Matching families are one of the major ingredients in the construction of locally decodable codes (LDCs) and the best known constructions of LDCs with a constant number of queries are based on matching families. The determination of the largest size of any matching family in Zm, where Zm is the ring of integers modulo m, is an interesting problem. In this paper, we show an upper bound of O((pq)...

In this note we shall investigate a topological version of the problem of Kurosh: “Is any algebraic algebra locally finite?” Kaplansky’s theorem concerning the local nilpotence of nil PI-algebras is well-known. We will prove a generalization of Kaplansky’s theorem to the class of locally compact rings. We use in the proof a theorem of A. I. Shirshov [8] concerning the height of a finitely gener...

Given two algebraic groups G, H over a field k, we investigate the representability of functor morphisms (of schemes) Hom(G,H) and subfunctor homomorphisms groups) Homgp(G,H). We show that is represented by group scheme, locally finite type, if k-vector space O(G) finite-dimensional; converse holds not étale. When G linearly reductive smooth, Homgp(G,H) smooth scheme M; moreover, every orbit ac...

Let G be a semisimple algebraic group of hermitίan type defined over Q, let £ = GR/K, where K c G is a maximal compact subgroup, be the symmetric domain associated to G, let Γ be an arithmetic subgroup of G, let (π, E) be a finite-dimensional representation of G defined over Q, let Φ:= Γ \ £ , and let £ be the locally constant sheaf over °tt associated to (π, E). Then under certain conditions o...

Since there are some important systems which have constraints not contained in their field algebras, we develop here in a C*-context the algebraic structures of these. The constraints are defined as a group G acting as outer automorphisms on the field algebra J^, α:Gι-> Aut $F, αG φ Inn J^, and we find that the selection of G-invariant states on 3F is the same as the selection of states ω on M(...

For a group G, we denote by G↔ the coarse space on G endowed with structure base {{(x,y)∈G×G:y∈xF}:F∈[G]<ω}, xF={z−1xz:z∈F}. Our goal is to explore interplays between algebraic properties of and asymptotic G↔. In particular, show that asdim G↔=0 if only G/ZG locally finite, ZG center G. an infinite subgroups discrete Dedekind group.

We prove the so-called Addition Theorem for algebraic entropy of actions cancellative right amenable monoids S on discrete abelian groups A by endomorphisms, under hypothesis that is locally monotileable (that is, admits a Følner sequence (Fn)n?N such Fn monotile Fn+1 every n?N). study in details class groups, also relation with already existing notions monotileability introduced B. Weiss and d...

We study the large-scale geometry of a large class of amenable locally compact groups comprising all solvable algebraic groups over a local field and their discrete subgroups. We show that the isoperimetric profile of these groups is in some sense optimal among amenable groups. We use this fact to compute the probability of return of symmetric random walks, and to derive various other geometric...

This article extracts the elements of algebra that play a central role in the design of efficient probabilistic verifiers or “probabilistically checkable proof systems (PCPs)”. The main algebraic elements are low-degree polynomials over finite fields. Their role can be broken up into three essential elements: 1. Their classical role in the design of error-correcting codes. 2. Their recently dis...

Abstract We present a systematic way to construct dynamical systems with a specific symmetry group G. Each symmetric strange attractor has a unique image attractor that is locally identical to it but different at the global topological level. Image attractors can be lifted to many inequivalent covering attractors. These are distinguished by an index that has related topological, algebraic and g...