Chinese Quarterly Journal of Mathematics

The Integral Representations and Their Applications on the Analytic Varieties of Bounded Domains in Stein Manifolds

CHEN Shu-jin

2021, 36(4): 331-355. doi:10.13371/j.cnki.chin.q.j.m.2021.04.001

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In this paper, we first obtain a unified integral representation on the analytic varieties of the general bounded domain in Stein manifolds (the two types bounded domains in [3] are regarded as its special cases). Secondly we get the integral formulas of the solution of \overline \partial-equation. And we use a new and unique method to give a uniform estimate of the solution of \overline \partial-equation, which is different from Henkin’s method.

An Efficient Algorithm for Low Rank Matrix Restoration Problem with Unknown Noise Level

JIN Zheng-fen, WANG Duo, SHANG You-lin, LV Jin-man

2021, 36(4): 356-368. doi:10.13371/j.cnki.chin.q.j.m.2021.04.002

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Recovering an unknown high dimensional low rank matrix from a small set of entries is widely spread in the fields of machine learning, system identification and image restoration, etc. In many practical applications, the few observations are always corrupted by noise and the noise level is also unknown. A novel model with nuclear norm and square root type estimator has been proposed, which does not rely on the knowledge or on an estimation of the standard deviation of the noise. In this paper, we firstly reformulate the problem to an equivalent variable separated form by introducing an auxiliary variable. Then we propose an efficient alternating direction method of multipliers(ADMM) for solving it. Both of resulting subproblems admit an explicit solution, which makes our algorithm have a cheap computing. Finally, the numerical results show the benefits of the model and the efficiency of the proposed method.

Triple Positive Solutions to a Third-Order m-Point Boundary Value Problem

SONG Xue, YANG Yun-rui, YANG Lu

2021, 36(4): 369-375. doi:10.13371/j.cnki.chin.q.j.m.2021.04.003

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An existence criterion of triple positive solutions to a third-order m-point boundary value problem is established by Avery-Peterson fixed point theorem. At the same time, a corresponding example is given to illustrate the result.

Uniqueness of Entire Functions Concerning Differences

QIU Shi-lin, LIU Dan

2021, 36(4): 376-389. doi:10.13371/j.cnki.chin.q.j.m.2021.04.004

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In this paper, we study the uniqueness of entire functions and prove the following theorem. Let f be a transcendental entire function of finite order. Then there exists at most one positive integer k, such that f ( z )∆ ^k_c f ( z ) −R ( z ) has finitely many zeros, where R ( z ) is a non-vanishing rational function and c is a nonzero complex number. Our result is an improvement of the theorem given by Andasmas and Latreuch [1].

N-Width for Some Classes of Periodic Functions

WANG Jia-wei, WU Ga-ridi

2021, 36(4): 390-394. doi:10.13371/j.cnki.chin.q.j.m.2021.04.005

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This paper considers the Kolmogorov width, the Gelfand width and the Linear width of the periodic smooth functions with boundary conditions in Orlicz space. Further, the relationships among these three types of width are obtained.

Phragmén-Lindelöf Alternative Result of the Forchheimer Equations

CHEN Xue-jiao, LI Yuan-fei

2021, 36(4): 395-404. doi:10.13371/j.cnki.chin.q.j.m.2021.04.006

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This paper investigates the spatial behavior of the solutions of the Forchheimer equations in a semi-infinite cylinder. Using the energy estimation method and the differential inequality technology, the differential inequality about the solution is derived. By solving this differential inequality, it is proved that the solutions grow polynomially or decay exponentially with spatial variables.

On Quasiconformal Mappings Between Hyperbolic Triangles

ZHAO Lei

2021, 36(4): 405-414. doi:10.13371/j.cnki.chin.q.j.m.2021.04.007

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Quasiconformal mappings between hyperbolic triangles are considered. We give an explicit estimate of the dilation of the quasiconformal mappings, which generalizes Bishop’s results.

Elementary C*-Algebras and Haagerup Tensor Products

HE Wei-jiao

2021, 36(4): 415-418. doi:10.13371/j.cnki.chin.q.j.m.2021.04.008

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In this note, we give a new characterization of elementary C*-algebras in terms of completely compact maps and Haagerup tensor products.

Updating Mass and Stiffness Matrices Using Eigenstructure Assignment Methods

LIU Li-na, YUAN Yu-ying, YUAN Yong-xin

2021, 36(4): 419-429. doi:10.13371/j.cnki.chin.q.j.m.2021.04.009

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A novel numerical method is presented to update mass and stiffness matrices simultaneously with measured vibration data by means of the combined acceleration and displacement output feedback. By the method, the required displacement and acceleration output feedback gain matrices are determined, and thus the optimal approximation mass matrix and stiffness matrix which satisfy the required orthogonality relation and eigenvalue equation are found. The proposed method is computationally efficient and the updated mass and stiffness matrices are also symmetric and have the compact expressions. The numerical example shows that the proposed method is reliable and attractive.

The Depth Distribution of Constacyclic Codes Over the Ring F_p+vF_p+v²F_p

LI Jian-hao, WU Hua-zhang

2021, 36(4): 430-440. doi:10.13371/j.cnki.chin.q.j.m.2021.04.010

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In this paper, we studied the depth spectrum and the depth distribution of constacyclic codes over the non-chain ring R = F_p+vF_p+v²F_p, where v³=v. By decomposing the linear codes C over R into the linear codes over the finite field Fp, three corresponding constacyclic codes C₁, C₂, C₃ over Fp were obtained. Furthermore, considering the depth spectrum of constacyclic codes over the finite filed Fp, and the relationship between constacyclic codes C₁, C₂, C₃ and C, the depth spectrum and the depth distribution of constacyclic codes over R were discussed.

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