覆盖族生成拓扑的约简

本文研究了在覆盖族产生的拓扑不变的条件下覆盖族的约简问题.利用拓扑学理论讨论覆盖广义粗糙集的约简理论,给出计算约简的方法,丰富了覆盖广义粗糙集理论.     

覆盖粗糙集属性约简的新算法

覆盖广义粗糙集是Pawlak粗糙集的重要推广,其属性约简是粗糙集理论中最重要的问题之一.Tsang等基于一种生成覆盖设计了覆盖信息系统属性约简算法,但并未明确指出其适用的覆盖粗糙集类型.在本文中,我们首先指出Tsang的属性约简算法适用的覆盖粗糙集是第五,第六和第七类.其次,我们通过建立覆盖与自反且传递的二元关系之间的等价关系,提出了一种时间复杂度更低的属性约简算法,并证明了本文中的属性约简方法就是Wang等所提出的一般二元关系属性约简的特例.本文不仅提出了属性约简的简化算法,还首次建立起覆盖属性约简与二元关系属性约简之间的联系,具有理论和实际的双重意义.     

基于最小描述交的覆盖广义粗糙集

针对Bonikowski覆盖广义粗糙集模型的不足,给出了基于最小描述史的覆盖上下近似算子.通过和Pawlak经典粗糙集以及Bonikowski的覆盖广义粗糙集比较,发现给出的覆盖上、下近似算子具有了对偶关系,并得到了相关重要性质;进一步讨论了在新定义下覆盖广义粗糙集的约简和公理化问题,丰富了覆盖广义粗糙条理论,并为覆盖广义粗糙集的应用提供了更确切的理论根据.     

覆盖信息系统一种新的约简方法北大核心

本文建立了一类基于元素最大描述的覆盖粗糙集,给出了与经典粗糙集理论相对应的覆盖粗糙集的基本性质,并讨论了不同覆盖生成相同覆盖近似算子的充要条件以及一个覆盖的约简。最后,通过构造区分矩阵来给出覆盖信息系统的约简与核心的判断定理,从而给出了求覆盖信息系统约简的一种方法。     

覆盖信息系统一种新的约简方法

本文建立了一类基于元素最大描述的覆盖粗糙集,给出了与经典粗糙集理论相对应的覆盖粗糙集的基本性质,并讨论了不同覆盖生成相同覆盖近似算子的充要条件以及一个覆盖的约简。最后,通过构造区分矩阵来给出覆盖信息系统的约简与核心的判断定理,从而给出了求覆盖信息系统约简的一种方法。     

基于覆盖粗糙集理论中的约简与求核

提出基于分辨矩阵的求覆盖粗糙集约简与核的方法,在Zakowski提出的覆盖粗糙集模型的基础上,利用分辨矩阵的一些性质,把文献[10]中的粗糙集理论中的约简与求核方法应用到基于覆盖的粗糙集理论中,既简化了覆盖粗糙集理论中的约简与求核过程,又推广了文献[10]的方法,最后举例说明此方法的有效性。     

集覆盖问题的粗糙集属性约简方法

集覆盖问题和粗糙集属性约简问题都是当前的研究热点,两者均有广泛的应用背景。目前,集覆盖理论与粗糙集理论的交叉研究还处于起步阶段。文章的工作主要是把集覆盖问题转化成测试代价敏感粗糙集属性约简问题,使得可应用粗糙集理论来研究集覆盖问题,目的在于丰富集覆盖理论与粗糙集理论的交叉研究。首先构造集覆盖的分辨矩阵,然后在该分辨矩阵上构造集覆盖对应的测试代价敏感信息系统模型,发现求解集合覆盖问题等价于求解对应测试代价敏感信息系统的最小测试代价约简。接着给出了基于正域正向近似加速器最小集覆盖问题的粗糙集解法。最后通过实例验证了该算法的可行性和有效性。     

粒度空间约简及其粗糙集模型研究

针对交可约粒度空间中覆盖、基和粒结构的关系,结合偏序关系的哈斯图,给出一种约简粒度空间的方法.另外,通过限定上、下近似算子的取值范围,重新定义了交可约粒度空间上的粗糙集模型,并讨论了其相关性质.     

基于优势关系的不协调区间值目标信息系统的分布约简

区间值信息系统是单值信息系统的的一种推广,知识约简是粗糙集理论的核心问题之一.在基于优势关系下的不协调区间值信息系统中引入了分布约简和最大分布约简的概念,进一步建立了分布约简和最大分布约简的判定定理和辨识矩阵,从而利用辨识矩阵给出了在优势关系下不协调区间值目标信息系统分布约简的具体方法.     

一致结构与一致覆盖族的非标准刻画

一致空间有三种等价描述方式,即通过满足各自条件的关系族,覆盖族和伪度量族.本文利用非标准分析的方法研究了一致空间,给出了一致结构与一致覆盖族的非标准刻画,并且利用（X×X）上的可滤单子化的等价关系构造了X上的一致结构.     

Asymptotic Solutions of Nonrelativistic Equations of Quantum Mechanics in Curved Nanotubes: I. Reduction to Spatially One-Dimensional Equations

We consider equations of nonrelativistic quantum mechanics in thin three-dimensional tubes (nanotubes). We suggest a version of the adiabatic approximation that permits reducing the original three-dimensional equations to one-dimensional equations for a wide range of energies of longitudinal motion. The suggested reduction method (the operator method for separating the variables) is based on the Maslov operator method. We classify the solutions of the reduced one-dimensional equation. In Part I of this paper, we deal with the reduction problem, consider the main ideas of the operator separation of variables (in the adiabatic approximation), and derive the reduced equations. In Part II, we will discuss various asymptotic solutions and several effects described by these solutions.     

Spectral computations: from operators to matrices

In this work we will address the problem of finding spectral values and bases for the corresponding spectral subspaces for a bounded operator on a Banach space. We will make a bridge between the spectral problem for the continuous operator and the computation of the eigenpairs of a matrix. This approximate problem results first from an approximation by a sequence of continuous operators with finite rank, followed by the reduction to a spectral problem for an operator whose domain as well as range are finite dimensional. Some discussion on defect correction procedures will be also presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

F格上的逼近算子

粗集理论是波兰学者Pawlak提出的知识表示新理论.Pawlak代数是粗集理论中粗集系统的抽象,其公理系统包含了知识粗表示所必须的全部性质.本文深入研究了F格上的逼近算子,建立了F格上弱逼近算子之间的某些代数运算,从而从理论上建立了各种知识粗表示之间的联系.我们还定义了逼近算子的闭包,进而用逼近算子导出拓扑,为信息系统的近似提供必要的数学基础.最后,作为特例,我们研究了粗集理论中由相似关系导出逼近算子的某些性质.     

模糊形式背景下的类近似算子

分别在形式背景和模糊形式背景下定义类下近似算子和类模糊下近似算子,并研究它们的性质.证明这两种算子分剐等价于形式背景和模糊形式背景下的*算子和模糊*算子.进一步给出类下近似算子与类模糊下近似算子的公理刺画.最后,对偶地讨论类上近似算子和类模糊上近似算子的定义和性质.     

The approximation properties determined by operator ideals

We introduce the notion of the right approximation property with respect to an operator ideal A and solve the duality problem for the approximation property with respect to an operator ideal A, that is, a Banach space X has the approximation property with respect to A ^d whenever X* has the right approximation property with respect to an operator ideal A. The notions of the left bounded approximation property and the left weak bounded approximation property for a Banach operator ideal are introduced and new symmetric results are obtained. Finally, the notions of the p-compact sets and the p-approximation property are extended to arbitrary Banach operator ideals. Known results of the approximation property with respect to an operator ideal and the p-approximation property are generalized.     

A modified Bogoliubov method applied to a simple boson model

We use a non-gauge-invariant modification of the exact Hamiltonian to obtain a new Hamiltonian-like operator for a simple exactly solvable boson model. The eigenvalues of the new operator are close to those of the original Hamiltonian. We make a one-body approximation of the new two-body operator in the spirit of the Bogoliubov approximation. Because only the number operator appears, the c-number approximation is not required individually for the creation or annihilation operators in the ground state. For the simple model, the results using the new approximation are closer to the exact results than the usual Bogoliubov results over a wide range of parameters. The improvement increases dramatically as the model interaction strength increases.     

Null space correction and adaptive model order reduction in multi-frequency Maxwell’s problem

A model order reduction method is developed for an operator with a non-empty null-space and applied to numerical solution of a forward multi-frequency eddy current problem using a rational interpolation of the transfer function in the complex plane. The equation is decomposed into the part in the null space of the operator, calculated exactly, and the part orthogonal to it which is approximated on a low-dimensional rational Krylov subspace. For the Maxwell’s equations the null space is related to the null space of the curl. The proposed null space correction is related to divergence correction and uses the Helmholtz decomposition. In the case of the finite element discretization with the edge elements, it is accomplished by solving the Poisson equation on the nodal elements of the same grid. To construct the low-dimensional approximation we adaptively choose the interpolating frequencies, defining the rational Krylov subspace, to reduce the maximal approximation error. We prove that in the case of an adaptive choice of shifts, the matrix spanning the approximation subspace can never become rank deficient. The efficiency of the developed approach is demonstrated by applying it to the magnetotelluric problem, which is a geophysical electromagnetic remote sensing method used in mineral, geothermal, and groundwater exploration. Numerical tests show an excellent performance of the proposed methods characterized by a significant reduction of the computational time without a loss of accuracy. The null space correction regularizes the otherwise ill-posed interpolation problem.     

基于支持向量机的算子逼近方法

Rn中连续算子的逼近问题的数值方法,一直是计算科学中研究的热点。本文引进了新兴的智能机器一支持向量机,以解决Rn中连续算子的逼近问题。在给出支持向量机用于算子逼近问题的详细数学表示之后,我们提出了分块逼近的算法,并通过具体的实例说明支持向量机在算子逼近问题中的有效性与优越性。     

Approximation by Bézier variants of the BBHK operators

In this work a new type of approximation operator—the Bézier variant of the BBHK operator—is introduced. Its approximation properties are studied. A convergence theorem for such approximation operators for locally bounded functions is established by means of some techniques of probability theory and analysis methods. This convergence theorem subsumes the approximation of functions of bounded variation as a special case.     

A new approximation operator of the Arato-Renyi type

In this paper a new approximation operator is introduced and its properties are studied. Special cases of this operator are the well-known Szàsz power-series approximation operator and its generalization by D. Leviatan. The behaviour of the new approximation operator at points of continuity and discontinuity is investigated by using probabilistic tools as the Chebishev inequality and Liapounov’s central limit theorem. Such probabilistic methods of proof simplify the proofs and give better understanding of the approximation mechanism.