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20 July 2009, Volume 29 Issue 4 Previous Issue    Next Issue

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TWO-DIMENSIONAL RIEMANN PROBLEMS: FROM SCALAR CONSERVATION LAWS TO COMPRESSIBLE EULER EQUATIONS LI Jie-Quan, CHENG Mo-Cheng, ZHANG Tong, ZHENG Yu-Xi Acta mathematica scientia,Series B. 2009, 29 (4):  777-802.  DOI: 10.1016/S0252-9602(09)60070-9 Abstract ( 2068 )   RICH HTML PDF (505KB) ( 2000 )   Save In this paper we survey the authors’ and related work on two-dimensional Rie-mann problems for hyperbolic conservation laws, mainly those related to the compressible Euler equations in gas dynamics. It contains four sections: 1. Historical review. 2. Scalar conservation laws. 3. Euler equations. 4. Simplified models. References | Related Articles | Metrics

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THE ACCELERATED SEARCH-EXTENSION METHOD FOR COMPUTING MULTIPLE SOLUTIONS OF SEMILINEAR PDEs LIU Ti-Wu, XIE Zi-Qing, CHEN Chuan-Miao Acta mathematica scientia,Series B. 2009, 29 (4):  803-816.  DOI: 10.1016/S0252-9602(09)60071-0 Abstract ( 1384 )   RICH HTML PDF (796KB) ( 2337 )   Save In this paper, we propose an accelerated search-extension method (ASEM)based on the interpolated coefficient finite element method, the search-extension method (SEM) and the two-grid method to obtain the multiple solutions for semilinear elliptic equations. This strategy is not only successfully implemented to obtain multiple solutions for a class of semilinear elliptic boundary value problems, but also reduces the expensive computation greatly. The numerical results in 1-D and 2-D cases will show the efficiency of our approach. References | Related Articles | Metrics

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MARKOWITZ STRATEGIES REVISED YAN Jia-An, ZHOU Xun-Yu Acta mathematica scientia,Series B. 2009, 29 (4):  817-828.  DOI: 10.1016/S0252-9602(09)60072-2 Abstract ( 1048 )   RICH HTML PDF (190KB) ( 1302 )   Save Continuous-time Markowitz’s mean-variance efficient strategies are modified by parameterizing a critical quantity. It is shown that these parameterized Markowitz strategies could reach the original mean target with arbitrarily high probabilities. This, in turn, motivates the introduction of certain stopped strategies where stock holdings are liquidated whenever the parameterized Markowitz strategies reach the present value of the mean target. The risk aspect of the revised Markowitz strategies are examined via expected discounted loss from the initial budget. A new portfolio selection model is suggested based on the results of the paper. References | Related Articles | Metrics

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POSITIVELY CURVED COMPLETE NONCOMPACT KÄHLER MANIFOLDS CHEN Bing-Long, SHU Xi-Beng Acta mathematica scientia,Series B. 2009, 29 (4):  829-845.  DOI: 10.1016/S0252-9602(09)60073-4 Abstract ( 911 )   RICH HTML PDF (238KB) ( 1293 )   Save In this paper we give a partial affirmative answer to a conjecture of Greene-Wu and Yau. Among other things, we prove that a complete noncompact K¨ahler surface with positive and bounded sectional curvature and with finite analytic Chern number c1(M)2 is biholomorphic to C2. References | Related Articles | Metrics

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ON THE 3D VISCOUS PRIMITIVE EQUATIONS OF THE LARGE-SCALE ATMOSPHERE GUO Bai-Ling, HUANG Dai-Wen Acta mathematica scientia,Series B. 2009, 29 (4):  846-866.  DOI: 10.1016/S0252-9602(09)60074-6 Abstract ( 1956 )   RICH HTML PDF (234KB) ( 1419 )   Save This paper is devoted to considering the three-dimensional viscous primitive equations of the large-scale atmosphere. First, we prove the global well-posedness for the primitive equations with weaker initial data than that in [11]. Second, we obtain the existence of smooth solutions to the equations. Moreover, we obtain the compact global attractor in V for the dynamical system generated by the primitive equations of large-scale atmosphere, which improves the result of [11]. References | Related Articles | Metrics

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AN UNGUIDED TOUR STARTED FROM CHIRALITY JIANG Ba-Ju, WANG Shi-Cheng Acta mathematica scientia,Series B. 2009, 29 (4):  867-880.  DOI: 10.1016/S0252-9602(09)60075-8 Abstract ( 937 )   RICH HTML PDF (225KB) ( 966 )   Save This survey article records an unguided mathematical tour by topologists at Peking University and their collaborators in the last ten years. The tour started from research on chirality and, attracted by questions around attractors, led to a zigzag path across topology and dynamics. People who joined us in this tour at various stages include Ding Fan, Liu Yi, Ni Yi, Pan Jianzhong, Yao Jiangang, Zheng Hao and Zhou Qing. Conversations with Robert Edwards, Wen Lan and others, added to the twists and turns that made the trip more fun. This article benefits from related lectures by these authors, in many conferences, universi-ties, as well as high schools. References | Related Articles | Metrics

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ON THE ASYMPTOTIC BEHAVIOR OF HOLOMORPHIC ISOMETRIES OF THE POINCARÉ|DISK INTO BOUNDED SYMMETRIC DOMAINS Ngaiming Mok Acta mathematica scientia,Series B. 2009, 29 (4):  881-902.  DOI: 10.1016/S0252-9602(09)60076-X Abstract ( 963 )   RICH HTML PDF (283KB) ( 1476 )   Save In this article we study holomorphic isometries of the Poincar´e disk into bounded symmetric domains. Earlier we solved the problem of analytic continuation of germs of holomorphic maps between bounded domains which are isometries up to normal-izing constants with respect to the Bergman metric, showing in particular that the graph V0 of any germ of holomorphic isometry of the Poincar´e disk Δ into an irreducible bounded symmetric domain  Ω   CN in its Harish-Chandra realization must extend to an affine-algebraic subvariety V ⊂ C×CN = CN+1, and that the irreducible component of V ∩(Δ×Ω) containing V0 is the graph of a proper holomorphic isometric embedding F : Δ → Ω. In this article we study holomorphic isometric embeddings which are asymptotically geodesic at a general boundary point b ∈ ∂Δ. Starting with the structural equation for holomor-phic isometries arising from the Gauss equation, we obtain by covariant differentiation an identity relating certain holomorphic bisectional curvatures to the boundary behavior of the second fundamental form σ of the holomorphic isometric embedding. Using the nonpositivity of holomorphic bisectional curvatures on a bounded symmetric domain, we prove that ||σ|| must vanish at a general boundary point either to the order 1 or to the order 1/2 , called a holomorphic isometry of the first resp. second kind. We deal with special cases of non-standard holomorphic isometric embeddings of such maps, showing that they must be asymptotically totally geodesic at a general boundary point and in fact of the first kind whenever the target domain is a Cartesian product of complex unit balls. We also study the boundary behavior of an example of holomorphic isometric embedding from the Poincaré disk into a Siegel upper half-plane by an explicit determination of the boundary behavior of holomorphic sectional curvatures in the directions tangent to the embedded Poincar´e disk, showing that the map is indeed asymptotically totally geodesic at a general boundary point and of the first kind. For the metric computation we make use of formulas for symplectic geometry on Siegel upper half-planes. References | Related Articles | Metrics

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MULTIPLE SOLUTIONS FOR THE p&q-LAPLACIAN PROBLEM WITH CRITICAL EXPONENT LI Gong-Bao, ZHANG Guo Acta mathematica scientia,Series B. 2009, 29 (4):  903-918.  DOI: 10.1016/S0252-9602(09)60077-1 Abstract ( 1027 )   RICH HTML PDF (222KB) ( 2387 )   Save In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p&q-Laplacian type involving the critical Sobolev exponent: −△pu − △qu = |u|p*−2u + μ|u|r−2u in Ω, u|∂Ω= 0, where Ω ⊂ RN is a bounded domain, N > p, p* = Np /N−p is the critical Sobolev exponent and μ > 0. We prove that if 1 < r < q < p < N, then there is a μ0 > 0, such that for any μ ∈ (0, μ0), the above mentioned problem possesses infinitely many weak solutions. Our result generalizes a similar result in [8] for p-Laplacian type problem. References | Related Articles | Metrics

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THE NAVIER-STOKES EQUATIONS WITH THE KINEMATIC AND VORTICITY BOUNDARY CONDITIONS ON NON-FLAT BOUNDARIES Gui-Qiang Chen, Dan Osborne, Zhongmin Qian Acta mathematica scientia,Series B. 2009, 29 (4):  919-948.  DOI: 10.1016/S0252-9602(09)60078-3 Abstract ( 1217 )   RICH HTML PDF (294KB) ( 1724 )   Save We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in Rn with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure p can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolute boundary conditions. Based on this, we first establish the well-posedness for an appropriate local linearized problem with the absolute boundary conditions and the initial condition (without the incompressibility condition), which establishes a velocity mapping. Then we develop apriori estimates for the velocity mapping, especially involving the Sobolev norm for the time-derivative of the mapping to deal with the complicated boundary conditions, which leads to the existence of the fixed point of the mapping and the existence of solutions to our initial-boundary value problem. Finally, we establish that, when the viscosity coefficient tends zero, the strong solutions of the initial-boundary value problem in Rn(n ≥ 3) with nonhomogeneous vorticity boundary condition converge in L2 to the corresponding Euler equations satisfying the kinematic condition. References | Related Articles | Metrics

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CLASSIFICATION OF POSITIVE SOLUTIONS FOR NONLINEAR DIFFERENTIAL AND INTEGRAL SYSTEMS WITH CRITICAL EXPONENTS Wenxiong Chen, Congming Li Acta mathematica scientia,Series B. 2009, 29 (4):  949-960.  DOI: 10.1016/S0252-9602(09)60079-5 Abstract ( 1098 )   RICH HTML PDF (186KB) ( 2086 )   Save We classify all positive solutions for the following integral system: ui(x) =∫Rn1/ |x − y|n−α fi(u(y))dy, x ∈ Rn, i = 1, · · · , m, 0 < α < n, and u(x) = (u1(x), u2(x), · · · , um(x)). Here fi(u), 1 ≤ i ≤ m, are real-valued functions of homogeneous degree n+α/ n−α and are monotone nondecreasing with respect to all the independent variables u1, u2, · · ·, um. In the special case n ≥ 3 and α = 2, we show that the above system is equivalent to the following elliptic PDE system: −△ui(x) = fi(u(x)), x ∈ Rn, i = 1, · · · , m, and u(x) = (u1(x), u2(x), · · · , um(x)). This system is closely related to the stationary Schr¨odinger system with critical exponents for Bose-Einstein condensate. References | Related Articles | Metrics

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N = 2 SCVA’s FROM A GENERALIZED CALABI-YAU MANIFOLD AND MIRROR SYMMETRY HU Sen, MA Wen-Ye, QIU Jing-Pei Acta mathematica scientia,Series B. 2009, 29 (4):  961-972.  DOI: 10.1016/S0252-9602(09)60080-1 Abstract ( 1110 )   RICH HTML PDF (199KB) ( 1096 )   Save We construct an N = 2 superconformal vertex algebra(SCVA) from a gener-alized Calabi-Yau manifold and compute the BRST cohomology of its associated topolog-ical vertex algebras. We show that the BRST cohomology coincides with the generalized Dobeault cohomology. We show that the two topological vertex algebras constructed from the N = 2 SCVA by A and B twist respectively are mirror pairs. References | Related Articles | Metrics

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STRONG SHOCKS IN NONISENTROPIC GAS DYNAMICS Constantine M. Dafermos Acta mathematica scientia,Series B. 2009, 29 (4):  973-979.  DOI: 10.1016/S0252-9602(09)60081-3 Abstract ( 1161 )   RICH HTML PDF (133KB) ( 1142 )   Save In nonisentropic gas dynamics, with general equations of state, strong com-pressive shocks satisfying the Liu E-condition may violate the Second Law of thermody-namics. References | Related Articles | Metrics

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EXACT CONTROLLABILITY FOR FIRST ORDER QUASILINEAR HYPERBOLIC SYSTEMS WITH VERTICAL CHARACTERISTICS LI Da-Qian, RAO Ba-Feng Acta mathematica scientia,Series B. 2009, 29 (4):  980-990.  DOI: 10.1016/S0252-9602(09)60082-5 Abstract ( 1144 )   RICH HTML PDF (156KB) ( 1547 )   Save We consider first order quasilinear hyperbolic systems with vertical charac-teristics. It was shown in [4] that such systems can be exactly controllable with the help of internal controls applied to the equations corresponding to zero eigenvalues. However, it is possible that, for physical or engineering reasons, we can not put any control on the equations corresponding to zero eigenvalues. In this paper, we will establish the ex-act controllability only by means of physically meaningful internal controls applied to the equations corresponding to non-zero eigenvalues. We also show the exact controllability for a very simplified model by means of switching controls. References | Related Articles | Metrics

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SYMBOLIC VERSOR COMPRESSION ALGORITHM Li Hongbo Acta mathematica scientia,Series B. 2009, 29 (4):  991-1004.  DOI: 10.1016/S0252-9602(09)60083-7 Abstract ( 930 )   RICH HTML PDF (187KB) ( 1180 )   Save In an inner-product space, an invertible vector generates a reflection with re-spect to a hyperplane, and the Clifford product of several invertible vectors, called a versor in Clifford algebra, generates the composition of the corresponding reflections, which is an orthogonal transformation. Given a versor in a Clifford algebra, finding another sequence of invertible vectors of strictly shorter length but whose Clifford product still equals the input versor, is called versor compression. Geometrically, versor compression is equivalent to decomposing an orthogonal transformation into a shorter sequence of reflections. This paper proposes a simple algorithm of compressing versors of symbolic form in Clifford algebra. The algorithm is based on computing the intersections of lines with planes in the corresponding Grassmann-Cayley algebra, and is complete in the case of Euclidean or Minkowski inner-product space. References | Related Articles | Metrics

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ASYMPTOTICS OF ORTHOGONAL POLYNOMIALS VIA THE RIEMANN-HILBERT APPROACH R. Wong, DIAO Yo-Qiu Acta mathematica scientia,Series B. 2009, 29 (4):  1005-1034.  DOI: 10.1016/S0252-9602(09)60084-9 Abstract ( 1072 )   RICH HTML PDF (307KB) ( 2105 )   Save In this survey we give a brief introduction to orthogonal polynomials, including a short review of classical asymptotic methods. Then we turn to a discussion of the Riemann-Hilbert formulation of orthogonal polynomials, and the Deift & Zhou method of steepest descent. We illustrate this new approach, and a modified version, with the Hermite polynomials. Other recent progress of this method is also mentioned, including applications to discrete orthogonal polynomials, orthogonal polynomials on curves, multiple orthogonal polynomials, and certain orthogonal polynomials with singular behavior. References | Related Articles | Metrics

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OPTIMAL CONVERGENCE RATES OF LANDAU EQUATION WITH EXTERNAL FORCING IN THE WHOLE SPACE YANG Tong, YU Hong-Bei Acta mathematica scientia,Series B. 2009, 29 (4):  1035-1062.  DOI: 10.1016/S0252-9602(09)60085-0 Abstract ( 1250 )   RICH HTML PDF (273KB) ( 1294 )   Save In this paper, we combine the method of constructing the compensating func-tion introduced by Kawashima and the standard energy method for the study on the Landau equation with external forcing. Both the global existence of solutions near the time asymptotic states which are local Maxwellians and the optimal convergence rates are obtained. The method used here has its own advantage for this kind of studies because it does not involve the spectrum analysis of the corresponding linearized operator. References | Related Articles | Metrics

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RITT-WU’S CHARACTERISTIC SET METHOD FOR ORDINARY DIFFERENCE POLYNOMIAL SYSTEMS WITH ARBITRARY ORDERING GAO Xiao-Shan, YUAN Chun-Meng, ZHANG Gui-Lin Acta mathematica scientia,Series B. 2009, 29 (4):  1063-1080.  DOI: 10.1016/S0252-9602(09)60086-2 Abstract ( 1097 )   RICH HTML PDF (242KB) ( 952 )   Save In this paper, a Ritt-Wu’s characteristic set method for ordinary difference systems is proposed, which is valid for any admissible ordering. New definition for irre-ducible chains and new zero decomposition algorithms are also proposed. References | Related Articles | Metrics

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ON THE BASIC REPRODUCTION NUMBER OF GENERAL BRANCHING PROCESSES LA Guo-Lie, MA Zhi-Meng, SUN Su-Yong Acta mathematica scientia,Series B. 2009, 29 (4):  1081-1094.  DOI: 10.1016/S0252-9602(09)60087-4 Abstract ( 1118 )   RICH HTML PDF (205KB) ( 1256 )   Save Under a very general condition (TNC condition) we show that the spectral radius of the kernel of a general branching process is a threshold parameter and hence plays a role as the basic reproduction number in usual CMJ processes. We discuss also some properties of the extinction probability and the generating operator of general branching processes. As an application in epidemics, in the final section we suggest a generalization of SIR model which can describe infectious diseases transmission in an inhomogeneous population. References | Related Articles | Metrics

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BOUND STATES FOR A STATIONARY NONLINEAR SCHRÖDINGER-POISSON SYSTEM WITH SIGN-CHANGING POTENTIAL IN R3 JIANG Yong-Sheng, ZHOU Huan-Song Acta mathematica scientia,Series B. 2009, 29 (4):  1095-1104.  DOI: 10.1016/S0252-9602(09)60088-6 Abstract ( 1176 )   RICH HTML PDF (190KB) ( 1495 )   Save We study the following Schr¨odinger-Poisson system  (Pλ)    −Δu + V (x)u + λφ(x)u = Q(x)up, x ∈R3,              −Δφ = u2, lim|x|→+∞ φ (x) = 0, u > 0, where λ > 0 is a parameter, 1 < p < +1, V (x) and Q(x) are sign-changing or non-positive functions in L∞(R3). When V (x) ≡ Q(x) ≡ 1, D. Ruiz [19] proved that (Pλ) with p ∈ (2, 5) has always a positive radial solution, but (Pλ) with p ∈ (1, 2] has solution only if λ > 0 small enough and no any nontrivial solution if λ ≥1/4 . By using sub-supersolution method, we prove that there exists λ0 > 0 such that (Pλ) with p ∈ (1,+∞) has always a bound state (H1(R3) solution) for λ ∈ [0, λ0) and certain functions V (x) and Q(x) in L∞(R3). Moreover, for every λ ∈ [0, λ0), the solutions uλ of (Pλ) converges, along a subsequence, to a solution of (P0) in H1 as λ → 0. References | Related Articles | Metrics

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A HARNACK TYPE INEQUALITY FOR SOME CONFORMALLY INVARIANT EQUATIONS ON HALF EUCLIDEAN SPACE Aobing Li, YanYan Li Acta mathematica scientia,Series B. 2009, 29 (4):  1105-1112.  DOI: 10.1016/S0252-9602(09)60089-8 Abstract ( 957 )   RICH HTML PDF (160KB) ( 1127 )   Save We establish a Harnack type inequality on half Euclidean space for general conformally invariant fully nonlinear elliptic equations of second order. References | Related Articles | Metrics