Este trabalho consiste na análise do comportamento do método WAMG (Wavelet-Based Algebraic Multigrid), método numérico de resolução de sistemas de equações lineares desenvolvido no LMAG-Laboratório de Eletromagnetismo Aplicado, com relação a diversos suavizadores. O fato dos vetores que compõem os operadores matriciais Pronlongamento e Restrição do método WAMG serem ortonormais viabiliza uma série de análises teóricas e de dados experimentais, permitindo visualizar características não permitidas nos outros métodos Multigrid (MG), englobando o Multigrid Geométrico (GMG) e o Multigrid Algébrico (AMG). O método WAMG V-Cycle com Filtro Haar é testado em uma variedade de sistemas de equações lineares variando o suavizador, o coeficiente de relaxação nos suavizadores Damped Jacobi e Sobre Relaxação Sucessiva (SOR), e a configuração de pré e pós-suavização. Entre os suavizadores testados, estão os métodos iterativos estacionários Damped Jacobi, SOR, Esparsa Aproximada a Inversa tipo Diagonal (SPAI-0) e métodos propostos com a característica de suavização para-otimizada. A título de comparação, métodos iterativos não estacionários são testados também como suavizadores como Gradientes Conjugados, Gradientes Bi-Conjugados e ICCG. Os resultados dos testes são apresentados e comentados. / This work is comprised of WAMG (Wavelet-Based Algebraic Multigrid) method behavioral analysis based on variety of smoothers, numerical method based on linear equation systems resolution developed at LMAG (Applied Electromagnetism Laboratory). Based on the fact that the vectors represented by WAMG Prolongation and Restriction matrix operators are orthonormals allows the use of a variety of theoretical and practical analysis, and therefore gain visibility of characteristics not feasible through others Multigrid (MG) methods, such as Geometric Multigrid (GMG) and Algebraic Multigrid (AMG). WAMG V-Cycle method with Haar Filter is tested under a variety of linear equation systems, by varying smoothers, relaxation coefficient at Damped Jacobi and Successive Over Relaxation (SOR) smoothers, and pre and post smoothers configurations. The tested smoothers are stationary iterative methods such as Damped Jacobi, SOR, Diagonal type-Sparse Approximate Inverse (SPAI-0) and suggested ones with optimized smoothing characteristic. For comparison purposes, the Conjugate Gradients, Bi-Conjugate Gradient and ICCG non-stationary iterative methods are also tested as smoothers. The testing results are formally presented and commented.

Vectorial Boolean functions are used as S-boxes in cryptosystems. To design inequivalent vectorial Boolean functions resistant to known attacks is one of the challenges in cryptography. Verifying whether two vectorial Boolean functions are equivalent or not is the final step in this challenge. Hence, finding a fast technique for determining whether two given vectorial Boolean functions are equivalent is an important problem. A special class of the equivalence called restricted extended affine (REA) equivalence is studied in this thesis. We study the verification complexity of REA-equivalence of two vectorial Boolean functions for some types, namely types I to VI. We first review the verification of the REA-equivalence types I to IV given in the recent work of Budaghyan and Kazymyrov (2012). Furthermore, we present the complexities of the verification of REA-equivalence types I and IV in the case basic simultaneous Gaussian elimination method is used. Next, we present two new REA-equivalence types V and VI with their complexities. Finally, we give the algorithms of each type I to VI with their MAGMA codes.

ABSTRACT EFFECTS OF GRAPHING CALCULATORS ON EIGHTH GRADE STUDENTS&amp / #8217 / ACHIEVEMENT IN GRAPHS OF LINEAR EQUATIONS AND CONCEPT OF SLOPE &Ouml / n&uuml / r, Yurdag&uuml / l M.S., Department of Elementary Science and Mathematics Education Supervisor: Assist. Prof. Dr. Ayhan K&uuml / rSat ERBAS May 2008, 76 pages The purpose of this study was to investigate the effects of graphing calculators on eight grade students&amp / #8217 / achievement in graphing linear equations and concept of slope. Pretest-posttest experimental-control group design was utilized in the study. While the students in experimental group (EG) received instruction about graphs of linear equations and concept of slope with graphing calculators, the students in control group (CG) was taught the same topics without using graphing calculators. There were 27 students (13 girls and 14 boys) in each group. Students in both EG and CG was administered an achievement test (i.e., MAT) consisting of questions related to graphing linear equations and slope concept before and after the instruction. Additionally, the teacher and six students from the EG were interviewed. The data obtained from students&amp / #8217 / post test scores of MAT were analyzed by Analysis of Variance (ANOVA). A statistically significant difference was found between the achievements of students in experimental and control groups. However, gender had no statistically significant effect on students&amp / #8217 / post test scores of MAT. Additionally, students&amp / #8217 / pre-test scores of MAT and their mathematics grades of the second semester of the seventh grade (MGS) were analysed by independent samples t-test. The results showed no statistically significant difference. On the other hand, the analysis of interview data revealed that graphing calculators affected students&amp / #8217 / attitudes towards mathematics in a positive way. Students had no considerable difficulty while using graphing calculators and they found studying with graphing calculators enjoyable. In summary, the results of the study showed that when graphing calculators used at elementary school level, they had positive effects on students&amp / #8217 / achievement and in some respects to their attitude. Consequently, integration of graphing calculators to elementary mathematics curriculum may be beneficial for students and teachers.

Submitted by Mateus Eduardo Boccardo null (mateuseboccardo@hotmail.com) on 2017-10-16T21:58:41Z No. of bitstreams: 1 SISTEMAS LINEARES- APLICAÇÕES E PROPOSTAS DE AULA USANDO A METODOLOGIA DE RESOLUÇÃO DE PROBLEMAS E O SOFTWARE GEOGEBRA.pdf: 1649568 bytes, checksum: c2692aee302e29bada55e36958c6599b (MD5) / Approved for entry into archive by Monique Sasaki (sayumi_sasaki@hotmail.com) on 2017-10-18T18:37:19Z (GMT) No. of bitstreams: 1 boccardo_me_me_sjrp.pdf: 1649568 bytes, checksum: c2692aee302e29bada55e36958c6599b (MD5) / Made available in DSpace on 2017-10-18T18:37:19Z (GMT). No. of bitstreams: 1 boccardo_me_me_sjrp.pdf: 1649568 bytes, checksum: c2692aee302e29bada55e36958c6599b (MD5) Previous issue date: 2017-09-25 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Sistemas Lineares, mais precisamente, Sistemas de Equações Lineares, é ferramenta útil para a resolução de vários problemas práticos e importantes, por exemplo, problemas relacionados a tráfego de veículos, balanceamento de equações químicas, cálculo de uma alimentação diária equilibrada, circuitos elétricos e interpolação polinomial. Neste trabalho abordamos o conteúdo Sistemas Lineares, seus métodos de resolução, algumas de suas inúmeras aplicações, bem como a interpretação geométrica do conjunto solução de sistemas lineares em duas ou três variáveis. Apresentamos também, uma análise de como esse assunto é tratado em alguns documentos oficiais de ensino. Por fim, são expostas duas Propostas de Aula que foram elaboradas para alunos do Ensino Básico, uma para ser desenvolvida usando a Resolução de Problemas como metodologia de ensino (na abordagem de problemas sobre sistemas lineares) e outra, sobre a Interpretação Geométrica do conjunto solução de Sistemas Lineares, para ser realizada na Sala de Informática, utilizando o software GeoGebra. / Linear System, more precisely, System of Linear Equations, is a useful tool for their solution of several practical and important problems, for example problems related to vehicle traffic, balancing of chemical equations, elaboration healthy daily diet, electrical circuits and polynomial interpolation. In this work, we study Linear System, its methods of resolution, some of its numerous applications, as well as the geometric interpretation of the solution set of linear system in two or three variables. We also present an analysis of how this subject is treated in some official teaching documents. Finally, we present two Class Proposals that are elaborated for Basic Education students, one to be developed using Problem Solving as a teaching methodology (in the approach to problems on linear system) and another, on the Geometric Interpretation of the solution set of Linear System, to be held in the Computer Laboratory, using GeoGebra software.

Orientador: Hugo Enrique Hernández-Figueroa / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação / Made available in DSpace on 2018-08-16T06:54:22Z (GMT). No. of bitstreams: 1 Claudio_Kleucio_D.pdf: 3139134 bytes, checksum: 900508bd03693258d7011b6af9debd55 (MD5) Previous issue date: 2010 / Resumo: O método de elementos finitos é largamente empregado na modelagem de problemas de eletromagnetismo. A modelagem implícita deste método recai em resolver sistemas de equações lineares esparsas, esta etapa é de alto custo computacional. Este trabalho propõe alternativas com o objetivo de melhorar o desempenho computacional das aplicações provenientes de formulações via elementos finitos, através do aproveitamento de soluções de sistemas de equações lineares por métodos direto e iterativo, para simular dispositivos ópticos com as características físicas alteradas constantemente. Na solução dos sistemas de equações, utilizou-se o método direto com Small Rank Adjustment e o método iterativo gradiente bi-conjugado estabilizado precondicionado com análises de reaproveitamento do precondicionador ILUT. Nos estudos desenvolvidos obteve-se um melhor desempenho computacional quando se utilizou o método iterativo. Estes resultados são de grande importância na área de otimização de dispositivos fotônicos tais como acopladores, filtros, demultiplexadores, etc, pois a otimização destes dispositivos consiste em avaliar várias configurações do espaço de busca, implicando em resolver vários sistemas de equações lineares similares provenientes do método de elementos finitos. / Abstract: The Finite Element Method is one of the most popular numerical tools in electromagnetics. Implicit schemes require the solution of sparse linear equation systems, this step demands a lot of computational time. This work proposes alternatives enhancements to obtain better computational performance of such implicit schemes. This was made through the improvement of direct and iterative methods, for problems which may be interpreted as perturbations of a given original one. This is very important specially in the optimization process of devices, due to the fact that one needs to solve many linear systems with little changes at each step, to explore the search space, so many perturbed linear systems are solved to obtain the optimum device. For direct methods the Small Rank Adjustment technique was used, while for iterative methods, the Preconditioned Gradient Stabilized Biconjugate Method reusing the preconditioner, were adopted. The applications were focused on the design of photonic devices, like couplers, filters, demultiplexers, etc. / Doutorado / Telecomunicações e Telemática / Doutor em Engenharia Elétrica

Introdução: Através da ferramenta estatística denominada análise de regressão linear múltipla se gerou equações preditivas de temperatura interna do ar de uma edificação com paredes e lajes compostas por painéis de concreto alveolar, com sistema de cobertura verde. Justificativa: Com equações preditivas é possível simular temperaturas internas de edificações utilizando uma pequena entrada de dados com uma precisão satisfatória. Utilizando tais equações é possível, também, corrigir erros de projetos antes de sua execução. Objetivos: Gerar equações preditivas para o período seco (outono e inverno) e para o período chuvoso (primavera e verão) para a edificação analisada. Metodologia: Foram selecionadas duas séries de dados, um referente ao período de característica seca e outro de característica chuvosa. Foram geradas equações preditivas de temperatura interna do ar máxima, média e mínima para os dois períodos, utilizando análise de regressão linear. Resultados: Foram geradas sete equações preditivas para o período seco e cinco para o período chuvoso. As diferenças máximas, em módulo, entre as temperaturas estimadas pelas equações e as monitoradas experimentalmente ficaram em aproximadamente 2°C. Conclusão: As equações preditivas geradas para os dois períodos considerados descrevem satisfatoriamente o comportamento térmico da edificação. / Introduction: Using a statistics tool called multiple linear regression, we created equations for predicting the indoor temperature in a building with walls and ceiling build from panels of alveolar concrete, with a green roof system. Explanation: Predictive equations enable simulations of indoor temperatures of buildings using a small number of data and with a satisfactory precision. They also allow corrections on project errors before they are put into effect. Objectives: Generate predictive equations for the building for the dry season (autumn and winter) and for the rainy season (spring and summer). Method: We selected two series of data, one for the dry and one for the rainy season. Using linear regression analysis we ran predictive equations for maximum, intermediate and minimum indoor temperatures of the air for both seasons. Results: We created seven predictive equations for the dry season and five for the wet season. The largest differences (in module) between the temperatures estimated using equations and monitored experimentally was approximately 2°C. Conclusion: The predictive equations generated for both periods described satisfactorily the thermal behavior of the building.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2015-01-16T13:17:29Z No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Dissertação - Marcelo Simplício de Lyra - 2014.pdf: 2216498 bytes, checksum: e17b32a5feca5b8118ebc97a0684a160 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-01-16T13:45:49Z (GMT) No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Dissertação - Marcelo Simplício de Lyra - 2014.pdf: 2216498 bytes, checksum: e17b32a5feca5b8118ebc97a0684a160 (MD5) / Made available in DSpace on 2015-01-16T13:45:49Z (GMT). No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Dissertação - Marcelo Simplício de Lyra - 2014.pdf: 2216498 bytes, checksum: e17b32a5feca5b8118ebc97a0684a160 (MD5) Previous issue date: 2014-07-02 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This research presents a new approach that aims to introduce linear programming into the high school taking into account the teaching techniques, teacher and student profiles, and the flexibility of the curriculum. The context of the linear programming involves problems with two or three variables, since problems with many variables cannot be easily considered in a high school curriculum, especially due to the time required to solve such problems. The approach is developed under an algebraic point of view, in which the linear problem‟s solutions are obtained by a resolution of systems of linear equations. The approach also considers a numerical and computer simulation software, denominated Octave®, in order to solve those systems of linear equations and, consequently, this software may be used as a tool that allows extending such approach to solve linear programming problems with several decision variables. / Esta pesquisa apresenta uma proposta de introdução da programação linear no ensino médio levando em consideração os métodos de ensino, o perfil profissional do professor, o perfil do estudante e a flexibilização do currículo escolar. O contexto da programação linear envolve problemas de duas ou três variáveis, uma vez que problemas com mais variáveis podem não se adequar ao currículo do ensino médio, em especial pelo fator tempo. Parte-se de um desenvolvimento algébrico, em que as soluções do problema são obtidas por meio da resolução de vários sistemas de equações lineares. A proposta também inclui utilizar um software de simulação numérica e computacional, denominado Octave®, para a resolução dos vários sistemas lineares e, consequentemente, ser usado como uma estratégia para estender a proposta para problemas de programação linear com várias variáveis de decisão.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Nesta dissertação, apresentamos os resultados de uma ampla pesquisa bibliográfica sobre as equações diofantinas e seus métodos de solução mais utilizados. A mais simples desta classe de equações é a da forma ax + by = c, com a, b e c números inteiros e ab 6= 0, chamada equação diofantina linear nas duas incógnitas x e y. No trabalho, expomos diversos métodos de resolução destas equações, em duas e três incógnitas. Para tanto, utilizamos conceitos de divisibilidade, divisão euclidiana, máximo divisor comum, números primos, dentre outros, que formam parte do currículo do Ensino Fundamental. No Brasil, as equações diofantinas não são comumente exploradas na Educação Básica, embora sejam perfeitamente compreensíveis nesse nível, como se mostra no texto do professor A. Guelfond, consultado na redação do trabalho. Na dissertação, incluímos, também, um capítulo sobre as contribuições de Diofanto para a Aritmética, que pode ser uma fonte de motivação para o estudo das equações diofantinas; e outro capítulo, ampliando as perspectivas sobre equações diofantinas não lineares. Esperamos que o trabalho seja uma fonte bibliográfica facilmente acessível aos professores da Educação Básica, e estimule seu interesse e criatividade para a introdução elementar desses conteúdos na prática docente e na preparação dos alunos para as Olimpíadas de Matemática. / In this dissertation, the results of a wide bibliographic research about Diophantine equations and their most used solution methods are exposed. The simplest equation of these class is the one in the form ax + by = c, with a, b and c integers numbers and ab 6= 0, called Diophantine linear equation in the unknowns x and y. Divers solutions methods for these equations, in two or three unknowns are discussed. Therefore, concepts like divisibility, Euclidean division, grated common divisor, prime numbers, among others, that are included in the Elementary Schools curriculum. In Brazil, Diophantine equations are not commonly exploited in Basic Education, even though they are perfectly understandable at this educational level, like Professor A. Guelfond shows in his book consulted in the redaction of the dissertation. There are also a chapter about Diophantuss contributions to Arithmetic, which can be a source of motivation to study the Diophantine equations; and another chapter, extending perspectives, about nonlinear Diophantine equations. We hope that the dissertation becomes a suitable easy accessible bibliographic font for Basic Education teachers and stimulates their interest and creativity for an elemental introducing of these contents in their teaching and in the students training for Math Olympiads.

L'objet de cette thèse est l'étude des équations de Schrödinger et des ondes, à la fois linéaires et non linéaires, dans des domaines extérieurs. Nous nous intéressons en particulier aux inégalités dites de Strichartz, qui sont une famille d'estimations dispersives mesurant la décroissance du flot linéaire, particulièrement utiles à l'étude des problèmes non linéaires correspondants. Dans des géométries dites non-captantes, c'est à dire où tous les rayons de l'optique géométrique partent à l'infini, de nombreux résultats montrent que de telles estimations sont aussi bonnes que dans l'espace libre. D'autre part, la présence d'une trajectoire captive induit nécessairement une perte au niveau d'une autre famille d'estimations à priori, les estimations d'effet régularisant et de décroissance locale de l'énergie, respectivement pour Schrödinger et pour les ondes. En contraste de quoi, nous montrons des estimations de Strichartz sans perte dans une géométrie captante instable : l'extérieur de plusieurs obstacles strictement convexes vérifiant la condition d'Ikawa. La seconde partie de cette thèse est dédiée à l'étude du comportement en temps long des équations non-linéaires sous-jacentes. Lorsque le domaine dans lequel elles vivent n'induit pas trop de concentration de l'énergie, on s'attend à ce qu'elles diffusent, c'est à dire se comportent de manière linéaire asymptotiquement en temps. Nous montrons un tel résultat pour les ondes non linéaires critiques à l'extérieur d'une classe d'obstacles généralisant la notion d'étoilé. A l'extérieur de deux obstacles strictement convexes, nous obtenons un résultat de rigidité concernant les solutions à flot compact, premier pas vers un résultat général. Enfin, nous nous intéressons à l'équation de Schrödinger non linéaire, dans l'espace libre, mais avec un potentiel. Nous montrons que les solutions diffusent si l'on prend un potentiel répulsif, ainsi qu'une somme de deux potentiels répulsifs ayant des surfaces de niveau convexes, ce qui fournit un exemple de diffusion dans une géométrie captante analogue à l'extérieur de deux convexes stricts. / We are concerned with Schrödinger and wave equations, both linear and non linear, in exterior domains. In particular, we are interested in the so-called Strichartz estimates, which are a family of dispersive estimates measuring decay for the linear flow. They turn out to be particularly useful in order to study the corresponding non linear equations. In non-captive geometries, where all the rays of geometrical optics go to infinity, many results show that Strichartz estimates hold with no loss with respect to the flat case. Moreover, the local smoothing estimates for the Schrödinger equation, respectively the local energy decay for the wave equation, which are another family of dispersive estimates, are known to fail in any captive geometry. In contrast, we show Strichartz estimates without loss in an unstable captive geometry: the exterior of many strictly convex obstacles verifying Ikawa's condition. The second part of this thesis is dedicated to the study of the long time asymptotics of the corresponding non linear equations. We expect that they behave linearly in large times, or scatter, when the domain they live in does not induce too much concentration effect. We show such a result for the non linear critical wave equation in the exterior of a class of obstacles generalizing star-shaped bodies. In the exterior of two strictly convex obstacles, we obtain a rigidity result concerning compact flow solutions, which is a first step toward a general result. Finally, we consider the non linear Schrödinger equation in the free space but with a potential. We prove that solutions scatter for a repulsive potential, and for a sum of two repulsive potentials with strictly convex level surfaces. This provides a scattering result in a framework similar to the exterior of two strictly convex obstacles.

In this thesis, we propose a novel hyper-rectangle cover theory which provides a new approach to analyzing mathematical problems with nonnegativity constraints on variables. In this theory, two fundamental concepts, cover order and cover length, are introduced and studied in details. In the same manner as determining the rank of a matrix, we construct a specific e ́chelon form of the matrix to obtain the cover order of a given matrix efficiently and effectively. We discuss various structures of the e ́chelon form for some special cases in detail. Based on the structure and properties of the constructed e ́chelon form, the concepts of non-negatively linear independence and non-negatively linear dependence are developed. Using the properties of the cover order, we obtain the necessary and sufficient conditions for the existence and uniqueness of the solutions for linear equations system with nonnegativity constraints on variables for both homogeneous and non-homogeneous cases. In addition, we apply the cover theory to analyze some typical problems in linear algebra and optimization with nonnegativity constraints on variables, including linear programming problems and non-negative least squares (NNLS) problems. For linear programming problem, we study the three possible behaviors of the solutions for it through hyper-rectangle cover theory, and show that a series of feasible solutions for the problem with the zero-cover e ́chelon form structure. On the other hand, we develop a method to obtain the cover length of the covered variable. In the process, we discover the relationship between the cover length determination problem and the NNLS problem. This enables us to obtain an analytical optimal value for the NNLS problem. / Thesis / Doctor of Philosophy (PhD)