Given a class of topological spaces and a class of mappings of topological spaces, the -résolvant of is denned to be the class of topological spaces all of whose -images lie in . Whenever is closed under composition and includes identity maps, is easily seen to be the largest class of spaces smaller than which is closed under -images.

Let F be a field, Mn(F) be the vector space of all w-square matrices with entries in F and a subset of Mn(F). It is of interest to determine the structure of linear maps T : Mn(F) →Mn(F) such that . For example: Let be GL(n, C), the group of all nonsingular n X n matrices over C [5]; the subset of all rank 1 matrices in MmXn(F) [4] (MmXn(F) is the vector space of all m X n matrices over F) ; the unitary group [2] ; or the set of all matrices X in Mn(F) such that det(X) = 0 [1]. Other results in this direction can be found in [3].

In this paper we generalize a theorem of Lyndon's [7], which states that a one-relator group G = F/(r) (F is free and r Ç F) has cohomological dimension cd (F/(r)) ≧ 2 if and only if the relator r is not a proper power in F. His proof relies on the Identity Theorem and recently he has shown [8] how a generalized version of this theorem and a generalized version of the Freiheitsatz can be simultaneously obtained by the methods of combinatorial geometry. These generalizations refer to a situation where the free group F is replaced by a free product of subgroups of the additive group of real numbers.

The Picard group Γ = PSL2 (Z(i)) is the group of linear transformations

with a, b, c, d Gaussian integers.

Γ is of interest both as an abstract group and in automorphic function theory [10]. In [10] Waldinger constructed a subgroup H of finite index which is a generalized free product, while in [1] Fine showed that T is a semidirect product with the subgroup H, contained as a subgroup of finite index in the normal factor.

In this paper we develop a theory of maximal and minimal operators and their duals associated with the system

Let M be a complex manifold of dimension n which admits a Kähler metric, and let D be a relatively compact domain on M whose boundary B is a C ∞ submanifold of M of real codimension one. The object of this paper is to use the potential theory associated with the Laplace-Beltrami operator on M to characterize the continuous functions on B which have holomorphic extensions to D.

The genus parameter for graphs has been studied extensively in recent years, with impetus given primarily by the Ringel-Youngs solution to the Heawood Map-coloring Problem [15]. This solution involved the determination of 𝛄(Kn), the genus of the complete graph Kn.

Beginning with Wall's introduction [19] of Z2-graded central simple algebras over a field K, a number of related generalizations of the Brauer group have been proposed. In [16] the field K was replaced by a commutative ring R, building upon the theory developed in [1]. The concept of a G-graded central simple K-algebra (G an abelian group) was first defined in [12]; this work and that of [16] was subsequently unified in [6] and [7] via the construction and computation of the graded Brauer group Bφ﹛R, G) (φ a bilinear form from G X G to U(R), the units of R).

In this paper we investigate the question of which continuous ﹛0, 1﹜-valued functions on a product space admit continuous extensions to where βXα is the Stone-Čech compactification of Xa and ﹛0, 1﹜ denotes the two point discrete space. This problem is clearly equivalent to determining which clopen subsets of have clopen closures in .

Brauer [1, p. 141] has discussed the question of which subgroups of a group can be determined from its character table. He mentions, referring to p-groups, that the Frattini subgroup can be determined. We show that for an arbitrary finite solvable group, the Frattini subgroup can be determined from the character table. Then we exhibit an infinite set of pairs of non-solvable groups such that both members of a given pair have the same character table but Frattini subgroups of different orders. All groups to be considered are finite.

In this paper, we will prove, as a consequence of the main theorem,

THEOREM A. (See Corollary 2.6). The group of an alternating knot, for which the leading coefficient of the knot polynomial is a prime power, is residually finite and solvable.

A C1-mapping ƒ from the oriented circle S1 into the oriented plane R2 such that f f’ (t) ≠ 0 for all t is called a regular immersion. We call a point p in Im f a double point if f-1(p) is a two element set with the corresponding tangent vectors being linearly independent. A regular immersion which is one-to-one except at a finite number of points whose images are double points is called a normal immersion. The work of Whitney [7], Titus [3] and Verhey [6] shows that the normal immersions form a dense open subset in the space of regular immersions with the usual C1-topology, and can be characterized up to diffeomorphic equivalence by a combinatorial invariant called the intersection sequence.

Throughout this paper, the Alexander-Spanier cohomology with compact supports will be used. Suppose X is a compact connected topological ra-manifold which admits an effective action of a compact connected Lie group G (m ≧ 19).

To construct a well behaved localization of a noetherian ring R at a semiprime ideal S, it seems necessary to assume that the set (S) of modulo S regular elements satisfies the Ore condition ; and it is convenient to require the Artin Rees property for the Jacobson radical of the quotient ring Rs in addition: one calls such 5 classical. To determine the classical semiprime ideals is no easy matter; it happens frequently that a prime ideal fails to be classical itself, but is minimal over a suitable classical semiprime ideal.

We investigate the structure of those non-separable metric spaces X, and their Stone-Čech compactifications, for which X has no closed discrete subspace of power equal to the weight of X. (Throughout this paper we denote the weight of X—the smallest power of a base for the topology of X—by the symbol wX.)

In his simple proof of the unsolvability of the word problem for groups [1], J. L. Britton proved a normal form theorem (Britton's Lemma) for groups obtained by the HNN construction. In the appendix to that paper he described a generalization of the HNN construction and sketched the proof of a generalization of Britton's Lemma for this new construction, Britton's Theorem A. In this note we demonstrate that all groups obtained by means of this generalized construction are in fact HNN groups; and it will follow that Theorem A is simply a restatement of Britton's Lemma. This argument makes it clear that while Theorem A can be (and has been) useful in various group theoretic situations, in practice, every application of this generalized construction can be replaced by a straightforward application of the HNN construction.

In this paper we introduce the notion of an almost Riemannian manifold. Briefly speaking, an almost-Riemannian structure on a Banach manifold is a generalization of the notion of a Riemannian structure on a Hilbert manifold, common examples would be manifolds of maps modelled on the Sobolev spaces Lkp. The successful use of weak Riemannian structures in hard problems has been given by Ebin [2] and Ebin and Marsden [10].

Nous allons montrer que tout système faible de Tchebycheff est voisin d'un système de Tchebycheff. Ce fait permet de prolonger aux systèmes faibles de Tchebycheff plusieurs résultats valides pour les systèmes de Tchebycheff, tels le théorème d'alternance et le théorème de de la Vallée- Poussin qui sont reliés au problème de la meilleure approximation uniforme par une fonction d'un système de Tchebycheff. Il arrive que certaines classes de fonctions splines forment un système faible de Tchebycheff. Pour ces classes, on pourra donc caractériser les meilleures approximations par des propriétés d'alternance de l'écart à la fonction à approcher.

Let G be a n affine algebraic group over the algebraically closed field k, and let V be an affine, normal algebraic variety over k on which G acts. Suppose that the ring of invariants k [F]G is finitely generated over k, and let W be the affine variety with k[W] = k[V]G. The purpose of this paper is to show that the induced homomorphism from the étale fundamental group of V to that of W is surjective, and to examine the consequences of this observation in terms of the relations between the Picard and divisor class groups of k[V] and k[W],

A method for constructing the product of two Schur functions was stated, but not proved in the most general case, by Littlewood and Richardson [1] in 1934. This method, which came to be known as the Littlewood-Richardson rule, was later proved completely by Robinson [2] in 1938. In this proof, Robinson describes an operation on a finite sequence of positive integers. It is this operation, set in a more general context, that is the subject of this paper.