text:
\section{Introduction}\label{s0}

Let $\{z_i\}_{i\in\calN}$ be a sequence of distinct points in a domain
$\Omega\subseteq \bC$, where $\calN\subseteq \bN$ with $|\calN|\ge 2$. If at
each $z_i$ we
place a particle with charge $a_i$ -- actually, an infinitely long thin rod
perpendicular to the plane with uniform charge distribution -- then up to
complex conjugation and a constant factor the resulting electrostatic force is
given by
\begin{equation}\label{n-f}
f(z)=\sum_{i\in\calN}\frac{a_i}{z-z_i}
\end{equation}
and the logarithmic potential generated by these charged particles is
\begin{equation}\label{n-pot}
U(z)=\sum_{i\in\calN}a_{i}\log\left|1-\frac{z}{z_i}\right|
\end{equation}
(cf.~\cite[p.~10]{Ke}). If $|\calN|=\infty$ then the series in~\eqref{n-f}
and~\eqref{n-pot} converge absolutely for all
$z\in\Omega\setminus\{z_i\}_{i\in\calN}$ provided that certain
growth conditions are satisfied (see \S \ref{s22} below). In this
case $f$ is meromorphic while $U$ is subharmonic in $\Omega$. Since
$\nabla U(z)=\overline{f(z)}$ the critical points of the
potential coincide with the equilibrium points of the field,
that is, points upon which a free electron (or rod) once placed would remain.
An important problem that goes back to Gauss is to describe
the distribution
and the relative geometry of the equilibrium points and charged particles. One
of the main purposes of this paper is to study this problem by means of the
{\em Hausdorff distance} and the notion of {\em weighted multivariate
majorization} that we define in \S \ref{s1} as an extension
of classical and matrix majorization. It is interesting to note that
although our results are purely analytic or geometric in nature, they
are obtained through a combination of the multilinear algebra and
operator theoretic methods developed in \cite{Ma1,Ma2} and \cite{Ma1,Ma2,P},
respectively. The fact that such geometric properties could be established
in this way was actually first noticed in {\em loc.~cit.}, where
long-standing conjectures on the geometry of polynomials were solved by
applying this type of techniques. Below we make extensive use of the key
observation made in the aforementioned papers, namely that studying
zeros and critical points of complex polynomials -- or, more generally,
poles and zeros of logarithmic potentials generated by finite positive charge
configurations -- is equivalent to studying spectra of normal matrices and
their compressions.

The weighted multivariate generalization of vector and matrix majorization
introduced in \S \ref{s1} is a natural preorder on finite sequences of vectors
in Euclidean space induced by the Choquet ordering for nonnegative Radon
measures (Definition~\ref{def-w}). We characterize this
new notion in terms of stochastic matrices and convex functions and we
recover in this way several related results on classical and multivariate
majorization.
In the process we use a Hahn-Banach type argument to give a new simple proof
of a fundamental theorem of Sherman (Theorem~\ref{gen-S}).

In \S \ref{s2} we state most of our main results. These describe the
distribution and the relative location of equilibrium points of logarithmic
potentials for discrete bounded planar configurations of positive
charges and are of three different types. First, we consider $n$ positive
charges and prove that the equilibrium points satisfy $\binom{n}{2}$ weighted
multivariate majorization relations and are in fact uniquely determined
by $n-1$ such relations (Theorems~\ref{t-1}--\ref{t-4}). The higher
de Bruijn-Springer relations thus obtained generalize some of the results of
\cite{Ma1,Ma2} to arbitrary (not necessarily equal) positive charges.
Second, we show that
the relative Hausdorff geometry of the extended set of equilibrium points and
the set of charged particles is controlled by the weighted standard deviation
of the latter (Theorems~\ref{t-5}--\ref{t-6}). For equal charges the bounds
given by these theorems
are much better and also more natural than those predicted by the famous
Sendov conjecture for complex polynomials and its converse,
see, e.g., \cite{Bo2}. As we explain in the sequel \cite{BP2} to this
paper, these results also seem to suggest that the Hausdorff geometry is
governed by natural dispersion measures associated with the given charge
configuration. Finally, we establish weighted
multivariate majorization relations for essentially any bounded discrete set
of positively charged particles with finite total charge for which
equilibrium points are known to exist (Theorems~\ref{t-7}--\ref{t-8}). We
further conjecture that equilibrium points actually always exist under some
mild assumptions (Conjectures~\ref{con-2}--\ref{new-c2} in \S \ref{s42}).

The proofs of the aforementioned results are given in \S \ref{s3}. As we
already pointed out, for finite charge configurations these are based on the
multilinear algebra and matrix theory techniques of \cite{Ma1,Ma2,P}. In the
process we are led to consider several natural related questions, such as a
hierarchy of inertia laws for finite systems of planar charged particles
that we propose and discuss in \S \ref{s-mom} (Conjecture~\ref{gen-mom}).
This would yield a whole new series of weighted generalized
de Bruijn-Springer relations. The techniques mentioned above actually extend
to the case of infinite discrete distributions of charge, which allows us to
make use of operator theoretic methods such as
compressions of compact normal operators on separable Hilbert space in order
to deal with this case. In this
way we arrive at a great many related questions on the interplay between value
distribution of meromorphic functions and variations of discrete spectra of
compact normal Hilbert space operators under finite-rank perturbations. In
\S \ref{s42}--\ref{s43} we give an operator theoretic approach to
the well-known Clunie-Eremenko-Rossi conjecture \cite{CER} dealing with the
existence of zeros of certain Borel series and we consider possible extensions
of this conjecture as well as their potential implications
(Conjectures~\ref{con-CER}--\ref{con-compl}).

\section{Weighted multivariate majorization}\label{s1}

The concept of majorization was first studied by economists
early in the twentieth century as a means for altering the unevenness of
distribution of wealth or income. The majorization preorder on $n$-tuples of
real numbers -- also known as the strong spectral order, vector majorization
or classical majorization -- was used
by Hardy, Littlewood and P\'olya in their study of analytic inequalities
\cite{HLP} and by Schur in his work on the spectra of positive semidefinite
Hermitian operators (cf.~\cite{DK}). Classical majorization has since become
an important tool in analysis, combinatorics and
statistics \cite{MO} and has recently found remarkable applications
to quantum mechanics and entanglement transformations in quantum
computation and information theory (see \cite{Bo1} and references therein).
The strong spectral order essentially quantifies the intuitive notion that
the components of a real $n$-vector $\mathbf x$ are ``less spread out'' than the
components of another such vector $\mathbf y$. Several matrix versions of this
notion have been proposed and studied in various contexts
(cf., e.g., \cite{Da} and \cite{MO}).

In this section we generalize classical and matrix
majorization by introducing a notion of
weighted multivariate majorization that we shall use throughout both this
paper and its sequel \cite{BP2}.

\begin{notation}\label{new-not1}
Let $k\in \bN$. For each $n\in \bN$ we define
{\allowdisplaybreaks
\begin{align}\label{sp-n}
&\calX_{n}^{k}=\left\{(\mathbf x_1,\ldots,\mathbf x_n)\,\,\big|\,\,\mathbf x_i\in \bR^k,
1\le i\le n\right\}\\
&\calA_{n}=\left\{(a_1,\ldots,a_n)\,\,\bigg|\,\,a_i\in (0,1),1\le i\le n,
\sum_{i=1}^{n}a_i=1\right\}\notag\\
&\bX_{n}^{k}=\calX_{n}^{k}\times \calA_{n}\notag
\end{align}
and set
\begin{equation}\label{sp}
\bX^{k}=\bigcup_{n=1}^{\infty}\bX_{n}^{k}.
\end{equation}
Given $X=(\mathbf x_1,\ldots,\mathbf x_m)\in \calX_{m}^k$ we denote by
$X^T=(\mathbf x_1,\ldots,\mathbf x_m)^T$ its ``transpose'', i.e., the ordered
$m$-tuple consisting of the vectors $\mathbf x_1,\ldots,\mathbf x_m$ written in column
form.} We identify $X^T$ with the $m\times k$ real matrix
whose $i$-th row consists of the coordinates of the vector $\mathbf x_i$ in the
standard basis of $\bR^k$. Let finally
$\bM_{m,n}^{rs}$ be the set of all row stochastic $m\times n$ matrices.
\end{notation}

The notion of weighted multivariate majorization defined below
(see Definition~\ref{def-w}) is motivated
by the following theorem of Sherman \cite{Sh}:

\begin{theorem}\label{gen-S}
Let $(X,\mathbf a)\in \bX_{m}^k$ and $(Y,\mathbf b)\in \bX_{n}^k$, where
$X=(\mathbf x_1,\ldots,\mathbf x_m)\in \calX_{m}^k$, $\mathbf a=(a_1,\ldots,a_m)\in \calA_m$,
$Y=(\mathbf y_1,\ldots,\mathbf y_n)\in \calX_{n}^k$ and $\mathbf b=(b_1,\ldots,b_n)\in \calA_n$.
The following conditions are equivalent:
\begin{itemize}
\item[(i)] For any (continuous) convex function $\Phi:\bR^k\rightarrow \bR$
one has
$$\sum_{i=1}^{m}a_{i}\Phi(\mathbf x_i)\le \sum_{j=1}^{n}b_{j}\Phi(\mathbf y_j).$$
\item[(ii)] There exists a matrix $R\in \bM_{m,n}^{rs}$
such that
$$\tilde{X}^T=R\tilde{Y}^T\text{ and }\,\mathbf b=\mathbf a R,$$
where $\tilde{X}^T$ is an $m\times k$ matrix
and $\tilde{Y}^T$ is an $n\times k$ matrix obtained by some (and then any)
ordering of the vectors in $X^T$ and $Y^T$, respectively.
\end{itemize}
\end{theorem}

The special case when $m=n$ and all weights are
equal in Theorem~\ref{gen-S} was long assumed to be an open question and
appears only as an implicit conjecture in \cite[p. 433]{MO}, which is the
definite reference on majorization theory. Although essentially correct,
Sherman's paper used rather complicated arguments and contained a series of
misprints that required several subsequent errata, which may explain why the
status of his proof seems to have remained unclear for such a long period.
Below we present a short and simple proof of Theorem~\ref{gen-S}.

Before embarking on the actual proof let us note that condition (i) in
Sherman's theorem may also be written in terms of the Choquet ordering for
positive measures, a notion that we shall actually use in~\cite{BP2} (see
Definition 9 in {\em loc.~cit.}). Indeed, any pair $(X,\mathbf a)\in\bX_{m}^{k}$
uniquely determines a discrete probability measure $\mu_{_{(X,\mathbf a)}}$ on
$\bR^k$ concentrating a mass $a_i$ at $\mathbf x_i$, $1\le i\le m$. Given two
positive measures $\mu_1$ and $\mu_2$ on $\bR^k$ one says that $\mu_2$
{\em dominates} $\mu_1$ {\em in the Choquet ordering} or that $\mu_2$ is a
{\em dilation} of $\mu_1$, denoted $\mu_1\prec \mu_2$,
if $\mu_1(\Phi)\le \mu_2(\Phi)$ for any continuous convex function $\Phi$ on
$\bR^k$. Theorem~\ref{gen-S} (i) may therefore be rewritten as
$\mu_{_{(X,\mathbf a)}}\prec \mu_{_{(Y,\mathbf b)}}$. We refer to~\cite{BR,CFM,EH,FH,Ph}
for a detailed account of the Choquet ordering and various comparisons of
nonnegative Radon measures and probability distributions in locally convex
separable topological spaces. In particular, in the latter setting
Sherman's theorem may in fact be viewed as a special case of the
Cartier-Fell-Meyer theorem \cite{CFM,Ph,BR} and its subsequent extension due
to Fischer-Holbrook \cite{FH}. The most general result of this type, namely a
basic representation theorem for convex domination of measures, seems to be
contained in \cite{EH}.

\begin{proof}[Proof of Theorem~\ref{gen-S}]
Since the implication (ii) $\Rightarrow$ (i) is an immediate consequence of
Jensen's inequality we shall only focus on the converse statement. Assume that
condition (i) in Theorem~\ref{gen-S} holds,
let $X=(\mathbf x_1,\ldots,\mathbf x_m)\in \calX_{m}^k$, $\mathbf a=(a_1,\ldots,a_m)\in\calA_m$,
$Y=(\mathbf y_1,\ldots,\mathbf y_n)\in \calX_{n}^k$, $\mathbf b=(b_1,\ldots,b_n)\in \calA_n$,
and define the following set
$$\calM(X,Y)=\left\{A=(a_{ij})\in \bM_{m,n}^{rs}\,\,\bigg|\,\,
\mathbf x_i=\sum_{j=1}^{n}a_{ij}\mathbf y_j,\,1\le i\le m\right\}.$$

\begin{lemma}\label{l1}
$\calM(X,Y)$ is a non-empty closed convex subset of $\bM_{m,n}^{rs}$.
\end{lemma}

\begin{proof}
It is clear that $\calM(X,Y)$ is closed and convex. To show that
$\calM(X,Y)\neq \emptyset$ it is enough to prove that if
Theorem~\ref{gen-S} (i) holds then $\mathbf x_i\in \co(\mathbf y_1,\ldots,\mathbf y_n)$ for
$1\le i\le m$, where $\co(\mathbf y_1,\ldots,\mathbf y_n)$
is the convex hull of the vectors $\mathbf y_1,\ldots,\mathbf y_n$. Supposing that
$\mathbf x_l\notin \co(\mathbf y_1,\ldots,\mathbf y_n)$ for some $l\in \{1,\ldots,m\}$ let $d$
denote the Euclidean distance in $\bR^k$ and consider the function
$\Phi:\bR^k\rightarrow \bR$ given by
$\Phi(\mathbf u)=d(\mathbf u,\co(\mathbf y_1,\ldots,\mathbf y_n))$.
Note that $\Phi(\mathbf x_l)>0$, $\Phi(\mathbf y_j)=0$ for $1\le j\le n$, and $\Phi$ is
a convex function since $\co(\mathbf y_1,\ldots,\mathbf y_n)$ is a non-empty convex subset
of $\bR^k$. It follows that
$\sum_{i=1}^{m}a_{i}\Phi(\mathbf x_i)>\sum_{j=1}^{n}b_{j}\Phi(\mathbf y_j)$,
which contradicts condition (i) of Theorem~\ref{gen-S}. Thus
 $\mathbf x_i\in \co(\mathbf y_1,\ldots,\mathbf y_n)$ for $1\le i\le m$, as required.
\end{proof}

Given $M=(m_{ij})\in \calM(X,Y)$ we define a vector
$\mathbf s(M)=(s_1(M),\ldots,s_n(M))\in \bR^n$ by
$s_j(M)=\sum_{i=1}^{m}a_im_{ij}$, $1\le j\le n$. Set
$$\calL(X,Y)=\{\mathbf s(M)\mid M\in\calM(X,Y)\}.$$

Note that $\al\mathbf s(M)+(1-\al)\mathbf s(N)=\mathbf s(\al M+(1-\al)N)$ for any
$M,N\in \calM(X,Y)$ and $\al\in [0,1]$, so that $\calL(X,Y)$ is a convex
subset of $\bR^n$. Moreover, since $\calM(X,Y)$ is closed the same must be
true for $\calL(X,Y)$.

\begin{lemma}\label{l2}
With the above notations one has $\mathbf b\in \calL(X,Y)$. In other words,
there exists a matrix $M_0\in \calM(X,Y)$ such that $\mathbf s(M_0)=\mathbf b$.
\end{lemma}

\begin{proof}
Assume that $\mathbf b\notin \calL(X,Y)$. Since $\calL(X,Y)$ is a closed convex set
it follows from the Hahn-Banach theorem that there exist $c\in \bR$ and
$(r_1,\ldots, r_n)\in \bR^n$ such that
\begin{equation}\label{h-b}
\sum_{j=1}^{n}r_js_j<c<\sum_{j=1}^{n}r_{j}b_j\text{ for }
(s_1,\ldots,s_n)\in \calL(X,Y).
\end{equation}
Given an arbitrary vector
$\mathbf y\in \co(\mathbf y_1,\ldots,\mathbf y_n)$ we define the simplex
$$W(\mathbf y)=\left\{(\la_1,\ldots,\la_n)\in [0,1]^n\,\,\bigg|\,\,
\mathbf y=\sum_{j=1}^{n}\la_j
\mathbf y_j,\sum_{j=1}^{n}\la_j=1\right\}.$$
Note that $W(\mathbf y)$ is compact for any $\mathbf y\in \co(\mathbf y_1,\ldots,\mathbf y_n)$
since it is complete and totally bounded. We may therefore define a function
$G:\co(\mathbf y_1,\ldots,\mathbf y_n)\rightarrow \bR$ by setting
$$G(\mathbf y)=\max\left\{\sum_{j=1}^{n}r_j\la_j\,\,\bigg|\,\,(\la_1,\ldots,\la_n)
\in W(\mathbf y)\right\}\text{ for }\mathbf y\in \co(\mathbf y_1,\ldots,\mathbf y_n).$$
Let $\la_j(\mathbf y)\in [0,1]$, $1\le j\le n$, be such that
$G(\mathbf y)=\sum_{j=1}^{n}r_j\la_j(\mathbf y)$. Since $\la_j(\mathbf y_i)=\de_{ij}$ one has
$G(\mathbf y_j)\ge r_j$ for $1\le j\le n$ and thus
\begin{equation}\label{h-b1}
\sum_{j=1}^{n}b_{j}G(\mathbf y_j)\ge \sum_{j=1}^{n}b_{j}r_{j}>c.
\end{equation}
On the other hand the left inequality in~\eqref{h-b} implies that for any
matrix $M=(m_{ij})\in \calM(X,Y)$ one has
\begin{equation}\label{h-b2}
c>\sum_{j=1}^{n}r_js_j(M)=\sum_{i=1}^{m}\sum_{j=1}^{n}a_{i}r_{j}m_{ij}.
\end{equation}
Let us now consider the extension $\tilde{G}$ of the function $G$ to $\bR^k$
given by $\tilde{G}(\mathbf y)=G(\mathbf y)$ if $\mathbf y\in \co(\mathbf y_1,\ldots,\mathbf y_n)$ and
$\tilde{G}(\mathbf y)=0$
for $\mathbf y\in \bR^k\setminus \co(\mathbf y_1,\ldots,\mathbf y_n)$. It is not difficult to
see that
$\tilde{G}$ is a concave function on $\bR^k$. By Lemma~\ref{l1} one has
$\mathbf x_i\in \co(\mathbf y_1,\ldots,\mathbf y_n)$, $1\le i\le m$, so that the numbers
$\la_j(\mathbf x_i)$ are well defined for $1\le i\le m$, $1\le j\le n$. Note that
$(\la_j(\mathbf x_i))\in \calM(X,Y)$ and that by~\eqref{h-b2} and~\eqref{h-b1} the
following holds
$$\sum_{i=1}^{m}a_{i}\tilde{G}(\mathbf x_i)
=\sum_{i=1}^{m}\sum_{j=1}^{n}a_{i}r_j\la_j(\mathbf x_i)
<\sum_{j=1}^{n}b_{j}r_{j}\le \sum_{j=1}^{n}b_{j}\tilde{G}(\mathbf y_j).$$
This contradicts however condition (i) of Theorem~\ref{gen-S} and thus we are
done. It follows that
$\mathbf b\in \calL(X,Y)$, which completes the proof of the lemma.
\end{proof}
The fact that (ii) $\Rightarrow$ (i) in Theorem~\ref{gen-S} is equivalent
to Lemma~\ref{l2}.
\end{proof}

\begin{remark}\label{r-thanks}
We are grateful to the referee for pointing out the similarity
between
the Hahn-Banach type argument used above and the one given in the original
proof of the Cartier-Fell-Meyer theorem \cite{CFM}.
\end{remark}

\begin{definition}\label{def-w}
The pair $(X,\mathbf a)\in \bX_{m}^k$ is said to be {\em weightily majorized} by
the pair $(Y,\mathbf b)\in \bX_{n}^k$, denoted $(X,\mathbf a)\prec (Y,\mathbf b)$, if the
conditions of Theorem~\ref{gen-S} are satisfied.
\end{definition}

\begin{remark}\label{bary}
If $(X,\mathbf a)\prec (Y,\mathbf b)$ then the $\mathbf a$-barycenter of $X$ must coincide with the
$\mathbf b$-barycenter of $Y$, that is,
$\sum_{i=1}^{m}a_{i}\mathbf x_i=\sum_{j=1}^{n}b_{j}\mathbf y_j$.
\end{remark}

It is clear from Definition~\ref{def-w} that the weighted majorization
relation is a preorder on $\bX^k$. Indeed, by Theorem~\ref{gen-S}
this relation is both reflexive and transitive. Moreover, if $m=n$ and
$a_i=b_j=\frac{1}{n}$, $1\le i,j\le n$, then
Theorem~\ref{gen-S} and Birkhoff's theorem \cite[Theorem A.2]{MO} imply that
the weighted multivariate majorization relation induces a partial ordering on
the set of unordered $n$-tuples of vectors in $\bR^k$. We shall refer to this
partial ordering as the {\em ordinary multivariate majorization} relation, for
which Sherman's theorem takes a particularly simple form:

\begin{corollary}\label{cor-po}
If $X=(\mathbf x_1,\ldots,\mathbf x_n)\in \calX_{n}^{k}$ and $Y=(\mathbf y_1,\ldots,\mathbf y_n)\in
\calX_{n}^{k}$ then the following conditions are equivalent:
\begin{itemize}
\item[(i)] The inequality $\sum_{i=1}^{n}\Phi(\mathbf x_i)\le
\sum_{j=1}^{n}\Phi(\mathbf y_j)$
holds for any convex function $\Phi:\bR^k\rightarrow \bR$.
\item[(ii)] There exists a doubly stochastic $n\times n$ matrix $S$ such that
$\tilde{X}^T=S\tilde{Y}^T$, where $\tilde{X}^T$ and $\tilde{Y}^T$ are
$n\times k$ matrices obtained by some ordering of the vectors in
$X^T$ and $Y^T$, respectively.
\end{itemize}
\end{corollary}

\begin{remark}
For $k=1$ Corollary~\ref{cor-po} amounts to a well-known
description of classical (vector) majorization due to Schur and to
Hardy-Littlewood-P\'olya \cite{HLP,MO}.
\end{remark}

\begin{remark}\label{ordi}
In the sequel we shall often consider weighted pairs of the form
$(X,\mathbf a)$ and $(Y,\mathbf b)$, where $X$ and $Y$ are unordered tuples of
complex numbers. In this
case we define the weighted majorization relation $(X,\mathbf a)\prec (Y,\mathbf b)$,
when appropriate, by identifying $\bC$ with $\bR^2$ in Theorem~\ref{gen-S}
and Definition~\ref{def-w}.
\end{remark}

Another interesting consequence of Sherman's theorem is as follows:

\begin{corollary}\label{cor-gs}
Assume that $m<n\le k$ and let $(\mathbf x_1,\ldots,\mathbf x_m)\in
\calX_{m}^k$, $(\mathbf y_1,\ldots,\mathbf y_n)\in \calX_{n}^k$,
$(a_1,\ldots,a_n)\in \calA_n$ and $(b_1,\ldots,b_n)\in \calA_n$. For
$m+1\le i\le n$ let $(c_{i1},\ldots,c_{in})\in \calA_n$ be such that
$b_{j}':=b_j-\sum_{i=m+1}^{n}a_{i}c_{ij}\neq 0$, $1\le j\le n$, and set
$\mathbf x_i=\sum_{j=1}^{n}c_{ij}\mathbf y_j$. If $\mathbf y_1,\ldots,\mathbf y_n$ are linearly
independent then the following conditions are equivalent:
\begin{itemize}
\item[(i)] $\sum_{i=1}^{n}a_{i}\Phi(\mathbf x_i)\le \sum_{j=1}^{n}b_{j}\Phi(\mathbf y_j)$
for all convex functions $\Phi:\bR^k\rightarrow \bR$.
\item[(ii)] $\sum_{i=1}^{m}a_i\Phi(\mathbf x_i)\le \sum_{j=1}^{n}b_{j}'\Phi(\mathbf y_j)$
for all convex functions $\Phi:\bR^k\rightarrow \bR$.
\end{itemize}
If either of these conditions is fulfilled then $b_{j}'>0$ for $1\le j\le n$.
\end{corollary}

\begin{proof}
The implication (ii) $\Rightarrow$ (i) being trivial let us show that
(i) $\Rightarrow$ (ii). Set $X=(\mathbf x_1,\ldots,\mathbf x_n)\in \calX_{n}^{k}$,
$X'=(\mathbf x_1,\ldots,\mathbf x_m)$, $Y=(\mathbf y_1,\ldots,\mathbf y_n)$, $\mathbf a=(a_1,\ldots,a_n)$
and $\mathbf b=(b_1,\ldots,b_n)$. Then condition (i) is equivalent to
$(X,\mathbf a)\prec (Y,\mathbf b)$. Note that
$\sum_{j=1}^{n}b_{j}'=\sum_{i=1}^{m}a_i=:\al\in (0,1)$
and let $\mathbf a'=(\al^{-1}a_1,\ldots,\al^{-1}a_m)\in\calA_m$ and
$\mathbf b'=(\al^{-1}b_{1}',\ldots,\al^{-1}b_{n}')$. Using Theorem~\ref{gen-S} and
the fact that $\mathbf y_1,\ldots,\mathbf y_n$ are linearly independent one can easily
show that $\mathbf b'\in \calA_n$ and $(X',\mathbf a')\prec (Y,\mathbf b')$, which
is equivalent to condition (ii).
\end{proof}

\begin{remark}
For an arbitrarily given convex function $\Phi:\bR^k\rightarrow \bR$
the inequality in Corollary~\ref{cor-gs} (i) is {\em \`a priori}
weaker than the one in (ii).
As a special case let us consider three complex
numbers $z_1,z_2,z_3$ with $z_2\bar{z}_3\notin \bR$ and $\al,\be,\ga\in (0,1)$
such that $\ga\neq (1-\al)\be$ and $1-\ga\neq (1-\al)(1-\be)$. Then
Corollary~\ref{cor-gs} implies that if
$$\al \Phi(z_1)+(1-\al)\Phi(\be z_2+(1-\be)z_3)\le
\ga \Phi(z_2)+(1-\ga)\Phi(z_3)$$
for all convex functions $\Phi:\bC\rightarrow \bR$ then $\ga>(1-\al)\be$,
$1-\ga>(1-\al)(1-\be)$, and the inequality
$$\al \Phi(z_1)+(1-\al)(\be \Phi(z_2)+(1-\be)\Phi(z_3))\le \ga \Phi(z_2)+
(1-\ga)\Phi(z_3)$$
holds for any convex function $\Phi:\bC\rightarrow \bR$.
\end{remark}

\begin{remark}
Theorem~\ref{gen-S} generalizes several
known results on vector and matrix majorization, many of which
seem in fact to be regularly rediscovered in various contexts,
see e.g.~\cite{Da} and references therein. Two new interesting
extensions of classical and multivariate majorization have
recently been introduced and studied in \cite{Ma2}.
\end{remark}

\section{Geometry of equilibrium points: statement of main
results}\label{s2}

In this section we formulate our main results. These describe the distribution
of equilibrium points of logarithmic potentials for discrete configurations
of positive charges (Theorems~\ref{t-1}--\ref{t-4} and
Theorems~\ref{t-7}--\ref{t-8}) and the Hausdorff geometry of the
extended set of equilibrium points for finite charge configurations
(Theorems~\ref{t-5}--\ref{t-6}).

\subsection{Weighted majorization relations for finite charge
configurations}\label{s21}

Let $z_i$, $1\le i\le n$, be distinct points in the complex plane carrying
positive charges $a_i$, $1\le i\le n$, respectively, where $n\ge 2$.
Throughout
this section we assume wlog that the $a_i$ are normalized so that
$\sum_{i=1}^{n}a_i=1$. The logarithmic potential generated by these charged
particles is given by
\begin{equation}\label{pot}
U(z)=\sum_{i=1}^{n}a_{i}\log\left|1-\frac{z}{z_i}\right|
\end{equation}
and the resulting electrostatic force is
\begin{equation}\label{f}
f(z)=\sum_{i=1}^{n}\frac{a_i}{z-z_i}.
\end{equation}

The class of functions of the form~\eqref{f} is essentially
the same as Sz-Nagy's class of generalized derivatives \cite{Sz}.
As explained in \S \ref{s0}, the critical points of $U(z)$ coincide with
the equilibrium points of the electrostatic field, that is, the zeros of
$f(z)$. By an argument reminiscent of the Gauss-Lucas theorem one
can see that all equilibrium points lie in $\co(z_1,\ldots,z_n)$,
cf., e.g., Proposition 3.1 in \cite{Ma2}. Remarkable generalizations of this
simple fact were recently obtained in {\em loc.~cit.}, where it was shown
that these points actually satisfy a whole sequence of majorization relations.
Theorems~\ref{t-1}--\ref{t-3} below further extend the aforementioned results.

{\allowdisplaybreaks
\begin{notation}
Let $w_1,\ldots,w_{n-1}$ be the critical points of $U(z)$ counted with
multiplicity and recall the set $\calA_m$ defined in~\eqref{sp-n}.
For $\la\in\bC\setminus\{0\}$ and $\mu\in\bC$ let
\begin{align}\label{not-1}
&W(\la,\mu)=(\la w_1+\mu,\ldots,\la w_{n-1}+\mu)\in\bC^{n-1},\\
&Z(\la,\mu)=(\la z_1+\mu,\ldots,\la z_n+\mu)\in\bC^{n},\notag\\
&\mathbf a=\left(\frac{1}{n-1},\ldots,\frac{1}{n-1}\right)\in\calA_{n-1},\quad
\mathbf b=\left(\frac{1-a_1}{n-1},\ldots,\frac{1-a_n}{n-1}\right)\in\calA_{n}.\notag
\end{align}
\end{notation}}

\begin{theorem}\label{t-1}
There exists a matrix $R\in \bM_{n-1,n}^{rs}$ such that
$W(\la,\mu)^T=RZ(\la,\mu)^T$ and $\mathbf b=\mathbf a R$ for any
$\la\in\bC\setminus\{0\}$ and $\mu\in\bC$, so that
$(W(\la,\mu),\mathbf a)\prec (Z(\la,\mu),\mathbf b)$. Equivalently, the inequality
\begin{equation}\label{f-t-1}
\sum_{j=1}^{n-1}\Phi(\la w_j+\mu)\le \sum_{i=1}^{n}(1-a_i)\Phi(\la z_i+\mu)
\end{equation}
is satisfied by any (continuous) convex function $\Phi:\bC\to \bR$.
\end{theorem}

\begin{remark}\label{pere-mala}
The special case of~\eqref{f-t-1} when $\la=1$, $\mu=0$ and $a_i=\frac{1}{n}$,
$1\le i\le n$, is already a considerable improvement of the Gauss-Lucas
theorem and was originally conjectured by de Bruijn and
Springer \cite{BrS}. This conjecture was recently proved
by S.~Malamud \cite{Ma1,Ma2} and R.~Pereira \cite{P} independently of each
other.
\end{remark}

\begin{remark}\label{r-3}
Theorem~\ref{t-1} implies in particular that the following ordinary
multivariate majorization relation holds in $\bR^2$:
$$(w_1,\ldots,w_{n-1},\ze)\prec (z_1,\ldots,z_{n-1},z_n),$$
where $\ze=\sum_{i=1}^{n}a_{i}z_i$ is the $(a_1,\ldots,a_n)$-barycenter of
$Z(1,0)$ (cf.~Remarks~\ref{bary} and~\ref{ordi}). This relation was previously
obtained in \cite[Proposition 4.3]{Ma2} and \cite[Theorem 5.4]{P}. Note though
that if $n\ge 3$ then even in the case of equal charges the above relation is
in general weaker than the corresponding one given by Theorem~\ref{t-1} since
the $z_i$ are linearly dependent (compare with Corollary~\ref{cor-gs}).
\end{remark}

As we shall now see, \eqref{f-t-1} is actually
but one relation in a sequence of $\binom{n}{2}$
weighted multivariate majorization relations satisfied by the zeros and poles
of $f(z)$. Moreover, the equilibrium points of $U(z)$ are in fact uniquely
determined by $n-1$ such relations. To formulate these results we need some
new notation.

\begin{notation}\label{new-not2}
Given $k\in \bN$ and $1\le m\le k$
let $\Pi_{k,m}$ denote the $m$th elementary symmetric function on $k$ symbols.
For $\la\in\bC\setminus \{0\}$, $\mu\in\bC$, $k\in\{1,\ldots,n-1\}$ and
$1\le m\le k$ set
{\allowdisplaybreaks
\begin{align}\label{not-4}
&W_{m}^{[k]}(\la,\mu)
=\Big(\om_{m}[r_1,\ldots,r_k](\la,\mu)\Big)_{1\le r_1<\ldots<r_k\le n-1}
\in \bC^{\binom{n-1}{k}},\\
&\text{where }\,\om_{m}[r_1,\ldots,r_k](\la,\mu)
=\Pi_{k,m}(\la w_{r_1}+\mu,\ldots,\la w_{r_k}+\mu),\notag\\
&Z_{m}^{[k]}(\la,\mu)
=\Big(\ze_{m}[s_1,\ldots,s_k](\la,\mu)\Big)_{1\le s_1<\ldots<s_k\le n}
\in \bC^{\binom{n}{k}},\notag\\
&\text{where }\,\ze_{m}[s_1,\ldots,s_k](\la,\mu)
=\Pi_{k,m}(\la z_{s_1}+\mu,\ldots,\la z_{s_k}+\mu),\notag\\
&\mathbf a^{[k]}=
\left(\binom{n-1}{k}^{-1},\ldots,\binom{n-1}{k}^{-1}\right)
\in \calA_{\binom{n-1}{k}},\notag\\
&\mathbf b^{[k]}=\left(\binom{n-1}{k}^{-1}
\!\!\left(1-\sum_{i=1}^{k}a_{s_i}\right)\right)_{1\le s_1<\ldots<s_k\le n}
\in \calA_{\binom{n}{k}}.\notag
\end{align}
Below we shall always assume that the components of $W_{m}^{[k]}(\la,\mu)$,
$Z_{m}^{[k]}(\la,\mu)$ and $\mathbf b^{[k]}$ are arranged
increasingly with respect to the lexicographic order on the set of all indices
consisting of $k$-tuples of increasing numbers in $\{1,\ldots,n\}$.
In particular,} for any $1\le m\le \binom{n}{k}$ the $m$th coordinates
of $Z_{m}^{[k]}(\la,\mu)$ and $\mathbf b^{[k]}$ are indexed by the same $k$-tuple
$(s_1,\ldots,s_k)$ satisfying $1\le s_1<\ldots<s_k\le n$. Note also that with
this ordering~\eqref{not-1} and~\eqref{not-4} imply that $\mathbf a^{[1]}=\mathbf a$,
$\mathbf b^{[1]}=\mathbf b$, $W_{1}^{[1]}(\la,\mu)=W(\la,\mu)$ and
$Z_{1}^{[1]}(\la,\mu)=Z(\la,\mu)$.
\end{notation}

Theorem 3.11 in \cite{Ma2} describes all possible sets of equilibrium points
for logarithmic potentials generated by a fixed configuration of particles
that are allowed to carry arbitrary positive charges. A natural related
inverse problem
is to describe geometrically the equilibrium points of a logarithmic potential
associated to a given set of particles with prescribed positive charges.
The following theorem solves this problem and provides a characterization of
the zeros of the
logarithmic potential $U(z)$ defined in~\eqref{pot} by means of
$n-1$ weighted majorization relations.

\begin{theorem}\label{t-3}
The following conditions are equivalent:
\begin{itemize}
\item[(i)] The zeros of $f(z)$ counted with multiplicity are
$w_1,\ldots,w_{n-1}$, that is,
\begin{equation}\label{cont}
\sum_{i=1}^{n}a_{i}\prod_{\substack{j=1\\j\neq i}}^{n}(z-z_j)
=\prod_{k=1}^{n-1}(z-w_k),\quad z\in\bC.
\end{equation}
\item[(ii)] There exist $\la\in\bC\setminus\{0\}$ and $\mu\in \bC$ such that
$$\left(W_{k}^{[k]}(\la,\mu),\mathbf a^{[k]}\right)\prec
\left(Z_{k}^{[k]}(\la,\mu),\mathbf b^{[k]}\right),\quad 1\le k\le n-1.$$
\item[(iii)] There exist $\la\in\bC\setminus\{0\}$ and $\mu\in \bC$ such that
for any $k\in\{1,\ldots,n-1\}$ one can find an
$\binom{n-1}{k}\times \binom{n}{k}$ row stochastic matrix $R_{k}(\la,\mu)$
satisfying
$$W_{k}^{[k]}(\la,\mu)^T=R_{k}(\la,\mu)Z_{k}^{[k]}(\la,\mu)^T\text{ and }
\mathbf b^{[k]}=\mathbf a^{[k]}R_{k}(\la,\mu).$$
\item[(iv)] There exist $\la\in\bC\setminus\{0\}$ and $\mu\in \bC$ such that
for any $k\in\{1,\ldots,n-1\}$ the inequality
\begin{equation*}
\begin{split}
\sum_{1\le r_1<\ldots<r_k\le n-1}&\Phi
\Big(\om_{k}[r_1,\ldots,r_k](\la,\mu)\Big)\\
\le
&\sum_{1\le s_1<\ldots<s_k\le n}\left(1-\sum_{i=1}^{k}a_{s_i}\right)
\Phi\Big(\ze_{k}[s_1,\ldots,s_k](\la,\mu)\Big)
\end{split}
\end{equation*}
holds for all (continuous) convex functions $\Phi:\bC\to \bR$.
\item[(v)] Either of conditions (ii)--(iv) is true for $\la=1$ and $\mu=0$.
\end{itemize}
If the above conditions are fulfilled then for any
$k\in\{1,\ldots,n-1\}$ there exists an
$\binom{n-1}{k}\times \binom{n}{k}$ row stochastic matrix $R_{k}$ such that
\begin{equation}\label{inv-m}
W_{k}^{[k]}(\la,\mu)^T=R_{k}Z_{k}^{[k]}(\la,\mu)^T\text{ and }
\mathbf b^{[k]}=\mathbf a^{[k]}R_{k}
\end{equation}
for all $\la\in\bC\setminus\{0\}$ and $\mu\in \bC$.
\end{theorem}

\begin{remark}\label{dBS}
Theorem~\ref{t-3} extends \cite[Theorem 4.7]{Ma2} to the case of arbitrary
(not necessarily equal) charges. If $a_i=\frac{1}{n}$, $1\le i\le n$, then
Theorem~\ref{t-3} reduces to the characterization of the zeros
and critical points of degree $n$ complex polynomials by means of $n-1$
majorization relations which was given in \cite[Theorem 4.10]{Ma2}.
\end{remark}

From Theorem~\ref{t-3} we deduce a number of corollaries.

\begin{corollary}\label{t-4}
For any $k\in\{1,\ldots,n-1\}$ there exists an
$\binom{n-1}{k}\times \binom{n}{k}$ row stochastic matrix $R_{k}$ such that
\begin{equation}\label{gen-m}
W_{m}^{[k]}(\la,\mu)^T=R_{k}Z_{m}^{[k]}(\la,\mu)^T\text{ and }
\mathbf b^{[k]}=\mathbf a^{[k]}R_{k}
\end{equation}
whenever $1\le m\le k$, $\la\in\bC\setminus\{0\}$ and $\mu\in \bC$. In
particular, for all such parameters $k$, $m$, $\la$, $\mu$ and any convex
function $\Phi:\bC\to \bR$ the following inequality holds:
\begin{equation}\label{g-conv-f}
\begin{split}
\sum_{1\le r_1<\ldots<r_k\le n-1}&\Phi
\Big(\om_{m}[r_1,\ldots,r_k](\la,\mu)\Big)\\
\le
&\sum_{1\le s_1<\ldots<s_k\le n}\left(1-\sum_{i=1}^{k}a_{s_i}\right)
\Phi\Big(\ze_{m}[s_1,\ldots,s_k](\la,\mu)\Big).
\end{split}
\end{equation}
\end{corollary}

\begin{remark}
Note that~\eqref{gen-m} is {\em \`a priori}
stronger than~\eqref{g-conv-f} since it
asserts that there exist matrices $R_k$ of the desired type that
simultaneously realize the weighted majorization relations between the
pairs $\left(W_{m}^{[k]}(\la,\mu),\mathbf a^{[k]}\right)$ and
$\left(Z_{m}^{[k]}(\la,\mu),\mathbf b^{[k]}\right)$ for {\em all}
admissible parameters $m$, $\la$ and $\mu$.
\end{remark}

Setting $m=1$ in Corollary~\ref{t-4} yields the following generalization of
Theorem~\ref{t-1}, which corresponds to the case when $k=1$ in~\eqref{f-t-2}
below.

\begin{corollary}\label{t-2}
For any $k\in\{1,\ldots,n-1\}$ there exists an
$\binom{n-1}{k}\times \binom{n}{k}$ row stochastic matrix
$R_k$ such that
$$W_{1}^{[k]}(\la,\mu)^T=R_{k}Z_{1}^{[k]}(\la,\mu)^T\text{ and }
\mathbf b^{[k]}=\mathbf a^{[k]}R_{k}$$
for any $\la\in\bC\setminus\{0\}$ and $\mu\in \bC$. Thus
$$\left(W_{1}^{[k]}(\la,\mu),\mathbf a^{[k]}\right)\prec
\left(Z_{1}^{[k]}(\la,\mu),\mathbf b^{[k]}\right),\quad 1\le k\le n-1,$$
or equivalently, for all convex functions $\Phi:\bC\to \bR$ and
$k\in\{1,\ldots,n-1\}$ one has
\begin{equation}\label{f-t-2}
\begin{split}
\sum_{1\le r_1<\ldots<r_k\le n-1}&\Phi
\Big(\om_{1}[r_1,\ldots,r_k](\la,\mu)\Big)\\
&\le
\sum_{1\le s_1<\ldots<s_k\le n}\left(1-\sum_{i=1}^{k}a_{s_i}\right)
\Phi\Big(\ze_{1}[s_1,\ldots,s_k](\la,\mu)\Big).
\end{split}
\end{equation}
\end{corollary}

\begin{remark}\label{la0}
The condition $\la\neq 0$ may actually be omitted from the above
inequa\-lities. However, if $\la=0$ the vectors defined in~\eqref{not-4}
are independent of the $z_i$ and
$w_j$ and the aforementioned inequalities become
equalities. Thus the corresponding weighted majorization relations
are trivially satisfied in this case.
\end{remark}

Taking $\la=1$, $\mu=0$ and $\Phi(z)=|z|^{\al}$ with $\al\ge 1$ in
Corollary~\ref{t-4} we obtain the following moment inequalities for the
charged particles and the equilibrium points.

\begin{corollary}\label{cor-mom}
For any $k\in\{1,\ldots,n-1\}$, $1\le m\le k$, and $\al\ge 1$ one has
\begin{equation*}
\begin{split}
\sum_{1\le r_1<\ldots<r_k\le n-1}&
\left|\Pi_{k,m}(w_{r_1},\ldots,w_{r_k})\right|^{\al}\\
&\le\sum_{1\le s_1<\ldots<s_k\le n}
\left|\Pi_{k,m}(z_{s_1},\ldots,z_{s_k})\right|^{\al},
\end{split}
\end{equation*}
where $\Pi_{k,m}$ denotes as before the $m$th elementary symmetric function on
$k$ symbols.
\end{corollary}

In particular, by setting $a_i=\frac{1}{n}$, $1\le i\le n$, and $m=1$ in
Corollary~\ref{cor-mom} we get
\begin{equation}\label{r-s}
\binom{n-1}{k}^{-1}\sum_{1\le r_1<\ldots<r_k\le n-1}
\left|\sum_{i=1}^{k}w_{r_i}\right|^{\al}
\le
\binom{n}{k}^{-1}\sum_{1\le s_1<\ldots<s_k\le n}
\left|\sum_{i=1}^{k}z_{s_i}\right|^{\al}
\end{equation}
for all $\al\ge 1$ and $k\in\{1,\ldots,n-1\}$. Note that for $k=2$ and
$\al\ge 1$ the inequalities in~\eqref{r-s} are quite similar to the
inequalities
$$\binom{n-1}{2}^{-1}\sum_{1\le r_1<r_2\le n-1}
\left|w_{r_1}-w_{r_2}\right|^{\al}
\le \binom{n}{2}^{-1}\sum_{1\le s_1<s_2\le n}
\left|z_{s_1}-z_{s_2}\right|^{\al}$$
that were conjectured by Rahman and Schmeisser in \cite[Remark 2.3.9]{RS}. In
\S \ref{s-mom} we conjecture a whole hierarchy of weighted
de Bruijn-Springer relations and inertia laws, see Conjecture~\ref{gen-mom}
below. The latter actually contains as special cases all types of moment
inequalities like those that we just described.

\subsection{Relative Hausdorff geometry}\label{s212}

The metric function between (closed) subsets of Euclidean space introduced by
Hausdorff makes use of the following notions.

\begin{definition}\label{d-HT}
The {\em relative} (alternatively, {\em directed} or {\em oriented})
{\em Hausdorff distance} from a subset $\Om_1$ of $\bC$ to another
such subset $\Om_2$ is given by
$$h(\Om_1,\Om_2)=\sup_{w\in\Om_1}\text{dist}(w,\Om_2)
=\sup_{w\in\Om_1}\inf_{z\in\Om_2}|w-z|.$$
Other frequent names and notations are the {\em forward Hausdorff distance}
from $\Om_1$ to $\Om_2$, $h_f(\Om_1,\Om_2):=h(\Om_1,\Om_2)$, and the
{\em backward Hausdorff distance} from $\Om_1$ to
$\Om_2$, $h_b(\Om_1,\Om_2):=h(\Om_2,\Om_1)$. The
{\em (symmetrized) Hausdorff distance} between $\Om_1$ and $\Om_2$ is then
given by
$$H(\Om_1,\Om_2)=\max\big(h(\Om_1,\Om_2),h(\Om_2,\Om_1)\big).$$
\end{definition}

Let $\calZ=\{z_1,\ldots,z_n\}$ be a set of charged particles and
$\calW=\{w_1,\ldots,w_{n-1}\}$ be the corresponding set of equilibrium points
of the logarithmic potential defined in~\eqref{pot}.
Theorems~\ref{t-1}--\ref{t-3} yield a
series of relations between the sets $\calW$ and $\calZ$ when all their
elements are considered simultaneously. These relations go much beyond the
obvious inclusion $\calW\subset \co(\calZ)$ and suggest that
no equilibrium point can actually lie too far away from the
particles. Our next result gives a precise meaning to this intuitive
fact and
shows that the relative Hausdorff distance $h(\calW,\calZ)$ is
dominated by the {\em $\mathbf a$-weighted standard deviation} of the
particles. The latter is given by the square root of the
{\em $\mathbf a$-weighted variance} of $\mathbf z=(z_1,\ldots,z_n)$, which we define as
\begin{equation}\label{wdev}
\si_2(\mathbf z;\mathbf a)^2=\inf_{\al\in\bC}\sum_{i=1}^{n}a_i|z_i-\al|^2
=\sum_{i=1}^{n}a_{i}|z_i-\ze|^2,
\end{equation}
where $\mathbf a=(a_1,\ldots,a_n)\in\calA_n$ and $\ze$ stands as before for the
$\mathbf a$-weighted barycenter
$\sum_{i=1}^{n}a_{i}z_i$ (cf.~Remark~\ref{r-3}). We shall actually prove
a stronger statement involving the {\em extended set of equilibrium points}
of the given system of positive charges, i.e., the set
\begin{equation}\label{ext}
\calW_e=\calW\cup \{\ze\}=\{w_1,\ldots,w_{n-1},\ze\}.
\end{equation}

\begin{theorem}\label{t-5}
In the above notation one has
$h(\calW,\calZ)\leq h(\calW_e,\calZ)\leq \si_2(\mathbf z;\mathbf a)$.
\end{theorem}

\begin{remark}\label{optim}
The bound given in Theorem~\ref{t-5} is sharp, as one can see for instance
by placing $n$ equal charges at the vertices of a regular $n$-gon.
\end{remark}

A natural question in this context is whether there is any analog of
Theorem~\ref{t-5} for the relative Hausdorff distance from $\calZ$
to $\calW_e$. The following
theorem provides an affirmative answer to this question when all
particles are assumed to lie on the same line and
shows that in this case the Hausdorff geometry of the set of
charged particles and
the extended set of equilibrium points is controlled by the weighted standard
deviation of the particles.

\begin{theorem}\label{t-6}
If $z_1,\ldots,z_n$ are collinear then
$H(\calZ,\calW_e)\leq \si_2(\mathbf z;\mathbf a)$.
\end{theorem}

\begin{remark}\label{optim-e}
Note that Theorem~\ref{t-6} is sharp and that a similar result cannot hold if
$\calW_e$ is replaced by $\calW$. To see this one may simply consider two
distinct points in the plane carrying unequal positive charges.
\end{remark}

\subsection{Infinite discrete distributions of charge}\label{s22}

As we point out in \S\ref{s42}, several difficulties occur when trying to
extend the results of \S \ref{s21} to logarithmic potentials generated by
an infinite discrete set of positive charges. To begin with, in this case
already the fundamental
question dealing with the {\em existence} of equilibrium points is quite
delicate and, as a matter of fact, still an open problem
(cf.~\cite{CER,ELR,LR}).

Nevertheless, it turns out that in many situations one
can obtain new information on the geometry of the electrostatic field through
natural infinite-dimensional versions of Theorem~\ref{t-1}. As we shall now
explain, this is for instance the case for collinear positive charges and,
more generally, for essentially any bounded discrete set of positively
charged particles with finite total charge for which equilibrium points are
known to exist. Let
$\{a_i\}_{i\in\bN}$ be a sequence of positive numbers and let
$\{z_i\}_{i\in\bN}$ be a sequence of distinct complex numbers satisfying the
conditions
\begin{equation}\label{i-hyp}
a_i>0\text{ and }|z_i|<\rho\text{ for }i\in\bN,\quad \sum_{i=1}^{\infty}a_i=1,
\quad \lim_{i\to\infty}z_i=\rho,
\end{equation}
where $\rho$ is a fixed positive number. Using the same electrostatic
interpretation as before -- that is, placing
a charge $a_i$ at $z_i$ for $i\in \bN$ -- we get the resulting force
\begin{equation}\label{i-f}
f(z)=\sum_{i=1}^{\infty}\frac{a_i}{z-z_i}
\end{equation}
and an associated logarithmic potential given by
\begin{equation*}
U(z)=\sum_{i=1}^{\infty}a_{i}\log\left|1-\frac{z}{z_i}\right|.
\end{equation*}
Set $\bD(\rho)=\{z\in\bC: |z|<\rho\}$ and
$\bD_{f}(\rho)=\bD(\rho)\setminus \{z_i\}_{i\in\bN}$. By~\eqref{i-hyp} one has
\begin{equation*}
\sum_{i=1}^{\infty}\frac{a_i}{|z_i|}<\infty,
\end{equation*}
so that the series in the right-hand side
of~\eqref{i-f} converges absolutely for all $z\in \bD_{f}(\rho)$ and thus
$f$ is
a meromorphic function in $\bD(\rho)$. One can see in similar fashion
that $U$
is subharmonic in $\bD(\rho)$. We have the following analog of
Theorem~\ref{t-1}.

\begin{theorem}\label{t-7}
If $\{z_i\}_{i\in\bN}$ is a real sequence satisfying~\eqref{i-hyp} then the
function $f$ defined by~\eqref{i-f} has an infinite discrete set of real
zeros $\{w_j\}_{j\in\bN}$ and the sequences $\{z_i\}_{i\in\bN}$ and
$\{w_j\}_{j\in\bN}$ interlace on the real axis. Moreover, there exists an
infinite matrix
$S=(s_{ij})_{i,j=1}^{\infty}$ such that for all $(i,j)\in\bN^2$ one has
\begin{equation*}
s_{ij}\ge 0,\quad \sum_{k=1}^{\infty}s_{ik}=1,\quad
\sum_{l=1}^{\infty}s_{lj}=1-a_j,\quad w_i=\sum_{k=1}^{\infty}s_{ik}z_k.
\end{equation*}
In particular, for any $\la\in\bC\setminus\{0\}$ and $\mu\in\bC$ the inequality
\begin{equation*}
\sum_{j=1}^{\infty}\Phi(\la w_j+\mu)\le
\sum_{i=1}^{\infty}(1-a_i)\Phi(\la z_i+\mu)
\end{equation*}
holds for any convex function $\Phi:\bC\to\bR$.
\end{theorem}

Our last main result is a natural analog of Theorem~\ref{t-7} for
all bounded discrete configurations of positive charges for
which equilibrium points are known to exist.

\begin{theorem}\label{t-8}
Let $\{z_i\}_{i\in\bN}$ be a sequence of distinct complex numbers
satisfying~\eqref{i-hyp} and let $f$ be the meromorphic function in
$\bD(\rho)$
defined by~\eqref{i-f}. If $f$ has a non-empty (discrete) set of
zeros $\{w_j\}_{j\in\bN}$ then the inequality
$$ \sum_{j=1}^{\infty}\Phi(\la w_j+\mu)\le
\sum_{i=1}^{\infty}(1-a_i)\Phi(\la z_i+\mu)$$
holds for all $\la\in\bC\setminus\{0\}$, $\mu\in\bC$ and any nonnegative
convex function $\Phi:\bC\to\bR$.
\end{theorem}

\begin{remark}\label{expl}
The fact that inequality~\eqref{bes-2} in \S \ref{s33} may be strict
explains why the
inequality in Theorem~\ref{t-8} holds in general only for {\em nonnegative}
convex functions.
The connection between majorization and such inequalities for nonnegative
convex functions was first found in \cite{FH}.
\end{remark}

\begin{remark}\label{kat}
Theorem~\ref{t-8} is reminiscent of Kato's bound for the
variation of discrete spectra of Hilbert space selfadjoint operators under
compact perturbations \cite{Ka1}.
\end{remark}

\section{Proof of main results}\label{s3}

\subsection{Proof of Theorems~\ref{t-1}--\ref{t-3}}\label{s31}

We shall use various methods from multilinear algebra and matrix analysis.
These require a few notations and preliminary results and we refer
to~\cite{Bh} and~\cite{MO} for background material.

\begin{notation}\label{not-proofs}
Given $m\in\bN$
and $1\le k\le m$
let $$Q_{k,m}=\{\mathbf i=(i_1,\ldots,i_k)\mid 1\le i_1<\ldots<i_k\le m\}.$$
If $B=(b_{ij})$ is an $m\times m$ matrix and $\mathbf i,\mathbf j\in Q_{k,m}$
we denote by
$$B(\mathbf i,\mathbf j)=B\binom{i_1,\ldots,i_k}{j_1,\ldots,j_k}$$
the $k\times k$ submatrix of $B$ lying in rows $i_1,\ldots,i_k$ and columns
$j_1,\ldots,j_k$. Below we shall always assume that the elements of $Q_{k,m}$
are arranged in lexicographic order, i.e., if $\mathbf i$ and $\mathbf j$ are distinct
$k$-tuples in $Q_{k,m}$ then $\mathbf i\ge \mathbf j$ if the first non-zero term in the
sequence $i_1-j_1,\ldots,i_k-j_k$ is positive.
\end{notation}

We fix an $n$-dimensional
complex Hilbert space $\calH$ with unitarily invariant scalar
product $\langle\cdot,\!\cdot\rangle$ and identity operator
$I=I_{\calH}=I_n\in L(\calH)$, where $L(\calH)$ is the
set of all linear operators on $\calH$. Let $A\in L(\calH)$ be a normal
operator with spectrum $\Si(A)=\{z_1,\ldots,z_n\}$ and choose an
orthonormal basis $(\mathbf{e}_1,\ldots,\mathbf{e}_n)$ of $\calH$ such that
$A\mathbf e_i=z_i\mathbf e_i$ for $1\le i\le n$. Define a unit vector
\begin{equation}\label{v-n}
\mathbf{v}_{n}=\sum_{i=1}^{n}\sqrt{a_i}\mathbf e_i
\end{equation}
and let $P$ be the orthogonal projection on the subspace
$\calK:=\mathbf{v}_{n}^{\perp}$ of $\calH$. The operator
$$A'=PAP|_{\calK}\in L(\calK)$$
is called the $P$-{\em compression} of $A$ \cite{D}. Recall the function $f$
defined in~\eqref{f}. The following statement may be found in
e.g.~\cite[formula (3.3)]{Ma2} and \cite[Lemma 2.2]{P}.

\begin{lemma}\label{g-comp}
With the above notations one has
$$\left\langle (A-zI_{\calH})^{-1}\mathbf{v}_n,\mathbf{v}_n\right\rangle=
\frac{\det(A'-zI_{\calK})}{\det(A-zI_{\calH})}=-f(z)\text{ for }z\in\bC
\setminus \Si(A).$$
\end{lemma}

\begin{proof}
Let $(\mathbf{v}_1,\ldots,\mathbf{v}_{n-1})$ be an orthonormal basis of
$\calK$. The
matrix representation of $A'$ in the basis $(\mathbf{v}_1,\ldots,
\mathbf{v}_{n-1})$ is given by
the $(n-1)\times (n-1)$ upper left-hand principal submatrix of the matrix
representation of $A$ in the orthonormal basis $(\mathbf{v}_1,\ldots,
\mathbf{v}_{n-1},\mathbf{v}_n)$ of
$\calH$. For any $z\in \bC
\setminus \Si(A)$ the $(n,n)$ entry of the
matrix representation of the (normal) operator $(A-zI_{\calH})^{-1}$ in
the basis $(\mathbf{v}_1,\ldots,\mathbf{v}_{n-1},\mathbf{v}_n)$ is given on
the one hand by
\begin{equation*}
\left\langle (A-zI_{\calH})^{-1}\mathbf{v}_n,\mathbf{v}_n\right\rangle
=\left\langle \sum_{i=1}^{n}\frac{\langle \mathbf{v}_n,\mathbf{e}_i\rangle}
{z_i-z}\mathbf{e}_i,
\sum_{i=1}^{n}\langle \mathbf{v}_n,\mathbf{e}_i\rangle \mathbf{e}_i
\right\rangle=-f(z).
\end{equation*}
On the other hand, by Cramer's rule the $(n,n)$ entry of the matrix
representation of $(A-zI_{\calH})^{-1}$ in
the basis $(\mathbf{v}_1,\ldots,\mathbf{v}_{n-1},\mathbf{v}_n)$ of
$\calH$ is given by the cofactor of the $(n,n)$
entry of the matrix representation of $(A-zI_{\calH})$ in the same
basis. Thus
$$\left\langle (A-zI_{\calH})^{-1}\mathbf{v}_n,\mathbf{v}_n\right\rangle=
\frac{\det(A'-zI_{\calK})}{\det(A-zI_{\calH})}$$
for $z\in \bC
\setminus \Si(A)$, which proves the lemma.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{t-1}]
By Lemma~\ref{g-comp} the zeros $w_1,\ldots,w_{n-1}$ of $f$ are the same
as the eigenvalues of the $P$-compression $A'$.
Let $(\mathbf v_1,\ldots,\mathbf v_{n-1})$ be an orthonormal
basis of $\calK$ that triangularizes $A'$. Then
\begin{equation}\label{f-1}
w_j=\mathbf v_{j}^{*}A'\mathbf v_j=\langle A'\mathbf v_j,\mathbf v_j\rangle=\langle A\mathbf v_j,\mathbf v_j\rangle
=\sum_{i=1}^{n}z_i|\langle \mathbf v_j,\mathbf e_i\rangle|^2
\end{equation}
since $\mathbf v_j=\sum_{i=1}^{n}\langle \mathbf v_j,\mathbf e_i\rangle\mathbf e_i$ for $1\le j\le
n-1$. Define the $(n-1)\times n$ matrix $R=(r_{ij})$ by setting $r_{ij}=
|\langle \mathbf v_i,\mathbf e_j\rangle|^2$ for $1\le i\le n-1$ and $1\le j\le n$. Using
the fact that $(\mathbf v_1,\ldots,\mathbf v_{n-1},\mathbf v_n)$ is an orthonormal basis of
$\calH$ one can easily check that
\begin{equation*}
\begin{split}
&\sum_{j=1}^{n}r_{ij}=\sum_{j=1}^{n}|\langle \mathbf v_i,\mathbf e_j\rangle|^2=||\mathbf v_i||^2
=1,\quad 1\le i\le n-1,\\
&\sum_{i=1}^{n-1}r_{ij}=\sum_{i=1}^{n-1}|\langle \mathbf v_i,\mathbf e_j\rangle|^2=
||\mathbf e_j||^2-|\langle \mathbf e_j,\mathbf v_n\rangle|^2=1-a_{j},\quad 1\le j\le n,
\end{split}
\end{equation*}
so that $R\in\bM_{n-1,n}^{rs}$ and $\mathbf b=\mathbf a R$. Note that by~\eqref{f-1}
one has
$$(w_1,\ldots,w_{n-1})^T=R(z_1,\ldots,z_n)^T.$$
Hence $(W(1,0),\mathbf a)\prec (Z(1,0),\mathbf b)$, which completes the proof
of the theorem in the case $\la=1$ and $\mu=0$. The general case follows
from this one by composing $f$ with a non-singular affine transformation of
the plane.
\end{proof}

\begin{remark}\label{normcompr}
The compression of a normal operator to a hyperplane
is not necessarily normal. For instance, Fan and Pall \cite{FP} showed that
principal submatrices of a given irreducible normal
matrix $B$ are normal if and only if $B$ has collinear eigenvalues.
\end{remark}

\begin{remark}
The proof of Theorem~\ref{t-1} given above is a slight modification of the
corresponding arguments in \cite{Ma2} and \cite{P}.
\end{remark}

We next prove Theorem~\ref{t-3}. Clearly, conditions (ii), (iii) and
(iv) in this theorem are equivalent by Definition~\ref{def-w} and
Theorem~\ref{gen-S}. Let us first show that (v) $\Rightarrow$ (i). Set
$$q(z)=\prod_{j=1}^{n-1}(z-w_j),\quad p_{i}(z)
=\prod_{\substack{j=1\\j\neq i}}^{n}(z-z_j),\quad 1\le i\le n,$$
and note that condition (i) is equivalent to
\begin{equation*}
q^{(n-1-k)}(0)=\sum_{i=1}^{n}a_{i}p_{i}^{(n-1-k)}(0),\quad 1\le k\le n-1.
\end{equation*}
Elementary computations show that
\begin{equation*}
\begin{split}
&q^{(n-1-k)}(0)=(-1)^{k}(n-1-k)!
\sum_{1\le r_1<\ldots<r_k\le n-1}\prod_{j=1}^{k}w_{r_j},\\
&\sum_{i=1}^{n}a_{i}p_{i}^{(n-1-k)}(0)=(-1)^{k}(n-1-k)!
\sum_{1\le s_1<\ldots<s_k\le n}\left(1-\sum_{i=1}^{k}a_{s_i}\right)
\prod_{i=1}^{k}z_{s_i},
\end{split}
\end{equation*}
for $1\le k\le n-1$. Thus condition (i) amounts to saying that if
$1\le k\le n-1$ then
\begin{equation}\label{id-k}
\sum_{1\le r_1<\ldots<r_k\le n-1}\prod_{j=1}^{k}w_{r_j}=
\sum_{1\le s_1<\ldots<s_k\le n}\left(1-\sum_{i=1}^{k}a_{s_i}\right)
\prod_{i=1}^{k}z_{s_i}.
\end{equation}
The fact that (v) $\Rightarrow$ (i) is now a
consequence of the following lemma.

\begin{lemma}\label{easy}
If condition (v) in Theorem~\ref{t-3} holds then~\eqref{id-k} is true for
$1\le k\le n-1$.
\end{lemma}

\begin{proof}
Using e.g.~condition (iii) in Theorem~\ref{t-3} with $\la=1$ and $\mu=0$
one gets
$$\mathbf a^{[k]}W_{k}^{[k]}(1,0)^T=\mathbf b^{[k]}Z_{k}^{[k]}(1,0)^T,\quad
1\le k\le n-1,$$
which is the same as the desired conclusion.
\end{proof}

We shall also need the following extension to normal matrices of a
well-known result of Schur for Hermitian matrices (cf.~\cite{Bh,MO}), see
e.g.~\cite[Proposition 3.9 (a)]{Ma2}.

\begin{lemma}\label{s-lem}
If $T=(t_{ij})$ is a normal $n\times n$ matrix with eigenvalues
$\tau_1,\ldots,\tau_n$ then
$$(t_{11},\ldots,t_{nn})\prec (\tau_1,\ldots,\tau_n),$$
i.e., $(t_{11},\ldots,t_{nn})^T=S(\tau_1,\ldots,\tau_n)^T$ for some
doubly stochastic $n\times n$ matrix $S$.
\end{lemma}

\begin{proof}
Let $U$ be a unitary matrix such that
$T=U\text{diag}(\tau_1,\ldots,\tau_n)U^{*}$ and set $S=U\circ \bar{U}$, where
$\circ$ denotes the Hadamard-Schur (entrywise) product of matrices and
$\bar{U}$ is
the complex conjugate of $U$. Then
$(t_{11},\ldots,t_{nn})^T=S(\tau_1,\ldots,\tau_n)^T$. It remains to note that
$S$ is a unitarily stochastic hence a doubly stochastic matrix.
\end{proof}

To complete the proof of Theorem~\ref{t-3} it will be convenient
to work with certain suitably chosen matrix representations of the operators
$A$ and $A'$ defined above. As in the proof of Theorem~\ref{t-1} let us fix
an orthonormal
basis $(\mathbf v_1,\ldots,\mathbf v_{n-1})$ of $\calK$ in
which the matrix representation of the $P$-compression $A'$ of $A$ is upper
triangular. For simplicity, denote the matrix representation of $A$ in the
basis $(\mathbf v_1,\ldots,\mathbf v_{n-1},\mathbf v_n)$ again by $A$ and let $A_i$,
$1\le i\le n$, be the degeneracy one principal submatrix of $A$ obtained by
deleting its $i$-th row and $i$-th column. Then clearly $A_n=A'$ and so by
Lemma~\ref{g-comp} the spectrum of $A_n$ is
$\Si(A_n)=\{w_1,\ldots,w_{n-1}\}$, where
$\Si(A_n)$ is viewed as a multiset whose elements occur as many times as their
algebraic multiplicities as eigenvalues of $A_n$. Thus $A$ is an $n\times n$
normal matrix with $\Si(A)=\{z_1,\ldots,z_n\}$ such that
\begin{equation}\label{form}
A=\begin{pmatrix}
A_{n} & \ast\\
\ast & \ze
\end{pmatrix},
\end{equation}
where $A_{n}$ is upper triangular with
$\Si(A_n)=\{w_1,\ldots,w_{n-1}\}$ and
$\ze=\langle A\mathbf v_n,\mathbf v_n\rangle=\sum_{i=1}^{n}a_{i}z_i$ is the
$(a_1,\ldots,a_n)$-barycenter of $Z(1,0)$ (cf.~Remark~\ref{r-3}). Note that
since the $w_j$ appear on the diagonal of $A_n$ Lemma~\ref{s-lem} actually
provides a new proof of Theorem~\ref{t-1}. Let $U=(u_{ij})$ be
the (unitary) change of basis matrix from $(\mathbf v_1,\ldots\mathbf v_n)$ to
$(\mathbf e_1,\ldots,\mathbf e_n)$ so that we may write
\begin{equation}\label{dec}
A=U\text{diag}(z_1,\ldots,z_n)U^{*}.
\end{equation}

\begin{proof}[Proof of Theorem~\ref{t-3}]
Let us first show that if~\eqref{cont} holds and $k\in\{1,\ldots,n-1\}$
then one can find an $\binom{n-1}{k}\times\binom{n}{k}$ row stochastic
matrix $R_k$ such that
\begin{equation}\label{want}
W_{k}^{[k]}(1,0)^T=R_{k}Z_{k}^{[k]}(1,0)^T
\text{ and }\mathbf b^{[k]}=\mathbf a^{[k]}R_k.
\end{equation}
This is clear for $k=1$ by Theorem~\ref{t-1} and so we may assume that
$k\ge 2$. Let
$$A^{(k)}=\left(\det A(\mathbf i,\mathbf j)\right)_{\mathbf i.\mathbf j\in Q_{k,n}}$$
be the $k$-th compound matrix of $A$ acting on the $k$-th Grassmann power
$\wedge^{k}\calH$ (recall that the elements of $Q_{k,n}$
and therefore also the
$\binom{n}{k}\times\binom{n}{k}$ determinants appearing in $A^{(k)}$
are arranged in lexicographic order). The Binet-Cauchy
formula for compound matrices \cite[Theorem 19.F.2]{MO} and~\eqref{dec}
imply that
$$A^{(k)}=U^{(k)}\text{diag}(z_1,\ldots,z_n)^{(k)}U^{*(k)},$$
which by an argument similar to the proof of Lemma~\ref{s-lem} leads to
\begin{equation}\label{stoc}
\left(\det A(\mathbf i,\mathbf i)\right)_{\mathbf i\in Q_{k,n}}^{T}=S_k Z_{k}^{[k]}(1,0)^T,
\quad S_k=U^{(k)}\circ \bar{U}^{(k)},
\end{equation}
where $\circ$ denotes as before the Hadamard-Schur product. Note that $S_k$
is an
$\binom{n}{k}\times\binom{n}{k}$ doubly stochastic matrix since $U^{(k)}$ is
unitary. Clearly, the diagonal elements of $A^{(k)}$ corresponding to the
$k\times k$
principal minors of $A_n$ are of the form
$\det A(\mathbf i,\mathbf i)$, $\mathbf i\in Q_{k,n-1}$. There are a total of $\binom{n-1}{k}$
such diagonal elements and these must actually coincide with the coordinates
of the vector $W_{k}^{[k]}(1,0)^T$ since $A_n$ is upper triangular. Let $R_k$
be the $\binom{n-1}{k}\times\binom{n}{k}$ row stochastic
matrix obtained from $S_k$ by deleting all elements $S_k(\mathbf i,\mathbf j)$ with
$\mathbf i\in Q_{k,n}\setminus Q_{k,n-1}$. We show that $R_k$
satisfies~\eqref{want}. For this one has to prove that for any
$\mathbf j=(j_1,\ldots,j_k)\in Q_{k,n}$ the identity
$$\sum_{1\le i_1<\ldots<i_{k-1}\le n-1}
S_{k}\binom{i_1,\ldots,i_{k-1},n}{j_1,\ldots,j_k}=\sum_{l=1}^{k}a_{j_l}$$
holds, which is equivalent to
\begin{equation}\label{want1}
\sum_{1\le i_1<\ldots<i_{k-1}\le n-1}
\left|\det U\binom{i_1,\ldots,i_{k-1},n}{j_1,\ldots,j_k}\right|^2
=\sum_{l=1}^{k}a_{j_l}.
\end{equation}
or
\begin{equation}\label{want2}
\sum_{\mathbf i\in Q_{k,n-1}}\left|\det U(\mathbf i,\mathbf j)\right|^2
=1-\sum_{l=1}^{k}a_{j_l}.
\end{equation}
By~\eqref{dec} one has $u_{nj}=\sqrt{a_j}$, $1\le j\le n$, so that by expanding
each determinant with respect to the last row we get
$$\det U\binom{i_1,\ldots,i_{k-1},n}{j_1,\ldots,j_k}=\sum_{m=1}^{k}(-1)^{k+m}
\sqrt{a_{j_m}}
\det U\binom{i_1,\ldots,i_{k-1}}{j_1,\ldots,\widehat{j_m},\ldots,j_k}$$
and thus
\begin{multline*}
\left|\det U\binom{i_1,\ldots,i_{k-1},n}{j_1,\ldots,j_k}\right|^2\\
=\sum_{r,s=1}^{k}(-1)^{r+s}\sqrt{a_{j_r}a_{j_s}}
\det U\binom{i_1,\ldots,i_{k-1}}{j_1,\ldots,\widehat{j_r},\ldots,j_k}
\det \bar{U}\binom{i_1,\ldots,i_{k-1}}{j_1,\ldots,\widehat{j_s},\ldots,j_k}
\end{multline*}
for $1\le i_1<\ldots<i_{k-1}\le n-1$. Therefore
\begin{equation*}
\sum_{1\le i_1<\ldots<i_{k-1}\le n-1}
\left|\det U\binom{i_1,\ldots,i_{k-1},n}{j_1,\ldots,j_k}\right|^2\\
=\sum_{r,s=1}^{k}(-1)^{r+s}\sqrt{a_{j_r}a_{j_s}}\al_{r,s}(\mathbf j),
\end{equation*}
where
\begin{equation*}
\al_{r,s}(\mathbf j)=\sum_{1\le i_1<\ldots<i_{k-1}\le n-1}
\det U\binom{i_1,\ldots,i_{k-1}}{j_1,\ldots,\widehat{j_r},\ldots,j_k}
\det \bar{U}\binom{i_1,\ldots,i_{k-1}}{j_1,\ldots,\widehat{j_s},\ldots,j_k}.
\end{equation*}
Let us consider the $k\times (n-1)$ matrix
$$M=U\binom{1,\ldots,n-1}{j_1,\ldots,j_k}^T$$
and the $k\times k$ matrix $B=(b_{ij})=MM^{*}$ as well as the
$k\times k$ matrix given by its $k-1$
Grassmann power $B^{(k-1)}=\left(b_{ij}^{(k-1)}\right)$. The Binet-Cauchy
formula again implies that
$$B^{(k-1)}=M^{(k-1)}M^{*(k-1)}$$
from which we deduce that
$$b_{k+1-r,k+1-s}^{(k-1)}=\al_{r,s}(\mathbf j),\quad 1\le r,s\le k.$$
Thus the left-hand side of~\eqref{want1} equals
$$\sum_{r,s=1}^{k}(-1)^{r+s}\sqrt{a_{j_r}a_{j_s}}b_{k+1-r,k+1-s}^{(k-1)}$$
and by further manipulating this expression we obtain
$$\sum_{1\le i_1<\ldots<i_{k-1}\le n-1}
\left|\det U\binom{i_1,\ldots,i_{k-1},n}{j_1,\ldots,j_k}\right|^2
=\sum_{\mathbf i\in Q_{k-1,k}}\det B(\mathbf i,\mathbf i)-k\det B$$
hence
$$\sum_{\mathbf i\in Q_{k,n-1}}\left|\det U(\mathbf i,\mathbf j)\right|^2
=1-\sum_{\mathbf i\in Q_{k-1,k}}\det B(\mathbf i,\mathbf i)+k\det B.$$
To show that the right-hand side of the above identity equals
the right-hand side of~\eqref{want2} we proceed as follows. Define the vector
$\mathbf u=(\sqrt{a_{j_1}},\ldots,\sqrt{a_{j_k}})$. Since $U$ is
unitary an easy computation shows that $B=I_k-\mathbf u^T \mathbf u$, so that $B$ depends
only on the numbers $a_{j_l}$, $1\le l\le k$. It follows that the expression
$$\sum_{1\le i_1<\ldots<i_k\le n-1}
\left|\det U\binom{i_1,\ldots,i_k}{j_1,\ldots,j_k}\right|^2
=\sum_{\mathbf i\in Q_{k,n-1}}\left|\det U(\mathbf i,\mathbf j)\right|^2$$
also depends exclusively on the numbers $a_{j_l}$, $1\le l\le k$, and is
therefore independent of the $z_i$, $1\le i\le n$. On the other hand, by
summing all coordinates corresponding to $\mathbf i\in Q_{k,n-1}$ in each side
of~\eqref{stoc} we get
\begin{equation}\label{n-id-k}
\begin{split}
\sum_{1\le r_1<\ldots<r_k\le n-1}\prod_{j=1}^{k}&w_{r_j}\\
=\sum_{1\le s_1<\ldots<s_k\le n}&\left(\sum_{1\le i_1<\ldots<i_k\le n-1}
\left|\det U\binom{i_1,\ldots,i_k}{s_1,\ldots,s_k}\right|^2\right)
\prod_{i=1}^{k}z_{s_l}.
\end{split}
\end{equation}
Obviously, \eqref{cont} defines the $w_j$, $1\le j\le n-1$, as continuous
functions of the $z_i$, $1\le i\le n$. By letting $z_i\to 0$ for
$i\in \{1,\ldots,n\}\setminus \{j_1,\ldots,j_k\}$ and $z_{j_l}\to 1$ for
$1\le l\le k$ in both~\eqref{n-id-k}
and~\eqref{id-k} and comparing coefficients in the resulting identities
we get~\eqref{want2}, which proves~\eqref{want}.

Finally, to prove the last statement of the theorem it is enough to consider
the function $f(\la^{-1}(z-\mu))$, where $\la\in\bC\setminus\{0\}$
and $\mu\in\bC$. The zeros and
poles of this function are the eigenvalues of
$\la A_n+\mu$ and $\la A+\mu$, respectively. These matrices satisfy
relations similar to~\eqref{form} and~\eqref{dec} while the unitary matrix
$U$ is unchanged.
One may therefore use the same arguments as above to get~\eqref{inv-m}.
\end{proof}

\begin{remark}\label{cred-Ma}
Modulo some technical modifications that have to deal with unequal weights,
the arguments given in the proof of Theorem~\ref{t-3} are the same as those
given by S.~Malamud in his proof of \cite[Theorem 4.7]{Ma2}.
\end{remark}

\begin{remark}\label{m-mat}
An interesting byproduct of the proof of Theorem~\ref{t-3}
is the following identity for a certain type of $M$-matrices. Let
$m\in \bN$, $\mathbf x=(x_1,\ldots,x_m)\in (0,1)^m$ with
$||\mathbf x||^2=\sum_{i=1}^{m}x_{i}^2<1$ and define the $m\times m$ matrix
$X=I_m-\mathbf x^T\mathbf x$. Clearly, $\mathbf x^T\mathbf x$ is a rank one matrix whose
eigenvalues $0$ (with multiplicity $m-1$) and $||\mathbf x||^2$ have modulus less
than $1$, so that $X$ is an $M$-matrix
(see, e.g.,\cite{MO}). By the arguments in the last part of the
proof of Theorem~\ref{t-3} we see that
$$\sum_{\mathbf i\in Q_{m-1,m}}\det X(\mathbf i,\mathbf i)-m\det X=\text{tr} (I_m-X),$$
both sides being equal to $||\mathbf x||^2$.
\end{remark}

Recall that if $C$ is an $n\times n$ matrix and $1\le k\le n$ one defines
the $k$-th additive compound matrix $\De_{k}(C)$ as the
$\binom{n}{k}\times\binom{n}{k}$ given by
$$\De_k(C)=\frac{d}{dt}(I_n+tC)^{(k)}\bigg|_{t=0}.$$
Equivalently, $\De_{k}(C)$ is the coefficient matrix of $t$ in the expansion
$(I_n+tC)^{(k)}=I_n+t\De_k(C)+\calO(t^2)$. As an operator, this is the same
as the restriction to $\wedge^{k}\calH$ of the operator
$$C\otimes I_n\otimes\cdots\otimes I_n+
I_n\otimes C\otimes I_n\otimes\cdots\otimes I_n
+\ldots +I_n\otimes I_n\otimes\cdots\otimes C$$
defined on $\otimes^{k}\calH$.
It is well known that if $C$ has eigenvalues $\la_1,\ldots,\la_n$ then
the eigenvalues of $\De_k(C)$ are $\la_{i_1}+\ldots+\la_{i_k}$,
$1\le i_1<\ldots<i_k\le n$ (cf., e.g., \cite{MO}).

\begin{proof}[Proof of Corollary~\ref{t-2}]
Using~\eqref{dec} and arguing as in the first part of the proof of
Theorem~\ref{t-3} we arrive at
$$\frac{d}{dt}(I_n+tA)^{(k)}\bigg|_{t=0}
=U^{(k)}\frac{d}{dt}
\left(\text{diag}(1+tz_1,\ldots,1+tz_n)^{(k)}\right)\bigg|_{t=0}U^{*(k)}.$$
Since $A_n$ is upper triangular the diagonal elements of $(I_n+tA)^{(k)}$
corresponding to the $k\times k$ principal minors indexed by multi-indices
$\mathbf i=(i_1,\ldots,i_k)\in Q_{k,n-1}$ have the form
$\prod_{j=1}^{k}(1+tw_{i_j})$. The conclusion now follows from
the obvious identities
$$\frac{d}{dt}\prod_{j=1}^{k}(1+tw_{i_j})\bigg|_{t=0}=\sum_{j=1}^{k}w_{i_j},
\quad \frac{d}{dt}\prod_{l=1}^{k}(1+tz_{s_l})=\sum_{l=1}^{k}z_{s_l}$$
combined with the arguments in the second
part of the proof of Theorem~\ref{t-3}.
\end{proof}

Corollary~\ref{t-2} may alternatively be proved by using~\eqref{inv-m} with
$\la=t$ and $\mu=1$ and differentiating with respect to $t$. Using this
alternative method we can actually prove Corollary~\ref{t-4}.

\begin{proof}[Proof of Corollary~\ref{t-4}]
Let $\la\in\bC\setminus\{0\}$ and $\mu\in\bC$.
By~\eqref{inv-m} and Remark~\ref{la0} there exists an
$\binom{n-1}{k}\times\binom{n}{k}$ row stochastic matrix $R_k$ such that
the relations
$$W_{k}^{[k]}(t\la,t\mu+1)^T=R_{k}Z_{k}^{[k]}(t\la,t\mu+1)^T\text{ and }
\mathbf b^{[k]}=\mathbf a^{[k]}R_{k}$$
hold for all $t\in\bC$. Thus
$$\frac{d^m}{dt^m}W_{k}^{[k]}(t\la,t\mu+1)^T\bigg|_{t=0}
=R_{k}\frac{d^m}{dt^m}Z_{k}^{[k]}(t\la,t\mu+1)^T\bigg|_{t=0}$$
for $m\in\{1,\ldots,k\}$, which is the same as~\eqref{gen-m}.
\end{proof}

\begin{remark}\label{rem-expl-distct}
The assumption that the points $z_i$, $1\le i\le n$, are {\em distinct} that
was made at the beginning of \S \ref{s21} was not actually used in the
above proofs. However, this condition is natural when viewing -- as we did --
these points as charged particles generating a resultant electrostatic force
given by~\eqref{f}. Clearly, the latter can be appropriately rewritten even
in the case of multiple points.
\end{remark}

\subsection{Proof of Theorems~\ref{t-5}--\ref{t-6}}\label{s32}

The inequality $h(\calW,\calZ)\leq h(\calW_e,\calZ)$ is trivially true since
$\calW\subseteq\calW_e$. To prove the interesting part of Theorem~\ref{t-5}
we make use of a result on spectral variations of normal matrices
which is usually referred to
as the Bauer-Fike theorem (cf., e.g., \cite[Theorem VI.3.3]{Bh}).

\begin{lemma}\label{b-f}
Let $A$ and $B$ be $n\times n$ complex matrices. If $A$ is normal then
$$h(\Si(B),\Si(A))\le ||A-B||,$$
where $h$ denotes the relative Hausdorff distance from $\Si(B)$ to $\Si(A)$.
\end{lemma}

Recall the orthonormal basis $(\mathbf v_1,\ldots,\mathbf v_n)$ of $\calH$ where the matrix
representation of the normal operator $A$ with $\Si(A)=\calZ$ is given
by~\eqref{form}, that is,
\begin{equation}\label{tri}
A=\begin{pmatrix}
A_{n} & \mathbf x\\
\mathbf y^{*} & \ze
\end{pmatrix}.
\end{equation}
In the above decomposition $A_n$ is an $(n-1)\times(n-1)$ upper triangular
matrix with $\Si(A_n)=\calW$,
$\ze=\langle A\mathbf v_n,\mathbf v_n\rangle=\sum_{i=1}^{n}a_{i}z_{i}$, and
$\mathbf x,\mathbf y\in\bC^{n-1}$. Set $B=A_n\oplus \ze$. An easy computation now
shows that
\begin{equation*}
\begin{split}
||A-B||^{2}&=||\mathbf x||^2=||\mathbf y||^2
=\sum_{i=1}^{n-1}|\langle A\mathbf v_{n},\mathbf v_i\rangle|^2
=||A\mathbf v_n||^2-|\langle A\mathbf v_n,\mathbf v_n\rangle|^2\\
&=\sum_{i=1}^{n}a_{i}|z_i|^2-|\ze|^2=\si_2(\mathbf z;\mathbf a)^2,
\end{split}
\end{equation*}
which combined with Lemma~\ref{b-f} yields the inequality stated in
Theorem~\ref{t-5}.

To prove Theorem~\ref{t-6} note first that by Theorem~\ref{t-5} it
remains to check that $h(\calZ,\calW_e)\le \si_2(\mathbf z;\mathbf a)$. The latter
inequality is invariant under non-singular affine transformations of
the complex plane and it is therefore enough to show that
$h(\calZ,\calW_e)\le \si_2(\mathbf z;\mathbf a)$ whenever $\mathbf z=(z_1,\ldots,z_n)\in\bR^n$.
Since in this case $A$ is a Hermitian operator
we get from~\eqref{tri} that $\mathbf x:=(x_1,\ldots,x_{n-1})^t=\mathbf y$ and
$A_n=\text{diag}(w_1,\ldots,w_{n-1})$. Suppose $z_i$ is
such that
\begin{equation}\label{ineq1}
\min_{1\le j\le n-1}|z_i-w_j|>\si_2(\mathbf z;\mathbf a)
\end{equation}
and let $\mathbf u=(u_1,\ldots,u_n)^t\in\bC^n$ be a unit eigenvector of $A$ with
eigenvalue $z_i$. Then by~\eqref{tri} we may rewrite the identity
$A\mathbf u=z_{i}\mathbf u$ as
\begin{equation}\label{ineq2}
(\ze-z_i)u_{n}=-\sum_{j=1}^{n-1}\bar{x}_ju_j\text{ and }
(w_j-z_i)u_j=x_ju_n,\quad 1\le j\le n-1.
\end{equation}
Note that $\si_2(\mathbf z;\mathbf a)>0$ since $z_1,\ldots,z_n$ are assumed to be
pairwise distinct.
Moreover, the fact that $||\mathbf u||=1$ combined with~\eqref{ineq2} implies that
$u_n\neq 0$. From~\eqref{ineq1} and the last $n-1$ identities in~\eqref{ineq2}
we get
$$|u_j|\le \si_2(\mathbf z;\mathbf a)^{-1}|x_j||u_n|,\quad 1\le j\le n-1,$$
and then using the first relation in~\eqref{ineq2} we deduce that
$$|\ze-z_i|\le \si_2(\mathbf z;\mathbf a)^{-1}||\mathbf x||^2=\si_2(\mathbf z;\mathbf a).$$
This shows that $h(\{z_i\},\calW_e)\le \si_2(\mathbf z;\mathbf a)$, which completes
the proof of Theorem~\ref{t-6}.

\subsection{Proof of Theorems~\ref{t-7}--\ref{t-8}}\label{s33}

It is clear that up to a rescaling we may assume that $\rho=1$ in both
Theorem~\ref{t-7}
and Theorem~\ref{t-8}. Below we shall use the Hilbert space of square
summable complex sequences $\calH=l^2(\bN)$
with standard scalar product $\langle \cdot,\cdot \rangle$ and standard
orthonormal basis $\{\mathbf e_i\}_{i\in\bN}$.

\begin{proof}[Proof of Theorem~\ref{t-7}]
We may assume wlog that $\{z_i\}_{i\in\bN}$ is an increasing sequence.
The fact that $f$ has infinitely many zeros $\{w_j\}_{j\in\bN}$ that interlace
the poles $\{z_i\}_{i\in\bN}$ on the real axis follows by noticing that all
zeros of $f$ are necessarily real and that $f$ is strictly decreasing on each
interval $(z_i,z_{i+1})$, $i\in\bN$. Alternatively, this may also be viewed as
a consequence of the operator theoretic interpretation given below combined
with Hochstadt's interlacing theorem for the eigenvalues of
rank one perturbations of compact selfadjoint Hilbert space operators
\cite{H}.

Let $A$ denote the unique bounded operator on $\calH$ satisfying
$A\mathbf e_i=(1-z_i)\mathbf e_i$, $i\in\bN$. Note that $A$ is injective,
selfadjoint and compact since the $z_i$ are real and $z_i\to 1$ as
$i\to\infty$. By assumption one has $\sum_{i=1}^{\infty}a_i=1$ and so we may
define a unit vector $\mathbf v\in\calH$ in the following way
\begin{equation}\label{i-v}
\mathbf v=\sum_{i=1}^{\infty}\sqrt{a_i}\mathbf e_i.
\end{equation}
Set $\calK=\mathbf v^{\perp}$ and let $P$ be the orthoprojection on $\calK$. Recall
from \cite[\S IV.6]{Ka2} that the Weinstein-Aronszajn determinant of the
second kind associated with $A$ and $P$ is given by
$\langle (A-zI)^{-1}\mathbf v,\mathbf v\rangle$. A simple computation shows that
\begin{equation}\label{wa1}
\langle (A-zI)^{-1}\mathbf v,\mathbf v\rangle=f(1-z),\quad z\in\bC\setminus
\{1-z_i\}_{i\in\bN}.
\end{equation}
Let us now consider the compression $A'$ of $A$ to $\calK$, that is, the
operator on $\calK$ given by $A'=PAP|_{\calK}$. Note that $A'$ is itself a
compact selfadjoint operator. We claim that the point spectrum of $A'$ consists
precisely of the zeros of $f(1-z)$.

\begin{lemma}\label{wa2}
The set of eigenvalues of $A'$ is $\{1-w_j\}_{j\in\bN}$.
\end{lemma}

\begin{proof}
If $\mathbf u\in\calK$ is an eigenvector of $A'$ with eigenvalue $\ze\in\bC$
(actually $\bR$) then
\begin{equation}\label{wa3}
(A-\ze I)\mathbf u=(I-P)A\mathbf u +(A'-\ze I)\mathbf u=\langle A\mathbf u,\mathbf v\rangle\mathbf v.
\end{equation}
From~\eqref{wa3} we see that $\langle A\mathbf u,\mathbf v\rangle\neq 0$ since otherwise
$\mathbf u$ would be an eigenvector of $A$, say $\mathbf u=\mathbf e_i$ for some $i\in\bN$,
which would then lead to the contradiction
$\langle A\mathbf u,\mathbf v\rangle=\langle A\mathbf e_i,\mathbf v\rangle=(1-z_i)\sqrt{a_i}\neq 0$.
By~\eqref{wa1} and~\eqref{wa3} we get
$$f(1-\ze)=\langle (A-\ze I)^{-1}\mathbf v,\mathbf v\rangle
=\langle A\mathbf u,\mathbf v\rangle^{-1}\langle \mathbf u,\mathbf v\rangle=0$$
so that $1-\ze=w_j$ for some $j\in\bN$. Conversely, let $w$ be such that
$f(1-w)=0$, that is, $\langle (A-wI)^{-1}\mathbf v,\mathbf v\rangle=0$, and set
$\mathbf u=(A-wI)^{-1}\mathbf v$. Then clearly $\mathbf u\neq 0$ and one has
$$\mathbf v=(A-wI)\mathbf u=\langle A\mathbf u,\mathbf v\rangle\mathbf v+(A'-wI)\mathbf u.$$
Since $(A'-wI)\mathbf u\in\calK$ and $\mathbf v\in\calK^{\perp}$ it follows that
$$(A'-wI)\mathbf u=(1-\langle A\mathbf u,\mathbf v\rangle)\mathbf v=0.$$
Thus $\mathbf u$ is an eigenvector of $A'$ with eigenvalue $w$.
\end{proof}

It is not difficult to see that $A'$ is injective. Indeed, if
$0\neq\mathbf u\in\calK$ is such that
$A'\mathbf u=0$ then from $PA\mathbf u=0$ and the fact that $A$ is injective we get
$A\mathbf u=\al\mathbf v$ for some $\al\in\bC\setminus\{0\}$. Hence
$$0=\langle \mathbf u,\mathbf v\rangle=\al\langle A^{-1}\mathbf v,\mathbf v\rangle=\al f(1),$$
which is a contradiction since the assumptions of the theorem (more
specifically, condition~\eqref{i-hyp} with $\rho=1$) imply that $f(1)>0$.

By the spectral theorem for compact selfadjoint operators
\cite[Corollary X.3.3.5]{DS} there exists a complete orthonormal
system $\{\mathbf f_j\}_{j\in\bN}$ in $\calK$ consisting of eigenvectors for $A'$.
Set $s_{jk}=|\langle \mathbf f_j,\mathbf e_k\rangle|^2$ for $j,k\in\bN$. Then
$$1-w_j=\langle A'\mathbf f_j,\mathbf f_j\rangle=\langle A\mathbf f_j,\mathbf f_j\rangle
=\sum_{k=1}^{\infty}s_{jk}(1-z_k).$$
Both $\{\mathbf v\}\cup\{\mathbf f_j\}_{j\in\bN}$ and $\{\mathbf e_k\}_{j\in\bN}$ are
complete orthonormal systems in $\calH$, so that
\begin{equation*}
\begin{split}
&\sum_{k=1}^{\infty}s_{jk}=\sum_{k=1}^{\infty}|\langle \mathbf f_j,\mathbf e_k\rangle|^2
=||\mathbf f_j||^2=1,\quad j\in\bN,\\
&\sum_{j=1}^{\infty}s_{jk}=\sum_{j=1}^{\infty}|\langle \mathbf f_j,\mathbf e_k\rangle|^2
=||\mathbf e_k||^2-|\langle \mathbf v,\mathbf e_k\rangle|^2=1-a_k,\quad k\in\bN,
\end{split}
\end{equation*}
which completes the proof of the theorem.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{t-8}]
Let $\{z_i\}_{i\in\bN}$ be a sequence of distinct complex numbers
satisfying~\eqref{i-hyp} with $\rho=1$ and let $A$ be the injective compact
normal operator on $\calH$ defined by $A\mathbf e_i=(1-z_i)\mathbf e_i$, $i\in\bN$. As
before we denote by $A'$ the compression $PAP|_{\calK}$ of $A$, where
$\calK=\mathbf v^{\perp}$ and $\mathbf v$ is given by~\eqref{i-v}. Clearly, $A'$ is a
compact operator on $\calK$. Moreover, since $\Re f(1)>0$ the arguments in
the proof of Theorem~\ref{t-7} show that $A'$ is injective. Note that
Lemma~\ref{wa2} remains valid in the present setting and that by assumption
$f$ has a discrete set of zeros $\{w_j\}_{j\in\bN}$. It follows that the
point spectrum of $A'$ is $\{1-w_j\}_{j\in\bN}$.
Denote by $\calK_{A'}$ the closed linear hull of the root subspaces of $A'$
and let $\widehat{A'}$ be the operator on $\calK_{A'}$ induced by $A'$. Recall
the operator version of Schur's lemma given in \cite[Lemma 4.1]{GK}.

\begin{lemma}\label{sch}
With the above notations there exists an orthonormal basis
$\{\mathbf f_j\}_{j\in\bN}$ of $\calK_{A'}$ in which the matrix of the operator
$\widehat{A'}$ has triangular form and
$\langle \widehat{A'}\mathbf f_j,\mathbf f_j\rangle=1-w_j$, $j\in\bN$.
\end{lemma}

Set $s_{jk}=|\langle \mathbf f_j,\mathbf e_k\rangle|^2$ for $j,k\in\bN$. By
Lemma~\ref{sch} we have
$$1-w_j=\langle \widehat{A'}\mathbf f_j,\mathbf f_j\rangle=\langle A\mathbf f_j,\mathbf f_j\rangle
=\sum_{k=1}^{\infty}s_{jk}(1-z_k).$$
It is clear that
\begin{equation}\label{bes-1}
\sum_{k=1}^{\infty}s_{jk}=\sum_{k=1}^{\infty}|\langle \mathbf f_j,\mathbf e_k\rangle|^2
=||\mathbf f_j||^2=1,\quad j\in\bN.
\end{equation}
On the other hand $\{\mathbf v\}\cup\{\mathbf f_j\}_{j\in\bN}$ is an orthonormal system
in $\calH$ so that by Bessel's inequality we get
\begin{equation}\label{bes-2}
\sum_{j=1}^{\infty}s_{jk}=\sum_{j=1}^{\infty}|\langle \mathbf f_j,\mathbf e_k\rangle|^2
\le ||\mathbf e_k||^2-|\langle \mathbf v,\mathbf e_k\rangle|^2=1-a_k,\quad k\in\bN.
\end{equation}
The inequality stated in the theorem by means of nonnegative convex
functions is an immediate consequence of~\eqref{bes-1}--\eqref{bes-2}.
\end{proof}

\section{A hierarchy of de Bruijn-Springer relations and inertia
laws}\label{s-mom}

Theorems~\ref{t-1}--\ref{t-3} suggest even deeper physical connections
between the set of equilibrium points and the given set of charged particles
in the case of finite planar systems, such as a whole new hierarchy of
inertia laws for planar solid bodies associated with these sets. In order to
make a precise statement
let us identify the complex plane $\bC$ with
the $x_1x_2$-plane in $\bR^3$ equipped with the standard coordinate system
$(O;x_1,x_2,x_3)$. For an integer $k\ge 2$ we denote by $\Xi_k$ the standard
simplex
$$\Xi_k=\left\{(t_1,\ldots,t_k)\in [0,1]^k\mid t_1+\ldots +t_k
=1\right\}.$$

\begin{definition}
Let $L$ be a line in $\bR^3$. Given a convex $k$-gon $\calK$ in $\bC$ with
vertices $v_1,\ldots,v_k$ and $\al\ge 1$ we define the $\al$-{\em moment
of} $\calK$ {\em with respect to} $L$ to be
$$I_{\al,L}(\calK)=\int\cdots\int_{\Xi_k}
d\!\left(t_{1}v_1+\ldots +t_{k}v_k,L\right)^{\al}dt_1\cdots dt_k,$$
where $d(\cdot,L)$ denotes the distance function associated with $L$.
\end{definition}

\begin{remark}
If $\calK$ is viewed as a planar solid body with uniform mass distribution then
$I_{\al,L}(\calK)$ may be interpreted as a higher moment of inertia (or
generalized angular momentum) of $\calK$ with respect to the rotation axis $L$.
\end{remark}

Let $\Si_m$ denote the symmetric group on $m$ elements. It is not difficult to
see that if $\calK$ is as above and $u_1,\ldots,u_m\in\bC$ are such that
$\calK=\co(u_1,\ldots,u_m)$ then
\begin{equation}\label{invar-m}
\int\cdots\int_{\Xi_m}d(t_{\pi(1)}u_1+\ldots +t_{\pi(m)}u_m,L)^{\al}
dt_1\cdots dt_m=I_{\al,L}(\calK)
\end{equation}
for any $\pi\in \Si_m$, so that $I_{\al,L}(\calK)$ depends only on the extreme
points of $\calK$.

The following conjecture is a powerful generalization of
Theorems~\ref{t-1}--\ref{t-3} and suggests a ``hierarchy'' of weighted
de Bruijn-Springer relations and inertia laws for planar polygons constructed
from the equilibrium points and the charged particles.

\begin{conjecture}\label{gen-mom}
Let $z_i$, $1\le i\le n$, be distinct points in $\bC$ and $a_i>0$,
$1\le i\le n$, be such that $\sum_{i=1}^{n}a_i=1$, where $n\ge 2$. Denote by
$w_j$, $1\le j\le n-1$, the zeros of
$$f(z)=\sum_{i=1}^{n}\frac{a_i}{z-z_i}$$
and let $\Phi:\bC\to \bR$ be an arbitrary convex function. Then for any
$k\in\{1,\ldots,n-1\}$, $1\le m\le k$, and $(t_1,\ldots,t_k)\in\bC^k$ one has
\begin{equation}\label{dbs-h}
\begin{split}
\sum_{1\le r_1<\ldots<r_k\le n-1}&\,\,\sum_{\pi\in \Si_k}
\Phi\left(\Pi_{k,m}(t_{\pi(1)}w_{r_1},\ldots,t_{\pi(k)}w_{r_k})\right)\\
\le \sum_{1\le s_1<\ldots <s_k\le n}&\left(1-\sum_{i=1}^{k}a_{s_i}\right)
\sum_{\pi\in \Si_k}
\Phi\left(\Pi_{k,m}(t_{\pi(1)}z_{s_1},\ldots,t_{\pi(k)}z_{s_k})\right),
\end{split}
\end{equation}
where $\Pi_{k,m}$ is the $m$th elementary symmetric function on $k$ symbols.
In particular, for any line $L$ in $\bR^3$ and $\al\ge 1$ one has
\begin{equation}\label{ang-m}
\begin{split}
\sum_{1\le r_1<\ldots<r_k\le n-1}
& I_{\al,L}\left(\tco(w_{r_1},\ldots,w_{r_k})\right)\\
& \le \sum_{1\le s_1<\ldots <s_k\le n}\left(1-\sum_{i=1}^{k}a_{s_i}\right)
I_{\al,L}\left(\tco(z_{s_1},\ldots,z_{s_k})\right).
\end{split}
\end{equation}
\end{conjecture}

The fact that~\eqref{dbs-h} implies~\eqref{ang-m} follows from~\eqref{invar-m}
by taking $m=1$ and integra\-ting over the simplex $\Xi_k$. Arguing as in the
proof of Corollary~\ref{t-4} one can see that it is actually enough to
prove~\eqref{dbs-h} for $m=k$ and $k\in\{1,\ldots,n-1\}$. Note also that in
the case of equal charges Conjecture~\ref{gen-mom} would provide a whole new
series of inequalities for averages of generalized inertia moments
involving
the critical points and the zeros of an arbitrary complex polynomial. These
would substantially improve both Corollary~\ref{cor-mom} and the
inequalities conjectured in \cite[Remark 2.3.9]{RS}.

\begin{remark}
In view of the proofs of \cite[Theorem 4.7]{Ma2} and Theorem~\ref{t-3} above,
a natural way of attacking Conjecture~\ref{gen-mom} would be to interpret
the arguments of $\Phi$ in~\eqref{dbs-h} as eigenvalues of some appropriately
defined multilinear power ``$\wedge^k A$'' of a normal matrix $A$ and one of
its principal submatrices, where ``$\wedge$'' is a suitable multilinear
functional on $\otimes^k \calH$. However, defining such a functional appears
to be a non-trivial question. Note for instance that an obvious polarization
of the usual antisymmetric tensor power, i.e., setting
$$A\wedge B=\frac{1}{2}\left[(A+B)^{(2)}-A^{(2)}-B^{(2)}\right]$$
for square matrices $A,B$ of the same order, cannot lead to the desired
conclusion since to begin with one does not get the correct dimensions (these
should agree with the number of $k$-tuples in the two sides of~\eqref{dbs-h}).
\end{remark}

\section{Zeros of Borel series with positive $l^1$-coefficients}\label{s42}

Theorem~\ref{t-8} applies essentially to any bounded discrete set of
positively charged particles with finite total charge for which equilibrium
points are known to exist. We conjecture that if the particles accumulate at
a point on the boundary of a disk containing all the particles then there
exist infinitely many equilibrium points.

\begin{conjecture}\label{con-2}
Let $\{a_i\}_{i\in\bN}$ be a sequence of positive numbers whose sum is finite
and $\{z_i\}_{i\in\bN}$ a sequence of distinct complex numbers
satisfying
$$|z_i|<\rho\text{ for }i\in\bN\text{ and }\lim_{i\to\infty}z_i
=\xi\text{ with }|\xi|=\rho$$
for some fixed positive number $\rho$. Then the function
\begin{equation}\label{i-ff}
f(z)=\sum_{i=1}^{\infty}\frac{a_i}{z-z_i}
\end{equation}
has an infinite set of zeros in the disk $\bD(\rho)=\{z\in\bC: |z|<\rho\}$.
\end{conjecture}

Note that the assumptions of Conjecture~\ref{con-2} imply
that condition~\eqref{i-hyp} holds and thus $f$ is meromorphic in
$\bD(\rho)$. Functions representable as a series of the form~\eqref{i-ff} are
known as Borel series \cite{W} and so Conjecture~\ref{con-2} may be rephrased
as follows: {\em every Borel series with positive $l^1$-coefficients whose
poles
lie in an open disk and accumulate at a point on the boundary of this disk
has infinitely many zeros}.

Conjecture~\ref{con-2} may also be translated in operator theoretic terms by
using the methods employed in the proofs of Theorems~\ref{t-7}--\ref{t-8}.
Note first that up to a similarity transformation of the complex plane we
may assume that $\xi=\rho=1$. Then one can show that
the above conjecture is equivalent to the following statement.

\begin{conjecture}\label{new-c2}
Let $\{z_n\}_{n\in\bN}$ be a sequence of distinct complex numbers in the open
unit disk converging to $1$ and denote by $A$ the (bounded) normal operator on
a separable infinite-dimensional complex Hilbert space $\calH$ satisfying
$A\mathbf e_n=z_{n}\mathbf e_n$, $n\in \bN$, where $\{\mathbf e_n\}_{n\in\bN}$ is an
orthonormal basis of $\calH$. If $P$ is a bounded operator on $\calH$ such
that $I-P$ is a rank one orthoprojection then the compression
$PAP|_{P\calH}$ has infinite point spectrum.
\end{conjecture}

\begin{remark}\label{more-gen}
The hypotheses in Conjecture~\ref{new-c2} imply that the normal operator
$I-A$ is compact, which may facilitate its study. However, it seems likely
that Conjecture~\ref{con-2} should actually
hold even under the less restrictive conditions $|z_i|<\rho$ for all
$i\in\bN$ and $\lim_{i\to\infty}|z_i|=\rho$. In other words, it is reasonable
to believe that the convergence condition in Conjecture~\ref{new-c2} may be
relaxed
as follows: $|z_n|<1$ for all $n\in\bN$ and $\lim_{n\to\infty}|z_n|=1$.
\end{remark}

\section{Operator versions of the Clunie-Eremenko-Rossi
conjecture}\label{s43}

Various results concerning the existence of equilibrium
points for Newtonian and logarithmic potentials have been obtained in
e.g.~\cite{CER}, \cite{ELR} and \cite{LR}. It should be emphasized though
that these deal almost exclusively with {\em unbounded} discrete charge
configurations. Note that Conjecture~\ref{con-2}
may actually be viewed as a natural analog for
{\em bounded} discrete charge configurations of the following well-known
conjecture
of Clunie-Eremenko-Rossi, see, e.g., \cite[Conjecture 2.7]{CER} and
\cite[Conjecture 1.1]{LR}.

\begin{conjecture}[Clunie-Eremenko-Rossi conjecture]\label{con-CER}
Let $\{a_i\}_{i\in\bN}$ be a sequence of positive numbers and
$\{z_i\}_{i\in\bN}$ a sequence of distinct complex numbers such that
$$z_i\to\infty\text{ as }i\to\infty\text{ and }
\sum_{z_i\neq 0}\frac{a_i}{|z_i|}<\infty.$$
Then the meromorphic function
$$f(z)=\sum_{i=1}^{\infty}\frac{a_i}{z-z_i}$$
has infinitely many zeros.
\end{conjecture}

A catchy albeit somewhat loose rephrasing of this conjecture is as follows:
{\em every flat universe has infinitely many resting points}.

Let us use the framework of \S \ref{s33} to give an operator theoretic
interpretation and generalization of Conjecture~\ref{con-CER}. Note first that
up to a translation of the variable $z$ and a rescaling of the coefficients
$a_n$ we may assume wlog that all $z_n$ are non-zero and satisfy the
normalization condition
\begin{equation}\label{i-norm}
\sum_{i=1}^{\infty}\frac{a_i}{|z_i|}=1.
\end{equation}
As in \S \ref{s33} we use the Hilbert space $\calH=l^2(\bN)$ with standard
inner
product $\langle\cdot,\cdot\rangle$ and standard complete orthonormal system
$\{\mathbf e_n\}_{n\in\bN}$. Set $z_n=r_n e^{i\te_n}$, $n\in\bN$, and let $A$ and
$B$ be the bounded normal operators on $\calH$ satisfying
\begin{equation}\label{ab1}
A\mathbf e_n=e^{i\te_n}\mathbf e_n,\quad B_n=r_{n}^{-1}\mathbf e_n,\quad n\in\bN.
\end{equation}
Clearly, $A$ is unitary and $B$ is compact. Condition~\eqref{i-norm} allows us
to define a unit vector $\mathbf v\in\calH$ by setting
$$\mathbf v=\sum_{n=1}^{\infty}\sqrt{\frac{a_n}{|z_n|}}\,\mathbf e_n$$
and one can easily verify that
\begin{equation}\label{ab2}
\langle (A-zB)^{-1}\mathbf v,\mathbf v\rangle=-f(z),\quad
z\in\bC\setminus \{z_n\}_{n\in\bN}.
\end{equation}
Set $\calK=\mathbf v^{\perp}$ and let $P$ be the orthoprojection on $\calK$. We use
the notation $A'$ and $B'$ for the compressions to $\calK$ of $A$ and $B$,
respectively, that is,
$$A'=PAP|_{\calK},\quad B'=P^BP|_{\calK}.$$
By definition, the set of generalized eigenvalues of $A'$ with respect to
$B'$ is
$$\Si_{_{B'}}(A')=\{z\in\bC\mid N(A'-zB')\neq \{0\}\},$$
where $N(T)$ denotes as usual the null space of an operator $T$. Let $\calZ(f)$
be the zero set of $f$.

\begin{lemma}\label{i-eig}
In the above notation one has $\calZ(f)=\Si_{_{B'}}(A')$.
\end{lemma}

\begin{proof}
We make an appropriate adaption of the argument used in the proof of
Lemma~\ref{wa2}. Using the fact that the $z_n$ are distinct
one can check that if $0\neq \mathbf u\in\calH$ then $\mathbf u\in N(A-\ze B)$ if and
only if $\ze=z_n$ and $\mathbf u=\mathbf e_n$ for some $n\in\bN$. However
$$\langle\mathbf e_n,\mathbf v\rangle=\sqrt{\frac{a_n}{|z_n|}}\neq 0,$$
which shows that $N(A-\ze B)\cap \calK=\{0\}$ for $\ze\in\bC$. Let now
$\mathbf u\in\calK$ be such that $(A'-\ze B')\mathbf u=0$ for some $\ze\in\bC$. From the
identity
$$(A-\ze B)\mathbf u=(I-P)(A-\ze B)\mathbf u+(A'-\ze B')\mathbf u
=\langle (A-\ze B)\mathbf u,\mathbf v\rangle\mathbf v$$
and $\langle (A-\ze B)\mathbf u,\mathbf v\rangle\neq 0$ it follows that
$(A-\ze B)^{-1}\mathbf v=\langle (A-\ze B)\mathbf u,\mathbf v\rangle^{-1}\mathbf u$.
By~\eqref{ab2} one gets
$$f(\ze)=-\langle (A-\ze B)\mathbf u,\mathbf v\rangle^{-1}\langle \mathbf u,\mathbf v\rangle=0,$$
so that $\ze\in\calZ(f)$. Conversely, let $w\in\calZ(f)$. Then
$w\notin\{z_n\}_{n\in\bN}$ and so $A-wB$ is invertible by~\eqref{ab1}. The
vector $\mathbf u:=(A-wB)^{-1}\mathbf v$ is clearly non-zero and one has
$\langle \mathbf u,\mathbf v\rangle=f(w)=0$ hence $\mathbf u\in\calK$. Thus
$$\mathbf v=(A-wB)\mathbf u=\langle (A-wB)\mathbf u,\mathbf v\rangle\mathbf v+(A'-wB')\mathbf u,$$
which shows that $(A'-wB')\mathbf u\in\mathbf v^{\perp}\cap\{\mathbf v\}=\{0\}$ and therefore
$w\in \Si_{_{B'}}(A')$.
\end{proof}

Recall that a vector with all non-zero coordinates in the standard
basis $\{\mathbf e_n\}_{n\in\bN}$ of $\calH$ is called totally non-zero. The
above discussion shows that Clunie-Eremenko-Rossi conjecture
(Conjecture~\ref{con-CER}) is in fact
equivalent to the following statement.

\begin{conjecture}\label{con-5}
Let $\{\al_n\}_{n\in\bN}$ and $\{\be_n\}_{n\in\bN}$ be a sequence of
unimodular complex numbers and a sequence of positive numbers converging to
$0$, respectively. Assume that $\al_{n}\be_{n}^{-1}$ are distinct and
define a unitary operator $A$ and a compact selfadjoint
operator $B$ on $\calH$ by setting
$$A\mathbf e_n=\al_n \mathbf e_n,\quad B\mathbf e_n=\be_n \mathbf e_n,\quad n\in\bN.$$
Let $\mathbf v$ be a totally non-zero unit vector in $\calH$, denote by $P$ the
orthoprojection on $\calK=\mathbf v^{\perp}$ and let $A'=PAP|_{\calK}$ and
$B'=PBP|_{\calK}$. Then $\left|\Si_{_{B'}}(A')\right|=\infty$, i.e.,
there exist infinitely many $z\in\bC$ such that $N(A'-zB')\neq\{0\}$.
\end{conjecture}

If the vector $\mathbf v$ in Conjecture~\ref{con-5} is represented in the standard
orthonormal basis of $\calH$ as
$\mathbf v=\sum_{n=1}^{\infty}v_n\mathbf e_n$ then Keldysh's value distribution theorem
for meromorphic functions implies that
Conjecture~\ref{con-5} is true provided that
$\sum_{n=1}^{\infty}\be_{n}^{-1}|v_n|^2<\infty$ (see, e.g., \cite{CER} and
\cite[Ch.~5, Theorem 6.2]{GO}.

Conjecture~\ref{con-5} suggests the following more general question: let $A$
be an injective compact normal Hilbert space operator
and let $B$ and $C$ be finite-rank operators. Is it true that
$\left|\Si_{_{I+C}}(A+B)\right|=\infty$, that is, are there
infinitely many $z\in\bC$ such that $N(A+B-z(I+C))\neq\{0\}$? It is not
difficult to see that an affirmative answer to this question would actually
imply the validity of the operator theoretic version of the
Clunie-Eremenko-Rossi
conjecture (Conjecture~\ref{con-5}). Indeed, in the notation of
Conjecture~\ref{con-5} we may write
\begin{equation*}
\begin{split}
A'-zB'=-zA\big\{A^{-1}B-A^{-1}[&PB(I-P)-(I-P)B]\\
&-z^{-1}[I-A^{-1}(PA(I-P)+(I-P)A)]\big\}.
\end{split}
\end{equation*}
It remains to note that under the assumptions of
Conjecture~\ref{con-5} the operator
$A^{-1}B$ is injective, compact and normal while both
$A^{-1}[PB(I-P)-(I-P)B]$
and $A^{-1}[PA(I-P)+(I-P)A]$ have rank at most two. However, the answer to
the question raised above is negative, as
one can see from the following example.

\begin{ex}
Let $J$ be the simple dissipative
Volterra operator on $L^{2}(0,1)$ defined by
$$(Jf)(x)=2i\int_{0}^{x}f(t)dt$$
and set $A=J+J^{*}$, $B=J-J^{*}$ and $C=0$. Then $A$ is
injective compact selfadjoint, $B$ is a one-dimensional operator while
the point spectrum of $A+B=2J$ is empty (cf.~\cite[p.~187]{GK}), so that
in this case $A+B-z(I+C)$ is injective for all $z\in\bC$.
\end{ex}

Let us finally mention that the existence of equilibrium points for some
discrete potentials generated by complex charges
was established in e.g.~\cite{LR}. We propose the following analog of
Conjecture~\ref{con-CER} for certain potentials of this type.

\begin{conjecture}\label{con-compl}
Let $\{a_n\}_{n\in\bN}$ and $\{z_n\}_{n\in\bN}$ be sequences of complex numbers
such that the $z_n$ are distinct and satisfy
$$a_{n}\bar{z}_n>0\text{ for }n\in\bN,\quad z_n\to\infty\text{ as }n\to\infty,
\quad \sum_{n=1}^{\infty}\left|\frac{a_n}{z_n}\right|<\infty.$$
Then the meromorphic function
$$f(z)=\sum_{n=1}^{\infty}\frac{a_n}{z-z_n}$$
has infinitely many zeros.
\end{conjecture}

As in the case of Conjectures~\ref{con-2} and~\ref{con-CER}, one can
reduce the conjecture stated above to a purely operator theoretic question.
Indeed, in the notations of Conjecture~\ref{con-compl} let $A$ denote the
injective compact normal operator on $\calH$ satisfying
$A\mathbf e_n=z_{n}^{-1}\mathbf e_n$, $n\in\bN$. Define a vector
$$\mathbf v=\sum_{n=1}^{\infty}\sqrt{\left|\frac{a_n}{z_n}\right|}\,\mathbf e_n\in\calH,$$
let $P$ the orthoprojection on $\mathbf v^{\perp}$ and set
$A'=PAP|_{\mathbf v^{\perp}}$. Then
$$\langle (A-z^{-1}I)^{-1}\mathbf v,\mathbf v\rangle=zf(z),\quad
z\in\bC^{*}\setminus \{z_n\}_{n\in\bN}.$$
Arguing as in the proof of Lemma~\ref{i-eig} one can show that
$\ze^{-1}\in\calZ(f)$ for any $\ze\in\Si_{p}(A')\setminus \{0\}$, where
$\calZ(f)$ is the zero set of $f$ and $\Si_{p}(A')$ denotes the point spectrum
of $A'$. For Conjecture~\ref{con-compl} it would therefore be enough to show
that $\left|\Si_{p}(A')\right|=\infty$.

\section*{Acknowledgements}

The author is grateful to Mihai Putinar, Nikolai Nikolski, Alexandre Eremenko,
Donald Sarason and Boris Shapiro for their interest in this work and
valuable remarks as well as to Semen Malamud for making available his
inspiring paper~\cite{Ma2}. Thanks are also due to the anonymous referee
for numerous constructive suggestions and for pointing out several useful
references.


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Yes, that's right, this is not another rumination on the eclipse. Talk about yesterday's news. On this innocuous Monday in August, the unshielded eyes of a nation were glued not on the heavens above, but on the more earthly constellation of Ms. Swift's social media accounts. After wiping out all previous record of her social media activities last week, the sultry chanteuse returned to the sharewaves with an unexpectedly sibilant silence.
"Silence" because the short video clip she posted to Facebook, Twitter, Tumblr, and Instagram has no sound. "Sibiliant" because a snaky hiss would likely accompany the image of a twitching reptilian tail if there were any audio. What does it all mean?! Only time (and Taylor) will tell. When you have 85 million followers on Twitter alone, you could probably cause a commotion simply by tweeting a photo of your breakfast bagel.
Given that this snakelike oddity follows on the heels of the aforementioned purge of all prior social media sharing, it's probably an early signal of Swift's first new recording since 2014. Rev up the publicity, and all of that. In an 2016 interview with Vogue that has now reentered the news cycle, Swift said, "This is the first time in 10 years that I haven't known what's next." Now, however, it would appear that she's figured out at least some of what the future may hold.
What actually caught our attention in all of this, however, is a further thought shared in that same interview: "I decided I was going to live my life a little bit," Swift said, "without the pressure on myself to create something." And can't we all relate to that? Whether you're the world's hottest female recording artist, or just a hard-working IT professional, it's important to remember that "work" is sometimes not the most important end in life.
Here at the IT Insights blog, we're all about encouraging our readers to push themselves farther and harder and faster. We love to encourage that drive to improve, to learn more, to continually climb the IT ladder, rising to new heights both personally and professionally. Challenge and hard work are good for the soul. It's also true, however, what they say about all work and no play. Or at least all work and no living one's life a little bit.
So while you're hustling along toward self-improvement and that next awesome job, don't forget to sometimes pull over to the side of the road, turn the car off, and maybe get out and wander around for a bit. Just enjoy being here. You can always turn into a snake — or a dragon, or a lizard — and create a blurry video to announce your return, whenever the ol' batteries have been recharged.


text:
Fatal Accident in Lock
In the evening of Sep 18, 2017, at 9.45 p.m. the "Walter", 15 ts (EU-No.: 04803370), a container barge collided with a protection wire in the Uelzen lock of the Elbe-Side Canal at Esterholz.
The ship was coming from Fallersleben and was headed to Hamburg. The 56-year old master lost consciousness shortly before entering the lock so the wheelhouse was not lowered down upon entering the lock.
The cabin slammed into the wire and was torn off and destroyed. The master was fatally injured. The alerted fire brigade freed him from the debris, but he died after initially having been resuscitated. Two other crew members who were on deck remained uninjured.
The lock was closed for ship traffic during the night hours. The damaged tug was pulled out of the lock on Sep 19 at 9.45 a.m. by another tug, and the lock could be reopened at 11 a.m.
Chad Administrative court fined oil consortium lead by Exxon with 74 billion USD
Wildlife Around The World Being Monitored From Space By Satellite Camera (VIDEO)
Bibby Offshore awarded with multi-million-dollar contract in North Sea
Newer PostBahrain Accuses Qatar of Seizing Three Vessels
Older PostNew UK Code Requires Tighter Security Against Cyber Attacks on Ships

redpajama_set_name: RedPajamaCommonCrawl


text:
import { mockTracking } from "Artsy/Analytics"
import { mount } from "enzyme"
import "jest-styled-components"
import React from "react"
import Waypoint from "react-waypoint"
import { ReadMoreButton, ReadMoreContainer } from "../ReadMoreButton"

jest.unmock("react-tracking")

describe("ReadMoreButton", () => {
  let props
  beforeEach(() => {
    props = {
      onClick: jest.fn(),
      referrer: "/article/this-cool-content",
    }
  })

  it("Calls onClick and tracking event on click", () => {
    const { Component, dispatch } = mockTracking(ReadMoreButton)
    const component = mount(<Component {...props} />)

    component
      .find(ReadMoreContainer)
      .at(0)
      .simulate("click")

    expect(dispatch).toBeCalledWith({
      action_type: "Clicked read more",
      context_module: "Read more",
      subject: "Read more",
    })
  })

  it("Calls a tracking impression", () => {
    const { Component, dispatch } = mockTracking(ReadMoreButton)
    const component = mount(<Component {...props} />)

    component
      .find(Waypoint)
      .getElement()
      .props.onEnter()

    expect(dispatch).toBeCalledWith({
      action_type: "Impression",
      context_module: "Read more",
      subject: "Read more",
    })
  })
})

redpajama_set_name: RedPajamaGithub