Characterization of Geodesic Flows on T 2 with and without Positive Topological Entropy

In the present work we consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus T ² for an arbitrary Riemannian metric. A natural non-negative quantity which measures the complexity of the geodesic flow is the topological entropy. In particular, positive topological entropy implies chaotic behavior on an invariant set in the phase space of positive Hausdorff-dimension (horseshoe). We show that in the case of zero topological entropy the flow has properties similar to integrable systems. In particular, there exists a non-trivial continuous constant of motion which measures the direction of geodesics lifted onto the universal covering \mathbbR²{\mathbb{R}^{2}} . Furthermore, those geodesics travel in strips bounded by Euclidean lines. Moreover, we derive necessary and sufficient conditions for vanishing topological entropy involving intersection properties of single geodesics on T ².     

Multiscale measures of equilibrium on finite dynamic systems

This article presents a new method for the study of the evolution of dynamic systems based on the notion of quantity of information. The system is divided into elementary cells and the quantity of information is studied with respect to the cell size. We have introduced an analogy between quantity of information and entropy, and defined the intrinsic entropy as the entropy of the whole system independent of the size of the cells. It is shown that the intrinsic entropy follows a Gaussian probability density function (PDF) and thereafter, the time needed by the system to reach equilibrium is a random variable. For a finite system, statistical analyses show that this entropy converges to a state of equilibrium and an algorithmic method is proposed to quantify the time needed to reach equilibrium for a given confidence interval level. A Monte-Carlo simulation of diffusion of A* atoms in A is then provided to illustrate the proposed simulation. It follows that the time to reach equilibrium for a constant error probability, t_e, depends on the number, n, of elementary cells as: t_e∝n^2.22_±0.06. For an infinite system size (n infinite), the intrinsic entropy obtained by statistical modelling is a pertinent characteristic number of the system at the equilibrium.     

Mean topological dimension

In this paper we present some results and applications of a new invariant for dynamical systems that can be viewed as a dynamical analogue of topological dimension. This invariant has been introduced by M. Gromov, and enables one to assign a meaningful quantity to dynamical systems of infinite topological dimension and entropy. We also develop an alternative approach that is metric dependent and is intimately related to topological entropy.     

Some results on information properties of coherent systems

This paper considers information properties of coherent systems when component lifetimes are independent and identically distributed. Some results on the entropy of coherent systems in terms of ordering properties of component distributions are proposed. Moreover, various sufficient conditions are given under which the entropy order among systems as well as the corresponding dual systems hold. Specifically, it is proved that under some conditions, the entropy order among component lifetimes is preserved under coherent system formations. The findings are based on system signatures as a useful measure from comparison purposes. Furthermore, some results on the system's entropy are derived when lifetimes of components are dependent and identically distributed. Several illustrative examples are also given.     

Information and dynamical systems: a concrete measurement on sporadic dynamics

We present a method for the study of dynamical systems based on the notion of quantity of information. Measuring the quantity of information of a string by using data compression algorithms, it is possible to give a notion of orbit complexity of dynamical systems. In compact ergodic dynamical systems, entropy is almost everywhere equal to orbit complexity. We have introduced a new compression algorithm called CASToRe which allows a direct estimation of the information content of the orbits in the 0-entropy case. The method is applied to a sporadic dynamical system (Manneville map).     

Entropy transmission in C1-dynamical systems

Three entropies of a state in C¹-dynamical systems are introduced and their relations and dynamical properties are studied. The entropy (information) transmission under a channel between two dynamical systems is considered. We find a condition under which our entropy becomes a dynamical invariant between two systems.     

Topological entropy of continuous functions on topological spaces

Adler, Konheim and McAndrew introduced the concept of topological entropy of a continuous mapping for compact dynamical systems. Bowen generalized the concept to non-compact metric spaces, but Walters indicated that Bowen’s entropy is metric-dependent. We propose a new definition of topological entropy for continuous mappings on arbitrary topological spaces (compactness, metrizability, even axioms of separation not necessarily required), investigate fundamental properties of the new entropy, and compare the new entropy with the existing ones. The defined entropy generates that of Adler, Konheim and McAndrew and is metric-independent for metrizable spaces. Yet, it holds various basic properties of Adler, Konheim and McAndrew’s entropy, e.g., the entropy of a subsystem is bounded by that of the original system, topologically conjugated systems have a same entropy, the entropy of the induced hyperspace system is larger than or equal to that of the original system, and in particular this new entropy coincides with Adler, Konheim and McAndrew’s entropy for compact systems.     

Minimal self-joinings and positive topological entropy

We show that the properties of almost minimal self-joinings and strong almost minimal self-joinings, introduced by del Junco in Topological Dynamics, are compatible with positive topological entropy, as opposed to the stronger property of minimal self-joinings. This is done both by proving existence theorems and by explicitly constructing some symbolic systems having these properties, which are modifications of the Chacón system. It is shown furthermore that these systems have no non-trivial factors with completely positive topological entropy.     

Relative entropy between quantum ensembles

Relative entropy between two quantum states, which quantifies to what extent the quantum states can be distinguished via whatever methods allowed by quantum mechanics, is a central and fundamental quantity in quantum information theory. However, in both theoretical analysis (such as selective measurements) and practical situations (such as random experiments), one is often encountered with quantum ensembles, which are families of quantum states with certain prior probability distributions. How can we quantify the quantumness and distinguishability of quantum ensembles? In this paper, by use of a probabilistic coupling technique, we propose a notion of relative entropy between quantum ensembles, which is a natural generalization of the relative entropy between quantum states. This generalization enjoys most of the basic and important properties of the original relative entropy. As an application, we use the notion of relative entropy between quantum ensembles to define a measure for quantumness of quantum ensembles. This quantity may be useful in quantum cryptography since in certain circumstances it is desirable to encode messages in quantum ensembles which are the most quantum, thus the most sensitive to eavesdropping. By use of this measure of quantumness, we demonstrate that a set consisting of two pure states is the most quantum when the states are 45° apart.     

Entropic order quantity (EnOQ) model for deteriorating items

Jaber et al. [M.Y. Jaber, R.Y. Nuwayhid, M.A. Rosen, Price-driven economic order systems from a thermodynamic point of view, Int. J. Prod. Res. 42 (24) (2004) 5167–5184] suggested that it might be possible to improve production systems performance by applying the first and second laws of thermodynamics to reduce system entropy (or disorder). They then used these laws to modify the economic order quantity (EOQ) model to derive an equivalent entropic order quantity (EnOQ). The results suggested that larger quantities should be ordered than is suggested by the classical EOQ model.     

Construction of Approximate Entropy Measure-Valued Solutions for Hyperbolic Systems of Conservation Laws

Entropy solutions have been widely accepted as the suitable solution framework for systems of conservation laws in several space dimensions. However, recent results in De Lellis and Székelyhidi Jr (Ann Math 170(3):1417–1436, 2009) and Chiodaroli et al. (2013) have demonstrated that entropy solutions may not be unique. In this paper, we present numerical evidence that state-of-the-art numerical schemes need not converge to an entropy solution of systems of conservation laws as the mesh is refined. Combining these two facts, we argue that entropy solutions may not be suitable as a solution framework for systems of conservation laws, particularly in several space dimensions. We advocate entropy measure-valued solutions, first proposed by DiPerna, as the appropriate solution paradigm for systems of conservation laws. To this end, we present a detailed numerical procedure which constructs stable approximations to entropy measure-valued solutions, and provide sufficient conditions that guarantee that these approximations converge to an entropy measure-valued solution as the mesh is refined, thus providing a viable numerical framework for systems of conservation laws in several space dimensions. A large number of numerical experiments that illustrate the proposed paradigm are presented and are utilized to examine several interesting properties of the computed entropy measure-valued solutions.     

Entropy flux-splittings for hyperbolic conservation laws part I: General framework

A general framework is proposed for the derivation and analysis of flux-splittings and the corresponding flux-splitting schemes for systems of conservation laws endowed with a strictly convex entropy. The approach leads to several new properties of the existing flux-splittings and to a method for the construction of entropy flux-splittings for general situations. A large family of genuine entropy flux-splittings is derived for several significant examples: the scalar conservation laws, the p-system, and the Euler system of isentropic gas dynamics. In particular, for the isentropic Euler system, we obtain a family of splittings that satisfy the entropy inequality associated with the mechanical energy. For this system, it is proved that there exists a unique genuine entropy flux-splitting that satisfies all of the entropy inequalities, which is also the unique diagonalizable splitting. This splitting can be also derived by the so-called kinetic formulation. Simple and useful difference schemes are derived from the flux-splittings for hyperbolic systems. Such entropy flux-splitting schemes are shown to satisfy a discrete cell entropy inequality. For the diagonalizable splitting schemes, an a priori L^∞ estimate is provided by applying the principle of bounded invariant regions. The convergence of entropy flux-splitting schemes is proved for the 2 × 2 systems of conservation laws and the isentropic Euler system. ©1995 John Wiley & Sons, Inc.     

Fine-grained and coarse-grained entropy in problems of statistical mechanics

We consider dynamical systems with a phase space Γ that preserve a measure μ. A partition of Γ into parts of finite μ-measure generates the coarse-grained entropy, a functional that is defined on the space of probability measures on Γ and generalizes the usual (ordinary or fine-grained) Gibbs entropy. We study the approximation properties of the coarse-grained entropy under refinement of the partition and also the properties of the coarse-grained entropy as a function of time. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 1, pp. 120–137, April, 2007.     

Coarse-grained entropy in dynamical systems

Let M be the phase space of a physical system. Consider the dynamics, determined by the invertible map T: M → M, preserving the measure µ on M. Let ν be another measure on M, dν = ρdµ. Gibbs introduced the quantity s(ρ) = ?∝ρ log ρdµ as an analog of the thermodynamical entropy. We consider a modification of the Gibbs (fine-grained) entropy the so called coarse-grained entropy. First we obtain a formula for the difference between the coarse-grained and Gibbs entropy. The main term of the difference is expressed by a functional usually referenced to as the Fisher information. Then we consider the behavior of the coarse-grained entropy as a function of time. The dynamics transforms ν in the following way: ν → ν _n , dν _n = ρ ○ T ^?n dµ. Hence, we obtain the sequence of densities ρ _n = ρ ○ T ^?n and the corresponding values of the Gibbs and the coarse-grained entropy. We show that while the Gibbs entropy remains constant, the coarse-grained entropy has a tendency to a growth and this growth is determined by dynamical properties of the map T. Finally, we give numerical calculation of the coarse-grained entropy as a function of time for systems with various dynamical properties: integrable, chaotic and with mixed dynamics and compare these calculation with theoretical statements.     

Some properties of entropy of order α and type β

In a recent paper, we defined entropy of order α and type β. For α = 1, it reduces to Renyi’s² entropy of order α, while for α = 1, β= 1 and for a complete probability distribution, it reduces to Shannon’s definition of entropy. In this paper we have studied some properties of this generalised entropy. We have also discussed the variability of Renyi’s entropy in the continuous case with co-ordinate systems and invariance of transinformation of order α and type β for linear transformations.     

Switching systems and entropy

Switching systems are non-autonomous dynamical systems obtained by switching between two or more autonomous dynamical systems as time goes on. They can be mainly found in control theory, physics, economy, biomathematics, chaotic cryptography and of course in the theory of dynamical systems, in both discrete and continuous time. Much of the recent interest in these systems is related to the emergence of new properties by the mechanism of switching, a phenomenon known in the literature as Parrondo's paradox. In this paper we consider a discrete-time switching system composed of two affine transformations and show that the switched dynamics has the same topological entropy as the switching sequence. The complexity of the switching sequence, as measured by the topological entropy, is fully transferred, for example, to the switched dynamics in this particular case.     

Local entropy theory for a countable discrete amenable group action

The local properties of entropy for a countable discrete amenable group action are studied. For such an action, a local variational principle for a given finite open cover is established, from which the variational relation between the topological and measure-theoretic entropy tuples is deduced. While doing this it is shown that two kinds of measure-theoretic entropy for finite Borel covers coincide. Moreover, two special classes of such an action: systems with uniformly positive entropy and completely positive entropy are investigated.     

同一水平下模糊规则信息熵的描述

给出了模糊知识系统及模糊决策逻辑公式的定义,在此基础上描述了模糊决策逻辑公式及模糊知识系统下模糊规则的信息熵,讨论了模糊规则信息熵的相关性质;其次,利用模糊规则信息熵对模糊规则进行了分类、评价,从而为建立合理的模糊系统提供了一种有效的判定方法;最后,通过实例验证了所提出理论的正确性.     

一个刚性守恒律方程组的全隐式差分方法

１．引言本文研究如下非线性刚性守恒律方程组的全隐式差分逼近． 方程（1．１）中的源项ｇ（ｕ,ｖ）定义为 ｇ（ｕ,ｖ）＝ｖ－（１－μ）ｆ（ｕ）,（１．２）其中ｆ是ｕ的一个给定函数,δ是一个小正参数,称为松弛时间,μ是参数．方程组（１．１）频繁出现于粘弹性力学中． 在零松弛时间限（δ→０）下,从（１．１）可得到如下方程组该方程组通常称为“平衡”模型,而方程组（１．１）称为“非平衡”模型． 文中将假设μ满足 ０＜ μ＜ １,（１．４）以便保证拟稳定性条件［１９,２０］和次特征条件［１１,２,３］： λ１≤λ＊…     

On a class of systems of quasilinear conservation laws

We consider hyperbolic conservation laws on matrix algebras. We describe entropies of such systems and study properties of generalized entropy solutions and strong generalized entropy solutions to the Cauchy problem. Bibliography: 15 titles. Translated from Problemy Matematicheskogo Analiza, No. 38, December 2008, pp. 73–91.