Orientador: Plamen Emilov Koshlukov / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatística e Computação Científica / Made available in DSpace on 2018-08-14T14:50:24Z (GMT). No. of bitstreams: 1 Freitas_JoseAntonioOliveirade_D.pdf: 1578135 bytes, checksum: a3352669dd5077f0f5949766026e7bb1 (MD5) Previous issue date: 2009 / Resumo: Nesta tese estudamos identidades polinomiais graduadas para certas álgebras. Inicialmente, estudamos identidades satisfeitas pelo produto tensorial Z2-graduado. Este estudo foi motivado pelo trabalho de Regev e Seeman com produtos tensoriais Z2-graduados. Eles provaram vários casos nos qual tal produto tensorial é PI equivalente a certas álgebras T-primas. Também conjeturaram que isto sempre ocorre. Trabalhamos com os demais casos e conseguimos provar que tal conjetura e verdadeira. Alêm disso provamos que para certas álgebras, quando consideramos corpos de característica positiva, o produto tensorial graduado ainda se comporta como o não graduado. Consideramos também o produto tensorial-graduado e suas identidades. Provamos que o Teorema A B de Regev continua válido no caso do produto tensorial-graduado quando as álgebras são graduadas por grupos abelianos nitos, e é um bicaracter antissimétrico. Também estudamos a PI equivalência do produto tensorial-graduado de álgebras T-primas. Em seguida estudamos identidades graduadas, descrevemos um conjunto de geradores para as identidades Z-graduadas da álgebra de Lie W1. A álgebra W1 é a álgebra das derivações do anel de polinômios K[t], e é conhecida como a álgebra de Witt. Provamos que se a característica do corpo for 0, então as identidades Z-graduadas de W1 são geradas por um conjunto de identidades de grau 2 e 3. Mais ainda, provamos que não é possível obter um conjunto nito de geradores para as identidades Z-graduadas de W1. / Abstract: In this PhD thesis we study graded polynomial identities for certain types of algebras. First, we study polynomial identities satised by the Z2-graded tensor products. This research was motivated by the paper of Regev and Seeman about the Z2-graded tensor products. They proved that in a series of cases such tensor products are PI equivalent to T-prime algebras. Then they conjectured that this is always the case. We deal here with the remaining cases and thus conrm Regev and Seeman's conjecture. Furthermore, we prove that for some algebras we can remove the restriction on the characteristic of the base eld, and we show that the behaviour of the corresponding graded tensor products is quite similar to that for the usual ungraded tensor products. We consider too the graded tensor products and their identities where is a skew symmetric bicharacter. We show that Regev's A B theorem holds for graded tensor products whenever the gradings are by nite abelian groups. Furthermore we study the PI equivalence of -graded tensor products of T-prime algebras. Afterwards we study the graded identities of the Lie algebra W1. We describe a set of generators of the Z-graded identities of W1. The algebra W1 is the algebra of derivation of the polynomial ring K[t], and it is known as the Witt algebra. We prove that if K is a eld of characteristic 0, then the Z-graded identities of W1 are consequences of a collection of polynomials of degree 2 and 3. Furthermore we prove that the Z-graded identities for W1 do not admit a nite basis. / Doutorado / Algebra / Doutor em Matemática

Noncommutative geometry has over the past four of decades grown into a rich field of study. Novel ideas and concepts are rapidly being developed, and a notable application of the theory outside of pure mathematics is quantum theory. This thesis will focus on a derivation-based approach to noncommutative geometry using the framework of real calculi, which is a rather direct approach to the subject. Due to their direct nature, real calculi are useful when studying classical concepts in Riemannian geometry and how they may be generalized to a noncommutative setting. This thesis aims to shed light on algebraic aspects of real calculi by introducing a concept of morphisms of real calculi, which enables the study of real calculi on a structural level. In particular, real calculi over matrix algebras are discussed both from an algebraic and a geometric perspective.Morphisms are also interpreted geometrically, giving a way to develop a noncommutative theory of embeddings. As an example, the noncommutative torus is minimally embedded into the noncommutative 3-sphere. / Ickekommutativ geometri har under de senaste fyra decennierna blivit ett etablerat forskningsområde inom matematiken. Nya idéer och koncept utvecklas i snabb takt, och en viktig fysikalisk tillämpning av teorin är inom kvantteorin. Denna avhandling kommer att fokusera på ett derivationsbaserat tillvägagångssätt inom ickekommutativ geometri där ramverket real calculi används, vilket är ett relativt direkt sätt att studera ämnet på. Eftersom analogin mellan real calculi och klassisk Riemanngeometri är intuitivt klar så är real calculi användbara när man undersöker hur klassiska koncept inom Riemanngeometri kan generaliseras till en ickekommutativ kontext. Denna avhandling ämnar att klargöra vissa algebraiska aspekter av real calculi genom att introducera morfismer för dessa, vilket möjliggör studiet av real calculi på en strukturell nivå. I synnerhet diskuteras real calculi över matrisalgebror från både ett algebraiskt och ett geometriskt perspektiv. Morfismer tolkas även geometriskt, vilket leder till en ickekommutativ teori för inbäddningar. Som ett exempel blir den ickekommutativa torusen minimalt inbäddad i den ickekommutativa 3-sfären.

In dieser Arbeit wird die analytische Fortsetzung von Quantenfeldtheorien auf dem nichtkommutativen Euklidischen Moyal-Raum mit kommutativer Zeit zu entsprechenden Moyal-Minkowski Raumzeit (Wick Rotation) erarbeitet. Dabei sind diese Moyal-Räume durch eine konstante Nichtkommutativiät gegeben. Einerseits wird die Wick Rotation im Kontext der algebraischen Quantenfeldtheorie, ausgehend von einer Arbeit von Schlingemann, hergeleitet. Von einem Netz Euklidischer Observablen wird die Lorentz’sche Theorie durch alle Bilder der fortgesetzten Poincare Gruppenwirkung auf der Zeit-Null Schicht erhalten. Dabei wird gezeigt, dass die Vorgänge der nichtkommutativen Deformation und der Wick Rotation kommutieren. Andererseits ist so eine analytische Fortsetzung ebenfalls für Quantenfeldtheorien, die durch einen Satz von Schwingerfunktionen definiert ist, möglich. Durch die Gültigkeit einer Kombination aus Wachstumsbedinungen, die aus der Wick Rotation von Osterwalder und Schrader bekannt sind, kann der Übergang zu einer deformierten Wightman-Theorie gezeigt werden. Abschließend beinhaltet diese Arbeit ergänzende Resultate zu den physikalischen Eigenschaften der Kovarianz und der Lokalität.

Indiana University-Purdue University Indianapolis (IUPUI) / In 1970, Serge Novikov made a statement which is now called, "The Novikov Conjecture" and is considered to be one of the major open problems in topology. This statement was motivated by the endeavor to understand manifolds of arbitrary dimensions by relating the surgery map with the homology of the fundamental group of the manifold, which becomes diffi cult for manifolds of dimension greater than two. The Novikov Conjecture is interesting because it comes up in problems in many different branches of mathematics like algebra, analysis, K-theory, differential geometry, operator algebras and representation theory. Yu later proved the Novikov Conjecture holds for all closed manifolds with discrete fundamental groups that are coarsely embeddable into a Hilbert space. The class of groups that are uniformly embeddable into Hilbert Spaces includes groups of Property A which were introduced by Yu. In fact, Property A is generally a property of metric spaces and is stable under quasi-isometry. In this thesis, a new version of Yu's Property A in the case of locally compact groups is introduced. This new notion of Property A coincides with Yu's Property A in the case of discrete groups, but is different in the case of general locally compact groups. In particular, Gromov's locally compact hyperbolic groups is of Property A.

This dissertation contains two parts: Commutative and Noncommutative Multidimensional ($d$-D) Linear Systems Theory. The first part focuses on the development of the interpolation theory to solve the $H^{\infty}$ control problem for $d$-D linear systems. We first review the classical discrete-time 1D linear system in the operator theoretical viewpoint followed by the formulations of the so-called Givone-Roesser and Fornasini-Marchesini models. Application of the $d$-variable $Z$-transform to the system of equations yields the transfer function which is a rational function of several complex variables, say $\mathbf{z} = (z_{1}, \dots, z_{d})$. We then consider the output feedback stabilization problem for a plant $P(\mathbf{z})$. By assuming that $P(\mathbf{z})$ admits a double coprime factorization, then a set of stabilizing controllers $K(\mathbf{z})$ can be parametrized by the Youla parameter $Q(\mathbf{z})$. By doing so, one can convert such a problem to the model matching problem with performance index $F(\mathbf{z})$, affine in $Q(\mathbf{z})$. Then, with $F(\mathbf{z})$ as the design parameter rather than $Q(\mathbf{z})$, one has an interpolation problem for $F(\mathbf{z})$. Incorporation of a tolerance level on $F(\mathbf{z})$ then leads to an interpolation problem of multivariable Nevanlinna-Pick type. We also give an operator-theoretic formulation of the model matching problem which lends itself to a solution via the commutant lifting theorem on the polydisk. The second part details a system whose time-axis is described by a free semigroup $\mathcal{F}_{d}$. Such a system can be represented by the so-called noncommutative Givone-Roesser, or noncommutative Fornasini-Marchesini models which are analogous to those in the first part. Application of a noncommutative $d$-variable $Z$-transform to the system of equations yields the transfer function expressed by a formal power series in several noncommuting indeterminants, say $T(z) = \sum_{v \in \mathcal{F}_{d}}T_{v}z^{v}$ where $z^{v} = z_{i_{n}} \dotsm z_{i_{1}}$ if $v = g_{i_{n}} \dotsm g_{i_{1}} \in \mathcal{F}_{d}$ and $z_{i}z_{j} \neq z_{j}z_{i}$ unless $i = j$. The concepts of reachability, controllability, observability, similarity, and stability are introduced by means of the state-space interpretation. Minimal realization problems for noncommutative Givone-Roesser or Fornasini-Marchesini systems are solved directly by a shift-realization procedure constructed from appropriate noncommutative Hankel matrices. This procedure adapts the ideas of Schützenberger and Fliess originally developed for "recognizable series" to our systems. / Ph. D.

No description available.

Attitude control remains one of the top engineering challenges faced by small satellite mission planning and design. Conventional methods for attitude control include propulsion, reaction wheels, magnetic torque coils, and passive stabilization mechanisms, such as permanent magnets that align with planetary magnetic fields. Drawbacks of these conventional attitude control methods for small satellites include size, power consumption, dependence on external magnetic fields, and lack of full control authority. This research investigates an alternative, novel approach to attitude-control method for small satellites, utilizing the noncommutative property of rigid body rotation sequences. Piezoelectric bimorph actuators are used to induce sinusoidal small-amplitude satellite oscillations on two of the satellites axes. While zero net change occurs on these signaled axes, the third axis can develop an average angular rate. This noncommutative attitude control methodology has several advantages over conventional methods, including scalability, power consumption, and operation outside of Earth's magnetic field. This research looks into the feasibility of such a system, and lays the foundation for a simple control system architecture.

In this thesis examples of spectral triples, which represent fractal sets, are examined and new insights into their noncommutative geometries are obtained. Firstly, starting with Connes' spectral triple for a non-empty compact totally disconnected subset E of {R} with no isolated points, we develop a noncommutative coarse multifractal formalism. Specifically, we show how multifractal properties of a measure supported on E can be expressed in terms of a spectral triple and the Dixmier trace of certain operators. If E satisfies a given porosity condition, then we prove that the coarse multifractal box-counting dimension can be recovered. We show that for a self-similar measure μ, given by an iterated function system S defined on a compact subset of {R} satisfying the strong separation condition, our noncommutative coarse multifractal formalism gives rise to a noncommutative integral which recovers the self-similar multifractal measure ν associated to μ, and we establish a relationship between the noncommutative volume of such a noncommutative integral and the measure theoretical entropy of ν with respect to S. Secondly, motivated by the results of Antonescu-Ivan and Christensen, we construct a family of (1, +)-summable spectral triples for a one-sided topologically exact subshift of finite type (∑{{A}} {{N}}, σ). These spectral triples are constructed using equilibrium measures obtained from the Perron-Frobenius-Ruelle operator, whose potential function is non-arithemetic and Hölder continuous. We show that the Connes' pseudo-metric, given by any one of these spectral triples, is a metric and that the metric topology agrees with the weak*-topology on the state space {S}(C(∑{{A}} {{N}}); {C}). For each equilibrium measure ν[subscript(φ)] we show that the noncommuative volume of the associated spectral triple is equal to the reciprocal of the measure theoretical entropy of ν[subscript(φ)] with respect to the left shift σ (where it is assumed, without loss of generality, that the pressure of the potential function is equal to zero). We also show that the measure ν[subscript(φ)] can be fully recovered from the noncommutative integration theory.

<p>The fractional quantum Hall effect (FQHE), discovered in 1982 in a two-dimensional electron system, has generated a wealth of successful theory and new concepts in condensed matter physics, but is still not fully understood. The possibility of having nonabelian quasiparticle statistics has recently attracted attention on purely theoretical grounds but also because of its potential applications in topologically protected quantum computing.</p><p>This thesis focuses on the quasiparticles using three different approaches. The first is an effective Chern-Simons theory description, where the noncommutativity imposed on the classical space variables captures the incompressibility. We propose a construction of the quasielectron and illustrate how many-body quantum effects are emulated by a classical noncommutative theory.</p><p>The second approach involves a study of quantum Hall states on a torus where one of the periods is taken to be almost zero. Characteristic quantum Hall properties survive in this limit in which they become very simple to understand. We illustrate this by giving a simple counting argument for degeneracy 2<i>n</i>^-1, pertinent to nonabelian statistics, in the presence of 2<i>n</i> quasiholes in the Moore-Read state and generalise this result to 2<i>n</i>-<i>k</i> quasiholes and <i>k </i>quasielectrons.</p><p>In the third approach, we study the topological nature of the degeneracy 2<i>n</i>^-1 by using a recently proposed analogy between the Moore-Read state and the two-dimensional spin-polarized p-wave BCS state. We study a version of this problem where one can use techniques developed in the context of high-<i>T</i>c superconductors to turn the vortex background into an effective gauge field in a Dirac equation. Topological arguments in the form of index theory gives the degeneracy 2<i>n</i>^-1 for 2<i>n</i> vortices.</p>

Efeitos térmicos em teorias de calibre em (2+1) dimensões são estudados em espaços onde as coordenadas podem ou não comutar. No caso comutativo, a dependência com a temperatura do tensor de polarização é calculada a um laço em teorias envolvendo tanto bósons quanto férmions. Como aplicação, são calculados os processos de blindagem em tais modelos, chegando ao interessante resultado de que cargas magnéticas não sofrem tais efeitos na QED3. Uma prova válida em qualquer ordem de perturbação é desenvolvida, confirmando este comportamento. Em teorias não comutativas, são estudadas as correções a um laço ao coeficiente de Chern-Simons, sendo encontrado que não existe o fenômeno da mistura UV/IR na teoria Chern-Simons-Higgs. O comportamento assintótico de tal coeficiente é analisado no regime de altas temperaturas. Vários outros aspectos envolvendo os efeitos térmicos em teorias de Chern-Simons são explorados. / Thermal effects in (2+1)-dimensional gauge theories are studied in both commutative as well as noncommutative manifolds. In the first situation, the finite temperature polarization tensor is computed a tone loop for fermionic and bosonic couplings. As an application, the screening masses are evaluated and it is found the surprising result that magnetic charges are not screened in QED3. It is demonstrated that this result holds to any order in pertubationtheory. In the noncommutative case, the one loop correction to the Chern-Simons coefficient is studied, and it is found that there is no UV/IR mixing in the Chern-Simons-Higgs model. The asymptotic behavior of such coefficient is analised in the high temperature regime. Several other interesting aspects involving thermal effects of Chern-Simons theories are also discussed.