Let F be a linear, but not necessarily closed, subspace of L2[X, dm], where (X,,m) is a σ-finite measure space with the Borel subsets of the locally compact space X. If u and v are measureable functions, then v is called a normalized contraction of u if and Assume that F is stable under normalized contractions, that is, if u ∈ F and v is a normalized contraction of u, then v ∈ F.

The object of this paper is to study some structural aspects of finite T0 + T4 and T0 + T5 spaces in order to establish certain recursion relations that can be used to obtain the number of (labelled as well as unlabelled) T0 + T5 topologies on a finite set.Here, as in [2], a topology 𝒥 is a T4(T5) space provided for any pair of disjoint closed sets A and B (separated sets A and B = A ∩ closure B = B ∩ closure A = 0) there exist disjoint open sets 0A and 0B 𝒥 such that A ⊆ 0A and B ⊆ 0B. An almost immediate consequence of these investigations is that the inherent simplicity of the connected T0 + T5 topologies ensures that they are reconstructable.

In this paper we derive those properties of topologically embedded curves and surfaces in E3 which can be obtained without use of Bing's Side Approximation Theorem [3] for surfaces. The local homology and homotopy properties studied classically play the largest role in the paper, but the final geometrization of some of the results requires theorems such as the PL Schoenflies Theorem, Dehn's Lemma, the Loop Theorem, the Sphere Theorem, and Waldhausen's generalization of the Loop Theorem (n.b., one application of Waldhausen's theorem (in (3.4)) requires use of the nontrivial normal subgroup in the statement of that theorem).

Soient A un anneau unitaire, G un groupe. L'anneau de groupe AG est le A-module libre ayant pour base les éléments de G, la multiplication étant définie par :

avec

Le premier résultat de cet article concerne les annulateurs des idéaux à gauche engendrés par les sous-groupes finis de G, il permet d'obtenir une démonstration facile de la caractérisation des anneaux de groupes parfaits à gauche.

In this paper, we extend the class of spaces to which the Σ and β theorems of Hodel apply, as well as the sum and subset theorems of [2]. Instead of the open cover definition of countable paracompactness, we utilize an equivalent formulation of countable paracompactness, due to Ishikawa [3].

The chromial P(M, λ) of a planar near-triangulation M has the leading term λv(M), where v(M) is the number of vertices of M. The problem of finding the number of rooted planar near-triangulations of a given class S, all supposed to have the same number of vertices, can be regarded as a special case of the problem of finding chromatic sums. We can sum P(M, λ) over the members of S, divide by the appropriate power of λ and let λ → ∞. We thus get the sum of the coefficient of the leading term of P(M, λ) for all M ∈ S, that is we get the number of members of S.

In this paper the character table of GL(3, q) (U(3, q2)), the group of all nonsingular n × n (unitary) matrices over GF(q) (GF(q2)), is used to obtain the character tables for the related subgroups SL(3, q), PSL(3, q) (SU(3, q2), PSU(3, q2)), the corresponding groups of matrices of determinant unity and the projective group respectively. There are very few abstract character tables which hold for entire families of groups. Such tables are of much greater value than tables for specific groups because, among other things, they enable one to discern various patterns common to the whole family.

In this paper, we are concerned with proving a formula for the computation of what is variously called the pattern inventory (e.g., see De Bruijn [2]) or the configuration counting series (e.g., see Harary [3]). Rather than redeveloping a large number of definitions, we shall assume the reader is already familiar with the terminology used by De Bruijn [2].

Polya, in a celebrated paper [4], proved a formula for computing the pattern inventory for all functions f defined on a set D (where D is acted on by a permutation group G), and mapping into a set R (which is called the store) for which the “store enumerator” in known.

Ce travail donne des conditions sous lesquelles un sousmodule pur de M est une sommante directe, non nécessairement de M luimême, mais d'un sous-module “proche” de celui-ci. (M est supposé sans torsion, dans un certain sens.) Ces conditions sont exprimées à partir d'un module H d'endomorphismes du mondule des fractions de M, étroitement lié à la notion d'“endomorphisme partiel” d'un module sans torsion, introduite par Fuchs [3] et que nous avons étudiée dans [5] et [6].

The lattices of all closed subspaces of separable, infinitedimensional Hilbert space (real, complex, and quaternionic) share the following purely lattice-theoretic properties. Each is complete, orthocomplemented, atomistic, irreducible, separable, M-symmetric, and orthomodular [2]. We will call any lattice possessing these seven properties a Hilbert lattice. The general situation which motivates the investigations of this paper concerns infinite-dimensional Hilbert lattices (the dimension of a Hilbert lattice being the cardinality of any maximal family of orthogonal atoms). There are several lattice theoretic properties, possessed by the three canonical lattices, whose only known proofs involve the analytic properties of the underlying Hilbert space, that is, there is no known purely lattice-theoretic proof of these properties.

By a Noetherian (Artinian) ring = (R; + , —, 0, ·) we mean an associative ring satisfying the ascending (descending) chain condition on left ideals. An arbitrary ring is said to be equationally compact if every system of ring polynomial equations with constants in is simultaneously solvable in provided every finite subset is. (The reader is referred to [2; 8; 13; 14] for terminology and relevant results on equational compactness, and to [4] for unreferenced ring-theoretical results.) In this report a characterization of equationally compact Artinian rings is given - roughly speaking, these are the finite direct sums of finite rings and Prüfer groups; as consequences it is shown that an equationally compact ring satisfying both chain conditions is always finite, as is any Artinian ring which is a compact topological ring.

Let R be an associative ring with identity, and let denote the category of unital left R-modules. It is known that if R is a commutative, Noetherian ring, then the maximal torsion radicals of correspond to the minimal prime ideals of R. In fact, Nӑstӑsescu and Popescu [15, Proposition 2.7] have given a more general result valid for arbitrary commutative rings. This paper investigates maximal torsion radicals over rings not necessarily commutative.

Let be a W*-algebra, A ( ) the group of all automorphisms of . In this paper we have determined the commutant G’ of a subgroup G of A () for certain classes of G and . The main results are as follows.

Theorem 1. If G is a locally compact abelian group acting by translation on the W*-algebra L∞(G), then the commutant of a dense subgroup of G is G itself.

Nous reprenons le problème des secrétaires tel que considéré en particulier par Chow, Moriguti, Robbins et Samuels [2]. Nous commençons par dégager une formule sur le rendement de certains temps d'arrêt dans le problème des secrétaires. Nous faisons voir comment une certaine fonction d'une infinité de variables est étroitement reliée aux temps d'arrêt optimaux. Enfin, nous étudions le comportement asymptotique du rendement de certains temps d'arrêt pour le problème des n secrétaires lorsque n tend vers l'infini.

Frobenius [2; 3] introduced the symbols

to specify partitions and the corresponding irreducible representations of the symmetric group Ss.

The general coefficient theorem [2] and the extended general coefficient theorem [3] state that the existence of certain quadratic differentials is a sufficient condition for a function to be a solution of certain extremum problems. The purpose of this paper is to show that in the case of simply connected regions this condition is also necessary.

We shall do this by a variational method of the Schiffer-Golusin-type. The main difficulty is, that the class of admissible functions for the general coefficient theorem is restricted and we must therefore have a method of variation with restrictions.

Let (respectively ) denote the ring of germs at 0 ∈ Rn of all C∞ functions (respectively Cμ functions) from Rn to R. For a given where is the space of all germs of C∞ mappings Rn → Rp, let J(φ) denote the ideal in generated by φ1, … , φp and the Jacobian determinants

where Let

Clearly, is an ideal in and where is the (unique) maximal ideal of .

In a previous paper [1], the authors introduced and studied the notion of special conformally flat spaces and quasi-umbilical hypersurfaces. In that paper, the authors proved that every conformally flat space of codimension one in a Euclidean space is special and, conversely, every special conformally flat space can be isometrically immersed in a Euclidean space as a quasi-umbilical hypersurface.

In the present paper, the authors study the conformally flat spaces of codimension 2 in a Euclidean space. (Manifolds, mappings, functions, etc. are assumed to be sufficiently differentiate and we shall restrict ourselves only to manifolds of dimension n > 3.)

Forelli in [1] has described the isometries of Hp(U) into Hp(U) for p≠2, 0 < p < ∞. We shall extend his methods to characterize the isometries of Hp(Un) onto Hp(Un).

The notation we shall use can be found in Rudin [3].

Let II represent a permutation that induces a map on functions of n complex variables by

Clearly II is an isometry of Hp(Un) onto Hp(Un).

This work grew out of a preliminary announcement (Notices of the Amer. Math. Soc. 18 (1971)). Here we modify the definition of residual finiteness given in [2]. This allows us, first of all, to consider a broader class of rings which are “essentially” residually finite and, secondly, to extend the notion to schemes. We then show that, for various topologies on the category of schemes, our notion of residual finiteness is local so that all relevant questions appear already at the ring level.