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Universidade Estadual de Campinas

1. Angelo Papa Neto. Rigid elements, valuations and structure of Witt rings.

Degree: Instituto de Matemática, Estatística e Computação Científica, 2007, Universidade Estadual de Campinas

An ordered field is an algebraic structure like the field of real numbers. However, while the field of real numbers have only one ordering, an arbitrary ordered field F may have more than one ordering, and also a infinite and uncountble number of orderings is allowed. To each element x Î F one can associate an binary quadratic form [1, x], called Pfister 1-fold form. The set of elements in F = F 0} which are represented by [1, x] is a group D[1,x], called value group of [1,x]. An element d Σ F is called rigid if D[1, d] = F2 U dF2, where F 2 denotes the subgroup of squares in F . An element d is called birigid if d and -d are both rigid. The main purpose of this thesis is to prove an structure theorem for Witt ring (of equivalence classes of quadratic forms) of an ordered field F with a rigid element which is not birigid and is negative in at least one ordering of F, that is, we get a decomposition of the Witt ring of F as a product of Witt rings of extensions H ˆ F and K ‰ F, both inside the quadratic closure of F. The Witt rings of H and K have a simpler structure than Witt ring of F. We get fields H and K by builting subgroups Rd and Sd associated to the rigid element d and making the addicional assumption that F = Rd·Sd holds. The field H is a henselization of F relative to a valuation ring (A;mA) of F such that Rd = (1 + mA) F2. The pythagorean field K has space of orderings XK homeomorphic to X/Sd, the space of orderings of F which contain Sd. Moreover, we settle an necessary and suficient condiction to decomposition F = Rd·Sd holds, relative to value group and residue field of valuation ring A.

Um corpo ordenado é uma estrutura algébrica similar à do corpo dos números reais. No entanto, ao contrário dos reais, um corpo arbitrário F pode admitir mais de uma ordem, inclusive um número infinito e não enumerável de ordens. A cada elemento x do corpo F podemos associar uma forma quadrática binária [1, x], chamada 1-forma de Pfister. Os elementos de F = F 0} representados por [1, x], constituem um grupo que chamamos grupo de valores da forma e denotamos por D[1,x]. Um elemento d Σ F é chamado rígido se D[1, d] = F2 U dF2 , onde F2 é o subgrupo de F formado pelos quadrados. Um elemento d é dito birígido se d e -d são rígidos. O presente trabalho tem como objetivo principal obter um teorema de estrutura para o anel de Witt (das classes de equivalência de formas quadráticas) de um corpo ordenado F admitindo um elemento rígido que não é birígido e que é negativo em relação à pelo menos uma das ordens do corpo. Mais precisamente, obtemos uma decomposição do anel de Witt de F como produto de anéis de Witt de duas extensões H ˆ F e K ‰ F, ambas contidas no fecho quadrático de F. Os anéis de Witt de H e K têm estrutura mais simples que a do anel de Witt de F. Obtemos os corpos H e K construindo subgrupos Rd e Sd associados ao elemento rígido d e exigindo que valha uma propriedade de decomposição: F = Rd· Sd. O corpo H é uma henselização de F relativa a um anel de valorização (A;mA) de F tal que Rd = (1 + mA) F2…

Advisors/Committee Members: Fernando Eduardo Torres Orihuela, Ires Dias, Dessislava Hristova Kochloukova, Rosali Brusamarello, Antonio Jose Engler.

Subjects/Keywords: Formas quadraticas; Formally real fields; Witt rings; Aneis de; Quadratic forms; Witt; Corpos formalmente reais

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APA (6th Edition):

Neto, A. P. (2007). Rigid elements, valuations and structure of Witt rings. (Thesis). Universidade Estadual de Campinas. Retrieved from http://libdigi.unicamp.br/document/?code=vtls000415821

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Neto, Angelo Papa. “Rigid elements, valuations and structure of Witt rings.” 2007. Thesis, Universidade Estadual de Campinas. Accessed July 15, 2020. http://libdigi.unicamp.br/document/?code=vtls000415821.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Neto, Angelo Papa. “Rigid elements, valuations and structure of Witt rings.” 2007. Web. 15 Jul 2020.

Vancouver:

Neto AP. Rigid elements, valuations and structure of Witt rings. [Internet] [Thesis]. Universidade Estadual de Campinas; 2007. [cited 2020 Jul 15]. Available from: http://libdigi.unicamp.br/document/?code=vtls000415821.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Neto AP. Rigid elements, valuations and structure of Witt rings. [Thesis]. Universidade Estadual de Campinas; 2007. Available from: http://libdigi.unicamp.br/document/?code=vtls000415821

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

2. Herlemont, Basile. Differential calculus on h-deformed spaces : Calcul différentiel sur des espaces h-déformés.

Degree: Docteur es, Physique Théorique et Mathématique, 2017, Aix Marseille Université

L'anneau \Diff(n) des opérateurs différentiels \h-déformés apparaît dans la théorie des algèbres de réduction.Dans cette thèse, nous construisons les anneaux des opérateurs différentiels généralisés sur les espaces vectoriels \h-déformés de type \gl. Contrairement aux espaces vectoriels q-déformés pour lequel l'anneau des opérateurs différentiels est unique \`a isomorphisme pr\`es, l'anneau généralisé des opérateurs différentiels \h-déformés \Diffs(n) est indexée par une fonction rationnelle σ en n variables, solution d'un syst\`eme d\'eg\'en\'er\'e d'\'equations aux diff\'erences finies. Nous obtenons la solution g\'en\'erale de ce syst\`eme. Nous montrons que le centre de \Diffs(n) est un anneau des polynômes en n variables. Nous construisons un isomorphisme entre des localisations de l'anneau \Diffs(n) et de l’algèbre de Weyl {W}n l’étendue par n indéterminés. Nous présentons des conditions irréductibilité des modules de dimension fini de \Diffs(n). Finalement, nous discutons des difficultés a trouver les constructions analogues pour l'anneau \Diff(n,N) correspondant \`a N copies de \Diff(n).

The ring \Diff(n) of \h-deformed differential operators appears in the theory of reduction algebras. In this thesis, we construct the rings of generalized differential operators on the \h-deformed vector spaces of \gl-type. In contrast to the q-deformed vector spaces for which the ring of differential operators is unique up to an isomorphism, the general ring of \h-deformed differential operators \Diffs(n) is labeled by a rational function σ in n variables, satisfying an over-determined system of finite-difference equations. We obtain the general solution of the system. We show that the center of \Diffs(n) is a ring of polynomials in n variables. We construct an isomorphism between certain localizations of \Diffs(n) and the Weyl algebra \Wn extended by n indeterminates. We present some conditions for the irreducibility of the finite dimensional \Diffs(n)-modules. Finally, we discuss difficulties for finding analogous constructions for the ring \Diff(n, N) formed by several copies of \Diff(n).

Advisors/Committee Members: Ogievetsky, Oleg (thesis director).

Subjects/Keywords: Opérateurs différentiels; Équation de Yang-Baxter; Algèbres de réduction; Algèbre enveloppante universelle; Théorie des représentations; Propriété de Poincaré – Birkhoff – Witt; Corps des fractions; Differential operators; Yang-Baxter equation; Reduction algebras; Universal enveloping algebra; Representation theory; Poincaré – Birkhoff – Witt; Rings of fractions; 530

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Herlemont, B. (2017). Differential calculus on h-deformed spaces : Calcul différentiel sur des espaces h-déformés. (Doctoral Dissertation). Aix Marseille Université. Retrieved from http://www.theses.fr/2017AIXM0377

Chicago Manual of Style (16th Edition):

Herlemont, Basile. “Differential calculus on h-deformed spaces : Calcul différentiel sur des espaces h-déformés.” 2017. Doctoral Dissertation, Aix Marseille Université. Accessed July 15, 2020. http://www.theses.fr/2017AIXM0377.

MLA Handbook (7th Edition):

Herlemont, Basile. “Differential calculus on h-deformed spaces : Calcul différentiel sur des espaces h-déformés.” 2017. Web. 15 Jul 2020.

Vancouver:

Herlemont B. Differential calculus on h-deformed spaces : Calcul différentiel sur des espaces h-déformés. [Internet] [Doctoral dissertation]. Aix Marseille Université 2017. [cited 2020 Jul 15]. Available from: http://www.theses.fr/2017AIXM0377.

Council of Science Editors:

Herlemont B. Differential calculus on h-deformed spaces : Calcul différentiel sur des espaces h-déformés. [Doctoral Dissertation]. Aix Marseille Université 2017. Available from: http://www.theses.fr/2017AIXM0377

3. Wright, Kayla. On Linked Quaternionic Pairings.

Degree: 2018, University of California – eScholarship, University of California

In this paper, we study linked bilinear pairings and their associated Witt rings. An open problem is to classify all linked quaterionic pairings. The Elementary Type Conjecture [1] asserts that every finite linked quaternionic pairing can be built from symplectic pairings using direct sums and group extensions iteratively. We investigate the validity of this conjecture by studying an infinite quaternionic pairing and its sub- pairings motivated by certain structures arising in Henselian dyadic valued fields.

Subjects/Keywords: Mathematics; Linked quaternionic pairings; quadratic forms; Witt rings

…Conjecture Definition 2.5.1. A direct sum of two Witt rings R1 , R2 is defined by the fiber… …iteration starting with basic Witt rings Z/2Z, Z, and the (finite) Witt ring W K of a… …known that all known finitely generated Witt rings are of elementary type. It was also proven… …by A. Carson and M. Marshall in Decomposition of Witt rings [8] that the… …Techniques and 12 Abstract Witt Rings III Fitzgerald [5]. This paper gives a four step… 

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Wright, K. (2018). On Linked Quaternionic Pairings. (Thesis). University of California – eScholarship, University of California. Retrieved from http://www.escholarship.org/uc/item/8z5020hg

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Wright, Kayla. “On Linked Quaternionic Pairings.” 2018. Thesis, University of California – eScholarship, University of California. Accessed July 15, 2020. http://www.escholarship.org/uc/item/8z5020hg.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Wright, Kayla. “On Linked Quaternionic Pairings.” 2018. Web. 15 Jul 2020.

Vancouver:

Wright K. On Linked Quaternionic Pairings. [Internet] [Thesis]. University of California – eScholarship, University of California; 2018. [cited 2020 Jul 15]. Available from: http://www.escholarship.org/uc/item/8z5020hg.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Wright K. On Linked Quaternionic Pairings. [Thesis]. University of California – eScholarship, University of California; 2018. Available from: http://www.escholarship.org/uc/item/8z5020hg

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

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