The Monte-Carlo algorithm for the solving of systems of linear algebraic equations by the Seidel method

The iteration algorithm is used to solve systems of linear algebraic equations by the Monte-Carlo method. Each next iteration is simulated as a random vector such that its expectation coincides with the Seidel approximation of the iteration process. We deduce a system of linear equations such that mutual correlations of components of the limit vector and correlations of two iterations satisfy them. We prove that limit dispersions of the random vector of solutions of the system exist and are finite.     

Flatness,plateau limit and limit length of single-plateau functions

In this paper we consider iteration of single-plateau functions, an important class of continuous functions with infinitely many forts, and investigate changes of number and length of plateaux under iteration.We use the indices flatness, plateau limit and limit length to formulate those changes. Furthermore, we compute the flatness, plateau limit and limit length for all the nine types of single-plateau functions.     

Semigroups of transcendental entire functions and their dynamics

We investigate the dynamics of semigroups of transcendental entire functions using Fatou–Julia theory. Several results of the dynamics associated with iteration of a transcendental entire function have been extended to transcendental semigroups. We provide some condition for connectivity of the Julia set of the transcendental semigroups. We also study finitely generated transcendental semigroups, abelian transcendental semigroups and limit functions of transcendental semigroups on its invariant Fatou components.     

Outer ellipsoidal approximations of the reachable set at infinity for linear systems

Outer ellipsoidal approximations to the reachable set at infinity for a linear control system with bounded scalar controls are obtained using a new method based on quadratic Lyapunov functions. These outer approximations are compared with those given by an algorithm due to Sabin and Summers, and also with certain tangential outer approximations, obtained using a fixed-point iteration scheme.This research was supported by the Natural Sciences and Engineering Research Council of Canada. The authors gratefully acknowledge the assistance of Mr. Ryan Davies, recipient of an NSERC Undergraduate Research Award.     

Dynamics of Mandelbrot cascades

Mandelbrot multiplicative cascades provide a construction of a dynamical system on a set of probability measures defined by inequalities on moments. To be more specific, beyond the first iteration, the trajectories take values in the set of fixed points of smoothing transformations (i.e., some generalized stable laws). Studying this system leads to a central limit theorem and to its functional version. The limit Gaussian process can also be obtained as limit of an “additive cascade” of independent normal variables.     

The Cascade Algorithm for the Numerical Computation of Refinable Functions

The cascade algorithm is a method which can be used for the computation of refinable functions. We prove here that in general the cascade algorithm will not inherit the accuracy and refinability of the limit refinable function. We provide conditions for which the cascade iteration does indeed preserve accuracy.     

Walsh函数的一个迭代方程体系

Shanks曾用迭代方程产生离散佩利编号Walsh函数。作者在[4]中,给出了产生离散沃尔什编号Wa1sh函数的迭代方程. 本文在上述基础上,提出了一个产生离散哈德玛编号Walsh函数的迭代方程,推出了离散哈德玛编号Walsh函数的表示式及其变换(FWHT)的快速计算公式. 上述三个极为类似的迭代方程,已构成了离散Walsh函数的迭代方程体系.连续的Walsh函数,也能用迭代方程这种形式来描述.     

Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps

We investigate limit theorems for Birkhoff sums of locally Hölder functions under the iteration of Gibbs-Markov maps. Aaronson and Denker have given sufficient conditions to have limit theorems in this setting. We show that these conditions are also necessary: there is no exotic limit theorem for Gibbs-Markov maps. Our proofs, valid under very weak regularity assumptions, involve weak perturbation theory and interpolation spaces. For L ² observables, we also obtain necessary and sufficient conditions to control the speed of convergence in the central limit theorem.     

Programmed Iteration Method in Differential Games with a Functional Target Set

A variant of the programmed iteration method is developed for solving game problems of realizing trajectories of a nonlinear conflict controlled system in a given set of functions. Solving these open-loop control problems yields an iterative process in a space of sets whose elements are functional positions, and the corresponding limit determines the set of successful solvability of the original problem in the class of set-valued quasi-strategies (nonanticipating responses to disturbances). For the conflict controlled system, the conditions of generalized uniqueness and uniform boundedness of programmed motions are assumed to hold. The use of an infinite-dimensional space of functional positions is essential, since the arising differential game generally does not satisfy the alternative solvability conditions in classes of feedback strategies.     

A variable dimension solution approach for the general spatial price equilibrium problem

Algorithms are presented that are specifically designed for solving general nonlinear multicommodity spatial price equilibrium problems, i.e., problems with nonlinear transportation cost functions, nonlinear supply and demand functions, inter-commodity congestion effects, intercommodity substitution and complementarity effects and interactions among transportation links and among spatially separated markets. The algorithms are specializations of an iterative method for solving nonlinear complementarity problems that requires solving a system of nonlinear equations at each iteration. The algorithms exploit the network structure of the problems to reduce the size of the system of equations to be solved at each iteration. The decision rules for determining which equations are to be included in the system at each iteration are extremely simple, and the remainder of the computational work is carried out by the nonlinear equation solver. Because of this, the algorithms are very easy to implement with readily available software. In addition, since the decision rules only require sign information, only the final system needs to be solved with precision.     

求解张量互补问题的一类光滑模系矩阵迭代方法

本文提出了求解张量互补问题的一类光滑模系矩阵迭代方法.其基本思想是,先将张量互补问题转化为等价的模系方程组,然后引入一个逼近的光滑函数进行求解.我们分析了算法的收敛性,并通过数值实验验证了所提出算法的有效性.     

Lyapunov iteration and equation characterizations for unobservable subspace

This note is concerned with the unobservable subspace of a linear system and some Lyapunov iteration and equations. It is shown that the unobservable subspace can be characterized by the Lyapunov iteration and equations defined in the paper. The results generalize some standard results on this topic and are expected to take fundamental functions in control system theory. Both continuous-time and discrete-time systems are considered. Numerical examples show the effectiveness of the proposed results.     

On Linear Inequality Systems with Smooth Coefficients

A linear inequality system with infinitely many constraints is polynomial [analytical] if its index set is a compact interval of the real line and all its coefficients are polynomial [analytical] functions of the index on this interval. This paper provides sufficient conditions for a given closed convex set to be the solution set of a certain polynomial or at least analytical system.The authors are indebted to Dr. J. M. Almira for valuable comments and suggestions.     

Separating the solution sets of analytical and polynomial systems

A linear inequality system with infinitely many constraints is polynomial (analytical) if its index set is a compact interval of the real line and all its coefficients are polynomial (analytical, respectively) functions of the index on this interval. This paper provides an example of analytical system whose solution set cannot be the solution set of any polynomial system. Research supported by DGES of Spain and FEDER of UE, Grant BFM2002-04114-C02-01. Research supported by CONACyT of Mexico, Grant 130036. Research partially supported by CONACyT of Mexico, Grant 44003.     

On solving the chaotic Chen system: a new time marching design for the variational iteration method using Adomian’s polynomial

This paper centres on the effectiveness of the variational iteration method and its modifications for numerically solving the chaotic Chen system, which is a three-dimensional system of ODEs with quadratic nonlinearities. This research implements the multistage variational iteration method with an emphasis on the new multistage hybrid of variational iteration method with Adomian polynomials. Numerical comparisons are made between the multistage variational iteration method, the multistage variational iteration method using the Adomian’s polynomials and the classic fourth-order Runge-Kutta method. Our work shows that the new multistage hybrid provides good accuracy and efficiency with a performance that surpasses that of the multistage variational iteration method.     

数值函数关于集值测度的积分的极限定理

本文给出了实值函数关于集值测度的积分序列的极限定理．     

An algorithm for maximizing entropy subject to simple bounds

An unusual form of the maximum entropy problem is considered, that includes simple bound constraints on the Fourier coefficients of the required image, as well as nonnegativity conditions on the image intensities. The algorithm avoids mixing these constraints by introducing a parameter into the objective function that is adjusted by an outer iteration. For each parameter value an inner iteration solves a large optimization calculation, whose constraints are just the simple bounds, by a combination of the conjugate gradient procedure and an active set method. An important feature is the ability to make many changes to the active set at once. The outer iteration includes a test for inconsistency of all the given constraints. The algorithm is described, a proof of convergence is given, and there are some second-hand remarks on numerical results.     

Asymptotic Behavior of the Attainable Sets of Linear Periodic Control Systems

This paper deals with the attainable sets of linear periodic control systems. The asymptotic behavior of the attainable sets over a long time interval is investigated in terms of shapes of the sets. The shape of a set stands for the totality of all its images under nonsingular linear transformations. It is shown that there exist limits of the shape of attainable sets corresponding to time instants with the same residue modulo the period of the system and that the limit shapes are different if the system includes a stable subsystem.     

A label semantics approach to linguistic hedges

We introduce a model for the linguistic hedges ‘very’ and ‘quite’ within the label semantics framework, and combined with the prototype and conceptual spaces theories of concepts. The proposed model emerges naturally from the representational framework we use and as such, has a clear semantic grounding. We give generalisations of these hedge models and show that they can be composed with themselves and with other functions, going on to examine their behaviour in the limit of composition.