Related papers: Solution of Belousov's problem

Let $\UT_n(q)$ denote the group of unipotent $n\times n$ upper triangular matrices over a field with $q$ elements. The degrees of the complex irreducible characters of $\UT_n(q)$ are precisely the integers $q^e$ with $0\leq e\leq \lfloor…

The inverse problem for representation functions takes as input a triple (X,f,L), where X is a countable semigroup, f : X --> N_0 \cup {\infty} a function, L : a_1 x_1 + ... + a_h x_h an X-linear form and asks for a subset A \subseteq X…

Let F be a p-adic field and n a positive integer. The local Langlands conjecture asserts the existence of a bijection between irreducible admissible representations of GL(n,F) and n-dimensional admissible representations of the Weil-Deligne…

We exhibit infinitely many natural numbers $n$ for which there exists at least one insolvable group of order $n$, and yet the holomorph of any solvable group of order $n$ has no insolvable regular subgroup. We also solve Problem 19.90 (d)…

Let $n$ be a positive integer and let $\mathbb F_{q^n}$ be the finite field with $q^n$ elements, where $q$ is a power of a prime. This paper introduces a natural action of the Projective Semilinear Group $\text{P}\Gamma \text{L}(2,…

We prove a version of both Jacobi's and Montel's Theorems for the case of continuous functions defined over the field $\mathbb{Q}_p$ of $p$-adic numbers. In particular, we prove that, if \[ \Delta_{h_0}^{m+1}f(x)=0 \ \ \text{for all}…

This paper aims at studying solvable-by-finite and locally solvable maximal subgroups of an almost subnormal subgroup of the general skew linear group $\GL_n(D)$ over a division ring $D$. It turns out that in the case where $D$ is…

In this paper, we deal with locally graded groups whose subgroups are either subnormal or soluble of bounded derived length, say d. In particular, we prove that every locally (soluble-by-finite) group with this property is either soluble or…

We consider time fractional parabolic equations in both divergence and non-divergence form when the leading coefficients $a^{ij}$ are measurable functions of $(t,x_1)$ except for $a^{11}$ which is a measurable function of either $t$ or…

We conjecture that if a system S \subseteq {x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} has only finitely many solutions in integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies |x_1|,...,|x_n| \leq…

Let $X$ be an algebraic variety over a finite field $\bF_q$, homogeneous under a linear algebraic group. We show that the number of rational points of $X$ over $\bF_{q^n}$ is a periodic polynomial function of $q^n$ with integer…

Let E_n={x_i=1, x_i+x_j=x_k, x_i*x_j=x_k: i,j,k \in {1,...,n}}. We prove: (1) there is an algorithm that for every computable function f:N-->N returns a positive integer m(f), for which a second algorithm accepts on the input f and any…

We introduce a class $\Lambda_{s}$ of functions with complicated local structure. Any function from the class belongs to one of three specifically defined types $f^s _k$, $f_+$, and $f^{-1} _+$ or is a specifically defined composition of…

For every $n \geq 1$, let $(\mathrm{FW}_n)$ denote the fixed-point property for median graphs of cubical dimension $n$ (or equivalently, for CAT(0) cube complexes of dimension $n$). In this article, we construct explicit examples of groups…

It is known that if every group satisfying an identity of the form yx ~ xU(x,y)y is abelian, so is every semigroup that satisfies that identity. Because a group has an identity element and the cancellation property, it is easier to show…

The main purpose of this work is to provide the general solutions of a class of linear functional equations. Let $n\geq 2$ be an arbitrarily fixed integer, let further $X$ and $Y$ be linear spaces over the field $\mathbb{K}$ and let…

A quantum Markov semigroup can be represented via classical diffusion processes solving a stochastic Schr\"odinger equation. In this paper we first prove that a quantum Markov semigroup is irreducible if and only if classical diffusion…

L. Klebanov proved the following theorem. Let $\xi_1, \dots, \xi_n$ be independent random variables. Consider linear forms $L_1=a_1\xi_1+\cdots+a_n\xi_n,$ $L_2=b_1\xi_1+\cdots+b_n\xi_n,$ $L_3=c_1\xi_1+\cdots+c_n\xi_n,$…

In the local, characteristic 0, non archimedean case, we consider distributions on GL(n+1) which are invariant under the adjoint action of GL(n). We prove that such distributions are invariant by transposition. This implies that an…

The main objective of this paper is the following two results. (1) There exists a computable bi-orderable group that does not have a computable bi-ordering; (2) There exists a bi-orderable, two-generated recursively presented solvable group…