Physical entropy, information entropy and their evolution equations

Inspired by the evolution equation of nonequilibrium statistical physics entropy and the concise statistical formula of the entropy production rate, we develop a theory of the dynamic information entropy and build a nonlinear evolution equation of the information entropy density changing in time and state variable space. Its mathematical form and physical meaning are similar to the evolution equation of the physical entropy: The time rate of change of information entropy density originates together from drift, diffusion and production. The concise statistical formula of information entropy production rate is similar to that of physical entropy also. Furthermore, we study the similarity and difference between physical entropy and information entropy and the possible unification of the two statistical entropies, and discuss the relationship among the principle of entropy increase, the principle of equilibrium maximum entropy and the principle of maximum information entropy as well as the connection between them and the entropy evolution equation.     

双曲型守恒律方程的熵稳定格式的一些讨论

本文讨论双曲型守恒律方程的熵稳定格式.对于给定的熵对,格式所满足的熵条件中的数值熵通量是不唯一的.Tadmor的充分条件可以唯一地确定标量方程的熵守恒通量,但不能唯一确定方程组的熵守恒通量,却可以给出方程组的空间一阶精度的熵守恒格式.也讨论了在熵守恒通量上添加数值粘性得到的显式熵稳定格式需要满足的条件及常见的时间离散对熵守恒和熵稳定的影响.     

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.     

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.     

Nonsymmetric entropy and maximum nonsymmetric entropy principle

Under the frame of a statistical model, the concept of nonsymmetric entropy which generalizes the concepts of Boltzmann’s entropy and Shannon’s entropy, is defined. Maximum nonsymmetric entropy principle is proved. Some important distribution laws such as power law, can be derived from this principle naturally. Especially, nonsymmetric entropy is more convenient than other entropy such as Tsallis’s entropy in deriving power laws.     

不完全信息下损失风险的研究

在获得损失分布不完全信息情况下,提出用方差和熵共同度量损失风险的方法.在不完全信息条件下,通过最大熵原理在最不确定的情况下得到最大熵损失分布,并获得了损失分布的熵函数值.用熵值度量损失分布对于均匀分布的离散程度,从而度量概率波动带来的风险;用方差度量损失对于均值的离散程度,从而度量状态波动带来的风险.由于熵是与损失变量更高阶矩信息相联系的,所以新方法是从更全面的角度对损失风险的预测.通过算例,进一步看出在获得高阶矩信息下,熵参与风险度量的必要性.     

On the mean square of the remainder for the euclidean lattice point counting problem

We study an invariant of dynamical systems called naive entropy, which is defined for both measurable and topological actions of any countable group. We focus on nonamenable groups, in which case the invariant is two-valued, with every system having naive entropy either zero or infinity. Bowen has conjectured that when the acting group is sofic, zero naive entropy implies sofic entropy at most zero for both types of systems. We prove the topological version of this conjecture by showing that for every action of a sofic group by homeomorphisms of a compact metric space, zero naive entropy implies sofic entropy at most zero. This result and the simple definition of naive entropy allow us to show that the generic action of a free group on the Cantor set has sofic entropy at most zero. We observe that a distal Γ-system has zero naive entropy in both senses, if Γ has an element of infinite order. We also show that the naive entropy of a topological system is greater than or equal to the naive measure entropy of the same system with respect to any invariant measure.     

On fundamental equation of statistical physics (II)

Some derivations based on the anomalous Langevin equation in Liouville space (i.e. Γ space) or its equivalent Liouville diffusion equation of time-reversal asymmetry are presented. The time rate of change, the balance equation, the entropy flow, the entropy production and the law of entropy increase of Gibbs nonequilibriurn entropy and Boltzmann nonequilibrium entropy are rigorously derived and presented here. Furthermore, a nonlinear evolution equation of Gibbs nonequilibrium entropy density and Boltzmann nonequilibrium entropy density is first derived. The evolution equation shows that the change of nonequilibrium entropy density originates from not only drift, but also typical diffusion and inherent source production. Contrary to conventional knowledge, the entropy production density σ?0 everywhere for all the inbomogeneous systems far from equilibrium cannot be proved. Conversely, σ may be negative in some local space of such systems.     

Entropy for semigroup actions on general topological spaces

This paper introduces both notions of topological entropy and invariance entropy for semigroup actions on general topological spaces. We use the concept of admissible family of open coverings to extending and studying the notions of Adler–Konheim–McAndrew topological entropy, Bowen topological entropy, and invariance entropy to the general theory of topological dynamics.     

区别度诱导的广义模糊熵

广义模糊熵是模糊熵在广义模糊补意义下的推广,本文从区别度的角度给出几个生成广义模糊熵的途径;通过一个具体的区别度公式得到了相应的一些广义模糊熵表达式,为实际使用广义模糊熵做了一些理论上的铺垫.     

Deng entropy

Dempster Shafer evidence theory has been widely used in many applications due to its advantages to handle uncertainty. However, how to measure uncertainty in evidence theory is still an open issue. The main contribution of this paper is that a new entropy, named as Deng entropy, is presented to measure the uncertainty of a basic probability assignment (BPA). Deng entropy is the generalization of Shannon entropy since the value of Deng entropy is identical to that of Shannon entropy when the BPA defines a probability measure. Numerical examples are illustrated to show the efficiency of Deng entropy.     

Entropy points and applications

First notions of entropy point and uniform entropy point are introduced using Bowen's definition of topological entropy. Some basic properties of the notions are discussed. As an application it is shown that for any topological dynamical system there is a countable closed subset whose Bowen entropy is equal to the entropy of the original system.

Then notions of C-entropy point are introduced along the line of entropy tuple both in topological and measure-theoretical settings. It is shown that each C-entropy point is an entropy point, and the set of C-entropy points is the union of sets of C-entropy points for all invariant measures.

     

关于有限马氏链相对熵密度和随机条件熵的一类极限定理

本文引进有限非齐次马链随机条件熵的概念，研究这个概念与相对熵密度的关系，并通过数列的绝对平均收敛的概念给出了有限非齐次马氏链的相对频率，相对熵密度和平均随机条件熵ａ．ｅ收敛于常数及有限非齐次马氏链熵率存在的条件。     

Bowen entropy for actions of amenable groups

Bowen introduced a definition of topological entropy of subset inspired by Hausdorff dimension in 1973 [1]. In this paper we consider the Bowen entropy for amenable group action dynamical systems and show that, under the tempered condition, the Bowen entropy of the whole compact space for a given Følner sequence equals the topological entropy. For the proof of this result, we establish a variational principle related to the Bowen entropy and the Brin–Katok local entropy formula for dynamical systems with amenable group actions.     

Fuzzy集的偏熵与关联熵

在文献[1]的基础上,推广模糊熵的概念。首次定义Fuzzy集的偏熵、关联熵和关联熵系数等新概念。对其主要性质进行讨论,并与模糊散度建立联系．得到一些重要结果。     

An axiomatization of entropy of capacities on set systems

We present an axiomatization of the entropy of capacities defined on set systems which are not necessarily the whole power set, but satisfy a condition of regularity. This entropy encompasses the definition of Marichal and Roubens for the entropy of capacities. Its axiomatization is in the spirit of the one of Faddeev for the classical Shannon entropy. In addition, we present also an axiomatization of the entropy for capacities proposed by Dukhovny.     

不等式组与变分不等式的极大熵函数方法

利用极大熵函数方法将不等式组及变分不等式的求解问题转化为近似可微优化问题，给出了不等式组及变分不等式问题近似解的可微优化方法，得到了不等式组和变分不等式问题的解集合的示性函数．     

变分不等式的极大熵函数方法

利用极大熵函数方法将不等式组及变分不等式的求解问题转化为近似可微优化问题,给出了不等式组及变分不等式问题近似解的可微优化方法,得到了不等式组和变分不等式问题的解集合的示性函数.     

MFCAV近似Riemann解法器在相容拉氏方法中的熵条件分析

 对基于MFCAV(Multi Fluid Channel on Averaged Volume)近似Riemann解法器的相容拉氏方法的熵条件进行了分析. 结果表明与满足声学形式Riemann解法器的熵不同, 前者只能在每个网格边界左、右两侧网格的熵随时间变化的和保证大于零, 即能保证整体熵增, 但不保证传统意义上的在每个网格中的熵增;而后者不仅保证整体熵增, 而且还满足传统意义上的熵增. 因此MFCAV的熵增相对声学形式解法器而言要弱一些, 由此表明其熵增可能要小些, 使得格式的耗散可能要小些.数值算例也验证了分析的正确性.     

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.