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AN APPLICABLE approximation method AND ITS APPLICATION

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Acta Mathematica Scientia 2015年第5期35卷 1189-1202页

作者：曹怀信陈峥立李莉余保民 College of Mathematics and Information Science Shaanxi Normal University School of Sciences Xi’an Polytechnic University College of Mathematics and Information Science Weinan Normal University

In this work, by choosing an orthonormal basis for the Hilbert space L^2[0, 1], an approximation method for finding approximate solutions of the equation （I ＋ K）x = y is proposed, called Haar wavelet approximation method （HWAM）. To prove the applicabifity of the HWAM, a more general applicability theorem on an approximation method （AM） for an operator equation Ax = y is proved first. As an application, applicability of the HWAM is obtained. Fhrthermore, four steps to use the HWAM are listed and three numerical examples are given in order to illustrate the effectiveness of the method.

关键词： Hilbert space applicability Haar wavelet approximation method operatorequation

Algebraic Independence by approximation method(Ⅱ)

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Acta Mathematica Sinica,English Series 2000年第3期16卷 395-398页

作者： Yaochen Zhu Institute of Applied Mathematics,Academy of Mathematics and Systems Sciences,Academia Sinica,Beijing 100080,P.R.China Fax:010-62541689 Institute of Applied Mathematics Academy of Mathematics and Systems Sciences Academia Sinica Beijing 100080 P. R. China

Let f（x）be a continued fraction with elements anx,where coefficients an are positive alge- braic *** the criterion of[1]for any nonzero real algebraic numbers α1,...,αs with distinct absolute values the algebraic independence of the values f（α1）,...,f（αs）is proved under certain as- sumption concerning only with an.For some transcendental numbers ζ the algebraic independence of values f（ζj）（j∈Z）is also established.

关键词： Algebraic independence Continued fraction approximation method

AN approximation method TO ESTIMATE THE HAUSDORFF MEASURE OF THE SIERPINSKI GASKET

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Analysis in Theory and Applications 2004年第2期20卷 158-166页

作者： RuanHuojun SuWeiyi ZhejiangUniversity China NanjingUniversity China

In this paper, we firstly define a decreasing sequence {Pn(S)} by the generation of the Sierpinski gasket where each Pn(S) can be obtained in finite steps. Then we prove that the Hausdorff measure Hs(S) of the Sierpinski gasket S can be approximated by {Pn(S)} with Pn(S)/(l + l/2n-3)s≤Hs(S)≤ Pn(S). An algorithm is presented to get Pn(S) for n ≤5. As an application, we obtain the best lower bound of Hs(S) till now: Hs(S)≥0.5631.

关键词： Hausdorff measure sierpinski gasket approximation method

Uniform modeling of parameter dependent nonlinear systems

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Journal of Zhejiang University-Science C(Computers and Electronics) 2012年第11期13卷 850-858页

作者： Najmeh EGHBAL Naser PARIZ Ali KARIMPOUR Department of Electrical Engineering Ferdowsi University of Mashhad

This paper addresses the problem of approximating parameter dependent nonlinear systems in a unified framework. This modeling has been presented for the first time in the form of parameter dependent piecewise affine systems. In this model, the matrices and vectors defining piecewise affine systems are affine functions of parameters. Modeling of the system is done based on distinct spaces of state and parameter, and the operating regions are partitioned into the sections that we call 'multiplied simplices'. It is proven that this method of partitioning leads to less complexity of the approximated model compared with the few existing methods for modeling of parameter dependent nonlinear systems. It is also proven that the approximation is continuous for continuous functions and can be arbitrarily close to the original one. Next, the approximation error is calculated for a special class of parameter dependent nonlinear systems. For this class of systems, by solving an optimization problem, the operating regions can be partitioned into the minimum number of hyper-rectangles such that the modeling error does not exceed a specified value. This modeling method can be the first step towards analyzing the parameter dependent nonlinear systems with a uniform method.

关键词： Parameter dependent nonlinear systems approximation method Parameter dependent piecewise affine systems Modeling

Flat limit of the de Sitter QFT in the rest frame vacuum

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Chinese Physics C 2021年第1期45卷 97-108页

作者： Ion I.Cotaescu West University of Timisoara V.Parvan Ave.4RO-300223 TimisoaraRomania

The problem of the flat limits of the scalar and spinor fields on the de Sitter expanding universe is considered in the traditional adiabatic vacuum and in the new rest frame vacuum we proposed recently,in which the frequencies are separated in the rest frames as in special *** is shown that only in the rest frame vacuum can the Minkowskian flat limit be reached naturally fbr any momentum,whereas in the adiabatic vacuum,this limit remains undefined in rest frames in which the momentum *** important role is played by the phases of the fundamental solutions in the rest frame vacuum,which must be regularized to obtain the desired Minkowskian flat *** procedure fixes the phases of the scalar mode functions and Dirac spinors,resulting in their definitive expressions derived *** physical consequenee is that,in the rest frame vacuum,the flat limits of the oneparticle operators are simply the corresponding operators of special relativity.

关键词： de Sitter space-time Klein-Gordon field Dirac field flat limit approximation method phase regularization

THE CONVERGENCE OF APPROACH PENALTY FUNCTION method FOR APPROXIMATE BILEVEL PROGRAMMING PROBLEM

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Acta Mathematica Scientia 2001年第1期21卷 69-76页

作者：万仲平周树民 Wuhan University Wuhan 430072 China Wuhan University of Science and Technology Wuhan 430070 China

In this paper, a new algorithm-approximate penalty function method is designed, which can be used to solve a bilevel optimization problem with linear constrained function. In this kind of bilevel optimization problem. the evaluation of the objective function is very difficult, so that only their approximate values can be obtained. This algorithm is obtained by combining penalty function method and approximation in bilevel programming. The presented algorithm is completely different from existing methods. That convergence for this algorithm is proved.

关键词： bilevel programming approximation method penalty function method convergence

Approximate method for Reliability Assessment of Complex Phased Mission Systems

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Journal of Shanghai Jiaotong university(Science) 2017年第2期22卷 247-251页

作者：周行李翔宇黄洪钟 Center for System Reliability and Safety University of Electronic Science and Technology of China School of Computer Engineering Chengdu Technological University

Phased-mission systems (PMSs) have wide applications in engineering practices, such as manmade satellites. Certain critical parts in the system, such as cold standby, hot standby and functional standby, are designed in redundancy architecture to achieve high reliability performance. State-space models such as Markov process have been used extensively in previous studies for reliability evaluation of PMSs with dynamic behaviors. The most popular way to deal with the dynamic behaviors is Markov process, but it is well known that Markov process is limited to exponential distribution. In practice, however, the lifetime of most machinery products can follow non-exponential distributions like the Weibull distribution which cannot be handled by the Markov process. In order to solve this kind of problem, we present a semi-Markov model combined with an approximation algorithm to analyze PMS reliability subjected to non-exponential failures. Furthermore, the accuracy of the approximation algorithm is investigated by comparing to an accurate solution, and a typical PMS (attitude and orbit control system) is analyzed to demonstrate the implementation of the method. © 2017, Shanghai Jiaotong University and Springer-Verlag Berlin Heidelberg.

关键词： phased mission system(PMS) dynamic behaviors approximation method semi-Markov process

A Comment the BooK“continuous-Time Markov Chains” by ***

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应用概率统计 1996年第1期 55-59页

作者： MUFA CHEN (Beijing Normal University, Beijing, 100875)

The book "Continuous-Time Markov Chains" by W. J. Anderson collects a large part of the development in the past thirty years. It is now a popular reference for the researchers on this subject or related fields. Unfortunately, due to a misunderstanding of the approximating methods, several results in the book are incorrectly stated or proved. Since the results are related to the present author’s work, to whom it may be a duty to correct the mistakes in order to avoid further confusion. We emphasize the approximating methods because they are useful in many situations.

关键词： Q-propess approximation method stochastically comparability.

An Arithmetic Property of Mahler's Series

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Acta Mathematica Sinica,English Series 1997年第3期13卷 407-412页

作者： Zhu Yaochen (Institute of Applied Mathematics,Academia Sinica,Beijing 100080,China) Institute of Applied Mathematics Academia Sinica Beijing China

In the present note the algebraic independence of values of several Mahler series with certain particular and different parameters at algebraic points is established by means of a typical approximation method handling... 详细信息

关键词： Algebraic independence Mahler’s series approximation method