The problems discussed in this thesis arise in the geometry of numbers, a part of the theory of numbers which originated under the inspiration of Minkowski at the turn of the last century. The first chapter is introductory and provides the background to the main part of the thesis. The well known theorems which are needed later are stated without proofs, references being given in the appendix. In each of the remaining three chapters a different problem is considered and is preceded by a short summary. The theorems are numbered according to the chapter in which they occur.

This thesis deals with various aspects of development in the field of both ordinary and basic hypergeometric functions. It comprises six chapters. The first chapter gives a brief survey of some of the recents developments in this field including the work done in the present thesis. The second chapter gives a systematic study of the transformations connected with the partial sums of generalised hypergeometric series, both ordinary and basic. The main theorem proved in the chapter gives the most general relation of its type. The third chapter is concerned with the development of the transformation theory of bilateral cognate trigonometrical series and generalises all the known results in that field. The fourth chapter gives the integral analogues of some of the transformations of basic series analogous to those for the ordinary series in Chapter VI of Bailey's Cambridge Tract. The fifth chapter deals with a systematic classification and study of two and three term relations between special kinds of well-poised series of the type and gives a new method, by integrals, of deducing these transformations easily. The last and the sixth chapter gives a number of identities involving basic analogues of Appell's hypergeometric functions of two variables and some associated functions.

After introducing basic notation and results in chapter one, we begin studying the model theory of the Peano axioms, P, proper in the second chapter where we give a proof of Rabin's theorem :- that P is not axiomatizable by any consistent set of [sigma]n sentences for any n[epsilon][omega], and also answer a question of Gaifman raised in Another problem, from the same article, is partially answered in chapter three, where we show every countable non-standard model, M, 'of P has an elementary equivalent end extension solving a Diophantine equation with coefficients in M, that was not solvable in M. In chapter four we investigate substructures of countable non-standard models of P, and show that every such model M, contains 2 substructures all isomorphic to M. Other related results are also proved. Chapter five contains theorems on indescernibles and omitting certain types in models of P. Chapter six is concerned with the following problem the set), of elementary substructures of M, is lattice ordered by inclusion. Which lattices are of the form for some We show that the pentagon lattice is of this form (answering a question suggested in [7] p. 280)and produce a class of non-modular lattices all of whose members are not of the form for any M = N, the standard model of P. Elementary co-final extensions of models of P are also investigated in this chapter. Finally, chapter seven concludes the thesis by posing some open problems suggested by the preceding text.

<p> We investigate the properties of finite tuples of commuting isometries that are constrained by a system of polynomial equations. More precisely, suppose <i>I</i> is an ideal in the ring of complex <i> n</i>-variable polynomials and that <i>I</i> determines an affine algebraic variety of dimension 1. Further, suppose that there are <i> n</i> commuting Hilbert space isometries <i>V</i>₁, . . . ,<i>V_n</i> with the property that <i>p</i>(<i> V₁</i>, . . . ,<i>Vⁿ</i>) = 0 for each <i>p</i> in the ideal <i>I.</i> Because the <i> n</i>-tuple (<i>V</i>₁, . . . ,<i>V_n</i>) can be decomposed as a direct sum of an <i>n</i>-tuple of unitary operators and a completely non-unitary <i>n</i>-tuple, we assume that the unitary summand is trivial. Under these assumptions, we can decompose the <i>n</i>-tuple as a finite direct sum of <i>n</i>-tuples of the form (<i>T</i>₁, . . . ,<i>T_n</i>), where each <i>T_i</i> either is multiplication by a scalar or is unitarily equivalent to a unilaterial shift of some multiplicity. We then focus on the special case in which <i>V</i>₁, . . . ,<i>V_n</i> are generalized shifts of finite multiplicity. In this case we are able to classify such <i>n</i>-tuples up to something we term &lsquo;virtual similarity&rsquo; using two pieces of data : the ideal of all polynomials p such that <i>p</i>(<i>V</i>₁, . . . ,<i>V_n</i>) = 0 and a finite tuple of positive integers.</p><p>

Abstract not available.

No description available.

No description available.

Cycle Systems

<p> In this study, we take a numerical modeling approach to investigate crustal-scale fluid flow in areas of crustal extension subjected to normal and/or detachment faulting. In areas subjected to continental extension, brittle normal faulting of the upper crust leads to steep topographic gradients that provide the driving force (head gradient) and pathways (fractures) to groundwater flow. Ductile extension in the lower crust is characterized by high heat fluxes, granitic intrusion, and migmatitic gneiss domes. When downward fluid flow reaches the detachment shear zone that separates the upper and lower crust, high heat flux combined with magmatic/metamorphic fluids cause density inversions leading to buoyancy-driven upward flow. Therefore, mid-crustal shear zones represent crustal-scale hydrothermal systems characterized by buoyancy-driven fluids convection. Several geochemical studies of North American core complexes show that circulation of meteoric fluids during the development of the detachment shear zone is ubiquitous. The circulation of fluids at lower crustal levels is the result of the interplay between rock type, temperature, porosity and permeability, and fluid pathways. </p><p> We present the results of finite-element numerical models using ABAQUS/Standard that simulate groundwater flow in an idealized cross-section of a metamorphic core complex. The simulations investigate the effects of (1) crust and fault permeability and porosity, (2) width of the faults, (3) depth of the faults and shear zone, and (4) topography (head gradient) on groundwater flow. Our results show that fluid migration to mid- to lower-crustal levels is significantly fault-controlled and depends primarily on the permeability contrast between the fault zone and the crustal rock as well as the presence of a permeable shear zone, and additionally, our simulations reveal that higher fault/crust permeability contrast leads to channelized flow in the fault zone and shear zone, while lower contrast allow leakage of the fluids in the crust. </p><p>

<p> In this work we investigate the global symmetries of six-dimensional superconformal field theories (6D SCFTs) via their description in F-theory. We provide computer algebra system routines determining global symmetry maxima for all known 6D SCFTs while tracking the singularity types of the associated elliptic fibrations. We tabulate these bounds for many CFTs including every 0-link based theory. The approach we take provides explicit tracking of geometric information which has remained implicit in the classifications of 6D SCFTs to date. We derive a variety of new geometric restrictions on collections of singularity collisions in elliptically fibered Calabi-Yau varieties and collect data from local model analyses of these collisions. The resulting restrictions are sufficient to match the known gauge enhancement structure constraints for all 6D SCFTs without appeal to anomaly cancellation and enable our global symmetry computations for F-theory SCFT models to proceed similarly. </p><p>