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A Boolean function can be represented in several different forms. These different representation have advantages and disadvantages of their own. The Algebraic Normal Form, truth table, and Walsh spectrum representations are widely studied in literature. In 1999, Claude Carlet and Phillippe Guillot introduced the Numerical Normal Form. NumericalNormal Form(NNF) of a Boolean function is similar to Algebraic Normal Form, with integer coefficients instead of coefficients from the two element field. Using NNF representation, just like the Walsh spectrum, characterization of several cryptographically important functions, such as resilient and bent functions, is possible. In 2002, Carlet had shown several divisibility results concerning resilient and correlation-immune functions using NNF. With these divisibility results, Carlet is able to give bounds concerning nonlinearity of resilient and correlation immune functions. In this thesis, following Carlet and Guillot, we introduce the Numerical Normal Form and derive the pairwise relations between the mentioned representations. Characterization of Boolean, resilient and bent functions using NNF is also given. We then review the divisibility results of Carlet, which will be linked to some results on the nonlinearity of resilient and correlation immune functions. We show the M&ouml / bius inversion properties of NNF of a Boolean function, using Gian-Carlo Rota&rsquo / s work as a guide. Finally, using a lot of the mentioned results, we prove a necessary condition on theWalsh spectrum of Boolean functions with given degree.

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2006-02-07Bitstream added on 2014-06-13T20:08:06Z : No. of bitstreams: 1 pereira_ms_me_sjrp.pdf: 2071399 bytes, checksum: 9f8844443f17c4fa7a041cc8bc621d54 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho classificamos problemas de bifurcação Z2-equivariantes de corank 2 até co- dimensão 3 via técnicas da Teoria de Singularidades. A abordagem para classificar tais problemas é baseada no processo de redução à forma normal de Birkhoff para estudar a interação de modos Hopf-Pontos de Equilíbrio. O comportamento geométrico das soluções dos desdobramentos das formas normais obtidas é descrito pelos diagramas de bifurcação e estudamos a estabilidade assintótica desses ramos. / In this work we classify the Z2-equivariant corank 2 bifurcation problems up to codimension 3 via Singularity Theory techniques. The approach to classify such problems is based on the Birkhoff normal form to study Hopf-Steady- State mode interaction. The geometrical behavior of the solutions of the unfolding of the normal forms is described by the bifurcation diagrams and we study the asymptotic stability of such branches.

Made available in DSpace on 2015-05-15T11:46:05Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 547480 bytes, checksum: 23748fbdacd76846a114baea058b21f6 (MD5) Previous issue date: 2013-05-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we study implicit differential equations. Following the Thom tranversality theorem and the singularity theory we find an open and dense subset of this equation class that present only good singularity. This singularity are of six kind well folded saddle, well folded node, well folded focus, elliptical gather, hyperbolic gather. Davydov,in [8] showed the normal forms of a IDE in the case of well folded saddle, well folded node, well folded focus. In the case of gathered singularities, Davydov showed that the normal forms of IDE contains functional moduli. For a special class of implicit differential equation, the binary differential equation (BDE), we study the normal forms in the case in that the discriminant is a Morse function. / Neste trabalho estudamos singularidades de equações diferenciais implícitas. Usando o Teorema de Transversalidade de Thom e a teoria das singularidades encontramos um subconjunto aberto e denso desta classe de equações que apresentam singularidades boas. Estas singularidades são apenas de seis tipos dobra-sela, dobra-nó, dobra-foco, cúspide elíptica e cúspide hiperbólica. Davydov, em [8], mostrou as formas normais da EDI nos casos de dobra-sela, dobra-nó e dobra-foco. No caso de cúspides, Davydov mostrou que as formas normais da EDI apresentam parâmetros e funções arbitrárias. Para uma classe especial de equações diferenciais implícitas, as equações diferenciais binária (EDB), estudamos a forma normal nos casos em que o discriminante é uma função de Morse.

Orientadores: Marco Antonio Teixeira, Ketty Abaroa de Rezende / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-10T14:10:31Z (GMT). No. of bitstreams: 1 Martins_RicardoMiranda_M.pdf: 921623 bytes, checksum: 8098f5c4875b6b586865b92ec6e474a0 (MD5) Previous issue date: 2008 / Resumo: A semelhança entre sistemas reversíveis e Hamiltonianos foi detectada nos primórdios do século passado por Birkhoff. Neste trabalho realizamos uma análise geométrica-qualitativa da dinâmica de um campo de vetores reversível em torno de um ponto de equilíbrio elíptico em R4. Especificamente, estudamos quando um campo reversível com tal tipo de equilíbrio é "equivalente" a um sistema Hamiltoniano. Como resultado, obtemos que tal sistema é Hamiltoniano, a menos de uma seqüência de mudanças de coordenadas e reescalonamentos do tempo. Prosseguindo a análise, impomos outra simetria ao campo e passamos a considerar sistemas bireversíveis. Classificamos completamente as possíveis simetrias que tornam um sistema bireversível por involuções gerando um grupo isomorfo a D4. Para tais sistemas, obtemos resultados um pouco mais fortes que os obtidos para sistemas reversíveis / Abstract: The similarity between reversible and Hamiltonian systems has been detected at the beginning of the past century by Birkhoff. In this project, we describe a geometrical-qualitative analysis of the dynamics of a reversible vector field around a elliptical singularity in R4. Specifically, we study when such a reversible vector field is "equivalent" to a Hamiltonian system. As a result, we obtain that such systems are always Hamiltonian, up to a sequence of changes of coordinates and time rescaling. Imposing another symmetry to the vector field, we work with bireversible systems. We completely classify all the possible symmetries which makes such systems bireversible by involutions generating a group isomorphic to D4. For these systems, we have obtained stronger results than in the reversible case / Mestrado / Sistemas Dinamicos / Mestre em Matemática

Orientador: Marco Antonio Teixeira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-17T14:36:33Z (GMT). No. of bitstreams: 1 Martins_RicardoMiranda_D.pdf: 1398633 bytes, checksum: 5ac0dfa4d175b6407f0b574a1aefd3e5 (MD5) Previous issue date: 2011 / Resumo: Na primeira parte desta tese, estudamos a semelhança entre sistemas dinâmicos reversíveis e Hamiltonianos, sob um ponto de vista formal. Nos restringimos a sistemas definidos ao redor de pontos de equilíbrio simples e simétricos. Mostramos que, sob algumas hipóteses, tais sistemas são formalmente orbitalmente equivalentes. Na segunda parte, estudamos a existência de conjuntos minimais em certas famílias de equações diferenciais. Especificamente, exibimos condições sob as quais existem cilindros e toros invariantes para sistemas de equações que são perturbações de sistemas reversíveis. / Abstract: In the first part of this thesis, we study the similarity between reversible and Hamiltonian dynamical systems, from a formal viewpoint. We restrict ourselves to systems defined around an isolated and symmetric equilibria. We show that, under some conditions, such systems are formally orbitally equivalent to Hamiltonian vector fields. In the second part, we study the existence of minimal sets for some families of diferential equations. We obtain conditions for the existence of the invariant cylinders and tori for perturbed reversible systems. / Doutorado / Sistemas Dinamicos / Doutor em Matemática

Inspired by problems in gauge field theory, this thesis is concerned with various aspects of infinite-dimensional differential geometry. In the first part, a local normal form theorem for smooth equivariant maps between tame Fréchet manifolds is established. Moreover, an elliptic version of this theorem is obtained. The proof these normal form results is inspired by the Lyapunov–Schmidt reduction for dynamical systems and by the Kuranishi method for moduli spaces, and uses a slice theorem for Fréchet manifolds as the main technical tool. As a consequence of this equivariant normal form theorem, the abstract moduli space obtained by factorizing a level set of the equivariant map with respect to the group action carries the structure of a Kuranishi space, i.e., such moduli spaces are locally modeled on the quotient by a compact group of the zero set of a smooth map. In the second part of the thesis, the theory of singular symplectic reduction is developed in the infinite-dimensional Fréchet setting. By refining the above construction, a normal form for momentum maps similar to the classical Marle–Guillemin–Sternberg normal form is established. Analogous to the reasoning in finite dimensions, this normal form result is then used to show that the reduced phase space decomposes into smooth manifolds each carrying a natural symplectic structure. Finally,the singular symplectic reduction scheme is further investigated in the situation where the original phase space is an infinite-dimensional cotangent bundle. The fibered structure of the cotangent bundle yields a refinement of the usual orbit-momentum type strata into so-called seams. Using a suitable normal form theorem, it is shown that these seams are manifolds. Taking the harmonic oscillator as an example, the influence of the singular seams on dynamics is illustrated. The general results stated above are applied to various gauge theory models. The moduli spaces of anti-self-dual connections in four dimensions and of Yang–Mills connections in two dimensions is studied. Moreover, the stratified structure of the reduced phase space of the Yang–Mills–Higgs theory is investigated in a Hamiltonian formulation after a (3 + 1)-splitting.

Bifurcations in Huang's chaotic chemical reactor system leading from simple dynamics into chaotic regimes are considered. Following the linear stability analysis, the periodic orbit resulting from a Hopf bifurcation of any of the six fixed points is constructed analytically by the method of multiple scales across successively slower time scales, and its stability is then determined by the resulting final secularity condition. Furthermore, we run numerical simulations of our chemical reactor at a particular fixed point of interest, alongside a set of parameter values that forces our system to undergo Hopf bifurcation. These numerical simulations then verify our analysis of the normal form.

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Airplane wings, turbine blades and other structures subjected to air or water flows, can undergo motions depending on their flexibility. As such, the performance of these systems depends strongly on their geometry and material properties. Of particular importance is the contribution of different nonlinear aspects. These aspects can be of two types: aerodynamic and structural. Examples of aerodynamic aspects include but are not lomited to flow separation and wake effects. Examples of structural aspects include but not limited to large deformations (geometric nonlinearities), concentrated masses or elements (inertial nonlinearities) and freeplay. In some systems, and depending on the parameters, the nonlinearities can cause multiple solutions. Determining the effects of nonlinearities of an aeroelastic system on its response is crucial. In this dissertation, different aeroelastic configurations where nonlinear aspects may have significant effects on their performance are considered. These configurations include: the effects of the wake on the flutter speed of a wing placed under different angles of attack, the impacts of the wing rotation as well as the aerodynamic and structural nonlinearities on the flutter speed of a rotating blade, and the effects of the recently proposed nonlinear energy sink on the flutter and ensuing limit cycle oscillations of airfoils and wings. For the modeling and analysis of these systems, we use models with different levels of fidelity as required to achieve the stated goals. We also use nonlinear dynamic analysis tools such as the normal form to determine specific effects of nonlinearities on the type of instability. / Ph. D.