In studying compact convex sets it is usually assumed that the compact convex set X is contained in a Hausdorff topological vector space L where the topology on X is the relative topology. Usually one assumes that L is locally convex. The reason for this is that most of the major theorems such as the Krein-Milman, Choquet-Bishop-de Leeuw, and most of the fixed point theorems require that there be enough continuous affine functions on X to separate points.

This paper answers affirmatively a question of Pavol Hell [2]: if a graph admits a full n-colouring for every finite n ≧ n0, does it admit an infinité full colouring? (A colouring is full if every pair of distinct colour classes is joined by at least one edge).

Let ω be the set of all real sequences a = ﹛an﹜n ≧0. Unless otherwise indicated operations on sequences will be coordinatewise. If any component of a has the entry oo the corresponding component of a-1 has entry zero. The convolution of two sequences s and q is given by s * q. The Toeplitz martix associated with sequence s is the lower triangular matrix defined by tnk = sn-k (n ≧ k), tnk = 0 (n < k). It can be seen that Ts(q) = s * q for each sequence q and that Ts is invertible if and only if s0 ≠ 0. We shall denote a diagonal matrix with diagonal sequence s by Ds.

The concept of an even cover is introduced early in elementary topology courses and is known to be valuable. Among other facts it is known that X is paracompact if and only if every open cover of X is even. In this paper we introduce the concept of an n-even cover and show its usefulness. Using n-even we define an embedding that on closed subsets is equivalent to collectionwise normal. We also give sufficient conditions for a point finite open cover to have a locally finite refinement and also sufficient conditions for this refinement to be even. Finally we show that the collection of all neighborhoods of the diagonal of X is a uniformity if and only if every even cover is normal. This last result is particularly interesting in light of the fact that every normal open cover is even.

In the following R will denote the real numbers, for a Hilbert space H, B(H) and L(H) will denote the collections of bounded linear operators on H and linear, but not necessarily bounded, operators on H respectively. Cosine Operator Functions, namely functions C:R ⟶ B(H) which satisfy D'Alembert's functional equation

(1)

and

(2)

Let ∑ = PG(3, q) denote 3-dimensional projective space over GF(q). A partial spread of ∑ is a collection W of pairwise skew lines in ∑. W is said to be maximal if it is not properly contained in any other partial spread. If every point of ∑ is contained in some line of W, then W is called a spread. Since every spread of PG(3, q) consists of q2 + 1 lines, the deficiency of a partial spread W is defined to be the number d = q2 + 1 — |W|. A maximal partial spread of ∑ which is not a spread is called a maximal strictly partial spread (msp spread) of ∑.

The maximal ideals in a commutative Banach algebra with identity have been elegantly characterized [5; 6] as those subspaces of codimension one which do not contain invertible elements. Also, see [1]. For a function algebra A, a closed separating subalgebra with constants of the algebra of complex-valued continuous functions on the spectrum of A, a compact Hausdorff space, this characterization can be restated: Let F be a linear functional on A with the property:

(*) For each ƒ in A there is a point s, which may depend on f, for which F(f) = f(s).

In [10], Witt laid the foundation for the study of quadratic forms over fields. Suppose Q is a quadratic form defined on a finite dimensional vector space V over a field of characteristic not equal to 2. Witt showed that non-zero vectors x and y in V satisfying Q(x) = Q(y) can be mapped into each other via an isometry of the vector space V. More generally, if τ : W ⟶ W’ is an isometry between subspaces of V, then τ extends to an isometry ϕ of V.

Throughout this paper, it is assumed that A is the complete, equicharacteristic, local ring of an algebraic curve at a one-branch singularity whose residue field is algebraically closed and contained in A. Hence, the domain A is dominated by only one valuation ring in its quotient held F, and if t is a uniformizing parameter, then the integral closure of A in F, denoted by Ā, is [[t]].

Many results concerning the automatic continuity of linear functions intertwining continuous linear operators on Banach spaces have been obtained, chiefly by B. E. Johnson and A. M. Sinclair [1; 2; 3; 5]. The purpose of this paper is essentially to extend this automatic continuity theory to the situation of Fréchet spaces. Our motive is partly to be able to handle the more general situation, since for example, questions about Fréchet spaces and LF spaces arise in connection with the functional calculus.

In the isomorphism theory of Bernoulli shifts, the or Hamming distance plays an important role. Two finite state stationary processes are close in the metric if, on the average, strings from one process can be matched with strings from the other process so that their outputs agree with high frequency. Since such agreement must hold in the arbitrarily distant future, it might at first seem difficult to control the pairing.

In [1], a number of “multilinear” functors Lpq, denned for finitely generated free modules, were introduced. They arose as cycles in a generic Koszul complex, and in turn gave rise to a large family of other generic complexes. One of the things we will do in this paper is study some of these new complexes in order to obtain new multilinear functors on free modules which appear as their cycles.

Main results. Let G be a group generated by two subnormal subgroups H and K. Denoting the class of nilpotent groups by 𝔑, and the limit of the lower central series by G𝔑, Wielandt showed in [14], for groups with a composition series that

(*)

Let D ⊂⊂ Cn be a bounded domain with smooth boundary ∂D, and let F be a bounded holomorphic function on D. A generalization of the classical theorem of Fatou says that the set E of points on ∂D at which F fails to have nontangential limits satisfies the condition σ (E) = 0, where a denotes surface area measure. We show in the present paper that this result remains true when σ is replaced by 1-dimensional Lebesgue measure on certain smooth curves γ in ∂D. The condition that γ must satisfy is that its tangents avoid certain directions.

The classical Pontrjagin-van Kampen structure theorem states that any locally compact abelian (LCA) group G can be written as the direct product of a vector group Rm (where R denotes the additive group of real numbers with the usual topology, and m is a non-negative integer) and an LCA group H which contains a compact open subgroup. This important theorem, which van Kampen deduced from the work of Pontrjagin, was first stated and proved in [5, p. 461].

Let I be an interval, and let for 1 ≦ j ≦ I < ∞ be abutted subintervals such that . Let τj be a linear differential expression defined on Ij. In this paper we study densely defined operators associated with

(0.1)

Volume decreasing properties of harmonic mappings of space forms were investigated by S. S. Chern and S. I. Goldberg [3] and the author. In a previous paper [6], a step toward generalization of the results was made proving the following theorem:

Theorem. Let ƒ: M —> N be a harmonic mapping of n-dimensional Riemannian manifolds, with C ≦ 0. Suppose the scalar curvature of M is not less than — S, and the Ricci curvature of N is not greater than —S/n, where S ≧ 0 and S > 0 are constants. Then, if u has a maximum on M,

i.e. ƒ is volume decreasing up to a constant.

L'objet de cet article est de donner quelques résultats concernant la structure des suites de fonctions mesurables sur un espace mesuré abstrait (X, ∑, μ), le théorème principal étant le suivant:

Theoreme (A). Soit (ƒn) une suite de fonctions mesurables sur un espace mesuré (X, ∑, μ) dont aucune sous-suite ne converge presque partout. Il existe alors un élément Y ∊ ∑, μ (Y) > 0, deux nombres r ∊ R, δ > 0 et une partie infinie M de N tels que, pour tout A ∊ ∑, ⊂ CF, μ(A) > 0 et pour toute partie infinie L CP.S. M (c'est-à-dire L\M est fini), on puisse trouver x et y ∊ A vérifiant ƒn(x) > r + δ et ƒn(y) < r pour une infinité d'indices n ∊ L.

In this article, we examine the relationship between the notions of thick groups and that of essentially finitely indecomposable (e.f.i.) groups. These groups are indecomposable objects in the quotient category of the category of abelian groups induced by the Serre-class of bounded groups.

There have been few published results concerning the relationship between the homology groups of branched and unbranched covering spaces of knots, despite the fact that these invariants are such powerful invariants for distinguishing knot types and have long been recognised as such [8]. It is well known that a simple relationship exists between these homology groups for cyclic covering spaces (see Example 3 in § 3), however for more complicated covering spaces, little has previously been known about the homology group, H1(M) of the branched covering space or about H1(U), U being the corresponding unbranched covering space, or about the relationship between these two groups.