For the class of meromorphically starlike functions of prescribed order, the concept of type has been introduced. A characterization of meromorphically starlike functions of order α and type β has been obtained when the coefficients in its Laurent series expansion about the origin are all positive. This leads to a study of coefficient estimates, distortion theorems, radius of convexity estimates, integral operators, convolution properties et cetera for this class. It is seen that the class considered demonstrates, in some respects, properties analogous to those possessed by the corresponding class of univalent analytic functions with negative coefficients.

We investigate the structure of the lattice of subalgebras of an infinite Boolean algebra; in particular, we make a contribution to the question as to when such a lattice is simple.

An analytic map h of type Bl from a Riemann surface R into another S, both having Greens functions, behaves well near the “boundaryr” of R. Let X stand for a family of holomorphic functions, and let f be holomorphic on S. We shall show, for several X′s, the following:

(i) f ∈ X(S) ⇔ foh ⇔ X(R);

‖foh‖ = ‖f‖.

Use is made of harmonic majoration of subharmonic functions on R and on S.

The Radon-Nikodým theorem and a sequential convergence result are given for integrals with respect to a Radon polymeasure on a finite product space.

We study integral transforms of functions belonging to the Jakubowski class S(m, M) and determine the range of values of the exponent for which the integral is a convex or a close to convex function.

The properties of Lebesgue outer measures embodied in the Vitali covering theorem, the Vitali-Carathéodory theorem, the Lusin theorem, the density theorem, outer regularity and inner regularity, and the relation between measurability and approximate continuity are studied in a general abstract space, called a topological Vitali measure space. The main theme is the mutual equivalence of these properties.

We consider in this paper the implicit complementarity problem imposed by quasi-variational inequalities and stochastic optimal control. The principal result is an existence theorem for the implicit complementarity problem in Hilbert spaces.

We relate the kernel of the cup product of 1-dimensional cohomology classes for a group G acting trivially on a field R to Hom(G2/G3,R), the space of group homomorphisms of the second stage of the lower central series for G into R, by means of explicit computations with cocycles. The precise result depends on whether the characteristic of the field is 0, an odd prime or 2.

A new technique, using the contraction mapping theorem, for solving quadratic equations in Banach space is introduced. The results are then applied to solve Chandrasekhar's integral equation and related equations without the usual positivity assumptions.

Nilpotent injectors exist in all finite groups.

For every Fitting class F of finite groups (see [2]), InjF(G) denotes the set of all H ≤ G such that for each N ⊴ ⊴ G , H ∩ N is an F -maximal subgroup of N (that is, belongs to F and i s maximal among the subgroups of N with this property). Let W and N* denote the Fitting class of all nilpotent and quasi-nilpotent groups, respectively. (For the basic properties of quasi-nilpotent groups, and of the N*-radical F*(G) of a finite group G3 the reader is referred to [5].,X. %13; we shall use these properties without further reference.) Blessenohl and H. Laue have shown in CJ] that for every finite group G, InjN*(G) = {H ≤ G | H ≥ F*(G) N*-maximal in G} is a non-empty conjugacy class of subgroups of G. More recently, Iranzo and Perez-Monasor have verified InjN(G) ≠ Φ for all finite groups G satisfying G = CG(E(G))E(G) (see [6]), and have extended this result to a somewhat larger class M of finite groups C(see [7]). One checks, however, that M does not contain all finite groups; for example, S5 ε M.

This paper deals with the interaction between the invariance group of a second order differential equation and its variational formulation. In particular I construct equivalent Lagrangians from all such group actions, thereby successfully completing an earlier attempt of mine which dealt with some traditionally important classes of actions.

In this thesis we study the following two types of hereditary optimal control problems: (i) problems governed by systems of ordinary differential equations with discrete time-delayed arguments appearing in both the state and the control variables; (ii) problems governed by parabolic partial differential equations with Neumann boundary conditions involving time-delays.