Calculating Topological Entropy

The attempt to find effective algorithms for calculating the topological entropy of piecewise monotone maps of the interval having more than three monotone pieces has proved to be a difficult problem. The algorithm introduced here is motivated by the fact that if f: [0, 1] → [0, 1] is a piecewise monotone map of the unit interval into itself, thenh(f)=lim_n→∞ (1/n) log Var(f ⁿ), where h(f) is the topological entropy off, and Var(f ⁿ) is the total variation off ⁿ. We show that it is not feasible to use this formula directly to calculate numerically the topological entropy of a piecewise monotone function, because of the slow convergence. However, a close examination of the reasons for this failure leads ultimately to the modified algorithm which is presented in this paper. We prove that this algorithm is equivalent to the standard power method for finding eigenvalues of matrices (with shift of origin) in those cases for which the function is Markov, and present encouraging experimental evidence for the usefulness of the algorithm in general by applying it to several one-parameter families of test functions.     

On the estimation of topological entropy

We study a method for estimating the topological entropy of a smooth dynamical system. Our method is based on estimating the logarithmic growth rates of suitably chosen curves in the system. We present two algorithms for this purpose and we analyze each according to its strengths and pitfalls. We also contrast these with a method based on the definition of topological entropy, using(n, )-spanning sets.     

Commodity predictability analysis with a permutation information theory approach

It is widely known that commodity markets are not totally efficient. Long-range dependence is present, and thus the celebrated Brownian motion of prices can be considered only as a first approximation. In this work we analyzed the predictability in commodity markets by using a novel approach derived from Information Theory. The complexity-entropy causality plane has been recently shown to be a useful statistical tool to distinguish the stage of stock market development because differences between emergent and developed stock markets can be easily discriminated and visualized with this representation space [L. Zunino, M. Zanin, B.M. Tabak, D.G. Pérez, O.A. Rosso, Complexity-entropy causality plane: a useful approach to quantify the stock market inefficiency, Physica A 389 (2010) 1891-1901]. By estimating the permutation entropy and permutation statistical complexity of twenty basic commodity future markets over a period of around 20 years (1991.01.02-2009.09.01), we can define an associated ranking of efficiency. This ranking is quantifying the presence of patterns and hidden structures in these prime markets. Moreover, the temporal evolution of the commodities in the complexity-entropy causality plane allows us to identify periods of time where the underlying dynamics is more or less predictable.     

Some variational formulas for Hausdorff dimension,topological entropy,and SRB entropy for hyperbolic dynamical systems

We derive several variational formulas for the topological entropy and SRB entropy of Axiom A flows on compact manifolds and for the Hausdorff dimension of basic sets for Axiom A diffeomorphisms on compact surfaces.     

油水两相流水包油流型多尺度排列熵分析

采用多尺度排列熵算法研究了垂直油水两相流水包油流型的多尺度动力学特性.首先,在内径为20 mm的垂直管道内采集了油水两相流水包油流型电导传感器波动信号,然后计算了不同流动工况下电导波动信号的多尺度排列熵值.研究发现多尺度排列熵率与均值可定量刻画水包油流型动力学复杂性;此外,提出了通过增量时间序列累积量与多尺度排列熵率联合分布识别三种不同水包油流型的新途径.     

An improved algorithm for computing topological entropy

A new algorithm is presented for computing the topological entropy of a unimodal map of the interval. The accuracy of the algorithm is discussed and some graphs of the topological entropy which are obtained using the algorithm are displayed.     

Fine-grained permutation entropy as a measure of natural complexity for time series

In a recent paper [2002 Phys. Rev. Lett. 88 174102], Bandt and Pompe propose permutation entropy (PE) as a natural complexity measure for arbitrary time series which may be stationary or nonstationary, deterministic or stochastic. Their method is based on a comparison of neighbouring values. This paper further develops PE, and proposes the concept of fine-grained PE (FGPE) defined by the order pattern and magnitude of the difference between neighbouring values. This measure excludes the case where vectors with a distinct appearance are mistakenly mapped onto the same permutation type, and consequently FGPE becomes more sensitive to the dynamical change of time series than does PE, according to our simulation and experimental results.     

基于变分模态分解与多尺度排列熵的生物组织变性识别

根据高强度聚焦超声(HIFU)治疗中超声散射回波信号的特点,本文利用变分模态分解(VMD)与多尺度排列熵(MPE)对生物组织变性识别进行了研究.首先对生物组织中的超声散射回波信号进行变分模态分解,根据各阶模态的功率谱信息熵值分离出噪声分量和有用分量;对分离出的有用信号进行重构并提取其多尺度排列熵;然后通过Gustafson-Kessel (GK)模糊聚类确定聚类中心,采用欧氏贴近度与择近原则对生物组织进行变性识别.将所提方法应用于HIFU治疗中超声散射回波信号实验数据,用遗传算法对多尺度排列熵的参数优化后,对293例未变性组织和变性组织的超声散射回波信号数据进行了多尺度排列熵分析,发现变性组织的超声散射回波信号的多尺度排列熵值要高于未变性组织;多尺度排列熵可以较好地识别生物组织是否变性.相对于EMD-MPE-GK模糊聚类以及VMD-小波熵(WE)-GK模糊聚类变性识别方法,本文所提方法中变性与未变性组织特征交叠区域数据点更少,聚类效果和分类性能更好;本实验环境下生物组织变性识别结果表明,该方法的识别率更高,高达93.81%.     

Topological phases of the two-leg Kitaev ladder

We study the phase diagram of the two-leg Kitaev model. Different topological phases can be characterized by either the number of Majorana modes for a deformed chain of the open ladder, or by a winding number related to the ‘h  -loop’ in the momentum space. By adding a three-spin interaction term to break the time-reversal symmetry, two originally different phases are glued together, so that the number of Majorana modes reduce to 0 or 1, namely, the topological invariant collapses to _Z2$Z_2$ from an integer Z. These observations are consistent with a recent general study [S. Tewari, J.D. Sau, arXiv:1111.6592v2].     

凝聚体中的拓扑量子态

探寻拓扑上非平庸的凝聚体物质状态,特别是其电子结构和输运性质,是当前凝聚体物理 学领域非常重要的前沿研究方向。本文讨论的大多数主题都与电子波函数的拓扑性质有关。全文 除简短的引言外,包括拓扑量子现象、各种拓扑相、拓扑性准粒子的异常输运性质、拓扑性集 体激发和耦合激发,以及继续发展的拓扑量子态研究五个章节。这些章节着重反映拓扑量子态研 究的各个侧面,汇总起来方可以凸显凝聚体中拓扑量子态的全貌。     

Topological entanglement entropy

We formulate a universal characterization of the many-particle quantum entanglement in the ground state of a topologically ordered two-dimensional medium with a mass gap. We consider a disk in the plane, with a smooth boundary of length L, large compared to the correlation length. In the ground state, by tracing out all degrees of freedom in the exterior of the disk, we obtain a marginal density operator rho for the degrees of freedom in the interior. The von Neumann entropy of rho, a measure of the entanglement of the interior and exterior variables, has the form S(rho) = alphaL - gamma + ..., where the ellipsis represents terms that vanish in the limit L --> infinity. We show that - gamma is a universal constant characterizing a global feature of the entanglement in the ground state. Using topological quantum field theory methods, we derive a formula for gamma in terms of properties of the superselection sectors of the medium.     

Permutation entropy and detrend fluctuation analysis for the natural complexity of cardiac heart interbeat signals

We compute fractal dimension and permutation entropy for healthy and people who have experienced heart failure. Our result shows that permutation entropy is a suitable approach as well as detrend fluctuation analysis (DFA). The result of DFA shows that the fractal dimensions for healthy and heart failure are different as well as the permutation entropy result. The fluctuation value for permutation entropy for an individual who has experienced heart failure is bigger than for a healthy person. There is some specific change in the interbeat signal of a person who has experienced heart failure, but there is not previous trend for a healthy person.     

Computing the topological entropy of maps of the interval with three monotone pieces

An algorithm is presented for computing the topological entropy of a piecewise monotone map of the interval having three monotone pieces. The accuracy of the algorithm is discussed and some graphs of the topological entropy obtained using the algorithm are displayed. Some of the ideas behind the algorithm have application to piecewise monotone functions with more than three monotone pieces.     

Practical considerations of permutation entropy

More than ten years ago Bandt and Pompe introduced a new measure to quantify complexity in measured time series. During these ten years, this measure has been modified and extended. In this review we will give a brief introduction to permutation entropy, explore the different fields of utilization where permutation entropy has been applied and provide a guide on how to choose appropriate parameters for different applications of permutation entropy.     

信息熵、玻尔兹曼熵以及克劳修斯熵之间的关系--兼论玻尔兹曼熵和克劳修斯熵是否等价

论述了信息熵、玻尔兹曼熵以及克劳修斯熵之间的关系;由不涉及具体系统的方法从玻尔兹曼关系、信息熵推导出了克劳修斯熵的表达式;指出玻尔兹曼熵与克劳修斯熵不是等价关系,而是玻尔兹曼熵包含克劳修斯熵,信息熵又包含玻尔兹曼熵。     

Topological phases: An expedition off lattice

Motivated by the goal to give the simplest possible microscopic foundation for a broad class of topological phases, we study quantum mechanical lattice models where the topology of the lattice is one of the dynamical variables. However, a fluctuating geometry can remove the separation between the system size and the range of local interactions, which is important for topological protection and ultimately the stability of a topological phase. In particular, it can open the door to a pathology, which has been studied in the context of quantum gravity and goes by the name of ‘baby universe’, here we discuss three distinct approaches to suppressing these pathological fluctuations. We complement this discussion by applying Cheeger’s theory relating the geometry of manifolds to their vibrational modes to study the spectra of Hamiltonians. In particular, we present a detailed study of the statistical properties of loop gas and string net models on fluctuating lattices, both analytically and numerically.     

A Topological Glass

We propose and study a model with glassy behavior. The state space of the model is given by all triangulations of a sphere with n nodes, half of which are red and half are blue. Red nodes want to have 5 neighbors while blue ones want 7. Energies of nodes with other numbers of neighbors are supposed to be positive. The dynamics is that of flipping the diagonal of two adjacent triangles, with a temperature dependent probability. We show that this system has an approach to a steady state which is exponentially slow, and show that the stationary state is unordered. We also study the local energy landscape and show that it has the hierarchical structure known from spin glasses. Finally, we show that the evolution can be described as that of a rarefied gas with spontaneous generation of particles and annihilating collisions.     

Topological dynamics of flipping Lorentz lattice gas models

We study the topological dynamics of the flipping mirror model of Ruijgrok and Cohen with one or an infinite number of particles. In particular we prove the topological transitivity and topological mixing up to a natural first integral for the one-particle model.     

Topological defects in cosmology

Present status of theories of topological defects in particle theory models of the early Universe is discussed. Various consequences of topological defects in cosmology, such as constraints on particle theory models, structure formation etc. are discussed.