Let $D$ denote the distance matrix for an $n+1$ point metric space $(X,d)$. In the case that $X$ is an unweighted metric tree, the sum of the entries in $D^{-1}$ is always equal to $2/n$. Such trees can be considered as affinely independent subsets o
f the Hamming cube $H_n$, and it was conjectured that the value $2/n$ was minimal among all such subsets. In this paper we confirm this conjecture and give a geometric interpretation of our result which applies to any subset of $H_n$.
Graham and Winkler derived a formula for the determinant of the distance matrix of a full-dimensional set of $n + 1$ points ${ x_{0}, x_{1}, ldots , x_{n} }$ in the Hamming cube $H_{n} = ( { 0,1 }^{n}, ell_{1} )$. In this article we derive a formula
for the determinant of the distance matrix $D$ of an arbitrary set of $m + 1$ points ${ x_{0}, x_{1}, ldots , x_{m} }$ in $H_{n}$. It follows from this more general formula that $det (D) ot= 0$ if and only if the vectors $x_{0}, x_{1}, ldots , x_{m}$ are affinely independent. Specializing to the case $m = n$ provides new insights into the original formula of Graham and Winkler. A significant difference that arises between the cases $m < n$ and $m = n$ is noted. We also show that if $D$ is the distance matrix of an unweighted tree on $n + 1$ vertices, then $langle D^{-1} mathbf{1}, mathbf{1} rangle = 2/n$ where $mathbf{1}$ is the column vector all of whose coordinates are $1$. Finally, we derive a new proof of Murugans classification of the subsets of $H_{n}$ that have strict $1$-negative type.
The periodic Benjamin-Ono equation is an autonomous Hamiltonian system with a Gibbs measure on $L^2({mathbb T})$. The paper shows that the Gibbs measures on bounded balls of $L^2$ satisfy some logarithmic Sobolev inequalities. The space of $n$-solito
n solutions of the periodic Benjamin-Ono equation, as discovered by Case, is a Hamiltonian system with an invariant Gibbs measure. As $nrightarrowinfty$, these Gibbs measures exhibit a concentration of measure phenomenon. Case introduced soliton solutions that are parameterised by atomic measures in the complex plane. The limiting distributions of these measures gives the density of a compressible gas that satisfies the isentropic Euler equations.
This paper analyses the periodic spectrum of Schrodingers equation $-f+qf=lambda f$ when the potential is real, periodic, random and subject to the invariant measure $ u_N^beta$ of the periodic KdV equation. This $ u_N^beta$ is the modified canonical
ensemble, as given by Bourgain ({Comm. Math. Phys.} {166} (1994), 1--26), and $ u_N^beta$ satisfies a logarithmic Sobolev inequality. Associated concentration inequalities control the fluctuations of the periodic eigenvalues $(lambda_n)$. For $beta, N>0$ small, there exists a set of positive $ u_N^beta$ measure such that $(pm sqrt{2(lambda_{2n}+lambda_{2n-1})})_{n=0}^infty$ gives a sampling sequence for Paley--Wiener space $PW(pi )$ and the reproducing kernels give a Riesz basis. Let $(mu_j)_{j=1}^infty$ be the tied spectrum; then $(2sqrt{mu_j}-j)$ belongs to a Hilbert cube in $ell^2$ and is distributed according to a measure that satisfies Gaussian concentration for Lipschitz functions. The sampling sequence $(sqrt{mu_j})_{j=1}^infty$ arises from a divisor on the spectral curve, which is hyperelliptic of infinite genus. The linear statistics $sum_j g(sqrt{lambda_{2j}})$ with test function $gin PW(pi)$ satisfy Gaussian concentration inequalities.
The nonlinear Schrodinger equation NLSE(p, beta), -iu_t=-u_{xx}+beta | u|^{p-2} u=0, arises from a Hamiltonian on infinite-dimensional phase space Lp^2(mT). For pleq 6, Bourgain (Comm. Math. Phys. 166 (1994), 1--26) has shown that there exists a Gibb
s measure mu^{beta}_N on balls Omega_N= {phi in Lp^2(mT) : | phi |^2_{Lp^2} leq N} in phase space such that the Cauchy problem for NLSE(p,beta) is well posed on the support of mu^{beta}_N, and that mu^{beta}_N is invariant under the flow. This paper shows that mu^{beta}_N satisfies a logarithmic Sobolev inequality for the focussing case beta <0 and 2leq pleq 4 on Omega_N for all N>0; also mu^{beta} satisfies a restricted LSI for 4leq pleq 6 on compact subsets of Omega_N determined by Holder norms. Hence for p=4, the spectral data of the periodic Dirac operator in Lp^2(mT; mC^2) with random potential phi subject to mu^{beta}_N are concentrated near to their mean values. The paper concludes with a similar result for the spectral data of Hills equation when the potential is random and subject to the Gibbs measure of KdV.
Negative type inequalities arise in the study of embedding properties of metric spaces, but they often reduce to intractable combinatorial problems. In this paper we study more quantitati
Every finite metric tree has generalized roundness strictly greater than one. On the other hand, some countable metric trees have generalized roundness precisely one. The purpose of this paper is to identify some large classes of countable metric tre
es that have generalized roundness precisely one. At the outset we consider spherically symmetric trees endowed with the usual combinatorial metric (SSTs). Using a simple geometric argument we show how to determine decent upper bounds on the generalized roundness of finite SSTs that depend only on the downward degree sequence of the tree in question. By considering limits it follows that if the downward degree sequence $(d_{0}, d_{1}, d_{2}...)$ of a SST $(T,rho)$ satisfies $|{j , | , d_{j} > 1 }| = aleph_{0}$, then $(T,rho)$ has generalized roundness one. Included among the trees that satisfy this condition are all complete $n$-ary trees of depth $infty$ ($n geq 2$), all $k$-regular trees ($k geq 3$) and inductive limits of Cantor trees. The remainder of the paper deals with two classes of countable metric trees of generalized roundness one whose members are not, in general, spherically symmetric. The first such class of trees are merely required to spread out at a sufficient rate (with a restriction on the number of leaves) and the second such class of trees resemble infinite combs.
