Related papers: On the limiting behaviour of arithmetic toral eige…

We define eventually symmetric functions to be those power series of bounded degree in infinitely many variables that are invariant under interchanging all the variables with large enough indices. We show how this ring $\tilde{\Lambda}$ is…

In a family of random variables, Taylor's law or Taylor's power law offluctuation scaling is a variance function that gives the variance $\sigma^{2}>0$ of a random variable (rv) $X$ with expectation $\mu >0$ as a powerof $\mu$: $\sigma…

We consider the spaces $A_p(\mathbb T^m)$ of functions $f$ on the $m$ -dimensional torus $\mathbb T^m$ such that the sequence of the Fourier coefficients $\hat{f}=\{\hat{f}(k), ~k \in \mathbb Z^m\}$ belongs to $l^p(\mathbb Z^m), ~1\leq…

Extending work of J. Raleigh, we compute polynomials $P_{n,F}(x)$ associated to certain families $F = \{f_m\}_{m = 3, 4, ...}$ of modular forms for Hecke groups $G(\lambda_m)$ with the property that $P_{n,F}(m)$ is the $n^{th}$ coefficient…

We develop a new method for studying the asymptotics of symmetric polynomials of representation-theoretic origin as the number of variables tends to infinity. Several applications of our method are presented: We prove a number of theorems…

In this article, we consider polynomials of the form $f(x)=a_0+a_{n_1}x^{n_1}+a_{n_2}x^{n_2}+\dots+a_{n_r}x^{n_r}\in \mathbb{Z}[x],$ where $|a_0|\ge |a_{n_1}|+\dots+|a_{n_r}|,$ $|a_0|$ is a prime power and $|a_0|\nmid |a_{n_1}a_{n_r}|$. We…

We analyze a general class of difference operators $H_\varepsilon = T_\varepsilon + V_\varepsilon$ on $\ell^2(\varepsilon \mathbb{Z}^d)$, where $V_\varepsilon$ is a one-well potential and $\varepsilon$ is a small parameter. We construct…

Consider an $L^2$-normalized Laplace-Beltrami eigenfunction $e_\lambda$ on a compact, boundary-less Riemannian manifold with $\Delta e_\lambda = -\lambda^2 e_\lambda$. We study eigenfunction triple products \[ \langle e_\lambda e_\mu, e_\nu…

In this article we study definable functions in tame expansions of algebraically closed valued fields. For a given definable function we have two types of results: of type (I), which hold at a neighborhood of infinity, and of type (II),…

Our work deals with symmetric rational functions and probabilistic models based on the fully inhomogeneous six vertex (ice type) model satisfying the free fermion condition. Two families of symmetric rational functions $F_\lambda,G_\lambda$…

On a countable tree $T$, allowing vertices with infinite degree, we consider an arbitrary stochastic irreducible nearest neighbour transition operator $P$. We provide a boundary integral representation for general eigenfunctions of $P$ with…

Given a matroid or flag of matroids we introduce several broad classes of polynomials satisfying Deletion-Contraction identities, and study their singularities. There are three main families of polynomials captured by our approach:…

A class of parametric functions formed by alternating compositions of multivariate polynomials and rectification style monomial maps is studied (the layer-wise exponents are treated as fixed hyperparameters and are not optimized). For this…

Gaussian quadrature rules are a classical tool for the numerical approximation of integrals with smooth integrands and positive weight functions. We derive and expicitly list asymptotic expressions for the points and weights of Gaussian…

Following work of Raleigh and Akiyama (\cite{raleigh1962fourier, akiyama1992note}), in \cite{interpolating} we considered (among other objects) families of weight zero meromorphic modular forms $J_m$ for Hecke groups $G(\lambda_m)$. We…

The class of differential-equation eigenvalue problems $-y''(x)+x^{2N+2}y(x)=x^N Ey(x)$ ($N=-1,0,1,2,3,...$) on the interval $-\infty<x<\infty$ can be solved in closed form for all the eigenvalues $E$ and the corresponding eigenfunctions…

Using the technique developed in approximation theory, we construct examples of exponential families of infinitely divisible laws which can be viewed as deformations of the normal, gamma, and Poisson exponential families. Replacing the…

For the family $P:=x^n+a_1x^{n-1}+\cdots +a_n$ of complex polynomials in the variable $x$ we study its {\em discriminant} $R:=$Res$(P,P',x)$, $R\in \mathbb{C}[a]$, $a=(a_1,\ldots ,a_n)$. When $R$ is regarded as a polynomial in $a_k$, one…

The principal aim of this work is to provide a family of unisolvent and minimal physical degrees of freedom, called weights, for N\'ed\'elec second family of finite elements. Such elements are thought of as differential forms $…

We define a class of "algebraic" random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner…