Neste trabalho, apresentamos um método computacional rigoroso para a demonstração de existência de órbitas periódicas de alguns sistemas de equações diferenciais ordinárias com campo autônomo do tipo polinomial. Mostraremos que o problema de encontrar órbitas periódicas para esses sistemas de equações é equivalente a buscar por raízes de certas funções definidas no espaço de Banach das sequências com decaimento algébrico. O método pode ser dividido em duas etapas. Na primeira, buscamos numericamente por soluções periódicas aproximadas. Na segunda, mostraremos a existência de uma órbita periódica numa vizinhança da curva encontrada numericamente. O rigor das verificações computacionais é garantido pelo uso de aritimética intervalar. / In this work, we present a rigorous computational method for proving the existence of periodic orbits of some systems of ordinary differential equations with autonomous vector field of polynomial type. We show that the problem of finding periodic orbits for these systems is equivalent to check for roots of certain functions defined in the Banach space of sequences with algebraic decay. The method can be divided into two steps. First, we seek, numerically, to approximated periodic solutions. Then, we show the existence of a periodic orbit in a neighborhood of the curve numerically found in the previous stage. The accuracy of the computational verifications is guaranteed by the use of interval arithmetic.

Neste trabalho, apresentamos um método computacional rigoroso para a demonstração de existência de órbitas periódicas de alguns sistemas de equações diferenciais ordinárias com campo autônomo do tipo polinomial. Mostraremos que o problema de encontrar órbitas periódicas para esses sistemas de equações é equivalente a buscar por raízes de certas funções definidas no espaço de Banach das sequências com decaimento algébrico. O método pode ser dividido em duas etapas. Na primeira, buscamos numericamente por soluções periódicas aproximadas. Na segunda, mostraremos a existência de uma órbita periódica numa vizinhança da curva encontrada numericamente. O rigor das verificações computacionais é garantido pelo uso de aritimética intervalar. / In this work, we present a rigorous computational method for proving the existence of periodic orbits of some systems of ordinary differential equations with autonomous vector field of polynomial type. We show that the problem of finding periodic orbits for these systems is equivalent to check for roots of certain functions defined in the Banach space of sequences with algebraic decay. The method can be divided into two steps. First, we seek, numerically, to approximated periodic solutions. Then, we show the existence of a periodic orbit in a neighborhood of the curve numerically found in the previous stage. The accuracy of the computational verifications is guaranteed by the use of interval arithmetic.

Consider an under-determined system of nonlinear equations F(x)=0, F:R^m→R^n, where F is continuously differentiable and m > n. This system appears in a variety of applications, including parameter-dependent systems, dynamical systems with periodic solutions, and nonlinear eigenvalue problems. Robust, efficient numerical methods are often required for the solution of this system. Newton's method is an iterative scheme for solving the nonlinear system of equations F(x)=0, F:R^n→R^n. Simple to implement and theoretically sound, it is not, however, often practical in its pure form. Inexact Newton methods and globalized inexact Newton methods are computationally efficient variations of Newton's method commonly used on large-scale problems. Frequently, these variations are more robust than Newton's method. Trust region methods, thought of here as globalized exact Newton methods, are not as computationally efficient in the large-scale case, yet notably more robust than Newton's method in practice. The normal flow method is a generalization of Newton's method for solving the system F:R^m→R^n, m > n. Easy to implement, this method has a simple and useful local convergence theory; however, in its pure form, it is not well suited for solving large-scale problems. This dissertation presents new methods that improve the efficiency and robustness of the normal flow method in the large-scale case. These are developed in direct analogy with inexact-Newton, globalized inexact-Newton, and trust-region methods, with particular consideration of the associated convergence theory. Included are selected problems of interest simulated in MATLAB.

This report explores the effect of interactive visualizations on the advancement of mathematics understanding. Not only do interactive visualizations aid mathematicians to expand the body of knowledge of mathematics but it also allows students an efficient way to process the information taught in schools. There are many concepts in mathematics that utilize interactive visualizations and examples of such concepts are illustrated within this report. / text

Submitted by Marlene Santos (marlene.bc.ufg@gmail.com) on 2016-07-13T20:13:32Z No. of bitstreams: 2 Dissertação - Mauricio Silva Louzeiro - 2016.pdf: 1453255 bytes, checksum: c23898f8b30d7250d9fc245034078281 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2016-07-14T13:28:30Z (GMT) No. of bitstreams: 2 Dissertação - Mauricio Silva Louzeiro - 2016.pdf: 1453255 bytes, checksum: c23898f8b30d7250d9fc245034078281 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2016-07-14T13:28:30Z (GMT). No. of bitstreams: 2 Dissertação - Mauricio Silva Louzeiro - 2016.pdf: 1453255 bytes, checksum: c23898f8b30d7250d9fc245034078281 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-03-04 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we will study a new strategy to minimize a convex function on a simplicial cone. This method consists in to obtain the solution of a minimization problem through the root of a semi-smooth equation associated to its optimality conditions. To nd this root, we use the semi-smooth version of the Newton's method, where the derivative of the function that de nes the semi-smooth equation is replaced by a convenient Clarke subgradient. For the case that the function is quadratic, we will see that it allows us to have weaker conditions for the convergence of the sequence generated by the semi-smooth Newton's method. Motivated by this new minimization strategy we will also use the semi-smooth Newton's method to nd roots of two special semi-smooth equations, one associated to x+ and the another one associated to jxj. / Neste trabalho, estudaremos uma nova estrat egia para minimizar uma fun c~ao convexa sobre um cone simplicial. Este m etodo consiste em obter a solu c~ao do problema de minimiza c~ao atrav es da raiz de uma equa c~ao semi-suave associada as suas condi c~oes de otimalidade. Para encontrar essa raiz, usaremos uma vers~ao semi-suave do m etodo de Newton, onde a derivada da fun c~ao que de ne a equa c~ao semi-suave e substitu da por um subgradiente de Clarke conveniente. Para o caso em que a fun c~ao e quadr atica, veremos que e poss vel obter condi c~oes mais fracas para a converg^encia da sequ^encia gerada pelo m etodo de Newton semi-suave. Motivados por esta nova estrat egia de minimiza c~ao tamb em usaremos o m etodo de Newton semi-suave para encontrar ra zes de dois tipos espec cos de equa c~oes semi-suaves, uma associada a x+ e a outra associada a jxj.

Submitted by Cássia Santos (cassia.bcufg@gmail.com) on 2017-03-16T12:01:01Z No. of bitstreams: 2 Dissertação - Stefan Alberto Gómez Guevara - 2017.pdf: 2201042 bytes, checksum: bd12be92bd41bae24c13758a1fc1a73d (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-03-20T13:11:14Z (GMT) No. of bitstreams: 2 Dissertação - Stefan Alberto Gómez Guevara - 2017.pdf: 2201042 bytes, checksum: bd12be92bd41bae24c13758a1fc1a73d (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-03-20T13:11:14Z (GMT). No. of bitstreams: 2 Dissertação - Stefan Alberto Gómez Guevara - 2017.pdf: 2201042 bytes, checksum: bd12be92bd41bae24c13758a1fc1a73d (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-03-08 / In this work we consider the problem of finding a singularity of a field of differentiable vectors X on a Riemannian manifold. We present a local analysis of the convergence of Newton's method to find a singularity of field X on an increasing condition. The analysis shows a relationship between the major function and the vector field X. We also present a semi-local Kantorovich type analysis in the Riemannian context under a major condition. The two results allow to unify some previously unrelated results. / Neste trabalho consideramos o problema de encontrar uma singularidade de um campo de vetores diferenciável X sobre uma variedade Riemanniana. Apresentamos uma análise local da convergência do método de Newton para encontrar uma singularidade do Campo X sobre uma condição majorante. A análise mostra uma relação entre a função majorante e o campo de vetores X. Também apresentamos uma análise semi-local do tipo Kantorovich no contexto Riemanniana sob uma condição majorante. Os dois resultados permitem unificar alguns resultados não previamente.

Neste trabalho, apresentamos um método computacional rigoroso para a demonstração de existência de órbitas periódicas de alguns sistemas de equações diferenciais ordinárias com campo autônomo do tipo polinomial. Mostraremos que o problema de encontrar órbitas periódicas para esses sistemas de equações é equivalente a buscar por raízes de certas funções definidas no espaço de Banach das sequências com decaimento algébrico. O método pode ser dividido em duas etapas. Na primeira, buscamos numericamente por soluções periódicas aproximadas. Na segunda, mostraremos a existência de uma órbita periódica numa vizinhança da curva encontrada numericamente. O rigor das verificações computacionais é garantido pelo uso de aritimética intervalar. / In this work, we present a rigorous computational method for proving the existence of periodic orbits of some systems of ordinary differential equations with autonomous vector field of polynomial type. We show that the problem of finding periodic orbits for these systems is equivalent to check for roots of certain functions defined in the Banach space of sequences with algebraic decay. The method can be divided into two steps. First, we seek, numerically, to approximated periodic solutions. Then, we show the existence of a periodic orbit in a neighborhood of the curve numerically found in the previous stage. The accuracy of the computational verifications is guaranteed by the use of interval arithmetic.

Knowlege of elektric power, power dissipation and voltage conditions in every single segments and elektric power system nodes is necessary for operations control and designing of another elektric power system expansion. In this Thesis, we will handle with steady state of system calculation using PAS-DAISY-OFF LINE software, solving various power network configurations and their verification.

Führungsgetriebe die durch Servomotoren angetrieben werden, benötigen für definierte Stellungen des Abtriebsglieds eine programmierte Funktion (elektronische Kurvenscheibe). Diese leitet sich aus dem möglicherweise fehlerbehafteten kinematischen Modell des Getriebes ab (inverse Kinematik). Zur Verbesserung der Genauigkeit der Führungsbewegung wird ein Verfahren zur Justierung der Übertragungsfunktion auf Basis des Newton-Verfahrens unter Nutzung der Singulärwertzerlegung vorgestellt. Dabei werden die realen Getriebeabmessungen anhand einer Messung berechnet und werden anschließend korrigiert zur Anpassung der Übertragungsfunktion verwendet.

Este trabalho propõe estudos de casos em sistemas de energia elétrica por meio do Fluxo de Potência Ótimo (FPO) e da Análise de Sensibilidade em diferentes cenários de operação. Para isso, foram obtidos dados teóricos, a partir de levantamento bibliográfico, que explicitaram os conceitos de otimização aplicados ao sistema estático de energia elétrica. A pesquisa fundamentou-se metodologicamente no método primal-dual barreira logarítmica e nas condições necessárias de primeira-ordem de Karush-Kuhn-Tucker (KKT) para o problema de FPO, e no teorema proposto por Fiacco (1976) para a Análise de Sensibilidade. Os sistemas de equações resultantes das condições de estacionaridade, da função Lagrangiana, foram resolvidos pelo método de Newton. Na implementação computacional foram usadas técnicas de esparsidade. Estudos de casos foram realizados nos sistemas 3, IEEE 14, 30, 118, 300 barras e no equivalente CESP 440 kV com 53 barras, em que foi verificada a eficiência das técnicas apresentadas. / This work proposes a study of cases in power systems by Optimal Power Flow (OPF) and Sensitivity Analysis in different operation scenarios. For this purpose, theoretical data were obtained, starting from a bibliographical review, which enlightened the optimization concepts applied to the static system of electrical energy. The research was methodologically based on the primal-dual logarithmic barrier method and in the first-order necessary Karush-Kuhn-Tucker conditions to the OPF problem and in the theorem proposed by Fiacco (1976) to the Sensitivity Analysis. The equation sets generated by the first-order necessary conditions of the Lagrangian function, were solved by Newton\'s method. In the computational implementation, sparsity techniques were used. Studies of cases were carried out in the 3, IEEE 14, 30, 118, 300 buses and in the equivalent CESP 440 kV 53 bus, where the efficiency of the presented techniques was verified.