K. Kunugi introduced the notion of ranked space as a generalization of that of metric spaces, (see [6]). In this note we define a metrizability of ranked spaces and study conditions under which a ranked space is metrizable.

Let the maximum degree d of a digraph D be the maximum of the set of all outdegrees and indegrees of the points of D. We prove that every digraph D of order P and maximum degree d has a d-regular superdigraph H with at most d + 1 more points, and that this bound, which is independent of p, is best possible.

In the paper "Finite complexes and integral representations" [Illinois Journal of Math, 26, (1982), p 442] an exact sequence relating homotopy types of (G, d)-complexes with objects of integral representation theory together with some known calculations seemed to imply that the group of homotopy types of (G, d)- complexes was always a subquotient of (ℤ|g|)*. This paper gives a new characterization of one of the terms of the above sequence that allows one to conclude that this is not generally true.

It is shown that Bernstein Polynomials do not diminish the total “twist” of space curves in contrast to their length and curvature diminishing properties. This phenomenon is shown to be related to the fact that Bernstein Polynomials of a plane curve may have more inflections than the curve possesses.

Two results are obtained in this paper. The first is a generalization of the imprimitivity theorem of Mackey, when the associated projection-valued measure is replaced by a commutative positive operator valued measure. The second is a necessary and sufficient condition for such a system of covariance to possess an overcomplete, covariant family of coherent states.

In this paper we define and study a class of compacta under the name of Fundamental Approximative Absolute Neighborhood Retracts. This class includes Borsuk’s Fundamental Absolute Neighborhood Retracts and Approximative Absolute Neighborhood Retracts in the sense of M. H. Clapp as proper subclasses. We also introduce the notion of q-strong movability and prove that Fundamental Approximative Absolute Neighborhood Retracts coincide with q-strongly movable compacta.

It is shown that in a certain extensive class of algebras one can associate with each Lie ideal a corresponding associative ideal which facilitates the study of Lie ideals, especially for simple algebras. We apply this construction to obtain new, simpler proofs of some known results of Herstein [10] and others on the Lie structure of associative rings.

Let A be a real n × n matrix and define the Hadamard ratio h(A) to be the absolute value of det A divided by the product of the Euclidean norms of the columns of A. It is shown that if A is a random variable whose distribution satisfies some simple symmetry properties then the random variable log h(A) has mean and variance . In particular, for each ε > 0, the probability that h(A) lies in the range tends to 1 as n tends to ∞.

Let α≥0 and β>— 1. The main result gives necessary and sufficient conditions for the sequence (εn) in order that the sequence (εnUn) will be absolutely summable by the Cesàro method Cβ for each sequence (Un) which is bounded or summable by the method Cα

Another theorem is proven when Cα and Cβ are replaced by triangular methods A = (ank) and B=(bnk) satisfying , where (ξnk) = (ank)-1.

A combinatorial proof is given for Uchimura’s identity

As a corollary to this proof we derive a formula for the sum of the nth powers of the divisors of m in terms of partitions of m.

In this paper we shall use matrix methods to obtain several generalizations of a well known result of Eneström and Kakeya about the location of the zeros of polynomials. We shall also obtain zero-free regions of polynomials having complex coefficients. Finally we prove some results concerning the zeros of a class of polynomials.

In the mid 1970's, Shelah formulated a weak version of ◊. This axiom Φ is a prediction principle for colorings of the binary tree of height ω1. Shelah and Devlin showed that Φ is equivalent to 2ℵ0 < 2ℵ1.

In this paper, we formulate Φp, a "Φ for partial colorings", show that both ◊* and Fleissner's “◊ for stationary systems” imply Φp, that ◊ does not imply Φp and that Φp does not imply CH.

We show that Φp implies that, in a normal first countable space, a discrete family of points of cardinality ℵ1 is separated.

It is proved that a function is nondecreasing if it is Baire one and Darboux and fulfills Lusin’s condition (N), and if its derivative is non-negative for almost every point at which the function is derivable. Using this result, a process to formulate various results on the existence and the valuation of indeterminate forms via various monotonicity theorems is illustrated. In particular, the ordinary Bernoulli-L’Hospital rule and some of its variations obtained recenty by A. M. Ostrowski are generalized.

If (X, β, μ) is a σ-finite, non-atomic measure space, and ϕ is an increasing non-negative concave function defined on the positive real numbers, we give a set of necessary and sufficient conditions for an additive functional T on the Lorentz space Nϕ to have an integral representation with a Caratheodory kernel. In the special case when T is statistical we classify the functional properties (enjoyed by the kernels) in terms of the Lorentz norm on the space.

For a connected open subset Ω of the plane and n a positive integer, let be the space introduced by Cowen and Douglas in their paper, “Complex geometry and operator theory”. Our main concern is the case n = 1, in which case we show the existence of a functional calculus for von Neumann operators in for which a spectral mapping theorem holds. In particular we prove that if the spectrum of , is a spectral set for T, and if , then σ(f(T)) = f(Ω)- for every bounded analytic function f on the interior of L, where L is compact, σ(T) ⊂ L, the interior of L is simply connected and L is minimal with respect to these properties. This functional calculus turns out to be nice in the sense that the general study of von Neumann operators in is reduced to the special situation where Ω is an open connected subset of the unit disc with .

A harmonic function on the unit disc is of Beurling type ω if its Fourier (or Taylor) coefficients grow no faster than exp ω(|n|) as |n|→∞, where ω is a given increasing, concave function with ω(x)/x ↓ 0 as x → ∞. These harmonic functions are characterized by the growth rate of their L1-norms on circles of radius r as r → 1. The classical Schwartz result follows as a corollary by taking ω(x) = log(1+x). The Gevrey case is also included in the general result if one uses ω(x) = xα, 0 < α < 1.

A primary group G is said to be a l.i.b. group if every idempotent endomorphism of every basic subgroup of G can be extended to an endomorphism of G. We establish the following characterization: A primary group is a l.i.b. group if and only if it is the direct sum of a torsion complete group and a divisible group. The technique used consists of a close analysis of certain subgroups of Prufer-like primary groups.

Les groupes p-torsion forment une classe de groupes abéliens mixtes dont les sous-groupes de p-base sont de torsion. Nous montrons ici que la généralisation naturelle à ces groupes de la notion de sous-groupe large développée pour les groupes primaires par R. S. Pierce, permet d'obtenir des résultats analogues. Ainsi nous caractérisons les sous-groupes p-larges d'un groupe p-torsion G en fonction des suites non-décroissantes d'entiers non-négatifs u = (ui) qui satisfont à la condition d'écart pour G. On obtient: un sous-groupe A du groupe p-torsion G est p-large si, et seulement si A est de la forme G(u) pour une suite u telle que pour tout x ∈ G, la suite (h(pix)) est plus grande presque partout que la suite u.

Nous déterminons aussi, les sous-groupes p-large de , le complété p-adique d'une somme directe de groupes cycliques non bornés B, ainsi que ceux des sous-groupes p-purs totalement invariants de engendrés par un élément.

We consider a finite group G with a group A acting on it in such a way as to induce a partition of G# (a situation which arises in the study of centralizer near-rings). With the additional hypothesis that (|Aω|, |G|) = 1, it is shown that either A is semiregular on G# or G is an irreducible module for A.

Let L be an a-implicative semilattice. We obtain a characterization of those elements which cover a. This gives a characterization of atoms in pseudocomplemented semilattices, and leads to various results on primes and irreducibles in semilattices. As an application, we prove that in a complete, atomistic lattice L, the following are equivalent (i) L is implicative (ii) L is (2, ∞) meet distributive (iii) each element of L is a meet of primes.