In literature Trefftz method normally has geometric (exponential) convergence. Recently many scholars have found that spectral method in some cases can converge faster than exponential, which is called super-geometric convergence. Since Trefftz method can be regarded as a kind of spectral method, we expect it might possess super-geometric convergence too. In this thesis, we classify all types of super-geometric convergence and compare their speeds. We develop a method to decide the convergent type of given error data. Finally we can observe in many numerical experiments the super-geometric convergence of Trefftz method to solve Helmholtz boundary value problems.

The purpose of this thesis is to develop a position control method for parallel manipulators so that the end effector can follow a desired trajectory specified in the task space where joint flexibility that occurs at the actuated joints is also taken into consideration. At the beginning of the study, a flexible joint is modeled, and the equations of motion of the parallel manipulators are derived for both actuator variables and joint variables by using the Lagrange formulation under three assumptions regarding dynamic coupling between the links and the actuators. These equations of motion are transformed to an input/output relation between the actuator torques and the actuated joint variables to achieve the trajectory tracking control. Moreover, the singular configurations of the parallel manipulators are explained. As a case study, a three degree of freedom, two legged planar parallel manipulator is simulated considering joint flexibility. The structural damping of the active joints, viscous friction at the passive joints and the rotor damping are also considered throughout the study. Matlab&reg / and Simulink&reg / softwares are used for the simulations. The results of the simulations reveal that steady state errors are negligibly small and good tracking performances can be achieved.

This thesis systematically derives an innovative approach, called FOMFE, for fingerprint ridge orientation modeling based on 2D Fourier expansions, and explores possible applications of FOMFE to various aspects of a fingerprint identification system. Compared with existing proposals, FOMFE does not require prior knowledge of the landmark singular points (SP) at any stage of the modeling process. This salient feature makes it immune from false SP detections and robust in terms of modeling ridge topology patterns from different typological classes. The thesis provides the motivation of this work, thoroughly reviews the relevant literature, and carefully lays out the theoretical basis of the proposed modeling approach. This is followed by a detailed exposition of how FOMFE can benefit fingerprint feature analysis including ridge orientation estimation, singularity analysis, global feature characterization for a wide variety of fingerprint categories, and partial fin gerprint identification. The proposed methods are based on the insightful use of theory from areas such as Fourier analysis of nonlinear dynamic systems, analytical operators from differential calculus in vector fields, and fluid dynamics. The thesis has conducted extensive experimental evaluation of the proposed methods on benchmark data sets, and drawn conclusions about strengths and limitations of these new techniques in comparison with state-of-the-art approaches. FOMFE and the resulting model-based methods can significantly improve the computational efficiency and reliability of fingerprint identification systems, which is important for indexing and matching fingerprints at a large scale.

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Diese Habilitationsschrift richtete auf zwei diskrete Strukturen aus: planare Strukturen und zufällige Graphen-Prozesse. Zunächst werden zufällige planare Strukturen untersucht, mit folgende Gesichtspunkte: - Wieviele planare Strukturen gibt es? - Wie kann effizient eine zufällige planare Struktur gleichverteilt erzeugt werden? - Welche asymptotischen Eigenschaften hat eine zufällige planare Struktur mit hoher Wahrscheinlichkeit? Um diese Fragen zu beantworten, werden die planaren Strukturen in Teile mit höherer Konnektivität zerlegt. Für die asymptotische Enumeration wird zuerst die Zerlegung als das Gleichungssystem der generierenden Funktionen interpretiert. Auf dem Gleichungssystem wird dann Singularitätenanalyse angewendet. Für die exakte Enumeration und zufällige Erzeugung wird die rekursive Methode verwendet. Für die typischen Eigenschaften wird die probabilistische Methode auf asymptotischer Anzahl angewendet. Des Weiteren werden zufällige Graphen-Prozesse untersucht. Zufällige Graphen wurden zuerst von Erdos und Renyi eingeführt und untersucht weitgehend seitdem. Ein zufälliger Graphen-Prozess ist eine Markov-Kette, deren Zustandsraum eine Menge der Graphen mit einer gegebenen Knotenmenge ist. Der Prozess fängt mit isolierten Konten an, und in jedem Ablaufschritt entsteht ein neuer Graph aus dem aktuellen Graphen durch das Hinzufügen einer neuen Kante entsprechend einer vorgeschriebenen Regel. Typische Fragen sind: - Wie ändert sich die Wahrscheinlichkeit, dass ein von einem zufälligen Graphen-Prozess erzeugter Graph zusammenhängend ist? - Wann erfolgt der Phasenübergang? - Wie groß ist die größte Komponente? In dieser Habilitationsschrift werden diese Fragen über zufällige Graphen-Prozesse mit Gradbeschränkungen beantwortet. Dafür werden probabilistische Methoden, insbesondere Differentialgleichungsmethode, Verzweigungsprozesse, Singularitätsanalyse und Fourier-Transformationen, angewendet. / This thesis focuses on two kinds of discrete structures: planar structures, such as planar graphs and subclasses of them, and random graphs, particularly graphs generated by random processes. We study first planar structures from the following aspects. - How many of them are there (exactly or asymptotically)? - How can we efficiently sample a random instance uniformly at random? - What properties does a random planar structure have, with high probability? To answer these questions we decompose the planar structures along the connectivity. For the asymptotic enumeration we interpret the decomposition in terms of generating functions and derive the asymptotic number, using singularity analysis. For the exact enumeration and the uniform generation we use the recursive method. For typical properties of random planar structures we use the probabilistic method, together with the asymptotic numbers. Next we study random graph processes. Random graphs were first introduced by Erdos and Renyi and studied extensively since. A random graph process is a Markov chain whose stages are graphs on a given vertex set. It starts with an empty graph, and in each step a new graph is obtained from a current graph by adding a new edge according to a prescribed rule. Recently random graph processes with degree restrictions received much attention. In the thesis, we study random graph processes where the minimum degree grows quite quickly with the following questions in mind: - How does the connectedness of a graph generated by a random graph process change as the number of edges increases? - When does the phase transition occur? - How big is the largest component? To investigate the random graph processes we use the probabilistic method, Wormald''s differential equation method, multi-type branching processes, and the singularity analysis.

Dans le contexte médico-chirurgical, la robotique peut être d’un grand intérêt pour des procédures plus sûres et plus précises. Les contraintes d’encombrement sont cependant très fortes et des mobilités complexes peuvent être nécessaires. A ce jour, la conception de mécanismes non conventionnels dédiés est alors difficile à réaliser faute d’outils génériques permettant une évaluation rapide de leurs performances. Cette thèse associe la continuation de haut-degré et la différentiation automatique pour répondre à cette problématique en introduisant une méthode de modélisation et un formalisme génériques pour la conception de mécanismes. Nos contributions concernent en particulier le développement d’outils numériques pour l’évaluation de l’espace de travail, et de la localisation et la nature des singularités d’un mécanisme, et une analyse de sensibilité de haut-degré. Ceux-ci sont évalués sur des mécanismes de référence. / In the medical and surgical background, robotics can be of great interest for safer and more accurate procedures. Size constraints are however strong and complex movements may be necessary. To date, the design of dedicated non-conventional mechanisms is then a difficult task because of a lack of generic tools allowing a fast evaluation of their performances. This thesis combines higher-order continuation and automatic differentiation to adress this issue through the introduction of a generic modelling method and a generic formalism for mechanism design. Our contributions especially concern the development of numerical tools for the evaluation of the workspace, of the singularity localization and nature, and for a higher-order sensitivity analysis. These tools are evaluated on reference mechanisms.

The present work was developed in the Postgraduate Program in Mathematics Education and Physics Teaching (PPGEM & EF) of the Federal University of Santa Maria (UFSM) and inserted in the line of research Teaching and Learning of Mathematics and its Philosophical, Historical and Epistemological Foundations. Based on interviews with the pedagogical advisor of the 8ª Regional Education Coordination (8ª CRE), responsible for the Special Education sector, with teachers of Mathematics from state schools of Basic Education and the analysis of official documents of the Brazilian legislation on Special Education and Inclusive Education, curricula and pedagogical projects of undergraduate courses in Mathematics, this research aimed to investigate how these Mathematics teachers organize their classes for teaching in classes that have students included. For the data collection, semi-structured interviews were carried out with five Mathematics teachers from the state public school system who work or have already worked with students included in the regular education and with the pedagogical advisor responsible for the Special Education sector of the 8ª CRE. We used as a theoretical methodological reference for conducting these interviews Oral History, based on the ideas of José Carlos Sebe Bom Meihy (1996) and, mainly, the works developed by the Oral History and Mathematics Education Research Group, coordinated by Antonio Vicente Marafioti Garnica (2005, 2007, 2011, 2013). The analysis of the interviews was performed - according to the work carried out by Maria Edneia Martins-Salandim - in two moments: analysis of singularities and analysis of convergences. In the analysis of singularities we try to record what is characteristic of each deponent in his narrative, its peculiarities and its particularities. In the analysis of convergences we seek to confront the narratives of our deponents with what is written in the official documents of the legislation on Special Education and Inclusive Education and with the grades and pedagogical projects of the degree courses in Mathematics of the institutions where the teachers interviewed concluded the course Training. For this, we list three units of analysis: initial training, continuing training and work with included students. We emphasize, from the analysis carried out, that the Mathematics teachers interviewed consider that the work carried out with students included in regular education is hard work and difficult to be developed, since they did not receive orientation in the initial training course to work with these students. In addition, they have taken part in a few continuing education courses related to this topic, and the 8ª CRE does not provide guidelines for this work that be aimed at teachers in areas other than Special Education. / O presente trabalho foi desenvolvido junto ao Programa de Pós-Graduação em Educação Matemática e Ensino de Física (PPGEM&EF) da Universidade Federal de Santa Maria (UFSM) e inserido á linha de pesquisa Ensino e Aprendizagem da Matemática e seus Fundamentos Filosóficos, Históricos e Epistemológicos. Com base em entrevistas com o assessor pedagógico da 8ª Coordenadoria Regional de Educação (8ª CRE), responsável pelo setor de Educação Especial, com professores de Matemática de escolas estaduais da Educação Básica e na análise de documentos oficiais da legislação brasileira sobre Educação Especial e Educação Inclusiva, de currículos e projetos pedagógicos de cursos de Licenciatura em Matemática, essa pesquisa teve como objetivo investigar como esses professores de Matemática organizam suas aulas para a docência em turmas que possuem alunos incluídos. Para a coleta dos dados foram realizadas entrevistas semiestruturadas com cinco professoras de Matemática da rede pública estadual de ensino que trabalham ou já trabalharam com alunos incluídos no ensino regular e com a assessora pedagógica responsável pelo setor de Educação Especial da 8ª CRE. Utilizamos como referencial teórico metodológico para a realização dessas entrevistas a História Oral, nos fundamentando nas ideias de José Carlos Sebe Bom Meihy (1996) e, principalmente, nos trabalhos desenvolvidos pelo grupo de pesquisa Grupo História Oral e Educação Matemática, coordenado por Antonio Vicente Marafioti Garnica (2005; 2007; 2011; 2013). A análise das entrevistas foi realizada – conforme o trabalho realizado por Maria Edneia Martins-Salandim – em dois momentos: análise de singularidades e análise de convergências. Na análise de singularidades procuramos registrar o que é característico de cada depoente em sua narrativa, suas peculiaridades e suas particularidades. Na análise de convergências buscamos confrontar as narrativas dos nossos depoentes com o que está escrito nos documentos oficiais da legislação sobre Educação Especial e Educação Inclusiva e com as grades e os projetos pedagógicos dos cursos de Licenciatura em Matemática das instituições onde as professoras entrevistadas concluíram o curso de formação inicial. Para isso, elencamos três unidades de análise: formação inicial, formação continuada e trabalho com alunos incluídos. Destacamos, a partir da análise realizada, que as professoras de Matemática entrevistadas consideram que o trabalho realizado com alunos incluídos no ensino regular é um trabalho árduo e difícil de ser desenvolvido, pois elas não receberam, no curso de formação inicial, orientações para trabalhar com esses alunos. Além disso, participaram de poucos cursos de formação continuada relacionados a esse tema e a 8ª CRE também não fornece orientações para esse trabalho que sejam voltadas aos professores de áreas que não seja a Educação Especial.

La combinatoire analytique étudie le comportement asymptotique des suites à travers les propriétés analytiques de leurs fonctions génératrices. Ce domaine a conduit au développement d’outils profonds et puissants avec de nombreuses applications. Au delà de la théorie univariée désormais classique, des travaux récents en combinatoire analytique en plusieurs variables (ACSV) ont montré comment calculer le comportement asymptotique d’une grande classe de fonctions différentiellement finies:les diagonales de fractions rationnelles. Cette thèse examine les méthodes de l’ACSV du point de vue du calcul formel, développe des algorithmes rigoureux et donne les premiers résultats de complexité dans ce domaine sous des hypothèses très faibles. En outre, cette thèse donne plusieurs nouvelles applications de l’ACSV à l’énumération des marches sur des réseaux restreintes à certaines régions: elle apporte la preuve de plusieurs conjectures ouvertes sur les comportements asymptotiques de telles marches,et une étude détaillée de modèles de marche sur des réseaux avec des étapes pondérées. / The field of analytic combinatorics, which studies the asymptotic behaviour ofsequences through analytic properties of their generating functions, has led to thedevelopment of deep and powerful tools with applications across mathematics and thenatural sciences. In addition to the now classical univariate theory, recent work in thestudy of analytic combinatorics in several variables (ACSV) has shown how to deriveasymptotics for the coefficients of certain D-finite functions represented by diagonals ofmultivariate rational functions. This thesis examines the methods of ACSV from acomputer algebra viewpoint, developing rigorous algorithms and giving the firstcomplexity results in this area under conditions which are broadly satisfied.Furthermore, this thesis gives several new applications of ACSV to the enumeration oflattice walks restricted to certain regions. In addition to proving several openconjectures on the asymptotics of such walks, a detailed study of lattice walk modelswith weighted steps is undertaken.