(u, v)(t, y, x) = \frac{G}{F} \left(-\frac{\partial}{\partial y}, \frac{\partial}{\partial x}\right) \eta(t, y, x)
\end{equation}
where $G$ represents the gravitational acceleration (in $meters/second^2$) and $F$ the Coriolis parameter (in $1/second$) at $30$\textdegree\ North.
The two functions of Equation~\eqref{eq:energy} are coded in the sample application, one of which is not respecting cyclic boundary conditions.
To numerically implement Equation~\eqref{eq:energy}, we discretize the integral in \eqref{eq:energy} and the partial derivatives in \eqref{eq:uv} by applying the method of finite differences and link it back to Equation~\eqref{eq:metamorphic_relation}.
Thus, the input data $\myvec{x}$ contains all discretized values of sea level $\eta$, of all coordinates $t$, $y$, $x$, and the physical constants $G$ and $F$.
A metamorphic transformation $g(x)$ could change all or some of the atomic data points in $\myvec{x}$ such that all $e(g(\myvec{x})) = e(\myvec{x})$ for all discrete values of time~$t$.
\begin{figure*}[htb]
\centering
\includegraphics[width=0.7\textwidth]{Fig_with_high.png}
\caption{Time series outputs of Kinetic energy of surface ocean calculated using the function not respecting the cyclic boundary conditions for (x,y) coordinates for multiple applied MRs.}
\label{fig:appied-MRs}
\end{figure*}
Fig.\ref{fig:appied-MRs} shows the time series output for multiple applied MRs calculated using the function in which the cyclic boundary condition is not implemented.
These MRs were manually identified based on the symmetries of the energy equation.
The bottom most plot is that of the original input data.
The applied MRs are multiplying $g$ and $h$ by the same constant, Keeping $G/F$ as constant, scaling $\eta, x$ and $y$ by a constant, transposing $x$ and $y$, reversing the direction of $x$, reversing the direction $y$, transposing the $x$ coordinates cyclically and transposing the $y$ coordinates cyclically.
We observe that for applied initial MRs the time series outputs are in accordance with the applied MRs.
Time series out puts for transposing the $x$ coordinates cyclically and transposing the $y$ coordinates cyclically differs from the rest as highlighted in Fig.~\ref{fig:appied-MRs} by red arrows.
\section{Problem Formulation}\label{s-problem}
For the given application domain, we may define the function under test $f: X \to Y$, where both $X$ and $Y$ are finite dimensional normed vector spaces.
Then a metamorphic relation $(g, h)$ satisfies
\begin{equation}
\label{eq:metamorphic_relation}
f(g(\myvec{x})) = h(f(\myvec{x}))
\end{equation}
where $\myvec{x} \in X$, $g: X \to X$ is the input mapping from source input to follow-up (morphed) input, and $h: Y \to Y$ is the output mapping from the source output to the follow-up output, that is the output associated with the morphed input.
Throughout this paper, we limit $h$ to be the identity map
\begin{equation}
\label{eq:metamorphic_relation_identiy_assumption}
f(g(\myvec{x})) = f(\myvec{x})
\end{equation}
and we constrain $g$ to be an affine transformation
\begin{eqnarray}
\label{eq:affine_transformation}
g(\myvec{x}) = \Gamma\cdot\myvec{x} + \myvec{\beta}
\end{eqnarray}
where $\Gamma$ is a matrix associated with an endormophism of $X$, and $\myvec{\beta} \in X$ is an offset.
Note that the limitation of $g$ being an affine transformation is motivated by retaining the ability of capturing physical properties related to symmetries of the governing equations, such as invariance under translation or rotation, while greatly reducing the soultion space.
\section{Proposed Method}\label{s-method}
There may exist an infinite number of possible $g$ satisfying \eqref{eq:metamorphic_relation_identiy_assumption} but one always is the identity map.
The task is then to find new $g$, or at least approximations to $g$, which differ from all $g$ known so far.
For this we use an iterative approach.
We represent the set of identified metamorphic relations associated with $f$ as
\begin{equation}
\label{eq:g}
G^f_{n-1} = \{g_0, g_1, \ldots, g_{n-1}\}
\end{equation}
We seek to find a new possible $g_n$ by minimizing a cost function that rewards a $g_n$ which minimizes the distance between source output and follow-up output $|f(g_n(\myvec{x})) - f(\myvec{x})|$ and penalizes if $g_n$ is already known.
Throughout the text, we take the norm $\left|\cdot\right|$ to be the Euclidean distance.
The cost function for a candidate $g_n$ and a given $G^f_{n-1}$ is defined by
\begin{equation}
\label{eq:cost_function}
J(g_n, G^f_{n-1}) = \int_{X}\frac{|f(g_n(\myvec{x})) - f(\myvec{x})|}{\epsilon + \prod_{g \in G^f_{n-1}}\left|g_n(\myvec{x}) - g(\myvec{x})\right|^2} dx
\end{equation}
where $g_n$ represents a potential morphing to $\myvec{x}$ such that it satisfies~\eqref{eq:metamorphic_relation_identiy_assumption} and $\epsilon$ is a small machine precision constant to avoid division by zero when $g_n$ is identical to a previously identified $g \in G^f_{n-1}$.
Note that in the denomiator we write the product of squared differences to ensure $\lim_{g_n \to g} J(g_n, G^f_{n-1}) = \infty$ which heavily penalizes $g_n$ for being close to an element of $G^f_{n-1}$.
We apply a modified algorithm loosely based on the genetic algorithm with a Monte Carlo optimization to find a $g_n$ that minimizes the cost function~\eqref{eq:cost_function}.
We start the iteration with $G^f_0 = \{\mathrm{id}_X\}$ containing only the identity map on $X$.
Initially, we assign random values to $\{\Gamma, \myvec{\beta}\}$ and in each step a mutation $\{\delta\Gamma, \myvec{\delta\beta}\}$ is proposed.
An acceptance threshold $p$ is set which determines at what rate a mutation that would increase the cost function is still not rejected.
If the cost after the application of the mutation is smaller than before the mutation, the applied mutation is always accepted.
The process is repeated to move towards a global minimum.
After converging to a minimum, we update the solution set $G^f_n = G^f_{n-1} \cap \{g_n\}$.
Then a new $g_{n+1}$ is created with randomly assigned $\{\Gamma, \myvec{\beta}\}$ and the process is restarted, this time minimizing $J(g_{n+1}, G^f_n)$.
With this method we are able to identify multiple metamorphic transformations which satisfy equation \eqref{eq:metamorphic_relation_identiy_assumption}.
\section{Challenges and Future Work}\label{s-future}
We identify multiple MRs using the approach proposed in Section \ref{s-method}.
We suspect the identified MRs are a chain of multiple elementary constituent MRs.
Hence, it will be challenging to explain identified MRs in terms of physical symmetries that hold in the real oceans.
We propose dimensionality reduction techniques to decompose the identified chained MRs.
The decomposed MRs will be closer to the constraints of the Physical Ocean and easier to interpret.
Since the resulting MRs will be distinct after decomposing, we expect test cases constructed with these MRs to be effective in uncovering defects.
They will also increase the test coverage of the application.
Since we assume input and output mapping relations, MRs corresponding to functional states of the software are not identified by the method proposed in Section \ref{s-method}.
Further work is to relax the restriction to $h$ in~\eqref{eq:metamorphic_relation} being the identity map.
Next steps will explore other optimization methods for the cost function constructed in Section\ref{s-method} including deep learning frameworks.
\section{Data Availability}\label{data_availability}
The sample application presented in Section~\ref{s-application} and the data generated are available in \cite{Rath2019}.
The published artifact contains a link to an interactive session where the example application can be executed.
\section*{Acknowledgments}\label{affiliations}
The first author is funded through the Helmholtz School for Marine Data Science (MarDATA), Grant No. HIDSS-0005.
We have developed the approach in collaboration of the Computer Science Department at Kiel University and the Ocean Circulation and Climate Dynamics Department at the GEOMAR Helmholtz Centre for Ocean Research Kiel.
\printbibliography
\end{document}
|
3,212,635,537,496 | arxiv | \section{Introduction}
Quantum state preparation is the first important step for any protocol that makes use of quantum resources. Examples of such protocols are quantum state teleportation and quantum key distribution which require entangled quantum states. In order to verify the integrity of the quantum state of the source prepared, one carries out \emph{quantum state tomography} on the source. Measurements are performed on a collection of quantum systems (electrons, photons, etc.) that are emitted from the source, that is, a \emph{quorum}. Then, the quantum state of the source is inferred from the measurement data obtained from this ensemble. The measurements are generically described by a set of positive operators $\Pi_j$ that compose a \emph{probability operator measurement} (POM). The procedure of state inference, which shall be our main focus in this article, is also known as \emph{quantum state estimation}. If the size of the ensemble is infinite, the estimation procedure will yield the unique \emph{true} quantum state of the source; this is the frequentist's definition of the true state, which we accept as the best description of what the source prepares. However, such an ensemble is never achievable in any laboratory setting, as one can only perform measurements on a \emph{finite} ensemble of quantum systems. As a result, the state estimator obtained will be different from the true state and depends on the details of the estimation procedure. To make statistical predictions, the corresponding operator $\hat\rho$ describing this estimator must be a \emph{statistical operator}, which is positive. This will ensure that the estimated probability $\hat p_j=\tr{\hat\rho\Pi_j}$ for an outcome $\Pi_j$ of \emph{any} set of POM is positive. We shall denote all estimated quantities with a ``hat'' symbol.
There are two popular methods for quantum state estimation: \emph{Bayesian} and \emph{maximum-likelihood} (ML). The Bayesian state estimation method \cite{bayesian1,bayesian2,bayesian3} constructs a state estimator from an integral average over all possible quantum states. The \emph{likelihood functional}, which yields the likelihood of obtaining a particular sequence of measurement detection with a given quantum state, serves as a weight for the average. This approach includes all the neighboring states near the maximum of the likelihood functional as possible guesses for the unknown $\rho_\text{true}$. These neighboring states are given especially significant weight when $N$ is small, in which case the likelihood functional is only broadly peaked at the maximum. However, the integral average unavoidably depends on how one measures volumes in the state space, and there is no universal and unambiguous method for that. The ML approach \cite{ml1,ml2,ml3,ml4}, on the other hand, simply chooses the estimator as the statistical operator that maximizes the likelihood functional. Rather than identifying a unique estimator, as the Bayesian approach always does, the ML method may only yield a convex set of estimators if the estimated probabilities $\hat p_j$ are consistent with more than one statistical operator. If the ML estimator is unique, and the quorum sufficiently large, both approaches give the same estimator since the likelihood functional peaks very strongly at the maximum.
When the measurement outcomes form an \emph{informationally complete} set, the measurement data obtained will contain maximal information about the source. Thus, a unique state estimator can be inferred with ML. Unfortunately, in tomography experiments performed on complex quantum systems with many degrees of freedom, it is not possible to implement such an informationally complete set of measurement outcomes. As a result, some information about the source will be missing and its quantum state cannot be completely characterized. The ML estimator obtained from these informationally incomplete data is no longer unique and there will in general be infinitely many other ML estimators which are consistent with the data. In Ref.~\cite{main}, we briefly reported an iterative algorithm (MLME) to estimate unknown quantum states from incomplete measurement data by maximizing the likelihood and \emph{von Neumann entropy} functionals. In that Letter, we assumed that the measurement detections are perfect with no detection losses, i.e. $\sum_j\Pi_j=1$. The application of this algorithm was illustrated with examples of homodyne tomography and we concluded that, together with a more objective Hilbert space truncation, this approach can serve as a reliable and statistically meaningful quantum state estimation with incomplete data.
In this article, we will present more details on the recently proposed MLME algorithm and apply it to various other situations. First, we give a brief review of the mathematical formalism for quantum state estimation in Sec.~\ref{sec:formalism} to set the stage for the subsequent discussions. Next, we derive the numerical MLME algorithms respectively for \emph{both} perfect and imperfect measurement detections in Sec.~\ref{sec:algo}, with the latter being particularly useful for actual experiments. We illustrate applications of the two algorithms with two examples in Sec.~\ref{sec:app} and finally conclude in Sec.~\ref{sec:conc}.
\section{Formalism of quantum state estimation}
\label{sec:formalism}
In a tomography experiment, an ensemble of $N$ copies of quantum systems, identically prepared, is measured using a POM which consists of positive measurement outcomes $\Pi_j$. For simplicity, we first assume that all measurement detections are perfect and hence $\sum_j\Pi_j=1$. The problem of imperfect detections will be dealt with in Sec.~\ref{subsec:algo_imperfect}. For each outcome, its number of occurrences is denoted by $n_j$ such that $\sum_jn_j=N$. The likelihood functional $\mathcal{L}(\{n_j\};\rho)$, for a particular sequence of independent detections, is then
\begin{equation}
\mathcal{L}(\{n_j\};\rho)=\prod_jp_j^{n_j}\,.
\label{like1}
\end{equation}
As a consequence of perfect measurement detections, $\sum_jp_j=1$. The ML procedure searches for the estimator $\hat\rho_\text{ML}$ which maximizes $\mathcal{L}(\{n_j\};\rho)$. For a $D$-dimensional Hilbert space, when a POM comprises $D^2$ or more measurement outcomes, of which $D^2$ of them are linearly independent, it is informationally complete. In this case, there exists a unique estimator $\hat\rho_\text{ML}$ for a given set of measurement data $\{n_j\}$. One can also define the outcome frequencies $f_j=n_j/N$ out of these measurement data such that $\sum_jf_j=1$. The corresponding functional $\mathcal{L}(\{n_j\};\rho)$ due to this informationally complete POM will peak at the unique global maximum $\hat\rho_\text{ML}$ over the space of $\rho$, whereby $\hat\rho_\text{ML}$ is solely determined by the frequencies $f_j$ and does not depend on the total number $N$ of measured copies.
The situation is different when the POM is informationally incomplete. In this case, there will be infinitely many ML estimators satisfying a smaller set of linearly independent constraints imposed by the incomplete measurement data. These ML estimators form a convex set of operators which maximize the convex functional $\mathcal{L}(\{n_j\};\rho)$. Geometrically, $\mathcal{L}(\{n_j\};\rho)$ possesses a convex plateau structure hovering over the space of $\rho$. The task, now, is to select one of these estimators for future statistical predictions. To do this, we adopt the well-known maximum-entropy (ME) principle advocated by Jaynes \cite{jaynes}. That is, we look for the estimator with the largest von Neumann entropy
\begin{equation}
S(\rho)=-\tr{\rho\log\rho}\,
\label{ent}
\end{equation}
among the convex set of ML estimators. This supplementary step introduces a small and smooth convex hill over the plateau structure so that a unique maximum can be obtained. The corresponding MLME estimator $\hat\rho_\text{MLME}$ is the least-bias estimator for the given set of incomplete measurement data; it can be regarded as the most conservative guess of the unknown quantum state out of the convex set of ML estimators.
At this point, we would like to comment on the distinction between this MLME technique and the conventional ME technique \cite{me1,me2}. The ME technique takes the outcome frequencies $f_j$ as \emph{bona fide} estimates for the probabilities $p_j$ and tries to search for the positive operator
\begin{equation}
\hat\rho_\text{ME}=\frac{\mathrm{e}^{\sum_j\lambda_j\Pi_j}}{\tr{\mathrm{e}^{\sum_j\lambda_j\Pi_j}}}
\end{equation}
that maximizes $S(\rho)$, subjected to the probability constraints which are mediated by the Lagrange multipliers $\lambda_j$. The fundamental problem with this scheme is that the $f_j$s cannot always be treated as probabilities since there may not be \emph{any} statistical operator $\rho$ for which $f_j=\tr{\rho\Pi_j}$. This is due to the statistical noise which is inherent in the outcome frequencies arising from measuring a finite ensemble of quantum systems. Therefore, in such cases, the ME technique fails as there simply is no positive operator which is consistent with the measurement data to begin with. The MLME algorithm, on the other hand, looks for the unique MLME estimator by confining the search within the plateau region inside the space of statistical operators. Thus, positivity is ensured. In cases where the $f_j$s are probabilities, both the ME and MLME schemes yield the same estimator by construction since the estimated probabilities $\hat p_j=f_j$ correspond to a statistical operator.
\section{The numerical algorithms}
\label{sec:algo}
\subsection{Perfect measurements}
\label{subsec:algo_perfect}
Assuming that the measurement detections are perfect, the likelihood functional $\mathcal{L}(\{n_j\};\rho)$ in Eq.~(\ref{like1}) gives a complete statistical description of all possible sequences of detections for the $N$ measured copies of quantum systems. Equivalently, one can consider the optimization of the \emph{normalized log-likelihood functional} $\log(\mathcal{L}(\{n_j\};\rho))/N$ to simplify the subsequent calculations, in view of the monotonic nature of the logarithmic function. The motivation for introducing the normalization will become clear soon. The MLME scheme can then be perceived as a standard constrained optimization problem: maximize $\log(\mathcal{L}(\{n_j\};\rho))/N$ subjected to the constraint that $S(\rho)$ takes the maximal value $S_\text{max}$. This is equivalent to maximizing $S(\rho)$ with the constraint that $\log(\mathcal{L}(\{n_j\};\rho))/N$ is maximal, as discussed above. The Lagrange functional for this optimization problem is defined as
\begin{equation}
\mathcal{I}(\lambda;\rho)=\lambda \bigl(S(\rho)-S_\text{max}\bigr)+\frac{1}{N}\log\mathcal{L}(\{n_j\};\rho)\,,
\end{equation}
where $\lambda$ is the Lagrange multiplier corresponding to the constraint for $S(\rho)$. We denote the estimator that maximizes $\mathcal{I}(\lambda;\rho)$ by $\hat\rho_{\text{I},\lambda}$. Incidently, the functional $\mathcal{I}(\lambda;\rho)$ is a sum of two different types of entropy, up to an irrelevant additive constant $\sum_jf_j\log f_j$: the von Neumann entropy $S(\rho)$ that quantifies the ``lack of information'', and the \emph{negative} of the \emph{relative entropy} $S(\{f_j\}|\{p_j\})=\sum_jf_j\log(f_j/p_j)$ that quantifies the ``gain of information'' from the measurement data. The scheme can now be interpreted as a simultaneous optimization of two complementary aspects of information, with an appropriately assigned constant relative weight $\lambda$. In addition, the normalization of $\log\mathcal{L}(\{n_j\};\rho)$ renders the optimal value of $\lambda$ to be independent of $N$.
When $\lambda=0$, we recover the Lagrange functional for the log-likelihood functional alone. Owing to the informational incompleteness of the measurement data, there exists a convex plateau structure for the log-likelihood functional. As $\lambda\rightarrow\infty$, the von Neumann entropy becomes increasingly more significant and the resulting estimator $\hat\rho_{\text{I},\lambda\rightarrow\infty}$ approaches the maximally-mixed state $1/D$. Naturally, when $\lambda$ takes on a very small positive value, the contribution from $\lambda S(\rho)$ becomes much smaller than $\log(\mathcal{L}(\{n_j\};\rho))/N$ and the effect of the von Neumann entropy functional is only significant over the plateau region in which the likelihood is maximal. Figure~\ref{fig:geom} illustrates all the aforementioned points.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{geom.eps}
\caption{Schematic diagrams of $\mathcal{I}(\lambda,\rho)$ on the space of statistical operators. The maximally-mixed state resides at the center of the square base which represents the Hilbert space. At the extremal points of $\lambda$, $\mathcal{I}(\lambda=0;\rho)=\log(\mathcal{L}(\{n_j\};\rho))/N$, with a convex plateau at the maximal value, and $\mathcal{I}(\lambda\rightarrow\infty;\rho)=\lambda S(\rho)$. Plot (c) shows the functional with an appropriate choice of value for $\lambda$ for MLME. An additional hill-like structure resulting from $S(\rho)$ is introduced over the plateau, so that the estimator with the largest entropy can be selected from the convex set of ML estimators within the plateau.}
\label{fig:geom}
\end{figure}
This means that, in general, $\lambda$ should be chosen so small that $S\left(\hat\rho_{\text{I},\lambda}\right)$ is very close to the minimum, and below which there are only very slight changes in the two entropy functionals \cite{main}.
Let us derive the iterative algorithm for maximizing $\mathcal{I}(\lambda\rightarrow 0;\rho)$ with respect to $\rho$. After varying $\mathcal{I}(\lambda\rightarrow 0;\rho)$, we have
\begin{equation}
\updelta\mathcal{I}(\lambda\rightarrow 0;\rho)=-\lambda\,\tr{\updelta\rho\log\rho}+\sum_j\frac{f_j}{p_j}\updelta p_j\,.
\label{var1}
\end{equation}
The variations $\updelta p_j$, or $\updelta\rho$, have to be such that $\rho$ stays positive after these variations. To choose their appropriate forms, we first parameterize the positive operator $\rho=\mathcal{A}^\dagger \mathcal{A}/\tr{\mathcal{A}^\dagger \mathcal{A}}$ with an auxiliary complex operator $\mathcal{A}$. Under this parametrization,
\begin{equation}
\updelta\rho=\frac{\updelta \mathcal{A}^\dagger \mathcal{A}+\mathcal{A}^\dagger\updelta \mathcal{A}-\rho\,\tr{\updelta \mathcal{A}^\dagger \mathcal{A}+\mathcal{A}^\dagger\updelta \mathcal{A}}}{\tr{\mathcal{A}^\dagger \mathcal{A}}}\,.
\label{varrho}
\end{equation}
Substituting $\updelta\rho$ in Eq.~(\ref{varrho}) into Eq.~(\ref{var1}), we have
\begin{equation}
\updelta\mathcal{I}(\lambda\rightarrow 0;\rho)=\tr{\frac{\updelta \mathcal{A}^\dagger \mathcal{A}}{\tr{\mathcal{A}^\dagger \mathcal{A}}}\mathfrak{R}+\mathfrak{R}\frac{\mathcal{A}^\dagger \updelta \mathcal{A}}{\tr{\mathcal{A}^\dagger \mathcal{A}}}}\,,
\label{var2}
\end{equation}
where
\begin{equation}
\mathfrak{R}=R-1-\lambda\bigl(\log\rho-\tr{\rho\log\rho}\bigr)
\end{equation}
with
\begin{equation}
R=\sum_j\frac{f_j}{p_j}\Pi_j\,.
\end{equation}
When $\mathcal{I}(\lambda\rightarrow 0;\rho)$ is maximal, we have $\updelta\mathcal{I}(\lambda\rightarrow 0;\rho)=0$ and the extremal equations
\begin{equation}
\rho\,\mathfrak{R}=\mathfrak{R}\rho=0
\label{exteqn1}
\end{equation}
are satisfied. Therefore, to solve these extremal equations numerically, we iterate the equation
\begin{equation}
\rho_\text{k+1}=\frac{\left(\mathcal{A}^\dagger_k+\updelta \mathcal{A}^\dagger_k\right)\left(\mathcal{A}_k+\updelta \mathcal{A}_k\right)}{\tr{\left(\mathcal{A}^\dagger_k+\updelta \mathcal{A}^\dagger_k\right)\left(\mathcal{A}_k+\updelta \mathcal{A}_k\right)}}\,
\label{itereqn1}
\end{equation}
starting from some statistical operator $\rho_1$, until $k=k'$ such that the norm of $\rho_{k'} \mathfrak{R}_{k'}$ is less than some pre-chosen value. We then take $\hat\rho_\text{MLME}\equiv\rho_{k'}$ as the MLME estimator. Maximizing $\mathcal{I}(\lambda\rightarrow 0;\rho)$ will require $\updelta\mathcal{I}(\lambda\rightarrow 0;\rho)$ to be positive whenever $\mathcal{I}(\lambda\rightarrow 0;\rho)$ is less than the maximal value. A straightforward way to enforce positivity is to set
\begin{equation}
\updelta \mathcal{A}_k\equiv\left(\updelta \mathcal{A}^\dagger_k\right)^\dagger\equiv\epsilon \mathcal{A}_k\mathfrak{R}_k\propto\epsilon\frac{\partial \mathcal{I}(\lambda;\rho)}{\partial \mathcal{A}_k}\,,
\label{itereqn2}
\end{equation}
with $\epsilon$ being a small positive constant. This is the \emph{steepest-ascent} method. We have thus established a numerical MLME scheme as a set of iterative equations (\ref{itereqn1}) and (\ref{itereqn2}) to search for the MLME estimator using the measurement data obtained from perfect measurement detections. More compactly, the relevant iterative equations are
\begin{align}
\rho_\text{k+1}&=\frac{\left(1+\epsilon \mathfrak{R}_k\right)\rho_k\left(1+\epsilon \mathfrak{R}_k\right)}{\tr{\left(1+\epsilon \mathfrak{R}_k\right)\rho_k\left(1+\epsilon \mathfrak{R}_k\right)}}\,,\nonumber\\
\mathfrak{R}_k&=R_k-1-\lambda\left(\log\rho_k-\tr{\rho_k\log\rho_k}\right)\,.
\label{mlmeiteralgo1}
\end{align}
We note that a more efficient algorithm, using the conjugate-gradient method, can be derived from this steepest-ascent algorithm, which is the subject of a separate discussion.
\subsection{Imperfect measurements}
\label{subsec:algo_imperfect}
In actual experiments, the measurement detections will usually be imperfect in the sense that the detection efficiency $\eta_j$ of a particular measurement outcome $\Pi_j$ is less than unity. In this case, the overall outcome probabilities
\begin{equation}
\tilde p_j\equiv\eta_jp_j
\end{equation}
will not sum to unity. Hence, we have a set of POM with outcomes $\tilde\Pi_j\equiv\eta_j\Pi_j$ such that $G\equiv\sum_j\tilde\Pi_j<1$. A consequence of this is that the true total number $M$ of copies received is not known, since only $N<M$ are detected ($N=M$ when all $\eta_j=1$ as in Sec.~\ref{subsec:algo_perfect}).
The likelihood functional that accounts for all $M$ copies of quantum systems in an experiment with imperfect detections is given by
\begin{equation}
\tilde{\mathcal{L}}(\{n_j\};\rho)=\frac{M!}{N!\,(M-N)!}\left(\prod_j\tilde p_j^{n_j}\right)\left(1-\eta\right)^{M-N}\,,
\label{like2}
\end{equation}
where $\eta=\sum_j\tilde p_j<1$. The additional combinatorial prefactor arises from the indistinguishability in the ordering of the detection sequence resulted from losses. With the help of Stirling's approximation for the factorials, the variation of the corresponding log-likelihood functional is given by
\begin{align*}
\updelta\log\tilde{\mathcal{L}}(\{n_j\};\rho)&=\tr{\left(N\tilde R-\frac{M-N}{1-\eta}G\right)\updelta\rho}\nonumber\\
&+\updelta M\log\left(\frac{(1-\eta)M}{M-N}\right)\,,
\end{align*}
where $\tilde R=\sum_jf_j\tilde\Pi_j/\tilde p_j$. Adopting the concept of maximum-likelihood, we derive an expression for $M$ such that $\log\tilde{\mathcal{L}}(\{n_j\};\rho)$ is maximized for any given $\rho$. This implies that the coefficient of the arbitrary $\updelta M$ must vanish and we have $M=N/\eta$ as the most-likely value of $M$. With this, the expression for $\tilde{\mathcal{L}}(\{n_j\};\rho)$ reduces to the simple form
\begin{equation}
\tilde{\mathcal{L}}(\{n_j\};\rho)=\prod_j\left(\frac{p_j}{\eta}\right)^{n_j}
\label{rel_like}
\end{equation}
up to an irrelevant multiplicative factor, with its corresponding logarithmic variation
\begin{equation}
\updelta\log\tilde{\mathcal{L}}(\{n_j\};\rho)=N\tr{\left(\tilde R-\frac{G}{\eta}\right)\updelta\rho}\,.
\label{varlike2}
\end{equation}
The additional term $-\updelta\rho G/\eta$ in the argument of the trace accounts for copies that have escaped detection.
Defining $\mathcal{I}(\lambda\rightarrow 0;\rho)$ for the new POM and its $\tilde{\mathcal{L}}(\{n_j\};\rho)$ in Eq.~(\ref{rel_like}), one can derive the iterative equations
\begin{align}
\rho_{k+1}&=\frac{\left(1+\epsilon \tilde{\mathfrak{R}}_k\right)\rho_k\left(1+\epsilon\tilde {\mathfrak{R}}_k\right)}{\tr{\left(1+\epsilon\tilde {\mathfrak{R}}_k\right)\rho_k\left(1+\epsilon \tilde {\mathfrak{R}}_k\right)}}\,,\nonumber\\
\tilde {\mathfrak{R}}_k&=\tilde R_k-\frac{G}{\eta^{(k)}}-\lambda\left(\log\rho_k-\tr{\rho_k\log\rho_k}\right)\,,
\label{iteralgo2}
\end{align}
with $\eta^{(k)}=\sum_j\tilde p^{(k)}_j$.
To highlight the importance of a proper treatment of imperfect measurement detections, we perform a simulation on $10^{3}$ randomly generated qubit states. Figure~\ref{fig:imp_det} compares the performance of the MLME algorithm derived in Sec.~\ref{subsec:algo_perfect}, with which we search for the MLME estimator by assuming that the measured data $\{n_j\}$ are all we have while ignoring the possible missing data, with that of the MLME algorithm derived in this section. The trace-class distance
\begin{equation}
\mathcal{D}_\text{tr}=\frac{1}{2}\tr{|\hat\rho_\text{MLME}-\rho_\text{true}|}
\end{equation}
is used as the figure of merit to quantify the distance between $\hat\rho_\text{MLME}$ and $\rho_\text{true}$. The lesson here is that if one neglects the consequence of imperfect measurements in performing state reconstruction, the quality of the resulting reconstructed state estimator will typically be much lower than that obtained from a scheme which accounts for this imperfection.
\begin{figure}[h!]
\centering
\includegraphics[width=0.4\textwidth]{imp_det.eps}
\caption{A comparison of two different schemes with $10^3$ random qubit true states distributed uniformly with respect to the Hilbert-Schmidt measure. Fifty experiments were simulated for every true state, with $N=5000$ for each experiment, and the respective average trace-class distances $\mathcal{D}^\text{avg}_\text{tr}$ were computed. The entire simulation was done with a set of randomly generated, informationally incomplete POM consisting of two imperfect measurement outcomes. The plot markers denoted by ``$+$'' represent reconstructed states using the algorithm in Eq.~(\ref{mlmeiteralgo1}) while ignoring the imperfection of the measurements, and those denoted by ``$\square$'' represent the reconstructed states using the algorithm in Eq.~(\ref{iteralgo2}) that accounts for this imperfection. The significant improvement in tomographic efficiency with the latter algorithm is a strong indication of the importance of a proper treatment of imperfect measurements.}
\label{fig:imp_det}
\end{figure}
\section{Applications}
\label{sec:app}
\subsection{Time-multiplexed detection tomography}
\label{subsec:tmd}
First, we apply the MLME technique to simulation experiments on \emph{time-multiplexed detection} (TMD) tomography \cite{TMD}. For experiments of this type, photon pulses, of a particular quantum state, containing more than one photon are sent through a series of beam splitters \cite{remark1}, each associated with a certain transmission probability. Behind each of the output ports of such a series is a single-photon detector that either registers a click from an incoming split photon pulse, with some detection efficiency, or does nothing. Thus, each output port has a certain overall efficiency $\tilde{\eta}_j$ which is related to the relevant transmission probabilities and detection efficiency (See Fig.~\ref{fig:tmd_diag}).
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{tmd_diag.eps}
\caption{A schematic diagram representing the time-multiplexed setup with $K+1$ output ports. The $T_j$s are the respective transmission probabilities for the $j$th beam splitter. The overall efficiency for, say, the $k$th port is given by $\tilde{\eta}_k=\eta_k(1-T_k+T_{K+1}\delta_{k,K+1})\prod^{k-1}_{j=1} T_j$.}
\label{fig:tmd_diag}
\end{figure}
As a consequence of this, the POM outcomes
\begin{equation}
\Pi_{j}=\sum_n\ket{n}c_{jn}\bra{n}
\end{equation}
will be a mixture of Fock states, with the coefficients $c_{jn}$ related to $\eta_j$ \cite{fiberloop}. If there are $N_\text{ports}$ output ports, where \emph{all} $\eta_j$s are different, there will be $2^{N_\text{ports}}$ distinct POM outcomes due to the binary nature of the single-photon detectors. In addition, $\sum^{2^{N_\text{ports}}}_{j=1}\Pi_j=1$ since the $2^{N_\text{ports}}$ binary sequences of detection configurations constitute all possible events. These POM outcomes commute and a measurement of these outcomes only gives information about the diagonal entries of the statistical operator of the true state in the Fock basis. In order to obtain information about the off-diagonal entries, one can, for instance, displace the current set of $2^{N_\text{ports}}$ POM outcomes in phase space with some complex value $\alpha_k$ away from the origin using the displacement operator
\begin{equation}
\mathcal{D}(\alpha_k)=\mathrm{e}^{\alpha_k A^\dagger-\alpha^*_k A}\,,
\end{equation}
where $A$ is the standard photon annihilation operator. Then, the new set of outcomes
\begin{equation}
\Pi_j(\alpha_k)=\frac{1}{\mathcal{N}}\mathcal{D}(\alpha_k)\Pi_j\mathcal{D}^\dagger(\alpha_k)\,,
\end{equation}
with $\mathcal{N}$ being the total number of such displaced set of $2^{N_\text{ports}}$ outcomes, do not commute with the undisplaced set. These displaced outcomes are suitable for a measurement that is designed to obtain information about the unknown true state by sampling over multiple $\alpha_k$s. Experimentally, these displaced POM outcomes can be realized with unbalanced homodyne detection \cite{unbalanced}.
In the simulations, four output ports, corresponding to a total of $2^4=16$ POM outcomes, are considered. Two different true states are selected to illustrate the results of MLME. The first true state is chosen to be a stationary state of a laser given by
\begin{equation}
\rho_{\text{ss}}=\mathrm{e}^{-\mu}\sum^{\infty}_{n=0}\ket{n}\frac{\mu^n}{n!}\bra{n}
\label{laser_ss}
\end{equation}
where $\mu$ is the mean number of photons \cite{laserss}. For the second true state, the statistical operator \mbox{$\rho_{\alpha'}=\ket{\textsc{m}(\alpha')}\bra{\textsc{m}(\alpha')}$}, where
\begin{equation}
\ket{\textsc{m}(\alpha')}=\frac{\ket{\alpha'}+\ket{-\alpha'}}{\sqrt{2\left(1+\mathrm{e}^{-2|\alpha'|^2}\right)}}\,
\label{schro_cat}
\end{equation}
is the superposition of the coherent states $\ket{\alpha'}$ and $\ket{-\alpha'}$, is chosen. The notation $\ket{\textsc{m}(\alpha')}$ is used to denote the ket for the \emph{male} ``Schr{\"o}dinger's cat'' state. See, for example, Ref.~\cite{catref} for a survey of the family of cat states. Statistical operators are first reconstructed from the simulated data. For this reconstruction, one has to decide on the dimension $D_\text{sub}$ of the truncated Hilbert space for the reconstructions. This procedure, also commonly known as \emph{state-space truncation}, depends on the prior information about the unknown state. In our case, suppose one knows that the mean number of photons of the source is $\mu\approx 4$, which is the value assigned in the simulation. Then, one may anticipate that all the relevant information about the true state should be contained in a Hilbert space of a dimension which is close to $\mu$. In fact, it is a common practice to choose $D_\text{sub}$, compatible with this information, such that the displaced operators form an informationally complete POM. Then, the standard ML method can be applied to state estimation. We shall compare the result of this approach with another, perhaps more objective, methodology in which we select a larger subspace compatible with this prior information and estimate the state with MLME.
After obtaining the reconstructed statistical operators, the Wigner functions $W(x,p)$ of the dimensionless position and momentum quadrature values, $x$ and $p$ respectively, are calculated in accordance with
\begin{align}
&W(x,p)\nonumber=2\mathrm{e}^{-|\alpha|^2}\sum^{\infty}_{m=0}\sum^{\infty}_{n=0}\bra{m}\rho\ket{n}\nonumber\\
\times &\left[(-1)^{j_<}\sqrt{\frac{2^{j_>}j_<!}{2^{j_<}j_>!}}(x+\mathrm{i}^{\,\text{sgn}(n-m)} p)^{|m-n|}L_{j_<}^{(|m-n|)}\left(2\,|\alpha|^2\right)\right]\,,
\label{wigner}
\end{align}
where $\alpha=x+\mathrm{i} p$ and $L_{n}^{(\nu)}(y)$ is the degree-$n$ \emph{associated Laguerre polynomial} in $y$ of order $\nu$, for all the statistical operators. Here, we define $j_<\equiv \min\{m,n\}$ and $j_>\equiv \max\{m,n\}$.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{tmd_density_ss.eps}
\caption{Density plots of the Wigner functions, in phase space, of various statistical operators for (a) the true state (20-dimensional stationary state of a laser, $\mu=4$) with $\tilde\tau\approx 0.394$, (b) the 5-dimensional ML estimator with $\tilde\tau\approx 0.921$ and (c) the 11-dimensional MLME estimator with $\tilde\tau\approx 0.489$. Here, brighter regions indicate the locations of larger Wigner function values, and vice versa. The statistical operator for (b) is obtained using ML by assuming a 5-dimensional subspace in which the displaced POM outcomes are informationally complete. The statistical operator for (c) is obtained by assuming a larger subspace of dimension 11 using MLME. Numerous artificial nonclassical features of the ML estimator, a signature of its highly oscillatory Wigner function, are manifested as an abnormally large value of $\tilde\tau$, an inevitable byproduct of state-space truncation. One can see that with MLME, extraneous artifacts of the Wigner function resulted from such a truncation can be largely removed.}
\label{fig:tmd_density}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{tmd_density_cat.eps}
\caption{Density plots of the Wigner functions, in phase space, of various statistical operators for (a) the true state ($\rho_{\alpha'}$, $\alpha'=5$), (b) the 8-dimensional ML estimator, (c) the 10-dimensional and (d) 15-dimensional MLME estimators. In this case, the Wigner function of the ML estimator differs greatly from that of the true state, an example of misleading information obtained via state-space truncation. A transition in the structure of the Wigner function occurs at $D_\text{sub}=10$, with the MLME estimator for $D_\text{sub}=15$ giving a more accurate estimated picture of the Wigner function of the true state.}
\label{fig:tmd_density_cat}
\end{figure}
To quantify the nonclassicality of the statistical operators, we make use of the concept of \emph{nonclassicality depth} introduced in Ref.~\cite{nonclassicalitydepth}. Let us define the function
\begin{equation}
\mathcal{R}(\alpha,\tau)=\frac{1}{\pi\tau}\int (\mathrm{d} w)\,\text{exp}\left(-\frac{|\alpha/\sqrt{2}-w|^2}{\tau}\right)P(w)\,,
\end{equation}
where $w$ is a complex variable, $(\mathrm{d} w)$ denotes the integral measure over the real and imaginary parts of $w$, $P(w)$ is the \emph{Glauber-Sudarshan $P$ function}, and the parameter $\tau$ is in the range $0\leq\tau\leq1$. From the above definition, it follows that $\mathcal{R}(\alpha,\tau)$ is a continuous interpolating function of $\tau$ from the typically singular, as well as non-positive, $P(\alpha/\sqrt{2})$ ($\tau\rightarrow 0$), to the Wigner function $W(\alpha)$ ($\tau=1/2$), and finally to the positive \emph{Husimi $Q$ function} $Q(\alpha)=\bra{\alpha}\rho\ket{\alpha}/2\pi$ ($\tau\rightarrow 1$).
The nonclassicality depth is then defined as the smallest value $\tau=\tilde\tau$, above which $\mathcal{R}(\alpha,\tau)\geq0$. Any mixture of coherent states is therefore a classical state since, in this case, $\tilde\tau=0$. A quantum state with $\tilde\tau>0$ is a nonclassical state. This measure of nonclassicality captures the nonclassical nature of quantum states through a one-parameter family of functions, which can otherwise be invisible to measures involving a fixed value of $\tau$, such as the conventional negativity of the Wigner function. Although quantifying nonclassicality with $\tilde\tau$ is a somewhat arbitrary procedure, we adopt it here as a measure of nonclassicality that is not worse than other proposals.
The generalization of \eqref{wigner} to arbitrary $\tau$ values,
\begin{align}
&\mathcal{R}(x,p,\tau)=\frac{\mathrm{e}^{-\frac{|\alpha|^2}{2\tau}}}{\tau}\sum^{\infty}_{m=0}\sum^{\infty}_{n=0}\bra{m}\rho\ket{n}\nonumber\\
\times \Bigg[&\,(-1)^{j_<}\sqrt{\frac{j_<!}{j_>!}}\left(\frac{1-\tau}{\tau}\right)^{j_>}\nonumber\\ &\,\left(\frac{x+\mathrm{i}^{\,\text{sgn}(n-m)} p}{\sqrt{2}(1-\tau)}\right)^{|m-n|}L_{j_<}^{(|m-n|)}\left(\frac{|\alpha|^2}{2\tau(1-\tau)}\right)\Bigg]\,,
\label{nonclass_eq}
\end{align}
is useful for the numerical computation of $\tilde\tau$. For the stationary state in Eq.~(\ref{laser_ss}), Eq.~(\ref{nonclass_eq}) simplifies to
\begin{align}
&\mathcal{R}_{\text{ss}}(x,p,\tau)\nonumber\\
=&\,\frac{\mathrm{e}^{-\frac{|\alpha|^2}{2\tau}-\mu}}{\tau}\sum^\infty_{n=0}(-1)^n\,\frac{\mu^n}{n!}\left(\frac{1-\tau}{\tau}\right)^nL_n\left(\frac{|\alpha|^2}{2\tau(1-\tau)}\right)\,.
\end{align}
The performances of both MLME and the standard ML method on the true states defined in Eqs.~(\ref{laser_ss}) and (\ref{schro_cat}) are illustrated by the Wigner function plots of the respective statistical operators obtained from both methods. These are shown in Figs.~\ref{fig:tmd_density} and \ref{fig:tmd_density_cat}. The respective nonclassicality depths are also computed for Fig.~\ref{fig:tmd_density}. For the state $\rho_{\alpha'}$, all the corresponding reconstructed statistical operators are highly nonclassical, with $\tilde\tau=1$ \cite{telenonclass} for all of them. Hence, rather than compare the $\tilde\tau$ values, the structure of the Wigner functions for various reconstruction subspaces will be briefly analyzed instead in Fig.~\ref{fig:tmd_density_cat}.
\subsection{Light-beam tomography}
\label{subsec:beam}
\begin{figure}
\includegraphics[width=0.8\columnwidth]{shdet}
\caption{Schematic diagram of the diffraction patterns of an incoming light beam that is obtained from a SH wave front sensor. The light beam is transformed by an array of microlenses (apertures). A CCD camera is placed at the rear focal plane of the array. The measurement data consist of the measured intensities of the beam. The intensity at the $j$th pixel, located at position $x_j$, behind the $k$th microlens aperture is denoted by $I_k(x_j)$.\label{fig:sh}}
\end{figure}
Finally, we make use of the MLME algorithm to reconstruct states of classical light beams that are measured using the Shack-Hartmann (SH) wave front sensor. An incoming light beam is transformed by a regular array of microlens apertures and detected in its rear focal plane by a charge-coupled device (CCD) camera (see Fig.~\ref{fig:sh}). A plane wave traversing in the transverse plane of the SH sensor gives rise to a detection, where the individual diffraction patterns are centered at the corresponding optical centers of the microlenses. For a distorted wave front, the observed diffraction pattern behind the $k$th microlens aperture will be deflected by an angle $\theta_k$. Since the set of angles $\theta_k$ is related to the local wave front tilts with respect to the transverse plane of the SH sensor, the shape of the wave front can be inferred. Clearly, this standard technique of wave front reconstruction fails in the presence of imperfect coherence, where the notions of ``wave front'' and ``optical phase'' are no longer well-defined and a more general description of the state of the light beam is necessary.
Recently, an alternative theory for SH detection, based on the principles of quantum state tomography, has been introduced. It was shown that a complete characterization of a beam of light is possible
from the measurement data obtained with the SH sensor under certain assumptions with regard to the aperture profiles \cite{coherence}. Analogously to quantum states, we can describe a coherent beam (mode), with a complex amplitude $\psi(x)$, by a ket $\ket{\psi}$, such that $\psi(x)=\langle x|\psi\rangle$. It should be understood, that this $\psi(x)$ is not a quantum mechanical probability amplitude, but a mathematical symbol with analogous properties that we exploit. At the focal plane of the $k$th microlens aperture, the amplitude $\psi'_k(x)$ of the transformed beam is given by
\begin{equation}
\label{transform}
\psi'_k(x)=\int\mathrm{d} x'\,h_k(x-x')a_k(x')\psi(x'),
\end{equation}
where $a_k(x)$ is the aperture function of the $k$th microlens aperture and the response function $h_k(x)$ describes the free propagation from the $k$th microlens to the SH sensor.
Now, suppose a generic \emph{partially} coherent beam is detected by the SH sensor. We can describe the state of such a beam with a \emph{coherence operator} $\rho_\text{coh}$. When using a computational basis of orthonormal modes $|\psi_n\rangle$, we have
\begin{equation}
\rho_\text{coh}\,=\,\sum_{mn}\ket{\psi_m}\rho^\text{coh}_{mn}\bra{\psi_n}.
\end{equation}
By defining the aperture operator
\begin{equation}
M^{(a)}_k=\int\mathrm{d} x'\,\ket{x'}a_k(x')\bra{x'}
\end{equation}
for the $k$th microlens aperture and the unitary propagation operator $U_k$, where $\bra{x}U_k\ket{x'}=h_k(x-x')$, that describes the free propagation from the $k$th microlens to the SH sensor, the representation of the corresponding transformed state $\rho'_\text{coh}$,
\begin{align}
\rho'_\text{coh}&=\,U_k\,M^{(a)}_k\,\rho_\text{coh}\,M^{(a)}_k\,U_k^\dagger\nonumber\\
&=\sum_{mn}\underbrace{U_k\,M^{(a)}_k\ket{\psi_m}}_{\equiv\ket{\psi'_m}}\rho^\text{coh}_{mn}\underbrace{\bra{\psi_n}M^{(a)}_k\,U_k^\dagger}_{\equiv\bra{\psi'_n}}\nonumber\\
&=\sum_{mn}\ket{\psi'_m}\rho^\text{coh}_{mn}\bra{\psi'_n}\,,
\end{align}
on the focal plane of the apertures follows from the linearity of optics transformations. The intensity $I_k(x_j)$ at position $x_j$ \cite{remark2} on the rear focal plane of the $k$th aperture is
\begin{equation}
\begin{split}
\label{intensity1}
I_k(x_j)&\equiv\langle x_j|\rho'_\text{coh}|x_j\rangle\\
&=\langle x_j| \bigg(\sum_{mn}\ket{\psi'_{m,j}}\rho^\text{coh}_{mn}\bra{\psi'_{n,k}}\bigg)
|x_j\rangle\\
&=\sum_{mn}\rho^\text{coh}_{mn}\,\psi'_{m,k}(x_j)\psi'_{n,k}(x_j)^*\,,
\end{split}
\end{equation}
where $\psi'_{n,k}(x_j)=\langle x_j|\psi'_{n,k}\rangle$ are the complex amplitudes of the transformed light beam obtained from the amplitudes $\psi_n(x_j)=\langle x_j|\psi_n\rangle$ of Eq.~\eqref{transform}. Since $\rho_\text{coh}$ possesses all the properties of a statistical operator, the MLME technique can be used to estimate the true coherence operator $\rho^\text{true}_\text{coh}$ of a partially coherent beam. To this end, we need to compute the corresponding POM describing the measurement outcomes of the SH sensor. By relating $I_k(x_j)$ to the corresponding probabilities of the outcomes $\Pi_k(x_j)\widehat{=}\sum_{mn}\ket{\psi'_m}\Pi_{k,nm}(x_j)\bra{\psi'_n}$, we have
\begin{equation}
\label{intensity2}
\begin{split}
I_k(x_j)&=\tr{\rho_\text{coh}\,\Pi_k(x_j)}\\
&=\sum_{mn}\rho^\text{coh}_{mn}\,\Pi_{k,nm}(x_j)\,.
\end{split}
\end{equation}
Comparing Eqs.~\eqref{intensity1} and \eqref{intensity2}, the positive operator describing the detection outcome at the
$j$th pixel of the CCD camera behind the $k$th aperture is given by
\begin{equation}
\label{POMelements}
\Pi_{k,nm}(x_j)=\psi'_{m,k}(x_j)\psi'_{n,k}(x_j)^*.
\end{equation}
As an illustrative example, the POM outcomes considered in this section are commuting operators in the infinite-dimensional Hilbert space with regard to the coherence operators. Equivalently, the aperture functions for the respective microlenses do not overlap in position. This is a \emph{special case} of a more general theory on Shack Hartmann detection, which will be discussed at length in another upcoming article.
\begin{figure}
\setlength{\unitlength}{0.8cm}
\begin{picture}(6,5)
\put(-4.5,-3){\includegraphics[width=1.35\columnwidth]{setup}}
\end{picture}
\caption{Experimental set-up involving a single-mode fiber (SMF), a spatial light modulator (SLM), an aperture stop (A) and a Shack-Hartmann (SH) sensor. \label{fig:setup}}
\end{figure}
In the experiment, a controlled preparation of optical beams is realized using the principles of digital holography \cite{setup}. Figure~\ref{fig:setup} shows the set-up. The essence of the beam preparation lies in the numerical construction of a digital hologram that is programmed to produce a superposition of a reference plane wave and a beam with the true state $\rho^\text{true}_\text{coh}$ of interest. This is achieved with the help of an amplitude spatial light modulator (OPTO SLM) with a resolution of 1024$\times$768 pixels. The hologram is then illuminated by the reference plane wave that is considered in the superposition. To approximately produce this plane wave, a collimated Gaussian beam is generated by placing the output of a single-mode fiber at the focal plane of a collimating lens. In this way, the digital hologram can be fully situated at the center of the collimated Gaussian beam of a larger beam waist, where this beam can then be approximated to be a plane wave with high accuracy. The resulting diffraction spectrum, after illuminating the digital hologram with the collimated Gaussian beam, involves several diffraction orders, of which only one contains useful information about $\rho^\text{true}_\text{coh}$. To filter out the unwanted diffraction orders, a 4-$f$ optical processor, with a small circular aperture stop placed at the rear focal plane of the second lens, is used for this purpose (the aperture stop in Fig.~\ref{fig:setup}). The resulting light beam with the state $\rho^\text{true}_\text{coh}$ is then focussed at the rear focal plane of the third lens. This completes the preparation stage.
The measurement of the light beam involves a Flexible Optical SH sensor with 128 microlenses that form a hexagonal array. Each microlens has a focal length of 17.9mm and a hexagonal aperture with a diameter of 0.3mm. The signal at the focal plane of the array is detected by a uEye CCD camera that has a resolution of 640$\times$480 pixels, with each pixel being 9.9$\upmu$m$\times$9.9$\upmu$m in size.
The aforementioned set-up is used for generating and analyzing low-order Laguerre-Gaussian (LG) modes. The LG modes can serve as important resources in quantum information processing \cite{Lgmodes}. In this experiment, only LG modes with no radial nodes are considered. Such modes form a one-parameter orthonormal basis, where the modes are specified by the orbital angular momentum quantum number $l$. In polar coordinates, the relevant part of the complex amplitude of a LG mode, for a fixed $l$, is given by
\begin{equation}
\langle s,\varphi|\text{LG}_l\rangle\propto s^l \mathrm{e}^{\mathrm{i} l \varphi} \mathrm{e}^{-s^2}\,.
\end{equation}
Nonzero values of $l$ give rise to helical wave fronts, for which each photon carries an orbital angular momentum of $l\hbar$.
For the source of light beams, we would like to prepare the state $\rho^\text{true}_\text{coh}=\rho^\text{sup}_\text{coh}=\ket{\psi_\text{sup}}\bra{\psi_\text{sup}}$, where
\begin{equation}
\label{statetrue}
\ket{\psi_\text{sup}}=\left(\ket{\text{LG}_0}-\ket{\text{LG}_1}\mathrm{i}-\ket{\text{LG}_2}\right)\frac{1}{\sqrt{3}}\,,
\end{equation}
using the OPTO SLM. In the presence of experimental imperfections, however, the true state $\rho^\text{true}_\text{coh}$ prepared this way will not be exactly the same as $\rho^\text{sup}_\text{coh}$. After measuring this beam with the SH sensor, the data are processed using the MLME algorithm in Eq.~(\ref{iteralgo2}) to obtain the estimator $\hat\rho^\text{MLME}_\text{coh}$ for $\rho^\text{true}_\text{coh}$, since $G<1$. To quantify the quality of $\hat\rho^\text{MLME}_\text{coh}$, we investigate the \emph{fidelity} between $\hat\rho^\text{MLME}_\text{coh}$ and $\rho^\text{sup}_\text{coh}$.
\begin{figure}
\includegraphics[width=0.9\columnwidth]{data012}
\caption{CCD image for the state $\rho^\text{true}_\text{coh}$. The relevant part of the SH readout used for the beam reconstruction is shown. Contributions from the individual SH apertures are indicated by bright spots, with each spot made up of multiple pixels. Note that the two void regions correspond to the phase singularities of the state $\rho^\text{sup}_\text{coh}$. This hints that $\rho^\text{true}_\text{coh}\approx\rho^\text{sup}_\text{coh}$.
\label{fig:data}}
\end{figure}
Figure~\ref{fig:data} shows the CCD image for the state $\rho^\text{true}_\text{coh}$. Each aperture gives rise to a bright spot in the CCD image. To maximize the signal-to-noise ratio, only the pixel with the highest intensity within each spot is selected as a measurement datum. The set of intensities, corresponding to maximum-intensity pixels, constitute the measurement data to be used for state reconstruction. In our case, the corresponding POM consists of $35$ linearly independent outcomes described by Eq.~\eqref{POMelements}. This measurement is, therefore, informationally complete for $D_\text{sub}\leq5$.
\begin{figure}
\includegraphics[width=\columnwidth]{reconstruction9}
\caption{MLME state estimation from informationally incomplete data for $D_\text{sub}=9$. The real (left) and imaginary (right) parts of the reconstructed coherence operator $\hat\rho^\text{MLME}_\text{coh}$ are shown. The reconstruction subspace is spanned by the modes $\text{LG}_l$, with $l=0,1,\ldots,8$. In this case, $56$ out of $91$ independent outcomes, required for complete characterization of $\rho^\text{true}_\text{coh}$, are not accessible, yet the MLME estimator $\hat\rho^\text{MLME}_\text{coh}$ is close to $\rho^\text{sup}_\text{coh}$, with a fidelity of $92\%$.
\label{fig:recon}}
\end{figure}
In cases where state reconstruction on informationally complete subspaces gives unsatisfactory results, the MLME approach can be used on the informationally incomplete data to give reasonable estimators on a larger subspace, as illustrated in Fig.~\ref{fig:recon}.
\begin{figure}[h!]
\vspace{10pts}
\includegraphics[angle=-90,width=0.75\columnwidth]{meanfid}
\caption{Average fidelities, computed over 50 random choices of computational bases, of the estimators for different dimensions $D_\text{sub}$ of the reconstruction subspace. The unfilled (filled) circular plot markers correspond to informationally complete (incomplete) tomography,
respectively.
\label{fig:meanfid}}
\end{figure}
So far, the procedure of state-space truncation is performed in the basis of the $\text{LG}_l$ modes. In this basis, when $\rho^\text{true}_\text{coh}$ is known to be quite close to $\rho^\text{sup}_\text{coh}$, the truncation of modes of higher orders will not result in a great loss of reconstruction information, as implied by the structure of $\rho^\text{sup}_\text{coh}$ in Eq.~(\ref{statetrue}). The situation will be very different when there is no such prior knowledge about $\rho^\text{true}_\text{coh}$, except for the fact that the possible values of $l$ lie in a certain range. In this situation, there is no appropriate strategy to choose a computational basis in which the state-space truncation can be done effectively and justifiably. More generally, estimating the unknown state $\rho^\text{true}_\text{coh}$ on a truncated subspace will, as a rule, result in missing important reconstruction information and this will lead to strongly biased estimators. A remedy for this problem is to perform state reconstruction on a sufficiently large subspace that is compatible with the knowledge about the range of values of $l$.
To emphasize this point, we simulate the following scenario:
\begin{itemize}
\item The set of measurement data, obtained from the CCD image shown in Fig.~\ref{fig:data}, is distributed to $50$ parties. The possible values of $l$ for the true state $\rho^\text{true}_\text{coh}$ are known to lie in the range $l\in[0,7]$.
\item Each party selects a computational basis and estimates the state of the beam for $D_\text{sub}=3,4,\ldots,8$ using either the ML (for $D_\text{sub}\leq5$) or the MLME algorithm (for $D_\text{sub}>5$).
\item The reconstructed estimators for the six values of $D_\text{sub}$ are reported by each party and the average fidelity of the estimators for every value of $D_\text{sub}$ are calculated.
\end{itemize}
A typical outcome of this scenario is shown in Fig.~\ref{fig:meanfid}. As can be seen, performing state-space truncations in order to reconstruct $\rho^\text{true}_\text{coh}$ with an informationally complete set of data generally leads to low fidelities in the estimators. Increasing the number of degrees of freedom and using the MLME algorithm to cope with the completeness issue seems to be a much better strategy.
\section{Conclusion}
\label{sec:conc}
We derived the iterative algorithms for informationally incomplete quantum state estimation respectively for perfect and imperfect measurements. Next, we applied these algorithms to time-multiplexed detection tomography and light-beam tomography. From these two applications, we learned that one should better not restrict the state reconstruction to a subspace in which the relevant measurements are informationally complete. Doing so can result in reconstruction artifacts that originate in the state-space truncation and may result in inaccurate estimators for the unknown true state. Instead, one should perform the reconstruction on a larger subspace, with additional unsampled degrees of freedom, that is compatible with any prior information about a given unknown state. Such a more objective way of state estimation results in a much better tomographic quality of the reconstructed estimator.
\section{Acknowledgements}
This work is supported by the NUS Graduate School for Integrative Sciences and Engineering and the Centre for Quantum Technologies, which is a Research Centre of Excellence funded by Ministry of Education and National Research Foundation of Singapore, as well as the Czech Ministry of Education, Project LC06007, IGA Project PRF\underline{\,\,\,}2011\underline{\,\,\,}005, and the Czech Ministry of Industry and Trade, Project FR-TI1/364.
|
3,212,635,537,497 | arxiv | \section{Introduction}
Let $V$ be the standard representation of $\ss$ (that is, $V$ is the hyperplane $\sum x_i=0$ in $\C^6$, with $\ss$ acting by permutation of the basis vectors).
The quartic hypersurfaces in $\P(V)\ (\cong \P^4)$ invariant under $\ss$ form the pencil
\[X_t:\ t\sum x_i^4-(\sum x_i^2)^2=0 \ ,\quad t\in \P^1\ .\]
This pencil contains the Burkhardt quartic (for $t=2 $) and the Igusa quartic ($t={4} $), which are both rational.
For $t\neq 0,2,4,6 $ and $\frac{10}{17}$, the quartic $X_t$ has exactly 30 nodes; the set of nodes $\mathcal{N}$ is the orbit under $\ss$ of $(1,1,\rho ,\rho ,\rho ^2,\rho ^2)$, with $\rho =e^{\frac{2\pi i}{3} }$ (\cite{vdG}, \S 4).
We will prove:
\begin{thm*}
For $t\neq 0,2,4,6 ,\frac{10}{7}$, $X_t$ is not rational.
\end{thm*}
The method is that of \cite{B} : we show that the intermediate Jacobian of a desingularization of $X_t$ is 5-dimensional and that the action of $\ss$ on its tangent space at $0$ is irreducible. From this one sees easily that this intermediate Jacobian cannot be a Jacobian or a product of Jacobians, hence $X_t$ is not rational by the Clemens-Griffiths criterion. We do not know whether $X_t$ is unirational.
\medskip
{\small I am indebted to A. Bondal and Y. Prokhorov for suggesting the problem, and to A. Dimca for explaining to me how to compute explicitly the defect of a nodal hypersurface.}
\bigskip
\section{The action of $\ss$ on $T_0(JX)$}
We fix $t\neq 0,2,4,6 ,\frac{10}{7}$, and denote by $X$ the desingularization of $X_t$ obtained by blowing up the nodes. The main ingredient of the proof is the fact that the action of $\ss$ on $JX$ is non-trivial. To prove this we consider the action of $\ss$ on the tangent space $T_0(JX)$, which is by definition $H^2(X,\Omega ^1_X)$.
\begin{lem}
Let $\mathcal{C}$ be the space of cubic forms on $\P(V)$ vanishing along $\mathcal{N}$. We have an isomorphism of $\ss$-modules $\mathcal{C}\cong V\oplus H^2(X,\Omega ^1_X)$.
\end{lem}
\pr The proof is essentially contained in \cite{C}; we explain how to adapt the arguments there to our situation. Let $b: P\rightarrow \P(V)$ be the blowing-up of $\P(V)$ along $\mathcal{N}$. The threefold $X$ is the strict transform of $X_t$ in $P$.
The exact sequence
\[0\rightarrow N^*_{X/P}\longrightarrow \Omega ^1_{P\, | X}\longrightarrow \Omega ^1_X\rightarrow 0 \]
gives rise to an exact sequence
\[0\rightarrow H^2(X,\Omega ^1_X)\longrightarrow H^3(X,N^*_{X/P}) \longrightarrow H^3(X,\Omega ^1_{P\, | X})\rightarrow 0\ . \] (\cite{C}, proof of theorem 1), which is $\ss$-equivariant. We will compute the two last terms.
The exact sequence
\[0\rightarrow \Omega ^1_P(-X)\longrightarrow \Omega ^1_P\longrightarrow \Omega ^1_{P\, | X}\rightarrow 0\]
provides an isomorphism $H^3(X,\Omega ^1_{P\, | X})\iso H^4(P, \Omega ^1_P(-X))$, and the latter space is isomorphic to
$H^4(\P(V), \Omega ^1_{\P(V)}(-4))$ (\cite{C}, proof of Lemma 3). By Serre duality $H^4(\P(V), \Omega ^1_{\P(V)}(-4))$ is dual to $H^0(\P(V), T_{\P(V)}(-1))\cong V$. Thus the $\ss$-module $H^3(X,\Omega ^1_{P\, | X})$ is isomorphic to $V^*$, hence also to $V$.
Similarly the exact sequence $\ 0\rightarrow \O_P(-2X)\longrightarrow \O_P(-X)\longrightarrow N^*_{X/P}\rightarrow 0\ $ and the vanishing of $H^i(P, \O_P(-X))$ (\cite{C}, Corollary 2) provide an isomorphism of $H^3(X,N^*_{X/P})$ onto $H^4(P, \O_P(-2X))$, which is naturally isomorphic to the dual of $\mathcal{C}$ (\cite{C}, proof of Proposition 2). The lemma follows.\qed
\begin{lem}
The dimension of $\mathcal{C}$ is $10$.
\end{lem}
\pr Recall that the \emph{defect} of $X_t$ is the difference between the dimension of $\mathcal{C}$ and its expected dimension, namely :
\[ \mathrm{def}(X_t):= \dim \mathcal{C} -( \dim H^0(\P(V), \O_{\P(V)}(3) )-\#\,\mathcal{N})\ .\]
Thus our assertion is equivalent to $\mathrm{def}(X_t)=5$.
To compute this defect we use the formula of \cite{D-S}, Theorem 1.5. Let $F=0$ be an equation of $X_t$ in $\P^4$; let $R:= \C[X_0,\ldots ,X_4]/(F'_{X_0},\ldots ,F'_{X_4})$ be the Jacobian ring of $F$, and let $R^{sm}$ be the Jacobian ring of a \emph{smooth} quartic hypersurface in $\P^4$. The formula is
\[ \mathrm{def}(X_t)= \dim R_7-\dim R^{sm}_7\ .\]
In our case we have $\dim R^{sm}_7=\dim R^{sm}_3=35-5=30$; a simple computation with Singular (for instance) gives $\dim R_7=35$. This implies the lemma.\qed
\bigskip
\begin{prop}
The $\ss$-module $H^2(X,\Omega ^1_X)$ is isomorphic to $V$.
\end{prop}
\pr
Consider the homomorphisms $a$ and $b$ of $\C^6$ into $H^0(\P(V), \O_{\P(V)}(3))$ given by $a(e_i)=x_i^3$, $b(e_i)=x_i\sum x_j^2$. They are both $\ss$-equivariant and map $V$ into $\mathcal{C}$; the subspaces $a(V)$ and $b(V)$ of $\mathcal{C}$ do not coincide, so we have $a(V)\cap b(V)=0$. By Lemma 2 this implies $\mathcal{C}=a(V)\oplus b(V)$, so
$H^2(X,\Omega ^1_X)$ is isomorphic to $V$ by Lemma 1.\qed
\medskip
\begin{rem}
Suppose $t=2,6$ or $\frac{10}{7} $. Then the singular locus of $X_t$ is $\mathcal{N}\cup\mathcal{N}'$, where $\mathcal{N}'$ is the $\ss$-orbit of the point $(1,-1,0,0,0,0)$ for $t=2$, $(-1,-1,-1,1,1,1)$ for $t=6$, $(-5,1,1,1,1,1)$ for $t=\frac{10}{7} $ \cite{vdG}. Since $x_1^3-x_0^3$ does not vanish on $\mathcal{N}'$, the space of cubics vanishing along $\mathcal{N}\cup\mathcal{N}'$ is strictly contained in $\mathcal{C}$.
By Lemma 1 it contains a copy of $V$, hence it is isomorphic to $V$; therefore $H^2(X,\Omega ^1_X)$ and $JX$ are zero in these cases. We have already mentioned that $X_2$ and $X_4$ are rational; we do not know whether this is the case for $X_6$ and $X_{\frac{10}{7} }$.
\end{rem}
\bigskip
\section{Proof of the theorem}
To prove that $X$ is not rational, we apply the Clemens-Griffiths criterion (\cite{CG}, Cor. 3.26): it suffices to prove that $JX$ is not a Jacobian or a product of Jacobians.
Suppose $JX\cong JC$ for some curve $C$ of genus $5$. By the Proposition $\ss$ embeds into the group of automorphisms of $JC$ preserving the principal polarization; by the Torelli theorem this group is isomorphic to $\Aut(C)$ if $C$ is hyperelliptic and $\Aut(C)\,\times \,\Z/2\ $ otherwise. Thus we find $\#\Aut(C)\geq \frac{1}{2}6! =360$. But this contradicts the Hurwitz bound $\ \#\Aut(C)\leq 84(5-1)=336$.
Now suppose that $JX$ is isomorphic to a product of Jacobians $J_1\times \ldots \times J_p$, with $p\geq 2$. Recall that such a decomposition is \emph{unique} up to the order of the factors: it
corresponds to the decomposition of the Theta divisor into irreducible components (\cite{CG}, Cor. 3.23). Thus the group $\ss$ permutes the factors $J_i$, and therefore acts on $[1,p]$; by the Proposition this action must be transitive. But we have $p\leq \dim JX=5$, so this is impossible.\qed
\bigskip
|
3,212,635,537,498 | arxiv | \section{Introduction}
Early computer vision approaches focused
on producing decent performance
on small datasets.
This often posed overwhelming difficulties,
so researchers seldom quantified
the prediction confidence.
An important mi\-le\-stone
was reached when generalization
was achieved
on realistic datasets
such as Pascal VOC \cite{everingham10ijcv},
CamVid \cite{brostow08eccv},
KITTI \cite{geiger13ijrr}, and
Cityscapes \cite{cordts15cvpr}.
These datasets assume
closed-world evaluation
\cite{scheirer13pami}
in which the training and test subsets
are sampled from the same distribution.
Such setup has been able to provide
a fast feedback on novel approaches
due to good alignment
with the machine learning paradigm.
This further accelerated development
and led us to the current state of research
where all these datasets are mostly solved,
at least in the strongly supervised setup.
Recent datasets further raise the bar
by increasing the number of classes
and image diversity.
However, despite this increased complexity,
the Vistas \cite{neuhold17iccv} dataset
is still an insufficient proxy
for real-life operation
even in a very restricted scenario
such as road driving.
New classes like bike racks
and ground animals were added,
however many important classes from
non-typical or worst-case images
are still absent.
These classes include
persons in non-standard poses,
crashed vehicles, rubble,
fallen trees etc.
Additionally, real-life images
may be affected by
particular image acquisition faults
including hardware defects,
covered lens etc.
This suggests that foreseeing every possible
situation may be an elusive goal,
and that our algorithms
should be designed to recognize
image regions which are foreign to
the training distribution.
The described deficiencies
emphasize the need
for a more robust approach
to dataset design.
First, an ideal dataset should identify
and target a set of explicit hazards
for the particular domain
\cite{zendel18eccv}.
Second (and more important),
an ideal dataset should endorse
open-set recognition paradigm
\cite{scheirer13pami}
in order to promote
detection of unforeseen hazards.
Consequently, the validation (val)
and test subsets should contain
various degrees of domain shift
with respect to the training distribution.
This should include
moderate domain shift factors
(e.g.\ adverse weather, exotic locations),
exceptional situations
(e.g.\ accidents, poor visibility, defects)
and outright outliers
(objects and entire images from other domains).
We argue that the WildDash dataset
\cite{zendel18eccv}
represents a step in the right direction,
although further development
would be welcome, especially
in the direction of enlarging
the negative part of the test dataset \cite{blum19arxiv}.
Models trained for open-set evaluation
can not be required
to predict an exact visual class
in outlier pixels.
Instead, it should suffice that
the outliers are recognized,
as illustrated in Fig.\,\ref{fig:approach}
on an image from the WildDash dataset.
\begin{figure}[htb]
\includegraphics[width=\textwidth]{simple_model.png}
\caption{
The proposed approach for simultaneous
semantic segmentation and outlier detection.
Our multi-task model predicts
i) a dense outlier map, and
ii) a semantic map with respect
to the 19 Cityscapes classes.
The two maps are merged to obtain
the final outlier-aware
semantic predictions.
Our model recognizes
outlier pixels (white)
on two objects
which are foreign to Cityscapes:
the ego-vehicle and the yellow forklift.
}
\label{fig:approach}
\end{figure}
This paper addresses simultaneous
semantic segmentation and
open-set outlier detection.
We train our models on inlier images
from two road-driving datasets:
Cityscapes \cite{cordts15cvpr}
and Vistas \cite{neuhold17iccv}.
We consider several
outlier detection approaches
from the literature
\cite{hendrycks17iclr,bevandic2018,hendrycks19iclr}
and validate their performance
on WildDash val (inliers),
LSUN val \cite{yu2015} (outliers),
and pasted objects from
Pascal VOC 2007 (outliers).
Our main hypotheses are
i) that training with noisy negatives
from a very large and diverse dataset
such as ImageNet-1k \cite{deng09cvpr}
can improve outlier detection,
and ii) that discriminative outlier detection and semantic
segmentation can share features
without significant deterioration of either task.
We confirm both hypotheses
by re-training our best models
on WildDash val, Vistas and ImageNet-1k,
and evaluating performance
on the WildDash benchmark.
\section{Related Work}
Previous approaches to
outlier detection in image data
are very diverse.
These approaches are based on
analyzing prediction uncertainty,
evaluating generative models,
or exploiting a broad secondary dataset
which contains both outliers and inliers.
Our approach is also related
to multi-task models
and previous work which explores
the dataset quality and dataset bias.
\subsection{Estimating Uncertainty
(or Confidence) of the Predictions}
Prediction confidence can be expressed
as the probability of the winning class
or max-softmax for short \cite{hendrycks17iclr}.
This is useful in image-wide
prediction of outliers,
although max-softmax
must be calibrated \cite{guo17icml}
before being interpreted as $P(\mathrm{inlier}|\mathbf{x})$.
The ODIN approach \cite{liang18iclr}
improves on \cite{hendrycks17iclr}
by pre-processing input images
with a well-tempered perturbation
aimed at increasing
the max-softmax activation.
These approaches are handy since they
require no additional training.
Some approaches model the uncertainty
with a separate head which learns either
prediction uncertainty
\cite{kendall17nips,lakshminarayanan17nips}
or confidence \cite{devries18arxiv}.
Such training is able to recognize examples
which are hard to classify
due to insufficient or inconsistent labels,
but is unable to deal with real outliers.
A principled information-theoretic approach
expresses the prediction uncertainty as
mutual information between
the posterior parameter distribution
and the particular prediction \cite{smith18uai}.
In practice, the required expectations
are estimated with Monte Carlo (MC) dropout
\cite{kendall17nips}.
Better results
have been achieved
with explicit ensembles
of independently trained models
\cite{lakshminarayanan17nips}.
However, both approaches
require many forward passes
and thus preclude real-time operation.
Prediction uncertainty
can also be expressed
by evaluating per-class
generative models
of latent features \cite{Lee2018ASU}.
However, this idea is not easily adaptable
for dense prediction in
which latent features
typically correspond to many classes
due to subsampling and dense labelling.
Another approach would be
to fit a generative model
to the training dataset and
to evaluate the likelihood of a given sample.
Unfortunately, this is very hard
to achieve with image data
\cite{nalisnick19iclr}.
\subsection{Training with Negative Data}
Our approach is most related
to three recent approaches
which train outlier detection
by exploiting a diverse
negative dataset \cite{torralba2011}.
The approach called outlier exposure
(OE) \cite{hendrycks19iclr}
processes the negative data
by optimizing cross entropy
between the predictions
and the uniform distribution.
Outlier detection has also been formulated
as binary classification \cite{bevandic2018}
trained to differentiate inliers
from the negative dataset.
A related approach \cite{Vyas2018OutofDistributionDU}
partitions the training data into K folds
and trains an ensemble of K
leave-one-fold-out classifiers.
However, this requires K forward passes.
while data partitioning
may not be straight-forward.
Negative training samples
can also be produced
by a GAN generator \cite{goodfellow14nips}.
Unfortunately, existing works
\cite{lee18iclr,sabokrou2018adversarially}
have been designed for
image-wide prediction in small images.
Their adaptation to dense prediction
on Cityscapes resolution
would not be straight-forward
\cite{brock19iclr}.
Soundness of training with negative data
has been challenged by \cite{shafaei2018}
who report under-average results
for this approach.
However, their experiments average results
over all negative datasets (including MNIST),
while we advocate for a very diverse
negative dataset such as ImageNet.
\subsection{Multi-task Training
and Dataset Design}
Multi-task models attach
several prediction heads
to shared features \cite{Caruana1997}.
Each prediction head has a distinct loss.
The total loss is usually expressed
as a weighted sum \cite{ngiam2011}
and optimized in an end-to-end fashion.
Feature sharing brings important advantages
such as cross-task enrichment
of training data \cite{bengio13pami}
and faster evaluation.
Examples of successful multi-task models
include combining
depth, surface normals and
semantic segmentation
\cite{Eigen2015PredictingDS},
as well as combining
classification, bounding box prediction
and per-class instance-level segmentation
\cite{mask}.
A map of task compatibility
with respect to knowledge transfer
\cite{zamir18cvpr}
suggests that many tasks are
suitable for multi-task training.
Dataset quality is
as a very important issue
in computer vision research.
Diverse negative datasets
have been used to reduce false positives
in several computer vision tasks
for a very long time \cite{torralba2011}.
A methodology for analyzing
the quality of road-driving datasets
has been proposed in \cite{zendel17ijcv}.
The WildDash dataset \cite{zendel18eccv}
proposes a very diverse validation dataset
and the first semantic segmentation benchmark
with open-set evaluation \cite{scheirer13pami}.
\begin{comment}
\begin{itemize}
\item outlier detection connected to uncertainty/confidence estimation:
\begin{itemize}
\item uncertainty estimated using max softmax \cite{hendrycks17iclr},
needs calibration \cite{guo17icml} because of overfitting
\item improve max softmax with ODIN \cite{liang18iclr}
\item learn to predict uncertainty
\cite{kendall17nips,lakshminarayanan17nips} or confidence
\cite{devries18arxiv}.
\item total uncertainity is a combination of different types of
uncertainties. \cite{kendall17nips} proposes two types of uncertainty:
aleatoric and epistemic.
\item \cite{smith18uai} proposes a way to calculate aleatoric and
epistemic uncertainty using output entropy and MC dropout
\item \cite{lakshminarayanan17nips} uses ensemble of several models
instead of MC dropout
\item \cite{Lee2018ASU} uses distance between features of the input sample
and the closest class-conditional Gaussian distribution (behaves similarly
to softmax on pixel level when applied to pixel level)
\item advantage of these approaches - no need to retrain models
\end{itemize}
\item outlier detection connected to generative models and single class classification:
\begin{itemize}
\item hard for high dimensional data, doesn't scale well, hard to easily
combine with existing classifiers
\item \cite{goodfellow14nips} introduces GANs, discriminator can be
used for outlier detection
\item \cite{sabokrou2018adversarially} autoencoder in place of the
generator, it is trained to denoise image. During evaluation the
autoencoder enhances inliers while distorting the outliers, making the
two more separable.
\item \cite{lee18iclr} train a GAN to generate images on the borders of
the distribution.
\end{itemize}
\item training with outliers:
\begin{itemize}
\item \cite{hendrycks19iclr} introduces outliers during training. For
outliers, KL divergence between model output and uniform distribution is
minimized
\item \cite{bevandic18arxiv} show that outlier pixels can be detected using a
binary classifier that was trained to differentiate between inliers and
and a larger, more diverse non-traffic scene dataset
\item \cite{Vyas2018OutofDistributionDU} train an ensemble of classifiers;
For each classifier some of the classes are ID and some are outlier
\end{itemize}
\item multitask DNNs:
\begin{itemize}
\item reduces computation, closer to real time application
\item \cite{Caruana1997} describes the task of learning multiple tasks
from the same representation
\item tasks usually done as a weighted sum of losses (e.g. \cite{Eigen2015PredictingDS},
\cite{ngiam2011})
\item \cite{zamir18cvpr} show that some visual tasks
are related and that related tasks can be solved in
a single system
\item \cite{dvornik2017blitznet} perform simultaneous
object detection and segmentation
\end{itemize}
\item since the task here is dense prediction in both cases, it is
similar to multilabel classification:
\begin{itemize}
\item \cite{mask}, Mask R-CNN, performs classification into classes and classification into objects and background
\end{itemize}
\item domain shift and fair evaluation:
\begin{itemize}
\item \cite{torralba2011} analyzes the problem of dataset bias
\item \cite{zendel17ijcv} analyze how to analyze the quality of a driving scenes
dataset
\item \cite{zendel18eccv} introduce WildDash based on previous
findings, wilddash validation is used as ID during testing
\item wilddash benchmark contains outlier samples
\item \cite{shafaei2018} define unbiased outlier detector evaluation - outlier
dataset used during training not used for outlier evaluation
\end{itemize}
\item improving generalization:
\begin{itemize}
\item \cite{kreso17cvrsuad}, \cite{orsic2019cvpr} show that imagenet pretraining improves
generalization
\item jittering during training, l2-regularization
\item generalization through model design (complicated models
prone to overfitting)
\end{itemize}
\end{itemize}
\end{comment}
\section{Simultaneous Segmentation and Outlier Detection}
\label{ss:model}
Our method combines two distinct tasks:
outlier detection and semantic segmentation,
as shown in Fig.\,\ref{fig:approach}.
We prefer to rely on shared features
in order to promote fast inference
and synergy between tasks \cite{bengio13pami}.
We assume that a large, diverse and noisy
negative dataset is available
for training purposes
\cite{hendrycks19iclr,bevandic2018}.
\subsection{Dense Feature Extractor}
Our models are based on
a dense feature extractor
with lateral connections
\cite{kreso17cvrsuad}.
The processing starts with a
DenseNet \cite{huang17cvpr} or
ResNet \cite{He2016DeepRL} backbone,
proceeds with spatial pyramid pooling (SPP)
\cite{he14eccv,zhao17cvpr}
and concludes with ladder-style upsampling
\cite{kreso17cvrsuad,lintsungyi17cvpr}.
The upsampling path consists of
three upsampling blocks (U1-U3)
which blend low resolution features
from the previous upsampling stage
with high resolution features from the backbone.
We speed-up and regularize the learning
with three auxiliary classification losses
(cf.~Fig.\,\ref{fig:model}).
These losses have soft targets corresponding
to ground truth distribution
across the corresponding window
at full resolution \cite{kreso19arxiv}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.9\textwidth]{arch2b4.png}
\caption{The proposed two-head model:
the classification head recovers semantic segmentation
while the outlier detection head
identifies pixels where
semantic segmentation may be wrong.
The output is produced by combining
these two dense prediction maps.
}
\label{fig:model}
\end{figure}
\subsection{Dense Outlier Detection}
There are four distinct approaches to formulate
simultaneous semantic segmentation and
dense outlier detection
over shared features.
The C-way multi-class approach
attaches the standard classification head
to the dense feature extractor
(C denotes the number of inlier classes).
The inlier probability
is formulated as max-softmax.
If a negative set is available,
this approach can be trained
to emit low max-softmax in outliers
by supplying a modulated
cross entropy loss term
towards uniform distribution
\cite{lee18iclr,hendrycks19iclr}.
The modulation factor
$\lambda_\textrm{KL}$
is a hyper-parameter.
Unfortunately, training on outliers
may compromise classification accuracy
and generate false positive outliers
at semantic borders.
The C-way multi-label approach
has C sigmoid heads.
The final prediction is the class
with maximum probability max-$\sigma$,
whereas the inlier probability
is formulated as max-$\sigma$.
Unfortunately, this formulation fails
to address the competition between classes,
which again compromises
classification accuracy.
The C+1-way multi-class approach
includes outliers as the C+1-th class,
whereas the inlier probability is
a 2-way softmax between
the max-logit over inlier classes
and the outlier logit.
To account for class disbalance
we modulate the loss due to outliers
with $\lambda_\textrm{C+1}$.
Nevertheless, this loss affects
inlier classification weights,
which may be harmful
when the negatives are noisy
(as in our case).
Finally, the two-head approach
complements the C-way classification head
with a head which directly predicts
the outlier probability
as illustrated in Fig.\,\ref{fig:model}.
The classification head is trained on inliers
while the outlier detection head is trained
both on inliers and outliers.
The outlier detection head uses
the standard cross entropy loss
modulated with hyper-parameter $\lambda_\textrm{TH}$.
We combine the resulting prediction maps
to obtain semantic segmentation
into C+1 classes.
The outlier detection head overrides
the classification head
whenever the outlier probability
is greater than a threshold.
Thus, the classification head is unaffected
by the negative data,
which provides hope to preserve
the baseline semantic segmentation accuracy
even when training on
extremely large negative datasets.
\subsection{Resistance to noisy outlier labels and sensitivity
to negative objects in positive context}
Training outlier detection on
a diverse negative dataset
has to confront noise
in negative training data.
For example, our negative dataset, ImageNet-1k,
contains several classes (e.g.\ cab, streetcar)
which are part of the Cityscapes ontology.
Additionally, several stuff
classes from Cityscapes
(e.g.\ building, terrain)
occur in ImageNet-1k backgrounds.
Nevertheless, these pixels
are vastly outnumbered by true inliers.
This is especially the case
when the training only considers
the bounding box of the object
which defines the ImageNet-1k class.
We address this issue
by training our models
on mixed batches
with approximately equal share
of inlier and negative images.
Thus, we perform many inlier epochs
during one negative epoch,
since our negative training dataset
is much larger than the inlier ones.
The proposed batch formation procedure
prevents occasional inliers from negative images
to significantly affect the training.
Additionally, it also favours
stable development of batchnorm statistics.
Hence, the proposed training approach
stands a fair chance to succeed.
\begin{comment}
We alleviate influence of noisy outliers
by forming batches with approximately
equal share of inlier and negative images.
Consequently, a relatively small number
of negative images with inlier pixels
can not significantly affect the training.
In practice, ImageNet-1k contains
several classes (e.g.\ cab, streetcar)
which are part of the Cityscapes ontology.
Additionally, several stuff
classes from Cityscapes
(e.g.\ building, terrain)
occur in ImageNet-1k backgrounds.
Nevertheless, these pixels
are vastly outnumbered by true outliers.
Hence, the proposed training approach
stands a fair chance to succeed.
\end{comment}
We promote detection
of outlier objects in inlier context
by pasting negative content
into inlier training images.
We first resize the negative image
to 5\% of the inlier image,
and then paste it at random.
We perform this before the cropping,
so some crops may contain only inlier pixels.
Unlike \cite{blum19arxiv},
we do not use the Cityscapes ignore class
since it contains many inliers.
\begin{comment}
In this paper we propose to combine the results of
two separate tasks - outlier detection and semantic segmentation -
into a single output. We show that the two tasks may share
convolutional features, as can be seen in Fig.
\ref{fig:approach}.
The model in Fig. \ref{fig:model} is based on ladder densenet
which comprises a backbone (e.g. densenet \cite{huang17cvpr} or
resnet \cite{He2016DeepRL}), spatial
pyramid pooling (SPP) module and an upsampling path.
The upsampling path contains three upsampling blocks.
One upsampling block is fed the outputs of a backbone block
and the previous upsampling block. These inputs are
processed and combined into a single output. The output of
the final upsampling block is propagated to outlier detector and
to semantic segmentator.
We use auxiliary losses for segmentation
along the upsampling path.
They have soft targets
where the expected value of a pixel
is a distribution calculated
for the corresponding pooling window
of the ground truth.
Semantic segmentation and outlier detection
can be performed by two heads
or by a single head.
The single head can be trained
only for segmentation.
Max-softmax of the segmentation
can be used for used as an estimate
of the confidence in the output
of the model. Low confidence can
indicate that a sample is an outlier.
Using outlier exposure, the model
can be explicitly trained to have low
max-softmax on outlier pixels.
This is done by minimizing
KL distance between uniform distribution
and the output of the model
for outlier pixels.
We can modify the single head by
swapping the softmax at the output of the
network with sigmods. In this setup,
it is possible for a pixel not to
have any class. Furthermore, for each class,
pixels of all of the other classes,
as well as the outlier images,
can be used as negative examples.
Another possible modification of
the segmentation head for outlier
detection is by adding an additional
"unknown" class.
The model with two heads separates the
two tasks. The outlier detection head can
be implemented as a binary classifier.
The binary head is trained using both
inlier and outlier images, while
the segmentation head ignores those
pixels.
\end{comment}
\section{Experiments}
We train most models on
Vistas inliers \cite{neuhold17iccv}
by mapping original labels
to Citysca\-pes \cite{cordts15cvpr} classes.
In some experiments we also use
Cityscapes inliers
to improve results
and explore influence of the domain shift.
We train all applicable models on outliers
from two variants of ImageNet-1k:
the full dataset (ImageNet-1k-full)
and the subset in which
bounding box annotations
are available (ImageNet-1k-bb).
In the latter case
we use only the bounding box
for training on negative images
(the remaining pixels are ignored)
and pasting into positive images.
We validate semantic segmentation
by measuring mIoU on WildDash val
separately from outlier detection.
We validate outlier detection
by measuring pixel level
average precision (AP)
in two different setups:
i) entire images are either
negative or positive, and
ii) appearance of negative
objects in positive context.
The former setup consists of
many assays
across WildDash val and random
subsets of LSUN images.
The LSUN subsets are dimensioned so that
the numbers of pixels in LSUN and WildDash val
are approximately equal.
Our experiments report mean and standard deviation
of the detection AP across 50 assays.
The latter setup involves WildDash val images
with pasted Pascal animals.
We select animals which take up
at least 1\% of the WildDash resolution,
and paste them at random in each WildDash image.
We normalize all images
with ImageNet mean and variance,
and resize them so that
the shorter side is 512 pixels.
We form training batches
with random 512$\times$512 crops
which we jitter with horizontal flipping.
We set the auxiliary loss weight to 0.4
and the classifier loss weight to 0.6.
We set $\lambda_\textrm{KL}$=0.2,
$\lambda_\textrm{C+1}$=0.05,
and $\lambda_\textrm{TH}$=0.2.
We use the standard Adam optimizer
and divide the learning rate
of pretrained parameters by 4.
All our models are trained
throughout 75 Vistas epochs,
which corresponds to
2 epochs of ImageNet-1k-full,
or 5 epochs of ImageNet-1k-bb.
We detect outliers by thresholding
inlier probability at $p_\textrm{IP}=0.5$.
Our models produce a dense index map
for C Cityscapes classes and 1 void class.
We obtain predictions at
the benchmark resolution
by bilinear upsampling.
We perform ODIN inference as follows.
First, we perform the forward pass
and the backward pass with respect
to the max-softmax activation
(we use temperature T=10).
Then we determine
the max-softmax gradient
with respect to pixels.
We determine the perturbation
by multiplying the sign of the gradient
with $\varepsilon$=0.001.
Finally, we perturb
the normalized input image,
perform another forward pass
and detect outliers according
to the max-softmax criterion.
\begin{comment}
Siniša moved this to section 3
When training with
outliers, we try to make sure that full outlier images make up
half of the batch. We do this by creating a batch with only
outlier images and then swap each of them with an inlier image
with a probability of 0.5. This inlier image can contain
only inlier pixels, but we can also
train with pasting. When training
that way, the ImageNet-1k image selected for swapping is
rescaled to make its area equal
to 5\% of a randomly selected inlier image area and then pasted
into a random location of that image.
Usages of ImageNet-1k-full and ImageNet-1k-bb also differ slightly. When
ImageNet-1k-full dataset is used, all of the pixels of the outlier
image are marked as outlier. If used with pasting
the entire image is pasted into an
inlier image. When ImageNet-1k-bb dataset is used, only the pixels
inside the bounding box are marked as outlier, the rest are set to
ignore. If ImageNet-1k-bb is used in tandem with pasting, only
the rectangle containing the object is pasted into the inlier image.
\end{comment}
\subsection{Evaluation on the WildDash Benchmark}
\label{ss:benchmark}
Table \ref{table:bench_results}
presents our results on
the WildDash semantic segmentation benchmark, and compares them to other submissions
with accompanying publications.
\setlength{\tabcolsep}{4pt}
\begin{table}[htb!]
\begin{center}
\caption{Evaluation of the semantic segmentation
models on WildDash bench}
\label{table:bench_results}
\begin{tabular}{|c||c|cccc|c|}
\hline
& \multicolumn{1}{c|}{Meta Avg}
& \multicolumn{4}{c|}{Classic}
& \multicolumn{1}{c|}{Negative}
\\
\cline{2-7}
Model
& mIoU
& mIoU
& iIoU
& mIoU
& iIoU
& mIoU\\
&cla &cla&cla &cat&cat&cla\\
\hline
\hline
\multicolumn{1}{|l||}{APMoE\_seg\_ROB \cite{kong2018pag}} & 22.2 & 22.5 & 12.6 & 48.1 & 35.2 & 22.8\\
\hline
\multicolumn{1}{|l||}{DRN\_MPC \cite{Yu2017}} & 28.3 & 29.1 & 13.9 & 49.2 & 29.2 & 15.9 \\
\hline
\multicolumn{1}{|l||}{DeepLabv3+\_CS \cite{chen2018encoder}} & 30.6 & 34.2 & 24.6 & 49.0 & 38.6 & 15.7\\
\hline
\multicolumn{1}{|l||}{LDN2\_ROB \cite{kreso18arxiv}} & 32.1 & 34.4 & 30.7 & 56.6 & 47.6 & 29.9\\
\hline
\multicolumn{1}{|l||}{MapillaryAI\_ROB \cite{bulo2017place}} &38.9 & 41.3 & \textbf{38.0} & 60.5 & \textbf{57.6} & 25.0\\
\hline
\multicolumn{1}{|l||}{AHiSS\_ROB \cite{meletis2018training}} & 39.0 & 41.0 & 32.2 & 53.9 & 39.3 & 43.6\\
\hline
\hline
\multicolumn{1}{|l||}{LDN\_BIN
(ours, two-head)} &
41.8 & \textbf{43.8} &
37.3 & 58.6 & 53.3 & \textbf{54.3}\\
\hline
\multicolumn{1}{|l||}{LDN\_OE
(ours, C$\times$ multi-class)} &
\textbf{42.7} & 43.3 &
31.9 & \textbf{60.7} & 50.3 & 52.8\\
\hline
\end{tabular}
\end{center}
\end{table}
Our two models use the same backbone
(DenseNet-169 \cite{huang17cvpr})
and different outlier detectors.
The LDN\_OE model has
a single C-way multi-class head.
The LDN\_BIN model has two heads
as shown in Figure \ref{fig:model}.
Both models have been trained on
Vistas train, Cityscapes train,
and WildDash val (inliers),
as well as on ImageNet-1k-bb
with pasting (outliers).
Our models significantly outperform
all previous submissions on negative images,
while also achieving the highest
meta average mIoU
(the principal benchmark metric)
and the highest mIoU for classic images.
We achieve the second-best iIoU score
for classic images, which indicates
underperformance on small objects.
This is likely due to the fact that
we train and evaluate our models
on half resolution images.
\subsection{Validation of Dense
Outlier Detection Approaches}
Table \ref{table:OOD_detection}
compares various approaches
for dense outlier detection.
All models are based on DenseNet-169,
and trained on Vistas (inliers).
The first section shows the results
of a C-way multi-class model
trained without outliers,
where outliers are detected
with max-softmax \cite{hendrycks17iclr}
and ODIN + max-softmax \cite{liang18iclr}.
We note that ODIN slightly improves
the results across all experiments.
\begin{table}[htb]
\centering
\caption{Validation of dense
outlier detection approaches.
WD denotes WildDash val.}
\label{table:OOD_detection}
\begin{tabular}{|c|c||c|c|c|}
\hline
Model & ImageNet &
\multicolumn{1}{c|}{AP WD-LSUN} &
\multicolumn{1}{c|}{AP WD-Pascal} &
\multicolumn{1}{c|}{mIoU WD}\\
\hline
\hline
\multicolumn{1}{|l|}{
C$\times$ multi-class} &
\ding{55} &$55.65 \pm 0.80$ & 6.01 & 49.07
\\
\hline
\multicolumn{1}{|l|}{
C$\times$ multi-class, ODIN} &
\ding{55} & $55.98 \pm 0.77$ & 6.92 & \textbf{49.77}
\\
\hline
\hline
\multicolumn{1}{|l|}{
C+1$\times$ multi-class}&
\ding{51} &
$98.92 \pm 0.06$ & 33.59 & 45.60
\\
\hline
\multicolumn{1}{|l|}{
C$\times$ multi-label} &
\ding{51} & $98.75 \pm 0.07$ & \textbf{57.31} & 42.72
\\
\hline
\multicolumn{1}{|l|}{
C$\times$ multi-class}
& \ding{51} & $ \textbf{99.49} \pm \textbf{0.04}$ & 41.72 & 46.69
\\
\hline
\multicolumn{1}{|l|}{two heads}
& \ding{51} & $99.25 \pm 0.04$ & 46.83 & 47.37
\\
\hline
\end{tabular}
\end{table}
\begin{comment}
The outlier exposure model is the segmentation
model trained using outlier exposure.
All of these models use max-softmax as the criterion
for detecting outliers.
C+1 is the segmentation model with the additional class
being the outlier class. The criteria for measuring
AP is the difference between the
probability of the (C+1)st class and max-sofmax.
The sigmoid model is trained using sigmoid instead of
softmax, which makes it possible for a network
not to classify a sample by having all of the
outputs below 0.5. The criterion for measuring AP
is max-sigmoid.
The model denoted as two head is the model
that has a second, binary head. The probability of
outlier is used for measuring AP.
\end{comment}
The second section of the table shows
the four dense outlier detection approaches
(cf.\ Section \ref{ss:model})
which we also train on
ImageNet-1k-bb with pasting (outliers).
Columns 3 and 4 clearly show
that training with noisy and diverse negatives
significantly improves outlier detection.
However, we also note a reduction
of the segmentation score
as shown in the column 5.
This reduction is lowest for
the C-way multi-class model
and the two-head model,
which we analyze next.
The two-head model is slightly worse
in discriminating WildDash val from LSUN,
which indicates that it is
more sensitive to domain shift
between Vistas train and WildDash val.
On the other hand, the two-head model achieves
better inlier segmentation
(0.7 pp, column 5),
and much better outlier detection
on Pascal animals (5 pp, column 4).
A closer inspection shows
that these advantages occur
since the single-head C-way approach
generates many false positive
outlier detections at semantic borders
due to lower max-softmax.
The C+1-way multi-class model performs the worst
out of all models trained with noisy outliers.
The sigmoid model performs
well on outlier detection
but underperforms on inlier segmentation.
\subsection{Validation of Dense
Feature Extractor Backbones}
Table \ref{table:backbone_val} explores
influence of different backbones
to the performance of our two-head model.
We experiment with ResNets and
DenseNets of varying depths.
The upsampling blocks are connected
with the first three DenseNet blocks,
as shown in Fig.\,\ref{fig:model}.
In the ResNet case, the upsampling blocks
are connected with the last addition
at the corresponding subsampling level.
We train on Vistas (inliers)
and ImageNet-1k-bb with pasting (outliers).
\begin{table}[htb]
\centering
\caption{Validation of backbones
for the two-head model.
WD denotes WildDash val.
}
\label{table:backbone_val}
\begin{tabular}{|c||c|c|c|}
\hline
Backbone &
\multicolumn{1}{c|}{AP WD-LSUN } &
\multicolumn{1}{c|}{AP WD-Pascal} &
\multicolumn{1}{c|}{mIoU WD}\\
\hline
\hline
\multicolumn{1}{|l||}{DenseNet-121} & $99.05 \pm 0.03$ & 55.84 & 44.75 \\
\hline
\multicolumn{1}{|l||}{DenseNet-169} & $\textbf{99.25} \pm \textbf{0.04}$ & 46.83 & 47.37\\
\hline
\multicolumn{1}{|l||}{DenseNet-201} & $98.34 \pm 0.07$ & 36.88 & \textbf{47.59} \\
\hline
\multicolumn{1}{|l||}{ResNet-34} & $97.19 \pm 0.07$ & 47.24 & 45.17\\
\hline
\multicolumn{1}{|l||}{ResNet-50} & $99.10 \pm 0.04$ & \textbf{56.18} & 41.65 \\
\hline
\multicolumn{1}{|l||}{ResNet-101} & $98.96 \pm 0.06$ & 52.02 & 43.67 \\
\hline
\end{tabular}
\end{table}
All models achieve very good
outlier detection in negative images.
There appears to be a trade-off between
detection of outliers at negative objects
and semantic segmentation accuracy.
We opt for better semantic segmentation results
since WildDash test does not have negative objects
in positive context.
We therefore use the DenseNet-169 backbone
in most other experiments due to
a very good overall performance.
\subsection{Influence of the Training Data}
Table \ref{table:dataset_results_in}
explores the influence of
inlier training data
to the model performance.
All experiments involve the two-head model
based on DenseNet-169, which was trained on outliers from ImageNet-1k-bb with pasting.
\begin{table}[htb]
\centering
\caption{Influence of
the inlier training dataset
to the performance of the two-head model
with the DenseNet-169 backbone.
WD denotes WildDash val.
}
\label{table:dataset_results_in}
\begin{tabular}{|c||c|c|c|}
\hline
Inlier training dataset &
AP WD-LSUN &
AP WD-Pascal &
mIoU WD\\
\hline
\hline
\multicolumn{1}{|l||}{Cityscapes} &
$66.57 \pm 0.86$ & 13.85 & 11.12\\
\hline
\multicolumn{1}{|l||}{Vistas} &
$99.25 \pm 0.04$ & 46.83 & 47.17 \\
\hline
\multicolumn{1}{|l||}{Cityscapes, Vistas} &
$\textbf{99.29} \pm \textbf{0.03}$ & \textbf{53.68} & \textbf{47.78} \\
\hline
\end{tabular}
\end{table}
The results suggest that there is
a very large domain shift between
Cityscapes and WildDash val.
Training on inliers from Cityscapes
leads to very low AP scores,
which indicates that many WildDash val pixels
are predicted as outliers
with respect to Cityscapes.
This suggests that Cityscapes
is not an appropriate training dataset
for real-world applications.
Training on inliers from Vistas
leads to much better results
which is likely due to greater variety
with respect to camera, time of day,
weather, resolution etc.
The best results across the board
have been achieved
when both inlier datasets
are used for training.
Table \ref{table:dataset_results_out}
explores the impact of
negative training data.
All experiments feature
the two-head model with DenseNet-169
trained on inliers from Vistas.
\begin{table}[htb]
\centering
\caption{Influence of the
outlier training dataset
to the performance of our two-head model
with the DenseNet-169 backbone.
WD denotes WildDash val.
}
\label{table:dataset_results_out}
\begin{tabular}{|l|c||c|c|}
\hline
\multirow{1}*{Outlier training dataset} &
outlier pasting &
\multicolumn{1}{c|}{AP WD-Pascal} &
\multicolumn{1}{c|}{mIoU WD}\\
\hline
\hline
\multicolumn{1}{|l|}{ImageNet-1k-full} &
no & 2.94 & 43.13\\
\hline
\multicolumn{1}{|l|}{ImageNet-1k-full} &
yes & 45.96 & 43.68\\
\hline
\multicolumn{1}{|l|}{ImageNet-1k-bb} &
yes & \textbf{46.83} & \textbf{47.17} \\
\hline
\end{tabular}
\end{table}
The table shows that training
with pasted negatives greatly improves
outlier detection on negative objects.
It is intuitively clear that
a model which never sees a border
between inliers and outliers
during training
does not stand a chance
to accurately locate such borders
during inference.
The table also shows that ImageNet-1k-bb
significantly boosts inlier segmentation,
while also improving outlier detection
on negative objects.
We believe that this occurs because
ImageNet-1k-bb has a smaller overlap
with respect to the inlier training data,
due to high incidence of Cityscapes classes
(e.g. vegetation, sky, road)
in ImageNet backgrounds.
This simplifies outlier detection
due to decreased noise in the training set,
and allows more capacity of
the shared feature extractor
to be used for the segmentation task.
The table omits outlier detection
in negative images,
since all models
achieve over 99 \% AP
on that task.
\subsection{Comparing the Two-Head and
C-way Multi-class Models}
\label{ss:comparison}
We now compare our two models
from Table \ref{table:bench_results}
in more detail.
We remind that the two models
have the same feature extractor
and are trained on the same data.
The two-head model performs better
in most classic evaluation categories
as well as in the negative category,
however it has a lower meta average score.
\begin{comment}
\begin{table}[htb]
\centering
\caption{Performance of the bin and oe model
on wilddash bench}
\label{table:bin_oe}
\begin{tabular}{|c||c|cccc|c|}
\hline
& \multicolumn{1}{c|}{Meta Avg}
& \multicolumn{4}{c|}{Classic}
& \multicolumn{1}{c|}{Negative}
\\
\cline{2-7}
Model
& mIoU
& mIoU
& iIoU
& mIoU
& iIoU
& mIoU
\\
&cla &cla&cla &cat&cat&cla\\
\hline
\hline
\multicolumn{1}{|l||}{two head} & 41.8 & \textbf{43.8} & \textbf{37.3} & 58.6 & \textbf{53.3} & \textbf{54.3}\\
\hline
\multicolumn{1}{|l||}{oe} & \textbf{42.7} & 43.3 & 31.9 & \textbf{60.7} & 50.3 & 52.8\\
\hline
\end{tabular}
\end{table}
\end{comment}
Table \ref{table:bin_oe_hazard} explores
influence of WildDash hazards
\cite{zendel18eccv}
on the performance of the two models.
The C-way multi-class model
has a lower performance drop
in most hazard categories.
The difference is especially large
in images with distortion and overexposure.
Qualitative experiments show
that this occurs since
the two-head model tends
to recognize pixels in
images with hazards as outliers
(cf.\ Fig.\,\ref{fig:bench_bin_oe}).
\begin{table}[htb]
\begin{center}
\caption{Impact of hazards
to performance of our
WildDash submissions.
The hazards are
image blur,
uncommon road coverage,
lens distortion,
large ego-hood,
occlusion,
overexposure,
particles,
dirty windscreen,
underexposure, and
uncommon variations.
LDN\_BIN denotes
the two-head model.
LDN\_OE denotes
the C-way
multi-class model.
}
\label{table:bin_oe_hazard}
\begin{tabular}{|c||cccccccccccc|}
\hline
\multirow{2}*{Model}& \multicolumn{10}{c|}{
Class mIoU drop across WildDash hazards
\cite{zendel18eccv}}
\\
\cline{2-11}
&blur
&cov
&dist
&hood
&occ.
&over
&part
&screen
&under
&\multicolumn{1}{c|}{var.}\\%iation}\\
\hline
\hline
\multicolumn{1}{|l||}{\textsc{ldn\_bin}} &-14\%&-14\%&-22\%&-14\%&\textbf{-3\%}&-35\%&-3\%&-9\%&\textbf{-25\%}&\multicolumn{1}{c|}{-8\%}\\
\hline
\multicolumn{1}{|l||}{\textsc{ldn\_oe}} & \textbf{-11\%}&\textbf{-13\%}&\textbf{-7\%}&\textbf{-10\%}&-5\%&\textbf{-24\%}& \textbf{0\%}&\textbf{-6\%}&-30\%&\multicolumn{1}{c|}{\textbf{-7\%}}\\
\hline
\end{tabular}
\end{center}
\end{table}
Fig.\,\ref{fig:bench_bin_oe} presents
a qualitative comparison of our two
submissions to the WildDash benchmark.
Experiments in rows 1 and 2 show
that the two-head model
performs better in classic images
due to better performance on semantic borders.
Furthermore, the two-head model is also better
in detecting negative objects in positive context
(ego-vehicle, the forklift, and the horse).
Experiments in row 3 show
that the two-head model
tends to recognize all pixels
in images with overexposure
and distortion hazards as outliers.
Experiments in rows 4 and 5 show that
the two-head model recognizes
entire negative images as outliers,
while the C-way single-head model
is able to recognize positive objects
(the four persons) in negative images.
These experiments suggest
that both models are able to detect
outliers at visual classes
which are (at least nominally)
not present in ImageNet-1k:
ego-vehicle, toy-brick construction,
digital noise, and text.
Fig.\,\ref{fig:bench2_bin_oe}
illustrates space for further improvement
by presenting a few
failure cases on WildDash test.
\newcommand{0.49\textwidth}{0.49\textwidth}
\begin{figure}[htb]
\centering
\includegraphics[width=0.49\textwidth]{wd0001final.jpg}
\hfill
\includegraphics[width=0.49\textwidth]{wd0074final.jpg}
\\[0.2em]
\includegraphics[width=0.49\textwidth]{wd0030final.jpg}
\hfill
\includegraphics[width=0.49\textwidth]{wd0044final.jpg}
\\[0.2em]
\includegraphics[width=0.49\textwidth]{wd0033final.jpg}
\hfill
\includegraphics[width=0.49\textwidth]{wd0056final.jpg}
\\[0.2em]
\includegraphics[width=0.49\textwidth]{wd0150final.jpg}
\hfill
\includegraphics[width=0.49\textwidth]{wd0153final.jpg}
\\[0.2em]
\includegraphics[width=0.49\textwidth]{wd0144final.jpg}
\hfill
\includegraphics[width=0.49\textwidth]{wd0147final.jpg}
\caption{Qualitative performance
of our two submissions
to the WildDash benchmark.
Each triplet contains
a test image (left),
the output of the two-head model (center),
and the output of the model trained
to predict uniform distribution in outliers (right).
Rows represent inlier images (1),
outlier objects in inlier context (2),
inlier images with hazards (3),
out-of-scope negatives (4),
and abstract negatives (5).
The two-head model produces
more outlier detections
while performing better
in classic images
(cf.~Table\,\ref{table:bench_results}).
}
\label{fig:bench_bin_oe}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=0.49\textwidth]{wd0002final.jpg}
\hfill
\includegraphics[width=0.49\textwidth]{wd0076final.jpg}
\\[0.2em]
\includegraphics[width=0.49\textwidth]{wd0013final.jpg}
\hfill
\includegraphics[width=0.49\textwidth]{wd0148final.jpg}
\caption{%
Failure cases on WD test
arranged as in
Fig.\,\ref{fig:bench_bin_oe}.
Top:
the two-head model predicts
outliers at trucks.
Bottom left:
both models
fail to accurately detect
birds on the road.
Bottom right:
both models exchange road and sky
due to position bias
(late experiments show this
can be improved with training
on smaller crops and scale jittering).
}
\label{fig:bench2_bin_oe}
\end{figure}
\subsection{Discussion}
Validation on LSUN and WildDash val suggests
that detecting entire outlier images
is an easy problem.
A useful practical application of this result
would be a module to detect
whether the camera is out of order
due to being covered, dirty or faulty.
Validation on pasted Pascal animals
suggests that detecting outliers
in inlier context
is somewhat harder but still
within reach of modern techniques.
Evaluation on WildDash test shows
that our models outperform
previous published approaches.
We note significant improvement
with respect to state of the art
in classic and negative images,
as well as in average mIoU score.
Our models successfully detect all
abstract and out-of-scope negatives \cite{zendel18eccv},
even though much of this content
is not represented by ImageNet-1k classes.
Future work should address further development
of open-set evaluation data\-sets
such as \cite{zendel18eccv,blum19arxiv}.
In particular, the community would benefit
from substantially larger negative test sets
which should include diverse non-ImageNet-1k content,
as well as outlier objects in inlier context
and inlier objects in outlier context.
\section{Conclusion}
We have presented an approach to combine
semantic segmentation and
dense outlier detection
without significantly deteriorating
either of the two tasks.
We cast outlier detection
as binary classification
on top of a shared convolutional re\-presentation.
This allows for solving both tasks
with a single forward pass
through a convolutional backbone.
We train on inliers from
standard road driving datasets
(Vista, Cityscapes),
and noisy outliers from a very diverse
negative dataset (ImageNet-1k).
The proposed training procedure tolerates
inliers in negative training images
and generalizes
to images with mixed content
(inlier background, outlier objects).
We perform extensive open-set validation
on WildDash val (inliers),
LSUN val (outliers),
and pasted Pascal objects (outliers).
The results confirm suitability
of the proposed training procedure.
The proposed multi-head model outperforms
the C-way multi-label model
and the C+1-way multi-class model,
while performing comparably to the
C-way multi-class model trained
to predict uniform distribution in outliers.
We apply our two best models to WildDash test
and set a new state of the art
on the WildDash benchmark.
\begin{comment}
We show that introducing outliers during training significantly
improves outlier detection . We show that ImageNet-1k
is a good choice for the outlier set to be used during
training, and are able to detect outlier pixels situated at
visual classes not found in ImageNet-1k. We show that a simple
use of pasting outlier samples into inlier images during training
is necessary for the ability of the model to detect
outlier patches.
We show the limits of using Cityscapes dataset for tackling
challenging examples, and show that Vistas is a better choice
due to its diversity of filming conditions.
We submitted the best to models to the WildDash
benchmark and achived the top spot.
\end{comment}
\bibliographystyle{splncs04}
|
3,212,635,537,499 | arxiv | \section{Introduction}
\label{section:1}
The three-player-game (3PG) is an interesting mathematical quiz. A 3PG involves three players and consists of a series of rounds, each zero-win round involves two out of the three players, at the end of the round the loser quits and the third player enters the ring and another round starts. The game terminates if all six win-lose relationships appear. During each round, two players win with equal probability.
For example, let Alice, Bob and Carole be three players. Alice and Bob play the first round, then Alice loses. Therefore the second round involves Bob and Carole. If Carole loses then Alice has to return to the ring. The game proceeds until every player has beaten the other two players for at least once.
One should take the win-lose situation of each round as the underlying infinite probability space and the number of rounds as a random variable. The target is to compute the expectation of this random variable.
We solve this question from scratch and give some additional analysis on generalization.
\section{Solving 3PG by Reduction}
\label{section:2}
\subsection{Formulation and Reduction Rules}
It is straightforward to observe that the sufficient statistics of any round in the game is the occurance of all possible win-lose relationships sofar and the current players in the ring. That is to say, at each stage of the game, the current situation is comprehensively described by six bits (indicate whether or not the six win-lose relationship appear) and one ternary bit (indicate the current players in the ring). We denote all possible states in the following form:
$$\mathcal{S}=\left\{\left(b_{1}b_{2}b_{3}b_{4}b_{5}b_{6}|t_{1}t_{2}\right)\right\},$$
where each $b_{i}\in\left\{0,1\right\}$ denotes whether a win-lose relationship appears or not. Whereas $t_{1}$ and $t_{2}$ in $\left\{1,2,3 \right\}$ are the indices for the players in battle. From $i=1,\cdots,6$, $b_{i}$ denotes the win-lose relationship of:
$$\text{Player 1 beats Player 2},$$
$$\text{Player 2 beats Player 1},$$
$$\text{Player 1 beats Player 3},$$
$$\text{Player 3 beats Player 1},$$
$$\text{Player 2 beats Player 3},$$
$$\text{Player 3 beats Player 2}.$$
For each $S\in\mathcal{S}$, let $f(S)$ be the expectation of the number of rounds of a modified 3PG begins with $S$. Now we are asked to compute, w.l.o.g.,:
$$f(000000|\left\{1,2\right\}).$$
It is natural to use conditional expectation to introduce the reduction rule, we begin with the definition:
$$\mathbb{E}[X]=\mathcal{P}(A)\mathbb{E}[X|A]+\mathcal{P}(\overline{A})\mathbb{E}[X|\overline{A}],$$
where $X$ is any r.v. and $A$ is an event. Let $X$ be \emph{the number of rounds of a 3PG begins with $S$} and $A$ be the event that $t_{1}$ wins over $t_{2}$ w.r.t. $S$, we have:
$$\mathbb{E}[X]=f(S),$$
$$\mathcal{P}(A)=\mathcal{P}(\overline{A})=\frac{1}{2},$$
$$\mathbb{E}[X|A]=1+f(S_{1}),$$
$$\mathbb{E}[X|\overline{A}]=1+f(S_{2}),$$
where $S$ reduces to state $S_{1}(S_{2})$ if $A$ does (not) happen. This expansion yields:
\begin{equation}
\label{equation:1}
f(S)=1+\frac{1}{2}\cdot f(S_{1})+\frac{1}{2}\cdot f(S_{2}),
\end{equation}
where $S_{1}$ and $S_{2}$ are two results from $S$ given $t_{1}$ or $t_{2}$ wins. \eqref{equation:1} would become the basis of reduction which finally solves the 3PG. We are going to see a dozen of examples in the coming section.
The reduction terminates at the basic states
$$f(111111|\left\{t_{1},t_{2}\right\})=0,$$
where $t_{1},t_{2}$ can be any players.
We define the \emph{order} of a state by the number of 0s in its binary parts. For example, $(000000|\left\{1,2\right\})$ is a state of order six.
The number of states $|S|$ is 192, however, some states is not going to appear with the root state $(000000|\left\{1,2\right\})$. Moreover, one is encouraged to evoke symmetry to further reduce computation. We define two states $S_{A}$ and $S_{B}$ to be symmetric if there exist a permutation $\pi$ on $\left\{1,2,3\right\}$ such that
$$\pi(S_{A})=S_{B},$$
where applying $\pi$ on a state $S\in\mathcal{S}$ changes both the order of binary indicators and the name of players in the ring. For $b$ indicates the relationship Player 1 beats Player 2 in $S_{A}$, its values also indicates whether the relationship Player $\pi (1)$ beats Player $\pi (2)$ in $S_{B}$. That is to say, $\pi$ on $\left\{1,2,3\right\}$ introduces a permutation on $\left\{1,2,3,4,5,6 \right\}$, for example, $\pi =(1,2)$ introduces (using group algebra notation):
$$(1,2)(3,5)(4,6)$$
While $\pi=(2,3)(1,2)$ introduces:
$$(1,4,5)(2,3,6)$$
on the first six binary bits on $\mathcal{S}$. Finally,
$$t_{B,i}=\pi (t_{A,i}),i=1,2.$$
For example, let
$$S_{A}=(101000|\left\{1,3 \right\})$$
$$S_{B}=(101000|\left\{1,2 \right\})$$
then $S_{A}$ and $S_{B}$ are symmetric by adopting
$$\pi=(2,3).$$
Naturally, symmetry is an equivalent relationship, and for symmetric states $S_{A}$ and $S_{B}$ we have:
$$f(S_{A})=f(S_{B}),$$
since their difference is only a matter of naming. This observation helps to reduce the number of states significantly. However, there is hardly any method to examine whether two states are symmetric other than checking all possible permutations.
\subsection{Preparations and Preprocessing}
Having obtained \eqref{equation:1}, one might eagerly argue that a simple recursion program would trivially solve the task:
Define:
$$\text{Compute}(S)=1+\frac{\text{Compute}(S_{1})}{2}+\frac{\text{Compute}(S_{1})}{2},$$
$$\text{Compute}(111111|\left\{t_{1},t_{2} \right\})=0.$$
Return $\text{Compute}(000000|\left\{1,2\right\})$.
With $\text{Compute}(\cdot)$ as an algorithmatic realization of $f(\cdot)$. Computing $\text{Compute}(000000|\left\{1,2\right\})$ is then done automatically by spanning a recursion tree (let $S_{1},S_{2}$ be two children of the node represents $S$), during which dynamic programming might help to reduce computation time \cite{cormen2009introduction}. However, this method is not determined to success since \eqref{equation:1} does not ensured that a state $S$ itself does not appear in the computing tree spanned by with $S$ as the root, which fact deadlocks this paradigm. At certain states, it is necessary to use the linear relationship between their expectations to solve $f(\cdot)$ and a naive recursion is far from enough.
This fact, together with the last observation from the previous section, indicates that instead of \emph{passively} spanning a computing tree, we should better \emph{aggressively} compute the leave states (those states with few 0s in their binary part) at first.
Before actually conducting reduction from $(000000|\left\{1,2\right\})$, we conduct preprocessing by computing $f(S)$ for some elementary states (states with small orders) beforehand. These computations are collected into a series of gradual propositions.
\textbf{Proposition A:}
$$f(111110|\left\{2,3 \right\})=4,$$
$$f(111110|\left\{1,2 \right\})=6,$$
$$f(111110|\left\{1,3 \right\})=6,$$
\textbf{Proof:} Let $x,y,z$ denote these three values respectively, we have according to \eqref{equation:1}:
$$x=1+\frac{y}{2},$$
$$y=1+\frac{z}{2}+\frac{x}{2},$$
$$z=1+\frac{y}{2}+\frac{x}{2}.$$
That is to say
$$
\begin{pmatrix}
x \\
y\\
z
\end{pmatrix}=
\begin{pmatrix}
0 & \frac{1}{2} & 0\\
\frac{1}{2} & 0 & \frac{1}{2}\\
\frac{1}{2} & \frac{1}{2} & 0\\
\end{pmatrix}
\begin{pmatrix}
x \\
y\\
z
\end{pmatrix}+
\begin{pmatrix}
1 \\
1\\
1
\end{pmatrix}
$$
This gives $x=4,y=6,z=6$ as the only solution. \qed
Proposition A finishes the computation of all states $S$ with five 1s in their binary part, i.e., all states of order one. Technically, let the only component as 0 be $t_{A}$ beats $t_{B}$ in a state $S=(\cdots|\left\{t_{1},t_{2}\right\})$, then if $\left\{t_{1},t_{2} \right\}=\left\{t_{A},t_{B} \right\}$ then the expectation of the corresponding state $f(S)$ is 4, otherwise it is 6. We now use this as the block of building estimation for states with order two.
Let us begin with states with two vacant relationships $t_{A}$ beats $t_{B}$ and $t_{B}$ beats $t_{A}$.
\textbf{Proposition B:}
$$f(111100|\left\{2,3 \right\})=7.$$
\textbf{Proof:} Applying \eqref{equation:1} to this state:
$$
\begin{aligned}
f(111100|\left\{2,3\right\})=1&+\frac{1}{2}\cdot f(111110|\left\{1,2\right\})\\
&+\frac{1}{2}\cdot f(111101|\left\{1,3\right\}).
\end{aligned}
$$
Applying Proposition A finishes the proof. \qed
\textbf{Proposition C:}
$$f(111100|\left\{1,3\right\})=9.$$
\textbf{Proof:}
Applying \eqref{equation:1} to this state:
$$
\begin{aligned}
f(111100|\left\{1,3\right\})=1&+\frac{1}{2}\cdot f(111100|\left\{1,2\right\})\\
&+\frac{1}{2}\cdot f(111100|\left\{2,3\right\}).
\end{aligned}
$$
Now considering $\pi =(2,3)$ then we have:
$$f(111100|\left\{1,3\right\})=f(111100|\left\{1,2\right\}),$$
combining this with Proposition B finishes the proof.\qed
Now if the only two vacant relationships are $t_{A}$ beats $t_{B}$ and $t_{B}$ beats $t_{A}$, we are ready to read the expectation of the state $S(\cdots|\left\{t_{1},t_{2}\right\})$. If $\left\{t_{1},t_{2}\right\}=\left\{t_{A},t_{B}\right\}$ then $f(S)$ is 7, else it is 9.
We then proceed to states with vacant relationships
\begin{itemize}
\item $t_{A}$ beats $t_{B}$ and $t_{A}$ beats $t_{C}$,
\item $t_{A}$ beats $t_{B}$ and $t_{C}$ beats $t_{B}$,
\item $t_{A}$ beats $t_{B}$ and $t_{B}$ beats $t_{C}$.
\end{itemize}
\textbf{Proposition D:}
$$f(010111|\left\{2,3 \right\})=8,$$
$$f(010111|\left\{1,2 \right\})=7,$$
\textbf{Proof:} Let $x,y$ denote $f(010111|\left\{2,3 \right\})$ and $f(010111|\left\{1,2 \right\})$, according to \eqref{equation:1} (one easily notes that $(010111|\left\{1,2 \right\})$ is symmetrical to $(010111|\left\{1,3 \right\})$):
$$x=1+\frac{y}{2}+\frac{y}{2},$$
$$y=1+\frac{f(110111|\left\{1,3 \right\})}{2}+\frac{x}{2}.$$ Applying Proposition A finishes the proof. \qed
\textbf{Proposition E:}
$$f(101011|\left\{2,3 \right\})=9,$$
$$f(101011|\left\{1,2 \right\})=8,$$
\textbf{Proof:} Let $x,y$ denote $f(101011|\left\{2,3 \right\})$ and $f(101011|\left\{1,2 \right\})$, using symmetry and Proposition A as in the proof of Proposition D:
$$x=1+y,$$
$$y=1+\frac{y}{2}+\frac{6}{2}.$$
This finishes the proof.\qed
\textbf{Proposition F:}
$$f(011101|\left\{1,2\right\})=\frac{38}{5},$$
$$f(011101|\left\{2,3\right\})=\frac{36}{5},$$
$$f(011101|\left\{1,3\right\})=\frac{42}{5}.$$
\textbf{Proof:} Let $x,y,z$ denote these three values, using \eqref{equation:1} and Proposition A:
$$x=1+\frac{6}{2}+\frac{y}{2},$$
$$y=1+\frac{4}{2}+\frac{z}{2},$$
$$z=1+\frac{x}{2}+\frac{y}{2}.$$
Solving this system yields
$$x=\frac{38}{5},y=\frac{36}{5},z=\frac{42}{5}.$$
\qed
Proposition B-F finish computing states of order two.
The number of states with three 1s/of order three is larger. There are at least $\frac{3*\binom{6}{3}}{3\!}=10$ unsymmetric states. Although unnecessary for the following sections, one is encouraged to compute all 13 independent states with three appeard win-loss relationships.
We are now ready to begin from $f(000000|\left\{1,2\right\})$ and hope that compution meets with the Propositions A-F before at an early stage of computation.
\subsection{The Main Reduction}
Attempting to solve this problem by reduction, applying \eqref{equation:1} onto $f(000000|\left\{1,2 \right\}).$:
$$
\begin{aligned}
f(000000|\left\{1,2\right\})=1&+\frac{1}{2}\cdot f(100000|\left\{1,3\right\})\\
&+\frac{1}{2}\cdot f(010000|\left\{2,3 \right\}).
\end{aligned}
$$
From now on, let $S_{1}$ be the state where the player with the smaller index winning the current round. Adopting
$$\pi =(1,2),$$
then $(100000|\left\{1,3\right\})$ and $(010000|\left\{2,3 \right\})$ are symmetric, so:
\begin{equation}
\label{equation:2}
f(000000|\left\{1,2\right\})=1+f(100000|\left\{1,3\right\}).
\end{equation}
Therefore we are left with the problem of computing $f(100000|\left\{1,3\right\})$. Now
\begin{equation}
\label{equation:3}
\begin{aligned}
f(100000|\left\{1,3\right\})=1&+\frac{1}{2}\cdot f(101000|\left\{1,2\right\})\\
&+\frac{1}{2}\cdot f(100100|\left\{2,3 \right\}).
\end{aligned}
\end{equation}
We first address $f(101000|\left\{1,2\right\})$ and then return to {\color{red}$f(100100|\left\{2,3\right\})$}. Since
$$
\begin{aligned}
f(101000|\left\{1,2\right\})=1&+\frac{1}{2}\cdot f(101000|\left\{1,3\right\})\\
&+\frac{1}{2}\cdot f(111000|\left\{2,3 \right\}).
\end{aligned}
$$
However, let
$$\pi =(2,3),$$
we conclude that
$$f(101000|\left\{1,2 \right\})=f(101000|\left\{1,3\right\}),$$
thus
\begin{equation}
\label{equation:4}
f(101000|\left\{1,2 \right\})=2+f(111000|\left\{2,3\right\}).
\end{equation}
Keep reducing:
$$
\begin{aligned}
f(111000|\left\{2,3\right\})=1&+\frac{1}{2}\cdot f(111010|\left\{1,2\right\})\\
&+\frac{1}{2}\cdot f(111001|\left\{1,3 \right\}).
\end{aligned}
$$
We are now meeting two states with four 1s and two 0s, $f(111010|\left\{1,2\right\})$ is symmetric to $f(010111|\left\{2,3\right\})$, hence its value is 8 according to Proposition D, while $f(111001|\left\{1,3\right\})$ addresses a state symmetric to $f(011101|\left\{2,3 \right\})$, whose value is $\frac{36}{5}$ according to Proposition F. Pluggin them back into \eqref{equation:4} gives:
$$f(101000|\left\{1,2\right\})=\frac{53}{5}.$$
To return to \eqref{equation:3}, we still need to compute $f(100100|\left\{2,3\right\})$.
\begin{equation}
\label{equation:5}
\begin{aligned}
f(100100|\left\{2,3\right\})=1&+\frac{1}{2}\cdot f(100110|\left\{1,2 \right\})\\
&+\frac{1}{2}\cdot f(100101|\left\{ 1,3\right\}).
\end{aligned}
\end{equation}
For
$$
\begin{aligned}
f(100110|\left\{1,2\right\})=1&+\frac{1}{2}\cdot f(100110|\left\{1,3 \right\})\\
&+\frac{1}{2}\cdot f(110110|\left\{2,3 \right\}).
\end{aligned}
$$
Where as one can easily observe the symmetry between $(100110|\left\{1,2\right\})$ and $(100110|\left\{1,3\right\})$, we have:
$$f(100110|\left\{1,2\right\})=2+f(110110|\left\{2,3\right\})=\frac{46}{5}.$$
The last term remained is $f(100101|\left\{1,3\right\})$. We begin with
$$
\begin{aligned}
f(100101|\left\{1,3\right\})=1&+\frac{1}{2}\cdot f(101101|\left\{1,2\right\})\\
&+\frac{1}{2}\cdot f(100101|\left\{2,3 \right\})\\
=\frac{9}{2}&+\frac{1}{2}\cdot f(100101|\left\{2,3\right\}).
\end{aligned}
$$
Finally, we have:
$$f(100101|\left\{2,3\right\})=\frac{24}{5}+\frac{1}{2}\cdot f(100101|\left\{1,3\right\}).$$
This gives:
$$f(100101|\left\{1,3\right\})=\frac{46}{5}.$$
Pluggin them into \eqref{equation:5} yields:
$$f(100100|\left\{2,3\right\})=\frac{51}{5}.$$
Now \eqref{equation:3} yields:
$$f(100000|\left\{1,3\right\})=11.4.$$
At length, pluggin this into \eqref{equation:2} yields:
$$f(000000|\left\{1,2\right\})=12.4.$$
Hitherto we have finished all the reduction.
\subsection{Simulation Results}
As for an empirical verification of the result, a straightforward Monte Carlo simulation was conducted (with 1,000 samples) and the result is shown as Figure. \ref{figure:1}, the mean of the number of rounds is 12.4287.
\begin{figure}[htb]
\centering
\includegraphics[width=0.5\textwidth] {./1.jpeg}
\caption{Monte Carlo simulation for 3PG with 1,000 samples.}
\label{figure:1}
\end{figure}
\section{The Second Order Analysis}
Given the expectation of the number of rounds in any states:
$$\left\{f(S):S\in\mathcal{S} \right\},$$
it is straightforward to compute the variance of the number of rounds by reduction. The bridge is:
$$\text{var}[X]=\text{var}[\mathbb{E}[X|Y]]+\mathbb{E}[\text{var}[X|Y]].$$
Now let $X$ be the random variable that denotes the number of rounds in the current state and $Y$ be the indicator of the current competition. Let $g(S)$ be the variance of the number of rounds of a 3PG begins from the state $S$, we have:
$$g(S)=\frac{(f(S_{1})-f(S_{2}))^{2}}{2}+\frac{g(S_{1})+g(S_{2})}{2}.$$
Thus given $\left\{f(S):S\in\mathcal{S} \right\}$ it is straightforward to compute $\left\{g(S):S\in\mathcal{S} \right\}$ (repeat what has been done in the sections before, \emph{reversely computing along the martingle!}) and deduce the variance of the number of rounds in a 3PG.
For example, consider the variances of the number of rounds for states $S_{1}=(111110|\left\{2,3 \right\})$, $S_{2}=(111110|\left\{1,2 \right\})$, $S_{3}=(111110|\left\{1,3 \right\})$. Using Proposition A, we have:
$$\begin{pmatrix}
g(S_{1}) \\
g(S_{2}) \\
g(S_{3})
\end{pmatrix}=
\begin{pmatrix}
0 & \frac{1}{2} & 0 \\
\frac{1}{2} & 0 & \frac{1}{2} \\
\frac{1}{2} & \frac{1}{2} & 0
\end{pmatrix}
\begin{pmatrix}
g(S_{1}) \\
g(S_{2}) \\
g(S_{3})
\end{pmatrix}
+
\begin{pmatrix}
18 \\
2 \\
2
\end{pmatrix}.
$$
Which yields:
$$\begin{pmatrix}
g(S_{1}) \\
g(S_{2}) \\
g(S_{3})
\end{pmatrix}=
\begin{pmatrix}
40 \\
44 \\
44
\end{pmatrix}.
$$
\section{The Probabilistic Framework}
The analysis so far is hardly relied on the probability space. The reason behind is that it is hard to establish the equivalence between an element in the probability space and the value of the random variable \cite{alon2004probabilistic}. Considering:
$$\Omega=\left\{+,- \right\}^{\infty}.$$
Where $+/-$ denotes the player with larger/smaller index winning the current round. To compute $\text{Pr}(X=n)$, where $X$ is the random variable that counts the number of rounds until termination. One has to find the number of $\left\{+,-\right\}^{n}$ sequences where all win-lose relationship appears until the final round. Although it is efficient to transcript a $\left\{+,-\right\}^{n}$ sequence into win-lose relationship sequences, it is hard to write down (be it exists) a tractable necessary and sufficient condition for $X=n$.
However, we could use the solution of 3PG to answer questions yielded from a more probabilistic perspective. For example:
\emph{Building up a string s with three characters \{`''a'',''b'',''c''\}, s begins with ''a'' and each character is followed by one different character with equal probability, s terminates until all six pairs appears in the string. What is the expected length of s?} This question is isomorphic to 3PG.
\section{Generalization}
Having finished the analysis of 3PG, we now proceed to a genelization study. The problem is, is it possible to find the asymptotical behavior of the solution to $n$-PG? The generalization of 3PG to $n$-PG is not unique, e.g., each round can still involve two players, and one random player enters the next round instead of the loser, or one can adopt ternary logic to mark the result of battles.
We study the general $n$-PG with two players participating each round, and a player is randomly (uniformly and independently) chosen to replace the loser of the current round in the ring.
First we try to address the states of order one, w.l.o.g., let the vacant relationship be Player 1 beats Player 2. There are four independent (unsymmetric) states with $\left\{1,2\right\}$, $\left\{1,3\right\}$, $\left\{2,3\right\}$ and $\left\{3,4 \right\}$ as the current pair of players in the ring (assuming $n \geq 4$). Let $x,y,z,w$ denote the corresponding expectations, then we have:
$$
\begin{pmatrix}
x\\
y\\
z\\
w
\end{pmatrix}=
A_{1}
\begin{pmatrix}
x\\
y\\
z\\
w
\end{pmatrix}+
\begin{pmatrix}
1\\
1\\
1\\
1
\end{pmatrix},
$$
with:
$$A_{1}=
\begin{pmatrix}
0 & 0 & \frac{1}{2} & 0\\
\epsilon & \frac{1}{2} & \epsilon & \frac{1}{2}\\
\epsilon & \epsilon & \frac{1}{2} & \frac{1}{2}\\
0 & \epsilon & \epsilon & 1-2\epsilon
\end{pmatrix},$$
where
$$\epsilon=\frac{1}{2(n-2)}.$$
Since we have:
$$
\begin{pmatrix}
x\\
y\\
z\\
w
\end{pmatrix}=(I-A_{1})^{-1}\cdot
\begin{pmatrix}
1\\
1\\
1\\
1
\end{pmatrix},
$$
as:
$$(I-A_{1})^{-1}=\sum_{i=0}^{\infty}A_{1}^{i},$$
the only task remained is to track the spectral radius of $A_{1}$ \cite{meyer2000matrix}, the trick here is to apply the Gerschgorin theorem to the last row of $A_{1}$, with yields that the largest eigenvalue of $A_{1}$ (assumed to be real) is no less than:
$$\lambda=1-4\epsilon.$$
Therefore the spectral radius of $(I-A)^{-1}$ is no less than:
$$\frac{1}{1-\lambda}\sim O(n).$$
So is the order of $x,y,z,w$.
Moving to states of order two is a similar case, let the independent states be $x',y',\cdots,w'$, we have:
$$
\begin{pmatrix}
x'\\
y'\\
\cdots\\
w'
\end{pmatrix}=A_{2}
\begin{pmatrix}
x'\\
y'\\
\cdots\\
w'
\end{pmatrix}+
\begin{pmatrix}
c_{x}\\
c_{y}\\
\cdots\\
c_{w}
\end{pmatrix},
$$
Where elements in $(c_{x},c_{y},\cdots,c_{w})^{\text{T}}$ are constants with value 1 or a multiple of $x,y,z,w$ that has been evaluated before. Hence the order of elements in $(c_{x},c_{y},\cdots,c_{w})^{\text{T}}$ is at most $O(n)$.
To measure the spectral radius of $A_{2}$, we resort to a similar line of reasoning: let $w$ be the expectation of the state where the current players on the ring is different from those players involved in the vacant win-lose relationships. Then the final row of $A_{2}$ has $1-\epsilon_{2}$ as the last component, where
$$\epsilon_{2}=\frac{4}{n-2}$$
in the most probable case. This yields the fact that the order of the spetral radius of $(I-A_{2})^{-1}$ be $O(n)$, hence the order of the expectations of states of order two turns out to be $O(n^{2})$.
In general, for states of order $\phi$, let $w_{\phi}$ be the expectation of the state where the current players on the ring are free from those $\phi$ vacant pairs, let $A_{\phi}$ be the transition matrix at that order. There are at most $2\phi$ players involve with the vacant pairs, hence the entry on the right-bottom most side of $A_{\phi}$ is at most:
$$1-\frac{2\phi}{n-2}.$$
That is to say, the spectral radius of $(I-A_{\phi})^{-1}$ is of order:
$$O\left(\frac{n}{\phi}\right).$$
Finally, counting all states of order $\phi=0,1,\cdots,n$, we have the order of the solution of an $n$-PG be:
$$O\left(\frac{n^{n}}{n!}\right)=O\left(\frac{\text{e}^{n}}{\sqrt{n}}\right).$$
Analogously, the variance for general $n$-PG can be approximated using the same framework. The $f(\cdot)$ for states of order $\phi$ is of order:
$$\frac{1}{\phi}\left(\frac{n\text{e}}{\phi} \right)^{\phi}.$$
We have that $g(\cdot)$ for states of order $\phi$ (denoted by $g_{\phi}$)is determined by the larger term in $f^{2}_{\phi}$ and $g_{\phi-1}$, so at least:
$$g_{\phi}\geq O\left(\frac{n}{\phi}\cdot f^{2}_{\phi}\right),$$
which is far less than the order of $f^{2}_{\phi}$, therefore we conjecture that the order of the variance in $n$-PG is:
$$O\left(\frac{\text{e}^{2n}}{n} \right).$$
In fact, we observe that the estimation on the expectation is possibly a rather slack one, this is due to the following facts:
\begin{itemize}
\item The estimation based on the Gerschgorin theorem on $A$ might significantly increase the spectral radius of $(I-A)^{-1}$.
\item The spectral radius might be involved with negligible terms so the speed of growth declines.
\end{itemize}
The simulation results of general $n$-PGs are illustrated as in Figure. \ref{figure:2}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.5\textwidth] {./2.jpeg}
\caption{Monte Carlo simulation for $n$-PG with 100 samples for each $n$.}
\label{figure:2}
\end{figure}
From which we might optimistically conjecture that the growth of the expectation of the number of rounds is only of order $n^{2}$, but there seems to be a vacancy in establishing this result.
\section{Conclusion}
This paper address the 3PG question. We attack this question with dynamic programming, highlight the necessary tricks that signicantly reduce redundant computation and analyze the ideas behind. The general case is also proposed and a rough bound is derived.
\section{Acknowledgement}
Haoran Ye for provided the 3PG question, Runbo Ni and his colleagues provided an early version of solution.
\bibliographystyle{ieeetr}
|
3,212,635,537,500 | arxiv | \section{Acknowledgments}
This research was supported in part by the Comisi\'on Interministerial de
Ciencia y Tecnolog\'{\i}a of Spain under contract MAT 94-0982-C02-01 and by
the Commission of European Communities under contract Ultrafast
CHRX-CT93-0133.
|
3,212,635,537,501 | arxiv | \section{Introduction}
A cornerstone of superstring theory is the formulation of perturbative closed string scattering amplitudes based on the path integral of two-dimensional supergravity coupling to matter fields on the worldsheet. The precise definition of the worldsheet path integral requires fixing the gauge redundancy of super-diffeomorphism and super-Weyl invariance. There are two standard gauge choices. The first one sets the worldsheet gravitino field to zero on local coordinate patches. The gauge choices for two different patches generally differ on the overlap by a superconformal transformation, giving rise to a super-Riemann surface (SRS) structure of the worldsheet \cite{DHoker:1988pdl, Witten:2012ga}. In this ``SRS formalism", the asymptotic closed string states are represented as punctures, of Neveu-Schwarz (NS) or Ramond (R) type, and their S-matrix elements are formulated as integrals of a form $\Omega$, defined through correlation functions of a superconformal field theory (SCFT) on the worldsheet SRS, over the supermoduli space of the latter. More precisely, for type II superstring considered in this paper, the integration of $\Omega$ is defined over a super contour $\mathfrak{S}$ of $\mathfrak{M}\times\overline{\mathfrak M}$, where $\mathfrak{M}$ and $\overline{\mathfrak M}$ are the supermoduli spaces of a pair of holomorphic and anti-holomorphic SRSs.
A second gauge choice is to set the worldsheet gravitino to a distribution supported at a finite set of points on the worldsheet. Up to regularization ambiguities that can be fixed by requirement of BRST invariance at the quantum level, integration over the gauge-inequivalent degrees of freedom of the gravitino gives rise to the insertions of picture-changing operators (PCOs) on the worldsheet equipped with an ordinary (bosonic) Riemann surface structure \cite{Friedan:1985ge, Verlinde:1987sd}. The space of states or vertex operators of the worldsheet SCFT is graded according to the picture number, of integer and half-integer values in the NS and R sectors respectively. The asymptotic string states are represented by BRST-closed vertex operators subject to Siegel constraint, in the NS sector of picture number $-1$, or in the R sector of picture number $-{1\over 2}$. The data of the worldsheet geometry together with the locations of PCOs are parameterized by a fiber bundle ${\cal Y}$ over the moduli space ${\cal M}$ of ordinary Riemann surfaces. The spacetime S-matrix elements are formulated as integrals of a form $\Omega$, defined through worldsheet SCFT correlators with PCO insertions, over a suitable contour ${\cal S}\subset{\cal Y}$. We refer to this as the PCO formalism, and will refer to ${\cal S}$ as the ``PCO contour".
Heuristically, the SRS and PCO formalisms of superstring amplitudes come from different gauge choices of the worldsheet path integral, and are anticipated to be equivalent by virtue of the Faddeev-Popov/BRST quantization procedure. However, as the path integral is only well-defined after gauge fixing, to establish such an equivalence requires demonstrating that the two formalisms are related by BRST-exact deformations at the quantum level, which is not at all obvious. Furthermore, the gauge choice that leads to PCOs may suffer from ``spurious singularities" that occur along complex codimension 1 loci in the fiber bundle ${\cal Y}$ \cite{Verlinde:1987sd, Atick:1987rk}, an issue that cast doubt on the consistency of the PCO formalism for over two decades.
The problem of spurious singularity was overcome in \cite{Sen:2014pia, Sen:2015hia} through the prescription of ``vertical integration". The basic idea is that the PCO contour ${\cal S}$ need not be a section of the bundle ${\cal Y}\to {\cal M}$, nor even a cycle; it suffices for ${\cal S}$ to be a chain that projects onto ${\cal M}$, such that the boundary of ${\cal S}$ consists of only ``vertical" components that extend in the fiber directions, on which a PCO moves along a closed path on a given Riemann surface. This allows for ${\cal S}$ to evade the loci of spurious singularities, while preserving the BRST-invariance and unitarity of superstring amplitudes.
The SRS formalism involves integrations over supermanifolds that, while in principle defined say via partition of unity, are often difficult to evaluate due to lack of convenient parameterizations of the supermoduli space \cite{Witten:2012bg, Donagi:2013dua}. In contrast, the PCO formalism gives a relatively straightforward algorithm for evaluating superstring amplitudes (numerically if necessary), and has been further extended to define off-shell amplitudes and vertices of superstring field theory \cite{Sen:2014pia, deLacroix:2017lif}. On the other hand, the choice of the PCO contour ${\cal S}$ is highly non-canonical. Nonetheless, it has been shown in \cite{Sen:2014pia, Sen:2015hia, Erler:2017dgr} that deformations of ${\cal S}$ do not affect on-shell amplitudes.
In this paper, we establish the equivalence between the two formalisms by showing that a specific choice of the super contour $\mathfrak{S}$, upon patchwise integration over fermionic coordinates, produces the PCO formalism. Our construction of $\mathfrak{S}$ consists of ``horizontal patches" in the supermoduli space that corresponds to PCOs at fixed locations, and interpolating ``vertical patches" that amount to vertical integration. In defining a patch of $\mathfrak{S}$ as the image of a map $\mathfrak{I}$ into the supermoduli space, and the integration of $\Omega$ via that of the pullback $\mathfrak{I}^*\Omega$, it is important to ensure that $\mathfrak{I}$ has non-singular Berezinian \cite{Witten:2012bg}. The latter is closely related to the evasion of spurious singularity in the PCO formalism.
In section \ref{sec:srs}, we review the notion of SRS, its supermoduli space, the meaning of a SCFT on a general SRS, and the formulation of type II superstring amplitudes as an integral over the supermoduli space. Section \ref{sec:pco} reviews the PCO, spurious singularities, and the integration contour used to define superstring amplitudes in the PCO formalism. In section \ref{sec:integration}, we explain the definition of integral over a supermanifold $\mathfrak{M}$ via partition of unity, and then introduce an alternative way of performing the integration by ``lifting and interpolation". The basic idea is that one can cover $\mathfrak{M}$ with super coordinate patches $\mathfrak{U}_\A$, each of which projects onto a bosonic patch ${\cal U}_\A$ of reduced space ${\cal M}$ via the projection map $\pi_\A: \mathfrak{U}_\A\to {\cal U}_\A$. Dividing ${\cal M}$ into cells ${\cal D}_\A\subset{\cal U}_\A$, one can reduce the integration over $\pi_\A^{-1}({\cal D}_\A)$ to a bosonic one by integrating out the fermionic fibers. However, the fermionic fibers of $\pi_\A$ and $\pi_\B$ generally do not agree on the boundary between adjacent cells, ${\cal D}_\A\cap {\cal D}_\B$. This is corrected for by an interpolating chain between $\pi_\A^{-1}({\cal D}_\A)$ and $\pi_\B^{-1}({\cal D}_\B)$ constructed as the image of an interpolation map $\mathfrak{I}_{\A\B}$. To recover the full integral over $\mathfrak{M}$, one must include the interpolating chains that account for the mismatch of fermionic fibers along all-codimensional interfaces between adjacent cells. An explicit construction of the super coordinate patches $\mathfrak{U}_\A$ of the supermoduli space $\mathfrak{M}$, and the transition maps between them, is given in section \ref{sec:lift}. The general structure of the interpolating chains for the supermoduli integration contour $\mathfrak{S}\subset \mathfrak{M}\times\overline{\mathfrak M}$ of type II superstring theory is described in section \ref{sec:supercontour}.
Section \ref{sec:mtos} is the core of the paper. In section \ref{sec:pcoemerge}, we explain how PCO insertions arise from the integration over the fiber of the projection map $\pi_\A$ of section \ref{sec:lift}. The interpolation map ${\mathfrak I}_{\A\B}$ between the fibers of adjacent patches is constructed in section \ref{sec:vertemerge}, and shown to reproduce the vertical integration in the codimension 1 case. This is then generalized to the higher codimension case in section \ref{sec:pcohighercodim}. An application of this general construction in the example of genus two SRSs will be discussed in a follow-up paper \cite{paper:g2}. Some concluding remarks are given in section \ref{sec:discuss}.
\section{The SRS formalism}
\label{sec:srs}
\subsection{SRS and its moduli space}
A super-Riemann surface (SRS) ${\mathfrak C}$ is a $1|1$ complex supermanifold, equipped with a totally non-integrable rank $0|1$ sub-bundle of its tangent bundle. It can be covered with super coordinate charts $U_i$, parameterized by bosonic and fermionic coordinates $(z_i,\theta_i)$, such that the transition map on the overlap $U_i\cap U_j$ takes form of a superconformal transformation
\ie\label{sctrans}
&z_i = f_{ij}(z_j) + \theta_j g_{ij}(z_j) h_{ij}(z_j),
\\
& \theta_i = g_{ij}(z_j) + \theta_j h_{ij}(z_j),~~~ {\rm with}~ h_{ij} = \pm\sqrt{\partial f_{ij} + g_{ij}\partial g_{ij}}.
\fe
Generally, the transition functions $f_{ij}$ and $g_{ij}$ can be deformed by a set of bosonic and fermionic parameters, while each $h_{ij}$ involves a choice of sign. Modulo superconformal diffeomorphism, the former give rise to the even and odd moduli, denoted by $\tau^m$ ($m=1,\cdots, d_{ e}$) and $\nu^a$ ($a=1,\cdots, d_{ o}$) respectively, whereas the latter give rise to a choice of spin structure, denoted by $\epsilon$. We will sometimes indicate the moduli dependence of the SRS with the notation $\mathfrak{C}_{t,\nu}$, or simply $\mathfrak{C}_\nu$ to indicate the odd moduli dependence.
By setting all the odd parameters in the transition maps that define $\mathfrak{C}$ to zero, which includes setting all of $g_{ij}$'s to zero, we obtain the reduced space $\Sigma$ which is an ordinary Riemann surface. Conversely, given an ordinary Riemann surface $\Sigma$ with a choice of spin structure $\epsilon$, there is a corresponding SRS $\mathfrak{C}_0$ with $g_{ij}= 0$, known as a split SRS.
For our purpose it is important to extend the definition of SRS to allow for punctures of NS and R type. The NS puncture amounts to marking a point, say $(z,\theta)=(0,0)$, on a super disc $D$ with coordinates $(z,\theta)$. The R puncture is defined by further splitting the disc $D$ into two wedges, parameterized by $(z,\theta)$ and $(z',\theta')$ such that the transition map between them takes the form $z'=z$, $\theta'=\pm\theta$, where the sign is the opposite on the two disjoint wedges of the overlap.
The string worldsheet in the superconformal gauge, where the worldsheet gravitino are set to zero patch wise, can be viewed as a diagonal subspace of $\mathfrak{C}\times \overline{\mathfrak{C}}'$, where $\overline{\mathfrak{C}}'$ is a SRS whose reduced space is $\overline\Sigma$, the complex conjugate Riemann surface of $\Sigma$. Note that the choice of spin structure of $\overline{\mathfrak{C}}'$ and the types of its punctures are a priori independent from those of $\mathfrak{C}$.
The SRS $\mathfrak{C}$ can be classified topologically according to the genus $g$ of its reduced space, the number of NS and R punctures, $n_{\rm NS}$ and $n_{\rm R}$, and the spin structure $\epsilon$. The supermoduli space of such SRS, denoted $\mathfrak{M}_{g, n_{\rm NS}, n_{\rm R}, \epsilon}$ or simply $\mathfrak{M}$, is a complex supermanifold of dimension $d_{ e}|d_{ o}$, where $d_{ e}=3g-3+n_{\rm NS} + n_{\rm R}$, $d_o = 2g-2 + n_{\rm NS} + {1\over 2}n_{\rm R}$. Its reduced space ${\cal M}_{g, n_{\rm NS}, n_{\rm R},\epsilon}$ is the moduli space of an ordinary Riemann surface of genus $g$, with $n_{\rm NS}+n_{\rm R}$ punctures and a choice of spin structure.
One should be cautious that a supermanifold is defined not as a set but as a locally ringed space. In particular, the operation of ``forgetting the fermionic coordinates" does not define a map from $\mathfrak{M}$ to its reduced space ${\cal M}$, as the pullback of a function under such an operation does not give a well-defined function on the supermanifold $\mathfrak{M}$. For our purpose, it is important to have a notion of integration of super forms on a supermanifold, that may be defined via partition of unity and respects Stokes' theorem, which we review in section \ref{sec:integration} (see also \cite{Witten:2012bg, Witten:2012ga}).
\subsection{SCFT on SRS }
\label{sec:srsscft}
Throughout this paper we will consider a superconformal field theory(SCFT) of central charge $c=0$ on the worldsheet, with holomorphic stress-energy tensor $T(z)$ and supercurrent $G(z)$ that obey the super-Virasoro algebra, as well as their anti-holomorphic counter parts. In particular, the singular part of the OPE of a pair of supercurrents takes the form
\ie\label{ggope}
G(z) G(0) \sim {2\over z} T(0).
\fe
With the absence of Weyl anomaly, the SCFT is a priori defined on a Riemann surface $\Sigma$ equipped with a spin structure $\epsilon$ that specifies the (anti-)periodicity of the supercurrent via transport along closed paths. Equivalently, we may view this as defining the SCFT on the corresponding split SRS $\mathfrak{C}_0$.
This notion of SCFT can be extended to that over an arbitrary SRS by deforming away from the split case, as follows. Given a set of super coordinate charts $U_i$ of $\mathfrak{C}$, parameterized by $(z_i, \theta_i)$, a deformation of the supermoduli amounts to a deformation of the transition maps (\ref{sctrans}). This is represented at the level of SCFT correlators by inserting
\ie\label{scdeform}
\delta{\cal T} = - \sum_{(ij)} \int_{C_{ij}} {[dz_i| d\theta_i ]\over 2\pi i}\, \mathbb{T}(z_i, \theta_i)\left. \big[ \delta z_i - (\delta \theta_i) \theta_i \big]\right|_{z_j,\theta_j},
\fe
where $\mathbb{T}(z,\theta) \equiv {1\over 2} G(z) + \theta T(z)$ is the super stress tensor, and $C_{ij}$ are a set of arcs defined as follows. Let $D_i$ be a closed domain contained within the bosonic part of $U_i$, such that $D_i$ and $D_j$ only meet along their boundary, and the union of $D_i$'s is the entirety of $\Sigma$. $C_{ij}$ is the arc along $D_i\cap D_j$, oriented such that $\sum_j C_{ij} = \partial D_i$. The superconformal covariance of (\ref{scdeform}) ensures that two different sets of coordinate charts and transition maps describe equivalent SRS if the corresponding super stress tensor insertions (\ref{scdeform}) are equivalent (in the SCFT sense) via contour deformations and OPEs as governed by the superconformal algebra.
A particularly useful parameterization of the odd moduli is through the gluing map of super discs, as follows. Starting with a split SRS $\mathfrak{C}_0$ with its reduced space $\Sigma$, we choose a set of points $z_a$ on $\Sigma$, $a=1,\cdots, d_o$, along with sufficient small discs $D_a$ centered at $z_a$. Denote by $U_a$ a coordinate patch of $\Sigma$ that contains $D_a$, which is further split into two patches $U_a = U_a'\cup D_a$, where $U_a'$ is an annulus that does not contain $z_a$. Each of these patches is promoted to a super coordinate chart of $\mathfrak{C}_0$, which we will denote by the same symbol. Let $(z,\theta)$ be the coordinates on the super chart of $U_a'$, and $(w,\eta)$ be the coordinates on the super chart of $D_a$, such that the transition map on their overlap in $\mathfrak{C}_0$ is simply $(w,\eta)=(z,\theta)$. Now we can construct a new SRS $\mathfrak{C}_\nu$ by deforming these transition maps to
\ie\label{superdiscglue}
& w = z - {\theta\nu^a\over z-z_a},
\\
& \eta = \theta - {\nu^a\over z-z_a},
\fe
where $\nu^a$ is a fermionic parameter, for each $a=1,\cdots,d_o$. In the language of SCFT, the deformation from $\mathfrak{C}_0$ to $\mathfrak{C}_\nu$ amounts to the insertion of
\ie\label{nvgdef}
\prod_a \left[ 1 - \oint_{\partial D_a} {dw\over 2\pi i} G(w) {\nu^a \over w-z_a} \right] = \prod_a \left[1+ \nu^a G(z_a) \right]
\fe
on $\Sigma$.
For generic choice of $z_1,\cdots, z_{d_o}$, the $\nu^a$'s give a set of non-degenerate local fermionic coordinates on the supermoduli space. Degeneration occurs if there is a weight $-{1\over 2}$ meromorphic differential $r(z) (dz)^{-{1\over 2}}$ with only $d_o$ simple poles at the $z_a$ with residue $r_a$, so that the linear combination of supercurrent insertions $\sum_a r_a G(z_a) = \oint_{\sqcup \partial D_a} {dz\over 2\pi i} r(z) G(z)$ vanishes by shrinking the contour $\sqcup_a \partial D_a$ away from $\{z_a\}$. This is precisely the condition for a spurious singularity (see section \ref{sec:pcointro}).
\subsection{Superstring amplitude}
\label{sec:superamp}
In type II superstring theory, the perturbative scattering amplitude of $n$ closed string asymptotic states, represented by BRST-closed vertex operators ${\cal V}_i$ that obey Siegel constraint, takes the general form
\ie
{\cal A}[{\cal V}_1,\cdots,{\cal V}_n] = \sum_{g\geq 0} {g_s^{2g-2}\over 2^{2g}} \sum_{\epsilon, \widetilde\epsilon} {\cal A}_{g,\epsilon, \widetilde\epsilon}[{\cal V}_1,\cdots,{\cal V}_n],
\fe
where $g$ is the genus and $\epsilon,\widetilde\epsilon$ label the holomorphic and anti-holomorphic spin structures, respectively. For simplicity of exposition we will focus on the case where all of the asymptotic string states are of (NS, NS) type. The distinction between type IIA and type IIB strings is reflected in the choice of overall sign of the odd spin structure contributions to the amplitude. We have
\ie\label{supermoduliintegral}
&{\cal A}_{g,\epsilon,\widetilde\epsilon}[{\cal V}_1,\cdots,{\cal V}_n] = \int_{\mathfrak{S}} \Omega,
\fe
where the integration contour $\mathfrak{S} \subset \mathfrak{M}_{g,n,\epsilon} \times \overline{{\mathfrak M}_{g,n,\widetilde\epsilon}}$ is a subspace of codimension $d_e|0$, whose reduced space is the diagonal subspace of ${\cal M}_{g,n,\epsilon} \times \overline{{\cal M}_{g,n,\widetilde\epsilon}}$. $\Omega$ is the super integral form defined in a neighborhood of $\mathfrak{S}$ in $\mathfrak{M}_{g,n,\epsilon} \times \overline{{\mathfrak M}_{g,n,\widetilde\epsilon}}$, given by the following correlator of the worldsheet SCFT on the SRS $\mathfrak{C}\times \overline{\mathfrak{C}'}$,
\ie\label{omegaintg}
& \Omega = \left\langle e^{\mathfrak{B}} \prod_{i=1}^n {\cal V}_i \right\rangle,
\fe
where $\mathfrak{B}$ is the 1-form
\ie
\mathfrak{B} =\sum_{k=1}^{2d_e} {\cal B}_{t^k} dt^k + \sum_{a=1}^{d_o} \delta(d\nu^a) \delta({\cal B}_{\nu^a}) + c.c.,
\fe
with
\ie{}
& {\cal B}_{t^k} = \sum_{(ij)} \int_{C_{ij}} {[dz_i|d\theta_i] \over 2\pi i} \, \mathbb{B}(z_i, \theta_i)\left. \left[ {\partial z_i\over \partial t^k} - {\partial \theta_i\over \partial t^k} \theta_i \right]\right|_{z_j,\theta_j},
\\
& {\cal B}_{\nu^a} = \sum_{(ij)} \int_{C_{ij}} {[dz_i|d\theta_i] \over 2\pi i} \, \mathbb{B}(z_i, \theta_i)\left. \left[ {\partial z_i\over \partial\nu^a} - {\partial \theta_i\over \partial\nu^a} \theta_i \right]\right|_{z_j,\theta_j} .
\fe
Here $\mathbb{B}=\B(z) + \theta b(z)$. The domain of integration is defined as in (\ref{scdeform}). Here $t^k=(\tau^m, \bar\tau^m)$ are bosonic moduli parameters. We emphasize that the superconformal transition maps need not depend holomorphically on $\tau^m$. We have also adopted the convention of \cite{Witten:2012bg} in which the 1-form $dt^k$ is Grassmann-odd, $d\nu^a$ is Grassmann-even, and the integral form $\delta(d\nu^a)$ is Grassmann-odd.
Note that ${\cal B}_{\nu^a}$ is a Grassmann-even object, and $\delta({\cal B}_{\nu^a})$ is generally not a local operator. The latter should be understood as a holomorphic distribution that serves to absorb the integration over zero modes of $\B$. Typically, ${\cal B}_{\nu^a}$ takes the form
\ie
{\cal B}_{\nu^a} = \int_C {dz\over 2\pi i} \B(z) g_a(z) +\cdots,
\fe
where $g_a(z)$ is a holomorphic vector field defined in a neighborhood of the path $C$, and $\cdots$ represents terms that involve odd moduli. The functional integration over $\B$ involves an integration over zero modes of the form $\B(z) =\sum_k \B_{(k)} \lambda^{(k)}(z)$, where $\lambda^{(k)}(z) (dz)^{3\over 2}$ are a basis of meromorphic weight ${3\over 2}$ forms that are compatible with the spin structure $\epsilon$ and have simple poles at the location of vertex operators ${\cal V}_1,\cdots,{\cal V}_n$. The result of the zero mode integral with $\delta({\cal B}_{\nu^a})$ insertion is
\ie
\int \prod_k d\B^{(k)} \prod_a \delta({\cal B}_{\nu^a}) = {1\over \det M_a^k},~~~~M_a^k = \int_C {dz\over 2\pi i} \lambda^{(k)}(z) g_a(z).
\fe
The matrix $M_a^k$ is non-degenerate provided that $\nu^a$ are non-degenerate fermionic coordinates on the supermoduli space. Consequently, the integrand $\Omega$ is non-singular at least away from the boundary of the moduli space.
\section{The PCO formalism}
\label{sec:pco}
\subsection{The PCO and spurious singularities}
\label{sec:pcointro}
Gauge fixing the worldsheet supergravity path integral by setting the gravitino to a sum over delta functions \cite{Verlinde:1987sd, DHoker:1988pdl}, upon suitable regularization, leads to the insertion of PCO, defined by
\ie\label{pcodef}
{\cal X}(z) &= - Q_B \cdot\Theta(\B(z)) = {1\over 2} \lim_{w\to z} \left[ G(w) \delta(\B(z)) - {1\over w-z} b(z) \delta'(\B(z)) \right] - {1\over 4} \partial b(z) \delta'(\B(z))
\fe
and similarly for its anti-holomorphic counter part $\widetilde{\cal X}(\bar z)$.
It is convenient to pass to an equivalent representation of the $\B\C$ system in terms of a linear dilaton field $\phi$ and Grassmann-odd fields $(\xi,\eta)$\footnote{The nontrivial OPEs among $\phi,\xi,\eta$ are $\phi(z)\phi(0)\sim -\ln z$, $\xi(z) \eta(0)\sim {1\over z}$.} \cite{Friedan:1985ge}, with $\B\simeq e^{-\phi}\partial\xi$, $\C\simeq \eta e^\phi$, $\delta(\B)\simeq e^\phi$, $\delta(\C)\simeq e^{-\phi}$. We adopt a cocycle convention in which $e^{\A\phi}$ is Grassmann odd for odd integer $\A$. In the $(\phi,\xi,\eta)$ representation, the PCO is written as
\ie
{\cal X}(z) &= Q_B \cdot \xi(z) = - {1\over 2} e^\phi G^{\rm matter} + c\partial \xi - {1\over 4} e^{2\phi} \partial\eta b - {1\over 4} \partial( e^{2\phi} \eta b).
\fe
The correlators of an arbitrary set of $\xi$, $\eta$, and $e^{\A\phi}$ insertions on any Riemann surface $\Sigma$ with a choice of spin structure are given in terms of theta functions on the Jacobian variety of $\Sigma$ and prime forms in \cite{Verlinde:1987sd}. An important feature of such correlators is the presence of so-called spurious singularities, which occur whenever there is a zero mode of $\C(z)$, that is, a weight $-{1\over 2}$ meromorphic differential with a simple pole at every $\delta(\B)$ insertion (or the location of a PCO) and a simple zero at every $\delta(\C)$ insertion (or location of a vertex operator for an NS string state).
Let us remark that the correlators of $\delta(\B)$ and $\delta(\C)$ can be used to construct half-integer weight meromorphic differentials of certain prescribed zeros and poles, which will be used extensively later. For instance, given a generic set of $n$ points $\{w_i\}$ and $2g-1+n$ points $\{z_a\}$ on $\Sigma$, there is a weight $-{1\over 2}$ meromorphic differential $\zeta = f(z) dz$ with simple poles at $z_a$ and zeros at $w_i$, that is unique up to constant rescaling, given by
\ie\label{meroweighthalf}
f(z) = {1\over \left\langle \delta(\C(z)) \prod_{i=1}^n \delta(\C(w_i)) \prod_{a=1}^{2g-1+n} \delta(\B(z_a)) \right\rangle_{\Sigma,\epsilon}}.
\fe
\subsection{The integration contour}
We will refer to the diagonal subspace ${\cal M}\subset {\cal M}_{g,n,\epsilon}\times \overline{{\cal M}_{g,n,\widetilde\epsilon}}$ (i.e. the reduced space of the supermanifold $\mathfrak{S}$ of section \ref{sec:superamp}) as the bosonic moduli space of the Riemann surface $\Sigma$ with holomorphic spin structure $\epsilon$ and anti-holomorphic spin structure $\widetilde\epsilon$. Let $\pi:{\cal Y}\to {\cal M}$ be the fiber bundle whose fiber
\ie
\pi^{-1}(x)\simeq (\Sigma_x\times \overline\Sigma_x)^{d_o}
\fe
is the space of the loci of $d_o$ pairs of holomorphic and anti-holomorphic PCOs inserted on the surface $\Sigma_x$ corresponding to the point $x\in {\cal M}$.
In the formalism of type II string amplitude based on PCOs \cite{Sen:2014pia, Sen:2015hia}, the genus $g$ amplitude takes the form
\ie\label{pcoamp}
{\cal A}_{g,\epsilon,\widetilde\epsilon}[{\cal V}_1,\cdots,{\cal V}_n] = \int_{\cal{S}} \widetilde\Omega.
\fe
Here $\widetilde\Omega$ is a form on ${\cal Y}$ defined as the worldsheet SCFT correlator
\ie\label{tildeomg}
\widetilde\Omega = \left\langle e^{\pi^*{\cal B}} \prod_{i=1}^n {\cal V}_i \prod_{a=1}^{d_o} \big[ {\cal X}(z_a)+d\xi(z_a)\big] \big[ \widetilde{\cal X}(\bar z_a)+d\widetilde\xi(\bar z_a) \big] \right\rangle_{\Sigma,\epsilon,\widetilde\epsilon},
\fe
where $(z_a,\bar z_a)$ are viewed as coordinates on the fiber of ${\cal Y}\to {\cal M}$. ${\cal B}$ is the 1-form on ${\cal M}$ defined by
\ie
{\cal B} = \sum_{(ij)} \int_{C_{ij}} {dz_i \over 2\pi i} \, b(z_i)\left. {\partial z_i\over \partial t^k}\right|_{z_j,\theta_j} dt^k + c.c.
\fe
The integration contour $\cal{S}$ in (\ref{pcoamp}) is a $d_e$-dimensional chain in ${\cal Y}$ that evades spurious singularities of $\widetilde\Omega$, such that $\pi(\cal{S})$ covers ${\cal M}$ once. Importantly, $\cal{S}$ need not be a cycle\footnote{In these considerations we ignore the boundary of the moduli space ${\cal M}$ whose proper treatment requires string field theory \cite{Sen:2014pia}.}; rather, its boundary $\partial \cal{S}$ consists of only ``vertical components", in the sense that
\ie\label{verticalholes}
\partial{\cal S} = \sum_k {\cal Q}_k,
\fe
where each component ${\cal Q}_k$ is a fiber bundle over a chain $\pi({\cal Q}_k) \subset {\cal M}$ of nonzero codimension, whose fiber is of the form $\C\times W \subset (\Sigma\times \overline\Sigma)^{d_o}$, where $\C$ is a 1-cycle in one of the $\Sigma$ or $\overline\Sigma$ factors, and $W$ is a chain in the product of the remaining $\Sigma$ and $\overline\Sigma$ factors.
Under a BRST variation of the string vertex operators, $\widetilde\Omega$ obeys
\ie
\widetilde\Omega[Q_B (\otimes_i {\cal V}_i)] = - d\widetilde\Omega[\otimes_i {\cal V}_i].
\fe
It follows that the BRST variation of the amplitude (\ref{pcoamp}),
\ie
{\cal A}_{g,\epsilon,\widetilde\epsilon}[Q_B (\otimes_i {\cal V}_i)] = - \int_{\cal S} d\widetilde\Omega = - \sum_k\int_{{\cal Q}_k} \widetilde\Omega
\fe
vanishes, since the integral of $\widetilde\Omega$ along each fiber of ${\cal Q}_k$ vanishes by virtue of $\oint_\C d\xi=0$ for 1-cycle $\C\subset \Sigma$ (and similarly for $\C\subset\overline\Sigma$ with $\xi$ replaced by $\widetilde\xi$).
In \cite{Sen:2015hia}, an explicit construction of ${\cal S}$ is given based on a dual triangulation of ${\cal M}$. We can describe a local model of this construction as follows. Let ${\cal D}_1, {\cal D}_2, \cdots, {\cal D}_{2d_e+1}$ be cells of a dual triangulation of ${\cal M}$ that meet at a point. We will denote ${\cal D}_{i_1\cdots i_{p+1}} = {\cal D}_{i_1}\cap \cdots \cap {\cal D}_{i_{p+1}}$ ($i_1<\cdots<i_{p+1}$) the $(2d_e-p)$-dimensional faces along which a subset of the cells meet. The restriction of ${\cal S}$ to the interior of $\pi^{-1}(\cup_i {\cal D}_i)$ is of the form
\ie\label{scontourcomp}
\left.{\cal S}\right|_{{\rm Int}(\pi^{-1}(\cup_i {\cal D}_i))} = \sum_{i_1<\cdots<i_{p+1}} {\cal S}_{i_1\cdots i_{p+1}},
\fe
where ${\cal S}_{i_1\cdots i_{p+1}}$ is a piecewise fiber bundle over ${\cal D}_{i_1\cdots i_{p+1}}$, whose fiber is a $p$-dimensional chain in $(\Sigma\times \overline\Sigma)^{d_o}$ that is the sum of $p$-cubes of the form $\C_1\times\cdots\times \C_p$, where each $\C_i$ is an arc in one of the $\Sigma$ or $\overline\Sigma$ factors. In the $p=0$ case, ${\cal S}_i$ is a section of ${\cal Y}|_{{\cal D}_i}$, which amounts to choosing the locations of $2d_o$ PCOs over the moduli domain ${\cal D}_i$. Moving along ${\cal D}_{i_1\cdots i_{p+1}}$, the fiber ${\cal S}_{i_1\cdots i_{p+1}}$ may jump only by $\C_i\to \C_i + \C'$, where $\C'$ is a 1-cycle in $\Sigma$ or $\overline\Sigma$ (as explained above, such jumps do not affect the integral of $\widetilde\Omega$ along the fiber). Furthermore, the components of (\ref{scontourcomp}) are subject to the matching condition
\ie{}
\left. \sum_{m=1}^{p+1}{\cal S}_{i_1\cdots i_{m-1} i_{m+1}\cdots i_{p+1}}\right|_{{\cal D}_{i_1\cdots i_p}} = - \partial_{ f} {\cal S}_{i_1\cdots i_p},
\fe
where $\partial_f$ stands for taking the boundary of the fiber of ${\cal S}_{i_1\cdots i_p}\to {\cal D}_{i_1\cdots i_p}$.
\section{Integration over the supermoduli space}
\label{sec:integration}
\subsection{Partition of unity}
The integration over a supermanifold $\mathfrak{M}$ of dimension $n_e|n_o$ can be defined via partition of unity, as follows.
We begin with a sufficiently fine covering of the reduced space ${\cal M}$ with open patches ${\cal U}_\A$, such that each patch ${\cal U}_\A$ can be lifted to a super coordinate patch $\mathfrak{U}_\A$ of ${\mathfrak M}$ via\footnote{Following \cite{Witten:2012bg}, we denote by $\mathbb{R}^{p|*q}$ the complex supersymmetric vector space parameterized by $p$ real bosonic coordinates and $q$ fermionic coordinates, emphasizing that there is no reality condition involving the fermionic coordinates.}
\ie
\varphi_\A: ~{\cal U}_\A\times \mathbb{R}^{0|*n_o} \to \mathfrak{U}_\A.
\fe
Let $\pi_\A: \mathfrak{U}_\A\to {\cal U}_\A$ be the corresponding projection map, namely $\pi_\A\circ\varphi_\A: {\cal U}_\A\times \mathbb{R}^{0|*n_o}\to {\cal U}_\A$ simply forgets the fermionic coordinates. Note that both the lift and projection are highly non-canonical.
Starting with a partition of unity on ${\cal M}$,
\ie
\sum_\A f_\A = 1,~~~~ {\rm Supp} (f_\A) \subset {\cal U}_\A,
\fe
we can construct a partition of unity on $\mathfrak{M}$ via
\ie\label{fapart}
F_\A = {\pi_\A^* f_\A \over \sum_\B \pi_\B^* f_\B},~~~~ \sum_\A F_\A = 1.
\fe
Note that the (fermionic) fibers of the projection $\pi_\A$ and $\pi_\B$ onto two patches ${\cal U}_\A$ and ${\cal U}_\B$ generally do not agree on the overlap ${\cal U}_\A\cup {\cal U}_\B$. Nonetheless, $\left.\sum_\B \pi_\B^* f_\B\right|_{{\mathfrak U}_\A}$ is equal to 1 plus quadratic and higher order terms in the fermionic coordinates and therefore invertible, ensuring that (\ref{fapart}) is well-defined.
The integration of a super form $\Omega$ over $\mathfrak{M}$ is then defined as
\ie
\int_{\mathfrak{M}}\Omega = \sum_\A \int_{{\cal U}_\A\times \mathbb{R}^{0|n_o}} \varphi_\A^*(F_\A\Omega).
\fe
Importantly, the resulting integral does not depend on the choice of partition of unity, and obeys the supermanifold version of Stokes' theorem \cite{Witten:2012bg}.
\subsection{Integration by lifting and interpolation}
\label{sec:inter}
An alternative way to perform the integration over the supermanifold $\mathfrak{M}$ is to begin with a cell decomposition of the reduced space ${\cal M} = \bigsqcup {\cal D}_\A$, say a dual triangulation, where each domain ${\cal D}_\A$ is contained in a patch ${\cal U}_\A$ that lifts to a super chart $\pi_\A: \mathfrak{U}_\A\to {\cal U}_\A$ as in the previous subsection, perform the integration over each $\pi_\A^{-1}({\cal D}_\A)$, and account for the mismatch between the fibers of $\pi_\A$ and $\pi_\B$ along ${\cal U}_\A\cap {\cal U}_\B$ by an ``interpolating integral". Such a construction amounts to patch-by-patch integration over a top dimensional chain that is homologically equivalent to $\mathfrak{M}$.
We begin by illustrating the interpolation scheme in a basic example. Consider a supermanifold of real dimension $1|n_o$ covered by two patches $\mathfrak{U}$ and $\widetilde{\mathfrak U}$, with coordinate maps
\ie\label{neighborpatches}
& \varphi: \{ (t, \nu^a) \in \mathbb{R}^{1|*n_o}: t<\delta\} \to^{\!\!\!\!\!\!\!\sim}\, {\mathfrak U},
\\
& \widetilde\varphi:\{ (\widetilde t, \widetilde\nu^a) \in \mathbb{R}^{1|*n_o}: \widetilde t>-\delta\} \to^{\!\!\!\!\!\!\!\sim}\, \widetilde {\mathfrak U} ,
\fe
where $\delta>0$, and the transition map $\widetilde\varphi^{-1}\circ\varphi$ takes the form
\ie\label{transitionmaptf}
\widetilde t = t + f(t, \nu),~~~~ \widetilde \nu^a = g^a(t,\nu),
\fe
for $t\in (-\delta,\delta)$, with $f(t,0)=0$. The reduced space is simply $\mathbb{R}$, covered by ${\cal U} = (-\infty,\delta)$ and $\widetilde{\cal U}=(-\delta,\infty)$. We denote by $\pi: \mathfrak{U}\to {\cal U}$ the projection map such that $\pi\circ \varphi$ simply forgets $\nu^a$, and similarly for $\widetilde\pi: \widetilde{\mathfrak U} \to \widetilde{\cal U}$.
\begin{figure}[h!]
\centering
\scalebox{.8}{
\begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {\includegraphics[width=.7\textwidth]{figures/fibers.pdf}};
\begin{scope}[shift={(image.south west)}, x={(image.south east)}, y={(image.north west)}]
\node[anchor=south, color=red] at (0.23, 0.85) {\large $\pi^{-1}(-\infty, 0)$};
\node[anchor=south, color=blue] at (0.77, 0.85) {\large $\widetilde\pi^{-1}(0, \infty)$};
\end{scope}
\end{tikzpicture}}
\scalebox{.8}{ \begin{tikzpicture}
\node[anchor=south west, inner sep=0] (image2) at (0, 0){\includegraphics[width=0.92\textwidth]{figures/interpolation_split.pdf}};
\begin{scope}[shift={(image2.south west)}, x={(image2.south east)}, y={(image2.north west)}]
\node[anchor=center] at (0.15, 1) {\large $\pi^{-1}(-\infty, 0)$};
\node[anchor=center] at (0.85, 1) {\large $\widetilde\pi^{-1}(0, \infty)$};
\node[anchor=center] at (0.5, 1) {\large $\mathfrak{I}_*$};
\end{scope}
\end{tikzpicture}
}
\caption{A schematic depiction of the interpolation scheme for integrating over $\mathfrak{M}$. The horizontal axis represents the bosonic coordinate $t$ while the vertical direction represents the fermionic coordinates $\nu^a$.
Top: The mismatched fibers coming from the projections $\pi$ (red, left) and $\widetilde{\pi}$ (blue, right).
Bottom: The splitting of the integration contour into integration over the fibers of $\pi$, the fibers of $\widetilde{\pi}$, and the image of the interpolation map $\mathfrak{I}$.
}
\end{figure}
The idea is to break the integration over $\mathfrak{U}\cup \widetilde{\mathfrak U}$ into the integration over $\pi^{-1}(-\infty,0)$ and $\widetilde\pi^{-1}(0,\infty)$, together with an integral over a super patch that interpolates between the fibers $\pi^{-1}(0)$ and $\widetilde\pi^{-1}(0)$. Namely, given a degree $1|n_o$ form $\Omega$, we can write
\ie\label{onedimbreak}
\int_{{\mathfrak U}\cup \widetilde{\mathfrak U}} \Omega = \int_{t<0} \varphi^*\Omega + \int_{\widetilde t>0} \widetilde\varphi^*\Omega + \int_{ [0,1]\times \mathbb{R}^{0|*n_o}} \mathfrak{I}^*\Omega,
\fe
where $\mathfrak{I}$ is an ``interpolation map"
\ie\label{intermap}
& \mathfrak{I} : [0,1]\times \mathbb{R}^{0|*n_o}\to {\mathfrak U}\cap \widetilde{\mathfrak U}
\fe
such that ${\cal J}(\{0\}\times \mathbb{R}^{0|*n_o})$ agrees with $\pi^{-1}(0)$, and that ${\cal J}(\{1\}\times \mathbb{R}^{0|*n_o})$ agrees with $\widetilde\pi^{-1}(0)$. The latter amounts to the matching condition
\ie\label{intermapbdry}
\varphi^{-1}\circ \mathfrak{I}\,(0,\nu) = (0,R(\nu)),~~~~ \widetilde\varphi^{-1}\circ \mathfrak{I}\, (1,\widetilde\nu) = (0, \widetilde R(\widetilde\nu)),
\fe
where $R$ and $\widetilde R$ are super-diffeomorphims from $\mathbb{R}^{0|*n_o}$ to itself. Importantly, we require that $\mathfrak{I}$ has non-singular Berezinian, so that the pullback $\mathfrak{I}^*\Omega$ is a well-defined form. An explicit construction of the interpolation map $\mathfrak{I}$ and the verification of (\ref{onedimbreak}) is given in Appendix \ref{sec:verifyint}.
To extend this construction to the integration over the supermanifold $\mathfrak{M}$, based on a dual triangulation of its reduced space ${\cal M}$, it suffices to describe a local model of the breakup of the integral. We begin with a local model of dual triangulation in $n_e$ dimensions, where $\mathbb{R}^{n_e}$ is divided into $n_e+1$ domains ${\cal D}_1,\cdots, {\cal D}_{n_e+1}$, defined as
\ie
{\cal D}_i = \{\vec t\in \mathbb{R}^{n_e}: t^k = \sum_{j\not=i} e_j^k x_j ,~ x_j\geq 0\},
\fe
where $e_j$ are a set of $n_e+1$ vectors in $\mathbb{R}^{n_e}$ that lie on the vertices of a simplex that contains the origin in its interior. We further define the codimension-$p$ faces
\ie
{\cal D}_{i_1\cdots i_{p+1}} = {\cal D}_{i_1}\cap \cdots \cap {\cal D}_{i_p} = \{ \vec t\in \mathbb{R}^{n_e}: t^k = \sum_{j\not=i_1,\cdots,i_{p+1}} e_j^k x_j, x_j\geq 0 \},
\fe
where the subscripts $i_1\cdots i_{p+1}$ are always assumed to be in increasing order. Let
\ie
\iota_j: {\cal D}_{i_1\cdots i_{m-1} j i_m\cdots i_{p}}\to {\cal D}_{i_1\cdots i_{p}}
\fe
be the obvious inclusion map.
Let ${\cal U}_i$ be the open charts containing ${\cal D}_i$,
\ie
{\cal U}_i = \{\vec t\in \mathbb{R}^{n_e}: t^k = \sum_{j\not=i} e_j^k x_j ,~ x_j\geq -\delta \},
\fe
for some $\delta>0$. Now consider a supermanifold covered by the super charts ${\mathfrak U}_1,\cdots, {\mathfrak U}_{n_e+1}$, each of which admits a coordinate map
\ie
\varphi_i: {\cal U}_i\times \mathbb{R}^{0|*n_o} \to^{\!\!\!\!\!\!\!\sim}\, {\mathfrak U}_i.
\fe
We can then break up the integral of a form $\Omega$ over the supermanifold according to
\ie\label{interform}
\int_{\cup_i {\mathfrak U}_i} \Omega = \sum_{p=0}^{n_e} \sum_{i_1<\cdots< i_{p+1}} \int_{\Delta^{p}\times {\cal D}_{i_1\cdots i_{p+1}} \times \mathbb{R}^{0|*n_o}} \mathfrak{I}_{i_1\cdots i_{p+1}}^*\Omega,
\fe
where $\Delta^p$ is a standard $p$-simplex. The interpolation maps
\ie{}
& \mathfrak{I}_{i_1\cdots i_{p+1}}: \Delta^{p}\times {\cal D}_{i_1\cdots i_{p+1}} \times \mathbb{R}^{0|*n_o} \to {\mathfrak U}_{i_1}\cap \cdots
\cap {\mathfrak U}_{i_{p+1}}
\fe
are required to have non-singular Berezinian, such that the following matching conditions hold:
\ie\label{intmatchcond}
& \mathfrak{I}_{i_1\cdots i_{p+1}}|_{\partial_m \Delta^{p}} = \mathfrak{I}_{i_1\cdots i_{m-1}i_{m+1}\cdots i_{p+1}} \circ R_{i_1\cdots i_{m-1}i_{m+1}\cdots i_{p+1}}^{(m)} \circ (\sigma_{m,p}\times \iota_{i_m}),~~~p\geq 1,
\\
& \mathfrak{I}_{i} = \varphi_i|_{{\cal D}_i\times \mathbb{R}^{0|*n_o}}.
\fe
Here $\partial_m \Delta^p$ is the $m$-th face ($1\leq m\leq p+1$) of $\Delta^p$, which is identified with the standard $(p-1)$-simplex via the map
\ie
\sigma_{m,p}: \partial_m \Delta^p \to^{\!\!\!\!\!\!\!\sim}\, \Delta^{p-1}.
\fe
$R_{i_1\cdots i_{p}}^{(m)}$ are super-diffeomorphisms on $\Delta^{p-1}\times {\cal D}_{i_1\cdots i_{p}} \times \mathbb{R}^{0|*n_o}$ of the form
\ie
R_{i_1\cdots i_{p}}^{(m)} (\vec s, \vec t, \nu) = (\vec s, \vec t, R_{i_1\cdots i_{p}}^{(m)}(s,t|\nu)),
\fe
where $R_{i_1\cdots i_{p}}^{(m)}(s,t|\nu)$ have non-singular Berezinian. For our application it will suffice to take $R_{i_1\cdots i_{p}}^{(m)}(s,t|\nu)$ to be a $GL(n_o,\mathbb{C})$ transformation on the $\nu^a$'s, at any given $\vec s\in \Delta^{p-1}$, $\vec t\in {\cal D}_{i_1\cdots i_p}$.
\subsection{Explicit construction of patches on the supermoduli space}
\label{sec:lift}
We will now give an explicit construction of the lifted super charts on the the supermoduli space $\mathfrak{M}_{g,n,\epsilon}$. Let $\{{\cal D}_\A\}$ be a sufficiently fine dual triangulation of the reduced space ${\cal M}_{g,n,\epsilon}$, such that each cell ${\cal D}_\A$ has an open neighborhood ${\cal U}_\A$ parameterizing a family of punctured Riemann surfaces $\Sigma$ equipped with spin structure $\epsilon$, on which we can specify a set of points $\{z_1,\cdots, z_{d_o}\}\subset\Sigma$ that are sufficiently generic in the sense described following (\ref{nvgdef}). Namely, we choose the points $z_a$ to be such that that there is no weight $-{1\over 2}$ meromorphic differential on $\Sigma$ that is compatible with the spin structure $\epsilon$, with only simple poles at $\{z_1,\cdots, z_{d_o}\}$ and simple zeros at the punctures.
The construction of (\ref{superdiscglue}) and (\ref{nvgdef}) then defines a family of SRS $\mathfrak{C}_\nu$ whose reduced space is $\Sigma$ with spin structure $\epsilon$, that depend on the fermionic parameters $\nu^a$, $a=1,\cdots,d_o$. Said equivalently, this construction give the coordinate map of a super chart ${\mathfrak U}_\A \subset \mathfrak{M}_{g,n,\epsilon}$,
\ie\label{phiamap}
\varphi_\A: {\cal U}_\A \times \mathbb{R}^{0|*d_o}\to {\mathfrak U}_\A ,
\fe
that takes $(\Sigma,\epsilon)\in{\cal U}_\A$ and $\nu\in\mathbb{R}^{0|*d_o}$ to $\mathfrak{C}_\nu$. Importantly, the genericness of $\{z_a\}$ in the sense defined above ensures that $\varphi_\A$ is non-degenerate. As before we will write $\pi_\A: \mathfrak{U}_\A \to {\cal U}_\A$ as the projection map, such that $\pi_\A\circ \varphi_\A^{-1}$ simply forgets the odd parameters.
On the overlap between two super charts ${\mathfrak U}_\A$ and ${\mathfrak U}_\B$, the coordinate maps $\varphi_\A$ and $\varphi_\B$ are constructed by choosing two different sets of points $\{z_a\}$ and $\{z_a' \}$ on $\Sigma$. To exhibit the transition map $\varphi_\B^{-1}\circ \varphi_\A$, it suffices to consider the case where $z_a'=z_a$ for $a=2,\cdots, d_o$, and only $z_1'$ differs from $z_1$, as follows.
The SRS that corresponds to $\varphi_\A(\Sigma,\epsilon,\nu)$ (omitting the anti-holomorphic data) is obtained from the split SRS $\mathfrak{C}_0$ by the insertion of
\ie\label{nwgdef}
\prod_{a=1}^{d_o} \left[1+ \nu^a G(z_a) \right] .
\fe
There is a weight $-{1\over 2}$ meromorphic differential $\zeta = f(z) (dz)^{-{1\over 2}}$ on $\Sigma$ (see (\ref{meroweighthalf})), with $d_o+1$ poles at $z_1, z_2, \cdots, z_{d_o}$ and $z_1'$, as well as simple zeros at each puncture of $\Sigma$, normalized such that the residue of $f(z)$ at $z=z_1$ is equal to 1. This allows us to express the insertion of $G(z_1)$ as a contour integral,
\ie\label{gcont}
G(z_1) &= \oint_{C_{z_1}} {dz\over 2\pi i} f(z) G(z),
\fe
where $C_{z_1}$ is a counterclockwise contour that encloses $z_1$. We will denote by $f^{(n)}(w)$ the coefficient of $(z-w)^n$ in the Laurent series of $f(z)$ around $z=w$ (which can be nonzero for $n\geq -1$). (\ref{nwgdef}) can be written equivalently as
\ie\label{gprodinter}
& \left[ 1 + \nu^1 \oint_{C_{z_1}} {dz\over 2\pi i} f(z) G(z) \right] \prod_{a=2}^{d_o} \left[1+ \nu^a G(z_a) \right]
\\
&= \left[ 1 - \nu^1 f^{(-1)}(z_1') G(z_1') - \nu^1 \sum_{b=2}^{d_o} \oint_{C_{z_b}} {dz\over 2\pi i} f(z) G(z) \right] \prod_{a=2}^{d_o} \left[1+ \nu^a G(z_a) \right]
\\
&= \left[ 1 - \nu^1 f^{(-1)}(z_1') G(z_1') \right] \prod_{a=2}^{d_o} \left[1+ \nu^a G(z_a) \right]
\\
&~~~~ - \nu^1\sum_{b=2}^{d_o} \left[ f^{(-1)}(z_b) G(z_b) - 2 \nu^b f^{(0)}(z_b) T(z_b) \right] \prod_{a=2, a\not=b}^{d_o} \left[1+ \nu^a G(z_a) \right] ,
\fe
where in deriving the last equality we have used (\ref{ggope}) and the absence of order $z^0$ term in the $G(z) G(0)$ OPE. We can then put (\ref{gprodinter}) back in the product form,
\ie
\left[ 1+ \nu'^1 G(z_1') \right] \prod_{a=2}^{d_o} \left[1+ \nu'^a G(z_a) \right] \left[1 + \sum_{b=2}^{d_o} 2 \nu^1 \nu^b f^{(0)}(z_b) T(z_b) \right],
\fe
where
\ie{}
& \nu'^1 = - \nu^1 f^{(-1)}(z_1'),
\\
& \nu'^a = \nu^a - \nu^1 f^{(-1)}(z_a) ,~~~a=2,\cdots,d_o.
\fe
From this we read off the transition map $\varphi_\B^{-1}\circ \varphi_\A$ on $({\cal U}_\A\cap {\cal U}_\B)\times \mathbb{R}^{0|*d_o}$,
\ie
\varphi_\B^{-1}\circ \varphi_\A: (\tau^m,\nu^a) \mapsto ( \tau'^m = \tau^m + \delta\tau^m, \nu'^a),
\fe
where the bosonic moduli shift $\delta\tau^m$ is generated by the insertion of
\ie
\sum_{b=2}^{d_o} 2 \nu^1 \nu^b f^{(0)}(z_b) T(z_b).
\fe
By moving points of $\{z_a\}$ one at a time, we can obtain the general transition maps between arbitrary pairs of super charts $\mathfrak{U}_\A$.
\subsection{The supermoduli integration contour $\mathfrak{S}$}
\label{sec:supercontour}
As explained in section \ref{sec:superamp}, we will be integrating not over the supermoduli space itself, but rather a complex codimension $d_e|0$ contour $\mathfrak{S} \subset \mathfrak{M}_{g,n,\epsilon} \times \overline{{\mathfrak M}_{g,n,\widetilde\epsilon}}$, whose reduced space is the diagonal subspace ${\cal M}\subset {\cal M}_{g,n,\epsilon} \times \overline{{\cal M}_{g,n,\widetilde\epsilon}}$. Different choices of $\mathfrak{S}$ are equivalent in homology, and result in the same integral of the form $\Omega$ which is closed for on-shell amplitudes.
Given a sufficiently fine dual triangulation $\{{\cal D}_\A\}$ of ${\cal M}$ and open patches ${\cal U}_\A\supset {\cal D}_\A$, we can construct the lifting map $\varphi_\A$ defined as in (\ref{phiamap}), based on the choice of a set of points $\{z_a^{(\A)} \}\subset \Sigma$, with $\varphi_\A({\cal U}_\A\times \mathbb{R}^{0|*d_o}) = \mathfrak{U}_\A\subset \mathfrak{M}_{g,n,\epsilon}$, and $\pi_\A(\mathfrak{U}_\A) = {\cal U}_\A$. Similarly, we can define the anti-holomorphic lifting map $\overline\varphi_\A$ that depends on $\{\bar z_a^{(\A)} \}\subset \overline\Sigma$, and the corresponding super chart $\overline{\mathfrak U}_\A\subset \overline{{\mathfrak M}_{g,n,\widetilde\epsilon}}$. Note that $\bar z_a^{(\A)}$ need not be complex conjugate to $z_a^{(\A)}$. Applying both the holomorphic and anti-holomorphic lifting map to a cell ${\cal D}_\A$ produces a patch
\ie\label{phipatch}
(\varphi_\A\times \overline\varphi_\A ) ({\cal D}_\A\times \mathbb{R}^{0|*d_o} \times \mathbb{R}^{0|*d_o}) = (\pi_\A^{-1}\times \overline \pi_\A^{-1}) ({\cal D}_\A)
\fe
of the contour $\mathfrak{S}$, which we will refer to as a ``horizontal" patch.
Next, we would like to construct interpolating ``vertical" patches that fill in the gaps between horizontal patches of the form (\ref{phipatch}) and close the chain $\mathfrak{S}$. This is analogous to (\ref{interform}) but applied to integration over a submanifold. Suppose the cells ${\cal D}_{\A_1},\cdots,{\cal D}_{\A_{p+1}}$ of the dual triangulation of ${\cal M}$ meet along the codimension-$p$ face ${\cal D}_{\A_1\cdots \A_{p+1}}$. We aim to construct a set of interpolation maps
\ie\label{intcontoumaps}
\mathfrak{I}_{\A_1\cdots \A_{p+1}}: \Delta^p\times {\cal D}_{\A_1\cdots \A_{p+1}}\times \mathbb{R}^{0|*d_o} \times \mathbb{R}^{0|*d_o} \to {\mathfrak U}_{\A_1\cdots \A_{p+1}} \times \overline{\mathfrak U}_{\A_1\cdots \A_{p+1}},
\fe
where ${\mathfrak U}_{\A_1\cdots \A_{p+1}}\equiv {\mathfrak U}_{\A_1} \cap\cdots \cap{\mathfrak U}_{\A_{p+1}}$, that obey matching conditions analogous to (\ref{intmatchcond}), of the form
\ie\label{supcontr}
& \mathfrak{I}_{\A_1\cdots \A_{p+1}}|_{\partial_m \Delta^{p}} = \mathfrak{I}_{\A_1\cdots \A_{m-1}\A_{m+1}\cdots \A_{p+1}} \circ R_{\A_1\cdots \A_{m-1}\A_{m+1}\cdots \A_{p+1}}^{(m)} \circ (\sigma_{m,p}\times \iota_{i_m}),~~~p\geq 1,
\\
& {\mathfrak I}_\A =\left. \varphi_\A\times \overline\varphi_\A \right|_{{\cal D}_\A\times \mathbb{R}^{0|*d_o} \times \mathbb{R}^{0|*d_o}}.
\fe
The contour $\mathfrak{S}$ is then built as
\ie
\mathfrak{S} = \sum_{p=0}^{2d_e} \sum_{\{\A_1,\cdots,\A_{p+1}\}} \mathfrak{I}_{\A_1\cdots \A_{p+1}}( \Delta^p\times {\cal D}_{\A_1\cdots \A_{p+1}}\times \mathbb{R}^{0|*d_o} \times \mathbb{R}^{0|*d_o}).
\fe
In the next section, we will describe a specific construction of (\ref{intcontoumaps}) that leads to precisely the vertical integration prescription.
\section{From the supermoduli contour to the PCO contour}
\label{sec:mtos}
\subsection{PCO from cutting a super disc}
\label{sec:pcoemerge}
We begin by considering the effect of integrating out a single fermionic modulus on a patch of the supermoduli space that parameterizes the gluing map of a super disc according to (a slight generalization of) (\ref{superdiscglue}). Namely, consider a super disc $D$ with coordinates $(w,\eta)$, and an overlapping super annulus $U'$ with coordinates $(z,\theta)$, with the transition map
\ie\label{transidisc}
& w = z - \frac{\theta \nu}{z - z_0(t)} ,
\\
& \eta = \theta - \frac{\nu}{z - z_0(t)}.
\fe
Here $\nu$ is an odd parameter, and the $z_0(t)$ can have arbitrary dependence on the bosonic moduli $t^k=(\tau^m, \bar\tau^m)$.
According to (\ref{nvgdef}) and (\ref{omegaintg}), the transition map (\ref{transidisc}) gives rise to the following $\nu$-dependent factor in the supermoduli integrand,
\ie\label{bbfact}
\left[ 1 + \nu \oint {dz\over 2\pi i} {G(z)\over z-z_0(t)} \right] \exp\left[ {\cal B}_{z_0(t)} \right] \delta(d\nu) \delta({\cal B}_\nu) ,
\fe
where the $z$-integral contour encircles $z_0(t)$. ${\cal B}_{z_0(t)}$ and ${\cal B}_\nu$ are given by
\ie
{\cal B}_{z_0(t)} &= \oint {dw\over 2\pi i} d\eta\, (\B + \eta b)\left. \left[ {\partial w\over \partial t^k} - {\partial\eta\over \partial t^k}\eta \right]\right|_{z,\theta} dt^k
\\
&= - 2 \nu \partial\beta(z_0(t)) dz_0(t)
\fe
and
\ie
{\cal B}_\nu &= \oint {dw\over 2\pi i} d\eta\, (\B + \eta b)\left. \left[ {\partial w\over \partial\nu} - {\partial\eta\over \partial\nu}\eta \right]\right|_{z,\theta}
\\
&= \oint {dw\over 2\pi i} d\eta\, (\B + \eta b) \left[ {2\eta\over w-z_0(t)} + {\nu\over (w-z_0(t))^2} \right]
\\
&= 2\B(z_0(t)) - \nu \partial b(z_0(t))
\fe
respectively. Thus, (\ref{bbfact}) can be written as
\ie{}
&\left[ 1 + \nu \oint {dz\over 2\pi i} {G(z)\over z-z_0(t)} \right] \Big[ 1 - 2\nu \partial\beta(z_0(t)) dz_0(t) \Big] \cdot \delta(d\nu)\left[ {1\over 2} \delta(\B(z_0(t))) - {1\over 4} \nu \partial b(z_0(t)) \,\delta'(\B(z_0(t))) \right]
\fe
Integrating this over $\nu$ gives
\ie{}
& {1\over 2}\oint \frac{\dif z}{2 \pi i} \frac{G(z)}{z - z_0(t)} \delta(\beta(z_0(t))) - \frac{1}{4}\partial b(z_0(t)) \,\delta'(\beta(z_0(t))) - dz_0(t) \partial\Theta(\B(z_0(t)))
\\
&= {\cal X}(z_0(t)) + d\xi(z_0(t)),
\fe
where $\xi=-\Theta(\B)$, and
\ie
{\cal X}(z) = {1\over 2}\oint {dz'\over 2\pi i} {G(z')\over z'-z} \delta(\B(z)) - {1\over 4} \partial b(z) \delta'(\B(z))
\fe
is precisely the PCO.
Applying this to all of the odd moduli parameters in the integration over the lifted patch $(\pi_\A^{-1}\times \overline\pi_\A^{-1})({\cal D}_\A)$ of section \ref{sec:lift}, where the points $\{z_a^{(\A)}\}\subset \Sigma$ and $\{\bar z_a^{(\A)}\}\subset \overline\Sigma$ are chosen to be independent of the local bosonic moduli parameter $t^k$, we obtain
\ie\label{tmpdd}
\int_{(\pi_\A^{-1}\times \overline\pi_\A^{-1})({\cal D}_\A)} \Omega = \int_{{\cal D}_\A} \left\langle e^{\cal B}\prod_{i=1}^n {\cal V}_i \prod_{a=1}^{d_o} \mathcal{X}(z_a^{(\A)}) \widetilde{\mathcal{X}}(\bar z_a^{(\A)}) \right\rangle.
\fe
This amounts to a ``horizontal patch" of the integration over the PCO contour (\ref{pcoamp}). We could of course slightly generalize (\ref{tmpdd}) to allow $z_a$ and $\bar z_a$ to depend on $t$, and recover an expression that involves the integrand (\ref{tildeomg}) including the $d\xi$, $d\widetilde\xi$ terms, but this is inessential for our consideration. In fact, simply allowing $z_a, \bar z_a$ to depend on $t$ would not enable us to close the chain $\mathfrak{S}$, as we must ensure that the former evade configurations that lead to spurious singularities. Instead, we will proceed by constructing the interpolating patches in the supermoduli space.
\subsection{The emergence of vertical integration}
\label{sec:vertemerge}
We now construct the interpolation map $\mathfrak{I}_{\A\B}$ between a pair of neighboring cells ${\cal D}_{\A},{\cal D}_{\B}\subset{\cal M}$ that meet along the codimension 1 face ${\cal D}_{\A\B}$, as the $p=1$ case of (\ref{intcontoumaps}). The lifting maps $\varphi_\A, \overline{\varphi}_\A$ are defined as in section \ref{sec:supercontour}, involving the choice of points $\{z_a^{(\A)}\}\subset\Sigma$, $\{\overline z_a^{(\A)}\}\subset\overline\Sigma$, and similarly for $\varphi_\B, \overline{\varphi}_\B$ with the points $\{z_a^{(\B)}\}$, $\{\bar z_a^{(\B)}\}$. $\mathfrak{I}_{\A\B}$ will be constructed as a composition (in the sense of homotopy) of a sequence of $2d_o$ interpolation maps
\ie\label{iibar}
& {\cal I}_{\bf a}: [0,1]\times {\cal D}_{\A\B} \times\mathbb{R}^{0|*d_o}\to {\mathfrak U}_{\A\B},
\\
& \overline{\cal I}_{\bf a}: [0,1]\times {\cal D}_{\A\B} \times\mathbb{R}^{0|*d_o}\to \overline{\mathfrak U}_{\A\B},
\fe
according to
\ie
& \mathfrak{I}_{\A\B}(s,t,\nu,\overline\nu) = ({\cal I}_{\bf a}(2d_o s-{\bf a}+1, t, \nu) , \overline\varphi_\A(t,\overline\nu)),~~~~{{\bf a}-1\over 2d_o} \leq s\leq {{\bf a}\over 2d_o},
\\
& \mathfrak{I}_{\A\B}(s,t,\nu,\overline\nu) = (\varphi_\B(t,\nu), \overline{\cal I}_{\bf a}(2d_o s-d_o-{\bf a}+1, t, \overline\nu) ),~~~~{{\bf a}-1\over 2d_o}+{1\over 2} \leq s\leq {{\bf a}\over 2d_o}+{1\over 2},
\fe
for ${\bf a}=1,\cdots,d_o$.
${\cal I}_{\bf a}(s,t,\nu)$ is defined as the SRS obtained from the split SRS $\mathfrak{C}_0$ over $(\Sigma_t,\epsilon)$ by
cutting $d_o+1$ super discs centered at
\ie\label{zloci}
z_1^{(\B)}, z_2^{(\B)}, \cdots, z_{\bf a}^{(\B)}, z_{\bf a}^{(\A)}, z_{{\bf a}+1}^{(\A)},\cdots, z_{d_o}^{(\A)}.
\fe
The gluing maps at the boundary of the first ${\bf a}-1$ discs, centered at $z_1^{(\B)},\cdots, z_{{\bf a}-1}^{(\B)}$, are the same as those that define the lifting map $\varphi_\B$, namely (\ref{superdiscglue}) with $z_a$ replaced with $z_a^{(\B)}$, that depend on the Grassmann-odd parameters $\nu^1,\cdots,\nu^{{\bf a}-1}$. The gluing maps at the boundary of the last $d_o-{\bf a}$ discs, centered at $z_{{\bf a}+1}^{(\A)},\cdots, z_{d_o}^{(\A)}$, are the same as those that define the lifting map $\varphi_\A$, that depend on $\nu^{{\bf a}+1},\cdots,\nu^{d_o}$. On the boundary of the discs centered at $z_{\bf a}^{(\B)}$ and $z_{\bf a}^{(\A)}$, however, we use a gluing map that interpolates between those that define $\varphi_\A$ and $\varphi_\B$ as follows.
Let $(w,\eta)$ be the coordinates on the super disc centered at $z_{\bf a}^{(\A)}$, and $(\widetilde w,\widetilde\eta)$ be the coordinates on the super disc centered at $z_{\bf a}^{(\B)}$. The gluing maps that depend on $\nu^a$ are
\ie\label{wetag}
w=z-{\theta (1-s) R(s) \nu^{\bf a}\over z- z_{\bf a}^{(\A)}},~~~~\eta = \theta - {(1-s) R(s)\nu^{\bf a}\over z- z_{\bf a}^{(\A)}},
\fe
and
\ie\label{wetatil}
\widetilde w=z-{\theta s \widetilde R(s)\nu^{\bf a}\over z- z_{\bf a}^{(\B)}},~~~~\widetilde\eta = \theta - {s \widetilde R(s)\nu^{\bf a}\over z- z_{\bf a}^{(\B)}},
\fe
where $R(s)$ and $\widetilde R(s)$ are nowhere vanishing complex valued functions on $[0,1]$. Note that our matching conditions do not require $R(0)$ and $\widetilde R(1)$ to be identity.
To first order in $\nu$, the deformation that defines the SRS ${\cal I}_{\bf a}(s,t,\nu)$ amounts to the insertion of
\ie\label{sumgs}
\sum_{b=1}^{{\bf a}-1} \nu^b G(z_b^{(\B)}) + \sum_{b'={\bf a}+1}^{d_o} \nu^{b'} G(z_{b'}^{(\A)}) + \nu^{\bf a} \left[ (1-s) R(s) G(z_{\bf a}^{(\A)}) + s \widetilde R(s) G(z_{\bf a}^{(\B)})\right]
\fe
on the worldsheet. For generic $\{z_a^{(\A)}\}$ and $\{z_a^{(\B)}\}$, up to constant rescaling there is a unique weight $-{1\over 2}$ meromorphic differential $\zeta = f(z) (dz)^{-{1\over 2}}$ with $d_o+1$ simple poles at (\ref{zloci}). By consideration of contour deformation similarly to that at the end of section \ref{sec:srsscft}, the variation of (\ref{sumgs}) under $\nu\to \nu+\delta\nu$ is degenerate if
\ie\label{srresid}
{s \widetilde R(s) \over (1-s) R(s)} = {f^{(-1)}(z_{\bf a}^{(\B)}) \over f^{(-1)} (z_{\bf a}^{(\A)})}
\fe
where $f^{(-1)}(z_0)$ stands for the residue of $f(z)$ at $z=z_0$, for some $s\in[0,1]$. With generic choices of complex $R(s)$ and $\widetilde R(s)$, (\ref{srresid}) can certainly be avoided, thereby ensuring that ${\cal I}_a$ has non-singular Berenzinian, as needed.
The anti-holomorphic counter part to ${\cal I}_{\bf a}$, which we denoted by $\overline{\cal I}_{\bf a}$ in (\ref{iibar}), is defined similarly via gluing maps on the complex conjugate SRS. Note that the points $\bar z_a^{(\A)}, \bar z_a^{(\B)}\in \overline\Sigma$ involved in this construction need not be the complex conjugates of $z_a^{(\A)}, z_a^{(\B)}\in\Sigma$.
The integration over the interpolation patch defined by $\mathfrak{I}_{\A\B}$ can be written as
\ie\label{iabint}
\int_{[0,1]\times {\cal D}_{\A\B}\times \mathbb{R}^{0|*d_o}\times \mathbb{R}^{0|*d_o}} \mathfrak{I}_{\A\B}^*\Omega = \int_{[0,1]\times {\cal D}_{\A\B}\times \mathbb{R}^{0|*d_o}\times \mathbb{R}^{0|*d_o}} \left[ \sum_{{\bf a}=1}^{d_o} ({\cal I}_{\bf a} \times \overline\varphi_\A)^*\Omega + \sum_{{\bf a}=1}^{d_o} (\varphi_\B\times\overline{\cal I}_{\bf a})^*\Omega \right]
\fe
The integrand $({\cal I}_{\bf a} \times \overline\varphi_\A)^*\Omega$, for instance, contains the following $s$ and $\nu^{\bf a}$ dependent factor
\ie\label{bbsfact}
{\cal B}_s \delta(d\nu^{\bf a}) \delta({\cal B}_{\nu^{\bf a}}),
\fe
where
\ie{}
& {\cal B}_s = - 2 \nu^{\bf a} \left[ \B(z_{\bf a}^{(\A)}) {d\over ds}((1-s)R(s)) + \B(z_{\bf a}^{(\B)}) {d\over ds}(s \widetilde R(s)) \right] ds
\\
& {\cal B}_{\nu^{\bf a}} = 2\left[ (1-s) R(s)\B(z_{\bf a}^{(\A)}) + s\widetilde R(s) \B(z_{\bf a}^{(\B)})\right] - \nu^{\bf a} \left[ ((1-s) R(s))^2 \partial b(z_a^{(\A)}) + (s\widetilde R(s))^2 \partial b( z_{\bf a}^{(\B)}) \right].
\fe
Note that we don't need to include insertion of $G$ associated with the $\nu^{\bf a}$-deformation in (\ref{bbsfact}) because ${\cal B}_s$ already saturates the $\nu^{\bf a}$-integral.
Integrating (\ref{bbsfact}) over $s$ and $\nu^{\bf a}$ results in
\ie\label{thetadiff}
&- \int_0^1 ds \left[ \B(z_{\bf a}^{(\A)}) {d\over ds}((1-s)R(s)) + \B(z_{\bf a}^{(\B)}) {d\over ds}(s \widetilde R(s)) \right]\delta\left( (1-s) R(s)\B(z_{\bf a}^{(\A)}) + s\widetilde R(s) \B(z_{\bf a}^{(\B)}) \right)
\\
&= -\int_0^1 ds\, \partial_s \Theta\left( (1-s) R(s)\B(z_{\bf a}^{(\A)}) + s\widetilde R(s) \B(z_{\bf a}^{(\B)}) \right)
\\
&= -\Theta\big(\widetilde R(1) \B(z_{\bf a}^{(\B)})\big) + \Theta\big(R(0) \B(z_{\bf a}^{(\A)})\big).
\fe
Recall that the holomorphic distribution $\Theta(\B)$ is homogenous in its argument, and thus the RHS of (\ref{thetadiff}) is the same as
\ie
\xi(z_{\bf a}^{(\B)}) - \xi(z_{\bf a}^{(\A)}).
\fe
This matches precisely the vertical integration prescription associated with moving a holomorphic PCO from $z_{\bf a}^{(\A)}$ to $z_{\bf a}^{(\B)}$.
(\ref{iabint}) is therefore equivalent to the sum over $2d_o$ vertical integrals, each of which involves moving a single holomorphic or anti-holomorphic PCO. The result can be written as
\ie{}
&\int_{[0,1]\times {\cal D}_{\A\B}\times \mathbb{R}^{0|*d_o}\times \mathbb{R}^{0|*d_o}} \mathfrak{I}_{\A\B}^*\Omega
\\
&=\int_{{\cal D}_{\A\B}} \Bigg\langle e^{\cal B} \prod_{i=1}^n {\cal V}_i \Bigg[ \sum_{{\bf a}=1}^{d_o} \prod_{b=1}^{{\bf a}-1} {\cal X}(z_b^{(\B)}) \big[ \xi(z_{\bf a}^{(\B)}) - \xi(z_{\bf a}^{(\A)}) \big] \prod_{b'={\bf a}+1}^{d_o} {\cal X}(z_{b'}^{(\A)}) \prod_{c=1}^{d_o} \widetilde{\cal X}(\bar z_c^{(\A)})
\\
&~~~~~~~~~~~~~~~ + \sum_{{\bf a}=1}^{d_o} \prod_{b=1}^{d_o}{\cal X}(z_b^{(\B)}) \prod_{c=1}^{{\bf a}-1} \widetilde{\cal X}(\bar z_c^{(\B)}) \big[ \widetilde\xi(\bar z_{\bf a}^{(\B)}) - \widetilde\xi(\bar z_{\bf a}^{(\A)}) \big] \prod_{c'={\bf a}+1}^{d_o} \widetilde{\cal X}(\bar z_{c'}^{(\A)}) \bigg] \Bigg\rangle.
\fe
\subsection{Recovering the full PCO contour ${\cal S}$}
\label{sec:pcohighercodim}
To complete the supermoduli integration contour $\mathfrak{S}$ outlined in section \ref{sec:supercontour}, we now extend the construction of the interpolation map of the previous subsection to $\mathfrak{I}_{\A_1 \cdots \A_{p+1}}$ supported at the codimension $p$ interface between ${\cal D}_{\A_1},\cdots,{\cal D}_{\A_{p+1}}\subset {\cal M}$. Each cell ${\cal D}_{\A_m}$ is contained in its open neighborhood ${\cal U}_{\A_m}$, the latter being lifted to a supermoduli patch $\mathfrak{U}_{\A_m}$ via the map $\varphi_{\A_m}$ specified through PCO locations $\{z_a^{(\A_1)}\},\cdots,\{z_a^{(\A_{p+1})}\}$ in the sense of section \ref{sec:pcoemerge}, along with their anti-holomorphic counter parts.
We begin with an auxiliary space
\ie
V = \Delta_1\times\cdots\times\Delta_{d_o},
\fe
where each $\Delta_a$ is a $p$-simplex, whose vertices are denoted $\{v_{a,1},\cdots, v_{a,p+1}\}$. We will refer to a tuple of vertices $(v_{1,x_1},\cdots,v_{d_o,x_{d_o}})\in V$, where $1\leq x_a\leq p+1$, as a vertex of $V$. Such vertices are in 1-1 correspondence with ``shuffled" PCO locations $\{z_a^{(\A_{x_a})}\}_{1\leq a\leq d_o}$, where the $a$-th PCO position is taken from that of the lifting map $\varphi_{\A_{x_a}}$. A sequence of PCO moves from $\{z_a^{(\A_{x_a})}\}$ to $\{z_a^{(\A_{y_a})}\}$ can be associated with a path along the edges of the embedded $d_o$-hypercube $\prod_{a=1}^{d_o}\overline{v_{a,x_a} v_{a,y_a}} \,\subset V$. For instance, the interpolation map $\mathfrak{I}_{\A_x \A_y}$ of section \ref{sec:vertemerge} is associated with the path
\ie
\Gamma_{xy}: (v_{1,x}, v_{2,x},\cdots,v_{d_o,x})\to (v_{1,y}, v_{2,x},\cdots,v_{d_o,x})\to \cdots \to (v_{1,y}, v_{2,y},\cdots,v_{d_o,y}),
\fe
which shall be viewed as a piecewise linear map from the standard 1-simplex to $V$.
Given three such paths, say $\Gamma_{\A_1\A_2}$, $\Gamma_{\A_2\A_3}$, $\Gamma_{\A_1\A_3}$ associated with $\mathfrak{I}_{\A_1\A_2}$, $\mathfrak{I}_{\A_2\A_3}$, $\mathfrak{I}_{\A_1\A_3}$, we can form a closed path as the piecewise linear map
\ie\label{gammabdry}
\partial\Gamma_{\A_1\A_2\A_3}: \partial\Delta^2 \to V
\fe
composed by joining the three paths, each of which is defined on a boundary segment $\partial_m \Delta^2$, $m=1,2,3$. Evidently, $\partial\Gamma_{\A_1\A_2\A_3}$ is contractible and can be extended to
\ie\label{gammathree}
\Gamma_{\A_1\A_2\A_3}: \Delta^2 \to V.
\fe
It is convenient to realize $\Gamma_{\A_1\A_2\A_3}$ as a homotopy between the path $\Gamma_{\A_1\A_2}\triangleright \Gamma_{\A_2\A_3}$ (here $\triangleright$ stands for ``joining") and $\Gamma_{\A_1\A_3}$, that is composed of a sequence of elementary homotopies of two types.
The first type of elementary homotopy, which we refer to as a ``merger", is a homotopy between two paths $\Gamma$ and $\Gamma'$ that differ only at the respective segments
\ie\label{mergeraa}
& v_{{\bf a},1} \to v_{{\bf a},2} \to v_{{\bf a},3} \subset \Gamma,
\\
& v_{{\bf a},1} \to v_{{\bf a},3} \subset \Gamma',
\fe
where we have exhibited only the movement between vertices in $\Delta_{\bf a}$, while the coordinates in each of the other simplex factors of $V$, namely $\Delta_1,\cdots,\Delta_{{\bf a}-1},\Delta_{{\bf a}+1},\cdots,\Delta_{d_o}$, are fixed to be at some vertex. The merger is defined as the homotopy associated with the obvious map from $\Delta^2$ to the 2-simplex spanned by $\{v_{{\bf a},1}, v_{{\bf a},2},v_{{\bf a},3}\}$ in $\Delta_{\bf a}$.
The second type of elementary homotopy, which we refer to as a ``square move", is a homotopy between $\Gamma$ and $\Gamma'$ that differ at the following segments,
\ie\label{squareab}
& (v_{{\bf a},2}, v_{{\bf b},1}) \to (v_{{\bf a},2}, v_{{\bf b},2}) \to (v_{{\bf a},3}, v_{{\bf b},2}) \subset\Gamma,
\\
& (v_{{\bf a},2}, v_{{\bf b},1}) \to (v_{{\bf a},3}, v_{{\bf b},1}) \to (v_{{\bf a},3}, v_{{\bf b},2}) \subset\Gamma',
\fe
where we have exhibited only the movement of a pair of vertices on $\Delta_{\bf a}$ and $\Delta_{\bf b}$, where $1\leq a<b\leq d_o$, while the coordinates on the other $\Delta_c$ factors for $c\not={\bf a},{\bf b}$ are at some fixed vertices along these segments of the paths. The square move is defined as the homotopy associated with the obvious map from $[0,1]^2$ to the square $\overline{v_{{\bf a},2} v_{{\bf a},3}}\times \overline{v_{{\bf b},1}v_{{\bf b},2}}$ in $\Delta_{\bf a}\times \Delta_{\bf b}$.
\begin{figure}
\centering
\begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {\includegraphics[width=.3\textwidth]{figures/square_triangle.pdf}};
\begin{scope}[shift={(image.south west)}, x={(image.south east)}, y={(image.north west)}]
\node[anchor=north east] at (0.05, 0.15) {$(v_{{\bf a}, 1}, v_{{\bf b}, 1})$};
\node[anchor=north] at (0.55, 0.04) {$(v_{{\bf a}, 1}, v_{{\bf b}, 3})$};
\node[anchor=north west] at (0.95, 0.12) {$(v_{{\bf a}, 3}, v_{{\bf b}, 3})$};
\node[anchor=west] at (0.79, 0.57) {$(v_{{\bf a}, 2}, v_{{\bf b}, 3})$};
\node[anchor=south] at (0.52,0.98) {$(v_{{\bf a}, 2}, v_{{\bf b}, 2})$};
\node[anchor=east] at (0.23,0.7) {$(v_{{\bf a}, 2}, v_{{\bf b}, 1})$};
\end{scope}
\end{tikzpicture}
\caption{An example of a construction of a homotopy between $\Gamma_{\A_1\A_2} \triangleright \Gamma_{\A_2\A_3}$ and $\Gamma_{\A_1\A_3}$ for a specific choice of $\Gamma_{xy}$ using a square move and two triangle moves. }
\end{figure}
To construct the interpolation map associated with (\ref{gammathree}), which is composed of a sequence of elementary homotopy moves, we first choose a set interpolation maps of the form (\ref{iibar}) associated with each edge of a path $\Gamma$ that appear in the homotopy sequence, and then construct the interpolation map associated with each merger or square move between the $\Gamma$'s, subject to compatibility conditions as follows.
We associate the merger move of (\ref{mergeraa}) with an interpolation map
\ie\label{segmerg}
{\cal I}^{\rm M}: \Delta^2\times {\cal D}_{\A_1\A_2\A_3}\times \mathbb{R}^{0|*d_o} \to \mathfrak{U}_{\A_1\A_2\A_3},
\fe
where ${\cal I}^{\rm M}(\vec s, t, \nu)$ is the SRS obtained by deforming the split SRS over $\Sigma_t$ with a set of $d_o+2$ superconformal gluing maps along boundaries of super discs that depend on the odd parameter $\nu^a$, $a=1,\cdots,d_o$. Among them, the $d_o-1$ gluing maps that involve $\nu^b$ for $b\not={\bf a}$ are those of the standard form (\ref{superdiscglue}), where the center of the super disc $z_b$ is the $b$-th PCO location as specified by the vertex in $\Delta_b$ of the segment appearing in (\ref{mergeraa}). The gluing maps that involve $\nu^{\bf a}$, on the other hand, are defined on the boundaries of three super discs $D_1, D_2, D_3$ centered at the PCO locations associated with $v_{{\bf a},1}$, $v_{{\bf a},2}$, $v_{{\bf a},3}$. Let $(w_i,\eta_i)$ be the super coordinates on $D_i$. The gluing maps are of the form
\ie
w_i = z - {\theta P_i(\vec s) \nu^{\bf a} \over z-z_{\bf a}^{(i)}}, ~~~~ \eta_i = z - {P_i(\vec s) \nu^{\bf a}\over z-z_{\bf a}^{(i)}},
\fe
for $i=1,2,3$, $\vec s\in \Delta^2$. Here $P_i$ are a set of three complex valued smooth functions on $\Delta^2$,
subject to the following conditions.
Let us label the three vertices of the domain $\Delta^2$ of ${\cal I}^{\rm M}$ as $\{1,2,3\}$, and the edges as $\{\overline{12},\overline{23},\overline{13}\}$. We demand that ${\cal I}^{\rm M}|_{\overline{ij}}$ agrees with the interpolation map ${\cal I}_{ij}$ of the form (\ref{iibar}) already chosen for the edge $\overline{v_{{\bf a},i}v_{{\bf a},j}}$ in $\Gamma$ or $\Gamma'$, up to rescaling $\nu^{\bf a}$ by a non-vanishing complex valued function $Q_i(s)$ on $[0,1]$. In other words, let $R_{ij}(s)$ and $\widetilde R_{ij}(s)$ be the complex valued functions appearing in the gluing maps (\ref{wetag}), (\ref{wetatil}) associated with the interpolation map ${\cal I}_{ij}$, we demand
\ie\label{pppq}
P_1|_{\overline{12}}(s) = (1-s) R_{12}(s) Q_3(s),~~~ P_2|_{\overline{12}}(s) = s \widetilde R_{12}(s) Q_3(s),~~~ P_3|_{\overline{12}}(s) =0,
\fe
and similar relations related by cyclic permutation of the vertices $\{1,2,3\}$. By virtue of the construction of ${\cal I}_{ij}$ as in section \ref{sec:vertemerge}, the compatibility condition (\ref{pppq}) already ensures that ${\cal I}^{\rm M}$ has non-singular Berezinian when restricted to $\partial \Delta^2$. However, we must ensure that this is the case when extending ${\cal I}^{\rm M}$ to the interior of its domain $\Delta^2$.
By a similar argument as below (\ref{sumgs}), the Berezinian of ${\cal I}^{\rm M}$ would be singular if there exists a weight $-{1\over 2}$ meromorphic differential $\zeta=f(z) dz^{-{1\over 2}}$ with $d_o+2$ poles at $\{z_b: 1\leq b\leq d_o, b\not={\bf a}\}\cup\{z_{\bf a}^{(i)}, i=1,2,3\}$, whose residue at $z_{\bf a}^{(i)}$ is equal to $P_i(\vec s)$ for some $\vec s\in\Delta^2$. This is equivalent to the locus
\ie\label{cpzero}
\sum_{i=1}^3 c_i P_i(\vec s) = 0
\fe
where $c_i$ are given by the inverse of the correlators $\big\langle\delta(\B(z_{\bf a}^{(i)}) \prod_{b\not={\bf a}} \delta(\B(z_b))\big\rangle$ of the $\B\C$ system. While we already know that $\sum_{i=1}^3 c_i P_i(\vec s)$ is non-vanishing on $\partial\Delta^2$, a priori the phase of the former could have a nonzero winding number around $\partial\Delta^2$ which would force (\ref{cpzero}) somewhere in the interior of $\Delta^2$. By choosing the functions $Q_i(s)$ appearing in (\ref{pppq}), however, such a winding number can be eliminated, which allows for extending $P_i(\vec s)$ to $\Delta^2$ while evading (\ref{cpzero}).
Having constructed the interpolation map for the merger move, a calculation similar to section \ref{sec:vertemerge} shows that it does not contribute to the integral of $\Omega$. We now turn to the interpolation map of the square move (\ref{squareab}), of the form
\ie{}
{\cal I}^{\rm Sq}: [0,1]^2\times {\cal D}_{\A_1 \A_2 \A_3}\times \mathbb{R}^{0|*d_o} \to \mathfrak{U}_{\A_1 \A_2 \A_3}.
\fe
${\cal I}^{\rm Sq}(\vec s,t,\nu)$ is the SRS defined by deforming gluing maps on the boundary of $d_o+2$ discs. Among them, $d_o-2$ gluing maps that involve $\nu^c$ for $c\not={\bf a}, {\bf b}$ are on the boundary of discs centered at $z_c$ as specified by the vertex in $\Delta_c$ of the segment of the paths appearing in (\ref{squareab}). The gluing maps that involve $\nu^{\bf a}$ and $\nu^{\bf b}$ are defined on the boundaries of four super discs $D_{ij}$, $i,j=1,2$, centered at PCO locations $\{z_{\bf a}^{(2)}, z_{\bf a}^{(3)},z_{\bf b}^{(1)},z_{\bf b}{}^{(2)} \}\equiv \{z_{11},z_{12},z_{21},z_{22}\}$ associated with the vertices $v_{{\bf a},2}$, $v_{{\bf a},3}$, $v_{{\bf b} ,1}$, $v_{{\bf b},2}$. Let $(w_{ij}, \eta_{ij})$ be the supercoordinates on $D_{ij}$. The gluing maps are of the form
\ie
w_{ij} = z - {\theta P_{ij}(\vec s)\cdot \nu\over z-z_{ij}},~~~~ \eta_{ij} = z-{ P_{ij}(\vec s)\cdot\nu\over z-z_{ij}}
\fe
for $i,j=1,2$, where we used the notation $\nu=(\nu^{\bf a},\nu^{\bf b})$. Each $P_{ij}(\vec s)\equiv (P_{ij1}(\vec s), P_{ij2}(\vec s))$ is a $\mathbb{C}^2$-valued function on $[0,1]^2$, subject to the boundary conditions
\ie\label{fourqs}
& P_{ij\ell} (0,s) = \begin{pmatrix} R_{11}(0) & 0 \\ (1-s) R_{21}(s) & s \widetilde R_{21}(s) \end{pmatrix}_{ij} (Q_1)_{i\ell}(s),
~~~P_{ij\ell}(1,s) = \begin{pmatrix} 0 & \widetilde R_{11}(1) \\ (1-s) R_{22}(s) & s \widetilde R_{22}(s) \end{pmatrix}_{ij}(Q_3)_{i\ell}(s) ,
\\
& P_{ij\ell} (s,0) = \begin{pmatrix} (1-s) R_{11}(s) & s \widetilde R_{11}(s) \\ R_{21}(0) & 0 \end{pmatrix}_{ij} (Q_2)_{i\ell}(s),
~~~ P_{ij\ell} (s,1) = \begin{pmatrix} (1-s) R_{12}(s) & s \widetilde R_{12}(s) \\ 0 & \widetilde R_{21}(1) \end{pmatrix}_{ij} (Q_4)_{i\ell}(s),
\fe
where the functions $R_{ij},\widetilde R_{ij}$ for $ij=\{11,21\}$ and $\{12,22\}$ are those appearing in the gluing maps (\ref{wetag}), (\ref{wetatil}) used in constructing the interpolation map attached to the edges $\{\overline{v_{{\bf a},2}v_{{\bf a},3}}, \overline{v_{{\bf b},1}v_{{\bf b},2}}\}$ in $\Gamma$ and $\Gamma'$, respectively. $Q_1,\cdots,Q_4$ are $GL(2,\mathbb{C})$-valued functions on $[0,1]$ that are chosen to allow for extending $P_{ij\ell}$ smoothly into the interior of $[0,1]^2$ while ensuring that ${\cal I}^{\rm Sq}$ has non-singular Berezinian.
\bigskip
\begin{figure}[h]
\centering
\begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {\includegraphics[width=.7\textwidth]{figures/pco_square.pdf}};
\begin{scope}[shift={(image.south west)}, x={(image.south east)}, y={(image.north west)}]
\node[anchor=north] at (0.000 + 0.063, 0.000 + 0.130) {$\scriptstyle 11$};
%
\node[anchor=south] at (0.000 + 0.063, 0.000 + 0.057) {$\scriptstyle 21$};
%
\node[anchor=north] at (0.396 + 0.063, 0.000 + 0.130) {$\scriptstyle 11$};
\node[anchor=north] at (0.396 + 0.145, 0.000 + 0.130) {$\scriptstyle 12$};
\node[anchor=south] at (0.396 + 0.063, 0.000 + 0.057) {$\scriptstyle 21$};
%
%
\node[anchor=north] at (0.792 + 0.145, 0.000 + 0.130) {$\scriptstyle 12$};
\node[anchor=south] at (0.792 + 0.063, 0.000 + 0.057) {$\scriptstyle 21$};
%
\node[anchor=north] at (0.000 + 0.063, 0.350 + 0.130) {$\scriptstyle 11$};
%
\node[anchor=south] at (0.000 + 0.063, 0.350 + 0.057) {$\scriptstyle 21$};
\node[anchor=south] at (0.000 + 0.145, 0.350 + 0.057) {$\scriptstyle 22$};
\node[anchor=north] at (0.396 + 0.063, 0.350 + 0.130) {$\scriptstyle 11$};
\node[anchor=north] at (0.396 + 0.145, 0.350 + 0.130) {$\scriptstyle 12$};
\node[anchor=south] at (0.396 + 0.063, 0.350 + 0.057) {$\scriptstyle 21$};
\node[anchor=south] at (0.396 + 0.145, 0.350 + 0.057) {$\scriptstyle 22$};
%
\node[anchor=north] at (0.792 + 0.145, 0.350 + 0.130) {$\scriptstyle 12$};
\node[anchor=south] at (0.792 + 0.063, 0.350 + 0.057) {$\scriptstyle 21$};
\node[anchor=south] at (0.792 + 0.145, 0.350 + 0.057) {$\scriptstyle 22$};
\node[anchor=north] at (0.000 + 0.063, 0.834 + 0.130) {$\scriptstyle 11$};
%
%
\node[anchor=south] at (0.000 + 0.145, 0.834 + 0.057) {$\scriptstyle 22$};
\node[anchor=north] at (0.396 + 0.063, 0.834 + 0.130) {$\scriptstyle 11$};
\node[anchor=north] at (0.396 + 0.145, 0.834 + 0.130) {$\scriptstyle 12$};
%
\node[anchor=south] at (0.396 + 0.145, 0.834 + 0.057) {$\scriptstyle 22$};
%
\node[anchor=north] at (0.792 + 0.145, 0.834 + 0.130) {$\scriptstyle 12$};
%
\node[anchor=south] at (0.792 + 0.145, 0.834 + 0.057) {$\scriptstyle 22$};
\node[anchor=south] at (0.5, 0.53) {\large $\displaystyle \mathcal{I}^{\rm Sq}$};
\node[anchor=south] at (0.5, 0.32) {\small $s_2 = 0$};
\node[anchor=north] at (0.5, 0.68) {\small $s_2 = 1$};
\node[anchor=south, rotate=-90] at (0.32, 0.5) {\small $s_1 = 0$};
\node[anchor=north, rotate=-90] at (0.68, 0.5) {\small $s_1 = 1$};
\node[anchor=south] at (0.272, 0.5) {$Q_1$};
\node[anchor=south] at (0.728, 0.5) {$Q_3$};
\node[anchor=east] at (0.5, 0.272) {$Q_2$};
\node[anchor=east] at (0.5, 0.728) {$Q_4$};
\end{scope}
\end{tikzpicture}
\caption{An illustration of the interpolation map ${\cal I}^{\rm Sq}$ associated with a square move. The shaded square at the center represents the domain of the interpolation parameters $(s_1,s_2)$. The neighboring shaded domains are those of other interpolation maps. In each region, the (up to four) circles represent the nontrivial gluing maps of the discs centered at $z_{ij}$ on the SRS. An arrow pointing between two circles indicates interpolation between the gluing maps as $s_1$ or $s_2$ varies, in accordance with the pattern of PCO movement. Matching the boundaries segments of ${\cal I}^{\rm Sq}$ to the neighoring segments of the integration contour involves further $GL(2,\mathbb{C})$ rotations on the fermionic coordinates $(\nu^{\bf a}, \nu^{\bf b})$ by $Q_i$.}
\end{figure}
More explicitly, the Berezinian of ${\cal I}^{\rm Sq}$ would be singular if the $2\times 2$ matrix
\ie\label{ijps}
A^m{}_\ell(\vec s) = \sum_{i,j=1}^2 c_{ij}^m P_{ij\ell}(\vec s)
\fe
is singular, where the coefficients $c_{ij}^m$ are such that $\sum_{i,j} c_{ij}^m f^{(-1)}(z_{ij})=0$ for all weight $-{1\over 2}$ meromorphic differentials $\zeta = f(z) dz^{-{1\over 2}}$ with $d_o+2$ poles at $\{z_c: 1\leq c\leq d_o, c\not={\bf a},{\bf b}\} \cup \{z_{ij}: i,j=1,2\}$. Up to overall rescaling, such $c^m_{ij}$ are related to $\B\C$ system correlators via
\ie
c_{ij}^1 c_{i'j'}^2 - c^2_{ij} c^1_{i'j'} = \Big\langle\delta(\B(z_{ij})) \delta(\B(z_{i'j'})) \prod_{c\not={\bf a},{\bf b}} \delta(\B(z_c))\Big\rangle^{-1}.
\fe
We can choose $Q_1,\cdots, Q_4$ in (\ref{fourqs}) to eliminate the winding number of $A^m{}_{\ell}(\vec s)$ in $GL(2,\mathbb{C})$ as $\vec s$ moves around the boundary of the square $[0,1]^2$. This then removes the obstruction in extending $P_{ij\ell}$ to $[0,1]^2$ while maintaining the non-degeneracy of (\ref{ijps}).
The integration of $({\cal I}^{\rm Sq})^*\Omega$ with respect to $\vec s\in [0,1]^2$ and $\nu^{\bf a}, \nu^{\bf b}$ can be performed analogously to the computation of section \ref{sec:vertemerge}, and produces the factor
\ie
\Big[ \xi(z_{\bf a}^{(3)}) - \xi(z_{\bf a}^{(2)}) \Big] \Big[ \xi(z_{\bf b}^{(2)}) - \xi(z_{\bf b}^{(1)}) \Big]
\fe
in precise agreement with the vertical integration prescription associated with moving a pair of holomorphic PCOs, over a codimension 2 locus in the bosonic moduli space.
The above construction can be generalized inductively to $\mathfrak{I}_{\A_1\A_2\cdots \A_{p+1}}$, which is associated with a ``$p$-path"
\ie\label{gammapis}
\Gamma_{\A_1\A_2\cdots \A_{p+1}}: \Delta^p \to V,
\fe
whose restriction to the $m$-th face $\partial_m\Delta^p$ agrees with $\Gamma_{\A_1\cdots \A_{m-1} i_{m+1}\cdots \A_{p+1}}$, $m=1,\cdots,p+1$. Viewed as a homotopy, (\ref{gammapis}) is composed of merger moves that do not contribute to the supermoduli integral, and the $p$-dimensional analog of square moves, or ``hypercube moves". Each hypercube move is associated with an embedded hypercube of the form
\ie
\prod_{m=1}^p\overline{v_{{\bf a}_m, p+1-m}v_{{\bf a}_m, p+2-m}} \,\subset \prod_{m=1}^p \Delta_{{\bf a}_m},
\fe
and gives rise to an interpolation map
\ie
{\cal I}^{\rm HC} : [0,1]^p\times {\cal D}_{\A_1\cdots\A_{p+1}} \times \mathbb{R}^{0|*d_o}
\to \mathfrak{U}_{\A_1\cdots\A_{p+1}},
\fe
such that the integral of $({\cal I}^{\rm HC})^*\Omega$ with respect to $\vec s\in [0,1]^p$ and the odd parameters $\nu^{{\bf a}_1},\cdots, \nu^{{\bf a}_p}$ gives rise to the factor
\ie
\prod_{m=1}^p \Big[ \xi(z_{{\bf a}_m}^{(p+2-m)}) - \xi(z_{{\bf a}_m}^{(p+1-m)}) \Big]
\fe
that amounts to the vertical integration associated with moving $p$ holomorphic PCOs.
\section{Concluding remarks}
\label{sec:discuss}
While the emergence of PCO from integrating over the fermionic moduli parameter in the gluing map of the super disc is well known (see e.g. \cite{Polchinski:1998rr}), a key point of the construction of section \ref{sec:mtos} is that the interpolation map $\mathfrak{I}_{\A\B}$ that defines the segment of the supermoduli integration contour at the interface between the cells ${\cal D}_\A$ and ${\cal D}_\B$ involves not actual continuous movement of the PCO location, but rather deforming simultaneously the gluing maps on a pair of discs corresponding to the initial and final positions of the PCO in the vertical integration. Furthermore, it is important that $\mathfrak{I}_{\A\B}$ need not serve as interpolation, in the sense of a continuous map, between the coordinate maps $\varphi_\A$ and $\varphi_\B$. Rather, we only need to demand that the boundary of the image of $\mathfrak{I}_{\A\B}$ to cancel against the boundaries of ${\cal D}_\A$ and ${\cal D}_\B$ lifted into the supermoduli space, so as to close the supermoduli contour $\mathfrak{S}$ into a cycle. This relaxation in the interpolation condition, characterized by the freedom of the super-diffeomorphism $R$ appearing in (\ref{intmatchcond}) or (\ref{supcontr}), allows for the construction of the interpolation maps over arbitrary codimension interfaces with non-singular Berezenian without obstruction.
In a follow-up paper \cite{paper:g2}, we will illustrate the general construction of section \ref{sec:mtos} in the example of genus two SRSs, for both odd and even spin structures, with explicit parameterization of the supermoduli space related to specific PCO configurations.
Given the understanding of this paper, one may hope to recast closed superstring field theory, currently formulated in the PCO language \cite{deLacroix:2017lif}, in terms of SRS and supermoduli integration. Another possible future direction is to find a local projection of the supermoduli space that recovers the bosonic moduli space integrand of the pure spinor formalism \cite{Berkovits:2000fe, Berkovits:2002zk, Berkovits:2013eqa}.
\section*{Acknowledgements}
XY thanks Ashoke Sen for conversations that motivated this work. We would like to thank Ashoke Sen for reading a preliminary draft, and Ted Erler for correspondence.
This work is supported in part by a Simons Investigator Award from the Simons Foundation, by the Simons Collaboration Grant on the Non-Perturbative Bootstrap, and by DOE grants DE-SC0007870.
|
3,212,635,537,502 | arxiv | \section{The Portable Parallelization Strategies project}
The High Energy Physics Center for Computational Excellence (HEP/CCE) is a pilot project whose mandate is to provide strategies for HEP experiments to adapt to using increasingly heterogeneous High Performance Computers. It is split into 4 parts, targeting portable parallelization strategies (PPS), fine-grained I/O and related storage issues (IOS), event generators (EG), and complex workflows (CW)~\cite{cce_webpage}. The PPS group is investigating various portability solutions that will permit single source code to be compiled for and executed on multiple different heterogeneous architectures. This is becoming an essential requirement, as each of these architectures use different languages and APIs, such as CUDA~\cite{cuda} for NVIDIA GPUs, SYCL~\cite{sycl} for Intel GPUs, HIP~\cite{hip} for AMD GPUs, and HLS~\cite{Duarte_2018} for FPGAs (see Fig. \ref{fig:matrix}). HEP experiments, which now have code bases in the million lines of source code, do not have the person power to port their CPU code to each back end. Furthermore, validating and maintaining multiple versions of algorithms written in different languages would be exceedingly onerous.
\begin{figure}
\begin{center}
\includegraphics[width=0.90\hsize]{architecture-technology-matrix.png}
\end{center}
\caption{Matrix of portability technologies supporting a variety of hardware architectures.}
\label{fig:matrix}
\end{figure}
The HEP/CCE-PPS group is currently evaluating Kokkos~\cite{kokkos,CarterEdwards20143202}, SYCL~\cite{sycl}, Alpaka~\cite{MathesP3MA2017,ZenkerAsHES2016,worpitz_2015_49768}, OpenMP/OpenACC~\cite{dagum1998openmp} and {\tt std::execution::parallel}~\cite{std_par} by porting a small number of HEP test beds taken from several different experiments to each portability layer (see Sec. \ref{sec:usecases}). These are
\begin{itemize}
\item Patatrack and P2R from CMS~\cite{:2008zzk} which perform pixel detector pattern recognition and tracking
\item The WireCell toolkit from DUNE~\cite{DUNE:2016hlj} which performs space point formation in the liquid argon time projection chamber (TPC)
\item FastCaloSim from ATLAS~\cite{Aad:2008zzm} which does a fast parameterized simulation of the liquid argon calorimeter
\item A pixel detector tracking workflow from the “A Common Tracking Software” project (ACTS)~\cite{andreas_salzburger_2022_6220148}
\end{itemize}
The HEP/CCE-PPS group is tightly integrated with a number of HEP experiments, with core developers from each experiment being represented in the group. Each port will be evaluated according to a set of metrics (see Sec.~\ref{sec:metrics}), and at the end of the process, the group will make recommendations back to the experiments and the HEP community in general as to the suitability of each technology that was investigated. It should be noted that no one best solution is likely to exist, as the needs and characteristics of each experiment are different.
\section{Metrics}
\label{sec:metrics}
With the goal to assist the process to make recommendations to HEP experiments and the HEP community, we designed a set of metrics that are of interest to the HEP communities and how scientific software is being developed, and evaluate all portability technologies in question using this set of properties. The metrics set aims at evaluating the whole programming experience for the developer/user using the portability solution, not just the specification or the capability of the solution. Hence we collected HEP use-case programs (see Sec.~\ref{sec:usecases}) for portability solutions from different sub-fields and implement them in different portability technologies. We get hands-on experience for applying a certain technology to HEP software problems by implementing our use cases in the portability technologies under consideration. This includes building, debugging, and adapting existing code to a given technology, which will be reflected in the evaluation of the metrics.
The metrics set will serve as a point-of-reference for the information about these portability solutions, which more often still lack the needed level of documentation, and help the HEP community make the informed decision when choosing the portability solution to work with.
The metrics are grouped according to the following categories and are documented here~\cite{metrics_doc}:
\begin{itemize}
\item Ease of learning for experts and novices
\item Ease of code conversion
\begin{itemize}
\item From CPU code to Accelerator (GPU, etc.) code
\item From low level (CUDA, etc.) to higher level portability code
\item From one portability framework to another
\end{itemize}
\item Impact on other existing code
\begin{itemize}
\item Extent of modifications to existing code: does it take over main(), does it affect the threading or execution model, etc.
\item Extent of modifications to Event Data Model (EDM): data transfer and access across different memory space, etc.
\end{itemize}
\item Impact on existing tool chain and build infrastructure
\begin{itemize}
\item Extent of modifications to build rules / system
\item Do we need to recompile the entire software stack?
\item CMake or make changes/integration
\end{itemize}
\item Hardware mapping
\begin{itemize}
\item Is the technology working on all current hardware architectures?
\item Support for new hardware features and new architectures
\end{itemize}
\item Feature availability
\begin{itemize}
\item Reductions, kernel chaining, callbacks, etc
\item Concurrent kernel execution
\item Support for interfacing to optimized math-heavy libraries (FFTs, etc.)
\end{itemize}
\item Ease of Debugging
\begin{itemize}
\item How easy is it to debug implementations of code in the technologies?
\end{itemize}
\item Address needs of all types of workflows
\begin{itemize}
\item Scaling with \# of kernels / application
\item Scaling with \# of developers
\item Support for users by portability technology developers
\end{itemize}
\item Long-term sustainability and code stability
\begin{itemize}
\item Support model of technologies, stability of implementation if underlying libraries (CUDA) change
\item CUDA is going to be around for a long time, what about the portability solutions?
\item Long term support for technologies by vendors
\end{itemize}
\item Compilation time
\begin{itemize}
\item Separate builds for different architectures?
\item Compatibility with experiment’s software distribution strategies
\end{itemize}
\item Performance: CPU and GPU
\begin{itemize}
\item Does the portable code version (CPU and GPU uses same code) degrade the CPU performance or use more memory?
\end{itemize}
\item Aesthetics
\begin{itemize}
\item compatibility with C++ standards
\end{itemize}
\item Interoperability
\begin{itemize}
\item Can you mix portability technologies in the same application? How are external packages treated if they are imported into experiment software stacks and use different portability technologies? (CMSSW~\cite{jones:2006,jones:2014,jones:2015,jones:2017,bocci:2020a} is using Kokkos, but Geant~\cite{1610988,AGOSTINELLI2003250,ALLISON2016186} is using Alpaka)
\item Interaction with existing thread pool on CPU/GPU back ends?
\end{itemize}
\end{itemize}
\section{Use Cases}
\label{sec:usecases}
In the following, we briefly describe the use cases that are being used in this study.
\begin{description}
\item[FastCaloSim] is a parametrized simulation of the ATLAS Liquid Argon Calorimeter~\cite{ATL-SOFT-PUB-2018-002}. The codebase was originally written in C++, then ported to CUDA. The CUDA implementation consists of 3 relatively small kernels, which perform a memory re-initialization, the main energy deposition simulation, and finally a stream compaction. It has been ported to Kokkos, SYCL, and std::par, targeting NVIDIA, Intel and AMD hardware.
\item[ACTS] is a track reconstruction toolkit for general HEP detectors~\cite{andreas_salzburger_2022_6220148}, which is based on C++. The R\&D lines for ACTS parallelization on heterogeneous architectures consist of a number of core algorithms for tracking on GPUs (traccc), a geometry offloading package designed explicitly for GPUs (detray), and a memory management layer (vecmem) that is architecture neutral. All tracking algorithms, which include hit clusterization, seeding and Kalman filtering (both simple and combinatorial) will be offloaded onto the GPU to minimize the data transfers between host and device.
\item[Wire Cell] The Wire-Cell Toolkit~\cite{wirecell_toolkit,Qian:2018qbv} is a C++ software library for the simulation, signal processing, reconstruction and visualization of Liquid Argon Time Projection Chamber (LArTPC) detectors for neutrino experiments, such as the planned Deep Underground Neutrino Experiments (DUNE~\cite{DUNE:2016hlj}). The use case we study is the LArTPC signal simulation module in Wire-Cell, which simulates the LArTPC detector response. So far the signal simulation module has been re-implemented in Kokkos~\cite{Dong:2022wxg}, and investigation with OpenMP is in progress.
\item[Patatrack] The Patatrack use case consists of CMS Heterogeneous Pixel Reconstruction~\cite{bocci:2020b,patatrack_standalone} code, that processes the raw pixel detector data up to pixel tracks and vertices, extracted into a standalone program. It includes a multi-threaded mock framework providing similar behavior as CMS software framework CMSSW~\cite{jones:2006,jones:2014,jones:2015,jones:2017,bocci:2020a}, and input data from CMS Open Data~\cite{ttbardata}. The original code was developed to run on NVIDIA GPUs with CUDA, accompanied with a simple translation header to allow compilation to CPU. We have ported the code to Kokkos and HIP, targeting NVIDIA and AMD GPUs.
\item[P2R] is a light-weight mini-app which performs the track propagation in radial direction and Kalman update kernels in track reconstruction~\cite{p2r}. With a simplified geometry and standalone setup, P2R can be used to test the performance of core tracking computation in various portability technologies in a shorter timescale. The original version is adapted from the mkFit project~\cite{mkfit}, which implements a parallel Kalman Filter Algorithm~\cite{kalman}, and has been re-implemented in CUDA, HIP, Kokkos, Alpaka, OpenACC and std::par.
\item[Random Number Generators] We have leveraged the SYCL programming model and its interoperability with third-party libraries to demonstrate cross-platform performance portability across heterogeneous resources. We have implemented NVIDIA and AMD random number generator extensions to the oneMKL open-source interfaces library~\cite{onemkl}. The utility of our extensions are exemplified in a real-world setting via a high-energy physics simulation application, showing the performance of implementations that capitalize on SYCL interoperability are at par with native implementations, attesting to the cross-platform performance portability of a SYCL-based approach to scientific codes.
\end{description}
\section{Preliminary Results}
\label{sec:prelresults}
We gathered some preliminary results from our studies of various portability solutions.
\subsection{Kokkos}
Kokkos is a programming model and a C++ library for portable performance applications~\cite{kokkos}. It provides high-level parallel algorithms, such as for, prefix scan, and reduction, that can be nested with some restrictions, as well as multidimensional array data types. The mapping of work of the algorithms and the default layout of the multidimensional array depend on the chosen back end. Currently (Kokkos 3.5) these back ends include CPU serial, CPU parallel with OpenMP or Posix Threads, and device parallel with CUDA, HIP, HPX~\cite{Kaiser2020}, or SYCL. The high abstraction level on both algorithms and data is expected to provide reasonably good out-of-the-box performance also on computing architectures beyond CPUs and GPUs.
We were able to express all the custom algorithms in the use cases (FastCaloSim, Wire Cell Toolkit, Patatrack, p2r) in the Kokkos’ programming model, but we experienced some challenges. Kokkos requires its run time library to be built for one set of host serial, host parallel, and device parallel back end at a time, and e.g. in case of CUDA the library can support exactly one major GPU architecture version. While this approach works fine for HPC codes that are typically compiled for a specific supercomputer, it poses challenges for HEP experiment frameworks for which a single build is expected to be used in about 200 data centers with different computer hardware. In Kokkos 3.5, the CPU serial back end is thread safe, but it cannot be efficiently used from multiple threads, limiting its current usefulness in multi-threaded applications that process multiple collision events concurrently. Kokkos developers are working to improve the performance for this use case. Much of the HEP data is structured, and multidimensional arrays are useful only in limited use cases, implying a need for a separate solution for data structures. Currently Kokkos does not provide a unified, portable interface to Fast Fourier Transform (FFT) algorithms (e.g. to optimize platform-specific implementations), but such interface is being worked on.
For the WireCell toolkit use case, we have seen moderate performance speedups from multi-core CPUs, AMD GPUs and NVIDIA GPUs, using Kokkos. We have also demonstrated that running multiple concurrent processes to share the GPUs can further improve the performance gains, setting a promising direction for efficient utilization of HPC systems in Wire-Cell. For other use cases we saw performance degradation for specific hardware platforms (AMD GPUs in case of FastCaloSim).
\subsection{SYCL}
SYCL is an open-standard C++-based programming model that facilitates parallel programming on heterogeneous platforms. It provides a single source programming model, enabling developers to write both host-side and kernel code in the same file. Employing C++-based template programming, developers can leverage higher-level programming features in writing accelerator-enabled applications with the ability to integrate the native acceleration API, when needed, by using different interoperability interfaces provided by SYCL. The latest specification, SYCL 2020, is based on ISO C++17 standard, and features standard programming with templates and lambda functions to develop optimized code which can be offloaded to special purpose compute accelerators such as GPUs, FPGA, or AI/ML accelerators. The SYCL specification is designed to be a higher level abstraction above low-level native acceleration APIs with interoperability between existing libraries and other parallel programming models and can be built on top of OpenMP, Vulkan~\cite{vulkan}, OpenCL~\cite{opencl}, Kokkos, Raja~\cite{raja}, or some other back end. The SYCL programming model offers performance portability across various vendor hardware and interoperability with both open-source and closed-source (proprietary) software. As SYCL evolves, HPC-critical features will continue to be incorporated into the specification.
The applicability of our SYCL-based Random Number Generators (RNGs)~\cite{2021arXiv210901329P,9652858} has been evaluated in a GPU port of FastCaloSim~\cite{10.3389/fdata.2021.665783}. The interfaces we developed enabled the seamless integration of SYCL RNGs into FastCaloSim with no code modification across the evaluated platforms. The SYCL 2020 interoperability functionality enabled custom kernels and vendor-dependent library integration to be abstracted out from the application, leading to improvement of maintainability of the application and reducing the source lines of code. Using our RNG interfaces, we achieve comparable performance with native solutions on different architectures. Whereas the original C++ version of FastCaloSim had two separate code bases, x86 and CUDA, our RNG work has enabled event processing on a variety of major vendor hardware from a single SYCL entry point. Hence, the SYCL RNG based integration facilitates the code maintainability by reducing the FastCaloSim code size without introducing any significant performance overhead.
\subsection{OpenMP}
OpenMP is a directive-based programming model that has evolved from a shared-memory programming model for multicore CPU architectures to one with rich features to support GPU accelerator offloading. The current version 5.1 of the OpenMP specification includes support for loop-level target offloading, memory management, and asynchronous CPU/GPU execution, all of which may be crucial for experimental HEP workflows. Major HPC hardware vendors such as AMD, HPE, Intel and NVIDIA are all onboard with the OpenMP model, and have been actively developing the compiler infrastructure to support the new and improved OpenMP features. However, as a directive-based programming model, it may not offer the same level of flexibility as language-based programming models such as Kokkos or SYCL. Nevertheless, the premise of only needing to add a few pragmas to the code and letting the compilers handle the low-level optimizations holds great promise for the future, when even more diverse and complex HPC architectures are to be expected. With the anticipated industrial support, OpenMP may become the portable programming model of choice in the future, and R\&D into how HEP software can take full advantage of this potential should be supported timely so that we don’t fall behind the rest of the scientific computing community.
Our initial investigations into the OpenMP target offloading features have shown that in addition to the simple loop parallelism, many performance-enhancing features are also well supported and relatively easy to use. These examples include asynchronous kernel executions, interoperability with optimized vendor libraries and the ability to specify memory spaces. Compiler support for OpenMP offloading has also been improving rapidly, with several functional compilers available on the market, such as the open-source LLVM Clang compiler, GNU C Compiler, HPE’s CCE compiler, along with NVIDIA’s, AMD’s and Intel’s compilers to support their own GPU architectures. While the performances with these compilers can vary quite a bit depending on the specific use case, they have all been steadily improving.
\subsection{std::par}
std::parallel::execution (std::par) has been part of the C++ standard since C++17, and offers a high level interface to execute the contents of loops concurrently. Until recently, the concurrent back ends have been limited to the host side, using libraries such as TBB~\cite{tbb} to execute on different CPU threads and cores. Recently, both Intel and NVIDIA have released compilers that can target GPU devices (oneapi::dpl and nvc++ respectively). Given their high level nature, most low level functions and optimizations of domain specific languages such as CUDA and SYCL are not available, resulting in a loss of overall performance. However, the entry bar to users is extremely low, and requires little knowledge of GPU programming. Both of these compilers are still rather immature, exhibiting a number of compiler bugs and lack of build system integration, so performance numbers should be taken lightly. The continued development of these compilers is however a good indication that vendors are seeking standards based solutions, with both low and high level APIs.
\section{Conclusions}
The portable parallelization strategies (PPS) project of the High Energy Physics Center for Computational Excellence (HEP/CCE) is investigating solutions to the changing hardware architecture landscape of today and in the future. With accelerators like GPUs becoming more mainstream, especially in HPC systems, the development of scientific code is at a crossroad leaving the convenient era of x86-only code. To continue writing scientific code efficiently with a large and not always professionally trained user community to run on all hardware architectures, we think we need community solutions for portability techniques that will allow the coding of an algorithm once, and the ability to execute it on a variety of hardware products from many vendors. To that effect, the PPS project is investigating the feasibility of currently available portability solutions and will compare them based on a defined set of metrics. The goal is to develop recommendations for potential users when to deploy the appropriate solution. Preliminary results show deploying portability solutions is currently far from the convenience of having a standard, but there are encouraging successes using these solutions. We think without them, the scientific success of our experiments and endeavors is in danger, as software development could be expert driven and costly to be able to run on available hardware infrastructure. We think the best solution for the community would be an extension to the C++ standard with a very low entry bar for users, supporting all hardware forms and vendors. We are very far from that ideal though. In the future, as a community, we need to request and work on portability solutions and strive to reach this ideal.
\bibliographystyle{JHEP}
|
3,212,635,537,503 | arxiv | \section{Introduction}
Coherency is a property of matter in quantum world which is most demonstrated in a double-slit experiment. This is also important to understand the dual character of quantum matters. Richard Feynman emphasized that the double-slit interference is at the heart of quantum phenomena: ''In reality, it contains the only mystery, the basic peculiarities of all of quantum mechanics'' \cite{Feynman}.
The double-slit diffraction pattern of a micro-particle is a distinguishing point between classical mechanics (CM) and quantum mechanics (QM). The interference fringes of wave-particles are described within QM. This follows from an objective interpretation of the wave function and by the property that, in QM, superpositions of states are possible, while this is not so in CM \cite{Zecca1, Zecca3}. Interference is resulted from the uncertainty principle between the momentum and the position of the quantum system. If we take the vacuum into account, it will require entanglement between complementary variables such as the momentum and the position or the angular momentum components \cite{Zimmermann}.
Many works have been done to explain double-slit experiment under different conditions such as interference for the macro-system in experimental and theoretical contexts. Nairz, Arndt and Zeilinger studied the interference pattern for the fullerene molecule as a large object similar to a classical system \cite{Nairz}. Hornberger $\it{et}$ $\it{al.}$ investigated the effect of environment on interference pattern of fullerenes as a macro-molecule \cite{Hornberger2}. Also, Hornberger and others studied quantum interferenece of the clusters in experiment \cite{Hornberger1}. Gerlich and others showed the quantum diffraction of large organic molecules \cite{Gerlich}.
The double-slit diffraction pattern can be illustrated by describing the incoming particle in some proper states such as Gaussian wave packets \cite {Merzbacher, Holland}. Zecca in many studies investigated theoretically one, two and $N$-slit diffraction patterns for the Gaussian wave packets \cite{Zecca4, Zecca5, Zecca7}. The use of Gaussian states is sufficiently general, because it includes the limit case of plane waves and that of the wave packets narrower than the slit width. Moreover, due to the development of experimental techniques, possible deviations from the standard form of the interference pattern can be better explained by Gaussian-like states \cite{Zecca2}.
In this study, considering a macroscopic quantum oscillator interacting with two micro-oscillating particles in the environment, we exactly obtain the wave function of the system in the ground state as a Gaussian wave packet. Then, the time evolution for the wave function of the $\it{central}$ system $\it{after}$ the slits is evaluated by considering the $y$-dependence of the wave packet, passing the slits, in two limiting cases. The slits are located on the $y$-axis, in a symmetric position with respect to the $x$-axis. The wave packet describing the incoming particle is factorized in its $x$ and $y$ dependences. It is assumed that $x$ and $y$ dependences of the wave function remain factorized during and after passing the slits \cite{Zecca2}. We show that when quantum character of the macro-system is evanesced, interference fringes are diminished and the diffraction pattern becomes more similar to a classical one.
The double-slit experiment is also a point where one can compare the predictions of QM with those of stochastic electrodynamics with spin (SEDS). The comparison of the two theories was first studied in \cite{Zecca5}. The SEDS approach is based on electromagnetic interactions \cite{cavalleri1}. Our results show that there is a clear difference between the predictions of QM and those of SEDS for a macroscopic quantum system coupled to the environmental particles which depends on the quantum behaviour of the macro-system.
The paper is organized as follows. In section 2, we describe principles of the standard form of the Schr\"{o}dinger equation which is called dimensionless analysis. In section 3, bilinear harmonic model of the environment is introduced. Then, the ground state is exactly evaluated for the system coupled to two particles of the environment. In section 4, the two- dimensional double-slit diffraction pattern is analyzed. Two limiting cases of the problem are illustrated and discussed afterwards, to show how the emergence of classicality in the system leads to the appearance of macro-type fringes in the interference patterns. In section 5, we analyze the double-slit interference pattern for macro-system based on SEDS and compare the results with QM. Finally, the results are discussed in the conclusion part.
\section{Dimensionless form of the Schr\"{o}dinger equation}
First, we introduce dimensionless parameters for an arbitrary quantum system. We define $R_0$ and $U_0$ as constant units of length and energy, respectively. Subsequently, for a particle of mass $M$, one can define the characteristic time $\tau_0$ as \cite{Taka}:
\begin{equation}
\label{eq1}
\tau_0=\frac{R_0}{(U_0/ M)^\frac{1}{2}}
\end{equation}
in which $U_0$ is in order of the kinetic energy of the system and the unit of momentum could be defined as $P_0=(U_0 M)^\frac{1}{2}$. Also, the conjugate variables $q$ and $p$, as well as the time $t$ in the dimensionless forms are defined as:
\begin{eqnarray}
\label{eq2}
q=\frac{R}{R_0}, ~~~ p=\frac{P}{P_0}, ~~~ t=\frac{T}{\tau_0}
\end{eqnarray}
\noindent where $R$, $P$ and $T$ are the conventional position, momentum and time, respectively. Then, the following relations for the potential energy $V$ and the Hamiltonian $H_s$ could be introduced in the dimensionless regime as:
\begin{eqnarray}
\label{eq3}
V(\hat q)=\frac{U(\hat R)}{U_0},~~~ {\hat{H}_s}=\frac{\hat{H}_S}{U_0}
\end{eqnarray}
\noindent where $U(\hat R)$ and $\hat{H}_S$ are the potential energy and the Hamiltonian in the ordinary Schr\"{o}dinger equation. Finally, the dimensionless form of the Schr\"{o}dinger equation can be written as:
\begin{equation}
\label{eq4}
i\bar h\frac{d\psi(t)}{dt}={\hat{H}_s}\psi(t)
\end{equation}
\noindent where
\begin{equation}
\label{eq5}
{\hat{H}_s}=\frac{\hat{p}^2}{2}+V(\hat{q})
\end{equation}
\noindent and
\begin{eqnarray}
\label{eq6}
[\hat{q},\hat{p}]=i\bar h
\end{eqnarray}
Here, we define
\begin{equation}
\label{eq7}
\bar h=\frac{\hbar}{U_0\tau_0}=\frac{\hbar}{{P_0}{R_0}}=\lbrace\frac{\hbar^2}{MU_0 R^2_0}\rbrace^\frac{1}{2}
\end{equation}
In equations (6) and (7), instead of Planck constant $\hbar$, a new dimensionless parameter $\bar h$ appears, measured in units of the action $U_0 \tau_0$, on which the quantum nature of the system depends. The situation where $\bar h\ll1$ is called the quasi-classical situation. The values of $\bar h$ between 0.01 to 0.1 could show the macroscopic trait of the proposed system \cite{Taka}. In many applications of double-well potentials, $R_0$ is defined as characteristic length of resonance between left and right counterparts of the potential $U(\hat R)$ . So, $\bar h$ in (7) can be rewritten as:
\begin{equation}
\label{eq8}
\bar h=\lambdabar_0/R_0
\end{equation}
\noindent where $\lambdabar_0=\lambda_0 / 2\pi$. Here, $\lambda_0$ denotes the de Broglie wavelength of the central system. For a macroscopic quantum system , $\lambda_0$ is too small compared to $R_0$ which the latter is nearly a fixed value for known models of potential. Thus, regarding the quasi-classical systems, the condition $\bar h<0.1$ seems suitable for our future purposes. For smaller values of $\bar h$, the macro-system shows more classical trait.
One can also define $\omega_0=\tau_0^{-1}= {P_0}/ {R_0 M}$ in relation with the $\it{particle}$ aspect of a system in a resonance situation. Accordingly, for the $\it{wave}$ aspect of the system, one can consider another unit of momentum $P_0'$ resulted from the phase velocity $v_0'=\omega_0/ k_0 (k_0=2\pi / \lambda_0)$, so that, $P_0'=M\lambdabar_0\omega_0$. Then, using the relation (8), we conclude that:
\begin{equation}
\label{9}
\bar h=\frac{P_0'}{P_0}
\end{equation}
This is another demonstration of how the value of $\bar h$ can display the classicality of the system. For a macroscopic quantum system, the wave character is weakened, so that $P_0' \ll P_0$.
\section{Bilinear-Harmonic model of the environment}
Suppose that the central system is a quantum harmonic oscillator. The entire system is composed of the system and two environmental micro-oscillators. The total Hamiltonian could be written as :
\begin{equation}
\label{eq10}
\hat{H}=\hat{H}_s+\hat{H}_e+\hat{H}_{se}
\end{equation}
\noindent where $\hat{H}_s$ is the Hamiltonian of the system and $\hat{H}_e$ and $\hat{H}_{se}$ are the environment and the interaction Hamiltonians, respectively.
In the dimensionless form, the following classes of Hamiltonian can be used \cite{Taka}:
\begin{equation}
\label{eq11}
{\hat{H}_s}=\frac{\hat{p}^2}{2}+V(\hat{q})
\end{equation}
\begin{equation}
\label{eq12}
\hat{H}_e={\sum_\alpha}[ \frac{\hat{p}^2_\alpha}{2}+\frac{\omega^2_\alpha}{2}{\hat x^2_\alpha-}\frac{\bar h\omega_\alpha}{2}]
\end{equation}
\begin{equation}
\label{eq13}
\hat H_{se}=-\sum_\alpha{\omega}^2_\alpha f_\alpha(\hat q)\hat x_\alpha+\frac{1}{2}\sum_\alpha \omega^2_\alpha \lbrace f_\alpha (\hat q)\rbrace ^2
\end{equation}
\noindent where ${\alpha}$ varies from 1 to 2, which ${\alpha}$ is the number of the environmental oscillators. In these relations, $ x_\alpha $, $ p_\alpha $ and $ \omega_\alpha $ denote the position, the momentum and the frequency of the environmental particles, respectively. The last constant term in (12), which merely displaces the origin of energy, has been added for later convenience. Hereafter, for simplicity, we assume that $f_\alpha(q)={\gamma_\alpha}q$, where $ 0<\gamma_\alpha<1$ denotes the strength of the interaction between the system and the environment. So, one can reach the conclusion that $\hat H_{se}$ is linear both to $ \hat x_\alpha$ and $\hat q$. So, the model is called $\it{bilinear}$.
Considering the bilinear condition, the total form of Hamiltonian can be presented as:
\begin{widetext}
\begin{equation}
\label{eq14}
H=\frac{p^2}{2}+V_1(q)+\sum_\alpha[\frac{p^2_\alpha}{2}+\frac{\omega^2_\alpha}{2}x^2_\alpha(1-\gamma_\alpha)-\frac{\bar h\omega_\alpha}{2}]+\frac{1}{2}\sum_\alpha \gamma_\alpha\omega^2_\alpha(x_\alpha-q)^2
\end{equation}
\end{widetext}
where
\begin{equation}
\label{eq15}
V_1(q)=V(q)-\frac{1}{2}\lbrace \sum_\alpha{\gamma_\alpha}(1-\gamma_\alpha)\omega^2_\alpha\rbrace q^2
\end{equation}
\noindent Here, $V(q)=\frac{1}{2}\omega_e ^2 q^2$ is the potential and $ \omega_e $ is the vibration frequency of the system. This shows that the system feels the efficient potential $V_1(q)$. Moreover, each environmental oscillator $\alpha$ with spring constant $(1-\gamma_\alpha)\omega^2_\alpha$ is coupled to the system with spring constant $\gamma_\alpha\omega^2_\alpha$.
Now, we obtain the wave function of the central system in the ground state. The total Hamiltonian can be written as:
\begin{equation}
\label{eq16}
H=\frac{p^2}{2}+\frac{\omega^2 }{2}q^2+\sum_\alpha[ \frac{p^2_\alpha}{2}+\frac{\omega^2_\alpha}{2}x^2_\alpha-\omega^2_\alpha\gamma_\alpha q x_\alpha]
\end{equation}
where
\begin{equation}
\label{eq17}
\omega=[\omega_e ^2+\sum_\alpha \omega^2_\alpha \gamma^2_\alpha]^\frac{1}{2}
\end{equation}
For decoupling the Hamiltonian, we define plus-minus position and momentum coordinates, $(x_+,x_-)$ and
$(p_+, p_-)$, respectively, by rotating of the position coordinates $(q',x_\alpha)$ and the momentums $(p',p_\alpha)$ as the following \cite{McDermott}:
\begin{equation}
\label{eq18}
\begin{pmatrix}
x_+ \\
x_-
\end{pmatrix}
\begin{matrix}\\\mbox{}\end{matrix}
=\begin{pmatrix} cos\theta & sin\theta \\ -sin\theta & cos\theta \end{pmatrix}
\begin{pmatrix} q' \\
x_\alpha
\end{pmatrix}
\end{equation}
\begin{equation}
\label{eq19}
\begin{pmatrix}
p_+ \\
p_-
\end{pmatrix}
\begin{matrix}\\\mbox{}\end{matrix}
=\begin{pmatrix} cos\theta & sin\theta \\ -sin\theta & cos\theta \end{pmatrix}
\begin{pmatrix} {p'} \\
{p}_\alpha
\end{pmatrix}
\end{equation}
\noindent where
\begin{equation}
\label{eq20}
p'=\frac{p}{N} , ~~~ q'=\frac{q}{N}
\end{equation}
and $N$ is the total number of the environmental oscillators, which in our study $N=2$. Now, we define:
\begin{equation}
\label{eq21}
\theta=\frac{1}{2}arctan[ \frac{\omega'^2_\alpha}{\omega^2 -\omega^2_\alpha}]
\end{equation}
\noindent where $\omega'_\alpha=i ( 2{\omega^2_\alpha}{\gamma_\alpha})^\frac{1}{2}$. According to above decoupling method, the total Hamiltonian can be written as the sum of the individual Hamiltonians indexing $\alpha$:
\begin{equation}
\label{eq22}
{H}={\sum_\alpha}{H_\alpha}
\end{equation}
\noindent where
\begin{equation}
\label{eq23}
H_\alpha=\frac{p'^2}{2}+\frac{\omega^2}{2} q'^2+ \frac{p^2_\alpha}{2}+\frac{\omega^2_\alpha}{2}x
^2_\alpha-\omega^2_\alpha \gamma_\alpha q' x_\alpha
\end{equation}
Under the rotation, the kinetic energy part in Hamiltonian (23) remaines invariant. Thus, decoupling of the Hamiltonian is obtained by diagonalizing the potential energy. The rotations transform the Hamiltonian to
\begin{equation}
\label{eq24}
H_\alpha=\frac{p_{+\alpha}^2}{2}+\frac{1}{2} \omega_{+\alpha}^2 x_{+\alpha}^2+\frac{p_{-\alpha}^2}{2}+\frac{1}{2} \omega^2_{-\alpha} x_{-\alpha}^2
\end{equation}
\noindent where $H_\alpha =H_{+\alpha}+H_{-\alpha}$.
\noindent Here, we define:
\begin{equation}
\label{eq25}
\omega_{+\alpha}=\lbrace \omega^2 \cos^2\theta+\omega^2_\alpha \sin^2\theta+\omega'^2_\alpha \sin\theta\cos\theta\rbrace^\frac{1}{2}
\end{equation}
and
\begin{equation}
\label{eq26}
\omega_{-\alpha}=\lbrace \omega^2 \sin^2\theta+\omega^2_\alpha \cos^2\theta-\omega'^2_\alpha \sin\theta\cos\theta\rbrace^\frac{1}{2}
\end{equation}
We also assume that for two particles of the environment, $\omega_\alpha$ and $\gamma_\alpha$ are nearly the same. Then, if $tan2\theta>0$ in (21), we should have $\omega^2<\omega^2_\alpha$. This means that
\begin{equation}
\label{eq27}
\omega^2_e<\omega^2_\alpha(1-\gamma^2 N)
\end{equation}
\noindent where $\omega^2_e=\omega^2-\omega^2_\alpha\gamma^2 N$ (see(17)) and $\gamma_\alpha=\gamma$. Yet, in (27), it is necessary that $(1-\gamma^2 N)>0$, or $N\gamma^2 <1$. In other words, the number of particles in the environment should restrict the strength of interaction $\gamma$, which is not a legitimate condition. On the other hand, if we take $tan2\theta<0$, it will be obtained from (9), (17) and (21) that
\begin{equation}
\label{eq28}
\lambda^2_0(1-\gamma^2 N)<\lambda^2_\alpha
\end{equation}
\noindent where $\lambda_\alpha$ is the wavelength of the environmental particles. For both small values of $N$ and $\gamma$ (so that $1-\gamma^2 N\approx1$), one concludes from (28) that
\begin{equation}
\label{eq29}
\bar h=\lambdabar_0/R_0<\lambdabar_\alpha/R_0
\end{equation}
\noindent which guarantees the quasi-classical behavior of the central system. Because, the characteristic wavelength of the macro-system $\lambda_0$ is much smaller than the corresponding wavelength $\lambda_\alpha$ of the micro-particles of the environment. We choose $0.01<\bar h=\lambdabar_0/R_0<0.1$ in (8) to reach the definite bound of $\bar h$ for a quasi-classical system, as mentioned before. Also, in (28), one can notice that $N\gamma^2 >(1-\lambda^2_\alpha / \lambda^2_0)$. If, one assumes that $\lambda_\alpha / \lambda_0<1$ (contrary to (29)), the values of $N$ and $\gamma$ will be again restricted to a positive constant value which is not reasonable, since they are independent parameters. So, for any value of $N$ and $0<\gamma<1$, $\lambda_\alpha / \lambda_0>1$, the condition of (29) is compelling. In effect, the situation for having a macroscopic quantum system is now ready. The key point is that the emergence of classicality, here, is due to the conditions the system interacts under which with the environment.
For calculating the wave function of the system, we define $a=\sin\theta$ and $b=\cos\theta$. Using these definitions, the plus-minus position coordinates of two particles of the environment can be presented as:
\begin{align}
\label{eq30}
x_{+i}&=bq'+a{x}_i \nonumber\\
x_{-i}&=-aq'+b{x}_i
\end{align}
\noindent where $i=1,2$. In this case, the normal ground state wave function for Hamiltonian (24) can be obtained as:
\begin{widetext}
\begin{equation}
\label{eq31}
\psi_0(x_{+1} , x_{-1},x_{+2},x_{-2})=(\frac{{\omega_{+1}} {\omega_{-1}}{\omega_{+2}}{\omega_{-2}}}{{\pi^4 \bar h^4}})^\frac{1}{4}\exp(\frac{-{\omega_{+1}}{x_{+1}^2}}{2\bar h})\exp(\frac{-{\omega_{-1}}{x_{-1}^2}}{2\bar h})\exp(\frac{-{\omega_{+2}}{x_{+2}^2}}{2\bar h})\exp(\frac{-{\omega_{-2}}{x_{-2}^2}}{2\bar h})
\end{equation}
\end{widetext}
Then, one gets the probability distribution for the position coordinate of the system, $P(q)$, by integrating the probability density over the spatial coordinates of the environmental oscillators ($x_{1} , x_{2}$).
\noindent Considering the relations (30) and (31), we obtain:
\begin{widetext}
\begin{align}
\label{eq32}
P(q)=&\frac{1}{\pi \bar h}(\omega_{+1} \omega_{-1}\omega_{+2}\omega_{-2})^\frac{1}{2}[(a^2\omega_{+1}+b^2\omega_{-1})(a^2\omega_{+2}+b^2\omega_{-2})]^\frac{-1}{2} \nonumber\\
&\times\exp\lbrace {\frac{-q^2 [(a^2\omega_{+1}\omega_{+2})(\omega_{-1}+\omega_{-2})+(b^2\omega_{-1}\omega_{-2})(\omega_{+1}+\omega_{+2})]}{\bar h(a^2\omega_{+1}+b^2\omega_{-1})(a^2\omega_{+2}+b^2\omega_{-2})}\rbrace}
\end{align}
\end{widetext}
Using the probability distribution (32), after some mathematical manipulation, one gets the normal wave function for the system as the following:
\begin{widetext}
\begin{align}
\label{eq33}
\psi(q)=&(\frac{1}{\pi \bar h})^\frac{1}{4}[(a^2\omega_{+1}+b^2\omega_{-1})(a^2\omega_{+2}+b^2\omega_{-2})]^\frac{-1}{4}
{[(a^2{\omega_{+1}}{\omega_{+2}})(\omega_{-1}+\omega_{-2})+(b^2{\omega_{-1}}{\omega_{-2}})(\omega_{+1}+\omega_{+2})]}^\frac{1}{4}\nonumber\\
&\times\exp\lbrace {\frac{-q^2 [(a^2\omega_{+1}\omega_{+2})(\omega_{-1}+\omega_{-2})+(b^2\omega_{-1}\omega_{-2})(\omega_{+1}+\omega_{+2})]}{2\bar h(a^2\omega_{+1}+b^2\omega_{-1})(a^2\omega_{+2}+b^2\omega_{-2})}\rbrace}
\end{align}
\end{widetext}
In the next section, we will formulate the double-slit diffraction pattern, using the wave function of the system in (33).
\section{Double-slit diffraction pattern based on QM approach}
Let us consider the system described in the previous section in two dimensions. We assume that the system has been in interaction with the environment only in $y$-direction, so that regarding the $x$-direction, the state of the system behaves like a Gaussian wave packet independent of any environmental effect. After the slits in both directions $x$ and $y$, the system can be viewed as a free particle, for which the wave function in the $y$-direction is defined as (33) at $t=0$ (just after the slits) where $q\equiv y$. The system has a macroscopic quantum character due to the conditions elaborated in section 3 (see relation (29)). Now, we analyze the two-slit diffraction pattern in two dimensions. For this purpose, first we define the region $R$ that is inaccessible to the particle, assumed to be a subset of the $(x, y)$ plane:
\begin{equation}
\label{eq34}
R= \lbrace(x, y): \mid x \mid<a, y\in (-\infty, -d'-b']\cup [-d', d]\cup[d+b, \infty)\rbrace
\end{equation}
\noindent where $b$ and $b'$ are the widths of the slits with depth $2a$ ($b,b'\ll 1$) and $d$ and $d'$ are the distances of the slits from the origin \cite{Zecca4}.
The Hamiltonian of the system at $t\geq 0$ is defined as:
\begin{equation}
\label{eq35}
H=H_0+V(x,y)
\end{equation}
where $V(x,y)$ is an infinite step potential which does not allow the particle to tunnel to the region $R$. For solving the Schr\"{o}dinger equation, which in general could not be separated in terms of $x$ and $y$ dependences, one can use factorized solutions as legitimate method. The diffraction pattern of a particle which is defined by (33), can be studied in the following way. We consider a Gaussian wave packet coming from the remote $x$ region with probability distribution centered on a point moving with velocity $\bar h k_{0x}$ on the $x$-axis ($y=0$) \cite{Zecca4, Zecca5}:
\begin{equation}
\label{eq36}
\psi(x, y, t)=\chi(x, t)\phi(y,t)
\end{equation}
\noindent where
\begin{widetext}
\begin{equation}
\label{eq37}
\chi(x, t)=[\frac{\zeta}{\pi^\frac{1}{2}(1+i\bar h\zeta^2t)}]^\frac{1}{2}\lbrace exp[-\frac{\zeta^2}{2}\frac{(x-x_0-k_{0x}t )^2}{1+i\bar h\zeta^2 t}+\frac{ik_{0x}}{\bar h}(x-x_0)-ik_{0x}^2t/2\bar h]\rbrace
\end{equation}
\end{widetext}
Here, $\zeta$ is of the order of $(\omega_e/ \bar h)^\frac{1}{2}$ and $k_{0x}=\lambdabar_0/\lambdabar$ where $\lambda$ is the wavelength of the wave packet in the $x$-direction $(\lambdabar=\lambda/ 2\pi)$. We have also:
\begin{equation}
\label{eq38}
\phi(y, t)=[\frac{\beta}{\pi^\frac{1}{2}(1+i\bar h\beta^2t)}]^\frac{1}{2}exp[-\frac{\beta^2}{2}\frac{(y-y_0)^2}{1+i\bar h\beta^2 t}]
\end{equation}
\noindent where for the wave function of the central system in (33) at $t=0$, $\beta$ can be defined as:
\begin{widetext}
\begin{equation}
\label{eq39}
\beta=\lbrace\frac{[(a^2{\omega_{+1}}{\omega_{+2}})(\omega_{-1}+\omega_{-2})+(b^2{\omega_{-1}}{\omega_{-2}})(\omega_{+1}+\omega_{+2})]}{\bar h(a^2\omega_{+1}+b^2\omega_{-1})(a^2\omega_{+2}+b^2\omega_{-2})}\rbrace ^\frac{1}{2}
\end{equation}
\end{widetext}
\noindent All parameters and variables in relations (37) to (39) are dimensionless. For evaluating the behavior of the wave packet after the slits, we assume that the part of the wave function $\psi(x, y, t)$ relative to the points $(x, y)$ (such that $\mid y \mid >b $ ) is reflected towards the negative $x$-regions by the barrier $R$, because no tunneling effect is possible with an infinite potential barrier. So, we assume that the wave packet after the slits, at initial time $t=0$, can be presented as:
\begin{equation}
\label{eq40}
\psi_I(x, y, 0)=\chi_a(x, 0)\phi_I(y, 0)
\end{equation}
\noindent where $\chi_a(x, 0)$ is $\chi(x, 0)$ in (37) with $x_0=a$ and $\phi_I(y, 0)$ is $\phi(y, 0)$ in (38), now defined in two-interval set $I=[d,d+b]\cup [-d'-b',-d']$. Since the initial wave function is separated by its $x$ and $y$ constituent functions and the particle freely moves after the slits, one can deduce that:
\begin{equation}
\label{eq41}
\psi_I(x, y, t)=\chi_a(x, t)\phi_I(y, t)
\end{equation}
\noindent where $\chi_a(x, t)$ is defined in (37) with $x_0=a$. On the other hand, after the slits, the wave function in the $y$-direction evolves as:
\begin{widetext}
\begin{align}
\label{eq42}
\phi_I(y, t)=\frac{1}{2\bar h}\frac{\beta^\frac{1}{2}}{\pi^\frac{5}{4}}\int_ {-\infty}^{+\infty} exp[\frac{i}{\bar h}(p_yy-\frac{p^2_yt}{2})]dp_y \int_ {I} exp[-\frac{i}{\bar h}p_y\xi-\frac{\beta^2}{2}(\xi-y_0)^2]d\xi
\end{align}
\end{widetext}
\noindent where the first integral is the Fourier transform of time evolution of the wave function in the momentum space and the last integral is the wave function of the particle in the momentum space at $t=0$.
Integrating over the variable $p_y$, the above wave function can be presented as:
\begin{widetext}
\begin{align}
\label{eq43}
\phi_I(y, t)=(\frac{\beta}{2\pi^\frac{3}{2}i\bar ht})^\frac{1}{2} exp[y^2\frac{i}{2\bar ht}-y_0^2\frac{\beta^2}{2}]\int_ {I} exp[-\xi^2(\frac{\beta^2}{2}-\frac{i}{2\bar ht})+\xi(y_0\beta^2-\frac{iy}{2\bar ht})]d\xi
\end{align}
\end{widetext}
In the next section, we will analyze two limiting cases of the diffraction pattern, resulting from the wave function in (43).
\subsection{Limiting cases of the problem}
In the following, we consider two different limiting situations:
1) We suppose that the wave packet reaching the slits is narrower than both the slits, i.e.,
\begin{equation}
\label{eq44}
\Delta y=\frac{1}{\beta\sqrt{2}}\ll b, b'
\end{equation}
\noindent The above condition shows that $\beta$ is very large and $b, b' \ll 1$. In this case, from (43) one can show that:
\begin{widetext}
\begin{align}
\label{eq45}
\phi_I(y, t) \phi_I^*(y, t)\cong \frac{\pi^\frac{-3}{2}\beta}{(1+\bar h^2t^2\beta^4)^\frac{1}{2}} exp[{-\frac{\beta^2(y-y_0)^2}{1+\bar h^2t^2\beta^4}}]
[\int_ {(y_0+d')\beta /\sqrt{2}}^{(y_0+b'+d')\beta /\sqrt{2}}exp(-t^2)dt+\int_ {(y_0-d-b)\beta /\sqrt{2}}^{(y_0-d)\beta /\sqrt{2}}exp(-t^2)dt]^2
\end{align}
\end{widetext}
\noindent This shows that $\phi_I \phi_I^*$ is a Gaussian-like distribution \cite{Zecca4, Zecca5}. Thus, the situation described in this case corresponds to an incident wave packet which after the slits, essentially remains undisturbed in its configuration or is reflected toward the negative $x$-direction, according to whether the incoming y-probability distribution is centered with regard to one of the slits or not. In this case, the diffraction pattern, for any value of $\bar h$ between $0.01<\bar h<0.1$ has a Gaussian form. No interference pattern is seen here, since the particle passes through one slit. The result is shown in Fig.1. For simplicity we have assumed that $y_0=0$.
\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{1.jpg}
\caption{Interference pattern for the case in which the incoming wave packet is too narrow with respect to both slits. }
\end{figure}
2) Now, we assume that the wave packet reaching the slits has very large uncertainty, depicted by the y-position probability distribution, i.e.,
\begin{equation}
\label{eq46}
\Delta y=\frac{1}{\beta\sqrt{2}}\gg b, b'
\end{equation}
\noindent From (46), it is evident that $\beta$ should be very small. By setting $\beta^2\approx 0$ in (43) and neglecting the term $i\xi^2 / 2\bar ht$ against $iy\xi / 2\bar ht$ for large values of $y$ (knows as far-field approximation), the integral term can be obtained as:
\begin{equation}
\label{eq47}
\frac{2\bar ht}{y}\lbrace exp[-\frac{iy}{\bar ht}(d+\frac{b}{2})] sin \frac{by}{2\bar ht}+exp[\frac{iy}{\bar ht}(d'+\frac{b'}{2})] sin \frac{b'y}{2\bar ht}\rbrace
\end{equation}
\noindent Assuming that $d=d'$, $b=b'$, from (43) and (47) one gets:
\begin{widetext}
\begin{equation}
\label{eq48}
\phi_I(y, t) \phi_I^*(y, t)\cong \frac{2b^2\beta}{\pi^\frac{3}{2}\bar ht} exp[-\beta^2 y_0^2] \frac{sin^2(by / 2\bar ht)}{(by/2\bar ht)^2} cos^2[\frac{y}{\bar ht} (d+\frac{b}{2})]
\end{equation}
\end{widetext}
As expected, this probability has a maximum at $y_0=0$. If the separation of the slits is of the order of the slit width $(d\cong b)$, the factor containing the \textit{cosine} will be practically negligible and the expression (48) essentially gives the elementary diffraction pattern of a plane wave passing through a single slit. If the separation of the slits is much greater than their width $(d\gg b)$, the relation (48) represents a high-frequency pattern modulated by elementary diffraction fringes. In this latter case, the resulting diffraction pattern is plotted in different situations with $\bar h=0.1$ and $\bar h=0.01$. The case for which $\bar h=0.1$ shows the quantum trait of the system and for $\bar h=0.01$, the pattern describes the classical behavior of the macro-system (see Fig.2). This is in agreement with what we expect for interference fringes of a macroscopic quantum system, when the value of $\bar h$ is sufficiently small ($\bar h < 0.1$) \cite{Zecca4, Zecca5}.
\begin{figure}
{\includegraphics[scale=0.5]{2.jpg}}
{\includegraphics[scale=0.5]{3.jpg}}
\caption{Interference pattern for the case in which the incoming wave packet has a great uncertainty $y$-direction with Left) $\bar h=0.1$ and Right) $\bar h=0.01$.}
\end{figure}
\section{Double-slit diffraction pattern based on SEDS approach}
In SEDS, the Schr\"{o}dinger equation is considered as a rough approximation for a stochastic process, which works well for the average of the trajectories of each state under some definite conditions (for example, for electrons bound in atoms), but not for single trajectories which occurs for the scattering process. A similar restriction holds for the diffraction of a beam of the particles passing through the two slits. The diffraction is due to the standing waves of the zero point field (ZPF) between the edges of the two slits that establishes itself across the clifts and depends on the classical spin motion with constant speed \cite{Zecca5, cavalleri2, cavalleri3}.
As a particle approaches one slit, its precession frequency increases and when it is equal to one of the standing ZPF waves between the two slits, it undergoes a transverse impulse from the ZPF with the following $y$-component of velocity:
\begin{equation}
\label{eq49}
v_y=\frac{\bar h\omega_n}{2c}
\end{equation}
where $c$ is the speed of light and $\omega_n$ is the angular velocity of the particle around the unit vector ($\hat n$) perpendicular to the plane of gyration orbit. The relation (49) is in dimensionless form. These random transversal impulses are maximum when $\omega_n$ coincides with a peak of the ZPF spectrum inside the slits. Notice that the transverse deviations should occur not only when the particle has equal probability to pass through either one or the other of the two slits, but also when it can pass through only one slit \cite{Zecca5}.
For the velocity vector, the deviation angle of the beam $\theta$ is defined as $sin\theta=\langle v_y^2\rangle^{1/2}/v$, where $v$ is the particle speed before and after crossing the slits. Using (49), one can obtain the following dimensionless relation
\begin{equation}
\label{eq50}
sin\theta=\pm \frac{\bar h\omega_n}{2cv}
\end{equation}
where $\bar h\omega_n$ is dimensionless energy per normal modes of the ZPF. The intensity of the deviated beam depends on the spatial density of modes allowed by the slits. The amplitudes of the ZPF waves are spatially uniform in space and zero on the wall of the slits. Consequently, the spatial Fourier transform of the ZPF amplitude is
\begin{equation}
\label{eq51}
F_s=\frac{1}{2b}\int_ {-b}^{b} exp[ik_yy]dy=\frac{sin(k_yb)}{k_yb}
\end{equation}
The corresponding spatial distribution of the energy modes allowed by the slits is proportional to
\begin{equation}
\label{eq52}
\rho_E \propto[sin(k_yb)/k_yb]^2
\end{equation}
which is familiar for ZPF waves and is also equivalent with what can be obtained from (48) for a wide beam (i.e., $\Delta y\gg b$) \cite{Zecca5}. The intensity maxima occur for
\begin{equation}
\label{eq53}
k_y=0 \quad \textrm{and} \quad k_yb=\pi (n+\frac{1}{2}), \quad n=1,2,3,....
\end{equation}
Since $k_y=\omega/c$, one can show that this corresponds to
\begin{equation}
\label{eq54}
sin\theta_m=0 \quad \textrm{and} \quad sin\theta_m=\pm \frac{\bar h\pi}{2bv}(n+\frac{1}{2})
\end{equation}
For a given $r$, where $r$ is the distance of the particle from the nearest edge of the slit and $\omega=v/r$, three maxima should appear: one for the central undeviated beam, and two others for two different points with opposite signs. This resembles the quantum prediction, specially when the open macro-system behaves more classically (see Fig.2 for $\bar h=0.01$). When the order of classicality is low (e.g., $\bar h=0.1$), the interference pattern has more details.
Yet, the interaction of the ZPF waves with the incident beam does not depend on the size of the slits as well as the width of the incident beam itself. So, the diffraction pattern should appear again, if the beam is much narrower than the slit width (i.e., $\Delta y\ll b$). This prediction is totally different with QM, because according to QM, no diffraction should occur in this condition (see the relation (45) and Fig.1). Therefore, the prediction of SEDS for a narrow beam of the particles with respect to the widths of the slits is that for each position of the entering beam, there are two angles of deviation given by (50). The average intensity of the two deviated beams is given by (52). Clearly, there should be always a central, non-deviated beam. By displacing gradually the position of the entering beam and registering the successive pairs of the opposite spots, the complete diffraction pattern can be observed for a wide beam (i.e., $\Delta y\gg b$), as mentioned before \cite{Zecca5,cavalleri1}.
We have drawn in Fig.3 the deviation angle $\theta$ against the parameter $\bar h$ and the widths of the slits $b$. As is clear, when the angle $\theta$ decreases, along with an increase in the widths of the slits $b$, the quantum nature of the macro-system is evanesced corresponding to smaller values of $\bar h$. In this situation, we expect that no interference occurs, but SEDS predict some fringes, as explained before. On the other hand, the same dependence of $\bar h$ could not be observed for the narrow slits, since the deviation angle $\theta$ increases. In this cases, SEDS predicts somehow an invariant interference pattern like for wide slits. Nevertheless, as we see in Fig.2, for an open quantum system, the interference fringes could be variant, depending on the nature of the system (illustrated by $\bar h$). This shows that there is a basic relationship between the quantum behaviour of the macro-system and the double-slit interference pattern predicted by SEDS.
\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{4.jpg}
\caption{The dependence of the deviation angle $\theta$ on the parameter $\bar h$ and the slits width $b$ for a macroscopic quantum system coupled to the environmental degrees of freedom. }
\end{figure}
Consequently, in the SEDS approach the interference fringes could be always present but with fixed patterns, since any particle passing through one of the two slits constantly feels the ZPF waves due to the boundary conditions. In other words, the spatial waves of the ZPF in (51) are always present between the two slits.
\section{Conclusion}
Interference patterns for different quantum systems have been considered for many decades. In recent years, however, the experts have encountered $\it{how}$ the quantum-to-classical transition occurrs when the system shows classical trait. Taking into account the effects of an interacting environment on a quantum harmonic system via a simple oscillating model, we have shown that when the quantumness of the system is evanesced (measured by the parameter $\bar h$ in (29)), interference fringes are diminished in accordance with known patterns observed for macro-molecules (see Fig. 2b) \cite{Nairz}. The environmental effects are not important when the incoming wave packet somehow describes the position state of the system in a given direction (see Fig.1). So, we have now a controllable parameter $\bar h$ by which we can follow and demonstrate the effects of the environment on quantum behavior of the system. This may open new door to the way one can better understand the emergence of classical appearance of the physical world in an interactive manner.
Moreover, we compared the double-slit interference patterns obtained by QM and SEDS for an open macroscopic quantum system. Our results show that, contrary to what expected for closed systems, the diffraction pattern predicted by SEDS is not totally equal to the quantum case when the beam of particles is transversally uniform and much larger than the slit width $b$. In the latter case, the interference pattern shows different fringes for various value of $\bar h$. However, in the case of a beam that is narrow with respect to $b$, the difference is more apparent, because in this situation too, the SEDS theory predicts an interference pattern, while in quantum approach we see no interference. These results show that there is a clear difference between the predictions of QM and those of SEDS for an open macroscopic quantum system.
|
3,212,635,537,504 | arxiv | \section{Introduction}\label{sec:intro}
The use of data driven approaches for material property prediction and material design have become exceedingly common. Because of the dependency between the quality and availability of input data, and the model predictions, different domains of the feature space are predicted with different accuracy levels. The need to identify such domains not only helps us figure out the predictions with high confidence levels, but also provides us with better models to be used for downstream tasks like material design. In this work, we use solubility prediction as a test case to develop an domain of applicability detection method, and provide analysis techniques to gain further insight on identified domains.
Solubility prediction plays a vital role in many disciplines including drug discovery, medicine, fertilizers and energy storage systems. Despite the efforts over many decades, scientists are yet to come up with a highly accurate predictive models for a wide variety of molecules.
The paper is outlined as follows. First we explain the outlier detection method used. Next, we try to identify distinctive properties of outliers using different methods. Finally, we introduce several methods to automatically identify outlier molecules.
In Table \ref{table:curr-preds}, we list the solubility prediction accuracies obtained by different machine learning models for our dataset.
\begin{table}[tbh]
\centering
\small
\caption{CV scores obtained for PNNL dataset by recent models.}
\begin{tabular}{lrr}
\toprule
method & rmse & r2 \\
\midrule
etr & 1.06 & 0.78 \\
cbm & 1.06 & 0.78 \\
rf1 & 1.07 & 0.78 \\
xgbm & 1.09 & 0.77 \\
lgbm & 1.10 & 0.77 \\
afpm & 1.11 & 0.76 \\
svm & 1.13 & 0.75 \\
gnn-fgrp & 1.15 & 0.75 \\
gnn-reg & 1.15 & 0.75 \\
mdm & 1.15 & 0.74 \\
gbr & 1.20 & 0.72 \\
\bottomrule
\end{tabular}
\label{table:curr-preds}
\end{table}
\section{Data}
Our dataset is composed of data in three prior datasets: PNNL Organic Solubility dataset, AQUASOL and Cui dataset.
Brief description of data.
\section{Outlier Detection Methods}\label{sec:methods}
\subsection{\textbf{Outlier Detection}}\label{sec:ol-detect}
We compare and combine two different approaches for outlier detection - structure-only outlier detection and solubility-aware outlier detection. These two approaches have different purposes. The detection of structural outliers can be used to identify molecules which appear structurally unusual without regard to the effect of that deviation on the solubility prediction. Such molecules are more likely to be difficult for the model due to lack of training examples, but are not necessarily so. The solubility-aware outlier detection on the other hand identifies outliers for which solubility prediction is a challenge. Such challenge may occur because of unusual molecular structure, measurement noise in the solubility values, error in the solubility annotations, or difficulties distinguishing the effect of small structural changes on solubility. By combining both structure-only and solubility-aware methods, we can distinguish some of these different causes for molecular property prediction errors.
\subsubsection{Structure-Only}
Describe and cite the main existing methods.
ABOD uses angles rather than distances between a data point and the other points\cite{Kriegel_2008}. In the CBLOF method an outlier factor is caclulated which is based on the distance to clusters identified using a clustering algorithm\cite{He_2003}. Isolation Forest method determines outliers by isolating each datapoint by partitioning the datapoints using a randomly chosen attribute. Outliers can be quickly isolated compared to inliers. Outliers are identified by finding the path length in the isolation tree\cite{Liu_2008}.
\subsubsection{Solubility-Aware}
Our solubility-aware outlier detection method is based on an algorithm described in \citet{Cao2017}. Here, we train 1000 machine learning models by splitting the entire dataset into random train and test sets. For each trained model, the errors made on the molecules in the test set are collected. Consequently, multiple error measurements exist for a given molecule. Molecules associated with large mean error, large variability in the error, or both can be identified as outliers.
The ML model we used in this work is ExtraTreesRegressor as implemented in the \textit{scikit-learn} package~\footnote{https://scikit-learn.org/stable/}. Training was performed using the default parameters, each time taking a different random train and test split. We then find the mean and standard deviation of prediction errors for the 22,100 molecules in our dataset. We find that each molecule has been predicted on average 100 times with a standard deviation of about 10 times. The minimum number of predictions made for any molecule is 63 providing sufficient signal to quantify the prediction error. In Figure \ref{fig:ol} we plot the mean and the standard deviation of the prediction errors.
Next, we have to define thresholds to separate outliers and inliers in this space. The method given in ref, involves dividing the mean error - standard deviation of the error space into four quadrants as shown in Figure \ref{fig:ol} a. Different from the original method, we first scale the mean errors and standard deviation of the errors so that these values are between 0 and 1. Then an arbitrary percentile value is chosen for mean error and standard deviation of the error.
The points which are enclosed by these thresholds correspond to inlier molecules. We then find the cross validated R2 score corresponding to the predicted and actual solubilities of these molecules. We repeat this process while reducing the percentile value from 95 to 65.
An alternative is to choose a percentile value for the scaled outlier-ness as shown in Figure \ref{fig:ol} b. Outlier-ness is defined as the euclidean distance to a point in the mean error - standard deviation in the error space. In this case the threshold value for outlierness is varied from 95 to 65 and the cross validated R2 of the resulting inliers were found. We find that this method chooses better inliers than the original method.
\textcolor{orange}{describe this method.}
\begin{figure*}[!t]
\centering
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[width=\textwidth]{images/mean_sdev_th.png}
\end{subfigure}
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[width=\textwidth]{images/olness_th.png}
\end{subfigure}
\caption{Mean error versus the standard deviation of the error for each molecule in the entire dataset. Molecules associated with large mean and standard deviation of the error are considered as outliers.}
\label{fig:ol}
\end{figure*}
\section{Results}
\subsection{Model Performance}
Next, we observe how the performance of the solubility prediction models changes after removing outliers selected using different methods. In Table \ref{table:cv_nol_diff_methods}, we list the accuracies obtained by existing anomaly detection methods and the solubility-aware outlier detection method. The number of outliers removed is approximately 15\% of the total number of molecules. It should be noted that it is not fair to compare the performance of the solubility-aware method with the structure-only methods in terms of model predictive performance, because the solubility-aware method used solubility error information in its outlier selection approach. We are not aiming to maximize the solubilitiy prediction performance using this method, which would not be possible to apply to new molecules which don't have solubilty labels. Instead, we aim to understand the types of molecules which are detected as outliers to understand the existing challenges in solubility prediction models.
We find that the removal of structural outliers has a significant effect on the performance of the predictive models, with an improvement from and RMSE of 1.01 to around 0.90. This shows that property prediction model performance is strongly driven by the structural diversity present in the test set used to evaluate the model performance. With the extra information available to it, the solubility-aware method achieves an RMSE of 0.56, showing that structural diversity alone does not explain the challenges in predictive performance that are experienced by the predictive models. Identifying the additional factors within the dataset that drive this performance gap will be crucial to achieving significant improvements in property prediction.
\begin{table}[tbh]
\centering
\footnotesize
\caption{CV scores obtained for inliers predicted by different anomaly detection methods. ExtratreesRegressor has been used as the ML model. \textcolor{red}{Add performance on inliers only when using the full training set (no outlier removal) }. \textcolor{orange}{don't quite understand this part.}
}
\begin{tabular}{lllr}
\toprule
methods & RMSE & R2 & ol\% \\
\midrule
pnnl-method & 0.56 \pm 0.01 & 0.92 \pm 0.0 & 15.58 \\
Isolation Forest & 0.86 \pm 0.0 & 0.82 \pm 0.0 & 15.58 \\
Principal Component Analysis (PCA) & 0.87 \pm 0.02 & 0.82 \pm 0.01 & 15.58 \\
One-class SVM (OCSVM) & 0.88 \pm 0.01 & 0.81 \pm 0.01 & 15.58 \\
K Nearest Neighbors (KNN) & 0.88 \pm 0.03 & 0.82 \pm 0.01 & 14.18 \\
Angle-based Outlier Detector (ABOD) & 0.89 \pm 0.02 & 0.83 \pm 0.0 & 16.86 \\
Histogram-base Outlier Detection (HBOS) & 0.89 \pm 0.02 & 0.81 \pm 0.01 & 15.58 \\
Feature Bagging & 0.89 \pm 0.02 & 0.82 \pm 0.01 & 14.66 \\
Local Outlier Factor (LOF) & 0.89 \pm 0.03 & 0.82 \pm 0.01 & 14.61 \\
Minimum Covariance Determinant (MCD) & 0.89 \pm 0.01 & 0.82 \pm 0.0 & 15.58 \\
Cluster-based Local Outlier Factor (CBLOF) & 0.89 \pm 0.02 & 0.81 \pm 0.01 & 15.58 \\
all\_data & 1.01 \pm 0.02 & 0.8 \pm 0.01 & - \\
\bottomrule
\end{tabular}
\label{table:cv_nol_diff_methods}
\end{table}
For the solubility-aware method, it is important to test the robustness of our results to selection of the machine learning model used to detect them. In Table \ref{table:cv-nol-models}, we show cross validated accuracies obtained by different machine learning models for the selected inliers. Note that the outliers and inliers were separated based on the prediction errors made by an ExtraTreesRegressor. This shows that similar performance improvements are shown across different machine learning modeling approaches even when a particular model is used for the outlier selection process.
\begin{table}[tbh]
\centering
\small
\caption{CV scores obtained using different ML models for inliers determined by solubility-aware method.}
\begin{tabular}{l|rr}
\toprule
\textbf{\small{Model}} & \textbf{\small{RMSE}} & \textbf{\small{$R^2$}} \\
\midrule
Random Forest & A & A \\
MLP & A & A \\
GNN & A & A \\
LGB & 0.61 & 0.91 \\
\bottomrule
\end{tabular}
\label{table:cv-nol-models}
\end{table}
\subsection{Outliers and Inliers}
In this section, we are going to analyse the proprties of outliers and inliners in detail.
There are two main reasons for a molecule to become an outlier: (1) its structure deviates vastly from most of the other structures in the dataset, (2) its solubility cannot be predicted by applying the same rules that are valid for most of the other molecules. This may occur when molecules have erroneous solubility labels in the dataset, when there is noise in the solubility measurements, or when structurally similar molecules have differing solubilities. We attempt to separate these two types of outliers in order to better understand the root causes for each type.
To identify structural outliers, we use the 10 unsupervised anomaly detection methods implemented in \textit{pyod}~\footnote{https://pyod.readthedocs.io/en/latest/} and listed in Table \ref{table:cv_nol_diff_methods}. The total number of outliers detected by all the methods is 8283 (\textcolor{red}{have to check whether this number is reproducible - if we find the outliers again, do we get the same outliers?} \textcolor{dgreen}{this number is not exactly reproducible, because some of the methods need random initial seeds. But the there should not be large variations in this number when we repeat the experiment. i will check this by repeating several times.}). The number of outliers that all the models agree to be as outliers is 367. Then we found the molecules that are common to the outliers detected by all the pyod methods and outliers detected by the solubility-aware method. We identify these 2347 common molecules as the \textit{structural outliers}. We should note that because we leverage the combination of the structure-only methods and the solubility-aware method, that these molecules are structurally unusual in a way that impacts the solubility prediction performance. If molecules are structurally unusual but the models still achieve sufficient predictive accuracy, they are not included in the outlier set.
\textcolor{red}{How many moleculare are detected by the unsupervised methods, but are not in the solubility-aware set? This may be an interesting set to look at.}There are 5936 molecules detected by the unsupervised methods, but are not in the solubility-aware set. There are 968. Any outliers detected by the solubility-aware method but which are not detected by the unsupervised methods are considered to be \textit{data outliers}. \textcolor{red}{How many data outliers?} There are 968 data outliers.
\subsubsection{Structural outliers and domain of applicability}
We aim to understand the relationship between outlier molecules and their interpretable molecular descriptors and structural properties. In Table \ref{table:ol_nol_diff1} we show the percentage of outliers and inliers having certain properties which we intuitively expect may distinguish outliers from inliers. For example, we might expect the number of atoms in the molecules as well as their complexity to be significant factors. We find that, indeed, outlier molecules are more likely to be large or complex compared with inlier molecules.
\begin{table}[tbh]
\centering
\small
\caption{Number of molecules in outliers and inliers, having different properties. The numbers are given as a percentage of the total number of molecules in each group. [updated]}
\begin{tabular}{l|r|r}
\toprule
Property & Outlier \% & # Inlier \% \\
\midrule
\ nAtom > 100 & 5.95 & 0.69 \\
\ BertzCT>1500 & 6.33 & 1.42 \\
\ chiral centers & 14.9 & 9.79 \\
\bottomrule
\end{tabular}
\label{table:ol_nol_diff1}
\end{table}
We can extend the idea in Table \ref{table:ol_nol_diff1} by studying the distributions of outliers and inliers of different molecular features. Here, we first calculate the difference between the extreme values in the outlier and inlier distributions. We aim to identify features for which the range of the values is much larger for outliers than inliers. For a given feature, the extreme (maximum or minimum) values could be either on the positive or negative side of the distribution. \textcolor{red}{Is it possible for it to be on both sides of the distribution?}. To find out in which direction the extreme values are, we calculate distance difference for both sides and chose the side corresponding to the maximum distance. The features are then ordered in the descending order of the distance. The features are standard scaled so that the distances are comparable across features and are not driven by the natural magnitude of the feature values. Figure \ref{fig:fbf} show the distributions of outliers and inliers of some features with largest distance values. We can identify these features as those that separate outliers and inliers. In other words, these features define the \textit{domain of applicability}. DOA can help improve prediction accuracy for a subset of molecules. \textcolor{red}{Does the accuracy improve the inliers alone? Or the does the accuracy improve because it no longer has to predict for the outliers? Can you look at the RMSE and R2 for inliers in the test set before and after removing outliers from the training set?}
In Table \ref{tbl:doa} we show the top 20 molecular descriptors that define the domain of applicability for our dataset. The `direction' says whether a descriptor's value should be less than or equal, or greater than or equal to the threshold value. For example, SsPH2 value of a molecule should be less than or equal to 0 in order for it to be in the domain of applicability. SsssB value of molecules in the domain of applicability should be $\geq$ 0 \textcolor{red}{Describe how threshold is selected}.
\begin{table}[!tbh]
\centering
\caption{Domain of applicability.}
\begin{tabular}{lrlrr}
\toprule
descriptor & threshold & direction & mean\_oln & \#ols \\
\midrule
NssssSn & 0.00 & == & 0.31 & 18 \\
Mpe & 0.88 & >= & 0.28 & 65 \\
IC5 & 6.31 & <= & 0.23 & 50 \\
n10AHRing & 0.00 & == & 0.23 & 8 \\
bpol & 55.38 & <= & 0.20 & 612 \\
n5FARing & 0.00 & == & 0.20 & 2 \\
SsssNH & 0.00 & >= & 0.20 & 3 \\
NsAsH2 & 0.00 & == & 0.19 & 1 \\
n5 & 0.00 & == & 0.19 & 582 \\
ATSC0v & 4387.78 & <= & 0.19 & 759 \\
TIC5 & 419.22 & <= & 0.19 & 730 \\
nFRing & 2.00 & <= & 0.19 & 30 \\
n6 & 0.00 & == & 0.18 & 229 \\
fMF & 0.72 & <= & 0.18 & 6 \\
nG12aRing & 0.00 & == & 0.18 & 2 \\
NdssSe & 0.00 & == & 0.18 & 2 \\
C1SP2 & 5.00 & <= & 0.18 & 115 \\
ATS1m & 7854.83 & <= & 0.18 & 845 \\
nRing & 6.00 & <= & 0.18 & 228 \\
SsPH2 & 0.00 & == & 0.18 & 1 \\
\bottomrule
\end{tabular}
\label{tbl:doa}
\end{table}
\textcolor{orange}{A breif discussion about some of the features in Table \ref{tbl:doa}}
\textcolor{orange}{We can show the molecules that have been removed at each step of the feature based filtering process (for some number of steps). And probably give a qualitative structural property or chemistry based discussion.}
\textcolor{red}{We should try some other methods of feature selection. For example, rather than comparing the extreme values only you could compare the full distributions using the KL divergence (or similar metric) and find the features where the distributions are most different between outliers and inliers.}
We can remove extreme values corresponding to the first $n$ features, build models using the rest of the molecules and check how the prediction accuracy changes. As shown in Figure \ref{rmse_fbf} the filter based filtering results in increase in the prediction accuracy. This shows that this method can be used to separate easy-to-predict molecules from difficult-to-predict molecules. This allows us to down select a set of molecules for which the solubilities can be predicted with high accuracy. One issue with this method is that a large amount of data has to be sacrificed to achieve a considerable increase in the accuracy.
\begin{figure}[!htb]
\centering
\includegraphics[width=1\textwidth]{images/sup/doa/fbf_sup.png}
\caption{Decrease in the prediction error with feature based filtering. \textcolor{red}{In addition to comparing with removing random molecules, let's compare with selecting random features and removing extreme values of those features.}
\textcolor{orange}{have to update this figure}
}
\label{rmse_fbf}
\end{figure}
\subsubsection{Structural Outlier Clustering}
To gain further insight on the outliers, we use K-means clustering to identify groups of structurally similar outliers. This method allows us to identify features that might have caused these molecules to become outliers.
In order to reduce the effect of the random nature of the clustering methods, we performed clustering several times and selected the molecules that get grouped together most frequently. This method is explained further in supporting information. In Figure \ref{fig:mol-clusters} we show the molecules belonging to each cluster found in this way.
For some of the clusters, it is easy to recognize their distinguishing characteristics (eg: Molecules having long carbon and fluorinated chains, circular ring systems). To identify the prominent features of others, we follow a method
described at \textcolor{orange}{add the reference}. Using a random forest classification model, which has the ability to identify feature importances, we train models to separately identify members of a selected cluster from the members of all the other clusters combined. The resulting feature importance values tell us which features played a major role in helping the model distinguish the selected cluster from the other clusters.
\begin{figure*}[!htb]
\centering
\includegraphics[width=1\textwidth]{images/sup/clusters/image_groups.png}
\caption{Clusters of outlier molecules.}
\label{fig:mol-clusters}
\end{figure*}
\begin{figure*}[!htb]
\centering
\includegraphics[width=1\textwidth]{images/sup/clusters/cluster_fimp.png}
\caption{Clusters feature importance}
\label{fig:mol-clusters-2}
\end{figure*}
Xp-3dv depends on valence electrons. BertzCT is a measure of the complexity of molecules. These have likely become outliers because of their large size.
The second group mainly consists of salt structures. JGI1 is the mean topological charge of order 1.
AATSC0d: averaged and centered moreau-broto autocorrelation of lag 0 weighted by sigma electrons,
ATSC1d:centered moreau-broto autocorrelation of lag 1 weighted by sigma electrons.
By looking at the feature distributions, we cannot find distinguishing features that uniquely identify clsuter3.
Cluster 4 also consists of salt structures. Some molecules contain metal atoms. TSRW10 is total walk count (leg-10, only self returning walk).
For cluster 5,
ATS1dv: moreau-broto autocorrelation of lag 1 weighted by valence electrons
ATS6v:moreau-broto autocorrelation of lag 6 weighted by vdw volume
ATSC0dv:centered moreau-broto autocorrelation of lag 0 weighted by valence electrons
GGI8: 8-ordered raw topological charge.
By looking at the feature distributions, we cannot find distinguishing features that uniquely identify clsuter6.
Cluster 7 consists of 2 molecules.
Tmo main distinguishing features of clsyter 8 are number of sCH3 and moreau-broto autocorrelation of lag 4 weighted by ionization potential. Outliers tend to have a large number of sCH3 and ionization potential (is this true?)
number of sssCH, number of aromatic rings, and 3-ordered neighborhood information content are the distinguishing features for cluster 9.
Cluster 10 can be identified using BertzCT, SRW06, number of double bonds in kekulized structure(nBondsKD) and number of aromatic bonds in non-kekulized structure(nBondsA).
Molecules in cluster 11 seem to contain long chains. number of ssCH2 (NssCH2), sum of ssCH2 and 1-ordered complementary information content are distinguishing features.
A noticeable feature of cluster 12 molecules is the presence of I atoms. In the feature importance plot, SsI and nI are sum of I atoms connected with single bonds and the total number of I atoms in the molecule respectively.
\subsubsection{Data outliers}
Data outliers are not large, complex molecules. However, they seem to have a very similar counterpart more often than the structural outliers.
\begin{table}[tbh]
\centering
\small
\caption{Number of molecules in outliers, inliers, structural outliers and data outliers having different properties. The numbers are given as a percentage of the total number of molecules in each group. \textcolor{red}{Add inliers.}}
\begin{tabular}{lrr}
\toprule
Property & Structural OL \% & Data OL \% \\
\midrule
Chiral centers & 15.72 & 8.47 \\
nAtom > 100 & 8.14 & 0.00 \\
BertzCT > 1500 & 9.20 & 0.00 \\
Have an isomer counterpart & 13.98 & 16.01 \\
\bottomrule
\end{tabular}
\label{table:CV_LIT}
\end{table}
Results shown in Figures \ref{fig:hyp-sim1} and \ref{fig:hyp-sim2} do not prove that having a very similar counterpart does not make a particular molecule an outlier. Because there are many inliners which are very similar to another molecule. However, data outliers are more likely to have a very similar counterpart compared to structural outliers.
\textcolor{red}{Check whether data outliers are more likely to from certain sources than structural outliers or inliers}
\begin{figure*}[!t]
\centering
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[width=\textwidth]{images/n_similar_pairs.png}
\end{subfigure}
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[width=\textwidth]{images/diff_vs_simth.png}
\end{subfigure}
\caption{ (a) Distribution of average solubility differences corresponding to molecule pairs that are >0.99 similar. (b) Mean solubility difference between nearest neighbors versus the similarity threshold.}
\label{fig:ol}
\end{figure*}
\begin{table}[tbh]
\centering
\small
\caption{References corresponding to data outliers with largerst outlierness}
\begin{tabular}{lr}
\toprule
ExpRef & olness \\
\midrule
LETINSKI.DJ ET AL. (2002) & 0.42 \\
Bulletin\_de\_la\_Societe\_Chimique\_de\_France;\_(195... & 0.39 \\
Staverman. A.J.. Ph. D. thesis. University of L... & 0.38 \\
YAWS.CL ET AL. (2001) & 0.37 \\
COATES.M ET AL. (1985) & 0.35 \\
Schreiber;\_Chemische\_Berichte;\_vol.\_57;\_(1924);... & 0.33 \\
MERCK INDEX (2013) & 0.27 \\
SEIDELL.A (1941) & 0.27 \\
TOMLIN.C (1997) & 0.26 \\
PARRISH.CF (1983) & 0.26 \\
\bottomrule
\end{tabular}
\label{table:ref_olness}
\end{table}
\section{Predicting Outliers}
In this section, we propose two supervised outlier detection methods. One is based on training a classifier and the other on the outlierness associated with molecules.
To demonstrate these processes, we first split our dataset into train and test folds. Prediction errors for each molecule in the train set are calculated as explained in the section A. Next we used the same method which we used in section A to separate outliers and inliers. Outlier percentage we consider is 20\% of the train-validation set.
\subsection{Training a classifier}
Once we have identified the outliers and inliers in the train, we can train a classifier to detect outliers in the test fold. We arbitrarily used ExtraTreesClassifier as the classifier. Once the outliers are detected, we can remove them from train and test folds and train a regression machine learning method to predict the solubilities of the remaining inliers. In Table \ref{table:remove-ols-pnnl}, we list the solubility prediction accuracies obtained when outliers are removed from both train and test sets, only from test set and only from the train set.
\begin{table}
\centering
\small
\caption{Solubility prediction accuracies when outliers are removed from train or test and train+test folds.}
\begin{tabular}{lrr}
\toprule
Outliers removed from & RMSE & R2 \\
\midrule
Train+Test & 0.89 & 0.81 \\
Test only & 0.85 & 0.82 \\
Train only & 1.19 & 0.73 \\
\bottomrule
\end{tabular}
\label{table:remove-ols-pnnl}
\end{table}
In order to compare our method with the supervised outlier detection methods, we removed outliers detected using these from train or test sets and predicted solubility using the ExtraTreesRegressors. It is somewhat complicated to compare the solubility prediction accuracies obtained by our method with those from unsupervised outlier detection methods. Unsupervised methods do not use solubility values. Therefore, we can use the entire dataset to select a certain percentage of outliers. However, our method relies entirely on the train set. Our classifier tries to predict the outliers in the unseen test data, based on the separation of outliers and inliers in the train set. Suppose that we decided to find $x\%$ of outliers in the train set using our method. If we now predict $y\%$ of outliers in the test set, the total number of outliers detected by our method is $x+y\%$. Therefore, we should also find $x+y\%$ of outliers using the unsupervised methods in order to do a fair comparison. Number of outliers from the entire dataset and the outliers in the test set are given in Table \ref{table:remove-ols-from-test}. The percentage of outliers detected in train-validation set is 19.13\% and using the classification method we found 0.0103\% outliers in the test set. The total number of outliers in 20.16\% of the entire set. Therefore, for unsupervised methods, we also used 20.16\% as the percentage of outliers we want to find. In Table \ref{table:remove-ols-from-test} we list the solubility prediction accuracies obtained after removing outliers from the test set and train+test sets.
Interestingly, we observe that removing outliers from train set harmful for solubility prediction for all the methods. However, when outliers are removed only from the test set, our method performs better than the unsupervised methods. We also notice that the number of outliers detected in the test set by our method is considerably lower than the outliers detected by unsupervised methods.
\begin{table*}
\centering
\small
\caption{RMSE and R2 for inlier test molecules selected by different methods.
K Nearest Neighbors (KNN), Principal Component Analysis (PCA), One-class SVM (OCSVM),Isolation Forest (IF), Angle-based Outlier Detector (ABOD), Cluster-based Local Outlier Factor (CBLOF), Feature Bagging (FB), Histogram-base Outlier Detection (HBOS), Minimum Covariance Determinant (MCD), Local Outlier Factor (LOF)
}
\begin{tabular}{lrr|rr}
\toprule
& \multicolumn{2}{c|}{Remove from test only} &\multicolumn{2}{c}{Remove from test+train} \\
meth & rmse & r2 & rmse & r2 \\
\midrule
pnnl & 0.85 & 0.82 & 0.89 & 0.81 \\
K Nearest Neighbors (KNN) & 0.86 & 0.84 & 0.88 & 0.83 \\
One-class SVM (OCSVM) & 0.86 & 0.81 & 0.88 & 0.80 \\
Isolation Forest & 0.86 & 0.80 & 0.88 & 0.80 \\
Principal Component Analysis (PCA) & 0.88 & 0.80 & 0.88 & 0.80 \\
Local Outlier Factor (LOF) & 0.89 & 0.82 & 0.91 & 0.81 \\
Histogram-base Outlier Detection (HBOS) & 0.89 & 0.80 & 0.91 & 0.79 \\
Angle-based Outlier Detector (ABOD) & 0.89 & 0.82 & 0.90 & 0.82 \\
Feature Bagging & 0.90 & 0.82 & 0.90 & 0.82 \\
Minimum Covariance Determinant (MCD) & 0.90 & 0.82 & 0.89 & 0.82 \\
Cluster-based Local Outlier Factor (CBLOF) & 0.90 & 0.79 & 0.91 & 0.79 \\
Average KNN & 0.95 & 0.81 & 0.98 & 0.80 \\
\bottomrule
\end{tabular}
\label{table:remove-ols-from-test}
\end{table*}
When we remove outliers detected by our method and supervised methods, we observe that the prediction accuracies can be further increased (see Table \ref{table:rem-ols-pnnl-pyod}).
\begin{table}
\small
\caption{RMSE and R2 when both pnnl and pyod outliers are removed from the test set.
}
\begin{tabular}{lrrr}
\toprule
Method & RMSE & R2 & \#ol\_test \\
\midrule
K Nearest Neighbors (KNN) & 0.77 & 0.84 & 396 \\
Angle-based Outlier Detector (ABOD) & 0.78 & 0.85 & 468 \\
Local Outlier Factor (LOF) & 0.78 & 0.84 & 420 \\
Feature Bagging & 0.78 & 0.84 & 424 \\
Minimum Covariance Determinant (MCD) & 0.80 & 0.84 & 421 \\
Average KNN & 0.81 & 0.83 & 296 \\
Cluster-based Local Outlier Factor (CBLOF) & 0.81 & 0.81 & 418 \\
One-class SVM (OCSVM) & 0.82 & 0.81 & 408 \\
Principal Component Analysis (PCA) & 0.82 & 0.81 & 403 \\
Histogram-base Outlier Detection (HBOS) & 0.82 & 0.81 & 382 \\
Isolation Forest & 0.83 & 0.81 & 376 \\
\bottomrule
\end{tabular}
\label{table:rem-ols-pnnl-pyod}
\end{table}
In Table \ref{table:cls-A}, we show the difference between outliers detected by the our method and the unsupervised methods in terms of number of atoms and BertzCT.
\subsection{Feature based filtering}
Another method to remove outliers is by leveraging the domain of applicability information. As we did with the full dataset in section A, we can use only the train set to find feature thresholds that define the domain of applicability. Then we can remove the test molecules which do not fall into the domain of applicability and use the rest to make predictions.
In Table \ref{table:doa-train}, we list the thresholds corresponding to top features which define the domain of applicability arrived at using the train set.
\begin{table}
\caption{Domain of applicability determined using the train set.}
\begin{tabular}{lrlr}
\toprule
prop & th & direction & mean\_oln \\
\midrule
Mpe & 0.87 & >= & 0.36 \\
IC2 & 5.45 & <= & 0.30 \\
NssssSn & 0.00 & == & 0.30 \\
IC3 & 6.03 & <= & 0.26 \\
n10HRing & 0.00 & == & 0.25 \\
n5FRing & 0.00 & == & 0.24 \\
bpol & 55.38 & <= & 0.24 \\
nFRing & 2.00 & <= & 0.23 \\
ATSC0v & 4387.78 & <= & 0.22 \\
n5 & 0.00 & == & 0.22 \\
TIC5 & 419.97 & <= & 0.22 \\
nRing & 6.00 & <= & 0.22 \\
nG12aHRing & 0.00 & == & 0.22 \\
NdssSe & 0.00 & == & 0.22 \\
n6 & 0.00 & == & 0.22 \\
C1SP2 & 5.00 & <= & 0.22 \\
fMF & 0.72 & <= & 0.21 \\
ATS1m & 7854.83 & <= & 0.21 \\
ATSC1dv & 106.88 & <= & 0.21 \\
\bottomrule
\end{tabular}
\label{table:doa-train}
\end{table}
Figure \ref{fig:fbf-test} shows how the solubility prediction accuracy improves as molecules that are not inside the domain of applicability are removed from the test set.
\begin{figure}[!htb]
\centering
\includegraphics[width=1\textwidth]{images/unsup/doa/fbf_sup.png}
\caption{How test RMSE reduces.}
\label{fig:fbf-test}
\end{figure}
\section*{Availability of data and materials
The source code is available on GitHub at https://github.com/pnnl/solubility-prediction-paper. The data set prepared by \citet{2021Gao} is accessible at https://figshare.com/s/6258a546a27a2373bf2a.
\section*{Author Contributions}
\section*{Conflicts of interest}
There are no conflicts to declare
\section*{Acknowledgements}
This work was supported by Energy Storage Materials Initiative (ESMI), which is a Laboratory Directed Research and Development Project at Pacific Northwest National Laboratory (PNNL). PNNL is a multiprogram national laboratory operated for the U.S. Department of Energy (DOE) by Battelle Memorial Institute under Contract no. DE- AC05-76RL01830.
\begin{acknowledgement}
Please use ``The authors thank \ldots'' rather than ``The
authors would like to thank \ldots''.
The author thanks Mats Dahlgren for version one of \textsf{achemso},
and Donald Arseneau for the code taken from \textsf{cite} to move
citations after punctuation. Many users have provided feedback on the
class, which is reflected in all of the different demonstrations
shown in this document.
\end{acknowledgement}
\begin{suppinfo}
This will usually read something like: ``Experimental procedures and
characterization data for all new compounds. The class will
automatically add a sentence pointing to the information on-line:
\end{suppinfo}
\subsection{Domains of Applicability}
\textcolor{nc}{To demonstrate how we can use the knowledge on in-doman and out-of-domain molecules gain more insight about them, we discuss two examples. In the first case, after visual inspection of some outliers, we hypothesized that the presence of intra-molecular Hydrogen bonds could be a cause for the existence of some data outliers. Kuhn and coworkers have provided a criterion to intra-molecular hydrogen bonds in organic molecules \cite{Kuhn2010}. We used the same criterion to detect the molecules with intra-molecular hydrogen bonds in our dataset.}
\textcolor{nc}{Another hypothesis is that as the length of the carbon chain increases, there is a possibility for them to fold. This phenomenon is observed in polymers and protein structures. Molecules with folded carbon chains can result in low solubilities. Such folding can occur in molecules when there are rotatable bonds. From Figure \ref{fig:feature-imp-difference-expl}, we can see that molecules in Group 5 consists of more rotational bonds compared to other structures. Out of the molecules we considered to obtain clusters in Figure \ref{fig:ood-groups}, 4\% of Group 10 molecules are inliers, 13\% are outliers, 4\% are structural outliers,
7\% are data outliers and 2\% are structural anomalies. Even though not conclusive, we think that this kind of analysis using the molecules in domains and subdomains can be helpful to figure out the causes of current limitations in our models.
}
\begin{figure*}[!t]
\centering
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{images/intraH/example_ab.png}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{images/intraH/example_c.png}
\caption{}
\end{subfigure}
\caption{Examples of molecules in which intra-molecular Hydrogen bonds can exist.}
\label{fig:}
\end{figure*}
\begin{table}[!htb]
\caption{Number of molecules with possible intra-molecular Hydrogen bonding sites
as percentage of all the train data.
}
\centering
\begin{tabular}{lc}
\toprule
Data type & \% of IntraH \\
\midrule
STR.OL & 0.13 \\
DATA.OL & 0.28 \\
STR.ANMLY & 0.05 \\
ALL.OL (NEW) & 0.46 \\
ALL.IL (NEW) & 0.57 \\
ALL.OL (OLD) & 0.41 \\
ALL.IL (OLD) & 0.63 \\
\bottomrule
\end{tabular}
\label{table:}
\end{table}
However, to confirm the validity of this test, we should have solubility values for the chosen `test' set. Hence, for this purpose, we used the PNNL test set which was not used in the DoA determining process.
\section{Introduction}\label{sec:intro}
Solubility prediction is an inevitable task in many disciplines including drug discovery, medicine, fertilizers and energy storage systems. Despite the efforts over many decades, scientists are yet to come up with a highly accurate prediction model for a wide variety of molecules. Even though there are studies that report r2 scores around 0.93, these results have been obtained for a small subset of the molecules for which the solubility data are available. In our previous work we used deep learning methods to predict solubilities of the molecules in the largest and the most complex (in terms of the structural properties) solubility dataset up to date. The best accuracy we could achieve for logS is rmse = 1.05, and r2= 0.77. One of the reasons for not being able to achieve high accuracies could be the lack of training data, particularly for some solubility ranges. In this work we try explore another possibility; are there some molecules that are intrinsically difficult to be predicted? If so, what are the characteristics of these molecules? Can we explain why the solubilities of these molecules are difficult to predicted? Can we filter out these molecules using machine learning?
The paper is outlined as follows. First we explain the outlier detection method used to identify molecules of which the solubilities are difficult to predicted. Next, we try to identify distinctive properties of outliers using different methods. Finally, we introduce several methods to automatically identify outlier molecules.
\section{Methods}\label{sec:methods}
\subsection{Outlier Detection}\label{sec:ol-detect}
Outlier detection was performed based on a method explained in \citet{Cao2017}. Here, N machine learning models are trained after splitting the whole data set into random train, validation and test sets. For each trained model, the errors made on the molecules in the test set are collected. Consequently, multiple error measurements exist for a given molecule. For each molecule, we can calculate the mean and the standard deviation of the error. In Figure \ref{fig:ol} we plot these two parameters for every molecule in the dataset. Molecules associated with large mean error and standard deviation in the error can be identified as outliers. We can define arbitrary thresholds to separate outliers from non-outliers. We chose threshold such that the R\textsuperscript{2} score calculated using non-outliers is at least 0.9. This threshold results in 2002 outlier molecules and B non-outlier molecules.
\begin{figure*}[!htb]
\centering
\includegraphics[width=.8\textwidth]{images/oldetect_main.png}
\caption{Mean error versus the standard deviation of the error for each molecule in the entire dataset. Molecules associated with large mean and standard deviation of the error are considered as outliers.}
\label{fig:ol}
\end{figure*}
\subsection{Differences between outliers and non-outliers}
Outlier detection alone is not very usefull unless we cannot identify the differences between outliers and non-outliers in terms of molecular properties. In this section we discuss several methods to analyse structural characteristics of outliers.
\subsubsection{Using Features Correlated with Outlierness}
We can quantify the ``outlier-ness'' of a molecule by calculating the distance from the origin to each point in Figure \ref{fig:ol}. We can then find which molecular properties are correlated with the outlierness. In Figure \ref{fig:ol_corr} we show the five most postively and negatively correlated features with outlierness. (As the correlation values are very low may be we should not talk too much about this.) As Lipinski is an descrete variable, it's correlation cannot be easily interpreted. Difficulty to predict low solubilities and the solubilities has been pointed out in previous work. n3 is the number of atoms in the 3rd concentric layer around the centroid of the molecule. (we can use these top correlated features for the feature-based filtering process.)
\begin{figure*}[!t]
\centering
\begin{subfigure}[b]{0.8\textwidth}
\centering
\includegraphics[width=\textwidth]{images/pos_corr_olness.png}
\end{subfigure}
\begin{subfigure}[b]{0.8\textwidth}
\centering
\includegraphics[width=\textwidth]{images/neg_corr_olness.png}
\end{subfigure}
\caption{Positively and negatively correlated features with outlierness}
\label{fig:ol_corr}
\end{figure*}
\subsubsection{Using Clustering}
Here, we used K-means clustering to group molecules based on their structural properties. We then checked whether any group is composed only of outlier or non-outlier molecules. There are two such groups. As shown in Figure \ref{fig:clst}, one group contains molecules consisting of long chains.
\begin{figure*}[!t]
\centering
\begin{subfigure}[b]{0.8\textwidth}
\centering
\includegraphics[width=\textwidth]{images/clst12.png}
\end{subfigure}
\begin{subfigure}[b]{0.8\textwidth}
\centering
\includegraphics[width=\textwidth]{images/clst5.png}
\end{subfigure}
\caption{Learning curves for the MDM model (left) and the GNN model (right) with and without pretraining.}
\label{fig:clst}
\end{figure*}
\subsubsection{Using Molecular linearity}
Molecular linearity is a phenomenological aspect of molecular structure that often arises when the behavior of intramolecular forces is considered. Often distinguished by a uniformity in the structure relative to each atom’s experienced steric environment, molecular linearity consists of a generally unbranched structural appearance of molecular bonding In these molecules, the average bond order may not necessarily be uniform, but the number of atomic neighbors often is. As such, we require the use of a rapid classifier for distinguishing the level of linearity present in a molecule that draws its information upon the number of, and proximity of nearest neighbor atoms, relating them to the entirety of the molecule’s atomic constituents.
\\
We propose a classifier herein that counts the shortest atomic distance from a given atom in a molecule to a terminal atom in the molecule, comparing the marginal differences as one incrementally chooses a different atom in the molecule, according to the atomic indices. In a linear molecule, the distances from primary (non-Hydrogen) atoms to each other changes constantly, as no atom is disproportionately separated by bond distances to a terminal atom. For instance, the opposing Carbon atoms in pentane are equally separated from one another by the number of carbons in between them. Any other carbon atom in the molecule would be separated from a terminal carbon according to the number of carbons, or bonds, between itself and the terminal point of reference. We may summarize this behavior of linearity as prescribed by our classifier to hold a directly proportional relationship between atomic index
\\
In nonlinear compounds, there is a large level of branching, owed principally to the presence of cyclic groups, and larger moieties tailored across tertiary carbons in the molecule. As such, a given atom in the molecule may by bond length alone possess a shorter distance to a terminal atom, despite its greater spatial separation from
\subsubsection{Intra- and Inter-molecular H bonds.}
We hypothesise that some molecules have to potential to form intra- and inter-molecular hydrogen bonds.
\subsection{Automatic Detection of Outliers}
In this section we try to develop two methods to detect outliers when the solubilities of the test set are not known. These methods enables us to (1) understand the applicability domain of our predictive models and (2) improve prediction accuracy for a subset of molecules. The first step in these approaches is to find outliers in train and validation sets according to the methos described in Section~\nameref{sec:ol-detect}.
\subsubsection{Feature based filtering}
The first method is based on identifying extreme values of molecular features. For each feature, we find the difference between the extreme values in the outlier and non-outlier distributions. For a given feature, the extreme values could be either on the positive or negative side. To find out in which direction the extreme values are, we calculate distance difference for both sides and chose the side corresponding to the maximum distance. The features are then ordered in the descending order of the distance ``a''.
\begin{figure*}[!htb]
\centering
\includegraphics[width=\textwidth]{images/ol_nol_diff.png}
\caption{Distributions of outlier and non-outlier properties }
\label{feature_based_filter}
\end{figure*}
The extreme values of features on the top of this list are supposed to be associated with the outlier molecules than those towards the end. We can remove extreme values corresponding to the first `n' features, build models using the rest of the molecules and check how the prediction accuracy changes. A value is considered as an extreme value or an outlier if it is greater than Q3 + 1.5$\times$IQR or less than Q1 - 1.5$\times$IQR, where Q1, Q2 and IQR stand for first quartile, second quartile and the inter-quartile range.
\begin{figure*}[!htb]
\centering
\includegraphics[width=\textwidth]{images/effect_of_ol_removal_rmse.png}
\caption{Decrease in the prediction error with feature based filtering.}
\label{rmse_fbf}
\end{figure*}
As shown in Figure \ref{rmse_fbf} the filter based filtering results in increase in the prediction accuracy. This shows that this method can be used to separate easy-to-predict molecules from difficult-to-predict molecules. This allows us to down select a set of molecules for which the solubilities can be predicted with high accuracy. One issue with this method is that a large amount of data has to be sacrificed to achieve a considerable increase in the accuracy.
\subsubsection{Training a classifier}
In the second method we train a machine learning model to classify outliers. We trained a random forest classifier using the outlier and non-outlier molecules we identified in the train set. The trained model can then be used to predict the non-outliers in the train, validation and test set. Using the predicted non outliers, we can now train a new model to predict the solubilities. Optionally, we can tune the hyperparameters. The accuracies of the predicted solubilities are given in Table A.
\subsubsection{Training a classifier for a feature correlated with solubility}
We have found that the feature FilterItLogs is correlated with logS with a coefficient of about 0.67. This is not surprising as FilterItLogs is a theoretical estimation of logS.
\begin{table}[t]
\caption{Effect on outlier removal}
\begin{center}
\begin{tabular}{lrr}
\toprule
{} & without outliers & with outliers \\
\midrule
RMSE & 0.947 & 1.030 \\
R2 & 0.799 & 0.763 \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\begin{figure*}[!htb]
\centering
\includegraphics[width=.8\textwidth]{images/fi_clf.png}
\caption{}
\label{}
\end{figure*}
We can also use the most important features for the classifer for feature based filtering.
When building solubility prediction models, we assumed that the experimental error associated with solubility data is independent source of the data. To verify whether this is true, we used the data source as an label-encoded feature for the classification task. Even though the classification accuracy did not change substantially, we find that the information about the source of the data happens to be the most important feature for the classification task. We then used label encoded data source in the prediction models. The accuracy of fully connected neural networks was not affected. However, the accuracy of the ensemble methods showed a significant increase.
Additionally, we can remove molecules if their core structures are found only in outlier molecules or they belong to a cluster that are only found in outlier molecules. However, these methods did not result in a significant reduction in the prediction error.
\begin{acknowledgement}
This work was supported by Energy Storage Materials Initiative (ESMI), which is a Laboratory Directed Research and Development Project at Pacific Northwest National Laboratory (PNNL). PNNL is a multiprogram national laboratory operated for the U.S. Department of Energy (DOE) by Battelle Memorial Institute under Contract no. DE- AC05-76RL01830
\end{acknowledgement}
\clearpage
\bibliographystyle{plainnat}
\section*{}
\vspace{-1cm}
\footnotetext{\textit{Pacific Northwest National Laboratory, Richland, WA, USA\\
E-mail:gihan.panapitiya@pnnl.gov (G.P.), emily.saldanha@pnnl.gov (E.S.)}}
\footnotetext{\dag~Electronic Supplementary Information (ESI) available: [details of any supplementary information available should be included here]. See DOI: 00.0000/00000000.}
\section{Introduction}\label{sec:intro}
The use of data driven approaches for material property prediction and material design have become exceedingly common. \textcolor{orange}{ quality of input experimental data affects predictions. knowledge about the domain of applicability is important to use the the models in a meaningful way. Expand these to justify the need for an outlier detection and DOA study.}
In this work, we use solubility prediction as a test case.
Solubility prediction plays a vital role in many disciplines including drug discovery, medicine, fertilizers and energy storage systems. Despite the efforts over many decades, scientists are yet to come up with a highly accurate predictive models for a wide variety of molecules.
In our previous work, we used deep learning methods to predict solubilities of the molecules in the largest and the most complex (in terms of the structural properties) solubility dataset up to date.
One of the reasons for not being able to achieve high accuracies could be the lack of training data, particularly for some solubility ranges. In this work we try to explore another possibility; are there some molecules that are intrinsically difficult to be predicted? If so, what are the characteristics of these molecules? Can we explain why the solubilities of these molecules are difficult to predict? Can we filter out these molecules using machine learning?
The paper is outlined as follows. First we explain the outlier detection method used. Next, we try to identify distinctive properties of outliers using different methods. Finally, we introduce several methods to automatically identify outlier molecules.
In Table \ref{table:curr-preds}, we list the solubility prediction accuracies obtained by different machine learning models for our dataset.
\begin{table}[tbh]
\centering
\small
\caption{CV scores obtained for PNNL dataset by recent models.}
\begin{tabular}{lrr}
\toprule
method & rmse & r2 \\
\midrule
etr & 1.06 & 0.78 \\
cbm & 1.06 & 0.78 \\
rf1 & 1.07 & 0.78 \\
xgbm & 1.09 & 0.77 \\
lgbm & 1.10 & 0.77 \\
afpm & 1.11 & 0.76 \\
svm & 1.13 & 0.75 \\
gnn-fgrp & 1.15 & 0.75 \\
gnn-reg & 1.15 & 0.75 \\
mdm & 1.15 & 0.74 \\
gbr & 1.20 & 0.72 \\
\bottomrule
\end{tabular}
\label{table:curr-preds}
\end{table}
\section{Data}
Our dataset is composed of data in three prior datasets: PNNL Organic Solubility dataset, AQUASOL and Cui dataset.
Brief description of data.
\section{Outlier Detection Methods}\label{sec:methods}
\subsection{\textbf{Outlier Detection}}\label{sec:ol-detect}
We compare and combine two different approaches for outlier detection - structure-only outlier detection and solubility-aware outlier detection. These two approaches have different purposes. The detection of structural outliers can be used to identify molecules which appear structurally unusual without regard to the effect of that deviation on the solubility prediction. Such molecules are more likely to be difficult for the model due to lack of training examples, but are not necessarily so. The solubility-aware outlier detection on the other hand identifies outliers for which solubility prediction is a challenge. Such challenge may occur because of unusual molecular structure, measurement noise in the solubility values, error in the solubility annotations, or difficulties distinguishing the effect of small structural changes on solubility. By combining both structure-only and solubility-aware methods, we can distinguish some of these different causes for molecular property prediction errors.
\subsubsection{Structure-Only}
Describe and cite the main existing methods.
\subsubsection{Solubility-Aware}
Our solubility-aware outlier detection method is based on an algorithm described in \citet{Cao2017}. Here, we train 1000 machine learning models by splitting the entire dataset into random train and test sets. For each trained model, the errors made on the molecules in the test set are collected. Consequently, multiple error measurements exist for a given molecule. Molecules associated with large mean error, large variability in the error, or both can be identified as outliers.
The ML model we used in this work is ExtraTreesRegressor as implemented in the \textit{scikit-learn} package~\footnote{https://scikit-learn.org/stable/}. Training was performed using the default parameters, each time taking a different random train and test split. We then find the mean and standard deviation of prediction errors for the 22,100 molecules in our dataset. We find that each molecule has been predicted on average 100 times with a standard deviation of about 10 times. The minimum number of predictions made for any molecule is 63 providing sufficient signal to quantify the prediction error. In Figure \ref{fig:ol} we plot the mean and the standard deviation of the prediction errors.
Next, we have to define thresholds to separate outliers and inliers in this space. The method given in ref, involves dividing the mean error - standard deviation of the error space into four quadrants as shown in Figure \ref{fig:ol} a. Different from the original method, we first scale the mean errors and standard deviation of the errors so that these values are between 0 and 1. Then an arbitrary percentile value is chosen for mean error and standard deviation of the error.
The points which are enclosed by these thresholds correspond to inlier molecules. We then find the cross validated R2 score corresponding to the predicted and actual solubilities of these molecules. We repeat this process while reducing the percentile value from 95 to 65.
An alternative is to choose a percentile value for the scaled outlier-ness as shown in Figure \ref{fig:ol} b. Outlier-ness is defined as the euclidean distance to a point in the mean error - standard deviation in the error space. In this case the threshold value for outlierness is varied from 95 to 65 and the cross validated R2 of the resulting inliers were found. We find that this method chooses better inliers than the original method.
\textcolor{orange}{describe this method.}
\begin{figure*}[!t]
\centering
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[width=\textwidth]{images/mean_sdev_th.png}
\end{subfigure}
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[width=\textwidth]{images/olness_th.png}
\end{subfigure}
\caption{Mean error versus the standard deviation of the error for each molecule in the entire dataset. Molecules associated with large mean and standard deviation of the error are considered as outliers.}
\label{fig:ol}
\end{figure*}
\section{Results}
\subsection{Model Performance}
Next, we observe how the performance of the solubility prediction models changes after removing outliers selected using different methods. In Table \ref{table:cv_nol_diff_methods}, we list the accuracies obtained by existing anomaly detection methods and the solubility-aware outlier detection method. The number of outliers removed is approximately 15\% of the total number of molecules. It should be noted that it is not fair to compare the performance of the solubility-aware method with the structure-only methods in terms of model predictive performance, because the solubility-aware method used solubility error information in its outlier selection approach. We are not aiming to maximize the solubilitiy prediction performance using this method, which would not be possible to apply to new molecules which don't have solubilty labels. Instead, we aim to understand the types of molecules which are detected as outliers to understand the existing challenges in solubility prediction models.
We find that the removal of structural outliers has a significant effect on the performance of the predictive models, with an improvement from and RMSE of 1.01 to around 0.90. This shows that property prediction model performance is strongly driven by the structural diversity present in the test set used to evaluate the model performance. With the extra information available to it, the solubility-aware method achieves an RMSE of 0.56, showing that structural diversity alone does not explain the challenges in predictive performance that are experienced by the predictive models. Identifying the additional factors within the dataset that drive this performance gap will be crucial to achieving significant improvements in property prediction.
\begin{table}[tbh]
\centering
\footnotesize
\caption{CV scores obtained for inliers predicted by different anomaly detection methods. ExtratreesRegressor has been used as the ML model. \textcolor{red}{Add performance on inliers only when using the full training set (no outlier removal) }. \textcolor{orange}{don't quite understand this part.}
}
\begin{tabular}{lllr}
\toprule
methods & RMSE & R2 & ol\% \\
\midrule
pnnl-method & 0.56 \pm 0.01 & 0.92 \pm 0.0 & 15.58 \\
Isolation Forest & 0.86 \pm 0.0 & 0.82 \pm 0.0 & 15.58 \\
Principal Component Analysis (PCA) & 0.87 \pm 0.02 & 0.82 \pm 0.01 & 15.58 \\
One-class SVM (OCSVM) & 0.88 \pm 0.01 & 0.81 \pm 0.01 & 15.58 \\
K Nearest Neighbors (KNN) & 0.88 \pm 0.03 & 0.82 \pm 0.01 & 14.18 \\
Angle-based Outlier Detector (ABOD) & 0.89 \pm 0.02 & 0.83 \pm 0.0 & 16.86 \\
Histogram-base Outlier Detection (HBOS) & 0.89 \pm 0.02 & 0.81 \pm 0.01 & 15.58 \\
Feature Bagging & 0.89 \pm 0.02 & 0.82 \pm 0.01 & 14.66 \\
Local Outlier Factor (LOF) & 0.89 \pm 0.03 & 0.82 \pm 0.01 & 14.61 \\
Minimum Covariance Determinant (MCD) & 0.89 \pm 0.01 & 0.82 \pm 0.0 & 15.58 \\
Cluster-based Local Outlier Factor (CBLOF) & 0.89 \pm 0.02 & 0.81 \pm 0.01 & 15.58 \\
all\_data & 1.01 \pm 0.02 & 0.8 \pm 0.01 & - \\
\bottomrule
\end{tabular}
\label{table:cv_nol_diff_methods}
\end{table}
For the solubility-aware method, it is important to test the robustness of our results to selection of the machine learning model used to detect them. In Table \ref{table:cv-nol-models}, we show cross validated accuracies obtained by different machine learning models for the selected inliers. Note that the outliers and inliers were separated based on the prediction errors made by an ExtraTreesRegressor. This shows that similar performance improvements are shown across different machine learning modeling approaches even when a particular model is used for the outlier selection process.
\begin{table}[tbh]
\centering
\small
\caption{CV scores obtained using different ML models for inliers determined by solubility-aware method.}
\begin{tabular}{l|rr}
\toprule
\textbf{\small{Model}} & \textbf{\small{RMSE}} & \textbf{\small{$R^2$}} \\
\midrule
Random Forest & A & A \\
MLP & A & A \\
GNN & A & A \\
LGB & 0.61 & 0.91 \\
\bottomrule
\end{tabular}
\label{table:cv-nol-models}
\end{table}
\subsection{Outliers and Inliers}
In this section, we are going to analyse the proprties of outliers and inliners in detail.
There are two main reasons for a molecule to become an outlier: (1) its structure deviates vastly from most of the other structures in the dataset, (2) its solubility cannot be predicted by applying the same rules that are valid for most of the other molecules. This may occur when molecules have erroneous solubility labels in the dataset, when there is noise in the solubility measurements, or when structurally similar molecules have differing solubilities. We attempt to separate these two types of outliers in order to better understand the root causes for each type.
To identify structural outliers, we use the 10 unsupervised anomaly detection methods implemented in \textit{pyod}~\footnote{https://pyod.readthedocs.io/en/latest/} and listed in Table \ref{table:cv_nol_diff_methods}. The total number of outliers detected by all the methods is 8283 (\textcolor{red}{have to check whether this number is reproducible - if we find the outliers again, do we get the same outliers?} \textcolor{dgreen}{this number is not exactly reproducible, because some of the methods need random initial seeds. But the there should not be large variations in this number when we repeat the experiment. i will check this by repeating several times.}). The number of outliers that all the models agree to be as outliers is 367. Then we found the molecules that are common to the outliers detected by all the pyod methods and outliers detected by the solubility-aware method. We identify these 2347 common molecules as the \textit{structural outliers}. We should note that because we leverage the combination of the structure-only methods and the solubility-aware method, that these molecules are structurally unusual in a way that impacts the solubility prediction performance. If molecules are structurally unusual but the models still achieve sufficient predictive accuracy, they are not included in the outlier set.
\textcolor{red}{How many moleculare are detected by the unsupervised methods, but are not in the solubility-aware set? This may be an interesting set to look at.}There are 5936 molecules detected by the unsupervised methods, but are not in the solubility-aware set. There are 968. Any outliers detected by the solubility-aware method but which are not detected by the unsupervised methods are considered to be \textit{data outliers}. \textcolor{red}{How many data outliers?} There are 968 data outliers.
\subsubsection{Structural outliers and domain of applicability}
We aim to understand the relationship between outlier molecules and their interpretable molecular descriptors and structural properties. In Table \ref{table:ol_nol_diff1} we show the percentage of outliers and inliers having certain properties which we intuitively expect may distinguish outliers from inliers. For example, we might expect the number of atoms in the molecules as well as their complexity to be significant factors. We find that, indeed, outlier molecules are more likely to be large or complex compared with inlier molecules.
\begin{table}[tbh]
\centering
\small
\caption{Number of molecules in outliers and inliers, having different properties. The numbers are given as a percentage of the total number of molecules in each group. [updated]}
\begin{tabular}{l|r|r}
\toprule
Property & Outlier \% & # Inlier \% \\
\midrule
\ nAtom > 100 & 5.95 & 0.69 \\
\ BertzCT>1500 & 6.33 & 1.42 \\
\ chiral centers & 14.9 & 9.79 \\
\bottomrule
\end{tabular}
\label{table:ol_nol_diff1}
\end{table}
We can extend the idea in Table \ref{table:ol_nol_diff1} by studying the distributions of outliers and inliers of different molecular features. Here, we first calculate the difference between the extreme values in the outlier and inlier distributions. We aim to identify features for which the range of the values is much larger for outliers than inliers. For a given feature, the extreme (maximum or minimum) values could be either on the positive or negative side of the distribution. \textcolor{red}{Is it possible for it to be on both sides of the distribution?}. To find out in which direction the extreme values are, we calculate distance difference for both sides and chose the side corresponding to the maximum distance. The features are then ordered in the descending order of the distance. The features are standard scaled so that the distances are comparable across features and are not driven by the natural magnitude of the feature values. Figure \ref{fig:fbf} show the distributions of outliers and inliers of some features with largest distance values. We can identify these features as those that separate outliers and inliers. In other words, these features define the \textit{domain of applicability}. DOA can help improve prediction accuracy for a subset of molecules. \textcolor{red}{Does the accuracy improve the inliers alone? Or the does the accuracy improve because it no longer has to predict for the outliers? Can you look at the RMSE and R2 for inliers in the test set before and after removing outliers from the training set?}
\begin{figure*}[!htb]
\centering
\includegraphics[width=1\textwidth]{images/ol_nol_subplots.png}
\caption{Cumulative distribution plots for outliers and inliers. These molecular properties have the largest differences between outlier - non-outlier extreme values. Molecular properties that are correlated with a Pearson correlation value greater than 0.95 are removed. \textcolor{red}{Are these in scaled or actual units? How many of these are just always zero except for a few outliers? That is maybe why the plots are not looking very informative.}}
\label{fig:fbf}
\end{figure*}
In Table \ref{tbl:doa} we show the top 20 molecular descriptors that define the domain of applicability for our dataset. The `direction' says whether a descriptor's value should be less than or equal, or greater than or equal to the threshold value. For example, SsPH2 value of a molecule should be less than or equal to 0 in order for it to be in the domain of applicability. SsssB value of molecules in the domain of applicability should be $\geq$ 0 \textcolor{red}{Describe how threshold is selected}.
\begin{table}[!tbh]
\centering
\caption{Domain of applicability.}
\begin{tabular}{lrlrr}
\toprule
descriptor & threshold & direction & mean\_oln & \#ols \\
\midrule
NssssSn & 0.00 & == & 0.31 & 18 \\
Mpe & 0.88 & >= & 0.28 & 65 \\
IC5 & 6.31 & <= & 0.23 & 50 \\
n10AHRing & 0.00 & == & 0.23 & 8 \\
bpol & 55.38 & <= & 0.20 & 612 \\
n5FARing & 0.00 & == & 0.20 & 2 \\
SsssNH & 0.00 & >= & 0.20 & 3 \\
NsAsH2 & 0.00 & == & 0.19 & 1 \\
n5 & 0.00 & == & 0.19 & 582 \\
ATSC0v & 4387.78 & <= & 0.19 & 759 \\
TIC5 & 419.22 & <= & 0.19 & 730 \\
nFRing & 2.00 & <= & 0.19 & 30 \\
n6 & 0.00 & == & 0.19 & 229 \\
fMF & 0.72 & <= & 0.18 & 6 \\
nG12aRing & 0.00 & == & 0.18 & 2 \\
NdssSe & 0.00 & == & 0.18 & 2 \\
C1SP2 & 5.00 & <= & 0.18 & 115 \\
ATS1m & 7854.83 & <= & 0.18 & 845 \\
nRing & 6.00 & <= & 0.18 & 228 \\
SsPH2 & 0.00 & == & 0.18 & 1 \\
\bottomrule
\end{tabular}
\label{tbl:doa}
\end{table}
\textcolor{orange}{A breif discussion about some of the features in Table \ref{tbl:doa}}
\textcolor{orange}{We can show the molecules that have been removed at each step of the feature based filtering process (for some number of steps). And probably give a qualitative structural property or chemistry based discussion.}
\textcolor{red}{We should try some other methods of feature selection. For example, rather than comparing the extreme values only you could compare the full distributions using the KL divergence (or similar metric) and find the features where the distributions are most different between outliers and inliers.}
We can remove extreme values corresponding to the first $n$ features, build models using the rest of the molecules and check how the prediction accuracy changes. As shown in Figure \ref{rmse_fbf} the filter based filtering results in increase in the prediction accuracy. This shows that this method can be used to separate easy-to-predict molecules from difficult-to-predict molecules. This allows us to down select a set of molecules for which the solubilities can be predicted with high accuracy. One issue with this method is that a large amount of data has to be sacrificed to achieve a considerable increase in the accuracy.
\begin{figure}[!htb]
\centering
\includegraphics[width=.45\textwidth]{images/effect_of_ol_removal_rmse.png}
\caption{Decrease in the prediction error with feature based filtering. \textcolor{red}{In addition to comparing with removing random molecules, let's compare with selecting random features and removing extreme values of those features.}
\textcolor{orange}{have to update this figure}
}
\label{rmse_fbf}
\end{figure}
\subsubsection{Structural Outlier Clustering}
To gain further insight on the outliers, we use K-means clustering to identify groups of structurally similar outliers. This method allows us to identify features that might have caused these molecules to become outliers. \textcolor{orange}{test with other clustering methods as well. maybe we will get better clusters.}
In order to reduce the effect of the random nature of the clustering methods, we performed clustering several times and selected the molecules that get grouped together most frequently. This method is explained further in supporting information. In Figure \ref{fig:mol-clusters} we show the molecules belonging to each cluster found in this way.
For some of the clusters, it is easy to recognize their distinguishing characteristics (eg: Molecules having long carbon and fluorinated chains, circular ring systems). To identify the prominent features of others, we follow a method
described at \textcolor{orange}{add the reference}. Using a random forest classification model, which has the ability to identify feature importances, we train models to separately identify members of a selected cluster from the members of all the other clusters combined. The resulting feature importance values tell us which features played a major role in helping the model distinguish the selected cluster from the other clusters.
\begin{figure*}[!htb]
\centering
\includegraphics[width=1\textwidth]{images/sup/clusters/image_groups.png}
\caption{Clusters of outlier molecules. Set 1. \textcolor{orange}{have to check whether we get similar clusters when we run this again. we will have different number of clusters, but we should get similar members grouped together. Also have to keep only the groups that are difficult to interpret in the main figures. have a seperate figure for others or move to SI.} [updated]}
\label{fig:mol-clusters}
\end{figure*}
\begin{figure*}[!htb]
\centering
\includegraphics[width=1\textwidth]{images/sup/clusters/cluster_fimp.png}
\caption{Clusters feature importance [updated] }
\label{fig:mol-clusters-2}
\end{figure*}
\subsubsection*{Other hypothesis to distinguish outliers}
~
\textbf{Are isomer structures more likely to become outliers? }
\textbf{Are certain core structures frequently found in outliers?}
\textbf{Intra- and Inter-molecular Interactions.} We hypothesise that some molecules have the potential to form inter- and intra-molecular interactions such as hydrogen bonding,(refs) pi-stacking,(refs) and ion-pairing.(refs) Give some notes/definitions of each.
\\
Molecules that form intermolecular H-bonds or pi-stack will be intrinsically difficult for solubility prediction as the intermolecular interactions necessitates that more than one molecule to be present and the machine learning model only investigates the properties of a single molecule. It is hypothesized that the presence of these intermolecular forces will decrease the calculated solubility as the molecules interact with one another, forming larger, oligomer(ref) and polymer(ref) like structures in solution. Ion-pairing is different in that the strength of the ion-pairing could shift the amount of hydrolyzed vs. ion-pairs that are present in solution.(ref) \textcolor{red}{Can we verify that the errors tend to be in this direction for this type of molecule?} Experimentally, both are measured as part of solubilized concentration, whereas computationally, only the solubility of the hydrolyzed ions is determined.(is this accurate)
\\
It is unknown how intramolecular H-bonds may impact solubility, but the formation of them can remove potential polar groups from interacting with water or restrict the movement/free rotation of molecule bonds. It is anticipated that these intramolecular interactions, within a single molecule, should be captured by the machine learning model.
\textbf{Long Alkyl Chains.} Long alkyl chains will undergo hydrophobic collapse when dissolved in water and as a result of the high number of rotatable bonds there will be 2(to the powerof)n number of possible conformations that could be present. However, the machine learning model does not take into account these different conformations when determining the solubility and is therefore looking at an inaccurate description of the molecule that is in solution. A common subset of these alkyl chains are perfluorinated alkanes, where the hydrogens have been replaced by fluorines. These can also undergo hydrophobic collapse. In addition they are known(ref) to be less soluble than their hydrocarbon counterparts in aqueous solution.
\textit{Chiral Molecules.} The difficulty in using machine learning to determine the solubility of chiral molecules is the data set that used used to train them. This is a result because the "real" data that is used...
\textcolor{red}{For all these cases we need to connect the hypothesized causes of outlierness with the actual detected outliers? Are these types of molecules more likely to be outliers?}
\subsubsection{Data outliers}
Data outliers are not large, complex molecules. However, they seem to have a very similar counterpart more often than the structural outliers.
\begin{table}[tbh]
\centering
\small
\caption{Number of molecules in outliers, inliers, structural outliers and data outliers having different properties. The numbers are given as a percentage of the total number of molecules in each group. \textcolor{red}{Add inliers.}}
\begin{tabular}{lrrrr}
\toprule
Property & Structural OL \% & Data OL \%\\
\midrule
\ Chiral centers & 17.87 & 12.04\\
\ nAtom > 100 & 8.45 & 0.00 \\
\ BertzCT > 1500 & 9.12 & 0.00\\
\ Have an isomer counterpart & 19.72 & 30.07 \\
\bottomrule
\end{tabular}
\label{table:CV_LIT}
\end{table}
Results shown in Figures \ref{fig:hyp-sim1} and \ref{fig:hyp-sim2} do not prove that having a very similar counterpart does not make a particular molecule an outlier. Because there are many inliners which are very similar to another molecule. However, data outliers are more likely to have a very similar counterpart compared to structural outliers.
\textcolor{red}{Check whether data outliers are more likely to from certain sources than structural outliers or inliers}
\begin{figure}[!htb]
\centering
\includegraphics[width=.45\textwidth]{images/n_similar_pairs.png}
\caption{Distribution of average solubility differences corresponding to molecule pairs that are >0.99 similar}
\label{fig:hyp-sim1}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[width=.45\textwidth]{images/diff_vs_simth.png}
\caption{Mean solubility difference between nearest neighbors versus the similarity threshold.}
\label{fig:hyp-sim2}
\end{figure}
\section{Predicting Outliers}
In this section, we propose two supervised outlier detection methods. One is based on training a classifier and the other on the outlierness associated with molecules.
To demonstrate these processes, we first split our dataset into train and test folds. Prediction errors for each molecule in the train set are calculated as explained in the section A. Next we used the same method which we used in section A to separate outliers and inliers. Outlier percentage we consider is 20\% of the train-validation set.
\subsection{Training a classifier}
Once we have identified the outliers and inliers in the train, we can train a classifier to detect outliers in the test fold. We arbitrarily used ExtraTreesClassifier as the classifier. Once the outliers are detected, we can remove them from train and test folds and train a regression machine learning method to predict the solubilities of the remaining inliers. In Table \ref{table:remove-ols-pnnl}, we list the solubility prediction accuracies obtained when outliers are removed from both train and test sets, only from test set and only from the train set.
\begin{table}
\centering
\small
\caption{Solubility prediction accuracies when outliers are removed from train or test and train+test folds.}
\begin{tabular}{lrr}
\toprule
Outliers removed from & RMSE & R2 \\
\midrule
Train+Test & 0.89 & 0.81 \\
Test only & 0.85 & 0.82 \\
Train only & 1.19 & 0.73 \\
\bottomrule
\end{tabular}
\label{table:remove-ols-pnnl}
\end{table}
In order to compare our method with the supervised outlier detection methods, we removed outliers detected using these from train or test sets and predicted solubility using the ExtraTreesRegressors. It is somewhat complicated to compare the solubility prediction accuracies obtained by our method with those from unsupervised outlier detection methods. Unsupervised methods do not use solubility values. Therefore, we can use the entire dataset to select a certain percentage of outliers. However, our method relies entirely on the train set. Our classifier tries to predict the outliers in the unseen test data, based on the separation of outliers and inliers in the train set. Suppose that we decided to find $x\%$ of outliers in the train set using our method. If we now predict $y\%$ of outliers in the test set, the total number of outliers detected by our method is $x+y\%$. Therefore, we should also find $x+y\%$ of outliers using the unsupervised methods in order to do a fair comparison. Number of outliers from the entire dataset and the outliers in the test set are given in Table \ref{table:remove-ols-from-test}. The percentage of outliers detected in train-validation set is 19.13\% and using the classification method we found 0.0103\% outliers in the test set. The total number of outliers in 20.16\% of the entire set. Therefore, for unsupervised methods, we also used 20.16\% as the percentage of outliers we want to find. In Table \ref{table:remove-ols-from-test} we list the solubility prediction accuracies obtained after removing outliers from the test set and train+test sets.
Interestingly, we observe that removing outliers from train set harmful for solubility prediction for all the methods. However, when outliers are removed only from the test set, our method performs better than the unsupervised methods. We also notice that the number of outliers detected in the test set by our method is considerably lower than the outliers detected by unsupervised methods.
\begin{table*}
\centering
\small
\caption{RMSE and R2 for inlier test molecules selected by different methods.
K Nearest Neighbors (KNN), Principal Component Analysis (PCA), One-class SVM (OCSVM),Isolation Forest (IF), Angle-based Outlier Detector (ABOD), Cluster-based Local Outlier Factor (CBLOF), Feature Bagging (FB), Histogram-base Outlier Detection (HBOS), Minimum Covariance Determinant (MCD), Local Outlier Factor (LOF)
}
\begin{tabular}{lrr|rr|}
\toprule
& \multicolumn{2}{c|}{Remove from test only} &\multicolumn{2}{c|}{Remove from test+train} \\
meth & rmse & r2 & rmse & r2 \\
\midrule
pnnl & 0.85 & 0.82 & 0.89 & 0.81 \\
K Nearest Neighbors (KNN) & 0.86 & 0.84 & 0.88 & 0.83 \\
One-class SVM (OCSVM) & 0.86 & 0.81 & 0.88 & 0.80 \\
Isolation Forest & 0.86 & 0.80 & 0.88 & 0.80 \\
Principal Component Analysis (PCA) & 0.88 & 0.80 & 0.88 & 0.80 \\
Local Outlier Factor (LOF) & 0.89 & 0.82 & 0.91 & 0.81 \\
Histogram-base Outlier Detection (HBOS) & 0.89 & 0.80 & 0.91 & 0.79 \\
Angle-based Outlier Detector (ABOD) & 0.89 & 0.82 & 0.90 & 0.82 \\
Feature Bagging & 0.90 & 0.82 & 0.90 & 0.82 \\
Minimum Covariance Determinant (MCD) & 0.90 & 0.82 & 0.89 & 0.82 \\
Cluster-based Local Outlier Factor (CBLOF) & 0.90 & 0.79 & 0.91 & 0.79 \\
Average KNN & 0.95 & 0.81 & 0.98 & 0.80 \\
\bottomrule
\end{tabular}
\label{table:remove-ols-from-test}
\end{table*}
When we remove outliers detected by our method and supervised methods, we observe that the prediction accuracies can be further increased (see Table \ref{table:rem-ols-pnnl-pyod}).
\begin{table}
\small
\caption{RMSE and R2 when both pnnl and pyod outliers are removed from the test set.
}
\begin{tabular}{lrrr}
\toprule
meth & rmse & r2 & oltest \\
\midrule
Feature Bagging & 0.77 & 0.84 & 441 \\
KNN) & 0.78 & 0.84 & 404 \\
ABOD) & 0.78 & 0.85 & 479 \\
LOF) & 0.78 & 0.84 & 441 \\
MCD) & 0.79 & 0.84 & 436 \\
CBLOF) & 0.81 & 0.81 & 431 \\
Average KNN & 0.82 & 0.83 & 301 \\
PCA) & 0.82 & 0.81 & 413 \\
HBOS) & 0.82 & 0.82 & 387 \\
OCSVM) & 0.82 & 0.81 & 417 \\
Isolation Forest & 0.83 & 0.81 & 389 \\
\bottomrule
\end{tabular}
\label{table:rem-ols-pnnl-pyod}
\end{table}
In Table \ref{table:cls-A}, we show the difference between outliers detected by the our method and the unsupervised methods in terms of number of atoms and BertzCT.
\begin{table*}
\small
\caption{difference between outliers detected by the our method and the unsupervised methods}
\begin{tabular}{lrrrrrrrr}
\toprule
meth & n\_comm & n\_diff & m-nA-com & m-nA-pnnl-only & m-BCT-com & m-BCT-pnnl-only & chr-comm & chr-pnnl-only \\
\midrule
ABOD & 111 & 116 & 64.81 & 44.45 & 852.17 & 586.14 & 22 & 28 \\
CBLOF & 121 & 106 & 69.23 & 37.48 & 921.98 & 481.35 & 31 & 19 \\
Feature Bagging & 110 & 117 & 61.97 & 47.29 & 785.25 & 651.33 & 23 & 27 \\
HBOS & 137 & 90 & 67.46 & 34.53 & 930.38 & 390.24 & 36 & 14 \\
Isolation Forest & 151 & 76 & 61.77 & 39.76 & 844.20 & 461.97 & 36 & 14 \\
KNN & 124 & 103 & 63.81 & 43.09 & 891.93 & 504.69 & 30 & 20 \\
LOF & 109 & 118 & 62.60 & 46.84 & 801.68 & 637.29 & 24 & 26 \\
MCD & 118 & 109 & 53.43 & 55.46 & 643.80 & 794.63 & 19 & 31 \\
OCSVM & 144 & 83 & 61.86 & 41.47 & 835.05 & 510.08 & 36 & 14 \\
PCA & 146 & 81 & 61.60 & 41.44 & 835.44 & 501.35 & 36 & 14 \\
\bottomrule
\end{tabular}
\label{tbl:pnnl-and-pyod-ols}
\end{table*}
\blindtext[2]
\subsection{Feature based filtering}
Another method to remove outliers is by leveraging the domain of applicability information. As we did with the full dataset in section A, we can use only the train set to find feature thresholds that define the domain of applicability. Then we can remove the test molecules which do not fall into the domain of applicability and use the rest to make predictions.
In Table \ref{table:doa-train}, we list the thresholds corresponding to top features which define the domain of applicability arrived at using the train set.
\begin{table}
\caption{Domain of applicability determined using the train set.}
\begin{tabular}{lrlr}
\toprule
prop & th & direction & mean\_oln \\
\midrule
Mpe & 0.87 & >= & 0.36 \\
IC2 & 5.45 & <= & 0.30 \\
NssssSn & 0.00 & == & 0.30 \\
IC3 & 6.03 & <= & 0.26 \\
n10HRing & 0.00 & == & 0.25 \\
n5FRing & 0.00 & == & 0.24 \\
bpol & 55.38 & <= & 0.24 \\
nFRing & 2.00 & <= & 0.23 \\
ATSC0v & 4387.78 & <= & 0.22 \\
n5 & 0.00 & == & 0.22 \\
TIC5 & 419.97 & <= & 0.22 \\
nRing & 6.00 & <= & 0.22 \\
nG12aHRing & 0.00 & == & 0.22 \\
NdssSe & 0.00 & == & 0.22 \\
n6 & 0.00 & == & 0.22 \\
C1SP2 & 5.00 & <= & 0.22 \\
fMF & 0.72 & <= & 0.21 \\
ATS1m & 7854.83 & <= & 0.21 \\
ATSC1dv & 106.88 & <= & 0.21 \\
\bottomrule
\end{tabular}
\label{table:doa-train}
\end{table}
Figure \ref{fig:fbf-test} shows how the solubility prediction accuracy improves as molecules that are not inside the domain of applicability are removed from the test set.
\begin{figure}[!htb]
\centering
\includegraphics[width=.45\textwidth]{images/fbf.png}
\caption{.}
\label{fig:fbf-test}
\end{figure}
\section*{Availability of data and materials
The source code is available on GitHub at https://github.com/pnnl/solubility-prediction-paper. The data set prepared by \citet{2021Gao} is accessible at https://figshare.com/s/6258a546a27a2373bf2a.
\section*{Author Contributions}
\section*{Conflicts of interest}
There are no conflicts to declare
\section*{Acknowledgements}
This work was supported by Energy Storage Materials Initiative (ESMI), which is a Laboratory Directed Research and Development Project at Pacific Northwest National Laboratory (PNNL). PNNL is a multiprogram national laboratory operated for the U.S. Department of Energy (DOE) by Battelle Memorial Institute under Contract no. DE- AC05-76RL01830.
\balance
\section{Introduction}
The use of data-driven approaches for material property prediction and material design have become increasingly common. Because of the dependency between the quality and availability of training data and the resulting model predictions, trained property prediction models will achieve different levels of accuracy on different regions of the material feature space. The ability to identify such domains helps with understanding the limitations of a given model. Such information is needed to ensure that the model is applied only where valid predictions can be obtained and that unwarranted trust is not placed in predictions for which the model cannot function well. In this work, we use the prediction of molecular solubility as a test case to develop a domain of applicability (DoA) detection method and provide analysis techniques to gain further insight into the identified domains.
Solubility prediction plays a vital role in many disciplines including drug discovery~\cite{Christel_2018, Lipinski_2001,Li_2012}, medicine~\cite{savjani2012, Cisneros2017}, fertilizers~\cite{Guo2020}, and energy storage systems~\cite{li_2015, Yan_2021}. Despite efforts over many decades, there are still limitations in the development of highly accurate predictive models that can function well across a diverse range of molecules. However, as is the case with any predictive task, there are certain regions of molecular structure space where the predictions do achieve high accuracy. The ability to identify the regions in the molecular descriptor space that have sufficient predictive accuracy has two main benefits:
\begin{enumerate}
\item \textbf{The reliability of predictions for artificially designed molecules can be assessed.} Machine learning-based molecular design and discovery pipelines rely on the application of predictive models to novel molecular structures. Without knowing the reliability of a model associated with a particular prediction, we are not in a position to make recommendations regarding the corresponding molecule - for example suggesting to proceed with synthesizing the molecule. DoA analysis provides the ability to assess the likelihood of erroneous predictions on novel molecular structures.
\item \textbf{Improvements to models can be targeted by understanding their current limitations.} By analyzing the regions of molecular feature space that are not well predicted by the model, the model developers can target additional data collection or modeling approaches to address these weaknesses.
\end{enumerate}
In this work we propose a pipeline for identifying domains in molecular feature space associated with different levels of predictive accuracy. The major contributions of our work include: (1) a method to find domains of applicability which we show to be effective for data not previously seen by the model, (2) an outlier-inlier separation method based on an \textit{outlierness} score, (3) clustering-based analysis methods to gain more insight into the detected domains, 4) insights into the relationships between applicability domains and predictive errors of the model.
We start with first identifying molecules that behave like outliers in terms of the model predictive accuracy. Such molecules are likely to identify the boundary regions of the model's domain of applicability. We define these DoA boundaries by comparing the distributions of molecular descriptors corresponding to outlier and non-outlier molecules to identify feature values corresponding to higher outlier likelihood. We show the validity of the identified domains by demonstrating that model accuracy improves as out-of-domain molecules are removed from the evaluation set. Finally, we identify a set of molecules which behave like outliers in terms of predictive accuracy but are not outside the domain of applicability in terms of their structural features. We perform analysis to understand the factors that lead to the anomalous predictive behavior for these molecules.
The term Domain of Applicability has been used in slightly different contexts in the literature. In one context, the applicability domain is identified as the range of feature space of the materials used to train a predictive model \cite{Jaworska2005, Sahigara2012}. According to this definition, a query material is identified as within the domain of applicability if it is structurally similar to those used in the training. Some authors only use independent physico-chemical properties for this comparison while some also include the prediction or independent target property \cite{Jaworska2005}. Various methods based on feature ranges, distance between data points, and probability density distribution have been used to find the domains of applicability~\cite{Jaworska2005, Sahigara2012, Sheridan2004, Tetko_2008}. In range based methods, a hyper-rectangle defining the domain is constructed using either raw features or transformed features such as principle components \cite{Jaworska2005}. In the distance-based methods, the distance from the test data point to a point representing the reference data is measured. The centroid of the reference data, distance to K-nearest neighbors in the reference data, and average distance between the test point and the data points in the reference data are among the approaches used to select the reference point \cite{Jaworska2005, Sahigara2012}. The test points with distances less than a predefined threshold value are considered to be part of the domain of the reference data.
Rather than simply identifying regions near the training data, our approach aims to identify domains in the feature space where models can achieve high prediction accuracy. Prior efforts have approached DoA identification in a similar context. For example, \citet{sutton_identifying_2020} presented a method based on subgroup discovery~\cite{Boley2012, van2012, nguyenV2015} to find such domains with high predictive accuracies based on an impact metric. In our work, we use a different strategy to identify the subgroups and rank order them in terms of their impact on the prediction accuracy. Our domain identification is based on the differences in the feature distributions of outlier and inlier molecules. Similarly to \citet{sutton_identifying_2020}, we define domains in the form of individual feature inequality constraints (i.e nAtom < 200 indicates that the molecules with less than 200 atoms are in-domain while larger molecules are out-of-domain). However, in our work, these feature inequalities are rank ordered using an \textit{outlier-ness} score of the molecules that do not satisfy the inequality. This rank ordering ensures that we can identify the features that can most effectively separate the regions of high model accuracy. We have released our code at https://github.com/pnnl/doa.
\section{Data}
Our dataset is composed of three prior datasets consisting of experimental aqueous solubility measurements: the SOMAS dataset (10,162 molecules)~\cite{gao_somas_2022, Panapitiya2022}, AqSolDB (4425 molecules) \cite{sorkun_2019} and the Cui dataset (7283 molecules) \cite{cui2020}. We first merge the three datasets and resolve duplicate molecules using a method described in the Supporting Information. Next, we divide the full dataset into train and test sets, where the train set is used to determine the rules to identify the domain of applicability and the test set is used to evaluate validity of the identified domains. The train and test sets consist of 20,229 and 1641 molecules respectively. To avoid confusion with another train set used in our method, we refer to this train set as the \textit{full train} set. For training machine learning models based on this data, we calculate 721 derived molecular descriptors using the Mordred package~\footnote{https://github.com/mordred-descriptor/mordred}\cite{Moriwaki_2018} and 87 3D and fragment-based descriptors using our own package \cite{Panapitiya2022, Panapitiya2018}.
\section{Methods}\label{sec:methods}
\subsection{\textbf{Outlier Detection}}\label{sec:ol-detect}
Our approach starts by first identifying the outlier molecules in the dataset where we consider outliers to be molecules for which the model can not predict well or consistently.
The outlier detection method is based on an algorithm described in \citet{Cao2017}. We perform multiple repetitions of training a machine learning model to predict solubility from the molecular features by splitting the \textit{full train} set into random train and validation sets multiple times. For each trained model, the resulting predictive errors measured using root mean squared error (RMSE) on the molecules in the validation set are collected. Consequently, multiple error results exist for each molecule as they each are sampled into the validation set multiple times. We define outlier molecules as those that are associated with large mean error, large variability in the error, or both. The machine learning algorithm we used as the prediction model is ExtraTreesRegressor as implemented in the \textit{scikit-learn} package~\footnote{https://scikit-learn.org/stable/}. Training was performed using the default parameters, each time taking a different random train and validation split. We perform 1000 such repetitions and find that each molecule has been predicted as part of the validation set on average 100 times with a standard deviation of about 10 times. The minimum number of predictions made for any molecule is 59 which provides sufficient signal to quantify the prediction error and its variability. The mean and the standard deviation of the prediction errors of all the molecules in the train set are shown in Figure \ref{fig:mean-sdev}.
The next task is to define a criteria to separate outliers and inliers. The method given in the original approach~\cite{Cao2017} involves dividing the mean error - standard deviation of the error space into four quadrants as shown in Figure~\ref{fig:mean-sdev} and labeling molecules belonging to farthest quadrants from the origin (colored in red in Figure \ref{fig:mean-sdev}) as outliers. In contrast, we seek to separate outliers by defining a single \textit{outlier-ness} metric which takes into account both the mean error and the error variability. The outlier-ness is found by first scaling the mean errors and error standard deviation so that these values are between 0 and 1 using MinMax scaling (Figure \ref{fig:mean-sdev}). \textit{Outlier-ness} is then calculated as the Euclidean distance from the origin to each data-point in the mean-variation space as shown by the red arrow in Figure \ref{fig:mean-sdev}. The distribution of the resulting \textit{outlier-ness} values for all the molecules are shown in Figure \ref{fig:dist-olness}.
Next we aim to define a threshold for \textit{outlier-ness} such that a certain proportion of molecules is selected as outliers (for example the red shaded region in Figure \ref{fig:dist-olness}). In Figure~\ref{fig:olp-vs-r2}, we plot the achieved $R^2$ of the inlier data as a function of what proportion of the data is removed as outliers. This approach allows us to associate the chosen inliers with a particular prediction accuracy. For this study, we selected 20\% as the outlier percentage which results in $> 0.92$ $R^2$ value for the corresponding inliers.
\begin{figure*}[!t]
\centering
\begin{subfigure}[b]{0.31\textwidth}
\centering
\includegraphics[width=\textwidth]{images/unsup/mean_sdev_th.png}
\caption{}
\label{fig:mean-sdev}
\end{subfigure}
\begin{subfigure}[b]{0.31\textwidth}
\centering
\includegraphics[width=\textwidth]{images/unsup/olness_th.png}
\caption{}
\label{fig:dist-olness}
\end{subfigure}
\begin{subfigure}[b]{0.31\textwidth}
\centering
\includegraphics[width=\textwidth]{images/unsup/R2_vs_olp.png}
\caption{}
\label{fig:olp-vs-r2}
\end{subfigure}
\caption{(a) Mean error versus the standard deviation of the error for each molecule in the entire dataset. (b) Distribution of \textit{outlier-ness} for all molecules. (c) The achieved cross-validated $R^2$ values as a function of the percent of molecules removed based on \textit{outlier-ness}.}
\label{fig:ol-detect}
\end{figure*}
\subsection{\textbf{Finding Domains of Applicability}}\label{sec:doa-detect}
Once we identify the outliers and inliers within the dataset, we are ready to identify the domains of applicability for the model. Rather than splitting the data into single set of out-of-domain (OOD) and in-domain (ID) molecules, we aim to find increasingly strict ID regions of molecular space for which we can expect to achieve increasing levels of model accuracy. These domain regions are defined in terms of threshold values for different molecular descriptors such that the cumulative application of additional thresholds leads to the smaller and smaller hyper-rectangular regions of molecular feature space.
To define the boundaries of these regions, we aim to identify thresholds beyond which molecules have a high likelihood of being outliers. To identify the relevant molecular descriptors, we first address the issue of high levels of correlation among the descriptors. If two descriptors are correlated with a Pearson coefficient greater than 0.95, we randomly remove one of them from the dataset to ensure that we identify the unique factors driving the model behavior. As shown in Table
\ref{table:doa-table}, our goal is to find a list of descriptors along with threshold values that separate ID and OOD molecules for each domain. Details regarding the definitions of the descriptors can be found in the Supporting Information, Mordred documentation \cite{Moriwaki_2018} and Reference \cite{Panapitiya2022}. In what follows, we describe the method of building this table.
First, we aim to select which descriptors distinguish the OOD molecules from the ID molecules. As an initial screening step, we identify descriptors for which the outlier molecules have a larger range of values than the inliers on either the high or the low end. The molecular descriptors are first scaled to zero mean and unit variance so that the differences are comparable across descriptors and are not driven by the natural magnitude of the descriptor values.
Next, for each descriptor $X$, we calculate the difference between maximum and minimum values of outlier values ($X_{out} = \{X_i \vert i \in \mathrm{outliers} \}$) and inlier values ($X_{in} = \{X_i \vert i \in \mathrm{inliers} \}$).
\begin{equation}
\begin{split}
\Delta_{max} & = \max X_{out} - \max X_{in} \\
\Delta_{min} & = \min X_{out} - \min X_{in} \\
\end{split}
\end{equation}
It should be noted that we only consider cases which satisfy $\Delta_{max} > 0$ or $\Delta_{min} < 0$. That is, when either the maximum outlier value exceeds that of the inliers or the minimum outlier value lies beneath that of the inliers.
For each descriptor that meets this initial filtering criteria, we then seek to define a threshold that will be used to divide ID molecules from OOD molecules. We use the distribution of the inliers to select the threshold, specifically leveraging the first (Q1) and third (Q3) quartiles and the interquartile range (IQR) of the entire descriptor value distribution (that is, the descriptor values of inliers and outliers). We set a threshold which is calculated as either (Q1+1.5 $\times$ IQR) or (Q3-1.5 $\times$ IQR) depending on whether the outlier values exceed or lie beneath the inliers (Figure \ref{fig:th-ths}). The threshold value, $X_{t}$ , for each descriptor is selected using the following criteria,
\begin{equation}
\begin{split}
X_{t} & = Q3+1.5 \times IQR, \text{if } |\Delta_{max}| > |\Delta_{min}|\\
X_{t} & = Q1 - 1.5 \times IQR, \text{if } |\Delta_{max}| < |\Delta_{min}|\\
\end{split}
\end{equation}
Once we determine the threshold for each descriptor, the next task is to find the relative importance of each descriptor in separating ID and OOD molecules. We achieve this by calculating the mean \textit{outlier-ness} of OOD molecules defined for each descriptor. For example, for the descriptor \textit{Mpe}, descriptor values greater than 0.875 correspond to OOD molecules. We calculate the mean \textit{outlier-ness} of these OOD molecules and use it as a measure of importance of \textit{Mpe} relative to the other descriptors. The descriptors in Table \ref{table:doa-table} are arranged in the descending order of the mean \textit{outlier-ness} of the OOD molecules.
The direction specifies whether the property value of the molecule should be equal to, less than or equal to, or greater than or equal to the threshold in order to consider it part of the domain of applicability. For example, we can define the first domain of applicability as molecules for which \textit{NssssSn} == 0, while the second domain of applicability is molecules for which \textit{NssssSn} == 0 and \textit{Mpe} $>=$ 0.875.
\begin{figure*}[!t]
\centering
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{images/threshold_detection.png}
\caption{}
\label{fig:th-difference}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{images/th_boxplot.png}
\caption{}
\label{fig:th-ths}
\end{subfigure}
\caption{Threshold selection. (a) We use the difference between extreme values (maximum and minimum) as an initial filtering step to determine which descriptors may be relevant for defining the DoA boundaries. (b) For the selected descriptors, we use the whiskers of the inlier descriptor distribution as the thresholds that separate ID and OOD regions.}
\label{fig:th-detect}
\end{figure*}
\begin{table}[!thb]
\caption{ DoA table. Descriptor thresholds defining the domain of applicability and resulting test set performance ($R^2$ and RMSE) after limiting the data to this domain. }
\centering
\begin{tabular}{cllccrr}
\toprule
Domain & Descriptor & Threshold & Direction & Test OOD \% & R\textsuperscript{2} & RMSE \\
\midrule
1 & NssssSn & 0.00 & == & 0.06 & 0.78 & 1.04 \\
2 & Mpe & 0.87 & >= & 0.18 & 0.78 & 1.04 \\
3 & IC2 & 5.45 & <= & 0.30 & 0.79 & 1.02 \\
4 & n10AHRing & 0.00 & == & 0.37 & 0.79 & 1.02 \\
5 & IC3 & 6.05 & <= & 0.55 & 0.79 & 1.02 \\
6 & bpol & 55.58 & <= & 2.86 & 0.80 & 0.97 \\
7 & n5FRing & 0.00 & == & 2.86 & 0.80 & 0.97 \\
8 & nFRing & 2.50 & <= & 2.86 & 0.80 & 0.97 \\
9 & SsssNH & 0.00 & >= & 2.86 & 0.80 & 0.97 \\
10 & nG12aHRing & 0.00 & == & 2.86 & 0.80 & 0.97 \\
11 & TIC5 & 420.58 & <= & 4.20 & 0.80 & 0.96 \\
12 & ATSC0v & 4399.51 & <= & 5.00 & 0.80 & 0.96 \\
13 & SsAsH2 & 0.00 & == & 5.00 & 0.80 & 0.96 \\
14 & n5 & 0.00 & == & 5.79 & 0.80 & 0.94 \\
15 & nG12AHRing & 0.00 & == & 5.91 & 0.80 & 0.94 \\
16 & n6 & 0.00 & == & 5.97 & 0.80 & 0.95 \\
17 & ATS1m & 7851.87 & <= & 6.52 & 0.80 & 0.94 \\
18 & fMF & 0.75 & <= & 6.52 & 0.80 & 0.94 \\
19 & nRing & 6.00 & <= & 6.70 & 0.80 & 0.94 \\
20 & ATS2m & 11514.37 & <= & 7.56 & 0.80 & 0.94 \\
21 & ATS0p & 127.37 & <= & 10.60 & 0.82 & 0.90 \\
\bottomrule
\end{tabular}
\label{table:doa-table}
\end{table}
The application of these threshold filters should define increasingly small regions of molecular space for which the model is able to make increasingly accurate predictions. In order to evaluate the validity of the domains, we observed the RMSE of the test set predictions after removing OOD molecules corresponding to each domain from the test set. This shows the expected level of accuracy the model can achieve when we limit its application to just those molecules falling within the selected applicability domain. The blue curve in Figure \ref{fig:fbf} shows our results. Each point corresponds to a domain in Table \ref{table:doa-table}. We observe that as we narrow the domain (as we go down the DoA table), we achieve better prediction accuracy on the remaining ID molecules. We compare this result with the case in which we remove sets of random molecules of the same size from the test set and make predictions on the remaining data. We find that predictive performance is not affected by this random removal. Additionally, to validate the \textit{outlier-ness}-based ordering of the descriptor threshold application, we also tested the effect of randomly sampling descriptors to define the domains. The resulting test RMSE values shown by the green curve shows that the \textit{outlier-ness}-based domain ranking is crucial.
\begin{figure*}[!t]
\centering
\includegraphics[width=.8\textwidth]{images/unsup/fbf_percent.png}
\caption{Reduction in the test set RMSE with the removal of data corresponding to different domains (blue) compared with the removal of random groups of molecules (red) and removal based on random selection of features (green). }
\label{fig:fbf}
\end{figure*}
\subsection{Subdomains}\label{sec:subdomains}
We next seek to qualitatively understand the types of molecules which are determined to be in-domain and out-of-domain by this method. While the descriptor thresholds provide information about the boundaries of the domains, they do not give us the full picture of the distributions of molecules within each group, especially for domains which leverage a small number of descriptors to define the domain boundaries. To gain more insight, we employ K-means clustering to identify subgroups inside the chosen domains. In order to reduce the effect of the random nature of the clustering method, we perform clustering several times and select the molecules that get grouped together most frequently. The details of this approach are explained further in Supporting Information. In Figure \ref{fig:ood-groups} we show examples of twelve molecular groups which are outside of Domain 21 in the DoA table (Table \ref{table:doa-table}) to provide a qualitative picture of the types of molecules which are out-of-domain for the model.
\begin{figure*}[!t]
\centering
\includegraphics[width=1\textwidth]{images/ood_clusters/ood_groups.png}
\caption{Groups of OOD molecules identified using clustering.}
\label{fig:ood-groups}
\end{figure*}
For some of the clusters, it is easy to recognize their distinguishing characteristics just by visual inspection. For example, Group 2 contains molecules with long carbon chains and Group 10 consists of molecules containing fluorinated carbon chains. However, for other groups, the distinguishing characteristics are less clear when only looking at a few examples. Therefore, we seek to quantitatively define the prominent descriptors that characterize each group. We leverage a classification method with the capability to rank descriptor importances to identify the molecular descriptors that distinguish the subdomain from the rest of the molecules. The classifier we used was the ExtraTreesClassifier as implemented in the \textit{scikit-learn} package with default parameters. The molecules belonging to a given group in Figure \ref{fig:ood-groups} are assigned as the positive class and all the other molecules in our \textit{full train} set (defined in Section \ref{sec:ol-detect}) are assigned as the negative class. While the ExtraTreesClassifier can identify which descriptors were most important in distinguishing the molecules, it cannot automatically identify how the in-group molecules differ from the others in terms of those features. Therefore, we also calculated the difference between the mean values of in-group and out-of-group descriptors ($\bar{X}_{in-group} - \bar{X}_{out-of-group}$) to identify whether the in-group molecules have higher or lower values of each feature than the typical molecule. In Figure \ref{fig:feature-difference-expl} we show five most important descriptors for each group (blue bars) determined by the classifier along with the corresponding mean descriptor differences (green bars). To aid subdomain interpretability for the purposes of this analysis, we focus on easy-to-interpret descriptors rather than more complex and theoretical descriptors. These easy-to-interpret descriptors were chosen by inspection. Before finding the feature differences, we have scaled each descriptor so that it has a zero mean and a standard deviation of 1 (standard scaling), such that the descriptor differences can be interpreted as standard deviations. The descriptor importances are also scaled so that the largest value is equal to the largest descriptor difference value. This scaling does not affect the interpretation of descriptor importances, as these values can only be compared relatively within the same group.
We will present several examples to demonstrate how we can use the information in Figure \ref{fig:feature-difference-expl} to gain more insight on each group. Such analysis can also be helpful to understand possible causes for the inability of the models to make highly accurate predictions for these groups.
The presence of iodine atoms is a dominant characteristic of Group 1 molecules as evidenced by the large descriptor importance and positive difference for the number of iodine atoms (\textit{nI}). These molecules are also distinguished having a larger molecular weight (MW) and higher than typical numbers of bromine atoms (\textit{nBr}). Group 2 molecules have a larger number of rotatable bonds (\textit{nRot}) compared to the others. Having large values for the number of bonds (\textit{nBonds}), the number of SP3 hybridized carbons bonded to two other carbons (\textit{C2SP3}), and the number of single bonds (\textit{nBondsS}) is indicative of the long carbon chains in these molecules which we can visually deduce. For Group 3, having higher atomic polarizability (\textit{apol}), high numbers of basic groups (\textit{nBase}) and higher numbers of acidic groups (\textit{nAcid}) are distinguishing characteristics. In combination, these factors provide an indication that these molecules have a tendency to interact with molecules of the same kind and with the solvent molecules in a solution. Therefore, one can argue that the prediction difficulty for Group 3 molecules is likely due to deficiencies in the current molecular representations to account for complex solute-solute and/or solute-solvent interactions resulting due to the large number of acidic and basic groups.
\begin{figure*}[!t]
\centering
\includegraphics[width=1\textwidth]{images/ood_clusters/ood_feature_imp_difference_expl_standard.png}
\caption{The distinguishing characteristics of each cluster of OOD molecules. The blue bars indicate the relative descriptor importances for the ExtraTreesClassifier when classifying the molecule groups in Figure \ref{fig:ood-groups} versus the other molecules. The green bars indicate the difference between mean descriptor values of molecule groups in Figure \ref{fig:ood-groups} and the other molecules.}
\label{fig:feature-difference-expl}
\end{figure*}
\subsection{\textbf{Measuring DoA Size}}\label{sec:doa-size}
Once we have established the DoA thresholds for a given property prediction model, we can leverage this information to determine the DoA on any molecular dataset for which we want to make predictions. To demonstrate the practical application of this technique, we sampled 10,000 random molecules from PubChem's collection of compounds~\footnote{https://ftp.ncbi.nlm.nih.gov/pubchem/Compound/Extras/CID-SMILES.gz}. We aim to demonstrate how the choice of training data for a property prediction model can have a significant impact on the size of the DoA of the resulting model. The size of the DoA is defined as the number of molecules in a target dataset that lie inside the domain of interest. This will be the number of molecules for which the model can be expected to achieve a target degree of accuracy. A model with a larger DoA is more generalizable and can be applied with confidence to larger number and diversity of molecular structures. For this purpose we leverage three training sets differing in size and diversity: our full training set (20,229 molecules), the Cui dataset (9933 molecules) and the Delaney dataset (1117 molecules).
The complexity differences in these datasets in terms of structural properties are given in Table S2 in the Supporting Information.
We would hypothesize that domains of applicability identified using a larger and more diverse training set such as our full training set or the Cui dataset would encompass a larger number of molecules than the domains identified using a smaller dataset.
We can observe this by measuring the number of PubChem molecules that are found to be in-domain for models trained using the three training sets. In Figure \ref{fig:domain-size-a}, we plot the number of in-domain PubChem molecules versus the domain index for the three datasets. As hypothesized, the number of PubChem molecules that fall inside the domains determined by the less diverse Delaney dataset is smaller than the number of molecules inside the domains determined using more diverse datasets.
Each subsequent domain in our DoA analysis is associated with an increasing degree of expected model accuracy. We can quantify the expected degree of accuracy for a given domain by measuring the model performance on a test set of molecules that fall within each domain. We aim to determine the relationship between the expected predictive accuracy on the in-domain data and the size of the in-domain data, as measured by the number of in-domain molecules from the PubChem sample. Using the full train set, the Cui dataset and the Delaney dataset as training sets, we trained three ExtraTreesRegressor models and made predictions for molecules in our test set.
In Figure \ref{fig:domain-size-c}, we plot the RMSE of our test molecules belonging to a certain domain versus the number of the 10K sample of PubChem molecules that fall in to the same domain. This shows the expected fraction of the external PubChem molecules which fall into different domains and for which we should expect a given degree of accuracy as measured by the RMSE.
For the region where the number of in-domain molecules is the same for all three datasets, in general, molecules selected by the full train set and the Cui domains have significantly higher prediction accuracies than the molecules selected by Delanay's DoA. This shows that models trained on larger more diverse training sets can both be applied across a broader range of molecules and be expected to achieve higher degrees of accuracy.
\begin{figure*}[!t]
\centering
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[width=\textwidth]{images/domain/percentage_in_pubchem.png}
\caption{}
\label{fig:domain-size-a}
\end{subfigure}
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[width=\textwidth]{images/domain/rmse_nmols_all_domains_pubchem_dom_pcnt.png}
\caption{}
\label{fig:domain-size-c}
\end{subfigure}
\caption{
(a) Percentage of in-domain molecules in the PubChem set based on the DoA determined using models trained on different datasets.
(b) RMSE of predictions on our test set by models trained on different train sets versus the percentage of total PubChem molecules which are in-domain for different DoAs.
}
\label{fig:domain-size}
\end{figure*}
\subsection{DoA Limitations}
Despite DoA being effective in identifying regions in descriptor space with improved prediction accuracies, we find that DoA analysis by itself cannot be used to detect all the outliers in the data. For example, removing 10\% of the data using our outlier-detection method results in a $R^2$ of about 0.90 for the remaining inliers (Figure \ref{fig:ol-detect}). However, when we remove the same amount of test set data based on the DoA approach, we achieve only a cross-validated $R^2$ of 0.82 (see Table \ref{table:doa-table}). This observation indicates that DoA analysis is not able to identify and remove all of the outlier molecules. In the next section we provide more insight into the types of outliers which cannot be identified using structural filters alone.
\subsection{\textbf{Structural and Data Outliers}}\label{sec:doa-select}
We assume that outliers are of mainly two types: \textit{structural outliers} and \textit{data outliers}. Structural outliers are molecules for which prediction is challenging due to being structurally unusual relative to the training data. Data outliers on the other hand are molecules which are not structurally unusual but still appear challenging for the solubility prediction models. This challenge may occur because of measurement noise in the solubility values, error in the solubility annotations, or difficulties distinguishing the effect of small structural changes on solubility. For these molecules, the target property apparently cannot be predicted accurately by applying the same rules that are valid for most of the other molecules. DoA analysis enables us to identify the molecular features driving the prediction challenges of structural outliers but cannot identify the data outliers because it only takes into account the molecular structure and not the measured property value. There is also a possibility that some molecules may be structurally unusual but are still able to be predicted accurately by the model as some types of structural features may prove less challenging than others. This fact introduces a third class of outliers - \textit{structural anomalies}. In Figure \ref{fig:oltypes} we depict how a dataset can consist of different types of outliers.
\begin{figure*}[!t]
\centering
\includegraphics[width=.9\textwidth]{images/str_data_ols/oltypes2.png}
\caption{Composition of a molecular dataset in terms of different types of outliers.}
\label{fig:oltypes}
\end{figure*}
To identify structural outliers, we employed 18 relatively fast unsupervised anomaly detection methods implemented in the \textit{pyod} package~\footnote{https://pyod.readthedocs.io/}. Refer to the supporting information for more details on the anomaly detection process. The molecules that get classified as outliers by two or more of the anomaly detection methods were considered as structural outliers. We found 2693 structural outliers which were detected by both the pyod method and by our method as outliers. As the number of outliers detected by our method is 4046, the remaining 1353 detected outliers from our method were then considered as data outliers since they do not appear to be structurally unusual but still behave as outliers. The 5765 which are flagged by \textit{pyod} but not detected as outliers by our method are considered to be the structural anomalies.
We next seek to understand the relationships between the different types of the outliers and the molecules that are detected as in-domain and out-of-domain using the DoA method. In Figure \ref{fig:olp-vs-doa-id}, we show the proportion of in-domain and out-of-domain molecules which fall into each of the outlier types for each domain. From this figure, we can see that in-domain molecules consist almost entirely of inliers and structural anomalies indicating that our method to define the domain boundaries is effective at separating well-predicted from poorly-predicted molecules.
Similarly, we find that the OOD molecules consist primarily of the structural outliers for the highest ranked domains but do start to incorporate some well-predicted structural anomalies as well as a small number of inliers for later domains. This trend indicates the effectiveness of the domain rank-ordering method. From Figure \ref{fig:olp-vs-doa-id} we see that the ability of the threshold values to separate outliers and inliers has reduced by a small amount for domain indices greater than 60. However, as we observed in Figure \ref{fig:fbf}, using only the first 20 domains, we can get rid of highly influential out-of-domain molecules to effect a significant increase in the prediction accuracy. We also demonstrate that the OOD molecules fail to incorporate the data outliers, which are poorly-predicted but not structurally distinct.
\begin{figure*}[!t]
\centering
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{images/domain/OL_types_in_IND.png}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{images/domain/OL_types_in_OOD.png}
\caption{}
\end{subfigure}
\caption{Proportion of molecules which are data outliers (DATA.OL), structural outliers (STR.OL), structural anomalies (STR.ANMLY), and inliers (IL) (a) inside and (b) outside the domain of applicability, for first 100 domains.
}
\label{fig:olp-vs-doa-id}
\end{figure*}
To get a better understanding of the characteristic features of the molecules in different outlier categories, we train a classification model to distinguish each type from the rest of the molecules. We plot the descriptor importance scores and mean feature differences in Figure \ref{fig:feat_diff-oltypes} using the same approach as we did in Section~\ref{sec:subdomains} to analyze the subdomains. We can see that, as expected, structural outliers and structural anomalies can be distinguished by being larger and more complex than the other molecules as well as having higher polarizability. Correspondingly, inlier molecules are found to be significantly smaller and less complex with lower polarizability and higher solubility. For data outliers, the differences in the structural descriptors are smaller than those of the other molecules and appear similar to the inliers. This fact reinforces our hypothesis that data outliers cannot be detected using structural descriptors.
\begin{figure*}[!t]
\centering
\includegraphics[width=1\textwidth]{images/oltypes/oltype_feature_imp_difference.png}
\caption{
The distinguishing characteristics of each type of outliers relative to all other molecules. The purple bars show the descriptor importance scores from the ExtraTreesClassifier when classifying the molecules belonging to each outlier category versus the rest of the molecules. The orange bars show the difference between the mean descriptor values of molecules belonging to each outlier category and the rest of the molecules}
\label{fig:feat_diff-oltypes}
\end{figure*}
Based on the findings of our previous work \cite{Panapitiya2022}, we hypothesize that one reason for the existence of the data outliers is limitations in the representation of isomer structures. In some cases even though the molecules are structurally very similar, their solubilities can have significant differences. In Figure \ref{fig:str-data-ols}(a) we plot the mean solubility difference between pairs of molecules, whose molecular descriptor cosine similarity is greater than a threshold value. In general, we see that the solubility differences between similar molecules follow a fixed ranking of Structural Outliers $>$ Data Outliers $>$ Structural Anomalies $>$ Inliers even as the similarity values approach 1 where the molecules become nearly identical. In Figure \ref{fig:str-data-ols}(b), we focus on the solubility differences of very similar pairs of molecules (cosine similarity > 0.99) and show the distribution of solubility differences for such pairs in each group. The number of pairs with close-to-zero solubility difference is largest for inliers and structural anomalies while structural outliers and data outliers have many similar pairs of molecules with significant solubility differences.
These results provide an explanation for the relative difficulties of making predictions for the different groups of molecules. For inliers, very similar molecules also tend to have very similar solubilities making it easy for the models to correctly predict their solubilities. A similar patterns holds for the structural anomalies. Even though they are structurally different from the majority, their structurally similar counterparts are likely to have similar solubilities, making it easy for the predictive models to extrapolate to the new structures. Finally, despite not being structurally distinctive compared with the rest of the molecules, the data outliers have significantly larger solubility differences with their structural neighbors than observed for the inliers, explaining the challenge of inferring their solubility. The ability to infer large solubility differences arising from fine-grained structural changes is a limitation of current data-driven predictive ML approaches. It is also possible that some of the large solubility differences observed between similar structures are the result of measurement noise or errors, which would also drive the observed prediction errors for these molecules.
\begin{figure*}[!t]
\centering
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{images/ol_logsol/diff_vs_simth.png}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{images/ol_logsol/log_sol_difference_of_isomers.png}
\caption{}
\end{subfigure}
\caption{ (a) Mean solubility difference versus pairwise cosine similarity for pairs of similar molecules. (b) Cumulative distribution of the proportion of pairs with cosine similarity $>=$ 0.99 having an absolute solubility difference less than a threshold value. }
\label{fig:str-data-ols}
\end{figure*}
\subsection{Non-linear detection of outliers}
One limitation of our DoA approach is that the domain boundaries are each defined using a single descriptor resulting in each domain being defined by a hyper-rectangle in feature space. However, in practice, regions of molecular space are likely described better by non-linear boundaries which are functions of multiple descriptors. We can use machine learning algorithms to understand how such non-linear detection methods might improve on the linear method at the cost of reduced interpretability. In Figure \ref{fig:oltype-classification} we show the classification accuracies of ExtraTreesClassifier models that attempt to distinguish each type of outliers from the rest of the molecules.
\begin{figure*}[!t]
\centering
\includegraphics[width=.8\textwidth]{images/nonlinear/ol_clsf_oltypes_and_others.png}
\caption{Classification scores corresponding to classifying different types of outliers versus the rest of the molecules using ExtraTreesClassifier models.}
\label{fig:oltype-classification}
\end{figure*}
The classification results shown in Figure \ref{fig:oltype-classification} indicate that structural outliers and anomalies are both structurally distinguishable from other molecules. Meanwhile, the data outliers are not able to be distinguished from other molecules using structural features alone, which validates our understanding of this type of outlier being related to a combination of structural and property information. Based on the predictability of the structural outliers, we can use the predicted outlierness as a measure to filter in-domain from out-of-domain molecules in a non-linear fashion. However, this approach has a downside of a lack of interpretability relative to our feature-based filtering approach.
The classifiers were trained to detect inliers and different types of outliers using the same train set which was used to find the DoA thresholds. The trained classifiers can be used to obtain the probabilities for a given molecule to get classified as an inlier or an outlier. The predicted probability of being an outlier is used as metric to separate in-domain (low probability of being an outlier) and out-of-domain (high probability of being an outlier) molecules. If the non-linear patterns detected by the classification model are more effective, we should see that this method can identify applicability domains of the same size but with higher accuracy than the linear DoA method. To test this, we remove the same percentage of molecules corresponding the domain sizes from the linear DoA method and then find the prediction accuracy for the remaining in-domain molecules. Our results are shown in Figure \ref{fig:ol-clsfy} as function of the number of molecules removed.
As evidenced by the decreasing trend in the RMSE, the classifiers trained using structural outliers and the combined set of outliers are effective in identifying molecules which are more difficult to predict.
Interestingly, we find that the linear DoA method performs similarly to the classifiers when removing small numbers of molecules. This provides an indication of the effectiveness of the feature selection of our DoA method at identifying the top set of relevant features for boundary definition. However, when larger numbers of molecules are filtered out to identify the very high-accuracy domains, the benefit of the non-linear filtering becomes clear as the structural outlier classifier begins to outperform the original method. The relative performance of the classification models on the different types of outliers shows that predicting structural outliers is more effective than predicting data outliers or structural anomalies. The failure of the data outlier classifier likely derives from the poor ability of the classifier to identify data outliers based on structural features alone. Meanwhile, the poor performance of the structural anomaly classifier confirms our understanding that structural outliers are structurally unusual but not difficult for the solubility prediction model.
\begin{figure*}[!t]
\centering
\includegraphics[width=1\textwidth]{images/nonlinear/classifier_res_all_points_multiple_all_molecules.png}
\caption{The improvement in solubility prediction RMSE due to the filtering certain numbers of molecules based on the linear DoA method and based on outlier class probabilities from the outlier classifier models. The error bars show the standard deviation of the RMSE values across five randomly initialized models. }
\label{fig:ol-clsfy}
\end{figure*}
\section{Conclusion}\label{sec13}
We presented a new method to find domains of applicability for which predictive models can be expected to achieve increasing degrees of accuracy. Our method specifies the boundaries of each domain using descriptors or features of the dataset. The effectiveness of our method was shown by calculating the prediction accuracies of the in-domain molecules corresponding to each domain. The accuracies were found to improve as the domains get more restricted. We also proposed methods to get more insight on different domains, which is important for identifying difficult-to-predict molecules and thereby improve model accuracies. We demonstrate that training models based on larger, more diverse datasets leads to increased size of the applicability domains in structural feature space and to improved generalizability of the models to new datasets. Finally, we explore the relationship between structurally-defined applicability domains and molecules which behave as predictive outliers and identify three classes of such outliers. These results indicate that structural information alone is not sufficient to identify the model failure modes. We demonstrate that training classifiers using the outlier groups is also an effective way to detect applicability domains and that the non-linear thresholding of such models can improve on the linear DoA determination method at the cost of reduced interpretability of the domain boundaries.
\section*{Supplementary information}
Data preparation details, KMeans clsutering, Unsupervised Anomaly Detection, Complexities of PNNL, Cui and Delaney datasets, Sample molecules from subdomains.
\section*{Acknowledgments}
This work was supported by Energy Storage Materials Initiative (ESMI), which is a Laboratory Directed Research and Development Project at Pacific Northwest National Laboratory (PNNL). PNNL is a multiprogram national laboratory operated for the U.S. Department of Energy (DOE) by Battelle Memorial Institute under Contract no. DE- AC05-76RL01830.
\bibliographystyle{unsrtnat}
|
3,212,635,537,505 | arxiv | \section{Introduction}
In this paper we consider the problem of tracking quantiles when data arrive sequentially (data stream). The problem has been considered for many applications like portfolio risk measurement in the stock market \cite{gilli2006application, abbasi2013bootstrap}, fraud detection \cite{zhang2008detecting}, signal processing and filtering \cite{stahl2000quantile}, climate change monitoring \cite{zhang2011indices}, SLA violation monitoring \cite{sommers2007accurate,sommers2010multiobjective}
and back-bone network monitoring \cite{choi2007quantile}.
Suppose that we are interested in estimating the quantile related to some probability $q$.
The most natural estimator is to use the $q$ quantile of the sample distribution.
Unfortunately, such a quantile estimator has clear disadvantages for data streams as computation time and memory requirement are linear to the number of samples received so far from the data stream. Such methods thus are infeasible for large data streams.
Several algorithms have been proposed to deal with those challenges. Most of the proposed methods fall under to the category of what can be called histogram or batch based methods. The methods are based on efficiently maintaining a histogram estimate of the data stream distribution such that only a small storage footprint is required. A thorough review of state-of-the-art histogram and batch methods is given in the related work section (Section \ref{sec:relwork}).
Another ally of methods are the so-called incremental update methods. The latter methods are based on performing small updates of the quantile estimate every time a new sample is received from the data stream. {\color{black} Generally, the current estimate is a convex combination of the estimate at the previous time step and a quantity depending on the current observation.} One of the first and prominent examples of this family of methods is the algorithm attributed to Tierney (1983) \cite{Tierney1983} which is based on the stochastic learning theory. A few modifications of the Tierney method have been suggested, see e.g. \cite{Chen2000, cao2010tracking, cao2009incremental, Chambers2006}.
In data stream applications, a common situation is that the distribution of the samples from the data stream varies with time. Such system or environment is referred to as a dynamical system in the literature. Given a dynamical system, two main problems are considered in the literature namely to i) dynamically update estimates of quantiles of all data received from the stream so far or ii) estimate quantiles of the current distribution of the data stream (tracking). Despite the importance of efficient tracking of statistical properties, the tracking problem ii) has been far less studied in the literature than problem i). Incremental methods are well suited to address the tracking problem ii) while histogram and batch methods mainly have been used to address problem i). Histogram and batch based methods are not well suited for the tracking problem ii) and incremental methods typically are the only viable lightweight alternatives \cite{cao2009incremental}.
Motivated by the lack of research on the tracking problem ii), the authors of this paper introduced the deterministic multiplicative incremental quantile estimator (DUMIQE) \cite{yazidi17multiplicative} given by
\begin{align}
\label{eq:1}
\begin{split}
\widetilde{Q}_{n+1}(q) &\leftarrow \widetilde{Q}_{n}(q) + \lambda q \widetilde{Q}_{n}(q) \hspace{16mm}\text{ if } x_n > \widetilde{Q_{n}}(q) \\
\widetilde{Q}_{n+1}(q) &\leftarrow \widetilde{Q}_{n}(q) - \lambda (1-q) \widetilde{Q}_{n}(q) \hspace{6mm}\text{ if } x_n \leq \widetilde{Q_{n}}(q)
\end{split}
\end{align}
The intuition behind the estimator is simple. If the received sample has a value below the current quantile estimate, the estimate is decreased. Alternatively, whenever the received sample has a value above the current quantile estimate, the estimate is increased. The ``weights'' $q$ and $1-q$ are included to ensure convergence to the true quantile. Even though the estimator is really simple, it can document state-of-the-art tracking performance \cite{yazidi17multiplicative}. However, as Eq. \eqref{eq:1} reveals, the estimator do not use the values of the received samples directly to update the estimate, but only whether the value of the samples are above or below some varying threshold. Intuitively, this seems like a waste of information received from the data stream. In this paper, we thus present an estimator that uses the values of the received samples directly. The estimator is such that the update step size is \textit{proportional} to the distance between the current estimate and the value of the sample. Thus if the current estimate is off-track compared to the data stream, the estimator will perform large jumps to rapidly get back on-track. A theoretical proof is provided to document the convergence properties of the estimator in addition to extensive simulation experiments. The experiments show that the estimator outperforms DUMQUE and several other legacy state-of-the-art quantile estimators.
The EWA of observations is known to be state-of-the-art estimator to track expectations of dynamically varying data streams \cite{gardner2006exponential}. Interestingly, we will show that the suggested quantile estimator in this paper is in fact an instance of a \textit{generalized} EWA such that \textit{quantiles} and \textit{not} expectations are tracked. To the best of our knowledge, this is the first EWA based quantile estimator found in the literature.
The paper is organized as follows. In Section \ref{sec:ewqe} we present the novel quantile estimator using an EWA of observations. In Section \ref{sec:qea}, we present a quantile estimation algorithms based on the estimator in Section \ref{sec:ewqe}. In Section \ref{sec:se}, we perform extensive experiments that document the superiority of the suggested algorithm. Finally, in Section \ref{sec:real-life} we apply the quantile estimator on real-life data related to the problem of efficient online control of indoor climate. More specifically the estimator is used to detect when a machine learning model should be retrained/updated which is commonly referred to as concept drift detection \cite{gama2014survey}.
\section{Related Work}
\label{sec:relwork}
In this Section, we shall review some of the related work on estimating quantiles from data streams. However, as we will explain later, these related works require some memory restrictions which renders our work radically distinct from them. In fact, our approach requires storing only one sample value in order to update the estimate.
The most representative work for this type of ``streaming'' quantile estimator is due to the seminal work of Munro and Paterson \cite{Munro1980}.
In \cite{Munro1980}, Munro and Paterson described a $p$-pass algorithm for selection using $O (n^{1/(2p)})$ space for any $p \ge 2$. Cormode and Muthukrishnan \cite{Cormode2005} proposed a more space-efficient data structure, called the Count-Min sketch, which is inspired by Bloom filters, where one estimates the quantiles of a stream as the quantiles of a random sample of the input. The key idea is to maintain a random sample of an appropriate size to estimate the quantile, where the premise is to select a subset of elements whose quantile approximates the true quantile. From this perspective, the latter body of research requires a certain amount of memory that increases as the required accuracy of the estimator increases \cite{Weide1978}.
Furthermore, in the case where the underlying distribution changes over time, those methods suffer from large bias in the summary information since the stored data might be stale \cite{Chen2000}. Examples of these works include \cite{Arasu2004, Weide1978, Munro1980, Greenwald2001, guha2009stream}. Guha and McGregor \cite{guha2009stream} advocate the use of random-order data models in contrast to adversarial-order models. They show that computing the median requires exponential number of passes in adversarial model while requiring $O (\log \log n)$ in random order model.
In \cite{Chen2000, cao2010tracking, cao2009incremental, Chambers2006}, the authors proposed modifications of the stochastic approximation algorithm \cite{Tierney1983}. While Tierney \cite{Tierney1983} uses a sample mean update from previous quantile estimates, \cite{Chen2000, cao2010tracking, cao2009incremental, Chambers2006} propose an exponential decay in the usage of old estiamtes. This modification is particularly helpful to track quantiles of non-stationary data stream distributions. Indeed, a ``weighted'' update scheme is applied to incrementally build local approximations of the distribution function in the neighborhood of the quantiles. More recent approaches in this direction is the Frugal algorithm by Ma et al. \cite{ma2013frugal}, which is an additive alternative to the multiplicative estimator in Eq. \eqref{eq:1}, and the DQTRE and DQTRSE algorithms by Tiwari and Pandey \cite{tiwari2018technique}. A nice property of the DUMIQE in Eq. \eqref{eq:1} and the estimator suggested in this paper is that the update size is automatically adjusted dependent on the scale/range of the data. This makes the estimators robust to substantial changes in the data stream. The DQTRE and DQTRSE aims to achieve the same by estimating the range of the data using peak and valley detectors. However, a disadvantage with these algorithms is that several tuning parameters are required to estimate the range making the algorithms challenging to tune.
In many network monitoring applications, quantiles are key indicators
for monitoring the performance of the system. For instance, system administrators are interested in monitoring the $95\%$ quantile of the response time of a web-server so that to hold it under a certain threshold. Quantile tracking is also useful for detecting abnormal events and in intrusion detection systems in general. However, the immense traffic volume of high speed networks impose some computational challenges: little storage and the fact that the computation needs to be ``one pass'' on the data.
It is worth mentioning that the seminal paper of Robbins and Monro \cite{robbins1951stochastic} which established the field of research called ``stochastic approximation'' \cite{kushner2003stochastic} have included an incremental quantile estimator as a proof of concept of the vast applications of the theory of stochastic approximation.
An extension of the latter quantile estimator which first appeared as example in \cite{robbins1951stochastic} was further developed in \cite{joseph2004efficient} in order to handle the case of ``extreme quantiles''. Moreover, the estimator provided by Tierney \cite{Tierney1983} falls under the same umbrella of the example given in \cite{robbins1951stochastic}, and thus can be seen as an extension of it.
As Arandjelovic remarks \cite{arandjelovic2015two}, most quantile estimation algorithms are not single-pass algorithms and thus are not applicable for streaming data. On the other hand, the single pass algorithms are concerned with the exact computation of the quantile and thus require a storage space of the order of the size of the data which is clearly an unfeasible condition in the context of big data stream.
Thus, we submit that all work on quantile estimation using more than one pass, or storage of the same order of the size of the observations seen so far is not relevant in the context of this paper.
When it comes to memory efficient methods that require a small storage footprint, histogram based methods form an important class. A representative work in this perspective is due to Schmeiser and Deutsch \cite{schmeiser1977quantile}. In fact, they proposed to use equidistant bins where the boundaries are adjusted online. Arandjelovic et al. \cite{arandjelovic2015two} use a different idea than equidistant bins by attempting to maintain bins in a manner that maximizes the entropy of the corresponding estimate of the historical data distribution. Thus, the bin boundaries are adjusted in an online manner.
Nevertheless, histogram based methods have problems addressing the problem of tracking quantiles of the current data stream distribution\cite{cao2009incremental} and are mainly used to recursively update quantiles for all data received so far.
In \cite{naumov2007exponentially}, the authors propose a memory efficient method for simultaneous estimation of several quantiles using interpolation methods and a grid structure where each internal grid point is updated upon receiving an observation. The application of this approach is limited for stationary data. The approximation of the quantiles relies on using linear and parabolic interpolations, while the tails of the distribution are approximated using exponential curves. It is worth mentioning that the latter algorithm is based on the $P^2$ algorithm \cite{jain1985p}.
In \cite{jain1985p}, Jain et al. resort to five markers so that to track the quantile, where the markers correspond to different quantiles and the min and max of the observations. Their concept is similar to the notion of histograms, where each marker has two measurements, its height and its position. By definition, each marker has some ideal position, where some adjustments are made to keep it in its ideal position by counting the number of samples exceeding the marker. In simple terms, for example, if the marker corresponds to the $80\%$ quantile, its ideal position will be around the point corresponding to $80\%$ of the data points below the marker. However, such approach does not handle the case of non-stationary quantile estimation as the position of the markers will be affected by stale data points. Then based on the position of the markers, quantiles are computed by supposing that the curve passing through three adjacent markers is parabolic and by using a piecewise parabolic prediction function.
It is worth mentioning that an important research direction that has received little attention in the literature revolves around updating the quantile estimates under the assumption that portions of the data are deleted. Such assumption is realistic in many real life settings where data needs to be deleted due to the occurrence of errors, or because the data samples are merely out-of-date and thus should be replaced. The deletion triggers a re-computation of the quantile \cite{Cao2009incrementalmultiple}, which is considered a complex operation. Note that the case of deleted data is more challenging than the case of insertion of new data.
In fact, the insertion can be handled easily using either sequential or batch updates, while quantile update upon deletion requires more complex forms of updates.
Finally, Lou et al. \cite{luo2016quantiles} perform extensive experiments to compare several of the algorithms described above.
\section{Quantile Estimator Using a Generalized Exponentially Weighted Average of Observations}
\label{sec:ewqe}
Let $X_n$ denote a stochastic variable representing the possible outcomes from a data stream at time $n$ and let $x_n$ denote a random sample (realization) of $X_n$. We assume that $X_n$ is distributed according to some distribution $f_n(x)$ that varies dynamically over time $n$. We denote the cumulative distribution of $X_n$ with $F_n(x)$, i.e. $P(X_n \leq x) = F_n(x)$. Further, let $Q_{n}(q)$ denote the quantile associated with probability $q$, i.e $P(X_n \leq Q_n(q)) = F_n(Q_n(q)) = q$.
A weakness of the state-of-the-art DUMIQE in Eq. \eqref{eq:1} is that the update step size is independent of the amount of the current error in the quantile estimate. We now propose an incremental quantile estimator where the update step size is \textit{proportional} to the distance between the received sample and current estimate. Thus, if the current estimate is off-track compared to the data stream, the estimator will initiate large jumps to rapidly get back on-track. The suggested estimator is described formally as follows
\begin{align}
\label{eq:12a}
\begin{split}
\widehat{Q}_{n+1}(q) &\leftarrow \widehat{Q}_{n}(q) + \lambda c_n \frac{q}{\mu_n^+ - \widehat{Q}_{n}(q)} \left|x_n - \widehat{Q}_{n}(q) \right| \hspace{10mm} \text{ if } x_n > \widehat{Q_{n}}(q) \\[2mm]
\widehat{Q}_{n+1}(q) &\leftarrow \widehat{Q}_{n}(q) - \lambda c_n \frac{1-q}{\widehat{Q}_{n}(q) - \mu_n^-} \left|x_n - \widehat{Q}_{n}(q) \right| \hspace{10mm} \text{ if } x_n \leq \widehat{Q_{n}}(q)
\end{split}
\end{align}
where $\mu^+ = E(X_n|X_n > \widehat{Q}_{n}(q))$ and $\mu^- = E(X_n|X_n < \widehat{Q}_{n}(q))$. Naturally, the conditional expectations satisfy the inequality
\begin{align*}
\mu^- < \widehat{Q}_{n}(q) < \mu^+
\end{align*}
such that $\mu_n^+ - \widehat{Q}_{n}(q) > 0$ and $\widehat{Q}_{n}(q) - \mu_n^- > 0$. The factors $q/(\mu_n^+ - \widehat{Q}_{n}(q))$ and $(1-q)/(\widehat{Q}_{n}(q) - \mu_n^-)$ are included to ensure that the estimator converges to the true quantile value.
The constants $c_n$ can be any sequence of positive and bounded values. The estimator performed well when the fractions in Eq. \eqref{eq:12a} were ``normalizied'' as follows
\begin{align}
\label{eq:30}
c_n = \left( \frac{q}{\mu_n^+ - \widehat{Q}_{n}(q)} + \frac{1-q}{\widehat{Q}_{n}(q) - \mu_n^-} \right )^{-1}
\end{align}
Substituting Eq. \eqref{eq:30} into Eq. \eqref{eq:12a} we get
\begin{align}
\label{eq:12b}
\begin{split}
\widehat{Q}_{n+1}(q) &\leftarrow \widehat{Q}_{n}(q) + \lambda a_n \left|x_n - \widehat{Q}_{n}(q) \right| \hspace{16mm}\text{ if } x_n > \widehat{Q_{n}}(q) \\
\widehat{Q}_{n+1}(q) &\leftarrow \widehat{Q}_{n}(q) - \lambda (1-a_n) \left|x_n - \widehat{Q}_{n}(q) \right| \hspace{6mm}\text{ if } x_n \leq \widehat{Q_{n}}(q)
\end{split}
\end{align}
where
\begin{align}
\label{eq:3}
a_n = \frac{q}{\mu_n^+ - \widehat{Q}_{n}(q)} \left/ \left( \frac{q}{\mu_n^+ - \widehat{Q}_{n}(q)} + \frac{1-q}{\widehat{Q}_{n}(q) - \mu_n^-} \right ) \right.
\end{align}
Please note that since $\mu_n^+ - \widehat{Q}_{n}(q) > 0$ and $\widehat{Q}_{n}(q) - \mu_n^- > 0$ we have that $0 < a_n < 1$. By factoring out $\widehat{Q}_{n}(q)$ and $x_n$ we get
\begin{align*}
\widehat{Q}_{n+1}(q) &\leftarrow (1 - \lambda a_n) \widehat{Q}_{n}(q) + \lambda a_n x_n \hspace{26mm}\text{ if } x_n > \widehat{Q_{n}}(q) \\
\widehat{Q}_{n+1}(q) &\leftarrow (1 - \lambda (1-a_n)) \widehat{Q}_{n}(q) + \lambda (1-a_n) x_n \hspace{6mm}\text{ if } x_n \leq \widehat{Q_{n}}(q)
\end{align*}
which can be written as
\begin{align}
\label{eq:2}
\widehat{Q}_{n+1}(q) &\leftarrow (1 - b_n) \widehat{Q}_{n}(q) + b_n x_n
\end{align}
where $b_n = \lambda\left(a_n + I\left(x_n \leq \widehat{Q}_{n}(q)\right)(1-2a_n)\right)$ and $I(A)$ the indicator function returning one (zero) if $A$ is true (false).
Now we will present a theorem that catalogs the properties of the estimator $\widehat{Q}_{n}(q)$ for a stationary data stream, i.e. $X_n = X \sim F(x), \,\, n=1,2,\ldots$.
\begin{theorem}
\label{thm:1}
Let $Q(q) = F^{-1}(q)$ be the true quantile to be estimated. Applying the updating rule in Eq. \eqref{eq:2}, we obtain:
\begin{align*}
\lim_{n \lambda \to \infty, \lambda \to 0} \widehat{Q}_{n}(q) = Q(q)
\end{align*}
\end{theorem}
\noindent The proof of the theorem can be found in Appendix \ref{app:proof}. Although the quantile estimator $\widehat{Q}_{n}(q)$ given in Eq. \eqref{eq:2} is designed to estimate quantiles for dynamic environments, it is an important requirement that the estimator converges to the true quantile for static data streams as verified by Theorem \ref{thm:1}.
We end this section with a remark.\\
\textit{Remark 1:} If the conditional expectations are symmetrically positioned on each side of the quantile estimate, then $\widehat{Q}_{n}(q) - \mu_n^- = \widehat{Q}_{n}(q) - \mu_n^-$ and $a_n = q$ which is equal to DUMIQE. In other words, we can interpret that $\widehat{Q}_{n}(q) - \mu_n^-$ and $\widehat{Q}_{n}(q) - \mu_n^-$ ensure that the update rules take into account the asymmetries of the data stream distribution on each side of the quantile.
\subsection{Connection to the EWA}
\label{sec:connecEWA}
A simple and intuitive approach to track the expectation of a data stream distribution, i.e. $\mu_n = E(X_n)$, is the weighted moving average
\begin{align}
\label{eq:19} \widehat{\mu}_n = \frac{1}{W_n} \sum_{i=0}^n w_i x_i
\end{align}
where $W_n = \sum_{j=1}^nw_j$. Using $w_{n-h} = \cdots = w_n = 1$ and the other weights equal to zero, Eq. \eqref{eq:19} reduces to the standard moving average. Intuitively, it seems more reasonable to use weights with decreasing values. The decrease should be more rapid than the standard sample mean $w_i = 1/i$ to be able to track the changes in the data stream.
Consider the following recursive update scheme
\begin{align}
\label{eq:20} \widehat{\mu}_0 &\leftarrow x_0 \\
\label{eq:21} \widehat{\mu}_{n+1} &\leftarrow (1 - \alpha) \widehat{\mu}_{n} + \alpha x_n
\end{align}
where the current estimate is a convex combination of the estimate at the previous time step and the observation. By substitution, we get
\begin{align}
\label{eq:40}
\widehat{\mu}_{n+1} &= \alpha (x_n + (1-\alpha)x_{n-1} + (1-\alpha)^2x_{n-2} + \cdots + (1-\alpha)^{n-1}x_{1}) + (1-\alpha)^n x_0
\end{align}
Interestingly, from Eq. \eqref{eq:40} we see that Eq. \eqref{eq:20} to Eq. \eqref{eq:21} can be interpreted as an EWA of observations. The estimator is highly popular and known to be the state-of-the-art approach to track expectations of dynamically varying data streams. Inspecting the incremental update form of our quantile estimator in Eq. \eqref{eq:2}, we see that it is identical to the update form of Eq. \eqref{eq:21}, except that the $0 < b_n < 1$ varies with time. Thus by keeping the weights constant as in Eq. \eqref{eq:21}, the estimator will track the expectation of the data stream distribution, while using the weights $0 <b_n < 1$ in Eq. \eqref{eq:2}, the estimator will track a quantile of the distribution.
\section{Quantile Estimation Algorithm}
\label{sec:qea}
The interpretation of the update rule in Eq. \eqref{eq:2} as an EWA of observations (recall Section \ref{sec:connecEWA}) and Theorem \ref{thm:1} constitute some intriguing theoretical results on the link between EMA and quantile estimation. However, the update rule in Eq. \eqref{eq:2} cannot be used directly since the conditional expectations, $\mu_n^+$ and $\mu_n^-$, are unknown and need to be estimated. Probably the most natural approach is to track conditional expectations using an EWA of observations as given in in Eq. \eqref{eq:20} to Eq. \eqref{eq:21}. This results in the following update rules:
\begin{align}
&\bullet\hspace{4mm} \label{eq:8}\widehat{Q}_{n+1}(q) \leftarrow (1 - \widehat{b}_n) \widehat{Q}_{n}(q) + \widehat{b}_n x_n \\[2mm]
&\notag \bullet\hspace{4mm} \text{If } x_n > \widehat{Q_{n}}(q)\\
&\hspace{8mm}\text{-}\hspace{4mm} \label{eq:6}\widehat{\mu}_{n+1}^+ \leftarrow \widehat{Q}_{n+1}(q) - \widehat{Q}_{n}(q) + (1-\gamma) \widehat{\mu}_{n}^+ + \gamma x_n\\
&\hspace{8mm}\text{-}\hspace{4mm} \label{eq:6b}\widehat{\mu}_{n+1}^- \leftarrow \widehat{Q}_{n+1}(q) - \widehat{Q}_{n}(q) + \widehat{\mu}_{n}^- \\
&\notag \bullet\hspace{4mm}\text{Else}\\
&\hspace{8mm}\text{-}\hspace{4mm} \label{eq:7b}\widehat{\mu}_{n+1}^+ \leftarrow \widehat{Q}_{n+1}(q) - \widehat{Q}_{n}(q) + \widehat{\mu}_{n}^+ \\
&\hspace{8mm}\text{-}\hspace{4mm} \label{eq:7}\widehat{\mu}_{n+1}^- \leftarrow \widehat{Q}_{n+1}(q) - \widehat{Q}_{n}(q) + (1-\gamma) \widehat{\mu}_{n}^- + \gamma x_n \\
&\bullet\hspace{4mm} \widehat{a}_{n+1} \leftarrow \frac{q}{\widehat{\mu}_{n+1}^+ - \widehat{Q}_{n+1}(q)} \left/ \left( \frac{q}{\widehat{\mu}_{n+1}^+ - \widehat{Q}_{n+1}(q)} + \frac{1-q}{\widehat{Q}_{n+1}(q) - \widehat{\mu}_{n+1}^-} \right ) \right.\\
&\bullet\hspace{4mm} \widehat{b}_{n+1} \leftarrow \lambda\left(\widehat{a}_{n+1} + I\left(x_n \leq \widehat{Q}_{n+1}(q)\right)(1-2\widehat{a}_{n+1})\right)
\end{align}
In each of the equations \eqref{eq:6} to \eqref{eq:7}, the part $\widehat{Q}_{n+1}(q) - \widehat{Q}_{n}(q)$ is included to ensure that the conditional expectation estimates are relative to the current quantile estimate $\widehat{Q}_{n+1}(q)$.
Thus Eq. \eqref{eq:8} tracks the overall trends of the dynamical data stream while Eq. \eqref{eq:6} to Eq. \eqref{eq:7} are responsible for estimating the conditional expectations \textit{relative} to the quantile estimate. Thus, for most dynamic data streams it is reasonable to use a value of the EWA tuning parameter, $\gamma$, that is on a smaller scale than $\lambda$ \cite{konda2004convergence}. This is verified in our experiments. In the rest of the paper, we denote this EWA quantile estimator approach for QEWA. We end this section with a remark.
\textit{Remark1:} We evaluated a second approach based on estimating the streaming distribution, $f_n(x)$, and computing the unknown conditional expectations from the estimated distribution. The streaming distribution where estimated by tracking several quantiles $Q_n(q_1), Q_n(q_2), \ldots, \ldots, Q_n(q_{K})$ and a linear spline were interpolated between the quantile estimates. However experiments showed that the QEWA approach performed better than this spline approach. The spline approach therefore is not followed any further in the paper.
\section{Experiments based on Synthetic Data}
\label{sec:se}
\begin{figure*}
\centering
\includegraphics[width = 0.9\textwidth]{traces}
\caption{Quantile estimates in every iteration using the DUMIQE and the suggested algorithm QEWA using ratio $\gamma/\lambda = 1/100$.}
\label{fig:10}
\end{figure*}
In this section we perform a thorough comparison of the performance of the suggested algorithm QEWA and other quantile estimators in the literature. Figure \ref{fig:10} shows tracking of the quantile with probability $q = 0.7$ for the suggested algorithm QEWA and DUMIQE. The true quantile is given as the dashed black line. The tuning parameters are adjusted such that the estimation error in the stationary parts after convergence is the same for the two algorithms. We see that the proposed algorithm QEWA tracks the true quantile more efficiently after a switch than the DUMIQE. For the suggested algorithm, the step size is proportional to the difference between the observations and the quantile estimate (recall Eq. \eqref{eq:12b}). After a switch, these differences are large, and our devised algorithm makes large steps to get back on-track. The DUMIQE, and the other state-of-the art incremental algorithms, use the same step size independent of the these difference, resulting in poorer tracking.
The results below show a more systematic evaluation of the performance of the suggested algorithm against seven state-of-the-art quantile estimators namely the DUMIQE and RUMIQE by Yazidi and Hammer \cite{yazidi17multiplicative}, the estimator due to Cao et al. \cite{cao2010tracking}, the Frugal approach by Ma et al. \cite{ma2013frugal}, the selection algorithm by Guha and McGregor \cite{guha2009stream} and the DQTRE and DQTRSE algorithms by Tiwari and Pandey \cite{tiwari2018technique}. For the DQTRE and DQTRSE algorithms we used values of the tuning parameters recommended in \cite{tiwari2018technique}, namely $\alpha = 0.1, \beta = (1 - \alpha)^{\lambda}, p_b = 1/10$ and $l = 1/4$ which performed well in our experiments.
The estimator in this paper is designed to perform well for dynamically changing data streams and the experiments will focus on such streams.
We consider four different cases where we assume that the data are outcomes from a normal distribution or from a $\chi^2$ distribution. For two of the cases we look at both a case where the data stream varies smoothly (periodic) or switches rapidly (switch). For the normal distribution periodic case, we assume that the expectation of the distribution varies with time
\begin{align*}
\mu_n = a \sin \left( \frac{2\pi}{T} n \right), \,\,\, n = 1,2,3, \ldots
\end{align*}
which is the sinus function with period $T$. For the switch case, the expectation jumps between values $a$ and $-a$.
\begin{align*}
\mu_n = \left \{
\begin{array}{ll}
a & \text{ if } n\,\text{mod}\,T \leq T/2 \\
-a & \text{ else }
\end{array}
\right.
\end{align*}
We assume that the standard deviation of the normal distribution does not vary with time and is equal to one.
For the $\chi^2$ distribution periodic case, we assume that the number of degrees of freedom varies with time as follows
\begin{align*}
\nu_n = a \sin \left( \frac{2\pi}{T} n \right) + b, \,\,\, n = 1,2,3, \ldots
\end{align*}
where $b > a$ such that $\nu_n > 0$ for all $n$. For the switch case, the number of degrees of freedom jumps between values $a + b$ and $-a + b$
\begin{align*}
\mu_n = \left \{
\begin{array}{ll}
a+b & \text{ if } n\,\text{mod}\,T \leq T/2 \\
-a+b & \text{ else }
\end{array}
\right.
\end{align*}
In the experiments we used $a = 2$ and $b=6$.
We estimated quantiles of both the normally and $\chi^2$ distributed data streams above using two different periods, namely $T=100$ (rapid variation) and $T=500$ (slow variation), i.e. in total eight different data streams. For each data stream we estimated the $50$, $70$ and $90\%$ quantiles ending up with a total of $24$ different estimation tasks.
To measure estimation error, we used the root mean squares error (RMSE) for each quantile given as:
\begin{align}
\label{eq:27}
\text{RMSE}\, = \sqrt{ \frac{1}{N}\sum_{n=1}^N \left(Q_n(q) - \widehat{Q}_n(q)\right)^2 }
\end{align}
where $N$ is the total number of samples in the data stream. In the experiments, we used $N = 10^6$ which efficiently removed any Monte Carlo errors in the experimental results. In order to obain a good overview of the performance of the algorithms, we measured the estimation error for a large set of different values of the tuning parameters of the algorithms.
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[width = 0.5\textwidth]{reserr_normal_RUMIQE_DUMIQE_cao_xmQ_q_0_5_periodic_tau_100_DQTRE} & \includegraphics[width = 0.5\textwidth]{reserr_normal_RUMIQE_DUMIQE_cao_xmQ_q_0_5_periodic_tau_500_DQTRE} \\
\includegraphics[width = 0.5\textwidth]{reserr_normal_RUMIQE_DUMIQE_cao_xmQ_q_0_7_periodic_tau_100_DQTRE} & \includegraphics[width = 0.5\textwidth]{reserr_normal_RUMIQE_DUMIQE_cao_xmQ_q_0_7_periodic_tau_500_DQTRE} \\
\includegraphics[width = 0.5\textwidth]{reserr_normal_RUMIQE_DUMIQE_cao_xmQ_q_0_9_periodic_tau_100_DQTRE} & \includegraphics[width = 0.5\textwidth]{reserr_normal_RUMIQE_DUMIQE_cao_xmQ_q_0_9_periodic_tau_500_DQTRE}
\end{tabular}
\caption{Normal distribution periodic case: The left and right columns show results for $T=100$ and $T=500$, respectively. The rows from top to bottom show results when estimating quantile $Q_n(q=0.5),\,\, Q_n(q=0.7)$ and $Q_n(q=0.9)$, respectively. Ratio refers to the ratio between the tuning parameters, i.e. ratio = $\gamma/\lambda$. The upper $x$ axis refers to the step size in the Frugal algorithms.}
\label{fig:2}
\end{figure}
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[width = 0.5\textwidth]{reserr_normal_RUMIQE_DUMIQE_cao_xmQ_q_0_5_switch_tau_100_DQTRE} & \includegraphics[width = 0.5\textwidth]{reserr_normal_RUMIQE_DUMIQE_cao_xmQ_q_0_5_switch_tau_500_DQTRE} \\
\includegraphics[width = 0.5\textwidth]{reserr_normal_RUMIQE_DUMIQE_cao_xmQ_q_0_7_switch_tau_100_DQTRE} & \includegraphics[width = 0.5\textwidth]{reserr_normal_RUMIQE_DUMIQE_cao_xmQ_q_0_7_switch_tau_500_DQTRE} \\
\includegraphics[width = 0.5\textwidth]{reserr_normal_RUMIQE_DUMIQE_cao_xmQ_q_0_9_switch_tau_100_DQTRE} & \includegraphics[width = 0.5\textwidth]{reserr_normal_RUMIQE_DUMIQE_cao_xmQ_q_0_9_switch_tau_500_DQTRE}
\end{tabular}
\caption{Normal distribution switch case: The left and right columns show results for $T=100$ and $T=500$, respectively. The rows from top to bottom show results when estimating quantile $Q_n(q=0.5),\,\, Q_n(q=0.7)$ and $Q_n(q=0.9)$, respectively. Ratio refers to the ratio between the tuning parameters, i.e. ratio = $\gamma/\lambda$. The upper $x$ axis refers to the step size in the Frugal algorithms.}
\label{fig:3}
\end{figure}
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[width = 0.5\textwidth]{reserr_chisq_RUMIQE_DUMIQE_cao_xmQ_q_0_5_periodic_tau_100_DQTRE} & \includegraphics[width = 0.5\textwidth]{reserr_chisq_RUMIQE_DUMIQE_cao_xmQ_q_0_5_periodic_tau_500_DQTRE} \\
\includegraphics[width = 0.5\textwidth]{reserr_chisq_RUMIQE_DUMIQE_cao_xmQ_q_0_7_periodic_tau_100_DQTRE} & \includegraphics[width = 0.5\textwidth]{reserr_chisq_RUMIQE_DUMIQE_cao_xmQ_q_0_7_periodic_tau_500_DQTRE} \\
\includegraphics[width = 0.5\textwidth]{reserr_chisq_RUMIQE_DUMIQE_cao_xmQ_q_0_9_periodic_tau_100_DQTRE} & \includegraphics[width = 0.5\textwidth]{reserr_chisq_RUMIQE_DUMIQE_cao_xmQ_q_0_9_periodic_tau_500_DQTRE}
\end{tabular}
\caption{$\chi^2$ distribution periodic case: The left and right columns show results for $T=100$ and $T=500$, respectively. The rows from top to bottom show results when estimating quantile $Q_n(q=0.5),\,\, Q_n(q=0.7)$ and $Q_n(q=0.9)$, respectively. Ratio refers to the ratio between the tuning parameters, i.e. ratio = $\gamma/\lambda$. The upper $x$ axis refers to the step size in the Frugal algorithms.}
\label{fig:4}
\end{figure}
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[width = 0.5\textwidth]{reserr_chisq_RUMIQE_DUMIQE_cao_xmQ_q_0_5_switch_tau_100_DQTRE} & \includegraphics[width = 0.5\textwidth]{reserr_chisq_RUMIQE_DUMIQE_cao_xmQ_q_0_5_switch_tau_500_DQTRE} \\
\includegraphics[width = 0.5\textwidth]{reserr_chisq_RUMIQE_DUMIQE_cao_xmQ_q_0_7_switch_tau_100_DQTRE} & \includegraphics[width = 0.5\textwidth]{reserr_chisq_RUMIQE_DUMIQE_cao_xmQ_q_0_7_switch_tau_500_DQTRE} \\
\includegraphics[width = 0.5\textwidth]{reserr_chisq_RUMIQE_DUMIQE_cao_xmQ_q_0_9_switch_tau_100_DQTRE} & \includegraphics[width = 0.5\textwidth]{reserr_chisq_RUMIQE_DUMIQE_cao_xmQ_q_0_9_switch_tau_500_DQTRE}
\end{tabular}
\caption{$\chi^2$ distribution switch case: The left and right columns show results for $T=100$ and $T=500$, respectively. The rows from top to bottom show results when estimating quantile $Q_n(q=0.5),\,\, Q_n(q=0.7)$ and $Q_n(q=0.9)$, respectively. Ratio refers to the ratio between the tuning parameters, i.e. ratio = $\gamma/\lambda$. The upper $x$ axis refers to the step size in the Frugal algorithms.}
\label{fig:5}
\end{figure}
Figures \ref{fig:2} to \ref{fig:5} illustrate the results of our experiments. For the normal distribution period case (Figure \ref{fig:2}), we see that the QEWA algorithm outperforms all the algorithms in the literature.
In accordance with the analysis in Section \ref{sec:qea}, the QEWA algorithm performed the best using a small value of the ratio $\gamma/\lambda$. The Cao et al. algorithm struggled with numerical problems for some choices of the tuning parameters and therefore some of the curves are short.
For the normal distribution switch case (Figure \ref{fig:3}), we see that the QEWA algorithm again outperforms all the algorithms in the literature. Again we see that the QEWA performs best using a small value of the ratio $\gamma/\lambda$.
For the $\chi^2$ distribution cases we see that the QEWA algorithm also here outperforms the other algorithms. For $q=0.9$, the QEWA algorithm documents competitive results to the best performing alternative algorithms. Also here a small value of the ratio $\gamma/\lambda$ is the preferable choice.
Among the alternative algorithms there are no consistency in which algorithm are closest to the performance of the QEWA, but overall the DUMIQE and DQTRE seem to be closes. However, all the alternative algorithms suffer with significantly poorer results than the QEWA for at least some cases. E.g. DQTRE performs poorly when estimating quanties in the tails ($q = 0.9$) and DUMIQE for the switch cases.
\begin{table}[h]
\centering
\begin{tabular}{cccc}
& $q = 0.5$ & $q = 0.7$ & $q = 0.9$ \\ \hline
$T=100$ & 1.4278 & 1.5279 & 1.7646\\
$T=500$ & 1.4233 & 1.5433 & 1.7342\\
\end{tabular}
\caption{Normal distribution periodic case: Root mean squared estimation error for the selection algorithm \cite{guha2009stream}.}
\label{tab:2}
\end{table}
\begin{table}[h]
\centering
\begin{tabular}{cccc}
& $q = 0.5$ & $q = 0.7$ & $q = 0.9$ \\ \hline
$T=100$ & 2.0541 & 2.3171 & 2.5479 \\
$T=500$ & 2.0947 & 2.3489 & 2.5427 \\
\end{tabular}
\caption{Normal distribution switch case: Root mean squared estimation error for the selection algorithm \cite{guha2009stream}.}
\label{tab:3}
\end{table}
\begin{table}[h]
\centering
\begin{tabular}{cccc}
& $q = 0.5$ & $q = 0.7$ & $q = 0.9$ \\ \hline
$T=100$ & 1.4441 & 1.7423 & 2.4316\\
$T=500$ & 1.4386 & 1.7273 & 2.6951\\
\end{tabular}
\caption{$\chi^2$ distribution periodic case: Root mean squared estimation error for the selection algorithm \cite{guha2009stream}.}
\label{tab:4}
\end{table}
\begin{table}[h]
\centering
\begin{tabular}{cccc}
& $q = 0.5$ & $q = 0.7$ & $q = 0.9$ \\ \hline
$T=100$ & 2.0367 & 2.3913 & 3.3717\\
$T=500$ & 2.0462 & 2.4137 & 3.1166\\
\end{tabular}
\caption{$\chi^2$ distribution switch case: Root mean squared estimation error for the selection algorithm \cite{guha2009stream}.}
\label{tab:5}
\end{table}
Tables \ref{tab:2} to \ref{tab:5} show results for the selection algorithm \cite{guha2009stream}. The algorithm does not have any tuning parameters and the results thus are presented in tables. We see that QEWA outperforms the selection algorithm with a clear margin for all the different cases.
In summary the QEWA algorithm outperforms all the different state-of-the-art algorithms from the literature. Best performance is achieved using a small value of the ratio $\gamma/\lambda$.
\section{Real-life Data Experiments -- Concept Drift Detection}
\label{sec:real-life}
In most challenging data prediction tasks, the relation between input and output data evolves over time. Thus if static relationships are assumed, prediction performance will degrade with time. In the field of machine learning and data mining this phenomenon is referred to as concept drift \cite{gama2014survey}. Different strategies have been suggested to detect when the performance of the predictive model degrades and thus should be retrained/updated \cite{gama2014survey}. Current state-of-the-art strategies monitor the \textit{average} predictive error, but for real-life applications it is often more relevant to control that the prediction error rarely \textit{goes above} some critical threshold. In this example we demonstrate how to perform concept drift detection and adaptation on such a critical threshold by tracking an upper quantile of the prediction error distribution, e.g. the 80\% quantile. As an application domain, we investigate the case of efficient control of indoor climate.
Heating, ventilation and air conditioning (HVAC) systems typically control indoor climate by reacting on the current room conditions such as indoor temperature. However, given the time required for a HVAC system to adjust to changes in the indoor climate, such strategies always will lag behind resulting in poor control of indoor climate and energy usage. This raises the need for building models that forecast future indoor climate temperature and use this as input to the HVAC system. Zamora-Mart{\'\i}nez et al. \cite{zamora2014line} propose to use artificial neural network (ANN) models to forecast future indoor temperature based on a total of 20 features including outdoor climate variables such as temperature and precipitation amounts and indoor climates variables such as CO$_2$ level. Since more observations are received with time and the relation between input and output may evolve with time, the model is retrained in an online manner. The authors however do not take advantage of concept drift detection in order to efficiently decide when to retrain the model.
We now demonstrate how the suggested quantile estimator in this paper can be used for concept drift detection for the online indoor temperature forecasting problem described above. We consider the same dataset as in \cite{zamora2014line} where new observation of input and output variables is received every 15 minutes. We forecasted indoor temperature 15 minutes into the future using an autoregressive (AR) model of order one. In addition to the current indoor temperature, the current value of the other 20 features were used as input to the forecasting model. Given the large number of features, regularization of the model parameters was required to get a reliable forecasts and we relied on LASSO regularization \cite{friedman2010regularization}\footnote[3]{This model is a simple and natural forecasting model, but other and more advanced machine learning models that predict on the continuous scale, like ANN models, could also be used.}.
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[width = 0.5\textwidth]{Demonstrate_room1} & \includegraphics[width = 0.5\textwidth]{Demonstrate_room2}
\end{tabular}
\caption{The left and right panels refer to dining room and bed room, respectively. The x-axis refers to the number of days since the observation started. The gray curves show the forecasting error predicting 15 minutes into the future. The red curves show the linear trends in the forecasting error.}
\label{fig:6}
\end{figure}
First, we trained the LASSO AR model based on eight days of observations and used the model to predict 15 minutes into the future each time a new observation was received. The results are shown in Figure \ref{fig:6}. The figure demonstrates that if the model is not retrained after day eight, the forecasting error gradually increases with time (the red line). In other words, the data is subject to concept drift and the forecasting model should be retrained as more observations are received. Instead of retraining the model regularly according to a fixed periodicity which is clearly ineffective, a sophisticated approach consists of retraining the model only if concept drift is detected.
We now build a concept drift and model retraining procedure based on quantile tracking. We required that the indoor temperature forecasting error rarely should go above two degrees centigrade. We used the QEWA estimator to track the 80\% quantile of the forecasting error data stream (the gray curves in Figure \ref{fig:6}). If the quantile estimate went above two degrees centigrade, the model was retrained. We trained the model for the first time after 24 hours of observations. The results are shown in Figure \ref{fig:7}. After the initial training after 24 hours of observations, the $80\%$ quantile estimate of the forecasting error distribution went above two degrees three times and each time the model was retrained. The results demonstrates that by a few selected retrainings of the model, the forecasting error is controlled, indicated by a horizontal linear trend (red curves) in Figure \ref{fig:7}.
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[width = 0.5\textwidth]{Retrainalg_room1} & \includegraphics[width = 0.5\textwidth]{Retrainalg_room2}
\end{tabular}
\caption{The left and right panels refer to dining room and bed room, respectively. The x-axis refers to the number of days since the observation started. The gray curves show the forecasting error predicting 15 minutes into the future. The blue curves show tracking of the 80\% quantiles of the forecasting error data streams. The black dots along the x-axis show when the model was retrained. The red curves show the linear trends in the forecasting error.}
\label{fig:7}
\end{figure}
In conclusion, the example demonstrates how the suggested quantile estimator can be useful for concept drift detection and model adaptation.
\section{Closing remarks}
The exponentially weighted moving average of observations is known to be the state-of-art estimator to track the expectation of dynamically varying data stream distributions. In this paper, we have presented an incremental quantile estimator that is in fact a generalized exponential weighted moving average estimator. To the best of our knowledge, this is the first quantile estimator in the literature that falls within this well-known class of efficient estimators. The experiments show that the estimator outperforms state-of-the-art quantile estimators in the literature.
We demonstrate how tracking of quantiles has application in the field of machine learning. More particularly, we show how the suggested estimator can be used for tracking quantiles of the prediction error distribution in order to detect when a machine learning model should be retrained.
A potential ally for future research is to extend the QEWA estimator to simultaneously track multiple quantiles. One could of course, just run the QEWA estimator for each quantile of interest, but this could potentially lead to a violation of the monotone property of quantiles. The monotone property of quantiles, refers to the requirement that an estimate of a higher quantile should be always bigger than an estimate of a lower quantile e.g. the 50\% quantile always be above the 30\% quantile.
\clearpage
\bibliographystyle{plain}
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