Katok formula for the induced measure-theoretic entropy

Entropy is undoubtedly one of the most essential characteristics of dynamical systems. In this article, we define a topological version of the induced measure-theoretic entropy and obtain its Katok entropy formula. We show that the induced measure-theoretic entropy coincides with Hausdorff dimension of the ergodic measure in a symbolic space and the BS dimensions of the ergodic measures can be characterized by the induced measure-theoretic entropies. As an application, we give a variational principle of the BS dimension by the induced measure-theoretic entropy.     

Some results on the entropy of non-autonomous dynamical systems

In this paper, we advance the entropy theory of discrete non-autonomous dynamical systems that was initiated by Kolyada and Snoha in 1996. The first part of the paper is devoted to the measure-theoretic entropy theory of general topological systems. We derive several conditions guaranteeing that an initial probability measure, when pushed forward by the system, produces an invariant measure sequence whose entropy captures the dynamics on arbitrarily fine scales. In the second part of the paper, we apply the general theory to the non-stationary subshifts of finite type, introduced by Fisher and Arnoux. In particular, we give sufficient conditions for the variational principle, relating the topological and measure-theoretic entropy, to hold.     

多重序列比对Alignment的信息度量准则

多重序列的比对（Alignment)的核心问题是对多重DNA（或RNA，蛋白质）序列，寻找它们的相似部分或稳定区域，但如何定义多重序列的相似度（或罚分函数）则是解决多重序列经对的前提。首先对多重序列相似度的合理性进行了讨论，给出了一组合理的相似度所必需具备的条件，再利用Shannon熵的特点，给出一个满足这些合理性条件的多重序列比对的优化准则。     

Tree network 1-median location with interval data: a parameter space-based approach

We consider a family of 1-median location problems on a tree network where the vertex weights are ranges rather than point values. We define a new framework for making sound decisions under uncertainty which is primarily based on the interplay between the points in the tree and the data that induce the family of problems. An important feature of this framework is that it provides a novel understanding of the problem under uncertainty by collectively handling all possible realizations of the weights. The key element is the notion of a region of a optimality. Based on the regions of optimality, we define three optimality criteria and give low-order polynomial methods to compute the associated solution sets.     

M-P逆在线性随机方程中的应用

本文证明了可料过程的M-P逆仍然可料.利用M-P逆讨论了线性随机方程可料解的结构,由此求出了扩散模型中的所有等价鞅测度,并给出了最小逆熵鞅测度.     

有限维线性系统参数辨识问题

本文考虑一维输入输出有限维线性系统参数辨识问题。该问题源于电机系统的参数辨识。首先我们对简单参数形式的线性时不变动态系统,给出了一种精确辨识方法,并给出了这种辨识方法的误差界。这种方法能够保证辨识出的参数是最佳的;而且不用求解对应的非线性最小二乘问题,只需求一元多项式的根,从而大大减少计算量。接着我们考虑了复合参数形式的线性时变系统参数辨识问题,给出了一种近似辨识方法,并导出了该方法的误差界;该方法本质是通过求解非线性最小二乘问题来辨识参数。对于测量存在误差、误差服从区间分布的线性系统,我们给出了其等价的确定性问题,并给出了几个算例。计算结果表明,本文给出的参数辨识方法是有效的。     

Asymptotic stability,Onsager fluxes and reaction kinetics

This paper constitutes a first step in the derivation of thermodynamics directly from the dynamics of physical systems. The existence of an asymptotically stable equilibrium point is used to construct a family of admissible entropy functions. These functions have nonnegative entropy production and assume an absolute maximum at the equilibrium point. A nonlinear generalization of the Onsager theory is then used to obtain a one-to-one correspondence between entropy production functions and the governing system of autonomous rate equations. The theory is applied to well stirred chemically reacting systems with constant temperature and pressure. This allows the derivation of chemical potentials, Gibbs' potentials and enthalpy for such systems. The rate equations for reaction kinetics and for classical thermodynamic reaction theory are obtained. Classic thermodynamic reaction theory is shown to give maximum entropy for constant enthalpy at the equilibrium point, while reaction kinetics gives this result only to within quadratic terms in the departure from equilibrium.     

Topological entropy dimension for noncompact sets

Similar with the fractal dimension, we introduce the concept of topological entropy dimension to classify the sets with entropy zero. We prove that the entropy dimension of the space in this article is not greater than that defined by De Carvalho, where he introduced the entropy dimension for the system, and give some examples indicating that such inequality is optimal. Some basic propositions of entropy dimension are discussed and it turns out that the entropy dimension is invariant under conjugacy. The property of the countable stability and a power rule for the entropy dimension of any set are obtained. It is shown that any set shares the same entropy dimension with its image set.     

Diagnostic entropy: a quantitative measure of the effects of signal incompleteness on system diagnosis

A set of measures quantifying the effects of signal incompleteness upon system diagnosis are defined and investigated. The most important is diagnostic entropy, a new quantitative measure of the effects of signal incompleteness upon system uncertainty. Diagnostic entropy is defined as the average uncertainty of a system when the system is indicated to be in an undesired state. This measure appears to be more useful for quantifying the difficulty of system diagnosis than conventional system entropy or conditional system entropy measures due to its relevance to the difficulty of human diagnosis of the system when it is in an undesired state. The magnitude of the diagnostic entropy is shown to be usually larger than that of the conditional entropy for typical highly reliable systems. This means that the uncertainty of the system is larger than usual when a reliable system is in an undersired state. We also suspect, but have not identified, the existence of a relationship between the diagnostic entropy of a system and the average physiological stress of human operators in diagnosing the system.     

Usc/fibred entropy structure and applications

For a given metrizable space X, we study continuity properties of the entropy as function not only of the measure but also of the dynamical system on X. We introduce the notion of robust tail entropy, which implies upper semicontinuity of the topological entropy but also stability of measures of maximal entropy (when the topological entropy is continuous). This gives, inter alia, simple proofs of results of Misiurewicz and Raith for multimodal interval maps. We also consider fibred entropy structures which allow us to investigate the symbolic extensions of (smooth) skew-product systems.     

Skew-cyclic codes

We generalize the notion of cyclic codes by using generator polynomials in (non commutative) skew polynomial rings. Since skew polynomial rings are left and right euclidean, the obtained codes share most properties of cyclic codes. Since there are much more skew-cyclic codes, this new class of codes allows to systematically search for codes with good properties. We give many examples of codes which improve the previously best known linear codes.     

Polynomial generating pairing and its criterion for optimal pairing

We define a polynomial generating pairing (PGP) and propose a method to construct a family of pairing friendly curves from PGP. We show that a bilinear map over the family is directly determined by the coefficients of the PGP and the map is non-degenerate under a minor condition which is satisfied with cryptographic parameters. Finally, we provide a criterion for PGP to obtain an optimal pairing.     

An unfolding method for directional spectrometers

The development of new spectrometers that are sensitive to the directional distribution of neutrons requires new unfolding methods that can determine the distribution of the neutron fluence as a function of energy and angle. Such information is needed to compute non-isotropic dosimetric quantities (e.g. personal dose equivalent and effective dose). We describe an unfolding method that applies the maximum entropy principle to this problem. It maximises the relative entropy, defined as the information-theory entropy of the distribution of the neutron fluence relative to a distribution that encodes prior knowledge, subject to constraints imposed by the measurements. We provide examples of the applicability of the method using data from two directional spectrometers of different design that have been developed in the context of EVIDOS, a project concerning mixed neutron-photon field analysis in the nuclear industry.     

A generalized planck distribution

We introduce a generalization of the standard Planck distribution discussed by Johnson and Kotz (1970, Section 33.6.1). This generalization results in a very flexible family which contains the gamma distribution as a particular case. In this paper we provide a comprehensive treatment of the mathematical properties of the family. We derive expressions for thenth moment, moment generating function, characteristic function, mean deviation about the mean, mean deviation about the median, Rényi entropy, Shannon entropy and the asymptotic distribution of the extreme order statistics. Estimation and simulation issues are also considered.     

Approximate structural reanalysis for optimization along a line

The need for multiple repeated analyses, which usually involve much computational effort, is one of the main difficulties in the optimization of large structures. This study deals with approximate structural reanalysis along a line in the design space, a problem common to many optimal design procedures. The problem is first stated in terms of a single independent variable. Explicit polynomial approximations for the nodal displacements are then presented. It is shown that Taylor series expansion of the displacements and a series based on a simple iteration procedure are equivalent. Furthermore, the series coefficients can readily be computed, providing efficient and high-degree polynomial approximations. To improve the quality of the approximations, two nonpolynomial series are proposed. The relationship between the various methods is discussed and simple numerical examples demonstrate applications. All the proposed procedures require only a single exact analysis to obtain the explicit expressions for any given line. The results obtained are very close to the exact solution and indicate that the proposed methods provide efficient and high-quality approximations for the nodal displacements.     

基于阻抗特性多项式拟合的直驱风电机组次同步振荡稳定判据

近年来,阻抗分析法已成为分析新能源发电并网系统稳定性问题的一种主要研究方法。以直驱风电机组并网系统为例,分析了现有的奈奎斯特（Nyquist）稳定判据的特点和适用范围。为了弥补现有阻抗稳定判据的不足,提出了一种基于阻抗特性分式多项式函数拟合的量化稳定判据,采用分式多项式函数等效拟合理论推导或实测的风电机组端口阻抗特性,在拟合频段内分式多项式与原阻抗特性等价。通过求取拟合多项式零点获得了系统振荡频率和阻尼水平,量化分析了系统稳定特性,拓展了阻抗稳定判据的适用范围。最后,通过理论分析和时域仿真验证了分式多项式拟合判据的正确性和有效性。     

Sequences of regressions and their independences

Ordered sequences of univariate or multivariate regressions provide statistical models for analysing data from randomized, possibly sequential interventions, from cohort or multi-wave panel studies, but also from cross-sectional or retrospective studies. Conditional independences are captured by what we name regression graphs, provided the generated distribution shares some properties with a joint Gaussian distribution. Regression graphs extend purely directed, acyclic graphs by two types of undirected graph, one type for components of joint responses and the other for components of the context vector variable. We review the special features and the history of regression graphs, prove criteria for Markov equivalence and discuss the notion of a?simpler statistical covering model. Knowledge of Markov equivalence provides alternative interpretations of a given sequence of regressions, is essential for machine learning strategies and permits to use the simple graphical criteria of regression graphs on graphs for which the corresponding criteria are in general more complex. Under the known conditions that a Markov equivalent directed acyclic graph exists for any given regression graph, we give a polynomial time algorithm to find one such graph.     

Shannon entropy of partially polarized and partially coherent light with Gaussian fluctuations

We propose to analyze Shannon entropy properties of partially coherent and partially polarized light with Gaussian probability distributions. It is shown that the Shannon entropy is a sum of simple functions of the intensity, of the degrees of polarization, and of the intrinsic degrees of coherence that have been recently introduced. This analysis clearly demonstrates the contribution of partial polarization and of partial coherence to the characterization of disorder of the light provided by the Shannon entropy, which is a standard measure of randomness. We illustrate these results on two simple examples.     

Boundary element modelling to solve the Grad–Shafranov equation as an axisymmetric problem

The Grad–Shafranov equation describes the magnetic flux distribution of plasma in an axisymmetric system like a tokamak-type nuclear fusion device. The equation is transformed into an equivalent boundary integral equation by expanding the inhomogeneous term related to the plasma current into a polynomial. In the present research, the singularity of the fundamental solution, which consists of two elliptic integrals, and the properties of singular integrals have been minutely investigated. The discontinuous quadratic boundary elements have been introduced to give an accurate solution with a small number of boundary elements. Ampere's circuital law has been applied to estimate the total plasma current from the boundary integral of the poloidal field, and based on this idea, a new scheme to calculate the eigenvalue has also been proposed.     

On necessary and sufficient conditions for differential flatness

This paper is devoted to the characterization of differentially flat nonlinear systems in implicit representation, after elimination of the input variables, in the differential geometric framework of manifolds of jets of infinite order. We extend the notion of Lie-B?cklund equivalence, introduced in Fliess et al. (IEEE Trans Automat Contr 44(5):922–937, 1999), to this implicit context and focus attention on Lie-B?cklund isomorphisms associated to flat systems, called trivializations. They can be locally characterized in terms of polynomial matrices of the indeterminate \fracddt{\frac{d}{dt}} , whose range is equal to the kernel of the polynomial matrix associated to the implicit variational system. Such polynomial matrices are useful to compute the ideal of differential forms generated by the differentials of all possible trivializations. We introduce the notion of a strongly closed ideal of differential forms, and prove that flatness is equivalent to the strong closedness of the latter ideal, which, in turn, is equivalent to the existence of solutions of the so-called generalized moving frame structure equations. Two sequential procedures to effectively compute flat outputs are deduced and various examples and consequences are presented.