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.

The following result is proved in [1, p. 6].

Theorem 1. Let X be a complete metric space, and let T and Tn(n = 1, 2,…)be contraction mappings of X into itself with the same Lipschitz constant k<1, and with fixed points u and un respectively. Suppose that limn → ∞ Tn(x) = T(x) for every x ∊ X. Then limn → ∞ un = u.

The following theorem is the principal result of this paper.

Let (M, d) be a metric space and T a self-mapping of M satisfying the condition for x,y ∊ M

1

where a, b, c, e,f are nonnegative and we set α=a+b+c+e+f.

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 Thomsen graph [2, p. 22] consists of six vertices partitioned into two classes of three each, with every vertex in one class connected to every vertex in the other; it is the graph of the “gas, water, and electricity” problem [1, p. 206]. (All graphs considered in this paper will be undirected, having neither loops nor multiple edges.)

The purpose of this paper is to investigate for graphs the existence of certain valuations which have some "magic" property. The question about the existence of such valuations arises from the investigation of another kind of valuations which are introduced in [1] and are related to cyclic decompositions of complete graphs into isomorphic subgraphs.

Throughout this paper the word graph will mean a finite undirected graph without loops or multiple edges having at least one edge. By G(m, n) we denote a graph having m vertices and n edges, by V(G) and E(G) the vertex-set and the edge-set of G, respectively. Both vertices and edges are called the elements of the graph.

There is a well-known elementary problem:

(S3) Given a triangle T with the vertices a1, a2, a3, to find in the plane of T the point p which minimize s the sum of the distances |pa1| + |pa2| + |pa3|.

p, called the Steiner point of T, is unique: if an angle of T is ≥ 2π/3 then p is its vertex, otherwise p lies inside T and the sides of T subtend at p the angle 2π/3. In the latter case p is called the S-point of T, and it can be found by the following simple construction: let a12 be the third vertex of the equilateral triangle whose other two vertices are a1 and a2, and whose interior does not overlap that of T, let C be the circle through a1, a2 a12; then p is the intersection of C and the straight segment a12a3. It is easily proved that any one of the three ellipses through p with two of the vertices of T as foci is tangent at p to the circle through p about the third vertex of T.

Suppose that

1.1

is a transcendental integral function. In this article we develop the theory initiated by Wiman [22, 23] and deepened by other writers including Valiron [18, 19, 20], Saxer [15], Clunie [4, 5] and Kövari [10, 11], which describes the local behaviour of f(z), near a point where | f(z) | is large, in terms of the power seriesf of f(z).

We study orthogonal polynomials for which the weight function is a linear combination of the Jacobi weight function and two delta functions at 1 and — 1. These polynomials can be expressed as 4F3 hypergeometric functions and they satisfy second order differential equations. They include Krall’s Jacobi type polynomials as special cases. The fourth order differential equation for the latter polynomials is derived in a more simple way.

Erdös has defined g(n) as the smallest integer such that any set of g(n) points in the plane, no three collinear, contains the vertex set of a convex n-gon whose interior contains no point of this set. Arbitrarily large sets containing no empty convex 7-gon are constructed, showing that g(n) does not exist for n≥l. Whether g(6) exists is unknown.

Let R be a ring with center Z, and S a nonempty subset of R. A mapping F from R to R is called centralizing on S if [x, F(x)] ∊ Z for all x ∊ S. We show that a semiprime ring R must have a nontrivial central ideal if it admits an appropriate endomorphism or derivation which is centralizing on some nontrivial one-sided ideal. Under similar hypotheses, we prove commutativity in prime rings.

We give a necessary and sufficient condition on weight functions u and v such that for l≤p≤q≤∞ there exists a constant C for which

A corresponding dual result is also given. This extends a result of B. Muckenhoupt which appeared in Studia Math., 34 (1972).

Throughout this paper we consider associative rings with unity elements. In §1 various results on the representation of rings by rings of sections of special rings are compared. In particular, it is shown that results enunciated by Dauns and Hofmann, Koh, and the present author may all be deduced from one statement, the proof of which appears in §3.

The primordial problems of linear algebra are the solution of a system of linear equations

and the solution of the eigenvalue problem

for the eigenvalues λk, and corresponding eigenvectors of a given matrix A.

The definition of strong Cesaro summability with respect to a modulus is extended to a definition of strong A -summability with respect to a modulus when A is a nonnegative regular matrix summability method. It is shown that if a sequence is strongly A-summable with respect to an arbitrary modulus then it is A-statistically convergent and that Astatistical convergence and strong A-summability with respect to a modulus are equivalent on the bounded sequences.

If R is a ring and S ⊆ R, a mapping f:R —> R is called strong commutativity- preserving (scp) on S if [x, y] = [f(x),f(y)] for all x,y € S. We investigate commutativity in prime and semiprime rings admitting a derivation or an endomorphism which is scp on a nonzero right ideal.

Beineke and Harary [l] recently showed that the maximum number of strong tournaments with k nodes that can be contained in a tournament with n nodes is

if 3 ≤ k ≤ n. The object of this note is to obtain some additional results of this type. We will use essentially the same terminology as was used in [ l ], so we will not repeat the standard definitions here.

We study the topology of the nontrivial component, , of self-adjoint Fredholm operators on a separable Hilbert space. In particular, if {Bt} is a path of such operators, we can associate to {Bt} an integer, sf({Bt}), called the spectral flow of the path. This notion, due to M. Atiyah and G. Lusztig, assigns to the path {Bt} the net number of eigenvalues (counted with multiplicities) which pass through 0 in the positive direction. There are difficulties in making this precise — the usual argument involves looking at the graph of the spectrum of the family (after a suitable perturbation) and then counting intersection numbers with y = 0.

We present a completely different approach using the functional calculus to obtain continuous paths of eigenprojections (at least locally) of the form . The spectral flow is then defined as the dimension of the nonnegative eigenspace at the end of this path minus the dimension of the nonnegative eigenspace at the beginning. This leads to an easy proof that spectral flow is a well-defined homomorphism from the homotopy groupoid of onto Z. For the sake of completeness we also outline the seldom-mentioned proof that the restriction of spectral flow to is an isomorphism onto Z.

Let R be a prime ring and T be a nontrivial automorphism of R. If xxT—xTx is in the center of the ring for every x in R, then R is a commutative integral domain.

I would like to express my thanks to the Canadian Mathematical Society for inviting me to present this lecture. I would also like to express my appreciation to C.J. Smyth for numerous helpful conversations during his visit this year at the University of British Columbia.

This paper follows reasonably closely the outline of the lecture presented in Ottawa. More details are given here though and a number of proofs which would not be otherwise accessible have been added as Appendices. The attentive reader will soon realize the appropriateness of the title.