This paper furthers the foundation of the theory of quasigroups obeying the law x(xy) = yx by studying their algebraic properties. Much information is obtained by analysing the cycle decomposition of left translations regarded as permutations, and other results are obtained by representation in terms of abelian groups with an operation.

In this paper we study the Cauchy problem for the second order nonlinear hyperbolic partial differential equation

with initial conditions

where

and |u|, |ux|, |uy| < ∞, y ≥ 0, r = r(x) ∈ C4(·), ν = ν(x) ∈ C4(·).

These conditions on k, H, f, r, and ν are assumed to be satisfied in some sufficiently small neighborhood of the segment I, y = 0, in the upper half-plane y > 0

This paper generalizes the results obtained by N.A. Lar'kin (Differencial'nye Uravnenija 8 (1972), 76–84), who has treated the special case H = H(x, y, u); that is, the quasi-linear hyperbolic equation (*).

A closure algebra is a set X with a closure operator C defined on it. It is possible to construct a topology τ on MX, the family of maximal, proper, closed subsets of X, and then to examine the relationship between the algebraic structure of (X, C) and the topological structure of the dual space (MX τ) This paper describes the algebraic conditions which are necessary and sufficient for the dual space to be separable metric and metric respectively.

Let T be a mapping from a set X into itself and let H(X) be a functional Hilbert space on the set X. Then the composition operator CT on H(X) induced by T is a bounded linear transformation from H(X) into itself defined by CTf = f ∘ T. In this paper composition operators are characterized in the case when H(X) = H2(π+) in terms of the behaviour of the inducing functions in the vicinity of the point at infinity. An estimate for the lower bound of ∥CT∥ is given. Also the invertibility of CT is characterized in terms of the invertibility of T.

This note investigates, for locally convex topological vector spaces, the question of how far the property of being barrelled is stable under small increase in the size of the dual space. If the dual F of a barrelled space E is enlarged by a finite dimensional vector space M, then E remains barrelled under the new Mackey topology τ(E, F+M). We discuss what happens when M has countable dimension.

A duality theorem, and a converse duality theorem, are proved for a nonlinear fractional program, where the numerator of the objective function involves a concave function, not necessarily differentiable, and also the support function of a convex set, and the denominator involves a convex function, and the support function of a convex set. Various known results are deduced as special cases.

It is shown that the inequality

holds for an infinity of integers p, q and that here the factor ½ may not be replaced by a smaller number. Corresponding best possible inequalities are given for the numbers e±2/t, where t is a positive integer.

A characterization of induced central extensions which gives an explicit relationship between induced central extensions and n-stem extensions is obtained. Using the characterization, necessary and sufficient conditions for a central extension of an abelian group by a nilpotent group of class n to be a Baer sum of an induced central extension and an extension of class n are obtained.

A class of locally convex spaces, a B-subfamily of finite order, is defined and the omega lemma for spaces belonging to this family is proved.

A graph G on p ≥ 2 vertices is said to be hypotraversable if G is not traversable but G − v is traversable, that is, G − v has an open trail which contains all the vertices and edges of G − v, for each vertex v of G. In this paper we first characterize hypotraversable graphs and then, using this characterization, obtain a complete enumeration of these graphs on p vertices for each positive integer p.

The concept of the perfect map on a convergence space (X, q), where q is a convergence function, is introduced and investigated. Such maps are not assumed to be either continuous or surjective. Some nontrivial examples of well known mappings between topological spaces, nontopological pretopological spaces and nonpseudotopological convergence spaces are shown to be perfect in this new sense. Among the numerous results obtained is a covering property for perfectness and the result that such maps are closed, compact, and for surjections almost-compact. Sufficient conditions are given for a compact (respectively almost-compact) map to be perfect. In the final section, a major result shows that if f: (X, q) → (Y, p) is perfect and g: (X, q) → (Z, s) is weakly-continuous into Hausdorff Z, then (f, g): (X, q) → (Y×Z, p×s) is perfect. This result is given numerous applications.

I this paper we extend such dynamical concepts as weak almost periodicity and simple equicontinuity of topological dynamics to the context of groups of *-automorphisms on a C*-algebra with unit. If (A, G) is a C* flow and if S(A) is the state space of A, we show that (A, G) is weakly almost periodic if and only if (S(A), G) is weakly almost periodic, and that, if A is a von Neumann algebra, then A is G-finite if and only if G is simple equicontinuous on the unit ball of A with respect to the weak *-topology.