The (logarithmic) capacity or transfinite diameter is originally defined for compact sets in the complex plane. An extension may be made by defining the capacity of a given arbitrary set in the plane as the supremum of the capacities of all compact sets contained in the given set. Convergence in capacity is defined analogously to convergence in measure. It is shown in this paper that properties of convergence in capacity are also analogous to those of convergence in measure.

It is proved that if G is a permutation group on a set Ω every orbit of which contains more than mn elements, then any pair of subsets of Ω containing m and n elements respectively can be separated by an element of G.

Weak duality and direct duality theorems are proved, under appropriate assumptions, for the following pair of programming problems in complex space:

The objective function may be nondifferentiable and the constraints are of a more general nature than those considered earlier by various authors. Several well-known results are shown to be special cases of the results proved here.

Let Qp and K be the rational p–adic field and an algebraic extension of Qp of finite degree, respectively, and let I and Ik be the subsets of Qp and of K consisting of the p–adic integers of these fields.

It is known that the continuous functions f: I → Qp can be written as

where this series converges uniformly on I, and that such continuous functions need not be differentiable at any point. We here study continuous functions F: IK → K which for all X on IK are the sum of a uniformly convergent series

It is proved that such functions F(X) have at every point of IK derivatives of all orders. In the special case when K is totally ramified, they cannot in general be developed into power series that converge everywhere on IK, but this is possible when K is not totally ramified.

In 1919, Adolf Hurwitz formed the quaternion ring R composed of elements whose coordinates were either all integers or halves of odd integers. The objective of this paper is to examine the (two-sided) ideal structure in the hurwitzean polynomial ring R[x], formed by taking all polynomials with coefficients in R. The maximal and prime ideals of R[x] will be characterized with results surprisingly analogous to those in Z[x]. In addition, a canonical basis, of the type developed by G. Szekeres, 1952, for polynomial domains, will be developed for the ideals of R[x].

Evidence is presented which suggests that the following assertions about a variety ⊻ of groups may be equivalent:

(a) ⊻ is locally finite,

(b) all ⊻-groups satisfying the maximal condition for normal subgroups are finite, and

(c) all ⊻-groups satisfying the minimal condition for normal subgroups are finite.

It is shown that all solutions of ẍ + 2x3 = p(t) are bounded, the notation indicating that p is periodic. It is not necessary to have a small parameter multiplying p.

The essential step is to show by appeal to Moser's theorem that, under the mapping (of the initial-value plane) which corresponds to the equation, there are invariant simple closed curves. This implies also that there is an uncountable infinity of almost-periodic solutions and, for each positive integer m, an infinity of periodic solutions of least period 2mπ (2π being taken as the least period of p ).

It is suggested that for a large class of equations the same attack would show all solutions of ẍ + g(x) = p(t) bounded. However, in order to show the method clearly, no generalisation is attempted here.

Let G be a locally finite group, k a field of characteristic p ≥ 0 and V a right kG-module. We say that V is an -module over kG, if each p′-subgroup H of G contains a finite subgroup F with the same fixed points as H in V. (By convention, 0′ is taken as the set of all primes.) Such modules arise as elementary abelian section of -groups, a class of locally finite groups similar in many ways to the class of finite soluble groups.

The main theorem is that if V is an -module over kG with trivial Frattini submodule, and G is almost abelian, then every composition factor of V is complemented. This is a crucial ingredient in Tomkinson's theory of prefrattini subgroups in a certain subclass of . An example is given to show that the theorem breaks down for metabelian G. This leads to an example of a -group in which there are no analogues of prefrattini subgroups - the first situation where one of the standard conjugacy classes of subgroups of finite soluble groups has no decent analogue in the whole class

Given a site T, that is, a category equipped with a fixed Grothendieck topology, we provide a definition of fibration for morphisms of the presheaves on T. We verify that the notion is well-behaved with respect to composition, base change, and exponentiation, and is trivial on the topos of sheaves. We compare our definition to that of Kan fibration in the semi-simplicial setting. Also we show how we can obtain a notion of fibration on our ground site T and investigate the resulting notion in certain ring-theoretic situations.

The complex nonlinear complementarity problem considered here is the following: find z such that

where S is a polyhedral cone in Cn, S* the polar cone, and g is a mapping from Cn into itself. We study the extent to which the existence of a z ∈ S with g(z) ∈ S* (feasible point) implies the existence of a solution to the nonlinear complementarity problem, and extend, to nonlinear mappings, known results in the linear complementarity problem on positive semi-definite matrices.