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We develop linear discretization of complex analysis, originally introduced by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We prove convergence of discrete period matrices and discrete Abelian integrals to their continuous…
We compute the compactly supported Euler characteristic of the space of degree $d$ irreducible polynomials in $n$ variables with real coefficients and show that the values are given by the digits in the so-called balanced binary expansion…
Given a set $\mathcal{S}$ of positive measure on the circle and a set of integers $\Lambda$, one may consider the family of exponentials $E\left(\Lambda\right):=\left\{ e^{i\lambda t}\right\}_{\lambda\in\Lambda}$ and ask whether it is a…
The Steinitz exchange lemma is a basic theorem in linear algebra used, for example, to show that any two bases for a finite-dimensional vector space have the same number of elements. The result is named after the German mathematician Ernst…
In this paper we investigate four concepts of exponential stability for difference equations in Banach spaces. Characterizations of these concepts are given. They can be considered as variants for the discrete-time case of the classical…
Mixture distributions arise in many parametric and non-parametric settings -- for example, in Gaussian mixture models and in non-parametric estimation. It is often necessary to compute the entropy of a mixture, but, in most cases, this…
Despite the recent advances in the theory of exponential Riesz bases, it is yet unknown whether there exists a set $S \subset \mathbb{R}^d$ which does not admit a Riesz spectrum, meaning that for every $\Lambda \subset \mathbb{R}^d$ the set…
We give improved bounds for the equidistribution of (multiparameter) nilsequences subject to any degree filtration. The bounds we obtain are single exponential in dimension, improving on double exponential bounds of Green and Tao. To obtain…
The purpose of this paper is threefold. First the natural extension of Riesz potentials to the context of quasi metric measure spaces for the class of upper doubling measures are studied on Lebesgue spaces, obtaining necessary and…
We prove some properties of positive polynomial mappings between Riesz spaces, using finite difference calculus. We establish the polynomial analogue of the classical result that positive, additive mappings are linear. And we prove a…
Discretization methods for ordinary differential equations based on the use of matrix exponentials have been known for decades. This set of ideas has come off age and acquired greater urgency recently, within the context of geometric…
Let $X_N$ be an $N$-dimensional subspace of $L_2$ functions on a probability space $(\Omega, \mu)$ spanned by a uniformly bounded Riesz basis $\Phi_N$. Given an integer $1\leq v\leq N$ and an exponent $1\leq q\leq 2$, we obtain universal…
Given a finite dimensional Banach space X with dimX = n and an Auerbach basis of X, it is proved that: there exists a set D of n + 1 linear combinations (with coordinates 0, -1, +1) of the members of the basis, so that each pair of…
The distances between flats of a Poisson $k$-flat process in the $d$-dimensional Euclidean space with $k<d/2$ are discussed. Continuing an approach originally due to Rolf Schneider, the number of pairs of flats having distance less than a…
We show that balls, circles and 2-spheres can be identified by generalized Riesz energy among compact submanifolds of the Euclidean space that are either closed or with codimension 0, where the Riesz energy is defined as the double integral…
In our investigation, we focus on the reverse Riesz transform within the framework of manifolds with ends. Such manifolds can be described as the connected sum of finite number of Cartesian products $\mathbb{R}^{n_i} \times \mathcal{M}_i$,…
The Wasserstein distance between probability measures on compact spaces provides a natural invariant quantitative measure of equidistribution, which is partly similar to the classical discrepancy appearing in Erd\"os-Tur\'an type…
In the function field setting with a fixed characteristic, it was proven by the second and third authors that the values $\log \big|L\big(\frac12, \chi_D\big)\big|$ as $D$ varies over monic and square-free polynomials are asymptotically…
The Erd\H{o}s-Anning theorem states that every point set in the Euclidean plane with integer distances must be either collinear or finite. More strongly, for any (non-degenerate) triangle of diameter~$\delta$, at most $O(\delta^2)$ points…
Kusner asked if $n+1$ points is the maximum number of points in $\mathbb{R}^n$ such that the $\ell_p$ distance $(1<p<\infty)$ between any two points is $1$. We present an improvement to the best known upper bound when $p$ is large in terms…