Latest content added for Digital Library Collection: UNT Theses and Dissertationshttps://digital.library.unt.edu/explore/collections/UNTETD/browse/?fq=untl_institution:UNT&fq=str_degree_department:Department+of+Mathematics2017-10-09T11:44:47-05:00UNT LibrariesThis is a custom feed for browsing Digital Library Collection: UNT Theses and DissertationsA General Approach to Buhlmann Credibility Theory2017-10-09T11:44:47-05:00https://digital.library.unt.edu/ark:/67531/metadc1011812/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc1011812/"><img alt="A General Approach to Buhlmann Credibility Theory" title="A General Approach to Buhlmann Credibility Theory" src="https://digital.library.unt.edu/ark:/67531/metadc1011812/small/"/></a></p><p>Credibility theory is widely used in insurance. It is included in the examination of the Society of Actuaries and in the construction and evaluation of actuarial models. In particular, the Buhlmann credibility model has played a fundamental role in both actuarial theory and practice. It provides a mathematical rigorous procedure for deciding how much credibility should be given to the actual experience rating of an individual risk relative to the manual rating common to a particular class of risks. However, for any selected risk, the Buhlmann model assumes that the outcome random variables in both experience periods and future periods are independent and identically distributed. In addition, the Buhlmann method uses sample mean-based estimators to insure the selected risk, which may be a poor estimator of future costs if only a few observations of past events (costs) are available. We present an extension of the Buhlmann model and propose a general method based on a linear combination of both robust and efficient estimators in a dependence framework. The performance of the proposed procedure is demonstrated by Monte Carlo simulations.</p>Crystallographic Complex Reflection Groups and the Braid Conjecture2017-10-09T11:44:47-05:00https://digital.library.unt.edu/ark:/67531/metadc1011877/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc1011877/"><img alt="Crystallographic Complex Reflection Groups and the Braid Conjecture" title="Crystallographic Complex Reflection Groups and the Braid Conjecture" src="https://digital.library.unt.edu/ark:/67531/metadc1011877/small/"/></a></p><p>Crystallographic complex reflection groups are generated by reflections about affine hyperplanes in complex space and stabilize a full rank lattice. These analogs of affine Weyl groups have infinite order and were classified by V.L. Popov in 1982. The classical Braid theorem (first established by E. Artin and E. Brieskorn) asserts that the Artin group of a reflection group (finite or affine Weyl) gives the fundamental group of regular orbits. In other words, the fundamental group of the space with reflecting hyperplanes removed has a presentation mimicking that of the Coxeter presentation; one need only remove relations giving generators finite order. N.V Dung used a semi-cell construction to prove the Braid theorem for affine Weyl groups. Malle conjectured that the Braid theorem holds for all crystallographic complex reflection groups after constructing Coxeter-like reflection presentations. We show how to extend Dung's ideas to crystallographic complex reflection groups and then extend the Braid theorem to some groups in the infinite family [G(r,p,n)]. The proof requires a new classification of crystallographic groups in the infinite family that fail the Steinberg theorem.</p>A Classification of the Homogeneity of Countable Products of Subsets of Real Numbers2017-10-09T11:44:47-05:00https://digital.library.unt.edu/ark:/67531/metadc1011753/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc1011753/"><img alt="A Classification of the Homogeneity of Countable Products of Subsets of Real Numbers" title="A Classification of the Homogeneity of Countable Products of Subsets of Real Numbers" src="https://digital.library.unt.edu/ark:/67531/metadc1011753/small/"/></a></p><p>Spaces such as the closed interval [0, 1] do not have the property of being homogeneous, strongly locally homogeneous (SLH) or countable dense homogeneous (CDH), but the Hilbert cube has all three properties. We investigate subsets X of real numbers to determine when their countable product is homogeneous, SLH, or CDH. We give necessary and sufficient conditions for the product to be homogeneous. We also prove that the product is SLH if and only if X is zero-dimensional or an interval. And finally we show that for a Borel subset X of real numbers the product is CDH iff X is a G-delta zero-dimensional set or an interval.</p>Numerical Values of the Hausdorff and Packing Measures for Limit Sets of Iterated Function Systems2017-10-09T11:44:47-05:00https://digital.library.unt.edu/ark:/67531/metadc1011825/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc1011825/"><img alt="Numerical Values of the Hausdorff and Packing Measures for Limit Sets of Iterated Function Systems" title="Numerical Values of the Hausdorff and Packing Measures for Limit Sets of Iterated Function Systems" src="https://digital.library.unt.edu/ark:/67531/metadc1011825/small/"/></a></p><p>In the context of fractal geometry, the natural extension of volume in Euclidean space is given by Hausdorff and packing measures. These measures arise naturally in the context of iterated function systems (IFS). For example, if the IFS is finite and conformal, then the Hausdorff and packing dimensions of the limit sets agree and the corresponding Hausdorff and packing measures are positive and finite. Moreover, the map which takes the IFS to its dimension is continuous. Developing on previous work, we show that the map which takes a finite conformal IFS to the numerical value of its packing measure is continuous. In the context of self-similar sets, we introduce the super separation condition. We then combine this condition with known density theorems to get a better handle on finding balls of maximum density. This allows us to extend the work of others and give exact formulas for the numerical value of packing measure for classes of Cantor sets, Sierpinski N-gons, and Sierpinski simplexes.</p>Results in Algebraic Determinedness and an Extension of the Baire Property2017-07-12T03:17:08-05:00https://digital.library.unt.edu/ark:/67531/metadc984214/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc984214/"><img alt="Results in Algebraic Determinedness and an Extension of the Baire Property" title="Results in Algebraic Determinedness and an Extension of the Baire Property" src="https://digital.library.unt.edu/ark:/67531/metadc984214/small/"/></a></p><p>In this work, we concern ourselves with particular topics in Polish space theory. We first consider the space A(U) of complex-analytic functions on an open set U endowed with the usual topology of uniform convergence on compact subsets. With the operations of point-wise addition and point-wise multiplication, A(U) is a Polish ring. Inspired by L. Bers' algebraic characterization of the relation of conformality, we show that the topology on A(U) is the only Polish topology for which A(U) is a Polish ring for a large class of U. This class of U includes simply connected regions, simply connected regions excluding a relatively discrete set of points, and other domains of usual interest. One thing that we deduce from this is that, even though C has many different Polish field topologies, as long as it sits inside another Polish ring with enough complex-analytic functions, it must have its usual topology. In a different direction, we show that the bounded complex-analytic functions on the unit disk admits no Polish topology for which it is a Polish ring.
We also study the Lie ring structure on A(U) which turns out to be a Polish Lie ring with the usual topology. In this case, we restrict our attention to those domains U that are connected. We extend a result of I. Amemiya to see that the Lie ring structure is determined by the conformal structure of U. In a similar vein to our ring considerations, we see that, again for certain domains U of usual interest, the Lie ring A(U) has a unique Polish topology for which it is a Polish Lie ring. Again, the Lie ring A(U) imposes topological restrictions on C. That is, C must have its usual topology when sitting inside any Polish Lie ring isomorphic to A(U).
In the last chapter, we introduce a new ideal of subsets of Polish spaces consisting of what we call residually null sets. From this ideal, we introduce an algebra consisting of what we call R-sets which is consistently a strict extension of the algebra of Baire property sets. We show that the algebra of R-sets is closed under the Alexandrov-Suslin operation and generalize Pettis' Theorem. From this, we provide new automatic continuity results and give a generalization of a result of D. Montgomery which shows that minimal assumptions on the continuity of group operations of an abstract group G with a Polish topology imply that G is actually a Polish group. We also see that many results pertaining to the algebra of Baire property sets generalize to the context of R-sets.</p>Partition Properties for Non-Ordinal Sets under the Axiom of Determinacy2017-07-12T03:17:08-05:00https://digital.library.unt.edu/ark:/67531/metadc984121/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc984121/"><img alt="Partition Properties for Non-Ordinal Sets under the Axiom of Determinacy" title="Partition Properties for Non-Ordinal Sets under the Axiom of Determinacy" src="https://digital.library.unt.edu/ark:/67531/metadc984121/small/"/></a></p><p>In this paper we explore coloring theorems for the reals, its quotients, cardinals, and their combinations. This work is done under the scope of the axiom of determinacy. We also explore generalizations of Mycielski's theorem and show how these can be used to establish coloring theorems. To finish, we discuss the strange realm of long unions.</p>A Decomposition of the Group Algebra of a Hyperoctahedral Group2017-02-19T19:42:09-06:00https://digital.library.unt.edu/ark:/67531/metadc955102/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc955102/"><img alt="A Decomposition of the Group Algebra of a Hyperoctahedral Group" title="A Decomposition of the Group Algebra of a Hyperoctahedral Group" src="https://digital.library.unt.edu/ark:/67531/metadc955102/small/"/></a></p><p>The descent algebra of a Coxeter group is a subalgebra of the group algebra with interesting representation theoretic properties. For instance, the natural map from the descent algebra of the symmetric group to the character ring is a surjective algebra homomorphism, so the descent algebra implicitly encodes information about the representations of the symmetric group. However, this property does not hold for other Coxeter groups. Moreover, a complete set of primitive idempotents in the descent algebra of the symmetric group leads to a decomposition of the group algebra as a direct sum of induced linear characters of centralizers of conjugacy class representatives. In this dissertation, I consider the hyperoctahedral group. When the descent algebra of a hyperoctahedral group is replaced with a generalization called the Mantaci-Reutenauer algebra, the natural map to the character ring is surjective. In 2008, BonnafĂ© asked whether a complete set of idempotents in the Mantaci-Reutenauer algebra could lead to a decomposition of the group algebra of the hyperoctahedral group as a direct sum of induced linear characters of centralizers. In this dissertation, I will answer this question positively and go through the construction of the idempotents, conjugacy class representatives, and linear characters required to do so.</p>Contributions to Descriptive Set Theory2017-02-19T19:42:09-06:00https://digital.library.unt.edu/ark:/67531/metadc955115/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc955115/"><img alt="Contributions to Descriptive Set Theory" title="Contributions to Descriptive Set Theory" src="https://digital.library.unt.edu/ark:/67531/metadc955115/small/"/></a></p><p>Assume AD+V=L(R). In the first chapter, let W^1_1 denote the club measure on \omega_1. We analyze the embedding j_{W^1_1}\restr HOD from the point of view of inner model theory. We use our analysis to answer a question of Jackson-Ketchersid about codes for ordinals less than \omega_\omega. In the second chapter, we provide an indiscernibles analysis for models of the form L[T_n,x]. We use our analysis to provide new proofs of the strong partition property on \delta^1_{2n+1}</p>Rankin-Cohen Brackets for Hermitian Jacobi Forms and Hermitian Modular Forms2017-02-19T19:42:09-06:00https://digital.library.unt.edu/ark:/67531/metadc955117/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc955117/"><img alt="Rankin-Cohen Brackets for Hermitian Jacobi Forms and Hermitian Modular Forms" title="Rankin-Cohen Brackets for Hermitian Jacobi Forms and Hermitian Modular Forms" src="https://digital.library.unt.edu/ark:/67531/metadc955117/small/"/></a></p><p>In this thesis, we define differential operators for Hermitian Jacobi forms and Hermitian modular forms over the Gaussian number field Q(i). In particular, we construct Rankin-Cohen brackets for such spaces of Hermitian Jacobi forms and Hermitian modular forms. As an application, we extend Rankin's method to the case of Hermitian Jacobi forms. Finally we compute Fourier series coefficients of Hermitian modular forms, which allow us to give an example of the first Rankin-Cohen bracket of two Hermitian modular forms. In the appendix, we provide tables of Fourier series coefficients of Hermitian modular forms and also the computer source code that we used to compute such Fourier coefficients.</p>Quantum Drinfeld Hecke Algebras2016-08-31T22:41:47-05:00https://digital.library.unt.edu/ark:/67531/metadc862764/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc862764/"><img alt="Quantum Drinfeld Hecke Algebras" title="Quantum Drinfeld Hecke Algebras" src="https://digital.library.unt.edu/ark:/67531/metadc862764/small/"/></a></p><p>Quantum Drinfeld Hecke algebras extend both Lusztig's graded Hecke algebras and the symplectic reflection algebras of Etingof and Ginzburg to the quantum setting. A quantum (or skew) polynomial ring is generated by variables which commute only up to a set of quantum parameters. Certain finite groups may act by graded automorphisms on a quantum polynomial ring and quantum Drinfeld Hecke algebras deform the natural semi-direct product. We classify these algebras for the infinite family of complex reflection groups acting in arbitrary dimension. We also classify quantum Drinfeld Hecke algebras in arbitrary dimension for the infinite family of mystic reflection groups of Kirkman, Kuzmanovich, and Zhang, who showed they satisfy a Shephard-Todd-Chevalley theorem in the quantum setting. Using a classification of automorphisms of quantum polynomial rings in low dimension, we develop tools for studying quantum Drinfeld Hecke algebras in 3 dimensions. We describe the parameter space of such algebras using special properties of the quantum determinant in low dimension; although the quantum determinant is not a homomorphism in general, it is a homomorphism on the finite linear groups acting in dimension 3.</p>