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Made available in DSpace on 2016-01-13T13:27:31Z (GMT). No. of bitstreams: 0 Previous issue date: 2015-06-19. Added 1 bitstream(s) on 2016-01-13T13:31:13Z : No. of bitstreams: 1 000855552.pdf: 610349 bytes, checksum: 8a368bbef1b3c15e173a4bf8451aa241 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O objetivo deste trabalho é apresentar um estudo sobre números primos. Trataremos de assuntos clássicos da Teoria dos Números: Congruências, O pequeno Teorema de Fermat, o Teorema de Wilson, a função φ de Euler e o Teorema de Euler. Utilizando estes resultados passaremos a investigar testes de primalidade, números primos especiais e funções que geram números primos / The aim of this work is a study of prime numbers. We will work with classical subjects of Number Theory, such as, Congruences, The little Fermat's Theorem, the Wilson's Theorem, the Euler's function φ and the Euler's Theorem. Using these results we will investigate primality tests, special prime numbers and functions defining prime numbers

Orientador: Jamil Viana Pereira / Banca: Sergio Henrique Monari Soares / Banca: Rawlilson de Oliveira Araujo / Resumo: O objetivo deste trabalho é apresentar um estudo sobre números primos. Trataremos de assuntos clássicos da Teoria dos Números: Congruências, O pequeno Teorema de Fermat, o Teorema de Wilson, a função φ de Euler e o Teorema de Euler. Utilizando estes resultados passaremos a investigar testes de primalidade, números primos especiais e funções que geram números primos / Abstract: The aim of this work is a study of prime numbers. We will work with classical subjects of Number Theory, such as, Congruences, The little Fermat's Theorem, the Wilson's Theorem, the Euler's function φ and the Euler's Theorem. Using these results we will investigate primality tests, special prime numbers and functions defining prime numbers / Mestre

We study certain aspects of the Möbius randomness principle and more specifically the Möbius disjointness conjecture of P. Sarnak. In paper A we establish this conjecture for all orientation preserving circle homeomorphisms and continuous interval maps of zero entropy. In paper B we show, that for all subshifts of finite type with positive topological entropy the Möbius disjointness does not hold. In paper C we study a class of three-interval exchange maps arising from a paper of Bourgain and estimate its Hausdorff dimension. In paper D we consider the Chowla and Sarnak conjectures and the Riemann hypothesis for abstract sequences and study their relationship. / <p>QC 20171016</p>

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. / Graduate Studies, College of (Okanagan) / Graduate

This thesis is concerned with the Langlands program; namely the global Langlands correspondence, Langlands functoriality, and a conjecture of Gross. In chapter 1, we cover the most important background material needed for this thesis. This includes material on reductive groups and their root data, the definition of automorphic representations and a general overview of the Langlands program, and Gross' conjecture concerning attaching l-adic Galois representations to automorphic representations on certain reductive groups G over &Qopf;. In chapter 2, we show that odd-dimensional definite unitary groups satisfy the hypotheses of Gross' conjecture and verify the conjecture in this case using known constructions of automorphic l-adic Galois representations. We do this by verifying a specific case of a generalisation of Gross' conjecture; one should still get l-adic Galois representations if one removes one of his hypotheses but with the cost that their image lies in ^CG(&Qopf;_l) as opposed to ^LG(&Qopf;_l). Such Galois representations have been constructed for certain automorphic representations on G, a definite unitary group of arbitrary dimension, and there is a map ^CG(&Qopf;_l) &rarr; ^LG(&Qopf;_l) precisely when G is odd-dimensional. In chapter 3, which forms the main part of this thesis, we show that G = U_n(B) where B is a rational definite quaternion algebra satisfies the hypotheses of Gross' conjecture. We prove that one can transfer a cuspidal automorphic representation &pi; of G to a &pi;' on Sp_2n (a Jacquet-Langlands type transfer) provided it is Steinberg at some finite place. We also prove this when B is indefinite. One can then transfer &pi;′ to an automorphic representaion of GL_2n+1 using the work of Arthur. Finally, one can attach l-adic Galois representations to these automorphic representations on GL_2n+1, provided we assume π is regular algebraic if B is indefinite, and show that they have orthogonal image.

Copy number variation, causing by the genome rearrangement, generally refers to the copy numbers increased or decreased of large genome segments whose lengths are more than 1kb. Such copy number variations mainly appeared as the sub-microscopic level of deletion and duplication. Copy number variation is an important component of genome structural variation, and is one of pathogenic factors of human diseases. Next generation sequencing technology is a popular CNV detection method and it has been widely used in various fields of life science research. It possesses the advantages of high throughput and low cost. By tailoring NGS technology, it is plausible to sequence individual cells. Such single cell sequencing can reveal the gene expression status and genomic variation profile of a single-cell. Single cell sequencing is promising in the study of tumor, developmental biology, neuroscience and other fields. However, there are two challenging problems encountered in CNV detection for NGS data. The first one is that since single-cell sequencing requires a special genome amplification step to accumulate enough samples, a large number of bias is introduced, making the calling of copy number variants rather challenging. The performances of many popular copy number calling methods, designed for bulk sequencings, are not consistent and cannot be applied on single-cell sequenced data directly. The second one is to simultaneously analyze genome data for multiple samples, thus achieving assembling and subgrouping similar cells accurately and efficiently. The high level of noises in single-cell-sequencing data negatively affects the reliability of sequence reads and leads to inaccurate patterns of variations. To handle the problem of reliably finding CNVs in NGS data, in this thesis, we firstly establish a workflow for analyzing NGS and single-cell sequencing data. The CNVs identification is formulated as a quadratic optimization problem with both constraints of sparsity and smoothness. Tailored from alternating direction minimization (ADM) framework, an efficient numerical solution is designed accordingly. The proposed model was tested extensively to demonstrate its superior performances. It is shown that the proposed approach can successfully reconstruct CNVs especially somatic copy number alteration patterns from raw data. By comparing with existing counterparts, it achieved superior or comparable performances in detection of the CNVs. To tackle this issue of recovering the hidden blocks within multiple single-cell DNA-sequencing samples, we present an permutation based model to rearrange the samples such that similar ones are positioned adjacently. The permutation is guided by the total variational (TV) norm of the recovered copy number profiles, and is continued until the TV-norm is minimized when similar samples are stacked together to reveal block patterns. Accordingly, an efficient numerical scheme for finding this permutation is designed, tailored from the alternating direction method of multipliers. Application of this method to both simulated and real data demonstrates its ability to recover the hidden structures of single-cell DNA sequences.

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The modularity theorem implies that for every elliptic curve E /Q there exist rational maps from the modular curve X_0(N) to E, where N is the conductor of E. These maps may be expressed in terms of pairs of modular functions X(z) and Y(z) that satisfy the Weierstrass equation for E as well as a certain differential equation. Using these two relations, a recursive algorithm can be constructed to calculate the q - expansions of these parameterizations at any cusp. These functions are algebraic over Q(j(z)) and satisfy modular polynomials where each of the coefficient functions are rational functions in j(z). Using these functions, we determine the divisor of the parameterization and the preimage of rational points on E. We give a sufficient condition for when these preimages correspond to CM points on X_0(N). We also examine a connection between the algebras generated by these functions for related elliptic curves, and describe sufficient conditions to determine congruences in the q-expansions of these objects.

<pre>This dissertation concerns two questions involving the injectivity of specialization homomorphisms for elliptic surfaces. We primarily focus on elliptic surfaces over the projective line defined over the rational numbers. The specialization theorem of Silverman proven in 1983 says that, for a fixed surface, all but finitely many specialization homomorphisms are injective. Given a subgroup of the group of rational sections with explicit generators, we thus ask the following.</pre><pre>Given some rational number, how can we effectively determine whether or not the associated specialization map is injective?</pre><pre>What is the set of rational numbers such that the corresponding specialization maps are injective?</pre><pre>The classical specialization theorem of Neron proves that there is a set S which differs from a Hilbert subset of the rational numbers by finitely many elements such that for each number in S the associated specialization map is injective. We expand this into an effective procedure that determines if some rational number is in S, yielding a partial answer to question 1. Computing the Hilbert set provides a partial answer to question 2, and we carry this out for some examples. We additionally expand an effective criterion of Gusic and Tadic to include elliptic surfaces with a rational 2-torsion curve.<br></pre>