Journal home Browse Featured articles

Featured articles

  • Select all
  • SURVEY ARTICLE
    Tao QIAN
    Frontiers of Mathematics in China, 0: 337-371. https://doi.org/10.1007/s11464-022-1014-1

    Positive-instantaneous frequency representation for transient signals has always been a great concern due to its theoretical and practical importance, although the involved concept itself is paradoxical. The desire and practice of uniqueness of such frequency representation (decomposition) raise the related topics in approximation. During approximately the last two decades there has formulated a signal decomposition and reconstruction method rooted in harmonic and complex analysis giving rise to the desired signal representations. The method decomposes any signal into a few basic signals that possess positive instantaneous frequencies. The theory has profound relations to classical mathematics and can be generalized to signals defined in higher dimensional manifolds with vector and matrix values, and in particular, promotes kernel approximation for multi-variate functions. This article mainly serves as a survey. It also gives two important technical proofs of which one for a general convergence result (Theorem 3.4), and the other for necessity of multiple kernel (Lemma 3.7).

    Expositorily, for a given real-valued signal f one can associate it with a Hardy space function F whose real part coincides with f. Such function F has the form F=f+ iHf, where H stands for the Hilbert transformation of the context. We develop fast converging expansions of F in orthogonal terms of the form

    F= k =1c kB k

    where Bk’s are also Hardy space functions but with the additional properties

    Bk( t)= ρk (t) ei θk (t),ρ k0 ,θ k' (t) 0,a.. e

    The original real-valued function f is accordingly expanded

    f= k =1ρ k(t)cosθ k (t)

    which, besides the properties of ρ k and θ k given above, also satisfies

    H( ρkcosθk( t)= ρk (t)sin θk(t).

    Real-valued functions f(t )=ρ(t)cosθ (t) that satisfy the condition

    ρ 0,θ' (t) 0,H(ρcosθ) (t)= ρ(t)sinθ( t)

    are called mono-components. If f is a mono-component, then the phase derivative θ '(t )is defined to be instantaneous frequency of f. The above described positive-instantaneous frequency expansion is a generalization of the Fourier series expansion. Mono-components are crucial to understand the concept instantaneous frequency. We will present several most important mono-component function classes. Decompositions of signals into mono-components are called adaptive Fourier decompositions (AFDs). We note that some scopes of the studies on the 1D mono-components and AFDs can be extended to vector-valued or even matrix-valued signals defined on higher dimensional manifolds. We finally provide an account of related studies in pure and applied mathematics.

  • 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.

  • RESEARCH ARTICLE
    Mu-Fa CHEN
    Frontiers of Mathematics in China, 2018, 13(6): 1267-1311. https://doi.org/10.1007/s11464-018-0716-x

    The main purpose of the paper is looking for a larger class of matrices which have real spectrum. The first well-known class having this property is the symmetric one, then is the Hermite one. This paper introduces a new class, called Hermitizable matrices. The closely related isospectral problem, not only for matrices but also for differential operators is also studied. The paper provides a way to describe the discrete spectrum, at least for tridiagonal matrices or one-dimensional differential operators. Especially, an unexpected result in the paper says that each Hermitizable matrix is isospectral to a birth–death type matrix (having positive sub-diagonal elements, in the irreducible case for instance). Besides, new efficient algorithms are proposed for computing the maximal eigenpairs of these class of matrices.

  • RESEARCH ARTICLE
    Yanping CHEN, Liwei WANG
    Frontiers of Mathematics in China, 2018, 13(5): 1013-1031. https://doi.org/10.1007/s11464-018-0718-8

    For bL ip (R n), the Calderón commutator with variable kernel is defined by [b , T1]f(x)=p.v.R nΩ(x,xy) |x y|n+1( b(x )b(y))f( y)dy.In this paper, we establish the L2( Rn) boundedness for [b, T1] with Ω( x, z') L (R n) ×Lq ( Sn 1)(q 2(n 1)/n) satisfying certain cancellation conditions. Moreover, the exponent q 2(n 1)/n is optimal. Our main result improves a previous result of Calderón.

  • RESEARCH ARTICLE
    Lijian AN
    Frontiers of Mathematics in China, 2018, 13(4): 763-777. https://doi.org/10.1007/s11464-018-0693-0

    Suppose that G is a nite p-group. If G is not a Dedekind group, then G has a non-normal subgroup. We use pM(G) and pm(G) to denote the maximum and minimum of the orders of the non-normal subgroups of G; respectively. In this paper, we classify groups G such that M(G)2m(G)1: As a by-product, we also classify p-groups whose orders of non-normal subgroups are pk and pk+1:

  • RESEARCH ARTICLE
    Mu-Fa CHEN
    Frontiers of Mathematics in China, 2018, 13(3): 509-524. https://doi.org/10.1007/s11464-018-0697-9

    This paper is a continuation of the author’s previous papers [Front. Math. China, 2016, 11(6): 1379–1418; 2017, 12(5): 1023–1043], where the linear case was studied. A shifted inverse iteration algorithm is introduced, as an acceleration of the inverse iteration which is often used in the non-linear context (the p-Laplacian operators for instance). Even though the algorithm is formally similar to the Rayleigh quotient iteration which is well-known in the linear situation, but they are essentially different. The point is that the standard Rayleigh quotient cannot be used as a shift in the non-linear setup. We have to employ a different quantity which has been obtained only recently. As a surprised gift, the explicit formulas for the algorithm restricted to the linear case (p = 2) is obtained, which improves the author’s approximating procedure for the leading eigenvalues in different context, appeared in a group of publications. The paper begins with p-Laplacian, and is closed by the non-linear operators corresponding to the well-known Hardy-type inequalities.

  • RESEARCH ARTICLE
    Haibin CHEN, Liqun QI, Yisheng SONG
    Frontiers of Mathematics in China, 2018, 13(2): 255-276. https://doi.org/10.1007/s11464-018-0681-4

    Stimulated by the study of sufficient matrices in linear complementarity problems, we study column sufficient tensors and tensor complementarity problems. Column sufficient tensors constitute a wide range of tensors that include positive semi-definite tensors as special cases. The inheritance property and invariant property of column sufficient tensors are presented. Then, various spectral properties of symmetric column sufficient tensors are given. It is proved that all H-eigenvalues of an even-order symmetric column sufficient tensor are nonnegative, and all its Z-eigenvalues are nonnegative even in the odd order case. After that, a new subclass of column sufficient tensors and the handicap of tensors are defined. We prove that a tensor belongs to the subclass if and only if its handicap is a finite number. Moreover, several optimization models that are equivalent with the handicap of tensors are presented. Finally, as an application of column sufficient tensors, several results on tensor complementarity problems are established.

  • RESEARCH ARTICLE
    Nguyen Minh CHUONG, Nguyen Thi HONG, Ha Duy HUNG
    Frontiers of Mathematics in China, 2018, 13(1): 1-24. https://doi.org/10.1007/s11464-017-0677-5

    We introduce the p-adic weighted multilinear Hardy-Cesàro operator. We also obtain the necessary and sufficient conditions on weight functions to ensure the boundedness of that operator on the product of Lebesgue spaces, Morrey spaces, and central bounded mean oscillation spaces. In each case, we obtain the corresponding operator norms. We also characterize the good weights for the boundedness of the commutator of weighted multilinear Hardy-Cesàro operator on the product of central Morrey spaces with symbols in central bounded mean oscillation spaces.

  • EDITORIAL
    Shmuel FRIEDLAND, Liqun QI, Yimin WEI, Qingzhi YANG
    Frontiers of Mathematics in China, 2017, 12(6): 1277. https://doi.org/10.1007/s11464-017-0669-5
  • RESEARCH ARTICLE
    Jianjun CHEN, Xiaofeng WANG, Jin XIA, Guangfu CAO
    Frontiers of Mathematics in China, 2017, 12(4): 769-785. https://doi.org/10.1007/s11464-017-0640-5

    The present paper mainly gives some applications of Berezin type symbols on the Dirichlet space of unit ball. We study the solvability of some Riccati operator equations of the form XAX+ XBCX= Drelated to harmonic Toeplitz operators on the Dirichlet space. Especially, the invariant subspaces of Toeplitz operators are also considered.