Related papers: Low dimensional pinned distance sets via spherical…
This note uses a simple example to show how moment inequality models used in the empirical economics literature lead to general minimax relative efficiency comparisons. The main point is that such models involve inference on a low…
The logarithmic Sobolev inequality is fundamental in mathematical physics. Associated stability estimates are equivalent to uncertainty principles. Via a second moment bound, $W^{1,1}$ estimates are obtained in one dimension and similar…
Let $T=t_0 ... t_{n-1}$ be a text and $P = p_0 ... p_{m-1}$ a pattern taken from some finite alphabet set $\Sigma$, and let $\dist$ be a metric on $\Sigma$. We consider the problem of calculating the sum of distances between the symbols of…
New upper bounds on the relative entropy are derived as a function of the total variation distance. One bound refines an inequality by Verd\'{u} for general probability measures. A second bound improves the tightness of an inequality by…
We compare the low-energy partial wave analyses $\pi N$ scattering with a high-energy data via finite energy sum rules. We construct a new set of amplitudes by matching the imaginary part from the low-energy analysis with the high-energy,…
The distance of an extremal of the P\'olya-Szeg\H{o} inequality from a translate of its symmetric decreasing rearrangement is controlled by the measure of the set of critical points.
In the current paper we present a new proof of the small ball inequality in two dimensions. More importantly, this new argument, based on an approach inspired by lacunary Fourier series, reveals the first formal connection between this…
This paper studies the properties of a new lower bound for the natural pseudo-distance. The natural pseudo-distance is a dissimilarity measure between shapes, where a shape is viewed as a topological space endowed with a real-valued…
Pinning of dislocations at nanosized obstacles like precipitates, voids and bubbles, is a crucial mechanism in the context of phenomena like hardening and creep. The interaction between such an obstacle and a dislocation is often explored…
We prove fixed point theorems in a space with a distance function that takes values in a partially ordered monoid. On the one hand, such an approach allows one to generalize some fixed point theorems in a broad class of spaces, including…
Adapting a technique of Pintz, we give an elementary demonstration of the Deuring phenomenon: a zero of \zeta(s) off the critical line gives a lower bound on L(1,\chi). The necessary tools are Dirichlet's 'method of the hyperbola', Euler…
We consider a simplified model of particles with effectively distance dependent statistics, that is particles coupled to a gauge field the Lagrangian of which contains the Chern-Simons term. We analyze the low-lying states of the…
The Whitney embedding theorem gives an upper bound on the smallest embedding dimension of a manifold. If a data set lies on a manifold, a random projection into this reduced dimension will retain the manifold structure. Here we present an…
We present several natural notions of distance between spectral density functions of (discrete-time) random processes. They are motivated by certain filtering problems. First we quantify the degradation of performance of a predictor which…
Let $\mathbb{F}_q^d$ be a $d$-dimensional vector space over a finite field $\mathbb{F}_q$ with $q$ elements. For $x\in \mathbb{F}_q^d$, let $\|x\| = x_1^2+\dots+x_d^2$. By abuse of terminology, we shall call $\|\cdot\|$ a norm on…
Positive semi-definite kernels are used to induce pseudo-metrics, or ``distances'', between measures. We write these as an expected quadratic variation of, or expected inner product between, a random field and the difference of measures.…
Given a random variable $X$ and considered a family of its possible distortions, we define two new measures of distance between $X$ and each its distortion. For these distance measures, which are extensions of the Gini's mean difference,…
The distance set $\Delta(E)$ of a set $E$ consists of all non-negative numbers that represent distances between pairs of points in $E$. This paper studies sparse (less than full-dimensional) Borel sets in $\mathbb R^d$, $d \geq 2$ with a…
Given a set $S$ consisting of $n$ points in $\mathbb{R}^d$ and one or two vantage points, we study the number of orderings of $S$ induced by measuring the distance (for one vantage point) or the average distance (for two vantage points)…
How much one has learned from an experiment is quantifiable by the information gain, also known as the Kullback-Leibler divergence. The narrowing of the posterior parameter distribution $P(\theta|D)$ compared with the prior parameter…