Let PR(X) denote the Pixley-Roy topology on the collection of all nonempty, finite subsets of a space X. For each cardinal κ, let κ* be the cardinal κ with the co-finite topology. We use PR(κ*) to obtain a partial solution in ZFC to F. Tall's question whether every normal, locally compact, metacompact space is paracompact [6]. W.S. Watson has answered this question affirmatively assuming V = L[7]. The question also has an affirmative answer if we assume either that the space is perfectly normal [1] or that it is locally connected [4].

A space X is said to be boundedly metacompact (boundedly paracompact) provided that for each open cover of X there is a positive integer n such that has a point finite (locally finite) open refinement of order n. As the main result of this paper, we show every normal, locally compact, boundedly metacompact space is paracompact.

In an earlier paper [5], the author defined a zeta function on the real sphere , whereas in the present paper it is proposed to define one on the unitary sphere where xi's are complex numbers and their complex conjugates. Following E. Cartan, harmonics on the unitary sphere are defined and then a zeta function formed just as in the case of a real sphere. The unitary sphere is seen to behave like an even-dimensional closed manifold, since results similar to the ones proved by the author and A. Pleijel [6] for closed manifolds (of even dimensions) are observed here also.

A Fuchsian group G is a discrete group of fractional linear transforms each of which preserve a disc (or half plane). We consider only groups which preserve the unit disc Δ = {z: |z| < 1} and none of whose transforms, except the identity, fix infinity (any Fuchsian group is conjugate to such a group).

Bernstein's classical integral representation theorem for completely monotonie functions can be proved most elegantly, on a commutative semigroup with identity, by the integral version of the Kreĭn-Milman theorem [2]. The key to this approach is the identification (as exponentials) of the extremal points of the normalized completely monotonie functions. Alternate proofs of this identification are given in § 1. The first (Corollary 1.3) is based on the Kreĭn-Milman theorem and the second (see remarks following Corollary 1.5) is derived from elementary analytic techniques. Other interesting facts about completely monotonie functions are mentioned in passing. For example, we observe that the normalized completely monotonie functions form a simplex (Corollary 1.4). In Corollary 1.6 we note that the product of completely monotonie functions corresponds to the convolution of their representing measures. Thus the normalized completely monotonie functions form an affine semigroup [3].

Throughout this paper L will be an orthomodular lattice and the set of all maximal Boolean subalgebras, also called blocks [4], of L. For every x ∈ L, C(x) will be the set of all elements of L which commute with x. Let n ≧ 1 be a natural number. In this paper we consider the following conditions for L:

An: L has at most n blocks,

Bn: there exists a covering of L by at most n blocks,

Cn: the set ﹛C(x)| x ∈ L﹜ has at most n elements,

Dn: out of any n + 1 elements of L at least two commute.

We also consider quantified versions of these statements, namely the statements A, B, C, D defined by: A ⇔ ∃ nAn, B ⇔ ∃ nBn, C ⇔ ∃ nCn and D ⇔ ∃ nDn.

We prove the consistency of

$$~~\text{add}\left( \mathcal{N} \right)<\operatorname{cov}\left( \mathcal{N} \right)<\mathfrak{p}\text{=}\mathfrak{s}\text{=}\mathfrak{g}< \text{add}\left( \mathcal{M} \right)=\text{cof}\left( \mathcal{M} \right)<\mathfrak{a}=\mathfrak{r}=\text{non}\left( N \right)=\mathfrak{c}$$

$\text{ZFC}$

In recent years considerable attention has been focused on non-linear hyperbolic differential equations with the object of establishing the existence of global solutions. It is our aim here to establish the existence of weak solutions of boundary value problems for non-linear equations of the form

(1-1)

where d is a real constant called the damping coefficient, u(t) is a vector-valued function defined on a subinterval of the real line into a space of complex-valued functions u(x) defined on a bounded domain Ω in the real Euclidean space EN of N dimensions, ut(t) ≡ du(t)/dt, and A(t) is the family of partial differential operators of order 2m (m = 1, 2, …) on Ω given in generalized divergence form by

(1-2)

with

We give sufficient conditions for the simplicity of reduced amalgamated products of C*-algebras. We show that in some situations a minimal projection in a unital C*-algebra A is minimal in a free product A *-cB. We show that in certain situations if a minimal projection in A were minimal in a particular reduced free product of A and B then the reduced free product would be a simple C*-algebra which has finite and infinite projections.

If (X, τ) is a topological space (with topology τ) and A is a subset of X, then the topology τ(A) = {U ⋃ (V ⋂ A)|U, V ∈ τ} is said to be a simple extension of τ. It seems that N. Levine introduced this concept in (4) and he proved, among other results, the following:

(A) If (X, τ) is a regular (completely regular) space and A is a closed subset of X, then (X, τ(A)) is a regular (completely regular) space.

(B) Let (X, τ) be a normal space, and A a closed subset of X. Then (X, τ(A)) is normal if and only if X — A is a normal subspace of (X, τ).

(C) Let (X, τ) be a countably compact (compact or Lindelöf) and A ∉ τ.

The class of finite groups G of the form ABA, where A and B are subgroups of G, is of interest since it includes the finite doubly transitive groups, which admit such a representation with A the subgroup fixing a letter and B of order 2. It is natural to ask for conditions on A and B which will imply the solvability of G. It is known that a group of the form AB is solvable if A and B are nilpotent. However, no such general result can be expected for ABA -groups, since the simple groups PSL(2,2n) admit such a representation with A cyclic of order 2n + 1 and B elementary abelian of order 2n. Thus G need not be solvable even if A and B are abelian.

In (3) Herstein and the author have shown that G is solvable if A and B are cyclic of relatively prime orders; and in (2) we have shown that G is solvable if A and B are cyclic and A possesses a normal complement in G.

A finite p-group G is said to be regular if for every pair of elements a, b in G,

1

where c is an element of the derived group of the subgroup generated by a and b. We shall say that G is nearly regular if (i) there is a central element z of order p such that G/⧼z⧽ is regular, and (ii) the subgroup generated by x and y is regular whenever x ∈ G and y ∈ G′, the derived group of G.

SL(2,C) is the group of all complex unimodular 2 × 2 matrices. A subgroup of SL(2, C) is said to be discrete if it does not contain any convergent sequence of distinct elements. A subgroup is said to be elementary if the commutator of any two elements of infinite order has trace 2. The discreteness condition which this note relates to is the following:

PROPOSITION 1. If two complex, unimodular 2 × 2 matrices X and Y generatea non-elementary, discrete group, then

We study Dupin hypersurfaces in ${{\mathbb{R}}^{5}}$ parametrized by lines of curvature, with four distinct principal curvatures. We characterize locally a generic family of such hypersurfaces in terms of the principal curvatures and four vector valued functions of one variable. We show that these vector valued functions are invariant by inversions and homotheties.

A classical theorem of n-dimensional projective geometry asserts: if Desargues' theorem holds (in particular, if n ≥ 3) then the points of the geometry can be assigned homogeneous coordinates (α1, … , αn+l) with αi not all zero in a suitable division ring (2, p. 104; 3, Theorems 10, 11, ex. 19, p. 204).

In (6)Scholz asked if the inequality

1.1

held for all positive integers q, where l(n)is the number of multiplications required to raise xto the nth power (a precise definition of l(n)in terms of addition chains is given in § 2). Soon afterwards, Brauer (2) showed, among other things, that l(n) ∼(log n)/(log2). This suggests the problem of Calculating

1.2

It can be deduced from (2) that θ≦ 1. If θ <1, (1.1) follows immediately for infinitely many q.My main result,Theorem 5 of § 4, merely shows that θ is slightly larger than ⅓.Actually, I know of no case where (1.1) is not in fact an equality; a tedious calculation verifies this for 1 ≦ q≦ 8.

We prove a new bound on collinear triples in subgroups of prime finite fields and use it to give some new bounds on exponential sums with trinomials.

1. Introduction. The following remarks concern two different parts of the theory of extreme quadratic forms. In §2, I shall give a new proof for the theorem of Voronoï (4), which asserts that a form is extreme if and only if it is both perfect and eutactic. (For the definitions see, e.g., Coxeter (2) or the text below.) There is indeed a comparatively simple proof in Bachmann's Zahlentheorie, IV, 2. For two reasons, however, it may not be useless to communicate another proof here.

In this note we discuss some properties of well distributed sequences. We take 0 ⩽ a < b ⩽ 1 and let I(x) denote the characteristic function of the interval [a, b], so that .

Sei ein (nicht notwendig kommutativer) Körper, sei ein (ebenfalls nicht notwendig kommutativer) echter Unterkörper von . Mit bezeichnen wir die projektive Gerade über , d.h. die Menge , wo ∞ ein neues (in die Betrachtung hereinkommendes) Element, genannt “unendlich”, darstellt. Ist entsprechend , so ist also (dasZeichen ⊂ bezeichne echtes Enthaltensein). Die Elemente von nennen wir Punkte, die Punktmengen die projektive Gruppe von (s. § 1) darstellt, nennen wir Ketten. Eine eineindeutige Abbildung der Menge der Punkte auf sich, die in beiden Richtungen Ketten in Ketten überführt, heiße eine Kettenverwandtschaft.Diese Forderung, daß nämlich Bilder und Urbilder von Ketten wieder Ketten sind, kann (s. § 4) (wenn im Zentrum von liegt) reduziert werden so, daß nur Ketten in Ketten übergehen sollen. Dann liegt nämlich bereits eine Kettenverwandtschaft vor.

We shall say that the sequence x = {sk} is summable- (Z, p) to the value s if

where p is a positive integer and

Ignoring the values of n, 1 — p ≤ n < 0, which are clearly irrelevant, the transformation (Z, p) coincides with the regular Nörlund transformation defined by the sequence ( 1 , 1 , . . . , 1, 0, 0 , . . .) containing p initial 1's. This class of methods was first studied systematically by Silverman and Szász (8).