Frontiers of Mathematics in China

ISSN 1673-3452

ISSN 1673-3576(Online)

CN 11-5739/O1

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, Volume 6 Issue 2

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RESEARCH ARTICLE
Learning rates for multi-kernel linear programming classifiers
Feilong CAO, Xing XING
Front Math Chin. 2011, 6 (2): 203-219.  
https://doi.org/10.1007/s11464-011-0103-3

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In this paper, we consider the learning rates of multi-kernel linear programming classifiers. Our analysis shows that the convergence behavior of multi-kernel linear programming classifiers is almost the same as that of multi-kernel quadratic programming. This is implemented by setting a stepping stone between the linear programming and the quadratic programming. An upper bound is presented for general probability distributions and distribution satisfying some Tsybakov noise condition.

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On uniqueness and existence of viscosity solutions to Hessian equations in exterior domains
Limei DAI, Jiguang BAO
Front Math Chin. 2011, 6 (2): 221-230.  
https://doi.org/10.1007/s11464-011-0109-x

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In this paper, we obtain the uniqueness and existence of viscosity solutions with prescribed asymptotic behavior at infinity to Hessian equations in exterior domains.

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On quantum cluster algebras of finite type
Ming DING
Front Math Chin. 2011, 6 (2): 231-240.  
https://doi.org/10.1007/s11464-011-0104-2

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We extend the definition of a quantum analogue of the Caldero-Chapoton map defined by D. Rupel. When Q is a quiver of finite type, we prove that the algebra ??|k|(Q) generated by all cluster characters is exactly the quantum cluster algebra ??|k|(Q).

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Regularity for weakly (K1,K2(x))-quasiregular mappings of several n-dimensional variables
Hongya GAO, Qiuhua HUANG, Fang QIAN
Front Math Chin. 2011, 6 (2): 241-251.  
https://doi.org/10.1007/s11464-011-0093-1

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The definition for weakly (K1,K2(x))-quasiregular mappings of several n-dimensional variables is given. A regularity property is obtained by using the stability result of Hodge decomposition, some analytical tools of Sobolev spaces, and differential geometry, which can be regarded as a generalization of the results due to T. Iwaniec and Hongya Gao.

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A lowest order divergence-free finite element on rectangular grids
Yunqing HUANG, Shangyou ZHANG
Front Math Chin. 2011, 6 (2): 253-270.  
https://doi.org/10.1007/s11464-011-0094-0

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It is shown that the conforming Q2,1;1,2-Q1 mixed element is stable, and provides optimal order of approximation for the Stokes equations on rectangular grids. Here, Q2,1;1,2=Q2,1×Q1,2, and Q2,1 denotes the space of continuous piecewise-polynomials of degree 2 or less in the x direction but of degree 1 in the y direction. Q1 is the space of discontinuous bilinear polynomials, with spurious modes filtered. To be precise, Q1 is the divergence of the discrete velocity space Q2,1;1,2. Therefore, the resulting finite element solution for the velocity is divergence-free pointwise, when solving the Stokes equations. This element is the lowest order one in a family of divergence-free element, similar to the families of the Bernardi-Raugel element and the Raviart-Thomas element.

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Spaces of type BLO on non-homogeneous metric measure
Haibo LIN, Dachun YANG
Front Math Chin. 2011, 6 (2): 271-292.  
https://doi.org/10.1007/s11464-011-0098-9

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Let (?,d,μ) be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. In this paper, we introduce the space RBLO(μ) and prove that it is a subset of the known space RBMO(μ) in this context. Moreover, we establish several useful characterizations for the space RBLO(μ). As an application, we obtain the boundedness of the maximal Calderón-Zygmund operators from L(μ) to RBLO(μ).

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A class of new braided Hopf algebras
Tianshui MA, Haiying LI, Shuanhong WANG
Front Math Chin. 2011, 6 (2): 293-308.  
https://doi.org/10.1007/s11464-011-0096-y

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We give the necessary and sufficient conditions for a general crossed product algebra equipped with the usual tensor product coalgebra structure to be a Hopf algebra. Furthermore, we obtain the necessary and sufficient conditions for the general crossed product Hopf algebra to be a braided Hopf algebra which generalizes some known results.

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A new characterization of Finsler metrics with constant flag curvature 1
Xiaohuan MO
Front Math Chin. 2011, 6 (2): 309-323.  
https://doi.org/10.1007/s11464-011-0099-8

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The purpose of this article is to derive an integral inequality of Ricci curvature with respect to Reeb field in a Finsler space and give a new geometric characterization of Finsler metrics with constant flag curvature 1.

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Collision local times of two independent fractional Brownian motions
Xiangjun WANG, Jingjun GUO, Guo JIANG
Front Math Chin. 2011, 6 (2): 325-338.  
https://doi.org/10.1007/s11464-011-0095-z

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In this paper, the collision local times for two independent fractional Brownian motions are considered as generalized white noise functionals. Moreover, the collision local times exist in L2 under mild conditions and chaos expansions are also given.

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Flows in 3-edge-connected bidirected graphs
Erling WEI, Wenliang TANG, Xiaofeng WANG
Front Math Chin. 2011, 6 (2): 339-348.  
https://doi.org/10.1007/s11464-011-0111-3

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It was conjectured by A. Bouchet that every bidirected graph which admits a nowhere-zero k-flow admits a nowhere-zero 6-flow. He proved that the conjecture is true when 6 is replaced by 216. O. Zyka improved the result with 6 replaced by 30. R. Xu and C. Q. Zhang showed that the conjecture is true for 6-edge-connected graph, which is further improved by A. Raspaud and X. Zhu for 4-edge-connected graphs. The main result of this paper improves Zyka’s theorem by showing the existence of a nowhere-zero 25-flow for all 3-edge-connected graphs.

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Quasineutral limit of bipolar quantum hydrodynamic model for semiconductors
Xiuhui YANG
Front Math Chin. 2011, 6 (2): 349-362.  
https://doi.org/10.1007/s11464-011-0102-4

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This paper is concerned with the quasineutral limit of the bipolar quantum hydrodynamic model for semiconductors. It is rigorously proved that the strong solutions of the bipolar quantum hydrodynamic model converge to the strong solution of the so-called quantum hydrodynamic equations as the Debye length goes to zero. Moreover, we obtain the convergence of the strong solutions of bipolar quantum hydrodynamic model to the strong solution of the compressible Euler equations with damping if both the Debye length and the Planck constant go to zero simultaneously.

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Singular values of nonnegative rectangular tensors
Yuning YANG, Qingzhi YANG
Front Math Chin. 2011, 6 (2): 363-378.  
https://doi.org/10.1007/s11464-011-0108-y

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The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. Some properties concerning the singular values of a real rectangular tensor were discussed by K. C. Chang et al. [J. Math. Anal. Appl., 2010, 370: 284-294]. In this paper, we give some new results on the Perron-Frobenius Theorem for nonnegative rectangular tensors. We show that the weak Perron-Frobenius keeps valid and the largest singular value is really geometrically simple under some conditions. In addition, we establish the convergence of an algorithm proposed by K. C. Chang et al. for finding the largest singular value of nonnegative primitive rectangular tensors.

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12 articles