The topic of this thesis is the didactics of mathematical analysis. The thesis describes selected observations from the reeducation in an individual tutoring environment of for- mal knowledge of university students in the field of calculus. The aim of the thesis is to describe what formal knowledge appeared, to describe and evaluate selected reeducation interventions and on this basis formulate appropriate methodological recommendation. In the first chapter we deal with the contradiction between definition and concept concept of students, we outline how to convey to students the purpose of definitions and we suggest how to teach students to work with definitions properly, including understanding quan- tified propositions. In the second chapter we present the theory of process and concept together with the generic model theory. In the third chapter we explain the methods of work with students and the methods of the analysis of videos from tutoring. In the fourth chapter we analyze cognitive processes of the concept of sequence limits. KEYWORDS reeducation, individual tutoring, mechanical knowledge, calculus, definitions, quantified proposition, infinity, sequence, limit 1

Untersucht wird das Cauchy Problem für degenerierte $p$-Evolutionsgleichungen. Dabei kann für Gleichungen höherer Ordnung in $D_t$, die nur von der Zeit abhängen, gezeigt werden, dass das Problem $H^\\infinity$ korrekt ist. Dafür werden gewisse Bedingungen an die Koeffizienten und deren erste Ableitungen gestellt. $H^\\infinity$ korrekt bedeutet dabei, dass die Anfangsdaten $u_0\\in H^s$, $u_1$ in einem dazugehörigen Sobolevraum und die Lösung bezüglich $x$ in $H^{s-s_0}$ liegen. Eine Notwendigkeit für die Bedingungen kann allerdings nicht gezeigt werden. Auch ist offen, ob der Regularitätsverlust wirklich eintritt. Später wird der Beweis erweitert um das Ergebniss für Koeffizienten zu zeigen, die in gewisser Weise auch vom Ort abhängen können. Im zweiten Teil der Dissertation geht es um Korrektheit für degenerierte $p$-Evolutionsgleichungen mit zeitabhängigen Koeffizienten und zweiter Ordnung in $D_t$. Gefordert werden Bedingungen an die Koeffizienten und die ersten beiden Ableitungen bezüglich der Zeit. Damit wird gezeigt, dass diese in Skalen von Sobolevräumen korrekt gestellt sind. Abschließend wird die Schärfe der Bedingungen und das tatsächliche Auftreten des Regularitätsverlustes in der Lösung bewiesen.

This Ph.D. thesis deals with E1-ring structures on the Hermitian K-theory in the motivic setting, more precisely, the existence of such structures on the motivic spectrum representing the hermitianK-theory is proven. The presence of such structure is established through two different approaches. In both cases, we consider the category of algebraic vector bundles over a scheme, with the usual requirements to do motivic homotopy theory. This category has two natural symmetric monoidal structures given by the direct sum and the tensor product, together with a duality coming from the functor represented by the structural sheaf. The first symmetric monoidal structure is the one that we are going to group complete along this text, and we will see that the second one, the tensor product, is preserved giving rise to an E1-ring structure in the resulting spectrum. In the first case, a classic infinite loop space machine applies to the hermitian category of the category of algebraic vector bundles over a scheme. The second approach abords the construction using a new hermitian infinite loop space machine which uses the language of infinity categories. Both assemblies applied to our original category have like output a presheaf of E1-ring spectra. To get an spectrum representing the hermitian K-theory in the motivic context we need a motivic spectrum, i.e, a P1-spectrum. We use a delooping construction at the end of the text to obtain a presheaf of E1-ring P1-spectra from the two presheaves of E1-ring spectra indicated above.

This dissertation advances tools for the certification of unmanned aircraft system (UAS) flight controllers. We develop two thrusts to this goal: (1) the validation and improvement of an uncertain UAS framework based on integral quadratic constraint (IQC) theory and (2) the development of novel IQC theorems which allow the analysis of uncertain systems having time-varying characteristics. Pertaining to the first thrust, this work improves and implements an IQC-based robustness analysis framework for UAS. The approach models the UAS using a linear fractional transformation on uncertainties and conducts robustness analysis on the uncertain system via IQC theory. By expressing the set of desired UAS flight paths with an uncertainty, the framework enables analysis of the uncertain UAS flying about any level path whose radius of curvature is bounded. To demonstrate the versatility of this technique, we use IQC analysis to tune trajectory-tracking and path-following controllers designed via H2 or H-infinity synthesis methods. IQC analysis is also used to tune path-following PID controllers. By employing a non-deterministic simulation environment and conducting numerous flight tests, we demonstrate the capability of the framework in predicting loss of control, comparing the robustness of different controllers, and tuning controllers. Finally, this work demonstrates that signal IQCs have an important role in obtaining IQC analysis results which are less conservative and more consistent with observations from flight test data. With regards to the second thrust, we prove a novel theorem which enables robustness analysis of uncertain systems where the nominal plant and the IQC multiplier are linear time-varying systems and the nominal plant may have a non-zero initial condition. When the nominal plant and the IQC multiplier are eventually periodic, robustness analysis can be accomplished by solving a finite-dimensional semidefinite program. Time-varying IQC multipliers are beneficial in analysis because they provide the possibility of reducing conservatism and are capable of expressing uncertainties that have unique time-domain characteristics. A number of time-varying IQC multipliers are introduced to better describe such uncertainties. The utility of this theorem is demonstrated with various examples, including one which produces bounds on the UAS position after an aggressive Split-S maneuver. / Doctor of Philosophy / This work develops tools to aid in the certification of unmanned aircraft system (UAS) flight controllers. The forthcoming results are founded on robust control theory, which allows the incorporation of a variety of uncertainties in the UAS mathematical model and provides tools to determine how robust the system is to these uncertainties. Such a foundation provides a complementary perspective to that obtained with simulations. Whereas simulation environments provide a probabilistic-type analysis and are oftentimes costly, the following results provide worst-case guarantees—for the allowable disturbances and uncertainties—and require far less computational resources. Here we take two approaches in our development of certification tools for UAS. First we validate and improve on an uncertain UAS framework that relies on integral quadratic constraint (IQC) theory to analyze the robustness of the UAS in the presence of uncertainties and disturbances. Our second approach develops novel IQC theorems that can aid in providing bounds on the UAS state during its flight trajectory. Though the applications in this dissertation are focused on UAS, the theory can be applied to a wide variety of physical and nonphysical problems wherein uncertainties in the mathematical model cannot be avoided.

Selected aspects of Nicholas of Cusa's cosmology and anthropology This thesis targets selected topics of theology of creation and of man, that is, theological cosmology and anthropology in the work of a Renaissance philosopher, theologian, mathematician and scientist Nicholas of Cusa. I shall first introduce Cusanus' curriculum vitae, it's historical context and basic characteristics of theological anthropology in the work of Cusanus. After this, I shall present in greater detail selected topics of his mathematical theology, including historical context of this specific discourse on God, world and man using language of mathematics. This includes mathematical description of both created world and of theological reality, e.g. God's transcendence and immanence, the place of man in the universe, or the role of universe which has a mediating role between God and man in the works of Cusanus. Mathematical language used by Cusanus is part of a wide stream of Neoplatonist philosophical and theological tradition. Cusanus is inspired not only by pre-Christian philosophical traditions (Pythagorean school, Plotinus' concept of One), but also by Greek patristic writers like Gregory of Nyssa, Maximus the Confessor, Pseudo-Dionysius the Areopagite. In his concept of "great conjecture" Cusanus presents a model of...

Les travaux présentés dans cette thèse ont été effectués au Centre de Recherche en Automatique de Nancy (CRAN). Ils portent sur l'observation et la commande des systèmes singuliers non linéaires. Dans un premier temps nous nous sommes intéressés à la synthèse d'observateur et au filtrage H infini des systèmes singuliers bilinéaires. Dans un deuxième temps, nous avons étudié la synthèse d'observateur pour les systèmes singuliers non linéaires Lipschitziens. La dernière partie de ce travail concerne la stabilisation et la commande basée observateurs des systèmes singuliers non linéaires. L'objectif de ce travail a été de proposer des résultats facilement implémentables et de couvrir une large classe de systèmes non linéaires. La contribution principale de ce mémoire a été de proposer des observateurs H infini pour les systèmes singuliers non linéaires, en utilisant le non biais de l'erreur d'estimation. Les paramètres de ces observateurs sont obtenus par la résolution des inégalités matricielles linéaires (LMIs). Le deuxième apport concerne la synthèse de commande stabilisante et l'utilisation d'un des observateurs proposés dans cette thèse pour la synthèse d'une commande basée observateur pour les systèmes singuliers non linéaires. Cette dernière est réalisée grâce à la réécriture des fonctions non linéaires sous des formes adéquates à l'application de la commande des systèmes / This thesis work is realized in the Research Center in Automatic Control of Nancy (CRAN). It concerns the observation and control of nonlinear singular systems. Firstly, we were interested in the observer design and H infinity filtering for singular bilinear systems. In a second step, we studied the observers design for Lipschitz nonlinear singular systems. The last part of this work relates to the stabilization and observer based controller for a classe of singular nonlinear systems. The objective is to develop a simple and straightforward results which covers a large class of nonlinear systems. The main contribution of this thesis is in the H infinity observers design for nonlinear singular systems. It is based on the parametrization of the solution of the constrained generalized Sylvester equation. The second contribution relates to the design of stabilizing control and using the proposed observer to design an obsever based controller for nonlinear singular systems. Solutions of these problems are obtained by using Linear Matrix Inequalities (LMI) Formulation

Made available in DSpace on 2015-03-04T18:50:44Z (GMT). No. of bitstreams: 1 Introducao.pdf: 140805 bytes, checksum: fc7ea84f193f6d764fa24f41af40d07f (MD5) Previous issue date: 2007-03-09 / Fundação Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro / Este trabalho trata do problema de controle H-infinito de uma classe de sistemas lineares com saltos Markovianos (MJLS) a tempo contínuo, onde a cadeia de Markov toma valores em um conjunto infinito enumerável. Um bounded real lemma (que chamamos JBRL) é desenvolvido, estabelecendo que a factibilidade de um conjunto infinito de desigualdades matriciais lineares (LMIs) interconectadas é necessária e suficiente para que um dado sistema seja estocasticamente estável (SS) e atenda a um desempenho H-infinito prescrito. O problema H-infinito estudado consiste na atenuação do efeito que perturbações estocásticas de energia finita causam na saída de um sistema, no pior caso. Neste problema, conhecido na literatura como "disturbance attenuation" (DA), assumimos ainda que o controlador somente tem acesso ao processo de saltos e a uma saída do sistema. Os controladores de interesse devem garantir que tanto a estabilidade (SS) quanto um desempenho H-infinito sejam observados no sistema em malha fechada - donde as condições impostas pelo JBRL são determinantes para a existência de soluções. Um importante aspecto dessa nova abordagem é que ferramentas tão fundamentais quanto o Complemento de Schur ou o Lema da Projeção, p.ex., não podem mais ser usados para manipular os conjuntos de LMIs infinitamente acopladas - tal dificuldade é contornada pela introdução de versões estendidas desses resultados, no início do trabalho. Um dos principais resultados deste trabalho caracteriza a existência de soluções através de dois problemas LMI complementares, um dos quais torna possível o design computacional de controladores. Por fim, são apresentados algoritmos para a construção prática de controladores, ótimos ou sub-ótimos, dando origem a um conjunto de ferramentas que, especialmente no caso em que a cadeia de Markov é finita, podem ser implementadas computacionalmente de maneira imediata. Mesmo no caso finito, os resultados da tese são mais fortes do que aqueles atualmente encontrados na literatura.

La stabilisation inertielle de ligne de visée est essentiellement un problème de rejet de perturbations : il faut rendre la ligne de visée de la caméra embarquée dans le viseur insensible aux mouvements du porteur. Les méthodes de commande robuste du type H-infini sont bien adaptées à la résolution de ce type de problème, et plus particulièrement l’approche Loop-Shaping qui repose sur des concepts de réglage de l’automatique fréquentielle classique. Cependant, les correcteurs obtenus via cette approche sont généralement d’ordre élevé et donc difficilement implémentables sur le calculateur embarqué du viseur.Dans cette thèse, nous avons proposé des méthodologies de synthèse de correcteurs robustes d’ordre réduit et/ou de structure fixée. Pour cela, nos travaux ont été axés sur :- L’optimisation pour la synthèse H-infini à ordre et/ou structure fixée. Tout d’abord nous avons exploré les possibilités offertes par l’optimisation sous contraintes LMI (Linear Matrix Inequalities). Celles-ci se sont avérées limitées, bien que de nombreux algorithmes aient été proposés dans ce cadre depuis le début des années 90. Ensuite, nous avons opté pour l’optimisation non lisse. En effet des outils numériques récemment développés rendent accessible cette approche, et leur efficacité s’est avéré indéniable.- L’adaptation au cadre particulier du critère H-infini Loop-Shaping.La structure particulière de ce critère de synthèse a été exploitée afin de mieux prendre en compte les pondérations, et d’améliorer la réduction d’ordre du correcteur final. Enfin, une approche basée uniquement sur le réglage graphique d’un gabarit de gain fréquentiel en boucle ouverte est proposée. Ces différentes méthodologies sont illustrées, tout au long de la thèse, sur un viseur dont le modèle a été identifié à partir de mesures expérimentales. / Inertial line of sight stabilization is a disturbance rejection problem: the goal is to hold steady in the inertial space, the line of sight of a camera, which is carried on a mobile vehicle. H-infinity robust control techniques are well suited for this type of problem, in particular the Loop-Shaping approach which relies on classical frequency domain concepts. However, this approach results in high order controllers which are hardly implementable on the real time embedded electronic unit of the sight system.In this thesis, fixed order and fixed structure controller design methodologies are proposed. This development follows two main axis: - Fixed order H-infinity Optimization. First, fixed order controllers have been investigated through the LMI (Linear Matrix Inequalities) optimization framework. However the numerical efficiency of this approach is still limited, despite the large amount of research in this area since the 90’s. Then, we used recently developed and more efficient tools that recast the fixed order H-infinity synthesis problem as a nonsmooth optimization problem.- Adaptation to the H-infinity Loop-Shaping frameworkWe adapted the 4 block H-infinity criterion in order to include the weighting filters in the fixed order controller optimization, which enhance the final controller order reduction. Then, we proposed a fixed order controller design approach, based only on graphically tuning a target open loop frequency gain.

Let F : Kn &rarr; Kp be a polynomial map, where K = R or C. Motivated by the characterization of the integral closure of ideals in the ring On by means of analytic inequalities proven by Lejeune-Teissier [46], we define the set Sp(F) of special polynomials with respect to F. The set Sp(F) can be considered as a counterpart, in the context of polynomial maps Kn &rarr; Kp, of the notion of integral closure of ideals in the ring of analytic function germs (~&lceil;+. In this work, we are mainly interested in the determination of the convex region S0(F) formed by the exponents of the special monomials with respect to F. Let us fix a convenient Newton polyhedron &lceil; + ~&sube; Rn. We obtain an approximation to S0</sub (F) when F is strongly adapted to ~&sube; +, which is a condition expressed in terms of the faces of ~&lceil;+ and the principal parts at infinity of F. The local version of this problem has been studied by Bivià-Ausina [4] and Saia [71]. Our result about the estimation of S0(F) allows us to give a lower estimate for the Lojasiewicz exponent at infinity of a given polynomial map with compact zero set. As a consequence of our study of ojasiewicz exponents at infinity we have also obtained a result about the uniformity of the ojasiewicz exponent in deformations of polynomial maps Kn &rarr; Kp. Consequently we derive a result about the invariance of the global index of real polynomial maps Rn &rarr; Rn. As particular cases of the condition of F being adapted to ~&lceil;+ there appears the class of Newton non-degenerate polynomial maps at infinity and pre-weighted homogeneous maps. The first class of maps constitute a natural extension for maps of the Newton non-degeneracy condition introduced by Kouchnirenko for polynomial functions. We characterize the Newton non-degeneracy at infinity condition of a given polynomial map F : Kn &rarr; Kp in terms of the set S0((F, 1)), where (F, 1) : Kn &rarr; Kp+1 is the polynomial map whose last component function equals 1. Motivated by analogous problems in local algebra we also derive some results concerning the multiplicity of F. / Seja F :Kn &rarr; Kp uma aplicação polinomial, onde K = C ou K = R. Motivados pela caracterização do fecho integral de ideais no anel On por meio de desigualdades analíticas provadas por Lejeune-Teissier [46], definimos o conjunto Sp(F) de polinomios especiais com respeito a F. O conjunto Sp(F) pode ser considerado como um homólogo, no contexto das aplicações polinomiais Kn &rarr; Kp, da noção de fecho integral de ideais no anel de germes de funções analíticas (Kn 0) &rarr; K. Neste trabalho, estamos interessados principalmente na determinação da região convexa S0 (F) formado pelos expoentes dos monômios especiais com respeito a F. Fixado um poliedro de Newton conveniente ~&lceil; + ~&sube; Rn, é obtida uma aproximação de S0(F), quando F é fortemente adaptada a &lceil; + o qual é uma condição expressada em termos das faces de ~&lceil; + e as partes principais no infinito de F. A versão local deste problema foi estudado por Bivià-Ausina [4] e Saia [71]. Nosso resultado sobre a estimativa de S0(F) nos permite dar uma estimativa inferior para o expoente Lojasiewicz no infinito de uma aplicação polinomial Kn &rarr; Kp, com conjunto F-1(0) compacto. Como uma consequência do estudo dos expoentes de Lojasiewicz no infinito também foi obtido um resultado sobre a uniformidade do expoente Lojasiewicz em deformações de aplicações polinomiais Kn &rarr; Kp e consequentemente, um resultado sobre a invariância do índice global de aplicações polinomiais reais Rn &rarr; Rn. Como casos particulares da condição de F ser adaptada a ~&lceil; + aparecem a classe de aplicações polinomiais Newton não degeneradas e as aplicações polinomiais pre-quase homogêneas. A primeira classe de aplicações constitui uma extensão natural da condição Newton não-degeneração introduzida por Kouchnirenko para funções polinomiais. Caracterizamos a condição Newton não-degeneração para uma determinada aplicação polinomial F : Kn &rarr; Kp em termos do conjunto S0((F, 1)), onde (F, 1) : Kn &rarr; Kp+1 é a aplicação polinomial cuja última função componente é igual a 1. Motivados por problemas análogos em álgebra local, também obtivemos alguns resultados sobre a multiplicidade de F.

La conception des systèmes d'entrainement de bande est étudiée depuis de nombreuses années. Ces systèmes sont très répandus dans l'industrie puisque le conditionnement sous forme de bobines simplifie le traitement de nombreux matériaux tels que le papier, le carton, les polymères, ... Ces systèmes regroupent un grand nombre de difficultés : ils sont de grande dimension, un fort couplage existe entre les grandeurs et de nombreux paramètres varient au cours du temps. La méthode d’optimisation classique consiste à considérer chaque partie du système sans tenir compte des autres. De plus, au sein d'un même sous-ensemble, l’optimisation est réalisée discipline par discipline. Une nouvelle approche est proposée et appliquée à la synthèse de la commande : il s’agit de considérer le système global en prenant en compte la robustesse paramétrique. Ensuite, le choix optimal du tracteur maître et de la technologie d'asservissement de la tension de bane associée sont étudiées. / The design of roll-to-roll systems is studied for several years. This kind of system is very common in industry because the wound roll packaging simplify the treatment of material such as paper, cardboard, polymers, metal ... The studied systems have a high number of difficulties: they are large scale systems, a high coupling between physical values exists and several parameters are time dependent. The classical optimisation method consists in considering separately each subsystem without taking into account interactions. Moreover, the classical optimisation is made for each scientific field. A new approach is developed and applied to controller synthesis: the controllers are synthesized considering the global system with parametric uncertainties. The optimal choice of the master roller position and the technology used to control web tension are then studied.