Related papers: Abel's problem, Gauss and Cartier congruences over…
It is known that the numbers which occur in Apery's proof of the irrationality of zeta(2) have many interesting congruence properties while the associated generating function satisfies a second order differential equation. We prove…
We prove the unique existence and exponential decay of global in time classical solutions to the special relativistic Boltzmann equation without any angular cut-off assumptions with initial perturbations in some weighted Sobolev spaces. We…
The functional-integral quantization of non-Abelian gauge theories is affected by the Gribov problem at non-perturbative level: the requirement of preserving the supplementary conditions under gauge transformations leads to a non-linear…
We give a new method for numerically solving Abel integral equations of first kind. An estimation for the error is obtained. The method is based on approximations of fractional integrals and Caputo derivatives. Using trapezoidal rule and…
Nowadays the Euler-Poisson-Darboux equation is extensively studied in several settings. The main questions on avery spaces are explicit solutions for the classical Cauchy problems with the second data null. In this note we will generalize…
A finite algebra $\bA=\alg{A;\cF}$ is \emph{dualizable} if there exists a discrete topological relational structure $\BA=\alg{A;\cG;\cT}$, compatible with $\cF$, such that the canonical evaluation map $e\_{\bB}\colon \bB\to \Hom(…
In this paper we continue the investigation of the Abel equation with the right part belonging to a Lebesgue weighted space. We have improved the previously known result - the uniqueness and existence theorem formulated in terms of the…
We show that if a finite, large enough subset A of an arbitrary abelian group satisfies the small doubling condition |A + A| < (log |A|)^{1 - epsilon} |A|, then A must contain a three-term arithmetic progression whose terms are not all…
We study the Abel differential equation x0 = A(t)x3 + B(t)x2 +C(t)x. Specifically, we find bounds on the number of its rational solutions when A(t), B(t) and C(t) are polynomials with real or complex coefficients; and on the number of…
We consider Abel maps for regular smoothing of nodal curves with values in the Esteves compactified Jacobian. In general, these maps are just rational, and an interesting question is to find an explicit resolution. We translate this problem…
We propose an efficient and flexible method for solving Abel integral equation of the first kind, frequently appearing in many fields of astrophysics, physics, chemistry, and applied sciences. This equation represents an ill-posed problem,…
Within the scope of elementary number theory, we prove that, as the main result, if $1 \leq x < y < z$ are integers such that at least one of $y, z, x+y$ is prime then $x^{n}+y^{n} \neq z^{n}$ for every odd integer $n \geq 3$. This result…
For the Cauchy problem for an operator differential equation of the form $y'(z) = Ay(z)$, where $A$ is a closed linear operator on a Banach space over the completion of an algebraic closure of the field of $p$-adic numbers, a criterion of…
We prove a new Elekes-Szab\'o type estimate on the size of the intersection of a Cartesian product $A\times B\times C$ with an algebraic surface $\{f=0\}$ over the reals. In particular, if $A,B,C$ are sets of $N$ real numbers and $f$ is a…
We give an overview of recent progress around a problem introduced by Elekes and R\'onyai. The prototype problem is to show that a polynomial $f\in \mathbb{R}[x,y]$ has a large image on a Cartesian product $A\times B\subset \mathbb{R}^2$,…
Osburn, Sahu and Straub introduced the numbers: \begin{align*} A_n^{(r,s,t)}=\sum_{k=0}^n{n\choose k}^r{n+k\choose k}^s{2k\choose n}^t, \end{align*} for non-negative integers $n,r,s,t$ with $r\ge 2$, which includes two kinds of Ap\'ery…
Behavior of solutions of $f''+Af=0$ is discussed under the assumption that $A$ is analytic in $\mathbb{D}$ and $\sup_{z\in\mathbb{D}}(1-|z|^2)^2|A(z)|<\infty$, where $\mathbb{D}$ is the unit disc of the complex plane. As a main result it is…
Let $\b$ be a Borel subalgebra of a simple Lie algebra $\g$ and let $\Ab$ denote the set of all Abelian ideals of $\b$. We consider $\Ab$ as poset with respect to inclusion, the zero ideal being the unique minimal element of $\Ab$. It was…
Albert algebras, a specific kind of Jordan algebra, are naturally distinguished objects among commutative non-associative algebras and also arise naturally in the context of simple affine group schemes of type $F_4$, $E_6$, or $E_7$. We…
We propose interconnections between some problems of PDE, geometry, algebra, calculus and physics. Uniqueness of a solution of the Dirichlet problem and of some other boundary value problems for the string equation inside an arbitrary…