In this paper we investigate the zeros of power series

1

for some functions of coefficients A. In particular, we derive upper and lower bounds for the number of zeros of f in its domain of analyticity.

The line spaces of J. Cantwell are characterized among the axiomatic convexity spaces defined by Kay and Womble. This characterization is coupled with a recent result of Doignon to give an intrinsic solution of the linearization problem.

We shall apply the spectral theorem for self adjoint operators in Hilbert space to study an operator version of the Stieltjes moment problem [1]. In the course of the work we shall make use of the Friedrichs extension theorem which states that any non-negative symmetric operator in Hilbert space has a non-negative self adjoint extension.

Given a resplendent model for Peano arithmetic there exists a full satisfaction class over , i.e. an assignment of truth-values, to all closed formulas in the sense of with parameters from , which satisfies the usual semantic rules. The construction is based on the consistency of an appropriate system of -logic which is proved by an analysis of standard approximations of nonstandard formulas.

It is shown that a nonstandard model of Peano arithmetic which has a full satisfaction class is necessarily recursively saturated.

Dans [8], nous avons étudié l'ordre de grandeur des sommes de la forme , pour ou k, où g(n) et f(n) appartiennent respectivement à une classe de fonctions multiplicatives et additives.

Suppose throughout that {pn} is a sequence of non-negative numbers with p0>0, that

and that {sn} is a sequence of real numbers.

There are a number of theories which assign to a function defined on the real line a measure that reflects somehow the variation of that function. The most familiar of these is, of course, the Lebesgue-Stieltjes measure associated with any monotonie function. The problem in general is to provide a construction of a measure from a completely arbitrary function in such a way that the values of this measure provide information about the total variation of the function over sets of real numbers and from which useful inferences can be drawn.

In this note we collect some observations on the outer measures ψf and ψf that have been introduced in [4] and which describe the total variation of the function f. These properties have direct applications to the study of the derivative and the relative derivative. For definitions and notation the reader is referred to [4].

A Hausdorff-Young theorem is given for Lp-valued analytic functions on the open unit disc and estimates on such functions and their derivatives are deduced.

Kakutani's Theorem states that every point convex and use multifunction ϕ defined on a compact and convex set in a Euclidean space has at least one fixed point. Some necessary conditions are given here which ϕ must satisfy if c is the unique fixed point of ϕ. It is e.g. shown that if the width of ϕ(c) is greater than zero, then ϕ cannot be lsc at c, and if in addition c lies on the boundary of ϕ(c), then there exists a sequence {xk} which converges to c and for which the width of the sets ϕ(xk) converges to zero. If the width of ϕ(c) is zero, then the width of ϕ(xk) converges to zero whenever the sequence {xk} converges to c, but in this case ϕ can be lsc at c.

D. F. Dawson has considered several questions of the following nature. Suppose T is a regular matrix summability method. If A is a regular matrix and x is a sequence having a finite limit point, then there exists a subsequence y of x such that each finite limit point of x is a T-limit point of Ay. In the present paper, we show the regularity condition for A may be replaced by the requirement that A be a limit preserving bv to c map. This leads to summability characterizations for several classes of sequences.

The aim of this paper is to prove the following theorem:

Let R be a semiperfect ring. Let Q be a left R -module satisfying (a) Q is R-projective and (b) J(Q) is small in Q. Then Q is projective.

Every locally convex space E of dimension at most c contains a hyperplane G with the following property: the linear hull of each bounded Banach disk in G is finite-dimensional.

It is shown that in a dense-in-itself Hausdorff space if every set having a dense interior is open, then every compact set is finite.