We present a novel theorem of Borel Combinatorics that sheds light on the types of continuous functions that can be defined on the Cantor space. We specifically consider the part X=F(2ᴳ) from the Cantor space, where the group G is the additive group of integer pairs ℤ². That is, X is the set of aperiodic {0,1} labelings of the two-dimensional infinite lattice graph. We give X the Bernoulli shift action, and this action induces a graph on X in which each connected component is again a two-dimensional lattice graph. It is folklore that no continuous (indeed, Borel) function provides a two-coloring of the graph on X, despite the fact that any finite subgraph of X is bipartite. Our main result offers a much more complete analysis of continuous functions on this space. We construct a countable collection of finite graphs, each consisting of twelve "tiles", such that for any property P (such as "two-coloring") that is locally recognizable in the proper sense, a continuous function with property P exists on X if and only if a function with a corresponding property P' exists on one of the graphs in the collection. We present the theorem, and give several applications.

Spreading models have become fundamental to the study of asymptotic geometry in Banach spaces. The existence of spreading models in every Banach space, and the so-called good sequences which generate them, was one of the first applications of Ramsey theory in Banach space theory. We use Ramsey theory and other techniques from infinitary combinatorics to examine some old and new questions concerning spreading models and good sequences. First, we consider the lp spreading model problem which asks whether a Banach space contains lp provided that every spreading model of a normalized block basic sequence of the basis is isometrically equivalent to lp. Next, using the Hindman-Milliken-Taylor theorem, we prove a new stabilization theorem for spreading models which produces a basic sequence all of whose normalized constant coefficient block basic sequences are good. When the resulting basic sequence is semi-normalized, all the spreading models generated by the above good sequences must be uniformly equivalent to lp or c0. Finally, we investigate the assumption that every normalized block tree on a Banach space has a good branch. This turns out to be a very strong assumption and is equivalent to the space being 1-asymptotic lp. We also show that the stronger assumption that every block basic sequence is good is equivalent to the space being stabilized 1-asymptotic lp.

Combinatorics is a growing topic in mathematics with widespread applications in a variety of fields. Because of this, it has become increasingly prominent in both K-12 and undergraduate curricula. There is a clear need in mathematics education for studies that address cognitive and pedagogical issues surrounding combinatorics, particularly related to students' conceptions of combinatorial ideas. In this study, I describe my investigation of students' thinking as it relates to counting problems. I interviewed a number of post-secondary students as they solved a variety of combinatorial tasks, and through the analysis of this data I defined and elaborated a construct that I call set-oriented thinking. I describe and categorize ways in which students used set-oriented thinking in their counting, and I put forth a model for relationships between the formulas/expressions, the counting processes, and the sets of outcomes that are involved in students' counting activity.

In this paper we explore coloring theorems for the reals, its quotients, cardinals, and their combinations. This work is done under the scope of the axiom of determinacy. We also explore generalizations of Mycielski's theorem and show how these can be used to establish coloring theorems. To finish, we discuss the strange realm of long unions.

With the publication of the famous Erdős-Ko-Rado Theorem in 1961, intersection problems became a popular area of combinatorics. A family of combinatorial objects is t-intersecting if any two of its elements mutually t-intersect, where the latter concept needs to be specified separately in each instance. This thesis is split into two parts; the first is concerned with intersecting injections while the second investigates intersecting posets. We classify maximum 1-intersecting families of injections from {1, ..., k} to {1, ..., n}, a generalisation of the corresponding result on permutations from the early 2000s. Moreover, we obtain classifications in the general t>1 case for different parameter limits: if n is large in terms of k and t, then the so-called fix-families, consisting of all injections which map some fixed set of t points to the same image points, are the only t-intersecting injection families of maximal size. By way of contrast, fixing the differences k-t and n-k while increasing k leads to optimal families which are equivalent to one of the so-called saturation families, consisting of all injections fixing at least r+t of the first 2r+t points, where r=|_ (k-t)/2 _|. Furthermore we demonstrate that, among injection families with t-intersecting and left-compressed fixed point sets, for some value of r the saturation family has maximal size . The concept that two posets intersect if they share a comparison is new. We begin by classifying maximum intersecting families in several isomorphism classes of posets which are linear, or almost linear. Then we study the union of the almost linear classes, and derive a bound for an intersecting family by adapting Katona's elegant cycle method to posets. The thesis ends with an investigation of the intersection structure of poset classes whose elements are close to the antichain. The overarching theme of this thesis is fixing versus saturation: we compare the sizes and structures of intersecting families obtained from these two distinct principles in the context of various classes of combinatorial objects.

Chip-firing is a deceptively simple game played on the vertices of a graph, which was independently discovered in probability theory, poset theory, graph theory, and statistical physics. In recent years, chip-firing has been employed in the development of a theory of divisors on graphs analogous to the classical theory for Riemann surfaces. In particular, Baker and Norin were able to use this set up to prove a combinatorial Riemann-Roch formula, whose classical counterpart is one of the cornerstones of modern algebraic geometry. It is now understood that the relationship between divisor theory for graphs and algebraic curves goes beyond pure analogy, and the primary operation for making this connection precise is tropicalization, a certain type of degeneration which allows us to treat graphs as “combinatorial shadows” of curves. The development of this tropical relationship between graphs and algebraic curves has allowed for beautiful applications of chip-firing to both algebraic geometry and number theory. In this thesis we continue the combinatorial development of divisor theory for graphs. In Chapter 1 we give an overview of the history of chip-firing and its connections to algebraic geometry. In Chapter 2 we describe a reinterpretation of chip-firing in the language of partial graph orientations and apply this setup to give a new proof of the Riemann-Roch formula. We introduce and investigate transfinite chip-firing, and chip-firing with respect to open covers in Chapters 3 and 4 respectively. Chapter 5 represents joint work with Arash Asadi, where we investigate Riemann-Roch theory for directed graphs and arithmetical graphs, the latter of which are a special class of balanced vertex weighted graphs arising naturally in arithmetic geometry.

A research report submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in partial fulfilment of the requirements for the degree of Master of Science / This research report investigates the application of Charron-Bost's measure of currency m to Milner's Calculus of Communicating Systems (CCS). The aim of this is twofold: first to evaluate the measure m in terms of criteria gathered from the literature: and second to determine the feasiblllty of measuring concurrency in CCS and hence provide a new tool for understanding concurrency using CCS. The approach taken is to identify the differences hetween the message-passing formalism in which the measure m is defined, and CCS and to modify this formalism to-enable the mapping of CCS agents to it. A software tool, the Concurrency Measurement Tool, is developed to permit experimentation with chosen CCS agents. These experiments show that the measure m, although intuitively appealing, is defined by an algebraic expression that is ill-behaved. A new measure is defined and it is shown that it matches the evaluation criteria better than m, although it is still not ideal. This work demonstrates that it is feasible to measure concurrency in CCS and that a methodology has been developed for evaluating concurrency measures. / Andrew Chakane 2018

O presente estudo discorre sobre o uso de atividades lúdicas como alternativa no processo de ensino-aprendizagem de Matemática, em destaque, um jogo de cartas como atividade de revisão no ensino de Análise Combinatória e Probabilidade. Atividades como jogos, desafios e outros são fundamentais para que o discente possa aprender de forma efetiva e satisfatória alguns conteúdos de Matemática, uma vez que o gosto por essa disciplina não é unânime entre os discentes. Atividades com jogos é uma importante ferramenta para motivar e auxiliar no estudo desta matéria. Objetivando esclarecer a influência desse jogo matemático no desenvolvimento do discente durante o processo de ensino-aprendizagem, neste estudo utilizamos um jogo de cartas na Escola Estadual Hercília Carvalho da Silva, localizada no Município de Gurupi – TO. Esse trabalho relata a aplicação de um jogo envolvendo cartas que foi aplicado no 2o ano do ensino médio, em um período de 2 meses, como uma forma de demonstrar a potencialidade da atividade lúdica enquanto metodologia inovadora das ações pedagógicas, buscando aulas mais diversificadas para conduzir os discentes envolvidos neste grupo a uma aprendizagem significativa nos conteúdos de Análise Combinatória e Probabilidade. Posteriormente foi aplicado um questionário ao grupo de discentes, este com 11 questões de múltipla escolha, a atividade (questionário) fora realizada no dia 14 de março de 2017, durante a aula de Matemática. Pelos dados coletados, fora possível verificar que a aplicação do jogo de cartas facilita o entendimento dos discentes em referência às matérias de Análise Combinatória e Probabilidade, o que não seria integralmente possível apenas com a exposição teórica e listas de exercícios. A pesquisa, quanto aos procedimentos, delineia como uma de campo e levantamento de dados. Quanto aos objetivos, teve como base o estudo descritivo e exploratório, focalizando em identificar a percepção que o discente possui acerca do conteúdo de Análise Combinatória e Probabilidade por meio de um jogo de cartas como atividade para revisar, aprofundar ou motivar o estudo destes conteúdos. / The present study discusses the use of play activities as a facilitator in the teachinglearning process of Mathematics, in particular, a game of cards as a review activity in the teaching of Combinatorial Analysis and Probability. Activities such as games, challenges and others are fundamental so that the student can learn effectively and satisfactorily some contents of Mathematics, since the taste for this discipline is not unanimous for the students. For this, the use of games activities is an important tool to motivate and assist in the study of this matter. In order to clarify the influence of this mathematical game on the student’s development during the teaching-learning process, in this study we used a game of cards at the Hercília Carvalho da Silva State School, located in the Municipality of Gurupi-TO. This paper reports on the application of a game involving letters that was applied in the second year of high school in a period of two months as a way of demonstrating the potential of playful activity as an innovative methodology of pedagogical actions, seeking more diversified classes to conduct The students involved in this group to a significant learning in the contents of Combinatorial Analysis and Probability. Later, a questionnaire was applied to the group of students, with eleven multiple choice questions, activity (research - questionnaire) was performed on March 14, 2017, during the Mathematics class. From the collected data, it was possible to verify that the application of the card game facilitates the understanding of the students in reference to the Matters of Combinatorial Analysis and Probability, which would not be entirely possible only with theoretical exposition and lists of exercises. The research, regarding the procedures, delineates as a field and data collection. Regarding the objectives, it was based on the descriptive and exploratory study, focusing on identifying the perception that the student has about the content of Combinatorial Analysis and Probability through a game of cards as an activity to review, deepen or motivate the study of these contents.

Made available in DSpace on 2016-04-27T16:57:36Z (GMT). No. of bitstreams: 1 Carlos Alberto de Miranda Pinheiro.pdf: 3726604 bytes, checksum: 7f0acb671f10ca6a3f9af822c2232603 (MD5) Previous issue date: 2015-03-17 / This paper presents the development of a research, which proposed to investigate the knowledge of Combinatorial Analysis studied in the Brazilian schools during 1895 and 2009. This research articulates inside a project called Educação Estatística e Educação Financeira na Escola Básica (Statistical Education and Financial Education in Basic School), linked to the research group PEA-MAT. We analysed some didactic aspects and mathematicians present in seven textbooks used in Brazilian schools during that time, we also had a bibliographical and documental study in presuppositions of content analysis. We wanted to answer the following investigation question:what characteristics of knowledge insertion of Combinatorial Analysis are identified in textbooks during 1895 and 2009? The first phase of content analysis consisted in collecting and studying texts that would inform us what textbooks presented the knowledge of Combinatorial Analysis and some aspects of historical moments of these books, since the foundation of the Pedro II School. With this, we identified that the program of 1895 was the one with more insertion of textbooks and new school contents, before the main educational reforms occurred in the first decades of the XX century. We also tried to identify on the website of the Programa Nacional do Livro Didático (PNLD-Ensino Médio) (National Program of Textbooks-High School) the Mathematic book more used in Belémdo Pará schools, in the first phase of the program, 2004-2009. The second phase of content analysis was the material exploration. In this phase we analysed based on the Anthropological Theory of Didactic, specifically about the praxeological organizations not only mathematical but also didactic, together with the notion of didactic models proposed by Josep Gáscon. Among the results observed, we highlight that while in the first books the focus was only on presentation of formulas deduction, some years later, the task of calculating the values from those formulas was introduced, modifying the approach from theoretical to technical or classic. We can also observe the change in the techniques to calculate the Arrange and the Permutation in the book produced based on the Movement of Modern Mathematics and in a more recent book, approved by the PNLD-Ensino Médio. The tasks and the techniques related to the calculation of numbers of simple combinations will not undergo transformations as time goes by. That is, the praxeological organization identified in the block task/technique/technology/theory changed from a theoretical to a technical approach during the whole period studied / Este trabalho apresenta o desenvolvimento de uma pesquisa que se propôs a investigar os saberes da Análise Combinatória estudada nas escolas brasileiras, no período entre 1895 e 2009. Esta pesquisa articula-se no interior do projeto Educação Estatística e Educação Financeira na Escola Básica , vinculado ao grupo de pesquisa PEA-MAT. Analisamos alguns aspectos didáticos e matemáticos presentes em sete livros didáticos que circularam nas escolas brasileiras, no período em questão, realizando um estudo bibliográfico e documental fundamentado em pressupostos da análise de conteúdo. Buscamos responder à seguinte questão de investigação: que características de inserção dos saberes da Análise Combinatória nos livros didáticos podem ser identificadas no período 1895-2009? A primeira fase da análise de conteúdo consistiu em reunir e estudar textos que nos informassem quais livros escolares apresentavam os saberes da Análise Combinatória e alguns aspectos do momento histórico desses livros, desde a fundação do Colégio Pedro II. Com isso, identificamos que o programa de 1895 foi o que teve a maior inserção de livros didáticos e de novos conteúdos escolares, antes das principais reformas educacionais ocorridas nas primeiras décadas do século XX. Também procuramos identificar no site do Programa Nacional do Livro Didático (PNLD-Ensino Médio) o livro de Matemática mais utilizado nas escolas de Belém do Pará, na primeira fase de implantação desse programa, 2004-2009. A segunda fase da análise de conteúdo foi a exploração do material. Nesta fase foi realizada uma análise à luz da Teoria Antropológica do Didático, especificamente sobre as organizações praxeológica tanto matemáticas como didáticas, juntamente com a noção de modelos didáticos, proposta por JosepGáscon. Entre os resultados observados, destacamos que, enquanto nos primeiros livros o enfoque era apenas a apresentação da dedução das fórmulas, ao longo do tempo foi inserida a tarefa de calcular os valores a partir dessas fórmulas, passando, assim, de uma abordagem puramente teoricista para uma abordagem tecnicista ou clássica. Observa-se, também, uma mudança nas técnicas para calcular o Arranjo e a Permutação no livro produzido à luz das ideias do Movimento de Matemática Moderna e no livro mais recente, aprovado no PNLD-Ensino Médio. As tarefas e as técnicas, relativas ao cálculo do número de combinações simples, não sofreram transformações, ao longo do tempo. Ou seja, a organização praxeológica identificada no bloco tarefa/técnica/tecnologia/teoria passou de uma abordagem teoricista para uma abordagem tecnicista ou clássica ao longo de todo o período estudado

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Instruções no link http://biblioteca.ufam.edu.br/servicos/elaboracao-de-ficha-catalografica Dúvidas? ddbc@ufam.edu.br on 2018-11-09T13:58:53Z (GMT) / Submitted by Gilvan Passos (gilvan.dspassos@gmail.com) on 2018-11-09T20:14:38Z No. of bitstreams: 4 GilvanTCC.pdf: 392056 bytes, checksum: c92e4c9757ada7893dc6f62a78267aa6 (MD5) IMG_20181102_131441.jpg: 1218636 bytes, checksum: 36aa8c31ec2aca115870ea2c2a9e278c (MD5) IMG_20181102_131427.jpg: 1672384 bytes, checksum: ef52fc665bf97e6c37ae0b3c0202ac2c (MD5) fichacatalografica.pdf: 5598 bytes, checksum: 78c21bd3648cbde20ad062f8314ad74d (MD5) / Approved for entry into archive by PPGM Matemática (ppgmufam@gmail.com) on 2018-11-13T14:28:28Z (GMT) No. of bitstreams: 4 GilvanTCC.pdf: 392056 bytes, checksum: c92e4c9757ada7893dc6f62a78267aa6 (MD5) IMG_20181102_131441.jpg: 1218636 bytes, checksum: 36aa8c31ec2aca115870ea2c2a9e278c (MD5) IMG_20181102_131427.jpg: 1672384 bytes, checksum: ef52fc665bf97e6c37ae0b3c0202ac2c (MD5) fichacatalografica.pdf: 5598 bytes, checksum: 78c21bd3648cbde20ad062f8314ad74d (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2018-11-13T18:08:41Z (GMT) No. of bitstreams: 4 GilvanTCC.pdf: 392056 bytes, checksum: c92e4c9757ada7893dc6f62a78267aa6 (MD5) IMG_20181102_131441.jpg: 1218636 bytes, checksum: 36aa8c31ec2aca115870ea2c2a9e278c (MD5) IMG_20181102_131427.jpg: 1672384 bytes, checksum: ef52fc665bf97e6c37ae0b3c0202ac2c (MD5) fichacatalografica.pdf: 5598 bytes, checksum: 78c21bd3648cbde20ad062f8314ad74d (MD5) / Made available in DSpace on 2018-11-13T18:08:41Z (GMT). No. of bitstreams: 4 GilvanTCC.pdf: 392056 bytes, checksum: c92e4c9757ada7893dc6f62a78267aa6 (MD5) IMG_20181102_131441.jpg: 1218636 bytes, checksum: 36aa8c31ec2aca115870ea2c2a9e278c (MD5) IMG_20181102_131427.jpg: 1672384 bytes, checksum: ef52fc665bf97e6c37ae0b3c0202ac2c (MD5) fichacatalografica.pdf: 5598 bytes, checksum: 78c21bd3648cbde20ad062f8314ad74d (MD5) Previous issue date: 2018-03-28 / This work aims to study combinatorial analysis, which is an important branch of mathematics which is not usually subtly treated and through many years was teached as the mechanical memorization, leaving aside the learning process, self-learning and logical construction. It is important to emphasize the application of combinatorial analysis in set theory and probabilities theory that are often present in problem solving. It is necessary to present to our students the potential and beauty of the logical construction of ideas of combinatorial analysis, not excluding formulas applications, that can be used when the concepts and structure is well assimilated. We present counting methods beyond those used in basic education such as repetition chaotic permutations combinations, inclusion and exclusion principles, Kaplansky and Dirichlet lemmas, but we also highlight basic methods such as simple arrangements, simple combinations, and simple permutations. Beyond that, we present a generalization of the factorial numbers through the Gamma function besides olympics problems resolutions. / Este trabalho tem por objetivo estudar Análise Combinatória, que é um importante ramo da matemática que normalmente não é tratado com sutileza e transmitida ao longo dos anos através de memorização mecânica deixando o processo aprendizagem, auto-aprendizagem e construção lógica de lado. É importante enfatizar a aplicação da Análise Combinatória nas teorias dos conjuntos e teoria das probabilidades que muitas vezes se fazem presentes nas resoluções de problemas. Se faz necessário apresentar para nossos alunos o potencial e a beleza da construção lógica de ideias que a Análise Combinatória proporciona não excluindo as aplicações de fórmulas mas que elas possam ser usadas quando os conceitos e a estrutura forem bem assimiladas. Apresentamos métodos de contagem além dos usados no ensino básico como permutações caóticas combinações com repetição, princípio da inclusão e exclusão, lemas de Kaplansky e de Dirichlet mas também destacamos os métodos básicos como arranjos simples, combinações simples e permutações simples. Além disso, para apresentamos uma generalização dos números fatoriais definida pela função Gama e resoluções de problemas de olimpíadas.