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We propose a new macroscopic model derived from the classical nonlinear Boltzmann equation. A set of partial differential equations is obtained easily. The unknowns depend on the time and space coordinates, and are related to the…
A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of…
In the spectral Petrov-Galerkin methods, the trial and test functions are required to satisfy particular boundary conditions. By a suitable linear combination of orthogonal polynomials, a basis, that is called the modal basis, is obtained.…
We realize that geometric polynomials and p-Bernoulli polynomials and numbers are closely related with an integral representation. Therefore, using geometric polynomials, we extend some properties of Bernoulli polynomials and numbers such…
In this paper, we introduce two new forms of the dual Hartwig-Spindelb{\"o}ck decomposition and employ them to derive explicit representations for several classes of dual generalized inverses. Building on these representations, we further…
We obtain a family of matrix integrals which decompose to a product of Gamma-functions (they have some relations with S.G.Gindikin 'Beta', but generally speaking essentially differ from it). We obtain Plancherel formula for Berezin…
We study approximation in the unit interval by rational numbers whose numerators are selected randomly with certain probabilities. Previous work showed that an analogue of Khintchine's Theorem holds in a similar random model and raised the…
A derivation is given of the Boltzmann-Vlasov equation beginning from multiple scattering considerations. The motivation for the discussion, which is purely pedagogical in nature, is the current interest in understanding the origins of…
The simple product formulae for derivatives of scalar functions raised to different powers are generalized for functions which take values in the set of symmetric positive definite matrices. These formulae are fundamental in derivation of…
A novel refinement of the conventional treatment of Kadanoff--Baym equations is suggested. Besides the Boltzmann equation another differential equation is used for calculating the evolution of the non-equilibrium two-point function.…
In this paper, we revisit foundations of umbral calculus using a straightforward approach based on an explicit matrix realization of binomial convolution. We construct an umbral duality of Wronskian type for rational curves in echelon form,…
We review some recent results on recursion relations which help evaluating arbitrary non-diagonal, radial hydrogenic matrix elements of $r^\lambda$ and of $\beta r^\lambda$ ($\beta$ a Dirac matrix) derived in the context of Dirac…
In this paper we discuss a generalization of the classica PBW-theorem to the case of Koszul algebras. Our result is a slight generalization of that obtained by A.Polischuk and L.Positselsky, but the proof is different and uses deformation…
The article is devoted to the issue of the polar form of octonions. This is a~continuation of the works initiated by Hahn and Snopek in their articles from 2011. The results presented in the article show errors made in previous…
The goal of this paper is to present an algebraic approach to the basic results of the theory of linear recurrence relations. This approach is based on the ideas from the theory of representations of one endomorphisms (a special case of…
An approximate formula for the partitions of Goldbach's Conjecture is derived using Prime Number Theorem and a heuristic probabilistic approach. A strong form of Goldbach's conjecture follows in the form of a lower bounding function for the…
The paper [GLZ] "L-functions of Carlitz modules, resultantal varieties and rooted binary trees" is devoted to a description of some resultantal varieties related to L-functions of Carlitz modules. It contains a conjecture that some of these…
In this short paper, we re-derive the Bochner formula for the Laplacian by considering local variations of volume. The derivation is rooted in the fact that the Laplacian of a function measures the volume variation along the flow of the…
We present a probabilistic argument supporting the application of polar duality, as discussed in our previous work, to express the indeterminacy principle of quantum mechanics. Our approach combines the properties of the Mahler volume of a…
We present a novel approach to Gaussian Berezin correlation functions. A formula well known in the literature expresses these quantities in terms of submatrices of the inverse matrix appearing in the Gaussian action. By using a recently…