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Frontiers of Mathematics in China

ISSN 1673-3452 (Print)
ISSN 1673-3576 (Online)
CN 11-5739/O1
Postal Subscription Code 80-964
2019 Impact Factor: 1.03

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, Volume 16 Issue 1

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SURVEY ARTICLE
Function characterizations via commutators of Hardy operator
Shanzhen LU
Front. Math. China. 2021, 16 (1): 1-12.  https://doi.org/10.1007/s11464-021-0894-9
Abstract   PDF (300KB)

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.

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RESEARCH ARTICLE
Singular integral operators on product domains along twisted surfaces
Ahmad AL-SALMAN
Front. Math. China. 2021, 16 (1): 13-28.  https://doi.org/10.1007/s11464-021-0911-z
Abstract   PDF (299KB)

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.

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Grothendieck rings of a class of Hopf algebras of Kac-Paljutkin type
Jialei CHEN, Shilin YANG, Dingguo WANG
Front. Math. China. 2021, 16 (1): 29-47.  https://doi.org/10.1007/s11464-021-0893-x
Abstract   PDF (337KB)

We construct the Grothendieck rings of a class of 2n2 dimensional semisimple Hopf Algebras H2n2,which can be viewed as a generalization of the 8 dimensional Kac-Paljutkin Hopf algebra H8.All irreducible H2n2-modules are classified. Furthermore, we describe the Grothendieck rings r(H2n2) by generators and relations explicitly.

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Exceptional sets in Waring-Goldbach problem for fifth powers
Zhenzhen FENG, Zhixin LIU
Front. Math. China. 2021, 16 (1): 49-58.  https://doi.org/10.1007/s11464-021-0899-4
Abstract   PDF (278KB)

We consider exceptional sets in the Waring-Goldbach problem for fifth powers. For example, we prove that all but O(N131/132) integers satisfying the necessary local conditions can be represented as the sum of 11 fifth powers of primes, which improves the previous results due to A. V. Kumchev [Canad. J. Math., 2005, 57: 298–327] and Z. X. Liu [Int. J. Number Theory, 2012, 8: 1247–1256].

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Dynamical behaviors of non-autonomous fractional FitzHugh-Nagumo system driven by additive noise in unbounded domains
Chunxiao GUO, Yiju CHEN, Ji SHU, Xinguang YANG
Front. Math. China. 2021, 16 (1): 59-93.  https://doi.org/10.1007/s11464-021-0896-7
Abstract   PDF (365KB)

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.

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Proper resolutions and Gorensteinness in extriangulated categories
Jiangsheng HU, Dondong ZHANG, Panyue ZHOU
Front. Math. China. 2021, 16 (1): 95-117.  https://doi.org/10.1007/s11464-021-0887-8
Abstract   PDF (317KB)

Let (,E,s) be an extriangulated category with a proper class ξ of E-triangles, and W an additive full subcategory of (,E,s). We provide a method for constructing a proper Wξ-resolution (resp., coproper Wξ- coresolution) of one term in an E-triangle in ξ from that of the other two terms. By using this way, we establish the stability of the Gorenstein category GWξ in extriangulated categories. These results generalize the work of Z. Y. Huang [J. Algebra, 2013, 393: 142{169] and X. Y. Yang and Z. C. Wang [Rocky Mountain J. Math., 2017, 47: 1013{1053], but the proof is not too far from their case. Finally, we give some applications about our main results.

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Fourier transform of anisotropic mixed-norm Hardy spaces
Long HUANG, Der-Chen CHANG, Dachun YANG
Front. Math. China. 2021, 16 (1): 119-139.  https://doi.org/10.1007/s11464-021-0906-9
Abstract   PDF (349KB)

Let a=(a1,...,an)[1,)n,p:=(p1,...pn)(0,1]n,Hap(n) be the anisotropic mixed-norm Hardy space associated with a defined via the radial maximal function, and let f belong to the Hardy space Hap(n). In this article, we show that the Fourier transform f^ coincides with a continuous function g on n in the sense of tempered distributions and, moreover, this continuous function g; multiplied by a step function associated with a; can be pointwisely controlled by a constant multiple of the Hardy space norm of f: These proofs are achieved via the known atomic characterization of Hap(n) and the establishment of two uniform estimates on anisotropic mixed-norm atoms. As applications, we also conclude a higher order convergence of the continuous function g at the origin. Finally, a variant of the Hardy{Littlewood inequality in the anisotropic mixed-norm Hardy space setting is also obtained. All these results are a natural generalization of the well-known corresponding conclusions of the classical Hardy spaces Hp(n) with p(0,1], and are even new for isotropic mixed-norm Hardy spaces on n.

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Bi-block positive semidefiniteness of bi-block symmetric tensors
Zheng-Hai HUANG, Xia LI, Yong WANG
Front. Math. China. 2021, 16 (1): 141-169.  https://doi.org/10.1007/s11464-021-0874-0
Abstract   PDF (367KB)

The positive definiteness of elasticity tensors plays an important role in the elasticity theory. In this paper, we consider the bi-block symmetric tensors, which contain elasticity tensors as a subclass. First, we define the bi-block M-eigenvalue of a bi-block symmetric tensor, and show that a bi-block symmetric tensor is bi-block positive (semi)definite if and only if its smallest bi-block M-eigenvalue is (nonnegative) positive. Then, we discuss the distribution of bi-block M-eigenvalues, by which we get a sufficient condition for judging bi-block positive (semi)definiteness of the bi-block symmetric tensor involved. Particularly, we show that several classes of bi-block symmetric tensors are bi-block positive definite or bi-block positive semidefinite, including bi-block (strictly) diagonally dominant symmetric tensors and bi-block symmetric (B)B0-tensors. These give easily checkable sufficient conditions for judging bi-block positive (semi)definiteness of a bi-block symmetric tensor. As a byproduct, we also obtain two easily checkable suffcient conditions for the strong ellipticity of elasticity tensors.

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Biquadratic tensors, biquadratic decompositions, and norms of biquadratic tensors
Liqun QI, Shenglong HU, Xinzhen ZHANG, Yanwei XU
Front. Math. China. 2021, 16 (1): 171-185.  https://doi.org/10.1007/s11464-021-0895-8
Abstract   PDF (284KB)

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.

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Convergence of complex martingale for a branching random walk in an independent and identically distributed environment
Xin WANG, Xingang LIANG, Chunmao HUANG
Front. Math. China. 2021, 16 (1): 187-209.  https://doi.org/10.1007/s11464-021-0882-0
Abstract   PDF (341KB)

We consider an d-valued discrete time branching random walk in an independent and identically distributed environment indexed by time n. Let Wn(z)(zd) be the natural complex martingale of the process. We show necessary and sufficient conditions for the Lα-convergence of Wn(z) for α>1, as well as its uniform convergence region.

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Boundedness of some integral operators and commutators on homogeneous Herz spaces with three variable exponents
Xia YU, Zongguang LIU
Front. Math. China. 2021, 16 (1): 211-237.  https://doi.org/10.1007/s11464-021-0897-6
Abstract   PDF (335KB)

We obtain the boundedness of some integral operators and commutators on homogeneous Herz spaces with three variable exponents K ˙p(),q()α(), such as some sublinear operators, the fractional integral and its commutator.

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Minimal period estimates on P-symmetric periodic solutions of first-order mild superquadratic Hamiltonian systems
Xiaofei ZHANG, Chungen LIU
Front. Math. China. 2021, 16 (1): 239-253.  https://doi.org/10.1007/s11464-021-0903-z
Abstract   PDF (316KB)

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.

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