Approximate Symmetries and Infinite Series Symmetry Reduction Solutions to Perturbed Kuramoto-Sivashinsky Equation

Starting from Lie symmetry theory and combining with the approximate symmetry method, and using the package LieSYMGRP proposed by us, we restudy the perturbed Kuramoto-Sivashinsky （KS） equation. The approximate symmetry reduction and the infinite series symmetry reduction solutions of the perturbed KS equation are constructed. Specially, if selecting the tanh-type travelling wave solution as initial approximate, we not only obtain the general formula of the physical approximate similarity solutions, but also obtain several new explicit solutions of the given equation, which are first reported here.

Approximate direct reduction method:infinite series reductions to the perturbed mKdV equation

The approximate direct reduction method is applied to the perturbed mKdV equation with weak fourth order dispersion and weak dissipation. The similarity reduction solutions of different orders conform to formal coherence, accounting for infinite series reduction solutions to the original equation and general formulas of similarity reduction equations. Painleve Ⅱ type equations, hyperbolic secant and Jacobi elliptic function solutions are obtained for zeroorder similarity reduction equations. Higher order similarity reduction equations are linear variable coefficient ordinary differential equations.

Local Kernel Dimension Reduction in Approximate Bayesian Computation

Approximate Bayesian Computation (ABC) is a popular sampling method in applications involving intractable likelihood functions. Instead of evaluating the likelihood function, ABC approximates the posterior distribution by a set of accepted samples which are simulated from a generating model. Simulated samples are accepted if the distances between the samples and the observation are smaller than some threshold. The distance is calculated in terms of summary statistics. This paper proposes Local Gradient Kernel Dimension Reduction (LGKDR) to construct low dimensional summary statistics for ABC. The proposed method identifies a sufficient subspace of the original summary statistics by implicitly considering all non-linear transforms therein, and a weighting kernel is used for the concentration of the projections. No strong assumptions are made on the marginal distributions, nor the regression models, permitting usage in a wide range of applications. Experiments are done with simple rejection ABC and sequential Monte Carlo ABC methods. Results are reported as competitive in the former and substantially better in the latter cases in which Monte Carlo errors are compressed as much as possible.

Approximate Similarity Reduction for Perturbed Kaup-Kupershmidt Equation via Lie Symmetry Method and Direct Method

The perturbed Kaup-Kupershmidt equation is investigated in terms of the approximate symmetry perturbationmethod and the approximate direct method.The similarity reduction solutions of different orders are obtainedfor both methods, series reduction solutions are consequently derived.Higher order similarity reduction equations arelinear variable coefficients ordinary differential equations.By comparison, it is find that the results generated from theapproximate direct method are more general than the results generated from the approximate symmetry perturbationmethod.

APPROXIMATE COMMON DIVISORS OF POLYNOMIALS AND DEGREE REDUCTION FOR RATIONAL CURVES

Abstract This paper deals with how to perturb a given set of polynomials so as to include a common linear factor. An algorithm is derived for determining such a set of perturbation polynomials which are subject to certain constrains at the endpoints of a prescribed parametric interval and minimized in a certain sense. This result can be combined with subdivision technique to obtain a continuous piecewise approximation to a rational curve.

Rough Set Based K-Exception Approach to Approximate Rule Reduction

There are rules refering to infrequent instances after the procession of attribute reductionand value reduction with traditional methods.A rough set RS based k-exception approach (RSKEA) torule reduction is presented.Its main idea lies in a two-phase RS based rule reduction.An ordinarydecision table is attained through general method of RS knowledge reduction in the first phase.Then a k-exception candidate set is nominated according to the decision table.RS rule reduction is employed forthe reformed source data set,which remove all the instances included in the k-exception set.We apply theapproach to the automobile database.Results show that it can reduce the number and complexity of ruleswith adjustable conflict rate,which contributes to approximate rule reduction.

Fusing Supervised and Unsupervised Measures for Attribute Reduction

It is well-known that attribute reduction is a crucial action of rough set.The significant characteristic of attribute reduction is that it can reduce the dimensions of data with clear semantic explanations.Normally,the learning performance of attributes in derived reduct is much more crucial.Since related measures of rough set dominate the whole process of identifying qualified attributes and deriving reduct,those measures may have a direct impact on the performance of selected attributes in reduct.However,most previous researches about attribute reduction take measures related to either supervised perspective or unsupervised perspective,which are insufficient to identify attributes with superior learning performance,such as stability and accuracy.In order to improve the classification stability and classification accuracy of reduct,in this paper,a novel measure is proposed based on the fusion of supervised and unsupervised perspectives:(1)in terms of supervised perspective,approximation quality is helpful in quantitatively characterizing the relationship between attributes and labels;(2)in terms of unsupervised perspective,conditional entropy is helpful in quantitatively describing the internal structure of data itself.In order to prove the effectiveness of the proposed measure,18 University of CaliforniaIrvine(UCI)datasets and 2 Yale face datasets have been employed in the comparative experiments.Finally,the experimental results show that the proposed measure does well in selecting attributes which can provide distinguished classification stabilities and classification accuracies.

Lower Approximation Reduction in Ordered Information System with Fuzzy Decision

Attribute reduction is one of the most important problems in rough set theory. This paper introduces the concept of lower approximation reduction in ordered information systems with fuzzy decision. Moreover, the judgment theorem and discernable matrix are obtained, in which case an approach to attribute reduction in ordered information system with fuzzy decision is constructed. As an application of lower approximation reduction, some examples are applied to examine the validity of works obtained in our works..

Stable Routh-Padé-type approximation in model reduction of interval systems

An interval Pade-type approximation is introduced and then Routh-Pade-type method （IRPTM） is presented to model reduction in interval systems. The denominator in reduced model is obtained from the stable Routh table, and its numerator is constructed by the interval Pade-type definition. Compared to the existing Routh-Pade method, IRPTM does not need to solve linear interval equations theoretical analysis shows that IRPTM has example is given to illustrate our method. Hence, we do not have to compute smaller computational cost than that interval division in the process. Moreover, of Routh-Pade method. A typical numerical

Methods for Lower Approximation Reduction in Inconsistent Decision Table Based on Tolerance Relation

It is well known that most of information systems are based on tolerance relation instead of the classical equivalence relation because of various factors in real-world. To acquire brief decision rules from the information systems, lower approximation reduction is needed. In this paper, the lower approximation reduction is proposed in inconsistent information systems based on tolerance relation. Moreover, the properties are discussed. Furthermore, judgment theorem and discernibility matrix are obtained, from which an approach to lower reductions can be provided in the complicated information systems.

Fast assignment reduction in inconsistent incomplete decision systems

This paper focuses on fast algorithm for computing the assignment reduct in inconsistent incomplete decision systems. It is quite inconvenient to judge the assignment reduct directly ac-cording to its definition. We propose the judgment theorem for the assignment reduct in the inconsistent incomplete decision system, which greatly simplifies judging this type reduct. On such basis, we derive a novel attribute significance measure and construct the fast assignment reduction algorithm （F-ARA）, intended for com-puting the assignment reduct in inconsistent incomplete decision systems. Final y, we make a comparison between F-ARA and the discernibility matrix-based method by experiments on 13 Univer-sity of California at Irvine （UCI） datasets, and the experimental results prove that F-ARA is efficient and feasible.

Infinite series symmetry reduction solutions to the modified KdV-Burgers equation

From the point of view of approximate symmetry, the modified Korteweg-de Vries-Burgers （mKdV-Burgers） equation with weak dissipation is investigated. The symmetry of a system of the corresponding partial differential equations which approximate the perturbed mKdV-Burgers equation is constructed and the corresponding general approximate symmetry reduction is derived; thereby infinite series solutions and general formulae can be obtained. The obtained result shows that the zero-order similarity solution to the mKdV-Burgers equation satisfies the Painleve II equation. Also, at the level of travelling wave reduction, the general solution formulae are given for any travelling wave solution of an unperturbed mKdV equation. As an illustrative example, when the zero-order tanh profile solution is chosen as an initial approximate solution, physically approximate similarity solutions are obtained recursively under the appropriate choice of parameters occurring during computation.

Classification and Approximate Solutions to a Class of Perturbed Nonlinear Wave Equations

A complete approximate symmetry classification of a class of perturbed nonlinear wave equations isperformed using the method originated from Fushchich and Shtelen.Moreover,large classes of approximate invariantsolutions of the equations based on the Lie group method are constructed.

Model reduction using the genetic algorithmand routh approxi mations

A new method of model reduction combining the genetic algorithm（GA） with the Routh approximation method is presented. It is suggested that a high-order system can be approximated by a low-order model with a time delay. The denominator parameters of the reduced-order model are determined by the Routh approximation method, then the numerator parameters and time delay are identified by the GAL. The reduced-order models obtained by the proposed method will always be stable if the original system is stable and produce a good approximation to the original system in both the frequency domain and time domain. Two numerical examples show that the method is cornputationally simple and efficient.

A matrix Padé-type-Routh model reduction method for multivariable linear systems

In the present paper, the formulae for matrix Padé-type approximation were improved. The mixed model reduction method of matrix Padé-type-Routh for the multivariable linear systems was presented. A well-known example was given to illustrate that the mixed method is efficient.

Uncertainty and Reduction in Intuitionistic Fuzzy Covering Based Rough Sets

This paper proposes a novel pair of induced IF covering approximation operators in an IF covering approximation space,and discusses some basic properties about definable IFSs.A measure is defined to describe the uncertainty of IFSs in IF covering approximation spaces.Finally,we study the properties of reductions of an IF covering respectively based on induced IF covering approximation operators and IF covering approximation operators.

A quadratic programming method for optimal degree reduction of Bézier curves with G^1-continuity

This paper presents a quadratic programming method for optimal multi-degree reduction of B6zier curves with G^1-continuity. The L2 and I2 measures of distances between the two curves are used as the objective functions. The two additional parameters, available from the coincidence of the oriented tangents, are constrained to be positive so as to satisfy the solvability condition. Finally, degree reduction is changed to solve a quadratic problem of two parameters with linear constraints. Applications of degree reduction of Bezier curves with their parameterizations close to arc-length parameterizations are also discussed.

不完备形式背景的三支近似概念约简

不完备形式背景是形式概念分析中常见的数据表现形式,它可以反映对象和属性间的“具有”,“不具有”和“不确定是否具有”三种不同关系.本文在概念约简的框架下,从对象三支近似概念和属性三支近似概念两个角度出发,提出了不完备形式背景的三支近似概念约简理论;进一步通过定义三支近似代表概念矩阵以及三支近似代表概念函数,给出了求解不完备形式背景所有三支近似概念约简的方法.

一维周期基础的带隙解析解及其优化设计研究

周期结构的带隙特征为土木工程隔震领域提供了新思路,其中一维周期基础结构因其构造简单且经济适用而倍受关注。本文通过研究一维周期基础结构的振动特性,推导了用于计算一维橡胶-混凝土周期基础带隙的近似解析解,并在此基础上提出了基于上部结构共振区的一维橡胶-混凝土周期基础优化设计方法。频域和时域数值算例表明,由该优化方法所设计的周期基础可保证其上部结构在一个宽且连续的频段内均有较好的减震效果。

基于改进近似动态规划的安全约束机组组合算法

针对电力系统的安全约束机组组合问题,提出了一种基于改进近似动态规划的求解算法。考虑到安全约束机组组合是一个多时段混合整数非线性规划问题,难以直接求解,所提算法将其建模成马尔可夫决策过程,并利用近似动态规划实现解耦求解。为了处理近似动态规划决策空间过大的问题,所提算法通过决策空间缩减技术提前获得大部分机组的开停机状态,有效提高求解效率。在IEEE 39节点系统和一个实际系统上进行了算例分析,验证了所提算法的有效性。