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UNT Theses and Dissertations
 A Comparative Study of Non Linear Conjugate Gradient Methods
 We study the development of nonlinear conjugate gradient methods, Fletcher Reeves (FR) and Polak Ribiere (PR). FR extends the linear conjugate gradient method to nonlinear functions by incorporating two changes, for the step length αk a line search is performed and replacing the residual, rk (rk=bAxk) by the gradient of the nonlinear objective function. The PR method is equivalent to FR method for exact line searches and when the underlying quadratic function is strongly convex. The PR method is basically a variant of FR and primarily differs from it in the choice of the parameter βk. On applying the nonlinear Rosenbrock function to the MATLAB code for the FR and the PR algorithms we observe that the performance of PR method (k=29) is far better than the FR method (k=42). But, we observe that when the MATLAB codes are applied to general nonlinear functions, specifically functions whose minimum is a large negative number not close to zero and the iterates too are large values far off from zero the PR algorithm does not perform well. This problem with the PR method persists even if we run the PR algorithm for more iterations or with an initial guess closer to the actual minimum. To improve the PR algorithm we suggest finding a better weighing parameter βk, using better line search method and/or using specific line search for certain functions and identifying specific restart criteria based on the function to be optimized. digital.library.unt.edu/ark:/67531/metadc283864/
 Centers of Invariant Differential Operator Algebras for Jacobi Groups of Higher Rank
 Let G be a Lie group acting on a homogeneous space G/K. The center of the universal enveloping algebra of the Lie algebra of G maps homomorphically into the center of the algebra of differential operators on G/K invariant under the action of G. In the case that G is a Jacobi Lie group of rank 2, we prove that this homomorphism is surjective and hence that the center of the invariant differential operator algebra is the image of the center of the universal enveloping algebra. This is an extension of work of Bringmann, Conley, and Richter in the rank 1case. digital.library.unt.edu/ark:/67531/metadc283833/
 On Groups of Positive Type
 We describe groups of positive type and prove that a group G is of positive type if and only if G admits a nontrivial partition. We completely classify groups of type 2, and present examples of other groups of positive type as well as groups of type zero. digital.library.unt.edu/ark:/67531/metadc277804/
 Polish Spaces and Analytic Sets
 A Polish space is a separable topological space that can be metrized by means of a complete metric. A subset A of a Polish space X is analytic if there is a Polish space Z and a continuous function f : Z —> X such that f(Z)= A. After proving that each uncountable Polish space contains a nonBorel analytic subset we conclude that there exists a universally measurable nonBorel set. digital.library.unt.edu/ark:/67531/metadc277605/
 Physical Motivation and Methods of Solution of Classical Partial Differential Equations
 We consider three classical equations that are important examples of parabolic, elliptic, and hyperbolic partial differential equations, namely, the heat equation, the Laplace's equation, and the wave equation. We derive them from physical principles, explore methods of finding solutions, and make observations about their applications. digital.library.unt.edu/ark:/67531/metadc277898/
 Graev Metrics and Isometry Groups of Polish Ultrametric Spaces
 This dissertation presents results about computations of Graev metrics on free groups and characterizes isometry groups of countable noncompact HeineBorel Polish ultrametric spaces. In Chapter 2, computations of Graev metrics are performed on free groups. One of the related results answers an open question of Van Den Dries and Gao. In Chapter 3, isometry groups of countable noncompact HeineBorel Polish ultrametric spaces are characterized. The notion of generalized tree is defined and a correspondence between the isomorphism group of a generalized tree and the isometry group of a HeineBorel Polish ultrametric space is established. The concept of a weak inverse limit is introduced to capture the characterization of isomorphism groups of generalized trees. In Chapter 4, partial results of isometry groups of uncountable compact ultrametric spaces are given. It turns out that every compact ultrametric space has a unique countable orbital decomposition. An orbital space consists of disjoint orbits. An orbit subspace of an orbital space is actually a compact homogeneous ultrametric subspace. digital.library.unt.edu/ark:/67531/metadc271898/
 Traveling Wave Solutions of the Porous Medium Equation
 We prove the existence of a oneparameter family of solutions of the porous medium equation, a nonlinear heat equation. In our work, with space dimension 3, the interface is a half line whose end point advances at constant speed. We prove, by using maximum principle, that the solutions are stable under a suitable class of perturbations. We discuss the relevance of our solutions, when restricted to two dimensions, to gravity driven flows of thin films. Here we extend the results of J. Iaia and S. Betelu in the paper "Solutions of the porous medium equation with degenerate interfaces" to a higher dimension. digital.library.unt.edu/ark:/67531/metadc271876/
 Determinacyrelated 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. digital.library.unt.edu/ark:/67531/metadc271913/
 Descriptive Set Theory and Measure Theory in Locally Compact and Nonlocally Compact Groups
 In this thesis we study descriptivesettheoretic and measuretheoretic 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 LevySteinitz 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 ?03complete. 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 countable union of Haar null sets in a Polish group is Haar null. We show the openly Haar null ideal may be distinct from the Haar null ideal, which resolves an uncertainty of Solecki. We show that compact sets are always Haar null in S? and in any countable product of locally compact noncompact groups, which extends the domain of a result of Dougherty. We show that any countable product of locally compact noncompact groups decomposes into the disjoint union of a meager set and a Haar null set, which gives a partial positive answer to a question of Darji. We display a translation property in the homeomorphism group Homeo+[0,1] which is impossible in any nontrivial locally compact group. Other related results are peppered throughout. digital.library.unt.edu/ark:/67531/metadc271792/
 Real Analyticity of Hausdorff Dimension of Disconnected Julia Sets of Cubic Parabolic Polynomials
 Consider a family of cubic parabolic polynomials given by for nonzero complex parameters such that for each the polynomial is a parabolic polynomial, that is, the polynomial has a parabolic fixed point and the Julia set of , denoted by , does not contain any critical points of . We also assumed that for each , one finite critical point of the polynomial escapes to the superattracting fixed point infinity. So, the Julia sets are disconnected. The concern about the family is that the members of this family are generally not even biLipschitz conjugate on their Julia sets. We have proved that the parameter set is open and contains a deleted neighborhood of the origin 0. Our main result is that the Hausdorff dimension function defined by is real analytic. To prove this we have constructed a holomorphic family of holomorphic parabolic graph directed Markov systems whose limit sets coincide with the Julia sets of polynomials up to a countable set, and hence have the same Hausdorff dimension. Then we associate to this holomorphic family of holomorphic parabolic graph directed Markov systems an analytic family, call it , of conformal graph directed Markov systems with infinite number of edges in order to reduce the problem of real analyticity of Hausdorff dimension for the given family of polynomials to prove the corresponding statement for the family . digital.library.unt.edu/ark:/67531/metadc271768/
 Nonparametric Estimation of Receiver Operating Characteristic Surfaces Via Bernstein Polynomials

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Receiver operating characteristic (ROC) analysis is one of the most widely used methods in evaluating the accuracy of a classification method. It is used in many areas of decision making such as radiology, cardiology, machine learning as well as many other areas of medical sciences. The dissertation proposes a novel nonparametric estimation method of the ROC surface for the threeclass classification problem via Bernstein polynomials. The proposed ROC surface estimator is shown to be uniformly consistent for estimating the true ROC surface. In addition, it is shown that the map from which the proposed estimator is constructed is Hadamard differentiable. The proposed ROC surface estimator is also demonstrated to lead to the explicit expression for the estimated volume under the ROC surface . Moreover, the exact mean squared error of the volume estimator is derived and some related results for the mean integrated squared error are also obtained. To assess the performance and accuracy of the proposed ROC and volume estimators, MonteCarlo simulations are conducted. Finally, the method is applied to the analysis of two real data sets. digital.library.unt.edu/ark:/67531/metadc177212/  Semisupervised and Selfevolving Learning Algorithms with Application to Anomaly Detection in Cloud Computing
 Semisupervised learning (SSL) is the most practical approach for classification among machine learning algorithms. It is similar to the humans way of learning and thus has great applications in text/image classification, bioinformatics, artificial intelligence, robotics etc. Labeled data is hard to obtain in real life experiments and may need human experts with experimental equipments to mark the labels, which can be slow and expensive. But unlabeled data is easily available in terms of web pages, data logs, images, audio, video les and DNA/RNA sequences. SSL uses large unlabeled and few labeled data to build better classifying functions which acquires higher accuracy and needs lesser human efforts. Thus it is of great empirical and theoretical interest. We contribute two SSL algorithms (i) adaptive anomaly detection (AAD) (ii) hybrid anomaly detection (HAD), which are self evolving and very efficient to detect anomalies in a large scale and complex data distributions. Our algorithms are capable of modifying an existing classier by both retiring old data and adding new data. This characteristic enables the proposed algorithms to handle massive and streaming datasets where other existing algorithms fail and run out of memory. As an application to semisupervised anomaly detection and for experimental illustration, we have implemented a prototype of the AAD and HAD systems and conducted experiments in an oncampus cloud computing environment. Experimental results show that the detection accuracy of both algorithms improves as they evolves and can achieve 92.1% detection sensitivity and 83.8% detection specificity, which makes it well suitable for anomaly detection in large and streaming datasets. We compared our algorithms with two popular SSL methods (i) subspace regularization (ii) ensemble of Bayesian submodels and decision tree classifiers. Our contributed algorithms are easy to implement, significantly better in terms of space, time complexity and accuracy than these two methods for semisupervised anomaly detection mechanism. digital.library.unt.edu/ark:/67531/metadc177238/
 Regular Semigroups
 This thesis describes semigroups and the properties of both regular and inverse semigroups. digital.library.unt.edu/ark:/67531/metadc163914/
 Borel Sets and Baire Functions
 This paper examines the relationship between Borel sets and Baire functions. digital.library.unt.edu/ark:/67531/metadc163964/
 Connectedness and Some Concepts Related to Connectedness of a Topological Space
 The purpose of this thesis is to investigate the idea of topological "connectedness" by presenting some of the basic ideas concerning connectedness along with several related concepts. digital.library.unt.edu/ark:/67531/metadc163958/
 Linear Algebras
 This paper is primarily concerned with the fundamental properties of a linear algebra of finite order over a field. A discussion of linear sets of finite order over a field is used as an introduction to these properties. digital.library.unt.edu/ark:/67531/metadc163844/
 A Classification of Regular Planar Graphs
 The purpose of this paper is the investigation and classification of regular planar graphs. The motive behind this investigation was a desire to better understand those properties which allow a graph to be represented in the plane in such a manner that no two edges cross except perhaps at vertices. digital.library.unt.edu/ark:/67531/metadc164029/
 Continuous Multifunctions
 This paper is a discussion of multifunctions, various types of continuity defined on multifunctions, and implications of continuity for the range and domain sets of the multifunctions. digital.library.unt.edu/ark:/67531/metadc164025/
 On the Stielitjes Integral
 This paper is a study of the Stieltjes integral, a generalization of the Riemann integral normally studied in introductory calculus courses. The purpose of the paper is to investigate many of the basic manipulative properties of the integral. digital.library.unt.edu/ark:/67531/metadc164008/
 Completing the Space of Step Functions
 In this thesis a study is made of the space X of all step functions on [0,1]. This investigation includes determining a completion space, X*, for the incomplete space X, defining integration for X*, and proving some theorems about integration in X*. digital.library.unt.edu/ark:/67531/metadc164014/
 Measure Functions
 This thesis examines measure functions. A measure function has as its domain of definition a class of sets. It also must satisfy a certain additive condition. To state a concise definition of a measure function, it is convenient to define set function and completely additive set function. digital.library.unt.edu/ark:/67531/metadc163875/
 Elliptic Geometry
 This thesis discusses elliptic geometry including the order and incidence properties, projective properties and congruence properties. digital.library.unt.edu/ark:/67531/metadc163883/
 Equivalence Classes of Subquotients of Pseudodifferential Operator Modules on the Line
 Certain subquotients of Vec(R)modules of pseudodifferential operators from one tensor density module to another are categorized, giving necessary and sufficient conditions under which two such subquotients are equivalent as Vec(R)representations. These subquotients split under the projective subalgebra, a copy of ????2, when the members of their composition series have distinct Casimir eigenvalues. Results were obtained using the explicit description of the action of Vec(R) with respect to this splitting. In the length five case, the equivalence classes of the subquotients are determined by two invariants. In an appropriate coordinate system, the level curves of one of these invariants are a pencil of conics, and those of the other are a pencil of cubics. digital.library.unt.edu/ark:/67531/metadc149627/
 Kleinian Groups in Hilbert Spaces
 The theory of discrete groups acting on finite dimensional Euclidean open balls by hyperbolic isometries was borne around the end of 19th century within the works of Fuchs, Klein and Poincaré. We develop the theory of discrete groups acting by hyperbolic isometries on the open unit ball of an infinite dimensional separable Hilbert space. We present our investigations on the geometry of limit sets at the sphere at infinity with an attempt to highlight the differences between the finite and infinite dimensional theories. We discuss the existence of fixed points of isometries and the classification of isometries. Various notions of discreteness that were equivalent in finite dimensions, no longer turn out to be in our setting. In this regard, the robust notion of strong discreteness is introduced and we study limit sets for properly discontinuous actions. We go on to prove a generalization of the BishopJones formula for strongly discrete groups, equating the Hausdorff dimension of the radial limit set with the Poincaré exponent of the group. We end with a short discussion on conformal measures and their relation with Hausdorff and packing measures on the limit set. digital.library.unt.edu/ark:/67531/metadc149579/
 Hochschild Cohomology and Complex Reflection Groups
 A concrete description of Hochschild cohomology is the first step toward exploring associative deformations of algebras. In this dissertation, deformation theory, geometry, combinatorics, invariant theory, representation theory, and homological algebra merge in an investigation of Hochschild cohomology of skew group algebras arising from complex reflection groups. Given a linear action of a finite group on a finite dimensional vector space, the skew group algebra under consideration is the semidirect product of the group with a polynomial ring on the vector space. Each representation of a group defines a different skew group algebra, which may have its own interesting deformations. In this work, we explicitly describe all graded Hecke algebras arising as deformations of the skew group algebra of any finite group acting by the regular representation. We then focus on rank two exceptional complex reflection groups acting by any irreducible representation. We consider indepth the reflection representation and a nonfaithful rotation representation. Alongside our study of cohomology for the rotation representation, we develop techniques valid for arbitrary finite groups acting by a representation with a central kernel. Additionally, we consider combinatorial questions about reflection length and codimension orderings on complex reflection groups. We give algorithms using character theory to compute reflection length, atoms, and poset relations. Using a mixture of theory, explicit examples, and calculations using the software GAP, we show that Coxeter groups and the infinite family G(m,1,n) are the only irreducible complex reflection groups for which the reflection length and codimension orders coincide. We describe the atoms in the codimension order for the groups G(m,p,n). For arbitrary finite groups, we show that the codimension atoms are contained in the support of every generating set for cohomology, thus yielding information about the degrees of generators for cohomology. digital.library.unt.edu/ark:/67531/metadc149591/
 On Steinhaus Sets, Orbit Trees and Universal Properties of Various Subgroups in the Permutation Group of Natural Numbers
 In the first chapter, we define Steinhaus set as a set that meets every isometric copy of another set at exactly one point. We show that there is no Steinhaus set for any fourpoint subset in a plane.In the second chapter, we define the orbit tree of a permutation group of natural numbers, and further introduce compressed orbit trees. We show that any rooted finite tree can be realized as a compressed orbit tree of some permutation group. In the third chapter, we investigate certain classes of closed permutation groups of natural numbers with respect to their universal and surjectively universal groups. We characterize twosided invariant groups, and prove that there is no universal group for countable groups, nor universal group for twosided invariant groups in permutation groups of natural numbers. digital.library.unt.edu/ark:/67531/metadc149691/
 Some Generalizations in the Theory of Summable Series
 It will be our purpose to study a generalized definition of sum of a series and the restrictions which must be placed upon it in order that it shall satisfy the generally accepted requirements of any generalized definition of sum of a series. We shall then proceed to investigate the possibilities of further generalizing this process. digital.library.unt.edu/ark:/67531/metadc130256/
 Some Properties of Dini Derivatives
 The purpose of this paper is to derive certain of the fundamental properties of the Dini derivatives of an arbitrary real function. To this end it will be necessary to investigate the properties of the limits superior and inferior of real functions and to prove the Vitali Covering Theorem as well as a fundamental theorem on the metric density of arbitrary point sets. digital.library.unt.edu/ark:/67531/metadc130342/
 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. digital.library.unt.edu/ark:/67531/metadc130370/
 Convergence Preserving Matrices
 This paper is the result of a study of triangular matrices with particular emphasis on those which are convergence preserving transformations. digital.library.unt.edu/ark:/67531/metadc130470/
 Basic Fourier Transforms
 The purpose of this paper is to develop some of the more basic Fourier transforms which are the outgrowth of the Fourier theorem. Although often approached from the standpoint of the series, this paper will approach the theorem from the standpoint of the integral. digital.library.unt.edu/ark:/67531/metadc130482/
 Some Properties of the Perron Integral
 The purpose of this thesis is to restate the definition of the integral as given by O. Perron, to establish some of the fundamental properties of the Perron integral, and to prove the equivalence between the Perron and Lebesgue integrals in the bounded case. digital.library.unt.edu/ark:/67531/metadc130430/
 The Structure of a Boolean Algebra
 The purpose of this chapter is to develop a form of a "free" Boolean algebra with Σ as a base, by imposing the usual Boolean operations on the set Σ and thus generating new elements freely within explicitly prescribed restrictions. digital.library.unt.edu/ark:/67531/metadc130610/
 The Laplace Transformation
 A set of definitions, theorems and proofs to describe the Laplace transformation. digital.library.unt.edu/ark:/67531/metadc130616/
 The Comparability of Cardinals
 The purpose of this composition is to develop a rigorous, axiomatic proof of the comparability of the cardinals of infinite sets. digital.library.unt.edu/ark:/67531/metadc130514/
 An Approximate Solution to the Dirichlet Problem
 In the category of mathematics called partial differential equations there is a particular type of problem called the Dirichlet problem. Proof is given in many partial differential equation books that every Dirichlet problem has one and only one solution. The explicit solution is very often not easily determined, so that a method for approximating the solution at certain points becomes desirable. The purpose of this paper is to present and investigate one such method. digital.library.unt.edu/ark:/67531/metadc130550/
 Compact Convex Sets in Linear Topological Spaces
 The purpose of this paper is to examine properties of convex sets in linear topological spaces with special emphasis on compact convex sets. digital.library.unt.edu/ark:/67531/metadc130516/
 Concerning linear spaces
 The basis for this thesis is H. S. Wall's book, Creative Mathematics, with particular emphasis on the chapter in that book entitled "More About Linear Spaces." digital.library.unt.edu/ark:/67531/metadc130586/
 Concerning the Convergence of Some Nets
 This thesis discusses the convergence of nets through a series of theorems and proofs. digital.library.unt.edu/ark:/67531/metadc130552/
 Topological Spaces, Filters and Nets
 Explores topological spaces, filters, and nets with definitions and examples. digital.library.unt.edu/ark:/67531/metadc130658/
 Random Sampling
 The purpose of this study is to show the use of random sampling in solving certain mathematical problems. The origin of random numbers to be used in sampling is discussed and methods of sampling from known distributions are then given together with an indication that the sampling procedures are unbiased. digital.library.unt.edu/ark:/67531/metadc130442/
 Random Variables of One Dimension
 This thesis examines random variables of one dimension. digital.library.unt.edu/ark:/67531/metadc130444/
 A Generalized Study of the Conjugate and InnerProduct Functions
 The usual practice in any discussion of an innerproduct space is to restrict the field over which the innerproduct space is defined to the field of complex numbers. In defining the innerproduct function, (x,y), a second function is needed; namely the conjugate function (x,y)* so that (x,y) ± (y,x)*. We will attempt to generalize this concept by investigating the existence of a conjugate function defined on fields other than the field of complex numbers and relate this function to an innerproduct function defined on a linear space L over these fields. digital.library.unt.edu/ark:/67531/metadc130818/
 Linear FirstOrder DifferentialDifference Equations of Retarded Type with Constant Coefficients
 This paper is concerned with equations in which all derivatives are ordinary rather than partial derivatives. The customary meanings of differential order and difference order of an equation are observed. digital.library.unt.edu/ark:/67531/metadc130872/
 Extensions of Modules
 This thesis discusses groups, modules, the module of homomorphisms, and extension of modules. digital.library.unt.edu/ark:/67531/metadc130830/
 Some Properties of Rings and Ideals
 The purpose of this paper will be to investigate certain properties of algebraic systems known as rings. digital.library.unt.edu/ark:/67531/metadc130536/
 Compact Topological Spaces
 The purpose of this paper is to investigate some properties of compact topological spaces and to relate these concepts to the separation properties. digital.library.unt.edu/ark:/67531/metadc130506/
 Spaces of HIntegrable Functions
 In this thesis we consider integrals of a certain class of interval functions. Specifically we consider a nondegenerate number interval [a,b], a real valued function m, defined and nondecreasing on [a,b], and the set Hm, of real valued functions f, defined on [a,b] such that: 1) f(a)=0; 2) for each subinterval [p,q] of [a,b], if m(q)m(p)=0, then f(q)f(p)=0; and 3) the set of all sums of the form Σ(Δf)2/Δm for subdivisions D of [a,b] is bounded above. digital.library.unt.edu/ark:/67531/metadc130970/
 Exhaustibility and Related Set Properties
 The purpose of this paper is to develop certain fundamental properties of exhaustible sets and their complements and to examine various set properties which are generalizations, with respect to exhaustible neglect, or wellknown set properties. digital.library.unt.edu/ark:/67531/metadc130236/
 The Order Topology on a Linearly Ordered Set
 The purpose of this paper is to investigate from two viewpoints an orderinduced topology on a set X. digital.library.unt.edu/ark:/67531/metadc131238/