Dissertations / Theses on the topic 'Number Theory and Field Theory'
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1
Rozario, Rebecca. "The Distribution of the Irreducibles in an Algebraic Number Field." Fogler Library, University of Maine, 2003. http://www.library.umaine.edu/theses/pdf/RozarioR2003.pdf.
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Rakotoniaina, Tahina. "Explicit class field theory for rational function fields." Thesis, Link to the online version, 2008. http://hdl.handle.net/10019/1993.
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Briggs, Matthew Edward. "An Introduction to the General Number Field Sieve." Thesis, Virginia Tech, 1998. http://hdl.handle.net/10919/36618.
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With the proliferation of computers into homes and businesses and the explosive growth rate of the Internet, the ability to conduct secure electronic communications and transactions has become an issue of vital concern. One of the most prominent systems for securing electronic information, known as RSA, relies upon the fact that it is computationally difficult to factor a "large" integer into its component prime integers. If an efficient algorithm is developed that can factor any arbitrarily large integer in a "reasonable" amount of time, the security value of the RSA system would be nullified. The General Number Field Sieve algorithm is the fastest known method for factoring large integers. Research and development of this algorithm within the past five years has facilitated factorizations of integers that were once speculated to require thousands of years of supercomputer time to accomplish. While this method has many unexplored features that merit further research, the complexity of the algorithm prevents almost anyone but an expert from investigating its behavior. We address this concern by first pulling together much of the background information necessary to understand the concepts that are central in the General Number Field Sieve. These concepts are woven together into a cohesive presentation that details each theory while clearly describing how a particular theory fits into the algorithm. Formal proofs from existing literature are recast and illuminated to clarify their inner-workings and the role they play in the whole process. We also present a complete, detailed example of a factorization achieved with the General Number Field Sieve in order to concretize the concepts that are outlined.Master of Science
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Hughes, Garry. "Distribution of additive functions in algebraic number fields." Title page, contents and summary only, 1987. http://web4.library.adelaide.edu.au/theses/09SM/09smh893.pdf.
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McLeman, Cameron William. "A Golod-Shafarevich Equality and p-Tower Groups." Diss., The University of Arizona, 2008. http://hdl.handle.net/10150/194026.
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Let K be a quadratic imaginary number field, let Kp^(infinity) the top of its p-class field tower for p an odd prime, and let G=Gal(Kp^(infinity)/K). It is known, due to a tremendous collection of work ranging from the principal results of class field theory to the famous Golod-Shafarevich inequality, that G is finite if the p-rank of the class group of K is 0 or 1, and is infinite if this rank is at least 3. This leaves the rank 2 case as the only remaining unsolved case. In this case, while finiteness is still a mystery, much is still known about G: It is a 2-generated, 2-related pro-p-group equipped with an involution that acts as the inverse modulo commutators, and is of one of three possible Zassenhaus types (defined in the paper). If such a group is finite, we will call it an interesting p-tower group. We further the knowledge on such groups by showing that one particular Zassenhaus type can occur as an interesting p-tower group only if the group has order at least p^24 (Proposition 8.1), and by proving a succinct cohomological condition (Proposition 4.7) for a p-tower group to be infinite. More generally, we prove a Golod-Shafarevich equality (Theorem 5.2), refining the famous Golod-Shafarevich inequality, and obtaining as a corollary a strict strengthening of previous Golod-Shafarevich inequalities (Corollary 5.5). Of interest is that this equality applies not only to finite p-groups but also to p-adic analytic pro-p-groups, a class of groups of particular relevance due to their prominent appearance in the Fontaine-Mazur conjecture. This refined version admits as a consequence that the sizes of the first few modular dimension subgroups of an interesting p-tower group G are completely determined by p and its Zassenhaus type, and we compute these sizes. As another application, we prove a new formula (Corollary 5.3) for the Fp-dimensions of the successive quotients of dimension subgroups of free pro-p-groups.
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Solomon, Y. J. "A critique of psychological theories of number development and a reorientation of the field." Thesis, Lancaster University, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.374154.
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Swanson, Colleen M. "Algebraic number fields and codes /." Connect to online version, 2006. http://ada.mtholyoke.edu/setr/websrc/pdfs/www/2006/172.pdf.
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Bamunoba, Alex Samuel. "Cyclotomic polynomials (in the parallel worlds of number theory)." Thesis, Stellenbosch : Stellenbosch University, 2011. http://hdl.handle.net/10019.1/17865.
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Thesis (MSc)--Stellenbosch University, 2011.ENGLISH ABSTRACT: It is well known that the ring of integers Z and the ring of polynomials A = Fr[T] over a finite field Fr have many properties in common. It is due to these properties that almost all the famous (multiplicative) number theoretic results over Z have analogues over A. In this thesis, we are devoted to utilising this analogy together with the theory of Carlitz modules. We do this to survey and compare the analogues of cyclotomic polynomials, the size of their coefficients and cyclotomic extensions over the rational function field k = Fr(T).
AFRIKAANSE OPSOMMING: Dit is bekend dat Z, die ring van heelgetalle en A = Fr[T], die ring van polinome oor ’n eindige liggaam baie eienskappe in gemeen het. Dit is as gevolg van hierdie eienskappe dat feitlik al die bekende multiplikative resultate wat vir Z geld, analoë in A het. In hierdie tesis, fokus ons op die gebruik van hierdie analogie saam met die teorie van die Carlitz module. Ons doen dit om ’n oorsig oor die analoë van die siklotomiese polinome, hul koëffisiënte, en siklotomiese uitbreidings oor die rasionele funksie veld k = Fr(T).
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Cipra, James Arthur. "Waring’s number in finite fields." Diss., Kansas State University, 2010. http://hdl.handle.net/2097/4152.
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Doctor of PhilosophyDepartment of Mathematics
Todd E. Cochrane
This thesis establishes bounds (primarily upper bounds) on Waring's number in finite fields. Let $p$ be a prime, $q=p^n$, $\mathbb F_q$ be the finite field in $q$ elements, $k$ be a positive integer with $k|(q-1)$ and $t= (q-1)/k$. Let $A_k$ denote the set of $k$-th powers in $\mathbb F_q$ and $A_k' = A_k \cap \mathbb F_p$. Waring's number $\gamma(k,q)$ is the smallest positive integer $s$ such that every element of $\mathbb F_q$ can be expressed as a sum of $s$ $k$-th powers. For prime fields $\mathbb F_p$ we prove that for any positive integer $r$ there is a constant $C(r)$ such that $\gamma(k,p) \le C(r) k^{1/r}$ provided that $\phi(t) \ge r$. We also obtain the lower bound $\gamma(k,p) \ge \frac {(t-1)}ek^{1/(t-1)}-t+1$ for $t$ prime. For general finite fields we establish the following upper bounds whenever $\gamma(k,q)$ exists: $$ \gamma(k,q)\le 7.3n\left\lceil\frac{(2k)^{1/n}}{|A_k^\prime|-1}\right\rceil\log(k), $$ $$ \gamma(k,q)\le8n \left\lceil{\frac{(k+1)^{1/n}-1}{|A^\prime_k|-1}}\right\rceil, $$ and $$ \gamma(k,q)\ll n(k+1)^{\frac{\log(4)}{n\log|\kp|}}\log\log(k). $$ We also establish the following versions of the Heilbronn conjectures for general finite fields. For any $\ve>0$ there is a constant, $c(\ve)$, such that if $|A^\prime_k|\ge4^{\frac{2}{\ve n}}$, then $\gamma(k,q)\le c(\varepsilon) k^{\varepsilon}$. Next, if $n\ge3$ and $\gamma(k,q)$ exists, then $\gamma(k,q)\le 10\sqrt{k+1}$. For $n=2$, we have $\gamma(k,p^2)\le16\sqrt{k+1}$.
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Blackhurst, Jonathan H. "Proven Cases of a Generalization of Serre's Conjecture." Diss., CLICK HERE for online access, 2006. http://contentdm.lib.byu.edu/ETD/image/etd1386.pdf.
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Kawaguchi, Yuki. "Near Miss abc-Triples in General Number Fields." Kyoto University, 2019. http://hdl.handle.net/2433/242576.
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Le, hung Bao Viet. "Modularity of some elliptic curves over totally real fields." Thesis, Harvard University, 2014. http://dissertations.umi.com/gsas.harvard:11464.
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In this thesis, we investigate modularity of elliptic curves over a general totally real number field, establishing a finiteness result for the set non-modular j-invariants. By analyzing quadratic points on some modular curves, we show that all elliptic curves over certain real quadratic fields are modular.Mathematics
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Simons, Lloyd D. "The structure of the Hilbert symbol for unramified extensions of 2-adic number fields /." Thesis, McGill University, 1986. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=74002.
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McCoy, Daisy Cox. "Irreducible elements in algebraic number fields." Diss., Virginia Tech, 1990. http://hdl.handle.net/10919/39950.
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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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Rada, Ion Kolster Manfred. "The Lichtenbaum conjecture at the prime 2 /." *McMaster only, 2002.
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Hymo, John A. "Problems involving relative integral bases for quartic number fields." Diss., Virginia Tech, 1990. http://hdl.handle.net/10919/39404.
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In this dissertation the question of whether or not a relative extension of number fields has a relative integral basis is considered. In Chapters 2 and 3 we use a criteria of Mann to determine when a cyclic quartic field or a pure quartic field has an integral basis over its quadratic subfield. In the final chapter we study the question: if the relative discriminant of an extension K / k is principal, where [K : k] = l such that l is an odd prime and k is either a quadratic or a normal quartic number field, does K / k have an integral basis?Ph. D.
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Milovic, Djordjo. "On the 16-rank of class groups of quadratic number fields." Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLS157/document.
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Nous démontrons deux nouveaux résultats de densité à propos du 16-rang des groupes des classes de corps de nombres quadratiques. Le premier des deux est que le groupe des classes de Q(sqrt{-p}) a un élément d'ordre 16 pour un quart des nombres premiers p qui sont de la forme a^2+c^4 avec c pair. Le deuxième est que le groupe des classes de Q(sqrt{-2p}) a un élément d'ordre 16 pour un huitième des nombres premiers p=-1 (mod 4). Ces résultats de densité sont intéressants pour plusieurs raisons. D'abord, ils sont les premiers résultats non triviaux de densité sur le 16-rang des groupes des classes dans une famille de corps de nombres quadratiques. Deuxièmement, ils prouvent une instance des conjectures de Cohen et Lenstra. Troisièmement, leurs preuves impliquent de nouvelles applications des cribles développés par Friedlander et Iwaniec. Quatrièmement, nous donnons une description explicite du sous-corps du corps de classes de Hilbert de degré 8 de Q(sqrt{-p}) lorsque p est un nombre premier de la forme a^2+c^4 avec c pair; l'absence d'une telle description explicite pour le sous-corps du corps de classes de Hilbert de degré 8 de Q(sqrt{d}) est le frein principal à l'amélioration des estimations de la densité des discriminants positifs d pour lesquels l'équation de Pell négative x^2-dy^2=-1 est résoluble. Dans le cas du deuxième résultat, nous donnons une description explicite d'un élément d'ordre 4 dans le groupe des classes de Q(sqrt{-2p}) et on calcule son symbole d'Artin dans le sous-corps du corps de classes de Hilbert de degré 4 de Q(sqrt{-2p}), généralisant ainsi un résultat de Leonard et Williams. Enfin, nous démontrons un très bon terme d'erreur pour une fonction de comptage des nombres premiers qui est liée au 16-rang du groupe des classes de Q(sqrt{-2p}), donnant ainsi des indications fortes contre une conjecture de Cohn et Lagarias que le 16-rang est contrôlé par un critère de type ChebotarevWe prove two new density results about 16-ranks of class groups of quadratic number fields. The first of the two is that the class group of Q(sqrt{-p}) has an element of order 16 for one-fourth of prime numbers p that are of the form a^2+c^4 with c even. The second is that the class group of Q(sqrt{-2p}) has an element of order 16 for one-eighth of prime numbers p=-1 (mod 4). These density results are interesting for several reasons. First, they are the first non-trivial density results about the 16-rank of class groups in a family of quadratic number fields. Second, they prove an instance of the Cohen-Lenstra conjectures. Third, both of their proofs involve new applications of powerful sieving techniques developed by Friedlander and Iwaniec. Fourth, we give an explicit description of the 8-Hilbert class field of Q(sqrt{-p}) whenever p is a prime number of the form a^2+c^4 with c even; the lack of such an explicit description for the 8-Hilbert class field of Q(sqrt{d}) is the main obstacle to improving the estimates for the density of positive discriminants d for which the negative Pell equation x^2-dy^2=-1 is solvable. In case of the second result, we give an explicit description of an element of order 4 in the class group of Q(sqrt{-2p}) and we compute its Artin symbol in the 4-Hilbert class field of Q(sqrt{-2p}), thereby generalizing a result of Leonard and Williams. Finally, we prove a power-saving error term for a prime-counting function related to the 16-rank of the class group of Q(sqrt{-2p}), thereby giving strong evidence against a conjecture of Cohn and Lagarias that the 16-rank is governed by a Chebotarev-type criterion
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Bauer-Price, Pia. "The Selberg Trace Formula for PSL(2, OK) for imaginary quadratic number fields K of arbitrary class number." Bonn : [s.n.], 1991. http://catalog.hathitrust.org/api/volumes/oclc/26531368.html.
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Banaszak, Grzegorz. "Algebraic K-theory of number fields and rings of integers and the Stickelberger ideal /." The Ohio State University, 1990. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487676261012829.
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Simmering-Best, Vanessa Renée. "Developing a magic number: the dynamic field theory reveals why visual working memory capacity estimates differ across tasks and development." Diss., University of Iowa, 2008. https://ir.uiowa.edu/etd/213.
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Many daily activities require the temporary maintenance and manipulation of information in working memory. A hallmark of this system is its limited capacity—research suggests that adults can actively maintain only about 3–4 items at once. A central question in working memory research is the source of such capacity limits. One approach to this question is to study the developmental origins of working memory capacity. Developmental research on working memory has documented a general increase in capacity throughout childhood and into adulthood. There have been few investigations, however, into the mechanisms behind these developmental changes, and proposals that have been put forth do not specify the processes underlying changes in capacity. One particularly puzzling finding that remains to be explained is an apparent decrease in visual working memory capacity over development, from an adult-like capacity of 3–4 items at 10 months to 1.5 items at 5 years. One probable source of this developmental discrepancy is that these two capacity estimates were derived from different tasks: preferential looking in infancy and change detection in later childhood. Although these tasks differ in many respects, existing theoretical explanations of the processes underlying performance in these tasks are underspecified, making it difficult to identify specify the origin of the regression in capacity over development.
To investigate the developmental discrepancy across tasks, I developed a unified model to capture how capacity-limits arise in preferential looking and change detection. I used this model to generate three specific behavioral predictions: 1) in preferential looking, children and adults should show higher capacity estimates than infants; 2) capacity estimates should be higher in preferential looking than in change detection when tested in the same individuals; and 3) although capacity estimates differ, performance should be correlated across tasks because both rely on the same underlying working memory system. In addition, I proposed a fourth prediction regarding the unified model: developmental changes in both tasks should be captured by a specific developmental mechanism, the Spatial Precision Hypothesis. Results from three experiments and model simulations confirmed all four predictions, providing strong support for a new Dynamic Field Theory of visual working memory.
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Hinkel, Dustin. "Constructing Simultaneous Diophantine Approximations Of Certain Cubic Numbers." Diss., The University of Arizona, 2014. http://hdl.handle.net/10150/338879.
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For K a cubic field with only one real embedding and α, β ϵ K, we show how to construct an increasing sequence {m_n} of positive integers and a subsequence {ψ_n} such that (for some constructible constants γ₁, γ₂ > 0): max{ǁm_nαǁ,ǁm_nβǁ} < [(γ₁)/(m_n^(¹/²))] and ǁψ_nαǁ < γ₂/[ψ_n^(¹/²) log ψ_n] for all n. As a consequence, we have ψ_nǁψ_nαǁǁψ_nβǁ < [(γ₁ γ₂)/(log ψ_n)] for all n, thus giving an effective proof of Littlewood's conjecture for the pair (α, β). Our proofs are elementary and use only standard results from algebraic number theory and the theory of continued fractions.
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Harper, John-Paul. "The class number one problem in function fields." Thesis, Stellenbosch : Stellenbosch University, 2003. http://hdl.handle.net/10019.1/53619.
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Thesis (MComm)--Stellenbosch University, 2003.ENGLISH ABSTRACT: In this dissertation I investigate the class number one problem in function fields. More precisely I give a survey of the current state of research into extensions of a rational function field over a finite field with principal ring of integers. I focus particularly on the quadratic case and throughout draw analogies and motivations from the classical number field situation. It was the "Prince of Mathematicians" C.F. Gauss who first undertook an in depth study of quadratic extensions of the rational numbers and the corresponding rings of integers. More recently however work has been done in the situation of function fields in which the arithmetic is very similar. I begin with an introduction into the arithmetic in function fields over a finite field and prove the analogies of many of the classical results. I then proceed to demonstrate how the algebra and arithmetic in function fields can be interpreted geometrically in terms of curves and introduce the associated geometric language. After presenting some conjectures, I proceed to give a survey of known results in the situation of quadratic function fields. I present also a few results of my own in this section. Lastly I state some recent results regarding arbitrary extensions of a rational function field with principal ring of integers and give some heuristic results regarding class groups in function fields.
AFRIKAANSE OPSOMMING: In hierdie tesis ondersoek ek die klasgetal een probleem in funksieliggame. Meer spesifiek ondersoek ek die huidige staat van navorsing aangaande uitbreidings van 'n rasionale funksieliggaam oor 'n eindige liggaam sodat die ring van heelgetalle 'n hoofidealgebied is. Ek kyk in besonder na die kwadratiese geval, en deurgaans verwys ek na die analoog in die klassieke getalleliggaam situasie. Dit was die beroemde wiskundige C.F. Gauss wat eerste kwadratiese uitbreidings van die rasionale getalle en die ooreenstemende ring van heelgetalle in diepte ondersoek het. Onlangs het wiskundiges hierdie probleme ook ondersoek in die situasie van funksieliggame oor 'n eindige liggaam waar die algebraïese struktuur baie soortgelyk is. Ek begin met 'n inleiding tot die rekenkunde in funksieliggame oor 'n eindige liggaam en bewys die analogie van baie van die klassieke resultate. Dan verduidelik ek hoe die algebra in funksieliggame geometries beskou kan word in terme van kurwes en gee 'n kort inleiding tot die geometriese taal. Nadat ek 'n paar vermoedes bespreek, gee ek 'n oorsig van wat alreeds vir quadratiese funksieliggame bewys is. In hierdie afdeling word 'n paar resultate van my eie ook bewys. Dan vermeld ek 'n paar resultate aangaande algemene uitbreidings van 'n rasionale funksieliggaam oor 'n eindige liggaam waar die van ring heelgetalle 'n hoofidealgebied is. Laastens verwys ek na 'n paar heurisitiese resultate aangaande klasgroepe in funksieliggame.
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Occhipinti, Thomas. "Mordell-Weil Groups of Large Rank in Towers." Diss., The University of Arizona, 2010. http://hdl.handle.net/10150/194213.
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Let k be the algebraic closure of the field with q elements. We build upon recent work of Ulmer and Berger to give examples of elliptic curves and higher dimensional abelian varieties over the field K=k(t) with the property that their ranks become arbitrarily large when dth roots of the variable t are adjoined to K for d varying across the integers relatively prime to q. We also give a first example of an elliptic curve whose rank under such extensions grows linearly in d, for those d prime to q.
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Byard, Kevin. "Qualified difference sets : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand." Massey University, 2009. http://hdl.handle.net/10179/1204.
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Qualified difference sets are a class of combinatorial configuration. The sets are related to the residue difference sets that were first discussed in detail in 1953 by Emma Lehmer. Qualified difference sets consist of a set of residues modulo an integer v and they possess attractive properties that suggest potential applications in areas such as image formation, signal processing and aperture synthesis. This thesis outlines the theory behind qualified difference sets and gives conditions for the existence and nonexistence of these sets in various cases. A special case of the qualified difference sets is the qualified residue difference sets. These consist of the set of nth power residues of certain types of prime. Necessary and sufficient conditions for the existence of qualified residue difference sets are derived and the precise conditions for the existence of these sets are given for n = 2, 4 and 6. Qualified residue difference sets are proved nonexistent for n = 8, 10, 12, 14 and 18. A generalisation of the qualified residue difference sets is introduced. These are the qualified difference sets composed of unions of cyclotomic classes. A cyclotomic class is defined for an integer power n and the results of an exhaustive computer search are presented for n = 4, 6, 8, 10 and 12. Two new families of qualified difference set were discovered in the case n = 8 and some isolated systems were discovered for n = 6, 10 and 12. An explanation of how qualified difference sets may be implemented in physical applications is given and potential applications are discussed.
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Nyqvist, Robert. "Algebraic Dynamical Systems, Analytical Results and Numerical Simulations." Doctoral thesis, Växjö : Växjö University Press, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:vxu:diva-1142.
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Freyhult, Lisa. "Aspects of Yang-Mills Theory : Solitons, Dualities and Spin Chains." Doctoral thesis, Uppsala : Acta Universitatis Upsaliensis : Univ.-bibl. [distributör], 2004. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-4498.
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Lavallee, Melisa Jean. "Dihedral quintic fields with a power basis." Thesis, University of British Columbia, 2008. http://hdl.handle.net/2429/2788.
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Cryptography is defined to be the practice and studying of hiding information
and is used in applications present today; examples include the security of ATM
cards and computer passwords ([34]). In order to transform information to make it
unreadable, one needs a series of algorithms. Many of these algorithms are based on
elliptic curves because they require fewer bits. To use such algorithms, one must find
the rational points on an elliptic curve. The study of Algebraic Number Theory, and
in particular, rare objects known as power bases, help determine what these rational
points are. With such broad applications, studying power bases is an interesting
topic with many research opportunities, one of which is given below.
There are many similarities between Cyclic and Dihedral fields of prime degree;
more specifically, the structure of their field discriminants is comparable. Since the
existence of power bases (i.e. monogenicity) is based upon finding solutions to the
index form equation - an equation dependant on field discriminants - does this imply
monogenic properties of such fields are also analogous?
For instance, in [14], Marie-Nicole Gras has shown there is only one monogenic
cyclic field of degree 5. Is there a similar result for dihedral fields of degree 5? The
purpose of this thesis is to show that there exist infinitely many monogenic dihedral
quintic fields and hence, not just one or finitely many. We do so by using a well-
known family of quintic polynomials with Galois group D₅. Thus, the main theorem
given in this thesis will confirm that monogenic properties between cyclic and dihedral
quintic fields are not always correlative.
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Angelakis, Athanasios. "Universal Adelic Groups for Imaginary Quadratic Number Fields and Elliptic Curves." Thesis, Bordeaux, 2015. http://www.theses.fr/2015BORD0180/document.
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Cette thèse traite de deux problèmes dont le lien n’est pas apparent (1) A` quoi ressemble l’abélianisé AK du groupe de Galois absolu d’un corps quadratique imaginaire K, comme groupe topologique? (2) A` quoi ressemble le groupe des points adéliques d’une courbe elliptique sur Q, comme groupe topologique? Pour la première question, la restriction au groupe de Galois abélianisé nous permet d’utiliser la théorie du corps de classes pour analyser AK . Les travaux précédents dans ce domaine, qui remontent à Kubota et Onabe, décrivent le dual de Pontryagin de AK en termes de familles in- finies d’invariants de Ulm à chaque premier p, très indirectement. Notre approche directe par théorie du corps de classes montre que AK con- tient un sous-groupe UK d’indice fini isomorphe au groupe des unités Oˆ* de la complétion profinie Oˆ de l’anneau des entiers de K, et décrit explicitement le groupe topologique UK , essentiellement indépendamment du corps quadratique imaginaire K. Plus précisément, pour tout corps quadratique imaginaire différent de Q(i) et Q(v-2),on a UK ∼= U = Zˆ2 × Y Z/nZ. (n=1) Le caractère exceptionnel de Q(v-2) n’apparaît pas dans les travaux de Kubota et Onabe, et leurs résultats doivent être corrigés sur ce point.Passer du sous-groupe universel UK à AK revient à un problème d’extension pour des groupes adéliques qu’il est possible de résoudre en passant à une extension de quotients convenables impliquant le quotient Zˆ-libre maximal UK/TK de UK . Par résoudre , nous entendons que, pour chaque K suffisamment petit pour permettre des calculs de groupe de classes explicites, nous obtenons un algorithme praticable décidant le comportement de cette extension. Si elle est totalement non-scindée, alors AK est isomorphe comme groupe topologique au groupe universel U . Réciproquement, si l’extension tensorisée par Zp se scinde pour un premier p impair, alors AK n’est pas isomorphe à U . Pour le premier 2, la situation est particulière, mais elle reste contrôlée grâce à l’abondance de résultats sur la 2-partie des groupes de classes de corps quadratiques.Nos expérimentations numériques ont permis de mieux comprendre la distribution des types d’isomorphismes de AK quand K varie, et nous conduisent à des conjectures telles que pour 100% des corps quadratiques imaginaires K de nombre de classes premier, AK est isomorphe au groupe universel U .Pour notre deuxième problème, qui apparaît implicitement dans [?, Section 9, Question 1] (dans le but de reconstruire le corps de nombres K à partir du groupe des points adéliques E(AK ) d’une courbe elliptique convenable sur K), nous pouvons appliquer les techniques usuelles pour les courbes elliptiques sur les corps de nombres, en suivant les mêmes étapes que pour déterminer la structure du groupe Oˆ* rencontré dans notre premier problème. Il s’avère que, dans le cas K = Q que nous traitons au Chapitre 4, le groupe des points adéliques de presque toutes les courbes elliptiques sur Q est isomorphe à un groupe universel E = R/Z × Zˆ × Y Z/nZ (n=1)de nature similaire au groupe U . Cette universalité du groupe des points adéliques des courbes elliptiques provient de la tendance qu’ont les représentations galoisiennes attachées (sur le groupe des points de torsion à valeurs dans Q) à être maximales. Pour K = Q, la représentation galoisienne est maximale si est seulement si la courbe E est une courbe de Serre, et Nathan Jones [?] a récemment démontré que presque toutes les courbes elliptiques sur Q sont de cette nature. En fait, l’universalité de E(AK ) suit d’hypothèses bien plus faibles, et il n’est pas facile de construire des familles de courbes elliptiques dont le groupe des points adéliques n’est pas universel. Nous donnons un tel exemple à la fin du Chapitre 4The present thesis focuses on two questions that are not obviously related. Namely,(1) What does the absolute abelian Galois group AK of an imaginary quadratic number field K look like, as a topological group?(2) What does the adelic point group of an elliptic curve over Q look like, as a topological group?For the first question, the focus on abelian Galois groups provides us with class field theory as a tool to analyze AK . The older work in this area, which goes back to Kubota and Onabe, provides a description of the Pontryagin dual of AK in terms of infinite families, at each prime p, of so called Ulm invariants and is very indirect. Our direct class field theoretic approach shows that AK contains a subgroup UK of finite index isomorphic to the unit group Oˆ∗ of the profinite completion Oˆ of the ring of integers of K, and provides a completely explicit description of the topological group UK that is almost independent of the imaginary quadratic field K. More precisely, for all imaginary quadratic number fields different from Q(i) and Q(√−2), we have UK ∼= U = Zˆ2 × Y Z/nZ. (n=1)The exceptional nature of Q(√−2) was missed by Kubota and Onabe, and their theorems need to be corrected in this respect.Passing from the ‘universal’ subgroup UK to AK amounts to a group extension problem for adelic groups that may be ‘solved’ by passing to a suitable quotient extension involving the maximal Zˆ-free quotientUK/TK of UK . By ‘solved’ we mean that for each K that is sufficiently small to allow explicit class group computations for K, we obtain a practical algorithm to compute the splitting behavior of the extension. In case the quotient extension is totally non-split, the conclusion is that AK is isomorphic as a topological group to the universal group U . Conversely, any splitting of the p-part of the quotient extension at an odd prime p leads to groups AK that are not isomorphic to U . For the prime 2, the situation is special, but our control of it is much greater as a result of the wealth of theorems on 2-parts of quadratic class groups.Based on numerical experimentation, we have gained a basic under- standing of the distribution of isomorphism types of AK for varying K, and this leads to challenging conjectures such as “100% of all imagi- nary quadratic fields of prime class number have AK isomorphic to the universal group U ”.In the case of our second question, which occurs implicitly in [?, Section 9, Question 1] with a view towards recovering a number field K from the adelic point group E(AK ) of a suitable elliptic curve over K, we can directly apply the standard tools for elliptic curves over number fields in a method that follows the lines of the determination of the structure of Oˆ∗ we encountered for our first question.It turns out that, for the case K = Q that is treated in Chapter 4, the adelic point group of ‘almost all’ elliptic curves over Q is isomorphic to a universal groupE = R/Z × Zˆ × Y Z/nZ (n=1)that is somewhat similar in nature to U . The reason for the universality of adelic point groups of elliptic curves lies in the tendency of elliptic curves to have Galois representations on their group of Q-valued torsion points that are very close to being maximal. For K = Q, maximality of the Galois representation of an elliptic curve E means that E is a so-called Serre-curve, and it has been proved recently by Nathan Jones [?] that ‘almost all’ elliptic curves over Q are of this nature. In fact, universality of E(AK ) requires much less than maximality of the Galois representation, and the result is that it actually requires some effort to construct families of elliptic curves with non-universal adelic point groups. We provide an example at the end of Chapter 4
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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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Hori, Kumiko, and Shigeo Yoshida. "Nonlocal memory effects of the electromotive force by fluid motion with helicity and two-dimensional periodicity." Taylor & Francis, 2008. http://hdl.handle.net/2237/13015.
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Idrees, Zunera. "Elliptic Curves Cryptography." Thesis, Linnéuniversitetet, Institutionen för datavetenskap, fysik och matematik, DFM, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-17544.
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In the thesis we study the elliptic curves and its use in cryptography. Elliptic curvesencompasses a vast area of mathematics. Elliptic curves have basics in group theory andnumber theory. The points on elliptic curve forms a group under the operation of addition.We study the structure of this group. We describe Hasse’s theorem to estimate the numberof points on the curve. We also discuss that the elliptic curve group may or may not becyclic over finite fields. Elliptic curves have applications in cryptography, we describe theapplication of elliptic curves for discrete logarithm problem and ElGamal cryptosystem.
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Smith, Benjamin Andrew. "Explicit endomorphisms and correspondences." University of Sydney, 2006. http://hdl.handle.net/2123/1066.
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Doctor of Philosophy (PhD)In this work, we investigate methods for computing explicitly with homomorphisms (and particularly endomorphisms) of Jacobian varieties of algebraic curves. Our principal tool is the theory of correspondences, in which homomorphisms of Jacobians are represented by divisors on products of curves. We give families of hyperelliptic curves of genus three, five, six, seven, ten and fifteen whose Jacobians have explicit isogenies (given in terms of correspondences) to other hyperelliptic Jacobians. We describe several families of hyperelliptic curves whose Jacobians have complex or real multiplication; we use correspondences to make the complex and real multiplication explicit, in the form of efficiently computable maps on ideal class representatives. These explicit endomorphisms may be used for efficient integer multiplication on hyperelliptic Jacobians, extending Gallant--Lambert--Vanstone fast multiplication techniques from elliptic curves to higher dimensional Jacobians. We then describe Richelot isogenies for curves of genus two; in contrast to classical treatments of these isogenies, we consider all the Richelot isogenies from a given Jacobian simultaneously. The inter-relationship of Richelot isogenies may be used to deduce information about the endomorphism ring structure of Jacobian surfaces; we conclude with a brief exploration of these techniques.
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Cunningham, John B. "Field Testing the Effects of Low Reynolds Number on the Power Performance of the Cal Poly Wind Power Research Center Small Wind Turbine." DigitalCommons@CalPoly, 2020. https://digitalcommons.calpoly.edu/theses/2249.
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This thesis report investigates the effects of low Reynolds number on the power performance of a 3.74 m diameter horizontal axis wind turbine. The small wind turbine was field tested at the Cal Poly Wind Power Research Center to acquire its coefficient of performance, Cp, vs. tip speed ratio, λ, characteristics. A description of both the wind turbine and test setup are provided. Data filtration and processing techniques were developed to ensure a valid method to analyze and characterize wind power measurements taken in a highly variable environment. The test results demonstrated a significant drop in the wind turbine’s power performance as Reynolds number decreased. From Re = 2.76E5 to Re = 1.14E5, the rotor’s Cp_max changed from 0.30 to 0.19. The Cp vs. λ results also displayed a clear change in shape with decreasing Reynolds number. The analysis highlights the influence of the rotor’s Cl /Cd characteristics on the Cp vs. λ curve’s Reynolds number dependency. By not accounting for the effects of varying Reynolds number below the critical value for a rotor operating at constant λ, the design of the rotor planform may overestimate the actual performance of the turbine in real-world conditions. This problem is more evident in distributed-scale wind turbines, compared to utility-scale ones, because of the significantly shorter chord lengths, and therefore increased wind speed range where this effect occurs. Lastly, the wind turbine’s future control method and annual energy production are evaluated using the test results.
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Tyler, Michael Peter. "On the birational section conjecture over function fields." Thesis, University of Exeter, 2017. http://hdl.handle.net/10871/31600.
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The birational variant of Grothendieck's section conjecture proposes a characterisation of the rational points of a curve over a finitely generated field over Q in terms of the sections of the absolute Galois group of its function field. While the p-adic version of the birational section conjecture has been proven by Jochen Koenigsmann, and improved upon by Florian Pop, the conjecture in its original form remains very much open. One hopes to deduce the birational section conjecture over number fields from the p-adic version by invoking a local-global principle, but if this is achieved the problem remains to deduce from this that the conjecture holds over all finitely generated fields over Q. This is the problem that we address in this thesis, using an approach which is inspired by a similar result by Mohamed Saïdi concerning the section conjecture for étale fundamental groups. We prove a conditional result which says that, under the condition of finiteness of certain Shafarevich-Tate groups, the birational section conjecture holds over finitely generated fields over Q if it holds over number fields.
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Ferreira, Luan Alberto. "Teoria de corpos de classe e aplicações." Universidade de São Paulo, 2012. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-05122012-102820/.
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Neste projeto, propomos estudar a chamada \"Teoria de Corpos de Classe,\" que oferece uma descrição simples das extensões abelianas de corpos locais e globais, bem como algumas de suas aplicações, como os teoremas de Kronecker-Weber e Scholz-ReichardtIn this work, we study the so called \"Class Field Theory\", which give us a simple description of the abelian extension of local and global elds. We also study some applications, like the Kronecker-Weber and Scholz-Reichardt theorems
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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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Souza, Vera Lúcia Graciani de. "Fatoração de inteiros e grupos sobre conicas." [s.n.], 2009. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306147.
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Orientador: Martinho da Costa AraujoDissertação (mestrado profissional) - Universidade Estadual de Campinas. Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Este trabalho tem por objetivo fatorar número inteiro utilizando pontos racionais sobre o círculo unitário. Igualmente pretende determinar alguns grupos sobre cônicas. A pesquisa inicia com os conceitos básicos de Álgebra e Teoria dos Números, que fundamentam que o conjunto de pontos racionais sobre o círculo unitário tem uma estrutura de grupo. Desse conjunto é possível estender a idéia de grupo de pontos racionais sobre o círculo para pontos racionais sobre cônicas. Para encontrar os pontos racionais sobre o círculo foi usada uma parametrização do círculo por funções trigonométricas. Para cada ponto sobre o círculo unitário está associado um ângulo com o eixo positivo das abscissas, portanto adicionar pontos sobre o círculo equivale adicionar seus ângulos correspondentes. Com a operação "adição" de pontos sobre o círculo é possível definir uma estrutura de grupo que é utilizada para fatorar números inteiros. Para a cônica, a operação "adição" é determinada algebricamente ao calcular o coeficiente angular da reta que passa por dois pontos dados e o elemento neutro dessa cônica, também justificada geometricamente. No trabalho foram determinados os grupos de pontos racionais sobre cônicas e demonstrado alguns resultados sobre esses grupos usando os resíduos quadráticos e finalizando com a dedução de alguns resultados sobre a soma das coordenadas dos pontos sobre uma cônica.
Abstract: The objective of this paper is to factorize integer number using rational points on the unitary circle. Also, it intends to determinate some groups on the conics. The research begins with the basic concepts of Algebra and Number Theory ensuring that the rational points set on the unitary circle has a structure of group. From this set is possible to extend the idea of rational points on the circle toward rational points on conics. In order to find the rational points on the circle a parametrization by trigonometric function on it was used. For each point on the unitary circle it is associated an angle with abscissa positive axis, therefore adding points on the circle equals to add its corresponding angles. With the operation of "addition" points on the circle it is possible to define a group structure that is used to factorize integer numbers. For the conic, the "addition" operation is algebraically determinated when the angle coeficient of the line is calculated that joins two given points and the neutral element of that conic, which is geometrically justified. In the research the rational points groups on the conics were determined, and some result on these groups using quadratic residues were demonstrated, and it was finalized with the deduction of some results concerning the coordinates sum of points on a conics.
Mestrado
Mestre em Matemática
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Lezowski, Pierre. "Questions d’euclidianité." Thesis, Bordeaux 1, 2012. http://www.theses.fr/2012BOR14642/document.
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Nous étudions l'euclidianité des corps de nombres pour la norme et quelques unes de ses généralisations. Nous donnons en particulier un algorithme qui calcule le minimum euclidien d'un corps de nombres de signature quelconque. Cela nous permet de prouver que de nombreux corps sont euclidiens ou non pour la norme. Ensuite, nous appliquons cet algorithme à l'étude des classes euclidiennes pour la norme, ce qui permet d'obtenir de nouveaux exemples de corps de nombres avec une classe euclidienne non principale. Par ailleurs, nous déterminons tous les corps cubiques purs avec une classe euclidienne pour la norme. Enfin, nous nous intéressons aux corps de quaternions euclidiens. Après avoir énoncé les propriétés de base, nous étudions quelques cas particuliers. Nous donnons notamment la liste complète des corps de quaternions euclidiens et totalement définis sur un corps de nombres de degré au plus deuxWe study norm-Euclideanity of number fields and some of its generalizations. In particular, we provide an algorithm to compute the Euclidean minimum of a number field of any signature. This allows us to study the norm-Euclideanity of many number fields. Then, we extend this algorithm to deal with norm-Euclidean classes and we obtain new examples of number fields with a non-principal norm-Euclidean class. Besides, we describe the complete list of pure cubic number fields admitting a norm-Euclidean class. Finally, we study the Euclidean property in quaternion fields. First, we establish its basic properties, then we study some examples. We provide the complete list of Euclidean quaternion fields, which are totally definite over a number field with degree at most two
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Gusmão, Ítalo Moraes de Melo. "Números p-ádicos." Universidade Federal da Paraíba, 2015. http://tede.biblioteca.ufpb.br:8080/handle/tede/9337.
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We introduce and de ne the p-adics integer numbers as a result of a search for solutions, for a congruences system that derives from a variable polynomial equation with rational coe cients. We evidence that the p-adic integers set is strictly larger than the integers. We present a criterion so that a rational that holds a correspondent in a p-adic integers set. We search for the possibility to represent irrational and complex numbers as p-adics integers. Algebraically, the p-adic integers set will be an integral domain and, from this, we search for the construction of p-adic integers quotient eld so that shall form the p-adic rationals eld, from a purely algebraically point of view. In the second part, we will expose the bases for the construction of a norm that's di erent from the usual, establishing so a new metric in the rational numbers set and the construction of a non-archimedian eld.
Apresentamos e de nimos os números inteiros p-ádicos como o resultado de uma busca por soluções, para um sistema de congruências, que parte de uma equação polinomial de uma variável, com coe cientes racionais. Constatamos que o conjunto dos inteiros p-ádicos é estritamente maior que os inteiros. Mostramos um critério para que um racional possua um correspondente num conjunto de inteiros p-ádicos. Buscamos a possibilidade de representarmos números irracionais e números complexos como inteiros p-ádicos. Algebricamente, o conjunto dos inteiros p-ádicos será um domínio de integridade e, partindo disto, buscamos a construção de um corpo de frações dos inteiros p-ádicos, que formarão, assim, o corpo dos racionais p-ádicos, de um ponto de vista puramente algébrico. Na segunda parte, vamos expor os fundamentos para a construção de uma norma diferente da habitual, estabelecendo assim uma nova métrica, no conjunto dos números racionais, e a construção de um corpo não-arquimediano.
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Villanueva, Gutiérrez José Ibrahim. "Sur quelques questions en théorie d'Iwasawa." Thesis, Bordeaux, 2017. http://www.theses.fr/2017BORD0637/document.
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Ce travail de thèse comporte l'étude des invariants logarithmiques le long des $l^{d}$-extensions et se compose de trois parties étroitement reliées. La première partie est un compendium sur les divers approches à l'arithmétique algorithmique, c'est à dire l'étude générale des invariants logarithmiques. En particulier on y présente quatre définitions équivalentes du groupe de classes logarithmiques et on y démontre leur équivalence. On donne aussi une preuve alternative d'un théorème d'Iwasawa de type logarithmique. La deuxième partie s'interprète comme un addendum historique sur l'étude du groupe de classes logarithmiques le long des $l$-extensions. On démontre que sous la conjecture de Gross-Kuz'min la théorie d'Iwasawa peut être bien employée pour l'étude du cas non-cyclotomique. Ainsi, on démontre des relations entre les invariants $mu$ et $lambda$ correspondant au $ell$-groupe de classes avec les invariants $ilde{mu}$ et $ilde{lambda}$ attachés aux groupes de classes logarithmiques. La troisième partie comporte l'étude du module d'Iwasawa logarithmique pour des $l^{d}$-extensions, c'est à dire du groupe de Galois $X=Gal(L_{d}/K_{d})$ de la $ell$-extension maximale abélienne logarithmiquement non-ramifiée du compositum $K_{d}$ des différentes $l$-extensions d'un corps de nombres $K$. On démontre sous la conjecture de Gross-Kuz'min, de façon analogue au cas classique, que $X$ est bien un module noethérien et de torsion sous l'algèbre d'Iwasawa de $K_{d}$. Ainsi, on déduit des relations entre les invariants logarithmiques $ilde{mu}$ et $ilde{lambda}$ des $l$-extensions de $K$ qui satisfont une hypothèse de décompositionThis work is concerned with the study of logarithmic invariants on $l^{d}$-extensions and is subdivided in three pieces, which are closely related to each other. The first part is a compendium of the different approaches to logarithmic arithmetic, that is the study of the logarithmic invariants. In particular we show the equivalence between the four definitions of the logarithmic class group existing in the literature. Also we give an alternative proof of an Iwasawa logarithmic result. The second part can be thought as an historic addendum on the study of the logarithmic class group over $l$-extensions. Assuming the Gross-Kuz'min conjecture we show that the logarithmic class group can be studied in the Iwasawa setting for non-cyclotomic extensions. We also give relations between the classical $mu$ and $lambda$ invariants and the logarithmic invariants $ilde{mu}$ and $ilde{lambda}$ attached to the logarithmic class groups. The third part studies the properties of the Iwasawa logarithmic module for $l^{d}$-extensions, that is the Galois group $X=Gal(L_{d}/K_{d})$ of the maximal abelian $ell$-extension logarithmically unramified of the compositum $K_{d}$ of the different $l$-extensions of a number field $K$. Assuming the Gross-Kuz'min conjecture we show that $X$ is a noetherian torsion module over the Iwasawa algebra of $K_{d}$. We also deduce relations between the logarithmic invariants $ilde{mu}$ and $ilde{lambda}$ of the $l$-extensions of $K$ which satisfy a splitting condition
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Guo, Charng Rang. "On analytic number theory." Thesis, University of Oxford, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.357394.
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Harbour, Daniel 1975. "Elements of number theory." Thesis, Massachusetts Institute of Technology, 2003. http://hdl.handle.net/1721.1/17581.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 2003.Includes bibliographical references (leaves 202-205).
The dissertation argues for the necessity of a morphosemantic theory of number, that is, a theory of number serviceable both to semantics and morphology. The basis for this position, and the empirical core of the dissertation, is the relationship between semantically based noun classification and agreement in Kiowa, an indigenous, endangered language of Oklahoma. The central claim is that Universal Grammar provides three number features, concerned with unithood, existence of homogeneous subsets, and properties of those subsets. The features are used to analyze a wide variety of data. Semantic topics include the difference between granular and non-granular mass nouns, collective, non-collective and distributive plurals, and cardinality. Syntactic topics include the structure of DP, noun marking, agreement and suppletion. Morphological topics include the inventory of morphological operations, the featural basis of complex syncretisms, the difference between agreement and suppletion, whether features are privative or binary, and the nature of the Kiowa/Tanoan inverse. Keywords: Kiowa-Tanoan, number, morphology, semantics, agreement, suppletion, inverse, noun class, singular, dual, plural, features, binary, privative.
by Daniel Harbour.
Ph.D.
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Ali, T. "String theory and conformal field theory." Thesis, University of Cambridge, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.595446.
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In this thesis we consider some aspects of two dimensional Conformal Field Theory and their connection to String Theory. We have also studied some aspects of supersymmetry of M-Theory on Ricci-flat seven manifolds with 4-form fluxes. We concentrate mainly on certain supersymmetric extensions of the coset models due to Goddard, Kent and Olive (GKO). These models are known as the Kazama-Suzuki (KS) models and they are characterized by their N = 2 superconformal symmetry. Two series of the KS models enjoy a duality called level-rank duality which can be described roughly as duality between the dimension of the target space and the level of coset. We believe that the path-integral approach is the closest in spirit to string theory. Therefore, we formulate the level-rank duality of KS models in the path-integral approach by using the realization of GKO cosets as gauged Wess-Zumino-Novikov-Witten (gauged-WZNW) models. We derive, for a class of KS models, an expression for the partition function which is symmetric in the parameters of the level-rank duality. We compute the central charge of the models from this expression which matches that of Kazama and Suzuki in the operator approach. We then work out the target space metric and the dilation of the gauged-WZNW model based on the GKO coset SU(3)/(SU(2) x U(1)). It turns out to be quite a complicated metric with a non-trivial dilation. We verify, as a consistency check, that they satisfy the appropriate string theory effective equations of motion. We then argue that this background can arise naturally in type II string theory compactified down to AdS3 space. We then turn to Eleven Dimensional Supergravity which is the low energy limit of M-theory. We adopt a metric ansatz which is a warped product of four dimensional Minkowski space and a (non-compact) seven manifold with 4-form fluxes turned on it. We derive the condition for unbroken supersymmetry with fluxes and non-trivial warp-factor. We show that the same condition implies that the seven manifold is conformal to a Ricci-flat manifold. We also point out the limitation of some naive ansatze about the structure of the Killing spinor. At this stage we are unable to give an explicit solution to the supersymmetry condition.
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Buchanan, Dan Matthews. "Analytic Number Theory and the Prime Number Theorem." Youngstown State University / OhioLINK, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1525451327211365.
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Alkauskas, Giedrius. "Several problems from number theory." Doctoral thesis, Lithuanian Academic Libraries Network (LABT), 2009. http://vddb.library.lt/obj/LT-eLABa-0001:E.02~2009~D_20091008_155751-23469.
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Doctoral thesis is devoted to investigation of three problems. The first one deals with the analytic properties and representation in closed or almost closed form of the Stieltjes tranform of the Minkowski question mark function (that is, the generating function of moments, the so called dyadic period function). The main result claims that the dyadic period function can be represented as a convergent series of rational functions with rational coefficients. In the proof the techniques from complex dynamics, analytic theory of continued fractions, the theory of several complex variables are being used. The second problem is dealing with functional equations associated with norm and other forms. It is shown that these functional equations sometimes have other solutions apart from the trivial ones. Finally, we present a new proof of Fermat’s little theorem.Daktaro disertacijoje sprendžiami trys uždaviniai. Pirmasis nagrinėja Minkovskio “klaustuko” funkcijos Stieltjes’o transformacijos (tai yra, šios funkcijos momentų generuojančios funkcijos, taip vadinamosios diadinės periodo funkcijos), analizines savybes ir jos išraišką uždara ar beveik uždara forma. Pagrindinis rezultatas teigia, kad diadinę periodo funkciją galima išreikšti racionaliųjų funkcijų su racionaliaisiais koeficientais konverguojančia eilute. Įrodyme naudojama kompleksinės dinamikos, analizinės grandininių trupmenų teorijos, kelių kompleksinių kintamųjų funkcijų teorijos technika. Antrasis uždavinys nagrinėja funkcines lygtis, susietas su norminėmis ir kitomis kelių kintamųjų formomis. Yra parodoma, kad šios funkcinės lygtys kartais turi kitų, netrivialiųjų sprendinių. Galiausiai, yra pateikiamas naujas mažosios Fermat teoremos įrodymas.
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Dyke, Steven Douglas. "Topics in analytic number theory." Thesis, University of Oxford, 1992. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.334817.
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Rowe, Paul Michael Dominic. "Contributions to metric number theory." Thesis, Royal Holloway, University of London, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.408263.
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Amirkhanyan, Gagik M. "Problems in combinatorial number theory." Diss., Georgia Institute of Technology, 2014. http://hdl.handle.net/1853/51865.
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The dissertation consists of two parts. The first part is devoted to results in Discrepancy Theory. We consider geometric discrepancy in higher dimensions (d > 2) and obtain estimates in Exponential Orlicz Spaces. We establish a series of dichotomy-type results for the discrepancy function which state that if the L¹ norm of the discrepancy function is too small (smaller than the conjectural bound), then the discrepancy function has to be very large in some other function space.The second part of the thesis is devoted to results in Additive Combinatorics. For a set with small doubling an order-preserving Freiman 2-isomorphism is constructed which maps the set to a dense subset of an interval. We also present several applications.
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Powell, Kevin James. "Topics in Analytic Number Theory." BYU ScholarsArchive, 2009. https://scholarsarchive.byu.edu/etd/2084.
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The thesis is in two parts. The first part is the paper “The Distribution of k-free integers” that my advisor, Dr. Roger Baker, and I submitted in February 2009. The reader will note that I have inserted additional commentary and explanations which appear in smaller text. Dr. Baker and I improved the asymptotic formula for the number of k-free integers less than x by taking advantage of exponential sum techniques developed since the 1980's. Both of us made substantial contributions to the paper. I discovered the exponent in the error term for the cases k=3,4, and worked the case k=3 completely. Dr. Baker corrected my work for k=4 and proved the result for k=5. He then generalized our work into the paper as it now stands. We also discussed and both contributed to parts of section 3 on bounds for exponential sums. The second part represents my own work guided by my advisor. I study the zeros of derivatives of Dirichlet L-functions. The first theorem gives an analog for a result of Speiser on the zeros of ζ'(s). He proved that RH is equivalent to the hypothesis that ζ'(s) has no zeros with real part strictly between 0 and ½. The last two theorems discuss zero-free regions to the left and right for L^{(k)}(s,χ).