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**Partner:**UNT Libraries

**Department:**Department of Mathematics

**Collection:**UNT Theses and Dissertations

### A Fundamental Study of Cardinal and Ordinal Numbers

**Date:**August 1966

**Creator:**Thornton, Robert Leslie

**Description:**The purpose of this paper is to present a discussion on the basic fundamentals of the theory of sets. Primarily, the discussion will be confined to the study of cardinal and ordinal numbers. The concepts of sets, classes of sets, and families of sets will be undefined quantities, and the concept of the class of all sets will be avoided.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130761/

### Fundamentals of Partially Ordered Sets

**Date:**August 1968

**Creator:**Compton, Lewis W.

**Description:**Gives the basic definitions and theorems of similar partially ordered sets; studies finite partially ordered sets, including the problem of combinatorial analysis; and includes the ideas of complete, dense, and continuous partially ordered sets, including proofs.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130984/

### G-domains, G-ideals, and Hilbert Rings

**Date:**August 1972

**Creator:**Draper, Ruben P.

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131539/

### A Generalization of Newton's Method

**Date:**1948

**Creator:**LeBouf, Billy Ruth

**Description:**It is our purpose here to investigate the method of solving equations for real roots by Newton's Method and to indicate a generalization arising from this method.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc75705/

### A Generalization of Sturmian Sequences: Combinatorial Structure and Transcendence

**Date:**August 1998

**Creator:**Risley, Rebecca N.

**Description:**We investigate a class of minimal sequences on a finite alphabet Ak = {1,2,...,k} having (k - 1)n + 1 distinct subwords of length n. These sequences, originally defined by P. Arnoux and G. Rauzy, are a natural generalization of binary Sturmian sequences. We describe two simple combinatorial algorithms for constructing characteristic Arnoux-Rauzy sequences (one of which is new even in the Sturmian case). Arnoux-Rauzy sequences arising from fixed points of primitive morphisms are characterized by an underlying periodic structure. We show that every Arnoux-Rauzy sequence contains arbitrarily large subwords of the form V^2+ε and, in the Sturmian case, arbitrarily large subwords of the form V^3+ε. Finally, we prove that an irrational number whose base b-digit expansion is an Arnoux-Rauzy sequence is transcendental.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc278440/

### A Generalization of the Weierstrass Approximation Theorem

**Date:**August 1972

**Creator:**Murchison, Jo Denton

**Description:**A presentation of the Weierstrass approximation theorem and the Stone-Weierstrass theorem and a comparison of these two theorems are the objects of this thesis.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131551/

### A Generalized Study of the Conjugate and Inner-Product Functions

**Date:**June 1967

**Creator:**Wright, Dorothy P.

**Description:**The usual practice in any discussion of an inner-product space is to restrict the field over which the inner-product space is defined to the field of complex numbers. In defining the inner-product 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 inner-product function defined on a linear space L over these fields.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130818/

### Generic Algebras and Kazhdan-Lusztig Theory for Monomial Groups

**Access:**Use of this item is restricted to the UNT Community.

**Date:**May 2006

**Creator:**Alhaddad, Shemsi I.

**Description:**The Iwahori-Hecke algebras of Coxeter groups play a central role in the study of representations of semisimple Lie-type groups. An important tool is the combinatorial approach to representations of Iwahori-Hecke algebras introduced by Kazhdan and Lusztig in 1979. In this dissertation, I discuss a generalization of the Iwahori-Hecke 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 R-polynomials, I show that this algebra determines a partial order on G(r,1,n) that generalizes the Chevalley-Bruhat order on the symmetric group. I also consider possible analogues of Kazhdan-Lusztig polynomials.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc5235/

### A Genesis for Compact Convex Sets

**Date:**May 1969

**Creator:**Ferguson, Ronald D.

**Description:**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?

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131089/

### Gibbs/Equilibrium Measures for Functions of Multidimensional Shifts with Countable Alphabets

**Date:**May 2011

**Creator:**Muir, Stephen R.

**Description:**Consider a multidimensional shift space with a countably infinite alphabet, which serves in mathematical physics as a classical lattice gas or lattice spin system. A new definition of a Gibbs measure is introduced for suitable real-valued functions of the configuration space, which play the physical role of specific internal energy. The variational principle is proved for a large class of functions, and then a more restrictive modulus of continuity condition is provided that guarantees a function's Gibbs measures to be a nonempty, weakly compact, convex set of measures that coincides with the set of measures obeying a form of the DLR equations (which has been adapted so as to be stated entirely in terms of specific internal energy instead of the Hamiltonians for an interaction potential). The variational equilibrium measures for a such a function are then characterized as the shift invariant Gibbs measures of finite entropy, and a condition is provided to determine if a function's Gibbs measures have infinite entropy or not. Moreover the spatially averaged limiting Gibbs measures, i.e. constructive equilibria, are shown to exist and their weakly closed convex hull is shown to coincide with the set of true variational equilibrium measures. It follows that the ...

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc68021/