Thesis (Ph.D.)--York University, 2007. Graduate Programme in Mathematics. / Typescript. Includes bibliographical references (leaves 113-116). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:NR29337

Yes / Grothendieck's bound is used in the context of a single quantum system, in contrast to previous work which used it for multipartite entangled systems and the violation of Bell-like inequalities. Roughly speaking the Grothendieck theorem considers a 'classical' quadratic form ${\cal C}$ that uses complex numbers in the unit disc, and takes values less than 1. It then proves that if the complex numbers are replaced with vectors in the unit ball of the Hilbert space, then the 'quantum' quadratic form ${\cal Q}$ might take values greater than 1, up to the complex Grothendieck constant $k_\mathrm G$. The Grothendieck theorem is reformulated here in terms of arbitrary matrices (which are multiplied with appropriate normalisation prefactors), so that it is directly applicable to quantum quantities. The emphasis in the paper is in the 'Grothendieck region' $(1,k_\mathrm G)$, which is a classically forbidden region in the sense that ${\cal C}$ cannot take values in it. Necessary (but not sufficient) conditions for ${\cal Q}$ taking values in the Grothendieck region are given. Two examples that involve physical quantities in systems with six and 12-dimensional Hilbert space, are shown to lead to ${\cal Q}$ in the Grothendieck region $(1,k_\mathrm G)$. They involve projectors of the overlaps of novel generalised coherent states that resolve the identity and have a discrete isotropy.

Esta dissertação tem como objetivos um estudo detalhado do módulo canônico e do funtor dualizante para anéis de Cohen-Macaulay locais e as demonstrações dos teoremas de dualidade de Grothendieck. Iniciamos com o caso Artiniano e depois estendemos ao caso geral. Analisamos a unicidade do funtor dualizante através da interveniência do módulo canônico, uma peça chave da álgebra comutativa moderna. Focamos, em especial, nos chamados anéis de Gorenstein, caracterizados, entre os anéis de Cohen-Macaulay, como aqueles que são seu próprio módulo canônico. Explicitamos o funtor dualizante. Analisamos o comportamento do módulo canônico sob o processo de localização e completamento. Por fim, trabalhamos nas demonstrações dos teoremas de dualidade de Grothendieck. _________________________________________________________________________________________ ABSTRACT: The main purpose of this dissertation is a detailed study of canonical module and the dualizing functor for local Cohen-Macaulay rings and the proof of the duality theorems of Grothendieck. We start with the Artinian case and later we extend to the general case. Using the canonical module, an important subject of modern commutative algebra, we examine the unicity of dualizing functor. We give special attention in Gorenstein rings, those whom are Cohen-Macaulay rings and have free canonical module. After, we make explicit the dualizing functor and analyze the behavior of the canonical module under the localization and completion process. We conclude with the duality theorems of Grothendieck.

Made available in DSpace on 2014-06-12T18:33:33Z (GMT). No. of bitstreams: 2 arquivo8716_1.pdf: 394933 bytes, checksum: 01ca084378f3790faafdbe0837235749 (MD5) license.txt: 1748 bytes, checksum: 8a4605be74aa9ea9d79846c1fba20a33 (MD5) Previous issue date: 2007 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Esta dissertação tem como objetivos um estudo detalhado do módulo canônico e do funtor dualizante para anéis de Cohen-Macaulay locais e as demonstrações dos teoremas de dualidade de Grothendieck. Iniciamos com o caso Artiniano e depois estendemos ao caso geral. Analisamos a unicidade do funtor dualizante através da interveniência do módulo canônico, uma peça chave da álgebra comutativa moderna. Focamos, em especial, nos chamados anéis de Gorenstein, caracterizados, entre os anéis de Cohen-Macaulay, como aqueles que são seu próprio módulo canônico. Explicitamos o funtor dualizante. Analisamos o comportamento do módulo canônico sob o processo de localização e completamento. Por fim, trabalhamos nas demonstrações dos teoremas de dualidade de Grothendieck

Grothendieck published an extraordinary paper entitled ”Resume de la theorie metrique des pro¬duits tensoriels topologiques” in 1953. The main result of this paper is the inequality which is commonly known as Grothendieck Inequality. Following Kirivine, in this article, we give the proof of Grothendieck Inequality. We refor¬mulate it in different forms. We also investigate the famous Grothendieck constant KG. The Grothendieck constant was achieved by taking supremum over a special class of matrices. But our attempt will be to investigate it, considering a smaller class of matrices, namely only the positive definite matrices in this class. Actually we want to use it to get a counterexample of Matsaev’s conjecture, which was proved to be right by Von Neumann in some specific cases. In chapter 1, we shall state and prove the Grothendieck Inequality. In chapter 2, we shall introduce tensor product of vector spaces and different tensor norms. In chapter 3, we shall formulate Grothendieck Inequality in different forms and use the notion of tensor norms for its equivalent formation .In the last chapteri.ein chapter4we shall investigate on the Grothendieck constant.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2000. / Includes bibliographical references (p. 83-85).

Submitted by Reginaldo Soares de Freitas (reginaldo.freitas@ufv.br) on 2015-12-07T13:51:38Z No. of bitstreams: 1 texto completo.pdf: 484378 bytes, checksum: b64c83697ba96794dde4a234137b0510 (MD5) / Made available in DSpace on 2015-12-07T13:51:38Z (GMT). No. of bitstreams: 1 texto completo.pdf: 484378 bytes, checksum: b64c83697ba96794dde4a234137b0510 (MD5) Previous issue date: 2015-02-25 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho apresentamos uma categorificação da álgebra de Weyl, a partir de um estudo introdutório da Teoria de Categorificação Algébrica, que envolve conceitos básicos de categorias e funtores, com o objetivo de construir os grupos a de Grothendieck. Fazemos também um estudo mais detalhado da categorificação e de representações polinomiais da álgebra de Weyl que são realizadas por funtores indução e restrição de categorias apropriadas. / In this dissertation we present a categorification of the Weyl algebra from an intro- ductory study of the Theory of Algebraic Categorification involving basic concepts of categories and functors, with the objective of constructing the Grothendieck groups. We also present a detailed study of the categorification of the polyno- mial representations of Weyl algebra which are done by induction and restriction functors of appropriate categories.

The objective of this thesis is to provide a detailed analysis of a new invariant for Grothendieck topoi in the special case of the topos of continuous G-sets and continuous G-equivariant maps. We use a well-known site to present the isotropy group in elementary terms, as systems of right cosets of open subgroups of G. We establish properties of the the isotropy group for an arbitrary topological group and use the developed theory to compute the isotropy group for the Schanuel topos.

O objetivo dessa Dissertação é detalhar resultados conhecidos de Cohomologia em Topos de Grothendieck. Para isso, apresentamos a Álgebra Homológica em seu contexto mais geral, através de Categorias Abelianas, introduzindo as principais noções da área como funtores derivados e sequências espectrais. Desenvolvemos também o essencial da Teoria de Topos, explicando como um topos de Grothendieck surge como uma certa generalização dos feixes de conjuntos e fornecemos aspectos lógicos dos topos elementares. Focamos sobretudo nos Topos de Grothendieck pois a partir deles podemos construir categorias abelianas com suficientes injetivos, as quais são necessárias para expressar os grupos de cohomologia. / The final objective of this Dissertation is to detail known results of Cohomology in Grothendieck Topos. For this, we present Homological Algebra in its more general context, through Abelian Categories, introducing the main notions of the area as derived functors and spectral sequences. We also develop the basics of the Topos Theory, explaining how a Grothendieck Topos arises as a certain generalization of sheafs and we provide logical aspects of the elementary topos. We focus mainly in the Grothendieck Topos because from them we can construct abelians categories.

Cette thèse est consacrée à l'étude des interactions entre les structures homotopiques en théorie des catégories et les modèles catégoriques de la théorie des types de Martin-Löf. Le mémoire s'articule selon trois axes: les bifibrationos de Quillen, les catégories homotopiques des bifibrations de Quillen, et les tribus généralisées. Le premier axe définit une nouvelle notion de bifibration classifiant les pseudo foncteurs avec de bonnes propriétés depuis un catégorie de modèles et à valeurs dans la 2-catégorie des catégories de modèles et adjonctions de Quillen entre elles. En particulier on montre comment équipper d'une structure de modèle la construction de Grothendieck d'un tel pseudo foncteur. Le théorème principal de cette partie est une caractérisation des bonnes propriétés qu'un pseudo foncteur doit posséder pour supporter cette structure de catégorie de modèles sur sa construction de Grothendieck. En ce sens, on améliore les deux théorèmes précédemment existants dans la littérature qui ne donnent que des conditions suffisantes alors que nous donnons des conditions nécessaires et suffisantes. Le second axe se concentre sur le foncteur induit entre les catégories homotopiques des catégories de modèles mises en oeuvre dans une bifibration de Quillen. On y prouve que cette localization peut se faire en deux étapes au moyen d'un quotient homotopique à la Quillen itéré. De manière à rendre cette opération rigoureuse, on a besoin de travailler dans un cadre légèrement plus large que celui imaginé par Quillen : en se basant sur le travail d'Egger, on utilise des catégories de modèles sans nécessairement tous les (co)égalisateurs. Le chapitre de prérequis sert précisément à reconstruire la théorie basique des l'algèbre homotopique à la Quillen dans ce cadre élargi. Les structures mis à nu dans cette partie imposent de considérer des versions "homotopique" des poussés en avant et des tirés en arrière qu'on trouve habituellement dans les (op)fibrations de Grothendieck. C'est le point de départ pour le troisième axe, dans lequel on définit une nouvelle structure, appelée tribu relative, qui permet d'axiomatiser des versions homotopiques de la notion de flèche cartésienne et cocartésienne. Cela est obtenu en réinterprétant les (op)fibrations de Grothendieck en termes de problèmes de relèvement. L'outil principal dans cette partie est une version relative des systèmes de factorisation stricts ou faibles usuels. Cela nous permet en particulier d'expérimenter un nouveau demodèle de la théorie des types dépendants intentionnelle dans lequelles types identités sont donnés par l'exact analogue homotopique du prédicat d'égalité dans les hyperdoctrines de Lawvere. / This thesis is concerned with the study of the interplay between homotopical structures and categorical model of Martin-Löf's dependent type theory. The memoir revolves around three big topics: Quillen bifibrations, homotopy categories of Quillen bifibrations, and generalized tribes. The first axis defines a new notion of bifibrations, that classifies correctly behaved pseudo functors from a model category to the 2-category of model categories and Quillen adjunctions between them. In particular it endows the Grothendieck construction of such a pseudo functor with a model structure. The main theorem of this section acts as a charaterization of the well-behaved pseudo functors that tolerates this "model Gothendieck construction". In that respect, we improve the two previously known theorems on the subject in the litterature that only give sufficient conditions by designing necessary and sufficient conditions. The second axis deals with the functors induced between the homotopy categories of the model categories involved in a Quillen bifibration. We prove that this localization can be performed in two steps, by means of Quillen's construction of the homotopy category in an iterated fashion. To that extent we need a slightly larger framework for model categories than the one originally given by Quillen: following Egger's intuitions we chose not to require the existence of equalizers and coequalizers in our model categories. The background chapter makes sure that every usual fact of basichomotopical algebra holds also in that more general framework. The structures that are highlighted in that chapter call for the design of notions of "homotopical pushforward" and "homotopical pullback". This is achieved by the last axis: we design a structure, called relative tribe, that allows for a homotopical version of cocartesian morphisms by reinterpreting Grothendieck (op)fibrations in terms of lifting problems. The crucial tool in this last chapter is given by a relative version of orthogonal and weak factorization systems. This allows for a tentative design of a new model of intentional type theory where the identity types are given by the exact homotopical counterpart of the usual definition of the equality predicate in Lawvere's hyperdoctrines