Dissertations / Theses on the topic 'Lie quadratic'

In this thesis, we defind a new invariant of quadratic Lie algebras and quadratic Lie superalgebras and give a complete study and classification of singular quadratic Lie algebras and singular quadratic Lie superalgebras, i.e. those for which the invariant does not vanish. The classification is related to adjoint orbits of Lie algebras o(m) and sp(2n). Also, we give an isomorphic characterization of 2-step nilpotent quadratic Lie algebras and quasi-singular quadratic Lie superalgebras for the purpose of completeness. We study pseudo-Euclidean Jordan algebras obtained as double extensions of a quadratic vector space by a one-dimensional algebra and 2-step nilpotent pseudo-Euclidean Jordan algebras, in the same manner as it was done for singular quadratic Lie algebras and 2-step nilpotent quadratic Lie algebras. Finally, we focus on the case of a symmetric Novikov algebra and study it up to dimension 7.

Dans cette thèse, nous définissons un nouvel invariant des algèbres de Lie quadratiques et des superalgèbres de Lie quadratiques et donnons une étude et classification complète des algèbres de Lie quadratiques singulières et des superalgèbres de Lie quadratiques singulières, i.e. celles pour lesquelles l’invariant n’est pas nul. La classification est en relation avec les orbites adjointes des algèbres de Lie o(m) et sp(2n). Aussi, nous donnons une caractérisation isomorphe des algèbres de Lie quadratiques 2-nilpotentes et des superalgèbres de Lie quadratiques quasi-singulières pour le but d’exhaustivité. Nous étudions les algèbres de Jordan pseudoeuclidiennes qui sont obtenues des extensions doubles d’un espace vectoriel quadratique par une algèbre d’une dimension et les algèbres de Jordan pseudo-euclidienne 2-nilpotentes, de la même manière que cela a été fait pour les algèbres de Lie quadratiques singulières et des algèbres de Lie quadratiques 2-nilpotentes. Enfin, nous nous concentrons sur le cas d’une algèbre de Novikov symétrique et l’étudions à dimension 7
In this thesis, we defind a new invariant of quadratic Lie algebras and quadratic Lie superalgebras and give a complete study and classification of singular quadratic Lie algebras and singular quadratic Lie superalgebras, i.e. those for which the invariant does not vanish. The classification is related to adjoint orbits of Lie algebras o(m) and sp(2n). Also, we give an isomorphic characterization of 2-step nilpotent quadratic Lie algebras and quasi-singular quadratic Lie superalgebras for the purpose of completeness. We study pseudo-Euclidean Jordan algebras obtained as double extensions of a quadratic vector space by a one-dimensional algebra and 2-step nilpotent pseudo-Euclidean Jordan algebras, in the same manner as it was done for singular quadratic Lie algebras and 2-step nilpotent quadratic Lie algebras. Finally, we focus on the case of a symmetric Novikov algebra and study it up to dimension 7

The Lie algebra associated to the lower central and p-central series of a group is an important invariant of the group but is difficult to compute. For a finitely presented group this Lie algebra, can be determined under a certain condition on the initial forms of the relators, namely that of strong freeness. We give an algorithm for the strong freeness of 4 quadratic Lie polynomials in 4 variables over an arbitrary field thus extending a result of Bush and Labute.

Dans cette thèse, on étudie la structure de quelques types d'algèbres (binaires et ternaires) munies d'une forme bilinéaire symétrique, non dégénérée et associative (ou invariante). La première partie de cette thèse est consacrée à l'étude des algèbres de Leibniz quadratiques. On montre que ces algèbres sont symétriques. De plus, on utilise la T*-extension et la double extension pour montrer plusieurs résultats sur ce type d'algèbres. Ensuite, on a remarqué que l'anti-commutativité du crochet de Lie donne naissance à de nouveaux types d'invariance pour les algèbres de Leibniz : L'invariance à gauche et l'invariance à droite. Alors, on s'est intéresse à l'étude des algèbres de Leibniz (gauche et droite) munies d'une forme bilinéaire symétrique, non dégénérée et invariante à gauche (et invariante à droite). On prouve que ces algèbres sont Lie admissibles. En second lieu, on s'intéresse aux systèmes triples de Lie et de Jordan. On débute la deuxième partie de cette thèse par la description inductive des systèmes triples de Lie quadratiques au moyen de la double extension. En plus, on introduit la T*extension des systèmes triples de Jordan pseudo-Euclidien. Finalement, on donne deux nouvelles caractérisations des systèmes triples de Jordan semi-simples parmi les systèmes triples de Jordan pseudo-Euclidiens
In this thesis, we study the stucture of several types of algebras endowed with Symmetric, non degenerate and invariant bilinear forms. In the first part, we study quadratic Leibniz algebras. First, we prove that these algebras are symmetric. Then, we use the T*-extension and the double extension to prove some properties of this type of Leibniz algebras. Besides, since we observe that the skew-symmetry of the Leibniz bracket gives rise to other types of invariance for a bilinear form on a Leibniz algebra: The left invariance and the right invariance. We focus on the study of left (resp. right) Leibniz algebras with symmetric, non degenerate and left (resp. right) invariant bilinear form. In particular, we prove that these algebras are Lie admissibles. The second part of this work is dedicated to the study of quadratic Lie triple systems and pseudo-euclidien Jordan triple systems. We start by giving an inductive description of quadratic Lie triple systems using double extension. Next, we introduce the T*-extension of Jordan triple systems. Finally, we give new caracterizations of semi-simple Jordan triple systems among pseudo-euclidian Jordan triple systems

A quadratic form on \(\mathbb{R}^n\) is a map of the form \(x \mapsto x^T M x\), where M is a symmetric \(n \times n\) matrix. A quadratic map from \(\mathbb{R}^n\) to \(\mathbb{R}^m\) is a map, all m of whose components are quadratic forms. One of the two central questions in this thesis is this: when is the image of a quadratic map \(Q: \mathbb{R}^n \rightarrow \mathbb{R}^m\) a convex subset of \(\mathbb{R}^m\)? This question has intrinsic interest; despite being only a degree removed from linear maps, quadratic maps are not well understood. However, the convexity properties of quadratic maps have practical consequences as well: underlying every semidefinite program is a quadratic map, and the convexity of the image of that map determines the nature of the solutions to the semidefinite program. Quadratic maps that map into \(\mathbb{R}^2\) and \(\mathbb{R}^3\) have been studied before (in (Dines, 1940) and (Calabi, 1964) respectively). The Roundness Theorem, the first of the two principal results in this thesis, is a sufficient and (almost) necessary condition for a quadratic map \(Q: \mathbb{R}^n \rightarrow \mathbb{R}^m\) to have a convex image when \(m \geq 4\), \(n \geq m\) and \(n \not= m + 1\). Concomitant with the Roundness Theorem is an important lemma: when \(n < m\), quadratic maps from \(\mathbb{R}^n\) to \(\mathbb{R}^m\)seldom have convex images. This second result in this thesis is a controllability condition for bilinear systems defined on direct products of the form \(\mathcal{G} \times\mathcal{G}\), where \(\mathcal{G}\) is a simple Lie group. The condition is this: a bilinear system defined on \(\mathcal{G} \times\mathcal{G}\) is not controllable if and only if the Lie algebra generated by the system’s vector fields is the graph of some automorphism of \(\mathcal{g}\), the Lie algebra of \(\mathcal{G}\).
Engineering and Applied Sciences

Thesis (Ph. D.)--Ohio State University, 2003.
Title from first page of PDF file. Document formatted into pages; contains xi, 231 p.; also includes graphics Includes bibliographical references (p. 224-229). Available online via OhioLINK's ETD Center

The purpose of this thesis in to introduce the systematic study of symmetries in binary differential equations (BDEs). We formalize the concept of a symmetric BDE, under the linear action of a compact Lie group. One of the main results establishes a formula that relates the algebraic and geometric effects of the occurrence of the symmetry in the problem. Using tools from invariant theory and representation theory for compact Lie groups we deduce the general forms of equivariant binary differential equations under compact subgroups of O(2). A study about the behavior of the invariant straight lines on the configuration of homogeneous BDEs of degree n is done with emphasis on cases in which n = 0 and n = 1. Also for the linear case (n = 1) the equivariant normal forms are presented. Symmetries of linear 1-forms are also studied and related with symmetries of tangent orthogonal vectors fields associated with it.
O objetivo desta tese é introduzir o estudo sistemático de simetrias em equações diferenciais binárias (EDBs). Neste trabalho formalizamos o conceito de EDB simétrica sobre a ação de um grupo de Lie compacto. Um dos principais resultados é uma fórmula que relaciona o efeito geométrico e algébrico das simetrias presentes no problema. Utilizando ferramentas da teoria invariante e de representação para grupos compactos deduzimos as formas gerais para EDBs equivariantes. Um estudo sobre o comportamento das retas invariantes na configuração de EDBs com coeficientes homogêneos de grau n é feito com ênfase nos casos de grau 0 e 1, ainda no caso de grau 1 são apresentadas suas formas normais. Simetrias de 1-formas lineares são também estudadas e relacionadas com as simetrias dos seus campos tangente e ortogonal.

[Truncated abstract] This thesis is concerned with geometrically-defined curves that can be used for interpolation in Riemannian or, more generally, semi-Riemannian manifolds. As in much of the existing literature on such curves, emphasis is placed on manifolds which are important in computer graphics and engineering applications, namely the unit 3-sphere S3 and the closely related rotation group SO(3), as well as other Lie groups and spheres of arbitrary dimension. All geometrically-defined curves investigated in the thesis are either higher order variational curves, namely critical points of cost functionals depending on (covariant) derivatives of order greater than 1, or defined by geometrical algorithms, namely generalisations to manifolds of algorithms from the field of computer aided geometric design. Such curves are needed, especially in the aforementioned applications, since interpolation methods based on applying techniques of classical approximation theory in coordinate charts often produce unnatural interpolants. However, mathematical properties of higher order variational curves and curves defined by geometrical algorithms are in need of substantial further investigation: higher order variational curves are solutions of complicated nonlinear differential equations whose properties are not well-understood; it is usually unclear how to impose endpoint derivative conditions on, or smoothly piece together, curves defined by geometrical algorithms. This thesis addresses these difficulties for several classes of curves. ... The geometrical algorithms investigated in this thesis are generalisations of the de Casteljau and Cox-de Boor algorithms, which define, respectively, polynomial B'ezier and piecewise-polynomial B-spline curves by dividing, in certain ratios and for a finite number of iterations, piecewise-linear control polygons corresponding to finite sequences of control points. We show how the control points of curves produced by the generalised de Casteljau algorithm in an (almost) arbitrary connected finite-dimensional Riemannian manifold M should be chosen in order to impose desired endpoint velocities and (covariant) accelerations and, thereby, piece the curves together in a C2 fashion. A special case of the latter construction simplifies when M is a symmetric space. For the generalised Cox-de Boor algorithm, we analyse in detail the failure of a fundamental property of B-spline curves, namely C2 continuity at (certain) knots, to carry over to M.

Ce mémoire est consacré à des résultats d'analyse harmonique réelle dans des cadres géométriques discrets (graphes) ou continus (groupes de Lie).Soit $\Gamma$ un graphe (ensemble de sommets et d'arêtes) muni d'un laplacien discret $\Delta=I-P$, où $P$ est un opérateur de Markov.Sous des hypothèses géométriques convenables sur $\Gamma$, nous montrons la continuité $L^p$ de fonctionnelles de Littlewood-Paley fractionnaires. Nous introduisons des espaces de Hardy $H^1$ de fonctions et de $1$-formes différentielles sur $\Gamma$, dont nous donnons plusieurs caractérisations, en supposant seulement la propriété de doublement pour le volume des boules de $\Gamma$. Nous en déduisons la continuité de la transformée de Riesz sur $H^1$. En supposant de plus des estimations supérieures ponctuelles (gaussiennes ou sous-gaussiennes) sur les itérées du noyau de l'opérateur $P$, nous obtenons aussi la continuité de la transformée de Riesz sur $L^p$ pour $10$, $1\leq p\leq+\infty$ et $1\leq q\leq +\infty$. Les résultats sont valables en croissance polynomiale ou exponentielle du volume des boules
This thesis is devoted to results in real harmonic analysis in discrete (graphs) or continuous (Lie groups) geometric contexts.Let $\Gamma$ be a graph (a set of vertices and edges) equipped with a discrete laplacian $\Delta=I-P$, where $P$ is a Markov operator.Under suitable geometric assumptions on $\Gamma$, we show the $L^p$ boundedness of fractional Littlewood-Paley functionals. We introduce $H^1$ Hardy spaces of functions and of $1$-differential forms on $\Gamma$, giving several characterizations of these spaces, only assuming the doubling property for the volumes of balls in $\Gamma$. As a consequence, we derive the $H^1$ boundedness of the Riesz transform. Assuming furthermore pointwise upper bounds for the kernel (Gaussian of subgaussian upper bounds) on the iterates of the kernel of $P$, we also establish the $L^p$ boundedness of the Riesz transform for $10$, $1\leq p\leq+\infty$ and $1\leq q\leq +\infty$.These results hold for polynomial as well as for exponential volume growth of balls

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
In this study, we evaluate if the congruence x2 a (mod m), where m is prime and (a;m) = 1, has or not solutions, highlighting the importance of Quadratic Residues and consequently the cooperation of the Legendre's Symbol, the Euler's Criterion and the Gauss' Lemma. Also, we demonstrate the Law of Quadratic Reciprocity generalizing situations for composite numbers, that is, the Jacobi's Symbol and its properties. We present some proposals of activities for the High School involving the subject matter and its possible applications, through an understandable language for students of this level.
Neste estudo, vamos avaliar se a congruência x2 a (mod m), onde m é primo e (a;m) = 1, apresenta ou não solução, destacando a importância dos Resíduos Quadráticos e, consequentemente da cooperação do Símbolo de Legendre, do Critério de Euler e do Lema de Gauss. Também, demonstraremos a Lei de Reciprocidade Quadrática generalizando situações para números compostos, ou seja, o Símbolo de Jacobi e suas propriedades. Apresentamos algumas propostas de atividades para o Ensino Médio envolvendo o assunto abordado e suas possíveis aplicações, através de uma linguagem compreensível aos alunos deste nível de ensino.

220 relevés par I.P.A. complétés par 3 dénombrements par cartographie des territoires ont été réalisés dans une succession de végétation. de la série du Chêne liège ainsi que dans des reboisements de Pin maritime et d'Eucalyptus, dans la région d'Annaba - EI-Kala (nord-est algérien). Les résultats obtenus font ressortir une nette différence entre la perception à priori du phénomène successionnel, à travers la vision qu'en a le forestier ou le phytoécologue, et qui nous a conduit à une stratification fine du milieu, et la perception de l'avifaune qui a révélé une indifférenciation de certains milieux, notamment les maquis. Cette indifférenciation est due à la grande amplitude d'habitat des espèces. L'analyse de la dynamique de l'avifaune le long de la succession, a permis de mettre en évidence 2 gradients : un gradient milieux ouverts (maquis) - milieux fermés (forêts) et, au sein des forêts, un gradient forêts de plaine-forêts d'altitude. La comparaison de la succession algérienne avec 4 successions européennes, a révélé une grande similitude des avifaunes des stades mûrs des 5 successions et une originalité des avifaunes des stades jeunes. Ce phénomène est en rapport avec l'histoire de la mise en place des avifaunes dans le bassin méditerranéen. Nous avons discuté par ailleurs quelques arguments qui nous ont permis de mettre en évidence un syndrome d'insularité biologique au Maghreb.

The quadratic class of time-frequency distributions (TFDs) forms a set of tools which allow to effectively extract important information from a nonstationary signal. To determine which TFD best represents the given signal, it is a common practice to visually compare different TFDs' time-frequency plots, and select as best the TFD with the most appealing plot. This visual comparison is not only subjective, but also difficult and unreliable especially when signal components are closely-spaced in the time-frequency plane. To objectively compare TFDs, a quantitative performance measure should be used. Several measures of concentration/complexity have been proposed in the literature. However, those measures by being derived with certain theoretical assumptions about TFDs are generally not suitable for the TFD selection problem encountered in practical applications. The non-existence of practically-valuable measures for TFDs' resolution comparison, and hence the non-existence of methodologies for the signal optimal TFD selection, has significantly limited the use of time-frequency tools in practice. In this thesis, by extending and complementing the concept of spectral resolution to the case of nonstationary signals, and by redefining the set of TFDs' properties desirable for practical applications, we define an objective measure to quantify the quality of TFDs. This local measure of TFDs' resolution performance combines all important signal time-varying parameters, along with TFDs' characteristics that influence their resolution. Methodologies for automatically selecting a TFD which best suits a given signal, including real-life signals, are also developed. The optimisation of the resolution performances of TFDs, by modifying their kernel filter parameters to enhance the TFDs' resolution capabilities, is an important prerequisite in satisfying any additional application-specific requirements by the TFDs. The resolution performance measure and the accompanying TFDs' comparison criteria allow to improve procedures for designing high-resolution quadratic TFDs for practical time-frequency analysis. The separable kernel TFDs, designed in this way, are shown to best resolve closely-spaced components for various classes of synthetic and real-life signals that we have analysed.

We begin by discussing motivation for our consideration of a structure called a Lie 2-algebra, in particular an important class of Lie 2-algebras are the Courant Algebroids introduced in 1990 by Courant. We wish to attach some natural definitions from operad theory, mainly the notion of a module over an algebra, to Lie 2-algebras and hence to Courant algebroids. To this end our goal is to show that Lie 2-algebras can be described as what are called \emph{homotopy algebras over an operad}. Describing Lie 2-algebras using operads also solves the problem of showing that the equations defining a Lie 2-algebra are consistent. Our technical discussion begins by introducing some notions from operad theory, which is a generalization of the theory of operations on a set and their compositions. We define the idea of a quadratic operad and a homotopy algebra over a quadratic operad. We then proceed to describe Lie 2-algebras as homotopy algebras over a given quadratic operad using a theorem of Ginzburg and Kapranov. Next we briefly discuss the structure of a braided monoidal category. Following this, motivated by our discussion of braided monoidal categories, a new structure is introduced, which we call a commutative 2-algebra. As with the Lie 2-algebra case we show how a commutative 2-algebra can be seen as a homotopy algebra over a particular quadratic operad. Finally some technical results used in previous theorems are mentioned. In discussing these technical results we apply some ideas about distributive laws and Koszul operads.

Tese de doutoramento em Matemática (Pré-Bolonha), Especialidade de Matemática Pura apresentada à Faculdade de Ciências e Tecnologia da Universidade de Coimbra
O objetivo fundamental desta dissertação é apresentar uma visão abrangente sobre rolamentos, sem deslize nem torção, de variedades diferenciáveis, contribuindo para aprofundar o conhecimento teórico nesta área e evidenciar potenciais aplicações. Começamos por apresentar uma definição de aplicação rolamento para o caso mais geral em que o movimento acontece dentro de espaços ambiente que são variedades pseudo-Riemannianas. Isto generaliza a definição clássica de Sharpe. A seguir, provamos algumas propriedades essenciais dos rolamentos e fazemos a ligação destes com o transporte paralelo de vetores. Dentro do contexto geral, analisamos os rolamentos das hiperquádricas de espaços pseudo-Euclidianos, com enfoque no caso dos espaços pseudo-hiperbólicos H_k^n (r). Apresentamos as equações da cinemáticas do rolamento de H_k^n (r) sobre o espaço afim associado ao espaço tangente num ponto. A obtenção de soluções explícitas destas equações é alcançada em dois casos particulares, destacando-se a situação em que o rolamento é feito ao longo de geodésicas. Rolamentos de um espaço pseudo-hiperbólico sobre outro e de pseudoesferas são igualmente tratados. Investigamos os rolamentos de grupos de Lie quadráticos sobre um espaço afim tangente. Também nestes casos se deduzem as equações da cinemática e se procuram soluções explícitas. A abordagem usada neste caso tem a preocupação de não destruir a estrutura matricial que caracteriza os elementos destes grupos matriciais. Estudamos a controlabilidade de rolamentos nos casos da hiperquádrica H_k^n (r) e dos grupos de Lie quadráticos principais, os grupos pseudo-ortogonais e os grupos simpléticos. Seguimos uma abordagem algébrica que passa por reescrever as equações da cinemática como um sistema de controlo afim a evoluir num grupo de Lie. Aplicamos os resultados obtidos anteriormente na resolução de problemas de interpolação suave em variedades e apresentamos um algoritmo interpolador. As propriedades dos rolamentos permitem transformar um problema de interpolação complicado, formulado numa variedade, num outro mais simples de resolver. São ainda fornecidos os ingredientes necessários para a implementação prática do algoritmo nos casos particulares de H_0^n (r) e H_1^n (r).
The primary goal of this dissertation is to present a comprehensive overview about rolling motions, subject to non-slip and non-twist constraints, of differentiable manifolds, contributing to deepen the theoretical knowledge in this area and to point out potential applications. We first present a definition of rolling map for the situation when the motion occurs inside an ambient space which is a pseudo-Riemannian manifold. This generalizes the classical definition of Sharpe. We then present several essential properties of rolling and make the connection between rolling motions and parallel transport of vectors. Within this general framework, we analyze the rolling of hyperquadrics embedded in pseudo-Euclidean spaces, focusing on the case of pseudo-hyperbolic spaces H_k^n (r). The kinematic equations of rolling H_k^n (r) on the affine space associated to the tangent space at a point is presented. Explicit solutions of these equations are obtained in two particular cases, with emphasis when the rolling is done along geodesics. Rolling of a pseudo-hyperbolic space on another and rolling of pseudo-spheres are equally treated. We investigate the rolling of quadratic Lie groups on an affine space tangent. We also derive the corresponding kinematic equations and look for explicit solutions. The approach used here is chosen so that the matrix structure that characterizes the elements of these matrix groups is not destroyed. We also address the controllability issue of rolling motions in the cases of hyperquadrics H_k^n (r) and of the most important quadratic Lie groups, pseudo-orthogonal groups and symplectic groups. We used an algebraic approach to controllability that requires rewriting the kinematic equations as a control system evolving on a Lie group. We apply the results previously obtained to solve problems of smooth interpolation on manifolds and present an interpolating algorithm. The properties of rolling enable to transform a complicated interpolation problem, formulated on a manifold, on another much simpler to solve. Ingredients needed to implement the algorithm are provided in the specific cases of H_0^n (r) and H_1^n (r).