More on the Laterally σ-Complete Reflection of an Archimedean Lattice-Ordered Group

The laterally -complete reflection is given a new, one-step construction, which, in some notable cases affords an explicit realization. For example, if X is a compact Hausdorff topological space, the laterally -complete reflection of C(X) consists of all real-valued functions on X which are countably Baire-piecewise continuous. This realization of the reflection also shows, easily, that it carries rings to rings and, for objects with a designated unit, that the singularity of the unit is preserved.     

Convergence of Averages in the Ergodic Theorem for Groups ?d

Some estimates of the rates of convergence in the ergodic theorem for actions of groups ^d are given. Moreover, a martingale-ergodic theorem for ^d is proved. This theorem may be considered as an ergodic theorem in which the exact initial coordinates of points in the phase space are gradually forgotten. Bibliography: 9 titles.     

The Decision Problems for Properties of Transformation Groups

In this paper the fundamental result about the decision problems for properties of FRT-groups (i. e. the groups each of which is isomorphic with a group generated by a finite number of recursive transformations) has been proved: Let p be any algebraic property for groups such that there is a FRT-group G_1 which has the property p, a FRT-group G_2 which has not the property p, and G_2 is not isomorphic with any subgroup of any FRT-group which has the property p. Then the problem of deoiding, for any given groap P genetated by a finite number of recursive transformations, whether or not the group G isomorphic with P has the property p is unsolvable.     

Recurrence Relationships for the Mean Number of Faces and Vertices for Random Convex Hulls

This paper studies the convex hull of n random points in R^d\mathsf{R}^{d} . A recently proved topological identity of the author is used in combination with identities of Efron and Buchta to find the expected number of vertices of the convex hull—yielding a new recurrence formula for all dimensions d. A recurrence for the expected number of facets and (d−2)-faces is also found, this analysis building on a technique of Rényi and Sulanke. Other relationships for the expected count of i-faces (1≤i<d) are found when d≤5, by applying the Dehn–Sommerville identities. A general recurrence identity (see (3) below) for this expected count is conjectured.     

Explicit Formulae for Values of Dedekind Zeta Functions of Two Kinds of Cyclotomic Fields

Let K = Q(m) denote the m-th cyclotomic field, and K+ its maximal real subfield, where m =exp is an m-th primary root of unity. Let K (s) denote the Dedekind zeta function ofK. For prime integers m = p, Fumio Hazama recently in [1] obtained formulae for calculating special values of K(s) and K+(s), i.e., calculating formulae of K+(1 - n) and for positive integers n, which are the newest results of a series of his work in many years (see [1-3]).Here we develop Hazama's work for prime integ…     

Hölder Conditions for Endomorphisms of Hyperbolic Groups

It is proved that an endomorphism φ of a hyperbolic group G satisfies a Hölder condition with respect to a visual metric if and only if φ is virtually injective and Gφ is a quasiconvex subgroup of G. If G is virtually free or torsion-free co-hopfian, then φ is uniformly continuous if and only if it satisfies a Hölder condition if and only if it is virtually injective.     

Modulation Spaces of Symbols for Representations of?Nilpotent Lie Groups

We investigate continuity properties of operators obtained as values of the Weyl correspondence constructed by Pedersen (Invent. Math. 118:1?C36, 1994) for arbitrary irreducible representations of nilpotent Lie groups. To this end we introduce modulation spaces for such representations and establish some of their basic properties. The situation of square-integrable representations is particularly important and in the special case of the Schr?dinger representation of the Heisenberg group we recover the classical modulation spaces used in the time-frequency analysis.     

On Eigenvalues of Group Elements in Representations of Simple Algebraic Groups and Finite Chevalley Groups

This is an expository paper discussing a number of problems on eigenvalues of elements in representations of algebraic groups and finite Chevalley groups. We stress on the algebraic group case which relates also to similar problems for Lie groups and Lie algebras.     

On the Maximality of Certain Orthogonal Groups Embedded in Symplectic and Unitary Groups resp

It is of interest to study the maximality of a classical group embedded in another. We have proved in [1] the maximality of special generalized symplectic (unitary, orthogonal resp.) groups in corresponding special linear groups. And this paper is a sketch of a section of [1] devoted to another two cases.     

An Analogue of Beurling’s Theorem for NA Groups

In this paper, we prove Beurling’s theorem for NA groups, from which we derive some other versions of uncertainty principles.     

On the Number of Conjugacy Classes in Semi-Products of Certain Finite Groups

Let V be a vector space of n-dimension over the field GF(p) of p elements,where p is a prime. V is also an elementary abelian p-group.Let G be a p'-group oflinear transformations on V.Theorem 1 Let π_v(a_1,a_2) be the number of the common fixed points of a_1and a_2 on V, a_1, a_2 ∈G. Let k(GV) be the number of conjugacy classes in thesemi-product GV (We also denote it by GV) of G and V. Then     

Gröbner–Shirshov Bases for Schreier Extensions of Groups

In this article, by using the Gröbner–Shirshov bases, we give characterizations of the Schreier extensions of groups when the group is presented by generators and relations. An algorithm to find the conditions of a group to be a Schreier extension is obtained. By introducing a special total order, we obtain the structure of the Schreier extension by an HNN group.     

Gelfand–Kirillov Dimension for Automorphism Groups of Relatively Free Algebras

Using ideas of our recent work on automorphisms of residually nilpotent relatively free groups, we introduce a new growth function for subgroups of the automorphism groups of relatively free algebras Fⁿ(V) over a field of characteristic zero and the related notion of Gelfand-Kirillov dimension, and study their behavior. We prove that, under some natural restrictions, the Gelfand-Kirillov dimension of the group of tame automorphisms of Fⁿ(V) is equal to the Gelfand-Kirillov dimension of the algebra Fⁿ(V). We show that, in some cases, the Gelfand-Kirillov dimension of the group of tame automorphisms of Fⁿ(V) is smaller than the Gelfand-Kirillov dimension of the whole automorphism group, and calculate the Gelfand-Kirillov dimension of the automorphism group of Fⁿ(V) for some important varieties V.Partially supported by Grant MM605/96 of the Bulgarian Foundation for Scientific Research.2000 Mathematics Subject Classification: primary 16R10, 16P90; secondary 16W20, 17B01, 17B30, 17B40     

The Hilbert–Poincaré Series for Some Algebras of Invariants of Cyclic Groups

Let be a linear representation of a finite group over a field of characteristic 0. Further, let R be the corresponding algebra of invariants, and let P (t) be its Hilbert–Poincaré series. Then the series P (t) represents a rational function (t)/(t). If R is a complete intersection, then (t) is a product of cyclotomic polynomials. Here we prove the inverse statement for the case where is an almost regular (in particular, regular) representation of a cyclic group. This yields an answer to a question of R. Stanley in this very special case. Bibliography: 3 titles.     

The Problem of Freeness for Euler Monoids and Möbius Groups

The Euler monoid E_n = {(a,b,t) epsilon Z³ : a² + b² = tⁿ, n S 1, is free if and only if n is odd (Theorem 1). We extend the results of Lyndon and Ullman, and Beardon concerning the set of those rational numbers mu epsilon (-2,2) for which the matrix Möbius group G_mu generated by A= and B = is not free (Theorems 2, 3, 4).     

The Σ1-Invariant for Some Artin Groups of Rank 3 Presentation

We classify the Bieri–Neumann–Strebel–Renz invariant Σ¹(G) for a class of Artin groups based on the full graph with 4 vertices.     

K(π 1)'s for Artin Groups of Finite Type

We construct K( 1)'s for Artin groups of type C _n and D _n , using the lattice of elements preceding a Coxeter element in the partial order defined by reflection length.