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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
Properties of an Integral of E.J. McShane
The problem with which this paper is concerned is that of investigating the properties of an integral which was first defined by E. J. McShane in lecture notes presented at the Conference on Modern Theories of Integration, held at the University of Oklahoma in June, 1969.
Date:
August 1973
Creator:
Dennis, Thomas B.
System:
The UNT Digital Library
A Partial Characterization of Upper Semi-Continuous Decompositions
The goal of this paper is to characterize, at least partially, upper semi-continuous decompositions of topological spaces and the role that upper semi-continuity plays in preserving certain topological properties under decomposition mappings. Attention is also given to establishing what role upper semi-continuity plays in determining conditions under which decomposition spaces possess certain properties. A number of results for non-upper semi-continuous decompositions are included to help clarify the scope of the part upper semi-continuity plays in determining relationships between topological spaces and their decomposition spaces.
Date:
December 1973
Creator:
Dennis, William Albert
System:
The UNT Digital Library
Some Properties of Derivatives
This paper is concerned with certain properties of derivatives and some characterizations of linear point sets with derivatives. In 1946, Zygmunt Zahorski published a letter on this topic listing a number of theorems without proof, and no proof of these assertions has been published. Some of the theorems presented here are paraphrases of Zahorski's statements, developed in a slightly different order.
Date:
1951
Creator:
Dibben, Philip W.
System:
The UNT Digital Library
On Ordered Pairs of Cardinal Numbers
This thesis is on ordered pairs of cardinal numbers.
Date:
1957
Creator:
Dickinson, John Dean
System:
The UNT Digital Library
Ádám's Conjecture and Its Generalizations
This paper examines idam's conjuecture and some of its generalizations. In terms of Adam's conjecture, we prove Alspach and Parson's results f or Zpq and ZP2. More generally, we prove Babai's characterization of the CI-property, Palfy's characterization of CI-groups, and Brand's result for Zpr for polynomial isomorphism's. We also prove for the first time a characterization of the CI-property for 1 SG, and prove that Zn is a CI-Pn-group where Pn is the group of permutation polynomials on Z,, and n is square free.
Date:
August 1990
Creator:
Dobson, Edward T. (Edward Tauscher)
System:
The UNT Digital Library
Some Fundamental Properties of Valuations Defined on a Field
The purpose of this thesis is to develop some properties of a special class of functions called valuations. The study begins with and examination of the properties of valuations defined on an arbitrary field, F, and later, consideration is given to valuations defined on the field of rational numbers. The concept of a pseud-valuation is introduced and an investigation is made of the properties of pseudo-valuations.
Date:
January 1969
Creator:
Doerr, James C.
System:
The UNT Digital Library
Properties of Semigroups
This paper is an introductory, algebraic study of semigroups.
Date:
June 1966
Creator:
Donnell, William Anthony
System:
The UNT Digital Library
Metric Half-Spaces
This paper is a study of some of the basic properties of the metric half-space topology, a topology on a set which is derived from a metric on the set. In the first it is found that in a complete inner product space, the metric half-space topology is the same as one defined in terms of linear functionals on the space. In the second it is proven that in Rn the metric half-space topology is the same as the usual metric topology. In the third theorem it is shown that in a certain sense the nature of the metric halfspace topology generated by a norm on the space determines whether the norm is quadratic, that is to say, whether or not there exists an inner product on the space with the property that |x|^2=(x,x) for all x in the space.
Date:
May 1972
Creator:
Dooley, Willis L.
System:
The UNT Digital Library
R₀ Spaces, R₁ Spaces, And Hyperspaces
The purpose of this paper is to further investigate R0 spaces, R1 spaces, and hyperspaces. The R0 axiom was introduced by N. A. Shanin in 1943. Later, in 1961, A. S. Davis investigated R0 spaces and introduced R1 spaces. Then, in 1975, William Dunham further investigated R1 spaces and proved that several well-known theorems can be generalized from a T2 setting to an R1 setting. In Chapter II R0 and R1 spaces are investigated and additional theorems that can be generalized from a T2 setting to an R1 setting are obtained.
Date:
December 1976
Creator:
Dorsett, Charles I.
System:
The UNT Digital Library
G-domains, G-ideals, and Hilbert Rings
The problem with which this investigation is concerned is that of determining the properties of the following: a particular type of integral domain, the G-domain; a type of prime ideal, the G-ideal; and a special type of ring, the Hilbert ring.
Date:
August 1972
Creator:
Draper, Ruben P.
System:
The UNT Digital Library
Invariants of Polynomials Modulo Frobenius Powers
Rational Catalan combinatorics connects various Catalan numbers to the representation theory of rational Cherednik algebras for Coxeter and complex reflection groups. Lewis, Reiner, and Stanton seek a theory of rational Catalan combinatorics for the general linear group over a finite field. The finite general linear group is a modular reflection group that behaves like a finite Coxeter group. They conjecture a Hilbert series for a space of invariants under the action of this group using (q,t)-binomial coefficients. They consider the finite general linear group acting on the quotient of a polynomial ring by iterated powers of the irrelevant ideal under the Frobenius map. Often conjectures about reflection groups are solved by considering the local case of a group fixing one hyperplane and then extending via the theory of hyperplane arrangements to the full group. The Lewis, Reiner and Stanton conjecture had not previously been formulated for groups fixing a hyperplane. We formulate and prove their conjecture in this local case.
Date:
May 2020
Creator:
Drescher, Chelsea
System:
The UNT Digital Library
On Uniform Convergence
In this paper, we will be concerned primarily with series of functions and a particular type of convergence which will be described. The purpose of this paper is to familiarize the reader with the concept of uniform convergence. In the main it is a compilation of material found in various references and revised to conform to standard notation.
Date:
1951
Creator:
Drew, Dan Dale
System:
The UNT Digital Library
Topologies on Complete Lattices
One of the more important concepts in mathematics is the concept of order, that is, the description or comparison of two elements of a set in terms of one preceding or being smaller than or equal to the other. If the elements of a set, as pairs, exhibit certain order-type characteristics, the set is said to be a partially ordered set. The purpose of this paper is to investigate a special class of partially ordered sets, called lattices, and to investigate topologies induced on these lattices by specially defined order related properties called order-convergence and star-convergence.
Date:
December 1973
Creator:
Dwyer, William Karl
System:
The UNT Digital Library
Around the Fibonacci Numeration System
Let 1, 2, 3, 5, 8, … denote the Fibonacci sequence beginning with 1 and 2, and then setting each subsequent number to the sum of the two previous ones. Every positive integer n can be expressed as a sum of distinct Fibonacci numbers in one or more ways. Setting R(n) to be the number of ways n can be written as a sum of distinct Fibonacci numbers, we exhibit certain regularity properties of R(n), one of which is connected to the Euler φ-function. In addition, using a theorem of Fine and Wilf, we give a formula for R(n) in terms of binomial coefficients modulo two.
Date:
May 2007
Creator:
Edson, Marcia Ruth
System:
The UNT Digital Library
Automorphism Groups
This paper will be concerned mainly with automorphisms of groups. The concept of a group endomorphism will be used at various points in this paper.
Date:
August 1965
Creator:
Edwards, Donald Eugene
System:
The UNT Digital Library
Overrings of an Integral Domain
This dissertation focuses on the properties of a domain which has the property that each ideal is a finite intersection of a π-ideal, the properties of a domain which have the property that each ideal is a finite product of π-ideal, and the containment relations of the resulting classes of ideals. Chapter 1 states definitions which are needed in later chapters. Chapters 2 and 3 focuses on domains which have the property that each ideal in D is a finite intersection of π-ideals while Chapter 4 focuses on domains with the property that each ideal is a finite product of π-ideals. Chapter 5 discusses the containment relations which occur as a result of Chapters 2 and 3.
Date:
August 1992
Creator:
Emerson, Sharon Sue
System:
The UNT Digital Library
Sequences of Real Numbers
The purpose of this thesis is to examine general properties, convergence, and limit points of sequences of real numbers.
Date:
January 1965
Creator:
Eskew, Mark F.
System:
The UNT Digital Library
Peripherally Continuous Functions, Graph Maps and Connectivity Maps
The purpose of this paper is to investigate some of the more basic properties of peripherally continuous functions, graph maps and connectivity maps.
Date:
August 1967
Creator:
Evans, Bret Edgar
System:
The UNT Digital Library
Applications in Fixed Point Theory
Banach's contraction principle is probably one of the most important theorems in fixed point theory. It has been used to develop much of the rest of fixed point theory. Another key result in the field is a theorem due to Browder, Göhde, and Kirk involving Hilbert spaces and nonexpansive mappings. Several applications of Banach's contraction principle are made. Some of these applications involve obtaining new metrics on a space, forcing a continuous map to have a fixed point, and using conditions on the boundary of a closed ball in a Banach space to obtain a fixed point. Finally, a development of the theorem due to Browder et al. is given with Hilbert spaces replaced by uniformly convex Banach spaces.
Date:
December 2005
Creator:
Farmer, Matthew Ray
System:
The UNT Digital Library
Strong Choquet Topologies on the Closed Linear Subspaces of Banach Spaces
In the study of Banach spaces, the development of some key properties require studying topologies on the collection of closed convex subsets of the space. The subcollection of closed linear subspaces is studied under the relative slice topology, as well as a class of topologies similar thereto. It is shown that the collection of closed linear subspaces under the slice topology is homeomorphic to the collection of their respective intersections with the closed unit ball, under the natural mapping. It is further shown that this collection under any topology in the aforementioned class of similar topologies is a strong Choquet space. Finally, a collection of category results are developed since strong Choquet spaces are also Baire spaces.
Date:
August 2011
Creator:
Farmer, Matthew Ray
System:
The UNT Digital Library
Euclidean Rings
The cardinality of the set of units, and of the set of equivalence classes of primes in non-trivial Euclidean domains is discussed with reference to the categories "finite" and "infinite." It is shown that no Euclidean domains exist for which both of these sets are finite. The other three combinations are possible and examples are given. For the more general Euclidean rings, the first combination is possible and examples are likewise given. Prime factorization is also discussed in both Euclidean rings and Euclidean domains. For Euclidean rings, an alternative definition of prime elements in terms of associates is compared and contrasted to the usual definitions.
Date:
May 1974
Creator:
Fecke, Ralph Michael
System:
The UNT Digital Library
A Genesis for Compact Convex Sets
This paper was written in response to the following question: what conditions are sufficient to guarantee that if a compact subset A of a topological linear space L^3 is not convex, then for every point x belonging to the complement of A relative to the convex hull of A there exists a line segment yz such that x belongs to yz and y belongs to A and z belongs to A? Restated in the terminology of this paper the question bay be given as follow: what conditions may be imposed upon a compact subset A of L^3 to insure that A is braced?
Date:
May 1969
Creator:
Ferguson, Ronald D.
System:
The UNT Digital Library
Existence of Many Sign Changing Non Radial Solutions for Semilinear Elliptic Problems on Annular Domains
The aim of this work is the study of the existence and multiplicity of sign changing nonradial solutions to elliptic boundary value problems on annular domains.
Date:
August 1998
Creator:
Finan, Marcel Basil
System:
The UNT Digital Library