It is shown that the nth center of a radical ring coincides with that of its adjoint group, from which a result of Jennings is sharpened and a conjecture of his is confirmed.

This paper concerns the interaction between group theory and classical mechanics in general and with the application of this theory to plasma physics in particular.

Mechanics has had a profound role in the historical development of mathematics. Leading original thinkers in pure mathematics such as Newton, Euler, Lagrange, Jacobi, Laplace, Cauchy, Gauss, Riemann, Poincaré, Hilbert, Birkhofï, Smale and Arnold were also great original thinkers in various facets of mechanics. The interaction between mechanics and pure mathematics remains one of the most active and flourishing areas of current research. It is this flavor which I hope to convey in this lecture.

The purpose of this exposition is threefold. Firstly, we wish to show that many concepts and arguments carry over from pre-ordered sets to categories. Secondly, we wish to make some propaganda for the notion of "subequalizer" of two functors, which appears to be more fundamental than Lawvere's so-called "commacategory", in the same sense in which equalizers are more fundamental than pullbacks. Thirdly, we wish to give simple proofs of the adjoint functor theorem and related theorems, which appear to be more economic than those in the literature. The author wishes to take this opportunity to refine some arguments that he has published earlier. He is indebted to Michael Barr, whose presentation of similar proofs in his course has provided the stimulation for preparing this note for publication, to John Isbell for his critical reading of the manuscript and to William Schelter for suggesting a shortcut in one of the proofs.

We give a simple and explicit example of a complex Banach space which is not isomorphic to its complex conjugate, and hence of two real-isomorphic spaces which are not complex-isomorphic.

This article deals with the distributions of the product and the quotient of two correlated exponential random variables. We consider here three types of bivariate exponential distributions: Marshall-Olkin's bivariate exponential distribution, Gumbel's Type I bivariate exponential distribution, and Gumbel's Type II bivariate exponential distribution.

A subset K of a lattice is said to be directed if for any a, b∊K there is c∊K with c ≥ a, b. A complete lattice L is called upper continuous if for every directed subset (aα) and every element b.

The following is a slight improvement of [4; Anmerkung 1. 11, p. 11].

Throughout this article p denotes a fixed number such that 1 ≤ p < ∞. The definition of a real Lp space associated with a measure space is well known. These spaces are Banach Spaces and, with the usual partial ordering of (equivalence classes of) functions, also Banach Lattices.

A set E of edges of a graph G is said to be a dominating set of edges if every edge of G either belongs to E or is adjacent to an edge of E. If the subgraph 〈E〉 induced by E is a trail T, then T is called a dominating trail of G. Dominating circuits are defined analogously. A sufficient condition is given for a graph to possess a spanning (and thus dominating) circuit and a sufficient condition is given for a graph to possess a spanning (and thus dominating) trail between each pair of distinct vertices. The line graph L(G) of a graph G is defined to be that graph whose vertex set can be put in one-to-one correspondence with the edge set of G in such a way that two vertices of L(G) are adjacent if and only if the corresponding edges of G are adjacent. The existence of dominating trails and circuits is employed to present results on line graphs and second iterated line graphs, respectively.

A graph G is a couple (X, R) where X is a set, R ⊂ X × X. If G is an undirected graph without loops (R a symmetric irreflexive relation), we can interpret G as a couple (X, R), where R is a set of two-element subsets of X, i.e. . This interpretation is generalized in the notion of society.

A society is a couple (X, R), where ; a k-society is a society (X, R) with |A| = k for each A ∊ R.

These pages summarize recent progress on the oscillation problem for semilinear elliptic partial differential equations of the form

(1)

in unbounded domains Ω in n-dimensional Euclidean space Rn. Our attention is restricted to the second order symmetric equation (1), and completeness is not attempted; the emphasis is on results obtained in the last five years.

The main result of this paper is to prove that a generalization of the Principle of Dependent Choices, introduced by A. Levy [2; see also 1, Chapter 8], is equivalent to a form of Zorn's Lemma.

In [A], Arhangel'skii showed that for any T2 space X, |X|≤2L(x)χ(x), where L(X) is the Lindelöf degree of X and χ(X) is the character of X.

In [B], Bell, Ginsburg and Woods improved this result, assuming normality, by showing that for T4 spaces X, |X|≤2wL(x)χ(x), where wL(X) is the weak Lindelöf degree of X.

We introduce below a new cardinal function aL(X), the almost Lindelöf degree of X, which agrees with L(X) on T3 spaces, but which is often smaller than L(X) on T2 spaces, and show that for T2 spaces X,

We construct a class of fermions (or bosons) by using a Clifford (or Weyl) algebra to get two families of irreducible representations for the extended affine Lie algebra $\widetilde{\mathfrak{g}{{\mathfrak{l}}_{N}}\left( {{\mathbb{C}}_{q}} \right)}$ of level (1, 0) (or (−1, 0)).

When I agreed to give the 1976 Jeffery-Williams Lecture to the Canadian Mathematical Congress, I did not envisage that it would be my lot to give what might now be called the first memorial lecture in this series of lectures. During the past year, Ralph Jeffery and Lloyd Williams both died, and their deaths brought forward many deserved tributes to these two wonderful men. My lecture today will forge a link with my mathematical past, a past in which both of these men played important roles.

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

We prove that for a contact process restricted to the cube [1,n]d and initially fully occupied, the time to die out, when it is suitably normalized, converges to an exponential random variable as n tends to infinity.

A local version of $\text{VMO}$ is defined, and the local Hardy space ${{h}_{1}}$ is shown to be its dual. An application to weak-$*$ convergence in ${{h}_{1}}$ is proved.

A partially ordered set P has the fixed point property if every orderpreserving mapping f of P to P has a fixed point, that is, f(a) = a for some aϵP; call P fixed point free if P does not have the fixed point property.

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, of conductor $N$ and without complex multiplication. For any positive integer $l$, let ${{\phi }_{1}}$ be the Galois representation associated to the $l$-division points of $E$. From a celebrated 1972 result of Serre we know that ${{\phi }_{1}}$ is surjective for any sufficiently large prime $l$. In this paper we find conditional and unconditional upper bounds in terms of $N$ for the primes $l$ for which ${{\phi }_{1}}$ is not surjective.

We construct several examples of higher-dimensional Calabi–Yau manifolds and prove their modularity.