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In the study of triple systems, one question faced is that of finding for what order a decomposition exists. We state and prove a necessary and sufficient condition for the existence of a bicyclic mixed triple system based on the three possible partial orientations of the 3-cycle with twice as many arcs as edges. We also explore the existence of rotational and reverse mixed triple systems. Our principal proof technique applied is the difference method. Finally, this work contains a result on packing of complete mixed graphs on v vertices, Mv, with isomorphic copies of two of the mixed triples and a possible leave structure.

Let A and B be real or complex C*-algebras. Let V and W be real or complex (right) full Hilbert C*-modules over A and B, respectively. Let T be a linear bijective map from V onto W. We show the following four statements are equivalent. (a) T is a unitary operator, i.e., there is a ∗-isomorphism £\ : A ¡÷ B such that <Tx,Ty> = £\(<x,y>), ∀ x,y∈ V ; (b) T preserves TRO products, i.e., T(x<y,z>) =Tx<Ty,Tz>, ∀ x,y,z in V ; (c) T is a 2-isometry; (d) T is a complete isometry. Moreover, if A and B are commutative, the four statements are also equivalent to (e) T is a isometry. On the other hand, if V and W are complex Hilbert C*-modules over complex C*-algebras, then T is unitary if and only if it is a module map, i.e., T(xa) = (Tx)£\(a), ∀ x ∈ V,a ∈ A.

Le projet de recherche s’inscrit dans un contexte d'imagerie moléculaire, où la modalité d'imagerie d'intérêt est la tomographie d'émission par positrons (TEP) appliquée en recherche sur les petits animaux. Afin de permettre l’observation de détails infimes, les plus récents développements sur ce genre de scanner ont constamment amélioré leur résolution spatiale, sans toutefois obtenir les mêmes progrès en terme de sensibilité. Parmi les méthodes étudiées afin de combler cette lacune, la récupération de coïncidences triples à l’aide d'un réseau de neurones artificiels semble être une technique viable. En effet, malgré une dégradation du contraste, celle-ci permet d'améliorer substantiellement la sensibilité de l’image. Cette technique n'est cependant pas prête à être intégrée aux protocoles de recherche, car son application est pour l’instant limitée à un traitement hors ligne des données d'acquisition d'un scanner. En conséquence, la faisabilité d'une telle approche en temps réel n'est donc pas garantie, car le flux de coïncidences d'un scanner est très important et ses ressources de calculs sont limitées. Dans l’intention d'inclure ce gain en sensibilité pendant une acquisition où le traitement est effectué en temps réel, ce projet de recherche propose une implémentation d'un réseau de neurones artificiels au sein d'une matrice de porte programmable (FPGA) pouvant récupérer en temps réel les coïncidences diffuses triples du scanner LabPET, version 4 cm. La capacité de traitement obtenue est 1 087 000 coïncidences triples par seconde en utilisant 23.1% des ressources d'unités logiques d'un FPGA de modèle XC2VP50. Comparativement à un programme équivalent à haute précision sur ordinateur personnel, l’analyse de validité prend la même décision dans 99.9% des cas et la même ligne de réponse est choisie dans 97.9% des cas. Intégrées à l’image, les coïncidences triples permettent une augmentation de sensibilité jusqu’à 39.7%, valeur qui est en deçà [de] celle obtenue des recherches antérieures, mais expliquée par des conditions d'acquisition différente. Au niveau de la qualité de l’image, la dégradation du contraste de 16,1% obtenu est similaire à celle observée antérieurement. En référence à ces résultats, les ressources limitées d'un scanner de tomographie d'émission par positrons sont avérées suffisantes pour permettre l’implémentation d'un réseau de neurones artificiels devant classifier en temps réel les coïncidences triples du scanner. En terme de contributions, l’implémentation en temps réel réalisée pour ce projet confirme la faisabilité de la technique et apporte une nouvelle approche concrète pour améliorer la sensibilité. Dans une autre mesure, la réussite du projet de recherche contribue à faire connaître la technique des réseaux de neurones artificiels dans le domaine de la tomographie d’émission par positrons. En effet, cette approche est pertinente à considérer en guise d'alternative aux solutions traditionnelles. Par exemple, les réseaux de neurones artificiels pourraient effectuer une évaluation correcte du phénomène des coïncidences fortuites.

The Resource Description Framework (RDF) is the logical data model of the Semantic Web. RDF encodes information as a directed graph using a set of labeled edges known formally as resource-property-value statements or, in common usage, as RDF triples or simply triples. Values recorded in RDF triple form are either Universal Resource Identifiers (URIs) or literals. The use of URIs allows links between distributed data sources, which enables a logical model of data as a graph spanning the Internet. SPARQL is a standard SQL-like query language on RDF triples. This report describes the translation of SPARQL queries to equivalent SQL queries operating on a relational representation of RDF triples, and the physical optimization of that representation using the IBM DB2 relational database management system. Performance was evaluated using the Berlin SPARQL Benchmark. The results show that the implementation can perform well on certain queries, but more work is required to improved overall performance and scalability. / text

This study concerns a teaching design with the purpose to facilitate the students’ objectification of the Pythagorean Theorem. Twelve 14-year old students (N=12) participated in the study before the theorem was introduced to them at school. The design incorporated ideas from the ‘embodied mind’ framework, history and realistic mathematics, linking ‘embodied verticality’ with ‘perpendicularity’. The qualitative analyses suggested that the participants were led to the conquest of the ‘first level of objectification’ (through numbers) of the Pythagorean Theorem, showing also evidence of appropriate ‘fore-conceptions’ of the ‘second level of objectification’ (through proof) of the theorem. The triangle the sides of which are associated with the Basic Triple (3,4,5) served as a primary instrument for the students’ objectification, mainly, by facilitating their ‘generic abstraction’ of the Pythagorean Triples.

Submitted by MARCOS EDER CUPAIOLI (marcoscupaioli@hotmail.com) on 2016-09-13T14:53:52Z No. of bitstreams: 1 Dissertação-MARCOS-EDER-CUPAIOLI-Matemática-Final Repositório.pdf: 3030917 bytes, checksum: 5fee5216541ccf70c5a9acb075b9976f (MD5) / Rejected by Felipe Augusto Arakaki (arakaki@reitoria.unesp.br), reason: Solicitamos que realize uma nova submissão seguindo as orientações abaixo: No campo “Versão a ser disponibilizada online imediatamente” foi informado que seria disponibilizado o texto completo porém no campo “Data para a disponibilização do texto completo” foi informado que o texto completo deverá ser disponibilizado apenas 6 meses após a defesa. Caso opte pela disponibilização do texto completo apenas 6 meses após a defesa selecione no campo “Versão a ser disponibilizada online imediatamente” a opção “Texto parcial”. Esta opção é utilizada caso você tenha planos de publicar seu trabalho em periódicos científicos ou em formato de livro, por exemplo e fará com que apenas as páginas pré-textuais, introdução, considerações e referências sejam disponibilizadas. Se optar por disponibilizar o texto completo de seu trabalho imediatamente selecione no campo “Data para a disponibilização do texto completo” a opção “Não se aplica (texto completo)”. Isso fará com que seu trabalho seja disponibilizado na íntegra no Repositório Institucional UNESP. Por favor, corrija esta informação realizando uma nova submissão. Agradecemos a compreensão. on 2016-09-14T21:55:41Z (GMT) / Submitted by MARCOS EDER CUPAIOLI (marcoscupaioli@hotmail.com) on 2016-09-15T01:47:57Z No. of bitstreams: 1 Dissertação-MARCOS-EDER-CUPAIOLI-Matemática-Final Repositório.pdf: 3030917 bytes, checksum: 5fee5216541ccf70c5a9acb075b9976f (MD5) / Approved for entry into archive by Juliano Benedito Ferreira (julianoferreira@reitoria.unesp.br) on 2016-09-15T18:14:38Z (GMT) No. of bitstreams: 1 cupaioli_me_me_sjrp.pdf: 3030917 bytes, checksum: 5fee5216541ccf70c5a9acb075b9976f (MD5) / Made available in DSpace on 2016-09-15T18:14:38Z (GMT). No. of bitstreams: 1 cupaioli_me_me_sjrp.pdf: 3030917 bytes, checksum: 5fee5216541ccf70c5a9acb075b9976f (MD5) Previous issue date: 2016-08-19 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Este trabalho aborda um conjunto de atividades experimentais com a finalidade de demonstrar um dos mais belos e importantes teoremas da Matemática: o Teorema de Pitágoras. São conhecidas mais de 400 demonstrações, aqui optamos por utilizar uma demonstração devido a Rudolf Wolf, por possibilitar uma abordagem geométrica lúdica através da dissecção de figuras planas. Inicialmente apresentamos o conceito geral de semelhança e áreas das figuras planas que utilizam propriedades e áreas de polígonos equidecomponíveis. Posteriormente, realizamos um breve resgate histórico sobre diversas demonstrações do Teorema e da vida de Pitágoras. Destacamos, também, uma maneira de achar algumas ternas pitagóricas, utilizando a sequência de Fibonacci. Por fim, foram propostas e desenvolvidas atividades experimentais em sala de aula com a utilização de moldes em EVA, explorando o Teorema de Pitágoras e algumas de suas aplicações. / This work contains a set of experimental activities in order to prove one of the most beautiful and important theorems in Mathematics: the Pythagorean Theorem. There are known more than 400 proofs, here we chose to use a proof due to Rudolf Wolf, by allowing a playful geometric approach by dissection of plane figures. Initially we present the general concept of similarity and areas of plane figures using properties and areas of equidecomposable polygons. Later, we do a brief historical review of some proofs of Theorem and Pythagoras's life. We also highlight a way to find some Pythagorean triples using the Fibonacci sequence. Finally, it was proposed and developed experimental activities in the classroom with the use of molds EVA, exploring the Pythagorean theorem and some of its applications.

La géométrie non commutative, du fait qu'elle permet de généraliser des objets géométriques sous forme algébrique, offre des perspectives intéressantes pour réunir la théorie quantique des champs et la relativité générale dans un seul cadre. Elle peut être abordée selon différents points de vue et deux d'entre eux sont présentés dans cette thèse. Le premier, le calcul différentiel basé sur les dérivations, nous a permis de construire une action de Yang-Mills-Higgs dans laquelle apparait des champs pouvant être interprétés comme des champs de Higgs. Avec le second, les triplets spectraux, on peut généraliser la notion de distance entre état et calculer des formules de distance. C'est ce que nous avons fait dans le cas de l'espace de Moyal et du tore non commutatif. / Noncommutative geometry offers interesting prospects to gather the quantum field theory and relativity in one general framework because it allows one to generalize geometric objects algebraically. It can be approached from different points of view and two of them are presented in this PhD. The first, calculus based on derivations, allowed us to construct a Yang-Mills-Higgs action which appears in fields that can be interpreted as Higgs fields. With the second, spectral triples, we can generalize the notion of distance between states. We calculated the distance formulas in the case of the Moyal space and the noncommutative torus.

This study concerns a teaching design with the purpose to facilitate the students’ objectification of the Pythagorean Theorem. Twelve 14-year old students (N=12) participated in the study before the theorem was introduced to them at school. The design incorporated ideas from the ‘embodied mind’ framework, history and realistic mathematics, linking ‘embodied verticality’ with ‘perpendicularity’. The qualitative analyses suggested that the participants were led to the conquest of the ‘first level of objectification’ (through numbers) of the Pythagorean Theorem, showing also evidence of appropriate ‘fore-conceptions’ of the ‘second level of objectification’ (through proof) of the theorem. The triangle the sides of which are associated with the Basic Triple (3,4,5) served as a primary instrument for the students’ objectification, mainly, by facilitating their ‘generic abstraction’ of the Pythagorean Triples.

In this thesis examples of spectral triples, which represent fractal sets, are examined and new insights into their noncommutative geometries are obtained. Firstly, starting with Connes' spectral triple for a non-empty compact totally disconnected subset E of {R} with no isolated points, we develop a noncommutative coarse multifractal formalism. Specifically, we show how multifractal properties of a measure supported on E can be expressed in terms of a spectral triple and the Dixmier trace of certain operators. If E satisfies a given porosity condition, then we prove that the coarse multifractal box-counting dimension can be recovered. We show that for a self-similar measure μ, given by an iterated function system S defined on a compact subset of {R} satisfying the strong separation condition, our noncommutative coarse multifractal formalism gives rise to a noncommutative integral which recovers the self-similar multifractal measure ν associated to μ, and we establish a relationship between the noncommutative volume of such a noncommutative integral and the measure theoretical entropy of ν with respect to S. Secondly, motivated by the results of Antonescu-Ivan and Christensen, we construct a family of (1, +)-summable spectral triples for a one-sided topologically exact subshift of finite type (∑{{A}} {{N}}, σ). These spectral triples are constructed using equilibrium measures obtained from the Perron-Frobenius-Ruelle operator, whose potential function is non-arithemetic and Hölder continuous. We show that the Connes' pseudo-metric, given by any one of these spectral triples, is a metric and that the metric topology agrees with the weak*-topology on the state space {S}(C(∑{{A}} {{N}}); {C}). For each equilibrium measure ν[subscript(φ)] we show that the noncommuative volume of the associated spectral triple is equal to the reciprocal of the measure theoretical entropy of ν[subscript(φ)] with respect to the left shift σ (where it is assumed, without loss of generality, that the pressure of the potential function is equal to zero). We also show that the measure ν[subscript(φ)] can be fully recovered from the noncommutative integration theory.