Search Results

Using Steepest Descent to Find Energy-Minimizing Maps Satisfying Nonlinear Constraints

Description: The method of steepest descent is applied to a nonlinearly constrained optimization problem which arises in the study of liquid crystals. Let Ω denote the region bounded by two coaxial cylinders of height 1 with the outer cylinder having radius 1 and the inner having radius ρ. The problem is to find a mapping, u, from Ω into R^3 which agrees with a given function v on the surfaces of the cylinders and minimizes the energy function over the set of functions in the Sobolev space H^(1,2)(Ω; R^3) having norm 1 almost everywhere. In the variational formulation, the norm 1 condition is emulated by a constraint function B. The direction of descent studied here is given by a projected gradient, called a B-gradient, which involves the projection of a Sobolev gradient onto the tangent space for B. A numerical implementation of the algorithm, the results of which agree with the theoretical results and which is independent of any strong properties of the domain, is described. In chapter 2, the Sobolev space setting and a significant projection in the theory of Sobolev gradients are discussed. The variational formulation is introduced in Chapter 3, where the issues of differentiability and existence of gradients are explored. A theorem relating the B-gradient to the theory of Lagrange multipliers is stated as well. Basic theorems regarding the continuous steepest descent given by the Sobolev and B-gradients are stated in Chapter 4, and conditions for convergence in the application to the liquid crystal problem are given as well. Finally, in Chapter 5, the algorithm is described and numerical results are examined.
Date: August 1994
Creator: Garza, Javier, 1965-
Partner: UNT Libraries

L'Hospital's Rule

Description: The purpose of this paper is to present proofs for six cases of L'Hospital's Rule for the evaluation of indeterminate forms. It is also a purpose to reduce to one of these six cases some other indeterminate forms to which L'Hospital's Rule is applicable. In the course of presenting these proofs several theorems and definitions will be used without proof.
Date: 1950
Creator: Spidell, William H.
Partner: UNT Libraries

Cycles and Cliques in Steinhaus Graphs

Description: In this dissertation several results in Steinhaus graphs are investigated. First under some further conditions imposed on the induced cycles in steinhaus graphs, the order of induced cycles in Steinhaus graphs is at most [(n+3)/2]. Next the results of maximum clique size in Steinhaus graphs are used to enumerate the Steinhaus graphs having maximal cliques. Finally the concept of jumbled graphs and Posa's Lemma are used to show that almost all Steinhaus graphs are Hamiltonian.
Date: December 1994
Creator: Lim, Daekeun
Partner: UNT Libraries

The Effects of Writing-to-learn Tasks on Achievement and Attitude in Mathematics

Description: The problem of this study was to determine the effects of implementing writing-to-learn tasks in mathematics instruction on fourth grade students' achievement and attitude toward mathematics. Also addressed in this study is whether or not achievement and attitude measures of female students and low achieving students are effected by the use of writing in mathematics.
Date: May 1994
Creator: Millican, Beverly Robinson
Partner: UNT Libraries

Property (H*) and Differentiability in Banach Spaces

Description: A continuous convex function on an open interval of the real line is differentiable everywhere except on a countable subset of its domain. There has been interest in the problem of characterizing those Banach spaces where the continuous functions exhibit similar differentiability properties. In this paper we show that if a Banach space E has property (H*) and B_E• is weak* sequentially compact, then E is an Asplund space. In the case where the space is weakly compactly generated, it is shown that property (H*) is equivalent for the space to admit an equivalent Frechet differentiable norm. Moreover, we define the SH* spaces, show that every SH* space is an Asplund space, and show that every weakly sequentially complete SH* space is reflexive. Also, we study the relation between property (H*) and the asymptotic norming property (ANP). By a slight modification of the ANP we define the ANP*, and show that if the dual of a Banach spaces has the ANP*-I then the space admits an equivalent Fréchet differentiability norm, and that the ANP*-II is equivalent to the space having property (H*) and the closed unit ball of the dual is weak* sequentially compact. Also, we show that in the dual of a weakly countably determined Banach space all the ANP-K'S are equivalent, and they are equivalent for the predual to have property (H*).
Date: August 1993
Creator: Obeid, Ossama A.
Partner: UNT Libraries

Applications of Rapidly Mixing Markov Chains to Problems in Graph Theory

Description: In this dissertation the results of Jerrum and Sinclair on the conductance of Markov chains are used to prove that almost all generalized Steinhaus graphs are rapidly mixing and an algorithm for the uniform generation of 2 - (4k + 1,4,1) cyclic Mendelsohn designs is developed.
Date: August 1993
Creator: Simmons, Dayton C. (Dayton Cooper)
Partner: UNT Libraries

Primitive Substitutive Numbers are Closed under Rational Multiplication

Description: 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)
Partner: UNT Libraries

Countable Additivity, Exhaustivity, and the Structure of Certain Banach Lattices

Description: The notion of uniform countable additivity or uniform absolute continuity is present implicitly in the Lebesgue Dominated Convergence Theorem and explicitly in the Vitali-Hahn-Saks and Nikodym Theorems, respectively. V. M. Dubrovsky studied the connection between uniform countable additivity and uniform absolute continuity in a series of papers, and Bartle, Dunford, and Schwartz established a close relationship between uniform countable additivity in ca(Σ) and operator theory for the classical continuous function spaces C(K). Numerous authors have worked extensively on extending and generalizing the theorems of the preceding authors. Specifically, we mention Bilyeu and Lewis as well as Brooks and Drewnowski, whose efforts molded the direction and focus of this paper. This paper is a study of the techniques used by Bell, Bilyeu, and Lewis in their paper on uniform exhaustivity and Banach lattices to present a Banach lattice version of two important and powerful results in measure theory by Brooks and Drewnowski. In showing that the notions of exhaustivity and continuity take on familiar forms in certain Banach lattices of measures they show that these important measure theory results follow as corollaries of the generalized Banach lattice versions. This work uses their template to generalize results established by Bator, Bilyeu, and Lewis.
Date: August 1999
Creator: Huff, Cheryl Rae
Partner: UNT Libraries

A Content Analysis of the Writing Assignments Contained in the Four Basal Mathematics Textbook Series Adopted by the State of Texas

Description: The purpose of this study was to identify and compare specific writing assignments provided in the four basal mathematics textbook series, grades six through eight, accepted by the state of Texas in 1990. The student and teachers' editions by each publisher were analyzed (1) for the total number and types of writing assignments provided, (2) to compare how the writing assignments compared with the four purposes of writing mandated in the English Language Arts Framework, Kindergarten through Grade 12 for the state of Texas, (3) to compare how the writing assignments compared with the recommendations for communication opportunities stated in the National Council of Teachers of Mathematics' Curriculum and Evaluation Standards for School Mathematics for grades five through eight, and (4) to compare the number and types of writing assignments among the four publishers. The total number of writing assignments varied among publishers ranging from 151 to 316 in the student editions and from 147 to 523 in the teacher's editions. The findings of this study indicate that from 80 to 98 percent of the writing assignments in the student editions and from 72 to 96 percent of the writing assignments in the teacher's editions corresponded to the Informative purpose of writing. Very few writing assignments were provided corresponding to the Literary, Expressive, and Persuasive purposes of writing. The writing assignments corresponding to the NCTM recommendations varied among publishers. Writing assignments dealing with modeling mathematical situations ranged from 14 to 66 percent in the student editions and from 24 to 39 percent in the teacher's editions. Writing assignments focusing on understanding and definitions ranged 15 to 61 percent in the student editions and from 31 to 53 percent in the teacher's editions. Writing assignments focusing on interpretation and application ranged from 5 to 29 percent in the student editions and ...
Date: May 1993
Creator: Irvin, Barbara Bando
Partner: UNT Libraries

[Who's Who in Mathematics: Bill Townsend, 1943]

Description: Photograph of Bill Townsend, the Who's Who in Mathematics for the Yucca yearbook in 1943. In the image Townsend is writing or solving an equation on a chalkboard inside a classroom. "Who's Who" was an honor to students who demonstrated collegiate leadership through the balance of both academic excellence and extracurricular involvement in their personal field.
Date: 1943
Partner: UNT Libraries Special Collections

The Effect of Professional Development in Performance Assessment on Mathematics Achievement and Attitude

Description: The problem of this study was to determine the effect of professional development in the use of performance assessment in fourth grade mathematics on student achievement and attitude toward mathematics. Achievement was measured by subtest and total mathematics scores on norm-referenced and criterion-referenced tests. Attitude was measured by a survey of student attitudes.
Date: May 1994
Creator: McAdoo, Penny Coyne
Partner: UNT Libraries

Factors Related to Student Retention in Community College Developmental Education Mathematics

Description: This study investigated the factors related to student retention in a comprehensive community college developmental education mathematics program. The purpose was to identify and describe these factors and to develop strategies for improving retention in the community college developmental education mathematics program. Tinto's 1975 model of institutional departure was employed to examine different factors relating to retention in developmental education mathematics courses. In accordance with established criteria, data were collected using the Institutional Integration Scale (IIS) and Students Existing Records (SER). The IIS survey instrument questionnaire was completed by 41 students from a sample of 56 developmental education students enrolled in college level mathematics, and the data thus collected were used for analysis. Data were analyzed using frequency count, percentage, and the chi-square statistical analysis with a significant level of 0.05. The analysis of the data showed that the responding sample was primarily white, females aged 18 to 45. Most of the respondents had high grade point averages, did not miss any developmental education mathematics classes, and attended extra curricular activities infrequently. More fathers than mothers of the sample population had received a college education. Academic goal commitment, institutional experience, academic involvement, and placement grades were not statistically significant factors influencing retention. Among the major findings were: Development education instructors appeared to make the difference, institutional experience, academic goal commitment, and placement grades did not appear to play a major role; the students' academic involvement beyond classes appeared negligible; age, gender, grade point average, and parental educational levels were not significant factors for student retention in developmental education mathematics courses. Although statistical evidence did not support reversal of the proposed null hypotheses, pertinent issues for further research were raised.
Date: August 1992
Creator: Umoh, Udoudo J. (Udoudo Jimmy)
Partner: UNT Libraries

A Comparison of the Attitude and Achievement in Mathematics of Algebra I Students Using Computer-based Instruction and Traditional Instructional Methods

Description: This study investigated the use of computer-based instruction as a means of teaching Algebra I, compared to the teaching of the same topics using traditional methodologies. The achievement level of the two groups, and three aspects of attitude toward mathematics were considered. Achievement and attitude differences by gender were also analyzed.
Date: December 1992
Creator: Wohlgehagen, Kathleen Shannon
Partner: UNT Libraries

National Mathematics Advisory Panel

Description: The U.S. Department of Education, in partnership with the Conference Board of Mathematical Sciences, hosted the first National Math Panel Forum on October 6-7, 2008. This page documents the Forum activities, which brought together various organizations and other interested parties.
Date: 2008
Creator: National Mathematics Advisory Panel
Partner: UNT Libraries Government Documents Department

Steepest Sescent on a Uniformly Convex Space

Description: This paper contains four main ideas. First, it shows global existence for the steepest descent in the uniformly convex setting. Secondly, it shows existence of critical points for convex functions defined on uniformly convex spaces. Thirdly, it shows an isomorphism between the dual space of H^{1,p}[0,1] and the space H^{1,q}[0,1] where p > 2 and {1/p} + {1/q} = 1. Fourthly, it shows how the Beurling-Denny theorem can be extended to find a useful function from H^{1,p}[0,1] to L_{p}[1,0] where p > 2 and addresses the problem of using that function to establish a relationship between the ordinary and the Sobolev gradients. The paper contains some numerical experiments and two computer codes.
Date: August 1995
Creator: Zahran, Mohamad M.
Partner: UNT Libraries

Existence of a Sign-Changing Solution to a Superlinear Dirichlet Problem

Description: We study the existence, multiplicity, and nodal structure of solutions to a superlinear elliptic boundary value problem. Under specific hypotheses on the superlinearity, we show that there exist at least three nontrivial solutions. A pair of solutions are of one sign (positive and negative respectively), and the third solution changes sign exactly once. Our technique is variational, i.e., we study the critical points of the associated action functional to find solutions. First, we define a codimension 1 submanifold of a Sobolev space . This submanifold contains all weak solutions to our problem, and in our case, weak solutions are also classical solutions. We find nontrivial solutions which are local minimizers of our action functional restricted to various subsets of this submanifold. Additionally, if nondegenerate, the one-sign solutions are of Morse index 1 and the sign-changing solution has Morse index 2. We also establish that the action level of the sign-changing solution is bounded below by the sum of the two lesser levels of the one-sign solutions. Our results extend and complement the findings of Z. Q. Wang ([W]). We include a small sample of earlier works in the general area of superlinear elliptic boundary value problems.
Date: August 1995
Creator: Neuberger, John M. (John Michael)
Partner: UNT Libraries

Continuous, Nowhere-Differentiable Functions with no Finite or Infinite One-Sided Derivative Anywhere

Description: 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)
Partner: UNT Libraries