Related papers: A short proof of solution formulas for the linear …
Differential equations parameterized by neural networks become expensive to solve numerically as training progresses. We propose a remedy that encourages learned dynamics to be easier to solve. Specifically, we introduce a differentiable…
In this article we study solutions to second order linear difference equations with variable coefficients. Under mild conditions we provide closed form solutions using finite continued fraction representations. The proof of the results are…
A number of authors have proven explicit versions of Lehmer's conjecture for polynomials whose coefficients are all congruent to 1 modulo m. We prove a similar result for polynomials f(X) that are divisible in (Z/mZ)[X] by a polynomial of…
We look at the number of solutions of an equation of the form f_1*f_2*...*f_k=a in a finite field, where each f_i is a multilinear polynomial. We use two methods to construct a solution of this problem for the cases a=0, a<>0, and we…
The purpose of this paper is to present a syatemic study of some familes of higher-order Euler numbers and polynomials. In particular, by using the basis property of higher-order Euler polynomials for the space of polynomials of degree less…
We consider a solution to a parametric family of the Cauchy problems for $m$th-order linear differential equations with constant coefficients. Parameters of the family are the coefficients of the differential equation and the initial values…
We present a short elementary proof of the well-known criterion for a cubic polynomial to have three real roots. The proof is based on Fermat's approach to calculus for polynomials. This approach illustrates the idea of a derivative…
We solve the difference equation with linear coefficients by the Momentenansatz to obtain explicit formulas for orthogonal polynomials.
Everyone knows that the Euler characteristic of a combinatorial manifold is given by the alternating sum of its numbers of simplices. It is shown that there are other linear combinations of the numbers of simplices which are combinatorial…
We extend our previous work on Poisson-like formulas for subresultants in roots to the case of polynomials with multiple roots in both the univariate and multivariate case, and also explore some closed formulas in roots for univariate…
Let $(G_n(x))_{n=0}^\infty$ be a $d$-th order linear recurrence sequence having polynomial characteristic roots, one of which has degree strictly greater than the others. Moreover, let $m\geq 2$ be a given integer. We ask for…
A symbolic computational algorithm which detects " linear "` solutions of nonlinear polynomial differential equations of single functions, is developed throughout this paper.
The roots of any polynomial of degree m with integer coefficients, can be computed by manipulation of sequences made from 2m distinct symbols and counting the different symbols in the sequences. This method requires only 'primitive'…
A novel method, connecting the space of solutions of a linear differential equation, of arbitrary order, to the space of monomials, is used for exploring the algebraic structure of the solution space. Apart from yielding new expressions for…
The method of characteristics is a classical method for gaining understanding in the solution of a partial differential equation. It has recently been applied to the adjoint equations of the 2D Euler equations and the first goal of this…
In this paper we prove the multiplicity of solutions for a class of quasilinear problems in $ \mathbb{R}^{N} $ involving variable exponents. The main tool used is in the proof are the direct methods, Ekeland's variational principle and some…
In this paper I present a kind of proof for classical Euclidean geometric problems which relies on both synthetic and analytic geometry. Using the elementary tools of polynomial algebra and multivariate calculus we manage to reduce the…
This note presents a simple proof of the characteristic function of Student's $t$-distribution. The method of proof, which involves finding a differential equation satisfied by the characteristic function, is applicable to many other…
By the classical Sturm's theorem, the number of distinct real roots of a given real polynomial $f(x)$ within any interval $(a,b]$ can be expressed by the number of variations in the sign of the Sturm chain at the bounds. Through…
This is a short review of some recent results obtained by the author. These results are related the problem of obtaining polynomial identities (computational formulas) for some matrix functions by means of the known polarization theorem,…