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A MODIFIED ALGORITHM OF FINDING AN ELEMENT OF CLARKE GENERALIZED GRADIENT FOR A SMOOTH COMPOSITION OF MAX-TYPE FUNCTIONS A MODIFIED ALGORITHM OF FINDING AN ELEMENT OF CLARKE GENERALIZED GRADIENT FOR A SMOOTH COMPOSITION OF MAX-TYPE FUNCTIONS

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Journal of Computational Mathematics 2000年第5期18卷 513-520页

作者： Yan Gao (department of basic Science, china university of Mining and Technology-beijing, beijing 100083, china) （北京）中国矿业大学北京 100083

This paper refers to Clarke generalized gradient for a smooth composition of max-type functions of the form: f(x) = g(x, maxj∈J1 f1j(x),''', maxj∈Jm fmj(x)), where x ∈Rn, Ji, i = 1,''',m are finite index sets, g and fij,j ∈ Ji, i = 1,... )m, are continuously differentiable on Rm+n and Rn, respectively. In a previous paper) we proposed an algorithm of finding an element of Clarke generalized gradient for f, at a point. In that paper, finding an element of Clarke generalized gradient for f, at a point, is implemented by determining the compatibilities of systems of linear inequalities many times. So its computational amount is very expensive. In this paper) we will modify the algorithm to reduce the times that the compatibilities of systems of linear inequalities have to be determined.

关键词： Nonsmooth optimization Clarke generalized gradient Max-type function.

Complete solving of explicit evaluation of Gauss sums in the index 2 case

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Science china Mathematics 2010年第9期53卷 2525-2542页

作者： YANG Jing1,2 & XIA LingLi3,1department of Mathematical Sciences,Tsinghua university,beijing 100084,china 2Division of Mathematical Sciences,School of Physical and Mathematical Sciences,Nanyang Technological university,637371,Singapore 3basic courses department,beijing union university,beijing 100101,china 1. department of Mathematical Sciences Tsinghua University Beijing 100084 China2. Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University 637371 Nanyang Singapore3. Basic Courses Department Beijing Union University Beijing 100101 China

Let p be a prime number,N be a positive integer such that gcd(N,p) = 1,q = pf where f is the multiplicative order of p modulo *** χ be a primitive multiplicative character of order N over finite field *** paper studies the problem of explicit evaluation of Gauss sums G(χ) in the "index 2 case"(i.e.[(Z/NZ):] = 2).Firstly,the classification of the Gauss sums in the index 2 case is ***,the explicit evaluation of Gauss sums G(χλ)(1 λ N-1) in the index 2 case with order N being general even integer(i.e.N = 2r·N0,where r,N0 are positive integers and N0 3 is odd) is ***,combining with the researches before,the problem of explicit evaluation of Gauss sums in the index 2 case is completely solved.

关键词： Gauss sum Stickelberger’s theorem Stickelberger congruence Davenport-Hasse lifting formula Davenport-Hasse product formula

A heuristic path-estimating algorithm for large-scale real-time traffic information calculating

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Science china(Technological Sciences) 2008年第S1期51卷 165-174页

作者： L WeiFeng1,ZHU TongYu1,WU DongDong2,DAI Hong3 & HUANG Jian1 1 State Key Laboratory for Software Development Environment,Beihang university,beijing 100083,china 2 beijing Transportation Information Center,beijing 100055,china 3 college of Arts & Science,beijing union university,beijing 100083,china 1. State Key Laboratory for Software Development Environment Beihang University Beijing 100083 China 2. beijing Transportation Information Center Beijing 100055 China 3. college of Arts & Science Beijing Union University Beijing 100083 China

As the original Global Position System (GPS) data in Floating Car Data have the accuracy problem,this paper proposes a heuristic path-estimating algorithm for large-scale real-time traffic information calculating. It uses the heuristic search method,imports the restriction with geometric operation,and makes comparison between the vectors composed of the vehicular GPS points and the special road network model to search the set of vehicular traveling route candidates. Finally,it chooses the most optimal one according to weight. Experimental results indicate that the algorithm has considerable efficiency in accuracy (over 92.7%) and com-putational speed (max 8000 GPS records per second) when handling the GPS tracking data whose sampling rate is larger than 1 min even under complex road network conditions.

关键词： path-estimating floating car data (FCD) road network vector GPS

Wave mechanics in quantum phase space: hydrogen atom

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Progress in Natural Science:Materials International 2007年第2期17卷 144-149页

作者： LU Jun ( department of Foundational Science, beijing union university, beijing 100101, china department of Computational Science, National university of Singapore, Singapore 117543, Singapore) [a]department of Foundational Science Beijing Union University Beijing China [b]department of Computational Science National university of Singapore Singapore Singapore

The rigorous solutions of the stationary Schrodinger equation for hydrogen atom are solved with the wave-mechanics method within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. The "Fourier-like" projection transformations of wave function from the phase space to position and momentum spaces are extended to three-dimensional systems. The eigenfunctions in general position and momentum spaces could be obtained through the transformations from eigenfunction in the phase space.

关键词： quantum phase space hydrogen atom projection transformation.

From quantum spectra to classical orbits: Varying rectangular billiards

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Progress in Natural Science:Materials International 2008年第8期18卷 927-930页

作者： Jun Lu * department of Foundational Science, beijing union university, beijing 100101, china Institute of Theoretical Physics, Chinese Academy of Sciences, beijing 100080, china department of Foundational Science Beijing Union University Beijing 100101 China Institute of Theoretical Physics Chinese Academy of Sciences Beijing 100080 China

The quantum-spectrum function is introduced by means of the eigenvalues and the eigenfunctions in the system of varying rectangular billiards. It is found that the Fourier transform of the quantum-spectrum function corresponds with the classical orbits in the system. These results give new evidence about the classical-quantum correspondence.

关键词： Quantum spectrum Classical-quantum correspondence Semiclassical physics

Gas kinetic algorithm for flows in Poiseuille-like microchannels using Boltzmann model equation

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Science china(Physics,Mechanics & Astronomy) 2005年第4期48卷 496-512页

作者： LI Zhihui1,2, ZHANG Hanxin1 & FU Song2 ⒈ National Laboratory for CFD, Hypervelocity Aerodynamics Institute,china Aerodynamics Research and Development Center,Mianyang 621000,china 2.department of Engineering Mechanics,Tsinghua university,beijing 100084,china 1. National Laboratory for CFD Hypervelocity Aerodynamics Institute China Aerodynamics Research and Development Center 621000 Mianyang China 2. department of Engineering Mechanics Tsinghua University 100084 Beijing China

The gas-kinetic unified algorithm using Boltzmann model equation have been extended and developed to solve the micro-scale gas flows in Poiseuille-like micro-channels from Micro-Electro-Mechanical Systems (MEMS). The numerical modeling of the gas kinetic boundary conditions suitable for micro-scale gas flows is presented. To test the present method, the classical Couette flows with various Knudsen numbers, the gas flows from short microchannels like plane Poiseuille and the pressure-driven gas flows in two-dimensional short microchannels have been simulated and compared with the approximate solutions of the Boltzmann equation, the related DSMC results, the modified N-S solutions with slip-flow boundary theory, the gas-kinetic BGK-Burnett solutions and the experimental data. The comparisons show that the present gas-kinetic numerical algorithm using the mesoscopic Boltzmann simplified velocity distribution function equation can effectively simulate and reveal the gas flows in microchannels. The numerical experience indicates that this method may be a powerful tool in the numerical simulation of micro-scale gas flows from MEMS.

关键词： Boltzmann model equation, discrete velocity ordinate method, finite-difference scheme, Couette flow, Poiseuille flow, gas flow in short-microchannel.

Density-functional formula for strongly correlated systems

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Progress in Natural Science:Materials International 2005年第5期15卷 395-401页

作者： WANG Huaiyu1*, HAN Rushan2 and CHEN Nanxian1(1. department of Physics, Tsinghua university, beijing 100084, china 2. department of Physics, Peking university, beijing 100871, china) department of Physics Tsinghua University Beijing 100084 China department of Physics Peking University Beijing 100871 China

Density functional method is applied for strongly correlated systems. Based on the assumption that the systems are composed of electrons in singly-occupied orbitals and those in doubly-occupied orbitals, a set of self-consistent equations are obtained by standard variation procedure. The equations consist of two parts. One part is to solve the wave functions of the electrons in singly-occupied orbitals and the other is to solve the wave functions of the electrons in doubly-occupied orbitals. The physical meanings of the terms appearing in the equations are discussed.

关键词： density-functional theory strongly correlated systems doubly-occupied orbital two-particle wave function self-consistent equations.

CONTROLLED RELEASE FROM PDMAEMA GELS PREPARED BY GAMMA RADIATION

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Chinese Journal of Polymer Science 2002年第5期20卷 409-412页

作者： Ning Liu Min Yi Shuang-ji Chen Hong-fei Ha college of Chemistry and Molecular Engineering Peking university beijing 100871, china department of Biochemistry college of Applied Arts and Science, beijing union university beijing 100083. china Peking Univ Coll Chem & Mol Engn Beijing 100871 Peoples R China beijing union Univ Coll Appl Arts & Sci Dept Biochem Beijing 100083 Peoples R China

Poly(N,N-dimethylaminoethyl methacrylate) (polyDMAEMA) hydrogels prepared by gamma-irradiation showed obvious temperature-sensitivity in a temperature range of 38-40degreesC and pH-sensitivity at pH=2.5. They also showed electric response behavior although it was not typical. The hydrogels were used in controlled release at different pH, temperature, and electric voltage. The release rates of methylene blue (MB) from the gels at 52degreesC and pH=1.24 were faster than those at 20degreesC and pH=10.56, respectively. In addition, the release rate at a field voltage of 5.0 was also faster than that without electric field.

关键词： poly(N,N-dimethylaminoethyl methacrylate) environmental sensitivity hydrogel controlled release

Constraints on smoothness parameter and dark energy using observational H(z) data

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Research in Astronomy and Astrophysics 2011年第2期11卷 125-136页

作者： Hao-Ran Yu1,Tian Lan1,Hao-Yi Wan2,Tong-Jie Zhang1,3 and Bao-Quan Wang4 1 department of Astronomy,beijing Normal university,beijing 100875,china 2 business Office,beijing Planetarium,beijing 100044,china 3 Center for High Energy Physics,Peking university,beijing 100871,china 4 department of Physics,Dezhou university,Dezhou 253023,china department of Astronomy Beijing Normal University Beijing 100875 China business Office Beijing Planetarium Beijing 100044 China Center for High Energy Physics Peking University Beijing 100871 China department of Physics Dezhou University Dezhou 253023 China

With large-scale homogeneity,the universe is locally inhomogeneous,clustering into stars,galaxies and larger *** property is described by the smoothness parameter α which is defined as the proportion of matter in the form of intergalactic *** we consider the inhomogeneities over a small scale,there should be modifications of the cosmological distances compared to a homogenous *** and Roeder developed a second-order ordinary differential equation（D-R equation） that describes the angular diameter distance-redshift relation for inhomogeneous cosmological ***,we may obtain the D-R equation for observational H（z） data（OHD）.The density-parameter ΩM,the state of dark energy ω,and the smoothness-parameter α are constrained by a set of OHD in a spatially flat XCDM universe as well as a spatially flat XCDM *** using a χ2 minimization method,we get α = 0.81＋0.19-0.20 and ΩM = 0.32＋0.12-0.06 at the 1σ confidence *** we assume a Gaussian prior of ΩM = 0.26 ± 0.1,we get α = 0.93＋0.07-0.19 and ΩM = 0.31＋*** the XCDM model,α is constrained to α≥ 0.80 but ω is weakly constrained around-1,where ω describes the equation of state of the dark energy（pX = ωρX）.We conclude that OHD constrains the smoothness parameter more effectively than the data of SNe Ia and compact radio sources.

关键词： cosmology -- dark energy -- cosmological parameters