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Davis, Tinka. "Forty two problems of first degree from Diophantus’ Arithmetica." Thesis, Wichita State University, 2010. http://hdl.handle.net/10057/5437.
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This work brings to the audience Diophantus' problems of first degree in a literal word for word English translation from Ver Eecke's French translation of Arithmetica. In addition, these problems are accompanied by commentary in modern notation, as well as some modern and general solutions to appropriate problems.Thesis (M.S.)--Wichita State University, College of Liberal Arts and Sciences, Dept. of Mathematics and Statistics.
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Zuin, Elenice de Souza Lodron. "Por uma nova arithmetica: o sistema métrico decimal como um saber escolar em Portugal e no Brasil oitocentistas." Pontifícia Universidade Católica de São Paulo, 2007. https://tede2.pucsp.br/handle/handle/11205.
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Previous issue date: 2007-05-28Conselho Nacional de Desenvolvimento Científico e Tecnológico
This study fits into the field of the History of School Disciplines. Our objective is to find how the introduction of the metric system into Brazil and Portugal in the second half of the nineteenth century came about. This new knowledge had to be integrated into the general education system in order to adhere to the legislation of both countries. The renovation led to changes in school Arithmetic, not only due to the inclusion of a new system of weights and measurements, but also to other content, such as decimal numbers. Our main sources were Portuguese and Brazilian school printed material published in the eighteen hundreds. With regard to the methods used to incorporate the metric decimal system, we can affirm that the period of study constitutes a transition phase during which diverse publications and methodologies abounded in an attempt to establish a model. We show that incorporation of the new knowledge does not occur in the same manner in all schools, even though these may follow the same guidelines and didactic texts, nor does it occur straight away due to the fact that school culture needs time to adapt to the changes imposed, giving it new meaning. We conclude that during the period studied, certain bases were established for the disciplinarization of the metric decimal system and for the changes which took place in the teaching of Arithmetic in primary schools
Este estudo se enquadra no campo da Historia das Disciplinas Escolares. Objetivamos verificar como ocorreu a introdução do sistema métrico em Portugal e no Brasil na segunda metade do século XIX. Esse era um novo saber que deveria se integrar à formação geral para o cumprimento da legislação nos dois países. A reforma provocou alterações na Aritmética escolar, não só pela inclusão do novo sistema de pesos e medidas, mas, também, de outros conteúdos, como os números decimais. Nossas principais fontes foram os impressos escolares portugueses e brasileiros publicados nos Oitocentos. Em relação ao modo de incorporar o sistema métrico decimal, constatamos que, o período estudado constituiu-se em uma fase de transição, na qual diversas publicações e meto-dologias distintas circularam na tentativa de se fixar um modelo. Comprovamos que a incorporação de um saber não ocorre da mesma maneira em todas as escolas, ainda que sejam seguidos os mesmos textos didáticos e as mesmas orientações, e nem se dá de forma imediata, porque a cultura escolar necessita de um tempo para apropriar-se do que lhe é imposto, dando-lhe novos significados. Concluímos que, no período estudado, se estabeleceram algumas bases para a escolarização do sistema métrico decimal e para as alterações que deveriam ocorrer no ensino de Aritmética nas escolas primárias
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Zuin, Elenice de Souza Ladron. "Por uma nova arithmetica: o sistema métrico decimal como um saber escolar em Portugal e no Brasil oitocentista." reponame:Repositório Institucional da UFSC, 2007. https://repositorio.ufsc.br/xmlui/handle/123456789/177674.
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Tese apresentada À PUC São Paulo, Programa de Educação Matemática, SP, 2007, il. 318f.Submitted by David Antonio Costa (david.costa@ufsc.br) on 2017-07-21T02:42:10Z No. of bitstreams: 1 Elenice de Souza Lodron Zuin.pdf: 5177683 bytes, checksum: 7bbb0b2835888394fffc6186a0f14349 (MD5)
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Este estudo se enquadra no campo da História das Disciplinas Escolares. Objetivamos verificar como ocorreu a introdução do sistema métrico em Portugal e no Brasil na segunda metade do século XIX. Esse era um novo saber que deveria se integrar à formação geral para o cumprimento da legislação nos dois países. A reforma provocou alterações na Arithmética escolar, não só pela inclusão do novo sistema de pesos e medidas, mas também de outros conteúdos como os números decimais. Nossas principais fontes foram os impressos escolares portugueses e brasileiros publicados nos Oitocentos. Em relação ao modo de incorporar o sistema métrico decimal, constatamos que, o período estudado constituiu-se em uma fase de transição, na qual diversas publicações e metodologias distintas circularam na tentativa de se fixar um modelo. Comprovamos que a incorporação de um saber não ocorre da mesma maneira em todas as escolas, ainda que sejam seguidos os mesmos textos didáticos e as mesmas orientações, e nem se dá de forma imediata, porque a cultura escolar necessita de um tempo para apropriar-se do que lhe é imposto, dando-lhe novos significados. Concluímos que, no período estudado, se estabeleceram algumas bases para a escolarização do sistema métrico decimal e para as alterações que deveriam ocorrer no ensino de Aritmética nas escolas primárias.
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Hilzendeger, Maria Aparecida Maia. ""Primeira arithmetica para meninos" e a constituição de masculinidades na província de São Pedro do Rio Grande do Sul." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2009. http://hdl.handle.net/10183/18262.
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Esta Dissertação tem como objetivos identificar e analisar os discursos da masculinidade presentes no livro didático "Primeira Arithmetica Para Meninos", organizado pelo engenheiro, educador e escritor brasileiro José Theodoro de Souza Lobo. Esse livro, editado pela Livraria Selbach & CIA, foi aprovado pelo Conselho de Instrução e por uma comissão da Escola Militar do Rio Grande do Sul para ser adotado nas escolas públicas e nas escolas particulares nessa Província. Com base nas teorizações sobre as relações de gênero, segundo o viés pós-estruturalista, metodologicamente, foi desenvolvido um movimento que denominei analítico-descritivo-analítico, documentando e sistematizando o conjunto de informações focadas nesse livro didático e ampliadas por demais fontes, entre elas, legislação da época, cartas-parecer e manual de civilidade. Essas fontes, necessárias ao objetivo em questão, foram tomadas como monumentos no sentido foucaultiano. A escolha do livro didático "Primeira Arithmetica Para Meninos" deu-se por considerá-lo um artefato cultural, compreendendo que nele circularam discursos que implicaram - direta ou indiretamente - a produção de identidades de gênero, de acordo com determinados modos de ser menino. Foram estabelecidos quatro focos de análise: O ensino de Matemática no livro didático "Primeira Arithmetica Para Meninos", em que descrevo e analiso alguns aspectos em relação aos conteúdos desenvolvidos nos três primeiros capítulos do livro; O nome de autor, em que examino de que maneira, em que condições e segundo quais regras foi produzido, utilizado e valorizado o livro didático "Primeira Arithmetica Para Meninos"; A inferiorização da identidade feminina, em que verifico que saberes foram produzidos e veiculados através do livro didático de Matemática sobre feminilidades, determinando "modos de ser mulher"; A constituição da masculinidade, em que analiso a contribuição desse livro didático no fortalecimento de uma educação diferenciada para meninos, problematizando as significações criadas sobre "modos de ser menino" dados como normais, corretos, naturais, únicos. Por fim, concluo que esse livro didático, através de um conhecimento matemático - Aritmética - desenvolvido para meninos, permitiu a circulação de discursos que contribuíram para a produção de masculinidades, em consonância com o proposto pela legislação e pelas orientações estabelecidas nos manuais de civilidade da época.This dissertation aims at both identifying and analyzing discourses of masculinity in the didactic book named "Primeira Arithmetica Para Meninos", organized by the engineer, educator and writer José Theodoro de Souza Lobo. This book, published by Livraria Selbach & CIA, was approved by Conselho de Instrução and a commission from Escola Militar do Rio Grande do Sul to be adopted in public and private schools in that province. Based on theories about gender relations, according to the post-structuralist view, in terms of methodology, I have developed a movement that I have characterized as analytical-descriptive-analytical, documenting and systematizing the set of information focused on that didactic book and broadened with the help of other sources, such as regulations, reports and civility handbooks that were current at the time of its publication. Those sources, which have been necessary to the achievement of the objective of this study, have been taken as monuments, in the Foucauldian sense. The didactic book "Primeira Arithmetica para Meninos" has been regarded as a cultural artifact that spread contents implying - both directly and indirectly - the production of gender identities, in accordance with certain ways of being a boy. Four focuses have been set for analysis: 1) Mathematics teaching in the didactic book "Primeira Arithmetica Para Meninos", in which I have described and analyzed some aspects of the contents developed in the first three chapters of the book; 2) The author's name, in which I have examined how, and under which conditions and regulations, the didactic book "Primeira Arithmetica Para Meninos" was produced, used and valued; 3) Lowering of the female identity, in which I have observed which knowledges were produced and spread through the Mathematics didactic book concerning femininities, thus determining "ways of being a woman"; and 4) The constitution of masculinity, in which I have both analyzed the contribution of this book towards the strengthening of a differentiated education for boys, and problematized significations created about "ways of being a boy" taken as normal, correct, natural, and unique. Finally, I have concluded that this didactic book, through a mathematical knowledge - Arithmetic - developed for boys, allowed for the circulation of discourses that contributed to the production of masculinities, in accordance with what was proposed by regulations and guidelines pointed out in civility handbooks from that time.
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Grion, Anna. "Martiani Capellae De Nuptiis Philosopiae et Mercurii liber VII. Introduzione, traduzione e commento." Doctoral thesis, Università degli studi di Trieste, 2013. http://hdl.handle.net/10077/9141.
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2011/2012La tesi verte sul settimo libro del De Nuptiis Philologiae et Mercurii del cartaginese Marziano Capella, dedicato all'aritmetica. Il lavoro presenta il testo latino, stabilito sulla base delle edizioni critiche esistenti, la traduzione e puntuali note di commento di tipo filologico e contenutistico. L'introduzione fornisce un inquadramento del libro all'interno dell'opera e presenta le fonti e i caratteri dell'Aritmetica di Marziano.
XXV Ciclo
1984
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Lopes, Gabriela Lucheze de Oliveira. "A criatividade matem?tica de John Wallis na obra Arithmetica Infinitorum: contribui??es para ensino de c?lculo diferencial e integral na licenciatura em matem?tica." PROGRAMA DE P?S-GRADUA??O EM EDUCA??O, 2017. https://repositorio.ufrn.br/jspui/handle/123456789/22700.
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A pesquisa que originou este texto de tese de doutorado teve como objetivo examinar de que forma as ideias de John Wallis, emergentes na obra Arithmetica Infinitorum, datada de 1656, apresentou inova??es que podem contribuir para o encaminhamento conceitual e did?tico de no??es b?sicas da componente curricular de C?lculo Diferencial e Integral, no curso de Licenciatura em Matem?tica. Nesse sentido, avaliamos o potencial pedag?gico da referida obra para subsidiar o ensino de conceitos matem?ticos, em particular as no??es de integrais, com vistas ao melhoramento do entendimento dos estudantes acerca dessas ideias matem?ticas, tratadas nos Cursos de Forma??o de Professores de Matem?tica. Por admitirmos que os alunos necessitam ampliar o n?mero de trajet?rias que levam ao desenvolvimento de uma ideia Matem?tica ? que, neste trabalho, nos propusemos a responder a seguinte quest?o: como a explora??o did?tica do exerc?cio criativo de um matem?tico na hist?ria pode contribuir na abordagem pedag?gica para o ensino de conte?dos de C?lculo e An?lise na Licenciatura em Matem?tica? Para tal, apoiamo-nos em princ?pios de criatividade elaborados por Mihaly Csikszentmihalyi, que prop?s um modelo para criatividade que leva em considera??o o contexto social e cultural. Por considerarmos fundamental a explica??o do ciclo do pensamento referente ? inven??o matem?tica, associamos a esses princ?pios os processos do Pensamento Matem?tico Avan?ado, proposto por Tommy Dreyfus, de modo que destacamos como esses processos se conectam com as no??es de criatividade. Assim, formulamos um modelo para examinarmos a obra Arithmetica Infinitorum, indicando seus potenciais pedag?gicos para subsidiar o ensino de conceitos matem?ticos baseado em um car?ter investigativo. De maneira que foi poss?vel estabelecermos uma proposta de conex?o entre conhecimento matem?tico desenvolvido historicamente por diferentes matem?ticos e seus potenciais conceituais epistemol?gicos, com a possibilidade de ser implementada na a??o do professor de Matem?tica formador de professores de Matem?tica, com vistas a desenvolver compet?ncias e habilidades para uma futura atua??o do professor em forma??o.
The research which arose this doctorate?s thesis had as purpose examining in which ways John Wallis? ideas, emerging in Arithmetica Infinitorum, dated 1656, has presented contributing innovations for the didactic and conceptual guiding of Differential and Integral Calculus? curricular components basic notions, in Mathematics Licentiate course. For that matter, we evaluated the production?s pedagogical potential to subsidize mathematical concepts? teaching, mainly integral notions, aiming theim provement of students? understanding about these mathematical ideas, which are contemplated in the Mathematics Teachers training course. Acknowledging that the students need to expand the number of paths which lead to the development of a Mathematical idea, in this study we propose to answer the following question: how can the didactic exploration of a mathematician?s creative exercise contribute to the pedagogical approach for the Calculus and Analysis teaching, in Mathematics Licentiate course? For that we leaned on the creativity criteria discussed by Mihaly Csikszentmihalyi, due to considering it substantial in the thinking cycle explanation regarding the Mathematics creation. We relate to these principles the processes developed by Advanced Mathematical Thinking, suggested by Tommy Dreyfus, in order to highlight how these processes attach to creativity notions. Therefore, we formulated a model to examine the writing Arithmetica Infinitorum pointing its pedagogical potential to subsidize mathematical concepts? teaching, based on aninvestigative character. This way, it was possible to establish a connection proposal between mathematical knowledge historically developed by different mathematicians and their conceptual and epistemological potentials, with a possibility of being implemented in Mathematics teacher?s actions, Mathematics teacher?s trainer, in order to grow expertise and abilities for a forthcoming actuation of the training teacher.
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Hanss, Michael. "Applied fuzzy arithmetic : an introduction with engineering applications /." Berlin [u.a.] : Springer, 2005. http://www.loc.gov/catdir/enhancements/fy0662/2004117177-d.html.
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Leibniz, Gottfried Wilhelm. "Disputatio arithmetica de complexionibus quam in illustri Academia Lipsiensi indultu amplissimæ Facultatis philosophicæ pro loco in ea obtinendo prima vice habebit M. Gottfredus Guilielmus Leibnüzius Lipsiensis...d. 7. Martii anno 1666." Strasbourg : SICD, 2007. http://imgbase-scd-ulp.u-strasbg.fr/displayimage.php?album=306&pos=1.
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Humphrey, Illo, and Boèce. "De institutione arithmetica et De institutione musica de Boèce : dans l'enseignement scientifique et philosophique du Haut Moyen âge en Neustrie : édition d'un manuscrit du IXe siècle (Paris, BNF, latin 14064), textes et gloses." Paris 10, 2004. http://www.theses.fr/2004PA100020.
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Cette recherche est consacrée entièrement à l'oeuvre scientifique et philosophique de Boèce (480[?]-Pavie). La thèse retrace l'évolution des différentes étapes de l'influence qu'exerça l'enseignement de Boèce sur la nouvelle branche de la culture intellectuelle carolingienne entre 814 et 877. Cet enseignement, appelé par Boèce lui-même "quadruvium" (De arithmecia I, 1 : éd. G. Friedlein, p. 9:28) est essentiellement d'origine pythagoricienne et platonicienne. Dans cette étude sont développés les thèmes suivants : 1° L'enseignement scientifique-philosophique de Boèce, lequel nous a été conservé intégralement dans ses deux traités De institutione arithmetica libri duo et De institutione musica libri quinque, a été destiné non seulement au futur Mathematicus, musicus, geometres, astronomus, mais à l'ensemble des élèves de l'époque de la "correctio", instituée par le célèbre capitulaire Admonitio generalis promulgué le 23 mars 789, quelle que fut la finalité de leurs études respectives. 2° L'art iconographique, architectural et, dans une certaine mesure, la confection des manuscrits, furent effectués selon le même principe fondamental qui régit la formation des intervalles musicaux, à savoir le Régime de l'Octave, caractérisé par le simple rapport diapasonal, c'est à dire binaire. 3° Le concept fondamental du Régime de l'Octave, qui révèle du domaine de l'Epistémologie, est en effet la base de la philosophie des nombres, celle des proportions et par conséquent celle de l'ars musica. 4° Etant donné la présence de gloses rédigées à la 1ere et à la 2e personne du singulier à la fin du prologue du De institutione arithmetica dans cinq manuscrits du IXe siècle, et la glose clé sous forme de subscriptio (". . . Legi opusculum meum") au début du De institutione arithmetica II, 1 (cf. Paris, BNF, latin 14064, f. 37:10-12), on est en droit de déduire que les gloses à la fin du prologue furent écrites à l'intention de Symmaque (i. E. Le dédicataire et "probator" du traité), ce qui prouverait que l'ensemble des gloses en question sont à l'origine de la main de Boèce lui-même (i. E. "auctor"), montrant ainsi l'importance majeure du manuscrit de base de cette étude, en l'occurrence : Paris, BNF, latin 14064 (IXe s. ). L'ouvrage est divisé en trois parties distinctes, réparties sur cinq tomes. Le tome I contient la Bibliographie des ouvrages imprimés et une Sitographie des ouvrages sur Internet, l'Introduction (une étude historique détaillée sur Boèce du triple point de vue intellectuel, prosographique et hagiographique), enfin, un Commentaire. Le tome II contient une Introduction, le Catalogue des manuscrits choisis, l'Edition princeps intégrale des gloses du De institutione arithmetica de Boèce tirées du manuscrit Paris, BNF, latin 14034, f. 1-84r°). Le tome III renferme les appendices critiques annexes : glossaires grec, latin, musicologique, tableaux pédagogiques sur la formation des intervalles musicaux, un glossaire critique de sténographie latine, l'étude paléographique exhaustive du manuscrit de base : Paris, BNF, latin 14064 (IXe s. , f. 1-84r°). Dans le tome III figurent également les planches et les cartes géographiques. Le tome IV contient les Index grec, français, latin ainsi que l'Index des manuscrits mentionnés. Quant au tome V, un Manuel pédagogique, il constitue un appendice contenant l'ensemble des descriptiones scientifiques-philosophiques (i. E. Diagrammes, figures, tableaux pédagogiques, miniatures) de l'ouvrage.
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Womack, David. "A New Pedagogical Model for Teaching Arithmetic." Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-81151.
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Young children’s ‘alternative’ notions of science are well documented but their unorthodox ideas about arithmetic are less well known. For example, studies have shown that young children initially treat numbers as position markers rather than size symbols. Also, children often hold a transformational view of operations; that is, they are reluctant to accept the commutativity of addition and multiplication.
This ‘alternative’ view of operations is often overlooked by teachers, keen to demonstrate the so called ‘laws’ of arithmetic. However, this paper argues that we should not be in any haste to replace these
primitive intuitions; instead, we should show that transformational operations actually reflect how objects behave when acted on in the physical world. The paper draws on earlier research of the writer
in which young children used signs for transformational arithmetic in game scenarios. In particular, it examines the feasibility of ‘sums’ in which the operator is distinguished from the operand. In short, this
paper presents the theory behind an entirely new way of teaching arithmetic, based on children’s ‘alternative’ intuitions about numbers and operations.
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Buss, Samuel R. "Bounded arithmetic /." Napoli : Bibliopolis, 1986. http://catalogue.bnf.fr/ark:/12148/cb35611934b.
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Dallaway, Richard. "Dynamics of arithmetic : a connectionist view of arithmetic skills." Thesis, University of Sussex, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.358186.
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Steel, Sylvia Kathleen. "The development of arithmetic in normal and arithmetic disabled children." Thesis, Royal Holloway, University of London, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.394027.
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Miles, Richard Craig. "Arithmetic dynamical systems." Thesis, University of East Anglia, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.323222.
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Devi, Roshni. "Modelling arithmetic strategies." Thesis, Open University, 1991. http://oro.open.ac.uk/56451/.
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This thesis examines children's arithmetic strategies and their relation to the concepts of commutativity and associativity. Two complementary methods were used in this research: empirical studies and computational models. Empirical studies were carried out to identify the strategies children used for solving problems like 3 + 4, and 3 + 4 + 7, and the conceptual knowledge associated with them. Their understanding of subtraction problems where the minuend is less than the subtrahend (e.g. 6-8) was also considered. A study with 105 subjects revealed a variety of strategies and information about children's knowledge of commutativity and associativity. Four levels of performance of commutativity were also identified. A longitudinal study was carried out with 12 children in order to obtain details of children's changes in strategy, and to double check the results obtained in the main study. The strategies observed to be used by children over a 20 month period parallel those found in previous studies, which show a general transition to more efficient methods. However, the longitudinal study revealed that development of such arithmetic strategies is a slow process. Furthermore, the studies indicated that knowledge of commutativity is a prerequisite for associativity. Models of the observed strategies have been implemented in the form of production rules in a computer program called PALM. The process of implementation highlighted features of children's problem solving that had not been 'detected during the studies. In addition to models that describe the space of strategies, a model of learning has been implemented for the transition from procedural knowledge of commutativity to that of associativity. The model is capable of generalizing its inbuilt knowledge, for instance, its ability to solve 2-term arithmetic expressions, to allow it to solve more complex problems, such as 3-term arithmetic expressions. A further model has been constructed for learning arithmetic strategies that are more efficient than those already represented in the program. It learns specific rules by adding conditions for efficient problem solving to its previous general rules.
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Weinstein, Madeleine. "Adinkras and Arithmetical Graphs." Scholarship @ Claremont, 2016. http://scholarship.claremont.edu/hmc_theses/85.
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Adinkras and arithmetical graphs have divergent origins. In the spirit of Feynman diagrams, adinkras encode representations of supersymmetry algebras as graphs with additional structures. Arithmetical graphs, on the other hand, arise in algebraic geometry, and give an arithmetical structure to a graph. In this thesis, we will interpret adinkras as arithmetical graphs and see what can be learned.
Our work consists of three main strands. First, we investigate arithmetical structures on the underlying graph of an adinkra in the specific case where the underlying graph is a hypercube. We classify all such arithmetical structures and compute some of the corresponding volumes and linear ranks.
Second, we consider the case of a reduced arithmetical graph structure on the hypercube and explore the wealth of relationships that exist between its linear rank and several notions of genus that appear in the literature on graph theory and adinkras.
Third, we study modifications of the definition of an arithmetical graph that incorporate some of the properties of an adinkra, such as the vertex height assignment or the edge dashing. To this end, we introduce the directed arithmetical graph and the dashed arithmetical graph. We then explore properties of these modifications in an attempt to see if our definitions make sense, answering questions such as whether the volume is still an integer and whether there are still only finitely many arithmetical structures on a given graph.
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Martinez, Metzmeier César. "Two problems in arithmetic geometry. Explicit Manin-Mumford, and arithmetic Bernstein-Kusnirenko." Thesis, Normandie, 2017. http://www.theses.fr/2017NORMC224/document.
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Dans la première partie de cette thèse, on présente des bornes supérieures fines pour le nombre de sous-variétés irréductibles de torsion maximales dans une sous-variété du tore complexe algébrique $(\mathbb{C}^{\times})^n$ et d'une variété abélienne. Dans les deux cas, on donne une borne explicite en termes du degré des polynômes définissants et la variété ambiante. De plus, la dépendance en le degré des polynômes est optimale. Dans le cas du tore complexe, on donne aussi une borne explicite en termes du degré torique de la sous-variété. En conséquence de ce dernier résultat, on démontre les conjectures de Ruppert, et Aliev et Smyth pour le nombre de points de torsion isolés dans une hypersurface. Ces conjectures bornent ce nombre en terme, respectivement, du multi-degré et du volume du polytope de Newton d'un polynôme définissant l'hypersurface.Dans la deuxième partie de cette thèse, on présente une borne supérieure pour la hauteur des zéros isolés, dans le tore, d'un système de polynômes de Laurent sur un corps adélique qui satisfait la formule du produit. Cette borne s'exprime en termes des intégrales mixtes des fonctions toit locales associées à la hauteur choisie et le système des polynômes de Laurent. On montre aussi que cette borne est presque optimale dans quelques familles d'exemples. Ce résultat est un analogue arithmétique du théorème de Bern\v{s}tein-Ku\v{s}nirenkoIn the first part of this thesis we present sharp bounds on the number of maximal torsion cosets in a subvariety of a complex algebraic torus $(\mathbb{C}^{\times})^n$ and of an Abelian variety. In both cases, we give an explicit bound in terms of the degree of the defining polynomials and the ambient variety. Moreover, the dependence on the degree of the polynomials is sharp. In the case of the complex torus, we also give an effective bound in terms of the toric degree of the subvariety. As a consequence of the latter result, we prove the conjectures of Ruppert, and Aliev and Smyth on the number of isolated torsion points of a hypersurface. These conjectures bound this number in terms of the multidegree and the volume of the Newton polytope of a polynomial defining the hypersurface, respectively.In the second part of the thesis, we present an upper bound for the height of isolated zeros, in the torus, of a system of Laurent polynomials over an adelic field satisfying the product formula. This upper bound is expressed in terms of the mixed integrals of the local roof functions associated to the chosen height function and to the system of Laurent polynomials. We also show that this bound is close to optimal in some families of examples. This result is an arithmetic analogue of the classical Bern\v{s}tein-Ku\v{s}nirenko theorem
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Sanders, Tom. "Topics in arithmetic combinatorics." Thesis, University of Cambridge, 2007. https://www.repository.cam.ac.uk/handle/1810/236994.
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This thesis is chiefly concerned with a classical conjecture of Littlewood's regarding the L¹-norm of the Fourier transform, and the closely related idem-potent theorem. The vast majority of the results regarding these problems are, in some sense, qualitative or at the very least infinitary and it has become increasingly apparent that a quantitative state of affairs is desirable. Broadly speaking, the first part of the thesis develops three new tools for tackling the problems above: We prove a new structural theorem for the spectrum of functions in A(G); we extend the notion of local Fourier analysis, pioneered by Bourgain, to a much more general structure, and localize Chang's classic structure theorem as well as our own spectral structure theorem; and we refine some aspects of Freiman's celebrated theorem regarding the structure of sets with small doubling. These tools lead to improvements in a number of existing additive results which we indicate, but for us the main purpose is in application to the analytic problems mentioned above. The second part of the thesis discusses a natural version of Littlewood's problem for finite abelian groups. Here the situation varies wildly with the underlying group and we pay special attention first to the finite field case (where we use Chang's Theorem) and then to the case of residues modulo a prime where we require our new local structure theorem for A(G). We complete the consideration of Littlewood's problem for finite abelian groups by using the local version of Chang's Theorem we have developed. Finally we deploy the Freiman tools along with the extended Fourier analytic techniques to yield a fully quantitative version of the idempotent theorem.
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Szanto, Gabriella. "Arithmetic disability of adults." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk2/tape16/PQDD_0010/NQ27773.pdf.
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Riis, Søren. "Independence in bounded arithmetic." Thesis, University of Oxford, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.386574.
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Malins, E. J. "Hard-wiring interval arithmetic." Thesis, University of Ulster, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.554235.
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Floating point arithmetic has become prevalent in virtually every scientific computation, yet suffers from rounding, truncation and catastrophic cancellation errors to the extent that the result may be completely inaccurate. Whilst modern processors do not have the capabilities to natively support interval arithmetic, it can be implemented in software though such a method of implementation results in larger program size and often more than doubles execution times. Embedded systems do not have the luxury of utilising software implementations of interval arithmetic and so a hardware based alternative must be sought. This thesis examines the algorithms traditionally used for interval multiplication and proposes a new method, free of the caveats which plague traditional multiplication computation. The inherent parallelism of these algorithms is examined to determine their throughput and latency characteristics, given a selection of platforms which exhibit variations in the number of input buses, FPUs and result buses. A software implementation of each method of interval multiplication is described, alongside their relative performances. Given the data-dependant nature of interval computation and the current trend towards vector processing architectures, these software implementations include reformulations of both tradi- tional methods of interval multiplication to provide vectorisation of the software and thus completely eliminate branching. Realised as co-processors to the Altera Nios Il, this thesis describes the design decisions, specification, area requirements, power utilisation and per-cycle performance of hardware implementations of the brute-force, nine case and herein presented integer-based interval multipliers.
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Green, B. "Topics in arithmetic combinatorics." Thesis, University of Cambridge, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.599660.
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Aghasi, Mansour. "Geometry of arithmetic surfaces." Thesis, Durham University, 1996. http://etheses.dur.ac.uk/5270/.
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In this thesis my emphasis is on the resolution of the singularities of fibre products of Arithmetic Surfaces. In chapter one as an introduction to my thesis some elementary concepts related to regular and singular points are reviewed and the concept of tangent cone is defined for schemes over a discrete valuation ring. The concept of arithmetic surfaces is introduced briefly in the end of this chapter. In chapter 2 my new procedures namely the procedure of Mojgan(_1) and the procedure of Mahtab(_2) and a new operator called Moje are introduced. Also the concept of tangent space is defined for schemes over a discrete valuation ring. In chapter 3 the singularities of schemes which are the fibre products of some surfaces with ordinary double points are resolved. It is done in two different methods. The results from both methods are consistent. In chapter 4, I have tried to resolve the singularities of a special class of arithmetic three-folds, namely those which are the fibre product of two arithmetic surfaces, which were very helpful to achieve my final results about the resolution of singularities of fibre products of the minimal regular models of Tate. Chapter 5 includes my final results which are about the resolution of singularities of the fibre product of two minimal regular models of Tate.
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Haili, Hailiza Kamarul. "Distributional problems in arithmetic." Thesis, University of Liverpool, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.366245.
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Bird, Alexander. "Arithmetic, grammar and ontology." Thesis, University of Cambridge, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.386989.
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Nasr, Entesar. "Distribution problems in arithmetic." Thesis, University of Liverpool, 2018. http://livrepository.liverpool.ac.uk/3022467/.
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In this thesis we use modern developments in ergodic theory and uniform distribution theory to investigate the distribution of polynomials, partial quotients of convergents, random and oscillatory sequences.
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Fiske, James Alexander Stuart. "A reconfigurable arithmetic processor." Thesis, Massachusetts Institute of Technology, 1988. http://hdl.handle.net/1721.1/14419.
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McLeod, John Angus. "Arithmetic hyperbolic reflection groups." Thesis, Durham University, 2013. http://etheses.dur.ac.uk/7743/.
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This thesis uses Vinberg’s algorithm to study arithmetic hyperbolic reflection groups which are contained in the groups of units of quadratic forms. We study two families of quadratic forms: the diagonal forms −dx_0^2 + x_1^2 + ... + x_n^2 ; and the forms whose automorphism groups contain the Bianchi groups. In the first instance we classify over Q the pairs (d,n) for which such a group can be found, and in some cases we can compute the volumes of the fundamental polytopes. In the second instance we use a combination of the geometric and number theoretic information to classify the reflective Bianchi groups by first classifying the reflective extended Bianchi groups, namely the maximal discrete extension of the Bianchi groups in PSL(2,C). Finally we identify some quadratic forms in the first instance and completely classify those in the second which have a quasi-reflective structure.
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Bamunoba, Alex Samuel. "Arithmetic of carlitz polynomials." Doctoral thesis, Stellenbosch : Stellenbosch University, 2014. http://hdl.handle.net/10019.1/95850.
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Bingham, Aram. "Commutative n-ary Arithmetic." ScholarWorks@UNO, 2015. http://scholarworks.uno.edu/td/1959.
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Motivated by primality and integer factorization, this thesis introduces generalizations of standard binary multiplication to commutative n-ary operations based upon geometric construction and representation. This class of operations are constructed to preserve commutativity and identity so that binary multiplication is included as a special case, in order to preserve relationships with ordinary multiplicative number theory. This leads to a study of their expression in terms of elementary symmetric polynomials, and connections are made to results from the theory of polyadic (n-ary) groups. Higher order operations yield wider factorization and representation possibilities which correspond to reductions in the set of primes as well as tiered notions of primality. This comes at the expense of familiar algebraic properties such as associativity, and unique factorization. Criteria for primality and a naive testing algorithm are given for the ternary arithmetic, drawing heavily upon modular arithmetic. Finally, connections with the theory of partitions of integers and quadratic forms are discussed in relation to questions about cardinality of primes.
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Royals, Robert. "Arithmetic and dynamical systems." Thesis, University of East Anglia, 2015. https://ueaeprints.uea.ac.uk/57191/.
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In this thesis we look at a number of topics in the area of the interaction between dynamical systems and number theory. We look at two diophantine approximation problems in local �fields of positive characteristic, one a generalisation of the Khintchine{Groshev theorem, another a central limit theorem. We also prove a P�olya{Carlson dichotomy result for a large class of adelicly perturbed rational functions. In particular we prove that for a finite set of primes S that the power series f(z) generated by the Fibonacci series with all primes in S removed has a natural boundary.
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Cheung, Chak-Chung Ray. "Customisable arithmetic hardware designs." Thesis, Imperial College London, 2007. http://hdl.handle.net/10044/1/11976.
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Wang, Shaoyun. "A CORDIC arithmetic processor /." Digital version accessible at:, 1998. http://wwwlib.umi.com/cr/utexas/main.
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Castelli, Vina Maureen. "Weak Fragments of Arithmetic." OpenSIUC, 2015. https://opensiuc.lib.siu.edu/theses/1738.
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The purpose of this thesis is to investigate which weak fragments of arithmetic are essential to proving certain theorems that students are required to use, understand, and/or prove according to the Common Core Standards. The strength of a fragment sometimes correlates to computational complexity and can serve as a proxy for cognitive difficulty. To determine which are the weak fragments of arithmetic are essential to a proof, we first interpret the chosen standard into formal mathematical language; this process generates a theorem to be proven. We proceed to prove this theorem using weak fragments of arithmetic such as IOPEN or $I\Delta_0$. After we have proven a theorem, we must then show that we used the minimal fragment possible in our proof. This has only been done for one theorem, namely The Remainder Theorem. We found a proof for The Remainder Theorem using $I\Delta_0$. To prove that this is the minimal fragment needed for the proof (that IOPEN would not suffice), we built a non-standard model of IOPEN using forcing in which The Remainder Theorem is not true.
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Feliu, Trijueque Elisenda. "On Higher Arithmetic Intersection Theory." Doctoral thesis, Universitat de Barcelona, 2007. http://hdl.handle.net/10803/658.
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The results of this thesis contribute to the program of developing a higher arithmetic intersection theory. These results constitute chapters 3 and 5. Chapters 2 and 4 consist of the preliminary results needed for chapters 3 and 5, in the area of homotopy theory of simplicial sheaves and algebraic K-theory.In chapter 3, we develop a higher intersection theory on arithmetic varieties, "à la Bloch". We construct a representative of the Beilinson regulator using the Deligne complex of differential forms. Next, we develop a theory of higher arithmetic Chow groups, for any arithmetic variety X over a field. We prove that the construction is functorial and that there is a commutative and associative product structure, compatible with the algebraic intersection product. Therefore, we provide an arithmetic intersection product for arithmetic varieties over a field.
Chapters 4 and 5 are devoted to the definition of Adams operations on higher arithmetic K-theory. By the nature of the definition of the higher arithmetic K-groups, it is apparently necessary to have a description of the Adams operations in algebraic K-theory in terms of a chain morphism, compatible with the representative of the Beilinson regulator "ch".
In chapter 4, we obtain a chain morphism inducing Adams operations on higher algebraic K-theory over the field of rational numbers. This definition is of combinatory nature. This chain morphism is designed to commute with the Beilinson regulator "ch" given by Burgos and Wang.
In chapter 5 it is shown that this chain morphism indeed commutes with the representative of the Beilinson regulator "ch" and we use this fact to define Adams operations on the rational higher arithmetic K-groups.
The development of this study required tools to compare morphisms from algebraic K-groups to a suitable cohomology theory or to the K-groups themselves. In chapter 2, we study these comparisons at a general level, providing theorems giving sufficient conditions for two morphisms to agree. The theory underlying the proofs is the homotopy theory of simplicial sheaves. As an application, we prove that the Adams operations defined by Grayson agree for any regular noetherian scheme of finite Krull dimension with the Adams operations defined by Gillet and Soulé by means of homotopy theory of sheaves. In particular, this implies that the Adams operations defined by Grayson's work.
TÍTOL DE LA TESI: "Sobre la teoria d'intersecció aritmètica superior"
TEXT:
Aquesta tesi s'emmarca en el programa de la geometria d'Arakelov que es basa en obtenir una teoria d'intersecció aritmètica seguint les passes de la teoria d'intersecció algebraica. Els resultats d'aquesta tesi contribueixen al programa de desenvolupar una teoria d'intersecció aritmètica superior. Aquests són els resultats que constitueixen els capítols 3 i 5. Els capítols 2 i 4 consisteixen en resultats preliminars que es necessiten pels capítols 3 i 5, en l'àrea de teoria homotòpica de feixos simplicials i K-teoria algebraica.
En el capítol 3, hem desenvolupat una teoria d'intersecció superior en varietats aritmètiques "à la" Bloch. És a dir, hem modificat els grups de Chow superiors definits per Bloch via una construcció explícita del regulador de Beilinson en termes de cicles algebraics.
Hem construït un representant del regulador de Beilinson usant el complex de Deligne de formes diferencials. Tot seguit, hem desenvolupat una teoria de grups de Chow aritmètics superiors, per a qualsevol varietat aritmètica X sobre un cos. Demostrem que hi ha un producte associatiu i commutatiu, compatible amb el producte d'intersecció algebraic. Per tant, donem un producte d'intersecció aritmètic per varietats aritmètiques sobre un cos.
Tot seguit ens vam centrar en la relació entre els grups de Chow aritmètics superiors definits i els K-grups aritmètics superiors. Per tal de seguir l'esquema algebraic, hauríem de tenir una descomposició dels grups Kn(X) racionals donada pels espais de vectors propis de les operacions Adams. Per la naturalesa de la definició de Kn(X), tant considerant la fibra homotòpica com els grups d'homotopia modificats de Takeda, és aparentment necessari tenir una descripció de les operacions d'Adams en K-teoria algebraica en termes d'un morfisme de cadenes, compatible amb el representant del regulador de Beilinson "ch".
En el capítol 4, obtenim un morfisme de cadenes que indueix les operacions d'Adams en K-teoria algebraica superior, sobre el cos dels nombres racionals. Aquesta definició és de naturalesa combinatòrica. A més, el morfisme està construït amb la idea en ment que hauria de commutar amb el regulador de Beilinson "ch" donat per Burgos i Wang.
En el capítol 5 demostrem que aquest morfisme de cadenes commuta amb "ch" i usem aquest fet per definir operacions d'Adams en els K-grups aritmètics superiors tensorialitzats amb Q.
El desenvolupament d'aquest treball requeria eines per comparar morfismes dels K-grups algebraics superiors a grups de cohomologia adequats o als mateixos K-grups. En el capítol 2, estudiem aquestes comparacions a un nivell general, donant teoremes que detallen condicions suficients per tal que dos morfismes coincideixin. La teoria en què es recolzen les demostracions és la teoria homotòpica de feixos simplicials.
Com a aplicació, demostrem que les operacions d'Adams definides per Grayson a coincideixen, per a tot esquema noeterià regular de dimensió de Krull finita, amb les operacions d'Adams definides per Gillet i Soulé. En particular, se segueix que les operacions d'Adams definides per Grayson satisfan les identitats usuals d'un lambda-anell, fet que no quedava demostrat en l'article de Grayson.
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Graumann, Günter. "Problem Fields in Elementary Arithmetic." Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-79913.
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Working with problems and making investigations is an activity one has to learn already very early. Therefore in primary school children should not only learn concepts and solve given tasks. They also should find out knowledge and reasons by themselves. Here you will find some problem fields in elementary arithmetic within children of primary school can make different investigations and find as well as give reasons for special statements. The topics concerned are partitions of numbers, sums of consecutive numbers, figured numbers, sequences and chains, table of hundred and numberwalls.
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Selander, Björn. "Arithmetic of three-point covers." Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-7497.
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Any cover of the Riemann sphere with rational branch points is known to be defined over the algebraic numbers. Hence the Galois group of the rationals acts on the category of such branched covers. Particulars about this action are still scarce, even in the simplest non-abelian case, the case with just three branch points. The first paper in this thesis describes a new algorithm, which uses modular form techniques in order to compute the equations for a cover of the Riemann sphere which is hyperelliptic as a curve. Given such equations one may easily determine the Galois orbit to which the cover belongs. We compute and discuss all covers of degree 6 and genus 2, and complete the case of covers of degree 7 and genus 1 as well. The second paper gives a proof of a formula for the number of three-point G-covers with a fixed special G-deformation datum (here G is a finite group which is strictly divisible by a prime number p). Since such a datum is an invariant for the action of the inertia group at p, this gives partial information about the action of this inertia group.
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Poon, Joseph Kin-Shing. "An arithmetic processor for robotics." Thesis, University of British Columbia, 1985. http://hdl.handle.net/2429/25133.
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An arithmetic processor chip for use in robotics has been designed in 4µm CMOS technology. The chip is intended to function as a slave processor to a general purpose microprocessor host and be able to perform robot arm coordinate transformation calculations for use in real-time control applications. A parallel-processing, multi-pipelined architecture has been used to produce a 45mm² chip for which a machine cycle time of 200ns appears possible. The general nature of the architecture of this microprogrammable processor renders it useful for a range of computational tasks in robotics in addition to coordinate transformation.Applied Science, Faculty of
Electrical and Computer Engineering, Department of
Graduate
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Ziyang, Wang. "Non-binary Distributed Arithmetic Coding." Thesis, Université d'Ottawa / University of Ottawa, 2015. http://hdl.handle.net/10393/32318.
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Distributed source coding (DSC) is a fundamental concept in information theory. It
refers to distributed compression of correlated but geographically separated sources. With the development of wireless sensor networks, DSC has attracted great research interest in the recent years [26].
Although many channel code based DSC schemes have been developed (e.g., those based on turbo codes [11]and LDPC codes [20]), this thesis focuses on the arithmetic coding based approaches, namely, Distributed Arithmetic Coding (DAC) due to its simplicity in encoding [8]. To date, most of the DAC approaches that have been proposed deal with binary sources and can not handle non-binary cases. Little research has been done to extend DAC for non-binary sources. This work aims at developing efficient DAC techniques for the compression of non-binary sources.
The key idea of DAC is representing the source symbols by overlapping intervals, as
opposed to the case of conventional arithmetic coding where the intervals representing the symbols do not overlap. However the design of the overlapping intervals has been completely of heuristic nature to date. As such, the first part of this work is a thorough study of various interval-overlapping rules in binary DAC so as to understand how these rules impact the performance of DAC. The insight acquired in this study is used in the second part of this work, where two DAC algorithms are proposed to compress non-binary non-uniform sources. The first algorithm applies a designed overlap structure in DAC process, while the second converts a non-binary sequence into a binary sequence by Huffman Coding and
encoding the result in binary DAC. Simulation studies are performed to demonstrate the efficiencies of the two proposed algorithms in a variety of source parameter settings.
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Selander, Björn. "Arithmetic of three-point covers /." Uppsala : Department of Mathematics, Univ. [distributör], 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-7497.
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Hamel, Mariah. "Arithmetic structures in random sets." Thesis, University of British Columbia, 2008. http://hdl.handle.net/2429/2838.
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We prove various results in additive combinatorics for subsets of random sets. In particular we extend Sarkozy's theorem and a theorem of Green on long arithmetic progressions in sumsets to dense subsets of random sets with asymptotic density 0. Our proofs require a transference argument due to Green and Green-Tao which enables us to apply known results for sets of positive upper density to subsets of random sets which have positive relative density. We also prove a density result which states that if a subset of a random set has positive relative density, then the sumset of the subset must have positive upper density in the integers.
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Chipeniuk, Karsten. "Structure and arithmetic in sets." Thesis, University of British Columbia, 2011. http://hdl.handle.net/2429/33674.
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We prove results in arithmetic combinatorics involving sums of prime numbers and also some variants of the Erdös-Szemerédi sum-product phenomenon. In particular, we prove nontrivial lower bounds on the density in the integers of the sumset of a positive relative density subset of the primes. The proof of this result uses Green and Green-Tao pseudorandomness arguments to reduce the problem to an analogous statement for relatively dense subsets of the multiplicative subgroup of integers modulo a large integer N. The latter statement is resolved with a combinatorial argument which bounds high moments of a representation function. We also show that if two distinct sets A and B of complex numbers have very small productset, then they produce maximally large iterated sumsets. This uses an algebraic concept of the multiplicative dimension of a finite set. As an application of the case A=B, we obtain a quantitative version of a result of Chang on sums and products of distinct complex elements.
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Kapoor, Vishaal. "Asymptotic formulae for arithmetic functions." Thesis, University of British Columbia, 2011. http://hdl.handle.net/2429/34018.
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In this work we will consider several questions concerning the asymptotic
nature of arithmetic functions. First, we conduct a finer analysis on the behavior
of λ(Euler's totient function(n)) in relation to λ(λ(n)), proving that log(λ(Euler's totient function(n))/λ(λ(n)))
is asymptotic to (log log n)(log log log n)for almost all n. Second, we establish
an asymptotic formula for sums of a generalized divisor function on the
Gaussian numbers. And third, for complex-valued multiplicative functions
that are suffciently close to 1 on the primes and bounded on prime powers,
we determine the average value over a short interval x < n ≤ x+w provided
the interval is suffciently long with respect to x.
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D'Aquino, Paola. "Exponentiation and fragments of arithmetic." Thesis, University of Oxford, 1992. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.317811.
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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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Flatters, Anthony. "Arithmetic Properties of recurrence sequences." Thesis, University of East Anglia, 2010. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.520281.
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Lee, Peter. "Hybrid-logarithmic arithmetic and applications." Thesis, University of Kent, 2011. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.633518.
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This thesis is a contribution to the existing body of research on logarithmic arithmetic and signal processing. The implementation of log2 arithmetic circuits using modem digital hardware has been an area of active research for over 40 years. In this time over 400 academic papers in journals and conferences have been published and more than 40 patent applications submitted. At the time of writing there are at least 6 different research groups around the world actively working on new algorithms for conversion to and from the logarithmic domain and using logarithmic arithmetic and logarithmic signal processing in a wide range of academic, industrial, consumer and scientific applications. This thesis is separated into two sections. The first section deals with algorithms for logarithmic and anti-logarithmic conversion. It includes an overview and comparison of existing conversion algorithms before presenting two new conversion architectures which are more computationally efficient and suitable for implementation in both ASIC and VLSI technologies. The second section presents material published by the author on two specific applications of logarithmic signal processing where a Hybrid-Logarithmic Number System (Hybrid-LNS or Hybrid-Log) approach has been used. The first is the analysis, design· and implementation of a Discrete Cosine Transform (and its inverse) architecture which has been optimised for use in image compression applications such as JPEG and MPEG. The second describes the TOTEM neural network processor before discussing its implementation in both full-custom IC and FPGA technologies. The concentration. on Hybrid-LNS solutions indicates that this thesis does not discuss in any significant detail the problem of performing addition and subtraction in the logarithnlic domain. There has been extensive research into this problem in recent years and it · is beyond the scope of this thesis. This work is intended to add to the continued debate about the advantages/disadvantages of Hybrid-LNS architectures over "pure" logarithmic or LNS processors.
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Thorne, Jack A. "The Arithmetic of Simple Singularities." Thesis, Harvard University, 2012. http://dissertations.umi.com/gsas.harvard:10339.
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We investigate some arithmetic orbit problems in representations of linear algebraic groups arising from Vinberg theory. We aim to give a description of the orbits in these representations using methods with an emphasis on representation theory rather than algebraic geometry, in contrast to previous works of other authors. It turns out that for the representations we consider, the orbits are related to the arithmetic of the Jacobians of certain algebraic curves, which appear as the smooth nearby fibers of deformations of simple singularities. We calculate these families of algebraic curves, and show that the 2-torsion in their Jacobians is canonically identified with the stabilizers of certain orbits in the corresponding representations.Mathematics
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Moss, Patrick Barry. "The arithmetic of realizable sequences." Thesis, University of East Anglia, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.396736.
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Crawley, David George. "Time optimal arithmetic for VLSI." Thesis, University of Cambridge, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.239081.
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