A mapping ϕ from one group, G, into another, H, is said to be a semi-homomorphism of G if ϕ(aba) = ϕ(a) ϕ(b) ϕa) for all a, b ∊G. Clearly any homomorphism or anti-homomorphism is a semi-homomorphism; the converse, however, need not be true in general. It is perfectly clear what one intends by a semi-isomorphism or semi-automorphism.

Our purpose here is to show that for a rather general situation a semi-homomorphism turns out to be a homomorphism or an anti-homomorphism. In (2) we proved that any semi-automorphism of a simple group which contains an element of order 4 must automatically be either an automorphism or an anti-automorphism.

In this paper, we construct two kinds of Boolean algebras with extreme cellular properties and nice embedding properties. The extreme cellular properties are a σ — j-linked but not σ — j + 1-linked and ccc but not σ — 2-linked. The nice embedding properties are that they are ZF-definable subalgebras of both P/F and R (see Preliminaries for notation). It is the author's opinion that R contains much of the “ZF-strength” of P/F.

The Gibbs phenomenon may be described, quite generally, as follows. Let a sequence {fn(x)} (n = 0, 1, 2, … ,) converge to a function f(x) for x in the interval x0 < x < x0+ h. We say that {fn(x)} displays the Gibbs phenomenon in a right-hand neighbourhood of the point X0, if

A similar definition holds for a left-hand neighbourhood. If {fn(x)} displays the Gibbs phenomenon at both sides of x0, we say simply that {fn(x)} displays Gibbs phenomenon at the point X0.

In this paper all rings are commutative and have a unity. All ring homomorphisms preserve the unity. We let L denote the standard language for rings with two distinct constants, 0 and 1, playing the role of the zero and the unity respectively. A ring is regular if it satisfies the axiom (∀r) (∃r′)(rr′r = r) and it is algebraically closed if, for each integer n ≧ 1, it satisfies the sentence

Given a stochastic process adapted to an increasing family of right-continuous σ-algebras, it is often useful for many purposes to enlarge the a-algebras. In the present paper we shall consider enlargements which involve embedding the process in a larger probability space. The first question investigated is what kinds of enlargements it might be useful to consider. To study stopping times, the least requirement needed to have a complete theory is that convergent sequences of stopping times converge to a function which is also a stopping time, and for this it is necessary to make the enlargement right continuous.

This paper is concerned with the presentation of certain elements of the group SL(n, K) as products of a minimal number of transvections. To explain the terminology, let V be an n-dimensional left vector space over a (not necessarily commutative) field K. The group of all non-singular linear transformations of V onto V (i.e. the group of all collineations of V) is the group GL(n, K). This group is generated by collineations leaving a hyperplane pointwise fixed. When n = 2 these collineations are called axial collineations and the invariant hyperplane (line) is then called an axis.

Existence of solutions to the nonlinear boundary value problem on the semi-infinite interval bounded on [0, ∞), are established. In the process we obtain new existence results for boundary value problems on compact intervals.

In [1; 2 ; 7] Gabriel, Goldman, and Silver have introduced the notion of a localization of a ring which generalizes the usual notion of a localization of a commutative ring at a prime. These rings may not be local in the sense of having a unique maximal ideal. If we are to obtain information about a ring R from one of its localizations, Qτ (R) say, it seems reasonable that Qτ(R) be a tractable ring. This, of course, is what Goldie, Jans, and Vinsonhaler [4; 3; 8] did in the special case for Q(R) the classical ring of quotients.

1. Introduction. A dissection of a rectangle R into a finite number n of non-overlapping squares is called a squaring of R of order n. The n squares are called the elements of the dissection. If there is more than one element and the elements are all unequal the squaring is called perfect and R is a perfect rectangle. (We use R to denote both a rectangle and a particular squaring of it). If a squared (perfect) rectangle is a square we call it a squared (perfect) square.

An explicit description of a hyperbolic canonical coordinate system for an expansive automorphism of a compact connected abelian group is given. These dynamical systems are factors of subshifts of finite type. Some properties of the associated crossed product C*-algebra are discussed. In these examples, the C* -algebras of Ruelle are crossed product algebras.

We prove many simultaneous congruences mod 2 for elliptic and Hilbert modular forms among forms with different Atkin–Lehner eigenvalues. The proofs involve the notion of quaternionic $S$-ideal classes and the distribution of Atkin–Lehner signs among newforms.

Nous traitons de la régularité d'une fonction holomorphe sur un ouvert de Cn à partir de celle de la fonction ou de la partie réelle connue sur une sous-variété totalement réelle maximale (de dimension n) incluse dans le bord de l'ouvert de définition.

Un théorème classique de Privalov assure qu'une fonction continue sur un disque fermé de C, holomorphe sur le disque ouvert, de partie réelle lipschitzienne d'ordre α sur le bord du disque vérifie une condition de même ordre α sur tout le disque. De façon analogue, le module de continuité d'une fonction holomorphe sur un ouvert borné de Cn, continue sur l'adhérence est contrôlé par celui de la fonction restreinte au bord [2] et [5], Le bord de l'ouvert peut être remplacé par la frontière de Shilov dans le cas d'un polyèdre analytique [4].

E. L. Stout [9] étudie la régularité sur une sous-variété totalement réelle maximale M de classe Cp, d'une fonction holomorphe f définie sur un ouvert Ω de classe Cq dont le bord contient M et telle que M soit l'ensemble sur lequel |f| est maximum; il démontre que f|M est Cp−0 lorsque q > p et Cp−1 lorsque q = p pour p supérieur ou égal à 3.

For an algebraic number field K there is a similarity between the additive characters defined on the Witt ring W(K), [20], [11], [17], [14, p. 131], and the local root numbers associated to a real orthogonal representation of the absolute Galois group of K, [18], [5]. Using results of Deligne and of Serre, [16], we shall derive in (5.3) a formula expressing the value, at a prime in K, of the additive character on a Witt class in terms of the rank modulo 2, the stable Hasse-Witt invariant and the local root number associated to the real quadratic character defined by the square class of the discriminant. Thus we are able to separate out the contributions made to the value of the additive character by each of the standard Witt class invariants.

Blow up a sphere in one of the interstices of a lattice until it is held rigidly. There will be no lattice points in the interior and sufficiently many on the boundary so that their convex hull is a solid figure. Such a sphere was called an empty sphere by B. N. Delone in 1924 when he introduced his method for lattice coverings [3, 4]. The circumscribed polytope is called an L-polytope. Our interest in such matters stems from the following result [6, Theorems 2.1 and 2.3]: With a list of the L-polytopes for lattices of dimension ≦n one can give a geometrical description of the possible sets of integer solutions of

where f satisfies the following condition (in which Z denotes the integers):

The multiplicity conjecture of Herzog, Huneke, and Srinivasan is verified for the face rings of the following classes of simplicial complexes: matroid complexes, complexes of dimension one and two, and Gorenstein complexes of dimension at most four. The lower bound part of this conjecture is also established for the face rings of all doubly Cohen–Macaulay complexes whose 1-skeleton's connectivity does not exceed the codimension plus one as well as for all $\text{(}d-1)$-dimensional $d$-Cohen–Macaulay complexes. The main ingredient of the proofs is a new interpretation of the minimal shifts in the resolution of the face ring $\mathbf{k}[\Delta ]$ via the Cohen–Macaulay connectivity of the skeletons of $\Delta $.

Let R be a commutative ring, G a finite abelian group of order n and exponent m, and assume n is a unit in R. In [10], F. W. Long defined a generalized Brauer group, BD(R, G), of algebras with a G-action and G-grading, whose elements are equivalence classes of G-Azumaya algebras. In this paper we investigate the automorphisms of a G-Azumaya algebra A and prove that if Picm(R) is trivial, then these automorphisms are all, in some sense, inner.

In fact, each of these “inner” automorphisms can be written as the composition of an inner automorphism in the usual sense and a “linear“ automorphism, i.e., an automorphism of the type

with r(σ) a unit in R. We then use these results to show that the group of gradings of the centre of a G-Azumaya algebra A is a direct summand of G, and thus if G is cyclic of order pr, A is the (smash) product of a commutative and a central G-Azumaya algebra.

L. Fuchs has posed the problem of identifying those abelian groups that can serve as the additive structure of an injective module over some ring [1, p. 179], and in particular of identifying those abelian groups which are injective as modules over their endomorphism rings [1, p. 112]. Richman and Walker have recently answered the latter question, generalized in a non-trivial way [7], and have shown that the groups in question are of a rather restricted structure.

In this paper we consider abelian groups which are quasi-injective over their endomorphism rings. We show that divisible groups are quasi-injective as are direct sums of cyclic p-groups. Quasi-injectivity of certain direct sums (products) is characterized in terms of the summands (factors). In general it seems that the answer to the question of whether or not a group G is quasinjective over its endomorphism ring E depends on how big HomE(H, G) is, with H a fully invariant subgroup of G.

In this paper, we shall construct a vector space, called the (G, σ) space, which generalizes the tensor space, the Grassman space, and the symmetric space. Then we shall determine a necessary and sufficient condition that the (G, σ) product of the vectors x1, x2, …, xn is zero.

1. Let G be a permutation group on I = {1, 2, …, n} and F, an arbitrary field. Let σ be a linear character of G, i.e., σ is a homomorphism of G into the multiplicative group F* of F.

For each i ∈ I, let Vi be a finite-dimensional vector space over F. Consider the Cartesian product W = V1 × V2 × … × Vn.

1.1. Definition. W is called a G-set if and only if Vi = Vg(i) for all i ∊ I, and for all g ∊ G.

Let $\psi$ be a generic Drinfeld module of rank $r\,\ge \,2$ . We study the first elementary divisor ${{d}_{1,\,\wp }}\,\left( \psi\right)$ of the reduction of $\psi$ modulo a prime $\wp $ , as $\wp $ varies. In particular, we prove the existence of the density of the primes $\wp $ for which ${{d}_{1,\,\wp }}\,\left( \psi\right)$ is fixed. For $r\,=\,2$ , we also study the second elementary divisor (the exponent) of the reduction of $\psi$ modulo $\wp $ and prove that, on average, it has a large norm. Our work is motivated by J.-P. Serre's study of an elliptic curve analogue of Artin's Primitive Root Conjecture, and, moreover, by refinements to Serre's study developed by the first author and M. R. Murty.

A matrix H of order n = 4t with all entries from the set ﹛1, —1﹜ is Hadamard if HHt = 4tI. The set of Hadamard matrices is . A matrix is of type I or is skew-Hadamard if H = S — I where St = —S (some authors also use H = S + I). The set of type I members is . A matrix P is a signed permutation matrix if each row and each column has exactly one non-zero entry, and that entry is from the set ﹛1, —1﹜.