Der Schwerpunkt der vorliegenden Arbeit liegt auf Methoden zur Lösung nichtlinearer Optimierungsprobleme mit der Anforderung, jedes globale Optimum garantiert zu finden und mit einer im Voraus festgesetzten Genauigkeit zu approximieren. Eng verbunden mit dieser deterministischen Optimierung ist die Berechnung von Schranken für den Wertebereich einer Funktion über einem gegebenen Hyperquader. Verschiedene Ansätze, z. B. auf Basis der Intervallarithmetik, werden vorgestellt und analysiert. Im Besonderen werden Methoden zur Schrankengenerierung für multivariate ganz- und gebrochenrationale Polynome mit Hilfe der Darstellung in der Basis der Bernsteinpolynome weiterentwickelt. Weiterhin werden in der Arbeit schrittweise die Bausteine eines deterministischen Optimierungsverfahrens unter Verwendung der berechneten Wertebereichsschranken beschrieben und Besonderheiten für die Optimierung polynomialer Aufgaben näher untersucht. Die Analyse und Bearbeitung einer Aufgabenstellung aus dem Entwicklungsprozess für Fahrzeuggetriebe zeigt, wie die erarbeiteten Ansätze zur Lösung nichtlinearer Optimierungsprobleme die Suche nach energieeffizienten Getrieben mit einer optimalen Struktur unterstützen kann. Kontakt zum Autor: [Nachname] [.] [Vorname] [@] gmx [.] de

Os polinômios possuem características e propriedades que os tornam bastante importantes, seja modelando problemas da natureza e do cotidiano ou servindo como ferramenta de resolução de problemas ou, ainda, para alcançar resultados matemáticos mais avançados. O Currículo do Estado de São Paulo sugere uma sequência de conteúdos para serem trabalhados, levando o aluno a um aprendizado dos polinômios tanto do ponto de vista teórico quanto de aplicações. O ensino de polinômios é feito em espiral do 7º ano do Ensino Fundamental até o 3º ano do Ensino Médio, isto é, seu conteúdo é trabalhado gradativamente no decorrer dos anos escolares, sempre sendo retomado e aprofundado de acordo com o tempo escolar adequado. Este trabalho tem como objetivo contribuir com a formação de professores de Matemática do Ensino Básico apresentando uma sólida teoria sobre os polinômios no que diz respeito a definição, propriedades, operações algébricas, funções polinomiais, traçado de gráfico de polinômios e, já em um nível mais avançado, derivada e integral de polinômios. Além disso, revisamos os conceitos de espaços vetoriais, independência linear, base, projeções e ortogonalidade. A teoria apresentada é então utilizada no estudo de aproximações de funções por polinômios. Entre as formas de aproximação, apresentamos o polinômio de Taylor, a Interpolação Polinomial e o ajuste polinomial pelo Método dos Mínimos Quadrados. Ao longo do texto apresentamos aplicações no cotidiano como o cálculo do polinômio que descreve uma corrida de táxi, a fórmula 95 para aposentadoria e a curva de lucro de uma sorveteria em função do preço de seus sorvetes. / Polynomials have characteristics and properties that make them very important, modelling nature and daily problems or serving as a tool to solve problems, or even to achieve more advanced mathematical results. The curriculum of São Paulo state suggests a sequence of contents to be worked, leading the student to learning about polynomials from both the theoretical and the applications point of views. The polynomials teaching is done in a spiral way from the 7th year of elementary school to the 3rd year of high school, that is, its contents are gradually worked during the school years, always being resumed and deepened in accordance with the appropriate school time. This work aims to contribute to the training of basic education Mathematics teachers introducing them a solid theory of polynomials concerning about definition, properties, algebraic operations, polynomial functions, polynomial graphics and already at a more advanced level, derivative and integral of polynomials. In addition, we review the concepts of vector spaces, linear independence, base, projections and orthogonality. The presented theory is then used in the study of function approximations by polynomials. Among the forms of approach, we present the Taylor polynomial, the polynomial interpolation and polynomial fit by the least square method. Throughout the text we present applications in daily life such as the calculation of the polynomial that describes a taxi ride, the 95 formula for retirement and the the profit curve of an ice cream shop due to the price of their ice cream.

Os polinômios possuem características e propriedades que os tornam bastante importantes, seja modelando problemas da natureza e do cotidiano ou servindo como ferramenta de resolução de problemas ou, ainda, para alcançar resultados matemáticos mais avançados. O Currículo do Estado de São Paulo sugere uma sequência de conteúdos para serem trabalhados, levando o aluno a um aprendizado dos polinômios tanto do ponto de vista teórico quanto de aplicações. O ensino de polinômios é feito em espiral do 7º ano do Ensino Fundamental até o 3º ano do Ensino Médio, isto é, seu conteúdo é trabalhado gradativamente no decorrer dos anos escolares, sempre sendo retomado e aprofundado de acordo com o tempo escolar adequado. Este trabalho tem como objetivo contribuir com a formação de professores de Matemática do Ensino Básico apresentando uma sólida teoria sobre os polinômios no que diz respeito a definição, propriedades, operações algébricas, funções polinomiais, traçado de gráfico de polinômios e, já em um nível mais avançado, derivada e integral de polinômios. Além disso, revisamos os conceitos de espaços vetoriais, independência linear, base, projeções e ortogonalidade. A teoria apresentada é então utilizada no estudo de aproximações de funções por polinômios. Entre as formas de aproximação, apresentamos o polinômio de Taylor, a Interpolação Polinomial e o ajuste polinomial pelo Método dos Mínimos Quadrados. Ao longo do texto apresentamos aplicações no cotidiano como o cálculo do polinômio que descreve uma corrida de táxi, a fórmula 95 para aposentadoria e a curva de lucro de uma sorveteria em função do preço de seus sorvetes. / Polynomials have characteristics and properties that make them very important, modelling nature and daily problems or serving as a tool to solve problems, or even to achieve more advanced mathematical results. The curriculum of São Paulo state suggests a sequence of contents to be worked, leading the student to learning about polynomials from both the theoretical and the applications point of views. The polynomials teaching is done in a spiral way from the 7th year of elementary school to the 3rd year of high school, that is, its contents are gradually worked during the school years, always being resumed and deepened in accordance with the appropriate school time. This work aims to contribute to the training of basic education Mathematics teachers introducing them a solid theory of polynomials concerning about definition, properties, algebraic operations, polynomial functions, polynomial graphics and already at a more advanced level, derivative and integral of polynomials. In addition, we review the concepts of vector spaces, linear independence, base, projections and orthogonality. The presented theory is then used in the study of function approximations by polynomials. Among the forms of approach, we present the Taylor polynomial, the polynomial interpolation and polynomial fit by the least square method. Throughout the text we present applications in daily life such as the calculation of the polynomial that describes a taxi ride, the 95 formula for retirement and the the profit curve of an ice cream shop due to the price of their ice cream.

Cette recherche s’inscrit dans la problématique de l’intégration de technologie dans le processus d’apprentissage et enseignement des mathématique. Particulièrement, ce travail a pour objet étudier quel est l’impact de la participation des professeurs dans la conception des tâches qui confortement un système d’évaluation en ligne en mathématique, dans ce thèse on étudie notamment las tâches sur les fonctions polynômes. Quand on parle d’impact, on l’écrit à double sens : l’impact sur l’utilisation des ressources et sur la valeur épistémique des ressources conçues pour les professeurs.Du point de vue méthodologique, cette thèse s’inscrit dans un travail de type qualitatif qui est complété avec des analyses quantitatives. Pour étudier l’impact dans l’utilisation des ressources de la plateforme on faite une comparaison selon les rôles des professeurs (concepteurs et utilisateurs) a partir des données générés pour la plateforme et complétées avec des analyses des enregistrements de classes et des entretiens aux professeurs concepteurs. On peut conclure que il n’y a pas de différence dans l’utilisation des ressources quand dans un campus s’il y a des concepteurs et utilisateurs, par contre, quand dans un campus il y a que des utilisateurs, la participation des étudiantes dans la plateforme est beaucoup plus faible. Pour caractériser la valeur épistémique des tâches de la plateforme on a utilisé comme principal cadre théorique les Espaces de Travail Mathématique (ETM). Après de observer certains phénomènes dans l’ensemble de tâches de la plateforme, en utilisant le même cadre théorique, on a caractérisé les tâches habituelles des professeurs pour finalement faire une comparaison entre l’ETM potentiel de la plateforme et l’ETM idoine des professeurs. On a utilisé comme sources des données les tâches de la plateforme sur fonctions polynômes, les enregistrements des classes des professeurs concepteurs pour caractériser les tâches habituelles et des entretiens d’explicitation pour comprendre les raisons des certains choix des professeurs. On a constaté qu’il y a certaines phénomènes observés dans les tâches de la plateforme qui son dû à la dimension instrumental et las limitations des logiciels, surtout les relatives à l’utilisation des graphiques. Néanmoins il y a autres phénomènes qui son influencés principalement pour l’ETM idoine. De ce travail se dégagent également des pistes d’amélioration pour l’enrichissement des ressources de la plateforme et faire évoluer l’ETM idoine des professeurs à partir d’un accompagnement didactique. / This research is part of the problem of integrating technology into the process of teaching and learning mathematics. Particularly, this work aims to study the impact of the participation of teachers in the design of tasks that make up an online assessment system in mathematics, specifically in this thesis are studied tasks on polynomial functions. When we speak of impact, this refers in two senses: impact on the use of resources and on the epistemic value of the resources created by teachers. From the methodological point of view, this thesis is part of a qualitative work complemented by some quantitative analyses. In order to study the impact on the use of the platform's resources, a comparison was made according to the roles of the teachers (designers and users), based on the data generated by the platform and complemented by analysis of the class register and interviews with the designer teachers. It can be concluded that there are no differences in the use of resources on a campus if there are designers and users, whereas when on a campus there are only users the participation of students on the platform is much lower. The Mathematical Workspaces (MTS) were used as a theoretical framework to characterize the epistemic value of platform tasks. After observing certain phenomena in the set of tasks of the platform, using the same theoretical framework, the usual tasks of the professors designers were characterized to finally make a comparison between the potential ETM of the platform and the suitable ETM of the professors. The platform tasks on polynomial functions, the class records of the design teachers were used as data sources to characterize the usual tasks and explanatory interviews to understand the reasons for certain teacher choices. It was concluded that certain phenomena observed in the tasks of the platforms are due to the instrumental dimension and the limitations of the software, especially those linked to graphics. However, there are other phenomena that are mainly influenced by the suitable ETM. This work also gives rise to suggestions for improvement to enrich the resources of the platform and to make the suitable ETM of the teachers evolve on the basis of a didactic accompaniment. It was concluded that certain phenomena observed in the tasks of the platforms are due to the instrumental dimension and limitations of software, especially those linked to graphics. However, there are other phenomena that are mainly influenced by the suitable ETM.This work also gives rise to suggestions for improvement to enrich the resources of the platform and to make the suitable MTE of the teachers evolve from a didactic accompaniment. / Esta investigación se inscribe en la problemática de la integración de tecnología en los proceso de enseñanza y aprendizaje de las matemáticas. Particularmente, este trabajo tiene por objetivo estudiar el impacto de la participación de profesores en el diseño de las tareas que conforman un sistema de evaluación en línea en matemáticas, específicamente en esta tesis se estudian las tareas sobre funciones polinómicas. Cuando se habla de impacto, esto se refiere en dos sentidos: impacto en la utilización de los recursos y sobre el valor epistémico de los recursos creados por los profesores.Desde el punto de vista metodológico, esta tesis se inscribe en un trabajo de tipo cualitativo complementados con algunos análisis cuantitativos.Para estudiar el impacto en la utilización de los recursos de la plataforma se hizo una comparación según los roles de los profesores (diseñadores y utilizadores) a partir de los datos generados por la plataforma et complementados con análisis del registro de clases y de entrevistas a los profesores diseñadores. Se puede concluir que no hay diferencias en la utilización de los recursos en un campus si hay diseñadores y utilizadores, en cambio, cuando en un campus hay sólo utilizadores la participación de los estudiantes en la plataforma es mucho más baja.Para caracterizar el valor epistémico de las tareas de la plataformas se utilizó como marco teórico los Espacios de Trabajo Matemático (ETM). Después de observar ciertos fenómenos en el conjunto de tareas de la plataforma, utilizando el mismo marco teórico, se caracterizaron las tareas habituales de los profesores diseñadores para finalmente hacer una comparación entre el ETM potencial de la plataforma y el ETM idóneo de los profesores. Se utilizaron como fuentes de datos las tareas de la plataforma sobre funciones polinómicas, los registros de clases de los profesores diseñadores para caracterizar alas tareas habituales y entrevistas de explicitación para comprender las razones de ciertas elecciones de los profesores.Se concluyó que ciertos fenómenos observados en las tareas de la plataformas son debido a la dimensión instrumental y las limitaciones de los programas informáticos, sobretodo los ligados a los gráficos. Sin embargo, hay otros fenómenos que son influenciados principalmente por el ETM idóneo.De este trabajo se desprenden también pistas de mejora para enriquecer los recursos de la plataforma et para hacer evolucionar el ETM idóneo de los profesores a partir de un acompañamiento didáctico.

Les fonctions de F_q dans lui-même sont des objets étudiés dans de divers domaines tels que la cryptographie, la théorie des codes correcteurs d'erreurs, la géométrie finie ainsi que la géométrie algébrique. Il est bien connu que ces fonctions sont en correspondance exacte avec les polynômes en une variable à coefficients dans F_q. Nous étudierons trois classes de polynômes particulières: les polynômes Presque Parfaitement Non linéaires (Almost Perfect Nonlinear (APN)), les polynômes planaires ou parfaitement non linéaire (PN) et les o-polynômes.Les fonctions APN sont principalement étudiées pour leurs applications en cryptographie. En effet, ces fonctions sont celles qui offre la meilleure résistance contre la cryptanalyse différentielle.Les polynômes PN et les o-polynômes sont eux liés à des problèmes célèbres de géométrie finie. Les premiers décrivent des plans projectifs et les seconds sont en correspondance directe avec les ovales et hyperovales de P^2(F_q). Néanmoins, leurs champ d'application a été récemment étendu à la cryptographie symétrique et à la théorie des codes correcteurs d'erreurs.L'un des moyens utilisé pour compléter la classification est de considérer les polynômes présentant l'une des propriétés recherchées sur une infinité d'extension de F_q. Ces fonctions sont appelées fonction APN (respectivement PN ou o-polynômes) exceptionnelles.Nous étendrons la classification des polynômes APN et PN exceptionnels et nous donneront une description complète des o-polynômes exceptionnels. Les techniques employées sont basées principalement sur la borne de Lang-Weil et sur des méthodes élémentaires. / Functions from F_q to itself are interesting objects arising in various domains such as cryptography, coding theory, finite geometry or algebraic geometry. It is well known that these functions admit a univariate polynomial representation. There exists many interesting classes of such polynomials with plenty of applications in pure or applied maths. We are interested in three of them: Almost Perfect Nonlinear (APN) polynomials, Planar (PN) polynomials and o-polynomials. APN polynomials are mostly used in cryptography to provide S-boxes with the best resistance to differential cryptanalysis and in coding theory to construct double error-correcting codes. PN polynomials and o-polynomials first appeared in finite geometry. They give rise respectively to projective planes and ovals in P^2(F_q). Also, their field of applications was recently extended to symmetric cryptography and error-correcting codes.A complete classification of APN, PN and o-polynomials is an interesting open problem that has been widely studied by many authors. A first approach toward the classification was to consider only power functions and the studies were recently extended to polynomial functions.One way to face the problem of the classification is to consider the polynomials that are APN, PN or o-polynomials over infinitely many extensions of F_q, namely, the exceptional APN, PN or o-polynomials.We improve the partial classification of exceptional APN and PN polynomials and give a full classification of exceptional o-polynomials. The proof technique is based on the Lang-Weil bound for the number of rational points in algebraic varieties together with elementary methods.

Submitted by Maike Costa (maiksebas@gmail.com) on 2016-03-30T11:12:00Z No. of bitstreams: 1 arquivo total.pdf: 1989591 bytes, checksum: c1b3f2740144367fd7ef458d0603ba20 (MD5) / Made available in DSpace on 2016-03-30T11:12:00Z (GMT). No. of bitstreams: 1 arquivo total.pdf: 1989591 bytes, checksum: c1b3f2740144367fd7ef458d0603ba20 (MD5) Previous issue date: 2015-02-24 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In several scienti c character problems, it is common to come across us with the need to obtain an approximate solution to nd roots of functions. At this point, this paper aims to conduct a study about some methods used to obtain an approximate solution of the functions of roots. The survey was made by means of a literature review, focusing on Numerical Methods Bisection, False Position, Fixed Point, Newton-Raphson and Secant. In order to illustrate the operation and application of these methods, numerical test problems taken from the literature were performed by implementing these. For each test performed were analyzed parameters that in uence each method and the convergence situation for the approximate solution of the analyzed problems. Although these methods do not always make available exact roots, they can be calculated with the precision that the problem needs. At this point, it is evident the importance of studying methods for nding such equations roots. Thus, the work is justi ed on the need to discuss the problems facing the nding roots of polynomial functions in the literature. In addition, this paper describes a comparison between the methods studied by applying mathematical problems. All this research material becomes adept and e ective for students and professionals from all areas that make use of them, or perhaps wish to extract it for enrichment of several sources of study. / Em diversos problemas de caráter cientí co, é comum depararmo-nos com a necessidade de obter uma solução aproximada para encontrar raízes de funções. Nesse ponto, este trabalho objetiva realizar um estudo acerca de alguns métodos utilizados para a obtenção de uma solução aproximada das raízes de funções. A pesquisa realizada deu-se por meio de uma revisão bibliográ ca, enfocando os Métodos Numéricos da Bisseção, Falsa Posição, Ponto Fixo, Newton-Raphson, Secante e Muller. Com o intuito de ilustrar o funcionamento e aplicação desses métodos, foram realizados testes numéricos de problemas extraídos da literatura por meio da implementação destes. Para cada teste realizado foram analisados os parâmetros que in uenciam cada método e a situação de convergência para a solução aproximada dos problemas analisados. Embora esses métodos, nem sempre, disponibilizem raízes exatas, estas poderão ser calculadas com a precisão que o problema necessite. Nesse ponto, ca evidente a importância de estudar métodos para encontrar tais raízes de equações. Diante disso, o trabalho se justi ca na necessidade de se discutir os problemas voltados a encontrar raízes de funções polinomiais, existentes na literatura. Além disso, o presente trabalho descreve um comparativo entre os métodos estudados mediante aplicação de problemas matemáticos. Todo esse material de pesquisa torna-se hábil e e caz para os estudantes e pro ssionais de todas as áreas que dele faça uso, ou, porventura, pretendam extraí-lo para enriquecimento de fontes diversas de estudo.

Der Schwerpunkt der vorliegenden Arbeit liegt auf Methoden zur Lösung nichtlinearer Optimierungsprobleme mit der Anforderung, jedes globale Optimum garantiert zu finden und mit einer im Voraus festgesetzten Genauigkeit zu approximieren. Eng verbunden mit dieser deterministischen Optimierung ist die Berechnung von Schranken für den Wertebereich einer Funktion über einem gegebenen Hyperquader. Verschiedene Ansätze, z. B. auf Basis der Intervallarithmetik, werden vorgestellt und analysiert. Im Besonderen werden Methoden zur Schrankengenerierung für multivariate ganz- und gebrochenrationale Polynome mit Hilfe der Darstellung in der Basis der Bernsteinpolynome weiterentwickelt. Weiterhin werden in der Arbeit schrittweise die Bausteine eines deterministischen Optimierungsverfahrens unter Verwendung der berechneten Wertebereichsschranken beschrieben und Besonderheiten für die Optimierung polynomialer Aufgaben näher untersucht. Die Analyse und Bearbeitung einer Aufgabenstellung aus dem Entwicklungsprozess für Fahrzeuggetriebe zeigt, wie die erarbeiteten Ansätze zur Lösung nichtlinearer Optimierungsprobleme die Suche nach energieeffizienten Getrieben mit einer optimalen Struktur unterstützen kann. Kontakt zum Autor: [Nachname] [.] [Vorname] [@] gmx [.] de