Repositório RCAAP

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

We study Elliptic Curves. Initially we describe an operation on the curve which makes the set of points of an elliptic curve, over any eld, an abelian group. We introduce the Nagell-Lutz theorem which shows the necessary conditions for a rational point to have nite order. Next, we prove Mordell\'s theorem for curves dened by y2 = x3 + ax2 + bx. This theorem says that the set of rational points on an elliptic curve is a nitely generated abelian group. On the proof of this result, an algorithm is constructed. With this algorithm, it is possible, in some cases, to calculate the rank of the elliptic curve. We use this algorithm and the Nagell-Lutz theorem to study the Mordell-Weil Group of Elliptic Curves of the form y2 = x3 - px, where p is a prime number.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

Fundação de Amparo a Pesquisa do Estado de Minas Gerais

Evaluation codes (also called order domain codes) are traditionally introduced as generalized one point geometric Goppa codes. In the present dissertation we will give a new point of view on evaluation codes by introducing them instead as particular nice examples of affine variety codes. Our study includes a reformulation of the usual methods to estimate the minimum distances of evaluation codes into the setting of affine variety codes. Finally we describe the connection to the theory of one point geometric Goppa codes.

Fundação de Amparo a Pesquisa do Estado de Minas Gerais

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

Steganography is a subject that became very important in the study of information security. When it was related to Coding Theory, which was well developed, the research on this matter rapidly increased. In this work, we will introduce Steganography and we will show how the Coding Theory an help in its study; perfect codes will be related to a kind of stegoscheme and we will see the effect of wet paper codes in Steganography.

The aim of this work is to formulate a demonstration for the Averaging Theorem using classical analysis methods. In order to do that we studied some results that was used as the theoretical basis, such as the Liapunov Stability Theorem. Moreover, we will apply the Averaging theorem to a physical system with two degrees of freedom, wich is composed by a single mass with nonlinear coupling and parametric excitation. Thus, the investigation of the existence and stability of periodic orbits in this system is reduced to studying the stability of equilibrium points of the average system. This study was made using the Routh?Hurwitz stability criterion.

From Fermat s last theorem we know that the equation X3 +Y 3 = Z3 does not have nontrivial integral solutions. In 1769 Euler conjectured that this result may be generalized increasing the powers and the number of variables. In this work we give a counterexample to Euler s conjecture in the case of n = 4 showing that the equation A4 + B4 + C4 = D4 has nontrivial integral solutions. To do that we study plane algebraic curves, elliptic curves and use results from number theory, especially those on quadratic reciprocity. The quadratic reciprocity is the key factor in the choice of a particular elliptic curve, such that a solution in that elliptic curve becomes a nontrivial solution of the Euler s equation with n = 4. Finally, the arithmetic of the elliptic curves allows us to find infinite integral solutions for A4 + B4 + C4 = D4.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

In this work we consider C1 − generic vector fields over a compact, boundaryless, compact, of finite dimension Riemann manifold. The idea is to investigate differential local properties of these vector fields in order to obtain global properties for the induced flow. More precisely, we show if a C1−generic vector field is such that the only singularities accumulated by periodic orbits are co-dimension one singularities then: Either the vector field has a point been accumulated by periodic orbits of different Morse index or the vector field is sectional-Axiom A. Moreover, we show that the existence of points been accumulated by periodic orbits of different indices does not happen for star vector fields having spectral decomposition, which implies these ones should be sectional-Axiom A.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior