Related papers: Equilibrium of charges and differential equations …
The question as to whether the integrality of the spectrum of observed electric charges is due to a quantum effext has fascinated theoretical physicists throughout the last century. It leads to unanswered questions at the heart of quantum…
We use classical Jacobi polynomials to identify the equilibrium configurations of charged particles confined to the unit circle. Our main result unifies two theorems from a 1986 paper of Forrester and Rogers.
We present a mathematically rigorous quantum-mechanical treatment of a two-dimensional nonrelativistic quantum dual theories (with oscillator and Coulomb like potentials) on a plane and compare their spectra and the sets of eigenfunctions.…
This monograph, written for educational purposes, serves as an introduction to the concept of integrability as it applies to systems of differential equations (both ordinary and partial) as well as to vector-valued fields. The general cases…
A formalism is given to count integer and rational solutions to polynomial equations with rational coefficients. These polynomials $P(x)$ are parameterized by three integers, labeling an elliptic curve. The counting of the rational…
We consider each of the three classes of representations of cyclic groups that arise in the study of rational sphere maps. We study the possible number of terms for invariant polynomials with non-negative coefficients that are constant on…
We describe a method to evaluate multivariate polynomials over a finite field and discuss its multiplicative complexity.
We introduce a notion of palindromicity of a natural number which is independent of the base. We study the existence and density of palindromic and multiple palindromic numbers, and we raise several related questions.
The paper addresses the calculation of correlation functions of permanental polynomials of matrices with random entries. By exploiting a convenient contour integral representation of the matrix permanent some explicit results are provided…
We present a new approach to solving polynomial ordinary differential equations by transforming them to linear functional equations and then solving the linear functional equations. We will focus most of our attention upon the first-order…
We show the finiteness of perfect powers in orbits of polynomial dynamical systems over an algebraic number field. We also obtain similar results for perfect powers represented by ratios of consecutive elements in orbits. Assuming the…
We establish the boundedness character of solutions of a system of rational difference equations with a variable coefficient
A unified approach, for solving a wide class of single and many-body quantum problems, commonly encountered in literature is developed based on a recently proposed method for finding solutions of linear differential equations. Apart from…
The series solution of the radial part of the Schr\"odinger equation for simultaneous coulomb and harmonic potential involves three-term recursion relation and is thus difficult to solve for bound states. We have suggested a simple method…
Enlarging on Parts I, II, and III we write more equations in the desired format of the extended abstract theory of composites. We focus on a multitude of equations involving higher order derivatives. The motivation is that results and…
A quantum kinetic theory for correlated charged-particle systems in strong time-dependent electromagnetic fields is developed. Our approach is based on a systematic gauge-invariant nonequilibrium Green's functions formulation. We…
We prove a recent conjecture of Lassalle about positivity and integrality of coefficients in some polynomial expansions. We also give a combinatorial interpretation of those numbers. Finally, we show that this question is closely related to…
We consider the scalar sector of a general renormalizable theory and evaluate the effective potential through three loops analytically. We encounter three-loop vacuum bubble diagrams with up to two masses and six lines, which we solve using…
We review some of the recent results in the context of the Coulomb interaction between dielectric surfaces which are randomly charged but remain net-neutral on the average. Such surfaces are found to interact in vacuum with attractive…
We give conditions on a finite set of series of rational numbers to ensure that they are algebraically independent. Specialising our results to polynomials of lower degree, we also obtain new results on irrationality and $mathbb{Q}$-linear…