信息熵方程求解算法及其应用

给定一个离散且有限随机变量的信息熵,求其对应的概率分布需要解多元非线性方程,文中提出了一个将n元信息熵方程化为至多(n-1)个一元非线性方程求解的算法,证明了算法的正确性,给出了算法误差估计;运用熵方程求解算法设计了一种基于信息熵的文本数字水印方案.     

Metric entropy of convex hulls in type spaces--The critical case

Given a precompact subset of a type Banach space , where , we prove that for every and all

holds, where is the absolutely convex hull of and denotes the dyadic entropy number. With this inequality we show in particular that for given and with for all the inequality holds true for all . We also prove that this estimate is asymptotically optimal whenever has no better type than . For this answers a question raised by Carl, Kyrezi, and Pajor which has been solved up to now only for the Hilbert space case by F. Gao.

     

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.     

Asymptotics of orthogonal polynomial’s entropy

This is a brief account on some results and methods of the asymptotic theory dealing with the entropy of orthogonal polynomials for large degree. This study is motivated primarily by quantum mechanics, where the wave functions and the densities of the states of solvable quantum-mechanical systems are expressed by means of orthogonal polynomials. Moreover, the uncertainty principle, lying in the ground of quantum mechanics, is best formulated by means of position and momentum entropies. In this sense, the behavior for large values of the degree is intimately connected with the information characteristics of high energy states. But the entropy functionals and their behavior have an independent interest for the theory of orthogonal polynomials. We describe some results obtained in the last 15 years, as well as sketch the ideas behind their proofs.     

The von Neumann entropy and unitary equivalence of quantum states

K. He, J. Hou and M. Li have recently given a sufficient and necessary condition for unitary equivalence of quantum states. This condition is based on the von Neumann entropy. In this note, we first give a short proof of their result, and then we improve it.     

Using the Renyi entropy to describe quantum dissipative systems in statistical mechanics

We develop a formalism for describing quantum dissipative systems in statistical mechanics based on the quantum Renyi entropy. We derive the quantum Renyi distribution from the principle of maximum quantum Renyi entropy and differentiate this distribution (the temperature density matrix) with respect to the inverse temperature to obtain the Bloch equation. We then use the Feynman path integral with a modified Mensky functional to obtain a Lindblad-type equation. From this equation using projection operators, we derive the integro-differential equation for the reduced temperature statistical operator, an analogue of the Zwanzig equation in statistical mechanics, and find its formal solution in the form of a series in the class of summable functions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 3, pp. 444–453, September, 2008.     

Applying the Linblad equation to quantum dissipative systems

For quantum systems with linear dissipation, we obtain the representation of the Linblad equation in the canonical form via Hermitian operators. Based on this representation, we derive equations for the entropy density and for the statistical projection operator. We consider the quantum harmonic oscillator with linear dissipation as an example. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 2, pp. 288–294, August, 2006. An erratum to this article is available at .     

Quantum entropy of Dirac fields in black holes

We use the brick-wall model to study the quantum entropy of the Dirac field in a static black hole with a global monopole or a cosmic string. We show that the entropy of the Dirac field contains a quadratically divergent term and two logarithmically divergent ones and it is not proportional to the entropy of the scalar field. The contribution of the logarithmic term to the entropy depends on the black-hole characteristics and is always negative. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 1, pp. 60–64, October, 2006.     

Barriola-Vilenkin黑洞的统计熵

利用量子统计方法，直接计算Barriola-Vilenkin黑洞背景下玻色场和费米场的配分函数， 然后利用砖墙膜模型计算和讨论黑洞背景下玻色场和费米场的熵.     

Rotation and entropy

For a given map and an observable rotation vectors are the limits of ergodic averages of We study which part of the topological entropy of is associated to a given rotation vector and which part is associated with many rotation vectors. According to this distinction, we introduce directional and lost entropies. We discuss their properties in the general case and analyze them more closely for subshifts of finite type and circle maps.

     

Structure of entropy solutions to the eikonal equation

In this paper, we establish rectifiability of the jump set of an S ¹–valued conservation law in two space–dimensions. This conservation law is a reformulation of the eikonal equation and is motivated by the singular limit of a class of variational problems. The only assumption on the weak solutions is that the entropy productions are (signed) Radon measures, an assumption which is justified by the variational origin. The methods are a combination of Geometric Measure Theory and elementary geometric arguments used to classify blow–ups.?The merit of our approach is that we obtain the structure as if the solutions were in BV, without using the BV–control, which is not available in these variationally motivated problems. Received June 24, 2002 / final version received November 12, 2002?Published online February 7, 2003     

矩阵代数收敛至球面与非交换度量几何

介绍了Rieffel定义的紧致量子度量空间与量子Gromov-Hausdorff距离和近来Latrémolière定义的量子Gromov-Hausdorff邻距,分别讨论了矩阵代数如何在这两种量子距离下收敛至球面.     

Maximum entropy interpretation of autoregressive spectral densities

A new proof is given of the maximum entropy characterization of autoregressive spectral densities as models for the spectral density of a stationary time series. The new proof is presented in parallel with a proof of the maximum entropy characterization of exponential models for probability densities. Concepts of entropy, cross-entropy and information divergence are defined for probability densities and for spectral densities.     

On the possible values of the entropy of undirected graphs

The entropy of a digraph is a fundamental measure that relates network coding, information theory, and fixed points of finite dynamical systems. In this article, we focus on the entropy of undirected graphs. We prove any bounded interval only contains finitely many possible values of the entropy of an undirected graph. We also determine all the possible values for the entropy of an undirected graph up to the value of four.     

关于Tsallis型非对称熵的若干研究

基于Tsallis熵和非对称熵,本文提出了Tsallis型非对称熵,该熵推广了Tsallis熵和非对称熵,证明了最大的Tsallis型非对称熵原理,并且从该原理中可以获得比Tsallis熵及非对称熵原理更多的分布,从而说明该原理的有用性.     

Position dependent mass Scarf Hamiltonians generated via the Riccati equation

The construction of position dependent mass Scarf Hamiltonians of the trigonometric as well as the hyperbolic types is addressed by means of the factorization method and the Riccati equation. These Hamiltonians are shown to be independent of the ordering parameter of the kinetic term. Additionally, new families of Hamiltonians with the Scarf spectrum are also determined by supersymmetry. Some examples for masses with and without singularities are considered to illustrate our results.     

Multicolor containers,extremal entropy,and counting

In breakthrough results, Saxton‐Thomason and Balogh‐Morris‐Samotij developed powerful theories of hypergraph containers. In this paper, we explore some consequences of these theories. We use a simple container theorem of Saxton‐Thomason and an entropy‐based framework to deduce container and counting theorems for hereditary properties of k‐colorings of very general objects, which include both vertex‐ and edge‐colorings of general hypergraph sequences as special cases. In the case of sequences of complete graphs, we further derive characterization and transference results for hereditary properties in terms of their stability families and extremal entropy. This covers within a unified framework a great variety of combinatorial structures, some of which had not previously been studied via containers: directed graphs, oriented graphs, tournaments, multigraphs with bounded multiplicity, and multicolored graphs among others. Similar results were recently and independently obtained by Terry.     

Negative entropy,zero temperature and Markov chains on the interval

We consider ergodic optimization for the shift map on the modified Bernoulli space σ: [0, 1]^? → [0, 1]^?, where [0, 1] is the unit closed interval, and the potential A: [0, 1]^? → ? considered depends on the two first coordinates of [0, 1]^?. We are interested in finding stationary Markov probabilities µ_∞ on [0, 1]^? that maximize the value ∫ Adµ, among all stationary (i.e. σ-invariant) probabilities µ on [0, 1]^?. This problem correspond in Statistical Mechanics to the zero temperature case for the interaction described by the potential A. The main purpose of this paper is to show, under the hypothesis of uniqueness of the maximizing probability, a Large Deviation Principle for a family of absolutely continuous Markov probabilities µ _β which weakly converges to µ_∞. The probabilities µ _β are obtained via an information we get from a Perron operator and they satisfy a variational principle similar to the pressure in Thermodynamic Formalism. As the potential A depends only on the first two coordinates, instead of the probability µ on [0, 1]^?, we can consider its projection ν on [0, 1]². We look at the problem in both ways. If µ_∞ is the maximizing probability on [0, 1]^?, we also have that its projection ν _∞ is maximizing for A. The hypothesis about stationarity on the maximization problem can also be seen as a transhipment problem. Under the hypothesis of A being C ² and the twist condition, that is,

$\frac{{\partial ^2 A}}{{\partial x\partial y}}(x,y) \ne 0, for all (x,y) \in [0,1]^2 ,$

we show the graph property of the maximizing probability ν on [0, 1]². Moreover, the graph is monotonous. An important result we get is: the maximizing probability is unique generically in Mañé’s sense. Finally, we exhibit a separating sub-action for A.