In this thesis, we study local mixture models with a Hilbert space structure. First, we consider the fibre bundle structure of local mixture models in a Hilbert space. Next, the spectral decomposition is introduced in order to construct local mixture models. We analyze the approximation error asymptotically in the Hilbert space. After that, we will discuss the convexity structure of local mixture models. There are two forms of convexity conditions to consider, first due to positivity in the $-1$-affine structure and the second by points having to lie inside the convex hull of a parametric family. It is shown that the set of mixture densities is located inside the intersection of the sets defined by these two convexities. Finally, we discuss the impact of the approximation error in the Hilbert space when the domain of mixing variable changes.

In the first three chapters of this thesis we study two conjectures relating arithmetic with geometry, namely Tate and Lang&rsquo / s conjectures, for a certain class of algebraic surfaces. The surfaces we are interested in are assumed to be defined over a number field, have irregularity two and admit a genus two fibration over an elliptic curve. In the final chapter of the thesis we prove the isomorphism of the Picard motives of an arbitrary variety and its Albanese variety.

A construction is given for which the Hausdorff measure and dimension of an arbitrary abstract compact metric space (X, d) can be encoded in a spectral triple. By introducing the concept of resolving sequence of open covers, conditions are given under which the topology, metric, and Hausdorff measure can be recovered from a spectral triple dependent on such a sequence. The construction holds for arbitrary compact metric spaces, generalizing previous results for fractals, as well as the original setting of manifolds, and also holds when Hausdorff and box dimensions differ---in particular, it does not depend on any self-similarity or regularity conditions on the space. The only restriction on the space is that it have positive s₀ dimensional Hausdorff measure, where s₀ is the Hausdorff dimension of the space, assumed to be finite. Also, X does not need to be embedded in another space, such as Rⁿ.

The entire thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file; a non-technical public abstract appears in the public.pdf file. Title from PDF of title page (University of Missouri--Columbia, viewed on January 14, 2010). Thesis advisor: Dr. Alex Iosevich. Includes bibliographical references.

The present thesis has three main topics: geometry of coamoebas, hypergeometric functions, and geometry of zeros. First, we study the coamoeba of a Laurent polynomial f in n complex variables. We define a simpler object, which we call the lopsided coamoeba, and associate to the lopsided coamoeba an order map. That is, we give a bijection between the set of connected components of the complement of the closed lopsided coamoeba and a finite set presented as the intersection of an affine lattice and a certain zonotope. Using the order map, we then study the topology of the coamoeba. In particular, we settle a conjecture of M. Passare concerning the number of connected components of the complement of the closed coamoeba in the case when the Newton polytope of f has at most n+2 vertices. In the second part we study hypergeometric functions in the sense of Gel'fand, Kapranov, and Zelevinsky. We define Euler-Mellin integrals, a family of Euler type hypergeometric integrals associated to a coamoeba. As opposed to previous studies of hypergeometric integrals, the explicit nature of Euler-Mellin integrals allows us to study in detail the dependence of A-hypergeometric functions on the homogeneity parameter of the A-hypergeometric system. Our main result is a complete description of this dependence in the case when A represents a toric projective curve. In the last chapter we turn to the theory of real univariate polynomials. The famous Descartes' rule of signs gives necessary conditions for a pair (p,n) of integers to represent the number of positive and negative roots of a real polynomial. We characterize which pairs fulfilling Descartes' conditions are realizable up to degree 7, and we provide restrictions valid in arbitrary degree.

This thesis describes a geometric approach to integrable systems. In the first part we describe the geometry of Drinfeld--Sokolov integrable hierarchies including the corresponding tau-functions. Motivated by a relation between Drinfeld--Sokolov hierarchies and certain physical partition functions, we define a dispersionless limit of Drinfeld--Sokolov systems. We introduce a class of solutions which we call string solutions and prove that the tau-functions of string solutions satisfy Virasoro constraints generalizing those familiar from two-dimensional quantum gravity. In the second part we explain how procedures of Hamiltonian and quasi-Hamiltonian reductions in symplectic geometry arise naturally in the context of shifted symplectic structures. All constructions that appear in quasi-Hamiltonian reduction have a natural interpretation in terms of the classical Chern-Simons theory that we explain. As an application, we construct a prequantization of character stacks purely locally. / text

published_or_final_version / abstract / toc / Computer Science and Information Systems / Master / Master of Philosophy

No description available.

This thesis is concerned with developing and implementing algorithms based upon the geometry of pictures. Spherical pictures have been used in many areas of combinatorial group theory, and particularly, they have shown to be a useful method when studying the second homotopy module, 1T2, of a presentation ([3],[4],[7],[12],[41] and [64]). Computational programs that implement picture theoretical and design algorithms could advance the areas in which picture theory can be used, due to the much faster time taken to derive results than that of manual calculations. A variety of algorithms are presented. A data structure has been devised to represent spherical pictures. A method is given that verifies that a given data structure represents a picture, or set of pictures, over a group presentation. This method includes a new planarity testing algorithm, which can be performed on any graph. A computational algorithm has been implemented that determines if a given presentation defines a group extension. This work is based upon the algorithm of Baik et al. [1] which has been developed using the theory of pictures. A 3-presentation for a group G is given by < P, s >, where P is a presentation for G and s is a set of generators for 1T2. The set s can be described in a number of ways. An algorithm is given that produces a generating set of spherical pictures for 1T2 when s is given in the form of identity sequences. Conversely, if s is given in terms of spherical pictures, then the corresponding identity sequences that describe 1T2 can be determined. The above algorithms are contained in the Spherical PIcture Editor (SPICE). SPICE is a software package that enables a user to manually draw pictures over group presentations and, for these pictures, call the algorithms described above. It also contains a library of generating pictures for the non abelian groups of order at most 30. Furthermore, a method has been implemented that automatically draws a spherical picture from a corresponding identity sequence. Again, this new graph drawing technique can be performed on any arbitrary graph.

Includes bibliography. / 281 p ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--Dept. of Pure Mathematics, University of Adelaide, 1978