# Frontiers of Mathematics in China

ISSN 1673-3452 (Print)
ISSN 1673-3576 (Online)
2017 Impact Factor: 0.377

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 Select General techniques for constructing variational integrators Melvin LEOK, Tatiana SHINGEL Front Math Chin    2012, 7 (2): 273-303.   https://doi.org/10.1007/s11464-012-0190-9 Abstract   HTML   PDF (425KB) The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable discrete Lagrangian. The exact discrete Lagrangian can either be characterized variationally, or in terms of Jacobi’s solution of the Hamilton–Jacobi equation. These two characterizations lead to the Galerkin and shooting constructions for discrete Lagrangians, which depend on a choice of a numerical quadrature formula, together with either a finite-dimensional function space or a one-step method. We prove that the properties of the quadrature formula, finite-dimensional function space, and underlying one-step method determine the order of accuracy and momentum-conservation properties of the associated variational integrators. We also illustrate these systematic methods for constructing variational integrators with numerical examples. Cited: Crossref(15) WebOfScience(15)
 Select Convergence analysis of Jacobi spectral collocation methods for Abel-Volterra integral equations of second kind Xianjuan LI, Tao TANG Front Math Chin    2012, 7 (1): 69-84.   https://doi.org/10.1007/s11464-012-0170-0 Abstract   HTML   PDF (219KB) This work is to analyze a spectral Jacobi-collocation approximation for Volterra integral equations with singular kernel ?(t, s) = (t - s)-μ. In an earlier work of Y. Chen and T. Tang [J. Comput. Appl. Math., 2009, 233: 938-950], the error analysis for this approach is carried out for 0<μ<1/2 under the assumption that the underlying solution is smooth. It is noted that there is a technical problem to extend the result to the case of Abel-type, i.e., μ = 1/2. In this work, we will not only extend the convergence analysis by Chen and Tang to the Abel-type but also establish the error estimates under a more general regularity assumption on the exact solution. Cited: Crossref(24) WebOfScience(29)
 Select Spectral Theory of Nonnegative Tensors Qingzhi YANG, Liping ZHANG, Tan ZHANG, Guanglu ZHOU Front Math Chin    2013, 8 (1): 1.   https://doi.org/10.1007/s11464-012-0273-7 Abstract   HTML   PDF (29KB) null Cited: WebOfScience(2)
 Select Selected Topics in Computational Mathematics Junping WANG Front Math Chin    2012, 7 (2): 197.   https://doi.org/10.1007/s11464-012-0195-4 Abstract   HTML   PDF (35KB) null
 Select Laplacian and signless Laplacian Z-eigenvalues of uniform hypergraphs Changjiang BU,Yamin FAN,Jiang ZHOU Front. Math. China    2016, 11 (3): 511-520.   https://doi.org/10.1007/s11464-015-0467-x Abstract   PDF (117KB) We show that a connected uniform hypergraph G is odd-bipartite if and only if G has the same Laplacian and signless Laplacian Z-eigenvalues. We obtain some bounds for the largest (signless) Laplacian Z-eigenvalue of a hypergraph. For a k-uniform hyperstar with d edges (2d≥k≥3), we show that its largest (signless) Laplacian Z-eigenvalue is d. Cited: Crossref(2) WebOfScience(2)
 Select Geometric simplicity of spectral radius of nonnegative irreducible tensors Yuning YANG, Qingzhi YANG Front Math Chin    2013, 8 (1): 129-140.   https://doi.org/10.1007/s11464-012-0272-8 Abstract   HTML   PDF (117KB) We study the real and complex geometric simplicity of nonnegative irreducible tensors. First, we prove some basic conclusions. Based on the conclusions, the real geometric simplicity of the spectral radius of an evenorder nonnegative irreducible tensor is proved. For an odd-order nonnegative irreducible tensor, sufficient conditions are investigated to ensure the spectral radius to be real geometrically simple. Furthermore, the complex geometric simplicity of nonnegative irreducible tensors is also studied. Cited: Crossref(5) WebOfScience(4)
 Select Lower bounds of principal eigenvalue in dimension one Mu-Fa CHEN Front Math Chin    2012, 7 (4): 645-668.   https://doi.org/10.1007/s11464-012-0223-4 Abstract   HTML   PDF (280KB) For the principal eigenvalue with bilateral Dirichlet boundary condition, the so-called basic estimates were originally obtained by capacitary method. The Neumann case (i.e., the ergodic case) is even harder, and was deduced from the Dirichlet one plus a use of duality and the coupling method. In this paper, an alternative and more direct proof for the basic estimates is presented. The estimates in the Dirichlet case are then improved by a typical application of a recent variational formula. As a dual of the Dirichlet case, the refine problem for bilateral Neumann boundary condition is also treated. The paper starts with the continuous case (one-dimensional diffusions) and ends at the discrete one (birth-death processes). Possible generalization of the results studied here is discussed at the end of the paper. Cited: Crossref(3) WebOfScience(4)
 Select Some results and problems on commutators Shanzhen LU Front Math Chin    2011, 6 (5): 821-833.   https://doi.org/10.1007/s11464-011-0157-2 Abstract   HTML   PDF (121KB) In this paper, the author introduces some late results and puts forward a few problems on commutators of many important operators in harmonic analysis, included the Bochner-Riesz operator below the critical index, the strongly singular integral operator, the pseudo-differential operator, a class of convolution operators with oscillatory kernel, the Marcinkiewicz integral operator, and the fractional integral operator with rough kernel. Cited: Crossref(3) WebOfScience(4)
 Select Complete noncompact manifolds with harmonic curvature Yawei CHU Front Math Chin    2012, 7 (1): 19-27.   https://doi.org/10.1007/s11464-012-0168-7 Abstract   HTML   PDF (128KB) Let (Mn, g) be an n-dimensional complete noncompact Riemannian manifold with harmonic curvature and positive Sobolev constant. In this paper, by employing an elliptic estimation method, we show that (Mn, g) is a space form if it has sufficiently small Ln/2-norms of trace-free curvature tensor and nonnegative scalar curvature. Moreover, we get a gap theorem for (Mn, g) with positive scalar curvature. Cited: Crossref(4) WebOfScience(5)
 Select Continuous Optimization and Combinatorial Optimization Liqun QI, Li-Zhi LIAO, Wenan ZANG, Guanglu ZHOU, Front. Math. China    2010, 5 (1): 1-2.   https://doi.org/10.1007/s11464-009-0044-2 Abstract   PDF (48KB)
 Select Linear Algebra and Multilinear Algebra Liqun QI,Yimin WEI,Changqing XU,Tan ZHANG Front. Math. China    2016, 11 (3): 509-510.   https://doi.org/10.1007/s11464-016-0540-0 Abstract   PDF (37KB) Cited: WebOfScience(1)
 Select Attractors for stochastic lattice dynamical systems with a multiplicative noise CARABALLO Tomás, LU Kening Front. Math. China    2008, 3 (3): 317-335.   https://doi.org/10.1007/s11464-008-0028-7 Abstract   HTML   PDF (209KB) In this paper, we consider a stochastic lattice differential equation with diffusive nearest neighbor interaction, a dissipative nonlinear reaction term, and multiplicative white noise at each node. We prove the existence of a compact global random attractor which, pulled back, attracts tempered random bounded sets. Cited: Crossref(53) WebOfScience(57)
 Select Augmentation quotients for complex representation rings of dihedral groups Shan CHANG, Hong CHEN, Guoping TANG Front Math Chin    2012, 7 (1): 1-18.   https://doi.org/10.1007/s11464-011-0162-5 Abstract   HTML   PDF (197KB) Denote by Dm the dihedral group of order 2m. Let ?(Dm) be its complex representation ring, and let Δ(Dm) be its augmentation ideal. In this paper, we determine the isomorphism class of the n-th augmentation quotient Δn(Dm)/Δn+1(Dm) for each positive integer n. Cited: Crossref(2) WebOfScience(2)
 Select Moderate deviations and central limit theorem for small perturbation Wishart processes Lei CHEN, Fuqing GAO, Shaochen WANG Front Math Chin    2014, 9 (1): 1-15.   https://doi.org/10.1007/s11464-013-0291-0 Abstract   HTML   PDF (151KB) Let X? be a small perturbation Wishart process with values in the set of positive definite matrices of size m, i.e., the process X? is the solution of stochastic differential equation with non-Lipschitz diffusion coefficient: dXt?=?Xt?dBt+dBt'?Xt?+ρImdt, X0 = x, where B is an m × m matrix valued Brownian motion and B′denotes the transpose of the matrix B. In this paper, we prove that {Xt?-Xt0/?h2(?),?>0} satisfies a large deviation principle, and (Xt?-Xt0)/? converges to a Gaussian process, where h(?)→+∞ and ?h(?)→0 as ?→0. A moderate deviation principle and a functional central limit theorem for the eigenvalue process of X? are also obtained by the delta method. Cited: Crossref(2) WebOfScience(9)
 Select A degenerate parabolic system with localized sources and nonlocal boundary condition Yongsheng MI, Chunlai MU Front Math Chin    2012, 7 (1): 97-116.   https://doi.org/10.1007/s11464-011-0163-4 Abstract   HTML   PDF (207KB) This paper deals with the blow-up properties of the positive solutions to a degenerate parabolic system with localized sources and nonlocal boundary conditions. We investigate the influence of the reaction terms, the weight functions, local terms and localized source on the blow-up properties. We will show that the weight functions play the substantial roles in determining whether the solutions will blow-up or not, and obtain the blow-up conditions and its blow-up rate estimate. Cited: Crossref(1) WebOfScience(1)
 Select Best rank one approximation of real symmetric tensors can be chosen symmetric Shmuel FRIEDLAND Front Math Chin    2013, 8 (1): 19-40.   https://doi.org/10.1007/s11464-012-0262-x Abstract   HTML   PDF (186KB) We show that a best rank one approximation to a real symmetric tensor, which in principle can be nonsymmetric, can be chosen symmetric. Furthermore, a symmetric best rank one approximation to a symmetric tensor is unique if the tensor does not lie on a certain real algebraic variety. Cited: Crossref(17) WebOfScience(20)
 Select J-dendriform algebras Dongping HOU, Chengming BAI Front Math Chin    2012, 7 (1): 29-49.   https://doi.org/10.1007/s11464-011-0160-7 Abstract   HTML   PDF (207KB) In this paper, we introduce a notion of J-dendriform algebra with two operations as a Jordan algebraic analogue of a dendriform algebra such that the anticommutator of the sum of the two operations is a Jordan algebra. A dendriform algebra is a J-dendriform algebra. Moreover, J-dendriform algebras fit into a commutative diagram which extends the relationships among associative, Lie, and Jordan algebras. Their relations with some structures such as Rota-Baxter operators, classical Yang-Baxter equation, and bilinear forms are given. Cited: Crossref(6) WebOfScience(7)
 Select Joint probability generating function for degrees of active/passive random intersection graphs Yilun SHANG Front Math Chin    2012, 7 (1): 117-124.   https://doi.org/10.1007/s11464-011-0165-2 Abstract   HTML   PDF (128KB) Correlations of active and passive random intersection graphs are studied in this paper. We present the joint probability generating function for degrees of Gactive(n, m, p) and Gpassive(n, m, p), which are generated by a random bipartite graph G?(n, m, p) on n + m vertices. Cited: Crossref(4) WebOfScience(4)
 Select Estimates for multilinear singular integral operators with nonsmooth kernels Guoen HU, Yan MENG Front Math Chin    2012, 7 (1): 51-67.   https://doi.org/10.1007/s11464-011-0169-y Abstract   HTML   PDF (203KB) Let (X, d, μ) be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, we consider the behavior on Lp1(X) × · · · × Lpm(X) for the m-linear singular integral operators with nonsmooth kernels which were first introduced by Duong, Grafakos and Yan. Cited: Crossref(1) WebOfScience(1)
 Select Nonnegative tensor factorizations using an alternating direction method Xingju CAI, Yannan CHEN, Deren HAN Front Math Chin    2013, 8 (1): 3-18.   https://doi.org/10.1007/s11464-012-0264-8 Abstract   HTML   PDF (752KB) The nonnegative tensor (matrix) factorization finds more and more applications in various disciplines including machine learning, data mining, and blind source separation, etc. In computation, the optimization problem involved is solved by alternatively minimizing one factor while the others are fixed. To solve the subproblem efficiently, we first exploit a variable regularization term which makes the subproblem far from ill-condition. Second, an augmented Lagrangian alternating direction method is employed to solve this convex and well-conditioned regularized subproblem, and two accelerating skills are also implemented. Some preliminary numerical experiments are performed to show the improvements of the new method. Cited: Crossref(6) WebOfScience(6)
 Select Recent Progress in Probability and Statistics in China Dayue CHEN Front Math Chin    2013, 8 (3): 477-478.   https://doi.org/10.1007/s11464-013-0301-2 Abstract   HTML   PDF (34KB) null
 Select Recent Development in Symmetries and Integrability of Difference Equations Xingbiao HU, Qingping LIU, Senyue LOU, Changzheng QU, Youjin ZHANG Front Math Chin    2013, 8 (5): 999-1000.   https://doi.org/10.1007/s11464-013-0326-6 Abstract   HTML   PDF (40KB) null
 Select A characterization of λ-central BMO space Fayou ZHAO, Shanzhen LU Front Math Chin    2013, 8 (1): 229-238.   https://doi.org/10.1007/s11464-012-0251-0 Abstract   HTML   PDF (106KB) We give a characterization of the λ-central BMO space via the boundedness of commutators of n-dimensional Hardy operators. Cited: Crossref(1) WebOfScience(1)
 Select Precise large deviations for generalized dependent compound renewal risk model with consistent variation Yu CHEN, Weiping ZHANG, Chun SU Front Math Chin    2014, 9 (1): 31-44.   https://doi.org/10.1007/s11464-013-0350-6 Abstract   HTML   PDF (134KB) We investigate the precise large deviations of random sums of negatively dependent random variables with consistently varying tails. We find out the asymptotic behavior of precise large deviations of random sums is insensitive to the negative dependence. We also consider the generalized dependent compound renewal risk model with consistent variation, which including premium process and claim process, and obtain the asymptotic behavior of the tail probabilities of the claim surplus process. Cited: Crossref(1) WebOfScience(1)
 Select ?-tensors and nonsingular ?-tensors Xuezhong WANG,Yimin WEI Front. Math. China    2016, 11 (3): 557-575.   https://doi.org/10.1007/s11464-015-0495-6 Abstract   PDF (179KB) The H-matrices are an important class in the matrix theory, and have many applications. Recently, this concept has been extended to higher order ?-tensors. In this paper, we establish important properties of diagonally dominant tensors and ?-tensors. Distributions of eigenvalues of nonsingular symmetric ?-tensors are given. An ?+-tensor is semi-positive, which enlarges the area of semi-positive tensor from ?-tensor to ?+-tensor. The spectral radius of Jacobi tensor of a nonsingular (resp. singular) ?-tensor is less than (resp. equal to) one. In particular, we show that a quasi-diagonally dominant tensor is a nonsingular ?-tensor if and only if all of its principal sub-tensors are nonsingular ?-tensors. An irreducible tensor Ais an ?-tensor if and only if it is quasi-diagonally dominant. Cited: Crossref(4) WebOfScience(3)
 Select Reduced-order extrapolation spectral-finite difference scheme based on POD method and error estimation for three-dimensional parabolic equation Jing AN,Zhendong LUO,Hong LI,Ping SUN Front. Math. China    2015, 10 (5): 1025-1040.   https://doi.org/10.1007/s11464-015-0469-8 Abstract   PDF (160KB) In this study, a classical spectral-finite difference scheme (SFDS) for the three-dimensional (3D) parabolic equation is reduced by using proper orthogonal decomposition (POD) and singular value decomposition (SVD). First, the 3D parabolic equation is discretized in spatial variables by using spectral collocation method and the discrete scheme is transformed into matrix formulation by tensor product. Second, the classical SFDS is obtained by difference discretization in time-direction. The ensemble of data are comprised with the first few transient solutions of the classical SFDS for the 3D parabolic equation and the POD bases of ensemble of data are generated by using POD technique and SVD. The unknown quantities of the classical SFDS are replaced with the linear combination of POD bases and a reducedorder extrapolation SFDS with lower dimensions and sufficiently high accuracy for the 3D parabolic equation is established. Third, the error estimates between the classical SFDS solutions and the reduced-order extrapolation SFDS solutions and the implementation for solving the reduced-order extrapolation SFDS are provided. Finally, a numerical example shows that the errors of numerical computations are consistent with the theoretical results. Moreover, it is shown that the reduced-order extrapolation SFDS is effective and feasible to find the numerical solutions for the 3D parabolic equation. Cited: Crossref(5) WebOfScience(7)
 Select lk,s-Singular values and spectral radius of partially symmetric rectangular tensors Hongmei YAO,Bingsong LONG,Changjiang BU,Jiang ZHOU Front. Math. China    2016, 11 (3): 605-622.   https://doi.org/10.1007/s11464-015-0494-7 Abstract   PDF (176KB) The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we first study properties of lk,s-singular values of real rectangular tensors. Then, a necessary and sufficient condition for the positive definiteness of partially symmetric rectangular tensors is given. Furthermore, we show that the weak Perron-Frobenius theorem for nonnegative partially symmetric rectangular tensor keeps valid under some new conditions and we prove a maximum property for the largest lk,s-singular values of nonnegative partially symmetric rectangular tensor. Finally, we prove that the largest lk,ssingular value of nonnegative weakly irreducible partially symmetric rectangular tensor is still geometrically simple.
 Select New method for general Kennaugh’s pseudoeigenvalue equation in radar polarimetry Sitao LING, Tongsong JIANG Front Math Chin    2012, 7 (1): 85-95.   https://doi.org/10.1007/s11464-011-0166-1 Abstract   HTML   PDF (141KB) Kennaugh’s pseudo-eigenvalue equation is a basic equation that plays an extremely important role in radar polarimetry. In this paper, by means of real representation, we first present a necessary and sufficient condition for the general Kennaugh’s pseudo-eigenvalue equation having a solution, characterize the explicit form of the solution, and then study the solution of Kennaugh’s pseudo-eigenvalue equation. At last, we propose a new technique for finding the coneigenvalues and coneigenvectors of a complex matrix under appropriate conditions in radar polarimetry. Cited: Crossref(1) WebOfScience(1)
 Select Linear convergence of an algorithm for largest singular value of a nonnegative rectangular tensor Liping ZHANG Front Math Chin    2013, 8 (1): 141-153.   https://doi.org/10.1007/s11464-012-0260-z Abstract   HTML   PDF (127KB) An algorithm for finding the largest singular value of a nonnegative rectangular tensor was recently proposed by Chang, Qi, and Zhou [J. Math. Anal. Appl., 2010, 370: 284–294]. In this paper, we establish a linear convergence rate of the Chang-Qi-Zhou algorithm under a reasonable assumption. Cited: Crossref(3) WebOfScience(5)
 Select Finite groups with transitive semipermutability WANG Lifang, WANG Yanming Front. Math. China    2008, 3 (1): 101-108.   https://doi.org/10.1007/s11464-008-0009-x Abstract   HTML   PDF (115KB) A group G is said to be a T-group (resp. PT-group, PST-group), if normality (resp. permutability, S-permutability) is a transitive relation. In this paper, we get the characterization of finite solvable PST-groups. We also give a new characterization of finite solvable PT-groups. Cited: Crossref(1)