15 January 2015, Volume 31 Issue 1

    

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  Functionals for Multilinear Fractional Embedding

  William BECKNER

  Acta Mathematica Sinica. 2015, 31(1): 1-28. https://doi.org/10.1007/s10114-015-4321-6

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  A novel representation is developed as a measure for multilinear fractional embedding. Corresponding extensions are given for the Bourgain-Brezis-Mironescu theorem and Pitt's inequality. New results are obtained for diagonal trace restriction on submanifolds as an application of the Hardy-Littlewood-Sobolev inequality. Smoothing estimates are used to provide new structural understanding for density functional theory, the Coulomb interaction energy and quantum mechanics of phase space. Intriguing connections are drawn that illustrate interplay among classical inequalities in Fourier analysis.
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  Quantum McKay Correspondence for Disc Invariants of Toric Calabi-Yau 3-orbifolds

  Hua-Zhong KE, Jian ZHOU

  Acta Mathematica Sinica. 2015, 31(1): 29-34. https://doi.org/10.1007/s10114-015-3281-1

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  We announce a result on quantum McKay correspondence for disc invariants of outer legs in toric Calabi-Yau 3-orbifolds, and illustrate our method in a special example [C³/Z₅(1, 1, 3)].
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  Compact Toeplitz Operators on the Weighted Dirichlet Space

  Yu Feng LU, Yin Yin HU, Liu LIU

  Acta Mathematica Sinica. 2015, 31(1): 35-43. https://doi.org/10.1007/s10114-015-3380-z

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  In this paper, we completely characterize the compactness of Toeplitz operators with continuous symbol on the weighted Dirichlet space.
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  Self-similar Solutions of the Navier-Stokes Equations on Weak Weighted Lorentz Spaces

  Hong Liang LI, Jie Cheng CHEN

  Acta Mathematica Sinica. 2015, 31(1): 44-60. https://doi.org/10.1007/s10114-015-3484-5

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  In the present paper, we prove the existence of global solutions for the Navier-Stokes equations in Rⁿ when the initial velocity belongs to the weighted weak Lorentz space Λ^n,∞(u) with a sufficiently small norm under certain restriction on the weight u. At the same time, self-similar solutions are induced if the initial velocity is, besides, a homogeneous function of degree -1. Also the uniqueness is discussed.
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  Nonpositively Curved Almost Hermitian Metrics on Product of Compact Almost Complex Manifolds

  Chengjie YU

  Acta Mathematica Sinica. 2015, 31(1): 61-70. https://doi.org/10.1007/s10114-015-4226-4

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  In this paper, we give a classification of almost Hermitian metrics with nonpositive holomorphic bisectional curvature on a product of compact almost complex manifolds. This generalizes previous results of Zheng [Ann. of Math. (2), 137(3), 671-673 (1993)] and the author [Proc. Amer. Math. Soc., 139(4), 1469-1472 (2011)].
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  Conditional Log-Laplace Functional for a Class of Branching Processes in Random Environments

  Hao WANG

  Acta Mathematica Sinica. 2015, 31(1): 71-90. https://doi.org/10.1007/s10114-015-3741-7

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  A conditional log-Laplace functional (CLLF) for a class of branching processes in random environments is derived. The basic idea is the decomposition of a dependent branching dynamic into a no-interacting branching and an interacting dynamic generated by the random environments. CLLF will play an important role in the investigation of branching processes and superprocesses with interaction.
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  Variable Selection for Fixed Effects Varying Coefficient Models

  Gao Rong LI, Heng LIAN, Peng LAI, Heng PENG

  Acta Mathematica Sinica. 2015, 31(1): 91-110. https://doi.org/10.1007/s10114-015-3159-2

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  We consider the problem of variable selection for the fixed effects varying coefficient models. A variable selection procedure is developed using basis function approximations and group nonconcave penalized functions, and the fixed effects are removed using the proper weight matrices. The proposed procedure simultaneously removes the fixed individual effects, selects the significant variables and estimates the nonzero coefficient functions. With appropriate selection of the tuning parameters, an asymptotic theory for the resulting estimates is established under suitable conditions. Simulation studies are carried out to assess the performance of our proposed method, and a real data set is analyzed for further illustration.
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  Commuting Structure Jacobi Operator for Real Hypersurfaces in Complex Two-plane Grassmannians

  Carlos J. G. MACHADO, Juan de Dios PÉREZ, Young Jin SUH

  Acta Mathematica Sinica. 2015, 31(1): 111-122. https://doi.org/10.1007/s10114-015-1765-7

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  We classify real hypersurfaces in complex two-plane Grassmannians whose structure Jacobi operator commutes either with any other Jacobi operator or with the normal Jacobi operator.
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  Nonlinear Degenerate Parabolic Equations with Time-dependent Singular Potentials for Baouendi-Grushin Vector Fields

  Jun Qiang HAN, Qian Qiao GUO

  Acta Mathematica Sinica. 2015, 31(1): 123-139. https://doi.org/10.1007/s10114-015-3757-z

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  In this paper, we are concerned with the following three types of nonlinear degenerate parabolic equations with time-dependent singular potentials:
  ∂u^q/∂t=▽_α·(‖z‖^-pγ|▽_αu|^p-2▽_αu)+V(z, t)u^p-1,
  ∂u^q/∂t=▽_α·(‖z‖^-2γ▽_αu^m)+V(z, t)u^m,
  ∂u^q/∂t=u^μ▽_α·(u^τ|▽_αu|^p-2▽_αu)+V(z, t)u^p-1+μ+τ
  in a cylinder Ω×(0, T) with initial condition u(z, 0)=u₀(z) ≥ 0 and vanishing on the boundary ∂Ω×(0, T), where Ω is a Carnot-Carathéodory metric ball in R^d+k and the time-dependent singular potential function is V(z, t) ∈ L_loc¹ (Ω×(0, T)). We investigate the nonexistence of positive solutions of these three problems and present our results on nonexistence.
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  Measure of Noncompactness and Semilinear Nonlocal Functional Differential Equations in Banach Spaces

  Qi Xiang DONG, Gang LI

  Acta Mathematica Sinica. 2015, 31(1): 140-150. https://doi.org/10.1007/s10114-015-3097-z

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  This paper is concerned with the measure of noncompactness in the spaces of continuous functions and semilinear functional differential equations with nonlocal conditions in Banach spaces. The relationship between the Hausdorff measure of noncompactness of intersections and the modulus of equicontinuity is studied for some subsets related to the semigroup of linear operators in Banach spaces. The existence of mild solutions is obtained for a class of nonlocal semilinear functional differential equations without the assumption of compactness or equicontinuity on the associated semigroups of linear operators.
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  Entropy and Renormalized Solutions for Nonlinear Elliptic Problem Involving Variable Exponent and Measure Data

  Mohamed Badr BENBOUBKER, Houssam CHRAYTEH, Mostafa EL MOUMNI, Hassane HJIAJ

  Acta Mathematica Sinica. 2015, 31(1): 151-169. https://doi.org/10.1007/s10114-015-3555-7

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  We give an existence result of entropy and renormalized solutions for strongly nonlinear elliptic equations in the framework of Sobolev spaces with variable exponents of the type:
  -div (a(x, u,▽u)+φ(u))+g(x, u,▽u)=μ,
  where the right-hand side belongs to L¹(Ω)+W^-1,p'(x)(Ω), -div(a(x, u,▽u)) is a Leray-Lions operator defined from W^-1,p'(x)(Ω) into its dual and φ∈C⁰(R,R^N). The function g(x, u,▽u) is a non linear lower order term with natural growth with respect to |▽u| satisfying the sign condition, that is, g(x, u,▽u)u ≥ 0.
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  The Cycle Structure for Directed Graphs on Surfaces

  Zhao Xiang LI

  Acta Mathematica Sinica. 2015, 31(1): 170-176. https://doi.org/10.1007/s10114-015-3452-0

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  In this paper, the cycle structures for directed graphs on surfaces are studied. If G is a strongly connected graph, C is a ∏-contractible directed cycle of G, then both of Int(C,∏) and Ext(C,∏) are strongly connected graph; the dimension of cycles space of G is identified. If G is a strongly connected graph, then the structure of MCB in G is unique. Let G be a strongly connected graph, if G has been embedded in orientable surface S_g with f_w(G) ≥ 2(f_w(G) is the face-width of G), then any cycle base of G must contain at least 2g noncontractible directed cycles; if G has been embedded in non-orientable surface N_g, then any cycle base of G must contain at least g noncontractible directed cycles.