Repositório RCAAP

Fundação de Amparo a Pesquisa do Estado de Minas Gerais

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

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

In this work we present a study of Möbius transformation sand fuchsian groups as the basis for the development of the theory of some specific quotient spaces. In these quotient spaces we consider geometrically uniform hyperbolic lattices that will be isometrically embedded in the sphere S8 c R9

The solution of a conservation law may develop discontinuities like shocks and rarefactions waves, even if the initial condition is a smooth function. Then, numerical schemes should be able to generate ecient approximations in order to reproduce the same behavior as the analytic solution. Besides, these schemes have to capture the physically correct solution or entropy solution. The goal of this master dissertation is to study non-oscillatory schemes applied to one-dimensional scalar hyperbolic conservation laws. In order to reach the is objective, it is necessary to understand some special methods, such as, upwind scheme, TVD schemes, conservative schemes and monotone schemes. The eectiveness of the methods will be veried through the comparison with the well-known classical solutions exhibited in literature: Advection Equation and Burgers' Equation. The characteristic equations will be employed for getting analytic solutions of conservation laws. We will derive numerical approximations for conservation laws using ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for space discretization, and Runge-Kutta TVD (Total Variation Dimishing) for time discretization.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

In this work is analyzed the dynamical behavior of a Rayleigh type self-excited autoparametric system. Using the Regular Perturbation Theory and the Averaging Method, one can give conditions on the parameters of the system in order to get stability of the semitrivial solution. Along with this analysis, some bifurcations results are obtained too. Some results about non trivial solutions are obtained from the analysis of the equilibrium points in the invariant manifolds u = 0 and ρ = 0.

In this work we present a proof that the Hilbert s Tenth Problem is unsolvable. This problem is to give a computing algorithm which will tell of a given polynomial Diophantine equation with integer coefficients whether or not it has a solution in integers. We start developing some topics of basic number theory, that will be useful at some time. In this part we prove only main results. After that, we study Diophantine equation as well as Diophantine functions. Then, we prove a serie of lemas that will be useful to proof that the exponential function is Diophantine. From there, we define the concept of recursive function and prove that a function is Diophantine if and only if it is recursive. Finally we prove the Universality Theorem. We use this last theorem to proof that the Hilbert s Problem is unsolvable.

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

The purpose of this Master Dissertation is to build an algorithm which executes the Fuzzy Rule-Based System (FRBS) procedures. To reach this aim, it will be rstly studied the basic concepts of fuzzy set theory along with all FRBS components. After that, the computer codes that exihibit step-by-step some techniques for solving several problems using this powerful tool are going to show. Such a tool is widespread in Matlab. Promoting and spreading the accessibility to that tool is the intention of this work. It will be made through the presentation of several codes, which were developed in this dissertation. These codes do not depend on a specic computer package. The algorithm can be implemented in any existing programming language like C, for example. The programming of the algorithm can be made in Octave environment, too. Octave is similar to MATLAB; besides, it is a free software. Making dynamic adjustments of the FRBS parameters is an advantage of the algorithm developed here compared with MATLAB one. Note that any adjustments can be easily made while the code is running.

Fundação de Amparo a Pesquisa do Estado de Minas Gerais

In this dissertation we study lattices in quotient spaces. The basic quotient spaces are: (1) n-dimensional euclidean tori, obtained from quotient of Rn by discrete groups of isometries ge- nerated by linearly independent translations and (2) hyperbolic °at g-tori (tori of genus g ¸ 2), obtained from quotient of hyperbolic plane by fuchsian groups. In the euclidean environment, the considered lattices are provided of the additive group Z2; while in the hyperbolic case the studied lattices are the geometrically uniform and the cyclic ones.

Describing mathematically natural phenomena in order to make predictions and decisions is one of the biggest challenges in mathematics. Several phenomena can be described through partial differential equations, however many of these phenomena present linguistic variables, i.e. vague and imprecise information. Such characteristics make phenomenon modeling through differential equations difficult due to their dependence on the precision of the parameters used. The purpose of this research is to demonstrate the feasibility and applicability of fuzzy partial systems (p-fuzzy) for the modeling of the phenomena described by partial differential equations. In particular, the method proposed made possible the mathematical treatment of the physical phenomenon of luminescence, therefore demonstrating the potential of fuzzy p systems in relation to those obtained without the fuzzy modeling approach. The p-fuzzy systems were obtained using the function ANFIS Matlab, which came from a data set and identifies the membership functions as well as the parameters of fuzzy rules based systems. Analyzing the results, it is possible to conclude that the p-fuzzy system is a useful tool for modeling specific phenomena that involve partial variation rates, including time evolution.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

In this work some existence and stability results of periodic orbits of van der Pol-Mathieu Equation are studied. By using the Averaging Theorem we are able to prove, under mild conditions, the existence of two asymptotically stable periodic orbits of this equation. Moreover, the existence of invariant quadrics can be settled in plane phase of this equation.