Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 1997. / Includes bibliographical references. Also available on the Internet.

Thesis (doctoral)--Universität Bonn, 1993. / Includes bibliographical references (p. 68-71).

A thesis submitted to the Faculty of Science, University of the Witwatersrand and to the University Cheikh Anta Diop of Dakar(UCAD) in fulfillment of the requirements for a Dual-degree for Doctor in Philosophy in Mathematics. November 6th, 2017. / This thesis is about the study of Diophantine equations involving binary recurrent sequences with arithmetic functions. Various Diophantine problems are investigated and new results are found out of this study. Firstly, we study several questions concerning the intersection between two classes of non-degenerate binary recurrence sequences and provide, whenever possible, effective bounds on the largest member of this intersection. Our main study concerns Diophantine equations of the form '(jaunj) = jbvmj; where ' is the Euler totient function, fungn 0 and fvmgm 0 are two non-degenerate binary recurrence sequences and a; b some positive integers. More precisely, we study problems involving members of the recurrent sequences being rep-digits, Lehmer numbers, whose Euler’s function remain in the same sequence. We prove that there is no Lehmer number neither in the Lucas sequence fLngn 0 nor in the Pell sequence fPngn 0. The main tools used in this thesis are lower bounds for linear forms in logarithms of algebraic numbers, the so-called Baker-Davenport reduction method, continued fractions, elementary estimates from the theory of prime numbers and sieve methods. / LG2018

Chan, Kam Fung. / "September 2010." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 61-62). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 1.1 --- Main result --- p.6 / Chapter 1.2 --- Useful definition --- p.7 / Chapter 1.3 --- Outline of proof of Theoreml.l --- p.11 / Chapter 2 --- Proof of Theorem1.2 and 13 --- p.12 / Chapter 2.1 --- The spaces \hV\ and \hW\ --- p.13 / Chapter 2.2 --- The space \hWloc\ --- p.19 / Chapter 2.3 --- The space \Wloc\ --- p.23 / Chapter 3 --- Proof of Theoreml4 --- p.26 / Chapter 3.1 --- Sheaves with category structure --- p.26 / Chapter 3.2 --- W° and hW° --- p.29 / Chapter 3.3 --- Armlets --- p.29 / Chapter 4 --- Proof of Theorem15 --- p.36 / Chapter 4.1 --- Homotopy colimit decompositions --- p.36 / Chapter 4.2 --- Introducing boundaries --- p.50 / Chapter 4.2.1 --- Proof of Theorem4.21 --- p.53 / Chapter 4.2.2 --- Proof of Lemma4.20 --- p.56 / Chapter 4.3 --- Using the Harer-Ivanov stabilization theorem --- p.58 / Bibliography --- p.61

Hypergeometric functions seem to be ubiquitous in mathematics. In this document, we present a couple of ways in which hypergeometric functions appear in arithmetic geometry. First, we show that the number of points over a finite field [mathematical symbol] on a certain family of hypersurfaces, [mathematical symbol] ([lamda]), is a linear combination of hypergeometric functions. We use results by Koblitz and Gross to find explicit relationships, which could be useful for computing Zeta functions in the future. We then study more geometric aspects of the same families. A construction of Dwork's gives a vector bundle of deRham cohomologies equipped with a connection. This connection gives rise to a differential equation which is known to be hypergeometric. We developed an algorithm which computes the parameters of the hypergeometric equations given the family of hypersurfaces. / text

Thesis (Ph. D.)--University of Texas at Austin, 2009. / Title from PDF title page (University of Texas Digital Repository, viewed on Sept. 9, 2009). Vita. Includes bibliographical references.

Multiplicative reasoning permeates many mathematical topics, for example fractions and functions. Hence there is consensus on the importance of acquiring multiplicative reasoning. Multiplication is typically introduced as repeated addition, but when it is extended to include multi-digits and decimals a more general view of multiplication is required. There are conflicting reports in previous research concerning students’ understandings of multiplication. For example, repeated addition has been suggested both to support students’ understanding of calculations and as a hindrance to students’ conceptualisation of the two-dimensionality of multiplication. The relative difficulty of commutativity and distributivity is also debated, and there is a possible conflict in how multiplicative reasoning is described and assessed. These inconsistencies are addressed in a study with the aim of understanding more about students’ understandings of multiplication when it is expanded to comprise multi-digits and decimals. Understanding is perceived as connections between representations of different types of knowledge, linked together by reasoning. Especially connections between three components of multiplication were investigated; models for multiplication, calculations and arithmetical properties. Explicit reasoning made the connections observable and externalised mental representations. Twenty-two students were recurrently interviewed during five semesters in grades five to seven to find answers to the overarching research question: What do students’ responses to different forms of multiplicative tasks in the domain of multi-digits and decimals reveal about their understandings of multiplication? The students were invited to solve different forms of tasks during clinical interviews, both individually and in pairs. The tasks involved story telling to given multiplications, explicit explanations of multiplication, calculation problems including explanations and justifications for the calculations and evaluation of suggested calculation strategies. Additionally the students were given written word problems to solve. The students’ understandings of multiplication were robustly rooted in repeated addition or equally sized groups. This was beneficial for their understandings of calculations and distributivity, but hindered them from fluent use of commutativity and to conceptualise decimal multiplication. The robustness of their views might be explained by the introduction to multiplication, which typically is by repeated addition and modelled by equally sized groups. The robustness is discussed in relation to previous research and the dilemma that more general models for multiplication, such as rectangular area, are harder to conceptualise than models that are only susceptible to natural numbers. The study indicated that to evaluate and explain others’ calculation strategies elicited more reasoning and deeper mathematical thinking compared to evaluating and explaining calculations conducted by the students themselves. Furthermore, the different forms of tasks revealed various lines of reasoning and to get a richly composed picture of students’ multiplicative reasoning and understandings of multiplication, a wide variety of forms of tasks is suggested. / <p>At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 3: Manuscript. Paper 4: Manuscript.</p>

This paper presents the results of a study of dyscalculia. It is a retrospective archival study implemented with a deductive approach. On the basis of established research and theory 18 analytical categories were formulated, before a deductive thematic analysis of empirical material, consisting of journal data of 17 students investigated for dyscalculia, 14 girls (82.4%) and 3 boys (17.6%). The purpose of this study was to investigate the relationship between the concepts formulated in research on dyscalculia and actual mathematical difficulties as those found in practice of students at school. All pupils had early and long-term difficulties with mathematics, while not showing any difficulties in other subjects. Most have had an unsatisfactory learning environment. All had normal intelligence but difficulty with certain cognitive, self-regulatory and linguistic features. Difficulties persisted despite numerous and protracted relief efforts at school. The study highlights that some difficulties are more prominent than others in connection with dyscalculia. This applies particularly to working memory, automation, activity control, spatial functions, certain linguistic abilities, concentration, and executive functions. Pedagogical action adaptation had been completed for most of the students but did not show to have any noticible effect. One question that requires further research would be “ why adaptation does not give the desired effect.” / I denna uppsats redovisas resultatet av en studie av dyskalkyli. Det är en retrospektiv arkivstudie med en deduktiv ansats som genomförts. Med utgångspunkt i etablerad forskning och teoribildning formulerades 18 analytiska kategorier, före en deduktiv tematisk analys, på ett empiriskt material bestående av journaldata för 17 elever utredda med avseende på dyskalkylidiagnos, 14 flickor (82,4 %) och 3 pojkar (17,6 %). Syftet med studien var att undersöka förhållandet mellan de begrepp forskningen formulerat om dyskalkyli och faktiska matematiksvårigheter så som sådana visar sig i praktiken hos elever i skolan. Samtliga elever hade tidiga och långvariga svårigheter i matematik, men vanligen inte i andra ämnen. De flesta hade haft en otillfredsställande inlärningsmiljö. Alla hade normal intelligens men svårigheter med vissa kognitiva, självreglerande och språkliga funktioner. Svårigheterna kvarstod trots många och långvariga hjälpinsatser i skolan. Studien lyfter fram att vissa svårigheter är mer framträdande än andra i samband med dyskalkyli. Det gäller framförallt arbetsminne, automatisering, aktivitetsreglering, spatiala funktioner, vissa språkliga förmågor, koncentration och exekutiva funktioner. Pedagogisk åtgärdsanpassning hade genomförts för de flesta av eleverna men verkade inte ha haft någon större effekt. Varför åtgärdsanpassning inte ger avsedd effekt är ett problem som behöver undersökas vidare.

This thesis consists of three chapters. Chapter 1 explains how the research problems considered in this thesis fit into the investigation of local-global principle in the diophantine geometry, as well as gives a unified sketch of the proofs of the two main results in this thesis. Chapter 2 establishes a similar conclusion to Theorem B of a paper by Poonen and Voloch in another settings. Chapter 3 relates to the object considered in the main result of Chapter 2 to an old conjecture proposed by Skolem and solves some cases of its analog. / text

Submitted by JOSIANE FAXINA null (josi_unesp@hotmail.com) on 2017-04-18T23:08:53Z No. of bitstreams: 1 Dissertação Josiane Faxina.pdf: 4519766 bytes, checksum: 3f6a823eee0633e12b7eb3aceaee7d0a (MD5) / Approved for entry into archive by Luiz Galeffi (luizgaleffi@gmail.com) on 2017-04-19T13:43:32Z (GMT) No. of bitstreams: 1 faxina_j_me_bauru.pdf: 4519766 bytes, checksum: 3f6a823eee0633e12b7eb3aceaee7d0a (MD5) / Made available in DSpace on 2017-04-19T13:43:32Z (GMT). No. of bitstreams: 1 faxina_j_me_bauru.pdf: 4519766 bytes, checksum: 3f6a823eee0633e12b7eb3aceaee7d0a (MD5) Previous issue date: 2017-02-22 / A presente pesquisa tem como objetivo investigar as percepções que o professor dos anos iniciais do Ensino Fundamental tem sobre o trabalho com resolução de problemas no ensino dos conceitos aritméticos. Entende-se que o ensino da Matemática via resolução de problemas possibilita ao aluno uma construção do conhecimento onde ele mesmo desenvolverá ideias e estratégias com sentido próprio. Foi aplicado um questionário semiestruturado a 21 professores da rede municipal da cidade de Bauru, interior de São Paulo, no qual puderam descrever suas percepções, práticas e dificuldades no ensino das quatro operações via resolução de problemas. A literatura estudada aponta questões sobre como a resolução de problemas muitas vezes fica limitada a uma oportunidade de se aplicar o conteúdo estudado previamente e que o ensino das operações aritméticas ainda está muito vinculado ao algoritmo, proporcionando-se assim, um ensino restrito a procedimentos e técnicas, ou seja, a supervalorização de contas armadas. Não se prioriza o ensino dos conceitos presentes em cada operação, o que é possível garantindo-se ao aluno uma variedade de situações. Na análise dos dados coletados pelo questionário, foi possível verificar que os professores têm boas ideias sobre o trabalho com resolução de problemas e principalmente da importância desse trabalho nos anos iniciais, porém o número desses professores não indicou a maioria. Também foi possível notar que as ideias sobre o ensino das quatro operações sempre estão relacionadas à situação do cotidiano, resolução de problemas, materiais concretos entre outros. Entretanto, quando analisadas as dificuldades apontadas pelos professores, constatou-se que, em geral, estão relacionadas com a compreensão do sistema de numeração decimal e dos algoritmos. Sendo assim, nota-se que as ideias dos professores que ensinam Matemática nos anos iniciais, em parte, são coniventes com uma prática de ensino que prioriza uma Matemática que faça sentido ao aluno, porém também apontam preocupação excessiva em fazer o aluno aprender procedimentos sem uma prévia compreensão conceitual de fato. A partir da análise dos dados, e da literatura estudada, um e-book foi elaborado como produto educacional com propostas didáticas que subsidiem o professor no ensino dos conceitos aritméticos. / This research aims to investigate the representation of the teacher in the early years of elementary school has about working with problems solving in teaching the concepts of the four arithmetic operations. It is understood that the teaching of mathematics through problem solving allows the student a knowledge building where he will develop ideas and strategies with proper sense. A semi-structured questionnaire to 21 teachers of the municipal town in the interior of São Paulo State was applied, in which they could describe their ideas, practices and difficulties in the four operations teaching by problem solving. The literature shows questions about problem solving often is limited to one opportunity to apply content previously studied and the teaching of arithmetic operations is still very much linked to the algorithm, providing thus, a limited teaching procedures and techniques. No priority to the teaching of the present concepts in each operation, it is possible to guarantee the student a variety of situations. In the analysis of data collected by questionnaire, we found that teachers have good ideas about working with problem solving and especially the importance of this work in the early years, but the number of these teachers did not indicate the major part. It was also noted that the ideas of the four teaching operations are always related to everyday situation, problem solving and concrete materials among others. However, when analyzing the difficulties indicated by teachers, it was found that, in general, they are related to the understanding of the decimal number system and algorithms. Thus, until the present moment of this research, it is clear that the ideas of teachers who teach mathematics in the early years are colluding with a teaching practice that prioritizes a mathematics that makes sense to the student, but also show excessive concern to the student learning procedures without prior conceptual understanding indeed. From the data analysis, and studied literature, an e-book is designed as an educational product with proposals for activities that support the teacher in the teaching of arithmetic concepts.