The notions of partial Lagrangians, partial Noether operators and partial Euler-Lagrange equations are used in the construction of first integrals for ordinary differential equations (ODEs) that need not be derivable from variational principles. We obtain a Noetherlike theorem that provides the first integral by means of a formula which has the same structure as the Noether integral. However, the invariance condition for the determination of the partial Noether operators is different as we have a partial Lagrangian and as a result partial Euler-Lagrange equations. In order to investigate the effectiveness of the partial Lagrangian approach, some models such as the oscillator systems both linear and nonlinear, Emden and Ermakov-pinnery equations and the Hamiltonian system with two degrees of freedom are considered in this work. We study a general linear system of two second-order ODEs with variable coefficients. Note that, a Lagrangian exists for the special case only but, in general, the system under consideration does not have a standard Lagrangian. However, partial Lagrangians do exist for all such equations in the absence of Lagrangians. Firstly, we classify all the Noether and partial Noether operators for the case when the system admits a standard Lagrangian. We show that the first integrals that result due to the partial Noether approach is the same as for the Noether approach. First integrals are then constructed by the partial Noether approach for the general case when there is in general no Lagrangian for the system of two second-order ODEs with variable coefficients. We give an easy way of constructing first integrals for such systems by utilization of a partial Noether’s theorem with the help of partial Noether operators associated with a partial Lagrangian. Furthermore, we classify all the potential functions for which we construct first integrals for a system with two degrees of freedom. Moreover, the comparison of Lagrangian and partial Lagrangian approaches for the two degrees of freedom Lagrangian system is also given. In addition, we extend the idea of a partial Lagrangian for the perturbed ordinary differential equations. Several examples are constructed to illustrate the definition of a partial Lagrangian in the approximate situation. An approximate Noether-like theorem which gives the approximate first integrals for the perturbed ordinary differential equations without regard to a Lagrangian is deduced. We study the approximate partial Noether operators for a system of two coupled nonlinear oscillators and the approximate first integrals are obtained for both resonant and non-resonant cases. Finally, we construct the approximate first integrals for a system of two coupled van der Pol oscillators with linear diffusive coupling. Since the system mentioned above does not satisfy a standard Lagrangian, the approximate first integrals are still constructed by invoking an approximate Noether-like theorem with the help of approximate partial Noether operators. This approach can give rise to further studies in the construction of approximate first integrals for perturbed equations without a variational principle.

Two methods of generalizing the classical Noetherian theory to modules over arbitrary rings are described in detail. The first is by extending the primary ideals and isolated components of Murdoch to modules. The second is by using the tertiary sub-modules of Lesieur and Croisot. The development is self-contained except for elementary notions of ring and module theory. The definition of primal submodules with some results is included for completeness. Some concrete examples are given as illustrations. / Science, Faculty of / Mathematics, Department of / Graduate

O objetivo deste trabalho é estabelecer uma formulação Hamiltoniana para sistemas contínuos cujas energias cinétitica e potencial ou suas lagrangeanas envolvam derivadas de ordem superior em relação aos parâmetros espaciais das variáveis de campo e de velocidade. Ainda nesta perspectiva, são obtidas leis de conservação, uma expressão geral dos Parênteses de Poisson e sua invariância com respeito ao grupo de transformações canônicas das variáveis de campo e uma forma generalizada do Teorema de Noether. Finalmente, alguns exemplos da Mecânica do Contínuo são analisados em detalhe visando não só as possíveis aplicações dos resultados obtidos como o manuseio destes resultados em problemas concretos.

Made available in DSpace on 2014-08-27T14:36:45Z (GMT). No. of bitstreams: 0 Previous issue date: 2013-08-02Bitstream added on 2014-08-27T15:57:03Z : No. of bitstreams: 1 000781211.pdf: 1283604 bytes, checksum: c5b6c668d08991eba34fc8fcdb9fdabd (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Nesta tese, tem-se por objetivo usar o princípio de ação quântica de Schwinger para estudar e caracterizar dois tipos de sistemas quânticos: o primeiro sendo um oscilador harmônico forçado, cujos parâmetros dependem explicitamente do tempo e o segundo, um conjunto de osciladores harmônicos que interagem linearmente. Mostra-se para o primeiro, que a forma funcional desse princípio, i.e. o operador que origina as variações generalizadas das variáveis dinâmicas do sistema, além de permitir a construção das funções de transformação para qualquer tipo de sistema quântico, ajuda à determinação das quantidades conservadas e, conseqüentemente, à dedução do seu espectro de energia e o conjunto de funções próprias quando existirem. Caso contrário, se o sistema é dependente do tempo, podem-se construir as álgebras dinâmicas que permitem estudá-lo de uma maneira alternativa. Da mesma forma, para o segundo sistema, são propostos dois conjuntos de estados e de operadores: um associado aos estados que cada elemento do sistema apresenta em presença da interação, conhecidos na literatura como estados vestidos e outro, que representa os modos normais do sistema como um todo. Ambos conjuntos de estados são usados na implementação do princípio deaçãoquântica,permitindoencontrar: as soluções exatas, o espectro de energia, as funções de onda e as amplitudes de transição entre quaisquer dois estados nos quais se possa encontrar o sistema. Em cada caso, serão dados alguns exemplos que se contrastarão com os resultados associados a outras abordagens teóricas / This thesis has the aim of using the Schwinger Quantum Action Principle to study and characterize two kind of quantum systems: the ?rst one is a forced harmonic oscillator whose parameters explicitly depend on time and the second one, a set of harmonic oscillators which interacts linearly. We show for the ?rst system that the functional form of this principle, i.e. the operator which causes the generalized variations of the dynamical variables of the system, besides allowing the construction of transformation functions of any kind of system, help to determine the associated conserved quantities and therefore to deduct the form of the spectrum and the set of the eigen-functions of the system, if they exist. Otherwise, if the system is time-dependent, the dynamical algebras which allows studying it in an alternative way can be constructed. Similarly, for the second system two sets of states and operators are proposed. The ?rst one associated with the quantum state of each element of the system in the presence of interaction, known in the literature as Dressed States and the second one, which represents the normal modes of the system as a whole. Both sets of states are used in the implementation of the Quantum Action Principle allowing to ?nd the exact solutions, the spectrum, wave functions and amplitudes between any two states in which the system can be found. In each case, a few examples will be given and the results are contrasted with results associated with other theoretical approaches

Orientador: Bruto Max Pimentel Escobar / Banca: Ademir Eugênio Santana / Banca: Diógenes Galetti / Banca: Julio Marny Hoff da Silva / Banca: Dmitry Vasilevich / Resumo: Nesta tese, tem-se por objetivo usar o princípio de ação quântica de Schwinger para estudar e caracterizar dois tipos de sistemas quânticos: o primeiro sendo um oscilador harmônico forçado, cujos parâmetros dependem explicitamente do tempo e o segundo, um conjunto de osciladores harmônicos que interagem linearmente. Mostra-se para o primeiro, que a forma funcional desse princípio, i.e. o operador que origina as variações generalizadas das variáveis dinâmicas do sistema, além de permitir a construção das funções de transformação para qualquer tipo de sistema quântico, ajuda à determinação das quantidades conservadas e, conseqüentemente, à dedução do seu espectro de energia e o conjunto de funções próprias quando existirem. Caso contrário, se o sistema é dependente do tempo, podem-se construir as álgebras dinâmicas que permitem estudá-lo de uma maneira alternativa. Da mesma forma, para o segundo sistema, são propostos dois conjuntos de estados e de operadores: um associado aos estados que cada elemento do sistema apresenta em presença da interação, conhecidos na literatura como estados vestidos e outro, que representa os modos normais do sistema como um todo. Ambos conjuntos de estados são usados na implementação do princípio deaçãoquântica,permitindoencontrar: as soluções exatas, o espectro de energia, as funções de onda e as amplitudes de transição entre quaisquer dois estados nos quais se possa encontrar o sistema. Em cada caso, serão dados alguns exemplos que se contrastarão com os resultados associados a outras abordagens teóricas / Abstract: This thesis has the aim of using the Schwinger Quantum Action Principle to study and characterize two kind of quantum systems: the first one is a forced harmonic oscillator whose parameters explicitly depend on time and the second one, a set of harmonic oscillators which interacts linearly. We show for the first system that the functional form of this principle, i.e. the operator which causes the generalized variations of the dynamical variables of the system, besides allowing the construction of transformation functions of any kind of system, help to determine the associated conserved quantities and therefore to deduct the form of the spectrum and the set of the eigen-functions of the system, if they exist. Otherwise, if the system is time-dependent, the dynamical algebras which allows studying it in an alternative way can be constructed. Similarly, for the second system two sets of states and operators are proposed. The first one associated with the quantum state of each element of the system in the presence of interaction, known in the literature as Dressed States and the second one, which represents the normal modes of the system as a whole. Both sets of states are used in the implementation of the Quantum Action Principle allowing to find the exact solutions, the spectrum, wave functions and amplitudes between any two states in which the system can be found. In each case, a few examples will be given and the results are contrasted with results associated with other theoretical approaches / Doutor

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.

Seit W. Diffie und M. Hellman im Jahr 1976 ihren Ansatz für einen sicheren kryptographischen Schlüsselaustausch vorgestellten, ist der sogenannte Diskrete Logarithmus zu einem zentrales Thema der Kryptoanalyse geworden. Dieser stellt eine Erweiterung des bekannten Logarithmus auf beliebige endliche Gruppen dar. In der vorliegenden Dissertation werden zwei von C. Diem eingeführte Algorithmen untersucht, mit deren Hilfe der diskrete Logarithmus in der Picardgruppe glatter, nichthyperelliptischer Kurven vom Geschlecht g > 3 bzw. g > 4 über endlichen Körpern berechnet werden kann. Beide Ansätze basieren auf der sogenannten Indexkalkül-Methode und benutzen zur Erzeugung der dafür benötigten Relationen spezielle Linearsysteme, welche durch Schneiden von ebenen Modellen der Kurve mit Geraden erzeugt werden. Um Aussagen zur Laufzeit der Algorithmen tätigen zu können, werden verschiedene Sätze über die Geometrie von Kurven bewiesen. Als zentrale Aussage wird zum einem gezeigt, dass ebene Modelle niedrigen Grades effizient berechnet werden können. Zum anderen wird bewiesen, dass sich bei genügend großem Grundkörper die Anzahl der vollständig über dem Grundkörper zerfallenden Geraden wie heuristisch erwartet verhällt. Für beide Aussagen werden dabei Familien von Kurven betrachtet und diese gelten daher uniform für alle glatten, nichthyperelliptischen Kurven eines festen Geschlechts. Die genannten Resultate führen schlussendlich zu dem Beweis einer erwarteten Laufzeit von O(q^(2-2/(g-1))) für den ersten der beiden Algorithmen, wobei q die Anzahl der Elemente im Grundkörper darstellt. Der zweite Algoritmus verbessert dies auf eine heuristische Laufzeit in O(q^(2-2/(g-2))), imdem er Divisoren von höherem Spezialiätsgrad erzeugt. Es wird bewiesen, dass dieser Ansatz für einen uniform gegen 1 konvergierenden Anteil an glatten, nichthyperelliptischen Kurven eines festen Geschlechts über Grundkörpern großer Charakteristik eine große Anzahl an Relationen erzeugt. Wiederum werden zum Beweis der zugrundeliegenden geometrischen Aussagen Familien von Kurven betrachtet, um so die Uniformität zu gewährleisten. Beide Algorithmen wurden zudem implementiert. Zum Abschluss der Arbeit werden die Ergebnisse der entsprechenden Experimente vorgestellt und eingeordnet.

Submitted by Diego Sa de Lima null (diegos.lima88@hotmail.com) on 2018-03-03T04:54:32Z No. of bitstreams: 1 Dissertacao_Teorias_duais_massivas_spin32.pdf: 569707 bytes, checksum: 2062a40c0ffdddb4c5801a33f1f13f9d (MD5) / Approved for entry into archive by Pamella Benevides Gonçalves null (pamella@feg.unesp.br) on 2018-03-05T18:32:28Z (GMT) No. of bitstreams: 1 lima_ds_me_guara.pdf: 569707 bytes, checksum: 2062a40c0ffdddb4c5801a33f1f13f9d (MD5) / Made available in DSpace on 2018-03-05T18:32:28Z (GMT). No. of bitstreams: 1 lima_ds_me_guara.pdf: 569707 bytes, checksum: 2062a40c0ffdddb4c5801a33f1f13f9d (MD5) Previous issue date: 2018-02-05 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Nesta dissertação serão analisados os dois modelos conhecidos na literatura que descrevem partículas massivas de spin 3/2 em D=2+1 dimensões. Essa análise será feita, assim como nos trabalhos relacionados aos bósons (spin 1, spin 2 e spin 3), via procedimento de Imersão de Calibre de Noether (ICN), Solda generalizada, análise de vínculos e condições de Fierz-Pauli. Através de argumentos de simetria, via ICN, apresentaremos uma forma de relacionar os dois modelos e mostraremos que é possível construir um novo modelo de terceira ordem em derivadas. Apresentaremos um modelo de dubleto de segunda ordem em derivadas de onde é possível obter os demais modelos auto-duais da teoria. A partir da aplicação da ICN no modelo de dubleto construiremos um novo modelo, de quarta ordem em derivadas, análogo a versão linearizada da chamada " New Massive Gravity". / In this master's thesis we will analyze the two known models in the literature wich describe massive spin 3/2 particles in D = 2 + 1 dimensions. This analysis will be done, as was previously done on works related to the bosons (spin- 1 , spin- 2 and spin- 3 ), via Noether gauge embedment (NGE) procedure, generalized soldering, hamiltonian constraints analysis and Fierz-Pauli conditions. Through symmetry arguments, by NGE, we will present a way of relating the two models and show that it is possible to construct a new model in third order derivatives. We will show a second order derivative doublet-model whence it is possible to obtain the other self-dual models of the theory. From the application of NGE in the dublet model we will construct a new model, wich has a fourth-order derivative term, analogue to the linearized version of the so-called "New Massive Gravity"

Neste trabalho, temos o objetivo de introduzir o Problema de Noether Clássico e sua versão não- comutativa introduzida por J. Alev e F. Dumas em [AD06]. Discutiremos os principais casos co- nhecidos nos quais os problemas têm solução positiva, observando um forte paralelo entre os casos comutativo e não-comutativo. Cobriremos os tópicos preliminares necessários para entendimento dos enunciados: álgebras de Weyl, anéis de operadores diferenciais, extensões de Ore, localização em domínios não-comutativos, e corpos de Weyl. No Capítulo 5 deste trabalho, o aluno apresenta duas contribuições originais, obtidas em colaboração com seu orientador V. Futorny e F. Eshmatov: o Teorema 5.5, que é um resultado folclórico sobre invariantes de ações livres de grupos finitos no anel de operadores diferenciais de variedades afins; e o Teorema 5.6, que até onde sabemos é iné- dito, sobre invariantes dos Corpos de Weyl sob a ação de grupos de pseudo-reflexão. Todo material algébrico preliminar para a demonstração destes dois teoremas é incluído no texto da dissertação: um básico de teoria de invariantes, vários resultados da teoria de grupos de pseudo-reflexão, alguns conceitos básicos de geometria algébrica e álgebra comutativa, e uma discussão detalhada do quo- ciente de variedades afins sob ação de grupos finitos. / In this work we aim to introduce the Classical Noether´s Problem, and its noncommutative version introduced by J. Alev and F. Dumas in [AD06]. We discuss the most well known cases of positive solution of these problems, pointing out a strong similarity between the cases of positive solution for the classical and noncommutative versions of the Problem. We cover the preliminary topics to understand the statement and solutions of these problems: Weyl algebras, differential operators rings, Ore extensions, noncommutative localization, and Weyl Skew-Fields. In the Chapter 5 of this dissertation, the student shows two original contributions, obtained in collaboration with his advisor V. Futorny and F. Eshmatov: Theorem 5.5, a result belonging to the folklore of the area of differential operators, describing its invariants under the free action of a finite group on an affine variety; and Theorem 5.6, about the invariants of the Weyl skew-fields under the action of pseudo-reflection groups. As far as we know, this result is new. All preliminary algebraic facts to prove these two facts are included in the body of this text. It includes some basic facts on invariant theory, many results about pseudo-reflection groups, some basic concepts of algebraic geometry and commutative algebra, and a detailed discussion of the quotient of an affine variety under the action of a finite group.

Diese Dissertation untersucht Brill-Noether-Theorie der algebraischen Kurven, unter besonderer Berücksichtigung von Kurven auf K3-Flächen und Del-Pezzo-Flächen. In Kapitel 2 studieren wir den Gieseker-Petri-Ort GP_g im Modulraum M_g der glatten irreduziblen Kurven vom Geschlecht g. Dieser Ort wird definiert durch Kurven mit einer Brill-Noether-Varietät G^r_d(C), die singulär ist oder deren Dimension größer als erwartet ist. Der Satz von Gieseker-Petri impliziert, dass GP_g mindestens Kodimension 1 hat, und es wurde vermutet, dass er von reiner Kodimension 1 ist. Wir beweisen diese Vermutung für Geschlecht höchstens 13. Dies wird dadurch ermöglicht, dass man für kleine Geschlechter die Dimension der irreduziblen Komponenten von GP_g mittels "ad hoc"-Beweisführungen untersuchen kann. Lazarsfelds Beweis des Gieseker-Petri-Theorems mittels Kurven auf allgemeninen K3-Flächen deutet darauf hin, dass die Brill-Noether-Theorie von K3-Schnitten wichtig ist, um den Gieseker-Petri-Ort besser zu verstehen. Linearscharen von Kurven, die auf K3-Flächen liegen, stehen in tiefgehender Beziehung zu sogenannten Lazarsfeld-Mukai-Vektorbündeln. In Kapitel 3 untersuchen wir die Stabilität der Lazarsfeld-Mukai-Vektorbündel vom Rang 3 auf einer K3-Fläche S, und wir zeigen, dass sie viele Informationen über Netze vom Typ g^2_d auf Kurven in S enthalten. Wenn d größ genug ist, erhalten wir eine obere Schranke für die Dimension der Varietät G^2_d(C). Wenn die Brill-Noether-Zahl negativ ist, beweisen wir, dass jedes g^2_d in einer von einem Geradenbündel induzierten Linearschar enthalten ist, wie von Donagi und Morrison vermutet wurde. Kapitel 4 befasst sich mit Syzygien einer gegebenen Kurve C, die auf einer Del-Pezzo-Fläche liegt. Wir insbesondere, dass C die Greens Vermutung erfüllt, die impliziert, dass die Existenz gewisser spezieller Linearscharen auf C von den Gleichungen ihrer kanonischen Einbettung abgelesen werden kann. / We investigate Brill-Noether theory of algebraic curves, with special emphasis on curves lying on $K3$ surfaces and Del Pezzo surfaces. In Chapter 2, we study the Gieseker-Petri locus GP_g inside the moduli space M_g of smooth, irreducible curves of genus g. This consists, by definition, of curves [C] in M_g such that for some r, d the Brill-Noether variety G^r_d(C), which parametrizes linear series of type g^r_d on C, either is singular or has some components exceeding the expected dimension. The Gieseker-Petri Theorem implies that GP_g has codimension at least 1 in M_g and it has been conjectured that it has pure codimension 1. We prove this conjecture up to genus 13; this is possible since, when the genus is low enough, one is able to determine the irreducible components of GP_g and to study their codimension by "ad hoc" arguments. Lazarsfeld''s proof of the Gieseker-Petri-Theorem by specialization to curves lying on general K3 surfaces suggests the importance of the Brill-Noether theory of K3-sections for a better understanding of the Gieseker-Petri locus. Linear series on curves lying on a K3 surface are deeply related to the so-called Lazarsfeld-Mukai bundles. In Chapter 3, we study the stability of rank-3 Lazarsfeld-Mukai bundles on a K3 surface S, and show it encodes much information about nets of type g^2_d on curves C contained in S. When d is large enough and C is general in its linear system, we obtain a dimensional statement for the variety G^2_d(C). If the Brill-Noether number is negative, we prove that any g^2_d is contained in a linear series which is induced from a line bundle on S, as conjectured by Donagi and Morrison. Chapter 4 concerns syzygies of any given curve C lying on a Del Pezzo surface S. In particular, we prove that C satisfies Green''s Conjecture, which implies that the existence of some special linear series on C can be read off the equations of its canonical embedding.