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.

The middle graph of a graph G=( V, E) is the graph M(G) = (V∪E, E′), in which two vertices u, v are adjacent if either M is a vertex in V and v is an edge in E containing u, or u and v are edges in E having a vertex in common. Middle graphs have been characterized in terms of line graphs by Hamada and Yoshimura [7], who also investigated their traversability and connectivity properties. In this paper another characterization of middle graphs is presented, in which they are viewed as a class of intersection (representative) graphs of hereditary hypergraphs. Graph theoretic parameters associated with the concepts of vertex independence, dominance, and irredundance for middle graphs are discussed, and equalities relating the chromatic number of a graph to these parameters are obtained.

We prove polytopality of the generalized associahedra introduced in [5].

By a tournament Tn on n vertices, we shall mean a directed graph on n vertices for which every pair of distinct vertices form the endpoints of exactly one directed edge (e.g., see [5]). If x and y are vertices of Tn we say that x dominates y if the edge between x and y is directed from x to y. In 1962, K. Schütte [2] raised the following question: Given k > 0, is there a tournament Tn(k) such that for any set S of k vertices of Tn(k) there is a vertex y which dominates all k elements of S. (Such a tournament will be said to have property Pk.)

In this note we prove two results about derivations. The first result holds in any ring, whereas the second one is valid in prime rings. From the nature of the results one would expect them to be known, but we have not been able to locate them in the literature.

A linear map A : C(T) → C(S) is called separating if f • g ≡ 0 implies Af • Ag = 0. We describe the general form of such maps and prove that any separating isomorphism is continuous.

We provide general results on the behaviour of the Krasnoselski-Mann iteration process for nonexpansive mappings in a variety of normed settings.

Let R be a prime ring and U be a nonzero ideal or quadratic Jordan ideal of R. If L is a nontrivial automorphism or derivation of R such thatuL(u)—L(u)u is in the center of R for every u in U, then the ring R is commutative.

If S is a semigroup, then an S-set AS is a set A together with a representation of S by mappings of A into itself. In this article, the theory of injective envelopes is carried from R-modules to S-sets. These results are known to hold in every Grothendieck category, but the category EnsS of (right) S-sets is not even additive.

The different results proved in this paper do not have very much in common. Since they all deal with the location of the zeros of a polynomial, we have decided to put them in one place. Improving upon a classical result of Cauchy we obtain in § 2 a circle containing all the zeros of a polynomial. In § 3 we obtain an extension of the well known theorem of Enestrőm and Kakeya concerning the zeros of a polynomial whose coefficients are non-negative and monotonie.

The following problem due to A. Boyd, has enjoyed a certain popularity in recent months with several mathematicians. A different solution to the one given here has been given independently by R. T. Bumby and J. Spencer.

The Problem, There are n ladies, and each one of them knows an item of scandal which is not known to any of the others. They communicate by telephone, and whenever two ladies make a call, they pass on to each other, as much scandal as they know at that time. How many calls are needed before all the ladies know all the scandal?

A (simple, undirected) graph G is vertex transitive if for any two vertices of G there is an automorphism of G that maps one to the other. Similarly, G is edge transitive if for any two edges [a, b] and [c, d] of G there is an automorphism of G such that {c, d} = {f(a), f(b)}. A 1-path of G is an ordered pair (a, b) of (distinct) vertices a and b of G, such that a and b are joined by an edge. G is 1-transitive if for any two 1-paths (a, b) and (c, d) of G there is an automorphism f of G such that c = f(a) and d = f(b). A graph is regular of valency d if each of its vertices is incident with exactly d of its edges.

Every weakly compact composition operator between weighted Banach spaces $H_{v}^{\infty }$ of analytic functions with weighted sup-norms is compact. Lower and upper estimates of the essential norm of continuous composition operators are obtained. The norms of the point evaluation functionals on the Banach space $H_{v}^{\infty }$ are also estimated, thus permitting to get new characterizations of compact composition operators between these spaces.

Let θ = θ(k) be the positive root of θ2 + (k-2)θ-k = 0. Let f(n) = [(n + l)θ]-[nθ] for positive integers n, where [x] denotes the greatest integer in x. Then the elements of the infinite sequence (f(l), f(2), f(3),…) can be rapidly generated from the finite sequence (f(l), f(2),…,f(k)) by means of certain shift operators. For k = 1 we can generate (the characteristic function of) the sequence [nθ] itself in this manner.

As its name implies, this paper consists of observations on various topics in graph theory that stem from the concept of Hamiltonian cycle. We shall mainly adopt the notation and terminology of Harary [5]. However, we use vertices and edges for what are called "points" and "lines" in [5]. V(G), E(G) respectively will denote the sets of vertices and edges of graph G, and |X| will denote the cardinal of the set X.|V(G)| is the order of G, and |E(G)| the size of G. Throughout n is reserved for the order of G.

Let $\Phi $ be an algebraically closed field of characteristic zero, $G$ a finite, not necessarily abelian, group. Given a $G$-grading on the full matrix algebra $A\,=\,{{M}_{n}}\left( \Phi \right)$, we decompose $A$ as the tensor product of graded subalgebras $A\,=\,B\,\otimes \,C,\,B\,\cong \,{{M}_{p}}\left( \Phi \right)$ being a graded division algebra, while the grading of $C\,\cong \,{{M}_{q}}\left( \Phi \right)$ is determined by that of the vector space ${{\Phi }^{n}}$. Now the grading of $A$ is recovered from those of $A$ and $B$ using a canonical “induction” procedure.

We introduce semi-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. We give examples, investigate the geometry of foliations that arise from the definition of a Riemannian submersion, and find necessary sufficient conditions for total manifold to be a locally product Riemannian manifold. We also find necessary and sufficient conditions for a semi-invariant submersion to be totally geodesic. Moreover, we obtain a classification for semi-invariant submersions with totally umbilical fibers and show that such submersions put some restrictions on total manifolds.

Taking into account the effects of patch structure and nonlinear density-dependent mortality terms, we explore a class of almost periodic Nicholson’s blowflies model in this paper. Employing the Lyapunov function method and differential inequality technique, some novel assertions are developed to guarantee the existence and exponential stability of positive almost periodic solutions for the addressed model, which generalize and refine the corresponding results in some recently published literatures. Particularly, an example and its numerical simulations are arranged to support the proposed approach.

We describe the structure of the irreducible highest weight modules for the twisted Heisenberg–Virasoro Lie algebra at level zero. We prove that either a Verma module is irreducible or its maximal submodule is cyclic.

It is shown that if v is an odd prime power, other than a prime of the form 22n + 1, then there exists a Room square of order v + 1.

A room square of order 2n, where n is a positive integer, is an arrangement of 2n objects in a square array of 2 side 2n - 1, such that each of the (2n - 1)2 cells of the array is either-empty or contains exactly two distinct objects; each of the 2n objects appears exactly once in each row and column; and each (unordered) pair of objects occurs in exactly one cell.