A graph G has a finite set V of points and a set X of lines each of which joins two distinct points (called its end-points), and no two lines join the same pair of points. A graph with one point and no line is trivial. A line is incident with each of its end-points. Two points are adjacent if they are joined by a line. The degree of a point is the number of lines incident with it. The line-graph L(G) of G has X as its set of points and two elements x, y of X are adjacent in L(G) whenever the lines x and y of G have a common end-point. A walk in G is an alternating sequence v1, x1, v2, x2, …, vn of points and lines, the first and last terms being points, such that xi is the line joining vi to vi+1 for i=1, …, n-1.

A graph with m(>1) vertices, each pair of distinct vertices connected by an edge, and also a graph obtained from such a graph by the process of subdividing edges through the insertion of new vertices of valency 2, will be denoted by ≪m, o≫. A graph obtained from a graph with m(>2) vertices in which each pair of distinct vertices are connected by an edge, by deleting n (≤ m-1) edges incident with one and the same vertex, and also a graph obtained from such a graph by the process of subdividing edges through the insertion of new vertices of valency 2, will be denoted by ≪m, n≫.

Broadly speaking, the Hardy-Landau summation formula * is given by

where r(n) is the number of integer solutions of the Diophantine equation x2 + y2 = n, and f(x) and g(x) are transforms with respect to the Watson kernel that is:

and

Some of the most important G-Structures of the first kind [1] are those defined by linear operators satisfying algebraic relations. If the linear operator J acting on the complexified space of a differentiable manifold V satisfies a relation of the form

where I is the identity operator, the manifold has an almost complex structure ([2] [3]). The structures defined by

are the almost product structures ([3] [4]).

Let K, k be fields and K|k a finite galois extension with galois group G. The multiplicative group K* of K is a G-module, that is, a module over the integral group ring ZG, the module action of an element σ ϵ G being its effect as an automorphism. It is shown in [2, p. 158] that the first cohomology group vanishes:

1

This expository note was prompted by some questions asked by Professor P. Hilton during his lectures "Catégories non-abétiennes" at the University of Montréal, July 1964.

The descriptions of set functions as one to one and as onto can be characterized in terms of set function composition. A set function is one to one iff it has the left cancellation property, that is, f · g = f · h implies g = h.

Let Ω be an index set and for each αϵΩ let fα:X → Xα be a function where X and Xα are sets. Assume that, for each α, a topology is given for Xα. Then, as is well-known, the functions fα and the topologies determine a topology for X. This is the so-called weak or initial topology, which is generated by .

In a paper by LaSalle [l] on linear time optimal control the following lemma is proved:

Let Ω be the set of all r-dimensional vector functions U(τ) measurable on [ 0, t] with |ui(τ)≦1. Let Ωo be the subset of functions uo(τ) with |uoi(τ) = 1. Let Y(τ) be any (n × r ) matrix function in L1([ 0, t]).

Recently B. Palczewski and W. Pawelski [l] have given some sufficient conditions of uniqueness for the solutions of the Darboux problem for equations of the form:

1

The criteria given there for equations of the form (1) are natural generalizations of the criteria given by Krasnosielski and Krein [2] in the corresponding ordinary differential case. The purpose of the present note is to give a further generalization of the above result and two other uniqueness conditions for the solutions of the Darboux problem of the same form.

The nth Taylor mean of order r of a sequence {sn} is given by

1.1

where

1.2

Cowling [l] has shown that this method is regular if and only if 0 ≦ r < 1. Since r = 0 corresponds to ordinary convergence, it will be assumed here that 0 < r < 1.

If (X, S,μ) is an arbitrary complemented measure space and X is σ-finite then (L1)* = L∞ or, more precisely, (L1)* is isometric and isomorphic to L∞ by the correspondence

It is well known that there exist non σ-finite spaces with (L1)* ≥ L∞.

The motivation for this paper lies in the following remarkable property of certain probability distributions. The distribution law of the r. v. (random variable) X is exactly the same as that of 1/ X, and in the case of a r. v. with p. d. f. (probability density function) f(x; a, b) where a, b are parameters, the p. d. f. of 1/X is f(x; b, a). In the latter case the p. d. f. of the reciprocal is obtained from the p. d. f. of X by merely switching the parameters. The existence of random variables with this property is perhaps familiar to statisticians, as is evidenced by the use of the classical 'F' distribution. The Cauchy law is yet another example which illustrates this property. It seems, therefore, reasonable to characterize this class of random variables by means of this rather interesting property.

For any group G and integer n, let Pn:G→G be the function defined by Pn(g) = gn for all g ϵ G. If G is abelian then Pn is a homomorphism for all n. Conversely, it is well known (and easy to show) that if P2 or P-1 is a homomorphism then G is abelian. As the groups Gn described below show, for every n other than 2 and -1 there exist non-abelian groups for which Pn is a homomorphism.

In this note we wish to present an alternative proof for the following well-known theorem [1, Theorem 16]: every convex polytope X in Euclidean n-dimensional space Rn is the intersection of a finite family of closed half-spaces. It will be supposed that the converse of this theorem has been verified by conventional arguments, namely: every bounded intersection of a finite family of closed half-spaces in Rn is a convex polytope [cf. 1, Theorem 15].

A graph Gn consists of n nodes some pairs of which are joined by a single edge. A complete k-graph has k nodes and edges. Erdos [1] proved that if a graph Gn has edges, then it contains at least complete 3-graphs if it contains any at all. The main object of this note is to extend this result to complete k-graphs.

The following question was asked by R. A. Hirschfeld in [1]: "E and F are Banach spaces, F reflexive, D is a subset of E and T: D→F a nonlinear contraction, i.e.,

Can T be extended to a contraction (for E = F = Hilbert space the answer is yes)."