Proefschrift--Utrecht.

Proefschrift--Utrecht.

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Η μεταπτυχιακή αυτή διπλωματική εργασία στοχεύει στην παρουσίαση κάποιων από τις δημοφιλέστερες μεθόδους βελτιστοποίησης μη γραμμικών συναρτήσεων. Εξετάζεται σε κάθε περίπτωση τόσο το θεωρητικό υπόβαθρο, όσο και η πρακτική λειτουργικότητα της εκάστοτε μεθόδου, τα είδη των προβλημάτων όπου επιτυγχάνεται η μέγιστη αποτελεσματικότητα, λεπτομέρειες σχετικά με το ρυθμό σύγκλισης, καθώς και κάποια συγκριτικά ως προς τις προαναφερθείσες μεθόδους σχόλια. Αρχικά υπενθυμίζονται βασικές έννοιες που χρησιμοποιούνται στην πορεία της επισκόπησης των μεθόδων. Ακολούθως, γίνεται εκτενής αναφορά στις πλέον διαδεδομένες τετραγωνικές μεθόδους μονοδιάστατης βελτιστοποίησης, πιο συγκεκριμένα στις μεθόδους Μέγιστης Μείωσης, Newton, Διχοτόμησης, Fibonacci και Αναζήτησης Χρυσής Τομής. Τις μεθόδους κλίσης ακολουθούν οι μέθοδοι συζυγών κατευθύνσεων, που επιχειρούν ταχύτερη σύγκλιση και μείωση της πολυπλοκότητας. Στη συνέχεια περιγράφονται μέθοδοι μεταβλητής μετρικής, όπως η τροποποιημένη μέθοδος Newton, η Secant και ένας συνδυασμός των μεθόδων Μέγιστης Μείωσης και Newton. Η εργασία ολοκληρώνεται με την παρουσίαση μη τετραγωνικών προτύπων, όπως οι Καμπυλόγραμμες Τροχιές, η μέθοδος Jacobson-Oksman καθώς και Κωνικές Μέθοδοι. / This master course essay presents some of the most popular non-linear optimization methods. It refers both to the theory and the practice of each method, describes when each method is most efficient to be used, offers some convergence information and provides some comments about the comparison of the methods. Firstly, there is a reference of basic optimization theory which is followed by a detailed description of the most widely known quadratic optimization methods, such as Steepest Descent, Newton, Interval Halving, Fibonacci and Golden Section Search. Next section refers to Conjugate Direction methods which tend to be more efficient and converge faster. These are followed by Quasi-Newton methods, such as variations of the Newton method, Secant and a combination of Steepest Descent and Newton. Finally, some non-quadratic methods are presented, such as Jacobson-Oksman method and conic methods.

Student difficulties learning algebra are well documented. Many mathematics education researchers (e.g., Bednarz&Janvier, 1996; Davis, 1985, 1989; Vergnaud, 1988) argued that the difficulties students encounter in algebra arose when students were expected to shift suddenly from arithmetic to algebraic reasoning and that the solution to the problem was to integrate opportunities for elementary school students to simultaneously develop both arithmetic and algebraic reasoning. The process of generalization, or describing the overall pattern underlying a set of mathematical data, emerged as a focal point for extending beyond arithmetic reasoning to algebraic reasoning (Kaput, 1998; Mason, 1996). Given the critical importance for students to have opportunities to develop understanding of the fundamental algebraic concepts of variable and relationship, one could argue that providing opportunities to explore linear functions, the first function studied in depth in a formal algebra course, should be a priority for elementary students in grades 4-5. This study informs this debate by providing data about connections between different representations of constant growth functions and student algebraic reasoning in a context open to individual construction of representations and reasoning approaches. Participants included 9 fifth graders from the same elementary class. Data shows that students can generate representations which are effective reasoning tools for finding particular cases of the function and generalizing the function but that this depends on features of the problem representation, most importantly the representation of the additive constant. I identified four categories of algebraic reasoning on the task to find the tenth term and found that only students who used reasoning approaches with the additive constant separate and functional reasoning to find the variable component were able to generalize the function. These instances occurred on a story problem and two geometric pattern problems. None of the students used such a reasoning approach or were able to generalize on the numeric sequence problem which did not represent the additive constant separately. Implications for future research and for teaching for conceptual understanding of variable and relationship are discussed.

This paper is a study of linear operators defined on normed linear spaces. A basic knowledge of set theory and vector spaces is assumed, and all spaces considered have real vector spaces. The first chapter is a general introduction that contains assumed definitions and theorems. Included in this chapter is material concerning linear functionals, continuity, and boundedness. The second chapter contains the proofs of three fundamental theorems of linear analysis: the Open Mapping Theorem, the Hahn-Banach Theorem, and the Uniform Boundedness Principle. The third chapter is concerned with applying some of the results established in earlier chapters. In particular, the concepts of compact operators and Schauder bases are introduced, and a proof that an operator is compact if and only if its adjoint is compact is included. This chapter concludes with a proof of an important application of the Open Mapping Theorem, namely, the Closed Graph Theorem.

Graphing is a fundamental topic in algebra that is notoriously difficult for students. Much of the past research has focused on conceptions and misconceptions. This study extends past research by looking at the mathematical practices of a practitioner, specifically one instructor of a function-based covariation-focused algebra class in the linear functions unit. Considering practices in addition to conception adds dramatically to our understanding of mathematical activity because it leads to explicit descriptions of normative purposes that are connected to particular situations or problems and also specifies how tools and symbols are coordinated to achieve these purposes. The results of this study are three levels of empirically proven practices associated with the conception of one advanced level of covariational reasoning, chunky continuous covariation. This study not only describes how practices may be described at different levels of complexity, but also demonstrates how smaller practices may be combined to form larger, more complex practices. These practices can be used to guide instruction of those who want to participate in and become practitioners in the community of teachers of function-based covariation-focused algebra curricula.

Univ., Diss.--Joensuu, 2006.

When it comes to teaching linear and quadratic functions, we observed that the Mathematics teachers, guided by current textbooks, explore the construction of graphs by connecting points on the Cartesian plane. This method, while valid, does not guarantee an outline safe and does not imply the observation of important properties of these functions. With this in mind, this study aims to verify the efficiency of a screenplay produced in a explanatory and methodological booklet form. This script is a practical and efficient way to plot the graphs of polynomial functions of the 1st and 2nd grade using simple materials and essentials such as paper and pencil, and also a digital resource, the software GeoGebra. To check the efficiency of this route, we take a student group composed of individuals who attend high school in public schools and participant the Institutional Program for Scientific Initiation Scholarships Junior (PIBIC-Jr). The methodology consisted of questionnaires - before, during and after - the teaching intervention: lecture, referring to related functions and quadratic functions, highlighting the construction of their graphs, and use of a printed script, developed in the form of a explanatory booklet which worked the construction of graphs, using the principle of graph paper and pencil, and at another time, a dynamic geometry software. Using this software, and making a sketch of faster graphics, it became understandable graphical changes incurred when changing the coefficients of the functions. As a theoretical framework for the research exposed in this work, we rely on the contribution of the Theory of Didactic Situations, Guy Brousseau (2008), and representative forms of a function in the view of Raymond Duval (2009), in his Theory of Representation Registers Semiotics. The ten students participating in the study, seven had a breakthrough in the development of the trace graph of functions considered and understanding of their properties. Thus, one comes to the conclusion that it is possible to study and observe the properties of linear and quadratic functions taking into account specific issues pertaining to its graphics, both on paper and, especially, on the computer. / Quando se trata do ensino de funções afim e quadrática, observamos que os professores de Matemática, orientados pelos livros didáticos atuais, exploram a construção de gráficos mediante a ligação de pontos no plano cartesiano. Este método, embora seja válido, não garante um esboço seguro e não favorece a observação de propriedades importantes das referidas funções. Pensando nisto, este trabalho objetiva verificar a eficiência de um roteiro produzido em forma de cartilha explicativa e metodológica. Neste roteiro encontra-se uma maneira prática e eficiente para traçar os gráficos de funções polinomiais do 1º e 2º graus fazendo uso de materiais simples e indispensáveis, como o papel e o lápis, e também de um recurso digital, o software GeoGebra. Para verificar a eficiência deste roteiro, tomamos um grupo de alunos composto por indivíduos que cursam o Ensino Médio em escolas públicas e que participam do Programa Institucional de Bolsas de Iniciação Científica Junior (PIBIC-Jr). A metodologia adotada consistiu na aplicação de questionários - antes, durante e depois – de nossa intervenção de ensino: aulas expositivas, referente às funções afins e funções quadráticas, destacando a construção de seus gráficos; e uso do roteiro impresso que trabalhou a construção dos gráficos, fazendo uso, a princípio, de lápis e papel quadriculado e, em outro momento, de um software de geometria dinâmica. A utilização deste software, além de tornar o esboço de gráficos mais rápidos, tornou compreensíveis as modificações gráficas sofridas quando muda os coeficientes das funções. Como fundamentação teórica para a pesquisa exposta neste trabalho, contamos com a contribuição da Teoria das Situações Didáticas, de Guy Brousseau (2008), e das formas representativas de uma função na visão de Raymond Duval (2009), na sua Teoria de Registros de Representação Semiótica. Dos dez alunos participantes na pesquisa, sete apresentaram um avanço no desenvolvimento do traçado gráfico das funções consideradas e na compreensão das suas propriedades. Desta maneira, chega-se à conclusão de que é possível estudar e observar as propriedades das funções afim e quadrática levando em consideração pontos específicos pertencentes a seus gráficos, tanto no papel quanto, e especialmente, no computador.

Submitted by Douglas Vinicius Rosato Costa null (douglas_vrc@yahoo.com) on 2018-03-13T02:13:44Z No. of bitstreams: 1 Dissertação Douglas Costa Versão Final.pdf: 1946320 bytes, checksum: 336b5ddb2211738253b95133f808aa2e (MD5) / Approved for entry into archive by Elza Mitiko Sato null (elzasato@ibilce.unesp.br) on 2018-03-13T19:05:51Z (GMT) No. of bitstreams: 1 costa_dvr_me_sjrp.pdf: 1946320 bytes, checksum: 336b5ddb2211738253b95133f808aa2e (MD5) / Made available in DSpace on 2018-03-13T19:05:51Z (GMT). No. of bitstreams: 1 costa_dvr_me_sjrp.pdf: 1946320 bytes, checksum: 336b5ddb2211738253b95133f808aa2e (MD5) Previous issue date: 2018-03-02 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho é apresentado um estudo sobre funções afim e quadrática e resolução de situações-problema através do uso de programação, tendo como público alvo os estudantes do último ano do ensino fundamental e primeiro ano do ensino médio. Atualmente, notamos no ensino básico apatia e desmotivação por parte dos estudantes, por não julgarem necessário o que aprendem na escola e, principalmente, ao encarar as dificuldades apresentadas em Matemática. Partindo dessa premissa, objetiva-se apontar uma ligação direta entre resolução de situações-problema e programação, e abordar de forma interativa e atraente uma maneira de adquirir as habilidades necessárias nessas duas áreas. Utilizando o software Scratch para resolver as atividades propostas sobre funções afim e quadrática, conseguimos cativar o interesse dos estudantes e atingimos maior participação em sala de aula, por meio de atividades diferenciadas e criativas. Incluem-se ainda os benefícios de aprender a programar, que é considerada uma habilidade essencial para o futuro. / In this work, it is presented a study on linear and quadratic functions and problem-solving through the use of programming, focusing on the students of the last year of elementary school and the first year of high school. Nowadays we notice apathy and demotivation from the students in the basic education due to their belief that what they learn in the school is unnecessary and, mainly, when facing the usual difficulties concerning Mathematics. Based on this premise, this work aims to point out a direct link between problem-solving and programming, interactively and attractively approaching a way to acquire the necessary skills in these two areas. Using the Scratch software to solve the proposed activities on linear and quadratic functions, we were able to captivate students' interest and achieve greater participation in the classroom through differentiated and creative activities. It also includes the benefits of learning how to program, which is considered an essential skill for the future.