La physique des très hautes énergies nécessite une description cohérente des quatre forces fondamentales. La géométrie non commutative représente un cadre mathématique prometteur qui a déjà permis d'unifier la relativité générale et le modèle standard, au niveau classique, grâce au principe de l'action spectrale. L'étude des théories quantiques de champs sur des espaces non commutatifs est une première étape vers la quantification de ce modèle. Celles-ci ne sont pas simplement obtenues en récrivant les théories commutatives sur des espaces non commutatifs. En effet, ces tentatives ont révélé un nouveau type de divergences, appelé mélange ultraviolet/infrarouge, qui rend ces modèles non renormalisables. H. Grosse et R. Wulkenhaar ont montré, sur un exemple, qu'une modification du propagateur restaure la renormalisabilité. L'étude de la généralisation de cette méthode est le cadre de cette thèse. Nous avons ainsi étudié deux modèles sur espace de Moyal qui ont permis de préciser certains aspects des théories non commutatives. En espace x, la principale difficulté technique est due aux oscillations de l'interaction. Nous avons donc généralisé les résultats de T. Filk afin d'exploiter au mieux ces oscillations. Nous avons pu ainsi distinguer deux types de mélange, renormalisable ou pas. Nous avons aussi mis en lumière la notion d'orientabilité : le modèle de Gross-Neveu non commutatif orientable est renormalisable sans modification du propagateur. L'adaptation de l'analyse multi-échelles à la base matricielle a souligné l'importance du graphe dual et représente un premier pas vers une formulation des théories de champs indépendante de l'espace sous-jacent.

Thesis (Ph. D.)--University of Washington, 2001. / Vita. Includes bibliographical references (p. 177-179).

Τα σώματα, κυρίως τα μη αντιμεταθετικά, γενικά κατασκευάζονται ως σώματα κλασμάτων δακτυλίων, εντούτοις δεν έχουν όλοι οι δακτύλιοι σώμα κλασμάτων και για δοθέντα δακτύλιο μπορεί να είναι αρκετά δύσκολο να αποφανθούμε αν υπάρχει σώμα κλασμάτων. Στο κεφάλαιο 1 θα αναφέρουμε ορισμένες γενικές παρατηρήσεις πάνω στο είδος των συνθηκών, οι οποίες χαρακτηρίζουν την εμβάπτιση δακτυλίου σε σώμα κλασμάτων και δίνουμε αναγκαίες συνθήκες οι οποίες σχετίζονται με την τάξη των ελεύθερων modules. Στη συνέχεια περιγράφουμε τις συνθήκες του Ore για την εμβάπτιση αυτή, όπου γενικεύεται η αντιμεταθετική περίπτωση. Στο δεύτερο κεφάλαιο εισάγουμε στοιχεία από τη γενική θεωρία τοπολογικών δακτυλίων και modules. Πιο συγκεκριμένα, παρουσιάζονται θεμελιώδεις έννοιες και βασικά αποτελέσματα πάνω στους τοπολογικούς δακτύλιους και τα τοπολογικά σώματα, δίνοντας ιδιαίτερη έμφαση στα στρεβλά σώματα. Πιο συγκεκριμένα εξετάζουμε φραγμένα σύνολα, τοπολογικούς μηδενοδιαιρέτες, τοπολογικά μηδενοδύναμα στοιχεία και minimal τοπολογίες. Αναφέρουμε, επίσης αρκετά παραδείγματα τοπολογιών επί δακτυλίων και modules. Στο κεφάλαιο 3 ορίζουμε διατιμήσεις επί των στρεβλών σωμάτων και ασχολούμαστε με το πρόβλημα ύπαρξης ψευδό-διατιμήσεων σε δακτυλίους και modules. / Fields, especially skew fields, are generally constructed as the field of fractions of some ring, but of course not every ring has a field of fractions and for a given ring it may be quite difficult to decide if a field of fractions exists. In chapter 1 we shall bring some general observations on the kind of conditions to expect and give some necessary conditions relating to the rank of free modules. On the other hand there is the Ore condition generalizing the commutative case. Chapter 2 provides fundamental concepts and basic results on topological rings, modules and especially on skew fields. Under detailed consideration are bounded subsets, topological divisors of zero, topologically nilpotent elements and minimal topologies. There are also many examples of topologies on rings and modules. In chapter 3 we define norms on skew fields and discuss the problem of the existence of real-valued pseudonorms rings and modules.

This paper presents a survey of concepts in commutative algebra that have applications to topology and graph theory. The primary algebraic focus will be on Stanley-Reisner rings, classes of polynomial rings that can describe simplicial complexes. Stanley-Reisner rings are defined via square-free monomial ideals. The paper will present many aspects of the theory of these ideals and discuss how they relate to important constructions in commutative algebra, such as finite generation of ideals, graded rings and modules, localization and associated primes, primary decomposition of ideals and Hilbert series. In particular, the primary decomposition and Hilbert series for certain types of monomial ideals will be analyzed through explicit examples of simplicial complexes and graphs.

This thesis suggests an approach to compute the short-time behaviour of the hypoelliptic heat kernel corresponding to sub-Riemannian structures on unimodular Lie groups of type I, without previously holding a closed form expression for this heat kernel. Our work relies on the use of classical non-commutative harmonic analysis tools, namely the Generalized Fourier Transform and its inverse, combined with the Trotter product formula from the theory of perturbation of semigroups. We illustrate our main results by computing, to our knowledge, a first expression in short-time for the hypoelliptic heat kernel on the Engel and the Cartan groups, for which there exist no closed form expression. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2011-10-08 01:32:32.896

Thesis (Ph. D.)--University of Wisconsin--Madison, 1960. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph.D.)--University of Nebraska-Lincoln, 2008. / Title from title screen (site viewed Oct. 31, 2008). PDF text: v, 54 p. : ill. ; 769 K. UMI publication number: AAT 3313102. Includes bibliographical references. Also available in microfilm and microfiche formats.

Title: Generalized injectivity and approximations Author: Serap S¸ahinkaya Department:Algebra Faculty of Mathematics and Physics, Charles University Supervisor: Prof. RNDr. Jan Trlifaj, DSc, Faculty of Mathematics and Physics, Charles University Abstract: Injective modules form a basic class studied in contemporary module theory. One of their generalizations, inspired by tilting theory, is the notion of a cotilting module. While tilting modules behave well with respect to localization, we show that colocalization is the correct approach when comparing the struc- ture of cotilting modules over commutative noetherian rings R with the structure of cotilting modules over their localizations Rm where m runs over the maximal spectrum of R. This is done in Chapter 2 of this Dissertation whose main results were published in the paper [33]. In Chapter 3, we investigate approximation properties of other classic generalizations of injective modules, the Ci- and quasi- injective modules, introduced by Harada et al. Suprisingly, we prove that these classes provide for approximations only in exceptional cases (when all Ci mod- ules are injective, or pure-injective). The Dissertation ends with a set of open problems. Keywords: Commutative noetherian ring, (co)tilting module, (generalized) injec- tive module. 1

In this thesis we look into Emmy Noether's life and works. An overview of Emmy Noether's life gives context to understanding her approach to mathematics which produced seminal works. In Invariante Varationsprobleme, Noether proved the connection between symmetry and conservation laws; Noether's theorem is the foundations of modern physics. In Idealtheorie in Ringberichen, she proved the ascending chain condition on ideals in an abstract setting; this work started the "algebrization of mathematics" in 20th century. Noether continued to produce phenomenal works and influenced numerous branches of mathematics. By understanding Emmy Noether's life and her works, one achieves a greater understanding to the foundations of 20th century mathematics.

Orientador: Paulo Roberto Brumatti / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Ciencia da Computação / Made available in DSpace on 2018-08-09T23:37:45Z (GMT). No. of bitstreams: 1 DiazNoguera_MaribeldelCarmen_M.pdf: 632573 bytes, checksum: fbcf2bd0092558fce4ba4d082d4c68c7 (MD5) Previous issue date: 2007 / Resumo: O principal objetivo desta dissertação é apresentar resultados centrais sobre derivações localmente nilpotentes no anel de polinômios B = k[x1, ..., xn], para n = 3 que foram apresentados por Daniel Daigle em [2 ], [3] e [4] .Para este propósito, introduziremos os conceitos básicos e fundamentais da teoria das derivações num anel e apresentaremos resultados em relação a derivações localmente nilpotentes num domínio de característica zero e de fatorização única. Entre tais resultados está a fórmula Jacobiana que usaremos para descrever o conjunto das derivações equivalentes e localmente nilpotentes de B = k[x, y, z] e o conjunto LND(B), com B = k[x,y]. Também, explicítam-se condições equivalentes para a existência de uma derivação ?-homogênea e localmente nilpotente de B = k[x, y, z] com núcleo k[¿, g], onde {¿}, {g} e B, mdc(?) = mdc(?(¿), ? (g)) = 1 / Abstract: In this dissertation we present centraIs results on locally nilpotents derivations in a ring of polynomials B = k[x1, ..., xn], for n = 3, which were presented by Daniel Daigle in [2], [3] and [4]. For this, we introduce basic fundamenta1 results of the theory of derivations in a ring and we present results on locally nilpotents derivations in a domain with characteristic zero and unique factorization. One of these results is the Jacobian forrnula that we use to describe the set of the equivalent loca11y nilpotents derivations of B = k[x, y, z] and the set LND(B) where B = k[x, y]. Moreover, we give equivalent conditions to the existence of a ?-homogeneous locally nilpotent derivation in the ring B = k[x, y, z] with kernel k[¿, g], {¿} and {g} e B, and mdc(?) = mdc(?(¿), ? (g)) = 1 / Mestrado / Algebra / Mestre em Matemática