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In the late 1960’s, E.A. Nordgren and J.V. Ryff studied composition operators on the Hardy space H2. They provided upper and lower bounds on the norms of general composition operators and gave the exact norm in the case where the symbol map is an inner function. Composition operators themselves, on various other spaces, have been studied by many authors since and much deep work has been done concerning them. Recently, however B.D. MacCluer and T. Kriete have developed the study of composition operators on very general weighted Bergman spaces of the unit disk in the complex plane. My starting point is this work. Composition operators serve well to link the two areas of analysis, operator theory and complex function theory. The products of this link lie deep in complex analysis and are diverse indeed. These include a thorough study of the Schr¨oeder functional equation and its solutions, see [16] and the references therein, in fact some of the well known conjectures can be linked to composition operators. Nordgren, [12], has shown that the Invariant Subspace Problem can be solved by classifying the minimal invariant subspaces of a certain composition operator on H2, and de Branges used composition operators to prove the Bieberbach conjecture. In this thesis, I use various methods from complex function theory to prove results concerning composition operators on weighted Bergman spaces of the unit disk, the main result is the confirmation of two conjectures of T. Kriete, which appeared in [7]. I also construct, in the final chapter, inner functions which map one arbitrary weighted Bergman space into another.

In this thesis I study decidability, complexity and structural properties of strong and weak bisimilarity with respect to two process algebras, Basic Process Algebras and Basic Parallel Process Algebras. The decidability of strong bisimilarity for both algebras is an established result. For the subclasses of normed BPA-processes and BPP there even exist polynomial decision procedures. The complexity of deciding strong bisimilarity for the whole class of BPP is unsatisfactory since it is not bounded by any primitive recursive function. Here we present a new approach that encodes BPP as special polynomials and expresses strong bisimulation in terms of polynomial ideals and then uses a theorem about polynomial ideals (Hilbert's Basis Theorem) and an algorithm from computer algebra (Gröbner bases) to construct a new decision procedure. For weak bisimilarity, Hirshfeld found a decision procedure for the subclasses of totally normed BPA-processes and BPP, and Esparza demonstrated a semidecision procedure for general BPP. The remaining questions are still unsolved. Here we provide some lower bounds on the computational complexity of a decision procedure that might exist. For BPP we show that the decidability problem is NP-hard (even for the class of totally normed BPP), for BPA-processes we show that the decidability problem is PSPACE-hard. Finally we study the notion of weak bisimilarity in terms of its inductive definition. We start from the relation containing all pairs of processes and then form a non-increasing chain of relations by eliminating pairs that do not satisfy a certain expansion condition. These relations are labelled by ordinal numbers and are called approximants. We know that this chain eventually converges for some a' such that =a' = =b' = = for all a' < b'. We study the upper and lower bounds on such ordinals a'. We prove that for BPA, a' => w w, and for BPPA, a' => w.2. For some restricted classes of BPA and BPPA we show that = = =w.2.

We study the topology of spaces of polygons in Euclidean space, viewed up to translations. The main results concern the structure of the homology groups and of the cohomology rings of the spaces. In particular, it is shown that the spaces are classified by their cohomology rings. A principal tool used in the proofs is a new lacunary principle for Morse-Bott functions, which may be of independent interest. Several applications are discussed.

We describe the non linear realizations of chiral symmetry group and study some of its implications in elementary particle phycics. In Chapter 1, the basic concepts of non linear realization techniques are introduced the way of reviewing the special case of the chiral SU(2)::SU(2) group. In Chapter 2, the general formalism for chiral SU(η)xSU(η) is developed. This part is wholly dependent on the work by Coleman, Wess and Zumino. In Chapter 3 the method is generalized for local chiral invariance to describe the non-linear gauge fields. The Chapter 4 illustrates the use of non linear realization techniques in conjunction with the phenomenological lagrangian. This chapter is introductory to the final Chapter, 5, in which we have attempted to use the i) phenomenological lagrangian with non linear realization of chiral SU(3)xSU(3) to calculate some low energy hadronic reactions. As an important addition, a description of broken chiral SU(3)xSU(3)' is given. This follows the general scheme put forward by Gell-Kann, Cakes and Renner.

This thesis surveys the links between mathematics and algebraic logic in England in the first half of the 19th century. In particular, we show the impact that De Morgan's work on the calculus of functions in 1836 had on the shaping of his logic of relations in 1860. Similarly we study Boole's background in D-operational methods and its impact on his calculus of logic in 1847. The starting point of the thesis is Lagrange's algebraic calculus and Laplace's analytical methods prominent in late 18th century French mathematics. Revival in mathematical research in early 19th century England was mainly effected through the diffusion of Lagrange's calculus of operations as further developed by Arbogast, Servois and others in the 1800's and of Laplace's theory of attractions. Lagrange's algebraic calculus and Laplace's methods in analysis – particularly on functional equations – were considerably developed by Herschel and Babbage during the period 1812-1820. Further research on the foundations of the calculus of operations and functions was provided by Murphy, De Morgan and Gregory in the late 1830's. Symbolic methods in analysis were further extended by Boole in 1844. Boole was followed by several analysts distinguished in their obsession in further vindicating these methods through applications on two differential equations which originally appeared in Laplace's planetary physics. We record the main issues of De Morgan's logic and their mathematical background. Special reference is given to his logic of relations and its connection with his foundational study of the calculus of functions. On similar lines we study Boole's algebraic cast of logic drawing consequently a comparison between his two major works on logic. Moreover we emphasise his epistemological views and his evaluation of symbolical methods within logic and analysis.