Related papers: Random zeros entire functions
We consider an analytic function $f$ whose zero set forms a unit intensity Poisson process on the real line. We show that repeated differentiation causes the zero set to converge in distribution to a random translate of the integers.
We give some theoretical and computational results on "random" harmonic sums with prime numbers, and more generally, for integers with a fixed number of prime factors.
The present research deals with generalizations of the Salem function with arguments defined in terms of certain alternating expansions of real numbers. The special attention is given to modelling such functions by systems of functional…
Provided a special function of one variable and some of its derivatives can be accurately computed over a finite range, a method is presented to build a series of polynomial approximations of the function with a defined relative error over…
We describe the limit zero distributions of sequences of polynomials with positive coefficients.
We prove strong clustering of k-point correlation functions of zeroes of Gaussian Entire Functions. In the course of the proof, we also obtain universal local bounds for k-point functions of zeroes of arbitrary nondegenerate Gaussian…
A classification is provided of functors, in particular polynomial ones, from a category with a zero object in which every object is a finite sum of copies of a generating object, into an abelian category. This classification is extended to…
Consider a universal Turing machine that produces a partial or total function (or a binary stream), based on the answers to the binary queries that it makes during the computation. We study the probability that the machine will produce a…
We study sampling properties of the zero set of the Gaussian entire function on Fock spaces. Firstly, we relax Seip and Wallst\'en's density and separation conditions for sampling sets on Fock spaces to obtain weighted inequalities for sets…
A semiring can be ``completed'' (i.e., embedded into a semiring in which all infinite sums are defined and satisfy some reasonable properties) iff this semiring can be naturally partially ordered. This construction is ``natural'' (a left…
We consider distributions on $\mathbb{R}$ that can be written as the sum of a non-zero discrete distribution and an absolutely continuous distribution. We show that such a distribution is quasi-infinitely divisible if and only if its…
For a set of nonnegative integers $A$, denote by $R_{A}(n)$ the number of unordered representations of the integer $n$ as the sum of two different terms from $A$. In this paper we partially describe the structure of the sets, which have…
In the paper I study properties of random polynomials with respect to a general system of functions. Some lower bounds for the mathematical expectation of the uniform and recently introduced integral-uniform norms of random polynomials are…
An inequality, which combines the concept of completely monotone functions with the theory of divided differences, is proposed. It is a straightforward generalization of a result, recently introduced by two of the present authors.
Lower bounds are given for the number of non-real zeros of a second order linear differential polynomial with constant coefficients in a real entire function with finitely many non-real zeros.
Randomness comes in two qualitatively different forms. Apparent randomness can result both from ignorance or lack of control of degrees of freedom in the system. In contrast, intrinsic randomness should not be ascribable to any such cause.…
By random complex zeroes we mean the zero set of a random entire function whose Taylor coefficients are independent complex-valued Gaussian variables, and the variance of the k-th coefficient is 1/k!. This zero set is distribution invariant…
We give different integral representations of the Lommel function $s_{\mu,\nu}(z)$ involving trigonometric and hypergeometric $_2F_1$ functions. By using classical results of Polya, we give the distribution of the zeros of $s_{\mu,\nu}(z)$…
Here we briefly discuss how negative numbers, or "negative probabilities", can naturally arise in probabilistic expressions and be given an operational interpretation. Like the use of negative numbers in arithmetical expressions, the use of…
First a formula for the number of zeros of the orthogonal polynomial in the intervals is presented. Then a criteria about the appearance of a zero in a gap is given. Finally a necessary and sufficient condition is derived such that the…