Related papers: A brief introduction to approximate groups
In this article we give a concept of ground subgroup for finite and countable groups. By our definition such a subgroup of a group depends on a given subset of the group and on a given partition of the subset. For finite and free groups we…
We revisit the notion of probably approximately correct implication bases from the literature and present a first formulation in the language of formal concept analysis, with the goal to investigate whether such bases represent a suitable…
The article is devoted to linear quasigroups and some of their generalizations. In the first part main definitions and notions of the theory of quasigroups are given. In the second part some elementary properties of linear quasigroups and…
We give new characterizations of sofic groups: -- A group $G$ is sofic if and only if it is a subgroup of a quotient of a direct product of alternating or symmetric groups. -- A group $G$ is sofic if and only if any system of equations…
In this article we introduce and study a class of finite groups for which the orders of normal subgroups satisfy a certain inequality. It is closely connected to some well-known arithmetic classes of natural numbers.
We introduce some classical concepts in the representation theory of compact groups, in order to use them for a new generalization of the Peter-Weyl Theorem. We mostly deal with functions on locally compact groups possessing large…
We survey recent work ranging around the question in how far a group, or a property of a group, is determined by the set of finite quotient groups. Our focus lies on $S$-arithmetic groups, branch groups, and their relatives.
We establish various results on the structure of approximate subgroups in linear groups such as SL_n(k) that were previously announced by the authors. For example, generalising a result of Helfgott (who handled the cases n = 2 and 3), we…
We extend the treatment of functional dependence, the basic concept of dependence logic, to include the possibility of dependence with a limited number of exceptions. We call this approximate dependence. The main result of the paper is a…
We introduce a class of Banach algebras of generalized matrices and study the existence of approximate units, ideal structure, and derivations of them.
The aim of this text is to provide a clear description of the theory of Infra-nilmanifolds and their fundamental groups, the almost-Bieberbach groups. For most of the proofs of the results, we refer to the literature. Nevertheless, at…
A new and simple method for quasi-convex optimization is introduced from which its various applications can be derived. Especially, a global optimum under constrains can be approximated for all continuous functions.
We present a new neural network to approximate convex functions. This network has the particularity to approximate the function with cuts and can be easily adapted to partial convexity. We give an universal approximation theorem in the full…
Given a definably amenable approximate subgroup $A$ of a (local) group in some first-order structure, there is a type-definable subgroup $H$ normalised by $A$ and contained in $A^4$ such that every definable superset of $H$ has positive…
In this paper we discuss generalized group, provides some interesting examples. Further we introduce a generalized module as a module like structure obtained from a generalized group and discuss some of its properties and we also describes…
Recently, a complete characterization of connected Lie groups with the Approximation Property was given. The proof used of the newly introduced property (T*). We present here a short proof of the same result avoiding the use of property…
We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.
Arithmetic groups are groups of matrices with integral entries. We shall first discuss their origin in number theory (Gauss, Minkowski) and their role in the "reduction theory of quadratic forms". Then we shall describe these groups by…
We define the notion of an approximate triangulation for a manifold $M$ embedded in euclidean space. The basic idea is to build a nested family of simplicial complexes whose vertices lie in $M$ and use persistent homology to find a complex…
In this paper, we present a comprehensive system for the treatment of the topic of limits--conceptually, computationally, and formally. The system addresses fundamental linguistic flaws in the standard presentation of limits, which attempts…