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Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of…
This note examines the infinite divisibility of density-based transformations of normal random variables. We characterize a class of density-based transformations of normal variables which produces non-infinitely divisible distributions. We…
In this work we study the asymptotic distribution of eigenvalues in one-dimensional open sets. The method of proof is rather elementary, based on the Dirichlet lattice points problem, which enable us to consider sets with infinite measure.…
We study the number of nodal components (connected components of the set of zeroes) of functions in the ensemble of arithmetic random waves, that is, random eigenfunctions of the Laplacian on the flat $d$-dimensional torus $\mathbb{T}^{d}$…
We consider real eigen-functions of the Schr\"odinger operator in 2-d. The nodal lines of separable systems form a regular grid, and the number of nodal crossings equals the number of nodal domains. In contrast, for wave functions of non…
We consider wave propagation problems over 2-dimensional domains with piecewise-linear boundaries, possibly including scatterers. We assume that the wave speed is constant, and that the initial conditions and forcing terms are radially…
We state some inequalities for m-divisible and infinite divisible characteristic functions. Basing on them we propose a statistical test for a distribution to be infinitely divisible. Keywords: infinite divisible distributions; statistical…
Let N(f) be a number of nodal domains of a random Gaussian spherical harmonic f of degree n. We prove that as n grows to infinity, the mean of N(f)/n^2 tends to a positive constant, and that N(f)/n^2 exponentially concentrates around that…
We investigate the measure of nodal sets for Robin and Neumann eigenfunctions in the domain and on the boundary of the domain. A polynomial upper bound for the interior nodal sets is obtained for Robin eigenfunctions in the smooth domain.…
We study the statistics of quantum transmission through a one-dimensional disordered system modelled by a sequence of independent scattering units. Each unit is characterized by its length and by its action, which is proportional to the…
The general relationship between an arbitrary frequency distribution and the expectation value of the frequency distributions of its samples is esablished. A set of combinations of expectation values whose value does not in general depend…
We present a method for two-scale model derivation of the periodic homogenization of the one-dimensional wave equation in a bounded domain. It allows for analyzing the oscillations occurring on both microscopic and macroscopic scales. The…
We present a theoretical framework for understanding the wavefunctions and spectrum of an extensively studied paradigm for quasiperiodic systems, namely the Fibonacci chain. Our analytical results, which are obtained in the limit of strong…
The statistical properties of overlap sums of groups of four eigenfunctions of the Anderson model for localization as well as combinations of four eigenenergies are computed. Some of the distributions are found to be scaling functions, as…
A theoretical framework is developed to describe the transformation that distributes probability density functions uniformly over space. In one dimension, the cumulative distribution can be used, but does not generalize to higher…
In this note, we consider a fixed vector field $V$ on $S^2$ and study the distribution of points which lie on the nodal set (of a random spherical harmonic) where $V$ is also tangent. We show that the expected value of the corresponding…
Many physical observables can be represented as a particle spending some random time within a given domain. For a broad class of transport-dominated processes, we detail how it is possible to express the moments of the number of particle…
Topological measurements are increasingly being accepted as an important tool for quantifying complex structures. In many applications, these structures can be expressed as nodal domains of real-valued functions and are obtained only…
The counting function on binary values is extended to the signed case in order to count the number of transitions between contiguous locations. A generalized subdifferential for the sign change counting function is given where classical…
In this short note, we propose to extend differentiability (with respect to a multidimensional parameter) of a normalized eigenfunction associated to the simple, dominating eigenvalue of the weighted transfer operator for a uniformly…