Self-similar factor approximants for evolution equations and boundary-value problems

The method of self-similar factor approximants is shown to be very convenient for solving different evolution equations and boundary-value problems typical of physical applications. The method is general and simple, being a straightforward two-step procedure. First, the solution to an equation is represented as an asymptotic series in powers of a variable. Second, the series are summed by means of the self-similar factor approximants. The obtained expressions provide highly accurate approximate solutions to the considered equations. In some cases, it is even possible to reconstruct exact solutions for the whole region of variables, starting from asymptotic series for small variables. This can become possible even when the solution is a transcendental function. The method is shown to be more simple and accurate than different variants of perturbation theory with respect to small parameters, being applicable even when these parameters are large. The generality and accuracy of the method are illustrated by a number of evolution equations as well as boundary value problems.     

Method of self-similar factor approximants

The method of self-similar factor approximants is completed by defining the approximants of odd orders, constructed from the power series with the largest term of an odd power. It is shown that the method provides good approximations for transcendental functions. In some cases, just a few terms in a power series make it possible to reconstruct a transcendental function exactly. Numerical convergence of the factor approximants is checked for several examples. A special attention is paid to the possibility of extrapolating the behavior of functions, with arguments tending to infinity, from the related asymptotic series at small arguments. Applications of the method are thoroughly illustrated by the examples of several functions, nonlinear differential equations, and anharmonic models.     

Self-similar approach to market analysis

A novel approach to analyzing time series generated by complex systems, such as markets, is presented. The basic idea of the approach is the Law of Self-Similar Evolution, according to which any complex system develops self-similarly. There always exist some internal laws governing the evolution of a system, say of a market, so that each of such systems possesses its own character regulating its behaviour. The problem is how to discover these hidden internal laws defining the system character. This problem can be solved by employing the self-similar approximation theory, which supplies the mathematical foundation for the law of self-similar evolution. In this report, the theoretical basis of the new approach to analyzing time series is formulated, with an accurate explanation of its principal points. Received 15 August 2000     

Calculation of critical exponents by self-similar factor approximants

The method of self-similar factor approximants is applied to calculating the critical exponents of the O(N)-symmetric ϕ⁴ theory and of the Ising glass. It is demonstrated that this method, being much simpler than other known techniques of series summation in calculating the critical exponents, at the same time, yields the results that are in very good agreement with those of other rather complicated numerical methods. The principal advantage of the method of self-similar factor approximants is the combination of its extraordinary simplicity and high accuracy.     

电子商务中人类活动的标度行为实证研究

应用去趋势波动分析法,对电子商务中人类网上购物行为进行研究,首次探讨了人类浏览及购买行为时间序列(数量波动)标度律.首先,研究发现人类网上购物行为呈现出明显的周期性,其时间序列的概率密度函数具有显著的双模态特征.其次,利用傅里叶变换方法分析浏览以及购买行为时间序列的功率谱,发现其演化过程不同于无关联的泊松过程.最后,基于功率谱过滤周期性趋势的影响,对去除周期趋势后的浏览和购买行为时间序列进行去趋势波动分析,发现其标度行为表明其具有较强的长程关联特性,且平均标度值近似为1,表明其具有自组织临界性.实证研究结果与其他领域如因特网交通流和金融市场价格波动的标度行为相似,有助于理解人类活动如何影响电子商务系统演化和提高在线商务活动效率,对分析电子商务中人类行为活动的机制和预测其波动趋势具有重要的启示作用.     

Local normalization: Uncovering correlations in non-stationary financial time series

The measurement of correlations between financial time series is of vital importance for risk management. In this paper we address an estimation error that stems from the non-stationarity of the time series. We put forward a method to rid the time series of local trends and variable volatility, while preserving cross-correlations. We test this method in a Monte Carlo simulation, and apply it to empirical data for the S&P 500 stocks.     

利用非线性脉冲预整形实现脉冲快速自相似放大

报道了一种通过预先对信号光脉冲非线性整形, 进而在光纤放大器中实现自相似演化的方法. 利用透射光栅对和普通单模光纤组成被动脉冲整形装置, 预先优化脉冲的时域宽度和光谱质量, 使脉冲在光纤放大器中快速演化到自相似子. 研究中首先通过数值模拟对比, 说明了非线性整形对脉冲实现快速自相似放大的关键作用, 提高了放大器输出脉冲质量, 减小了去啁啾脉冲宽度. 实验中, 通过优化非线性整形输出, 在2.2 m 长的掺Yb³⁺光纤中, 在一定抽运功率范围内均实现了脉冲自相似放大, 去啁啾后得到脉冲宽度60 fs 的变换极限脉冲输出. 这种非线性脉冲预整形方法有效减小了 自相似演化所需光纤长度, 同时降低了自相似放大对种子源质量的要求, 首次将全正色散光纤锁模激光器成功用于自相似放大, 促进了当前自相似放大系统的全光纤化. 关键词： 非线性整形 光纤放大器 自相似放大 飞秒激光     

多变量时间序列最大李雅普诺夫指数的计算

根据单变量时间序列计算最大Lyapunov指数的算法思想，本文提出了一种于多变量时间序列最大Lyapunov指数计算的方法.针对原有算法需要使用重构相空间的特点，推广算法给出了多变量时间序列相空间重构参数的选择方法，并采用多变量重构相空间进行最大Lyapunov指数计算.经耦合Rssler系统产生多变量时间序列的仿真计算，验证了该算法的有效性，推广算法的计算结果表明多变量时间序列的计算结果优于单变量的结果，且更加接近理论计算结果. 关键词： 多变量时间序列 相空间重构 Lyapunov指数     

Model equations from a chaotic time series

We present a method for obtaining a set of dynamical equations for a system that exhibits a chaotic time series. The time series data is first embedded in an appropriate phase space by using the improved time delay technique of Broomhead and King (1986). Next, assuming that the flow in this space is governed by a set of coupled first order nonlinear ordinary differential equations, a least squares fitting method is employed to derive values for the various unknown coefficients. The ability of the resulting model equations to reproduce global properties like the geometry of the attractor and Lyapunov exponents is demonstrated by treating the numerical solution of a single variable of the Lorenz and Rossler systems in the chaotic regime as the test time series. The equations are found to provide good short term prediction (a few cycle times) but display large errors over large prediction time. The source of this shortcoming and some possible improvements are discussed.     

基于平均场支持向量机的混沌时间序列预测

提出了一种新的基于支持向量机的混沌时间序列预测方法，该方法利用平均场理论使支持向量机的学习过程变得简单高效。同时由于该方法将支持向量机的参数近似为高斯分布的，因此采用平均场理论能够容易的求解这些参数，这样获得的支持向量机的参数比传统的基于二次规划的算法更加精确，而且学习速度更快。最后利用该方法对嵌入维数与模型的泛化能力关系进行了探讨，并利用Mackey-Glass时间序列对该方法进行了验证，结果表明：该预测方法能精确地预测混沌时间序列,而且在混沌时间序列的嵌入维数未知时也能取得比较好的预测效果.这一结论预示着平均场支持向量机是一种研究混沌时间序列的有效方法.     

Three-dimensional mixed convection flow with variable thermal conductivity and frictional heating

In this article,three-dimensional mixed convection flow over an exponentially stretching sheet is investigated.Energy equation is modelled in the presence of viscous dissipation and variable thermal conductivity.Temperature of the sheet is varying exponentially and is chosen in a form that facilitates the similarity transformations to obtain self-similar equations.Resulting nonlinear ordinary differential equations are solved numerically employing the Runge-Kutta shooting method.In order to check the accuracy of the method,these equations are also solved using bvp4c built-in routine in Matlab.Both solutions are in excellent agreement.The effects of physical parameters on the dimensionless velocity field and temperature are demonstrated through various graphs.The novelty of this analysis is the self-similar solution of the threedimensional boundary layer flow in the presence of mixed convection,viscous dissipation and variable thermal conductivity.     

基于极端学习机的多变量混沌时间序列预测

针对多变量混沌时间序列预测问题, 提出了一种基于输入变量选择和极端学习机的预测模型. 其基本思想是 对多变量混沌时间序列进行相空间重构后, 采用互信息方法选择与预测输出统计相关最高的重构输入变量, 借助极端学习机的通用逼近能力建立多变量混沌时间序列的预测模型. 为进一步提高预测精度, 采用模型选择算法选择具有最小期望风险的极端学习机预测模型. 基于Lorenz, Rössler多变量混沌时间序列及Rössler超混沌时间序列的仿 真结果证明所提方法的有效性.     

Subdominant critical singularities in the bcc Ising model

In order to analyze coalescing singularities using series expansion, a modification of the method introduced by Baker and Hunter is proposed. The Padé approximants on the Mellin transform are computed simultaneously for the initial series and its derivatives, allowing unbiased estimates for the critical parameters. The method is applied to the series generated by Nickel for various values of the spin in the bcc Ising model. We show that even with these longer series, the subdominant indices display large variation with the spin, and remain too small compared to the universal renormalization group prediction. However, the method does not detect other singularities in the temperature plane which are responsible for the observed discrepancy. Therefore the discrepancy is not significant.     

Suppression of Output Power and NF Excursions in Cascades of Highly Inverted EDFAs with Packet-Switched Traffic

Significant output power excursions in cascades of erbium-doped fiber amplifiers (EDFAs) can cause serious problems in wavelength division multilexing (WDM) packet-switched burst-mode networks. Signal power excursions more serious than those induced by channel addition/removal in circuit switched networks can arise when data on the WDM channels is highly variable in nature. Self-similar traffic can be subject to large variation in EDFA gain. In order to prevent unacceptable error bursts, due, for example, to channel power becoming too low to preserve adequate eye opening or exceeding thresholds for optical nonlinearities, the channel power should be maintained constant. In this letter, it is shown that the sizable power and noise figure swings arising in a cascade of EDFAs with WDM burst-mode packet-switched networks with self-similar traffic can be effectively suppressed when highly inverted amplifiers are employed. The analysis is based on the application of a numerical model, which solves the transcendental equation for length averaged metastable level population of an EDFA.     

价电子轨道能量拓扑指数与Y、Bi系超导电性的关系

利用价电子轨道能量Ei和价电子层数ni,构建了新的原子点价δi,由点价δi构建价电子轨道能量拓扑指数mE。研究了Y系、Bi系高温超导氧化物与mE的0阶指数的关系,发现有较好的规律性。因此文中提出用价电子轨道能量拓扑指数mE作为Y系、Bi系高温超导氧化物超导电性的一个新判据,对今后制备出具有更高TC的Y系、Bi系超导氧化物有很好的指导意义。     

基于优化核极限学习机的风电功率时间序列预测

针对时间序列预测,在单隐层前馈神经网络的基础上,基于进化计算的优化策略,提出了一种优化的核极限学习机(optimized kernel extreme learning machine,O-KELM)方法.与极限学习机(extreme learning machine,ELM)方法相比,核极限学习机(kernel extreme learning machine,KELM)方法无须设定网络隐含层节点的数目,以核函数表示未知的隐含层非线性特征映射,通过正则化最小二乘算法计算网络的输出权值,它能以极快的学习速度获得良好的推广性.在KELM的基础上,分别将遗传算法、模拟退火、微分演化三种进化算法用于模型的结构输入选择、正则化系数以及核参数的优化选取,以进一步提高网络的性能.将O-KELM方法应用于标准Mackey-Glass混沌时间序列预测及某地区的风电功率时间序列预测实例中,在同等条件下,还与优化的极限学习机(optimized extreme learning machine,O-ELM)方法进行比较.实验结果表明,所提出的O-KELM方法在预测精度上优于O-ELM方法,表明了其有效性.     

远场模型方程的同伦近似对称约化

以同伦近似对称法为理论依据研究了远场模型方程, 通过归纳各阶相似约化解和各阶相似约化方程的通式构造相应的同伦级数解. 各阶相似约化方程均为线性变系数常微分方程, 并且可以从零阶开始依次求解. 同伦模型中的辅助参数影响同伦级数解的收敛性. 关键词： 同伦近似对称法 远场模型方程 同伦级数解     

Suppression of Output Power and NF Excursions in Cascades of Highly Inverted EDFAs with Packet-Switched Traffic

Significant output power excursions in cascades of erbium-doped fiber amplifiers (EDFAs) can cause serious problems in wavelength division multilexing (WDM) packet-switched burst-mode networks. Signal power excursions more serious than those induced by channel addition/removal in circuit switched networks can arise when data on the WDM channels is highly variable in nature. Self-similar traffic can be subject to large variation in EDFA gain. In order to prevent unacceptable error bursts, due, for example, to channel power becoming too low to preserve adequate eye opening or exceeding thresholds for optical nonlinearities, the channel power should be maintained constant. In this letter, it is shown that the sizable power and noise figure swings arising in a cascade of EDFAs with WDM burst-mode packet-switched networks with self-similar traffic can be effectively suppressed when highly inverted amplifiers are employed. The analysis is based on the application of a numerical model, which solves the transcendental equation for length averaged metastable level population of an EDFA.     

Relaxation in cooperative systems: Use of mixture virial coefficients

The coefficients in power series in the variable time that describe relaxation in a cooperative system can be calculated using a combinatorial approach where one considers how many ways one can introduce a given number of properly defined events in a system. The coefficients obtained in this manner can be related to the equilibrium virial coefficients for a mixture. If one assumes rapid internal equilibration, the relaxation process can be expressed completely in terms of the viral coefficients for a mixture with at most one solute particle, or, in some cases, just the virial coefficients for a single-component system. Thus, equilibrium virial coefficients can give useful information about the time evolution of processes in cooperative systems.     

基于条件熵扩维的多变量混沌时间序列相空间重构

提出一种多变量混沌时间序列相空间重构的条件熵扩维方法.首先使用互信息法求解每个变量的时间延迟,其次按条件熵最大原则逐步扩展相空间的嵌入维数,使得重构坐标从低维到高维的转换保持较强的独立性,最终的重构相空间具有较低的冗余度,为多变量时间序列的预测构造了有效的模型输入向量.通过对几个经典多变量混沌时间序列进行数值实验,结果表明该方法比单变量预测和已有多变量预测方法具有更好的预测效果,说明了该重构方法的有效性. 关键词： 多变量混沌时间序列 相空间重构 条件熵 神经网络预测