The existence and uniqueness of strong global solutions of initial-boundary value problems for the quasilinear equation utt—∂σi(uxi)/∂xi—ΔNut= f is established for functions σi(ξ), i = 1, …, N, satisfying: σi,(ξ) ∊ C1(-∞, ∞), σi(0) = 0 and for some constant K0.

In the definition of the Pn-integral [2] there is a difficulty with the condition Bn-2 ([2], p. 150) since it is not linear on the set of major and minor functions. As a result, the proof of Lemma 5.1 [2] fails since the difference Q(x)—q(x) need not satisfy the conditions of Theorem 4.2, [2].

The natural modes of a small planar transversal vibration of a fixed string of unit length and tension are determined by the eigenvalues and associated eigenfunctions of the differential equation

(1)

subject to the boundary condition

(2)

where the non-negative function p describes the mass distribution of the string. That the distribution of mass on the string influences the modes of vibration, may be reflected by observing that the eigenvalues determined by the system (1–2) may be considered functions of the density p, λn(p), where λ1(p)<λ2(p)<….

Any term or symbol undefined in this paper is defined in [5]. For graphs F and G, G > F means G contains a subgraph isomorphic to F and E(G) denotes the edge set of G. If E ⊆ E(G), 〈E〉 is the subgraph of G whose edge set is E and whose vertex set is that subset of vertices of G which are incident with edges in E.

This note answers affirmatively a question of the author [4, p. 456], by producing an example of an integrally closed quasi-local non valuation domain of global dimension 3, each of whose overrings is a goingdown ring. Although [4, Proposition 4.5] shows that such an example cannot be constructed by means of restrained power series, an approach via the more general D+M construction succeeds.

Let Un denote the nth ménage number. Within the class of order n matrices of zeros and ones with at most two zeros in every row and column the minimum permanent is Un when n is even and–1+Un when n is odd.

The purpose of this paper is to show that by a reorganization of the proofs of the main results concerning Radon-Nikodym derivatives in a von Neumann algebra of Pedersen and Takesaki in [12] and of Connes in paragraphs 1.1 and 1.2 of [5], considerable technical simplification can be achieved. Roughly speaking, the analytic vector techniques developed by these authors for the study of weights on a von Neumann algebra can be replaced, to a large extent, by the tensor product methods introduced by Connes, which are essentially algebraic in nature. In the exposition which follows, analytic vectors are not used at all (see, however, 4.4).

Consider approximation of elements of C[0, 1] with respect to the sup-norm by a non-empty subset V of C[0, 1]. Of interest in recent years are subsets V called suns. As C[0, 1] is an MS-space [1, 5], the suns V of C[0, 1] are precisely those subsets V for which each local best approximation is a global best approximation.

Let G be a finite group of automorphisms of the ring R, and let RG denote the ring of fixed points of G in R; that is, RG={x∊R|Xg = x,∀∊ G}. Let |G| denote the order of G. In this note, we prove the following:

Theorem.Assume that R has no nilpotent ideals and no |G|-torsion. Then if RG is semi-simple Artinian, R is semi-simple Artinian.

Limit vector spaces ('limitierte Vektorräume') were defined by Fischer [1] and concepts such as continuity, compactness, etc. were introduced and studied by him and others, e.g. by Kent [3]. In this paper the concepts of precompactness and boundedness in limit vector spaces are studied. It is shown that most of their properties in topological vector spaces hold in limit vector spaces.

The following theorem is a well-known tool in the study of measurable functions:

Theorem. Let (M; μ) be a finite measure space and let (xn) be a sequence of functions in L1(M; μ) so that xn→0 in the norm of L1(M; μ). Then there is a sub-sequence so that xnk →0 pointwise almost everywhere on M.

Consider a random power series Σ0∞ cn zn, that is, with coefficients {cn}0∞ chosen independently at random from the complex plane. What is the radius of convergence of such a series likely to be?

One approach to this question is to let the {cn}0∞ be independent random variables on some probability space. It turns out that, with probability one, the radius of convergence is constant. Moreover, if the cn are symmetric and have the same distribution, then the circle of convergence is almost surely a natural boundary for the analytic function given by the power series (See [1, Ch. IV, Section 3]). Our treatment of the question will be elementary and will not use these facts.

Il est bien connu que l'ensemble des fonctions de Baire de ℝ dans ℝ est égal à l'ensemble des fonctions boréliennes de ℝ dans ℝ. Dans cet article, nous généraliserons ce théorème de deux façons: premièrement, en étudiant plutöt des fonctions de X à Y, deux espaces topoiogiques normaux; et deuxièmement, en utilisant les suites généralisées de cardinalité donnée.

Let R be a ring with identity not equal to zero. A right R-module is said to be quasi-injective (pseudo-injective) if for every submodule N of M, every R-homomorphism (R-monomorphism) of N into M can be extended to an R-endomorphism of M [7] ([13]).

Some extensions of the open mapping and closed graph theorem are proved for certain classes of commutative topological semigroups, namely those embeddable as open subsets of topological groups. Preliminary results of independent interest include investigations of properties which “lift” from embeddable semigroups to the groups in which they are embedded, and from semigroup homomorphisms to homomorphisms of the groups.

The Ultrapower construction, which builds a new structure AI/D from a relational structure A and an ultrafilter D on a set I, is by now a familiar tool in Model Theory and many other branches of mathematics. In this note we present a result that belongs in the theory of ordered sets, i.e. where the relational structure A has just a single binary relation <, which satisfies the axioms for a strict linear order. We assume that the reader is familiar with the definition and standard notation for ultrapowers, as may be found, for example, in [1]. We differ from [1] only in our eschewing of Gothic capitals.

Let X be a metrizable topological space and f:X→X a continuous selfmapping such that for every x ∈ X the sequence of iterates {fn(x)} converges. It is proved that under these conditions the following two statements are equivalent:

1. There is a metrization of X relative to which f is contractive in the sense of Edelstein.

2. For any nonempty f-invariant compact subset Y of X the intersection of all iterates fn(Y) is a one-point set. The relation between this type of contractivity and the Banach contraction principle is also discussed.

In previous papers (see [1, 2, 3, 4]), we solved the following functional equation:

1

wheref=f(z) is an entire function of a complex variable z and x, y are complex variables.

It is well-known that odd dimensional spheres carry a normal contact structure [6], Blair, Ludden and Yano [2] have recently studied a more general structure whose non-trivial example is an even dimensional sphere. Recently, the present author introduced the notion of almost contingent structures [8] with a view to develop a unified theory of various existing structures on a differentiable manifold. It is the purpose of this paper to show that even as well as odd dimensional spheres carry an almost contingent structure. In the sequel, each manifold introduced is C∞, arcwise connected and satisfies the second axiom of countability.

It is shown that the operator of integration Fy(x) = ∫x0y(t) dt defined on the space C(—∞, ∞) of all continuous real valued functions on (—∞, ∞) is a contraction relative to a certain family of seminorms generating the topology of uniform convergence on compacta. However, as a contrast to this it is proved that F is not contractive with respect to any metric on C(—∞, ∞) inducing the above topology on C(—∞, ∞).