Entropy Computation on the Unit Disc of a Meromorphic Map

We propose a new definition of entropy based on both topological and metric entropy for the meromorphic maps. The entropy is then computed on the unit disc of a meromorphic map, which is called the extended Blaschke function, and is a nonlinear extension of the normalized Lorentz transformation. We nd that the de ned entropy is computable and observe several interested results, such as maximal entropy, entropy overshoot due to topological transition, entropy reduction to zero, and scaling invariance in conjunction with parameter space.     

Minimum entropy control of chaos via online particle swarm optimization method

One of the recently developed approaches for control of chaos is the minimum entropy (ME) control technique. In this method an entropy function based on the Shannon definition, is defined for a chaotic system. The control action is designed such that the entropy as a cost function is minimized which results in more regular pattern of motion for the system trajectories. In this paper an online optimization technique using particle swarm optimization (PSO) method is developed to calculate the control action based on ME strategy. The method is examined on some standard chaotic maps with error feedback and delayed feedback forms. Considering the fact that the optimization is online, simulation results show very good effectiveness of the presented technique in controlling chaos.     

Multi-dimensional scalar balance laws with discontinuous flux

We consider scalar balance laws with a dissipative source term. The flux function may be discontinuous with respect to both the space variable x and the unknown quantity u. We formulate the definition of entropy weak solutions and provide existence and uniqueness to the considered problem. The problem is formulated in the framework of multi-valued mappings. The notion of entropy measure-valued solutions is used to prove the so-called contraction principle and comparison principle.     

Quadratic entropy of uncertain sets

Uncertain set is a set-valued function on an uncertainty space, and attempts to model unsharp concepts. Firstly, a definition of quadratic entropy to characterize the uncertainty of uncertain sets resulting from information deficiency is proposed. Secondly, some properties of quadratic entropy for uncertain sets are given, and the relation between quadratic entropy and Liu’s entropy of uncertain sets is discussed. Finally, a quadratic cross entropy for uncertain sets is investigated.     

Topological entropy of maps on the real line

The aim of this paper is to introduce a definition of topological entropy for continuous maps such that, at least for continuous real maps, it keeps the following general philosophy: positive topological entropy implies that the map has a complicated dynamical behaviour. Besides, we pursue that our definition keeps some properties which are hold by the classic definition of topological entropy introduced for compact sets.     

Some remarks on conditional entropy

Summary Using a definition of conditional entropy given by Hanen and Neveu [5, 10, 11] we discuss in this paper some properties of conditional entropy and mean entropy, in particular an integral representation of conditional entropy (§ 2), and the decomposition theorem of the KolmogorovSina¯i invariant (§ 3) (see also [6–8] and [12]).There is an essential difference between Jacob's proof of the last called theorem [6] and the proof given below. The definition of a Lebesgue space, given by Rokhlin [7–9] and [12], is not used in this paper.     

Entropy solutions and flux identification of a scalar conservation law modelling centrifugal sedimentation

Centrifugal sedimentation of an ideal suspension in a rotating tube or basket can be modelled by an initial-boundary-value problem for a scalar conservation law with a nonconvex flux function. The sought unknown is the volume fraction of solids as function of radial distance and time for constant initial data. The method of characteristics is used to construct entropy solutions. The qualitatively different solutions, which depend on the initial value and the vessel radial coordinates, are presented in detail along with numerical simulations. Based on the entropy solutions, a new method of flux identification, which does not require any prescribed functional expression, is presented and illustrated with synthetic data.     

Loss Coefficients in Laminar Flows: Essential for the Design of Micro Flow Systems

The concept of head loss coefficients K for the determination of losses in conduit components is discussed in detail. While so far it has mainly been applied to fully turbulent flows it is extended here to also cover the laminar flow regime which is relevant for micro systems due to the low Reynolds numbers of these flows. Specific numbers of K can be determined by integration of the entropy generation field (second law analysis) obtained from a numerical simulation. It will be shown that a definition of K based on entropy generation is superior to a widely used definition that refers to a pressure drop caused by the conduit component. With the second law analysis details of the physics become available. For example it can be shown that often the main part of the entropy generation occurs downstream of the component. This aspect becomes important when several conduit components are combined in close proximity, like two 90 degree bends that are close to each other. Often in such situations the combination as a whole has to be looked upon as one new complex component. The general approach is discussed and illustrated for various conduit components and combinations of them. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

二维模糊向量可信性分布的联合熵及其性质

基于可信性测度, 定义了二维离散型模糊向量和二维连续型模糊向量的联合熵, 并研究了它们的极小性、极大性、单调性, 以及二维简单模糊向量的扩展性和严格凹性.     

A stochastic probing algorithm for global optimization

Recently, simulated annealing methods have proven to be a valuable tool for global optimization. We propose a new stochastic method for locating the global optimum of a function. The proposed method begins with the subjective specification of a probing distribution. The objective function is evaluated at a few points sampled from this distribution, which is then updated using the collected information. The updating mechanism is based on the entropy of a move selecting distribution and is loosely connected to some notions in statistical thermodynamics. Examples of the use of the proposed method are presented. These indicate its superior performance as compared with simulated annealing. Preliminary considerations in applying the method to discrete problems are discussed.     

Fuzzy Entropy： Axiomatic Definition and Neural Networks Model

The measure of uncertainty is adopted as a measure of information. The measures of fuzziness are known as fuzzy information measures. The measure of a quantity of fuzzy information gained from a fuzzy set or fuzzy system is known as fuzzy entropy. Fuzzy entropy has been focused and studied by many researchers in various fields. In this paper, firstly, the axiomatic definition of fuzzy entropy is discussed. Then, neural networks model of fuzzy entropy is proposed, based on the computing capability of neural networks. In the end, two examples are discussed to show the efficiency of the model.     

CONVERGENCE OF THE LAX-FRIEDRICHS SCHEME AND STABILITY FOR CONSERVATION LAWS WITH A DISCONTINUOUS SPACE-TIME DEPENDENT FLUX

The authors give the first convergence proof for the Lax-Friedrichs finite differencescheme for non-convex genuinely nonlinear scalar conservation laws of the formu_t f(k(x, t), u)_x = 0,where the coefficient k(x, t) is allowed to be discontinuous along curves in the (x, t)plane. In contrast to most of the existing literature on problems with discontinuouscoefficients, here the convergence proof is not based on the singular mapping approach,but rather on the div-curl lemma (but not the Young measure) and a Lax type en-tropy estimate that is robust with respect to the regularity of k(x, t). Following [14],the authors propose a definition of entropy solution that extends the classical Kruzkovdefinition to the situation where k(x, t) is piecewise Lipschitz continuous in the (x, t)plane, and prove the stability (uniqueness) of such entropy solutions, provided that theflux function satisfies a so-called crossng condition, and that strong traces of the solu-tion exist along the curves where k(x, t) is disco     

On the fuzzy minimum entropy control to stabilize the unstable fixed points of chaotic maps

This paper presents a fuzzy algorithm for controlling chaos in nonlinear systems via minimum entropy approach. The proposed fuzzy logic algorithm is used to minimize the Shannon entropy of a chaotic dynamics. The fuzzy laws are determined in such a way that the entropy function descends until the chaotic trajectory of the system is replaced by a regular one. The Logistic and the Henon maps as two discrete chaotic systems, and the Duffing equation as a continuous one are used to validate the proposed scheme and show the effectiveness of the control method in chaotic dynamical systems.     

On Data-Driven Computation of Information Transfer for Causal Inference in Discrete-Time Dynamical Systems

In this paper, we provide a novel approach to capture causal interaction in a dynamical system from time series data. In Sinha and Vaidya (in: IEEE conference on decision and control, pp 7329–7334, 2016), we have shown that the existing measures of information transfer, namely directed information, Granger causality and transfer entropy, fail to capture the causal interaction in a dynamical system and proposed a new definition of information transfer that captures direct causal interactions. The novelty of the information transfer definition used in this paper is the fact that it can differentiate between direct and indirect influences Sinha and Vaidya (2016). The main contribution of this paper is to show that the proposed definition of information transfers in Sinha and Vaidya (2016) and Sinha and Vaidya (in: Indian control conference, pp 303–308, 2017) can be computed from time series data, and thus, the direct influences in a dynamical system can be identified from time series data. We use transfer operator theoretic framework, involving Perron–Frobenius and Koopman operators for the data-driven approximation of the system dynamics and computation of information transfer. Several examples, involving linear and nonlinear system dynamics, are presented to verify the efficiency of the developed algorithm.

     

Entropy boundary and jump conditions in the theory of sedimentation with compression

We present an initial–boundary value problem of a quasilinear degenerate parabolic equation for the settling and consolidation of a flocculated suspension. The corresponding definition of generalized solutions is formulated. It is based on an entropy integral inequality in the sense of Kružkov. From this definition, jump and entropy conditions that have to be satisfied at discontinuities, and an entropy condition valid on one boundary of the computational domain are derived. The latter implies a set-valued reformulation of the original boundary condition. It is interpreted geometrically and characterized by the solution of an auxiliary hyperbolic Riemann problem. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons, Ltd.     

An approach to constructing models of multiphase elastic porous media

A novel approach to describing the behaviour of multiphase elastic porous media is proposed. The average values of the physical quantities needed to describe the motions of porous media are formulated using an integral relation. The validity of this relation is taken as the fundamental hypothesis. The integral definition of the average values enables integral relations to be devised for the average values from the integral laws of conservation of mass, momentum and energy and the increase in entropy. Along with the average values, the integral relations contain new variables that can be identified with generalized thermodynamic forces, which can be used to take into account the phase interaction in a porous medium. The integral relations are used to derive differential equations for the rate of entropy change and Gibbs relations for a porous medium as a basis for obtaining the constitutive relations. Relationships between the thermomechanical parameters of the model are established from the Gibbs relations under additional assumptions. The equation for the rate of entropy change can be used to establish relations between the generalized thermodynamic forces and fluxes. A complete system of differential equations in the defining parameters, which describes the motion of multiphase elastic porous media, is finally obtained.     

广义概率对偶犹豫模糊语言幂集结算子在多属性决策中的应用

在对偶犹豫模糊语言集、概率对偶犹豫模糊集和广义幂集结算子的基础上,研究了在概率对偶犹豫模糊语言环境下的广义幂集结算子问题。首先,给出了概率对偶犹豫模糊语言集的定义、运算规则、得分函数、精确函数、距离测度、熵。然后,定义了广义概率对偶犹豫模糊语言幂集结算子,并研究其具有的性质。其次,提出了一种决策方法来解决集结数据之间存在相互关系的概率对偶犹豫模糊语言多属性决策问题。最后,结合相关案例验证了该方法的有效性和可行性。     

Topological entropy for local processes

The aim of this article is to formalize definition of chaos (in terms of topological entropy) for dynamics of processes described by nonautonomous differential equations. We state a formal definition of topological entropy in this setting and provide tools for estimation of its value (its upper or lower bounds) in terms of Poincaré sections.     

Non-parametric estimation of the generalized past entropy function with censored dependent data

The generalized past entropy function introduced by Gupta and Nanda (2002) is viewed as a dynamic measure of uncertainty in past life. This measure finds applications in modeling past life time data. In the present work we provide non-parametric kernel-type estimator for the generalized past entropy function based on censored data. Asymptotic properties of the estimator are established under suitable regularity conditions. Simulation studies are carried out using the Monte Carlo method.     

Scalar conservation laws: Initial and boundary value problems revisited and saturated solutions

We revisit the classical theory of multidimensional scalar conservation laws. We reformulate the notion of the classical Kruzkov entropy solutions and study some new properties as well as the well-posedness of the initial value problem with inhomogeneous fluxes and general initial data. We also consider Dirichlet boundary value problems. We put forward a new and transparent definition for solutions and give a simple proof for their well-posedness in domains with smooth boundaries. Finally, we introduce the notion of saturated solutions and show that it is well-posed.