Relevant bibliographies by topics / Arithmetic Progression

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Arithmetic Progression'

Author: Grafiati
Published: 4 June 2021
Last updated: 1 February 2022

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Journal articles on the topic "Arithmetic Progression"

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Ginat, David. "No arithmetic progression." ACM Inroads 5, no. 3 (September 5, 2014): 42–43. http://dx.doi.org/10.1145/2655759.2655772.

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Dinneen, Michael J., Nan Rosemary Ke, and Masoud Khosravani. "Arithmetic Progression Graphs." Universal Journal of Applied Mathematics 2, no. 8 (October 2014): 290–97. http://dx.doi.org/10.13189/ujam.2014.020803.

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Sim, Kai An, and Kok Bin Wong. "Magic Square and Arrangement of Consecutive Integers That Avoids k-Term Arithmetic Progressions." Mathematics 9, no. 18 (September 14, 2021): 2259. http://dx.doi.org/10.3390/math9182259.

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In 1977, Davis et al., proposed a method to generate an arrangement of [n]={1,2,…,n} that avoids three-term monotone arithmetic progressions. Consequently, this arrangement avoids k-term monotone arithmetic progressions in [n] for k≥3. Hence, we are interested in finding an arrangement of [n] that avoids k-term monotone arithmetic progression, but allows k−1-term monotone arithmetic progression. In this paper, we propose a method to rearrange the rows of a magic square of order 2k−3 and show that this arrangement does not contain a k-term monotone arithmetic progression. Consequently, we show that there exists an arrangement of n consecutive integers such that it does not contain a k-term monotone arithmetic progression, but it contains a k−1-term monotone arithmetic progression.
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Bremner, Andrew, and Samir Siksek. "Squares in arithmetic progression over cubic fields." International Journal of Number Theory 12, no. 05 (May 10, 2016): 1409–14. http://dx.doi.org/10.1142/s179304211650086x.

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Euler showed that there can be no more than three integer squares in arithmetic progression. In quadratic number fields, Xarles has shown that there can be arithmetic progressions of five squares, but not of six. Here, we prove that there are no cubic number fields which contain five squares in arithmetic progression.
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Sanna, Carlo. "Covering an arithmetic progression with geometric progressions and vice versa." International Journal of Number Theory 10, no. 06 (August 14, 2014): 1577–82. http://dx.doi.org/10.1142/s1793042114500456.

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We show that there exists a positive constant C such that the following holds: Given an infinite arithmetic progression [Formula: see text] of real numbers and a sufficiently large integer n (depending on [Formula: see text]), there is a need of at least Cn geometric progressions to cover the first n terms of [Formula: see text]. A similar result is presented, with the role of arithmetic and geometric progressions reversed.
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Buchholz, R. H., and J. A. MacDougall. "Heron quadrilaterals with sides in arithmetic or geometric progression." Bulletin of the Australian Mathematical Society 59, no. 2 (April 1999): 263–69. http://dx.doi.org/10.1017/s0004972700032883.

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We study triangles and cyclic quadrilaterals which have rational area and whose sides form geometric or arithmetic progressions. A complete characterisation is given for the infinite family of triangles with sides in arithmetic progression. We show that there are no triangles with sides in geometric progression. We also show that apart from the square there are no cyclic quadrilaterals whose sides form either a geometric or an arithmetic progression. The solution of both quadrilateral cases involves searching for rational points on certain elliptic curves.
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MacDougall, Jim. "79.45 Some Arithmetic Progression Identities." Mathematical Gazette 79, no. 485 (July 1995): 390. http://dx.doi.org/10.2307/3618327.

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Heule, Marijn J. H. "Avoiding triples in arithmetic progression." Journal of Combinatorics 8, no. 3 (2017): 391–422. http://dx.doi.org/10.4310/joc.2017.v8.n3.a1.

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Bagemihl, Frederick, and F. Bagemihl. "ORDINAL NUMBERS IN ARITHMETIC PROGRESSION." Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 38, no. 1 (1992): 525–28. http://dx.doi.org/10.1002/malq.19920380148.

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Bourgain, J. "On Triples in Arithmetic Progression." Geometric And Functional Analysis 9, no. 5 (December 1, 1999): 968–84. http://dx.doi.org/10.1007/s000390050105.

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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.
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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 convenablement
This 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 > 0
This 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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Books on the topic "Arithmetic Progression"

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Milne, William J. Progressive arithmetic: Third book. Toronto: Morang Educational Co., 1995.

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Milne, William J. Progressive arithmetic: First book. Toronto: Morang Educational Co., 1995.

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Milne, William J. Progressive arithmetic: Second book. Toronto: Morang Educational Co., 1995.

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White, J. Progressive problems in arithmetic for fourth classes in public schools and candidates for entrance to high schools and collegiate institutes. Toronto: Copp, Clark, 1994.

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A course in analytic number theory. Providence, Rhode Island: American Mathematical Society, 2014.

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Florian, Luca, ed. Analytic number theory: Exploring the anatomy of integers. Providence, R.I: American Mathematical Society, 2012.

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Fish, Daniel W. Robinson's Progressive Practical Arithmetic. CreateSpace Independent Publishing Platform, 2016.

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Knafo, Emmanuel Robert. Variance of distribution of almost primes in arithmetic progressions. 2006.

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John, Friedlander, ed. Oscillation theorems for primes in arithmetic progressions and for sifting functions. Toronto: Dept. of Mathematics, University of Toronto, 1989.

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Leslie, John. The Philosophy of Arithmetic: Exhibiting a Progressive View of the Theory and Practice of Calculation, with Tables for the Multiplication of Numbers as Far as One Thousand. Franklin Classics, 2018.

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Book chapters on the topic "Arithmetic Progression"

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Gelfand, Israel M., and Alexander Shen. "The sum of an arithmetic progression." In Algebra, 79–80. Boston, MA: Birkhäuser Boston, 2004. http://dx.doi.org/10.1007/978-1-4612-0335-3_40.

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Feret, Jérôme. "The Arithmetic-Geometric Progression Abstract Domain." In Lecture Notes in Computer Science, 42–58. Berlin, Heidelberg: Springer Berlin Heidelberg, 2005. http://dx.doi.org/10.1007/978-3-540-30579-8_3.

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Mitzenmacher, Michael. "Arithmetic Progression Hypergraphs: Examining the Second Moment Method." In 2019 Proceedings of the Sixteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO), 127–34. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2019. http://dx.doi.org/10.1137/1.9781611975505.14.

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Wright, Steve. "Quadratic Residues and Non-Residues in Arithmetic Progression." In Lecture Notes in Mathematics, 227–71. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-45955-4_9.

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Ayoub, Raymond. "Dirichlet’s theorem on primes in an arithmetic progression." In Mathematical Surveys and Monographs, 1–36. Providence, Rhode Island: American Mathematical Society, 2006. http://dx.doi.org/10.1090/surv/010/01.

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Rosen, Michael. "Dirichlet L-Series and Primes in an Arithmetic Progression." In Graduate Texts in Mathematics, 33–43. New York, NY: Springer New York, 2002. http://dx.doi.org/10.1007/978-1-4757-6046-0_4.

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Gelfand, Israel M., and Alexander Shen. "Arithmetic progressions." In Algebra, 77–79. Boston, MA: Birkhäuser Boston, 2004. http://dx.doi.org/10.1007/978-1-4612-0335-3_39.

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Landman, Bruce, and Aaron Robertson. "Arithmetic progressions (𝑚𝑜𝑑𝑚)." In The Student Mathematical Library, 183–201. Providence, Rhode Island: American Mathematical Society, 2014. http://dx.doi.org/10.1090/stml/073/06.

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Harzheim, Egbert. "Almost Arithmetic Progressions." In Numbers, Information and Complexity, 17–20. Boston, MA: Springer US, 2000. http://dx.doi.org/10.1007/978-1-4757-6048-4_2.

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Murty, M. Ram. "Primes in Arithmetic Progressions." In Problems in Analytic Number Theory, 211–45. New York, NY: Springer New York, 2001. http://dx.doi.org/10.1007/978-1-4757-3441-6_12.

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Conference papers on the topic "Arithmetic Progression"

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Tarigan, Regina Ayunita, and Chun-Yen Shen. "Szemerédi's regularity lemma application on 3-term arithmetic progression." In THE 4TH INDOMS INTERNATIONAL CONFERENCE ON MATHEMATICS AND ITS APPLICATION (IICMA 2019). AIP Publishing, 2020. http://dx.doi.org/10.1063/5.0017419.

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Malonia, Meeta, and Surendra Kumar Agarwal. "Digital Image Watermarking using Discrete Wavelet Transform and Arithmetic Progression technique." In 2016 IEEE Students' Conference on Electrical, Electronics and Computer Science (SCEECS). IEEE, 2016. http://dx.doi.org/10.1109/sceecs.2016.7509352.

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Diop, I., S. M. Farssi, and H. B. Diouf. "Construction of codes protographes LDPC quasi-cycliques based on an arithmetic progression." In 2012 Second International Conference on Innovative Computing Technology (INTECH). IEEE, 2012. http://dx.doi.org/10.1109/intech.2012.6457749.

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Zhang, Xue, Lu Peng, Chenyan Li, and Qing Li. "Construct Parity-check Matrix H of QC LDPC Codes via Arithmetic Progression." In the Fifth International Conference. New York, New York, USA: ACM Press, 2016. http://dx.doi.org/10.1145/3033288.3033344.

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Ren, Zhixiong, Kefeng Zhang, Cong Li, Zhenglin Liu, Xiaofei Chen, Dongsheng Liu, and Xuecheng Zou. "On-chip transformer using multipath technique with arithmetic-progression step sub-path width." In 2015 IEEE International Conference on Electron Devices and Solid-State Circuits (EDSSC). IEEE, 2015. http://dx.doi.org/10.1109/edssc.2015.7285185.

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Fan, Zexuan, and Xiaolong Xu. "APDPk-Means: A New Differential Privacy Clustering Algorithm Based on Arithmetic Progression Privacy Budget Allocation." In 2019 IEEE 21st International Conference on High Performance Computing and Communications; IEEE 17th International Conference on Smart City; IEEE 5th International Conference on Data Science and Systems (HPCC/SmartCity/DSS). IEEE, 2019. http://dx.doi.org/10.1109/hpcc/smartcity/dss.2019.00238.

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BIANCHI, M., A. GILLIO, and P. P. PÁLFY. "A NOTE ON FINITE GROUPS IN WHICH THE CONJUGACY CLASS SIZES FORM AN ARITHMETIC PROGRESSION." In Proceedings of the Conference. WORLD SCIENTIFIC, 2011. http://dx.doi.org/10.1142/9789814350051_0003.

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Vantsevich, Vladimir V., Bhargav H. Joshi, and Gianantonio Bortolin. "Transmission Gear Ratio vs Fuel Consumption: Retrospective Analysis for Future Terrain Vehicle Applications." In ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/detc2012-70478.

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Abstract:
For decades, the technical problem of selecting optimal transmission gear ratio has been researched for various ground vehicles based on different selection criteria depending on a particular vehicle application; criteria of terrain mobility, traction and acceleration performance, engine power utilization and fuel efficiency have been widely in use. Innumerable analytical and experimental research results and data were implemented in thousand different transmission designs. Today, this unique information about the number of gears in transmission, and value of gear ratios make a field for (i) a research analysis of engineering efficiency of different transmission designs, which were in operation for decades, and (ii) developing more efficient analytical methodologies to select the number of gears and transmission gear ratios and, furthermore, continuously, in-real time control power transfer from the vehicle energy source to the wheels. This paper considers the first, of the above-listed problems in detail with applications to various terrain vehicles and then specifically to off-road wheeled vehicles. The analysis presented in the paper, envelops simple random samples of up to 50 vehicles. It starts from WW2 military vehicles, goes to off-road trucks of 1980s and finally compares modern dump trucks and other terrain vehicles of several major world OEMs. The paper presents an analytical method, computational algorithm and results of a study in which, the efficiency of conventional analytical methodologies are evaluated using actual data on fuel consumption and characteristics of transmissions, vehicle engines, driveline and running gear systems and payloads. To serve this purpose, actual data of each vehicle is compared with analytical data of the vehicle, computed using the conventional methods, with focus on gear/velocity ratios and average fuel consumption at each transmission gear. The fuel consumption analysis was carried out by computing vehicle transport capacity as a function of the average velocity and mass of the payload for each vehicle. The result shows a distinct change of behavior in gear design methodology between post war and present day vehicles. It was a determined divergence from the initial trends, which were based on either the geometrical progression method or arithmetic method for selecting the number of transmission speeds and the values of gear ratios. This resulted in not only having a wide range of speed characteristics of automatic transmission over a few manual gears, but also, as discovered in this study, lead to increased fuel consumption of some vehicles in all range of speeds. The WW2 vehicles designed with manual transmission have gear ratios are closely aligned to analytically calculated geometric progression. Same behavior is observed in the off-road vehicles of 1980’s. Here, with a manual transmission, the trend is more towards less number of gears and with large interval between speed ratios. This of course gives a better fuel efficiency, but leads to trade off in lower average vehicle velocity. The transmission design for modern day dump trucks is also very close to the geometric progression approach. The other modern off-road trucks, as discovered in the analysis, follow an arithmetic progression. Although this results in smooth transmission, but fuel efficiency is compromised significantly, compared to dump trucks. It is important to note that a design based on geometric progression, would result in same speed distribution with less number of gears and better fuel efficiency. For a modern day terrain trucks, to have an optimum combination of both characteristics, it is important to consider all the parameters affecting velocity ratios and fuel consumption and incorporate an efficient analytical methodology to stay competitive, in the rapidly evolving market of all terrain vehicles.
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Ladkany, George S., and Mohamed B. Trabia. "Incorporating Twinkling in Genetic Algorithms for Global Optimization." In ASME 2008 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2008. http://dx.doi.org/10.1115/detc2008-49256.

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Genetic algorithms have been extensively used as a reliable tool for global optimization. However these algorithms suffer from their slow convergence. To address this limitation, this paper proposes a two-fold approach to address these limitations. The first approach is to introduce a twinkling process within the crossover phase of a genetic algorithm. Twinkling can be incorporated within any standard algorithm by introducing a controlled random deviation from its standard progression to avoiding being trapped at a local minimum. The second approach is to introduce a crossover technique: the weighted average normally-distributed arithmetic crossover that is shown to enhance the rate of convergence. Two possible twinkling genetic algorithms are proposed. The performance of the proposed algorithms is successfully compared to simple genetic algorithms using various standard mathematical and engineering design problems. The twinkling genetic algorithms show their ability to consistently reach known global minima, rather than nearby sub-optimal points with a competitive rate of convergence.
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Coppersmith, D., and S. Winograd. "Matrix multiplication via arithmetic progressions." In the nineteenth annual ACM conference. New York, New York, USA: ACM Press, 1987. http://dx.doi.org/10.1145/28395.28396.

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