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**Department:**Department of Mathematics

**Collection:**UNT Theses and Dissertations

### Interpolation and Approximation

**Date:**May 1977

**Creator:**Lal, Ram

**Description:**In this paper, there are three chapters. The first chapter discusses interpolation. Here a theorem about the uniqueness of the solution to the general interpolation problem is proven. Then the problem of how to represent this unique solution is discussed. Finally, the error involved in the interpolation and the convergence of the interpolation process is developed. In the second chapter a theorem about the uniform approximation to continuous functions is proven. Then the best approximation and the least squares approximation (a special case of best approximation) is discussed. In the third chapter orthogonal polynomials as discussed as well as bounded linear functionals in Hilbert spaces, interpolation and approximation and approximation in Hilbert space.

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### Complete Ordered Fields

**Date:**August 1977

**Creator:**Arnold, Thompson Sharon

**Description:**The purpose of this thesis is to study the concept of completeness in an ordered field. Several conditions which are necessary and sufficient for completeness in an ordered field are examined. In Chapter I the definitions of a field and an ordered field are presented and several properties of fields and ordered fields are noted. Chapter II defines an Archimedean field and presents several conditions equivalent to the Archimedean property. Definitions of a complete ordered field (in terms of a least upper bound) and the set of real numbers are also stated. Chapter III presents eight conditions which are equivalent to completeness in an ordered field. These conditions include the concepts of nested intervals, Dedekind cuts, bounded monotonic sequences, convergent subsequences, open coverings, cluster points, Cauchy sequences, and continuous functions.

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### The Wallman Spaces and Compactifications

**Date:**December 1976

**Creator:**Liu, Wei-kong

**Description:**If X is a topological space and Y is a ring of closed sets, then a necessary and sufficient condition for the Wallman space W(X,F) to be a compactification of X is that X be T1 andYF separating. A necessary and sufficient condition for a Wallman compactification to be Hausdoff is that F be a normal base. As a result, not all T, compactifications can be of Wallman type. One point and finite Hausdorff compactifications are of Wallman type.

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### Valuations on Fields

**Date:**May 1977

**Creator:**Walker, Catherine A.

**Description:**This thesis investigates some properties of valuations on fields. Basic definitions and theorems assumed are stated in Capter I. Chapter II introduces the concept of a valuation on a field. Real valuations and non-Archimedean valuations are presented. Chapter III generalizes non-Archimedean valuations. Examples are described in Chapters I and II. A result is the theorem stating that a real valuation of a field K is non-Archimedean if and only if $(a+b) < max4# (a), (b) for all a and b in K. Chapter III generally defines a non-Archimedean valuation as an ordered abelian group. Real non-Archimedean valuations are either discrete or nondiscrete. Chapter III shows that every valuation ring identifies a non-Archimedean valuation and every non-Archimedean valuation identifies a valuation ring.

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### Subdirectly Irreducible Semigroups

**Date:**December 1978

**Creator:**Winton, Richard Alan

**Description:**Definition 1.1. The ordered pair (S,*) is a semi-group iff S is a set and * is an associative binary operation (multiplication) on S. Notation. A semigroup (S,*) will ordinarily be referred to by the set S, with the multiplication understood. In other words, if (a,b)e SX , then *[(a,b)] = a*b = ab. The proof of the following proposition is found on p. 4 of Introduction to Semigroups, by Mario Petrich. Proposition 1.2. Every semigroup S satisfies the general associative law.

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### Integrability, Measurability, and Summability of Certain Set Functions

**Date:**December 1977

**Creator:**Dawson, Dan Paul

**Description:**The purpose of this paper is to investigate the integrability, measurability, and summability of certain set functions. The paper is divided into four chapters. The first chapter contains basic definitions and preliminary remarks about set functions and absolute continuity. In Chapter i, the integrability of bounded set functions is investigated. The chapter culminates with a theorem that characterizes the transmission of the integrability of a real function of n bounded set functions. In Chapter III, measurability is defined and a characterization of the transmission of measurability by a function of n variables is provided, In Chapter IV, summability is defined and the summability of set functions is investigated, Included is a characterization of the transmission of summability by a function of n variables.

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### Hyperspaces

**Date:**December 1976

**Creator:**Voas, Charles H.

**Description:**This paper is an exposition of the theory of the hyperspaces 2^X and C(X) of a topological space X. These spaces are obtained from X by collecting the nonempty closed and nonempty closed connected subsets respectively, and are topologized by the Vietoris topology. The paper is organized in terms of increasing specialization of spaces, beginning with T1 spaces and proceeding through compact spaces, compact metric spaces and metric continua. Several basic techniques in hyperspace theory are discussed, and these techniques are applied to elucidate the topological structure of hyperspaces.

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### Fundamental Issues in Support Vector Machines

**Date:**May 2014

**Creator:**McWhorter, Samuel P.

**Description:**This dissertation considers certain issues in support vector machines (SVMs), including a description of their construction, aspects of certain exponential kernels used in some SVMs, and a presentation of an algorithm that computes the necessary elements of their operation with proof of convergence. In its first section, this dissertation provides a reasonably complete description of SVMs and their theoretical basis, along with a few motivating examples and counterexamples. This section may be used as an accessible, stand-alone introduction to the subject of SVMs for the advanced undergraduate. Its second section provides a proof of the positive-definiteness of a certain useful function here called E and dened as follows: Let V be a complex inner product space. Let N be a function that maps a vector from V to its norm. Let p be a real number between 0 and 2 inclusive and for any in V , let ( be N() raised to the p-th power. Finally, let a be a positive real number. Then E() is exp(()). Although the result is not new (other proofs are known but involve deep properties of stochastic processes) this proof is accessible to advanced undergraduates with a decent grasp of linear algebra. Its ...

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### Maximum Likelihood Estimation of Logistic Sinusoidal Regression Models

**Date:**December 2013

**Creator:**Weng, Yu

**Description:**We consider the problem of maximum likelihood estimation of logistic sinusoidal regression models and develop some asymptotic theory including the consistency and joint rates of convergence for the maximum likelihood estimators. The key techniques build upon a synthesis of the results of Walker and Song and Li for the widely studied sinusoidal regression model and on making a connection to a result of Radchenko. Monte Carlo simulations are also presented to demonstrate the finite-sample performance of the estimators

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### Polynomial Isomorphisms of Cayley Objects Over a Finite Field

**Date:**December 1989

**Creator:**Park, Hong Goo

**Description:**In this dissertation the Bays-Lambossy theorem is generalized to GF(pn). The Bays-Lambossy theorem states that if two Cayley objects each based on GF(p) are isomorphic then they are isomorphic by a multiplier map. We use this characterization to show that under certain conditions two isomorphic Cayley objects over GF(pn) must be isomorphic by a function on GF(pn) of a particular type.

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### Hausdorff, Packing and Capacity Dimensions

**Date:**August 1989

**Creator:**Spear, Donald W.

**Description:**In this thesis, Hausdorff, packing and capacity dimensions are studied by evaluating sets in the Euclidean space R^. Also the lower entropy dimension is calculated for some Cantor sets. By incorporating technics of Munroe and of Saint Raymond and Tricot, outer measures are created. A Vitali covering theorem for packings is proved. Methods (by Taylor and Tricot, Kahane and Salem, and Schweiger) for determining the Hausdorff and capacity dimensions of sets using probability measures are discussed and extended. The packing pre-measure and measure are shown to be scaled after an affine transformation. A Cantor set constructed by L.D. Pitt is shown to be dimensionless using methods developed in this thesis. A Cantor set is constructed for which all four dimensions are different. Graph directed constructions (compositions of similitudes follow a path in a directed graph) used by Mauldin and Willjams are presented. Mauldin and Williams calculate the Hausdorff dimension, or, of the object of a graph directed construction and show that if the graph is strongly connected, then the a—Hausdorff measure is positive and finite. Similar results will be shown for the packing dimension and the packing measure. When the graph is strongly connected, there is a constant so that ...

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### The Maximum Size of Combinatorial Geometries Excluding Wheels and Whirls as Minors

**Date:**August 1989

**Creator:**Hipp, James W. (James William), 1956-

**Description:**We show that the maximum size of a geometry of rank n excluding the (q + 2)-point line, the 3-wheel W_3, and the 3-whirl W^3 as minor is (n - 1)q + 1, and geometries of maximum size are parallel connections of (q + 1)-point lines. We show that the maximum size of a geometry of rank n excluding the 5-point line, the 4-wheel W_4, and the 4-whirl W^4 as minors is 6n - 5, for n ≥ 3. Examples of geometries having rank n and size 6n - 5 include parallel connections of the geometries V_19 and PG(2,3).

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### Minimization of a Nonlinear Elasticity Functional Using Steepest Descent

**Date:**August 1988

**Creator:**McCabe, Terence W. (Terence William)

**Description:**The method of steepest descent is used to minimize typical functionals from elasticity.

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### Bounded, Finitely Additive, but Not Absolutely Continuous Set Functions

**Date:**May 1989

**Creator:**Gurney, David R. (David Robert)

**Description:**In leading up to the proof, methods for constructing fields and finitely additive set functions are introduced with an application involving the Tagaki function given as an example. Also, non-absolutely continuous set functions are constructed using Banach limits and maximal filters.

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### Applications of Graph Theory and Topology to Combinatorial Designs

**Date:**December 1988

**Creator:**Somporn Sutinuntopas

**Description:**This dissertation is concerned with the existence and the isomorphism of designs. The first part studies the existence of designs. Chapter I shows how to obtain a design from a difference family. Chapters II to IV study the existence of an affine 3-(p^m,4,λ) design where the v-set is the Galois field GF(p^m). Associated to each prime p, this paper constructs a graph. If the graph has a 1-factor, then a difference family and hence an affine design exists. The question arises of how to determine when the graph has a 1-factor. It is not hard to see that the graph is connected and of even order. Tutte's theorem shows that if the graph is 2-connected and regular of degree three, then the graph has a 1-factor. By using the concept of quadratic reciprocity, this paper shows that if p Ξ 53 or 77 (mod 120), the graph is almost regular of degree three, i.e., every vertex has degree three, except two vertices each have degree tow. Adding an extra edge joining the two vertices with degree tow gives a regular graph of degree three. Also, Tutte proved that if A is an edge of the graph satisfying the above conditions, ...

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### Dynamics of One-Dimensional Maps: Symbols, Uniqueness, and Dimension

**Date:**May 1988

**Creator:**Brucks, Karen M. (Karen Marie), 1957-

**Description:**This dissertation is a study of the dynamics of one-dimensional unimodal maps and is mainly concerned with those maps which are trapezoidal. The trapezoidal function, f_e, is defined for eΣ(0,1/2) by f_e(x)=x/e for xΣ[0,e], f_e(x)=1 for xΣ(e,1-e), and f_e(x)=(1-x)/e for xΣ[1-e,1]. We study the symbolic dynamics of the kneading sequences and relate them to the analytic dynamics of these maps. Chapter one is an overview of the present theory of Metropolis, Stein, and Stein (MSS). In Chapter two a formula is given that counts the number of MSS sequences of length n. Next, the number of distinct primitive colorings of n beads with two colors, as counted by Gilbert and Riordan, is shown to equal the number of MSS sequences of length n. An algorithm is given that produces a bisection between these two quantities for each n. Lastly, the number of negative orbits of size n for the function f(z)=z^2-2, as counted by P.J. Myrberg, is shown to equal the number of MSS sequences of length n. For an MSS sequence P, let H_ϖ(P) be the unique common extension of the harmonics of P. In Chapter three it is proved that there is exactly one J(P)Σ[0,1] such that the ...

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### Operators on Continuous Function Spaces and Weak Precompactness

**Date:**August 1988

**Creator:**Abbott, Catherine Ann

**Description:**If T:C(H,X)-->Y is a bounded linear operator then there exists a unique weakly regular finitely additive set function m:-->L(X,Y**) so that T(f) = ∫Hfdm. In this paper, bounded linear operators on C(H,X) are studied in terms the measure given by this representation theorem. The first chapter provides a brief history of representation theorems of these classes of operators. In the second chapter the represenation theorem used in the remainder of the paper is presented. If T is a weakly compact operator on C(H,X) with representing measure m, then m(A) is a weakly compact operator for every Borel set A. Furthermore, m is strongly bounded. Analogous statements may be made for many interesting classes of operators. In chapter III, two classes of operators, weakly precompact and QSP, are studied. Examples are provided to show that if T is weakly precompact (QSP) then m(A) need not be weakly precompact (QSP), for every Borel set A. In addition, it will be shown that weakly precompact and GSP operators need not have strongly bounded representing measures. Sufficient conditions are provided which guarantee that a weakly precompact (QSP) operator has weakly precompact (QSP) values. A sufficient condition for a weakly precomact operator to be strongly ...

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### Existence of a Solution for a Wave Equation and an Elliptic Dirichlet Problem

**Date:**May 1988

**Creator:**Sumalee Unsurangsie

**Description:**In this paper we consider an existence of a solution for a nonlinear nonmonotone wave equation in [0,π]xR and an existence of a positive solution for a non-positone Dirichlet problem in a bounded subset of R^n.

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### Dually Semimodular Consistent Lattices

**Date:**May 1988

**Creator:**Gragg, Karen E. (Karen Elizabeth)

**Description:**A lattice L is said to be dually semimodular if for all elements a and b in L, a ∨ b covers b implies that a covers a ∧ b. L is consistent if for every join-irreducible j and every element x in L, the element x ∨ j is a join-irreducible in the upper interval [x,l]. In this paper, finite dually semimodular consistent lattices are investigated. Examples of these lattices are the lattices of subnormal subgroups of a finite group. In 1954, R. P. Dilworth proved that in a finite modular lattice, the number of elements covering exactly k elements is equal to the number of elements covered by exactly k elements. Here, it is established that if a finite dually semimodular consistent lattice has the same number of join-irreducibles as meet-irreducibles, then it is modular. Hence, a converse of Dilworth's theorem, in the case when k equals 1, is obtained for finite dually semimodular consistent lattices. Several combinatorial results are shown for finite consistent lattices similar to those already established for finite geometric lattices. The reach of an element x in a lattice L is the difference between the rank of x*, the join of x and all ...

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### Invertible Ideals and the Strong Two-Generator Property in Some Polynomial Subrings

**Date:**May 1987

**Creator:**Chapman, Scott T. (Scott Thomas)

**Description:**Let K be any field and Q be the rationals. Define K^1[X] = {f(X) e K[X]| the coefficient of X in f(X) is zero} and Q^1β[X] = {f(X) e Q[X]| the coefficent of β1(X) in the binomial expansion of f(X) is zero}, where {β1(X)}^∞ i=0 are the well-known binomial polynomials. In this work, I establish the following results: K^1[X] and Q^1β[X] are one-dimensional, Noetherian, non-Prüfer domains with the two-generator property on ideals. Using the unique factorization structure of the overrings K[X] and Q[X], the nonprincipal ideal structures of both rings are characterized, and from this characterization, necessary and sufficient conditions are found for a nonprincipal ideal to be invertible. The nonprincipal invertible ideals are then characterized in terms of the coefficients of the generators, and an explicit formula for the inverse of any proper invertible ideal is found. Finally, the class groups of both rings are shown to be torsion free abelian groups. Let n be any nonnegative integer. Results similar to the above are found in the generalizations of these two rings, K^n[X] and q^nβ[X], where the coefficients on the first n nonconstant basis elements are zero. For the domains K^1[X] and Q^1β[X], the property of strong two-generation is ...

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### Containment Relations Between Classes of Regular Ideals in a Ring with Few Zero Divisors

**Date:**May 1987

**Creator:**Race, Denise T. (Denise Tatsch)

**Description:**This dissertation focuses on the significance of containment relations between the above mentioned classes of ideals. The main problem considered in Chapter II is determining conditions which lead a ring to be a P-ring, D-ring, or AM-ring when every regular ideal is a P-ideal, D-ideal, or AM-ideal, respectively. We also consider containment relations between classes of regular ideals which guarantee that the ring is a quasi-valuation ring. We continue this study into the third chapter; in particular, we look at the conditions in a quasi-valuation ring which lead to a = Jr, sr - f, and a = v. Furthermore we give necessary and sufficient conditions that a ring be a discrete rank one quasi-valuation ring. For example, if R is Noetherian, then ft = J if and only if R is a discrete rank one quasi-valuation ring.

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### Weakly Dense Subsets of Homogeneous Complete Boolean Algebras

**Date:**August 1990

**Creator:**Bozeman, Alan Kyle

**Description:**The primary result from this dissertation is following inequality: d(B) ≤ min(2^< wd(B),sup{λ^c(B): λ < wd(B)}) in ZFC, where B is a homogeneous complete Boolean algebra, d(B) is the density, wd(B) is the weak density, and c(B) is the cellularity of B. Chapter II of this dissertation is a general overview of homogeneous complete Boolean algebras. Assuming the existence of a weakly inaccessible cardinal, we give an example of a homogeneous complete Boolean algebra which does not attain its cellularity. In chapter III, we prove that for any integer n > 1, wd_2(B) = wd_n(B). Also in this chapter, we show that if X⊂B is κ—weakly dense for 1 < κ < sat(B), then sup{wd_κ(B):κ < sat(B)} = d(B). In chapter IV, we address the following question: If X is weakly dense in a homogeneous complete Boolean algebra B, does there necessarily exist b € B\{0} such that {x∗b: x ∈ X} is dense in B|b = {c € B: c ≤ b}? We show that the answer is no for collapsing algebras. In chapter V, we give new proofs to some well known results concerning supporting antichains. A direct consequence of these results is the relation c(B) < wd(B), ...

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### Radially Symmetric Solutions to a Superlinear Dirichlet Problem in a Ball

**Date:**August 1987

**Creator:**Kurepa, Alexandra

**Description:**In this paper we consider a radially symmetric nonlinear Dirichlet problem in a ball, where the nonlinearity is "superlinear" and "superlinear with jumping."

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### Uniqueness of Positive Solutions for Elliptic Dirichlet Problems

**Date:**December 1990

**Creator:**Ali, Ismail, 1961-

**Description:**In this paper we consider the question of uniqueness of positive solutions for Dirichlet problems of the form - Δ u(x)= g(λ,u(x)) in B, u(x) = 0 on ϑB, where A is the Laplace operator, B is the unit ball in RˆN, and A>0. We show that if g(λ,u)=uˆ(N+2)/(N-2) + λ, that is g has "critical growth", then large positive solutions are unique. We also prove uniqueness of large solutions when g(λ,u)=A f(u) with f(0) < 0, f "superlinear" and monotone. We use a number of methods from nonlinear functional analysis such as variational identities, Sturm comparison theorems and methods of order. We also present a regularity result on linear elliptic equation where a coefficient has critical growth.

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