Baer has shown that, for a group G, finiteness of G=Zi(G) implies finiteness of ɣi+1(G). In this paper we will show that the converse is true provided that G=Zi(G) is finitely generated. In particular, when G is a finite nilpotent group we show that |G=Zi(G)| divides |ɣi+1(G)|d′ i(G), where d′i(G) =(d( G /Zi(G)))i.

we prove that if $m$ is a monoid and $a$ a finite set with more than one element‎, ‎then the residual finiteness of $m$ is equivalent to that of the monoid consisting of all cellular automata over $m$ with alphabet $a$‎.

Definitions of finiteness are set-theoretical properties P such that, if the axiom of choice (AC) is assumed, a set satisfies P if and only if it is finite. It is well known that the equivalence of many definitions of finiteness (to one another, and to “ordinary” finiteness, defined below) requires some (perhaps weaker) form of AC. Thus, it is of interest to study how various definitions of fin...

I present a criterion for all-order finiteness in N =1 SYM theories. Three applications are given; they yield all-order finite N=1 SYM models with global symmetries of the superpotential. 1. INTRODUCTION The aim of this paper is to present applications of the criterion for all-order finiteness in N = 1 SYM of [ 1]. All-order finiteness is here meant in the sense of exact vanishing of the pertur...

Abstract We discuss the finiteness of (two-term) silting objects. First, we investigate new triangulated categories without object. Second, study two classes $\tau$ -tilting-finite algebras and give numbers their two-term Finally, explore when -tilting-finiteness implies representation-finiteness obtain several in which a algebra is representation-finite.

The intersection graph $mathbb{Int}(A)$ of an $S$-act $A$ over a semigroup $S$ is an undirected simple graph whose vertices are non-trivial subacts of $A$, and two distinct vertices are adjacent if and only if they have a non-empty intersection. In this paper, we study some graph-theoretic properties of $mathbb{Int}(A)$ in connection to some algebraic properties of $A$. It is proved that the fi...

Some notions in mathematics can be considered relative. Relative is a term used to denote when the variation in the position of an observer implies variation in properties or measures on the observed object. We know, from Skolem theorem, that there are first-order models where R is countable and some where it is not. This fact depends on the position of the observer and on the instrument/langua...

8. Introduction Critical points of distance functions Toponogov's theorem; first application:a Background on finiteness theorems Homotopy Finiteness Appendix. Some volume estimates Betti numbers and rank Appendix: The generalized Mayer-Vietoris estimate Rank, curvature and diameter Ricci curvature, volume and the Laplacian Appendix. The maximum principle Ricci curvature, diameter growth and fin...

We analyze the periodicity of optimal long products of matrices. A set of matrices is said to have the finiteness property if the maximal rate of growth of long products of matrices taken from the set can be obtained by a periodic product. It was conjectured a decade ago that all finite sets of real matrices have the finiteness property. This conjecture, known as the “finiteness conjecture”, is...

We analyze the periodicity of optimal long products of matrices. A set of matrices is said to have the finiteness property if the maximal rate of growth of long products of matrices taken from the set can be obtained by a periodic product. It was conjectured a decade ago that all finite sets of real matrices have the finiteness property. This conjecture, known as the “finiteness conjecture”, is...