This note establishes estimates for lower bounds of linear forms at algebraic points of E-functions and G-functions defined over a general algebraic number field.

Let k be a complex number and let u be an element of a finite group G. Suppose that u does not belong to O(G), the maximal normal subgroup of G of odd order. It is shown that G satisfies X(1) – X(u) = k for every complex nonprincipal irreducible character X in the principal 2-block of G if and only if G/O(G) is isomorphic either to C2, a cyclic group of order 2, or to PSL (2, 2n), n ≧ 2.

If there is a W (n, p), then there is a symmetric W (n2, p2).

In this note we answer the following question: Given C(X) the latticeordered ring of real continuous functions on the compact Hausdorff space X and T an averaging operator on C(X), under what circumstances can X be decomposed into a topological product such that supports a measure m and Tf = h where By an averaging operator we mean a linear transformation T on C(X) such that: 1. T is positive, that is, if f>0 (f(x) ≧ 0 for all x ∈ and f(x) > 0 for some a ∈ X), then Tf>0. 2. T(fTg) = (Tf)(Tg). 3. T l = 1 where l(x) = 1 for all x ∈ X.

A (ν, κ, λ)-graph is defined in [3] as a graph on ν points, each of valency κ, and such that for any two points P and Q there are exactly λ points which are joined to both. In other words, if Si is the set of points joined to Pi, then

Si has k elements for any i

SiSj has λ elements if i≠j

The sets Si are the blocks of a (v, k, λ)-configuration, so a necessary condition on v, k, and λ that a graph should exist is that a (v, k, λ)- configuration should exist. Another necessary condition, reported by Bose (see [1]) and others, is that there should be an integer m satisfying have equal parity. We shall prove that these conditions are not sufficient.

This paper is devoted to the study of the normal (tangential) regularity of a closed set and the subdifferential (directional) regularity of its distance function in the context of Riemannian manifolds. The Clarke, Fréchet and proximal subdifferentials of the distance function from a closed subset in a Riemannian manifold are represented by corresponding normal cones of the set.

The symbol IG(X) denotes the inverse semigroup, under composition of functions, of all homeomorphisms between open subsets of a T1 topological space X. The first result is that two such semigroups IG(X) and IG(Y) are isomorphic if and only if the spaces X and Y are homeomorphic. Ideals of IG(X) are next examined and it is shown that for many spaces X the semigroup IG(X) is 0-simple. We also look at congruences on IG(X); one result is that we determine a congruence which in many instances is the largest proper congruence on IG(X).

We obtain a fully explicit quantitative version of the Eisenstein theorem on algebraic power series which is more suitable for certain applications than the existing version due to Dwork, Robba, Schmidt and van der Poorten. We also treat ramified series and Laurent series, and we demonstrate some applications; for instance, we estimate the discriminant of the number field generated by the coefficients.

Let N denote the discrete space of all natural numbers. A space X is N-compact if it is homeomorphic with some closed subspace of a product of copies of N. In this paper, N-compact spaces are characterized as homeomorphs of inverse limit space of inverse systems of copies of subsets of N. Also, it is shown that a space X is N-compact if and only if the space (X) of all non-empty compact subsets of X with the finite topology is N-compact.

A variety of groups is an equationally defined class of groups: equivalently, it is a class of groups closed under the operations of taking cartesian products, subgroups, and quotient groups. If and are varieties, then is the class of all groups G with a normal subgroup N in such that G/N is in ; is a variety, called the product of and . We denote by the variety generated by the unit group, and by the variety of all groups. We say that a variety is indecomposable if , and cannot be written as a product , with both and One of the basic results in the theory of varieties of groups is that the set of varieties, excluding , and with multiplication of varieties as above, is a free semi-group, freely generated by the indecomposable varieties. Thus one would like to be able to decide whether a given variety is indecomposable or not. In connection with this question, Hanna Neumann raises the following problem (as part of Problem 7 in her book [7]): Problem 1. Ifandprove that [] is indecomposable unless bothandhave a common non-trivial right hand factor.

Problem 24 of Hanna Neumann's book [3] reads: Does there exist, for a given integer n > 0, a Cross variety that is generated by its k-generator groups and contains (k+n)-generator critical groups? In such a variety, is every critical group that needs more than k generators a factor of a k-generator critical group, or at least of the free group of rank k? In a recent paper [1], R. G. Burns pointed out that the answer to the first question is an easy affirmative, and asked instead the question which presumably was intended: Given two positive integers k, l, does there exist a variety 23 generated by k-generator groups and also by a set S of critical groups such that S contains a group G minimally generated by k+l elements and S/{G} does not generate B? The purpose of this note is to record a simple example which shows that the answer to the question of Burns is affirmative at least for k = 2, l = 1, and also that the answer to the second question of Hanna Neumann's Problem 24 is negative.

In [1], the following theorem is proved:THEOREM. If f ∈C2π, α is a positive integer, and then Where , .

We speak of rigidity, if partial information about the prime decomposition in an extension of number fields K¦k determines the decomposition law completely (and hence the zeta function ζK), or even fixes the field K itself. Several concepts of rigidity, depending on the degree of information we start from, are introduced and studied. The strongest concept (absolute rigidity) was only known to hold for the ground field and all quadratic extensions. Here a complete list of all Galois quartic extensions which are absolutely rigid is given. For the weaker concept of rigidity, all rigid situations among the fields of degree up to 8 are determined.

Let G be a compact group. A sequence of functions in L∞ (G) is said to be a Rudin-Shapiro sequence (briefly, an RS-sequence) if the following conditions hold: (1) (2) (3) The main purpose here is to prove the following theorem: Theorem: Theorem. Let G be an infinite compact group. Then G has an RS-sequence consisting of trigonometric polynomials.

Groups, all proper factor-groups of which are hypercentral of finite torsion-free rank, are studied in this article.

We state and verify up to weight 172 a conjecture on the existence of a certain generating set for spaces of classical Siegel modular forms. This conjecture is particularly useful for calculations involving Fourier expansions. Using this generating set, we verify the Böcherer conjecture for nonrational eigenforms and discriminants with class number greater than one. As a further application we verify another conjecture for weights up to 150 and investigate an analog of the Victor–Miller basis. Additionally, we describe some arithmetic properties of the basis we found.

We extend various characterizations of scalar-valued lower semicontinuity and determine their relationship to the continuity of vector-valued convex functions. Order completeness of the range space is not assumed.

Suppose that the finite group G acts faithfully and irreducibly on the finite G-module V of characteristic p not dividing |G|. The well-known k(GV)-problem states that in this situation, if k(G V) is the number of conjugacy classes of the semidirect product GV, then k(GV) ≤ |V|. For p—solvable groups, this is equivalent to Brauer's famous k(B)-problem. In 1996, Robinson and Thompson proved the k(GV) problem for large p. This ultimately led to a complete proof of the k(GV)-problem. In this paper, we present a new proof of the k(G V)-problem for large p.

In this paper we investigate the following problem proposed by Lausch and Nöbauer: Let be a variety of universal algebras, B an algebra of and A a subalgebra of B. If a system of algebraic equations over A is solvable in , is it then solvable over B? We show that the answer is affirmative in certain varieties but negative in the general case.

A class of groups forms a (subnormal) coalition class, or is (subnormally) coalescent, if whenever H and K are subnormal -subgroups of a group G then their join <H, K> is also a subnormal -subgroup of G. Among the known coalition classes are those of finite groups and polycylic groups (Wielandt [15]); groups with maximal condition for subgroups (Baer [1]); finitely generated nilpotent groups (Baer [2]); groups with maximal or minimal condition on subnormal subgroups (Robinson [8], Roseblade [11, 12]); minimax groups (Roseblade, unpublished); and any subjunctive class of finitely generated groups (Roseblade and Stonehewer [13]).