In der Finanzmathematik hat der Besitzer einer amerikanische Option das Recht aber nicht die Pflicht, eine Aktie innerhalb eines bestimmten Zeitraums, für einen bestimmten Preis zu kaufen oder zu verkaufen. Die Bewertung einer amerikanische Option wird als so genanntes optimale stopping Problem formuliert. Erfolgt die Modellierung des Aktienkurses durch eine geometrische Brownsche Bewegung, wird der Wert einer amerikanischen Option durch ein deterministisches freies Randwertproblem (FRWP), oder einer äquivalenten Variationsungleichung (VU) auf ganz R in gewichteten Sobolev Räumen gegeben. Um Standardmethoden der Numerischen Mathematik anzuwenden, wird das unbeschränkte Gebiet zu einem beschränkten Gebiet abgeschnitten. Mit Hilfe der Fourier-Transformation wird eine Integraldarstellung der Lösung die den freien Rand explizit beinhaltet, hergeleitet. Mittels dieser Integraldarstellung werden Abschneidefehlerschranken bewiesen. Danach werden gewichtete Poincare Ungleichungen mit expliziten Konstanten bewiesen. Der Abschneidefehler und die gewichtete Poincare Ungleichung ermöglichen, einen zuverlässigen a posteriori Fehlerschätzer zwischen der exakten Lösung der VU und der semidiskreten Lösung des penalisierten Problems auf R herzuleiten. Eine hinreichend glatte Lösung der VU garantiert die Konvergenz der Lösung des penaltisierten Problems zur Lösung der VU. Ein a priori Fehlerschätzer für den Fehler zwischen der exakten Lösung der VU und der semidiskreten Lösung des penaltisierten Problems beendet die numerische Analysis. Die eingeführten aposteriori Fehlerschätzer motivieren einen Algorithmus für adaptive Netzverfeinerung. Numerische Experimente zeigen die verbesserte Konvergenz des adaptiven Verfahrens gegenüber der uniformen Verfeinerung. Der zuverlässige a posteriori Fehlerschätzer ermöglicht es, den Abschneidepunkt so zu wählen, dass der Gesamtfehler (Diskretisierungsfehler plus Abschneidefehler) kleiner als eine gegebenen Toleranz ist. / Among the central concerns in mathematical finance is the evaluation of American options. An American option gives the holder the right but not the obligation to buy or sell a certain financial asset within a certain time-frame, for a certain strike price. The valuation of American options is formulated as an optimal stopping problem. If the stock price is modelled by a geometric Brownian motion, the value of an American option is given by a deterministic parabolic free boundary value problem (FBVP) or equivalently a non-symmetric variational inequality (VI) on weighted Sobolev spaces on R. To apply standard numerical methods, the unbounded domain R is truncated to a bounded one. Applying the Fourier transform to the FBVP yields an integral representation of the solution including the free boundary explicitely. This integral representation allows to prove explicit truncation errors. Since the VI is formulated within the framework of weighted Sobolev spaces, we establish a weighted Poincare inequality with explicit determined constants. The truncation error estimate and the weighted Poncare inequality enable a reliable a posteriori error estimate between the exact solution of the VI and the semi-discrete solution of the penalised problem on R. A sufficient regular solution provides the convergence of the solution of the penalised problem to the solution of the VI. An a priori error estimate for the error between the exact solution of the VI and the semi-discrete solution of the penalised problem concludes the numerical analysis. The established a posteriori error estimates motivates an algorithm for adaptive mesh refinement. Numerical experiments show the improved convergence of the adaptive algorithm compared to uniform mesh refinement. The reliable a posteriori error estimate including explicit truncation errors allows to determine a truncation point such that the total error (discretisation and truncation error) is below a given error tolerance.

In dieser Arbeit untersuchen wir das Critical Slowing down der Gitter-QCD Simulationen. Wir führen eine Vorstudie in der quenched Approximation durch, in der wir feststellen, dass unsere Schätzung der exponentiellen Autokorrelation wie $\tauexp(a) \sim a^{-5} $ skaliert, wobei $a$ der Gitterabstand ist. In unquenched Simulationen mit O(a)-verbesserten Wilson-Fermionen finden wir ein ähnliches Skalierungsgesetz. Die Diskussion wird von einem gro\ss{}en Satz an Ensembles sowohl in reiner Eichtheorie als auch in der Theorie mit zwei entarteten Seequarks unterstützt. Wir haben darüber hinaus die Wirkung von langsamen algorithmischen Modi in der Fehleranalyse des Erwartungswertes von typischen Gitter-QCD-Observablen (hadronische Matrixelemente und Massen) untersucht. Im Kontext der Simulationen, die durch langsame Modi betroffen sind, schlagen wir vor und testen eine Methode, um zuverlässige Schätzungen der statistischen Fehler zu bekommen. Diese Methode soll in dem typischen Simulationsbereich der Gitter-QCD helfen, nämlich dann, wenn die gesamte erfasste Statistik O(10)\tauexp ist. Dies ist der typische Fall bei Simulationen in der Nähe des Kontinuumslimes, wo der Rechenaufwand für die Erzeugung von zwei unabhängigen Datenpunkten sehr gro\ss{} sein kann. Schlie\ss{}lich diskutieren wir die Skalenbestimmung in N_f=2-Simulationen mit der Kaon Zerfallskonstante f_K als experimentellem Input. Die Methode wird zusammen mit einer gründlichen Diskussion der angewandten Fehleranalyse erklärt. Eine Beschreibung der öffentlich zugänglichen Software, die für die Fehleranalyse genutzt wurde, ist eingeschlossen. / In this work we investigate the critical slowing down of lattice QCD simulations. We perform a preliminary study in the quenched approximation where we find that our estimate of the exponential auto-correlation time scales as $\tauexp(a)\sim a^{-5}$, where $a$ is the lattice spacing. In unquenched simulations with O(a) improved Wilson fermions we do not obtain a scaling law but find results compatible with the behavior that we find in the pure gauge theory. The discussion is supported by a large set of ensembles both in pure gauge and in the theory with two degenerate sea quarks. We have moreover investigated the effect of slow algorithmic modes in the error analysis of the expectation value of typical lattice QCD observables (hadronic matrix elements and masses). In the context of simulations affected by slow modes we propose and test a method to obtain reliable estimates of statistical errors. The method is supposed to help in the typical algorithmic setup of lattice QCD, namely when the total statistics collected is of O(10)\tauexp. This is the typical case when simulating close to the continuum limit where the computational costs for producing two independent data points can be extremely large. We finally discuss the scale setting in Nf=2 simulations using the Kaon decay constant f_K as physical input. The method is explained together with a thorough discussion of the error analysis employed. A description of the publicly available code used for the error analysis is included.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This paper proposes to investigate the difficulties presented by the students of the 9th year of elementary school in activities that involve the operations with fractions applied in a related function. The study is based on a qualitative approach and the idea that the analysis of students' written production can be worked as a teaching methodology (CURY, 2008). Data collection was performed through three questionnaires and a review activity applied to the students of a school in the interior of the state of Sergipe. A second approach was made around error analysis as a teaching methodology, by presenting and discussing with students the mistakes made by them in Questionnaire 1, trying to make them understand the reasoning that led them to make such mistakes, and from this overcome their difficulties, which was analyzed in the following questionnaires. / Este trabalho se propõe a investigar quais são as dificuldades apresentadas pelos estudantes do 9º ano do Ensino Fundamental em atividades que envolvem as operações com frações aplicadas numa função afim. O estudo está fundamentado numa abordagem qualitativa e na ideia de que a análise da produção escrita dos alunos pode ser trabalhada como metodologia de ensino (CURY, 2008). A coleta de dados foi realizada por meio de três questionários e uma Atividade de Análise do Erro aplicados aos alunos de uma escola do interior do estado de Sergipe. Foi feita uma segunda abordagem em torno da análise de erro como metodologia de ensino, ao apresentar e discutir com os estudantes os erros cometidos por eles no Questionário 1, procurando fazer com que eles entendam o raciocínio que os levou a cometer tais erros e, dessa forma, superarem suas dificuldades, o que foi analisado nos questionários seguintes. / Itabaiana, SE

L’enrichissement des éléments finis standard est un outil performant pour améliorer la qualité d’approximation. L’idée principale de cette approche est d’ajouter aux fonctions de base un ensemble de fonctions censées améliorer la qualité des solutions approchées. Le choix de ces dernières est crucial et est en grande partie basé sur la connaissance a priori de quelques informations telles que les caractéristiques de la solution, de la géométrie du problème à résoudre, etc. L’efficacité de cette approche pour résoudre une équation aux dérivées partielles dans un maillage fixe, sans avoir recours au raffinement, a été prouvée dans de nombreuses applications dans la littérature. La clé de son succès repose principalement sur le bon choix des fonctions de base et plus particulièrement celui des fonctions d’enrichissement. Une question importante se pose alors : quelles conditions faut-il imposer sur les fonctions d’enrichissement afin qu’elles génèrent des éléments finis bien définis ?Dans cette thèse sont abordés différents aspects d’une approche générale d’enrichissement d’éléments finis. Notre première contribution porte principalement sur l’enrichissement de l’élément fini du type Q_1. Par contre, notre seconde contribution, certainement la plus importante, met l’accent sur une approche plus générale pour enrichir n’importe quel élément fini qu’il soit P_k, Q_k ou autres, conformes ou non conformes. Cette approche a conduit à l’obtention des versions enrichies de l’élément de Han, l’élément de Rannacher-Turek et l’élément de Wilson, qui font maintenant partie des codes d’éléments finis les plus couramment utilisés en milieu industriel. Pour établir ces extensions, nous avons eu recours à l’élaboration de nouvelles formules de quadrature multidimensionnelles appropriées généralisant les formules classiques bien connues en dimension 1, dites du “point milieu,” des “trapèzes” et de leurs versions perturbées, ainsi que la formule de Simpson. Elles peuvent être vues comme des extensions naturelles de ces formules en dimension supérieure. Ces dernières, en plus de leurs tests numériques implémentés sous MATLAB, version R2016a, ont fait l’objet de notre troisième contribution. Nous mettons particulièrement l’accent sur la détermination explicite des meilleures constantes possibles apparaissant dans les estimations d’erreur pour ces formules d’intégration. Enfin, dans la quatrième contribution nous testons notre approche pour résoudre numériquement le problème d’élasticité linéaire à l’aide d’un maillage rectangulaire. Nous effectuons l’analyse numérique aussi bien l’analyse de l’erreur d’approximation et résultats de convergence que l’analyse de l’erreur de consistance. Nous montrons également comment cette dernière peut être établie à n’importe quel ordre, généralisant ainsi certains travaux menés dans le domaine. Nous réalisons la mise en œuvre de la méthode et donnons quelques résultats numériques établis à l’aide de la bibliothèque libre d’éléments finis GetFEM++, version 5.0. Le but principal de cette partie sert aussi bien à la validation de nos résultats théoriques, qu’à montrer comment notre approche permet d’élargir la gamme de choix des fonctions d’enrichissement. En outre, elle permet de montrer comment cette large gamme de choix peut aider à avoir des solutions optimales et également à améliorer la validité et la qualité de l’espace d’approximation enrichie. / The enrichment of standard finite elements is a powerful tool to improve the quality of approximation. The main idea of this approach is to incorporate some additional functions on the set of basis functions. These latter are requested to improve the accuracy of the approximate solution. Their best choice is crucial and is based on the knowledge of some a priori information, such as the characteristics of the solution, the geometry of the problem to be solved, etc. The efficiency of such an approach for finding numerical solutions of partial differential equations using a fixed mesh, without recourse to refinement, was proved in numerous applications in the literature. However, the key to its success lies mainly on the best choice of the basis functions, and more particularly those of enrichment functions.An important question then arises: How to suitably choose them, in such a way that they generate a well-defined finite element ?In this thesis, we present a general approach that enables an enrichment of the finite element approximation. This was the subject of our first contribution, which was devoted to the enrichment of the classical Q_1 element, as a first step. As a second step, in our second contribution, we have developed a more general framework for enriching any finite element either P_k, Q_k or others, conforming or nonconforming. As an illustration of how to use this framework to build new enriched finite elements, we have introduced the extensions of some well-known nonconforming finite elements, notably, Han element, Rannacher-Turek element and Wilson element, which are now part of the main code of finite element methods. To establish these extensions, we have introduced a new family of multivariate versions of the classical trapezoidal, midpoint and Simpson rules. These latter, in addition to their numerical tests under MATLAB, version R2016a, have been the subject of our third contribution. They may be viewed as an extension of the well-known trapezoidal, midpoint and Simpson’s one-dimensional rules to higher dimensions. We particularly pay attention to the explicit expressions of the best possible constants appearing in the error estimates for these cubatute formulas. Finally, in the fourth contribution we apply our approach to numerically solving the linear elasticity problem based on a rectangular mesh. We carry out the numerical analysis of the approximation error and also for the consistency error, and show how the latter can be established to any order. This constitutes a generalization of some work already done in the field. In addition to our theoretical results, we have also made some numerical tests, which were achieved by using the GetFEM++ library, version 5.0. The aim of this contribution was not only to confirm our theoretical predictions, but also to show how the new developed framework allows us to expand the range of choices of enrichment functions. Furthermore, we have shown how this wide choices range can help us to improve some approximation properties and to get the optimal solutions for the particular problem of elasticity.

No description available.

Résumé L’hypothèse de la période critique, émise par Lenneberg dans les années 60, affirmait qu’un enfant pouvait acquérir une langue seconde, sans difficulté, environ jusqu’à l’âge de la puberté. Après cette période, l’apprentissage d’un autre idiome serait difficile, dû à la latéralisation du cerveau. En même temps, les travaux de Chomsky enrichirent cette théorie avec l’idée de la Grammaire universelle, laquelle établit que nous possédons tous, dès la naissance, les éléments linguistiques universels qui nous permettent d’acquérir une langue maternelle. Tant que la Grammaire universelle est active, notre langue maternelle se développe et c’est pourquoi, si nous apprenons une autre langue pendant cette période, l’acquisition de celle-ci se produit de manière presque naturelle. Pour cette raison, plus une langue est apprise tôt, plus elle sera maîtrisée avec succès. En nous appuyant sur ce cadre théorique ainsi que sur l’Analyse d’erreurs, outil qui permet au professeur de prédire quelques erreurs avec la finalité de créer des stratégies d’apprentissage d’une langue seconde, nous tenterons de vérifier dans le présent travail si l’âge est un facteur qui influence positivement ou négativement l’apprentissage d’une langue seconde, l’espagnol dans ce cas-ci, par le biais de l’analyse comparative des prépositions a/ en dans deux groupes d’étudiants différents. / Abstract According to Leeneberg's critical period hypothesis from the sixties, a child can acquire a second language approximately until puberty, for afterwards the lateralization of the brain takes place and the acquisition of second language becomes more difficult. Further on, Chomsky’s research would contribute to this theory with the Universal Grammar idea or the inborn universal linguistic elements that we use in order to acquire mother tongue. While the Universal Grammar is still on, the already acquired Mother Tongue continues to develop. As a result, the acquisition of another language during this period becomes a natural process, leading to the conclusion that the earlier a child starts learning a second language, the better its acquisition will be. Therefore and for the purpose of this research, a theoretical framework and the error analysis, the tool allowing the teacher to predict some error in order to create the adequate language strategies for effective second language acquisition, are used. I will try to verify the age-factor influence positively or negatively in the acquisition of a second language, in this special case the Spanish language, by means of the comparative analysis of the a/ en prepositional pair with two different groups of apprentices.

En examinant les difficultés représentatives rencontrées par les étudiants chinois au cours de leur apprentissage du français langue étrangère (FLE), la présente thèse a pour objectif d’inventorier et analyser leurs erreurs récurrentes à l'écrit, à différents niveaux, en vue de proposer des procédés ou des suggestions qui pourraient éventuellement améliorer l’enseignement du FLE en Chine. La thèse est divisée en sept chapitres reposant sur l’analyse des erreurs à partir d'un point de grammaire précis - la voix passive, et puis, au niveau micro, sur différents aspects : perceptuo - mémoriels, métalinguistique et sémantico - pragmatiques, avec un bilan général à la fin. À l’issue de cette analyse, nous avons pu constater que les difficultés que les étudiants chinois rencontrent au cours de leur apprentissage sont dues principalement à l’écart entre le français et le chinois, leur langue maternelle, dont l'influence est partout sous-jacent. La présente recherche s’oriente vers la linguistique structurale et une analyse contrastive. En examinant cas par cas les différentes imperfections repérées dans notre corpus spécialement constitué à cette fin, nous essayons de découvrir les causes qui les sous-tendent. La deuxième étape est réservée à une étude de stratégie didactique. En nous appuyant sur le résultat de l'analyse précédente, nous procédons à une description statistique de notre matériel pour connaître la répartition des différentes catégories d’erreurs. En partant de cette image statistique, nous pouvons esquisser enfin des recommandations didactiques plus adéquates. / By examining the typical difficulties faced by Chinese students in their learning of French as foreign language (FLE in French), this thesis aims to list, categorize, and analyze their recurring errors at different levels in writing, in order to propose some proper suggestions that could possibly help improve the teaching of French as foreign language in China. The thesis is divided into seven chapters based on error analysis from a specific grammar point - the passive voice, and then, the thesis, at the micro level, on different aspects: perceptuo - memorial, meta-linguistics and semantic - pragmatic, analyzes the various types of written errors Chinese students make in their French learning, with a general assessment at the end. With previous analysis, the thesis concludes that the main causes of the difficulties Chinese students encounter in the French learning lie in the differences between French language and their mother tongue while the influence of the mother tongue is ubiquitous. It is with the reference of structural linguistics and contrastive linguistics in our analyzing process of the error cases, one by one from the materials specially gathered for this purpose, that the research has found the main causes that underlie the errors. And it is by using a statistical analysis of the corpus collected, that the research has also found the distribution of different categories of the errors, so that it provides a objective basis for the further improvement of French teaching in China.

Made available in DSpace on 2016-04-27T16:59:06Z (GMT). No. of bitstreams: 1 Jose Zucco.pdf: 7309797 bytes, checksum: e352e60fa42fc2fb548de2f8c6f4bc2f (MD5) Previous issue date: 2010-07-27 / Secretaria da Educação do Estado de São Paulo / The goal of this research is to analyze the performance of 20 high school seniors in the State Public School System in the resolution of four problems involving the concept of monotonic functions (increasing and decreasing). The selected problems were proposed in the Public School Mathematical Olympiads (OBMEP) of 2005, 2006, 2007, and 2008. The study has a qualitative approach. The theoretical reference for analyzing the performance was Raymond Duval s theory of Registers of Semiotic Representation (2003). The analyzed protocols contained both the resolution of the problems and the justifications presented by the students for their responses. The analyses indicated difficulties in interpreting the problem statement, as well as in the activities related to handling and changing the Registers of Semiotic Representation. This study also performed an analysis of the errors, which allowed the questions to be classified in groups based on the types of answers. The results revealed that most of the students were not familiar with the proposed problems, which required them to move among the different representations of a single mathematical object / A presente pesquisa tem por objetivo analisar o desempenho de 20 alunos do 3º ano do Ensino Médio da Rede Pública Estadual na resolução de quatro questões envolvendo o conceito de função monotônica (crescente e decrescente). As questões escolhidas foram propostas nas Olimpíadas de Matemática das Escolas Públicas (OBMEP) de 2005, 2006, 2007 e 2008. É uma pesquisa de abordagem qualitativa. A análise do desempenho teve por referência teórica a teoria dos Registros de Representação Semiótica de Raymond Duval (2003). Nos protocolos analisados constavam tanto a resolução das questões quanto as justificativas apresentadas pelos alunos para suas respostas. As análises indicaram dificuldades na interpretação dos enunciados das questões bem como nas ações relacionadas aos tratamentos e mudança de Registros de Representação Semiótica. Este estudo também efetivou análise dos erros, a qual possibilitou classificar as questões em grupos quanto aos tipos de respostas. Os resultados revelaram que a maioria dos alunos não estava familiarizada com as questões propostas as quais exigiam transitar pelas diferentes representação de um mesmo objeto matemático

Abstract not available

La ricerca prende le mosse dalla constatazione che gli studenti di madrelingua romena delle medie commettono molti errori nell'inglese scritto. Anche facendo tesoro della esperienza condotta come insegnante d'inglese in Romania, ho scelto come tema della Tesi di Dottorato un'analisi dell'errore con lo scopo di scoprirne le ragioni e di proporre dei miglioramenti per l'insegnamento e l'apprendimento. La ricerca si propone due obiettivi: 1) la messa a punto di un quadro teorico relativo all'analisi contrastiva, all'analisi dell'errore e i modelli di analisi dell'errore; 2) la ricerca sulle origini dell'errore, identificando, descrivendo, classificando e diagnosticando gli errori nell'inglese scritto in cui incorrono solitamente gli studenti romeni. Lo studio ha preso in esame terminologie relative a errori interlinguistici, transfer negativo/interferenza, interlanguage , errori intralinguistici, gravità degli errori e valutazione degli errori. Lo scopo ultimo è di dimostrare fino a che punto il transfer negativo/interferenza sia la causa di errori nella comunicazione scritta degli studenti romeni e fino a che punto questi errori siano causati dalla lingua target. I risultati hanno dimostrato che gli errori interlinguistici e quelli intralinguistici sono piuttosto bilanciati. Trova altresì conferma l'ipotesi dell'utilità di stabilire una gerarchia degli errori basata sulla loro frequenza, così da suggerire agli insegnanti contromisure mirate. l'unico fattore che incide. Trova altresì conferma l'ipotesi dell'utilità di stabilire una gerarchia degli errori basata sulla loro frequenza, così da suggerire agli insegnanti contromisure mirate. / As lower secondary school students commit many errors in writing when English is the target language and Romanian is the native language, I have decided to conduct an error analysis in order to find out what the sources of their errors are, and to suggest improvements for teaching and learning. The present study is intended to provide a theoretical background for contrastive analysis, error analysis, models for error analysis and sources of errors by identifying, describing, categorising, and diagnosing Romanian students' errors in English written compositions. It will also examine related terms such as interlingual errors, negative L1 transfer/interference, interlanguage errors, intralingual errors, error gravity and error evaluation. The ultimate goal of this paper is to demonstrate to what extent negative L1 transfer/interference is the cause for errors in my students' English writing and to what extent these errors are caused by the target language. The findings have demonstrated that interlingual (negative transfer) errors and intralingual (developmental) errors are quite balanced; while interference from the mother tongue plays a role, it is not the only interfering factor. The hypothesis regarding the importance of establishing an error hierarchy based on error frequency has also been validated.