Related papers: An electrostatic model for the roots of discrete c…

Classical solutions to PDEs with discrete state-dependent delay are studied. We prove the well-posedness in a set $X_F$ which is an analogous to the solution manifold used for ordinary differential equations with state-dependent delay. We…

Working over the split octonions over an algebraically closed field, we solve all polynomial equations in which all the coefficients but the constant term are scalar. As a consequence, we calculate the n-th roots of an octonion.

The goal of this work is to characterize all second order difference operators of several variables that have discrete orthogonal polynomials as eigenfunctions. Under some mild assumptions, we give a complete solution of the problem.

A class of variational schemes for the hydrodynamic-electrodynamic model of lossless free-electron gas in a quasineutral background is developed for high-quality simulations of surface plasmon polaritons. The Lagrangian density of lossless…

We consider static, spherically symmetric configurations in general relativity, supported by nonlinear electromagnetic fields with gauge-invariant Lagrangians depending on the single invariant $f = F_{\mu\nu} F^{\mu\nu}$. After a brief…

This paper is devoted to several new results concerning (standard) octonion polynomials. The first is the determination of the roots of all right scalar multiples of octonion polynomials. The roots of left multiples are also discussed,…

We examine the merits of using a family of polynomials that are orthogonal with respect to a non-classical weight function to discretize the speed variable in continuum kinetic calculations. We consider a model one-dimensional partial…

We describe a method for calculating the roots of special functions satisfying second order linear ordinary differential equations. It exploits the recent observation that the solutions of a large class of such equations can be represented…

We show that the method used in the Schwarzschild black hole for finding the elementary solution of the electrostatic equation in closed form cannot extend in higher dimensions. By contrast, we prove the existence of static, spherically…

Coupled discrete models abound in several areas of physics. Here we provide an extensive set of exact quasiperiodic solutions of a number of coupled discrete models in terms of Lam\'e polynomials of arbitrary order. The models discussed are…

We revisit the classical problem of construction of a fundamental system of solutions to a linear ODE whose elements remain analytic and linearly independent for all values of the roots of the characteristic polynomial.

Polynomial dynamical systems describing interacting particles in the plane are studied. A method replacing integration of a polynomial multi--particle dynamical system by finding polynomial solutions of a partial differential equations is…

Let $V$ be a set of isolated points in $\RR^d$. Define a linear functional $\CL$ on the space of real polynomials restricted on $V$, $\CL f = \sum_{x \in V} f(x)\rho(x)$, where $\rho$ is a nonzero function on $V$. Polynomial subspaces that…

We obtain the strong asymptotics of polynomials $p_n(\lambda)$, $\lambda\in\mathbb{C}$, orthogonal with respect to measures in the complex plane of the form $$ e^{-N(|\lambda|^{2s}-t\lambda^s-\overline{t\lambda}^s)}dA(\lambda), $$ where $s$…

As showed in (Fiedler, 1990), any polynomial can be expressed as a characteristic polynomial of a complex symmetric arrowhead matrix. This expression is not unique. If the polynomial is real with only real distinct roots, the matrix can be…

We present two methods for solving the electrostatics of point charges and multipoles on the surface of a sphere, \textit{i.e.} in the space $\mathcal{S}_{2}$, with applications to numerical simulations of two-dimensional polar fluids. In…

The non-linear second order Born-Infeld equation is reduced to a simpler first order complex equation, which can be trivially solved for the coordinates as functions of the field. Each solution is determined by the choice of a holomorphic…

We introduce a new class of polynomials of multiple orthogonality with respect to the product of $r$ classical discrete weights on integer lattices with noninteger shifts. We give explicit representations in the form of the Rodrigues…

We present here a probabilistic approach to the generation of new polynomials in two discrete variables. This extends our earlier work on the 'classical' orthogonal polynomials in a previously unexplored direction, resulting in the…

Classical orthogonal polynomials have widespread applications including in numerical integration, solving differential equations, and interpolation. Changing basis between classical orthogonal polynomials can affect the convergence,…