In this paper, we discuss questions of the existence and calculation of periodic, steady flows over periodic streambeds. There are some surprises.

Problems such as this, of flows in running streams, are free-surface problems, and part of the difficulty is that the domain occupied by the fluid is not completely known a priori.

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 purpose of this expository article is to familiarize the reader with one of the fundamental problems in the theory of infinite groups. We give an up-to-date account of the so-called Fox problem which concerns the identification of certain normal subgroups of free groups arising out of certain ideals in the free group rings. We assume that the reader is familiar with the elementary concepts of algebra.

It is known that every torsion-free soluble group G of finite Hirsch number hG is countable, and its homological and cohomological dimensions over the integers and rationals satisfy the inequalities

We prove that G must be finitely generated if the equality hG = cdQG holds. Moreover, we show that if G is a countable soluble group of finite Hirsch number, but not necessarily torsion-free, and if hG = cdQG, then hḠ = cdQḠ for every homomorphic image Ḡ of G.

Let p > l, r≠1, and let f(x) be a non-negative function defined in [0, ∞). The following inequality is due to G. H. Hardy [5, Ch. IX]:

1.1

where according as r>1 or r < l.

An ultrafilter completion is constructed for a nearness space. It is shown to preserve the Tl separation axiom. Characterizing conditions are given for it to be topological or for its topology to be compact. It is shown to have the simple extension topology and for a given Hausdorff space a compatible nearness structure is found for which its ultrafilter completion is homeomorphic to the Katetov H-closed extension.

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.

Let G be a lattice-ordered group (l-group), and let t, u∈ G+. We write tπu if t ∧ g = 1 is equivalent to u ∧ g = 1, and say that a tight Riesz order T on G is π-full if t ∈ T and t π U imply u∈T. We study the set of π-full tight Riesz orders on an l-permutation group (G, Ω), Ω a totally ordered set.

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.

We show that if a flow (R, X, π) on a separable metric space (X, d) satisfies (i) the transition mapping π(t, •): X → X is non-expansive for every t ≥ 0; (ii) X contains a globally uniformly asymptotically stable compact invariant subset, then the flow (R, X, π) is linear in the sense that it can be topologically and equivariantly embedded into a flow () on the Hilbert space l2 for which all of the transition mappings are linear operators on l2.

Suppose throughout that {sn} is a sequence of real numbers, λ > - 1, a > 0, and β is real. Let N be any non-negative integer such that αN + β > l.

We are concerned primarily with the logarithmic summability method L and the Borel-type method (B, α, β). Some known results involve the Abel-type summability method Aγ.

A homeomorphism f of a space M is pointwise periodic if for each m ∈ M there exists an integer k such that fk(m) =m, where fk is the kth iterate of f. Montgomery proves [5] that if M is a connected topological manifold, then f is periodic; i.e., there exists an integer n such that fn = id. Noting this, Weaver [7] proves that if M is an orientable 2-manifold of class C1, U⊆M open and C ⊆ U a compact connected set and if g : U → M of class C1 is such that (i) g(C) = C and (ii) whenever the derivative of g at points x∈C has rank 2 it is orientation preserving, then f = g|c : C → C periodic implies that all but a finite number of points of C have as least period the period of f.

Let ∑ be a class of maps in a category An object I of is ∑-injective if is an epimorphism for all σ ∈ ∑. This paper is concerned with the question of finding “enough” S-injectives in a functorial way.

Banchoff and Pohl [3] have proved the following generalization of the isoperimetric inequality.

Theorem. If γ is a closed, not necessarily simple, planar curve of length L, and w(p) is the winding number of a variable point p with respect to γ, then

1

with equality holding if and only if γ is a circle traversed a finite number of times in the same sense.

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.

Consider the differential equation

1

where n is even and f(t, y) is subject to the following conditions:

(a) f(t, y) is continuous on [0, ∞)× R;

(2) (b) f(t, y) is nondecreasing in y for each fixed t∈[0,∞);

(c) yf(t, y ) > 0 for y ≠ 0 and t∈[0,∞).

In this paper define the spaces l∞(Δ), c(Δ), and c0(Δ), where for instance l∞(Δ) = {x=(xk):supk |xk -xk + l|< ∞}, and compute their duals (continuous dual, α-dual, β-dual and γ-dual). We also determine necessary and sufficient conditions for a matrix A to map l∞(Δ) or c(Δ) into l∞ or c, and investigate related questions.

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

There has been a great deal of work recently concerning the relationship between the commutativity of a ring JR and the existence of certain specified derivations of R. Bell, Herstein, Procesei, Schacher, Ligh, Martindale, Putcha, Wilson, and Yaqub [1, 2, 6, 8, 9, 10, 11, 12, 14] have studied conditions on commutators which imply the commutativity of rings.

Let l and k be positive integers. Then for each integer n ≥ 1, define d(n; l, k) to be the number of (positive) divisors of n which lie in the arithmetic progression I mod k. Note that d(n;1,1) = d(n), the ordinary divisor function.