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 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/
 The MooreSmith Limit
 It is the purpose of this thesis to indicate in more detail how various limits defined in analysis, as well as other concepts not ordinarily defined as limits, may be obtained as special cases of the MooreSmith limit. digital.library.unt.edu/ark:/67531/metadc107833/
 Generic Algebras and KazhdanLusztig Theory for Monomial Groups

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The IwahoriHecke algebras of Coxeter groups play a central role in the study of representations of semisimple Lietype groups. An important tool is the combinatorial approach to representations of IwahoriHecke algebras introduced by Kazhdan and Lusztig in 1979. In this dissertation, I discuss a generalization of the IwahoriHecke algebra of the symmetric group that is instead based on the complex reflection group G(r,1,n). Using the analogues of Kazhdan and Lusztig's Rpolynomials, I show that this algebra determines a partial order on G(r,1,n) that generalizes the ChevalleyBruhat order on the symmetric group. I also consider possible analogues of KazhdanLusztig polynomials. digital.library.unt.edu/ark:/67531/metadc5235/  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/
 Integration of Vector Valued Functions
 This paper develops an integral for Lebesgue measurable functions mapping from the interval [0, 1] into a Banach space. digital.library.unt.edu/ark:/67531/metadc131526/
 RModules for the Alexander Cohomology Theory
 The Alexander Wallace Spanier cohomology theory associates with an arbitrary topological space an abelian group. In this paper, an arbitrary topological space is associated with an Rmodule. The construction of the Rmodule is similar to the Alexander Wallace Spanier construction of the abelian group. digital.library.unt.edu/ark:/67531/metadc131595/
 Understanding Ancient Math Through Kepler: A Few Geometric Ideas from The Harmony of the World
 Euclid's geometry is wellknown for its theorems concerning triangles and circles. Less popular are the contents of the tenth book, in which geometry is a means to study quantity in general. Commensurability and rational quantities are first principles, and from them are derived at least eight species of irrationals. A recently republished work by Johannes Kepler contains examples using polygons to illustrate these species. In addition, figures having these quantities in their construction form solid shapes (polyhedra) having origins though Platonic philosophy and Archimedean works. Kepler gives two additional polyhedra, and a simple means for constructing the “divine” proportion is given. digital.library.unt.edu/ark:/67531/metadc3269/
 The History of the Calculus
 The purpose of this essay is to trace the development of the concepts of the calculus from their first known appearance, through the formal invention of the method of the calculus in the second half of the seventeenth century, to our own day. digital.library.unt.edu/ark:/67531/metadc75389/
 Comparison of Some Mappings in Topology
 The main purpose of this paper is the study of transformations in topological space and relationships between special types of transformations. digital.library.unt.edu/ark:/67531/metadc108253/
 Uniqueness Results for the Infinite Unitary, Orthogonal and Associated Groups
 Let H be a separable infinite dimensional complex Hilbert space, let U(H) be the Polish topological group of unitary operators on H, let G be a Polish topological group and φ:G→U(H) an algebraic isomorphism. Then φ is a topological isomorphism. The same theorem holds for the projective unitary group, for the group of *automorphisms of L(H) and for the complex isometry group. If H is a separable real Hilbert space with dim(H)≥3, the theorem is also true for the orthogonal group O(H), for the projective orthogonal group and for the real isometry group. The theorem fails for U(H) if H is finite dimensional complex Hilbert space. digital.library.unt.edu/ark:/67531/metadc6136/
 Infinite Planar Graphs
 How many equivalence classes of geodesic rays does a graph contain? How many bounded automorphisms does a planar graph have? Neimayer and Watkins studied these two questions and answered them for a certain class of graphs. Using the concept of excess of a vertex, the class of graphs that Neimayer and Watkins studied are extended to include graphs with positive excess at each vertex. The results of this paper show that there are an uncountable number of geodesic fibers for graphs in this extended class and that for any graph in this extended class the only bounded automorphism is the identity automorphism. digital.library.unt.edu/ark:/67531/metadc2545/
 A Method for Approximating the Distributed Loads of an Airplane by Sets of Point Loads
 This paper gives the derivation of a method for determining the forces to be applied to these points which will simulate the load distributed over all the airplane. digital.library.unt.edu/ark:/67531/metadc107936/
 Uniformly σFinite Disintegrations of Measures
 A disintegration of measure is a common tool used in ergodic theory, probability, and descriptive set theory. The primary interest in this paper is in disintegrating σfinite measures on standard Borel spaces into families of σfinite measures. In 1984, Dorothy Maharam asked whether every such disintegration is uniformly σfinite meaning that there exists a countable collection of Borel sets which simultaneously witnesses that every measure in the disintegration is σfinite. Assuming Gödel’s axiom of constructability I provide answer Maharam's question by constructing a specific disintegration which is not uniformly σfinite. digital.library.unt.edu/ark:/67531/metadc84165/
 Complemented Subspaces of Bounded Linear Operators
 For many years mathematicians have been interested in the problem of whether an operator ideal is complemented in the space of all bounded linear operators. In this dissertation the complementation of various classes of operators in the space of all bounded linear operators is considered. This paper begins with a preliminary discussion of linear bounded operators as well as operator ideals. Let L(X, Y ) be a Banach space of all bounded linear operator between Banach spaces X and Y , K(X, Y ) be the space of all compact operators, and W(X, Y ) be the space of all weakly compact operators. We denote space all operator ideals by O. digital.library.unt.edu/ark:/67531/metadc4349/
 Level Curves of the Angle Function of a Positive Definite Symmetric Matrix

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Given a real N by N matrix A, write p(A) for the maximum angle by which A rotates any unit vector. Suppose that A and B are positive definite symmetric (PDS) N by N matrices. Then their Jordan product {A, B} := AB + BA is also symmetric, but not necessarily positive definite. If p(A) + p(B) is obtuse, then there exists a special orthogonal matrix S such that {A, SBS^(1)} is indefinite. Of course, if A and B commute, then {A, B} is positive definite. Our work grows from the following question: if A and B are commuting positive definite symmetric matrices such that p(A) + p(B) is obtuse, what is the minimal p(S) such that {A, SBS^(1)} indefinite? In this dissertation we will describe the level curves of the angle function mapping a unit vector x to the angle between x and Ax for a 3 by 3 PDS matrix A, and discuss their interaction with those of a second such matrix. digital.library.unt.edu/ark:/67531/metadc28376/  NearRings
 The primary objective of this work is to discuss some of the elementary properties of nearrings as they are related to rings. This study is divided into three subdivisions: (1) Basic Properties and Concepts of NearRings; (2) The Ideal Structure of NearRings; and (3) Homomorphism and Isomorphism of NearRings. digital.library.unt.edu/ark:/67531/metadc131500/
 Completely 0Simple Semigroups
 The purpose of this thesis is to explore some of the characteristics of 0simple semigroups and completely 0simple semigroups. digital.library.unt.edu/ark:/67531/metadc130974/
 TFunctions
 The main purpose of this paper is to make a detailed study of a certain class T of complex functions. The functions of the class T have a special mapping property and are meromorphic in every region. As an application of this study, certain elementary functions are defined and studied in terms of a special Tfunction. digital.library.unt.edu/ark:/67531/metadc108077/
 Product and Function Spaces
 In this paper the Cartesian product topology for an arbitrary family of topological spaces and some of its basic properties are defined. The space is investigated to determine which of the separation properties of the component spaces are invariant. digital.library.unt.edu/ark:/67531/metadc131391/
 Properties of Order Relations and Certain Partly Ordered Systems
 The purpose of this paper is to present a study of partly ordered sets. It includes a rigorous development of relations based on the notion of a relation as a set, lattices, and theorems concerning the lattice of subgroups of a group. digital.library.unt.edu/ark:/67531/metadc108130/
 MycielskiRegular Measures
 Let μ be a Radon probability measure on M, the ddimensional Real Euclidean space (where d is a positive integer), and f a measurable function. Let P be the space of sequences whose coordinates are elements in M. Then, for any point x in M, define a function ƒn on M and P that looks at the first n terms of an element of P and evaluates f at the first of those n terms that minimizes the distance to x in M. The measures for which such sequences converge in measure to f for almost every sequence are called Mycielskiregular. We show that the selfsimilar measure generated by a finite family of contracting similitudes and which up to a constant is the Hausdorff measure in its dimension on an invariant set C is Mycielskiregular. digital.library.unt.edu/ark:/67531/metadc84171/
 A Set of Axioms for a Topological Space
 Axioms for a topological space are generally based on neighborhoods where "neighborhood" is an undefined term. Then, limit points are defined in terms of neighborhoods. However, limit points seem to be the basic concept of a topological space, rather than neighborhoods. For this reason, it will be attempted to state a set of axioms for a topological space, using limit point as the undefined concept, and to delete the idea of neighborhoods from the theory. digital.library.unt.edu/ark:/67531/metadc108089/
 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. digital.library.unt.edu/ark:/67531/metadc96984/
 On Sets and Functions in a Metric Space
 The purpose of this thesis is to study some of the properties of metric spaces. An effort is made to show that many of the properties of a metric space are generalized properties of R, the set of real numbers, or Euclidean nspace, and are specific cases of the properties of a general topological space. digital.library.unt.edu/ark:/67531/metadc131457/
 Lebesgue Linear Measure
 This paper discusses the concept of a general definition of measure, and shows that the Lebesgue measure satisfies the requirements set forth for the ideal definition. digital.library.unt.edu/ark:/67531/metadc75587/
 Compactness and Equivalent Notions
 One of the classic theorems concerning the real numbers states that every open cover of a closed and bounded subset of the real line contains a finite subcover. Compactness is an abstraction of that notion, and there are several ideas concerning it which are equivalent and many which are similar. The purpose of this paper is to synthesize the more important of these ideas. This synthesis is accomplished by demonstrating either situations in which two ordinarily different conditions are equivalent or combinations of two or more properties which will guarantee a third. digital.library.unt.edu/ark:/67531/metadc130821/
 Dimensions in random constructions.
 We consider random fractals generated by random recursive constructions, prove zeroone 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. digital.library.unt.edu/ark:/67531/metadc3160/
 Some Theorems and Product Spaces
 This thesis is a study of some axioms and theorems, and product spaces. digital.library.unt.edu/ark:/67531/metadc107937/
 Metric Spaces
 This thesis covers fundamental properties of metric spaces, as well as completeness, compactness, and separability of metric spaces. digital.library.unt.edu/ark:/67531/metadc107938/
 Algebraic Integers
 The primary purpose of this thesis is to give a substantial generalization of the set of integers Z, where particular emphasis is given to number theoretic questions such as that of unique factorization. The origin of the thesis came from a study of a special case of generalized integers called the Gaussian Integers, namely the set of all complex numbers in the form n + mi, for m,n in Z. The main generalization involves what are called algebraic integers. digital.library.unt.edu/ark:/67531/metadc131119/
 Means and Mean Value Theorems
 This study covers means, mean value theorems of the differential calculus, and mean value theorems of integral calculus. digital.library.unt.edu/ark:/67531/metadc96987/
 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/
 The RiemannComplete Integral
 The problem with which this paper is concerned is that of defining the RiemannComplete Integral and comparing it with the Riemann and the Lebesgue Integrals. digital.library.unt.edu/ark:/67531/metadc131504/
 Some Results of Two Topological Spaces
 This thesis explores some results of two topological spaces. digital.library.unt.edu/ark:/67531/metadc130437/
 The Study of Translation Equivalence on Integer Lattices
 This paper is a contribution to the study of countable Borel equivalence relations on standard Borel spaces. We concentrate here on the study of the nature of translation equivalence. We study these known hyperfinite spaces in order to gain insight into the approach necessary to classify certain variables as either being hyperfinite or not. In Chapter 1, we will give the basic definitions and examples of spaces used in this work. The general construction of marker sets is developed in this work. These marker sets are used to develop several invariant tilings of the equivalence classes of specific variables . Some properties that are equivalent to hyperfiniteness in the certain space are also developed. Lastly, we will give the new result that there is a continuous injective embedding from certain defined variables. digital.library.unt.edu/ark:/67531/metadc4345/
 Certain Properties of Functions Related to Exhaustibility
 In this thesis, we shall attempt to present a study of certain properties of real functions related to the set property exhaustible. digital.library.unt.edu/ark:/67531/metadc107821/
 A Study of Functions on Metric Spaces
 This thesis describes various forms of metric spaces and establishes some of the properties of functions defined on metric spaces. No attempt is made in this paper to examine a particular type of function in detail. Instead, some of properties of several kinds of functions will be observed as the functions are defined on various forms of metric spaces such as connected spaces, compact spaces, complete spaces, etc. digital.library.unt.edu/ark:/67531/metadc130890/
 Abstract Measure
 This study of abstract measure covers classes of sets, measures and outer measures, extension of measures, and planer measure. digital.library.unt.edu/ark:/67531/metadc107950/
 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 multiagent 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 onetoone correspondence with the standard time series complexity measure sample entropy. digital.library.unt.edu/ark:/67531/metadc3702/
 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/
 Borel Determinacy and Metamathematics
 Borel determinacy states that if G(T;X) is a game and X is Borel, then G(T;X) is determined. Proved by Martin in 1975, Borel determinacy is a theorem of ZFC set theory, and is, in fact, the best determinacy result in ZFC. However, the proof uses sets of high set theoretic type (N1 many power sets of ω). Friedman proved in 1971 that these sets are necessary by showing that the Axiom of Replacement is necessary for any proof of Borel Determinacy. To prove this, Friedman produces a model of ZC and a Borel set of Turing degrees that neither contains nor omits a cone; so by another theorem of Martin, Borel Determinacy is not a theorem of ZC. This paper contains three main sections: Martin's proof of Borel Determinacy; a simpler example of Friedman's result, namely, (in ZFC) a coanalytic set of Turing degrees that neither contains nor omits a cone; and finally, the Friedman result. digital.library.unt.edu/ark:/67531/metadc3061/
 A Computation of Partial Isomorphism Rank on Ordinal Structures
 We compute the partial isomorphism rank, in the sense Scott and Karp, of a pair of ordinal structures using an EhrenfeuchtFraisse game. A complete formula is proven by induction given any two arbitrary ordinals written in Cantor normal form. digital.library.unt.edu/ark:/67531/metadc5387/
 Some Properties of Negligible Sets
 In the study of sets of points certain sets are found to be negligible, especially when applied to the theory of functions. The purpose of this paper is to discuss three of these "negligible" types, namely, exhaustible sets, denumerable sets, and sets of Lebesgue measure zero. We will present a complete existential theory in qspace for the three set properties mentioned above, followed by a more restricted discussion in the linear continuum by use of interval properties. digital.library.unt.edu/ark:/67531/metadc83358/
 Quadratic Forms
 This paper shall be mostly concerned with the development and the properties of three quadratic polynomials. The primary interest will by with nary quadratic polynomials, called forms. digital.library.unt.edu/ark:/67531/metadc108034/
 Polynomial Curve and Surface Fitting
 The main problems of numerical analysis involve performing analytical operations, such as integration, differentiation, finding zeroes, interpolation, and so forth, of a function when all the data available are some samples of the function. Therefore, the purpose of this paper is to investigate the following problem: given a set of data points (x[sub i], y[sub i]) which are samples of some function, determine an approximating function. Further, extend the problem to that of determining an approximating function for a surface given some samples (x[sub i], y[sub j], z[sub ij]) of the surface. digital.library.unt.edu/ark:/67531/metadc130895/
 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 Wave Equation in One Dimension
 It is intended that this paper present an acceptable proof of the existence of a solution for the wave equation. digital.library.unt.edu/ark:/67531/metadc108114/
 Linear Spaces
 The purpose of this paper is to present the results of a study of linear spaces with special emphasis of linear transformations, norms, and inner products. digital.library.unt.edu/ark:/67531/metadc108094/
 Topological Groups
 The notion of a topological group follows naturally from a combination of the properties of a group and a topological space. Since a group consists of a set G of elements which may be either finite or infinite and since this is also common to a topological space, a question is opened as to whether or not it is possible to assign a topology to a set of elements which form a group under a certain operation. Now it is possible to assign a topology to any set of elements if no restriction is placed on the topology assigned and hence this study would be of little value from the standpoint of the group itself. If however it is required that the group operation be continuous in the topological space then a very interesting theory is developed. digital.library.unt.edu/ark:/67531/metadc108079/
 Completeness Axioms in an Ordered Field
 The purpose of this paper was to prove the equivalence of the following completeness axioms. This purpose was carried out by first defining an ordered field and developing some basic theorems relative to it, then proving that lim [(u+u)*]^n = z (where u is the multiplicative identity, z is the additive identity, and * indicates the multiplicative inverse of an element), and finally proving the equivalence of the five axioms. digital.library.unt.edu/ark:/67531/metadc131462/