Dissertations / Theses on the topic '010101 Algebra and Number Theory'
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1
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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Sordo, Vieira Luis A. "ON P-ADIC FIELDS AND P-GROUPS." UKnowledge, 2017. http://uknowledge.uky.edu/math_etds/43.
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The dissertation is divided into two parts. The first part mainly treats a conjecture of Emil Artin from the 1930s. Namely, if f = a_1x_1^d + a_2x_2^d +...+ a_{d^2+1}x^d where the coefficients a_i lie in a finite unramified extension of a rational p-adic field, where p is an odd prime, then f is isotropic. We also deal with systems of quadratic forms over finite fields and study the isotropicity of the system relative to the number of variables. We also study a variant of the classical Davenport constant of finite abelian groups and relate it to the isotropicity of diagonal forms. The second part deals with the theory of finite groups. We treat computations of Chermak-Delgado lattices of p-groups. We compute the Chermak-Delgado lattices for all p-groups of order p^3 and p^4 and give results on p-groups of order p^5.
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Alnominy, Madai Obaid. "Monomial Progenitors and Related Topics." CSUSB ScholarWorks, 2018. https://scholarworks.lib.csusb.edu/etd/619.
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The main objective of this project is to find the original symmetric presentations of some very important finite groups and to give our constructions of some of these groups. We have found the Mathieu sporadic group M11, HS × D5, where HS is the sporadic group Higman-Sim group, the projective special unitary group U(3; 5) and the projective special linear group L2(149) as homomorphic images of the monomial progenitors 11*4 :m (5 :4), 5*6 :m S5 and 149*2 :m D37. We have also discovered 24 : S3 × C2, 24 : A5, (25 : S4), 25 : S3 × S3, 33 : S4 × C2, S6, 29: PGL(2,7), 22 • (S6 : S6), PGL(2,19), ((A5 : A5 × A5) : D6), 6 • (U4(3): 2), 2 • PGL(2,13), S7, PGL (2,8), PSL(2,19), 2 × PGL(2,81), 25 : (S6 × A5), 26 : S4 × D3, U(4,3), 34 : S4, 32 :D6, 2 • (PGL(2,7) :PSL(2,7), 22 : (S5 : S5) and 23 : (PSL3(4) : 2) as homomorphic images of the permutation progenitors 2*8 : (2 × 4 : 2), 2*16: (2 × 4 :C2 × C2), 2*9: (S3 × S3), 2*9: (S3 × A3), 2*9: (32 × 23) and 2*9: (33 × A3). We have also constructed 24: S3 × C2, 24 : A5, (25: S4), 25 : S3 × S3,: 33: S4 × C2, S6, M11 and U (3,5) by using the technique of double coset enumeration. We have determined the isomorphism types of the most of the images mentioned in this thesis. We demonstrate our work for the following examples: 34 : (32 * 23) × 2, 29 : PGL(2,7), 2•S6, (54 : (D4 × S3)), and 3: •PSL(2,19) ×2.
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Salt, Brittney M. "MONOID RINGS AND STRONGLY TWO-GENERATED IDEALS." CSUSB ScholarWorks, 2014. https://scholarworks.lib.csusb.edu/etd/31.
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This paper determines whether monoid rings with the two-generator property have the strong two-generator property. Dedekind domains have both the two-generator and strong two-generator properties. How common is this? Two cases are considered here: the zero-dimensional case and the one-dimensional case for monoid rings. Each case is looked at to determine if monoid rings that are not PIRs but are two-generated have the strong two-generator property. Full results are given in the zero-dimensional case, however only partial results have been found for the one-dimensional case.
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Constable, Jonathan A. "Kronecker's Theory of Binary Bilinear Forms with Applications to Representations of Integers as Sums of Three Squares." UKnowledge, 2016. http://uknowledge.uky.edu/math_etds/35.
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In 1883 Leopold Kronecker published a paper containing “a few explanatory remarks” to an earlier paper of his from 1866. His work loosely connected the theory of integral binary bilinear forms to the theory of integral binary quadratic forms. In this dissertation we discover the statements within Kronecker's paper and offer detailed arithmetic proofs. We begin by developing the theory of binary bilinear forms and their automorphs, providing a classification of integral binary bilinear forms up to equivalence, proper equivalence and complete equivalence.
In the second chapter we introduce the class number, proper class number and complete class number as well as two refinements, which facilitate the development of a connection with binary quadratic forms.
Our third chapter is devoted to deriving several class number formulas in terms of divisors of the determinant. This chapter also contains lower bounds on the class number for bilinear forms and classifies when these bounds are attained.
Lastly, we use the class number formulas to rigorously develop Kronecker's connection between binary bilinear forms and binary quadratic forms. We supply purely arithmetic proofs of five results stated but not proven in the original paper. We conclude by giving an application of this material to the number of representations of an integer as a sum of three squares and show the resulting formula is equivalent to the well-known result due to Gauss.
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Gopalan, Parikshit. "Computing with Polynomials over Composites." Diss., Georgia Institute of Technology, 2006. http://hdl.handle.net/1853/11564.
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In the last twenty years, algebraic techniques have been applied with great success to several areas in theoretical computer science. However, for many problems involving modular counting, there is a huge gap in our understanding depending on whether the modulus is prime or composite. A prime example is the problem of showing lower
bounds for circuits with Mod gates in circuit complexity. Proof techniques that work well for primes break down over composites. Moreover, in some cases, the problem for composites turns
out to be very different from the prime case. Making progress on these problems seems to require a better understanding of polynomials over
composites. In this thesis, we address some such "prime vs. composite" problems from algorithms, complexity and combinatorics, and the surprising connections between them.
We consider the complexity-theoretic problem of computing Boolean functions using polynomials modulo composites. We show that symmetric polynomials can viewed as simultaneous communication protocols. This equivalence
allows us to use techniques from communication complexity and number theory to prove degree bounds. We use these to give the first tight
degree bounds for a number of Boolean functions.
We consider the combinatorial problem of explicit construction of Ramsey graphs. We present a simple construction of such graphs using polynomials modulo composites. This approach gives a unifying view of many known constructions,and explains why they all achieve the same bound.We show that certain approaches to this problem cannot give better bounds.
Finally, we consider the algorithmic problem of interpolation for polynomials modulo composites. We present the first query-efficient algorithms for interpolation and learning under a distribution. These results rely on some new structural results about such polynomials.
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Munoz, Susana L. "A Fundamental Unit of O_K." CSUSB ScholarWorks, 2015. https://scholarworks.lib.csusb.edu/etd/133.
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In the classical case we make use of Pells equation to compute units in the ring OF. Consider the parallel to the classical case and the quadratic field extension that creates the ring OK. We use the generalized Pell's equation to find the units in this ring since they are solutions. Through the use of continued fractions we may further characterize this ring and compute its units.
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Fávaro, Eduardo Rogério [UNESP]. "Corpos cujo condutor é potência de primo: caracterização e reticulados ideais associados." Universidade Estadual Paulista (UNESP), 2012. http://hdl.handle.net/11449/100063.
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favaro_er_dr_sjrp.pdf: 449730 bytes, checksum: 66f6b6e8876e035dcd2e6aa8db337bbd (MD5)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)
Este trabalho esta relacionado com a Teoria Algébrica dos Números e aplicações em Reticulados Ideais. Descrevemos os corp os cujo condutor e potência de primo. Quando o primo e dois, descrevemos tamb em o anel de inteiros. Quando o primo e mpar calculamos o discriminante de um modo alternativo ao existente na literatura. Neste caso, e quando o corpo tem como grau o pr oprio primo mpar, descrevemos o anel de inteiros com uma base integral e a forma traço associada, além do mínimo euclidiano. Com isso, obtemos uma família de reticulados ideais de dimensão prima ímpar
This work is relate to Algebric Number Theory and applications in Ideal Lattices. We describ e numb er elds with p ower prime conductor. In the case prime two, we showed the ring of integers. For o dd prime, we give a new pro of for formula of discrimanate. In the case that the the degree of the eld is the o dd prime, we describ e the ring of integers, the trace form asso ciated and the Euclidean minimum. With this, we have a family of ideal lattices in odd prime dimension
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Shaughnessy, John F. "Finding Zeros of Rational Quadratic Forms." Scholarship @ Claremont, 2014. http://scholarship.claremont.edu/cmc_theses/849.
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In this thesis, we introduce the notion of quadratic forms and provide motivation for their study. We begin by discussing Diophantine equations, the field of p-adic numbers, and the Hasse-Minkowski Theorem that allows us to use p-adic analysis determine whether a quadratic form has a rational root. We then discuss search bounds and state Cassels' Theorem for small-height zeros of rational quadratic forms. We end with a proof of Cassels' Theorem and suggestions for further reading.
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Samson, Duncan Alistair. "An analysis of the influence of question design on pupils' approaches to number pattern generalisation tasks." Thesis, Rhodes University, 2008. http://hdl.handle.net/10962/d1003302.
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This study is based on a qualitative investigation framed within an interpretive paradigm, and aims to investigate the extent to which question design affects the solution strategies adopted by children when solving linear number pattern generalisation tasks presented in pictorial and numeric contexts. The research tool comprised a series of 22 pencil and paper exercises based on linear generalisation tasks set in both numeric and 2-dimensional pictorial contexts. The responses to these linear generalisation questions were classified by means of stage descriptors as well as stage modifiers. The method or strategy adopted was carefully analysed and classified into one of seven categories. A meta-analysis focused on the formula derived for the nth term in conjunction with its justification. The process of justification proved to be a critical factor in being able to accurately interpret the origin of the sub-structure evident in many of these responses. From a theoretical perspective, the central role of justification/proof within the context of this study is seen as communication of mathematical understanding, and the process of justification/proof proved to be highly successful in providing a window of understanding into each pupil’s cognitive reasoning. The results of this study strongly support the notion that question design can play a critical role in influencing pupils’ choice of strategy and level of attainment when solving pattern generalisation tasks. Furthermore, this study identified a diverse range of visually motivated strategies and mechanisms of visualisation. An awareness and appreciation for such a diversity of visualisation strategies, as well as an understanding of the importance of appropriate question design, has direct pedagogical application within the context of the mathematics classroom.
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Souza, Romario Sidrone [UNESP]. "Equações diofantinas lineares, quadráticas e aplicações." Universidade Estadual Paulista (UNESP), 2017. http://hdl.handle.net/11449/149949.
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Este trabalho é resultado de uma pesquisa bibliográfica sobre Diofanto e as equações que levam seu nome, as equações diofantinas. Mais especificamente, apresentamos as equações diofantinas lineares e alguns casos particulares das equações diofantinas quadráticas. Ainda, abordamos um estudo sobre alguns tópicos de teoria dos números e frações contínuas, afim de facilitar o entendimento sobre os teoremas e resultados acerca do tema central deste trabalho.
This work is the result of a bibliographical research about Diophantus and the equations that take his name, the Diophantine equations. More specifically, we present the linear diophantine equations and some particular cases of the quadratic diophantine equations. We have also studied topics about number theory and continuous fractions, in order to facilitate the understanding of theorems and results that are related to the central theme of this work.
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Truman, Paul James. "Hopf-Galois module structure of some tamely ramified extensions." Thesis, University of Exeter, 2009. http://hdl.handle.net/10036/71817.
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We study the Hopf-Galois module structure of algebraic integers in some finite extensions of $ p $-adic fields and number fields which are at most tamely ramified. We show that if $ L/K $ is a finite unramified extension of $ p $-adic fields which is Hopf-Galois for some Hopf algebra $ H $ then the ring of algebraic integers $ \OL $ is a free module of rank one over the associated order $ \AH $. If $ H $ is a commutative Hopf algebra, we show that this conclusion remains valid in finite ramified extensions of $ p $-adic fields if $ p $ does not divide the degree of the extension. We prove analogous results for finite abelian Galois extensions of number fields, in particular showing that if $ L/K $ is a finite abelian domestic extension which is Hopf-Galois for some commutative Hopf algebra $ H $ then $ \OL $ is locally free over $ \AH $. We study in greater detail tamely ramified Galois extensions of number fields with Galois group isomorphic to $ C_{p} \times C_{p} $, where $ p $ is a prime number. Byott has enumerated and described all the Hopf-Galois structures admitted by such an extension. We apply the results above to show that $ \OL $ is locally free over $ \AH $ in all of the Hopf-Galois structures, and derive necessary and sufficient conditions for $ \OL $ to be globally free over $ \AH $ in each of the Hopf-Galois structures. In the case $ p = 2 $ we consider the implications of taking $ K = \Q $. In the case that $ p $ is an odd prime we compare the structure of $ \OL $ as a module over $ \AH $ in the various Hopf-Galois structures.
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Silva, Paulo Roberto da [UNESP]. "Tópicos de teoria dos números algébricos e aplicações em reticulados e equações diofantinas." Universidade Estadual Paulista (UNESP), 2015. http://hdl.handle.net/11449/134025.
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000856863.pdf: 777534 bytes, checksum: 909775f1499bec6c0e032d80b0a95b99 (MD5)Neste trabalho é feito um estudo sobre tópicos de Teoria dos Números Algébricos como extensão de corpos, decomposição de ideais primos, corpos quadráticos e ciclotômicos, número de classe e unidade. Nosso principal objetivo é apresentar uma aplicação dessa teoria na construção de reticulados e solução de equações diofantinas
This work presents a study of topics in algebraic number theory as eld extensions, prime ideal decomposition, quadratic and cyclotomic elds, class number and units. Our main goal is to present an application of this theory in the construction of lattices and solution of Diophantine equations
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Silva, Alexsandro BelÃm da. "FamÃlias infinitas de corpos quadrÃticos imaginÃrios." Universidade Federal do CearÃ, 2010. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=5664.
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FundaÃÃo de Amparo à Pesquisa do Estado do CearÃCoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior
Seja ℓ > 3 um primo Ãmpar. Sejam So, S+, S_ conjuntos finitos mutuamente disjuntos de primos racionais. Para qualquer nÃmero real suficientemente grande X > 0, baseando-nos em [16], damos neste trabalho, um limite inferior do nÃmero de corpos quadrÃticos imaginÃrios k que satisfazem as seguintes condiÃÃes: o discriminante de k à maior que -X o nÃmero de classe de k à nÃo divisÃvel por ℓ, todo q â So se ramifica, todo q â S+ se decompÃe e todo q â S_ à inerte em k, respectivamente.
Let ℓ > 3 be an odd prime. Let So, S+, S_ be mutually disjoint finite sets of rational primes. For any suficiently large real number X > 0, basing ourselves on [16], we give this paper a lower bound of the number of imaginary quadratic fields k which satisfy the following conditions: the discriminant of k is greater than -X, the class number ok is not divisible by ℓ, every q â So ramifies, every q â S+ splits and every q â S_ is inert in k, respectively.
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Santos, Jefferson Marques. "Altura e equidistribuição de pontos algébricos." Universidade Federal de Goiás, 2017. http://repositorio.bc.ufg.br/tede/handle/tede/7564.
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The concept of roots of a polynomial is quite simple but has several applications. This concept extends more generally to the case of "small" algebraic points sequences in a curve. This dissertation aims to estimate the size of algebraic numbers by means of Weil height. In addition to showing that they are distributed evenly around the unit circle, through Bilu Equidistribution Theorem.
O conceito de raízes de um polinômio é bastante simples mas possui várias aplicações. Este conceito se estende de forma mais geral para o caso de sequências de pontos algébricos “pequenos” em uma curva. Esta dissertação tem por objetivo estimar o tamanho de números algébricos por meio da altura de Weil. Além de mostrar que os mesmos se distribuem uniformemente em torno do círculo unitário, por meio do Teorema de Equidistribuição de Bilu.
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Rezola, Nolberto. "Unique Prime Factorization of Ideals in the Ring of Algebraic Integers of an Imaginary Quadratic Number Field." CSUSB ScholarWorks, 2015. https://scholarworks.lib.csusb.edu/etd/205.
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The ring of integers is a very interesting ring, it has the amazing property that each of its elements may be expressed uniquely, up to order, as a product of prime elements. Unfortunately, not every ring possesses this property for its elements. The work of mathematicians like Kummer and Dedekind lead to the study of a special type of ring, which we now call a Dedekind domain, where even though unique prime factorization of elements may fail, the ideals of a Dedekind domain still enjoy the property of unique prime factorization into a product of prime ideals, up to order of the factors. This thesis seeks to establish the unique prime ideal factorization of ideals in a special type of Dedekind domain: the ring of algebraic integers of an imaginary quadratic number field.
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Svanström, Fredrik. "Properties of a generalized Arnold’s discrete cat map." Thesis, Linnéuniversitetet, Institutionen för matematik (MA), 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-35209.
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After reviewing some properties of the two dimensional hyperbolic toral automorphism called Arnold's discrete cat map, including its generalizations with matrices having positive unit determinant, this thesis contains a definition of a novel cat map where the elements of the matrix are found in the sequence of Pell numbers. This mapping is therefore denoted as Pell's cat map. The main result of this thesis is a theorem determining the upper bound for the minimal period of Pell's cat map. From numerical results four conjectures regarding properties of Pell's cat map are also stated. A brief exposition of some applications of Arnold's discrete cat map is found in the last part of the thesis.
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Oliveira, Silvio Barbosa de. "As equações diofantinas lineares e o livro didático de matemática para o ensino médio." Pontifícia Universidade Católica de São Paulo, 2006. https://tede2.pucsp.br/handle/handle/11059.
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Previous issue date: 2006-05-24This work involves a qualitative study of how the theme of linear Diophantine equations is approached in mathematics textbooks for high school students. Using the methods associated with content analysis (Bardin, 1977), I search for references, in both explicit and implicit forms, to these equations in two different sets of high school mathematics textbooks, both of which had been approved in the last PNLEM (a national project for the assessment of high school textbooks). Although elementary number theory has been highlighted by researchers in mathematics education, such as Campbell and Zazkis (2002), as a subject apt for the introduction and development of fundamental mathematical ideas in compulsory education, the results of this investigation indicate that it receives little attention in the textbooks analysed
Neste trabalho apresento um estudo qualitativo sobre a abordagem dada pelo livro didático do Ensino Médio ao tema equações diofantinas lineares . Por meio de uma análise de conteúdo, segundo Bardin (1977), busquei o assunto em sua forma explícita e implícita em duas coleções de Matemática para o Ensino Médio, aprovadas no último PNLEM. Embora a Teoria Elementar dos Números venha sendo tratada por pesquisadores de Educação Matemática, como Campbell e Zazkis (2002), como assunto propício para a introdução e desenvolvimento de idéias matemáticas fundamentais, no Ensino Básico, os resultados desta investigação indicam a pouca exploração do assunto por parte das coleções analisadas
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Costa, Eduardo Sad da. "As Equações Diofantinas Lineares e o Professor de Matemática do Ensino Médio." Pontifícia Universidade Católica de São Paulo, 2007. https://tede2.pucsp.br/handle/handle/11124.
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Previous issue date: 2007-05-21Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
This work involves a qualitative study about whether and how mathematics High-School teachers work with their students the trouble-situations regarding linear Diophantine equations. The study was performed by means of analyzing semi-structured interviews applied on six mathematics teachers from the states of São Paulo and Minas Gerais, teaching at high-school level. The Numbers Elementary Theory has been treated by several researchers on Mathematical Education, as Campbell e Zazkis (2002), Resende (2007), as an adequate subject for the introduction and development of fundamental Mathematical ideas in High- School. However, the results of such investigation show that, although the interviewed teachers affirmed that they did work with problems of discreet mathematics that can be modeled through linear Diophantine equations, none of them seemed to work with their students using the knowledge of these equations properties in order to decide whether they have solution, and what these solutions would be
Neste trabalho apresento um estudo qualitativo sobre se, e como, professores de Matemática do Ensino Médio trabalham com seus alunos situações-problema que recaem em equações diofantinas lineares. O estudo foi feito por meio da análise de entrevistas semi-estruturadas realizadas com seis professores de Matemática dos estados de São Paulo e Minas Gerais que lecionam no Ensino Médio. A Teoria Elementar dos Números vem sendo tratada por diversos pesquisadores de Educação Matemática, como Campbell & Zazkis (2002), Resende (2007), como assunto propício para a introdução e desenvolvimento de idéias Matemáticas fundamentais no Ensino Básico. No entanto os resultados desta investigação indicam que embora os professores entrevistados afirmassem trabalhar com problemas de matemática discreta modeláveis via equação diofantina linear, nenhum deles deu indícios de trabalhar com seus alunos utilizando conhecimentos das propriedades dessas equações para decidir se as mesmas tem solução e quais seriam essas soluções
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Amorós, Carafí Laia. "Images of Galois representations and p-adic models of Shimura curves." Doctoral thesis, Universitat de Barcelona, 2016. http://hdl.handle.net/10803/471452.
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The Langlands program is a vast and unifying network of conjectures that connect the world of automorphic representations of reductive algebraic groups and the world of Galois representations.
These conjectures associate an automorphic representation of a reductive algebraic group to every n-dimensional representation of a Galois group, and the other way around: they attach a Galois representation to any automorphic representation of a reductive algebraic group. Moreover, these correspondences are done in such a way that the automorphic L-functions attached to the two objects coincide.
The theory of modular forms is a field of complex analysis whose main importance lies on its connections and applications to number theory. We will make use, on the one hand, of the arithmetic properties of modular forms to study certain Galois representations and their number theoretic meaning. On the other hand, we will use the geometric meaning of these complex analytic functions to study a natural generalization of modular curves. A modular curve is a geometric object that parametrizes isomorphism classes of elliptic curves together with some additional structure depending on some modular subgroup. The generalization that we will be interested in are the so called Shimura curves. We will be particularly interested in their p-adic models.
In this thesis, we treat two different topics, one in each side of the Langlands program. In the Galois representations' side, we are interested in Galois representations that take values in local Hecke algebras attached to modular forms over finite fields. In the automorphic forms' side, we are interested in Shimura curves: we develop some arithmetic results in definite quaternion algebras and give some results about Mumford curves covering p-adic Shimura curves.
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Sofo, Anthony. "Summing series using residues." Thesis, 1998. https://vuir.vu.edu.au/15695/.
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Kent, Richard Peabody. "Geometry and algebra of hyperbolic 3-manifolds." Thesis, 2006. http://hdl.handle.net/2152/2732.
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Jonck, Elizabeth. "The path partition number of a graph." Thesis, 2012. http://hdl.handle.net/10210/7118.
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Ph.D.The induced path number p(G) of a graph G is defined as the minimum number of subsets into which the vertex set V(G) of G can be partitioned such that each subset induces a path. In this thesis we determine the induced path number of a complete £-partite graph. We investigate the induced path number of products of complete graphs, of the complement of such products and of products of cycles. For a graph G, the linear vertex arboricity lva(G) is defined as the minimum number of subsets into which the vertex set of C can be partitioned so that each subset induces a linear forest. Since each path is a linear forest, Iva(G) p(G) for each graph C. A graph G is said to be uniquely rn-li near- forest- partition able if lva(C) = in and there is only one partition of V(G) into m subsets so that each subset induces a linear forest. Furthermore, a graph C is defined to be nz- Iva- saturated if Iva(G) < in and lva(C + e) > iii for each e E We construct graphs that are uniquely n2-linear-forest-partitionable and in-lva-saturated. We characterize those graphs that are uniquely m-linear-forest-partitionable and rn-lvasaturated. We also characterize the orders of uniquely in- path- partitionable disconnected, connected and rn-p-saturated graphs. We look at the influence of the addition or deletion of a vertex or an edge on the path partition number. If C is a graph such that p(G) = k and p(G - v) = k - 1 for every v E V(G), then we say that C is k-minus-critical. We prove that if C is a connected graph consisting of cyclic blocks Bi with p(B1 ) = b, for i = 1,2, ... ,n where ii > 2 and k bi - n+ 1, then C is k- minus- critical if and only if each of the blocks B1 is a bj- minus- critical graph.
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Smith, Harold Justin. "Fractions of Numerical Semigroups." 2010. http://trace.tennessee.edu/utk_graddiss/750.
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Let S and T be numerical semigroups and let k be a positive integer. We say that S is the quotient of T by k if an integer x belongs to S if and only if kx belongs to T. Given any integer k larger than 1 (resp., larger than 2), every numerical semigroup S is the quotient T/k of infinitely many symmetric (resp., pseudo-symmetric) numerical semigroups T by k. Related examples, probabilistic results, and applications to ring theory are shown.
Given an arbitrary positive integer k, it is not true in general that every numerical semigroup S is the quotient of infinitely many numerical semigroups of maximal embedding dimension by k. In fact, a numerical semigroup S is the quotient of infinitely many numerical semigroups of maximal embedding dimension by each positive integer k larger than 1 if and only if S is itself of maximal embedding dimension. Nevertheless, for each numerical semigroup S, for all sufficiently large positive integers k, S is the quotient of a numerical semigroup of maximal embedding dimension by k. Related results and examples are also given.
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Samson, Duncan Alistair. "An analysis of the influence of question design on pupils' approaches to number pattern generalisation tasks /." 2007. http://eprints.ru.ac.za/1121/.
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(9010904), Tyler R. Billingsley. "Effective Injectivity of Specialization Maps for Elliptic Surfaces." Thesis, 2020.
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This dissertation concerns two questions involving the injectivity of specialization homomorphisms for elliptic surfaces. We primarily focus on elliptic surfaces over the projective line defined over the rational numbers. The specialization theorem of Silverman proven in 1983 says that, for a fixed surface, all but finitely many specialization homomorphisms are injective. Given a subgroup of the group of rational sections with explicit generators, we thus ask the following.
Given some rational number, how can we effectively determine whether or not the associated specialization map is injective?
What is the set of rational numbers such that the corresponding specialization maps are injective?
The classical specialization theorem of Neron proves that there is a set S which differs from a Hilbert subset of the rational numbers by finitely many elements such that for each number in S the associated specialization map is injective. We expand this into an effective procedure that determines if some rational number is in S, yielding a partial answer to question 1. Computing the Hilbert set provides a partial answer to question 2, and we carry this out for some examples. We additionally expand an effective criterion of Gusic and Tadic to include elliptic surfaces with a rational 2-torsion curve.
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(11186268), Razan Taha. "p-adic Measures for Reciprocals of L-functions of Totally Real Number Fields." Thesis, 2021.
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We generalize the work of Gelbart, Miller, Pantchichkine, and Shahidi on constructing p-adic measures to the case of totally real fields K. This measure is the Mellin transform of the reciprocal of the p-adic L-function which interpolates the special values at negative integers of the Hecke L-function of K. To define this measure as a distribution, we study the non-constant terms in the Fourier expansion of a particular Eisenstein series of the Hilbert modular group of K. Proving the distribution is a measure requires studying the structure of the Iwasawa algebra.
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Irick, Brian C. "On the Irreducibility of the Cauchy-Mirimanoff Polynomials." 2010. http://trace.tennessee.edu/utk_graddiss/707.
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The Cauchy-Mirimanoff Polynomials are a class of polynomials that naturally arise in various classical studies of Fermat's Last Theorem. Originally conjectured to be irreducible over 100 years ago, the irreducibility of the Cauchy-Mirimanoff polynomials is still an open conjecture.
This dissertation takes a new approach to the study of the Cauchy-Mirimanoff Polynomials. The reciprocal transform of a self-reciprocal polynomial is defined, and the reciprocal transforms of the Cauchy-Mirimanoff Polynomials are found and studied. Particular attention is given to the Cauchy-Mirimanoff Polynomials with index three times a power of a prime, and it is shown that the Cauchy-Mirimanoff Polynomials of index three times a prime are irreducible.
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Amberker, B. B. "Large-Scale Integer And Polynomial Computations : Efficient Implementation And Applications." Thesis, 1996. http://etd.iisc.ernet.in/handle/2005/1678.
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(11204136), Chris Karl Neuffer. "Genera of Integer Representations and the Lyndon-Hochschild-Serre Spectral Sequence." Thesis, 2021.
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There has been in the past ten to fifteen years a surge of activity concerning the cohomology of semi-direct product groups of the form $\mathbb{Z}^{n}\rtimes$G with G finite. A problem first stated by Adem-Ge-Pan-Petrosyan asks for suitable conditions for the Lyndon-Hochschild-Serre Spectral Sequence associated to this group extension to collapse at second page of the Lyndon-Hochschild-Serre spectral sequence. In this thesis we use facts from integer representation theory to reduce this problem to only considering representatives from each genus of representations, and establish techniques for constructing new examples in which the spectral sequence collapses.
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Elzinga, Randall J. "The Minimum Witt Index of a Graph." Thesis, 2007. http://hdl.handle.net/1974/682.
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An independent set in a graph G is a set of pairwise nonadjacent vertices, and the maximum size, alpha(G), of an independent set in G is called the independence number.
Given a graph G and weight matrix A of G with entries from some field F, the maximum dimension of an A-isotropic subspace, known as the Witt index of A, is an upper bound on alpha(G). Since any weight matrix can be used, it is natural to seek the minimum upper bound on the independence number of G that can be achieved by a weight matrix. This
minimum, iota_F^*(G), is called the minimum Witt index of G over F, and the resulting bound, alpha(G)<= iota_F^*(G), is called
the isotropic bound.
When F is finite, the possible values of iota_F^*(G) are determined and the graphs that attain the isotropic bound are characterized. The characterization is given in terms of graph classes CC(n,t,c)
and CK(n,t,k) constructed from certain spanning subgraphs called C(n,t,c)-graphs and K(n,t,k)-graphs. Here t is the term rank
of the adjacency matrix of G.
When F=R, the isotropic bound is known as the Cvetkovi\'c bound. It is shown that it is sufficient to consider a finite number of
weight matrices A when determining iota_R^*(G) and that, in many cases, two weight values suffice. For example, if the vertex set of G can be covered by alpha(G) cliques, then G attains the Cvetkovi\'c bound with a weight matrix with two weight values.
Inequalities on alpha and iota_F^* resulting from graph operations such as sums, products, vertex deletion, and vertex identification are examined and, in some cases, conditions that imply equality are proved. The equalities imply that the problem of determining whether or not alpha(G)=iota_F^*(G) can be reduced to that of determining iota_F^*(H) for certain crucial graphs H found from G.Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2007-09-04 15:38:47.57
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(10717026), Heng Du. "Arithmetic Breuil-Kisin-Fargues modules and several topics in p-adic Hodge theory." Thesis, 2021.
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Let K be a discrete valuation field with perfect residue field, we study the functor from weakly admissible filtered (φ,N,GK)-modules over K to the isogeny category of Breuil- Kisin-Fargues GK-modules. This functor is the composition of a functor defined by Fargues-Fontaine from weakly admissible filtered (φ,N,GK)-modules to GK-equivariant modifications of vector bundles over the Fargues-Fontaine curve XFF , with the functor of Fargues-Scholze that between the category of admissible modifications of vector bundles over XFF and the isogeny category of Breuil-Kisin-Fargues modules. We characterize the essential image of this functor and give two applications of our result. First, we give a new way of viewing the p-adic monodromy theorem of p-adic Galois representations. Also we show our theory provides a universal theory that enable us to compare many integral p-adic Hodge theories at the Ainf level.
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(8782541), Dongming She. "Local Langlands Correpondence for the twisted exterior and symmetric square epsilon-factors of GL(N)." Thesis, 2020.
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In this paper, we prove the equality of the local arithmetic and analytic epsilon- and L-factors attached to the twisted exterior and symmetric square representations of GL(N). We will construct the twisted symmetric square local analytic gamma- and L-factor of GL(N) by applying Langlands-Shahidi method to odd GSpin groups. Then we reduce the problem to the stablity of local coefficients, and eventually prove the analytic stabitliy in this case by some analysis on the asymptotic behavior of certain partial Bessel functions.
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(10732485), Clinton W. Bradford. "Square Forms Factoring with Sieves." Thesis, 2021.
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Square Form Factoring is an O(N1/4) factoring algorithm developed by D. Shanks using certain properties of quadratic forms. Central to the original algorithm is an iterative search for a square form. We propose a new subexponential-time algorithm called SQUFOF2, based on ideas of D. Shanks and R. de Vogelaire, which replaces the iterative search with a sieve, similar to the Quadratic Sieve.