Invariant measures for Markov maps of the interval

There is a theorem in ergodic theory which gives three conditions sufficient for a piecewise smooth mapping on the interval to admit a finite invariant ergodic measure equivalent to Lebesgue. When the hypotheses fail in certain ways, this work shows that the same conclusion can still be gotten by applying the theorem mentioned to another transformation related to the original one by the method of inducing.Partially supported by NSF MCS74-19388. A01     

Chaotic properties of systems with Markov dynamics

We present a general approach for computing the dynamic partition function of a continuous-time Markov process. The Ruelle topological pressure is identified with the large deviation function of a physical observable. We construct for the first time a corresponding finite Kolmogorov-Sinai entropy for these processes. Then, as an example, the latter is computed for a symmetric exclusion process. We further present the first exact calculation of the topological pressure for an N-body stochastic interacting system, namely, an infinite-range Ising model endowed with spin-flip dynamics. Expressions for the Kolmogorov-Sinai and the topological entropies follow.     

Applications of Markov spectra for the weighted partition network by the substitution rule

The weighted self-similar network is introduced in an iterative way. In order to understand the topological properties of the self-similar network, we have done a lot of research in this field.Firstly, according to the symmetry feature of the self-similar network, we deduce the recursive relationship of its eigenvalues at two successive generations of the transition-weighted matrix.Then, we obtain eigenvalues of the Laplacian matrix from these two successive generations.Finally, we calculate an accurate expression for the eigentime identity and Kirchhoff index from the spectrum of the Laplacian matrix.     

Entanglement in quantum integrable and chaotic systems with Markov dynamics

Entanglement evolution is studied in open systems represented by rings of qubits with the Ising interaction and variously oriented external field and with Markov environments. The effect of thermal or dephasing environment is manifested as exponential decrease of the entanglement superposed on its dynamics in the isolated system.     

Locality in quantum and Markov dynamics on lattices and networks

We consider gapped systems governed by either quantum or Markov dynamics, with the low-lying states below the gap being approximately degenerate. For a broad class of dynamics, we prove that ground or stationary state correlation functions can be written as a piece decaying exponentially in space plus a term set by matrix elements between the low-lying states. The key to the proof is a local approximation to the negative energy, or annihilation, part of an operator in a gapped system. Applications to numerical simulation of quantum systems and to networks are discussed.     

A generalized Kolmogorov-Sinai-like entropy under Markov shifts in symbolic dynamics

In this paper, a generalized Kolmogorov-Sinai-like entropy ( entropy) in the sense of Tsallis is proposed with a nonextensive parameter q under Markov shifts, which contains the classical Kolmogorov-Sinai (KS) entropy and the Rényi entropy as well as Bernoulli shifts as special cases. To verify the formula of this entropy, a one-dimensional iterative system is chosen as an example of Markov shifts, and its entropy is evaluated by a new refinement method of symbolic dynamics called symbolic refinement which differs from the conventional numerical method. The numerical results show that this entropy is monotonically decreasing as q increases.     

Temporal dynamics of pitch strength in regular interval noises.

The pitch strength of rippled noise and iterated rippled noise has recently been fitted by an exponential function of the height of the first peak in the normalized autocorrelation function [Yost, J. Acoust. Soc. Am. 100, 3329-3335 (1996)]. The current study compares the pitch strengths and autocorrelation functions of rippled noise (RN) and another regular-interval noise, "AABB." RN is generated by delaying a copy of a noise sample and adding it to the undelayed version. AABB with the same pitch is generated by taking a sample of noise (A) with the same duration as the RN delay and repeating it to produce AA, and then concatenating many of these once-repeated sequences to produce AABBCCDD.... The height of the first peak (h1) in the normalized autocorrelation function of AABB is 0.5, identical to that of RN. The current experiments show the following: (1) AABB and RN can be discriminated when the pitch is less than about 250 Hz. (2) For these low pitches, the pitch strength of AABB is greater than that for RN whereas it is about the same for pitches above 250 Hz. (3) When RN is replaced by iterated rippled noise (IRN) adjusted to match the pitch strength of AABB, the two are no longer discriminable. The pitch-strength difference between AABB and RN below 250 Hz is explained in terms of a three-stage, running-autocorrelation model. It is suggested that temporal integration of pitch information is achieved in two stages separated by a nonlinearity. The first integration stage is implemented as running autocorrelation with a time constant of 1.5 ms. The second model stage is a nonlinear transformation. In the third model stage, the output of the nonlinear transformation is long-term averaged (second integration stage) to provide a measure of pitch strength. The model provides an excellent fit to the pitch-strength matching data over a wide range of pitches.     

Chaotic dynamics,Markov partitions,and Zipf's law

A chaotic dynamics model creating Markovian strings of symbols as well as sequences of words is presented, and its possible relevance to Zipf's law is discussed.     

Induced maps,Markov extensions and invariant measures in one-dimensional dynamics

A way to study ergodic and measure theoretic aspects of interval maps is by means of the Markov extension. This tool, which ties interval maps to the theory of Markov chains, was introduced by Hofbauer and Keller. More generally known are induced maps, i.e. maps that, restricted to an element of an interval partition, coincide with an iterate of the original map.We will discuss the relation between the Markov extension and induced maps. The main idea is that an induced map of an interval map often appears as a first return map in the Markov extension. For S-unimodal maps, we derive a necessary condition for the existence of invariant probability measures which are absolutely continuous with respect to Lebesgue measure. Two corollaries are given.     

Asymptotics of the partition function of a general Markov random field on an infinite rectangular lattice

We investigate theoretically and numerically the asymptotics of the partition function of a general Markov random field (MRF) on an infinite rectangular lattice. We first propose the general local energy function (LEF)-parameterized MRF. Then we prove that the thermodynamic limit of the free energy of the MRF can be exactly characterized by the Perron root of the fundamental transfer matrix of a particular Markov additive process (MAP). This matrix possesses a special structure and many interesting properties that enable parallel computation of the Perron root and may be beneficial for deriving an analytical form of the free energy. We also develop another transfer matrix for numerical computation of the desired Perron root. Specifically, the former is a site-to-site transfer matrix on a twisted cylindrical lattice, while the latter is the one associated with a row-to-row transition on a vertical strip. Numerical results show that our methods exhibit consistent finite-size scaling behavior even for small values of the lattice width. This study reveals that the fundamental transfer matrix is an alternative direction of research on the analysis of the partition function of general MRFs within the scope of matrix algebra.     

Markov switching of the electricity supply curve and power prices dynamics

Regime-switching models seem to well capture the main features of power prices behavior in deregulated markets. In a recent paper, we have proposed an equilibrium methodology to derive electricity prices dynamics from the interplay between supply and demand in a stochastic environment. In particular, assuming that the supply function is described by a power law where the exponent is a two-state strictly positive Markov process, we derived a regime switching dynamics of power prices in which regime switches are induced by transitions between Markov states.In this paper, we provide a dynamical model to describe the random behavior of power prices where the only non-Brownian component of the motion is endogenously introduced by Markov transitions in the exponent of the electricity supply curve. In this context, the stochastic process driving the switching mechanism becomes observable, and we will show that the non-Brownian component of the dynamics induced by transitions from Markov states is responsible for jumps and spikes of very high magnitude. The empirical analysis performed on three Australian markets confirms that the proposed approach seems quite flexible and capable of incorporating the main features of power prices time-series, thus reproducing the first four moments of log-returns empirical distributions in a satisfactory way.     

由速度间隔不变性直接推导相对论动力学

证明了若采用Lorentz速度变换的微分形式—速度间隔不变性，即可由经典力学直接导出狭义相对论的动力学规律。     

Cortical dynamics and awareness state: an interpretation of observed interstimulus interval dependence in apparent motion

In a recent paper on Cortical Dynamics, Francis and Grossberg raise the question how visual forms and motion information are integrated to generate a coherent percept of moving forms? In their investigation of illusory contours (which are, like Kanizsa squares, mental constructs rather than stimuli on the retina) they quantify the subjective impression of apparent motion between illusory contours that are formed by two subsequent stimuli with delay times of about 0.2 s (called the interstimulus interval ISI). The impression of apparent motion is due to a back referral of a later experience to an earlier time in the conscious representation. A model is developed which describes the state of awareness in the observer in terms of a time dependent Schroedinger equation to which a second order time derivative is added. This addition requires as boundary conditions the values of the solution both at the beginning and after the process. Satisfactory quantitative agreement is found between the results of the model and the experimental results.We recall that in the von Neumann interpretation of the collapse of the quantum mechanical wave-function, the collapse was associated with an observer’s awareness. Some questions of causality and determinism that arise from later-time boundary conditions are touched upon.     

截断误差导致的非双曲不动点邻域拓扑变异

提供一个关于截断误差使简单系统复杂化的直接实验证据,以此证明存在混沌抗退化机理.分别构造了一个一维圆弧迭代系统和一个一维抛物线迭代系统,两者均有一个非双曲不动点,其迭代序列被证明是简单极限序列,数字计算实验显示这两个迭代系统都存在可以越过不动点的序列.采用计算"元胞"分析方法清晰地展示了截断误差导致非双曲不动点邻域拓扑变异:形成第I类阵发混沌通道,或产生纹波分岔.     

Energy partition in nuclear fission

A scission point model (two spheroid model TSM) including semi-empirical, temperature-dependent shell correction energies for deformed fragments at scission is presented. It has been used to describe the mass-asymmetry-dependent partition of the total energy release on both fragments from spontaneous and induced fission. Characteristic trends of experimental fragment energy and neutron multiplicity data as function of incidence energy in the Th — Cf region of fissioning nuclei are well reproduced. Based on model applications, information on the energy dissipated during the descent from second saddle of fission barrier to scission point have been deduced.