Related papers: Linear equations with unknowns from a multiplicati…

A system of linear equations is said underdetermined when there are more unknowns than equations. Such systems may have infinitely many solutions. In this case, it is important to single out solutions possessing special features. A well…

A planar node set $\mathcal X,$ with $|\mathcal X|=\binom{n+2}{2}$ is called $GC_n$ set if each node possesses fundamental polynomial in form of a product of $n$ linear factors. We say that a node uses a line $Ax+By+C=0$ if $Ax+By+C$…

A completion of an m-by-n matrix A with entries in {0,1,*} is obtained by setting all *-entries to constants 0 or 1. A system of semi-linear equations over GF(2) has the form Mx=f(x), where M is a completion of A and f:{0,1}^n --> {0,1}^m…

A Cullen number is a number of the form $m2^m+1$, where $m$ is a positive integer. In 2004, Luca and St\u anic\u a proved, among other things, that the largest Fibonacci number in the Cullen sequence is $F_4=3$. Actually, they searched for…

Let $C$ be the classical middle third Cantor set. It is well known that $C+C = [0,2]$ (Steinhaus, 1917). (Here $+$ denotes the Minkowski sum.) Let $U$ be the set of $z \in [0,2]$ which have a unique representation as $z = x + y$ with $x, y…

Let p be a prime number. Let G be a finite abelian p-group of exponent n (written additively) and A be a non-empty subset of $]n[:= \{1,2,..., n\}$ such that elements of A are incongruent modulo p and non-zero modulo p. Let $k \geq…

We discuss two conjectures. (I) For each x_1,...,x_n \in R (C) there exist y_1,...,y_n \in R (C) such that \forall i \in {1,...,n} |y_i| \leq 2^{2^{n-2}} \forall i \in {1,...,n} (x_i=1 \Rightarrow y_i=1) \forall i,j,k \in {1,...,n}…

Let n >1 be an integer, and G a doubly transitive subgroup of the symmetric group on X={1,...,n}. In this paper we find all linear group representations rho of G on an euclidean vector space V which contains a set of equiangular vector…

Let $(A_i)_{0\leq i\leq k}$ be generic matrices over $\mathbb{Q}$, the field of rational numbers. Let $K=\mathbb{Q}(E)$, where $E$ denotes the entries of the $(A_i)_i$, and let $\overline{K}$ be the algebraic closure of $K$. We show that…

This article considers the minimal non-zero (= indecomposable) solutions of the linear congruence $1\cdot x_1 + \cdots + (m-1)\cdot x_{m-1} \equiv 0 \pmod m$ for unknown non-negative integers $x_1, \ldots, x_n$, and characterizes the…

In this paper we consider combinatorial numbers $C_{m, k}$ for $m\ge 1$ and $k\ge 0$ which unifies the entries of the Catalan triangles $ B_{n, k}$ and $ A_{n, k}$ for appropriate values of parameters $m$ and $k$, i.e., $B_{n,…

Generalising Solomon's theorem, C. Gordon and F. Rodriguez-Villegas have proven recently that, in any group, the number of solutions to a system of coefficient-free equations is divisible by the order of this group whenever the rank of the…

A countable group G is called k-linear sofic (for some 0 <k \le 1) if finite subsets of G admit "approximate representations" by complex invertible matrices in the normalized rank metric, so that non-identity elements are k-away from the…

We study connections between linear equations over various semigroups and recursively enumerable sets of positive integers. We give variants of the universal Diophantine representation of recursively enumerable sets of positive integers…

Given a group $G$, we write $g^G$ for the conjugacy class of $G$ containing the element $g$. A theorem of B. H. Neumann states that if $G$ is a group in which all conjugacy classes are finite with bounded size, then the commutator subgroup…

We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers $r,n\geq 0$, we conjecture that $L_n^{(-1-n-r)}(x) = \sum_{j=0}^n \binom{n-j+r}{n-j}x^j/j!$ is a…

We will study the solutions to the equation $f(n) - g(n) = c$, where $f$ and $g$ are multiplicative functions and $c$ is a constant. More precisely, we prove that the number of solutions does not exceed $c^{1-\epsilon}$ when $f, g$ and…

This paper aims to study linear sets of minimum size in the projective line, that is $\mathbb{F}_q$-linear sets of rank $k$ in $\mathrm{PG}(1,q^n)$ admitting one point of weight one and having size $q^{k-1}+1$. Examples of these linear sets…

Let $\mathbb G = (G, +)$ be a group (either abelian or not). Given $X, Y \subseteq G$, we denote by $\langle Y \rangle$ the subsemigroup of $\mathbb G$ generated by $Y$, and we set $$\gamma(Y) := \sup_{y_0 \in Y} \inf_{y_0 \ne y \in Y} {\rm…

In this paper we prove that any finite group of order $n$ can be viewed as the group of the solutions of a certain matrix equation $XB=BY$, where the unknowns $X,Y$ are two permutation matrices of order $n$ and $(1+k)n+2 $ respectively and…