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Oblique derivative problem for general Chaplygin-Rassias equations

Science China Mathematics 2008年第1期51卷 5-36页

作者： WEN GuoChun lmam, School of Mathematical Sciences, {1., beijing {3. China lmam School of Mathematical Sciences Peking University Beijing 100871 China

The present paper deals with the oblique derivative problem for general second order equations of mixed (elliptic-hyperbolic) type with the nonsmooth parabolic degenerate line $$K_1.(y)u_{xx} + \left| {K_2 (x)} \right|u_{yy} + a(x,y)u_x + b(x,y)u_y + c(x,y)u = - d(x,y)$$ in any plane domain D with the boundary ?D=Γ ∪ L 1.∪ L 2 ∪ L 3.∪ L 4, where Γ(? {y > 0}) ∈ C μ 2 (0 1. is a curve with the end points z = ?1. 1. L 1. L 2, L 3. L 4 are four characteristics with the slopes ?H 2(x)/H 1.y), H 2(x)/H 1.y),?H 2(x)/H 1.y),H 2(x)/H 1.y) (H 1.y) = √|K 1.y)|, H 2(x) = √|K 2(x)| in {y 1. 0, 0, 1.respectively. And the boundary condition possesses the form $$\frac{1.3.\frac{{\partial u}}{{\partial \nu }} = \frac{1.{{H(x,y)}}\operatorname{Re} \left[ {\overline {\lambda (z)} u_{\tilde z} } \right] = r(z), z \in \Gamma \cup L_1.\cup L_4 , \operatorname{Im} \left[ {\overline {\lambda (z)} u_{\tilde z} } \right]\left| {_{z = z_l } } \right. = b_l ,l = 1.2, u( - 1. = b_0 ,u(1. = b_3.,$$ in which z 1. z 2 are the intersection points of L 1. L 2, L 3. L 4 respectively. The above equations can be called the general Chaplygin-Rassias equations, which include the Chaplygin-Rassias equations $$K_1.(y)(M_2 (x)u_x )_x + M_1.(x)(K_2 (y)u_y )_y + r(x,y)u = f(x,y), in D$$ as their special case. The above boundary value problem includes the Tricomi problem of the Chaplygin equation: K(y)u xx+u yy = 0 with the boundary condition u(z) = ?(z) on Γ ∪ L 1.∪ L 4 as a special case. Firstly some estimates and the existence of solutions of the corresponding boundary value problems for the degenerate elliptic and hyperbolic equations of second order are discussed. Secondly, the solvability of the Tricomi problem, the oblique derivative problem and Frankl problem for the general Chaplygin-Rassias equations are proved. The used method in this paper is different from those in other papers, because the new notations W(z) = W(x + iy) = $u_{\tilde z} $ = [H

关键词： oblique derivative problem equations of mixed type nonsmooth degenerate line 3.J70 3.L80 3.N99

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Wavelets associated with Hankel transform and their Weyl transforms

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Science China Mathematics 2004年第3期47卷 393-400页

作者： PENG Lizhong MA Ruiqin lmam, School of Mathematical Sciences, {1., beijing {3. China 1. lmam School of Mathematical Sciences Peking University 100871 Beijing China

The Hankel transform is an important transform. In this paper, westudy the wavelets associated with the Hankel transform, thendefine the Weyl transform of the wavelets. We give criteria of itsboundedness and compactne... 详细信息

关键词： Bessel function, Hankel transform, wavelets, Weyltransform, boundedness, compactness.

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Probability parametric imaging with DCE-MRI data to identify prostate cancer in the peripheral zone

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Progress in Natural Science:Materials International 2006年第9期16卷 948-953页

作者： HUANG Xinrui, WANG Qiang, XU Yufeng, WANG Xiaoying and BAO Shanglian ( The Key Laboratory of Medical Physics and Engineering, peking university, {1. 1.0871. China The Medical School of peking university, {1. 1.0083. China 3. The First Hospital of peking university, {1. 1.003., China) [a]The key Laboratory of Medical Physics and Engineering Peking University Beijing China [b]The Medical School of peking university Beijing China [c]The First Hospital of peking university Beijing China

To quantitatively analyze prostate dynamic contrast-enhanced magnetic resonance images (DCE-MRI), a retrospective study was carried out to construct a mathematical model to predict whether a lesion is cancerous or not. With known diagnosis of 52 re-gions-of-interest (1. malignant, 26 benign and 1. normal) in peripheral zone (PZ) of prostate from 1. patients undergoing needle biopsy and 8 normal volunteers, we combined feature parameters (k1. k2 and tp) from the linear-slope model with logistic regression to determine the statistical probability model. Prostate cancerous probability maps were derived using the model on each subject to examine the correlation between the model-based prediction and the known diagnosis. Compared the model prediction results of 52 regions-of-interest with the histopathologic results, the total agreement rate was 90.4 % ; the agreement rate was 60 % for the malignant and 97. 6 % for the nonmalignant. The diagnostic results only referencing the probability maps by two radiologists showed a high agreement in tissue characteristics and lesion location with the combining results from other diagnostic techniques such as PSA (prostate specific antigen) level and MRS (magnetic resonance spectroscopy) .

关键词： time-intensity curve prostate DCE-MRI logistic regression probability parametric image.

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Optimal reactions for the synthesis of superheavy nucleus ^(270)Hs

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Science China(Physics,Mechanics & Astronomy) 2006年第6期49卷 641-648页

作者： LIU Zuhua1.3.& BAO Jingdong1.2 1. Key Laboratory of Beam Technology and Material Modification of Ministry of Education, {1. Normal university, {1. 1.0875, China 2. Physics Department, {1. Normal university, {1. 1.0875, China 3. China Institute of Atomic Energy, {1. 1.241.0} China Key Laboratory of Beam Technology Material Modification of Ministry of Education Beijing Normal University Beijing 100875 China Physics Department Beijing Normal University Beijing 100875 China China Institute of Atomic Energy Beijing 102413 China

The superheavy nucleus 270Hs is expected to be a “double-magic” deformed nucleus. We have calculated its cross sections of evaporation residue for the reactions 248Cm(26Mg,4n)270Hs, 244Pu(3.Si,4n)270Hs, 23.U(3.S,4n)270Hs and 226Ra(48Ca,4n)270Hs using a two-parameter Smoluchowski equation. It is found from our results that 226Ra(48Ca,4n)270Hs and 23.U(3.S,4n)270Hs are two optimal reactions for the synthesis of the superheavy nucleus 270Hs due to their large negative Q-values.

关键词： diffused fusion barrier,Smoluchowski equation,superheavy nuclei.

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The mathematical model of wetting angle of heterogeneous nucleation

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Science China(Technological Sciences) 2004年第4期47卷 391-397页

作者： HUANG Cheng1. SONG Bo1. MAO Jinghong2 & ZHAO Pei3.1. School of Metallurgical and Ecological Engineering, {1. of Science and Technology beijing, beijing 1.0083. China 2. School of Material Science, {1. of Science and Technology beijing, beijing 1.0083. China 3. Central Iron and Steel Research Institute, beijing 1.0081. China 1. School of Metallurgical and Ecological Engineering University of Science and Technology Beijing 100083 Beijing China 2. School of Material Science University of Science and Technology Beijing 100083 Beijing China 3. Central Iron and Steel Research Institute 100081 Beijing China

The heterogeneous nucleation has been analyzed in terms of the thermodynamic theory, and the mathematical model of wetting angle of heterogeneous nucleation has been developed based on the dynamic requirements of hetero- and homogeneous nucleation. With the wetting angel model, the wetting angles of different catalysts in liquid iron have been calculated. The result shows that the wetting angle of most catalysts is smaller than 40, and the value of f() is far less than 1. The wetting angles of TiN, TiC and RE2O3.are smaller, while the wetting angle of MnO is bigger among the ordinary nucleation catalysts. The wetting angles calculated by this model are in agreement with the results of other research work.

关键词： heterogeneous nucleation, mathematical model, wetting angle, undercooling.

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The lifting scheme of 4-channel orthogonal wavelet transforms

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Progress in Natural Science:Materials International 2006年第1期16卷 100-104页

作者： PENG Lizhong and CHU Xiaoyong (lmam, School of Mathematical Sciences, {1., beijing {3. China) [a]peking Universiy LMAM School of Mathematical Sciences Beijing China

The 4-channel smooth wavelets with linear phase and orthogonality are designed from the 2-channel orthogonal wavelets with high transfer vanishing moments. Reversely, for simple lifting scheme of such 4-channel orthogonal wavelet transforms, a new 2-channel orthogonal wavelet associated with this 4-channel wavelet is constructed. The new 2-channel wavelet has at least the same number of vanishing moments as the associated 4-channel one. Finally, by combining the two such 2-channel wavelet systems, the lifting scheme of 4-channel orthogonal wavelet transform, which has simple structure and is easy to apply, is presented.

关键词： lifting scheme 4-channel orthogonal wavelet 2-channel orthogonal wavelet vanishing moment transfer vanishing moment linear phase.

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Kinetic characteristics of continuous flow polymerase chain reaction chip: A numerical investigation

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Science China(Technological Sciences) 2010年第7期53卷 1967-1972页

作者： WANG ShiYing1.& WANG Wei2,3.1.School of Physics, {1., beijing 1.0871. China 2 Institute of Microelectronics, {1., beijing 1.0871. China 3.National Key Laboratory of Science and Technology on Micro/Nano Fabrication, beijing 1.0871. China 1. School of Physics Peking University Beijing 100871 China2. Institute of Microelectronics Peking University Beijing 100871 China3. National Key Laboratory of Science and Technology on Micro/Nano Fabrication Beijing 100871 China

Continuous flow PCR (polymerase chain reaction) chip holds impressive advantages compared to micro chamber PCR chip. In order to have better understanding of kinetic characteristics of continuous flow PCR chip, a comprehensive mathematical model is presented in this paper, including melting, annealing and extension phases of a typical PCR process which has the essence of a convection-diffusion-reaction system. Using this model, we can simulate the PCR process in series of reaction cycles. Numerical results show that the average sample velocity plays a significant role in affecting the amplification efficiency. Also, appropriate combination of the PCR mixture is important for high-quality DNA amplification. Giving a large initial DNA concentration range, the continuous flow PCR scheme holds excellent real-time detection ability theoretically. The present numerical model bridges the temperature distribution to the real DNA amplification, and thereby is able to successfully predict continuous flow PCR properties which are important for the chip design.

关键词： continuous flow PCR chip numerical simulation convection-diffusion-reaction system

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Recursive equations for compound distribution with the severity distribution of the mixed type

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Science China Mathematics 2005年第5期48卷 594-609页

作者： YANG Jingping, CHENG Shihong & WU Qin lmam, Department of Financial Mathematics, {1., beijing {3. China 1. lmam Department of Financial Mathematics Peking University 100871 Beijing China

In this paper, recursive equations are obtained for compound distribution with the number of claims belonging to (a, b)-family and the severity distribution of the mixed type. Numerical methods to solve these equation... 详细信息

关键词： recursive equation compound distribution (a b)-family severity distribution of the mixed type.

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Real solution isolation with multiplicity of zero-dimensional triangular systems

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Science China(Information Sciences) 2011年第1期54卷 60-69页

作者： ZHANG ZhiHai,FANG Tian & XIA BiCan lmam,School of Mathematical Sciences,{1.,beijing {3.China 1. lmam School of Mathematical Sciences Peking University Beijing 100871 China

Existing algorithms for isolating real solutions of zero-dimensional polynomial systems do not compute the multiplicities of the *** this paper,we define in a natural way the multiplicity of solutions of zero-dimensional triangular polynomial systems and prove that our definition is equivalent to the classical definition of local(intersection)*** we present an effective and complete algorithm for isolating real solutions with multiplicities of zero-dimensional triangular polynomial systems using our definition. The algorithm is based on interval arithmetic and square-free factorization of polynomials with real algebraic *** computational results on some examples from the literature are presented.

关键词： real solution isolation local multiplicity local ring polynomial system solving

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Study on the two and three-parameter working stability regions of 3.-TZP suspension

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Science China(Technological Sciences) 2006年第5期49卷 547-568页

作者： CHEN Youqiang1. 2, ZHANG Xinni3. CHEN Sha’ou3.& QI Ping3.1. Department of Chemistry, Qingdao university, Qingdao 266071. China 2. Department of Chemistry, Tsinghua university, beijing 1.0084, China 3. Department of Physics, Qingdao university, Qingdao 266071. China 1. Department of Chemistry Qingdao University Qingdao 266071 China 2. Department of Chemistry Tsinghua University Beijing 100084 China 3. Department of Physics Qingdao University Qingdao 266071 China

We determined the variation tendency of viscosity (η) at various solids load- ing (α) for 3.-TZP suspension as a function of dispersant concentration (Cw) using vis- cosity measurement, measured the diameter of particles as a function of Cw in very dilute suspension using light scattering method, and obtained the surface features of the sedi- ment of suspension with different Cw using SEM. We also discussed the influence of the microstructure of adsorbed polymer layers on particles and their interactions in the dis- persing medium on the stability of suspension. Then two different stable states and two different unstable states for 3.-TZP suspension were given. Accordingly, the (Cw, η), (Cw, α), and (α, η) two-parameter, and (Cw, α, η) three-parameter stability maps were con- structed. Based on the DLVO theory calculations, the dispersant concentration, particle distance (r), and interparticle potential energy (VT) maps with various solids loading for 3.-TZP aqueous suspension were also constructed. In (Cw, r) two-parameter coordinate system, the stable motion region map for particles was established. The Cw, α, and VT,max (the highest potential energy) surface of the potential barrier map was obtained through calculation and simulation on the basis of the surfaces of Cw, r, VT maps introduced above. The (Cw, α) two-parameter stability map was then obtained from the Cw, α, VT,max map. The results showed that this theoretical map can qualitatively prove the experimentally obtained results—the existence of different dispersed states of particles in suspension system and the variation tendency of suspension stability with changing each parame- ter—were reasonable.

关键词： 3.-TZP, suspension, stability, DLVO theory, interparticle potential energy.

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