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  • RESEARCH ARTICLE
    Xiaofei ZHANG, Chungen LIU
    Frontiers of Mathematics in China, 2021, 16(1): 239-253. https://doi.org/10.1007/s11464-021-0903-z

    With the aid of P-index iteration theory, we consider the minimal period estimates on P-symmetric periodic solutions of nonlinear P-symmetric Hamiltonian systems with mild superquadratic growth.

  • RESEARCH ARTICLE
    Liqun QI, Shenglong HU, Xinzhen ZHANG, Yanwei XU
    Frontiers of Mathematics in China, 2021, 16(1): 171-185. https://doi.org/10.1007/s11464-021-0895-8

    Biquadratic tensors play a central role in many areas of science. Examples include elastic tensor and Eshelby tensor in solid mechanics, and Riemannian curvature tensor in relativity theory. The singular values and spectral norm of a general third order tensor are the square roots of the M-eigenvalues and spectral norm of a biquadratic tensor, respectively. The tensor product operation is closed for biquadratic tensors. All of these motivate us to study biquadratic tensors, biquadratic decomposition, and norms of biquadratic tensors. We show that the spectral norm and nuclear norm for a biquadratic tensor may be computed by using its biquadratic structure. Then, either the number of variables is reduced, or the feasible region can be reduced. We show constructively that for a biquadratic tensor, a biquadratic rank-one decomposition always exists, and show that the biquadratic rank of a biquadratic tensor is preserved under an independent biquadratic Tucker decomposition. We present a lower bound and an upper bound of the nuclear norm of a biquadratic tensor. Finally, we define invertible biquadratic tensors, and present a lower bound for the product of the nuclear norms of an invertible biquadratic tensor and its inverse, and a lower bound for the product of the nuclear norm of an invertible biquadratic tensor, and the spectral norm of its inverse.

  • RESEARCH ARTICLE
    Chunxiao GUO, Yiju CHEN, Ji SHU, Xinguang YANG
    Frontiers of Mathematics in China, 2021, 16(1): 59-93. https://doi.org/10.1007/s11464-021-0896-7

    The regularity of random attractors is considered for the nonautonomous fractional stochastic FitzHugh-Nagumo system. We prove that the system has a pullback random attractor that is compact in Hs(n)×L2(n) and attracts all tempered random sets of Ls(n)×L2(n) in the topology of Hs(n)×L2(n) with s(0,1). By the idea of positive and negative truncations, spectral decomposition in bounded domains, and tail estimates, we achieved the desired results.

  • SURVEY ARTICLE
    Shanzhen LU
    Frontiers of Mathematics in China, 2021, 16(1): 1-12. https://doi.org/10.1007/s11464-021-0894-9

    This paper is a summary of the research on the characterizations of central function spaces by the author and his collaborators in the past ten years. More precisely, the author gives some characterizations of central Campanato spaces via the boundedness and compactness of commutators of Hardy operator.

  • RESEARCH ARTICLE
    Xiaofeng CHEN, Jianlong CHEN
    Frontiers of Mathematics in China, 2020, 15(6): 1089-1104. https://doi.org/10.1007/s11464-020-0880-7

    We first prove that if a is both left (b; c)-invertible and left (c; b)- invertible, then a is both (b; c)-invertible and (c; b)-invertible in a *-monoid, which generalizes the recent result about the inverse along an element by L. Wang and D. Mosić [Linear Multilinear Algebra, Doi.org/10.1080/03081087. 2019.1679073], under the conditions (ab)* = ab and (ac)* = ac: In addition, we consider that ba is (c; b)-invertible, and at the same time ca is (b; c)-invertible under the same conditions, which extend the related results about Moore- Penrose inverses studied by J. Chen, H. Zou, H. Zhu, and P. Patrício [Mediterr J. Math., 2017, 14: 208] to (b; c)-inverses. As applications, we obtain that under condition (a2)* = a2; a is an EP element if and only if a is one-sided core invertible, if and only if a is group invertible.

  • RESEARCH ARTICLE
    Ze LI
    Frontiers of Mathematics in China, 2020, 15(5): 923-957. https://doi.org/10.1007/s11464-020-0857-6

    We prove the asymptotic stability of solitary waves to 1D nonlinear Schrödinger equations in the subcritical case with symmetry and spectrum assumptions. One of the main ideas is to use the vector fields method developed by S. Cuccagna, V. Georgiev, and N. Visciglia [Comm. Pure Appl. Math., 2013, 6: 957–980] to overcome the weak decay with respect to t of the linearized equation caused by the one dimension setting and the weak nonlinearity caused by the subcritical growth of the nonlinearity term. Meanwhile, we apply the polynomial growth of the high Sobolev norms of solutions to 1D Schrödinger equations obtained by G. Staffilani [Duke Math. J., 1997, 86(1): 109–142] to control the high moments of the solutions emerging from the vector fields method.

  • RESEARCH ARTICLE
    Xiequan FAN, Haijuan HU, Quansheng LIU
    Frontiers of Mathematics in China, 2020, 15(5): 891-914. https://doi.org/10.1007/s11464-020-0868-3

    Let {Zn, n0}be a supercritical branching process in an independent and identically distributed random environment. We prove Cramér moderate deviations and Berry-Esseen bounds for log(Zn+n0/Zn0 ) uniformly in n0 ,which extend the corresponding results by I. Grama, Q. Liu, and M. Miqueu [Stochastic Process. Appl., 2017, 127: 1255–1281] established for n0= 0. The extension is interesting in theory, and is motivated by applications. A new method is developed for the proofs; some conditions of Grama et al. are relaxed in our present setting. An example of application is given in constructing confidence intervals to estimate the criticality parameter in terms of log(Zn+n0/Zn0 ) and n.

  • RESEARCH ARTICLE
    Hui LIU, Hui ZHANG
    Frontiers of Mathematics in China, 2020, 15(6): 1155-1173. https://doi.org/10.1007/s11464-020-0885-2

    Let k>2 be an integer and P be a 2n×2n symplectic orthogonal matrix satisfying Pk = I2n and ker(Pj - I2n) = 0; 1≤j <k: For any compact convex hypersurface 2n with n≥2 which is P-cyclic symmetric, i.e., x implies Px ; we prove that if is (r;R)-pinched with R/r<(2k+2)/k,then there exist at least n geometrically distinct P-cyclic symmetric closed characteristics on for a broad class of matrices P:

  • RESEARCH ARTICLE
    Ahmad AL-SALMAN
    Frontiers of Mathematics in China, 2021, 16(1): 13-28. https://doi.org/10.1007/s11464-021-0911-z

    We introduce a class of singular integral operators on product domains along twisted surfaces. We prove that the operators are bounded on Lp provided that the kernels satisfy weak conditions.

  • RESEARCH ARTICLE
    Mu-Fa CHEN, Jin-Yu LI
    Frontiers of Mathematics in China, 2020, 15(5): 867-889. https://doi.org/10.1007/s11464-020-0859-4

    The first aim of the paper is to study the Hermitizability of secondorder differential operators, and then the corresponding isospectral operators. The explicit criteria for the Hermitizable or isospectral properties are presented. The second aim of the paper is to study a non-Hermitian model, which is now well known. In a regular sense, the model does not belong to the class of Hermitizable operators studied in this paper, but we will use the theory developed in the past years, to present an alternative and illustrated proof of the discreteness of its spectrum. The harmonic function plays a critical role in the study of spectrum. Two constructions of the function are presented. The required conclusion for the discrete spectrum is proved by some comparison technique.

  • RESEARCH ARTICLE
    Wen-Xiu MA
    Frontiers of Mathematics in China, 2019, 14(3): 619-629. https://doi.org/10.1007/s11464-019-0771-y

    Abundant exact interaction solutions, including lump-soliton, lumpkink, and lump-periodic solutions, are computed for the Hirota-Satsuma-Ito equation in (2+1)-dimensions, through conducting symbolic computations with Maple. The basic starting point is a Hirota bilinear form of the Hirota-Satsuma-Ito equation. A few three-dimensional plots and contour plots of three special presented solutions are made to shed light on the characteristic of interaction solutions.

  • RESEARCH ARTICLE
    Linsong WANG, Yun GAO, Naihuan JING
    Frontiers of Mathematics in China, 2019, 14(2): 421-433. https://doi.org/10.1007/s11464-019-0760-1

    We give a recursive algorithm to compute the multivariable Zassenhaus formula eX1+X2+...+Xn=eX1eX2...eXnΠk=2eWk and derive ane effective recursion formula of Wk.

  • RESEARCH ARTICLE
    Yonghong HUANG, Shanzhong SUN
    Frontiers of Mathematics in China, 2020, 15(1): 91-114. https://doi.org/10.1007/s11464-020-0823-3

    We prove that there do not exist quasi-isometric embeddings of connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics into a metric measure space satisfying the curvaturedimension condition RCD(0;N) with N 2 R and N>1: In fact, we can prove that a sub-Riemannian manifold whose generic degree of nonholonomy is not smaller than 2 cannot be bi-Lipschitzly embedded in any Banach space with the Radon-Nikodym property. We also get that every regular sub-Riemannian manifold do not satisfy the curvature-dimension condition CD(K;N); where K;N 2 R and N>1: Along the way to the proofs, we show that the minimal weak upper gradient and the horizontal gradient coincide on the Carnot-Carathéodory spaces which may have independent interests.

  • RESEARCH ARTICLE
    Yafang KONG
    Frontiers of Mathematics in China, 2022, 17(6): 1001-1013. https://doi.org/10.1007/s11464-022-1029-7

    In this paper, we study the multivariate linear equations with arbitrary positive integral coefficients. Under the Generalized Riemann Hypothesis, we obtained the asymptotic formula for the linear equations with more than five prime variables. This asymptotic formula is composed of three parts, that is, the first main term, the explicit second main term and the error term. Among them, the first main term is similar with the former one, the explicit second main term is relative to the non-trivial zeros of Dirichlet L-functions, and our error term improves the former one.

  • RESEARCH ARTICLE
    Ke GUO, Deren HAN, David Z. W. WANG, Tingting WU
    Frontiers of Mathematics in China, 2017, 12(5): 1139-1162. https://doi.org/10.1007/s11464-017-0631-6

    For solving minimization problems whose objective function is the sum of two functions without coupled variables and the constrained function is linear, the alternating direction method of multipliers (ADMM) has exhibited its efficiency and its convergence is well understood. When either the involved number of separable functions is more than two, or there is a nonconvex function, ADMM or its direct extended version may not converge. In this paper, we consider the multi-block separable optimization problems with linear constraints and absence of convexity of the involved component functions. Under the assumption that the associated function satisfies the Kurdyka- Lojasiewicz inequality, we prove that any cluster point of the iterative sequence generated by ADMM is a critical point, under the mild condition that the penalty parameter is sufficiently large. We also present some sufficient conditions guaranteeing the sublinear and linear rate of convergence of the algorithm.

  • SURVEY ARTICLE
    Liangxia WAN
    Frontiers of Mathematics in China, 2023, 18(1): 1-14. https://doi.org/10.3868/S140-DDD-023-005-X

    The alternating links give a classical class of links. They play an important role in Knot Theory. Ozsváth and Szabó introduced a quasi-alternating link which is a generalization of an alternating link. In this paper we review some results of alternating links and quasi-alternating links on the Jones polynomial and the Khovanov homology. Moreover, we introduce a long pass link. Several problems worthy of further study are provided.

  • RESEARCH ARTICLE
    Sibei YANG, Dachun YANG, Wen YUAN
    Frontiers of Mathematics in China, 2019, 14(1): 177-201. https://doi.org/10.1007/s11464-019-0744-1

    We establish a new characterization of the Musielak–Orlicz–Sobolev space on n; which includes the classical Orlicz–Sobolev space, the weighted Sobolev space, and the variable exponent Sobolev space as special cases, in terms of sharp ball averaging functions. Even in a special case, namely, the variable exponent Sobolev space, the obtained result in this article improves the corresponding result obtained by P. Hästö and A. M. Ribeiro [Commun. Contemp. Math., 2017, 19: 1650022] via weakening the assumption fL1(n) into fL 1(n), which was conjectured to be true by Hästö and Ribeiro in the aforementioned same article.

  • SURVEY ARTICLE
    Guohua QIAN
    Frontiers of Mathematics in China, 2023, 18(1): 15-32. https://doi.org/10.3868/S140-DDD-023-006-X

    For an irreducible character χ of a finite group G, we define its codegree as cod(χ)=|G:kerχ|χ(1). In this paper, we introduce some known results and unsolved problems about character codegrees in finite groups.

  • RESEARCH ARTICLE
    Qinghua XU,Ting YANG,Taishun LIU,Huiming XU
    Frontiers of Mathematics in China, 2015, 10(6): 1461-1472. https://doi.org/10.1007/s11464-015-0496-5

    Let K be the familiar class of normalized convex functions in the unit disk. Keogh and Merkes proved the well-known result that max?fK|a3λa22|max?{1/3,|λ1|},λ?, and the estimate is sharp for each λ. We investigate the corresponding problem for a subclass of quasi-convex mappings of type B defined on the unit ball in a complex Banach space or on the unit polydisk in ?n. The proofs of these results use some restrictive assumptions, which in the case of one complex variable are automatically satisfied.

  • RESEARCH ARTICLE
    Yannan CHEN, Shenglong HU, Liqun QI, Wennan ZOU
    Frontiers of Mathematics in China, 2019, 14(1): 1-16. https://doi.org/10.1007/s11464-019-0748-x

    Third order three-dimensional symmetric and traceless tensors play an important role in physics and tensor representation theory. A minimal integrity basis of a third order three-dimensional symmetric and traceless tensor has four invariants with degrees two, four, six, and ten, respectively. In this paper, we show that any minimal integrity basis of a third order three-dimensional symmetric and traceless tensor is also an irreducible function basis of that tensor, and there is no syzygy relation among the four invariants of that basis, i.e., these four invariants are algebraically independent.

  • SURVEY ARTICLE
    Xiaoqiang SUN, Jiguang BAO
    Frontiers of Mathematics in China, 2023, 18(2): 75-94. https://doi.org/10.3868/S140-DDD-023-0011-X

    Life activities are extremely complex phenomena in nature. From molecular signaling regulation to multi-cellular tissue formation and so on, the biological system consists of multiple temporal, spatial and functional scales. Multiscale mathematical models have extensive applications in life science research due to their capacity of appropriately simulating the complex multiscale biological systems. Many mathematical methods, including deterministic methods, stochastic methods as well as discrete or rule-based methods, have been widely used for modeling biological systems. However, the models at single scale are not sufficient to simulate complex biological systems. Therefore, in this paper we give a survey of two multiscale modeling approaches for biological systems. One approach is continuous stochastic method that couples ordinary differential equations and stochastic differential equations; Another approach is hybrid continuous-discrete method that couples agent-based model with partial differential equations. We then introduce the applications of these multiscale modeling approaches in systems biology and look ahead to the future research.

  • SURVEY ARTICLE
    Dongdong JIA, Yuebo SHEN, Gengsheng ZHANG
    Frontiers of Mathematics in China, 2023, 18(5): 301-312. https://doi.org/10.3868/s140-DDD-023-0024-x

    A combinatorial batch code has strong practical motivation in the distributed storage and retrieval of data in a database. In this survey, we give a brief introduction to the combinatorial batch codes and some progress.

  • RESEARCH ARTICLE
    Sanzheng QIAO, Yimin WEI
    Frontiers of Mathematics in China, 2018, 13(6): 1427-1445. https://doi.org/10.1007/s11464-018-0731-y

    For an n×n complex matrix A with ind(A) = r; let AD and Aπ = I-AAD be respectively the Drazin inverse and the eigenprojection corresponding to the eigenvalue 0 of A: For an n×n complex singular matrix B with ind(B) =s; it is said to be a stable perturbation of A; if I(BπAπ)2 is nonsingular, equivalently, if the matrix B satisfies the condition R(Bs) R(Bs)N(Ar)={0} and N(Bs)R(Ar)={0}, introduced by Castro-Gonz

  • RESEARCH ARTICLE
    Jie YANG,Weidong ZHAO
    Frontiers of Mathematics in China, 2016, 11(6): 1625-1643. https://doi.org/10.1007/s11464-016-0504-9

    This paper is concerned with numerical simulations for the GBrownian motion (defined by S. Peng in Stochastic Analysis and Applications, 2007, 541–567). By the definition of the G-normal distribution, we first show that the G-Brownian motions can be simulated by solving a certain kind of Hamilton-Jacobi-Bellman (HJB) equations. Then, some finite difference methods are designed for the corresponding HJB equations. Numerical simulation results of the G-normal distribution, the G-Brownian motion, and the corresponding quadratic variation process are provided, which characterize basic properties of the G-Brownian motion. We believe that the algorithms in this work serve as a fundamental tool for future studies, e.g., for solving stochastic differential equations (SDEs)/stochastic partial differential equations (SPDEs) driven by the G-Brownian motions.

  • RESEARCH ARTICLE
    Zhongqing LI
    Frontiers of Mathematics in China, 2023, 18(1): 43-50. https://doi.org/10.3868/S140-DDD-023-002-X

    The existence of bounded weak solutions, to a class of nonlinear elliptic equations with variable exponents, is investigated in this article. A uniform a priori L estimate is obtained by the De Giorgi iterative technique. Thanks to the weak convergence method and Minty's trick, the existence result is proved through limit process.

  • RESEARCH ARTICLE
    Mengyan XIE, Qing-Wen WANG
    Frontiers of Mathematics in China, 2020, 15(5): 1047-1070. https://doi.org/10.1007/s11464-020-0865-6

    We establish necessary and sufficient conditions for the existence of the reducible solution to the quaternion tensor equation A*NX*NB=Cvia Einstein product using Moore-Penrose inverse, and present an expression of the reducible solution to the equation when it is solvable. Moreover, to have a general solution, we give the solvability conditions for the quaternion tensor equation A1*NX1*MB1+A1*NX2*MB2+A2*NX3*MB2=C, which plays a key role in investigating the reducible solution to A*NX*NB=C. The expression of such a solution is also presented when the consistency conditions are met. In addition, we show a numerical example to illustrate this result.

  • RESEARCH ARTICLE
    Yue ZHANG, Wenjuan LI
    Frontiers of Mathematics in China, 2022, 17(6): 1181-1200. https://doi.org/10.1007/s11464-021-0983-9

    We focus on the Lp(R2) theory of the fractional Fourier transform (FRFT) for 1 ≤ p ≤ 2. In L1(R2), we mainly study the properties of the FRFT via introducing the two-parameter chirp operator. In order to get the point-wise convergence for the inverse FRFT, we introduce the fractional convolution and establish the corresponding approximate identities. Then the well-defined inverse FRFT is given via approximation by suitable means, such as fractional Gauss means and Able means. Furthermore, if the signal Fα,βf is received, we give the process of recovering the original signal f with MATLAB. In L2(R2), the general Plancherel theorem, direct sum decomposition, and the general Heisenberg inequality for the FRFT are obtained.

  • RESEARCH ARTICLE
    Liqun QI, Hui-Hui DAI, Deren HAN
    Frontiers of Mathematics in China, 2009, 4(2): 349-364. https://doi.org/10.1007/s11464-009-0016-6

    The strong ellipticity condition plays an important role in nonlinear elasticity and in materials. In this paper, we de?ne M-eigenvalues for an elasticity tensor. The strong ellipticity condition holds if and only if the smallest M-eigenvalue of the elasticity tensor is positive. If the strong ellipticity condition holds, then the elasticity tensor is rank-one positive de?nite. The elasticity tensor is rank-one positive de?nite if and only if the smallest Z-eigenvalue of the elasticity tensor is positive. A Z-eigenvalue of the elasticity tensor is an M-eigenvalue but not vice versa. If the elasticity tensor is second-order positive de?nite, then the strong ellipticity condition holds. The converse conclusion is not right. Computational methods for ?nding M-eigenvalues are presented.

  • RESEARCH ARTICLE
    Chune SHI, Jingshi XU
    Frontiers of Mathematics in China, 2013, 8(4): 907-921. https://doi.org/10.1007/s11464-012-0248-8

    The Herz type Besov and Triebel-Lizorkin spaces with variable exponent are introduced. Then characterizations of these new spaces by maximal functions are given.

  • RESEARCH ARTICLE
    Zerong HE, Nan ZHOU, Mengjie HAN
    Frontiers of Mathematics in China, 2023, 18(1): 51-62. https://doi.org/10.3868/S140-DDD-023-004-X

    We propose a class of new hierarchical model for the evolution of two interacting age-structured populations, which is a system of integro-partial differential equations with global feedback boundary conditions and may describe the interactions such as competition, cooperation and predator-prey relation. Based upon a group of natural conditions, the existence and uniqueness of solutions on infinite time interval are proved by means of fixed point and extension principle, and the continuous dependence of the solution on the initial age distribution is established. These results lay a sound basis for the investigation of stability, controllability and variable optimal control problems.

  • RESEARCH ARTICLE
    Xinya YANG
    Frontiers of Mathematics in China, 2023, 18(1): 63-74. https://doi.org/10.3868/S140-DDD-023-003-X

    In this paper, we study the continuous dependence of eigenvalue of Sturm-Liouville differential operators on the boundary condition by using of implicit function theorem. The work not only provides a new and elementary proof of the above results, but also explicitly presents the expressions for derivatives of the n-th eigenvalue with respect to given parameters. Furthermore, we obtain the new results of the position and number of the generated double eigenvalues under the real coupled boundary condition.

  • SURVEY ARTICLE
    Yiguang YANG
    Frontiers of Mathematics in China, 2023, 18(5): 313-326. https://doi.org/10.3868/s140-DDD-023-0022-x

    Determining the search direction and the search step are the two main steps of the nonlinear optimization algorithm, in which the derivatives of the objective and constraint functions are used to determine the search direction, the one-dimensional search and the trust domain methods are used to determine the step length along the search direction. One dimensional line search has been widely discussed in various textbooks and references. However, there is a less-known technique—arc-search method, which is relatively new and may generate more efficient algorithms in some cases. In this paper, we will survey this technique, discuss its applications in different optimization problems, and explain its potential improvements over traditional line search method.

  • RESEARCH ARTICLE
    Xuezhong WANG,Yimin WEI
    Frontiers of Mathematics in China, 2016, 11(3): 557-575. https://doi.org/10.1007/s11464-015-0495-6

    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.

  • RESEARCH ARTICLE
    Xutao LI, Michael K. NG
    Frontiers of Mathematics in China, 2015, 10(3): 649-680. https://doi.org/10.1007/s11464-014-0377-3

    We study iterative methods for solving a set of sparse non-negative tensor equations (multivariate polynomial systems) arising from data mining applications such as information retrieval by query search and community discovery in multi-dimensional networks. By making use of sparse and non-negative tensor structure, we develop Jacobi and Gauss-Seidel methods for solving tensor equations. The multiplication of tensors with vectors are required at each iteration of these iterative methods, the cost per iteration depends on the number of non-zeros in the sparse tensors. We show linear convergence of the Jacobi and Gauss-Seidel methods under suitable conditions, and therefore, the set of sparse non-negative tensor equations can be solved very efficiently. Experimental results on information retrieval by query search and community discovery in multi-dimensional networks are presented to illustrate the application of tensor equations and the effectiveness of the proposed methods.

  • SURVEY ARTICLE
    Shanzhen LU
    Frontiers of Mathematics in China, 2013, 8(6): 1237-1251. https://doi.org/10.1007/s11464-013-0323-9

    The aim of this paper is to state some conjectures and problems on Bochner-Riesz means in multiple Fourier series and integrals. The progress on these conjectures and problems are also mentioned.

  • RESEARCH ARTICLE
    Haixia YU, Junfeng LI
    Frontiers of Mathematics in China, 2018, 13(2): 449-457. https://doi.org/10.1007/s11464-018-0685-0

    We obtain the operator norms of the n-dimensional fractional Hardy operator Hα(0αN) from weighted Lebesgue spaces L|x|pp(?n) to weighted weak Lebesgue spaces L|x|βq,(?n).

  • RESEARCH ARTICLE
    Kinkar Ch. DAS,Kexiang XU,Junki NAM
    Frontiers of Mathematics in China, 2015, 10(3): 567-582. https://doi.org/10.1007/s11464-015-0431-9

    The first Zagreb index M1(G) is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index M2(G) is equal to the sum of the products of the degrees of pairs of adjacent vertices of the underlying molecular graph G. In this paper, we obtain lower and upper bounds on the first Zagreb index M1(G) of G in terms of the number of vertices (n), number of edges (m), maximum vertex degree (Δ), and minimum vertex degree (δ). Using this result, we find lower and upper bounds on M2(G). Also, we present lower and upper bounds on M2(G) +M2(G) in terms of n, m, Δ, and δ, where G denotes the complement of G. Moreover, we determine the bounds on first Zagreb coindex M1(G) and second Zagreb coindex M2(G). Finally, we give a relation between the first Zagreb index and the second Zagreb index of graph G.

  • LIU Zhaoli, WANG Zhi-Qiang
    Frontiers of Mathematics in China, 2008, 3(2): 221-238. https://doi.org/10.1007/s11464-008-0014-0
    In this survey article, we recall some known results on existence and multiplicity of sign-changing solutions of elliptic equations. Methods for obtaining sign-changing solutions developed in the last two decades will also be brief
  • RESEARCH ARTICLE
    Meiqi YAN, Hailou YAO
    Frontiers of Mathematics in China, 2020, 15(6): 1265-1293. https://doi.org/10.1007/s11464-020-0877-2

    Let Λ(0,0)=(AANBBNAB) be a Morita ring, where the bimodule homomorphisms ϕand ψ are zero. We study the finite presentedness, locally coherence, pure projectivity, pure injectivity, and FP-injectivity of modules over Λ(0,0). Some applications are then given.

  • RESEARCH ARTICLE
    Guiyu YANG
    Frontiers of Mathematics in China, 2015, 10(6): 1473-1481. https://doi.org/10.1007/s11464-015-0498-3

    We investigate the properties of nil-Coxeter algebras and nil-Ariki-Koike algebras. To be precise, from the view of standardly based algebras introduced by J. Du, H. Rui [Trans. Amer. Math. Soc, 1998, 350: 3207–3235], we give a description of simple modules of nil-Coxeter algebras and nil-Ariki-Koike algebras. Then we determine the representation type of nil-Coxeter algebras and nil-Ariki-Koike algebras. We also give a description of the center of nil-Ariki-Koike algebras.

  • SURVEY ARTICLE
    Jun-Ming XU, Meijie MA
    Frontiers of Mathematics in China, 2009, 4(2): 217-252. https://doi.org/10.1007/s11464-009-0017-5

    To ?nd a cycle (resp. path) of a given length in a graph is the cycle (resp. path) embedding problem. To ?nd cycles of all lengths from its girth to its order in a graph is the pancyclic problem. A stronger concept than the pancylicity is the panconnectivity. A graph of order n is said to be panconnected if for any pair of different vertices x and y with distance d there exist xy-paths of every length from d to n. The pancyclicity or the panconnectivity is an important property to determine if the topology of a network is suitable for some applications where mapping cycles or paths of any length into the topology of the network is required. The pancyclicity and the panconnectivity of interconnection networks have attracted much research interest in recent years. A large amount of related work appeared in the literature, with some repetitions. The purpose of this paper is to give a survey of the results related to these topics for the hypercube and some hypercube-like networks.

  • RESEARCH ARTICLE
    Qingyan WU,Zunwei FU
    Frontiers of Mathematics in China, 2016, 11(1): 155-172. https://doi.org/10.1007/s11464-015-0508-5

    In the setting of the Heisenberg group, based on the rotation method, we obtain the sharp (p, p) estimate for the Hardy operator. It will be shown that the norm of the Hardy operator on Lp(Hn) is still p/(p−1). This goes some way to imply that the Lp norms of the Hardy operator are the same despite the domains are intervals on ℝ, balls in ℝn, or ‘ellipsoids’ on the Heisenberg group Hn. By constructing a special function, we find the best constant in the weak type (1,1) inequality for the Hardy operator. Using the translation approach, we establish the boundedness for the Hardy operator from H1 to L1. Moreover, we describe the difference between Mp weights and Ap weights and obtain the characterizations of such weights using the weighted Hardy inequalities.

  • RESEARCH ARTICLE
    Yonghong LIU, Ting ZHANG, Yiheng TANG
    Frontiers of Mathematics in China, 2022, 17(6): 1015-1024. https://doi.org/10.1007/s11464-022-1030-1

    In this paper, we investigate functional limit problem for path of a Brownian sheet, Chung's functional law of the iterated logarithm for a Brownian sheet is obtained. The main tool in the proof is large deviation and small deviation for a Brownian sheet.

  • RESEARCH ARTICLE
    Simin SONG, Lifang YANG, Gengsheng ZHANG
    Frontiers of Mathematics in China, 2023, 18(1): 33-42. https://doi.org/10.3868/S140-DDD-023-001-X

    A generalized strongly regular graph of grade p, as a new generalization of strongly regular graphs, is a regular graph such that the number of common neighbours of both any two adjacent vertices and any two non-adjacent vertices takes on p distinct values. For any vertex v of a generalized strongly regular graph of grade 2 with parameters (n,k;a1,a2;c1,c2), if the number of the vertices that are adjacent to v and share ai(i=1,2) common neighbours with v, or are non-adjacent to v and share ci(i=1,2) common neighbours with v is independent of the choice of the vertex v, then the generalized strongly regular graph of grade 2 is free. In this paper, we investigate the generalized strongly regular graph of grade 2 with parameters (n,k;k1,a2;k1,c2) and provide the sufficient and necessary conditions for the existence of a family of free generalized strongly regular graphs of grade 2.

  • CARABALLO Tomás, LU Kening
    Frontiers of Mathematics in China, 2008, 3(3): 317-335. https://doi.org/10.1007/s11464-008-0028-7
    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.
  • Research articles
    Zunwei FU, Shanzhen LU,
    Frontiers of Mathematics in China, 2010, 5(3): 531-539. https://doi.org/10.1007/s11464-010-0015-7
    The operator norms of weighted Hardy operators onMorrey spaces are worked out. The other purpose of this paper is to establish a sufficient and necessary condition on weight functions which ensures the boundedness of the commutators of weighted Hardy operators (with symbols in BMO("Graphic")) on Morrey spaces.
  • RESEARCH ARTICLE
    Xiaoqian SUN, Xuelin YONG, Jianwei GAO
    Frontiers of Mathematics in China, 2020, 15(5): 1001-1009. https://doi.org/10.1007/s11464-020-0870-9

    Based on the Lie symmetry method, we derive the explicit optimal invest strategy for an investor who seeks to maximize the expected exponential (CARA) utility of the terminal wealth in a defined-contribution pension plan under a constant elasticity of variance model. We examine the point symmetries of the Hamilton-Jacobi-Bellman (HJB) equation associated with the portfolio optimization problem. The symmetries compatible with the terminal condition enable us to transform the (2+ 1)-dimensional HJB equation into a (1+ 1)-dimensional nonlinear equation which is linearized by its infinite-parameter Lie group of point transformations. Finally, the ansatz technique based on variables separation is applied to solve the linear equation and the optimal strategy is obtained. The algorithmic procedure of the Lie symmetry analysis method adopted here is quite general compared with conjectures used in the literature.

  • RESEARCH ARTICLE
    Guangkun SUN,Shuaiqi ZHANG,Guoxin LIU
    Frontiers of Mathematics in China, 2015, 10(6): 1433-1447. https://doi.org/10.1007/s11464-015-0492-9

    This article deals with the ruin probability in a Sparre Andersen risk process with the inter-claim times being Erlang distributed in the framework of piecewise deterministic Markov process (PDMP). We construct an exponential martingale by virtue of the extended generator of the PDMP to change the measure. Some results are derived for the ruin probabilities, such as the general expressions for ruin probability, Lundberg bounds, Cramér-Lundberg approximations, and finite-horizon ruin probability.

  • RESEARCH ARTICLE
    Tingting WANG, Wenpeng ZHANG
    Frontiers of Mathematics in China, 2011, 6(3): 557-563. https://doi.org/10.1007/s11464-011-0132-y

    In this paper, we use the elementary and analytic methods to study the computational problem of one kind mean value involving the classical Dedekind sums and two-term exponential sums, and give two exact computational formulae for them.

  • SURVEY ARTICLE
    King Fai LAI
    Frontiers of Mathematics in China, 2022, 17(2): 171-225. https://doi.org/10.1007/s11464-022-1008-z

    We discuss the role of differential equations in Lie group representation theory. We use Kashiwara’s pentagon as a reference frame for the real representation theory and then report on some work arising from its p-adic analogue by Emerton, Kisin, Patel, Huyghe, Schmidt, Strauch using Berthelot’s theory of arithmetic D-modules and Schneider–Stuhler theory of sheaves on buildings.