text:
\begin{document}
\title[]{The Classical Inverse Problem for Multi-Particle Densities in
the Canonical Ensemble Formulation}
\author{}
\address{}
\email{}
\thanks{}
\author{Irina Navrotskaya}
\curraddr{}
\email{}
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\keywords{}
\thanks{}

\begin{abstract}
We provide sufficient conditions for the solution of the classical
inverse problem in the canonical distribution for multi-particle
densities.  Specifically, we show that there exists a unique potential
in the form of a sum of $m$-particle ($m\geq 2$)
interactions producing a given
$m$-particle density.  The existence and uniqueness of the solution to the multi-particle inverse problem is essential for the numerical simulations of matter using effective potentials derived from structural data.  Such potentials are often employed in coarse-grained modeling.
The validity of the multi-particle inverse conjecture also has implications for liquid state theory.  For example, it provides the first step in proving the existence of the
hierarchy of generalized Ornstein-Zernike relations.
For the grand canonical distribution, the multi-particle inverse problem has been solved by Chayes and Chayes \cite{ChCh84}.  However, the setting of the canonical ensemble presents unique challenges arising from the impossibility of uncoupling interactions when the number of particles is fixed.
\end{abstract}

\maketitle

\section{Introduction\label{intro}}

Consider a system of $N$ identical particles with coordinates $x_{1},...,x_{N}$ in
some complete $\sigma $-finite measure space $(\Lambda; dx)$ with non-zero measure $dx$.  In the simplest case, $\Lambda \subset \mathbb{R}^{n},n=1,2$, or $3$, $x_{i}$ is the position of
the $i^{th}$ particle, and $dx$ is the Lebesgue measure.  However, the  results here are not limited to this case and apply whenever the conditions stated in the paper are satisfied.  Particularly,
$(\Lambda; dx)$ may be phase space and may include internal variables.  The completion of the product measure
$dx^{\otimes k}$ on
$\Lambda^{k}$ for $1\leq k\leq N$, is denoted by $d^{k}x$.

We begin with a somewhat formal description of the problem.
The total potential of the system has the form $W+U$, where
$W(x_{1},...,x_{N})$ is a fixed scalar internal potential,
and $U$ is an additional internal or external potential.  Since the particles in the system are identical, $W$ and $U$ are required to be symmetric functions.

For $1\leq m \leq N$, the classical $m$-particle density in the canonical distribution is defined (up to a multiplicative constant) as \cite{Hansen86}
\begin{equation}
\rho^{(m)}_{U}(x_{1},...,x_{m})=\frac{\int_{\Lambda
^{N-m}}e^{-W-U}dx_{m+1}\cdots dx_{N}}{Z(U)},  \label{1}
\end{equation}
where
\begin{equation}
Z(U)=\int_{\Lambda ^{N}}e^{-W-U}d^{N}x  \label{2}
\end{equation}
is the canonical partition function.  (The conditions ensuring that
$0 < Z(U) < \infty $ will be specified later.)  Throughout the paper the inverse temperature $\beta$ is taken to be $1$.

The $m$-particle inverse problem investigated here is whether, given a symmetric and positive function  $\rho^{(m)}$ on $\Lambda ^{m}$ with $\int_{\Lambda^{m}}\rho^{(m)}d^{m}x=1$, there exists a unique\footnote{The uniqueness of $U$ implies uniqueness of $u$.  See Theorem \ref{th_uniq}. } potential $U$ of the form
\begin{equation}
U(x_{1},...,x_{N})=\sum_{1\leq i_{1}<\cdots <i_{m}\leq
N}u(x_{i_{1}},...,x_{i_{m}})\quad\text{a.e.},  \label{3}
\end{equation}
such that $\rho^{(m)}_{U}=\rho^{(m)}$ a.e. on $\Lambda^m$.
In (\ref{3}), $u$ is a symmetric function on $\Lambda ^{m}$, so $U$ is sought as a sum of symmetric $m$-particle interactions.  A generalization of this problem to systems of several species (mixtures) will be addressed in a separate paper.

When $m=1$ the inverse problem originates from density functional theory for inhomogeneous fluids \cite{Evans79}.  It was solved for this case
by Chayes et al.
\cite[Sections 2 and 8]{ChChLi84}, for both the canonical and grand canonical
ensembles.  When $m\geq 2,$ but \textit{only} in the grand canonical ensemble, it was solved by Chayes and Chayes \cite{ChCh84}
when $\Lambda $ has finite $dx$ measure.

An immediate application of the multi-particle ($m\geq 2$)
inverse problem is to the
numerical modeling of liquid solutions and soft matter, particularly
coarse-grained (CG) modeling.  In a typical CG simulation, the structural data on CG sites (such as a set of pair or higher-order structural correlation functions
\cite{Noid13}) are "inverted" to find two- or multi-particle interactions between CG sites.  The existence and uniqueness of such interactions have been implicitly assumed by the long established numerical methods designed for calculating effective potentials from structural data, such as iterative Boltzmann inversion (IBI) \cite{Muller02} and inverse Monte Carlo (IMC) \cite{LyLa95}.  (For the description of basic methods for deducing CG potentials and a survey of literature on the subject, see a recent review by Noid and numerous references therein \cite{Noid13}.)

However, to the best of our knowledge, the inversion procedure has
never been justified for the canonical ensemble, even though numerical
simulations are often performed in this setting.  Instead, it has been erroneously assumed that the conclusions of \cite{ChCh84} are equally valid for the canonical distribution.
This is a serious misconception that has been persisting in literature for decades.  The setting of the grand canonical ensemble eliminates major difficulties that arise when the number $N$ of particles is fixed.

The existence (and differentiability) of the map between $m$-particle densities and $m$-particle interaction potentials  also has interesting applications to liquid state theory.  For example, it implies the existence of a hierarchy of generalized Ornstein-Zernike (OZ) relations connecting
$2m$-particle "direct correlation function" and $2m$-,...,$m$-particle densities.  (This will be the subject of a future paper.)

In this article, we provide sufficient conditions for the existence and uniqueness of
the solutions to the inverse problem in the canonical distribution when $m\leq N$.  These results are summarized in Theorem \ref{Th_ex_un_sol}.  (Note, that for $m=N$, the solution is trivial.  Namely, $U=-\log \rho^{(N)} -W+C$, where $C\in\Bbb{R}$ is any constant.)
To the best of our knowledge, there is no mathematical treatment of
the $m\geq 2$ case in that setting.  The uniqueness
-but not existence- was proved by Henderson \cite{He74} when $m=2.$

Similarly to Chayes et al. \cite{ChCh84,ChChLi84} we use a variational procedure.  Some of the arguments are adopted (after a suitable modification) from \cite[Sections 2 and 6]{ChChLi84}.  However, a number of issues have to be treated differently due to difficulties arising from the coupling of interactions.  In addition to the variational arguments, the verification
of the multi-particle inverse conjecture for the canonical ensemble involves non-trivial measure-theoretical problems.  These problems were addressed by
Navrotskaya and Rabier in \cite{Rabier15} where they
were formulated as measure-theoretical properties of
generalized $N$-means from $U$-statistics.  We will refer to
the results established in \cite{Rabier15} when needed.

\section{Preliminaries\label{prelim}}

In this section we introduce some terminology and state the assumptions.

All functions considered take values in the extended real line $\overline{\Bbb{R}}$, i.e. in $[-\infty,+\infty]$, unless specified otherwise.  The complement of a set $E\subset \Lambda^k$ is denoted $E^c=\Lambda^k\setminus E$, and $\left | E \right | =\int_{E} d^k x$ is the
$d^{k} x$ measure of a measurable set $E\subset\Lambda^{k}$.  "Almost everywhere" ("a.e.") is always understood relative to the measure $d^k x$, with $k$ obvious from the context, and the same is true regarding the measurability of functions.  Subsets of $\Lambda^k$ of $d^k x$ measure $0$ will be called \textit{null} sets.   Since reference will be very often made to the complements of the
null sets, we will call such complements \textit{co-null}.

We shall assume that $\rho^{(m)}$ is the $m$-variable reduction to $\Lambda^{m}$ of some a.e. positive and symmetric probability density $P$ on $\Lambda ^{N}$, i.e. $P>0$ and symmetric a.e., $\int_{\Lambda ^{N}}Pd^{N}x=1$, and
\begin{equation}
\rho^{(m)}(x_{1},...,x_{m})=\int_{\Lambda
^{N-m}}P(x_{1},...,x_{N})dx_{m+1}\cdots dx_{N}~\text{a.e. on $\Lambda^m$.}
\label{4}
\end{equation}
It follows from the properties of $P$ that $\rho^{(m)}$ is a.e. positive and symmetric, $\int_{\Lambda^{m}} \rho^{(m)}d^{m}x=1$, and if integration in (\ref{4}) is performed with respect to any set of $N-m$ variables, the function so obtained is
$\rho^{(m)}$ evaluated at the remaining $m$ variables.

The fixed internal potential $W$ is assumed to be an a.e. finite and
symmetric measurable function on $\Lambda^{N}$.  Particularly,
$| W^{-1}(\infty) |=0$,
which means that our conditions exclude hard core interactions.
The absence of hard cores is also a main assumption
in \cite{ChChLi84}, but they were allowed in \cite{ChCh84}.

We will also require that
\begin{equation}
(W+\log P)_{+}\in L^{1}(\Lambda ^{N};Pd^{N}x).  \label{7a}
\end{equation}
The quantity $\int_{\Lambda^N}(W+\log P) P d^{N}x$ is an
analogue (in the canonical ensemble) of one of the two
key functionals employed in the density functional
theory \cite{Evans79}.  Therefore, condition (\ref{7a})
assures that this functional is well-defined in some sense.
On these grounds a similar condition was imposed in \cite{ChCh84}.
\section{Existence and Uniqueness}
\label{sec:ex_uniq}

Similarly to \cite{ChCh84,ChChLi84}, a variational method is used to prove the existence of solutions to the inverse problem.  Consider the functional
\begin{equation}
\mathcal{F}_{P}(V):=\frac{e^{-\int_{\Lambda ^{N}}VPd^{N}x}}{Z(V)},  \label{8}
\end{equation}
with $Z$ as in (\ref{2}), defined on a set
\begin{multline}
\mathcal{V}_{P}:=\{V\in L^{1}(\Lambda ^{N};Pd^{N}x),e^{-V}\in L^{1}(\Lambda
^{N};e^{-W}d^{N}x),  \label{9} \\
V(x_{1},...,x_{N})=\sum_{1\leq i_{1}<\cdots <i_{m}\leq
N}v(x_{i_{1}},...,x_{i_{m}})\text{ a.e. on $\Lambda^{N}$},\\
\text{where $v$ is a symmetric real-valued measurable function on $\Lambda^m$}\}.
\end{multline}

Note that
$V\in\mathcal{V}_P$ is a.e. finite and symmetric.

The appearance of a functional in the form of (\ref{8}) in a
variational proof of the inverse problem is not accidental.  Disregarding the terms independent
of $V$, $-\log \mathcal{F}_P(V)$ is the
Kullback and Leibler mean information for discrimination between two probability distributions \cite{Kullback51}.  As applied to (\ref{8}), the mean information is a measure of overlap between the distributions characterized by probability densities $P$ and $e^{-W-V}/Z(V)$.  This quantity is also often referred to as \textit{relative entropy} \cite{Sh08}.
Murtola et al. \cite{Murtola09} were the first to notice the connection between the functionals used by
Chayes et al. in \cite{ChCh84,ChChLi84} and the relative entropy considered by Shell in \cite{Sh08}.

Before applying a variational procedure, we need to ensure that $\mathcal{V}_P\neq \emptyset$.
If $\exp(-W)\in L^{1}(\Lambda^N; d^{N} x)$, then $\mathcal{V}_{P}\neq \emptyset$ because it contains all constant functions.  (Particularly, $\mathcal{V}_{P}\neq \emptyset$ when $W=0$ and $|\Lambda|<\infty$, which is usually the case in the CG simulations described in the introduction.)
However, not to be limited by this option, we will assume, more generally, that $\mathcal{V}_{P}\neq \emptyset$.

In this section we will prove that there exists a unique maximizer (up to an additive constant) of $\mathcal{F}_{P}$ on $\mathcal{V}_{P}$.  Moreover, every such maximizer solves the inverse problem.  This settles the question about the existence of solutions on $\mathcal{V}_{P}$.  However, it is not clear why every solution of the inverse problem on $\mathcal{V}_{P}$ should be a maximizer of $\mathcal{F}_{P}$, and so such a solution may not be unique.  The uniqueness problem is more easily solved on a smaller set
$\mathcal{V}_{\rho _{(m)}}\subset \mathcal{V}_{P}$ (the inclusion will be proven shortly) defined by
\begin{multline}
\label{10}
\mathcal{V}_{\rho _{(m)}}:=\{V:e^{-V}\in L^{1}(\Lambda
^{N};e^{-W}d^{N}x),\\
V(x_{1},...,x_{N})=
\sum_{1\leq i_{1}<\cdots <i_{m}\leq N}v(x_{i_{1}},...,x_{i_{m}})\text{ a.e. on $\Lambda^{N}$},\\
\text{where $v$ is a symmetric real-valued measurable function on $\Lambda^m$,
$v\in L^{1}(\Lambda ^{m};\rho^{(m)}d^{m}x)$}\}.
\end{multline}

Note that
by Fubini-Tonelli theorem \cite[Theorem 2.39]{Folland99}:
\begin{multline}
\int_{\Lambda^N} V P d^N x=\sum_{1\leq i_{1}<\cdots <i_{m}\leq N} \int_{\Lambda^m} \rho^{(m)}(x_{i_1},...,x_{i_m})
v(x_{i_1},...,x_{i_m})dx_{i_1}\cdots dx_{i_m}\\=
\binom{N}{m}\int_{\Lambda^m} \rho^{(m)} v d^m x \in \Bbb{R}.
\label{eq15}
\end{multline}
Thus, $\mathcal{V}_{\rho _{(m)}}\subset \mathcal{V}_{P}$.  It is important to note that this inclusion is proper in general.  A counterexample is provided in \cite{Rabier15}.

We will see that (assuming $\mathcal{V}_{\rho^{(m)}}\neq \emptyset$) every maximizer of $\mathcal{F}_P$ on $\mathcal{V}_{\rho^{(m)}}$ also solves the inverse problem, and if it exists, this maximizer is unique.  Moreover, every solution of the inverse problem on $\mathcal{V}_{\rho^{(m)}}$ is a maximizer of $\mathcal{F}_P$ on this set.  Thus, the uniqueness of a solution on $\mathcal{V}_{\rho^{(m)}}$ (if one exists) is guaranteed.  However, it seems difficult to prove the existence of maximizers of $\mathcal{F}_P$ on $\mathcal{V}_{\rho^{(m)}}$, and so this set may not contain solutions.  Nevertheless, we will prove that $\mathcal{V}_{\rho^{(m)}}=\mathcal{V}_P$ under an additional assumption about the probability density $P$.    Thus, with this additional assumption, the inverse problem has a unique solution on $\mathcal{V}_{\rho^{(m)}}=\mathcal{V}_P$.  This result is summarized in Theorem \ref{Th_ex_un_sol}.

The functional $\mathcal{F}_P$ is well-defined on $\mathcal{V}_P$ as a positive real number.  Indeed, the numerator in (\ref{8}) is finite and positive, and $Z(V)<\infty$ by definition of $\mathcal{V}_P$.  Also, $V\in L^{1}(\Lambda^{N};P d^{N}x)$, and so $V$ is a.e. finite (using $P>0$ a.e.)  Thus, $Z(V)>0$ since $W$ is a.e. finite as well.
\begin{rem}
\rm{We mention here in passing that the requirement of $W$ being finite a.e. is actually implied by our other assumptions.  In fact, if $W=\infty$ on a set of positive measure, then $P>0$ a.e. implies that $(W+\log P)_{+}=\infty$ a.e. on this set, and therefore $(W+\log P)_{+}\notin L^{1}(\Lambda^{N};P d^{N}x)$.  Also, if $V\in\mathcal{V}_P$, then  $V$ is a.e. finite.  Thus, if $W=-\infty$ on a set of positive measure, then $e^{-V}\notin L^{1}(\Lambda^N;e^{-W}d^{N}x)$, and so
$\mathcal{V}_P=\emptyset$.}
\end{rem}

\begin{lem}
\label{V_P_Convex}
The set $\mathcal{V}_P$ is convex, and $\log \mathcal{F}_P$ is concave on $\mathcal{V}_P$.  More precisely,
\begin{equation}

\log \mathcal{F}_P(\lambda V_1+ (1-\lambda)V_0) \geq \lambda
\log \mathcal{F}_P(V_1)+(1-\lambda)\log \mathcal{F}_P(V_0)
\label{fp_conc}
\end{equation}
for every $\lambda\in(0,1)$ and every $V_0, V_1 \in \mathcal{V}_P$, with equality
if and only if $V_1-V_0$ is a constant a.e.  In particular, if $U_0, U_1\in \mathcal{V}_P$ are two maximizers of $\mathcal{F}_P$, then $U_1-U_0$ is a constant a.e.  The same statement holds true with $\mathcal{V}_P$ substituted with $\mathcal{V}_{\rho^{(m)}}$.
\label{lm5}
\end{lem}
\begin{proof}
Let $V_0, V_1\in \mathcal{V}_P$ and $\lambda\in (0,1)$.  Then $\lambda V_1 +(1-\lambda)V_0$ is in $L^1(\Lambda^N; Pd^N x)$ and has the "sum" structure required for membership in $\mathcal{V}_P$.  By Holder's inequality,
\begin{equation}
\label{Holder}
\int_{\Lambda^N} e^{-\lambda V_1 - (1-\lambda)V_0} e^{-W} d^N x \leq ||e^{-V_1}e^{-W}||_{1}^{\lambda} ||e^{-V_0}e^{-W}||_{1}^{(1-\lambda)}< \infty.
\end{equation}
Thus, $\lambda V_1 +(1-\lambda)V_0 \in \mathcal{V}_P$, and so the set
$\mathcal{V}_P$ is convex.

From (\ref{8}), $\log \mathcal{F}_P(V)=-\int_{\Lambda^N} V P d^N x-\log Z(V)$.  Since the first term is linear in $V$, inequality (\ref{fp_conc}) holds if and only if \newline $\log Z (\lambda V_1 + (1-\lambda) V_0 )\leq \lambda \log Z(V_1) + (1-\lambda) \log Z(V_0)$.  But this inequality is equivalent to (\ref{Holder}).  Moreover,  (\ref{Holder}) is an equality if and only if $V_1-V_0$ is a.e. a
constant \cite[Theorem 6.2]{Folland99}.  (We have used the fact that measures $d^{N}x$ and $e^{-W}d^{N}x$ are absolutely continues with respect to each other.)  The proof for $\mathcal{V}_{\rho^{(m)}}$ is the same.
\end{proof}

\subsection{Existence and uniqueness of maximizers of $\mathcal{F}_P$ on $\mathcal{V}_P$}
This subsection is devoted to the proof of the existence of maximizers of the functional $\mathcal{F}_P$ on the set $\mathcal{V}_P$, except for a single (crucial) issue resolved in \cite{Rabier15}.  We begin with the proof that $\mathcal{F}_{P}$ is bounded.
\begin{lem}
\label{lmF_Pbdd} The functional $\mathcal{F}_{P}$ is bounded on $\mathcal{V}_{P}.$ More precisely, $0<\mathcal{F}_{P}(V)
\leq e^{\int_{\Lambda ^{N}}(W+\log P)_{+}Pd^{N}x}$ for every $V\in \mathcal{V}_{P}.$
\end{lem}

\begin{proof}
By (\ref{7a}) and Jensen's inequality for the measure $P d^N x$ \cite[Theorem 3.3]{RudinRCA},
\begin{multline}
e^{-\int_{\Lambda^N} (V+(W+\log P)_{+})P d^N x}\leq \int_{\Lambda^N} e^{-V-(W+\log P)_{+}}P d^N x \\
\leq\int_{\Lambda^N} e^{-V-W-\log P} P d^N x=\int_{\Lambda^N} e^{-V-W} d^N x.
\end{multline}
Thus, $0<\mathcal{F}_{P}(V)\leq e^{\int_{\Lambda ^{N}}(W+\log P)_{+}Pd^{N}x}<\infty$.
\end{proof}
From now on, we set
\begin{equation}
M:=\sup_{V\in \mathcal{V}_{P}}\mathcal{F}_{P}(V)\in (0,\infty ).  \label{M}
\end{equation}
Let $(V_n)\in \mathcal{V}_P$ be a maximizing sequence for  $\mathcal{F}_{P}$, i.e. $\lim_{n\rightarrow\infty}\mathcal{F}_{P}(V_n)=M$.  (Recall that, by definition of $\mathcal{V}_P$ in (\ref{9}),
$V_n(x_1,...,x_N)=\sum_{1\leq i_{1}<\cdots <i_{m}\leq
N}v_n(x_{i_{1}},...,x_{i_{m}})$ a.e.,
where $v_n$ is a symmetric real-valued measurable function on
$\Lambda^m$.)
If $V\in \mathcal{V}_{P}$ and $C\in\Bbb{R}$, then \newline
${V+C=\sum_{1\leq i_1<\cdots<i_m\leq N}\left [v(x_{i_1},...,x_{i_m}) + {\binom{N}{m}}^{-1}C\right ]\in \mathcal{V}_{P}}$, and $\mathcal{F}_{P}(V+C)=\mathcal{F}_{P}(V)$.  Therefore, by adding a suitable constant to each $V_n$, it can be assumed that $Z(V_n)=1$.  Thus ${1=\int_{\Lambda^N} e^{-V_n}   e^{-W} d^N x}$, i.e. $e^{-V_n/2}\in L^2(\Lambda^N; e^{-W} d^N x)$, and
$|| e^{-V_n/2}||_{2,e^{-W} d^N x}  =1$.  Therefore, by reflexivity of $L^2(\Lambda^N; e^{-W} d^N x)$, there is a subsequence (still denoted by $(V_n)$) and $\Pi \in L^2(\Lambda^N; e^{-W} d^N x)$ such that $(e^{-V_n/2})$ converges weakly to $\Pi$ \cite[Theorem 3.18]{Brezis11}.

\begin{lem}
\label{Pi_nConL2}
$e^{-V_n/2}\rightarrow \Pi$ in $L^2(\Lambda^N; e^{-W} d^N x)$.  (In particular, $\int_{\Lambda^N} \Pi^2 e^{-W}=1$.)
\end{lem}
\begin{proof}
For convenience, let us define $\Pi_n:=e^{-V_n/2}$.  It suffices to show that ${1=\lim_{n\rightarrow\infty}||\Pi_n ||_{2,e^{-W}d^N x}= ||\Pi ||_{2,e^{-W}d^N x}}$.  The weak convergence implies
\begin{equation}
1=\lim_{n\rightarrow\infty}||\Pi_n ||_{2,e^{-W}d^N x}\geq ||\Pi ||_{2,e^{-W}d^N x}.
\label{npi_leq1}
\end{equation}
Let $\varepsilon >0$ be fixed.  There is $n_0\in \Bbb{N}$ such that for each $n\geq n_0$,
\begin{equation}
e^{-\int_{\Lambda^N} V_n P d^N x}=\mathcal{F}_P(V_n) > M(1-\varepsilon).
\label{FgM}
\end{equation}
By Mazur's theorem \cite[Theorem 3.13]{RudinFA}, there is a sequence of convex combinations $(\widetilde{\Pi}_n | n\geq n_0)$, i.e.
\begin{equation}
\widetilde{\Pi}_n=\sum_{k=n_0}^n \lambda_k^{(n)}\Pi_k,\quad
\lambda_k^{(n)}\geq 0\quad\forall ~n_0\leq k\leq n,\quad\text{and
$\sum_{k=n_0}^n \lambda_k^{(n)}=1$},
\end{equation}
such that ${\lim_{n\rightarrow\infty} || \widetilde{\Pi}_n -\Pi ||_{2,e^{-W} d^N x}}=0$.
For every $n\geq n_0$ choose $j_{n},k_{n}\in \{n_0,...,n\}$ such that
$\langle\Pi _{j_{n}},\Pi _{k_{n}}\rangle\leq \langle\Pi _{j},\Pi _{k}\rangle$ for every $j,k\in
\{n_0,...,n\},$ where $\langle\cdot ,\cdot \rangle$ denotes the inner product of $ L^{2}(\Lambda ^{N};e^{-W}d^{N}x).$  Then,
\begin{multline}
\label{tildePn_norm}
||\widetilde{\Pi} _{n}||_{2,e^{-W}d^{N}x}^{2}=\\
\sum_{j,k=n_0}^{n}\lambda_j^{(n)}\lambda _{k}^{(n)}\langle\Pi _{j},\Pi _{k}\rangle\geq \left( \sum_{j,k=n_0}^{n}\lambda
_{j}^{(n)}\lambda _{k}^{(n)}\right) \langle\Pi _{j_{n}},\Pi _{k_{n}}\rangle=\langle\Pi _{j_{n}},\Pi
_{k_{n}}\rangle.
\end{multline}
Since $\langle\Pi _{j_{n}},\Pi _{k_{n}}\rangle=\int_{\Lambda ^{N}}e^{-\frac{
V_{j_{n}}+V_{k_{n}}}{2}-W}d^{N}x,$ (\ref{tildePn_norm}) reads
\begin{equation}
||\widetilde{\Pi} _{n}||_{2,e^{-W}d^{N}x}^{2}\geq Z(Y_{n}),  \label{tPiZY}
\end{equation}
where $Y_{n}:=\frac{V_{j_{n}}+V_{k_{n}}}{2}\in \mathcal{V}_P$ (because $\mathcal{V}_P$ is convex by Lemma \ref{lm5}).  Therefore, using (\ref{FgM}),
\begin{equation}
M\geq \frac{e^{-\int_{\Lambda^N} (V_{j_{n}}+V_{k_{n}})/2}P d^N x}{Z(Y_n)}=
\frac{\left (\mathcal{F}_P(V_{j_{n}})\mathcal{F}_P(V_{k_{n}})\right )^{\frac{1}{2}}}{Z(Y_n)}>\frac{M(1-\varepsilon)}{Z(Y_n)}.
\label{MgM}
\end{equation}
Inequalities (\ref{tPiZY}) and (\ref{MgM}) imply that ${||{\widetilde{\Pi}} _{n}||_{2,e^{-W}d^{N}x}^{2}\geq Z(Y_{n})>1-\varepsilon}$. Therefore, using (\ref{npi_leq1}), ${1\geq ||{\Pi}||_{2,e^{-W}d^{N}x}^{2}=\lim_{n\rightarrow\infty} ||\widetilde{\Pi} _{n}||_{2,e^{-W}d^{N}x}^{2}\geq 1-\varepsilon}$ for any $\varepsilon>0$.  Thus,
$||{\Pi}||_{2,e^{-W}d^{N}x}||=1$.
\end{proof}
It follows from Lemma \ref{Pi_nConL2} (and the fact that  $d^N x$ and $e^{-W} d^N x$ have the same zero measure
sets) that there is a subsequence (still denoted $(V_n)$) such that ${e^{-V_{n}/2}\rightarrow\Pi}$ a.e. on $\Lambda^N$ \cite[Theorem 3.12]{RudinRCA}.  Let us define a co-null set
\begin{multline}
\label{E_def}
\tilde{E}:=\{(x_1,...,x_N)\in\Lambda^N:
\Pi(x_1,...,x_N)\in \Bbb{R}, \\
e^{-V_{n}(x_1,...,x_N)/2}\rightarrow\Pi(x_1,...,x_N)\}.
\end{multline}
In particular, $\Pi\in [0,\infty)$ on $\tilde{E}$.  Let us also define a function $U$ on $\Lambda^N$ as
\begin{equation}
\label{defU}
U(x):=\left\{\begin{array}{ll}
-2\log \Pi(x) &\text{if $x\in \tilde{E}\cap\{y\in\Lambda^N: \Pi(y)>0$}\},\\
\infty & \text{if $x\in \tilde{E}\cap\{y\in\Lambda^N: \Pi(y)=0$}\},\\
0 & \text{if $x\in \tilde{E}^{c}$}.
\end{array} \right .
\end{equation}
It is easy to confirm that $U$ is measurable, and $V_n\rightarrow U$ on $\tilde{E}$.

We will prove that $U\in \mathcal{V}_P$, and $\mathcal{F}_P(U)=M$.  We have
\begin{equation}
1=\int_{\Lambda^N}\Pi^2 e^{-W}d^N x=\int_{\tilde{E}} e^{-U}e^{-W}d^N x=\int_{\Lambda^N} e^{-U}e^{-W}d^N x.
\label{UinVP_1}
\end{equation}
Therefore, $e^{-U}\in L^1(\Lambda^N; e^{-W} d^N x)$.  It remains to prove that $U\in L^1(\Lambda^N; P d^N x)$, $U(x_1,...,x_N)=\sum_{1\leq i_1<\cdots<i_m\leq N} u(x_{i_1},...,x_{i_m})$ a.e. for some finite and symmetric measurable function $u$ on $\Lambda^m$ (i.e. $U\in \mathcal{V}_P$), and $\mathcal{F}_P(U)=M$.
\begin{lem}
\label{U_inL^1}
The function $U$ defined in (\ref{defU})  satisfies $U_{-} \in L^1(\Lambda^N; P d^N x)$.
\end{lem}
\begin{proof}
It follows from (\ref{UinVP_1}) that
\begin{equation}
1=\int_{\Lambda^N} e^{-U-W}d^N x = \int_{\Lambda^N} e^{-U-W-\log P}Pd^N x \geq \int_{\Lambda^N} e^{-U-(W+\log P)_{+}}P d^N x.
\label{UinVP_2}
\end{equation}
The relations $e^t\geq t_{+}$ and $(-t)_{+}=t_{-}$ for $t\in \overline{\Bbb{R}}$ together with (\ref{UinVP_2}) imply that \newline $\int_{\Lambda^N} \left ( U + (W+\log P)_{+}\right )_{-} P d^N x \leq 1$.  Particularly, \newline
$\left (U + (W+\log P)_{+}\right )_{-} \in L^1(\Lambda^N; P d^N x)$.  Next, using $(t+s)_{-}\leq t_{-}+s_{-}$ for $t\in \overline{\Bbb{R}}$ and $s\in{\Bbb{R}}$, with $t= U + (W+\log P)_{+}$ and $s=-(W+\log P)_{+}$, we obtain $U_{-}\leq \left (U + (W+\log P)_{+}\right )_{-} +
(W+\log P)_{+}$.  Therefore, by (\ref{7a}) and the above, $U_{-} \in L^1(\Lambda^N; P d^N x)$.
\end{proof}
\begin{lem}
\label{U_inL^1_2}
The function $U$ defined in (\ref{defU})  is in $\mathcal{V}_P$.  Moreover, $\mathcal{F}_P(U)=M$ with $M$ from (\ref{M}).
\end{lem}
\begin{proof}
Since $U_{-}\in L^1(\Lambda^N;P d^N x)$ by Lemma \ref{U_inL^1}, $U\in L^1(\Lambda^N;P d^N x)$ if and only if $U_{+}\in L^1(\Lambda^N;P d^N x)$.
This is proved below through an estimate that will also yield $\mathcal{F}_P(U)=M$ (after using Theorem 2.3 in
\cite{Rabier15} to show that $U\in \mathcal{V}_P$).

Let $S\subset\Lambda^N$ be the co-null set on which $W\in\Bbb{R}$ and $P\in (0,\infty)$.  For $k,\ell \in \Bbb{N},$ define
\begin{equation}
U^{k}:=\min (U,k),\qquad \phi^{\ell }:=\chi_{S}\min (W+\log P,\ell ).  \label{UkPhil}
\end{equation}
We have that
\begin{equation}
\label{Phi^ell_in_L^1}
0\leq W+\log P-\phi^{\ell} \leq (W+ \log P)_{+} \quad \text{on $S$}
\end{equation}
because on this set
$W+\log P-\phi^{\ell}=0$ when $W +\log P\leq\ell$, and
$0<W+\log P-\phi^{\ell}=(W+\log P)_{+}-\ell < (W+\log P)_{+}$
when $W +\log P>\ell$.
Particularly, $W+\log P-\phi^{\ell}\in L^1(\Lambda^N; P d^N x )$ by (\ref{7a}).
Moreover,

\begin{equation}
\lim_{\ell \rightarrow \infty} \int_{\Lambda^N}(W+\log P-\phi^{\ell})P=0
\label{eqn1}
\end{equation}
by dominated convergence.  Since also $U^{k}=-U_{-}+U_{+}^{k}\in L^1(\Lambda^N; P d^N x )$ by Lemma (\ref{U_inL^1}),
it follows that
$(U^{k}-V_n)/2 -W-\log P + \phi^{\ell }\in L^1(\Lambda^N; P d^N x )$.  Therefore, by Jensen's inequality
\begin{multline}
e^{-\int_{\Lambda^N} \frac{V_n}{2} P d^N x}e^{\int_{\Lambda^N}\left (\frac{U^{k}}{2}-W-\log P + \phi^{\ell }\right ) P d^N x}\leq \\
\int_{\Lambda^N} e^{-\left (\frac{(V_n-U^{k})}{2} +W+\log P - \phi^{\ell }\right )} P d^N x
=
\int_{\Lambda^N} e^{-\frac{V_n}{2}}e^{\frac{U^{k}}{2}+\phi ^{\ell }}e^{-W} d ^N x.
\label{eqn2}
\end{multline}
Next, we will show that $e^{\frac{U^{k}}{2}+\phi _{\ell }}\in L^2(\Lambda^N;e^{-W} d^N x)$.  Indeed, $\phi^{\ell }\leq W + \log P$ on $S$ implies
$e^{\phi^{\ell }-W}\leq P$ a.e., and therefore, $e^{\phi ^{\ell }-W}\in L^1(\Lambda^N; d^N x)$.  Also, $\phi ^{\ell }\leq \ell$,
$U^k\leq k$ imply $U^k +2 \phi ^{\ell }-W\leq k+\ell + \phi ^{\ell }-W$.  Therefore, $e^{U^k+2\phi ^{\ell }-W}\leq e^{k+\ell}e^{\phi ^{\ell }-W}\in
L^1(\Lambda^N; d^N x)$.  Equivalently,
\begin{equation}
e^{\frac{U^k}{2}+\phi ^{\ell }} \in L^2(\Lambda^N; e^{-W} d^N x).
\label{eqn3}
\end{equation}

With $k,\ell$ being held fixed, let $n\rightarrow\infty$.  Then, the leftmost side of (\ref{eqn2}) converges to
$\sqrt{M}e^{\int_{\Lambda^N}(\frac{U^k}{2}-W-\log P +\phi ^{\ell })P d^N x}$.  By (\ref{eqn3}) and weak convergence, the rightmost side of
(\ref{eqn2}) converges to
\begin{equation}
\int_{\Lambda^N} e^{-\frac{(U-U^k)}{2}}e^{\phi ^{\ell }-W} d^N x=\int_{\Lambda^{N}} e^{-\frac{(U-U^k)}{2}}e^{-(W+\log P -\phi ^{\ell })} P d^N x\leq 1,
\label{eqn4}
\end{equation}
where the last inequality follows by $U-U^k\geq 0$, $W+\log P -\phi _{\ell }\geq 0$, and $\int_{\Lambda^N} P=1$.    This yields
\begin{equation}
\label{eqn5}
\sqrt{M}e^{\int_{\Lambda^N}\frac{U^k}{2} P d^N x} e^{-\int_{\Lambda^N} (W+\log P-\phi _{\ell } )P d^N x}\leq 1.
\end{equation}

With $k$ being held fixed, let $\ell\rightarrow\infty$ in (\ref{eqn5}).  Then, by (\ref{eqn1}), we obtain
$\sqrt{M}e^{\int_{\Lambda^N}\frac{U^k}{2} P d^N x}\leq 1$.  Equivalently, using $U^k=-U_{-}+U_{+}^k$,
\begin{equation}
M\leq e^{-\int_{\Lambda^N}U^kP d^N x}=e^{-\int_{\Lambda^N}U_{+}^k P d^N x}e^{\int_{\Lambda^N}U_{-} P d^N x}.
\label{eqn6}
\end{equation}
Now, $U_{+}^k$ is increasing to $U_{+}$ pointwise.  Thus, by monotone convergence, \newline
$\lim_{k\rightarrow \infty}\int_{\Lambda^N}U_{+}^k P d^N x=\int_{\Lambda^N}U_{+}P d^N x$.  Taking $k\rightarrow\infty$ in (\ref{eqn6}) results into
\begin{equation}
0<M\leq e^{-\int_{\Lambda^N}U_{+} P d^N x}e^{\int_{\Lambda^N}U_{-} P d^N x},
\label{eqn7}
\end{equation}

which together with Lemma \ref{U_inL^1} shows that
$U_{+}$ (and therefore $U$) belongs to \newline $L^1(\Lambda^N; P d^N x)$.  In particular, since $P>0$ a.e.,
$U$ is a.e. finite.

To finish the proof, we need to show that $U\in \mathcal{V}_P$.  Since $V_n\rightarrow U$ a.e., it follows from Theorem 2.3 in \cite{Rabier15} that there is function
$u$ on $\Lambda^m$ such that $v_n\rightarrow u$ a.e.
on $\Lambda^m$.   In particular, $u$ is measurable, and
(using Lemma 2.1 in \cite{Rabier15})
$U(x_1,...,x_N)=\sum_{1\leq i_1<\cdots<i_m\leq N} u(x_{i_1},...,x_{i_m})$ a.e. on $\Lambda^N$.  Moreover, redefining $u$ on a set
of zero measure, it can be assumed that $u$ is symmetric
real-valued function on $\Lambda^m$.  Thus, $U$ has the sum structure required for membership
in $\mathcal{V}_P$, and $U\in\mathcal{V}_P$.  Then,
(\ref{eqn7})  and (\ref{UinVP_1})
imply that $\mathcal{F}_P(U)=e^{-\int_{\Lambda^N}U P d^N x}=M$.
\end{proof}

We can now finish the proof of the existence and uniqueness of maximizers of $\mathcal{F}_P$ on $\mathcal{V}_P$.  In fact, all the steps of this proof are already completed, and we essentially just need to cite the previous results.  For convenience, we repeat all the assumptions needed for the validity of the next theorem.  Note that the symmetry of $P$ and $W$ has not been used yet, and will not be needed in the reminder of the proof of the existence and uniqueness of maximizers.
\begin{theor}
\label{th_ex_un}
Let $\rho^{(m)}$ be the $m$-variable reduction of an a.e. positive probability density $P$, i.e. $\rho^{(m)}$ is given by (\ref{4}), and let $W$ be an a.e. finite measurable potential on $\Lambda^N$.  Suppose that
$(W+\log{P})_{+}\in L^{1}(\Lambda^N; Pd^N x)$.  If $\mathcal{V}_P\neq \emptyset$, there is $U\in\mathcal{V}_P$ (with
$U(x_1,...,x_N)=\sum_{1\leq i_1<\cdots<i_m\leq N} u(x_{i_1},...,x_{i_m})$ a.e. ) such that
$\mathcal{F}_P(V)\leq\mathcal{F}_P(U)$ for every $V\in\mathcal{V}_P$.
Furthermore, if $U_1$ and $U_0$ are two maximizers of $\mathcal{F}_P$ on $\mathcal{V}_P$ (or on $\mathcal{V}_{\rho^{(m)}}$, if they exist), then $U_1-U_0$ is a constant a.e., and the same is true for $u_1-u_0$.
\end{theor}
\begin{proof}
By Lemma \ref{U_inL^1_2}, there is $U\in\mathcal{V}_P$ such that
\begin{equation}
\mathcal{F}_P(V)\leq\mathcal{F}_P(U)~\text{for every $V\in\mathcal{V}_P$.}
\label{FPV_L_FPU}
\end{equation}

Suppose, there are $U_0$, $U_1\in\mathcal{V}_P$ (or $\mathcal{V}_{\rho^{(m)}}$) satisfying (\ref{FPV_L_FPU}).  Then, by Lemma \ref{lm5}, there is $C\in\Bbb{R}$ such that
$U_1-U_0=C$ a.e. on $\Lambda^{N}$.  Therefore, by Corollary 2.4 in \cite{Rabier15}, $u_1-u_0={\binom{N}{m}}^{-1}C$ a.e. on $\Lambda^m$.
\end{proof}
\subsection{Maximizers of $\mathcal{V}_P$ and solutions of the inverse problem}
\label{sec_max_sol}
Now we are ready to prove the existence of solutions to the inverse problem.
\begin{theor}
\label{th_max_sol}
Let $P$ be an a.e. positive and symmetric probability density, and $W$ be measurable and a.e. finite and symmetric potential on $\Lambda^N$.  Suppose that for some $U\in\mathcal{V}_P$, $\mathcal{F}_P(V)\leq\mathcal{F}_P(U)$ for every $V\in\mathcal{V}_P$.  Then, $\rho_U^{(m)}=\rho^{(m)}$ a.e.
\end{theor}
\begin{rem}
\rm{According to Theorem \ref{th_max_sol}, every maximizer of $\mathcal{F}_P(V)$ on $\mathcal{V}_P$ is a solution.  Since such maximizers exist by Theorem \ref{th_ex_un}, the solutions of the inverse problem exist under the assumptions of Theorem \ref{th_ex_un}, with the additional requirement for $P$ and $W$ to be a.e. symmetric.}
\end{rem}
\begin{proof}[of Theorem \ref{th_max_sol}]
Let $\xi\in L^{\infty}(\Lambda^m; d^m x)$ be a symmetric real-valued function.
For every $(x_1,...,x_N)\in\Lambda^N$, set
\begin{equation}
\Xi (x_{1},...,x_{N}):=\sum_{1\leq i_{1}<\cdots <i_{m}\leq N}\xi
(x_{i_{1}},...,x_{i_{m}}).  \label{Xi}
\end{equation}
By definition of the norm of $L^{\infty }(\Lambda ^{m};d^{m}x)$, there is a
co-null set $T_{m}$ of $\Lambda ^{m}$ such that $|\xi (x_{1},...,x_{m})|\leq
||\xi ||_{\infty,d^{m}x}$ for every $(x_{1},...x_{m})\in T_{m}$.  By Lemma 2.1 in \cite{Rabier15},
$| \Xi (x_{1},...,x_{N})| \leq\binom{N}{m} ||\xi ||_{\infty,d^{m}x}$ for every $(x_1,...,x_N)$ in some co-null set $T_N\subset\Lambda^N$, and so $\Xi\in L^{\infty }(\Lambda ^{N};d^{N}x)$.

For every $t\in\Bbb{R}$, $U+t\Xi\in\mathcal{V}_P$, and therefore, $\mathcal{F}_P(U+t\Xi)\leq\mathcal{F}_P(U)$.
The function $t\mapsto\mathcal{F}_P(U+t\Xi)$ is smooth.  This is obvious for the numerator in (\ref{8}).  The same property for the denominator follows by a theorem on
differentiation of parameter-dependent integrals \cite[Theorem 2.27]{Folland99}.   ($\Xi\in L^{\infty }(\Lambda ^{N};d^{N}x)$ is used here.)
Thus, $\frac{d}{dt}\mathcal{F}_{P}(U+t\Xi )_{|t=0}=0.$  This gives
\begin{equation}
\label{eq_3.2_1}
\mathcal{F}_P(U)\left[ \frac{\int_{\Lambda^N} e^{-U-W} \Xi d^{N}x}{Z(U)}-
\int_{\Lambda^N} \Xi P d^{N} x   \right ]=0.
\end{equation}
By symmetry, (\ref{eq_3.2_1}) amounts to
$\int_{\Lambda^{m}} \xi (\rho_U^{(m)}-\rho^{(m)} ) d^{m} x=0$.  Further, the set $S_m\subset\Lambda^m$ on which
$\rho^{(m)}$ and $\rho_U^{(m)}$ are both finite and symmetric is co-null.  Choosing
$\xi=\chi_{S_m}\times\func{sign}(\rho_U^{(m)}-\rho^{(m)} )$
(with $\func{sign}(0):=0 $), we obtain
$\int_{\Lambda^{m}} |\rho_U^{(m)}-\rho^{(m)} | d^{m} x=0$,
and so
$\rho_U^{(m)}=\rho^{(m)}$ a.e. on $\Lambda^m$.
\end{proof}
\begin{rem}
\rm{Theorem \ref{th_max_sol} is still true if $\mathcal{V}_P$ is replaced by $\mathcal{V}_{\rho^{(m)}}$, as can be easily verified by simply repeating the arguments in the proof.}
\end{rem}

\subsection{Uniqueness of the solutions to the inverse problem}
It remains to resolve the problem of the uniqueness of solutions.  So far, we have shown that solutions exist on $\mathcal{V}_P$.  However, as was mentioned in the introduction, they may not be unique.  It turns out though,
that, under an additional assumption about $P$, the sets $\mathcal{V}_P$ and $\mathcal{V}_{\rho^{(m)}}$ coincide.  Thus, with this assumption, $\mathcal{V}_{\rho^{(m)}}$ contains solutions.
  According to the next theorem, every solution on $\mathcal{V}_{\rho^{(m)}}$ is a maximizer of $\mathcal{F}_P$, and therefore is unique up to an additive constant by Theorem \ref{th_ex_un}.
\begin{theor}
\label{th_uniq}
Suppose that $U\in\mathcal{V}_{\rho^{(m)}}$, where \newline $U(x_1,...,x_N)=
\sum_{1\leq i_1<...<i_m\leq N} u(x_{i_1},...,x_{i_m})$ a.e.
Suppose also that $\rho_U^{(m)}=\rho^{(m)}$ a.e.  Then, $\mathcal{F}_P(V)\leq\mathcal{F}_P(U)$ for every $V\in\mathcal{V}_{\rho^{(m)}}$.
Consequently, if $U_0, U_1\in \mathcal{V}_{\rho^{(m)}}$,
and $\rho_{U_0}^{(m)}=\rho_{U_1}^{(m)}=\rho^{(m)}$ a.e.,
then $U_1-U_0$ is a.e. a constant, and the same is true for
$u_1-u_0$.
\end{theor}
\begin{proof}
The "Consequently" part of the conclusion follows directly from Theorem \ref{th_ex_un}.  To prove the rest, we need the following lemma:
\begin{lem}
\label{uniq_lemma}
Let us call the functions in the form of (\ref{Xi}) "admissible," for brevity.
Then, $\mathcal{F}_P(U+\Xi)\leq\mathcal{F}_P(U)$ for every admissible $\Xi$.
\end{lem}
\begin{proof}
Since, $\rho_U^{(m)}=\rho^{(m)}$ a.e.,
$\int_{\Lambda^m} \xi (\rho_U^{(m)}-\rho^{(m)})d^m x=0$.  Thus, $\frac{d}{dt}\mathcal{F}_{P}(U+t\Xi )_{|t=0}=0$ (by reversing the steps in the proof of Theorem \ref{th_max_sol}).  Moreover, by Lemma \ref{V_P_Convex},
$t\mapsto \log \mathcal{F}_P(U+t\Xi)$ is a concave (smooth) function on $\Bbb{R}$.  Therefore,
$\mathcal{F}_P(U+t\Xi)$ attains its maximum at $t=0$.  Taking $t=1$, gives
$\mathcal{F}_P(U+\Xi)\leq\mathcal{F}_P(U)$.
\end{proof}

Next, we will show that for every $V\in V_{\rho^{(m)}}$ and every $\varepsilon>0$ there is an admissible $\Xi$ such that $\mathcal{F}_P(V)\leq\mathcal{F}_P(U+\Xi)+\varepsilon$, and so Lemma \ref{uniq_lemma} implies that $\mathcal{F}_P(V)\leq\mathcal{F}_P(U)$ for every $V\in V_{\rho^{(m)}}$.

Let $\varepsilon>0$, and $V\in V_{\rho^{(m)}}$, where $V(x_1,...,x_N)=\sum_{1\leq i_1<\cdots<i_m\leq N} v(x_{i_1},...,x_{i_m})$ a.e.  Define $d_n=\max(v-u,-n)$, so $d_n$ is symmetric, bounded below, and $d_n+u\rightarrow v$ pointwise  and in
$L^{1}(\Lambda^{m};\rho^{(m)} d^m x)$ (by dominated convergence, because $|d_n+u|\leq |v-u|+|u|$).  Let  \newline
$\Delta_n(x_1,...,x_N)=\sum_{1\leq i_1<\cdots<i_m\leq N} d_n(x_{i_1},...,x_{i_m})$.  Then,
\begin{multline}
\int_{\Lambda^N} (U+\Delta_n) P d^N x= \binom{N}{m} \int_{\Lambda^m} (u+d_n) \rho^{(m)} d^{m} x\\
\rightarrow \binom{N}{m} \int_{\Lambda^m} v \rho^{(m)} d^{m} x=\int_{\Lambda^N} V P d^{N} x .
\label{uniq_one}
\end{multline}
Since $e^{-U-\Delta_n}$ is increasing to $e^{-V}$ a.e., the monotone convergence gives
\begin{equation}
\int_{\Lambda^N}e^{-U-\Delta_n-W}d^{N}x\rightarrow
\int_{\Lambda^N}e^{-V-W}d^{N}x.
\label{uniq_two}
\end{equation}
By (\ref{uniq_one}), (\ref{uniq_two}) and (\ref{8}), there is $n_0\in\Bbb{N}$ such that
$|\mathcal{F}_P(V)-\mathcal{F}_P(U+\Delta_{n_0})|<\varepsilon/2$.

Define $\xi_n=\min(d_{n_0},n)$.  Then, $\xi_n$ is symmetric, bounded, and
$u+\xi_n\rightarrow u+d_{n_o}$ pointwise and in $L^{1}(\Lambda^{m};\rho^{(m)} d^m x)$ (by dominated convergence, because $|u+\xi_{n} |\leq |u|+|v-u|$).  Define $\Xi_n(x_1,...,x_N)=\sum_{1\leq i_1<\cdots<i_m\leq N} \xi_n(x_{i_1},...,x_{i_m})$.  Therefore, similarly to (\ref{uniq_one}),
\begin{multline}
\int_{\Lambda^N} (U+\Xi_n) P d^N x= \binom{N}{m} \int_{\Lambda^m} (u+\xi_n) \rho^{(m)} d^{m} x\\
\rightarrow \binom{N}{m} \int_{\Lambda^m}( u+d_{n_0} )\rho^{(m)} d^{m} x=\int_{\Lambda^N} (U+\Delta_{n_0} )P d^{N} x .
\label{uniq_three}
\end{multline}
Because $e^{-U-\Xi_n}\rightarrow e^{-U-\Delta_{n_0}}$ a.e.,
and $e^{-U-\Xi_n}\leq e^{-U+\binom{N}{m}n_0}\in L^{1}(\Lambda^N;e^{-W}d^{N} x)$, the dominated convergence yields
\begin{equation}
\int_{\Lambda^N} e^{-U-\Xi_n} d^{N}x\rightarrow \int_{\Lambda^N} e^{-U-\Delta_{n_0}} d^{N}x.
\label{uniq_four}
\end{equation}
By (\ref{uniq_three}), (\ref{uniq_four}), and (\ref{8}), there is $n_1\in\Bbb{N}$ such that $|\mathcal{F}_P(U+\Delta_{n_0})-\mathcal{F}_P(U+\Xi_{n_1})|<\varepsilon/2$.
Taking $\Xi:=\Xi_{n_1}$, the triangle inequality gives $|\mathcal{F}_P(V)-\mathcal{F}_P(U+\Xi )|<\varepsilon$.
\end{proof}
\begin{rem}
\rm{The above argument will not work if it is only known that $V\in \mathcal{V}_P$ because $(\Delta_n)_{+}\geq (V - U)_{+}$, and so $U+\Delta_n$ may fail to be in $L^{1}(\Lambda^N;P d^{N}x)$.  If we were to define
$\Delta_n=\max(V-U,-n)$, then $(\Delta_n)_{+}=(V - U)_{+}$, but $\Delta_n$ would not have the required sum structure to construct $\Xi$.}
\end{rem}
\begin{rem}
\rm{The uniqueness of solutions on $\mathcal{V}_{\rho^{(m)}}$ can also be proved by the argument used by Chayes et al. \cite[Theorem 2.4]{ChChLi84}.  (The same argument
has also been used by Henderson \cite{He74}.)
However, Theorem \ref{th_uniq} gives us more information because
$\mathcal{V}_{\rho^{(m)}}$ may not contain maximizers, and therefore uniqueness of solutions on $\mathcal{V}_{\rho^{(m)}}$ does
not imply
that every solution is a maximizer, as Theorem \ref{th_uniq} asserts.}
\end{rem}

\subsection{Existence and uniqueness of the solutions to the inverse problem}
As was mentioned earlier in this section, the inclusion
$\mathcal{V}_{\rho^{(m)}}\subset\mathcal{V}_{P}$ is proper in general.  Provided that $u\in \mathcal{V}_{P}$, $u\in
\mathcal{V}_{\rho^{(m)}}$ if and only if $u\in L^{1}(\Lambda^m; \rho^{(m)} d^m x)$.  Theorem 4.5 in \cite{Rabier15} gives us a sufficient condition on $P$ under which $U$ in the form of (\ref{3}) and $U\in L^{1}(\Lambda^N; P d^N x)$ imply that $u\in L^{1}(\Lambda^m; \rho^{(m)} d^m x)$.  Adding this condition to the assumptions made previously, we are able to formulate the following theorem on the existence and uniqueness of solutions to the inverse problem.
Since the solution for $m=N$ is trivial, we will assume
that $m<N$.
\begin{theor}
\label{Th_ex_un_sol}
Let $N\geq 2$, $1\leq m\leq N-1$ be integers, and $\rho^{(m)}$ be an $m$-variable reduction of some a.e. positive and symmetric probability density $P$.  That is, $\rho^{(m)}$ is defined by (\ref{4}).  Suppose that for a.e. $x_N$ in some
subset $B\subset\Lambda$ of positive $dx$ measure, there
is a constant $\gamma(x_N)>0$ such that
\begin{equation}
\label{cond_on_P_sec3}
P(\cdot, x_{N} )\geq \gamma(x_N) \rho^{(N-1)}~ \text{a.e. on $\Lambda^{N-1}$}.
\end{equation}
Let $W$ be an a.e. finite and symmetric measurable potential on $\Lambda^N$.  Then, $\mathcal{V}_P=\mathcal{V}_{\rho^{(m)}}$.  If, in addition, $\mathcal{V}_P\neq\emptyset$ and $(W+\log{P})_{+}\in L^{1}(\Lambda^N; Pd^N x)$, there is $U\in\mathcal{V}_P$ (with
$U=\sum_{1\leq i_1<\cdots<i_m\leq N} u(x_{i_1},...,x_{i_m})$ a.e.)
such that $\rho^{(m)}_{U}=\rho^{(m)}$ a.e., where
$\rho^{(m)}_{U}$ is defined by (\ref{1}) and (\ref{2}).  Moreover, if $U_0$, $U_1\in\mathcal{V}_P$ and $\rho^{(m)}_{U_0}=\rho^{(m)}_{U_1}=\rho^{(m)}$ a.e., then $U_1-U_0$ is a constant a.e., and the same is true for $u_1-u_0$.
\end{theor}

\begin{rem}
\rm{Condition (\ref{cond_on_P_sec3}) holds for arbitrarily close perturbations in $L^{1}(\Lambda^{N} ; d^{N}x)$ of any
symmetric probability density $P$.  Moreover, if the
measure $dx$ is finite, then any bounded symmetric
probability density can be approximated in
$L^{\infty}(\Lambda^{N} ; d^{N}x)$ by bounded symmetric
probability densities satisfying (\ref{cond_on_P_sec3}).
See Theorems 4.7 and 4.8 in \cite{Rabier15}.}
\end{rem}

\begin{proof}[Proof of Theorem \ref{Th_ex_un_sol}]
According to Theorems \ref{th_ex_un} and \ref{th_max_sol}, there is
a unique maximizer of $\mathcal{F}_P$ on
$\mathcal{V}_P$, and this maximizer is a solution of the inverse problem.  Theorem 4.5 in \cite{Rabier15} implies
that if $P$ satisfies
(\ref{cond_on_P_sec3}) then
$\mathcal{V}_P=\mathcal{V}_{\rho^{(m)}}$.  In its turn,
Theorem \ref{th_uniq} asserts that every solution of the inverse
problem on $\mathcal{V}_{\rho^{(m)}}$ is a maximizer.  Thus,
uniqueness of solutions follows.
\end{proof}

\section*{Acknowledgements}

The author is deeply indebted to Patrick J. Rabier for his generous contributions to this work.  The author is glad to express her gratitude to William Noid for first introducing her to the inverse problem.


TITLE: Number of point of intersection of $y=(x+2)$ and $y=(x+2)^2$
QUESTION [1 upvotes]: The solution curve of the differential equation $\displaystyle (x^2+xy+4x+2y+4)\frac{dy}{dx}-y^2=0,x>0$
Passes through the point $(1,3).$ Then
$(1)$ number of point of intersection of  $y=x+2$
$(2)$ number of point of intersection of $y=(x+2)^2$

Try: $$\bigg[(x+2)^2+y(x+2)\bigg]\frac{dy}{dx}=y^2$$
$$y^2\frac{dx}{dy}=(x+2)^2+y(x+2)$$
Could some help me to solve it , thanks

REPLY [1 votes]: $$y^2 \frac{dx}{dy} = (x+2)^2 + y(x+2)\\
\frac{1}{(x+2)^2}\frac{dx}{dy}-\frac{1}{y(x+2)} = \frac{1}{y^2}$$
Observe it is a linear differential equation by treating $\frac{-1}{x+2}$ as some variable $t$. Integration factor will be $\frac{1}{y}$ and solution is
$$\frac{-1}{(x+2)y} = \int \frac{dy}{y^3} = \frac{-1}{2y^2}-c\\
2y=(x+2)(cy^2+1)$$
I think you can proceed.


TITLE: If $\mu(f>0)<1$ then $\lim\limits_{p\to 0^+}||f||_p=0$
QUESTION [2 upvotes]: Show that if $\mu(f>0)<1$ then $\lim\limits_{p\to 0^+}||f||_p=0$
Hint: Use Hölder's inequality.
But I can't see where I should use it.
I'm trying to use it in $\displaystyle\int |f|^p\,d\mu = \int |f|^{p-q}|f|^{q}$, but it didn't help.

REPLY [2 votes]: You're leaving out a hypothesis - as stated the problem is false, it can happen that $||f||_p=\infty$ for every $p>0$.
If you assume that there exists $p>0$ with $||f||_p<\infty$ then yes, the limit is $0$ (not $1$ as has been claimed), and yes this follows from Holder's inequality, even though we're talking about $p<1$ and Holder only applies to $p\ge1$. The $p$ in $||f||_p$ is not the $p$ we're going to use in Holder.
Also any proof that doesn't use the fact that $\mu(f>0)<1$ is wrong; without that assumption the limit could be more or less anything (in general the limit is $\exp(\int\log f)$).
I'm not going to do the whole problem, just an illustration of how the proof is going to work. Say $E$ is the set where $f>0$. Assume that $||f||_1<\infty$.
Note that $f^{1/2} = \Bbb 1_E f^{1/2}$. The Cauchy-Schwarz inequality shows that $$\int f^{1/2}=\int\Bbb 1_Ef^{1/2}\le||\Bbb1_E||_2||f^{1/2}||_2
=\mu(E)^{1/2}\left(\int f\right)^{1/2}.$$Square both sides and you get $$||f||_{1/2}\le\mu(E)||f||_1.$$
Note how we proved something about $||f||_{1/2}$ using Cauchy-Schwarz, which only talks about $p=2$. You can prove similar but more general inequalities using Holder, which will show that $||f||_p\to0$. (The fact that $\mu(E)<1$ is going to be important...)


TITLE: Non-comparison based sorting algorithm based on "Number of unique integer in random generated arrays"
QUESTION [2 upvotes]: I have asked questions about Number of unique integer in randomly generated arrays. Suppose we have $10^6$ random generated numbers, we should have about $~6*10^5$ unique numbers. I wrote a custom sort algorithm based on this assumption.
Pseudocode :
P := 0

RecursiveSort(Array A):
  If length of A <= 1:
    return A

  Find maximum and minimum of A.
  Create Array B of length A.

  P := P + length A                       // For evaluating purpose

  For each number I in A:
    I := (I - min) / (max - min)          // Rescaling I based on max and min of A
    K := I multiply by the length of A        // 0<=K<=Length of A
    K := round(K)                         // Make K an index in Array B
    Put I in B[K].                        // Each B[K] may hold several I

  Create Array Result of length 0.
  For each array L in B:
    For each number I in RecursiveSort(Array L):
       Put I in Result

  Return Result

After several runs, I noticed that  P / length Awas approximately 1.73 for uniform distribution and 2.19 for normal distribution. Why is that so?
For uniform distribution, after each iterations the size of Array A  reduce by $e$ based on the assumption, shouldn't P / length A be approximately 1.58 ($ 1 + e^{-1} + e^{-2} + e^{-3} + .. + e^{-n}$) ?
Here is actual python code:
import numpy as np
import random
A = []
lim = 1000000
#Generating normal distribution array.
# a_normal = np.random.normal(lim * 1000 / 2, lim * 1000 / 6, lim)
# a_normal = np.random.normal(lim * 1000 / 6, lim * 1000 / 4, lim)
# a_normal = np.random.normal(0, lim , lim)
# A = a_normal.tolist()

for _ in range(lim):
  A.append(random.randrange(lim * 100, lim * 1000 - 1))

P = 0

def custom_sort(A):
  if (len(A)) <= 1:
    return A

  mi = min(A)
  ma = max(A)

  global P
  P += len(A)

  if ma == mi: # if all numbers are the same
    return A

  B = [[] for _ in range(len(a))]
  for I in A:
    K = int((i - mi) / (ma - mi) * (len(a) - 1))
    B[K].append(I)

  result = []
  for L in B:
    for I in custom_sort(L):
      result.append(I)
  return result

sort(A)
print(P / len(A))

REPLY [1 votes]: This is a bucket sort with $k$ (the number of buckets) equal to the input length. It runs in linear time if the input data is uniformly distributed, since all the buckets are small after the first pass.
You can optimize a bucket sort for any input distribution by using that distribution's CDF as the bucketing function; if the input matches the expected distribution, then the CDF values will be uniformly distributed between 0 and 1.
So e.g. if you expect input data to be normally distributed, you could start by computing the mean and variance of the data and then bucketing using a normal CDF with those parameters.


TITLE: How to check if circles intersect each other?
QUESTION [2 upvotes]: I'm trying to do trilateration where I know the coordinates of three known points and the estimates of the radii - not guaranteed to be really precise. My question is, how can I check if the circles actually intersect each other? Does the checking step mentioned in this tutorial make sense, considering the estimate values?

REPLY [1 votes]: If by the checking step you mean the first three lines in the tutorial, it does make sense.  It sounds like your center points are known well, even if the distances are not.  For two circles, each with a range of radii, you can do two checks-one with the minimum radii and one with the maximum.  For the most part, you will then know if they intersect none, some, or all of the time the radii are in that range.  There are pathological cases where the intersection will disappear in the middle.  One example would be center $(0,0)$, distance $(\frac{5}{8},\frac{7}{8})$ and center $(1,0)$ distance $(\frac{1}{2},\frac{7}{4})$.  If the error in you distance estimates is small, this is unlikely.


text:
We are interested in a parameter $\theta_0 \in \Theta \subset \mathcal{R}^p$ that can be characterized as a minimizer of a population loss function
\begin{equation*}
    \theta_0 = \arg \min_{\theta \in \Theta} \E L (w,\theta,g_0),
\end{equation*}
where  $w \in \mathcal{W}$ is a random element and $g_0=g_0(z), z \subset w$ is a nuisance parameter. We focus on a  high-dimensional sparse regime of $\theta$, i.e., we allow its dimension $p$ to exceed the sample size $n$: $p \gg n$, but require $\theta_0$ to be sparse: $$ k = \|\theta_0\|_0 := |\{j: \theta_{0,j} \neq 0\}| \ll n. $$ Given an i.i.d sample $(w_i)_{i=1}^{2n}$, we propose Plug-In Regularized Estimator for $\theta$:
\begin{align*}
    \widehat{\theta} = \arg \min_{\theta \in \mathcal{T}} \frac{1}{n} \sum_{i=1}^n L(w_i,\theta,\widehat{g}) + \lambda \| \theta \|_1,
\end{align*}
where the first-stage estimate $\widehat{g}$ is obtained on an auxiliary sample $S'=(w_i)_{i=n+1}^{2n}$, $\mathcal{T} \subset \mathcal{R}^p$ is a search set and $\lambda$ is $\ell_1$-regularization penalty.  We construct the loss function $L(w,\theta,g_0)$ for a wide class of nonlinear semiparametric models and  establish finite-sample bounds on $\ell_2$ and $\ell_1$ rates of $\widehat{\theta}$.

The \textit{first} main contribution of this paper is to construct the $M$-estimator  loss function for a class of conditional moment restrictions (CMRs). Specifically, the CMR can  depend on the target parameter $\theta$ only via the single index $\Lambda(z)'\theta$, where $\Lambda(z)$ is a $p$-vector function of the covariates $z$ that are being conditioned on. We construct the  loss function in two steps. First, we transform the original CMR into an orthogonal CMR, thereby making it insensitive with respect to the biased estimation of the first stage.  Importantly, we show that the orthogonal CMR preserves the single index structure of the original CMR. Second, the orthogonal $M$-estimator loss is taken to be an indefinite integral of the orthogonal CMR, where the variable of integration is a one-dimensional single index (\eqref{eq:ode}).

The \textit{second} main contribution of this paper is the $\ell_1$-regularized $M$-estimator that converges at the oracle rate, where the oracle knows the first stage nuisance parameter and solves a  parametric problem. We establish the finite-sample bound
\begin{align}
\label{eq:rate}
   \|\widehat{\theta}-\theta_0\|_2 \lesssim_{P} \sqrt{k} O\left(\max\left\{\sqrt{\frac{\log(2p)}{2n}}, g_{n}^2\right\}\right),
\end{align}
where   $g_{n}$ is the $\ell_{2}$ and/or $\ell_{\infty}$-bound on the error of the first stage nuisance parameter. If the plug-in estimate converges with rate $g_{n}=o\left(\frac{\log p}{n}\right)^{1/4},$  the estimator attains the oracle rate. Otherwise, the estimator converges faster than the "na\"ive" plug-in $M$-estimator, whose convergence rate bound scales linearly in $g_n$.

We estimate the target parameter $\theta_0$ differently depending on whether the original moment function is monotone in single index. If monotonicity holds, the orthogonal loss is a convex function of $\theta$.  Then,
we define the estimator as the global minimum of  $\ell_1$-regularized orthogonal loss with the penalty parameter $\lambda$ defined as
\begin{equation}
\label{eq:lambda}
    \lambda \gtrsim_P \sqrt{\frac{2 \log p}{2n})}+ g_n^2
\end{equation}   (Algorithm \ref{def:lasso:intro}).  If monotonicity does not hold, we use the local minimum of the orthogonal loss with an oracle choice of penalty \eqref{eq:lambda} and the search set $\mathcal{T}$ being equal to a shrinking neighborhood of a preliminary estimator (Algorithm \ref{def:lasso:prelim}). We construct the preliminary estimator as a local minimum of the $\ell_1$-regularized orthogonal loss with an aggressive choice of penalty parameter $\lambda \sim g_n$ and the search set being equal to a bounded subset of $\mathcal{R}^p$.

We discover a remarkable property of the $\ell_1$-regularization norm that, unlike other norms, makes it compatible with the first stage regularization bias and sampling error. In the convex case, the $\ell_1$-regularization norm and the parameter's sparsity ensure that the global minimum of the loss $\widehat{\theta}$ belongs to a restricted low-dimensional cone of $\mathcal{R}^p$. On this cone, the first stage bias scales with the sparsity index $k$, not with the total dimension $p$. Therefore, the first stage effect on the curvature (and thus the admissible regularization penalty) is negligible.  In the non-convex case,  the $\ell_1$-penalty ensures that the first stage effect on the curvature has a product form $g_n R$, where $R$ is the radius of the search set in the $\ell_1$-norm. Thus,
the we achieve the second-order dependence by searching for the estimator in a shrinking neighborhood of $\theta$ with $R(n)=O(g_n)$.

 We  apply our theory to estimation of  conditional moment models with missing data, as studied in
\cite{carroll:95,carroll:91,chen:08,lee:95,stepanski:93} (Section~\ref{sec:selection}), agent utilities in games of incomplete information (Section~\ref{sec:strategic}), (e.g. see \cite{bajari:10}
and \cite{bajari:13} among others) and heterogeneous treatment effects with a nonlinear link function. In all these settings, we extend prior work (e.g. \cite{graham:2011}, \cite{doubleml2016},  \cite{bajari:13}) from the low-dimensional target parameter setting to its sparse high-dimensional counterpart. For all three applications, we establish concrete conditions on the complexity of the nuisance parameters under which our two-stage estimator achieves oracle convergence rate.

\paragraph{Organization of the paper} In Section \ref{sec:moment}, we construct an orthogonal $M$-estimator loss function for a class of single-index  CMRs and provide the finite-sample bounds for the Plug-In Regularized Estimator. In Section \ref{sec:applications}, we provide the concrete primitive conditions for the conditional moment models with missing data,  games of incomplete information, and heterogeneous treatment effects with a nonlinear link function.  In Appendix \ref{sec:app_proof}, we provide sufficient conditions for a generic  extremum estimation problem that contains $M$-estimation as a special case. In Appendix \ref{sec:proofs_moment} and \ref{sec:proof-apps}, we verify the conditions of Appendix \ref{sec:app_proof} to prove Theorems \ref{cor:non-ortho-rate} and \ref{thm:main-non-convex} and their Corollaries. In  Appendix \ref{sec:montecarlo}, we demonstrate the performance of our method in Monte-Carlo experiments.

\paragraph{Literature Review}

We build on two broad lines of research: orthogonal semiparametric estimation and  high-dimensional parametric models.

The first line of research  obtains a $\sqrt{N}$-consistent and asymptotically normal estimator of a low-dimensional target parameter  $\theta$ in the presence of a  nonparametric nuisance parameter (\cite{Neyman1959},  \cite{Neyman:1979}, \cite{HardleStoker1989},\cite{bickel:93}, \cite{NeweyStoker},  \cite{Andrews:1994},\cite{Newey1994}, \cite{Robins},  \cite{van:00},
\cite{doubleml2016}, \cite{LRSP},\cite{chernozhukov2018riesz}, \cite{Oprescu:18}).  It is common to estimate the target parameter in two stages, where a first stage estimate of the nuisance parameter is plugged into a sample analog of a mathematical relation that identifies the target, such as a moment condition, a likelihood function,  or the boundary of an identified set (\cite{Semenova:18}). A statistical procedure is called orthogonal (\cite{Neyman1959}, \cite{Neyman:1979}) if it is locally insensitive with respect to the estimation error of the first stage nuisance parameter.  Combining orthogonality and sample splitting, the Double Machine Learning framework of \cite{doubleml2016} and \cite{LRSP} has derived a $\sqrt{N}$-consistent asymptotically normal estimator of the target parameter, while allowing the nuisance parameters to be estimated by modern machine learning methods. We extend this framework to estimate high-dimensional targets and show how to construct an $M$-estimator loss function whose gradient is an orthogonal moment. We also contribute to the literature that derives  orthogonal and efficient scores in semiparametric models. Specifically, we construct a bias correction term for a nuisance parameter that is identified by a general conditional exogeneity restriction, covering
conditional expectation (\cite{Newey1994}) and conditional quantile (\cite{IchimuraNewey:17}) as special cases.

The second line of research establishes oracle rates for a high-dimensional parameter (\cite{belloni:11}, \cite{zhu:17,Zhu2017},\cite{nie2017}). We contribute to that literature by allowing the loss function to depend on a high-dimensional nuisance parameter and highlighting how the need to estimate this  parameter affects sufficient conditions. Building on \cite{Negahban2012}  in the convex case, we argue that the $\ell_1$-regularization norm, unlike other norms the authors considered, is compatible with the first stage error. Building on \cite{loh:13} and \cite{loh2017} in the non-convex case,  we show that the convergence of the local optimum of $\ell_1$-regularized $M$-estimator loss continues to hold in a shrinking neighborhood of the true parameter.

config: arxiv


TITLE: Topology: if for every $A,B \in T$, either $A \subset B$ or $B \subset A$, then the arbitrary union of elements of T is an element of T
QUESTION [4 upvotes]: I'm trying to prove that if $T$ is a collection of subsets of $X$, such that for  every $A,B\in T$, either $B \subset A$ or $A \subset B$, then $\bigcup_{i\in I}A_i \in T$
I know that $B\cup A=A$ if   $B \subset A$ and $B\cup A=B$  if $A \subset B$
but I don't know how to prove the above statement and I can't find a counterexample.

REPLY [2 votes]: Consider the collection of intervals on the real line of the form $(-\infty,q)$, where $q$ is a rational number. This is a countable collection which is isomorphic to the rational numbers as far as ordering goes.
But clearly it isn't closed under arbitrary unions, since taking any irrational number $r$ and consider the intervals bounded by $r$ from this collection.
Perhaps a simpler example is as above, but taking $q$ to be any real number. Then the union of the whole collection is the real numbers, which is not a bounded interval.


TITLE: Negation of "If n is a perfect square, then all of it's p-levels, for a prime factor, is even"
QUESTION [1 upvotes]: I am trying to negate the statement "If $n$ is a perfect square, then all of its p-levels, for a prime factor, is even"
My guess is "If $n$ is not a perfect square, then for at least one of its p-levels, for a prime factor, is odd"
What do you guys think?

REPLY [0 votes]: "If $P$ then $Q$" is like a conditional promise. It is a true promise as long as it is not broken. The only way to break a promise is to have the condition $P$ be true and yet the promised $Q$ be false.


TITLE: Round to n decimal places
QUESTION [0 upvotes]: So, let's say we want to evaluate $\pi$ using  $\sum\limits^\infty_{n=0} \frac{4(-1)^n}{2n+1}$
(or any other series) correct to four decimal places.
(i) Do we compare the series result to 3.1415 or 3.1416 or to the exact value of $\pi$ ?
(ii)    Let's say based on calculation we find $\pi$ correct to four decimal places to be $\sum\limits^{136115}_{n=0} \frac{4(-1)^n}{2n+1}$ . Now, do we round this answer to four decimal places and compare it to one of the above cases in (i) or do we not round the sum and we look at the first four decimal digits of the sum as the result without rounding, which is 3.1415?
(iii)   Which of the following is the correct setup if we apply the alternating series remainder formula $(|R_n|\leq b_{n+1})$  to get the smallest n such that the sum gives $\pi$ correct to four decimal places?
$b_{n+1}<10^{-4}\qquad,\qquad b_{n+1}\leq10^{-4}\qquad,\qquad b_{n+1}<\frac{5}{10^5}\qquad,\qquad b_{n+1}\leq \frac{5}{10^5}$
(iv) Now by using $b_{n+1}\leq10^{-4}$ the smallest number for n is n=39999. Is $\pi$ correct to four decimal places equal to $\pi \approx3.1415$ or $\pi \approx3.1416$  knowing that  $\sum\limits^{39999}_{n=0} \frac{4(-1)^n}{2n+1}\pi \approx3.14156765$
(v) Is n=136115 the smallest number that gives $\pi$ correct to four decimal places? because it is the smallest number after which the sum of the series' first four decimal digits never change?

REPLY [1 votes]: (i) Since the exact value of $\pi$ is unknown (and for a general constant for which, we aim to approximate), the error function is the one determining how many terms of the sequence should be regarded to keep the error below a threshold. Most often, an error bound is achievable instead of an exact error function. In this case
$$
{\pi=\sum_{n=0}^\infty 4\frac{(-1)^n}{2n+1}
\\
e(N)=\sum_{n=N+1}^\infty 4\frac{(-1)^n}{2n+1}
}
$$
hence
$$
e(2N-1){=\sum_{n=2N}^\infty 4\frac{(-1)^n}{2n+1}
\\=
\sum_{n=N}^\infty 4\left[\frac{1}{4n+1}-\frac{1}{4n+3}\right]
\\<
\sum_{n=N}^\infty 4\left[\frac{1}{4n-1}-\frac{1}{4n+3}\right]
\\=\frac{4}{4N-1}
\\\implies |e(N)|<\frac{4}{2N+1}
}
$$
In this case, we wish $|e(N)|<5\times 10^{-5}$ which leads to a sufficient condition of $\frac{4}{2N+1}<5\times 10^{-5}$ or $N\ge 40000$.
(ii and iii)
For small errors and large $N$s, such round-ups are seldom important. However, to measure the error, no round-up is used and $b_{n+1}\leq \frac{5}{10^5}$ is preferred among all.


TITLE: Permutations of two identical particles in two dimensions
QUESTION [3 upvotes]: In three spatial dimensions there are only two possible statistics: Bose-Einstein and Fermi-Dirac. This is the fact related with the statement that first homotopic group of 3-dimensional configuration space is permutations group. In a case of two spatial dimensions there are many types of statistics, which is related to fact that first homotopic group in two dimensions is braid group.
These are the words from Weinberg QFT Vol. 1. Can someone explain how explicitly appear many types of statistic in a case of two dimensions as the consequence of the statements above?

REPLY [3 votes]: The standard treatment of identical particles starts with one-particle states and then imposes symmetry conditions. This is kind of backwards. If you look at a formula like $$|x_1,x_2\rangle = \frac{1}{\sqrt 2}(|x_1\rangle | x_2\rangle \pm |x_1\rangle| x_2\rangle)$$
you are saying that the state is a linear combination of states where particle A is at $x_1$ and particle B is at $x_2$ or vice versa. But what you want to implement is that there is no such thing as particle A and particle B, there's just two particles.
The more honest way is like this. Our particles move in $\mathbb R^k$, so we have $(\mathbb R^k)^n$ for $n$ particles. To make them indistinguishable, identify all points that just differ by permuting coordinates. This is taking the quotient with $S_n$, the permutation group for $n$ objects. For this quotient to be nice1, we have to delete points where two or more particles are in the same place. That is, our configuration space is $$M = ((\mathbb R^k)^n - \Delta)/S_n.$$
Now suppose that we have two particles in the state $|\psi(t=0)\rangle = |x_1,x_2\rangle$. Remember, this means that one particle is at $x_1$ and one particle is at $x_2$. The state evolves with the path integral, $$\langle \phi|\psi(t=T)\rangle = \int D\psi\, \exp(-i\textstyle\int_0^T \mathcal L \, dt)$$
where the integral is of course over all paths.
The normalization of the path integral is set by the initial condition $$\langle\phi|x_1,x_2\rangle = \delta_{\phi,[x_1,x_2]} \tag{1}.$$
Now it is possible to argue that for short times, contributions from paths homotopic to the classical path $\gamma_c$ (i.e. the solution to the Euler-Lagrange equations) dominate. Therefore, the initial condition (1) only tells you about paths homotopic to the classical trajectory. You can define a different propagator by $$\langle \phi|\psi(t=T) \rangle = \int D\psi\, \sum_\alpha \chi(\alpha) \exp(-i\textstyle\int_0^T \mathcal L \, dt)$$
where $\alpha$ runs over the homotopy classes of paths between the initial and final points and $\chi(\alpha)$ are some complex weights. These paths can be labeled (not uniquely, but the ambiguity is inessential) with the fundamental group of $M$. Because of unitarity and the semigroup property of the propagator, we must have $|\chi(\alpha)| = 1$ and $\chi(\alpha)\chi(\beta) =\chi(\alpha\beta)$ the multiplication on the right being in the fundamental group. This is precisely the statement that the $\chi(\alpha)$ are a unitary representation of the fundamental group. This argument is presented in more detail (mostly it's about how to be careful about associating paths with elements of the fundamental group) in Ref. [1].
Since $\Delta$ consists of points verifying conditions like $x_1 = y_1, \ldots, x_k = y_k$, $\Delta$ is the union of spaces of codimension $k$. To affect the fundamental group, you need codimension $2$, therefore $(\mathbb R^k)^n - \Delta$ is simply connected for $k \ge 3$. (See Ref. [2] for a better argument.) Thus in three or more dimensions, $M$ is the quotient of a simply connected space, meaning that its fundamental group is $S_n$. The only unitary representations of $S_n$ are the familiar bosonic $$\chi(\alpha) = 1$$
and fermionic $$\chi(\alpha) = \begin{cases}+ 1 & \alpha \text{ is odd} \\ -1 & \alpha \text{ is even}\end{cases}$$
representations.
For $k = 2$, this argument does not work, and the fundamental group of $M$ is a braid group. Again, see Ref. [2] for a full argument that holds also when the particles move on a general manifold.
To get back to the elementary treatment, in more than two dimensions, $F = (\mathbb R^k)^n - \Delta$ is the universal cover of $M$. Any loop in $M$ lifts to one in $F$, unique up to choice of starting point, with endpoint dependent only on the homotopy class. Thus consider moving our particles around in a homotopically non-trivial way but returning to the original state with one particle at $x_1$ and one at $x_2$. Lift this path so that it starts at $(x_1,x_2)$ (ordered pair!) and ends at $(x_2, x_1)$. Since on the actual configuration space $M$, this path enters the path integral with a $\pm 1$, we can get away with not considering $M$ if we have $|(x_1, x_2)\rangle = \pm |(x_2,x_1)\rangle$.
References

Laidlaw, M. G. G. & DeWitt, C. M. Feynman Functional Integrals for
Systems of Indistinguishable Particles. Phys. Rev. D 3, 1375-1378 (1971).
Birman, J. S. On Braid Groups. Communications on Pure and Applied
Mathematics 22, 41-72 (1969).

1 Here nice means that $M$ is a manifold. If you don't delete $\Delta$ you get something called an orbifold which is a much less nice class of objects.