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Crystallographic Complex Reflection Groups and the Braid Conjecture
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.
Date:
August 2017
Creator:
Puente, Philip C
System:
The UNT Digital Library
Cycles and Cliques in Steinhaus Graphs
In this dissertation several results in Steinhaus graphs are investigated. First under some further conditions imposed on the induced cycles in steinhaus graphs, the order of induced cycles in Steinhaus graphs is at most [(n+3)/2]. Next the results of maximum clique size in Steinhaus graphs are used to enumerate the Steinhaus graphs having maximal cliques. Finally the concept of jumbled graphs and Posa's Lemma are used to show that almost all Steinhaus graphs are Hamiltonian.
Date:
December 1994
Creator:
Lim, Daekeun
System:
The UNT Digital Library
The D-Variant of Transfinite Hausdorff Dimension
In this lecture we introduce a new transfinite dimension function for metric spaces which utilizes Henderson's topological D-dimension and ascribes to any metric space either an ordinal number or the symbol Ω. The construction of our function is motivated by that of Urbański's transfinite Hausdorff dimension, tHD. Henderson's dimension is a topological invariant, however, like Hausdorff dimension and tHD the function presented will be invariant under bi-Lipschitz continuous maps and generally not under homeomorphisms. We present some original results on D-dimension and build the general theory for the D-variant of transfinite Hausdorff dimension, \mathrm{t}_D\mathrm{HD}. In particular, we will show for any ordinal number α, existence of a metrizable space which has \mathrm{t}_D\mathrm{HD} greater than or equal to α and less than or equal to \omega_\tau, where τ is the least ordinal which satisfies α < \omega_\tau.
Date:
May 2022
Creator:
Decker, Bryce
System:
The UNT Digital Library
A Decomposition of the Group Algebra of a Hyperoctahedral Group
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, Bonnafé 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.
Date:
December 2016
Creator:
Tomlin, Drew E
System:
The UNT Digital Library
Definable Structures on the Space of Functions from Tuples of Integers into 2
We give some background on the free part of the action of tuples of integers into 2. We will construct specific structures on this space, and then show that certain other structures cannot exist.
Date:
May 2023
Creator:
Olsen, Cody James
System:
The UNT Digital Library
Descriptions and Computation of Ultrapowers in L(R)
The results from this dissertation are an exact computation of ultrapowers by measures on cardinals $\aleph\sb{n},\ n\in w$, in $L(\IR$), and a proof that ordinals in $L(\IR$) below $\delta\sbsp{5}{1}$ represented by descriptions and the identity function with respect to sequences of measures are cardinals. An introduction to the subject with the basic definitions and well known facts is presented in chapter I. In chapter II, we define a class of measures on the $\aleph\sb{n},\ n\in\omega$, in $L(\IR$) and derive a formula for an exact computation of the ultrapowers of cardinals by these measures. In chapter III, we give the definitions of descriptions and the lowering operator. Then we prove that ordinals represented by descriptions and the identity function are cardinals. This result combined with the fact that every cardinal $<\delta\sbsp{5}{1}$ in $L(\IR$) is represented by a description (J1), gives a characterization of cardinals in $L(\IR$) below $\delta\sbsp{5}{1}. Concrete examples of formal computations are shown in chapter IV.
Date:
August 1995
Creator:
Khafizov, Farid T.
System:
The UNT Digital Library
Descriptive Set Theory and Measure Theory in Locally Compact and Non-locally Compact Groups
In this thesis we study descriptive-set-theoretic and measure-theoretic properties of Polish groups, with a thematic emphasis on the contrast between groups which are locally compact and those which are not. The work is divided into three major sections. In the first, working jointly with Robert Kallman, we resolve a conjecture of Gleason regarding the Polish topologization of abstract groups of homeomorphisms. We show that Gleason's conjecture is false, and its conclusion is only true when the hypotheses are considerably strengthened. Along the way we discover a new automatic continuity result for a class of functions which behave like but are distinct from functions of Baire class 1. In the second section we consider the descriptive complexity of those subsets of the permutation group S? which arise naturally from the classical Levy-Steinitz series rearrangement theorem. We show that for any conditionally convergent series of vectors in Euclidean space, the sets of permutations which make the series diverge, and diverge properly, are ?03-complete. In the last section we study the phenomenon of Haar null sets a la Christensen, and the closely related notion of openly Haar null sets. We identify and correct a minor error in the proof of Mycielski that a …
Date:
May 2013
Creator:
Cohen, Michael Patrick
System:
The UNT Digital Library
A Detailed Proof of the Prime Number Theorem for Arithmetic Progressions
We follow a research paper that J. Elstrodt published in 1998 to prove the Prime Number Theorem for arithmetic progressions. We will review basic results from Dirichlet characters and L-functions. Furthermore, we establish a weak version of the Wiener-Ikehara Tauberian Theorem, which is an essential tool for the proof of our main result.
Date:
May 2004
Creator:
Vlasic, Andrew
System:
The UNT Digital Library
Determinacy of Schmidt's Game and Other Intersection Games
Schmidt's game, and other similar intersection games have played an important role in recent years in applications to number theory, dynamics, and Diophantine approximation theory. These games are real games, that is, games in which the players make moves from a complete separable metric space. The determinacy of these games trivially follows from the axiom of determinacy for real games,ADR, which is a much stronger axiom than that asserting all integer games are determined, AD. One of our main results is a general theorem which under the hypothesis AD implies the determinacy of intersection games which have a property allowing strategies to be simplified. In particular, we show that Schmidt's (α,β,ρ) game on R is determined from AD alone, but on Rn for n≥3 we show that AD does not imply the determinacy of this game. We then give an application of simple strategies and prove that the winning player in Schmidt's (α,β,ρ) game on R has a winning positional strategy, without appealing to the axiom of choice. We also prove several other results specifically related to the determinacy of Schmidt's game. These results highlight the obstacles in obtaining the determinacy of Schmidt's game from AD
Date:
May 2020
Creator:
Crone, Logan
System:
The UNT Digital Library
Determinacy-related Consequences on Limit Superiors
Laczkovich proved from ZF that, given a countable sequence of Borel sets on a perfect Polish space, if the limit superior along every subsequence was uncountable, then there was a particular subsequence whose intersection actually contained a perfect subset. Komjath later expanded the result to hold for analytic sets. In this paper, by adding AD and sometimes V=L(R) to our assumptions, we will extend the result further. This generalization will include the increasing of the length of the sequence to certain uncountable regular cardinals as well as removing any descriptive requirements on the sets.
Date:
May 2013
Creator:
Walker, Daniel
System:
The UNT Digital Library
Determining Properties of Synaptic Structure in a Neural Network through Spike Train Analysis
A "complex" system typically has a relatively large number of dynamically interacting components and tends to exhibit emergent behavior that cannot be explained by analyzing each component separately. A biological neural network is one example of such a system. A multi-agent model of such a network is developed to study the relationships between a network's structure and its spike train output. Using this model, inferences are made about the synaptic structure of networks through cluster analysis of spike train summary statistics A complexity measure for the network structure is also presented which has a one-to-one correspondence with the standard time series complexity measure sample entropy.
Date:
May 2007
Creator:
Brooks, Evan
System:
The UNT Digital Library
Development of a Geometry from a Set of Axioms
The purpose of this paper is to develop a geometry based on fourteen axioms and four undefined terms.
Date:
May 1973
Creator:
Glasscock, Anita Louise
System:
The UNT Digital Library
A Development of a Set of Functions Analogous to the Trigonometric and the Hyperbolic Functions
The purpose of this paper is to define and develop a set of functions of an area in such a manner as to be analogous to the trigonometric and the hyperbolic functions.
Date:
August 1954
Creator:
Allen, Alfred I.
System:
The UNT Digital Library
A Development of the Exponential and Logarithmic Functions
This thesis discusses a development of the exponential and logarithmic functions.
Date:
1953
Creator:
Mackey, Benford B.
System:
The UNT Digital Library
The Development of the Natural Numbers by Means of the Peano Postulates
This thesis covers the development of the natural numbers by means of the peano postulates.
Date:
1951
Creator:
Baugh, Orvil Lee
System:
The UNT Digital Library
A Development of the Peano Postulates
The purpose of this paper is to develop the Peano postulates from a weaker axiom system than the system used by John L. Kelley in General Topology. The axiom of regularity which states "If X is a non-empty set, then there is a member Y of X such that the intersection of X and Y is empty." is not assumed in this thesis. The axiom of amalgamation which states "If X is a set, then the union of the elements of X is a set." is also not assumed. All other axioms used by Kelley relevant to the Peano postulates are assumed. The word class is never used in the thesis, though the variables can be interpreted as classes.
Date:
May 1963
Creator:
Peek, Darwin Eugene
System:
The UNT Digital Library
A Development of the Real Number System
The purpose of this paper is to construct the real number system. The foundation upon which the real number system will be constructed will be the system of counting numbers.
Date:
August 1961
Creator:
Matthews, Ronald Louis
System:
The UNT Digital Library
A Development of the Real Number System by Means of Nests of Rational Intervals
The system of rational numbers can be extended to the real number system by several methods. In this paper, we shall extend the rational number system by means of rational nests of intervals, and develop the elementary properties of the real numbers obtained by this extension.
Date:
1949
Creator:
Williams, Mack Lester
System:
The UNT Digital Library
Differentiable Functions
The primary purpose of this thesis is to carefully develop and prove some of the fundamental, classical theorems of the differential calculus for functions of two real variables.
Date:
June 1966
Creator:
McCool, Kenneth B.
System:
The UNT Digital Library
Differentiation in Banach Spaces
This thesis investigates the properties and applications of derivatives of functions whose domain and range are Banach spaces.
Date:
December 1972
Creator:
Heath, James Darrell
System:
The UNT Digital Library
Dimension spectrum and graph directed Markov systems.
Access:
Use of this item is restricted to the UNT Community
In this dissertation we study graph directed Markov systems (GDMS) and limit sets associated with these systems. Given a GDMS S, by the Hausdorff dimension spectrum of S we mean the set of all positive real numbers which are the Hausdorff dimension of the limit set generated by a subsystem of S. We say that S has full Hausdorff dimension spectrum (full HD spectrum), if the dimension spectrum is the interval [0, h], where h is the Hausdorff dimension of the limit set of S. We give necessary conditions for a finitely primitive conformal GDMS to have full HD spectrum. A GDMS is said to be regular if the Hausdorff dimension of its limit set is also the zero of the topological pressure function. We show that every number in the Hausdorff dimension spectrum is the Hausdorff dimension of a regular subsystem. In the particular case of a conformal iterated function system we show that the Hausdorff dimension spectrum is compact. We introduce several new systems: the nearest integer GDMS, the Gauss-like continued fraction system, and the Renyi-like continued fraction system. We prove that these systems have full HD spectrum. A special attention is given to the backward continued fraction …
Date:
May 2006
Creator:
Ghenciu, Eugen Andrei
System:
The UNT Digital Library
Dimension Theory
This paper contains a discussion of topological dimension theory. Original proofs of theorems, as well as a presentation of theorems and proofs selected from Ryszard Engelking's Dimension Theory are contained within the body of this endeavor. Preliminary notation is introduced in Chapter I. Chapter II consists of the definition of and theorems relating to the small inductive dimension function Ind. Large inductive dimension is investigated in Chapter III. Chapter IV comprises the definition of covering dimension and theorems discussing the equivalence of the different dimension functions in certain topological settings. Arguments pertaining to the dimension o f Jn are also contained in Chapter IV.
Date:
August 1986
Creator:
Frere, Scot M. (Scot Martin)
System:
The UNT Digital Library
Dimensions in Random Constructions.
We consider random fractals generated by random recursive constructions, prove zero-one laws concerning their dimensions and find their packing and Minkowski dimensions. Also we investigate the packing measure in corresponding dimension. For a class of random distribution functions we prove that their packing and Hausdorff dimensions coincide.
Date:
May 2002
Creator:
Berlinkov, Artemi
System:
The UNT Digital Library
Dimensions of statistically self-affine functions and random Cantor sets
The subject of fractal geometry has exploded over the past 40 years with the availability of computer generated images. It was seen early on that there are many interesting questions at the intersection of probability and fractal geometry. In this dissertation we will introduce two random models for constructing fractals and prove various facts about them.
Date:
May 2023
Creator:
Jones, Taylor
System:
The UNT Digital Library