Lie symmetries of geodesic equations and projective collineations

We prove a theorem which relates the Lie symmetries of the geodesic equations in a Riemannian space with the collineations of the metric. We apply the results to Einstein spaces and spaces of constant curvature. Finally with examples we show the use of the results.     

Conductance and Noncommutative Dynamical Systems

In this work, we introduce the notion of conductance in the context of Cuntz–Krieger C^∗-algebras. These algebras can be seen as a noncommutative version of topological Markov chains. Conductance is a useful notion in the theory of Markov chains to study the approach of a system to the equilibrium state. Our goal is twofold. On one hand, conductance can be used to measure the complexity of dynamical systems, complementing topological entropy. On the other hand, using C^∗-algebras, we can give a natural framework to analyze the path space of a finite graph associated to a Markov shift.     

Entropy for Symbolic Dynamics with Overlapping Alphabets

We consider shift spaces in which elements of the alphabet may overlap nontransitively. We define a notion of entropy for such spaces and show that it is equal to a limit of entropies of standard (non-overlapping) shifts when the underlying shift is of finite type. When a shift space with overlaps arises as a model for a discrete dynamical system with a finite set of overlapping neighborhoods, the entropy gives a lower bound for the topological entropy of the dynamical system.     

Identifying ill-behaved nonlinear processes without metrics: use of symbolic dynamics

Given ill-behaved psychological data that are unlikely to satisfy metric axioms, the use of encoding in symbolic dynamics, and hence leading into Markov analyses, is explored. Various measures of entropy are calculated. The tractability of entropic measures for categorizing the trajectories of nonlinear dynamics that may be present and chaotic is considered, with a focus on the case where there are two attractors and at least one heteroclinic orbit between them. Fast/slow dynamics are treated as a special case. The problem of identification is in other contexts the problem of diagnosis in time-varying pathologies. Some real data, selected for their psychological relevance in clinical, forensic and psychophysical processes, that are apparently edge-of chaos and nonstationary, are for comparison analysed both as metric and discrete and in symbolic encoding.     

Dynamics on certain sets of stochastic matrices

We study iteration of polynomials on symmetric stochastic matrices. In particular, we focus on a certain one-parameter family of quadratic maps which exhibits chaotic behavior for a wide range of the parameters. The well-known dynamical behavior of the quadratic family on the interval, and its dependence on the parameter, is reproduced on the spectrum of the stochastic matrices. For certain subclasses of stochastic matrices the referred dynamical behavior is also obtained in the matrix entries. Since a stochastic matrix characterizes a Markov chain, we obtain a discrete dynamical system on the space of reversible Markov chains. Therefore, depending on the parameter, there are initial conditions for which the corresponding reversible Markov chains will lead under iteration to a fixed point, to a periodic point, or to an aperiodic point. Moreover, there are sensitivity to initial conditions and the coexistence of infinite repulsive periodic orbits, both features of chaos.     

Minimal Spaces with Cyclic Group of Homeomorphisms

There are two main subjects in this paper. (1) For a topological dynamical system \((X,T)\) we study the topological entropy of its “functional envelopes” (the action of \(T\) by left composition on the space of all continuous self-maps or on the space of all self-homeomorphisms of \(X\)). In particular we prove that for zero-dimensional spaces \(X\) both entropies are infinite except when \(T\) is equicontinuous (then both equal zero). (2) We call Slovak space any compact metric space whose homeomorphism group is cyclic and generated by a minimal homeomorphism. Using Slovak spaces we provide examples of (minimal) systems \((X,T)\) with positive entropy, yet, whose functional envelope on homeomorphisms has entropy zero (answering a question posed by Kolyada and Semikina). Finally, also using Slovak spaces, we resolve a long standing open problem whether the circle is a unique non-degenerate continuum admitting minimal continuous transformations but only invertible: No, some Slovak spaces are such, as well.     

Poisson limit theorem for countable Markov chains in Markovian environments

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Fokker–Planck Equations for a Free Energy Functional or Markov Process on a Graph

The classical Fokker–Planck equation is a linear parabolic equation which describes the time evolution of the probability distribution of a stochastic process defined on a Euclidean space. Corresponding to a stochastic process, there often exists a free energy functional which is defined on the space of probability distributions and is a linear combination of a potential and an entropy. In recent years, it has been shown that the Fokker–Planck equation is the gradient flow of the free energy functional defined on the Riemannian manifold of probability distributions whose inner product is generated by a 2-Wasserstein distance. In this paper, we consider analogous matters for a free energy functional or Markov process defined on a graph with a finite number of vertices and edges. If N ≧ 2 is the number of vertices of the graph, we show that the corresponding Fokker–Planck equation is a system of N nonlinear ordinary differential equations defined on a Riemannian manifold of probability distributions. However, in contrast to stochastic processes defined on Euclidean spaces, the situation is more subtle for discrete spaces. We have different choices for inner products on the space of probability distributions resulting in different Fokker–Planck equations for the same process. It is shown that there is a strong connection but there are also substantial discrepancies between the systems of ordinary differential equations and the classical Fokker–Planck equation on Euclidean spaces. Furthermore, both systems of ordinary differential equations are gradient flows for the same free energy functional defined on the Riemannian manifolds of probability distributions with different metrics. Some examples are also discussed.     

Morse Decomposition of Semiflows on Topological Spaces

This paper studies Morse decompositions of discrete and continuous-time semiflows on compact Hausdorff topological spaces. We extend two classical results which are well-known facts for flows on compact metric spaces: the characterization of the Morse decompositions through increasing sequences of attractors and the existence of Lyapunov functions.     

The embedded theorem for family of quasi metric spaces in Menger space and its application

In this paper the notion of embedding for family of quasi metric spaces in Menger spaces is introduced and its properties are investigated. A common fixed point theorem for sequence of continuous mappings in Menger spaces is proved. These mappings are assumed to satisfy some generalizations of the contraction condition. The proving technique herein seems to be new even for mappings in Menger spaces.     

Vorticity and curvature at a stream surface

If S is a stream surface in a flow, we show the relationship between the three components of the vorticity field on S and the curvatures of the streamlines (geodesic torsion, normal curvature and geodesic curvature).     

On Estimates of the Boltzmann Collision Operator with Cut-off

We present new estimates of the Boltzmann collision operator in weighted Lebesgue and Bessel potential spaces. The main focus is put on hard potentials under the assumption that the angular part of the collision kernel fulfills some weighted integrability condition. In addition, the proofs for some previously known -estimates have been considerably shortened and carried out by elementary methods. For a class of metric spaces, the collision integral is seen to be a continuous operator into the same space. Furthermore, we give a new pointwise lower bound as well as asymptotic estimates for the loss term without requiring that the entropy is finite.     

Global Classical Solutions for Partially Dissipative Hyperbolic System of Balance Laws

The basic existence theory of Kato and Majda enables us to obtain local-in-time classical solutions to generally quasilinear hyperbolic systems in the framework of Sobolev spaces (in x) with higher regularity. However, it remains a challenging open problem whether classical solutions still preserve well-posedness in the case of critical regularity. This paper is concerned with partially dissipative hyperbolic system of balance laws. Under the entropy dissipative assumption, we establish the local well-posedness and blow-up criterion of classical solutions in the framework of Besov spaces with critical regularity with the aid of the standard iteration argument and Friedrichs’ regularization method. Then we explore the theory of function spaces and develop an elementary fact that indicates the relation between homogeneous and inhomogeneous Chemin–Lerner spaces (mixed space-time Besov spaces). This fact allows us to capture the dissipation rates generated from the partial dissipative source term and further obtain the global well-posedness and stability by assuming at all times the Shizuta–Kawashima algebraic condition. As a direct application, the corresponding well-posedness and stability of classical solutions to the compressible Euler equations with damping are also obtained.     

Fixed points of nonexpansive mappings on star-shaped subsets of a convex metric space

In this paper, we give some characteristic properties of star-shaped sets which include a subset of a convex metric space. Using the characteristic properties, we discuss the existence problems of fixed points of nonexpansive type mappings on star-shaped subsets of convex metric spaces, which generalize the recent results obtained by Ding Xie-ping, Beg and Azam. Finally, we give an example which shows that our generalizations are essential.     

Generating and Adding Flows on Locally Complete Metric Spaces

As a generalization of a vector field on a manifold, the notion of an arc field on a locally complete metric space was introduced in Bleecker and Calcaterra (J Math Anal Appl, 248: 645–677, 2000). In that paper, the authors proved an analogue of the Cauchy–Lipschitz Theorem, i.e they showed the existence and uniqueness of solution curves for a time independent arc field. In this paper, we extend the result to the time dependent case, namely we show the existence and uniqueness of solution curves for a time dependent arc field. We also introduce the notion of the sum of two time dependent arc fields and show existence and uniqueness of solution curves for this sum.     

From wrinkling to global buckling of a ring on a curved substrate

We present a combined analytical approach and numerical study on the stability of a ring bound to an annular elastic substrate, which contains a circular cavity. The system is loaded by depressurizing the inner cavity. The ring is modeled as an Euler–Bernoulli beam and its equilibrium equations are derived from the mechanical energy which takes into account both stretching and bending contributions. The curvature of the substrate is considered explicitly to model the work done by its reaction force on the ring. We distinguish two different instabilities: periodic wrinkling of the ring or global buckling of the structure. Our model provides an expression for the critical pressure, as well as a phase diagram that rationalizes the transition between instability modes. Towards assessing the role of curvature, we compare our results for the critical stress and the wrinkling wavelength to their planar counterparts. We show that the critical stress is insensitive to the curvature of the substrate, while the wavelength is only affected due to the permissible discrete values of the azimuthal wavenumber imposed by the geometry of the problem. Throughout, we contrast our analytical predictions against finite element simulations.     

Chain recurrence,semiflows, and gradients

This paper is a study of chain recurrence and attractors for maps and semiflows on arbitrary metric spaces. The main results are as follows. (i) C. Conley's characterization of chain recurrence in terms of attractors holds for maps and semiflows on any metric space. (ii) An alternative definition of chain recurrence for semiflows is given and is shown to be equivalent to the usual definition. The alternative definition uses chains formed of orbit segments whose lengths are at least 1, while in the usual definition these lengths are required to be arbitrarily long. (iii) The chain recurrent set of a continuous semiflow is the same as the chain recurrent set of its time-one map. (iv) Conditions on a real-valued function are given that ensure that the semiflow generated by its gradient has only equilibria in its chain recurrent set. An example is given (onR ³) showing that a gradient flow may have nonequilibrium chain recurrent points if these conditions are violated.     

BASIC THEORY AND APPLICATIONS OF PROBABILISTIC METRIC SPACES (Ⅰ)

This paper is devoted to the study of the basic theory and applications of probabilistic metric spaces (PM-space). In this paper the topological structure and properties for PM-space are considered. The conditions of metrization and the form of metric functions for PM-spaces. Menger PM-spaces and probabilistic normed linear spaces (PN-space) are given and the characterizations of various probabilistically bounded sets are presented. As applications we utilize these results obtained in this paper to study the linear operator theory and fixed point theory on PM-spaces. Projects supported by the Science Fund of the Chinese Academy of Sciences.     

A measure of complexity based on the order patterns

Cumulative Tsallis entropy (CE) is a recently introduced entropy metric to quantify the uncertainty of time series, and its expressions of continuous random variable and discrete random variable are consistents. So far, it has proved to have a good performance in the characteristics of time series. This paper presents a new method to measure the complexity and similarity of systems—cumulative Tsallis entropy based on the dispersion entropy (DCE). It is different from the traditional PE method to simply symbolize the sequence. Instead, the complexity of the system is characterized by focusing on the amplitude information of the time series and considering the influence of past events. We applied DCE to two kinds of simulation data and six global financial time series. The results show that DCE can be used as a diagnostic model to classify global financial data according to regional characteristics, financial background and government policies. In addition, as a classical method of non-stationary time series, we combine the MSE method with DCE to observe the financial market from different time scales and obtain rich intrinsic properties.

     

Projection-algebraic approximation of linear and nonlinear operator differential equations in Banach spaces

We develop a Calogero-type projection-algebraic method of discrete approximations for linear differential equations in Banach spaces and analyze the convergence of finite-dimensional approximations based on the functional-analytic approach to discrete approximations and methods of operator theory in Banach spaces. Applications of the obtained results to the functional-interpolation scheme of the projection-algebraic method of discrete approximations are considered. Based on a generalized Leray–Schauder-type theorem, we consider the projection-algebraic scheme of discrete approximations and analyze its solvability and convergence for a special class of nonlinear operator equations.