Let X be a topological space, a finite covering of X (the words ‘covering’ and ‘cover’ are used interchangeably). We say that has the almost fixed point property for a class of continuous maps f : X → X if for all there is an x ∈ X and such that x ∈ U and f(x) ∈ U, or, equivalently, if there is a such that .

A *-ring is an associative ring R with an anti-automorphism * of period 2 (involution). Call x ∈ R skew (symmetric) if x = - x* (x = x*) and let K(S) be the additive subgroup of all skews (symmetries). If [a, b] denotes the Lie product of a, b ∈ R (that is, ab — ba) and if [A, B] denotes the Lie product of the additive subgroups A and B of R (that is, the additive subgroup generated by [a, b], a and b ranging over A and B) then clearly [K, K] is an additive subgroup contained in K.

Soit Dn = ﹛z = (z1, . . . , zn) ∈ Cn, |zi| <> 1﹜ le polydisque de Cn et ƛn la mesure de Lebesgue de Cn normalisée sur Dn. Pour b > 0, .on définit les espaces de Bergman Ap(ƛn) de la manière suivante:

Ap(ƛn) est l'espace des fonctions analytiques dans Dn telles que:

Several authors [2; 3; 10; 12] have noticed the similarities between the theory of codes and the theory of Euclidean lattices. It is interesting to compare the two theories since they share a common problem, viz. the sphere packing problem. In the theory of codes one would like to find a code over Fp, i.e. a subspace of Fpn, such that non-intersecting spheres with respect to a given metric, centered at the code vectors, pack Fpn densely.

The Orlicz-Pettis type theorems are essentially statements about the equivalence of strong and weak subseries convergence of series in topological groups. The corresponding problem of the equivalence of subfamily summability for summable families was examined for instance in [2] and (16). We give here another uncountable generalization of the Orlicz-Pettis theorem in the sense that we assume less than subfamily summability and obtain straightforward generalizations of the classical case, which include or imply in a trivial manner—at least for polar topologies on groups—all known theorems “from weak to strong σ-additivity”.

In this paper we will be concerned with the determinants of matrices whose elements are 0, 1 or —1, 1. Accordingly, let Sn,k be the set of n X n 0, 1 matrices with exactly k ones in each row and column; and let Hn be the set of n X n — 1, 1 matrices. Let J = Jn denote (as usual) the n X n matrix all of whose elements are one.

The idea of one surface covering another is a useful one in the study of regular maps. Coxeter and Moser discuss a particular instance [3, p. 115] and the maps produced by Sherk [6] and Garbe [4] are formed by such coverings, though neither paper mentions that fact. In [5], Sherk explicitly constructs coverings of regular maps on the sphere.

It is known that the Pn-tegral as originally defined is not additive on abutting intervals. This paper offers a slight modification in the definition of the integral and develops necessary and sufficient conditions for the integral to be additive.

In an effort to unify the arithmetic of cardinal and ordinal numbers, Garrett Birkhoff [2; 3; 4; 5] (cf. [6]) defined several operations on partially ordered sets of which at least one, (cardinal) exponentiation, is of considerable independent interest: for partially ordered sets P and Q let PQ denote the set of all order-preserving maps of Q to P partially ordered by f ≦ g if and only if f(x) ≦ g(x) for each x ∈ Q.

During the early nineteenth century much effort was spent on attempts to find a “universal comparison test“: i.e., a sequence in l1 that dominates every other member of l1. The nonexistence of such a series converging at a minimal rate was demonstrated by Abel, et al. [1; 4; 7; 9, pp. 298-304].

Let Λ be an artin algebra, that is, an artin ring that is a finitely generated module over its center C which is also an artin ring. We denote by mod Λ the category of finitely generated left Λ-modules. We recall that the category of finitely generated modules modulo projectives is the category given by the following data: the objects are the finitely generated Λ-modules.

Throughout this paper the word “Artinian (Noetherian) ring” means an associative ring with minimum (maximum) condition on left ideals. According to C. Hopkins, an Artinian ring is Noetherian if it contains a left or right identity [3, p. 728]. However we shall consider Artinian rings without the assumption of existence of such an identity, and the theorem of Hopkins will be reproved.

Let the group be presented on three generators a, b, c with three defining relations

where p, q, r, s, t, u are integers, the “parameters” of the presentation.

Let p be a prime. The algebraic K-theory of Z/p2 is unknown. However it is easy to show that Ki(Z/p2) is finite if i > 0 and that it differs only in its K-torsion from Ki(Z/p) which was computed in [2]. To proceed further one surely needs the mod p (co-)homology of GLZ/p2. There is an exact sequence

Blocking sets in projective spaces have been of interest for quite some time, having applications to game theory (see [6; 7]) as well as finite nets and partial spreads (see [5]). In [4] Bruen showed that if B is a blocking set in a projective plane of order n, then

Let Ω C Cn be a bounded domain; let H∞ (Ω) be the uniform algebra of bounded analytic functions on 12; and let ∑ (Ω) be the maximal ideal space of H∞ (Ω). In the weak-* topology of (H∞ (Ω))*, ∑ (Ω) is a compact Hausdorf space in which Ω is embedded in a natural fashion, so that to every g ∈ H∞ (Ω) there corresponds the Gelfand transform ĝ ∈ C(∑ (Ω)); ĝ|Ω = g.

This paper originated in an attempt to carry over the results of [3] from the case of a field of characteristic different from two to that of semilocal rings. To carry this out, we reverse the point of view of [3] and do assume a full knowledge of the theory of Witt rings of classes of nondegenerate symmetric bilinear forms over semilocal rings as given, for example, in [10; 11]. It turns out that the rings WT of [3] are just the residue class rings of W(C), the Witt ring of a semilocal ring C, modulo certain intersections of prime ideals.