175 research outputs found

    On the Necessary and Sufficient Condition for a Set of Matrices to Commute and Some Further Linked Results

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    Es reproducción del documento publicado en http://dx.doi.org/10.1155/2009/650970This paper investigates the necessary and sufficient condition for a set of (real or complex) matrices to commute. It is proved that the commutator [A,B]=0 for two matrices A and B if and only if a vector v(B) defined uniquely from the matrix B is in the null space of a well-structured matrix defined as the Kronecker sum A⊕(−A∗), which is always rank defective. This result is extendable directly to any countable set of commuting matrices. Complementary results are derived concerning the commutators of certain matrices with functions of matrices f(A) which extend the well-known sufficiency-type commuting result [A,f(A)]=0.Ministerio de Educación DPI2006-00714 ; Gobierno Vasco GIC07143-IT-269-07 y SAIOTEK S-PE08UN1

    Necessary and Sufficient Condition for a Set of Matrices to Commute

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    The necessary and suffcient condition for a set of matrices to commute is given and proven

    Locating the sets of exceptional points in dissipative systems and the self-stability of bicycles

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    Sets in the parameter space corresponding to complex exceptional points have high codimension and by this reason they are difficult objects for numerical location. However, complex EPs play an important role in the problems of stability of dissipative systems where they are frequently considered as precursors to instability. We propose to locate the set of complex EPs using the fact that the global minimum of the spectral abscissa of a polynomial is attained at the EP of the highest possible order. Applying this approach to the problem of self-stabilization of a bicycle we find explicitly the EP sets that suggest scaling laws for the design of robust bikes that agree with the design of the known experimental machines

    Explicit integration of some integrable systems of classical mechanics

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    The main objective of the thesis is the analytical and geometrical study of several integrable finite-dimentional dynamical systems of classical mechanics, which are closely related, namely: - the classical generalization of the Euler top: the Zhukovski-Volterra (ZV) system describing the free motion of a gyrostat, i.e., a rigid body carrying a symmetric rotator whose axis is fixed in the body; - the Steklov-Lyapunov integrable case of the Kirchhoff equations describing the motion of a rigid body in an ideal incompressible liquid; - a nontrivial integrable generalization of the Steklov-Lyapunov system found by V.Rubanovskii: it describes the motion of a gyrostat in an ideal fluid in presence of a non-zero circulation. In our study we obtained explicit solution of the Zhukovski-Volterra ([2] and the Steklov-Lyapunov systems in terms of sigma- or theta-functions, and performed a bifurcation analysis of these systems, as well as of the Rubanovskii generalization. One should note that the solution of the ZV system was first given by V. Volterra, who, however, presented only its structure, but not the explicit formulas. The thesis gives a new alternative solution of this system by using an algebraic parametrization of the angular momentum. This allowed us to find poles and zeros of angular momentum in an algebraic way. The parametrization was also used to find an explicit solution for the Euler precession angle, and, as a consequence, to solve the Poisson equations describing the motion of the gyrostat in space. Similarly, by giving a geometric interpretation of the separating variables, and using the Weierstrass root functions, we reconstructed the thetafunctional solution of the Steklov-Lyapunov systems, which was first given by F. Kötter in 1899 without a derivation ([3]). In the study of bifurcations and singularities of the ZV system we used its bi-Hamiltonian structure ([1]. According the new method, the solution is critical, if there exist a parameter of corresponding family of Poisson brackets, for wich the rang of the brackets with this parameter drops. Applying new technics, based on the property of the system of being bi-Hamiltonian, we construct the bifurcation diagram of the ZV system. We also find the equilibrium points of the system, check the non-degeneracy condition for such points in the sense of the singularity theory of Hamiltonian systems, determine the types of equilibria points, and verify whether they are stable or not. We also describe the topological type of common levels of the first integrals of the ZV system. Similar problems have been discussed in many papers, but the goal of our work is to study the system and demonstrate the above techniques. It is a remarkable fact that using the bi-Hamiltonian property makes it possible to answer all the above questions practically without any difficult computations. The same method is applied to construct the bifurcation diagram for the Steklov-Lyapunov system, describe the zones of real motion, and analyze stability of critical periodic solutions. Then the bifurcation analysis is extended to the Rubanovskii generalizaton. Here the main difficulty is that the number of different types of the bifurcation diagram is quite high, so we only describe general properties of the bifurcation curves, do stability analysis for closed trajectories, and equilibria.El objetivo principal de la tesis es el estudio analítico y geométrico de varios sistemas integrables dinámicos y finito-dimensionales de la mecánica clásica que están estrechamente vinculados, a saber: -La generalización clásica de Euler top: el sistema Zhukovski-Volterra (ZV) que describe el movimiento libre de un giróstato, es decir, un cuerpo rígido que lleva un rotor simétrico cuyo eje es fijo al cuerpo. - El caso del sistema integrable de Steklov-Lyapunov de las ecuaciones de Kirchhoff que describen el movimiento de un cuerpo rígido en un líquido incompresible ideal; - Una generalización no trivial del sistema integrable de Steklov-Lyapunov encontrado por V. Rubanovskii que describe el movimiento de un giróstato en un fluido ideal en presencia de una circulación distinta de cero. En nuestro estudio hemos obtenido una solución explícita de los sistemas de Zhukovski-Volterra [2] y de Steklov-Lyapunov en términos de funciones sigma- o theta y hemos realizado un análisis de la bifurcación de estos sistemas, así como de la generalización de Rubanovskii. Hay que señalar que la solución del sistema de ZV fue dado por primera vez por V. Volterra, que, sin embargo, presenta sólo su estructura, pero no las fórmulas explícitas. La tesis ofrece una nueva solución alternativa de este sistema mediante el uso de una parametrización algebraica del momento angular. Esto nos ha permitido encontrar polos y ceros del momento angular en forma algebraica. La parametrización también se utilizó para encontrar una solución explícita para el ángulo de precesión de Euler, y, en consecuencia, para resolver las ecuaciones de Poisson que describen el movimiento de un giróstato en el espacio. Del mismo modo, al dar una interpretación geométrica de las variables de separación, y utilizando las funciones de las raíces Weierstrass, hemos reconstruido la solución thetafunctional de los sistemas de Steklov-Lyapunov, que fue dado por primera vez por F. Kotter en 1899 sin una derivación ([3]). En el estudio de las bifurcaciones y las singularidades del sistema ZV hemos utilizado su estructura bi-Hamiltoniana ([1]). Según el nuevo método, la solución es crítica, si existe un parámetro de la familia correspondiente del paréntesis de Poisson, para que el rango de las paréntesis con este parámetro se disminuye. Aplicando las nuevas técnicas, basadas en la propiedad del sistema de ser bi-Hamiltoniana, construimos el diagrama de bifurcación del sistema ZV. También hemos encontrado los puntos de equilibrio del sistema, verificando la condición de no-degeneración de estos puntos, en el sentido de la teoría de singularidad de los sistemas hamiltonianos, determinando los tipos de puntos de equilibrio, y comprobando si son estables o no. También hemos descrito el tipo topológico de los niveles comunes de los primeros integrales del sistema de ZV. Problemas similares se han discutido en muchas obras, pero el objetivo de nuestro trabajo es estudiar el sistema y demostrar las técnicas anteriormente mencionadas. Es un hecho notable que el uso de la propiedad bi-Hamilton permite responder a todas las preguntas anteriores, prácticamente sin ningún cálculo difícil. El mismo método se aplica para construir el diagrama de bifurcación para el sistema de Steklov-Lyapunov, describir las zonas de movimiento real, y analizar la estabilidad de soluciones periódicas críticas. A continuación, el análisis de bifurcación se extiende a la generalización Rubanovskii. Aquí la principal dificultad consiste en que el número de diferentes tipos del diagrama de bifurcación es bastante alto, por lo que sólo se describen las propiedades generales de las curvas de bifurcación, y el análisis de estabilidad se hace para trayectorias cerradas, y equilibrios

    Anosov actions: classification and the Zimmer Program

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    Consider a volume preserving Anosov C∞C^\infty action α\alpha on a compact manifold XX by semisimple Lie groups with all simple factors of real rank at least 2. More precisely we assume that some Cartan subgroup AA of GG (or equivalently GG) contains a dense set of elements which act normally hyperbolically on MM with respect to the orbit foliation of AA. We show that α\alpha is C∞C^\infty-conjugate to an action by left translations of a bi-homogeneous space M\H/ΛM \backslash H / \Lambda, where MM is a compact subgroup of a Lie group HH and Λ\Lambda is a uniform lattice in HH. Crucially to our arguments, we introduce the notion of leafwise homogeneous topological Anosov Rk\mathbb R^k actions for k≥2k \geq 2 and provide their C0C^0 classification, again by left translations actions of a homogeneous space. We then use accessibility properties, the invariance principle of Avila and Viana, cohomology properties of partially hyperbolic systems by Wilkinson and lifting to a suitable fibration to obtain the classification of Anosov GG actions from the classification of topological Anosov Rk\mathbb R^k actions.Comment: The proof of Lemma 11.11 of v1 is was incomplete. We complete it in this version, the main results in Section 2 remain unchanged. We also clarify better the arguments of some lemmas and fix some typo

    Cartan Actions of Higher Rank Abelian Groups and their Classification

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    We study transitive Rk×Zℓ\mathbb{R} ^k \times \mathbb{Z}^\ell actions on arbitrary compact manifolds with a projectively dense set of Anosov elements and 1-dimensional coarse Lyapunov foliations. Such actions are called totally Cartan actions. We completely classify such actions as built from low-dimensional Anosov flows and diffeomorphisms and affine actions, verifying the Katok-Spatzier conjecture for this class. This is achieved by introducing a new tool, the action of a dynamically defined topological group describing paths in coarse Lyapunov foliations, and understanding its generators and relations. We obtain applications to the Zimmer program
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