Frontiers of Mathematics in China


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Riccati equations and Toeplitz-Berezin type symbols on Dirichlet space of unit ball
Jianjun CHEN, Xiaofeng WANG, Jin XIA, Guangfu CAO
Front. Math. China    2017, 12 (4): 769-785.   DOI: 10.1007/s11464-017-0640-5
Abstract   PDF (192KB)

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.

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Infinite-dimensional 3-Lie algebras and their connections to Harish-Chandra modules
Ruipu BAI, Zhenheng LI, Weidong WANG
Front. Math. China    2017, 12 (3): 515-530.   DOI: 10.1007/s11464-017-0606-7
Abstract   PDF (186KB)

We construct two kinds of infinite-dimensional 3-Lie algebras from a given commutative associative algebra, and show that they are all canonical Nambu 3-Lie algebras. We relate their inner derivation algebras to Witt algebras, and then study the regular representations of these 3-Lie algebras and the natural representations of the inner derivation algebras. In particular, for the second kind of 3-Lie algebras, we find that their regular representations are Harish-Chandra modules, and the inner derivation algebras give rise to intermediate series modules of the Witt algebras and contain the smallest full toroidal Lie algebras without center.

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Large sieve inequality with sparse sets of moduli applied to Goldbach conjecture
Front. Math. China    2017, 12 (2): 261-280.   DOI: 10.1007/s11464-016-0527-x
Abstract   PDF (211KB)

We use the large sieve inequality with sparse sets of moduli to prove a new estimate for exponential sums over primes. Subsequently, we apply this estimate to establish new results on the binary Goldbach problem where the primes are restricted to given arithmetic progressions.

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Constructions of derived equivalences for algebras and rings
Changchang XI
Front. Math. China    2017, 12 (1): 1-18.   DOI: 10.1007/s11464-016-0593-0
Abstract   PDF (217KB)

In this article, we shall survey some aspects of our recent (or related) constructions of derived equivalences for algebras and rings.

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Generalized T3-plot for testing high-dimensional normality
Mingyao AI,Jiajuan LIANG,Man-Lai TANG
Front. Math. China    2016, 11 (6): 1363-1378.   DOI: 10.1007/s11464-016-0535-x
Abstract   PDF (578KB)

A new dimension-reduction graphical method for testing highdimensional normality is developed by using the theory of spherical distributions and the idea of principal component analysis. The dimension reduction is realized by projecting high-dimensional data onto some selected eigenvector directions. The asymptotic statistical independence of the plotting functions on the selected eigenvector directions provides the principle for the new plot. A departure from multivariate normality of the raw data could be captured by at least one plot on the selected eigenvector direction. Acceptance regions associated with the plots are provided to enhance interpretability of the plots. Monte Carlo studies and an illustrative example show that the proposed graphical method has competitive power performance and improves the existing graphical method significantly in testing high-dimensional normality.

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Topics in Metric Riemannian Geometry
Fuquan FANG, Xiaochun RONG, Wilderich TUSCHMANN, Yihu YANG
Front. Math. China    2016, 11 (5): 1097-1098.   DOI: 10.1007/s11464-016-0581-4
Abstract   PDF (49KB)
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Ring Theory and Related Topics
Nanqing DING,Tai Keun KWAK,Fang LI,Masahisa SATO
Front. Math. China    2016, 11 (4): 763-764.   DOI: 10.1007/s11464-016-0568-1
Abstract   PDF (48KB)
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Linear Algebra and Multilinear Algebra
Liqun QI,Yimin WEI,Changqing XU,Tan ZHANG
Front. Math. China    2016, 11 (3): 509-510.   DOI: 10.1007/s11464-016-0540-0
Abstract   PDF (37KB)
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Classification of Bott towers by matrix
Qifeng BAI,Fang LI
Front. Math. China    2016, 11 (2): 255-268.   DOI: 10.1007/s11464-015-0511-x
Abstract   PDF (146KB)

A criterion for the classification of Bott towers is presented, i.e., two ABott towers B?(A) and B?(A') are isomorphic if and only if the matrices A and A' are equivalent. The equivalence relation is defined by two operations on matrices. And it is based on the observation that any Bott tower B?(A) is uniquely determined by its structure matrix A, which is a strictly upper triangular integer matrix. The classification of Bott towers is closely related to the cohomological rigidity problem for both Bott towers and Bott manifolds.

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Weighted norm inequalities with multiple-weight for singular integral operators with non-smooth kernels
Rui BU,Houyu JIA
Front. Math. China    2016, 11 (1): 1-19.   DOI: 10.1007/s11464-015-0505-8
Abstract   PDF (183KB)

By sharp maximal function, we establish a weighted estimate with multiple-weight for the multilinear singular integral operators with non-smooth kernels.

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