Assuming the axiom of determinacy, we give a new proof of the strong partition relation on ω1. Further, we present a streamlined proof that J<λ+(a) (the ideal of sets which force cof Π α < λ) is generated from J<λ+(a) by adding a singleton. Combining these results with a polarized partition relation on ω1

The purpose of this thesis is to present a study of the Lebesgue definite integral, defined in four different ways.

The subject of this master's degree thesis is the proving in the civil proceedings. The reason why I chose this subject is primarily the fact that it is a crucial part of any civil trial. The proving is a process provided for by the law by which the court gains the factual findings that are essential for its decision. In the proving process I have focused on the evidentiary evaluation and particularly on the aspect of lawfulness of proofs because it is a very interesting and relatively little described area, which due to a technological progress will be more and more topical issue in a law practice. The aim of this thesis is to show the way general rules of illegal proofs affect a judicial practice of courts.

Por meio de acompanhamento realizado junto a quatro professores da rede estadual de ensino de São Paulo, realizamos um estudo de caso, com foco na abordagem das provas matemáticas no Ensino Médio. O texto descreve o acompanhamento das aulas, motivações e os obstáculos existentes para o desenvolvimento do tema em sala de aula. Para o desenvolvimento da pesquisa, utilizamos como referencial teórico, os seguintes trabalhos: Thompson (1992), sobre concepções docentes; as tipologias e funções das provas matemáticas, de Balacheff e De Villiers, respectivamente. Foram utilizados os seguintes instrumentos para a coleta de dados: observação direta, anotações de campo e entrevistas. Os resultados obtidos apontam para uma prática pedagógica utilitarista sem a participação ativa dos alunos. Quanto às provas matemáticas, constatamos a sua abordagem de forma intencional e planejada, sendo abordadas, no entanto, apenas em turmas específicas e ligadas ao interesse pessoal do professor e ainda, em geral, sem a participação ativa dos alunos no processo. Acreditamos não ser este o ambiente ideal para o desenvolvimento das provas matemáticas em sala de aula, que deveria ocorrer num espaço voltado à argumentação, levantamento de hipóteses, elaboração de conjecturas de modo a permitir o avanço dos alunos nos níveis das provas elaboradas. Constatamos ainda a influência de fatores como: interesse das turmas, indisciplina, cobranças internas (organização da sala, comportamento dos alunos em sala, abordagem dos conteúdos previstos) e externas (desempenho satisfatório nas avaliações internas e externas das quais a escola participa). / Through monitoring carried out with four teachers of the state of São Paulo teaching, we conducted a case study with a focus on addressing the mathematical proofs in high school. The text describes the monitoring of classes, existing motivations and obstacles to the issue of development in the classroom. For the development of research, we used as a theoretical reference, the following work: Thompson (1992) on teachers conceptions; the types and functions of mathematical proofs of Balacheff and De Villiers, respectively. The instruments for data collection were used: direct observation, field notes and interviews. The results point to a utilitarian pedagogical practice without the active participation of students. As for mathematical proofs, found his approach intentionally and planned, being addressed, however, only in specific classes and linked to the staff of teacher interest and also, in general, without the active participation of students in the process. We believe this is not the ideal environment for the development of mathematical proofs in the classroom, which should occur in an area facing the argument, raise hypotheses, conjectures preparing to allow the advancement of students in levels of elaborate tests. Still found the influence of factors such as interest groups, lack of discipline, internal charges (room organization, students\' behavior in class, approach the expected content) and external (satisfactory performance in internal and external ratings of which the school participates).

In this thesis, we provide constructive proofs of serveral generalizations of Sperner's Lemma, a combinatorial result which is equivalent to the Brouwer Fixed Point Theorem. This lemma makes a statement about the number of a certain type of simplices in the triangulation of a simplex with a special labeling. We prove generalizations for polytopes with simplicial facets, for arbitrary 3-polytopes, and for polygons. We introduce a labeled graph which we call a nerve graph to prove these results. We also suggest a possible non-constructive proof for a polytopal generalization.

This paper introduces Denotational Proof Languages (DPLs). DPLs are languages for presenting, discovering, and checking formal proofs. In particular, in this paper we discus type-alpha DPLs---a simple class of DPLs for which termination is guaranteed and proof checking can be performed in time linear in the size of the proof. Type-alpha DPLs allow for lucid proof presentation and for efficient proof checking, but not for proof search. Type-omega DPLs allow for search as well as simple presentation and checking, but termination is no longer guaranteed and proof checking may diverge. We do not study type-omega DPLs here. We start by listing some common characteristics of DPLs. We then illustrate with a particularly simple example: a toy type-alpha DPL called PAR, for deducing parities. We present the abstract syntax of PAR, followed by two different kinds of formal semantics: evaluation and denotational. We then relate the two semantics and show how proof checking becomes tantamount to evaluation. We proceed to develop the proof theory of PAR, formulating and studying certain key notions such as observational equivalence that pervade all DPLs. We then present NDL, a type-alpha DPL for classical zero-order natural deduction. Our presentation of NDL mirrors that of PAR, showing how every basic concept that was introduced in PAR resurfaces in NDL. We present sample proofs of several well-known tautologies of propositional logic that demonstrate our thesis that DPL proofs are readable, writable, and concise. Next we contrast DPLs to typed logics based on the Curry-Howard isomorphism, and discuss the distinction between pure and augmented DPLs. Finally we consider the issue of implementing DPLs, presenting an implementation of PAR in SML and one in Athena, and end with some concluding remarks.

This paper presents an algorithm for simplifying NDL deductions. An array of simplifying transformations are rigorously defined. They are shown to be terminating, and to respect the formal semantis of the language. We also show that the transformations never increase the size or complexity of a deduction---in the worst case, they produce deductions of the same size and complexity as the original. We present several examples of proofs containing various types of "detours", and explain how our procedure eliminates them, resulting in smaller and cleaner deductions. All of the given transformations are fully implemented in SML-NJ. The complete code listing is presented, along with explanatory comments. Finally, although the transformations given here are defined for NDL, we point out that they can be applied to any type-alpha DPL that satisfies a few simple conditions.

As the cloud computing paradigm has gained prominence, the need for verifiable computation has grown urgent. Protocols for verifiable computation enable a weak client to outsource difficult computations to a powerful, but untrusted, server. These protocols provide the client with a (probabilistic) guarantee that the server performed the requested computations correctly, without requiring the client to perform the computations herself. / Engineering and Applied Sciences

Por meio de acompanhamento realizado junto a quatro professores da rede estadual de ensino de São Paulo, realizamos um estudo de caso, com foco na abordagem das provas matemáticas no Ensino Médio. O texto descreve o acompanhamento das aulas, motivações e os obstáculos existentes para o desenvolvimento do tema em sala de aula. Para o desenvolvimento da pesquisa, utilizamos como referencial teórico, os seguintes trabalhos: Thompson (1992), sobre concepções docentes; as tipologias e funções das provas matemáticas, de Balacheff e De Villiers, respectivamente. Foram utilizados os seguintes instrumentos para a coleta de dados: observação direta, anotações de campo e entrevistas. Os resultados obtidos apontam para uma prática pedagógica utilitarista sem a participação ativa dos alunos. Quanto às provas matemáticas, constatamos a sua abordagem de forma intencional e planejada, sendo abordadas, no entanto, apenas em turmas específicas e ligadas ao interesse pessoal do professor e ainda, em geral, sem a participação ativa dos alunos no processo. Acreditamos não ser este o ambiente ideal para o desenvolvimento das provas matemáticas em sala de aula, que deveria ocorrer num espaço voltado à argumentação, levantamento de hipóteses, elaboração de conjecturas de modo a permitir o avanço dos alunos nos níveis das provas elaboradas. Constatamos ainda a influência de fatores como: interesse das turmas, indisciplina, cobranças internas (organização da sala, comportamento dos alunos em sala, abordagem dos conteúdos previstos) e externas (desempenho satisfatório nas avaliações internas e externas das quais a escola participa). / Through monitoring carried out with four teachers of the state of São Paulo teaching, we conducted a case study with a focus on addressing the mathematical proofs in high school. The text describes the monitoring of classes, existing motivations and obstacles to the issue of development in the classroom. For the development of research, we used as a theoretical reference, the following work: Thompson (1992) on teachers conceptions; the types and functions of mathematical proofs of Balacheff and De Villiers, respectively. The instruments for data collection were used: direct observation, field notes and interviews. The results point to a utilitarian pedagogical practice without the active participation of students. As for mathematical proofs, found his approach intentionally and planned, being addressed, however, only in specific classes and linked to the staff of teacher interest and also, in general, without the active participation of students in the process. We believe this is not the ideal environment for the development of mathematical proofs in the classroom, which should occur in an area facing the argument, raise hypotheses, conjectures preparing to allow the advancement of students in levels of elaborate tests. Still found the influence of factors such as interest groups, lack of discipline, internal charges (room organization, students\' behavior in class, approach the expected content) and external (satisfactory performance in internal and external ratings of which the school participates).

Mathematical proofs are a part of mathematics that involves thinking and reasoning, rather than computation. The conceptions of Ontario high school mathematics teachers, of what they consider to be mathematical proofs and the role proofs have in their teaching practice, were examined through the use of individual interviews (60 minutes per participant) and a focus group discussion (one 90 minute session). The transcripts were each analyzed through emergent coding before themes were formed from comparing codes across transcripts. The interpretive lens included looking at teacher beliefs on the nature of mathematics, roles of proofs, and mathematical authority. The participants distinguished their university experiences with mathematical proofs from their high school teaching experiences. They saw proofs through the Mathematical Process Expectation, Reasoning and Proving, and they also used proof-related words when describing how they would enact Reasoning and Proving. The participants valued the development of argumentation and sense-making, based on logic and reasoning, as an enduring life-skill, and outcome of school mathematics. The perspectives of the participants provided insight on how teachers inform their teaching practice with the Ontario Mathematics Curriculum. It also revealed some thoughts, desires, values, and struggles teachers may face when teaching mathematics.