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Dissertations / Theses on the topic 'Arithmetic Progression'
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Lee, June-Bok. "Integral solutions in arithmetic progression for elliptic curves." Diss., The University of Arizona, 1991. http://hdl.handle.net/10150/185436.
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Integral solutions to y² = X³ + k, where either the x's or the y's, or both, are in arithmetic progression are studied. When both the x's and the y's are in arithmetic progression, then this situation is completely solved. One set of solutions where the y's formed an arithmetic progression of length 4 have already been constructed. In this dissertation, we construct infinitely many set of solutions where there are 4 x's in arithmetic progression and we also disprove Mohanty's Conjecture[8] by constructing infinitely many set of solutions where there are 4, 5 and 6 y's in arithmetic progression.
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Marchetto, Raquel. "O uso do software GeoGebra no estudo de progressões aritméticas e geométricas, e sua relação com funções afins e exponenciais." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2017. http://hdl.handle.net/10183/172105.
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Esta pesquisa teve por objetivo verificar como é que o aluno consegue por si próprio manipular os recursos, tais como gráficos disponibilizados pelo software GeoGebra, para auxiliar nas práticas diárias de sala de aula, mais especificamente no que tange a construir a conexão entre as progressões aritméticas e as funções afins, bem como entre as progressões geométricas e as funções exponenciais. Este software possibilita fazer análises a partir de diferentes registros tais como: gráficos, tabelas e registros algébricos, seguindo a teoria dos registros semióticos de Duval. Como metodologia, desenvolvemos roteiros de atividades com duas turmas do 2º ano do Ensino Médio do Colégio Estadual Visconde de Bom Retiro. Os alunos foram convidados a construir, verificar e interpretar seus próprios resultados, refletindo e analisando estratégias para responder à questão: Quais relações os alunos conseguem evidenciar, através da comparação entre gráficos (obtidos com o GeoGebra) de funções afins e exponenciais, com progressões aritméticas e geométricas, respectivamente? Ao final da pesquisa, os registros coletados possibilitaram a validação qualitativa da proposta, mostrando que os alunos avançaram na compreensão dos conteúdos abordados.The aim of this research was to verify how the student can himself manipulate the resources, such as plots made available by the GeoGebra software, to aid in the daily classroom practices, specifically in the construction of the connection between arithmetic progressions and linear functions, as well as between geometric progressions and exponential functions. This software makes possible to analyze from different registers such as: plots, tables and algebraic records, following the theory of semiotic records of Duval. Our methodology consisted in developing activity scripts with students of two classes of the 2nd year of the High School of Visconde de Bom Retiro State College. More specifically, they were asked to build, verify and interpret their own results, speculating and analyzing strategies to answer the question: What relations are the students able to highlight through comparing plots (obtained with GeoGebra) of linear and exponential functions, with arithmetic progressions and geometric, respectively? At the end of the research, the collected records made possible the qualitative validation of the proposal, showing that the students improved their understanding of the focused contents.
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Cergoli, Daniel. "Ensino de logaritmos por meio de investigações matemáticas em sala de aula." Universidade de São Paulo, 2016. http://www.teses.usp.br/teses/disponiveis/45/45135/tde-29112017-171710/.
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Neste trabalho são apresentadas duas propostas de sequências didáticas para ensino de logaritmos. A primeira delas é destinada ao aperfeiçoamento de professores de Matemática e a outra, para alunos de Ensino Médio. Tais sequências foram desenvolvidas com base em pesquisas realizadas pelo Prof. João Pedro da Ponte sobre o processo de investigação matemática. A sequência didática para professores foi aplicada no Centro de Aperfeiçoamento do Ensino de Matemática do Instituto de Matemática e Estatística da Universidade de São Paulo (CAEM IME USP). Já a sequência para alunos foi aplicada em uma escola da rede estadual situada no município de São Paulo. Ambas foram analisadas sob os pontos de vista da eficiência e adequação, bem como da clareza das ideias apresentadas. As sequências didáticas têm como ponto de partida a observação das propriedades comuns a várias tabelas, cada uma contendo uma progressão geométrica ao lado de uma progressão aritmética. Tais propriedades caracterizam o que virá a ser definido como logaritmo. Essa introdução ao conceito de logaritmo é diferente da usual, que se baseia na solução de uma equação exponencial. O processo de investigação matemática visa a um aprendizado eficaz por parte do aluno, proporcionado por atividades que conduzam o aluno, de forma gradual, a fazer descobertas, formular conjecturas e buscar validações. Tais investigações são coordenadas e supervisionadas pelo professor, cujo papel é fundamental no processo de construção do conhecimento.This dissertation presents two didactic sequences for teaching and learning logarithms. One of them aims at Mathematics teachers and is designed for improving their knowledge. The other sequence is meant to be used on high school students. Both didactic sequences were developed based upon research carried out by Professor João Pedro da Ponte on Mathematical Investigations. The didactic sequence for teachers was applied at CAEM IME USP. The one for students was applied at a state school in the city of São Paulo. They were analysed from the points of view of efficiency and of adequacy, as well as of the clarity of the presented ideas. The didactic sequences start with the observation of properties common to multiple tables, each containing a geometric progression side by side with an arithmetic progression. The observed properties characterize what will be later defined as logarithm. Such introduction to the concept of logarithm is different from the usual, which is based on the solution of an exponential equation. The Mathematical Investigation process aims at an effective learning by the students, which is provided by activities that lead the student to gradually make discoveries, formulate conjectures, and search for validations. These investigations are coordinated and supervised by the teacher, whose role in the knowledge construction process is fundamental.
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Costa, Antonio Carlos de Lima. "O estudo de Induções e Recorrências - uma abordagem para o Ensino Médio." Universidade Federal da Paraíba, 2015. http://tede.biblioteca.ufpb.br:8080/handle/tede/8064.
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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
This work presents a research on the Induction Principles and the Recurrences, its history, mathematical concepts and applications used. Were developed some statements of Induction Principle and the Recurrences using some basic concepts from Number Theory, Combinatorics, Geometry and Set Theory that can be explored in high school. Was submitted a few activities that can be applied in the classroom of the high school in the development of mathematical concepts such as Arithmetic Progression, Geometric Progression, Geometry, Combinatorial Analysis and Numeric Sets among other topics.
Este trabalho apresenta uma pesquisa sobre os Princípios de indução e Recorrências, sua história, conceitos matemáticos utilizados e aplicações. Foram desenvolvidas algumas demonstrações do Princípio de indução e Recorrências, utilizando-se alguns conceitos básicos de Teoria dos Números, Análise Combinatória, Teoria dos Conjuntos e Geometria que podem ser explorados no Ensino Médio. Foram apresentadas algumas atividades que podem ser aplicadas em sala de aula do Ensino Médio no desenvolvimento de conceitos matemáticos como Progressão Aritmética, Progressão Geométrica, Geometria, Combinação e Conjuntos Numéricos entre outros temas.
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Brinsfield, Joshua Sol. "The Factoradic Integers." Thesis, Virginia Tech, 2016. http://hdl.handle.net/10919/71451.
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The arithmetic progressions under addition and composition satisfy the usual rules of arithmetic with a modified distributive law. The basic algebra of such mathematical structures is examined; this leads to the consideration of the integers as a metric space under the "factoradic metric", i.e., the integers equipped with a distance function defined by d(n,m)=1/N!, where N is the largest positive integer such that N! divides n-m. Via the process of metric completion, the integers are then extended to a larger set of numbers, the factoradic integers. The properties of the factoradic integers are developed in detail, with particular attention to prime factorization, exponentiation, infinite series, and continuous functions, as well as to polynomials and their extensions. The structure of the factoradic integers is highly dependent upon the distribution of the prime numbers and relates to various topics in algebra, number theory, and non-standard analysis.Master of Science
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Young, Catherine. "Adaptation of the mathematics recovery programme to facilitate progression in the early arithmetic strategies of Grade 2 learners in Zambia." Thesis, Rhodes University, 2017. http://hdl.handle.net/10962/4977.
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Research indicates that many children finish primary school in Southern Africa still reliant on inefficient counting strategies. This study extends the research of the South African Numeracy Chair project to early mathematics intervention with Grade 2 learners. It investigated the possible adaptation of the Mathematics Recovery programme to facilitate learner progression in early arithmetic strategies. This study aimed to investigate the possibility of adapting the Mathematics Recovery programme for use in a whole class setting, and to research the effectiveness of such an adapted programme. This study also aimed to investigate the extent of the phenomenon of unit counting and other early arithmetic strategies used in the early years in Zambia. This study was conducted from an emergent perspective. A review of the literature indicated that children who become stuck using unit counting face later mathematical difficulties, and that teacher over-emphasis on unit counting in the early years of schooling may be a contributing factor. This study used a qualitative design research methodology that consisted of a preparation phase, teaching experiment and retrospective analysis. The context of this teaching experiment was a seven week after-school intervention with a class of Grade 2 learners aged seven to eight in a rural Zambian primary school. Data collection and analysis focused on video recordings of a sample of 6 learners. The experimental teaching content focused on the Early Arithmetic Strategies aspect of the Mathematics Recovery programme. Although limited by time and research focus, this study found that all learners made some progress in early arithmetic strategies, and indicates that the Mathematics Recovery programme has potential for adaptation for early intervention in whole class teaching to address the mathematical education challenges in Zambia and beyond. This study also found that unit counting predominated in the sample learners, but that strategies were not yet entrenched, indicating this was a suitable age for early intervention. This study makes methodological contributions to a growing body of research into the adaptation of the Mathematics Recovery in Southern African contexts and suggests avenues for possible further research.
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Farhangi, Sohail. "On Refinements of Van der Waerden's Theorem." Thesis, Virginia Tech, 2016. http://hdl.handle.net/10919/73355.
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Feutrie, David. "Sur deux questions de crible." Electronic Thesis or Diss., Université de Lorraine, 2019. http://www.theses.fr/2019LORR0173.
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Cette thèse se divise en deux grandes parties. Le premier chapitre porte sur l’étude du nombre des entiers n’excédant pas x et n’admettant aucun diviseur dans une progression arithmétique a(mod q) donnée. Nous améliorons ici un résultat de Narkiewicz et Radziejewski de 2011 en fournissant une expression différente et plus simple du terme principal et en précisant le terme d’erreur. Les outils principaux sont la méthode de Selberg-Delange et le contour de Hankel. Nous étudions plus en détail le cas particulier où a n’est pas un résidu quadratique modulo q. Nous étendons également notre résultat aux entiers n’admettant aucun diviseur dans un ensemble fini de classes résiduelles modulo q. Le second chapitre est consacré aux entiers ultrafriables dans les progressions arithmé- tiques. Un entier y-ultrafriable est un entier dont toutes les puissances de nombres premiers qui le divisent sont inférieures à y. Nous commençons par étudier la fonction de comptage des ces entiers lorsqu’ils sont premiers à un entier q. Nous donnons ensuite des formules asymptotiques sur le nombre d’entiers y-ultrafriables inférieurs à un entier x et dans une progression arith- métique a modulo q, où q est un module y-friable, c’est-à-dire sans facteur premier supérieur à y. Nos résultats sont valables pour des entiers q, x, y tels que log x « y < x, q < yc/ log log y, où c > 0 est une constante choisie convenablementThis thesis is divided into two main parts. In the first chapter, we consider the number of integers not exceeding x and admitting no divisor in an arithmetic progression a(mod q) where q is fixed. We improve here a result of Narkiewicz and Radziejewski published in 2011 by providing a different main term with a simpler expression, and we specify the term error. The main tools are the Selberg-Delange method and the Hankel contour. We also study in detail the particular case where a is a quadratic nonresidue modulo q. We also extend our result to the integers which admit no divisor in a finite set of residual classes modulo q. In the second chapter, we study the ultrafriable integers in arithmetic progressions. An integer is said to be y-ultrafriable if no prime power which divide it exceeds y. We begin with the studying of the counting function of these integers when they are coprime to q. Then we give an asymptotic formula about the number of y-ultrafriable integers which don’t exceed a number x and in an arithmetic progression a modulo q, where q is a y-friable modulus, which means that it is without a prime divisor exceeding y. Our results are valid when q, x, y are integers which verify log x « y < x, q < yc/ log log y, where c > 0 is a suitably chosen constant
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Darreye, Corentin. "Sur la répartition des coefficients des formes modulaires de poids demi-entier." Thesis, Bordeaux, 2020. http://www.theses.fr/2020BORD0171.
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Cette thèse traite de certains aspects analytiques liés aux coefficients de Fourier des formes modulaires de poids demi-entier. On étudie en particulier deux problèmes a priori bien différents mais que l’on reliera.Tout d’abord, on s’intéresse aux sommes des coefficients d’une forme cuspidale de poids demi-entier dans les progressions arithmétiques. Un tel problème fut étudié précédemment dans un article de Fouvry, Ganguly, Kowalski et Michel mais dans le cas d’une forme de poids entier. Les auteurs montrent notamment que, dans un certain régime de convergence, on a une équirépartition gaussienne des sommes des coefficients dans des progressions arithmétiques de module fixé.Dans ce travail, on prouve un résultat analogue lorsque la forme modulaire est de poids demi-entier. On verra que, dans un régime de convergence plus fin, les sommes des coefficients en progression arithmétique s’équirépartissent selon une loi qui est différente de la loi normale obtenue par Fouvry, Ganguly, Kowalski et Michel en poids entier.Dans un deuxième temps, on étudiera les signes des coefficients d’une forme de poids demi-entier f et des possibles minorations en valeur absolue de ces derniers. En utilisant certaines techniques issues du premier problème ainsi que des résultats classiques de la théorie des formes de poids demi-entier, comme la correspondance de Shimura, la formule de Waldspurger ou encore la récente théorie des formes nouvelles, on établie une borne inférieure sur le nombre de coefficients normalisés f(n) tels que n le x, où n est pris dans une progression arithmétique, et f(n) > n^{−alpha} avec alpha > 0This thesis deals with some analytic aspects of Fourier coefficients of half-integral weight modular forms. We study in particular two different problems which will be nonetheless connected.On one hand, we are interested in sums of coefficients of half-integral weight cusp forms in arithmetic progressions. Such a problem was studied in a previous paper of Fouvry, Ganguly, Kowalski and Michel for an integral weight cusp form. They showed that, in acertain range of convergence, there is a Gaussian equidistribution of sums of coefficientsin arithmetic progressions of fixed modulus.In this work, we prove an analogous result in the case of a half-integral weight cusp form. We will see that, in a more restricted range of convergence, the sums of coefficients in arithmetic progressions equidistribute with respect to a distribution which is different from the normal distribution obtained by Fouvry, Ganguly, Kowalski and Michel in the integral weight case.On the other hand, we study the signs of Fourier coefficients of a half-integral weight cusp form f and we provide lower bounds for these coefficients. Using techniques from the previous problem and classical results from the theory of half-integral weight modular forms, such as Shimura’s correspondence, Waldspurger’s formula and the recent theory of newforms, we establish a lower bound for the number of normalized coefficients f(n) such that n le x, where n is taken in an arithmetic progression and f(n) > n^{−alpha} for positive alpha
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Shiu, Daniel Kai Lun. "Prime numbers in arithmetic progressions." Thesis, University of Oxford, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.318815.
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Rimanić, Luka. "Arithmetic progressions, corners and loneliness." Thesis, University of Bristol, 2018. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.761230.
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Carvalho, César Augusto Sverberi. "O aluno do ensino médio e a criação de uma fórmula para o termo geral da progressão aritmética." Pontifícia Universidade Católica de São Paulo, 2008. https://tede2.pucsp.br/handle/handle/11321.
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Previous issue date: 2008-08-08Secretaria da Educação do Estado de São Paulo
This work presents a qualitative research which is guided by the objective of investigating whether if it s possible to create conditions so that High School students generalise terms of arithmetic progressions and, if so, whether this generalization allows students to build a formula of general term of this type of sequence. The relevance of this research is justified by the importance of the work with observation and generalization of patterns, identified by researchers like Mason (1996), Lee (1996) and Vale and Pimentel (2005) as a resource to students express algebraic thinking and create algebraic expressions, giving sense to the use of symbols. For methodological procedures, there were used stages of Didactic Engineering, described by Artigue (1996), to develop, implement and analyze a didactic sequence for the students of the Grade 10. The analysis about the present resolutions in the protocols and the recordings that were made during some sessions indicated that many of the students succeeded in generalizing terms, but that didn t allow some of them to use formal algebraic notation to represent the generality
Este trabalho apresenta uma pesquisa qualitativa orientada pelo objetivo de investigar se é possível criar condições para que alunos do Ensino Médio generalizem termos de progressões aritméticas e, em caso afirmativo, se esta generalização permite que os alunos construam uma fórmula para o termo geral deste tipo de seqüência. A relevância desta pesquisa se justifica pela importância do trabalho com observação e generalização de padrões, apontado por pesquisadores como Mason (1996), Lee (1996) e Vale e Pimentel (2005) como recurso para que alunos manifestem o pensamento algébrico e criem expressões algébricas, dando sentido à utilização dos símbolos. Para os procedimentos metodológicos foram utilizadas fases da Engenharia Didática, descrita por Artigue (1996), para elaborar, aplicar e analisar uma seqüência didática para alunos de uma 1ª série do Ensino Médio. As análises das resoluções presentes nos protocolos e gravações feitas durante algumas sessões indicam que grande parte dos alunos conseguiu generalizar os termos, mas isso não possibilitou que algum deles utilizasse notação algébrica formal para representar a generalidade
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Solis, Alexandre. "Argumentação e prova no estudo de progressões aritméticas com o auxílio do Hot Potatoes." Pontifícia Universidade Católica de São Paulo, 2008. https://tede2.pucsp.br/handle/handle/11330.
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Previous issue date: 2008-10-09Secretaria da Educação do Estado de São Paulo
This research work deals with the theme Argumentation and Proof in the study of Arithmetic Progressions. Its purpose is to investigate the cognitive development of students in building concepts and knowledge related to the Numerical Sequence and Arithmetic Progression (AP), and in developing argumentation and proof-related competencies. Such purposes resulted from the experience in meetings with the research group of the project Argumentation and Proof in School Mathematics (AProvaME) at Pontifícia Universidade Católica de São Paulo (PUC/SP). As research methodology, there were used some principles of Didactical Engineering, and, for developing this work, a sequence of nineteen activities was prepared, which was applied to a group of eight students of the first year of the Brazilian High School Program from a Public School of the State of São Paulo. An authoring software known as Hot Potatoes was used for preparing the nine activities of the sequence. The JCloze tool of this software showed to be proper for the teacher, because it allowed the easy building of activities. As regards the students, it was possible to check the answers given, allowing more autonomy for solving the activities. The trial analyses showed that the sequence of activities permitted students to build concepts related to the Numerical Sequence and Arithmetic Progression, as well as competencies in mathematical argumentation and proof, more specifically, in the development of deductive reasoning that led them to determine the generalization of numerical sequences and to deduct the General Term Formula for AP. This research had a significant impact on my education and my understanding about the importance of argumentation and proof in the teaching practice
Este trabalho de pesquisa versa sobre o tema Argumentação e Prova no estudo de Progressões Aritméticas. Tem por objetivo a investigação do desenvolvimento cognitivo dos alunos na construção de conceitos e conhecimentos relacionados à Seqüência Numérica e Progressão Aritmética (PA), e no desenvolvimento de habilidades de argumentação e prova. Tais objetivos resultaram da experiência nos encontros com o grupo de pesquisa do projeto Argumentação e Prova na Matemática Escolar (AProvaME) na PUC/SP. Como metodologia de pesquisa foi utilizado alguns princípios da Engenharia Didática e para o desenvolvimento deste trabalho foi elaborada uma seqüência de dezenove atividades, aplicada a um grupo de oito alunos do primeiro ano do Ensino Médio de uma Escola Pública do Estado de São Paulo. Na elaboração de nove atividades da seqüência, utilizou-se um software de autoria, conhecido por Hot Potatoes. A ferramenta JCloze desse software mostrou-se adequada para o professor, pois permitiu a fácil construção de atividades. Com relação ao aluno houve a possibilidade de verificação das respostas dadas, permitindo uma maior autonomia na resolução das atividades. As análises da experimentação mostraram que a seqüência de atividades propiciou ao aluno a construção de conceitos relacionados à Seqüência Numérica e Progressão Aritmética, como também habilidades em argumentação e prova matemática, mais especificamente, no desenvolvimento de raciocínios dedutivos que o levou a determinar a generalização de seqüências numéricas e a construção da Fórmula do Termo Geral da PA. Esta pesquisa teve um impacto significativo na minha formação e no meu entendimento sobre a importância da argumentação e prova na prática docente
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Moreira, Nunes Ramon. "Problèmes d’équirépartition des entiers sans facteur carré." Thesis, Paris 11, 2015. http://www.theses.fr/2015PA112123/document.
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Cette thèse concerne quelques problèmes liés à la répartition des entiers sans facteur carré dansles progressions arithmétiques. Ces problèmes s’expriment en termes de majorations du terme d’erreurassocié à cette répartition.Les premier, deuxième et quatrième chapitres sont concentrés sur l’étude statistique des termesd’erreur quand on fait varier la progression arithmétique modulo q. En particulier on obtient une formuleasymptotique pour la variance et des majorations non triviales pour les moments d’ordre supérieur. Onfait appel à plusieurs techniques de théorie analytique des nombres comme les méthodes de crible et lessommes d’exponentielles, notamment une majoration récente pour les sommes d’exponentielles courtesdue à Bourgain dans le deuxième chapitre.Dans le troisième chapitre on s’intéresse à estimer le terme d’erreur pour une progression fixée. Onaméliore un résultat de Hooley de 1975 dans deux directions différentes. On utilise ici des majorationsrécentes de sommes d’exponentielles courtes de Bourgain-Garaev et de sommes d’exponentielles torduespar la fonction de Möbius dues à Bourgain et Fouvry-Kowalski-MichelThis thesis concerns a few problems linked with the distribution of squarefree integers in arithmeticprogressions. Such problems are usually phrased in terms of upper bounds for the error term relatedto this distribution.The first, second and fourth chapter focus on the satistical study of the error terms as the progres-sions varies modulo q. In particular we obtain an asymptotic formula for the variance and non-trivialupper bounds for the higher moments. We make use of many technics from analytic number theorysuch as sieve methods and exponential sums. In particular, in the second chapter we make use of arecent upper bound for short exponential sums by Bourgain.In the third chapter we give estimates for the error term for a fixed arithmetic progression. Weimprove on a result of Hooley from 1975 in two different directions. Here we use recent upper boundsfor short exponential sums by Bourgain-Garaev and exponential sums twisted by the Möbius functionby Bourgain et Fouvry-Kowalski-Michel
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Dyer, A. K. "Applications of sieve methods in number theory." Thesis, Bucks New University, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.384646.
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Coleman, Mark David. "Topics in the distribution of primes." Thesis, University of Cambridge, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.293491.
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Eduardo, Antonio Carlos. "Contextos para argumentar: uma abordagem para iniciacao a prova no EM utilizando P.A." Pontifícia Universidade Católica de São Paulo, 2007. https://tede2.pucsp.br/handle/handle/11253.
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Previous issue date: 2007-10-15Secretaria da Educação do Estado de São Paulo
This research invests in the conception of learning environments aimed to contribute to the creation of a culture of argumentation, proving and proof in mathematics classrooms. It developed within the context of the project AProvaME as part of the exploration of how to initiate students into aspects of the proving process in relation to the topic of Arithmetic Progressions. In designing this learning environment, we sought contributions from the studies of Bordenave from the areas of the Communication Science and in the field of Mathematics Education, from research relatied to argumentation and in particular the work of Bolite Frant and Castro and of Maher. These contributions enabled the elaboration of an interactive environment for the mediation of mathematical ideas. One of the roles of mediation within the study focuses, in the light of Communication, on the action of the teacher during the negotiation of the mathematics presented in the classroom. Aspects related to socialisation, interaction and mediation were inspired by the constructionist proposal of Papert and other constructionist thinkers. On the basis of these studies, an approach was adopted to the use of technology involving the conception of visual objects embedded within activities aiming to support the development of certain habits of mathematical thinking delineated by Goldenberg. This qualitative study made use of didactic resources such as as the dynamic of games and the use of the computer to promote interaction and the emergence of scenarios for medication. The instruments used in the collection of data Blogs and video recordings valorised the interpretation of the dialogs which occurred within these scenarios. The use of Blogs, still not well documented in research in Mathematics Education, served to show the evolution of mathematical fluency in the arguments produced by the students and also acted as a parameter on the practice of the educator. Editing of the videos collected, permitted the formatting of fragments of registers from the dialogs in the form of cartoon strips, which came to represent a product with a wide range of possible uses both in the interpretation of dialogs and in reflections about the role of the teacher. The results obtained in this study led to recommendations for the creation of new contexts for argumentation
Esta pesquisa investe na proposição de ambiente de aprendizagem como possibilidade de criar uma cultura na sala de aula que promova / favoreça a argumentação. No transcorrer do projeto APROVAME1 surgiu a opção em explorar tópicos do conceito Progressão Aritmética para auxiliar no desenvolvimento de processos de iniciação à prova. Na implementação deste ambiente de aprendizagem buscamos contribuições advindas dos estudos de Ciência da Comunicação através de Bordenave, da Educação Matemática pelos estudos de alguns pesquisadores voltados à argumentação, dentre os quais: Bolite Frant e Castro, e estudos sobre desenvolvimento de provas de Maher. Estas contribuições possibilitaram a elaboração de um ambiente interativo e propício à prática da mediação. Um dos papéis de mediação exercido durante este estudo é apresentado à luz da Comunicação, focando na ação do professor durante a negociação matemática que se apresenta em sala de aula. Corroboram para estes aspectos socializáveis do ambiente, interação e mediação, a proposta construcionista de Papert, valorizada pela contribuição de outros estudiosos do construcionismo. Através desses estudos, um dos usos da tecnologia nesta pesquisa volta-se à elaboração de objetos visuais e possibilita o design das atividades sob a ótica do desenvolvimento de alguns hábitos de pensamento matemáticos, segundo Goldenberg. Este estudo qualitativo, emprega recursos didáticos como a dinâmica do jogo e o uso do computador, para promover a interação e o surgimento de cenários de mediação. Os instrumentos de coleta de dados vídeo e blog valorizam a interpretação dos diálogos surgidos nesses cenários. O uso do blog, ainda pouco difundido entre pesquisas da Educação Matemática, serve para mostrar a evolução da fluência matemática na argumentação dos alunos, e também atua como parâmetro da prática do educador. A edição do vídeo permitiu a formatação dos registros de fragmentos dos diálogos na forma de quadrinhos, o que veio a se constituir num produto com amplas possibilidades de uso, tanto no tocante à interpretação dos diálogos, quanto na reflexão sobre a postura do educador. Os resultados obtidos por este estudo recomendam a criação de novos Contextos para Argumentar
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Cheung, King-kwong. "Prime solutions in arithmetic progressions of some linear ternary equations." Click to view the E-thesis via HKUTO, 2000. http://sunzi.lib.hku.hk/hkuto/record/B42575874.
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張勁光 and King-kwong Cheung. "Prime solutions in arithmetic progressions of some linear ternary equations." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2000. http://hub.hku.hk/bib/B42575874.
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White, Christopher J. "Finding primes in arithmetic progressions and estimating double exponential sums." Thesis, University of Bristol, 2016. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.707745.
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Hart, Derrick N. "Finite Field Models of Roth's Theorem in One and Two Dimensions." Thesis, Georgia Institute of Technology, 2006. http://hdl.handle.net/1853/11516.
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Recent work on many problems in additive combinatorics, such as Roth's Theorem, has shown the usefulness of first studying the problem in a finite field environment. Using the techniques of Bourgain to give a result in other settings such as general abelian groups, the author gives a walk through, including proof, of Roth's theorem in both the one dimensional and two dimensional cases (it would be more accurate to refer to the two dimensional case as Shkredov's Theorem). In the one dimensional case the argument is at its base Meshulam's but the structure will be essentially Green's. Let Ϝⁿ [subscript p], p ≠ 2 be the finite field of cardinality N = pⁿ. For large N, any subset A ⊂ Ϝⁿ [subscript p] of cardinality ∣A ∣≳ N ∕ log N must contain a triple of the form {x, x + d, x + 2d} for x, d ∈ Ϝⁿ [subscript p], d ≠ 0. In the two dimensional case the argument is Lacey and McClain who made considerable refinements to this argument of Green who was bringing the argument to the finite field case from a paper of Shkredov. Let Ϝ ⁿ ₂ be the finite field of cardinality N = 2ⁿ. For all large N, any subset A⊂ Ϝⁿ ₂ × Ϝⁿ ₂ of cardinality ∣A ∣≳ N ² (log n) − [superscript epsilon], ε <, 1, must contain a corner {(x, y), (x + d, y), (x, y + d)} for x, y, d ∈ Ϝⁿ₂ and d ≠ 0.
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樊家榮 and Ka-wing Fan. "Prime solutions in arithmetic progressions of some quadratic equationsand linear equations." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2000. http://hub.hku.hk/bib/B31225962.
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Vlasic, Andrew. "A Detailed Proof of the Prime Number Theorem for Arithmetic Progressions." Thesis, University of North Texas, 2004. https://digital.library.unt.edu/ark:/67531/metadc4476/.
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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.
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SOUZA, Carla Maria Pinto de. "Contrato didático : negociações, rupturas e renegociações a partir de uma sequência didática sobre progressão aritmética." Universidade Federal Rural de Pernambuco, 2011. http://www.tede2.ufrpe.br:8080/tede2/handle/tede2/5817.
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This study aimed to investigate how a teacher negotiates the contract with teaching students in the 2nd year of high school, in the implementation of a didactic sequence previously established for the teaching of Arithmetic Progression (AP). The didactic sequence was designed considering the type of didactic situations proposed by Brousseau (action, formulation, validation and institutionalization). Thus, the activities aimed at teaching sequence that enable your application to be performed according to a didactic contract type approximation, which is one in which value is an active student in constructing knowledge. The observed results showed that although we proposed a sequence to be applied as a kind of didactic contract approximate; negotiations, renegotiations and breaks the rules of the didactic contract were made during the development of the sequence. We believe that these breaches of the rules were motivated by previous marks of didactic contract, or implicit and explicit rules by which teacher and students were accustomed.
Esse estudo teve por objetivo investigar como uma professora negocia o contrato didático com alunos do 2º ano do Ensino Médio, na aplicação de uma sequência didática previamente elaborada para o ensino de Progressão Aritmética (P.A.). A sequência didática foi idealizada contemplando a tipologia das situações didáticas proposta por Brousseau (ação, formulação, validação e institucionalização). Com isso, as atividades dessa sequência didática visaram possibilitar que sua aplicação fosse realizada de acordo com um contrato didático do tipo aproximativo, que é aquele em que se valoriza uma postura ativa do aluno na construção do conhecimento. Os resultados observados evidenciaram que embora tivéssemos proposto uma sequência para ser aplicada conforme um contrato didático do tipo aproximativo, negociações, rupturas e renegociações de regras de contrato didático foram feitas ao longo do desenvolvimento da sequência. Acreditamos que essas rupturas das regras estabelecidas foram motivadas por marcas de contrato didático anteriores, ou seja, pelas regras implícitas e explícitas, que professora e alunos estavam habituados.
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Fan, Ka-wing. "Prime solutions in arithmetic progressions of some quadratic equations and linear equations /." Hong Kong : University of HOng Kong, 2000. http://sunzi.lib.hku.hk/hkuto/record.jsp?B23540308.
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Carnovale, Marc. "Arithmetic Structures in Small Subsets of Euclidean Space." The Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1555657038785892.
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Montrezor, Camila Lopes. "Funções aritméticas." Universidade de São Paulo, 2017. http://www.teses.usp.br/teses/disponiveis/55/55136/tde-25072017-082655/.
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Neste estudo, apresentamos conteúdos matemáticos adaptáveis tanto para os anos finais do ensino fundamental quanto para o ensino médio. Iniciamos com um conjunto de ideias preliminares: indução matemática, triângulo de Pascal, Binômio de Newton e relações trigonométricas, para a obtenção de fórmulas de somas finitas, em que os valores das parcelas são computados sobre números inteiros consecutivos, e da técnica de transformação de soma finita em telescópica. Enunciamos Progressões Aritméticas e Geométricas como sequências numéricas e suas propriedades, obtendo a soma de seus n primeiros termos, associando com propriedades do triângulo de Pascal. Por fim, descrevemos Funções Aritméticas, Funções Aritméticas Totalmente Multiplicativas e Fortemente Multiplicativas, como sequências de números naturais, com suas operações e propriedades, direcionando ao objetivo de calcular o número de divisores naturais de n, a soma de todos os divisores naturais de n, e assim por diante. Como consequência, exibimos a fórmula de contagem do número de polinômios mônicos irredutíveis.In this study, we present mathematical content that is adaptable to both of the final years of elementary school and to high school. We start with a set of preliminary ideas: mathematical induction, Pascal\'s triangle, Newton\'s binomial and trigonometric relations, to obtain finite sum formulas, where the parts are computed on consecutive integers, and the technique for transforming a finite sum in telescopic one. We state the Arithmetic and Geometric Progressions as numerical sequences and study their properties, obtaining the sum of their n first terms, associating with properties of the Pascal\'s triangle. Finally, we describe the Arithmetic, Totally Multiplicative and Strongly Multiplicative Arithmetic Functions, as sequences of natural numbers, with their operations and properties, as a way to calculating the number of natural divisors of n, the sum of all natural divisors of n, and so on. As a consequence, we obtain the counting formula of the number of irreducible mononical polynomials.
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Carvalho, Marcelly Mingorancia de. ""São Paulo Faz Escola": muda a abordagem de progressões na sala de aula?" Pontifícia Universidade Católica de São Paulo, 2010. https://tede2.pucsp.br/handle/handle/11451.
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Previous issue date: 2010-05-14Secretaria da Educação do Estado de São Paulo
This study reports a research aimed study which objective was investigate the changes that have occurred in relation to the work of teachers in the High School of the State chain of São Paulo, face to the material sent by the Education Department of the Sao Paulo State in 2008.This material provides a prior work with pattern generalization activities of several sequences before presenting progressions themselves The relevance of this research justify itself by the importance of the topic: "Observation of regularities and generalization of patterns," pointed out by researchers such as Vale and Pimentel (2005), Vale et al (2005) and Lee (1996), as a device that is not only attract and motivate students during the course of these activities but also promote the development and manifestation of students' algebraic thinking. To collect the data I have elaborated and performed semi-structured interviews with five teachers from state schools, based on the ideas of the Didactic Engineering, as Machado (2008).The interview analysis showed that it was unanimous the acceptance and approval of teachers concerning to this material, as well as provided significant changes in the teachers classes, and have been influenced teachers who ignored the issue at hand
O presente estudo relata uma pesquisa qualitativa cujo objetivo foi investigar as mudanças que ocorreram em relação ao trabalho dos professores do primeiro ano do Ensino Médio da rede estadual paulista, frente ao material sobre Progressões integrante da Proposta Curricular do Estado de São Paulo de 2008. Esse material traz um trabalho prévio com atividades de generalização de padrão de diversas sequências antes de apresentar progressões propriamente ditas. A relevância desta pesquisa se justifica pela importância do tema: Observação de regularidades e generalização de padrões , apontado por pesquisadores como Vale e Pimentel (2005), e Lee (1996), como um recurso que não só motiva e atrai o aluno como também promove o desenvolvimento e a manifestação do pensamento algébrico do estudante. Para a coleta de dados elaborei e realizei entrevistas semi-estruturadas, com cinco professores da rede estadual de ensino, inspiradas nas ideias da Engenharia Didática, conforme Machado (2008). A análise das entrevistas mostraram que foi unânime a aceitação e aprovação dos professores quanto a esse material, além de ter proporcionado mudanças significativas nas aulas dos docentes e de ter sensibilizado os professores que desconheciam o tema: Observação de regularidades e generalização de padrões
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Melo, Marcelo de Souza. "Progressões aritméticas na linha construtivista." Universidade Federal de Goiás, 2018. http://repositorio.bc.ufg.br/tede/handle/tede/9058.
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This work shows how the content Arithmetic Progressions can be approached following the constructivist line of teaching, making the students have more active participation in the construction of their knowledge. It is veri_ed that using this model, one can improve students' understanding by introducing in the initial classes one or more problem situations in order to raise previous knowledge for the later acquisition of new knowledge. There are some arguments of professors / educators on this subject and also the practical application of classes structured in the constructivist line on arithmetic progressions, for students of the second year of high school in a public school in the Federal District. The observations about this style of class were made not only by the teacher who applied the activity proposed in class, but also by the students who answered questions that allowed to express the impressions about the activity.
Este trabalho mostra como o conteúdo Progressões Aritméticas pode ser abordado seguindo a linha construtivista de ensino, fazendo com que os alunos tenham participa ção mais ativa na construção do seu conhecimento. É veri_cado que utilizando esse modelo, pode-se melhorar a compreensão dos discentes, introduzindo nas aulas iniciais, uma ou mais situações-problema, com o intuito de levantar conhecimentos prévios para a aquisição posterior do novo saber. Existem algumas argumentações de professores/ educadores consagrados sobre esse tema e também a aplicação prática de aulas estruturadas na linha construtivista sobre progressões aritméticas, para alunos do segundo ano do ensino médio de uma escola pública do Distrito Federal. As observações sobre este estilo de aula foram feitas não somente pelo professor que aplicou a atividade proposta em sala aula, mas também pelos discentes que responderam questões que permitiam expressar as impressões sobre a atividade.
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Archilia, Sebastião. "Construção do termo geral da progressão aritmética pela observação e generalização de padrões." Pontifícia Universidade Católica de São Paulo, 2008. https://tede2.pucsp.br/handle/handle/11318.
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Previous issue date: 2008-06-30Secretaria da Educação do Estado de São Paulo
Today it has been disclosed the poor performance of High School students in learning the Algebra. On the other hand, the results of researches, such as Vale e Pimentel (2005) and Machado (2006) among others, emphasized the importance of working with the observation and generalization of patterns to develop the algebraic thinking, which can help to overcome this problem. This situation and the suggestion help me to decide to investigate if high school students in a situation of patterns observation and generalization could construct an algebraic formulation of a general term of an arithmetic progression. To collect data drafted a didactic sequence based on the assumptions of Didactic Engineering as described by Machado (2008). The didactic sequence occurred in tree sessions with the participation of some of my students, all volunteers. For the conclusion I took into account only the results of the data analysis of 11 students present at all sessions. The results led me to conclude that, although students have expressed in natural language a formula for the general term, it was not enough to convert this result for the symbolic algebraic way
Tem sido amplamente divulgada o mau desempenho dos alunos do Ensino Médio em relação a questões de Matemática e especialmente da Álgebra. Por outro lado, os resultados de pesquisas, como os de Vale e Pimentel (2005) e de Machado (2006), entre outros, enfatizam a importância do trabalho com a observação e generalização de padrões para o desenvolvimento do pensamento algébrico, o que pode auxiliar na superação desse problema. Essa situação e a sugestão me levou a investigar se alunos da segunda série do Ensino Médio frente a atividades de observação e generalização de padrões de seqüências constroem uma fórmula para o termo genérico de uma Progressão Aritmética. Para a coleta de dados, elaborei uma seqüência didática embasada nos pressupostos da Engenharia Didática, conforme descrita por Machado (2008). Realizei três sessões com a participação de alguns de meus alunos, todos voluntários. Para a conclusão levei em conta somente os resultados das análises do desempenho de 11 alunos que estiveram presentes em todas as 3 sessões. Os resultados me levaram a concluir que, embora os alunos tenham expressado em linguagem natural uma fórmula para o termo geral, isso não foi suficiente para converterem esse resultado para uma forma simbólica algébrica
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Mantovani, Haroldo. "Atividades sobre progressões aritméticas através do reconhecimento de padrões." Universidade Federal de São Carlos, 2015. https://repositorio.ufscar.br/handle/ufscar/7112.
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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)
The importance of the development of sequences that are arithmetic progressions in the high school was observed in learning situations which the students could investigate and identify patterns in numerical and geometric sequences, building the algebraic language to describe them. This work contributes for the student to build the idea of algebra as language to express irregularities, that is one of the proposed issues by the National Curricular Parameters (PCNs) for the math teaching in the fourth cycle of high school, which contrasts, nowadays, with the scarcity of activities involving observed arithmetic progressions in at least twelve years of experience as a math teacher in these cycles. The elaboration of a teaching product, in the way of activity sheets that, through the recognizing of numerical and geometric patterns, takes the student to the comprehension of the concept of arithmetic progression which could be tested through the application of these activity sheets in two classrooms of the ninth year of high school in a public municipally school. The obtained results of these applications were analyzed and compared to the previous analyzes in raised hypothesis during the elaboration of the activity sheets, using, as investigation methodology, the Didactic Engineering. The students did the activities in groups of two or three, were well motivated and participated as principal character during the application of all steps proposed in the paper, which guaranteed the good development of the activity. In according to the evaluation learning, the students reaching the proposed goals and noting that the produced teaching material works. It is believed that the elaborated material can be useful for other teachers who want to develop, in their classes, arithmetic progressions through the recognizing of patterns, adapting it to the reality of their classrooms. This work contributes hugely to the author, bringing a big professional evolution that starts with the issue choice, continued in the elaboration of the didactic sequence and the application of the activity sheets and finished with the reflection of what have been done and is registered here.
A importância do desenvolvimento de progressões aritméticas que são sequências no ensino fundamental foi observada em situações de aprendizagem que os alunos puderam investigar e identificar padrões em sequências numéricas e geométricas, construindo a linguagem algébrica para descrevê-las. Esse trabalho contribui para que o aluno construa a ideia de álgebra como uma linguagem para expressar regularidades, que é um dos conteúdos propostos pelos Parâmetros Curriculares Nacionais (PCNs) para o ensino de Matemática no quarto ciclo do ensino fundamental, o qual contrasta, atualmente, com a escassez de atividades envolvendo progressões aritméticas constatada em pelo menos doze anos de experiência como professor de matemática nesses ciclos. A elaboração de um produto de ensino, na forma de folhas de atividades que, através do reconhecimento de padrões numéricos ou geométricos levam o estudante à compreensão do conceito de progressão aritmética pôde ser conferida através da aplicação dessas folhas de atividades em duas salas de 9º ano do ensino fundamental de uma escola municipal. Os resultados obtidos dessas aplicações foram analisados e comparados com as análises prévias em hipóteses levantadas durante a elaboração das folhas de atividades, usando, como metodologia de investigação, a Engenharia Didática. Os alunos realizaram as atividades em duplas ou em trios, se sentiram bem motivados e participaram como protagonistas durante a aplicação de todas as etapas propostas nas folhas, o que garantiu o bom desenvolvimento das atividades. De acordo com a avaliação do aprendizado, os alunos atingiram os objetivos propostos e constatou-se que o material de ensino produzido e aplicado funciona. Acredita-se que o material elaborado possa ser útil a outros professores que desejarem desenvolver, em suas aulas, progressões aritméticas através do reconhecimento de padrões, podendo adaptá-lo à realidade de suas turmas. Este trabalho contribuiu enormemente ao autor, trazendo uma grande evolução profissional que se iniciou na escolha do tema, permeou pela elaboração da sequência didática e pela aplicação das folhas de atividades e terminou pela reflexão sobre o que foi feito e se encontra registrado aqui.
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Henriot, Kevin. "Structures linéaires dans les ensembles à faible densité." Thèse, Paris 7, 2014. http://hdl.handle.net/1866/11116.
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Réalisé en cotutelle avec l'Université Paris-Diderot.Nous présentons trois résultats en combinatoire additive, un domaine récent à la croisée de la combinatoire, l'analyse harmonique et la théorie analytique des nombres. Le thème unificateur de notre thèse est la détection de structures additives dans les ensembles arithmétiques à faible densité, avec un intérêt particulier pour les aspects quantitatifs. Notre première contribution est une estimation de densité améliorée pour le problème, initié entre autres par Bourgain, de trouver une longue progression arithmétique dans un ensemble somme triple. Notre deuxième résultat consiste en une généralisation des bornes de Sanders pour le théorème de Roth, du cas d'un ensemble dense dans les entiers à celui d'un ensemble à faible croissance additive dans un groupe abélien arbitraire. Finalement, nous étendons les meilleures bornes quantitatives connues pour le théorème de Roth dans les premiers, à tous les systèmes d'équations linéaires invariants par translation et de complexité un.
We present three results in additive combinatorics, a recent field at the interface of combinatorics, harmonic analysis and analytic number theory. The unifying theme in our thesis is the detection of additive structure in arithmetic sets of low density, with an emphasis on quantitative aspects. Our first contribution is an improved density estimate for the problem, initiated by Bourgain and others, of finding a long arithmetic progression in a triple sumset. Our second result is a generalization of Sanders' bounds for Roth's theorem from the dense setting, to the setting of small doubling in an arbitrary abelian group. Finally, we extend the best known quantitative results for Roth's theorem in the primes, to all translation-invariant systems of equations of complexity one.
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Oskuii, Saeeid Tahmasbi. "Design of Low-Power Reduction-Trees in Parallel Multipliers." Doctoral thesis, Norwegian University of Science and Technology, Faculty of Information Technology, Mathematics and Electrical Engineering, 2008. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-1958.
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Multiplications occur frequently in digital signal processing systems, communication systems, and other application specific integrated circuits. Multipliers, being relatively complex units, are deciding factors to the overall speed, area, and power consumption of digital computers. The diversity of application areas for multipliers and the ubiquity of multiplication in digital systems exhibit a variety of requirements for speed, area, power consumption, and other specifications. Traditionally, speed, area, and hardware resources have been the major design factors and concerns in digital design. However, the design paradigm shift over the past decade has entered dynamic power and static power into play as well.
In many situations, the overall performance of a system is decided by the speed of its multiplier. In this thesis, parallel multipliers are addressed because of their speed superiority. Parallel multipliers are combinational circuits and can be subject to any standard combinational logic optimization. However, the complex structure of the multipliers imposes a number of difficulties for the electronic design automation (EDA) tools, as they simply cannot consider the multipliers as a whole; i.e., EDA tools have to limit the optimizations to a small portion of the circuit and perform logic optimizations. On the other hand, multipliers are arithmetic circuits and considering arithmetic relations in the structure of multipliers can be extremely useful and can result in better optimization results. The different structures obtained using the different arithmetically equivalent solutions, have the same functionality but exhibit different temporal and physical behavior. The arithmetic equivalencies are used earlier mainly to optimize for area, speed and hardware resources.
In this thesis a design methodology is proposed for reducing dynamic and static power dissipation in parallel multiplier partial product reduction tree. Basically, using the information about the input pattern that is going to be applied to the multiplier (such as static probabilities and spatiotemporal correlations), the reduction tree is optimized. The optimization is obtained by selecting the power efficient configurations by searching among the permutations of partial products for each reduction stage. Probabilistic power estimation methods are introduced for leakage and dynamic power estimations. These estimations are used to lead the optimizers to minimum power consumption. Optimization methods, utilizing the arithmetic equivalencies in the partial product reduction trees, are proposed in order to reduce the dynamic power, static power, or total power which is a combination of dynamic and static power. The energy saving is achieved without any noticeable area or speed overhead compared to random reduction trees. The optimization algorithms are extended to include spatiotemporal correlations between primary inputs. As another extension to the optimization algorithms, the cost function is considered as a weighted sum of dynamic power and static power. This can be extended further to contain speed merits and interconnection power. Through a number of experiments the effectiveness of the optimization methods are shown. The average number of transitions obtained from simulation is reduced significantly (up to 35% in some cases) using the proposed optimizations.
The proposed methods are in general applicable on arbitrary multi-operand adder trees. As an example, the optimization is applied to the summation tree of a class of elementary function generators which is implemented using summation of weighted bit-products. Accurate transistor-level power estimations show up to 25% reduction in dynamic power compared to the original designs.
Power estimation is an important step of the optimization algorithm. A probabilistic gate-level power estimator is developed which uses a novel set of simple waveforms as its kernel. The transition density of each circuit node is estimated. This power estimator allows to utilize a global glitch filtering technique that can model the removal of glitches in more detail. It produces error free estimates for tree structured circuits. For circuits with reconvergent fanout, experimental results using the ISCAS85 benchmarks show that this method generally provides significantly better estimates of the transition density compared to previous techniques.
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Hsieh, Wan-Chen, and 謝宛真. "Arithmetic Progression Segmented Extended Frequency-directed Run-length Coding for Test Data Compression." Thesis, 2017. http://ndltd.ncl.edu.tw/handle/5353gt.
Full textAbstract:
碩士元智大學
資訊工程學系
105
Due to advances in VLSI technology, a huge amount of test data complex the test process and also increase the testing time. Test data compression is a common and efficient method for the purpose of saving test data and testing time. In this thesis, we use statistical based approach to rearrange the AP-EFDR code such that the most frequency occurring test patterns can get shorter codes and the total amount of test data can thus be reduced significantly. We use ISCAS89 academic circuits to demonstrate the proposed approach. The experimental results confirmed that the proposed method can improve the compression ratio at a low cost of extra hardware compare to classical EFDR compressor.
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Fu, Yun-Ting, and 傅筠庭. "The Study of Course of Arithmetic Progression and Series in Junior High School." Thesis, 2019. http://ndltd.ncl.edu.tw/handle/5gkvau.
Full textAbstract:
碩士中原大學
應用數學研究所
107
This study aims to solve the problem and important concepts of Arithmetic Progression and Se- ries in jenior high school. Based on Mathematics Learning Area in Grade 7-12 Curriculum, teaching materials on Arithmetic Progression and Series for these students were designed to make them under- stand the essence of mathematics questions. In addition, the teaching materials provided the students with various ways to solve mathematics questions, developed students'' logical thinking ability, and further increased their interest in learning mathematics. This part of the thesis discusses the findings which emerged from the statistical analysis presented in the literature reviews and the analysis of tests, as follows: 1. The teaching materials of Arithmetic Progression and Series in jenior high school can allow stu- dents to continue to think about problem-solving strategies, in answer validation process, to develop their ability to analyze graphics and diverse thinking. 2. To support students with mathematical formula of Arithmetic Progression and Series in jenior high school, let students could set up to do thinking integration. 3. It is inductived that the stragtegies of problem solving gave a variety of solutions shown below, (1)Oberved the regular pattern of sequences. (2)The mathematical formula is composed of graphics and quantity. (3)The regular mathematical symbols analysis, then express as the mathematical formula.
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Cheng, Chun-Yueh, and 鄭淳月. "The Analysis of Problem Solving Process in Arithmetic Progression among Eighth Graders in Nantou Area." Thesis, 2019. http://ndltd.ncl.edu.tw/cgi-bin/gs32/gsweb.cgi/login?o=dnclcdr&s=id=%22107NCHU5507021%22.&searchmode=basic.
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Wu, Yi-Sung, and 吳怡松. "A Study on Integrating Information Technology into Instruction Model Combining Knowledge Structure Theories and Bayesian Networks－Taking the Unit of Arithmetic Progression as an Example." Thesis, 2010. http://ndltd.ncl.edu.tw/handle/52755612819361693276.
Full textAbstract:
碩士國立臺中教育大學
數學教育學系
98
The research field is “The unit of Arithmetic Progression” in “Grade 1-9 Curriculum Guidelines- The fourth phase in Mathematics Area” promulgated by Department of Ministry in 2003. The study analyzes the Competence Indicators in this unit and the relevant studies in order to structure profession knowledge structure in the unit of Arithmetic Progression and combine the knowledge structure and Bayesian networks to compile some materials of instruction. By using the instruction materials and Bayesian Networks based computerized adaptive diagnostic testing(BNAT) as tools, the present study can design some ways to do Computer- Technology Math Teaching and Make-up Teaching. Next, the present study evaluates the effect of the design by doing some teaching experiments. Besides, it discusses the influence on the effects of make-up teaching and learning efficiency of students who owns different attitudes toward computer technology in Computer-Technology Math Teaching. In addition, the present study discusses the students’ opinions on the Computer-Technology Math Teaching and the effect of computerized diagnosis exams. The results are as follows: 1.The computerized diagnosis tests based on the knowledge structure and Bayesian networks can shorten the test time and predict precisely the diagnosis. It can be used on practical teaching diagnosis exams. 2.About the learning efficiency, integrating information technology into instruction model, using the teaching materials compiled by researchers, is better than the traditional classroom teaching. As for the learning efficiency of students who owns different attitudes toward computer technology in Computer-Technology Math Teaching, there are no distinct differences. 3.About the learning efficiency of Make-up Teaching, integrating information technology into instruction model, using the teaching materials compiled by researchers, is better than the traditional classroom teaching. As for the learning efficiency of students who owns different attitudes toward computer technology in Computer-Technology Math Teaching, there are no distinct differences. 4.Integrating information technology into instruction model can make students think positively about the computers and the use of them. 5.After taking the Make-up Teaching, students in the group of integrating information technology into instruction are better than those in traditional classroom teaching on correcting mistakes and improving the secondary skills. 6.Over 95% of the students in the experimental group have non-negative opinions on integrating information technology into math teaching and the ways of evaluation. It shows that the students are fond of them. 7.Students with different attitudes toward computer technology share the opinions on integrating information technology into instruction and the ways of evaluation. On the two items “I hope teachers can take advantage of computers to teach us math more often” and “I think the computer test system can truly show my math ability”, “Positive-computer-attitude group” tend to own positive ones, while the “ Negative -computer-attitude group” tend to own non-positive ones. However, they can still accept them and won’t deny their functions. And the “Negative-computer-attitude group” agree more on that “Teachers’ lecture” are helpful ways to learn than “Positive-computer- attitude group”. Except for the above, there are no distinct differences on the opinions between students who own different attitudes toward computer technology.
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KUČEROVÁ, Renata. "Anylýza řešení úloh 2. kola 55. ročníku MO v Jihočeském kraji." Master's thesis, 2009. http://www.nusl.cz/ntk/nusl-51054.
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The aim of this diploma work is to analyse problems solving of the second round of the 55th year of the Mathematical Olympiad in South Bohemian region and to serve as a study material for further Mathematical Olympiads or as a collection of problems for talented students.
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Vijay, Sujith. "Arithmetic progressions combinatorial and number-theoretic perspectives." 2007. http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.13838.
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Johannson, Karen R. "Variations on a theorem by van der Waerden." 2007. http://hdl.handle.net/1993/321.
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The central result presented in this thesis is van der Waerden's theorem on arithmetic progressions. Van der Waerden's theorem guarantees that for any integers k
and r, there is an n so that however the set {1, 2, ... , n} is split into r disjoint partition classes, at least one partition class will contain a k-term arithmetic progression. Presented here are a number of variations and generalizations of van der Waerden's theorem that utilize a wide range of techniques from areas of mathematics including combinatorics, number theory, algebra, and topology.May 2007
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Knafo, Emmanuel Robert. "Variance of distribution of almost primes in arithmetic progressions /." 2006. http://link.library.utoronto.ca/eir/EIRdetail.cfm?Resources__ID=442465&T=F.
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Srikanth, Cherukupally. "Number Theoretic, Computational and Cryptographic Aspects of a Certain Sequence of Arithmetic Progressions." Thesis, 2016. http://etd.iisc.ernet.in/2005/3741.
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This thesis introduces a new mathematical object: collection of arithmetic progressions with elements satisfying the inverse property, \j-th terms of i-th and (i+1)-th progressions are multiplicative inverses of each other modulo (j+1)-th term of i-th progression".
Such a collection is uniquely de ned for any pair (a; d) of co-prime integers. The progressions of the collection are ordered. Thus we call it a sequence rather than a collection. The results of the thesis are on the following number theoretic, computational and cryptographic aspects of the defined sequence and its generalizations.
The sequence is closely connected to the classical Euclidean algorithm. Precisely, certain consecutive progressions of the sequence form \groupings". The difference between the common differences of any two consecutive progressions of a grouping is same. The number of progressions in a grouping is connected to the quotient sequence of the Euclidean algorithm on co-prime input pairs. The research community has studied extensively the behavior of the Euclidean algorithm. For the rst time in the literature, the connection (proven in the thesis) shows what the quotients of the algorithm signify. Further, the leading terms of progressions within groupings satisfy a mirror image symmetry property, called \symmetricity". The property is subject to the quotient sequence of the Euclidean algorithm and divisors of integers of the form x2 y2 falling in specific intervals.
The integers a, d are the primary quantities of the defined sequence in a computational sense. Given the two, leading term and common difference of any progression of the sequence can be computed in time quadratic in the binary length of d. On the other hand, the inverse computational question of finding (a; d), given information on some terms of the sequence, is interesting. This problem turns out to be hard as it requires finding solutions to an nearly-determined system of multivariate polynomial equations. Two sub-problems arising in this context are shown to be equivalent to the problem of factoring integers. The reduction to the factoring problem, in both cases, is probabilistic.
Utilizing the computational difficulty of solving the inverse problem, and the sub-problems (mentioned above), we propose a symmetric-key cryptographic scheme (SKCS), and a public key cryptographic scheme (PKCS). The PKCS is also based on the hardness of the problem of finding square-roots modulo composite integers. Our proposal uses the same algorithmic and computational primitives for effecting both the PKCS and SKCS. In addition, we use the notion of the sequence of arithmetic progressions to design an entity authentication scheme.
The proof of equivalence between one of the inverse computational problems (mentioned above) and integer factoring led us to formulate and investigate an independent problem concerning the largest divisor of integer N bounded by the square-root of N. We present some algorithmic and combinatorial results.
In the course of the above investigations, we are led to certain open questions of number theoretic, combinatorial and algorithmic nature. These pertain to the quotient sequence of the Euclidean algorithm, divisors of integers of the form x2 y2 p in specific intervals, and the largest divisor of integer N bounded by N.
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Poirier, Antoine. "Les progressions arithmétiques dans les nombres entiers." Thèse, 2012. http://hdl.handle.net/1866/6931.
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Le sujet de cette thèse est l'étude des progressions arithmétiques dans les nombres entiers. Plus précisément, nous nous intéressons à borner inférieurement v(N), la taille du plus grand sous-ensemble des nombres entiers de 1 à N qui ne contient pas de progressions arithmétiques de 3 termes. Nous allons donc construire de grands sous-ensembles de nombres entiers qui ne contiennent pas de telles progressions, ce qui nous donne une borne inférieure sur v(N). Nous allons d'abord étudier les preuves de toutes les bornes inférieures obtenues jusqu'à présent, pour ensuite donner une autre preuve de la meilleure borne. Nous allons considérer les points à coordonnés entières dans un anneau à d dimensions, et compter le nombre de progressions arithmétiques qu'il contient. Pour obtenir des bornes sur ces quantités, nous allons étudier les méthodes pour compter le nombre de points de réseau dans des sphères à plusieurs dimensions, ce qui est le sujet de la dernière section.The subject of this thesis is the study of arithmetic progressions in the integers. Precisely, we are interested in the size v(N) of the largest subset of the integers from 1 to N that contains no 3 term arithmetic progressions. Therefore, we will construct a large subset of integers with no such progressions, thus giving us a lower bound on v(N). We will begin by looking at the proofs of all the significant lower bounds obtained on v(N), then we will show another proof of the best lower bound known today. For the proof, we will consider points on a large d-dimensional annulus, and count the number of integer points inside that annulus and the number of arithmetic progressions it contains. To obtain bounds on those quantities, it will be interesting to look at the theory behind counting lattice points in high dimensional spheres, which is the subject of the last section.
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Fiorilli, Daniel. "Irrégularités dans la distribution des nombres premiers et des suites plus générales dans les progressions arithmétiques." Thèse, 2011. http://hdl.handle.net/1866/8333.
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Le sujet principal de cette thèse est la distribution des nombres premiers dans les progressions arithmétiques, c'est-à-dire des nombres premiers de la forme $qn+a$, avec $a$ et $q$ des entiers fixés et $n=1,2,3,\dots$ La thèse porte aussi sur la comparaison de différentes suites arithmétiques par rapport à leur comportement dans les progressions arithmétiques. Elle est divisée en quatre chapitres et contient trois articles.
Le premier chapitre est une invitation à la théorie analytique des nombres, suivie d'une revue des outils qui seront utilisés plus tard. Cette introduction comporte aussi certains résultats de recherche, que nous avons cru bon d'inclure au fil du texte.
Le deuxième chapitre contient l'article \emph{Inequities in the Shanks-Rényi prime number
race: an asymptotic formula for the densities}, qui est le fruit de recherche conjointe avec le professeur Greg Martin. Le but de cet article est d'étudier un phénomène appelé le <The main subject of this thesis is the distribution of primes in arithmetic progressions, that is of primes of the form $qn+a$, with $a$ and $q$ fixed, and $n=1,2,3,\dots$ The thesis also compares different arithmetic sequences, according to their behaviour over arithmetic progressions. It is divided in four chapters and contains three articles. The first chapter is an invitation to the subject of analytic number theory, which is followed by a review of the various number-theoretic tools to be used in the following chapters. This introduction also contains some research results, which we found adequate to include. The second chapter consists of the article \emph{Inequities in the Shanks-Rényi prime number race: an asymptotic formula for the densities}, which is joint work with Professor Greg Martin. The goal of this article is to study <
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Silva, Ana Rita Correia da. "Comparação dos Desempenhos de Crianças com Dislexia e Crianças com Progressão Normal em Leitura em Diferentes Domínios do Conhecimento Aritmético." Master's thesis, 2010. http://hdl.handle.net/10348/653.
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Dissertação de Mestrado em Psicologia ClínicaO presente estudo visa examinar as competências aritméticas das crianças com dislexia. Neste sentido encontra-se dividido em duas partes fundamentais. A primeira parte consiste numa revisão da literatura científica acerca dos desempenhos aritméticos dos indivíduos com dislexia e das relações entre os níveis de leitura, competências fonológicas, memória, velocidade de processamento e aritmética. Descreve-se, ainda, na primeira parte, a discalculia e os seus critérios de diagnóstico diferencial com a dislexia e com as dificuldades aritméticas não específicas. A segunda parte consiste num estudo empírico, elaborado a partir do enquadramento teórico anterior, cujo objectivo foi comparar os desempenhos de crianças com dislexia e crianças com progressão normal em leitura, da mesma idade, em diferentes domínios do conhecimento aritmético. Os resultados revelaram que as crianças com dislexia apresentaram mais dificuldades do que as crianças do grupo controlo em tarefas aritméticas verbais mas não nas tarefas aritméticas consideradas não verbais. Verificou-se, também, que as crianças com dislexia foram mais lentas a responder, do que as do grupo controlo, na maioria das tarefas aritméticas e em todas as tarefas de linguagem. São discutidas possíveis explicações e implicações clínicas dos resultados encontrados.
This study aims to examine the arithmetic skills of children with dyslexia. It is divided into two main parts. The first part consists in a review of the scientific literature about the arithmetic performance of individuals with dyslexia and the relationship between reading level, phonological skills, memory, processing speed and arithmetic. It is also discussed in the first part, dyscalculia and its differential diagnostic criteria with dyslexia and with arithmetic nonspecific difficulties. The second part is an empirical study, elaborated from the previous theoretical framework, which aims to compare the performance of children with dyslexia and children with normal progression in reading, having the same age, in different fields of arithmetic knowledge. The results showed that children with dyslexia had more difficulties than children in the control group in verbal arithmetic tasks but not in the arithmetic tasks considered nonverbal. It was found also that children with dyslexia were slower to respond than those in the control group to most of the arithmetic tasks and to all the language tasks. Tentative explanations and clinical implications of results are discussed.
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Freiberg, Tristan. "Strings of congruent primes in short intervals." Thèse, 2010. http://hdl.handle.net/1866/4556.
Full textAbstract:
Soit $p_1 = 2, p_2 = 3, p_3 = 5,\ldots$ la suite des nombres premiers, et soient $q \ge 3$ et $a$ des entiers premiers entre eux. R\'ecemment, Daniel Shiu a d\'emontr\'e une ancienne conjecture de Sarvadaman Chowla. Ce dernier a conjectur\'e qu'il existe une infinit\'e de couples $p_n,p_$ de premiers cons\'ecutifs tels que $p_n \equiv p_{n+1} \equiv a \bmod q$. Fixons $\epsilon > 0$. Une r\'ecente perc\'ee majeure, de Daniel Goldston, J\`anos Pintz et Cem Y{\i}ld{\i}r{\i}m, a \'et\'e de d\'emontrer qu'il existe une suite de nombres r\'eels $x$ tendant vers l'infini, tels que l'intervalle $(x,x+\epsilon\log x]$ contienne au moins deux nombres premiers $\equiv a \bmod q$. \'Etant donn\'e un couple de nombres premiers $\equiv a \bmod q$ dans un tel intervalle, il pourrait exister un nombre premier compris entre les deux qui n'est pas $\equiv a \bmod q$. On peut d\'eduire que soit il existe une suite de r\'eels $x$ tendant vers l'infini, telle que $(x,x+\epsilon\log x]$ contienne un triplet $p_n,p_{n+1},p_{n+2}$ de nombres premiers cons\'ecutifs, soit il existe une suite de r\'eels $x$, tendant vers l'infini telle que l'intervalle $(x,x+\epsilon\log x]$ contienne un couple $p_n,p_{n+1}$ de nombres premiers tel que $p_n \equiv p_{n+1} \equiv a \bmod q$. On pense que les deux \'enonc\'es sont vrais, toutefois on peut seulement d\'eduire que l'un d'entre eux est vrai, sans savoir lequel.
Dans la premi\`ere partie de cette th\`ese, nous d\'emontrons que le deuxi\`eme \'enonc\'e est vrai, ce qui fournit une nouvelle d\'emonstration de la conjecture de Chowla. La preuve combine des id\'ees de Shiu et de Goldston-Pintz-Y{\i}ld{\i}r{\i}m, donc on peut consid\'erer que ce r\'esultat est une application de leurs m\'thodes. Ensuite, nous fournirons des bornes inf\'erieures pour le nombre de couples $p_n,p_{n+1}$ tels que $p_n \equiv p_{n+1} \equiv a \bmod q$, $p_{n+1} - p_n < \epsilon\log p_n$, avec $p_{n+1} \le Y$.
Sous l'hypoth\`ese que $\theta$, le \og niveau de distribution \fg{} des nombres premiers, est plus grand que $1/2$, Goldston-Pintz-Y{\i}ld{\i}r{\i}m ont r\'eussi \`a d\'emontrer que $p_{n+1} - p_n \ll_{\theta} 1$ pour une infinit\'e de couples $p_n,p_$. Sous la meme hypoth\`ese, nous d\'emontrerons que $p_{n+1} - p_n \ll_{q,\theta} 1$ et $p_n \equiv p_{n+1} \equiv a \bmod q$ pour une infinit\'e de couples $p_n,p_$, et nous prouverons \'egalement un r\'esultat quantitatif.
Dans la deuxi\`eme partie, nous allons utiliser les techniques de Goldston-Pintz-Yldrm pour d\'emontrer qu'il existe une infinit\'e de couples de nombres premiers $p,p'$ tels que $(p-1)(p'-1)$ est une carr\'e parfait. Ce resultat est une version approximative d'une ancienne conjecture qui stipule qu'il existe une infinit\'e de nombres premiers $p$ tels que $p-1$ est une carr\'e parfait. En effet, nous d\'emontrerons une borne inf\'erieure sur le nombre d'entiers naturels $n \le Y$ tels que $n = \ell_1\cdots \ell_r$, avec $\ell_1,\ldots,\ell_r$ des premiers distincts, et tels que $(\ell_1-1)\cdots (\ell_r-1)$ est une puissance $r$-i\`eme, avec $r \ge 2$ quelconque. \'Egalement, nous d\'emontrerons une borne inf\'erieure sur le nombre d'entiers naturels $n = \ell_1\cdots \ell_r \le Y$ tels que $(\ell_1+1)\cdots (\ell_r+1)$ est une puissance $r$-i\`eme. Finalement, \'etant donn\'e $A$ un ensemble fini d'entiers non-nuls, nous d\'emontrerons une borne inf\'erieure sur le nombre d'entiers naturels $n \le Y$ tels que $\prod_ (p+a)$ est une puissance $r$-i\`eme, simultan\'ement pour chaque $a \in A$.Let $p_1 = 2, p_2 = 3, p_3 = 5,\ldots$ be the sequence of all primes, and let $q \ge 3$ and $a$ be coprime integers. Recently, and very remarkably, Daniel Shiu proved an old conjecture of Sarvadaman Chowla, which asserts that there are infinitely many pairs of consecutive primes $p_n,p_{n+1}$ for which $p_n \equiv p_{n+1} \equiv a \bmod q$. Now fix a number $\epsilon > 0$, arbitrarily small. In their recent groundbreaking work, Daniel Goldston, J\`anos Pintz and Cem Y{\i}ld{\i}r{\i}m proved that there are arbitrarily large $x$ for which the short interval $(x, x + \epsilon\log x]$ contains at least two primes congruent to $a \bmod q$. Given a pair of primes $\equiv a \bmod q$ in such an interval, there might be a prime in-between them that is not $\equiv a \bmod q$. One can deduce that \emph{either} there are arbitrarily large $x$ for which $(x, x + \epsilon\log x]$ contains a prime pair $p_n \equiv p_{n+1} \equiv a \bmod q$, \emph{or} that there are arbitrarily large $x$ for which the $(x, x + \epsilon\log x]$ contains a triple of consecutive primes $p_n,p_{n+1},p_{n+2}$. Both statements are believed to be true, but one can only deduce that one of them is true, and one does not know which one, from the result of Goldston-Pintz-Y{\i}ld{\i}r{\i}m. In Part I of this thesis, we prove that the first of these alternatives is true, thus obtaining a new proof of Chowla's conjecture. The proof combines some of Shiu's ideas with those of Goldston-Pintz-Y{\i}ld{\i}r{\i}m, and so this result may be regarded as an application of their method. We then establish lower bounds for the number of prime pairs $p_n \equiv p_{n+1} \equiv a \bmod q$ with $p_{n+1} - p_n < \epsilon\log p_n$ and $p_{n+1} \le Y$. Assuming a certain unproven hypothesis concerning what is referred to as the `level of distribution', $\theta$, of the primes, Goldston-Pintz-Y{\i}ld{\i}r{\i}m were able to prove that $p_{n+1} - p_n \ll_{\theta} 1$ for infinitely many $n$. On the same hypothesis, we prove that there are infinitely many prime pairs $p_n \equiv p_{n+1} \equiv a \bmod q$ with $p_{n+1} - p_n \ll_{q,\theta} 1$. This conditional result is also proved in a quantitative form. In Part II we apply the techniques of Goldston-Pintz-Y{\i}ld{\i}r{\i}m to prove another result, namely that there are infinitely many pairs of distinct primes $p,p'$ such that $(p-1)(p'-1)$ is a perfect square. This is, in a sense, an `approximation' to the old conjecture that there are infinitely many primes $p$ such that $p-1$ is a perfect square. In fact we obtain a lower bound for the number of integers $n$, up to $Y$, such that $n = \ell_1\cdots \ell_r$, the $\ell_i$ distinct primes, and $(\ell_1 - 1)\cdots (\ell_r - 1)$ is a perfect $r$th power, for any given $r \ge 2$. We likewise obtain a lower bound for the number of such $n \le Y$ for which $(\ell_1 + 1)\cdots (\ell_r + 1)$ is a perfect $r$th power. Finally, given a finite set $A$ of nonzero integers, we obtain a lower bound for the number of $n \le Y$ for which $\prod_{p \mid n}(p+a)$ is a perfect $r$th power, simultaneously for every $a \in A$.