Continuous Transformation Groups Research Articles

Singular spectrum time–series analysis and continuous transformation groups

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Nilsson Martin 2004Singular spectrum time–series analysis and continuous transformation groupsProc. R. Soc. Lond. A.460929–938http://doi.org/10.1098/rspa.2003.1197SectionRestricted accessResearch articleSingular spectrum time–series analysis and continuous transformation groups Martin Nilsson Martin Nilsson Nordic Institute of Theoretical Physics (NORDITA), Blegdamsvej 17, 2100 Copenhagen, Denmark () Google Scholar Find this author on PubMed Search for more papers by this author Martin Nilsson Martin Nilsson Nordic Institute of Theoretical Physics (NORDITA), Blegdamsvej 17, 2100 Copenhagen, Denmark () Google Scholar Find this author on PubMed Search for more papers by this author Published:08 March 2004https://doi.org/10.1098/rspa.2003.1197AbstractThis paper is a study of continuous–time singular spectrum analysis (SSA). We show that the principal eigenfunctions are solutions to a set of linear ordinary differential equations (ODEs) with constant coefficients. We also introduce a natural generalization of SSA, based on using local (Lie–) transformation groups to construct the trajectory function. The time translations (delay coordinates) used in standard SSA is a special case. A general relation between the transformation group used in the construction of the trajectory matrix and the principal eigenfunctions is demonstrated. The eigenfunctions satisfy a simple type of linear ODE with time–dependent coefficient, determined by the infinitesimal generator of the transformation group. Finally, more general one–parameter mappings are considered. Previous Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Golyandina N and Zhigljavsky A (2020) Introduction Singular Spectrum Analysis for Time Series, 10.1007/978-3-662-62436-4_1, (1-20), . This Issue08 March 2004Volume 460Issue 2043 Article InformationDOI:https://doi.org/10.1098/rspa.2003.1197Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/03/2004Published in print08/03/2004 License: Citations and impact KeywordsTime–Series AnalysisSpectral MethodsPrincipal Component AnalysisSpectral AnalysisSingular Spectrum AnalysisLie Groups

On the Similarity Solutions of the Inhomogeneous Nonlinear Diffusion Equation via Symmetry Method

The invariance under continuous group of transformations of the nonlinear diffusion equation in an inhomogeneous medium f ( x ) u t =( g ( x ) u n u x ) x with arbitrary thermal coefficients f ( x ) and g ( x ) is studied. The infinitesimals, similarity variables, dependent variables, and reduction to quadrature or exact solutions of this equation for physically realizable forms of f ( x ), g ( x ) and n are obtained too. Some interesting results of this study are new exact solutions, that do not seem to have been reported in the literature.

Invariant functionals and the uniqueness of invariant norms

Let τ be a representation of a compact group G on a Banach space ( X,||·||). The question we address is whether X carries a unique invariant norm in the sense that ||·|| is the unique norm on X for which τ is a representation. We characterize the uniqueness of norm in terms of the automatic continuity of the invariant functionals in the case when X is a dual Banach space and τ is a σ(X,X ∗) -continuous representation of G on X such that τ( G) consists of σ(X,X ∗) -continuous operators. We illustrate the usefulness of this characterization by studying the uniqueness of the norm on the spaces L p(Ω) , where Ω is a locally compact Hausdorff space equipped with a positive Radon measure and G acts on Ω as a group of continuous invertible measure-preserving transformations.

Projective Transformations of Pseudo-Riemannian Manifolds

A projective transformation of a pseudo-Riemannian manifoldM is an automorphism of the induced Riemannian connection of the projective structure that takes geodesics in M to geodesics again. For the first time, the problem of determining Riemannian spaces admitting continuous transformation groups preserving geodesics was considered by Sophus Lie for the case of two-dimensional surfaces (see [306]). However, as Fubini wrote in the preface to [262], “the famous mathematician did not succeed in solving this problem” (which Fubini called the “Lie problem”). Having criticized Lie’s method, which, to Fubini’s opinion, could not be used for the general solution of the problem, Fubini developed his own approach based on the infinitesimal calculus, later named the “Lie differentiation.” The subsequent development of the theory of projective transformations on linear connection spaces is connected with the names of E. Cartan, Eisenhart, Thomas, Knebelman, Schouten, Yano, Egorov, Vranceanu, Kobayashi, and others. It is known that in the spaces of constant curvature S, when considered in the small, the complete projective group coincides with the projective group of pseudo-Euclidean space, i.e., with the group of bilinear substitutions, and depends on n(n+ 2) parameters. In the spaces V n of nonconstant curvature, the order of the complete projective group does not exceed n(n−2)−5 (Egorov [91]), and, moreover, in the majority of cases, this group consists of similarity transformations (homotheties) or isometries. In 1903, in “Turin Academy Notes,” Fubini’s paper “Groups of geodesic transformations” was published [262], which as mentioned above, laid the foundations for a systematic definition and study of Riemannian spaces admitting infinitesimal projective transformations. Later on, Solodovnikov [156–158] continued Fubini’s research and completely solved the problem posed; the works of Fubini and Solodovnikov contain a classification of the Riemannian spaces V , n ≥ 3, in terms of (local) groups of projective transformations which are larger than homothety groups. The conclusions of Fubini and Solodovnikov are based on the assumption that the metrics considered are positive definite. Taking a given signature as a condition considerably complicates the problem and requires a basically new approach for its solution. It was proposed in the papers of the author [5– 11], where the problem of defining all pseudo-Riemannian manifolds with Lorentz signature (+ − . . .−) (Lorentz manifolds) of dimension n ≥ 3 admitting nonhomothetic infinitesimal projective and affine transformations was solved, and for each of them, the maximal projective and affine Lie algebras, together with the homothetic and isometric subalgebras, were defined. This paper includes a survey of the results in the theory of projective transformations of pseudoRiemannian manifolds, in particular, the solution of the classical geometrical problem of determining the pseudo-Riemannian metrics with corresponding geodesics (Sec. 5) and the Lie problem (Sec. 6). By using the technique of skew-normal frames developed by the author in [31] (see Sec. 1), all two-dimensional

On certain classes of exact solutions of Einstein equations for rotating fields in conventional and nonconventional form

Using the symmetry reduction approach we have herein examined, under continuous groups of transformations, the invariance of Einstein exterior equations for stationary axisymmetric and rotating case, in conventional and nonconventional forms, that is a coupled system of nonlinear partial differential equations of second order. More specifically, the said technique yields the invariant transformation that reduces the given system of partial differential equations to a system of nonlinear ordinary differential equations (nlodes) which, in the case of conventional form, is reduced to a single nlode of second order. The first integral of the resulting nlode has been obtained via invariant-variational principle and Noether’s theorem and involves an integration constant. Depending upon the choice of the arbitrary constant two different forms of the exact solutions are indicated. The generalized forms of Weyl and Schwarzschild solutions for the case of no spin have also been deduced as particular cases. Investigation of nonconventional form of Einstein exterior equations has not only led to the recovery of solutions obtained through conventional form but it also yields physically important asymptotically flat solutions. In a particular case, a single third order nlode has been derived which evidently opens up the possibility of finding many further interesting solutions of the exterior field equations.

Group properties and group-invariant solutions for infinitesimal transformations of the non-linearly traveling string

The purpose of this paper is to focus on the asymptotic stability and the construction of group-invariant solutions for a second-order hyperbolic wave equation involving an axially transported speed and the non-linear terms. We first characterize the most general expression for the infinitesimal generator of the continuous transformation groups. The Lie groups admitted by the system are found and exploited to describe general properties of families of solutions. Then, utilizing the invariant surface of the transformation groups, the governing equation is written as to obtain the corresponding ordinary differential equations whose integrations allow us further discussing stability and convergence of group invariant solutions.

Neutrinos in a spherical de sitter universe: Exclusive properties of isotropic geodesics

The behavior of neutrinos as particles with spin 1/2 and with a strictly zero rest mass is studied. The qualitative differences between the case in which a particle's mass is small and that in which it is strictly zero are considered. In the former, a spin-1/2 particle must be described by a spinor, while in the latter, it can be described by a semispinor. These differences are manifested not only at the quantum but also at the classical level. It is essential that the case of a particle with a zero rest mass must be treated as strictly relativistic at any level. The exclusive properties of isotropic geodesic lines are considered. The chief one is their inherent conformal invariance. Because of it, the conformal angular momentum of a particle with zero rest mass is conserved in a Riemann universe that admits of a continuous group of conformal transformations. This is a 15-parameter group in a spherical de Sitter universe. Accordingly, 15 linearly independent, conformal momenta are conserved in it. The exclusive properties of isotropic geodesics in a spherical de Sitter universe are also manifested in the fact that they are straight in a five-dimensional, pseudo-Euclidean space enveloping that universe.

Lie generators for computing steerable functions

We present a computational, group-theoretic approach to steerable functions. The approach is group-theoretic in that the treatment involves continuous transformation groups for which elementary Lie group theory may be applied. The approach is computational in that the theory is constructive and leads directly to a procedural implementation. For functions that are steerable with n basis functions under a k-parameter group, the procedure is efficient in that at most nk+1 iterations of the procedure are needed to compute all the basis functions. Furthermore, the procedure is guaranteed to return the minimum number of basis functions. If the function is not steerable, a numerical implementation of the procedure could be used to compute basis functions that approximately steer the function over a range of transformation parameters. Examples of both applications are described.

On the electrodynamic group: The relativistic radiation reaction force

The relativistic force exerted by a classical electromagnetic field on an electron is examined from the point of view that associates the differential geometry of the space–time trajectory with a continuous group of transformations. The group relevant to this particular case is the group generated by the electromagnetic field tensor (electrodynamic group). This group is studied in some detail by treating separately the two parts of the Liénard–Wiechert expression for the tensor. The intrinsic differential equations of the space–time trajectories associated with the electrodynamic group are obtained by the Cartan method of the moving frame. These equations show that the curvature of the trajectory is associated only with the generalized Coulomb field and the hypertorsion with the radiation field. According to these equations, the relativistic force may have two components along orthogonal spacelike directions: The first, an apparent force, reduces to the Coulomb force in the nonrelativistic limit. The other is orthogonal to the four-momentum and to the apparent force; this is the radiation reaction force. In the nonrelativistic limit it manifests itself as a frictional force reducing the momentum. The approach also offers a point of view of the conventional treatment of radiation reaction in which the nonlocality in time emerges from the geometry of the problem. The implications pertaining to other areas of physics of the causal connection found between the hypertorsion and the radiation field are briefly considered.

Application of the Group-Theoretical Method to Physical Problems

The concept of the theory of continuous groups of transformations has attracted the attention of applied mathematicians and engineers to solve many physical problems in the engineering sciences. Three applications are presented in this paper. The first one is the problem of time-dependent vertical temperature distribution in a stagnant lake. Two cases have been considered for the forms of the water parameters, namely water density and thermal conductivity. The second application is the unsteady freeconvective boundary-layer flow on a non-isothermal vertical flat plate. The third application is the study of the dispersion of gaseous pollutants in the presence of a temperature inversion. The results are found in closed form and the effect of parameters are discussed.

Conservation laws in nonhomogeneous plane elastostatics

Conservation laws in nonhomogeneous elastostatics are studied by means of a special version of Noether's theorem on invariant variational problems. The investigation is restricted to isotropic, linearly elastic bodies with smoothly varying elastic moduli and subjected to plane strain deformation fields. By applying Lie's infinitesimal criterion on the invariance of the action integral under a continuous transformation group, it is found that the materials admitting conservation laws are those for which the elastic parameters satisfy a first-order linear partial differential equation. While the general solution to this equation is readily available, a special emphasis is placed on extending Rice's J-integral to certain classes of materials with varying Young's modulus in the direction of a crack line. The extended integrals are related to stress intensity factors at the crack tip, and their use is illustrated by applying them to two simple examples. Also, the circumstances under which these integrals can be generalized to the three-dimensional case are briefly discussed.

FINITE DIFFERENCE MODELS ENTIRELY INHERITING CONTINUOUS SYMMETRY OF ORIGINAL DIFFERENTIAL EQUATIONS

The present paper is concerned with continuous groups of transformations in a space of discrete variables. The criterion of invariance of difference equations together with difference grid is discussed. One simple method to construct finite-difference equations and grids entirely inheriting admitted Lie groups of transformations of initial differential models is developed.

DYNAMICAL INTEGRATION

In this letter we show how to use a new form of integration, called dynamical integration, that utilizes the dynamics of a system defined by an ODE to construct a map that is in effect a one-step integrator. This method contrasts sharply with classical numerical methods that utilize polynomial or rational function approximations to construct integrators. The advantages of this integrator is that it uses only one step while preserving important dynamical properties of the solution of the ODE: First, if the ODE is conservative, then the one-step integrator is measure preserving. This is significant for a system having a highly nonlinear component. Second, the one-step integrator is actually a one-parameter family of one-step maps and is derived from a continuous transformation group as is the set of solutions of the ODE. If each element of the continuous transformation group of the ODE is topologically conjugate to its inverse, then so is each member of the one-parameter family of one-step integrators. If the solutions of the ODE are elliptic, then for sufficiently small values of the parameter, the one-step integrator is also elliptic. In the limit as the parameter of the one-step family of maps goes to zero, the one-step integrator satisfies the ODE exactly. Further, it can be experimentally verified that if the ODE is chaotic, then so is the one-step integrator. In effect, the one-step integrator retains the dynamical characteristics of the solutions of the ODE, even with relatively large step sizes, while in the limit as the parameter goes to zero, it solves the ODE exactly. We illustrate the dynamical, in contrast to numerical, accuracy of this integrator with two distinctly different examples: First we use it to integrate the unforced Van der Pol equation for large ∊, ∊≥10 which corresponds to an almost continuous square-wave solution. Second, we use it to obtain the Poincaré map for two different versions of the periodically forced Duffing equation for parameter values where the solutions are chaotic. The dynamical accuracy of the integrator is illustrated by the reproduction of well-known strange attractors. The production of these attractors is eleven times longer when using a conventional fourth-order predictor-corrector method. The theory presented here extends to higher dimensions and will be discussed in detail in a forthcoming paper. However, we caution that the theory we present here is not intended as a line of research in numerical methods for ODEs.

Some generating functions of modified Jacobi polynomials

The present note deals with the derivation of novel estensions of some bilateral and mixed trilateral generating functions of modified Jacobi polynomials Pn ^a ’ (x) from the stand point of one parameter group of continuous transformations. Some particular cases of interest as wel as the applicatîons of our results are laid down.

Variational characterization of topological entropy of continuous transformation groups. Case of actions of Rn

Variational characterization of topological entropy of continuous transformation groups. Case of actions of Rn

Continuous transformation groups on spaces

A differentiable group is a group in the category of (reduced and nonreduced) differentiable spaces. Special cases are the rationals ℚ, Lie groups, formal groups over ℝ or ℂ; in general there is some mixture of those types, the general structure, however,

Lie group for the Jordan and Brans-Dicke cosmology

A method based on the invariace under a continuous Lie group of transformations is worked out to reduce the problem of finding solutions to the cosmological equations of Jordan and Brans-Dicke theory of gravitation for the Robertson-Walker metrics and the cases of the dust universe and the vacuum universe. The reduction consists in a first-order differential equation and a quadrature for each case. Previously known cosmological solutions are re-obtained. In particular, it becomes apparent during the development of this scheme that the flat-space solutions are indeed the general solution.

Discontinuous solutions of nonlinear wave phenomena by similarity approach

One-parameter continuous groups of transformations are applied to reduce a system of conservation laws with discontinuous initial conditions to a nonlinear ordinary differential equation with corresponding initial conditions and the associated Rankine-Hugoniot relations. Solutions of the similarity representation so obtained are obtained both for a linear and a nonlinear case. The similarity technique is further applied to the analysis of the Riemann problem for a nonlinear elastic rod.

Group properties of the non-linear dynamic equations of elastic strings

Using the machinery of Lie group analysis, the non-linear system of partial differential equations governing the plane motion of an extensible string or cable which has undergone a large deformation is studied. The continuous transformation groups admitted by the system are found and used to describe general properties of families of solutions. Using the invariants of the transformation groups, the system is rewritten in various ways as a system of ordinary differential equations. This uncovers other interesting aspects of the system and allows it to be more easily handled computationally.

Regularization of actions of groups and groupoids on measured equivalence relations

The paper deals with the regularization problem for locally compact groups of non-strict automorphisms of measured equivalence relations. It is shown that by means of inessential reorganization of the equivalence relation and the group action one can make all the automorphisms of the given action to be strict with respect to the equivalence relation. A similar problem is solved for an action of a measure groupoid which leaves invariant mod 0 a measured equivalence relation. 1. The study of full groups introduced by H. A. Dye [5] as well as a consideration of outer conjugacy of subgroups of the full group normalizer (see the paper by A. Connes and W. Krieger [4] and the subsequent works [1], [2], [3], [8]) forces one to deal with maps and families of transformatio ns that behave properly only almost everywhere. This as a rule does not cause any difficulties when the transformation groups in question are nonsingular and countable. One needs only to discard a null set in order to make the action of such a group regular. However, in the case of continuous transformation groups this method of regularizatio n is inappropriate in general. The first step in studying the regularization of actions of continuous groups was made by G. Mackey in [10], where the existence and uniqueness of a point realization for an action of a locally compact group G as automorphisms of a Boolean algebra were established. In order to form the point realization of a Boolean G-space, the author applies the properties of a universal G-space on which the G-action is regular. This approach got its complete basis later in works of A. Ramsay [12], [14]. Similar problems were considered by A. M. Vershik [15] with another technique used. The present paper contains a solution of the regularization problem for groups of non-strict automorphisms of measured equivalence relations based on results of [12], [14] (§2). It is shown that by means of inessential reorganization of the equivalence relation and the group