Stability of Oseledets spaces is equivalent to stability of Lyapunov exponents

The objects of the multiplicative ergodic theorem (Lyapunov exponents and Oseledets spaces for products of random matrices) play a key role in the local theory of random dynamical systems. In this paper their behaviour under perturbations is investigated. It is shown that stability of Oseledets spaces is equivalent to stability of Lyapunov exponents.     

Symmetry and chaos in the complex Ginzburg-Landau equation - I. Reflectional symmetries

The complex Ginzburg-Landau (CGL) equation on a one-dimensional domain with periodic boundary conditions has a number of different symmetries. Solutions of the CGL equation may or may not be fixed by the action of these symmetries. We investigate the stability of chaotic solutions with some reflectional symmetry to perturbations which break that symmetry. This can be achieved by considering the isotypic decomposition of the space and finding the dominant Lyapunov exponent associated with each isotypic component. Our numerical results indicate that for most parameter values, chaotic solutions that have been restricted to lie in invariant subspaces are unstable to perturbations out of these subspaces, leading us to conclude that for these parameter values arbitrary initial conditions will generically evolve to a solution with the minimum amount of symmetry allowable. We have also found a small region of parameter space in which chaotic solutions that are even are stable with respect to odd perturbations.     

Non-degenerate Linear Codes Determined by their Weight Enumerators

 In [1], Homma-Kim-Yoo showed that a code C such that the support of the associated projective system is a union of two linear subspaces, is uniquely determined up to equivalence by its weight enumerator. Let W _C (z) be the weight enumerator of a code C. Assuming certain conditions on the coefficients of W _C (z) for some range, we prove that the support of the associated projective system is a union of two linear subspaces. Received: February 27, 2002; revised version: October 2, 2002 Keywords: Linear code, Projective system, Weight enumerator. MSC2000: 94B27, 94B65, 51E20 Partially supported by Grant-in-Aid for Scientific Research, Ministry of Education, #12640180     

RESPONSE TO ALI S. HADI

In many large-dimensional multivariate problems, it is useful to reduce the number of variates. One method of reducing the number of dimensions is to project the original data onto a subspace. The statistical analysis is then carried out in this subspace. Principal-component regression is an example of such a technique. For these applications it is useful to have a measure of the distance between subspaces and to study the sampling stability of such subspaces. To solve these problems, we use a metric on subspaces and bootstrap techniques. The techniques are applied to seven-dimensional vectors of upwelling radiances from the current meteorological satellites. We study the subspaces spanned by the principal components based on a sample categorized by location and surface type.     

Projective Systems Whose Support is the Union of Two Linear Subspaces with Nonempty Intersection

We consider the code corresponding to a projective system whose support is the union of two linear subspaces with nonempty intersection, and prove that the code is uniquely determined up to equivalence by its weight enumerator. Also we obtain the same conclusion for the complement of the union of two linear subspaces.The authors are supported by Korea Research Foundation Grant (KRF-2001-005-D00002).     

Enhancement of continuous estimation of distribution algorithms by density ensembles

In this article, two novel density ensembles methods – the resampling method and the subspaces method – are proposed for enhancing existing continuous Estimation of Distribution Algorithms (EDAs). In ‘resampling continuous EDAs’, a population of densities of the selected promising solutions is obtained by iteratively using the resampling operator and the density estimation operator, and new candidate solutions at the next generation are reproduced through sampling from all obtained densities of promising solutions. In ‘subspaces continuous EDAs’, a population of densities is obtained by randomly choosing a subset of all variables and estimating the density of all selected high-quality solutions in this subspace. The above steps iterate and many densities of high-quality solutions in different subspaces can be obtained. New candidate solutions at the next generation are reproduced through perturbing the old promising solutions by sampling from the densities in different subspaces. The results upon convergence with different numbers of variables and the effects of parameters on the performance of the density ensembles methods for continuous EDAs are studied based on the experimental results.     

Subspaces and polynomial factorizations over finite fields

Recently Niederreiter described a new method for factoring polynomials over finite fields. As with the Berlekamp technique, the method requires the construction of a linear subspace whose dimension is precisely the number of irreducible factors of the polynomial being considered. This paper explores the connection between these subspaces and gives a characterization of other subspaces having properties which are similar.     

一种新的二次插值模型算法

本文中,通过利用随算法表现出来的问题的局部信息,我们构造了几种新的搜索子空间,然后对二次插值模型在这些子空间中进行求解。目的是利用前面的迭代信息构造问题更有可能下降的方向。实验证明我们的方法对于大多数问题都可以有效的减少函数值的运算次数。     

Fuzzy rules extraction using a hybrid of base‐n‐number‐coded GA and SVD‐QR

Abstract

A hybrid of a base‐n‐number‐coded genetic algorithm (base‐n‐number‐coded GA) and an SVD‐QR is proposed to construct a fuzzy system directly from some gathered input‐output data of the identified system. Each individual in the base‐n‐number‐coded GA is applied to determine the fuzzy sets in each input variable. However, the grid‐type fuzzy partition by the fuzzy sets associated with each input variable may generate some redundant fuzzy subspaces. Therefore, an SVD‐QR method is applied to remove the redundant fuzzy subspaces to efficiently describe the behavior of the identified system so that the premise part of the fuzzy system is determined. Then, the recursive least‐squares method is used to determine the consequent part of the fuzzy system. Subsequently, a fitness function is defined such that it can guide the search procedure to select an appropriate fuzzy system that not only maintains a good performance but also has relevant fuzzy rules. Finally, two nonlinear system identification problems are used to illustrate the efficiency of the proposed method.     

一种用阈值提取子空间的多步匹配场反演方法

在分析已有的匹配场反演方法的基础上,构造了一种用阈值提取子空间的多步匹配场反演方法。它根据一定反演环境下参数的不同敏感性将参数划分为子集（子空间）,并依次在各敏感子空间内反演。反演时用一定的阈值将目标函数优于阈值的参数区域提取出,最后在提取出的已相对缩减的区域和最后一个子空间（通常是不敏感参数子空间）内联合反演全部参数,求得最优值。这样既可减少反演参数空间又能可靠地保证精确度,避免了已有的子空间方法反演结果受非反演参数失配影响的问题。仿真研究结果表明,本算法比已有的两类算法性能上有明显提高。     

Clustering in globally coupled oscillators

In this paper we describe approaches to clustering in systems of globally coupled identical oscillators. The first of these approaches is based entirely on the symmetry of such systems, and gives information about the behaviour of the systems near degenerate bifurcation points. We summarize existing results from such analysis, and indicate why further techniques are required to augment the symmetry-based methods. This leads to a second approach based on constructing certain reduced models. This modelling approach relies indirectly on symmetry, using the fact that the systems in question have many invariant subspaces as a result of their symmetry. It is shown how knowledge of behaviour on certain subspaces can be used to predict behaviour on others subspaces, even when their dimensions are different. In applications, this approach can be used to predict the stable clustering behaviour that cannot be predicted by other approaches and may be hard to find numerically. All results are illustrated with examples.     

Symmetry-breaking bifurcations on multidimensional fixed point subspaces

Symmetry-breaking bifurcations associated with fixed point subspaces of dimension greater than one are considered, for maximal isotropy subgroups, using techniques of blowing-up and degree theory. The leading non-linear term in the Taylor expansion of the bifurcation mapping restricted to the fixed point subspace, when satisfying a certain traversality condition or the non-vanishing of an appropriate index, governs the branching. Numerous primary symmetry-breaking branches are obtained and their stability is investigated. Applications involving gradient leading order terms have been calculated using Grobner bases and computer algebra. A general result on symmetry-breaking from O(3) is presented.     

基于子空间划分的多项式回归多色分色算法

提出了一种新的分色算法。方法首先对多色打印机进行色域分区,再将每个分区划分成若干个子空间,在每个子空间内运用最小二乘法建立多项式模型,对目标色度值进行分色时,搜索目标色度值所在最优子空间,求解对应的多项式方程,即可得到分色结果。实验表明,这种分色算法具有较高的精度和稳定性,能够满足多色高精度印刷的要求。     

一种基于子空间划分的LCD显示器光谱特征化模型

目的为实现LCD显示器的光谱特征化,提出一种基于子空间划分的BP神经网络结合PCA的光谱特征化模型。方法对显示器色空间进行子空间划分,在各子空间中进行模型的训练与检测。结果子空间划分后模型的色度和光谱精度,较未进行子空间划分的模型有明显提高,PCA在不影响模型精度的同时,降低了光谱维度,提高了算法的运行效率。结论该模型是一种高精度显示器特征化模型。     

Lyapunov exponents and the evolution of normals

Differential equations are derived which determine the smallest Lyapunov exponents. These o.d.e. arise from relationships describing the evolution of normal tensors associated with the subspaces of an n-dimensional Euclidean phase space. The derived equations are an expression in E_n of relationships in E₃ arising in continuum kinematics. Numerical studies indicate their utility.     

Hierarchical subspace evolution method for super large parallel computing: A linear solver and an eigensolver as examples

Solving linear equations and finding eigenvalues are essential tasks in many simulations for engineering applications, but these tasks often cause performance bottlenecks. In this work, the hierarchical subspace evolution method (HiSEM), a hierarchical iteration framework for solving scientific computing problems with solution locality, is proposed. In HiSEM, the original problem is converted to a corresponding minimization function. The problem is decomposed into a series of subsystems. Subspaces and their weights are established for the subsystems and evolve in each iteration. The subspaces are calculated based on local equations and knowledge of physical problems. A small-scale minimization problem determines the weights of the subspaces. The solution system can be hierarchically established based on the subspaces. As the iterations continue, the degrees of freedom gradually converge to an accurate solution. Two parallel algorithms are derived from HiSEM. One algorithm is designed for symmetric positive definite linear equations, and the other is designed for generalized eigenvalue problems. The linear solver and eigensolver performance is evaluated using a series of benchmarks and a tower model with a complex topology. Algorithms derived from HiSEM can solve a super large-scale problem with high performance and good scalability.     

可压缩欧拉方程在不变子空间中的精确解

欧拉方程是流体力学中非常重要的模型,被广泛应用于许多领域.构造它的精确解是数学物理中非常有意义的工作.精确解可以为理解它的非线性现象和物理意义提供具体的例子.本文旨在通过不变子空间方法构造可压缩欧拉方程的精确解.在变量变换意义下,由不变条件给出与可压缩方程相关的不变子空间;在这些不变子空间中,它被约化为一阶常微分方程组;通过求解这些常微分方程组,最终得到可压缩欧拉方程的一些精确解.     

基于划分子空间的数码相机颜色空间转换方法研究

基于数码相机的印刷品质量检测是未来发展的方向,在颜色检测领域需要首先解决色空间转换的精度问题。采用分子空间的多项式回归法实现了从RGB 颜色空间到CIEL* a* b* 颜色空间的转换,首先把RGB 颜色空间划分成若干个子空间,然后在每个子空间中运用最小二乘法建立多项式模型,对任意RGB 颜色值根据其所在子空间求解对应的多项式方程,即可得到L*a*b*值。实验表明,该方法的转换精度有了很大程度的提高,能够满足数码相机色空间转换的基本要求,为基于数码相机的印刷品质量检测奠定基础。     

An algebraic two‐level preconditioner for asymmetric,positive‐definite systems

A two‐level, linear algebraic solver for asymmetric, positive‐definite systems is developed using matrices arising from stabilized finite element formulations to motivate the approach. Supported by an analysis of a representative smoother, the parent space is divided into oscillatory and smooth subspaces according to the eigenvectors of the associated normal system. Using a mesh‐based aggregation technique, which relies only on information contained in the matrix, a restriction/prolongation operator is constructed. Various numerical examples, on both structured and unstructured meshes, are performed using the two‐level cycle as the basis for a preconditioner. Results demonstrate the complementarity between the smoother and the coarse‐level correction as well as convergence rates that are nearly independent of the problem size. Copyright © 2001 John Wiley & Sons, Ltd.