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Lie Algebra (lie + algebra)
Selected AbstractsLie Algebra and the Mobility of Kinematic ChainsJOURNAL OF FIELD ROBOTICS (FORMERLY JOURNAL OF ROBOTIC SYSTEMS), Issue 8 2003J. M. Rico This paper deals with the application of Lie Algebra to the mobility analysis of kinematic chains. It develops an algebraic formulation of a group-theoretic mobility criterion developed recently by two of the authors of this publication. The instantaneous form of the mobility criterion presented here is based on the theory of subspaces and subalgebras of the Lie Algebra of the Euclidean group and their possible intersections. It is shown using this theory that certain results on mobility of over-constraint linkages derived previously using screw theory are not complete and accurate. The theory presented provides for a computational approach that would allow efficient automation of the new group-theoretic mobility criterion. The theory is illustrated using several examples. © 2003 Wiley Periodicals, Inc. [source] Structured Condition Numbers of Multiple EigenvaluesPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2006María José Peláez We analyze the influence of matrix structure on the condition number of multiple, possibly defective eigenvalues. We show that the structured and unstructured Hölder condition numbers coincide for multiple eigenvalues of matrices belonging to certain classes of structured matrices, which can be characterized as either Jordan or Lie Algebras. We do this by explicitly finding a specific perturbation matrix, analogous to the classical Wilkinson perturbation, which attains the maximal variation within the class of structured matrices. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] String theory: exact solutions, marginal deformations and hyperbolic spacesFORTSCHRITTE DER PHYSIK/PROGRESS OF PHYSICS, Issue 2 2007D. Orlando Abstract This thesis is almost entirely devoted to studying string theory backgrounds characterized by simple geometrical and integrability properties. The archetype of this type of system is given by Wess-Zumino-Witten models, describing string propagation in a group manifold or, equivalently, a class of conformal field theories with current algebras. We study the moduli space of such models by using truly marginal deformations. Particular emphasis is placed on asymmetric deformations that, together with the CFT description, enjoy a very nice spacetime interpretation in terms of the underlying Lie algebra. Then we take a slight detour so to deal with off-shell systems. Using a renormalization-group approach we describe the relaxation towards the symmetrical equilibrium situation. In he final chapter we consider backgrounds with Ramond-Ramond field and in particular we analyze direct products of constant-curvature spaces and find solutions with hyperbolic spaces. [source] Ricci flows and infinite dimensional algebrasFORTSCHRITTE DER PHYSIK/PROGRESS OF PHYSICS, Issue 6-7 2004I. Bakas The renormalization group equations of two-dimensional sigma models describe geometric deformations of their target space when the world-sheet length changes scale from the ultra-violet to the infra-red. These equations, which are also known in the mathematics literature as Ricci flows, are analyzed for the particular case of two-dimensional target spaces, where they are found to admit a systematic description as Toda system. Their zero curvature formulation is made possible with the aid of a novel infinite dimensional Lie algebra, which has anti-symmetric Cartan kernel and exhibits exponential growth. The general solution is obtained in closed form using Bäcklund transformations, and special examples include the sausage model and the decay process of conical singularities to the plane. Thus, Ricci flows provide a non-linear generalization of the heat equation in two dimensions with the same dissipative properties. Various applications to dynamical problems of string theory are also briefly discussed. Finally, we outline generalizations to higher dimensional target spaces that exhibit sufficient number of Killing symmetries. [source] Lie-Poisson integrators: A Hamiltonian, variational approachINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 13 2010Zhanhua Ma Abstract In this paper we present a systematic and general method for developing variational integrators for Lie-Poisson Hamiltonian systems living in a finite-dimensional space ,,*, the dual of Lie algebra associated with a Lie group G. These integrators are essentially different discretized versions of the Lie-Poisson variational principle, or a modified Lie-Poisson variational principle proposed in this paper. We present three different integrators, including symplectic, variational Lie-Poisson integrators on G×,,* and on ,,×,,*, as well as an integrator on ,,* that is symplectic under certain conditions on the Hamiltonian. Examples of applications include simulations of free rigid body rotation and the dynamics of N point vortices on a sphere. Simulation results verify that some of these variational Lie-Poisson integrators are good candidates for geometric simulation of those two Lie-Poisson Hamiltonian systems. Copyright © 2009 John Wiley & Sons, Ltd. [source] Nonlinear systems possessing linear symmetryINTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 1 2007Daizhan Cheng Abstract This paper tackles linear symmetries of control systems. Precisely, the symmetry of affine nonlinear systems under the action of a sub-group of general linear group GL(n,,). First of all, the structure of state space (briefly, ss) symmetry group and its Lie algebra for a given system is investigated. Secondly, the structure of systems, which are ss-symmetric under rotations, is revealed. Thirdly, a complete classification of ss-symmetric planar systems is presented. It is shown that for planar systems there are only four classes of systems which are ss-symmetric with respect to four linear groups. Fourthly, a set of algebraic equations are presented, whose solutions provide the Lie algebra of the largest connected ss-symmetry group. Finally, some controllability properties of systems with ss-symmetry group are studied. As an auxiliary tool for computation, the concept and some properties of semi-tensor product of matrices are included. Copyright © 2006 John Wiley & Sons, Ltd. [source] Analysis of a class of potential Korteweg-de Vries-like equationsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 2 2010R. M. Edelstein Abstract We analyze a class of third-order evolution equations, i.e. ut = f(x, ux, uxx) uxxx+g(x, ux, uxx) via the method of preliminary group classification. This method is a systematic means of analyzing the equation for symmetries. We find explicit forms of f and g, which allow for a larger dimensional Lie algebra of point symmetries. Copyright © 2009 John Wiley & Sons, Ltd. [source] A new look at the quantum mechanics of the harmonic oscillatorANNALEN DER PHYSIK, Issue 7-8 2007H.A. Kastrup Abstract In classical mechanics the harmonic oscillator (HO) provides the generic example for the use of angle and action variables and I > 0 which played a prominent role in the "old" Bohr-Sommerfeld quantum theory. However, already classically there is a problem which has essential implications for the quantum mechanics of the (,,I)-model for the HO: the transformation is only locally symplectic and singular for (q,p) = (0,0). Globally the phase space {(q,p)} has the topological structure of the plane ,2, whereas the phase space {(,,I)} corresponds globally to the punctured plane ,2 -(0,0) or to a simple cone with the tip deleted. From the properties of the symplectic transformations on that phase space one can derive the functions h0 = I, h1 = Icos , and h2 = - Isin , as the basic coordinates on {(,,I)}, where their Poisson brackets obey the Lie algebra of the symplectic group of the plane. This implies a qualitative difference as to the quantum theory of the phase space {(,,I)} compared to the usual one for {(q,p)}: In the quantum mechanics for the (,,I)-model of the HO the three hj correspond to the self-adjoint generators Kj, j = 0,1,2, of certain irreducible unitary representations of the symplectic group or one of its infinitely many covering groups, the representations being parametrized by a (Bargmann) index k > 0. This index k determines the ground state energy of the (,,I)-Hamiltonian . For an m -fold covering the lowest possible value for k is k = 1/m, which can be made arbitrarily small by choosing m accordingly! This is not in contradiction to the usual approach in terms of the operators Q and P which are now expressed as functions of the Kj, but keep their usual properties. The richer structure of the Kj quantum model of the HO is "erased" when passing to the simpler (Q,P)-model! This more refined approach to the quantum theory of the HO implies many experimental tests: Mulliken-type experiments for isotopic diatomic molecules, experiments with harmonic traps for atoms, ions and BE-condensates, with charged HOs in external electric fields and the (Landau) levels of charged particles in external magnetic fields, with the propagation of light in vacuum, passing through strong external electric or magnetic fields. Finally it may lead to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant. [source] THE EIGENFUNCTION EXPANSION METHOD IN MULTI-FACTOR QUADRATIC TERM STRUCTURE MODELSMATHEMATICAL FINANCE, Issue 4 2007Nina Boyarchenko We propose the eigenfunction expansion method for pricing options in quadratic term structure models. The eigenvalues, eigenfunctions, and adjoint functions are calculated using elements of the representation theory of Lie algebras not only in the self-adjoint case, but in non-self-adjoint case as well; the eigenfunctions and adjoint functions are expressed in terms of Hermite polynomials. We demonstrate that the method is efficient for pricing caps, floors, and swaptions, if time to maturity is 1 year or more. We also consider subordination of the same class of models, and show that in the framework of the eigenfunction expansion approach, the subordinated models are (almost) as simple as pure Gaussian models. We study the dependence of Black implied volatilities and option prices on the type of non-Gaussian innovations. [source] | |||||||||||||