Table of Content

30 March 2021, Volume 36 Issue 1

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Buildings and Groups III

LAI King-fai

2021, 36(1):  1-31.  doi:10.13371/j.cnki.chin.q.j.m.2021.01.001

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This is the third part of a pedagogical introduction to the theory of buildings of Jacques Tits. We describe the construction and properties of the Bruhat-Tits building of a reductive group over a local field.

Viaritional Formulas for Translating Solitons with Density

LIU Hua-qiao

2021, 36(1):  32-40.  doi:10.13371/j.cnki.chin.q.j.m.2021.01.002

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In this paper, we introduce a kind of submanifold called translating solitons with density, and obtain two variational formulas for it, and show some geometric quantities of it.


New Complex Solutions of the Coupled KdV Equations

FAN Jun-jie , LI Guang-fang , Taogetusang

2021, 36(1):  41-48.  doi:10.13371/j.cnki.chin.q.j.m.2021.01.003

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 In this paper, new infinite sequence complex solutions of the coupled KdV equations are constructed with the help of function transformation and the second kind of elliptic equation. First of all, according to the function transformation, the coupled KdV equations are changed into the second kind of elliptic equation. Secondly, the new solutions and B¨acklund transformation of the second kind of elliptic equation are applied to search for new infinite sequence complex solutions of the coupled KdV equations. These solutions include new infinite sequence complex solutions composed by Jacobi elliptic
function, hyperbolic function and triangular function.

Existence and Multiplicity of Positive Solutions for a Coupled Fourth-Order System of Kirchhoff Type

LI Zhen-hui, XU Li-ping

2021, 36(1):  49-66.  doi:10.13371/j.cnki.chin.q.j.m.2021.01.004

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In this paper, we study a coupled fourth-order system of Kirchhoff type. Under appropriate hypotheses of Vi(x) for i=1,2, f and g, we obtained two main existence theorems of weak solutions for the problem by variational methods. Some recent results
are extended.

Spatial Asymptotic Properties of a System Wave Equations with Nonlinear Damping and Source Terms

LI Yuan-fei

2021, 36(1):  67-78.  doi:10.13371/j.cnki.chin.q.j.m.2021.01.005

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In this paper, the wave equation defined in a semi-infinite cylinder is considered,
in which the nonlinear damping and source terms is included. By setting an arbitrary
parameter greater than zero in the energy expression, the fast growth rate or decay rate
of the solution with spatial variables is obtained by using energy analysis method and
differential inequality technique. Secondly, we obtain the asymptotic behavior of the
solution on the external domain of the sphere. In addition, in this paper we also give
some useful remarks which show that our results can be extended to more models.


On the Diophantine Inequality Problem

SONG Yan-bo

2021, 36(1):  79-89.  doi:10.13371/j.cnki.chin.q.j.m.2021.01.006

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In this paper, we deal with a Diophantine inequality involving a prime, two squares of primes and one k-th power of a prime which give an improvement of the result given by Alessandro Gambini.


Nested Alternating Direction Method of Multipliers to Low-Rank and Sparse-Column Matrices Recovery

SHEN Nan , JIN Zheng-fen , WANG Qiu-yu

2021, 36(1):  90-110.  doi:10.13371/j.cnki.chin.q.j.m.2021.01.007

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The task of dividing corrupted-data into their respective subspaces can be well
illustrated, both theoretically and numerically, by recovering low-rank and sparse-column
components of a given matrix. Generally, it can be characterized as a matrix and a
`2,1-norm involved convex minimization problem. However, solving the resulting problem
is full of challenges due to the non-smoothness of the objective function. One of the
earliest solvers is an 3-block alternating direction method of multipliers (ADMM) which
updates each variable in a Gauss-Seidel manner. In this paper, we present three variants
of ADMM for the 3-block separable minimization problem. More preciously, whenever
one variable is derived, the resulting problems can be regarded as a convex minimization
with 2 blocks, and can be solved immediately using the standard ADMM. If the inner
iteration loops only once, the iterative scheme reduces to the ADMM with updates in a
Gauss-Seidel manner. If the solution from the inner iteration is assumed to be exact, the
convergence can be deduced easily in the literature. The performance comparisons with a
couple of recently designed solvers illustrate that the proposed methods are effective and
competitive.