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A group is called matricial field (MF) if it admits finite dimensional approximate unitary representations which are approximately faithful and approximately contained in the left regular representation. This paper provides a new class of…

The main purpose of this work is to characterize derivations through functional equations. This work consists of five chapters. In the first one, we summarize the most important notions and results from the theory of functional equations.…

Let $\A$ be an algebra and let $f(x_1,...,x_d)$ be a multilinear polynomial in noncommuting indeterminates $x_i$. We consider the problem of describing linear maps $\phi:\A\to \A$ that preserve zeros of $f$. Under certain technical…

Let $G$ be a group. A function $G\rightarrow G$ of the form $x\mapsto x^{\alpha}g$ for a fixed automorphism $\alpha$ of $G$ and a fixed $g\in G$ is called an affine map of $G$. In this paper, we study finite groups $G$ with an affine map of…

This paper contains a survey of recent developments in investigation of word equations in simple matrix groups and polynomial equations in simple (associative and Lie) matrix algebras along with some new results on the image of word maps on…

Let F be a commutative field of characteristic 0, G_n: F^n \times F^n -> F, G_n((x_1,...,x_n),(y_1,...,y_n))=(x_1-y_1)^2+...+(x_n-y_n)^2. We say that g:R^n->F^n preserves distance d>=0 if for each x,y \in R^n |x-y|=d implies…

We revisit the entire framework of group classification of differential equations. After introducing the notion of weakly similar classes of differential equations, we develop the mapping method of group classification for such classes,…

Given $n$ random variables, when does the matrix of pairwise $f$-mutual informations define a PSD kernel over variables? For convex finite generators $f:(0,\infty)\to\mathbb{R}$ with $f(1)=0$ and finite boundary value $f(0)$, we give a…

A countable group $G$ is said to be \emph{matricial field} (MF) if it admits a strongly converging sequence of approximate homomorphisms into matrices; i.e, the norms of polynomials converge to those in the left regular representation. $G$…

We construct the 12-dimensional exceptional field theory associated to the group $\mathrm{SL}(2) \times \mathbb{R}^+$ . Demanding the closure of the algebra of local symmetries leads to a constraint, known as the section condition, that…

Let $B$ be a M\"obius band and $f:B \to \mathbb{R}$ be a Morse map taking a constant value on $\partial B$, and $\mathcal{S}(f,\partial B)$ be the group of diffeomorphisms $h$ of $B$ fixed on $\partial B$ and preserving $f$ in the sense…

Given a function $f$ in a finite field ${\mathbb F}_q$ of $q$ elements, we define the functional graph of $f$ as a directed graph on $q$ nodes labelled by the elements of ${\mathbb F}_q$ where there is an edge from $u$ to $v$ if and only if…

Let $f:G\rightarrow H$ be a homomorphism of groups, we construct a topological space $X_f$ such that its group of homeomorphisms is isomorphic to $G$, its group of homotopy classes of self-homotopy equivalences is isomorphic to $H$ and the…

We give a criterion for maps on ultrametric spaces to be surjective and to preserve spherical completeness. We show how Hensel's Lemma and the multi-dimensional Hensel's Lemma follow from our result. We give an easy proof that the latter…

This paper addresses the question: given a scalar group, can we determine all the additions that transform this scalar group into a (near-)field? A key approach to addressing this problem involves transporting (near-)field structures via…

It is known that if $f$ is an analytic self map of the complex upper half-plane which also maps $\mathbb{R}\cup\{\infty\}$ to itself, and $f(i)=i$, then $f$ preserves the Cauchy distribution. This note concerns three results related to the…

We show how to efficiently count and generate uniformly at random finitely generated subgroups of the modular group $\textsf{PSL}(2,\mathbb{Z})$ of a given isomorphism type. The method to achieve these results relies on a natural map of…

In this paper we study the structure of the graphs associated with the iterations of the map $x \mapsto x+x^{-1}$ over finite fields of characteristic five.

We construct a new class of affine complements ${\mathbb P}^M\setminus S$ with the trivial group of automorphisms, where $S\subset {\mathbb P}^M$ is a rational hypersurface, $M$ is odd and $M\geqslant 5$.

We exhibit a map f between aspherical spaces X and Y such that f induces an isomorphism on homotopy groups but, with natural topologies, X and Y fail to have homeomorphic fundamental groups. Thus the topological fundamental group has the…