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Density estimation is an important technique for characterizing distributions given observations. Much existing research on density estimation has focused on cases wherein the data lies in a Euclidean space. However, some kinds of data are…
In this article, we study several probabilistic properties of polynomials defined over the ring of $p$-adic integers under the Haar measure. First, we calculate the probability that a monic polynomial is separable, generalizing a result of…
We consider the set of monic irreducible polynomials $P$ over a finite field $\mathbb{F}_q$ such that the multiplicative order modulo $P$ of some a in $\mathbb{F}_q(T)$ is divisible by a fixed positive integer $d$. Call $R_q(a,d)$ this set.…
We study moduli of holomorphic vector bundles on non-compact varieties. We discuss filtrability and algebraicity of bundles and calculate dimensions of local moduli. As particularly interesting examples, we describe numerical invariants of…
A polynomial complexity algorithm is designed which tests whether a point belongs to a given tropical linear variety.
We study how well a real number can be approximated by sums of two or more rational numbers with denominators up to a certain size.
We establish tight lower and upper bounds on the number of edges in traceable graphs in several classes of dense graphs. A graph is traceable if it has a Hamiltonian path. We show that the bound is: - quadratic for the class of graphs of…
Hypercomplex numbers are unital algebras over the real numbers. We offer a short demonstration of the practical value of hypercomplex analytic functions in the field of partial differential equations.
We give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds…
We present counting methods for some special classes of multivariate polynomials over a finite field, namely the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), and the relatively irreducible…
We define sets of orthogonal polynomials satisfying the additional constraint of a vanishing average. These are of interest, for example, for the study of the Hohenberg-Kohn functional for electronic or nucleonic densities and for the study…
This article presents some interesting and novel results concerning the average modulus of random polynomials on the unit circle and the unit disc, with coefficients distributed as standard normal variates. The paper also introduces new…
We study the dynamics of cubic polynomials restricted to their basins of infinity, and we enumerate topological conjugacy classes with given combinatorics.
In this paper we provide new methodology for inference of the geometric features of a multivariate density in deconvolution. Our approach is based on multiscale tests to detect significant directional derivatives of the unknown density at…
The spectral density of random graphs with topological constraints is analysed using the replica method. We consider graph ensembles featuring generalised degree-degree correlations, as well as those with a community structure. In each case…
We show that under very mild conditions on a measure $\mu$ on the real line, the span of $\{x^n\}_{n=j}^{\infty}$ is dense in $L^2(\mu)$ for any $j\in\mathbb{N}$. We also present a slightly weaker result with an interesting proof that uses…
We study the distribution of integral points on log varieties.
The wrapped normal distribution arises when a the density of a one-dimensional normal distribution is wrapped around the circle infinitely many times. At first look, evaluation of its probability density function appears tedious as an…
Graph polynomials are polynomials assigned to graphs. Interestingly, they also arise in many areas outside graph theory as well. Many properties of graph polynomials have been widely studied. In this paper, we survey some results on the…
We investigate density of various subalgebras of regular generalized functions in the special Colombeau algebra of generalized functions.