Related papers: Local criteria for the unit equation and the asymp…

First, we define some concepts similar to the local compactoidity or the c-compactness, and study relationships between these concepts and the original ones. As a result, we find a characterization of the local compactoidity when its…

In this paper we consider the existence of positive solutions for a singular elliptic problem involving an asymtotically linear nonlinearity and depending on one positive parameter. Using variational methods, together with comparison…

An alternative form of Fermats equation[1] is proposed. It represents a portion of the identity that includes three terms of Fermats original equation. This alternative form permits an elementary and compact proof of the first case of…

Let $d\in\mathbb{N}$ and let $\varphi\colon(0,1)\to[0,d]$. We prove that there exists a set $F\subset\mathbb{R}^d$ such that $\operatorname{dim}_A^\theta F=\varphi(\theta)$ for all $\theta\in(0,1)$ if and only if for every…

Given any collection F of computable functions over the reals, we show that there exists an algorithm that, given any L_F-sentence \varphi containing only bounded quantifiers, and any positive rational number \delta, decides either "\varphi…

In this work, we prove that every SFT, sofic shift, and strongly irreducible shift on locally finite groups has strong dynamical properties. These properties include that every sofic shift is an SFT, every SFT is strongly irreducible, every…

We investigate existence and uniqueness of solutions to a class of fractional parabolic equations satisfying prescribed pointwise conditions at infinity (in space), which can be time- dependent. Moreover, we study the asymptotic behaviour…

In this paper, we consider the regularity of weak solutions (in an appropriate space) to the elliptic partial differential equation \begin{equation*} (-\Delta_{p})^{s} u + (-\Delta_{q})^{s} u = f(x) \quad \text{in} \quad \mathbb{R}^{N},…

The theme of this article is to provide some sufficient conditions for the asymptotic property and oscillation of all solutions of third-order half-linear differential equations with advanced argument of the form…

We establish an aysmptotic formula for the number of points with coordinates in $\mb{F}_q[t]$ on a complete intersection of degree $d$ defined over $\mb{F}_q[t]$, with explicit error term, provided that the characteristic of $\mb{F}_q$ is…

For a given point P in the group of K-rational points E(K) of an elliptic curve, we consider the sequence of values (F_1(P),F_2(P),F_3(P),...) of the division polynomials of E at P. If K is a finite field, we prove that the sequence is…

In [2], an exhaustive construction is achieved for the class of all 4-dimensional unital division algebras over finite fields of odd order, whose left nucleus is not minimal and whose automorphism group contains Klein's four-group. We…

In this paper we prove new cases of the asymptotic Fermat equation with coefficients. This is done by solving remarkable $S$-units equations and applying a method of Frey-Mazur.

In this paper, we prove that for each number field $F$ there exists a uniform bound on the prime levels $p$ of elliptic curves $E/F$ for which $F(E[p])=F(\zeta_p)$. Under the Generalized Riemann Hypothesis, we also give uniform bounds on…

We prove that a graph G is asymptotically isomorphic to the ray if and only if G is uniformly spherically bounded and is of bounded local degrees. This problem arouse in combinatorics and was posed in [3] (Problem 10.1).

A basic version of Abhyankar's Lemma states that for two finite extensions $L$ and $F$ of a local field $K$, if $L|K$ is tamely ramified and if the ramification index of $L|K$ divides the ramification index of $F|K$, then the compositum…

We consider nonlinear Schr\"odinger equations in dimension 3 or higher. We prove that symmetric finite energy solutions close to orbitally stable ground states converge asymptotically to a sum of a ground state and a dispersive wave…

A finite group $G$ is said to be admissible over a field $F$ if there exists a division algebra $D$ central over $F$ with a maximal subfield $L$ such that $L/F$ is Galois with group $G$. In this paper we give a complete characterization of…

Let $A$ be a finite-dimensional local commutative algebra over $R$, $\dim_RA=n$. In this work we consider compact manifolds over $A$, and prove that the real part of an $A$-differentiable function is constant. Also we find estimates for the…

Let $p$ be a prime number and let $S=\{x^p+c_1,\dots,x^p+c_r\}$ be a finite set of unicritical polynomials for some $c_1,\dots,c_r\in\mathbb{Z}$. Moreover, assume that $S$ contains at least one irreducible polynomial over $\mathbb{Q}$. Then…