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Dimension Theory
This paper contains a discussion of topological dimension theory. Original proofs of theorems, as well as a presentation of theorems and proofs selected from Ryszard Engelking's Dimension Theory are contained within the body of this endeavor. Preliminary notation is introduced in Chapter I. Chapter II consists of the definition of and theorems relating to the small inductive dimension function Ind. Large inductive dimension is investigated in Chapter III. Chapter IV comprises the definition of covering dimension and theorems discussing the equivalence of the different dimension functions in certain topological settings. Arguments pertaining to the dimension o f Jn are also contained in Chapter IV.
Date:
August 1986
Creator:
Frere, Scot M. (Scot Martin)
System:
The UNT Digital Library
Weak and Norm Convergence of Sequences in Banach Spaces
We study weak convergence of sequences in Banach spaces. In particular, we compare the notions of weak and norm convergence. Although these modes of convergence usually differ, we show that in ℓ¹ they coincide. We then show a theorem of Rosenthal's which states that if {𝓍ₙ} is a bounded sequence in a Banach space, then {𝓍ₙ} has a subsequence {𝓍'ₙ} satisfying one of the following two mutually exclusive alternatives; (i) {𝓍'ₙ} is weakly Cauchy, or (ii) {𝓍'ₙ} is equivalent to the unit vector basis of ℓ¹.
Date:
December 1993
Creator:
Hymel, Arthur J. (Arthur Joseph)
System:
The UNT Digital Library
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 non-trivial partition. We completely classify groups of type 2, and present examples of other groups of positive type as well as groups of type zero.
Date:
August 1995
Creator:
Moore, Monty L.
System:
The UNT Digital Library
Continuous, Nowhere-Differentiable Functions with no Finite or Infinite One-Sided Derivative Anywhere
In this paper, we study continuous functions with no finite or infinite one-sided derivative anywhere. In 1925, A. S. Beskovitch published an example of such a function. Since then we call them Beskovitch functions. This construction is presented in chapter 2, The example was simple enough to clear the doubts about the existence of Besicovitch functions. In 1932, S. Saks showed that the set of Besicovitch functions is only a meager set in C[0,1]. Thus the Baire category method for showing the existence of Besicovitch functions cannot be directly applied. A. P. Morse in 1938 constructed Besicovitch functions. In 1984, Maly revived the Baire category method by finding a non-empty compact subspace of (C[0,1], || • ||) with respect to which the set of Morse-Besicovitch functions is comeager.
Date:
December 1994
Creator:
Lee, Jae S. (Jae Seung)
System:
The UNT Digital Library
Primitive Substitutive Numbers are Closed under Rational Multiplication
Lehr (1991) proved that, if M(q, r) denotes the set of real numbers whose expansion in base-r is q-automatic i.e., is recognized by an automaton A = (Aq, Ar, ao, δ, φ) (or is the image under a letter to letter morphism of a fixed point of a substitution of constant length q) then M(q, r) is closed under addition and rational multiplication. Similarly if we let M(r) denote the set of real numbers α whose base-r digit expansion is ultimately primitive substitutive, i.e., contains a tail which is the image (under a letter to letter morphism) of a fixed point of a primitive substitution then in an attempt to generalize Lehr's result we show that the set M(r) is closed under multiplication by rational numbers. We also show that M(r) is not closed under addition.
Date:
August 1998
Creator:
Ketkar, Pallavi S. (Pallavi Subhash)
System:
The UNT Digital Library
Plane Curves, Convex Curves, and Their Deformation Via the Heat Equation
We study the effects of a deformation via the heat equation on closed, plane curves. We begin with an overview of the theory of curves in R3. In particular, we develop the Frenet-Serret equations for any curve parametrized by arc length. This chapter is followed by an examination of curves in R2, and the resultant adjustment of the Frenet-Serret equations. We then prove the rotation index for closed, plane curves is an integer and for simple, closed, plane curves is ±1. We show that a curve is convex if and only if the curvature does not change sign, and we prove the Isoperimetric Inequality, which gives a bound on the area of a closed curve with fixed length. Finally, we study the deformation of plane curves developed by M. Gage and R. S. Hamilton. We observe that convex curves under deformation remain convex, and simple curves remain simple.
Date:
August 1998
Creator:
Debrecht, Johanna M.
System:
The UNT Digital Library
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.
Date:
August 1995
Creator:
Thompson, Jeremy R. (Jeremy Ray)
System:
The UNT Digital Library
Intuition versus Formalization: Some Implications of Incompleteness on Mathematical Thought
This paper describes the tension between intuition about number theory and attempts to formalize it. I will first examine the root of the dilemma, Godel's First Incompleteness Theorem, which demonstrates that in any reasonable formalization of number theory, there will be independent statements. After proving the theorem, I consider some of its consequences on intuition, focusing on Freiling's "Dart Experiment" which is based on our usual notion of the real numbers as a line. This experiment gives an apparent refutation of the Axiom of Choice and the Continuum Hypothesis; however, it also leads to an equally apparent paradox. I conclude that such paradoxes are inevitable as the formalization of mathematics takes us further from our initial intuitions.
Date:
August 1994
Creator:
Lindman, Phillip A. (Phillip Anthony)
System:
The UNT Digital Library
A Generalization of Sturmian Sequences: Combinatorial Structure and Transcendence
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.
Date:
August 1998
Creator:
Risley, Rebecca N.
System:
The UNT Digital Library
The Exponential Functions
The purpose of this paper is to clarify the problems of exponents by introducing notation not customarily used and by demonstrating certain theorems in regard to the properties of the exponential functions.
Date:
1947
Creator:
Sloan, Robert S.
System:
The UNT Digital Library
A Generalization of Newton's Method
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.
Date:
1948
Creator:
LeBouf, Billy Ruth
System:
The UNT Digital Library
The Analogues for t-Continuity of Certain Theorems on Ordinary Continuity
This study investigates the relationship between ordinary continuity and t-continuity.
Date:
1941
Creator:
Parrish, Herbert C.
System:
The UNT Digital Library
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.
Date:
1945
Creator:
Ashburn, Andrew
System:
The UNT Digital Library
Some Fundamental Properties of Gamma and Beta Functions
This paper consists of a discussion of the properties and applications of certain improper integrals, namely the gamma function and the beta function. There are also specific examples of application of these functions in certain fields of applied science.
Date:
1941
Creator:
Nolen, Robert L.
System:
The UNT Digital Library
The Dyadic Operator Approach to a Study in Conics, with some Extensions to Higher Dimensions
The discovery of a new truth in the older fields of mathematics is a rare event. Here an investigator may hope at best to secure greater elegance in method or notation, or to extend known results by some process of generalization. It is our purpose to make a study of conic sections in the spirit of the above remark, using the symbolism developed by Josiah Williard Gibbs.
Date:
1940
Creator:
Shawn, James Loyd
System:
The UNT Digital Library
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.
Date:
1940
Creator:
Beeman, William Edwin
System:
The UNT Digital Library
Some Properties and Applications of Elliptic Integrals
The object of this paper is to present the properties and some of the applications of the Elliptic Integrals.
Date:
1944
Creator:
Townsend, Bill B.
System:
The UNT Digital Library
A Detailed Proof of the Prime Number Theorem for Arithmetic Progressions
We follow a research paper that J. Elstrodt published in 1998 to prove the Prime Number Theorem for arithmetic progressions. We will review basic results from Dirichlet characters and L-functions. Furthermore, we establish a weak version of the Wiener-Ikehara Tauberian Theorem, which is an essential tool for the proof of our main result.
Date:
May 2004
Creator:
Vlasic, Andrew
System:
The UNT Digital Library
Lyapunov Exponents, Entropy and Dimension
We consider diffeomorphisms of a compact Riemann Surface. A development of Oseledec's Multiplicative Ergodic Theorem is given, along with a development of measure theoretic entropy and dimension. The main result, due to L.S. Young, is that for certain diffeomorphisms of a surface, there is a beautiful relationship between these three concepts; namely that the entropy equals dimension times expansion.
Date:
August 2004
Creator:
Williams, Jeremy M.
System:
The UNT Digital Library
Thermodynamical Formalism
Thermodynamical formalism is a relatively recent area of pure mathematics owing a lot to some classical notions of thermodynamics. On this thesis we state and prove some of the main results in the area of thermodynamical formalism. The first chapter is an introduction to ergodic theory. Some of the main theorems are proved and there is also a quite thorough study of the topology that arises in Borel probability measure spaces. In the second chapter we introduce the notions of topological pressure and measure theoretic entropy and we state and prove two very important theorems, Shannon-McMillan-Breiman theorem and the Variational Principle. Distance expanding maps and their connection with the calculation of topological pressure cover the third chapter. The fourth chapter introduces Gibbs states and the very important Perron-Frobenius Operator. The fifth chapter establishes the connection between pressure and geometry. Topological pressure is used in the calculation of Hausdorff dimensions. Finally the sixth chapter introduces the notion of conformal measures.
Date:
August 2004
Creator:
Chousionis, Vasileios
System:
The UNT Digital Library
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=b-Axk) 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 …
Date:
August 2013
Creator:
Pathak, Subrat
System:
The UNT Digital Library
Some Properties of Noetherian Rings
This paper is an investigation of several basic properties of noetherian rings. Chapter I gives a brief introduction, statements of definitions, and statements of theorems without proof. Some of the main results in the study of noetherian rings are proved in Chapter II. These results include proofs of the equivalence of the maximal condition, the ascending chain condition, and that every ideal is finitely generated. Some other results are that if a ring R is noetherian, then R[x] is noetherian, and that if every prime ideal of a ring R is finitely generated, then R is noetherian.
Date:
May 1986
Creator:
Vaughan, Stephen N. (Stephen Nick)
System:
The UNT Digital Library
The Cohomology for the Nil Radical of a Complex Semisimple Lie Algebra
Let g be a complex semisimple Lie algebra, Vλ an irreducible g-module with high weight λ, pI a standard parabolic subalgebra of g with Levi factor £I and nil radical nI, and H*(nI, Vλ) the cohomology group of Λn'I ⊗Vλ. We describe the decomposition of H*(nI, Vλ) into irreducible £1-modules.
Date:
May 1994
Creator:
Sawyer, Cameron C. (Cameron Cunningham)
System:
The UNT Digital Library
Algorithms of Schensted and Hillman-Grassl and Operations on Standard Bitableaux
In this thesis, we describe Schensted's algorithm for finding the length of a longest increasing subsequence of a finite sequence. Schensted's algorithm also constructs a bijection between permutations of the first N natural numbers and standard bitableaux of size N. We also describe the Hillman-Grassl algorithm which constructs a bijection between reverse plane partitions and the solutions in natural numbers of a linear equation involving hook lengths. Pascal programs and sample output for both algorithms appear in the appendix. In addition, we describe the operations on standard bitableaux corresponding to the operations of inverting and reversing permutations. Finally, we show that these operations generate the dihedral group D_4
Date:
August 1983
Creator:
Sutherland, David C. (David Craig)
System:
The UNT Digital Library