Related papers: A small value estimate for Ga x Gm

Let X be the set of integer points in some polyhedron. We investigate the smallest number of facets of any polyhedron whose set of integer points is X. This quantity, which we call the relaxation complexity of X, corresponds to the smallest…

A nonnegative matrix $A$ is said to be {\it primitive} if for some positive integer $m$, entries in $A^m$ are positive, notationally represented as $A^m>0.$ The smallest such $m$ is called the {\it exponent} of $A$, denoted $exp(A).$ For…

Motivated by a paradigm shift towards a hyper-connected world, we develop a computationally tractable small-gain theorem for a network of infinitely many systems, termed as infinite networks. The proposed small-gain theorem addresses…

Let $s \geq 3$ be a natural number, let $\psi(x)$ be a polynomial with real coefficients and degree $d \geq 2$, and let $A$ be some large, non-empty, finite subset of real numbers. We use $E_{s,2}(A)$ to denote the number of solutions to…

Let $G$ be a permutation group on a set $\Omega$. A base for $G$ is a subset of $\Omega$ whose pointwise stabiliser is trivial, and the base size of $G$ is the minimal cardinality of a base. If $G$ has base size $2$, then the corresponding…

The spectrum $\omega(G)$ of a finite group $G$ is the set of orders of elements of $G$. We present a polynomial-time algorithm that, given a finite set $\mathcal M$ of positive integers, outputs either an empty set or a finite simple group…

We present estimates on the small singular values of a class of matrices with independent Gaussian entries and inhomogeneous variance profile, satisfying a broad-connectedness condition. Using these estimates and concentration of measure…

Let $P_N(R)$ be the space of all real polynomials in $N$ variables with the usual inner product $<, >$ on it, given by integrating over the unit sphere. We start by deriving an explicit combinatorial formula for the bilinear form…

It is a classical problem to compute a minimal set of invariant polynomial generating the invariant ring of a finite group as an algebra. We present here an algorithm for the computation of minimal generating sets in the non-modular case.…

Let $\alpha_1,\alpha_2$ be non-zero algebraic numbers such that $\frac{\log \alpha_2}{\log\alpha_1}\notin\mathbb{Q}$ and let $\beta$ be a quadratic irrational number. In this article, we prove that the values of two relatively prime…

Let $(G,+)$ be an abelian group and consider a subset $A \subseteq G$ with $|A|=k$. Given an ordering $(a_1, \ldots, a_k)$ of the elements of $A$, define its {\em partial sums} by $s_0 = 0$ and $s_j = \sum_{i=1}^j a_i$ for $1 \leq j \leq…

We present large sample results for partitioning-based least squares nonparametric regression, a popular method for approximating conditional expectation functions in statistics, econometrics, and machine learning. First, we obtain a…

Let $(\xi,\eta)$ be a pair of jointly stationary, ergodic random measures of equal finite intensity. A balancing allocation is a translation-invariant (equivariant) map $T:\mathbb{R}^d\to\mathbb{R}^d$ such that the image measure of $\xi$…

A base for a finite permutation group $G \le \mathrm{Sym}(\Omega)$ is a subset of $\Omega$ with trivial pointwise stabiliser in $G$, and the base size of $G$ is the smallest size of a base for $G$. Motivated by the interest in groups of…

The present paper studies the existence of valuative interpolation on the local ring of an irreducible analytic subvariety at singular points. We firstly develop the concepts and methods of Zhou weights and Tian functions near singular…

The Gaussian and Laguerre orthogonal ensembles are fundamental to random matrix theory, and the marginal eigenvalue distributions are basic observable quantities. Notwithstanding a long history, a formulation providing high precision…

Building on work of Davenport and Schmidt, we mainly prove two results. The first one is a version of Gel'fond's transcendence criterion which provides a sufficient condition for a complex or $p$-adic number $\xi$ to be algebraic in terms…

We revisit the main result of Carmosino et al \cite{CILM18} which shows that an $\Omega(n^{\omega/2+\epsilon})$ size noncommutative arithmetic circuit size lower bound (where $\omega$ is the matrix multiplication exponent) for a…

We provide a new global small-gain theorem for feedback interconnections of monotone input-output systems with multi-valued input-state characteristics. This extends a recent small-gain theorem of Angeli and Sontag for monotone systems with…

We obtain a tail bound for the least non-zero singular value of $A-z$ when $A$ is a random matrix and $z$ is an eigenvalue of $A$ in a neighbourhood of a given point $z_0$ in the bulk of the spectrum. The argument relies on a resolvent…