Dissertationen: „Algebraic fields“

1

Hartsell, Melanie Lynne. „Algebraic Number Fields“. Thesis, University of North Texas, 1991. https://digital.library.unt.edu/ark:/67531/metadc501201/.

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This thesis investigates various theorems on polynomials over the rationals, algebraic numbers, algebraic integers, and quadratic fields. The material selected in this study is more of a number theoretical aspect than that of an algebraic structural aspect. Therefore, the topics of divisibility, unique factorization, prime numbers, and the roots of certain polynomials have been chosen for primary consideration.

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2

Lötter, Ernest C. „On towers of function fields over finite fields /“. Link to the online version, 2007. http://hdl.handle.net/10019.1/1283.

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3

Ganz, Jürg Werner. „Algebraic complexity in finite fields /“. Zürich, 1994. http://e-collection.ethbib.ethz.ch/show?type=diss&nr=10867.

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4

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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5

Rovi, Carmen. „Algebraic Curves over Finite Fields“. Thesis, Linköping University, Department of Mathematics, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-56761.

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This thesis surveys the issue of finding rational points on algebraic curves over finite fields. Since Goppa's construction of algebraic geometric codes, there has been great interest in finding curves with many rational points. Here we explain the main tools for finding rational points on a curve over a nite eld and provide the necessary background on ring and field theory. Four different articles are analyzed, the first of these articles gives a complete set of table showing the numbers of rational points for curves with genus up to 50. The other articles provide interesting constructions of covering curves: covers by the Hemitian curve, Kummer extensions and Artin-Schreier extensions. With these articles the great difficulty of finding explicit equations for curves with many rational points is overcome. With the method given by Arnaldo García in [6] we have been able to nd examples that can be used to define the lower bounds for the corresponding entries in the tables given in http: //wins.uva.nl/~geer, which to the time of writing this Thesis appear as "no information available". In fact, as the curves found are maximal, these entries no longer need a bound, they can be given by a unique entry, since the exact value of N_q(g) is now known.

At the end of the thesis an outline of the construction of Goppa codes is given and the NXL and XNL codes are presented.

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6

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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7

Alm, Johan. „Universal algebraic structures on polyvector fields“. Doctoral thesis, Stockholms universitet, Matematiska institutionen, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-100775.

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The theory of operads is a conceptual framework that has become a kind of universal language, relating branches of topology and algebra. This thesis uses the operadic framework to study the derived algebraic properties of polyvector fields on manifolds.The thesis is divided into eight chapters. The first is an introduction to the thesis and the research field to which it belongs, while the second chapter surveys the basic mathematical results of the field.The third chapter is devoted to a novel construction of differential graded operads, generalizing an earlier construction due to Thomas Willwacher. The construction highlights and explains several categorical properties of differential graded algebras (of some kind) that come equipped with an action by a differential graded Lie algebra. In particular, the construction clarifies the deformation theory of such algebras and explains how such algebras can be twisted by Maurer-Cartan elements.The fourth chapter constructs an explicit strong homotopy deformation of polynomial polyvector fields on affine space, regarded as a two-colored noncommutative Gerstenhaber algebra. It also constructs an explicit strong homotopy quasi-isomorphism from this deformation to the canonical two-colored noncommmutative Gerstenhaber algebra of polydifferential operators on the affine space. This explicit construction generalizes Maxim Kontsevich's formality morphism.The main result of the fifth chapter is that the deformation of polyvector fields constructed in the fourth chapter is (generically) nontrivial and, in a sense, the unique such deformation. The proof is based on some cohomology computations involving Kontsevich's graph complex and related complexes. The chapter ends with an application of the results to properties of a derived version of the Duflo isomorphism.The sixth chapter develops a general mathematical framework for how and when an algebraic structure on the germs at the origin of a sheaf on Cartesian space can be "globalized" to a corresponding algebraic structure on the global sections over an arbitrary smooth manifold. The results are applied to the construction of the fourth chapter, and it is shown that the construction globalizes to polyvector fields and polydifferential operators on an arbitrary smooth manifold.The seventh chapter combines the relations to graph complexes, explained in chapter five, and the globalization theory of chapter six, to uncover a representation of the Grothendieck-Teichmüller group in terms of A-infinity morphisms between Poisson cohomology cochain complexes on a manifold.Chapter eight gives a simplified version of a construction of a family of Drinfel'd associators due to Carlo Rossi and Thomas Willwacher. Our simplified construction makes the connections to multiple zeta values more transparent--in particular, one obtains a fairly explicit family of evaluations on the algebra of formal multiple zeta values, and the chapter proves certain basic properties of this family of evaluations.

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8

Bode, Benjamin. „Knotted fields and real algebraic links“. Thesis, University of Bristol, 2018. http://hdl.handle.net/1983/8527a201-2fba-4e7e-8481-3df228051413.

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This doctoral thesis offers a new approach to the construction of arbitrarily knotted configurations in physical systems. We describe an algorithm that given any link $L$ finds a function $f:mathbb{C}^2tomathbb{C}$, a polynomial in complex variables $u$, $v$ and the conjugate $overline{v}$, whose vanishing set $f^{-1}(0)$ intersects the unit three-sphere $S^3$ in $L$. These functions can often be manipulated to satisfy the physical constraints of the system in question. The explicit construction allows us to make precise statements about properties of these functions, such as the polynomial degree and the number of critical points of $arg f$. Furthermore, we prove that for any link $L$ in an infinite family, namely the closures of squares of homogeneous braids, the polynomials can be altered into polynomials from $mathbb{R}^4$ to $mathbb{R}^2$ with an isolated singularity at the origin and $L$ as the link of that singularity. Links for which such polynomials exist are called real algebraic links and our explicit construction is a step towards their classification. We also study the crossing numbers of composite knots and relate them to crossing numbers of spatial graphs. The resulting connections are expected to lead to a new approach to the conjecture of the additivity of the crossing number.

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9

McCoy, Daisy Cox. „Irreducible elements in algebraic number fields“. Diss., Virginia Tech, 1990. http://hdl.handle.net/10919/39950.

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10

Alnaser, Ala' Jamil. „Waring's problem in algebraic number fields“. Diss., Manhattan, Kan. : Kansas State University, 2009. http://hdl.handle.net/2097/2207.

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11

Lotter, Ernest Christiaan. „On towers of function fields over finite fields“. Thesis, Stellenbosch : University of Stellenbosch, 2007. http://hdl.handle.net/10019.1/1283.

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Thesis (PhD (Mathematical Sciences))--University of Stellenbosch, 2007.
Explicit towers of algebraic function fields over finite fields are studied by considering their ramification behaviour and complete splitting. While the majority of towers in the literature are recursively defined by a single defining equation in variable separated form at each step, we consider towers which may have different defining equations at each step and with arbitrary defining polynomials. The ramification and completely splitting loci are analysed by directed graphs with irreducible polynomials as vertices. Algorithms are exhibited to construct these graphs in the case of n-step and -finite towers. These techniques are applied to find new tamely ramified n-step towers for 1 n 3. Various new tame towers are found, including a family of towers of cubic extensions for which numerical evidence suggests that it is asymptotically optimal over the finite field with p2 elements for each prime p 5. Families of wildly ramified Artin-Schreier towers over small finite fields which are candidates to be asymptotically good are also considered using our method.

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12

Buyruk, Dilek. „On Algebraic Function Fields With Class Number Three“. Phd thesis, METU, 2011. http://etd.lib.metu.edu.tr/upload/12613080/index.pdf.

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Let K/Fq be an algebraic function field with full constant field Fq and genus g. Then the divisor class number hK of K/Fq is the order of the quotient group, D0K /P(K), degree zero divisors of K over principal divisors of K. The classification of the function fields K with hK = 1 is done by MacRea, Leitzel, Madan and Queen and the classification of the extensions with class number two is done by Le Brigand. Determination of the necessary and the sufficient conditions for a function field to have class number three is done by H¨
ulya T¨
ore. Let k := Fq(T) be the rational function field over the finite field Fq with q elements. For a polynomial N &isin
Fq[T], we construct the Nth cyclotomic function field KN. Cyclotomic function fields were investigated by Carlitz, studied by Hayes, M. Rosen, M. Bilhan and many other mathematicians. Classification of cyclotomic function fields and subfields of cyclotomic function fields with class number one is done by Kida, Murabayashi, Ahn and Jung. Also the classification of function fields with genus one and classification of those with class number two is done by Ahn and Jung. In this thesis, we classified all algebraic function fields and subfields of cyclotomic function fields over finite fields with class number three.

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13

RoÌˆer, Andrew. „On vector fields on singular algebraic surfaces“. Thesis, University of Warwick, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.415254.

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14

Alaca, Saban Carleton University Dissertation Mathematics and Statistics. „P-Integral bases of algebraic number fields“. Ottawa, 1994.

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15

Han, Ilseop. „Tractibility of algebraic function fields in one variable over global fields /“. Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 1999. http://wwwlib.umi.com/cr/ucsd/fullcit?p9944223.

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16

Voloch, J. F. „Curves over finite fields“. Thesis, University of Cambridge, 1985. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.355283.

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17

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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18

Jogia, Danesh Michael Mathematics &amp Statistics Faculty of Science UNSW. „Algebraic aspects of integrability and reversibility in maps“. Publisher:University of New South Wales. Mathematics & Statistics, 2008. http://handle.unsw.edu.au/1959.4/40947.

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We study the cause of the signature over finite fields of integrability in two dimensional discrete dynamical systems by using theory from algebraic geometry. In particular the theory of elliptic curves is used to prove the major result of the thesis: that all birational maps that preserve an elliptic curve necessarily act on that elliptic curve as addition under the associated group law. Our result generalises special cases previously given in the literature. We apply this theorem to the specific cases when the ground fields are finite fields of prime order and the function field $mathbb{C}(t)$. In the former case, this yields an explanation of the aforementioned signature over finite fields of integrability. In the latter case we arrive at an analogue of the Arnol'd-Liouville theorem. Other results that are related to this approach to integrability are also proven and their consequences considered in examples. Of particular importance are two separate items: (i) we define a generalization of integrability called mixing and examine its relation to integrability; and (ii) we use the concept of rotation number to study differences and similarities between birational integrable maps that preserve the same foliation. The final chapter is given over to considering the existence of the signature of reversibility in higher (three and four) dimensional maps. A conjecture regarding the distribution of periodic orbits generated by such maps when considered over finite fields is given along with numerical evidence to support the conjecture.

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19

Kosick, Pamela. „Commutative semifields of odd order and planar Dembowski-Ostrom polynomials“. Access to citation, abstract and download form provided by ProQuest Information and Learning Company; downloadable PDF file, 104 p, 2010. http://proquest.umi.com/pqdweb?did=1992491941&sid=3&Fmt=2&clientId=8331&RQT=309&VName=PQD.

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20

Riccomagno, Eva M. „Algebraic geometry in experimental design and related fields“. Thesis, University of Warwick, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.263314.

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21

Berardini, Elena. „Algebraic geometry codes from surfaces over finite fields“. Thesis, Aix-Marseille, 2020. http://www.theses.fr/2020AIXM0170.

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Nous proposons, dans cette thèse, une étude théorique des codes géométriques algébriques construits à partir de surfaces définies sur les corps finis. Nous prouvons des bornes inférieures pour la distance minimale des codes sur des surfaces dont le diviseur canonique est soit nef soit anti-strictement nef et sur des surfaces sans courbes irréductibles de petit genre. Nous améliorons ces bornes inférieures dans le cas des surfaces dont le nombre de Picard arithmétique est égal à un, des surfaces sans courbes de petite auto-intersection et des surfaces fibrées. Ensuite, nous appliquons ces bornes aux surfaces plongées dans P3. Une attention particulière est accordée aux codes construits à partir des surfaces abéliennes. Dans ce contexte, nous donnons une borne générale sur la distance minimale et nous démontrons que cette estimation peut être améliorée en supposant que la surface abélienne ne contient pas de courbes absolument irréductibles de petit genre. Dans cette optique nous caractérisons toutes les surfaces abéliennes qui ne contiennent pas de courbes absolument irréductibles de genre inférieur ou égal à 2. Cette approche nous conduit naturellement à considérer les restrictions de Weil de courbes elliptiques et les surfaces abéliennes qui n'admettent pas de polarisation principale
In this thesis we provide a theoretical study of algebraic geometry codes from surfaces defined over finite fields. We prove lower bounds for the minimum distance of codes over surfaces whose canonical divisor is either nef or anti-strictly nef and over surfaces without irreducible curves of small genus. We sharpen these lower bounds for surfaces whose arithmetic Picard number equals one, surfaces without curves with small self-intersection and fibered surfaces. Then we apply these bounds to surfaces embedded in P3. A special attention is given to codes constructed from abelian surfaces. In this context we give a general bound on the minimum distance and we prove that this estimation can be sharpened under the assumption that the abelian surface does not contain absolutely irreducible curves of small genus. In this perspective we characterize all abelian surfaces which do not contain absolutely irreducible curves of genus up to 2. This approach naturally leads us to consider Weil restrictions of elliptic curves and abelian surfaces which do not admit a principal polarization

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22

Wuria, Muhammad Ameen Hussein. „Invariant algebraic surfaces in three dimensional vector fields“. Thesis, University of Plymouth, 2016. http://hdl.handle.net/10026.1/4417.

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This work is devoted to investigating the behaviour of invariant algebraic curves for the two dimensional Lotka-Volterra systems and examining almost a geometrical approach for finding invariant algebraic surfaces in three dimensional Lotka-Volterra systems. We consider the twenty three cases of invariant algebraic curves found in Ollagnier (2001) of the two dimensional Lotka-Volterra system in the complex plane and then we explain the geometric nature of each curve, especially at the critical points of the mentioned system. We also investigate the local integrability of two dimensional Lotka-Volterra systems at its critical points using the monodromy method which we extend to use the behaviour of some of the invariant algebraic curves mentioned above. Finally, we investigate invariant algebraic surfaces in three dimensional Lotka- Volterra systems by a geometrical method related with the intersection multiplicity of algebraic surfaces with the axes including the lines at infinity. We will classify both linear and quadratic invariant algebraic surfaces under some assumptions and commence a study of the cubic surfaces.

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23

Abbott, John Anthony. „On the factorization of polynomials over algebraic fields“. Thesis, University of Bath, 1988. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.234672.

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24

Kribs, Richard A. „Fields with minimal discriminants : an empirical study“. Virtual Press, 2005. http://liblink.bsu.edu/uhtbin/catkey/1314333.

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Modern algebra is a ubiquitous topic within mathematics and has many large and interconnected branches. Within algebra there is a branch known as field theory. Two measures of the complexity of a number field are its discriminant and signature. This paper correlates these two measures for fields of degree 4— 8 for which the discriminant is as small as possible. In studying the different ways minimal discriminants were located, an exponential relationship was noticed between the absolute value of the minimal discriminant and the signature. The actual absolute minimal discriminants and the exponential trends were then compared to and consistent with lower bounds for the absolute minimal discriminants previously estimated by Andrew Odlyzko. As the area becomes more saturated with computational findings, relating newly-discovered facts to previous estimates is useful for refining current estimates and generating new conjectures.
Department of Mathematical Sciences

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25

曾紹祺 und Shiu-kei Tsang. „A survey on Golomb's Conjectures and Costas Arrays“. Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1999. http://hub.hku.hk/bib/B42575345.

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26

Tsang, Shiu-kei. „A survey on Golomb's Conjectures and Costas Arrays“. Click to view the E-thesis via HKUTO, 1999. http://sunzi.lib.hku.hk/hkuto/record/B42575345.

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27

Boulanger, Nicolas. „Algebraic aspects of gravity and higher spsin gauge fields“. Doctoral thesis, Universite Libre de Bruxelles, 2003. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/211290.

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28

Panario, Rodriguez Daniel Nelson. „Combinatorial and algebraic aspects of polynomials over finite fields“. Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk2/tape16/PQDD_0016/NQ28297.pdf.

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29

Grimm, David [Verfasser]. „Sums of Squares in Algebraic Function Fields / David Grimm“. Konstanz : Bibliothek der Universität Konstanz, 2011. http://d-nb.info/1024034984/34.

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30

Cobbe, Alessandro. „Steinitz classes of tamely rami ed Galois extensions of algebraic number fields“. Doctoral thesis, Scuola Normale Superiore, 2009. http://hdl.handle.net/11384/85661.

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31

Minardi, John. „Iwasawa modules for [p-adic]-extensions of algebraic number fields /“. Thesis, Connect to this title online; UW restricted, 1986. http://hdl.handle.net/1773/5742.

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32

Pizzato, Marco. „Some Problems Concerning Polynomials over Finite Fields, or Algebraic Divertissements“. Doctoral thesis, Università degli studi di Trento, 2013. https://hdl.handle.net/11572/367913.

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In this thesis we consider some problems concerning polynomials over finite fields. The first topic is the action of some groups on irreducible polynomials. We describe orbits and stabilizers. Next, we consider transformations of irreducible polynomials by quadratic and cubic maps and study the irreducibility of the polynomials obtained. Finally, starting from PN functions and monomials, we generalize this concept, introducing k-PN monomials and classifying them for small values of k and for fields of order p, p^2 and p^4.

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33

Pizzato, Marco. „Some Problems Concerning Polynomials over Finite Fields, or Algebraic Divertissements“. Doctoral thesis, University of Trento, 2013. http://eprints-phd.biblio.unitn.it/1121/1/PizzatoPhDThesisbis.pdf.

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In this thesis we consider some problems concerning polynomials over finite fields. The first topic is the action of some groups on irreducible polynomials. We describe orbits and stabilizers. Next, we consider transformations of irreducible polynomials by quadratic and cubic maps and study the irreducibility of the polynomials obtained. Finally, starting from PN functions and monomials, we generalize this concept, introducing k-PN monomials and classifying them for small values of k and for fields of order p, p^2 and p^4.

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34

Beyronneau, Robert Lewis. „The solvability of polynomials by radicals: A search for unsolvable and solvable quintic examples“. CSUSB ScholarWorks, 2005. https://scholarworks.lib.csusb.edu/etd-project/2700.

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35

Ducet, Virgile. „Construction of algebraic curves with many rational points over finite fields“. Thesis, Aix-Marseille, 2013. http://www.theses.fr/2013AIXM4043/document.

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L'étude du nombre de points rationnels d'une courbe définie sur un corps fini se divise naturellement en deux cas : lorsque le genre est petit (typiquement g<=50), et lorsqu'il tend vers l'infini. Nous consacrons une partie de cette thèse à chacun de ces cas. Dans la première partie de notre étude nous expliquons comment calculer l'équation de n'importe quel revêtement abélien d'une courbe définie sur un corps fini. Nous utilisons pour cela la théorie explicite du corps de classe fournie par les extensions de Kummer et d'Artin-Schreier-Witt. Nous détaillons également un algorithme pour la recherche de bonnes courbes, dont l'implémentation fournit de nouveaux records de nombre de points sur les corps finis d'ordres 2 et 3. Nous étudions dans la seconde partie une formule de trace d'opérateurs de Hecke sur des formes modulaires quaternioniques, et montrons que les courbes de Shimura associées forment naturellement des suites récursives de courbes asymptotiquement optimales sur une extension quadratique du corps de base. Nous prouvons également qu'alors la contribution essentielle en points rationnels est fournie par les points supersinguliers
The study of the number of rational points of a curve defined over a finite field naturally falls into two cases: when the genus is small (typically g<=50), and when it tends to infinity. We devote one part of this thesis to each of these cases. In the first part of our study, we explain how to compute the equation of any abelian covering of a curve defined over a finite field. For this we use explicit class field theory provided by Kummer and Artin-Schreier-Witt extensions. We also detail an algorithm for the search of good curves, whose implementation provides new records of number of points over the finite fields of order 2 and 3. In the second part, we study a trace formula of Hecke operators on quaternionic modular forms, and we show that the associated Shimura curves of the form naturally form recursive sequences of asymptotically optimal curves over a quadratic extension of the base field. Moreover, we then prove that the essential contribution to the rational points is provided by supersingular points

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36

Aubertin, Bruce Lyndon. „Algebraic numbers and harmonic analysis in the p-series case“. Thesis, University of British Columbia, 1986. http://hdl.handle.net/2429/30282.

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For the case of compact groups G = Π∞ j=l Z(p)j which are direct products of countably many copies of a cyclic group of prime order p, links are established between the theories of uniqueness and spectral synthesis on the one hand, and the theory of algebraic numbers on the other, similar to the well-known results of Salem, Meyer et al on the circle. Let p ≥ 2 be a prime and let k{x⁻¹} denote the p-series field of formal Laurent series z = Σhj=₋∞ ajxj with coefficients in the field k = {0, 1,…, p-1} and the integer h arbitrary. Let L(z) = - ∞ if aj = 0 for all j; otherwise let L(z) be the largest index h for which ah ≠ 0. We examine compact sets of the form [Algebraic equation omitted] where θ ε k{x⁻¹}, L(θ) > 0, and I is a finite subset of k[x]. If θ is a Pisot or Salem element of k{x⁻¹}, then E(θ,I) is always a set of strong synthesis. In the case that θ is a Pisot element, more can be proved, including a version of Bochner's property leading to a sharper statement of synthesis, provided certain assumptions are made on I (e.g., I ⊃ {0,1,x,...,xL(θ)-1}). Let G be the compact subgroup of k{x⁻¹} given by G = {z: L(z) < 0}. Let θ ɛ k{x⁻¹}, L(θ) > 0, and suppose L(θ) > 1 if p = 3 and L(θ) > 2 if p = 2. Let I = {0,1,x,...,x²L(θ)-1}. Then E = θ⁻¹Ε(θ,I) is a perfect subset of G of Haar measure 0, and E is a set of uniqueness for G precisely when θ is a Pisot or Salem element. Some byways are explored along the way. The exact analogue of Rajchman's theorem on the circle, concerning the formal multiplication of series, is obtained; this is new, even for p = 2. Other examples are given of perfect sets of uniqueness, of sets satisfying the Herz criterion for synthesis, and sets of multiplicity, including a class of M-sets of measure 0 defined via Riesz products which are residual in G. In addition, a class of perfect M₀-sets of measure 0 is introduced with the purpose of settling a question left open by W.R. Wade and K. Yoneda, Uniqueness and quasi-measures on the group of integers of a p-series field, Proc. A.M.S. 84 (1982), 202-206. They showed that if S is a character series on G with the property that some subsequence {SpNj} of the pn-th partial sums is everywhere pointwise bounded on G, then S must be the zero series if SpNj → 0 a.e.. We obtain a strong complement to this result by establishing that series S on G exist for which Sn → 0 everywhere outside a perfect set of measure 0, and for which sup |SpN| becomes unbounded arbitrarily slowly.
Science, Faculty of
Mathematics, Department of
Graduate

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37

Aleem, Hosam Abdel. „An algebraic approach to modelling the regulation of gene expression“. Thesis, University of Manchester, 2011. https://www.research.manchester.ac.uk/portal/en/theses/an-algebraic-approach-to-modelling-the-regulation-of-gene-expression(d5d400b5-690e-4f32-9fd6-c80e4db455f3).html.

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Biotechnology is witnessing a remarkable growth evident both in the types of new products and in the innovative new processes developed. More efficient process design, optimisation and troubleshooting can be achieved through a better understanding of the underlying biological processes inside the cell; a key one of which is the regulation of gene expression. For engineers such understanding is attained through mathematical modelling, and the most commonly used models of gene expression regulation are those based on differential equations, as they give quantitative results. However, those results are undermined by several difficulties including the large number of parameters some of which, such as kinetic constants, are difficult to determine. This prompted the development of qualitative models, most notably Boolean models, based on the assumption that biological variables are binary in nature, e.g. a gene can be on or off and a chemical species present or absent. There are situations however, where different actions take place in the cell at different threshold values of the biological variables, and hence the binary assumption no longer holds.The purpose of this study was to develop a method for modelling gene regulatory functions where the variables can be thought of as taking more than two discrete values. A method was developed, where, with the appropriate assumptions the biological variables can be regarded as elements of an algebraic structure known as a finite field, in which case the regulatory function can be considered as a function on such a field.The formulation was adopted from electronic engineering, and leads to a polynomial known as the Reed-Muller expansion of the discrete function.The model was first developed for the more familiar binary case. It was given three different algebraic interpretations each enabling the study of a different biological problem, albeit related to gene regulation. The first interpretation is as a function on a Boolean algebra, but using the Exclusive OR (XOR) operation instead of the OR operation. The discriminating superiority of the XOR allows a more efficient determination of the gene regulatory function from the data, a problem known as reverse engineering.The second interpretation is as a polynomial on a finite field, where analogy with the Taylor series expansion of a real valued function allowed the coefficients of the expansion to be thought of as conveying sensitivity information. Furthermore a method was devised to detect mutation in the cell by regarding the problem as detecting a fault in a digital circuit.The third interpretation is as a transform on a discrete function space, which was demonstrated to be useful in synthetic biology design. The method was then extended to the multiple-valued case and demonstrated with modelling the gene regulation of a well known example system, the bacteriophage lambda.

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Sorolla, Bardají Jordi. „On the algebraic limit cycles of quadratic systems“. Doctoral thesis, Universitat Autònoma de Barcelona, 2005. http://hdl.handle.net/10803/3089.

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39

Lotter, Ernest Christiaan. „Explicit constructions of asymptotically good towers of function fields“. Thesis, Stellenbosch : Stellenbosch University, 2003. http://hdl.handle.net/10019.1/53417.

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Thesis (MSc)--Stellenbosch University, 2003
ENGLISH ABSTRACT: A tower of global function fields :F = (FI, F2' ... ) is an infinite tower of separable extensions of algebraic function fields of one variable such that the constituent function fields have the same (finite) field of constants and the genus of these tend to infinity. A study can be made of the asymptotic behaviour of the ratio of the number of places of degree one over the genus of FJWq as i tends to infinity. A tower is called asymptotically good if this limit is a positive number. The well-known Drinfeld- Vladut bound provides a general upper bound for this limit. In practise, asymptotically good towers are rare. While the first examples were non-explicit, we focus on explicit towers of function fields, that is towers where equations recursively defining the extensions Fi+d F; are known. It is known that if the field of constants of the tower has square cardinality, it is possible to attain the Drinfeld- Vladut upper bound for this limit, even in the explicit case. If the field of constants does not have square cardinality, it is unknown how close the limit of the tower can come to this upper bound. In this thesis, we will develop the theory required to construct and analyse the asymptotic behaviour of explicit towers of function fields. Various towers will be exhibited, and general families of explicit formulae for which the splitting behaviour and growth of the genus can be computed in a tower will be discussed. When the necessary theory has been developed, we will focus on the case of towers over fields of non-square cardinality and the open problem of how good the asymptotic behaviour of the tower can be under these circumstances.
AFRIKAANSE OPSOMMING: 'n Toring van globale funksieliggame F = (FI, F2' ... ) is 'n oneindige toring van skeibare uitbreidings van algebraïese funksieliggame van een veranderlike sodat die samestellende funksieliggame dieselfde (eindige) konstante liggaam het en die genus streef na oneindig. 'n Studie kan gemaak word van die asimptotiese gedrag van die verhouding van die aantal plekke van graad een gedeel deur die genus van Fi/F q soos i streef na oneindig. 'n Toring word asimptoties goed genoem as hierdie limiet 'n positiewe getal is. Die bekende Drinfeld- Vladut grens verskaf 'n algemene bogrens vir hierdie limiet. In praktyk is asimptoties goeie torings skaars. Terwyl die eerste voorbeelde nie eksplisiet was nie, fokus ons op eksplisiete torings, dit is torings waar die vergelykings wat rekursief die uitbreidings Fi+d F; bepaal bekend is. Dit is bekend dat as die kardinaliteit van die konstante liggaam van die toring 'n volkome vierkant is, dit moontlik is om die Drinfeld- Vladut bogrens vir die limiet te behaal, selfs in die eksplisiete geval. As die konstante liggaam nie 'n kwadratiese kardinaliteit het nie, is dit onbekend hoe naby die limiet van die toring aan hierdie bogrens kan kom. In hierdie tesis salons die teorie ontwikkel wat benodig word om eksplisiete torings van funksieliggame te konstrueer, en hulle asimptotiese gedrag te analiseer. Verskeie torings sal aangebied word en algemene families van eksplisiete formules waarvoor die splitsingsgedrag en groei van die genus in 'n toring bereken kan word, sal bespreek word. Wanneer die nodige teorie ontwikkel is, salons fokus op die geval van torings oor liggame waarvan die kardinaliteit nie 'n volkome vierkant is nie, en op die oop probleem aangaande hoe goed die asimptotiese gedrag van 'n toring onder hierdie omstandighede kan wees.

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Gould, Miles. „Coherence for categorified operadic theories“. Connect to e-thesis, 2008. http://theses.gla.ac.uk/689/.

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Thesis (Ph.D.) - University of Glasgow, 2008.
Ph.D. thesis submitted to the Faculty of Information and Mathematical Sciences, Department of Mathematics, University of Glasgow, 2008. Includes bibliographical references. Print version also available.

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Hinkelmann, Franziska Babette. „Algebraic theory for discrete models in systems biology“. Diss., Virginia Tech, 2011. http://hdl.handle.net/10919/28509.

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This dissertation develops algebraic theory for discrete models in systems biology. Many discrete model types can be translated into the framework of polynomial dynamical systems (PDS), that is, time- and state-discrete dynamical systems over a finite field where the transition function for each variable is given as a polynomial. This allows for using a range of theoretical and computational tools from computer algebra, which results in a powerful computational engine for model construction, parameter estimation, and analysis methods. Formal definitions and theorems for PDS and the concept of PDS as models of biological systems are introduced in section 1.3. Constructing a model for given time-course data is a challenging problem. Several methods for reverse-engineering, the process of inferring a model solely based on experimental data, are described briefly in section 1.3. If the underlying dependencies of the model components are known in addition to experimental data, inferring a "good" model amounts to parameter estimation. Chapter 2 describes a parameter estimation algorithm that infers a special class of polynomials, so called nested canalyzing functions. Models consisting of nested canalyzing functions have been shown to exhibit desirable biological properties, namely robustness and stability. The algorithm is based on the parametrization of nested canalyzing functions. To demonstrate the feasibility of the method, it is applied to the cell-cycle network of budding yeast. Several discrete model types, such as Boolean networks, logical models, and bounded Petri nets, can be translated into the framework of PDS. Section 3 describes how to translate agent-based models into polynomial dynamical systems. Chapter 4, 5, and 6 are concerned with analysis of complex models. Section 4 proposes a new method to identify steady states and limit cycles. The method relies on the fact that attractors correspond to the solutions of a system of polynomials over a finite field, a long-studied problem in algebraic geometry which can be efficiently solved by computing GrÃ¶bner bases. Section 5 introduces a bit-wise implementation of a GrÃ¶bner basis algorithm for Boolean polynomials. This implementation has been incorporated into the core engine of Macaulay2. Chapter 6 discusses bistability for Boolean models formulated as polynomial dynamical systems.
Ph. D.

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Hall-Seelig, Laura. „Asymptotically good towers of global function fields and bounds for the Ihara function“. Amherst, Mass. : University of Massachusetts Amherst, 2009. http://scholarworks.umass.edu/dissertations/AAI3372263/.

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43

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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Krapp, Lothar Sebastian [Verfasser]. „Algebraic and Model Theoretic Properties of O-minimal Exponential Fields / Lothar Sebastian Krapp“. Konstanz : KOPS Universität Konstanz, 2019. http://d-nb.info/1202012558/34.

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Mohamed, Mostafa Hosni [Verfasser]. „Algebraic decoding over finite and complex fields using reliability information / Mostafa Hosni Mohamed“. Ulm : Universität Ulm, 2018. http://d-nb.info/1150781041/34.

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46

Kurtaran, Ozbudak Elif. „Results On Some Authentication Codes“. Phd thesis, METU, 2009. http://etd.lib.metu.edu.tr/upload/2/12610350/index.pdf.

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In this thesis we study a class of authentication codes with secrecy. We obtain the maximum success probability of the impersonation attack and the maximum success probability of the substitution attack on these authentication codes with secrecy. Moreover we determine the level of secrecy provided by these authentication codes. Our methods are based on the theory of algebraic function fields over finite fields. We study a certain class of algebraic function fields over finite fields related to this class of authentication codes. We also determine the number of rational places of this class of algebraic function fields.

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Oi, Masao. „On ramifications of Artin-Schreier extensions of surfaces over algebraically closed fields of positive characteristic I“. 京都大学 (Kyoto University), 2014. http://hdl.handle.net/2433/193564.

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JSIAM Letters Vol. 6 (2014) p.33-36
Kyoto University (京都大学)
0048
新制・課程博士
博士(理学)
甲第18639号
理博第4018号
新制||理||1579(附属図書館)
31553
京都大学大学院理学研究科数学・数理解析専攻
(主査)教授池田保, 教授雪江明彦, 教授上田哲生
学位規則第4条第1項該当

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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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49

Jacobs, G. Tony. „Reduced Ideals and Periodic Sequences in Pure Cubic Fields“. Thesis, University of North Texas, 2015. https://digital.library.unt.edu/ark:/67531/metadc804842/.

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The “infrastructure” of quadratic fields is a body of theory developed by Dan Shanks, Richard Mollin and others, in which they relate “reduced ideals” in the rings and sub-rings of integers in quadratic fields with periodicity in continued fraction expansions of quadratic numbers. In this thesis, we develop cubic analogs for several infrastructure theorems. We work in the field K=Q(), where 3=m for some square-free integer m, not congruent to ±1, modulo 9. First, we generalize the definition of a reduced ideal so that it applies to K, or to any number field. Then we show that K has only finitely many reduced ideals, and provide an algorithm for listing them. Next, we define a sequence based on the number alpha that is periodic and corresponds to the finite set of reduced principal ideals in K. Using this rudimentary infrastructure, we are able to establish results about fundamental units and reduced ideals for some classes of pure cubic fields. We also introduce an application to Diophantine approximation, in which we present a 2-dimensional analog of the Lagrange value of a badly approximable number, and calculate some examples.

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Solanki, Nikesh. „Uniform companions for expansions of large differential fields“. Thesis, University of Manchester, 2014. https://www.research.manchester.ac.uk/portal/en/theses/uniform-companions-for-expansions-of-large-differential-fields(a565a0d0-24b5-40a6-a414-5ead1631bc8d).html.

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Dissertationen zum Thema „Algebraic fields“

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