结构方程模型简化方法及在人机行为研究中的应用

为解决研究中结构方程模型过度复杂的问题,提出一种简化方法.分析了多维测量模型直接引入整体模型的局限性,对高阶验证性因子分析方法和项目组合法在模型简化中的适用性进行了分析,提出了首先检验简化必要性再使用高阶验证性因子分析方法和项目组合法对整体模型进行简化的两阶段方法.以电子学习系统持续使用行为的研究数据为例对该简化方法的有效性进行检验,简化前后拟合数据的对比分析表明该方法可以显著简化整体模型并获得理想的拟合优度.     

MULTILEVEL AUGMENTATION METHODS FOR SOLVING OPERATOR EQUATIONS

We introduce multilevel augmentation methods for solving operator equations based on direct sum decompositions of the range space of the operator and the solution space of the operator equation and a matrix splitting scheme. We establish a general setting for the analysis of these methods, showing that the methods yield approximate solutions of the same convergence order as the best approximation from the subspace. These augmentation methods allow us to develop fast, accurate and stable nonconventional numerical algorithms for solving operator equations. In particular, for second kind equations, special splitting techniques are proposed to develop such algorithms. These algorithms are then applied to solve the linear systems resulting from matrix compression schemes using wavelet-like functions for solving Fredholm integral equations of the second kind. For this special case, a complete analysis for computational complexity and convergence order is presented. Numerical examples are included to demonstra     

A survey of randomized algorithms for control synthesis and performance verification

In this paper, we present an overview of probabilistic techniques based on randomized algorithms for solving “hard’’ problems arising in performance verification and control of complex systems. This area is fairly recent, even though its roots lie in the robustness techniques for handling uncertain control systems developed in the 1980s. In contrast to these deterministic techniques, the main ingredient of the methods discussed in this survey is the use of probabilistic concepts. The introduction of probability and random sampling permits overcoming the fundamental tradeoff between numerical complexity and conservatism that lie at the roots of the worst-case deterministic methodology. The simplicity of implementation of randomized techniques may also help bridging the gap between theory and practical applications.     

Applying numerical continuation to the parameter dependence of solutions of the Schrödinger equation

In molecular reactions at the microscopic level, the appearance of resonances has an important influence on the reactivity. It is important to predict when a bound state transitions into a resonance and how these transitions depend on various system parameters such as internuclear distances. The dynamics of such systems are described by the time-independent Schrödinger equation and the resonances are modeled by poles of the S-matrix.Using numerical continuation methods and bifurcation theory, techniques which find their roots in the study of dynamical systems, we are able to develop efficient and robust methods to study the transitions of bound states into resonances. By applying Keller’s Pseudo-Arclength continuation, we can minimize the numerical complexity of our algorithm. As continuation methods generally assume smooth and well-behaving functions and the S-matrix is neither, special care has been taken to ensure accurate results.We have successfully applied our approach in a number of model problems involving the radial Schrödinger equation.     

Investigating Algebraic and Logical Algorithms to Solve Hopf Bifurcation Problems in Algebraic Biology

Symbolic methods to investigate Hopf bifurcation problems of vector fields arising in the context of algebraic biology have recently obtained renewed attention. However, the symbolic investigations have not been fully algorithmic but required a sequence of symbolic computations intervened with ad hoc insights and decisions made by a human. In this paper we discuss the use of algebraic and logical methods to reduce questions on the existence of Hopf bifurcations in parameterized polynomial vector fields to quantifier elimination problems over the reals combined with the use of the quantifier elimination over the reals and simplification techniques available in REDLOG. We can reconstruct most of the results given in the literature within a few seconds of computation time and extend the investigations on these systems to previously not analyzed related systems. Especially we discuss cases in which one suspects that no Hopf bifurcation fixed point exists for biologically relevant values of parameters and system variables. Here we focus on logical and algebraic techniques of finding subconditions being inconsistent with the hypothesis of the existence of Hopf bifurcation fixed points.      

Variational integrators for electric circuits

Variational integrators are symplectic-momentum preserving integrators that are based on a discrete variational formulation of the underlying system. So far, variational integrators have been mainly developed and used for a wide variety of mechanical systems. In this work, we develop a variational integrator for the simulation of electric circuits. An appropriate variational formulation is presented to model the circuit from which the equations of motion are derived. Finally, a corresponding time-discrete variational formulation provides an iteration scheme for the simulation of the electric circuit. In this way, a variational integrator is constructed that gains several advantages. A comparison to standard integration techniques shows that even for simple LCR circuits a better long-time energy behavior and frequency preservation can be obtained. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mathematical Problems in Circuit Simulation

Circuit simulation is a standard task for the computer-aided design of electronic circuits. The transient analysis is well understood and realized in powerful simulation packages for conventional circuits. However, further developments in production engineering lead to new classes of circuits, which cause difficulties for the numerical integration. The dimension of circuit models can be quite large (10 5 equations). The complexity of the models demands a higher level of ion. Parasitic effects become dominant. The signal to noise ratio becomes smaller. In this paper, we want to draw attention to three essential problems from a mathematical point of view, the DAE-index, consistent initial values, and asymptotic stability. These topics have been extensively analyzed only recently. We shall illustrate them by some simple examples.     

Design of $n$-dimensional multi-scroll Jerk chaotic system and its performances

Based on three-order Jerk and high-order Jerk chaotic systems, a general approach is proposed to generate $n$-dimensional multi-scroll Jerk chaotic attractors via nonlinear control. Dynamics of the $n$-dimensional multi-scroll Jerk chaotic systems are analyzed by means of the largest Lyapunov exponent and multi-scale permutation entropy complexity. As an experimental verification, four-dimensional Jerk chaotic attractors are implemented by analog circuits. Results of the numerical simulation are consistent with that of the hardware experiments. It shows that the method of obtaining complex Jerk chaotic attractors is effective.     

Bounded rationality, strategy simplification, and equilibrium

It is frequently suggested that predictions made by game theory could be improved by considering computational restrictions when modeling agents. Under the supposition that players in a game may desire to balance maximization of payoff with minimization of strategy complexity, Rubinstein and co-authors studied forms of Nash equilibrium where strategies are maximally simplified in that no strategy can be further simplified without sacrificing payoff. Inspired by this line of work, we introduce a notion of equilibrium whereby strategies are also maximally simplified, but with respect to a simplification procedure that is more careful in that a player will not simplify if the simplification incents other players to deviate. We study such equilibria in two-player machine games in which players choose finite automata that succinctly represent strategies for repeated games; in this context, we present techniques for establishing that an outcome is at equilibrium and present results on the structure of equilibria.     

The Computational Complexity of the Chow Form

We present a bounded probability algorithm for the computation of the Chowforms of the equidimensional components of an algebraic variety. In particular, this gives an alternative procedure for the effective equidimensional decomposition of the variety, since each equidimensional component is characterized by its Chow form. The expected complexity of the algorithm is polynomial in the size and the geometric degree of the input equation system defining the variety. Hence it improves (or meets in some special cases) the complexity of all previous algorithms for computing Chow forms. In addition to this, we clarify the probability and uniformity aspects, which constitutes a further contribution of the paper. The algorithm is based on elimination theory techniques, in line with the geometric resolution algorithm due to M. Giusti, J. Heintz, L. M. Pardo, and their collaborators. In fact, ours can be considered as an extension of their algorithm for zero-dimensional systems to the case of positive-dimensional varieties. The key element for dealing with positive-dimensional varieties is a new Poisson-type product formula. This formula allows us to compute the Chow form of an equidimensional variety from a suitable zero-dimensional fiber. As an application, we obtain an algorithm to compute a subclass of sparse resultants, whose complexity is polynomial in the dimension and the volume of the input set of exponents. As another application, we derive an algorithm for the computation of the (unique) solution of a generic overdetermined polynomial equation system.     

Fuzzy computational control for real Chua circuit

We present a computational procedure to control an experimental chaotic system by applying the occasional proportional feedback (OPF) method. The method implementation uses the fuzzy theory to relate the variable correction to the necessary adjustment in the control parameter. As an application we control the chaotic attractors of the Chua circuit. We present the developed circuits and algorithms to implement this control in real time. To simplify the used procedure, we use a low resolution analog to digital converter compensated for a lowpass filter that facilitates similar applications to control other systems.     

Classification of Structural Complexity for Mine Ventilation Networks

The mine ventilation system is most important and technical measure for ensuring safety production in mines. The structural complexity of a mine ventilation network can directly affect the safety and reliability of the underground mining system. Quantitatively justifying the degree of complexity can contribute to providing a deeper understanding of the essential characteristics of a network. However, so far, there is no such a model which is able to simply, practically, reasonably, and quantitatively determine or compare the structural complexity of different ventilation networks. In this article, by analyzing some typical parameters of a mine ventilation network, we conclude that there is a linear functional relationship among five key parameters (number of ventilation network branches, number of nodes, number of independent circuits, number of independent paths, and number of diagonal branches). Correlation analyses for the main parameters of ventilation networks are conducted based on SPSS. Based on these findings, a new evaluation model for the structural complexity of ventilation network (which is represented by C) has been proposed. By combining SPSS classification analyses results with the characteristics of mine ventilation networks, standards for the complexity classification of mine ventilation systems are put forward. Using the developed model, we carried out analyses and comparisons for the structural complexity of ventilation networks for typical mines. Case demonstrations show that the classification results correspond to the actual situations. © 2014 Wiley Periodicals, Inc. Complexity 21: 21–34, 2015     

A combinatorial problem related to sparse systems of equations

Nowadays sparse systems of equations occur frequently in science and engineering. In this contribution we deal with sparse systems common in cryptanalysis. Given a cipher system, one converts it into a system of sparse equations, and then the system is solved to retrieve either a key or a plaintext. Raddum and Semaev proposed new methods for solving such sparse systems common in modern ciphers which are combinations of linear layers and small S-boxes. It turns out that the solution of a combinatorial MaxMinMax problem provides an upper bound on the average computational complexity of those methods. In this paper we initiate the study of a linear algebra variation of the MaxMinMax problem. The complexity bound proved in this paper significantly overcomes conjectured complexity bounds for Gröbner basis type algorithms.     

信号处理中一类非线性方程组的快速求解

在声纳和雷达信号处理中,需要求解一类维数可变的非线性方程组,这类方程组具有混合三角多项式方程组形式.由于该问题有很多解,且其对应的最小二乘问题有很多局部极小点,用牛顿法等传统的迭代法很难找到有物理意义的解.若把它化为多项式方程组,再用解多项式方程组的符号计算方法或现有的同伦方法求解,由于该问题规模太大而不能在规定的时间内求解,而当考虑的问题维数较大时,利用已有的方法甚至根本无法求解.综合利用我们提出的解混合三角多项式方程组的混合同伦方法和保对称的系数参数同伦方法,我们给出该类问题一种有效的求解方法.利用这种方法,可以达到实时求解的目的,满足实际问题的需要.     

Asymptotic Stability Properties of $\theta$-Methods for the Multi-Pantograph Delay Differential Equation

This paper deals with the asymptotic stability analysis of $\theta$–methods for multi-pantograph delay differential equationHere $\lambda, μ_1,μ_2, ... , μ_l, u_0 \in C$.In recent years stability properties of numerical methods for this kind of equation has been studied by numerous authors. Many papers are concerned with meshes with fixed stepsize. In general the developed techniques give rise to non-ordinary recurrence relation. In this work, instead,we study constrained variable stpesize schemes, suggested by theoretical and computational reasons, which lead to a non-stationary difference equation. A general theorem is presented which can be used to obtain the characterization of the stability regions of $\theta$–methods.     

算子方程近似解直接方法的优化和信息复杂性研究进展

算子方程近似解直接方法的优化和信息复杂性是80年代发展起来的连续复杂性理论的两个主要方面,是计算机科学和数学的交叉研究领域．本文拟就这两个方面的研究进展做一简要介绍．     

A hybrid computer for X-ray crystallography

A hybrid computer for structure factor calculations in X-ray crystallography is described. The computer can calculate three-dimensional structure factors of up to 24 atoms in a single run and can generate the scatter functions of well over 100 atoms using Vandet al., or Forsyth and Wells approximations. The computer is essentially a digital computer with analog function generators, thus combining to advantage the economic data storage of digital systems and simple computing circuitry of analog systems. The digital part serially selects the data, computes and feeds the arguments into specially developed high precision digital-analog function generators, the outputs of which being d.c. voltages, are further processed by analog circuits and finally the sequential adder, which employs a novel digital voltmeter circuit, converts them back into digital form and accumulates them in a dekatron counter which displays the final result. The computer is also capable of carrying out 1-, 2-, or 3-dimensional Fourier summation, although in this case, the lack of sufficient storage space for the large number of coefficients involved, is a serious limitation at present.     

A structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equation

In this paper, we propose a structure-preserving doubling algorithm (SDA) for the computation of the minimal nonnegative solution to the nonsymmetric algebraic Riccati equation (NARE), based on the techniques developed for the symmetric cases. This method allows the simultaneous approximation to the minimal nonnegative solutions of the NARE and its dual equation, requiring only the solutions to two linear systems and several matrix multiplications per iteration. Similar to Newton's method and the fixed-point iteration methods for solving NAREs, we also establish global convergence for SDA under suitable conditions, using only elementary matrix theory. We show that sequences of matrices generated by SDA are monotonically increasing and quadratically convergent to the minimal nonnegative solutions of the NARE and its dual equation. Numerical experiments show that the SDA algorithm is feasible and effective, and outperforms Newton's iteration and the fixed-point iteration methods. This research was supported in part by RFDP (20030001103) & NSFC (10571007) of China and the National Center for Theoretical Sciences in Taiwan. This author's research was supported by NSFC grant 1057 1007 and RFDP grant 200300001103 of China.     

Numerical solution of transient heat conduction equation with variable thermophysical properties by the Tau method

In this article, we proposed the operational approach to the Tau method for solving linear and nonlinear one‐dimensional transient heat conduction equations with variable thermophysical properties which can involve heat generation term. To solve heat conduction equation, first we recall the Tau method to obtain a matrix form of the governing differential equation. Then boundary and initial conditions are transformed into a matrix form. Finally the resulting systems of linear or nonlinear algebraic equations are given. Afterwards, efficient error estimation is also introduced for this method. Some numerical examples are given to illustrate the efficiency and high accuracy of the proposed method and also results are compared with solutions obtained by other methods. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 964–977, 2014     

The General Theory of Linear Difference Equations over the Max-Plus Semi-Ring

We present the mathematical theory underlying systems of linear difference equations over the max-plus semi-ring. The result provides an analog of isomonodromy theory for ultradiscrete Painlevé equations, which are extended cellular automata, and provide evidence for their integrability. Our theory is analogous to that developed by Birkhoff and his school for linear q -difference equations, but stands independently of the latter. As an example, we derive linear problems in this algebra for ultradiscrete versions of the symmetric P_IV equation and show how it is a necessary condition for isomonodromic deformation of a linear system.