1,720,959 research outputs found
On the Computation and Composition of Belyiˇ Maps and Dessins D\u27enfants
No full textThis dissertation centers on computing with dessins d\u27enfants, in the form of constellations, and on the monodromy group of compositions of Belyiˇ maps. To begin, a discussion of known effective and efficient algorithms for computing constellations, Belyiˇ Maps, and dessins d\u27enfants from one another is presented. Following this is an analysis of how to use double cosets in an optimal way to count equivalence classes of constellations. In addition, class multiplication coefficients are used to count trees with certain passports, culminating in a new proof of a result of Mednykh. The method given by Wood for computing the constellation of a composition of Belyiˇ maps is further developed and extended to allow Belyiˇ maps which are defined over the complex numbers. By utilizing the fact that the monodromy group of the composition of Belyiˇ maps β о γ is a subgroup of a wreath product, generators of the monodromy group of β о γ are found by a simple algorithm. Additionally, a group is determined from β alone which allows one to find the monodromy group of β о γ , for any γ, simply by applying the monodromy representation of γ. Finally, using the previous results, a cryptographic protocol utilizing compositions of Belyiˇ maps is proposed. A probabilistic method for efficiently deciding if the monodromy group of a Belyiˇ map is either the alternating or symmetric group is discussed. Although the protocol, in its current form, is not efficient enough for practical use, it demonstrates the ability to design a cryptographic protocol around the problem of computing Belyiˇ maps
Mod 4 galois representations from elliptic curves, a Brauer-Severi variety and a certain Brauer type embedding problem
No full textWe determine the lattice of subfields contained in the fixed field [special characters omitted], where [special characters omitted] is a continuous, surjective mod 4 Galois representation. Let ρE,4 be a surjective Galois representation, induced by the action of G K on the 4-torsion points of an elliptic curve E, with invariant j0. We show that the unique S4 field extension contained in K( E[4]) is the splitting field of the principal quartic [special characters omitted]. We proceed to study principal quartic extensions L/K, and show that L/K is principal precisely when a certain Brauer-Severi variety has a K-rational point. When L/K is principal, and M/K is its normal closure, we show that the solvability of the embedding problem [special characters omitted] is completely determined by the discriminant dL/K. Moreover, we will show that if L/K is principal and the Hilbert symbol (–2,–dL/K) is trivial, then the embedding problem [special characters omitted] is solvable. Finally we will consider a theorem by Holden, in which he shows the existence of of mod 4 representations that do not arise from the 4-torsion of any elliptic curve E/B. We will show that the result is incorrect as stated and provide counterexamples
Minimal Models of Rational Elliptic Curves with Non-trivial Torsion
Get PDFThis dissertation concerns the formulation of an explicit modified Szpirobconjecture and the classification of minimal discriminants of rational elliptic curves with non-trivial torsion subgroup. The Frey curve y2=x( x+a) ( x-b) is a two-parameter family of elliptic curves which comes equipped with an easily computable minimal discriminant which helped pave the mathematical bridge that led to the proof of Fermat\u27s Last Theorem. In this dissertation, we extend the ideas of the Frey curve by considering two- and three- parameter families of elliptic curves which parameterize all rational elliptic curves with non-trivial torsion. First, we use these families to give a new proof of a classic result of Frey, Flexor, and Oesterlé which pertains to the primes at which an elliptic curve over a number field can have additive reduction. While our proof gives a weaker variant of the original statement, it is explicit and does not require the Néron model of an elliptic curve. As a consequence of this new proof, we attain our classification of minimal discriminants of rational elliptic curves with non-trivial torsion. In addition, we give necessary and sufficient conditions for when a rational elliptic curve with non-trivial torsion has additive reduction at a given prime. We also study the connection between torsion structure of a rational elliptic curve and the possible reduced minimal models The second theme of this dissertation concerns the modified Szpiro conjecture, which is equivalent to the ABC Conjecture. Roughly speaking, the modified Szpiro conjecture states that certain elliptic curves, known as good elliptic curves, are rare in nature. Masser gave a non-constructive proof which showed that there were infinitely many good Frey curves. In this dissertation, we give a constructive proof of Masser\u27s assertion. We then extend this result by proving that for each of the fifteen torsion subgroups T allowed by Mazur\u27s Torsion Theorem, there are infinitely many good elliptic curves E with torsion subgroup isomorphic to T. This proof is also constructive and allows for the construction of a database which consists of 13870964 good elliptic curves. We provide an analysis of these good elliptic curves to parallel the work done by the ABC@Home project concerning the ABC Conjecture and good ABC triples. The data obtained is then used to formulate an explicit version of the modified Szpiro conjecture. We then show that this explicit formulation allows for the construction of databases of elliptic curves which are exhaustive up to a given conductor. Lastly, we use the classification of minimal discriminants to study the local data of rational elliptic curves at a given prime via Tate\u27s Algorithm. These results and a study of the naive height of an elliptic curve allow us to prove that there is a lower bound on the modified Szpiro ratio which depends only on the torsion structure of an elliptic curve
Ranks of elliptic curves and Selmer groups
No full textThis dissertation concerns the computation of m-Selmer groups of elliptic curves via the number field method of descent. We lay out a strategy for explicitly representing cohomology classes ξ∈Sel (m)(E/K) by those m-th power classes of an étale K-algebra belonging to a naturally defined group and subject to finitely many additional local conditions. Under suitable hypotheses, we relate this group to the m-torsion in the class group and m-th power classes of the unit group of an appropriate subring of the étale K-algebra. We work out the strategy in detail when m =2$, getting the cubic étale algebra used in the classical number field method of 2-descent. When K = Q and this algebra splits as three copies of Q, we recast the theory of complete 2-descent in terms of the Frey-Hellegouarch correspondence between solutions to A + B + C = 0 and elliptic curves with E[2] ⊆ E(Q). We classify those cases with at most three local conditions to compute, and describe Sel(2)(E/ Q) explicitly for such curves
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Get PDFWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
Dispelling the Myths Behind First-author Citation Counts
Get PDFWe conducted a full-scale evaluative citation analysis study of scholars in the XML research field to explore just how different from each other author rankings resulting from different citation counting methods actually are, and to demonstrate the capability of emerging data and tools on the Web in supporting more realistic citation counting methods. Our results contest some common arguments for the continued
use of first-author citation counts in the evaluation of scholars, such as high correlations between author rankings by first-author citation counts and other citation
counting methods, and high costs of using more realistic citation counting methods that are not well-supported by the ISI databases. It is argued that increasingly available digital full text research papers make it possible for citation analysis studies to go beyond what the ISI databases have directly supported and to employ more
sophisticated methods
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