This work is concerned with the characterization of certain classes of stochastic processes via duality formulae. In particular we consider reciprocal processes with jumps, a subject up to now neglected in the literature. In the first part we introduce a new formulation of a characterization of processes with independent increments. This characterization is based on a duality formula satisfied by processes with infinitely divisible increments, in particular Lévy processes, which is well known in Malliavin calculus. We obtain two new methods to prove this duality formula, which are not based on the chaos decomposition of the space of square-integrable function- als. One of these methods uses a formula of partial integration that characterizes infinitely divisible random vectors. In this context, our characterization is a generalization of Stein’s lemma for Gaussian random variables and Chen’s lemma for Poisson random variables. The generality of our approach permits us to derive a characterization of infinitely divisible random measures. The second part of this work focuses on the study of the reciprocal classes of Markov processes with and without jumps and their characterization. We start with a resume of already existing results concerning the reciprocal classes of Brownian diffusions as solutions of duality formulae. As a new contribution, we show that the duality formula satisfied by elements of the reciprocal class of a Brownian diffusion has a physical interpretation as a stochastic Newton equation of motion. Thus we are able to connect the results of characterizations via duality formulae with the theory of stochastic mechanics by our interpretation, and to stochastic optimal control theory by the mathematical approach. As an application we are able to prove an invariance property of the reciprocal class of a Brownian diffusion under time reversal. In the context of pure jump processes we derive the following new results. We describe the reciprocal classes of Markov counting processes, also called unit jump processes, and obtain a characterization of the associated reciprocal class via a duality formula. This formula contains as key terms a stochastic derivative, a compensated stochastic integral and an invariant of the reciprocal class. Moreover we present an interpretation of the characterization of a reciprocal class in the context of stochastic optimal control of unit jump processes. As a further application we show that the reciprocal class of a Markov counting process has an invariance property under time reversal. Some of these results are extendable to the setting of pure jump processes, that is, we admit different jump-sizes. In particular, we show that the reciprocal classes of Markov jump processes can be compared using reciprocal invariants. A characterization of the reciprocal class of compound Poisson processes via a duality formula is possible under the assumption that the jump-sizes of the process are incommensurable. / Diese Arbeit befasst sich mit der Charakterisierung von Klassen stochastischer Prozesse durch Dualitätsformeln. Es wird insbesondere der in der Literatur bisher unbehandelte Fall reziproker Klassen stochastischer Prozesse mit Sprungen untersucht. Im ersten Teil stellen wir eine neue Formulierung einer Charakterisierung von Prozessen mit unabhängigen Zuwächsen vor. Diese basiert auf der aus dem Malliavinkalkül bekannten Dualitätsformel für Prozesse mit unendlich oft teilbaren Zuwächsen. Wir präsentieren zusätzlich zwei neue Beweismethoden dieser Dualitätsformel, die nicht auf der Chaoszerlegung des Raumes quadratintegrabler Funktionale beruhen. Eine dieser Methoden basiert auf einer partiellen Integrationsformel fur unendlich oft teilbare Zufallsvektoren. In diesem Rahmen ist unsere Charakterisierung eine Verallgemeinerung des Lemma fur Gaußsche Zufallsvariablen von Stein und des Lemma fur Zufallsvariablen mit Poissonverteilung von Chen. Die Allgemeinheit dieser Methode erlaubt uns durch einen ähnlichen Zugang die Charakterisierung unendlich oft teilbarer Zufallsmaße. Im zweiten Teil der Arbeit konzentrieren wir uns auf die Charakterisierung reziproker Klassen ausgewählter Markovprozesse durch Dualitätsformeln. Wir beginnen mit einer Zusammenfassung bereits existierender Ergebnisse zu den reziproken Klassen Brownscher Bewegungen mit Drift. Es ist uns möglich die Charakterisierung solcher reziproken Klassen durch eine Dualitätsformel physikalisch umzudeuten in eine Newtonsche Gleichung. Damit gelingt uns ein Brückenschlag zwischen derartigen Charakterisierungsergebnissen und der Theorie stochastischer Mechanik durch den Interpretationsansatz, sowie der Theorie stochastischer optimaler Steuerung durch den mathematischen Ansatz. Unter Verwendung der Charakterisierung reziproker Klassen durch Dualitätsformeln beweisen wir weiterhin eine Invarianzeigenschaft der reziproken Klasse Browscher Bewegungen mit Drift unter Zeitumkehrung. Es gelingt uns weiterhin neue Resultate im Rahmen reiner Sprungprozesse zu beweisen. Wir beschreiben reziproke Klassen Markovscher Zählprozesse, d.h. Sprungprozesse mit Sprunghöhe eins, und erhalten eine Charakterisierung der reziproken Klasse vermöge einer Dualitätsformel. Diese beinhaltet als Schlüsselterme eine stochastische Ableitung nach den Sprungzeiten, ein kompensiertes stochastisches Integral und eine Invariante der reziproken Klasse. Wir präsentieren außerdem eine Interpretation der Charakterisierung einer reziproken Klasse im Rahmen der stochastischen Steuerungstheorie. Als weitere Anwendung beweisen wir eine Invarianzeigenschaft der reziproken Klasse Markovscher Zählprozesse unter Zeitumkehrung. Einige dieser Ergebnisse werden fur reine Sprungprozesse mit unterschiedlichen Sprunghöhen verallgemeinert. Insbesondere zeigen wir, dass die reziproken Klassen Markovscher Sprungprozesse vermöge reziproker Invarianten unterschieden werden können. Eine Charakterisierung der reziproken Klasse zusammengesetzter Poissonprozesse durch eine Dualitätsformel gelingt unter der Annahme inkommensurabler Sprunghöhen.

The aim of this thesis is to analyze textbooks and on their basis to compile preparations for teaching of divisibility in elementary schools. The thesis is divided into six chapters including the introduction and conclusion. The second chapter contains the theoretical background of the literature studied. The next chapter offers an anylysis of selected textbooks. The fourth chapter describes the preparations for teaching. The following chapter contains an evaluation of teaching according to the preparations.

Submitted by Natalia Ojeda Mastronicola null (naty.mastronicola@yahoo.com.br) on 2016-03-14T01:37:28Z No. of bitstreams: 1 Dissertação Natalia Ojeda Mastronicola com ficha catalografica.pdf: 2397628 bytes, checksum: dcd1a480f60fabba28224e1631cdfc2c (MD5) / Approved for entry into archive by Ana Paula Grisoto (grisotoana@reitoria.unesp.br) on 2016-03-15T12:16:55Z (GMT) No. of bitstreams: 1 mastronicola_no_me_sjrp.pdf: 2397628 bytes, checksum: dcd1a480f60fabba28224e1631cdfc2c (MD5) / Made available in DSpace on 2016-03-15T12:16:55Z (GMT). No. of bitstreams: 1 mastronicola_no_me_sjrp.pdf: 2397628 bytes, checksum: dcd1a480f60fabba28224e1631cdfc2c (MD5) Previous issue date: 2016-02-15 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho, utilizamos aplicativos para smartphones e tablets (apps) no ensino da Aritmética, abordando tópicos como divisibilidade através da decomposição em fatores primos; mínimo múltiplo comum e máximo divisor comum. Este trabalho foi desenvolvido junto aos alunos do Ensino Fundamental. Além disso, tratamos também de temas normalmente não trabalhados no Ensino Básico como Teorema de Bézout e Função de Euler. O uso desses aplicativos aproveita-se dessa crescente tecnologia em poder dos alunos, auxiliando a aprendizagem de forma inovadora e tornando-a mais atraente. / In this work, we use some special apps for smartphones and tablets to teach Arithmetic, covering topics such as divisibility, prime decomposition of numbers, least common multiple and greatest common divisor. This study was developed with the students of elementary school. We also treat topics which are not normally worked in basic Education as Bézout's theorem and Euler function. We notice the use of these apps in the classroom brought more enthusiasm for students.

Tesis por compendio / [EN] The study of Complex Systems is one of the scientific fields that has had the highest productivity in recent decades and has not ceased to fascinate the community dedicated to studying its properties. In particular, Network Science has proven to be one of the most prolific areas within Complex Systems. In recent years, his methods have been applied to model multiple phenomena in real life, both naturally generated, such as in biology, and due to the actions and interactions of man, such as social networks or communication networks. Recently, it has been seen how the methods of Network Science can be applied in the context of mathematics, as is the case of Number Theory. One of the most studied cases is networks whose elements are numbers and which are related through the divisibility relation. The main objective of this thesis is to extend these studies to other sets of numbers. On the one hand, we study the divisibility in natural numbers when we obtain these from Pascal matrices of increasing size, which allows us to extract non-sequential sets of numbers with non-constant increments between them. On the other hand, we study the case of the divisibility relation of rational numbers. Cantor's diagonal argument provides a way to order all rational numbers, which allows us to check to what extent some of the properties observed for the divisibility of natural numbers are extensible to a more general context. The thesis is divided into 4 Chapters. Chapter 1 contains a general introduction to the thesis and it is structured into 6 sections. In Sections 1.1 and 1.2, we briefly introduce Network Science, show some application examples, and motivate the study of networks of numbers generated from the divisibility property. In Section 1.3, we define the objectives of this PhD thesis and its scope. In Section 1.4, we present the notion of network, its representations, and some measures that can be calculated on them, such as nodes degrees, their distribution, the assortativity and the clustering coefficients. In another hand, in Section 1.5, we review the best-known network models such as Erdo¿s and Re'nyi random networks, Watts and Strogatz small-world networks, Baraba'si and Albert scale-free networks, and hierarchical networks. Finally, at the end of this Chapter 1, we show in Section 1.6 a review of various studies carried out in order to apply Network Science methods to problems and properties that arise in Number Theory, such as divisibility networks or networks generated from Collatz's Conjecture. or Goldbach's Strong Conjecture. In Chapters 2 and 3, we show the results obtained and that have been published to date. Finally, in Chapter 4, we summarize the conclusions obtained and indicate some related problems that we consider of interest to address in the future. / [ES] El estudio de los Sistemas Complejos es uno de los campos científicos que ha tenido mayor productividad en las últimas décadas y no ha dejado de fascinar a la comunidad que se dedica al estudio de sus propiedades. En particular, la Ciencia de Redes se ha mostrado como una de las áreas más prolíficas dentro de los Sistemas Complejos. En los últimos años, sus métodos han sido aplicados para modelar múltiples fenómenos de la vida real tanto generados de manera natural, como puede ser en el caso de la biología, como debidos a las acciones e interacciones del hombre, como puede ser el caso de las redes sociales o las redes de comunicaciones. Recientemente, se ha visto cómo los métodos de la Ciencia de Redes pueden ser aplicados en el contexto de las matemáticas, como es el caso de la Teoría de Números. Uno de los casos que más se han estudiado es el de las redes cuyos elementos son números y que se relacionan mediante la relación de la divisibilidad. El objetivo principal de esta tesis es extender estos estudios a otros conjuntos de números. Por una parte, estudiamos la divisibilidad en los números naturales cuando obtenemos estos a partir de subconjuntos de números naturales extraídos de matrices de Pascal de orden creciente, lo que nos permite extraer conjuntos de números de manera no secuencial y con incrementos no constantes entre ellos. Por otra parte, estudiamos el caso de la relación de divisibilidad de los números racionales, dado que a partir del argumento diagonal de Cantor se pueden ordenar, lo que nos permite comprobar hasta qué punto algunas de las propiedades observadas para la divisibilidad de los números naturales son extensibles a un contexto más general. La tesis se divide en 4 capítulos. El capítulo 1 contiene una introducción general a la tesis y está estructurado en 6 secciones. En las secciones 1.1 y 1.2, presentamos brevemente la Ciencia de Redes, mostrando algunos ejemplos de aplicación y motivamos el estudio de redes de números generadas a partir de la propiedad de divisibilidad. En la Section 1.3, definimos los objetivos de esta tesis doctoral y su alcance. En la sección 1.4, presentamos la noción de red, sus formas de representación y algunas medidas que se pueden calcular sobre ellas, como son los grados de los nodos, la distribución de estos grados, la asortatividad y los coeficientes de clustering. Por otro lado, en la Sección 1.5, revisamos los modelos de redes más conocidos como son las redes aleatorias de Erdös y Rényi, las redes de pequeño mundo de Watts y Strogatz, las redes libres de escala de Barabási y Albert y las redes jerárquicas. Mostramos en la Sección 1.6, una revisión de diversos estudios realizados con el fin de aplicar métodos de la Ciencia de Redes a problemas y propiedades que surgen en la Teoría de Números, como son las redes de divisibilidad o redes generadas a partir de la Conjetura de Collatz o la Conjetura Fuerte de Goldbach. En los Capítulos 2 y 3, mostramos los resultados obtenidos y que han sido publicados hasta la fecha y, finalmente, en el Capítulo 4, resumimos las conclusiones obtenidas e indicamos algunos problemas relacionados que consideramos de interés abordar en un futuro. / [CAT] L'estudi dels Sistemes Complexos és un dels camps científiques que ha tingut major productivitat en les últimes dècades i no ha deixat de fascinar a la comunitat que es dedica a l'estudi de les seues propietats. En particular, la Ciència de Xarxes s'ha mostrat com una de les àrees més prolífica dins dels Sistemes Complexos. En els últims anys, els seus mètodes han sigut aplicats per a modelar múltiples fenòmens de la vida real tant generats de manera natural, com pot ser en el cas de la biologia, com deguts a les accions i interaccions de l'home, com pot ser el cas de les xarxes socials o les xarxes de comunicacions. Recentment, s'ha vist com els mètodes de la Ciència de Xarxes poden ser aplicats en el context de les matemàtiques, com és el cas de la Teoria de Números. Un dels casos que més s'han estudiat és el de les xarxes els elements de les quals són números i que es relacionen mitjançant la relació de la divisibilitat. L'objectiu principal d'aquesta tesi és estendre aquests estudis a altres conjunts de números. D'una banda, estudiem la divisibilitat en els nombres naturals quan obtenim aquests a partir de matrius de Pascal de grandària creixent, la qual cosa ens permet extraure conjunts de números de manera no sequëncial i amb increments no constants entre ells. D'altra banda, estudiem el cas de la relació de divisibilitat dels nombres racionals, atés que a partir de l'argument diagonal de Cantor es poden ordenar, la qual cosa ens permet comprovar fins a quin punt algunes de les propietats observades per a la divisibilitat dels nombres naturals són extensibles a un context més general. La tesi es troba dividida en 4 Capítols. El capítol 1, conté una introducció general a la tesi i está estructurat en 6 seccions. En les seccions 1.1 i 1.2, presentem breument la Ciència de Xarxes, mostrant alguns exemples d'aplicació i motivem l'estudi de xarxes de números generades a partir de la propietat de divisibilitat. En la Section 1.3, definim els objectius d'aquesta tesi doctoral y el seu abast. En la Secció 1.4, presentem la noció de xarxa, les seves formes de representació i algunes mesures que es poden calcular sobre elles, com són els graus dels nodes, la distribució d'aquests graus, la asortatividad i els coeficients de clustering. En la Sección 1.5, revisem els models de xarxes més coneguts com són les xarxes aleatòries de Erdös i Renyi, les xarxes de xicotet món de Watts i Strogatz, les xarxes lliures d'escala de Barabási i Albert i les xarxes jeràrquiques. Mostrem en la Sección 1.6 una revisió de diversos estudis realitzats amb la finalitat d'aplicar mètodes de la Ciència de Xarxes a problemes i propietats que sorgeixen en la Teoria de Números, com són les xarxes de divisibilitat o xarxes generades a partir de la Conjectura de Collatz o la Conjectura Forta de Goldbach. En els Capítols 2 i 3, vam mostrar els resultats obtinguts i que han sigut publicats fins hui i, finalment, en el Capítol 4, resumim les conclusions obtingudes i indiquem alguns problemes relacionats que considerem d'interés abordar en un futur. / Solares Hernández, PA. (2021). Sets of numbers from complex networks perspective [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/176015 / TESIS / Compendio

Made available in DSpace on 2016-04-27T16:56:59Z (GMT). No. of bitstreams: 1 Joice DAlmeida.pdf: 3241823 bytes, checksum: a8924fcb87b7c723dfad1184f181afc5 (MD5) Previous issue date: 2010-08-25 / Secretaria da Educação do Estado de São Paulo / This work presents a qualitative research whose goal is to investigate how its approach to the issue of divisibility and other matters of Elementary Theory Numbers in the Collection of Professor of Mathematics at the 8th grade, distributed by the Department of State Education São Paulo, in the 2008 and 2009 years. The relevance of studies involving the Elementary Theory of Numbers is the fact that this is a field ripe for the introduction and development of fundamental mathematical ideas, and to chance the solidification of mathematical thinking. To achieve this purpose, I use ideas for methodological content analysis described by Bardin (2009) and consider like essential topics of the Elementary Theory of Numbers to be studied in Primary Education those listed by Resende (2007) in his thesis. The analysis of the material indicate the presence of activities that promote the study of the issue of severability, and other topics in the Elementary Theory of Numbers, featuring the same way, innovative approaches to develop some content / O presente trabalho traz uma pesquisa qualitativa, cujo objetivo foi investigar como a abordagem dada à questão da divisibilidade e a outros temas da Teoria Elementar dos Números, nos Cadernos do Professor de Matemática da 7ª série (8º ano), distribuídos pela Secretaria da Educação do Estado de São Paulo, nos anos de 2008 e 2009. A relevância de estudos envolvendo a Teoria Elementar dos Números repousa no fato deste ser um campo propício para a introdução e desenvolvimento de ideias matemáticas fundamentais, além de oportunizar a solidificação do pensamento matemático. Para atingir o objetivo proposto, são utilizadas as ideias metodológicas para análise de conteúdo descrita por Bardin (2009), considero como tópicos essenciais da Teoria Elementar dos Números a serem estudados no Ensino Básico aqueles listados por Resende (2007) em sua tese. As análises do material indicam a presença de atividades que favorecem o estudo da questão da divisibilidade e de outros tópicos da Teoria Elementar dos Números, apresentando, da mesma forma, abordagens inovadoras para o desenvolvimento de alguns conteúdos

Made available in DSpace on 2016-04-27T16:57:09Z (GMT). No. of bitstreams: 1 Andre Henrique Pizysieznig.pdf: 2148950 bytes, checksum: e98b671a3a07d9dee100a4a9d809f88a (MD5) Previous issue date: 2011-10-18 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This work is part of the problem that questions the process of building major mathematical concepts from Elementary Number Theory for Basic Education. In this context, this study aimed to investigate the conception of divisibility of the K-5 students in School Elementary or through an approach with calculator. The main theoretical references were sought in Resende (2007) and Zazkis and Campbell (2002) regarding the Elementary Theory of Numbers and Sfard (1991) to cognitive processes, Silva et al (1990), Borba and Penteado (2007) and Bianchini and Machado (2010) served as reference to discuss the use of the calculator in the classroom. The qualitative research methodology is based mainly in the Engineering Curriculum. Two sessions with students from the 6th year of public schools in Sao Paulo concluded that among the four concepts focused on the proposed activities: multiple and divisor a natural number, prime numbers and operations division, the students also used a division the calculator and mental arithmetic, half the students showed an operational concept in the process of internalization of the conception of multiple , showed no students meet the mathematical meaning of the term divisor and mostly did not recognize a divisor given representation in prime factors an integer. The calculator was used for all the subjects to calculate and / or validate their responses, being used by some students uncritically, combining alternately with mental arithmetic and algorithmic / Este trabalho se insere na problemática que questiona o processo da construção dos principais conceitos matemáticos da Teoria Elementar dos Números durante a Educação Básica. Neste contexto, este estudo teve como objetivo investigar a concepção de divisibilidade de alunos do 6° ano do Ensino Fundamental por meio de uma abordagem com calculadora. Os principais referenciais teóricos foram buscados em Resende(2007) e em Zazkis e Campbell (2002) no que tange à Teoria Elementar dos Números e em Sfard (1991) para processos cognitivos; Silva et al (1990), Borba e Penteado (2007) e em Bianchini e Machado (2010) serviram de referência para discutir o uso da calculadora em sala de aula. A pesquisa de cunho qualitativo se embasou principalmente na metodologia da Engenharia Didática. Duas sessões com alunos do 6° ano da rede pública estadual de São Paulo permitiram concluir que dentre os quatro conceitos focalizados pelas atividades propostas: divisor e múltiplo de um número natural, números primos e operação de divisão, os alunos para realizar uma divisão recorrem igualmente a calculadora e ao cálculo mental, metade dos alunos mostraram uma concepção operacional em fase de interiorização do conceito de múltiplo , nenhum aluno mostrou conhecer o significado matemático do termo divisor e em sua maioria não reconheceu um divisor dada representação em fatores primos de um número inteiro. A calculadora foi utilizada por todos os sujeitos da pesquisa para calcular e/ou validar suas respostas, sendo utilizada por alguns alunos de forma acrítica, conciliando alternadamente com o cálculo mental e algorítmico

Made available in DSpace on 2016-04-27T16:57:47Z (GMT). No. of bitstreams: 1 EDM - Ednaldo Jose Leandro.pdf: 1331999 bytes, checksum: 6d297c02813a0be36a1ca7ac08bb6ce5 (MD5) Previous issue date: 2006-10-20 / This work focuses on the mathematical object factorial. It is part of the project Argumentation and Proof in School Mathematics (AprovaME), which involves a survey of the conceptions of Brazilian students. For this survey, two questionnaires were developed, one related to the domain of algebra and the other geometry and administered to a sample composed of 2012 students aged between 14 and 16 years, studying in the 8th grade or the 1st year of High School of schools located in the state of São Paulo. The questions analyzed for this study were included in the algebra questionnaire. Following a descriptive analysis of the data collected, which indicated that the students had considerable difficulties in constructing valid mathematical arguments, the data set was subjected to a multidimensional analysis using the software CHIC. The results obtained from this analysis evidenced three distinct groups of students within the sample: those who were unable to respond to questions involving the notion of factorial; students who privileged the use of numeric calculations in their responses; and students who focused on the properties of the factorial in constructing their justifications. It was also possible to identify those students whose response profiles most contributed to the formation of these groups. In a second phase of analysis, some of these students were interviews in order to obtain additional data related to factors motivating their responses. In this phase the questionnaire was also administered to mathematics teachers in schools that made up the sample. In general, the results, both quantitative and qualitative, suggest that the question of argumentation and proof, at least in relation to multiplication and division, is not being contemplated with these students. Calculations were the principle tools used by those who managed to respond to the questions and few students were able to justify their responses using mathematical properties, such as, for example, referring to the inverse relationship between multiplication and division / Este trabalho trata do objeto matemático fatorial. Ele visa contribuir com o projeto Argumentação e Prova na Matemática Escolar (AProvaME), que tem como uma das metas elaborar um levantamento das concepções sobre argumentação e provas de estudantes brasileiros. Para este levantamento, foram elaborados dois questionários, um de Álgebra e outro de Geometria, aplicados a uma amostra composta por 2012 alunos na faixa etária entre 14 e 16 anos, matriculados na 8ª série do Ensino Fundamental ou 1ª série do Ensino Médio em escolas no Estado de São Paulo. As questões que analisamos estão inseridas no questionário de álgebra. Depois de uma análise descritiva dos dados coletados, que indicou consideráveis dificuldades dos alunos em construir argumentos válidos, uma análise multidimensional foi efetuada, utilizando o software CHIC. Com os resultados dessa análise foi possível identificar principalmente três grupos distintos de alunos os que não conseguiram resolver as questões com a noção do fatorial; os alunos que privilegiaram o uso de cálculos numéricos nas suas respostas e os alunos que enfocaram propriedades do fatorial na construção de suas justificativas. Também foi possível identificar aqueles alunos cujos perfis de respostas mais contribuíram para a formação de tais grupos. Numa segunda fase, alguns desses alunos foram entrevistados para a obtenção de mais informação em relação às motivações de suas respostas. Nessa fase, o questionário também foi aplicado aos professores de escolas participantes da amostra. Em geral, nossas análises, tanto quantitativas quanto qualitativas, sugerem que a questão de argumentação e provas, pelo menos em relação à multiplicação e divisão, não estão sendo contempladas com esses alunos. Os que conseguiram responder às questões privilegiaram o cálculo como a principal ferramenta e poucos foram os que justificaram suas respostas com o uso de propriedades, por exemplo, citando a inversa relação entre multiplicar e dividir

This study is an attempt to explicate Gottfried Wilhelm von Leibniz&rsquo / s search for a way out of the labyrinth of the continuum in his early years of philosophizing. The main motive of the study is the belief that it would be worthwhile to see how Leibniz initially goes into the labyrinth and comes across with the riddles contained in it. Accordingly, this thesis is intended to discuss what the problem of the composition of the continuum is for the young Leibniz, which concepts and metaphysical problems are associated with the labyrinth, and what particular difficulties challenge Leibniz in his struggle. More importantly, the study seeks to delineate how Leibniz responds to these difficulties, what kinds of solutions he suggests, and how and why he changes his mind and offers different accounts concerning the composition of the continuum in his early writings. In this search for a way out of the labyrinth, some of the early writings of Leibniz written between 1666 and 1675 were studied with a particular emphasis on those directly related with the labyrinth of the continuum. During the study, the differences and transitions between geometrical, physical, and metaphysical accounts concerning the problem of the composition of the continuum were examined with a special focus on the bridging role of &lsquo / motion&rsquo / and the notion of &lsquo / conatus.&rsquo

The main aim of this diploma thesis is to make a collection of problems out of the thematic topic of number and variable, and divisibility of natural numbers which integrates the chosen curriculum in the educational area of Mathematics and its Applications and Man and the Society. The work is divided into two parts. The first part focuses on the theoretical background of the topic. The second, practical part includes the chosen curriculum (in terms of divisibility of natural numbers) from the point of view of the educational areas of Mathematics and its Applications and Man and the Society. The practical part can be used as a material for interdisciplinary teaching. There are also solutions for each piece of the material.

Submitted by JÚLIO HEBER SILVA (julioheber@yahoo.com.br) on 2016-12-12T15:53:35Z No. of bitstreams: 2 Dissertação - Débora Danielle Alves Moraes Priebe - 2016.pdf: 1557477 bytes, checksum: 54ff96d96b239797a8305d6ff67e2f12 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Jaqueline Silva (jtas29@gmail.com) on 2016-12-13T19:31:20Z (GMT) No. of bitstreams: 2 Dissertação - Débora Danielle Alves Moraes Priebe - 2016.pdf: 1557477 bytes, checksum: 54ff96d96b239797a8305d6ff67e2f12 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2016-12-13T19:31:20Z (GMT). No. of bitstreams: 2 Dissertação - Débora Danielle Alves Moraes Priebe - 2016.pdf: 1557477 bytes, checksum: 54ff96d96b239797a8305d6ff67e2f12 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-12-07 / This paper aims to present an educational proposal of some topics of arithmetic, also called Number Theory, for the final grades of elementary school, focusing on solving problems to challenge and entertain students with the range of possibilities arising from properties of Number Theory and develop their thinking skills through interesting problems that will give a new life to the subject . The reader will find in this work topics of divisibility, primes, Greatest Common Divisor, Least Common Multiple, Euclidean Algorithm, congruences, decimal representation, divisibility tests, as well as several examples, challenging problems and also curiosities about the congruence module 9. / Este trabalho tem como objetivo apresentar uma proposta de ensino de alguns tópicos de Aritmética, também denominada de Teoria dos Números, às séries finais do Ensino Fundamental, com foco na resolução de problemas, visando desafiar e fascinar os alunos com a gama de possibilidades oriunda das propriedades da Teoria dos Números e desenvolver sua capacidade de raciocínio através de problemas interessantes que darão uma nova vida ao assunto. O leitor encontrará neste trabalho tópicos de divisibilidade, primos, Máximo Divisor Comum, Mínimo Múltiplo Comum, Algoritmo de Euclides, congruências, representação decimal, testes de divisibilidade, além de diversos exemplos, problemas desafiadores e também curiosidades acerca da congruência módulo 9.