On travaille dans ZF, théorie des ensembles sans Axiome du Choix. En considérant des formes plus faibles de l'Axiome du choix, comme l'axiome de Hahn-Banach HB : "Toute forme linéaire sur un sous-espace vectoriel d'un espace vectoriel E, majorée par une forme sous-linéaire p se prolonge en une forme linéaire sur E majorée par p'', ou encore l'axiome de Tychonov T2 : "Un produit de compacts séparés est compact'', on étudie l'existence d'états dans les groupes ordonnés avec unité d'ordre. On poursuit l'étude en établissant des liens entre idéaux à gauche et états sur les C*-algèbres. / We work in ZF, set theory without Axiom of Choice. Using weak forms of Axiom of Choice, for example Hahn-Banach axiom HB : "Every linear form on a vector subspaceof a vector space E, increased by a sublinear form p can be extended to a linear form on E increased by p", or Tychonov axiom T2 : "Every product of compact Haussdorf is compact, we study the existence of states on ordered groups with order unit. We continue giving links between left ideals and states on C*-algebras.

O presente trabalho trata da importÃncia de se utilizar o MÃtodo de InduÃÃo MatemÃtica em demonstraÃÃes no Ensino BÃsico da MatemÃtica, pois, atualmente, percebe-se que essa tÃcnica de prova à raramente abordada nesse nÃvel de ensino. Para isso, percorremos um caminho que vai desde a importÃncia das demonstraÃÃes, seguido de uma seÃÃo na qual procuramos mostrar a diferenÃa entre induÃÃo e induÃÃo matemÃtica, passando pela definiÃÃo e explicaÃÃo desse mÃtodo. Mostramos tambÃm, as equivalÃncias entre as diversas formas do PrincÃpio da InduÃÃo e o PrincÃpio da Boa OrdenaÃÃo. ConcluÃmos com a resoluÃÃo de vÃrios exemplos, seguidos da sugestÃo de alguns problemas que visam facilitar o entendimento e a aplicaÃÃo do MÃtodo de InduÃÃo MatemÃtica no Ensino MÃdio. / This dissertation deals with the importance of using Mathematical Induction Method demonstrations in Basic Mathematics Teaching, because, currently, it is clear that this proof technique is rarely approached at that level of education. For this, we pursue a path that goes from the importance of the demonstrations, followed by a section in which we show the difference between induction and mathematical induction, including the definition and explanation of this method. We also show the equivalence between different forms of the Induction Principle and the Well Ordering Principle. We conclude with the resolution of some examples, followed by suggestions of some problems to facilitate the understanding and application of mathematics Induction Method in High School.

A teoria de conjuntos por vezes deixada de lado em algumas escolas de ensino mÃdio, constitui-se em um elemento primordial para o entendimento das funÃÃes, em especial. A nÃo abordagem, ou a sua abordagem superficial, deixa no estudante uma lacuna difÃcil de ser suprida em estudos posteriores. AliÃs, a lacuna deixada pode dificultar o desempenho do estudante no ensino superior. Diante desta constataÃÃo, à objetivo principal desta dissertaÃÃo fazer uma leitura dos principais tÃpicos ligados à Teoria de Conjuntos do ensino mÃdio, ao mesmo tempo em que faz uma ponte entre estes e outros pontos nÃo menos importantes, tratando conjuntos em uma linguagem mais acadÃmica. SerÃo abordados desde as propriedades e teoremas relacionados a conjuntos finitos, atà a sua generalizaÃÃo para conjuntos infinitos, culminando com o teorema de Cantor-Schroeder-Bernstein, o Axioma da Escolha, e o Lema de Zorn. Para tantos, realizaram-se pesquisas bibliogrÃficas em fontes variadas. / The set theory sometimes left out in some high schools, is in a key element for understanding the functions in particular. Failure to address this issue or its superficial approach leaves the student a difficult gap to be filled in later studies. Incidentally, the left gap may hinder student performance in higher education. If this is so, is the main objective of this work to a reinterpretation of the main topics linked to the high school set theory, while making a bridge between these and other equally important points dealing with sets in a more academic language. Will be covered from the properties and theorems related to finite sets up its generalization to infinite sets, culminating in the Cantor-Schroeder-Bernstein theorem, the Axiom of Choice and Zornâs Lemma. To this end, there were literature searches in various sources.

O presente trabalho tem como objetivo apresentar a construÃÃo dos nÃmeros naturais e a definiÃÃo axiomÃtica no que diz respeito Ãs quatro operaÃÃes fundamentais para alunos e professores do ensino fundamental. Para isso foi apresentado uma sequÃncia abordando inicialmente as consideraÃÃes sobre o estudo da MatemÃtica, o conceito de MatemÃtica, o saber matemÃtico e um breve histÃrico matemÃtico para se perceber como teorias e prÃticas matemÃticas foram criadas, desenvolvidas e utilizadas num contexto especÃfico de cada Ãpoca. No segundo momento foi descrita a construÃÃo dos nÃmeros naturais atravÃs dos axiomas de Peano, prosseguindo com a definiÃÃo rigorosa de cada operaÃÃo e finalizando com a relaÃÃo de ordem no conjunto dos nÃmeros naturais. / This paper aims to present the construction of the natural numbers and the axiomatic definition with respect to the four fundamental operations for students and teachers of elementary school.To this was presented a sequence initially addressing on the study of mathematics, the concept of mathematics, mathematical knowledge and a mathematical brief history to see how mathematical theories and practices are designed, developed and used in a specific context of each era. The second moment was described the construction of natural numbers through the Peano axioms, continuing with the rigorous definition of each operation and ending with the order relation in the set of natural numbers.

According to The Librarian’s axiom, the library loans and visits will increase when unemployment rates rise. In this bachelor thesis we want to investigate if the axiom was applicable in Sweden and Finland during the years 2008 until 2016. If we can find that TheLibrarian’s axiom can be confirmed, the libraries in Sweden and Finland can have stronger arguments against budget cuts and can prepare for increases in library usage. Furthermore no major studies on this topic have been executed in Scandinavia. The purpose of this study is to test a hypothesis, namely whether there is a connection between the economic situation and the library usage in Sweden and Finland. We also want to study the similarities and differences between the countries and try to explain them. We gathered our empirical data by using and analyzing official statistics from several databases. The variables we examined are physical library visits, loans of paper books, unemployment rates and Gini-coefficient. Our findings are presented in figures which are further analyzed with help of cross tabulations and correlation measures. The results of our study show that there is no strong connection between the library usage and the economic situation in neither Sweden nor Finland. There were little differences between the countries. The connections in Finland were a little bit stronger, but still not strong enough to confirm the axiom. To further explain our findings, there is a need for more research on this topic.

We introduce the Partition Principle PP, an axiom introduced by Russell in the context of its similarities and differences with the Axiom of Choice AC. We start by proving some properties of PP, and AC, and show that AC, entails PP. To address the problem of whether the converse holds, we develop the Zermelo-Fraenkel ZF set theory and examine its consistency and build a model in which AC, fails. We follow this with a discussion of forcing, a technique introduced by Paul Cohen to build new models of set theory from existing ones, which have differing properties from the starting model. We conclude by examining candidate models called permutation models where AC, fails, which may be useful as candidate models for forcing a model in which PP, holds but AC, does not. We conjecture that such a model exists, and that PP, does not entail AC. / Thesis / Master of Science (MSc)

Yongming Yanshou has been defined in previous scholarship as a "Chan Master," though I contend this designation does little to clarify the type of Buddhism he professed. In this thesis I argue that Yanshou viewed the Chan tradition as a movement completely integrated with the scriptural-based Chinese Buddhist traditions of his day, and Chan lineage, a primary feature around which the Song Chan tradition would base themselves, was only of peripheral concern. Instead, Yanshou took the Chan teachings and the scriptural traditions present in the mid-tenth century and organized them all under the "axiom of the one-mind" (yixin zong). This axiom formed the ontological foundation on which all of Yanshou's Buddhist theory and concepts are based, and through an investigation centering around the concept of li ("principle") in the extant writings of both Yanshou and Zongmi, I argue that Yanshou equated the one-mind (yixin) with li in a way that Zongmi never did, and li for Yanshou became synonymous with the axiom of the one-mind as Yanshou's ontic basis. / Thesis / Master of Arts (MA)

<p>In this thesis we prove a strong conceptual completeness result for first-order continuous logic. Strong conceptual completeness was proved in 1987 by Michael Makkai for classical first-order logic, and states that it is possible to recover a first-order theory T by looking at functors originating from the category Mod(T) of its models. </p> <p> We then give a brief account of simple theories in continuous logic, and give a proof that the characterization of simple theories using dividing holds in continuous structures. These results are a specialization of well established results for thick cats which appear in [Ben03b] and in [Ben03a].</p> <p> Finally, we turn to the study of non-archimedean Banach spaces over non-trivially valued fields. We give a natural language and axioms to describe them, and show that they admit quantifier elimination, and are N0-stable. We also show that the theory of non-archimedean Banach spaces has only one N 1-saturated model in any cardinality. </p> / Thesis / Doctor of Philosophy (PhD)

Med dagens digitaliserade samhälle används sociala medier i hög grad för att kommunicera med varandra. Detta öppnar upp möjligheten för kriminella att enklare kommunicera med varandra, och även utföra brott. Ett vanligt brott på sociala medier är utnyttjandet av barn för sexuell posering där Instagram används i en stor del av fallen. Med Instagrams funktion Vanish Mode (försvinnaläge) ökar möjligheterna att dölja sin verksamhet, vilket innebär en utmaning för det IT-forensiska arbetet.Uppsatsen undersöker möjligheten att återskapa meddelanden skickade med Instagrams försvinnaläge genom verktyget Magnet Axiom och manuellt. För att automatisera den manuella återskapningsprocessen skrivs också ett program som återskapar och strukturerar meddelandena. Experimentet visar att Magnet Axiom inte återskapar några meddelanden från försvinnaläget, men att det går att manuellt återskapa dem från Instagrams databaser. Meddelanden skapade i försvinnaläget återfinns i WAL-filen tillhörande den databas som lagrar chattmeddelande frånanvändaren. Genom att använda en hexläsare för att studera databasens rådata går det att tyda vilken data som lagras i den. I resultatet beskrivs även hur programmet går till väga för att automatiskt extrahera meddelandena från databaserna.

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