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A central aim of theoretical physics is to account for the structure of matter at the most elementary level as underlying the Standard Model of particle physics, and ideally also as a basis for a substantial dark sector, as distributed in…
It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential…
The method of constructing approximate solutions of the first boundary value problem for linear differential equations based on incomplete (even and odd) trigonometric splines is considered. The theoretical positions are illustrated by…
Bilinear systems of equations are defined, motivated and analyzed for solvability. Elementary structure is mentioned and it is shown that all solutions may be obtained as rank one completions of a linear matrix polynomial derived from…
In this paper we introduce elementary and completely explicit formulas for the derivative of any order of any function of the type 1/p, where p is a polynomial with known zeros.
This paper establishes the basis of the quaternionic differential geometry ($\mathbbm H$DG) initiated in a previous article. The usual concepts of curves and surfaces are generalized to quaternionic constraints, as well as the curvature and…
We give a shorter simpler proof of a result of Szalay on the equation $2^a + 2^b + 1 = z^2$. We give an elementary proof of a result of Luca on the equation of the title for prime $p > 2$. The elementary treatment is made possible by a…
The number of linear independent algebraic relations among elementary symmetric polynomial functions over finite fields is computed. An algorithm able to find all such relations is described. It is proved that the basis of the ideal of…
Using model theory and differential algebra, we give necessary conditions for algebraic ordinary differential equations to have a complex Pfaffian solution on some complex domain. These tools also allow us to give many examples of algebraic…
This paper presents an algebraic-geometric construction of the derivative developed initially within the class of polynomial functions without introducing limits at the initial stage. Tangency is characterized by an algebraic condition: the…
Asymptotic expansions for generalised trigonometric integrals are obtained in terms of elementary functions, which are valid for large values of the parameter $a$ and unbounded complex values of the argument. These follow from new…
We introduce and study the arithmetic function E_m(n), defined as the sum of the remainders of n when divided by the first m positive integers. Although the definition is elementary, the function encodes rich arithmetic structure. In this…
The main aim of this note is to provide characterization theorems concerning real derivations. Among others the following implication will be verified: Assume that $\xi\colon \mathbb{R}\to \mathbb{R}$ is a given differentiable function and…
The present text is a collection of notes about differential geometry prepared to some extent as part of tutorials about topics and applications related to tensor calculus. They can be regarded as continuation to the previous notes on…
The Conclusive Theorem has been established to determine the dependence of the three-axes positive-definite Finsleroid metric functions $F$ on the Finsleroid azimuthal angle $\theta$ in the three-dimensional case $N=3$, provided that the…
Most of theoretical physics is based on the mathematics of functions of a real or a complex variable; yet we frequently are drawn to try extending our reach to include quaternions. The non-commutativity of the quaternion algebra poses…
This paper describes the foundations of a differential geometry of a quaternionic curves. The Frenet-Serret equations and the evolutes and evolvents of a particular quaternionic curve are accordingly determined. This new formulation takes…
Given any two rational numbers $r_1$ and $r_2$, a necessary and sufficient condition is established for the three numbers $1$, $\cos (\pi r_1)$, and $\cos (\pi r_2)$ to be rationally independent. Extending a classical fact sometimes…
Integration of polynomials over the classical groups of unitary, orthogonal and symplectic matrices can be reduced to basic building blocks known as Weingarten functions. We present an elementary derivation of these functions.
This note provides a new approach to a result of Foregger and related earlier results by Keilson and Eberlein. Using quite different techniques, we prove a more general result from which the others follow easily. Finally, we argue that the…