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A Posterior Estimates in the Finite Element Methods E Wei-nan, MU Mo, HUANG Hong-ci Chinese Quarterly Journal of Mathematics    1988, 3 (1): 97-107.   Abstract （963）      PDF（pc） （588KB）（485）       Knowledge map    Save Related Articles | Metrics

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On Some Recent Progress in Complex Geometry|the Area Related to Homogeneous Manifolds GUAN Daniel Chinese Quarterly Journal of Mathematics    2020, 35 (2): 111-144.   DOI: 10.13371/j.cnki.chin.q.j.m.2020.02.001 Abstract （699）      PDF（pc） （1154KB）（400）       Knowledge map    Save In this article, we give a survey of some progress of the complex geometry, mostly related to the Lie group actions on compact complex manifolds and complex homogeneous spaces in the last thirty years. In particular, we explore some works in the special area in Dierential Geometry, Lie Group and Complex Homogeneous Space. Together with the special area in nonlinear analysis on complex manifolds, they are the two major aspects of my research interests. Related Articles | Metrics

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Nested Alternating Direction Method of Multipliers to Low-Rank and Sparse-Column Matrices Recovery SHEN Nan , JIN Zheng-fen , WANG Qiu-yu Chinese Quarterly Journal of Mathematics    2021, 36 (1): 90-110.   DOI: 10.13371/j.cnki.chin.q.j.m.2021.01.007 Abstract （542）      PDF（pc） （427KB）（469）       Knowledge map    Save 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. Related Articles | Metrics

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Some Applications of Surface Curvatures in Theoretical Physics YANG Yi-song Chinese Quarterly Journal of Mathematics    2023, 38 (3): 221-253.   DOI: 10.13371/j.cnki.chin.q.j.m.2023.03.001 Abstract （536）      PDF（pc） （496KB）（144）       Knowledge map    Save In this survey article, we present two applications of surface curvatures in theoretical physics. The first application arises from biophysics in the study of the shape of cell vesicles involving the minimization of a mean curvature type energy called the Helfrich bending energy. In this formalism, the equilibrium shape of a cell vesicle may present itself in a rich variety of geometric and topological characteristics. We first show that there is an obstruction, arising from the spontaneous curvature, to the existence of a minimizer of the Helfrich energy over the set of embedded ring tori. We then propose a scale-invariant anisotropic bending energy, which extends the Canham energy, and show that it possesses a unique toroidal energy minimizer, up to rescaling, in all parameter regime. Furthermore, we establish some genus-dependent topological lower and upper bounds, which are known to be lacking with the Helfrich energy, for the proposed energy. We also present the shape equation in our context, which extends the Helfrich shape equation. The second application arises from astrophysics in the search for a mechanism for matter accretion in the early universe in the context of cosmic strings. In this formalism, gravitation may simply be stored over a two-surface so that the Einstein tensor is given in terms of the Gauss curvature of the surface which relates itself directly to the Hamiltonian energy density of the matter sector. This setting provides a lucid exhibition of the interplay of the underlying geometry, matter energy, and topological characterization of the system. In both areas of applications, we encounter highly challenging nonlinear partial differential equation problems. We demonstrate that studies on these equations help us to gain understanding of the theoretical physics problems considered. Related Articles | Metrics

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Buildings and Groups III LAI King-fai Chinese Quarterly Journal of Mathematics    2021, 36 (1): 1-31.   DOI: 10.13371/j.cnki.chin.q.j.m.2021.01.001 Abstract （460）      PDF（pc） （533KB）（426）       Knowledge map    Save 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. Related Articles | Metrics

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Buildings and Groups II CHAO Kuok Fai, LAI King Fai Chinese Quarterly Journal of Mathematics    2020, 35 (3): 221-254.   DOI: 10.13371/j.cnki.chin.q.j.m.2020.03.001 Abstract （382）      PDF（pc） （1238KB）（429）       Knowledge map    Save This is the second part of a pedagogical introduction to the theory of buildings of Jacques Tits. We define a(B,N) pair and construct a building out of it. Then we give a description of Chevalley groups, their(B,N) pair and the associated buildings. We illustrates this theory with many examples from classical groups.  Related Articles | Metrics

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On the Distribution of the Values of a Class of Arithmetical Functions SHAO Pin-tsung Chinese Quarterly Journal of Mathematics    1987, 2 (2): 27-43.   Abstract （325）      PDF（pc） （895KB）（469）       Knowledge map    Save 对于数义在自然数集 N 上的数论函数（Arithmetical function）f（n）, n\inN,取值可以是实数, 也可以是复数（一般为实数）, 例如著名的 Euler 函数 \varphi(n)=\Sigma_{(a,n)=1,a<n}1,  除数函数τ（n）=\Sigma_{d/n}1,以及除数和函数σ（n）=\Sigma_{d/n}d  Related Articles | Metrics

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On the Sum of Powers of Natural Numbers CHEN Jing-run, LI Jian-yu Chinese Quarterly Journal of Mathematics    1987, 2 (1): 1-18.   Abstract （315）      PDF（pc） （2527KB）（306）       Knowledge map    Save Related Articles | Metrics

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L\acute{e}vy-Prohorov Metric on the Measure Space QU Li-min, ZHU Ji-yun Chinese Quarterly Journal of Mathematics    2017, 32 (1): 42-48.   DOI: 10.13371/j.cnki.chin.q.j.m.2017.01.005 Abstract （311）      PDF（pc） （324KB）（160）       Knowledge map    Save Under the premise of infinitely many pure strategies, by defining the new LP* metric, striking an equivalence of topology and weak* topology, we prove that the existence of the essential component.  Related Articles | Metrics

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On Relationship between Lower-order of Coefficients and Growth of Solutions of Complex Differential Equations near a Singular Point LIU Yuan-zhu, LONG Jian-ren, ZENG San-gui Chinese Quarterly Journal of Mathematics    2020, 35 (2): 163-170.   DOI: 10.13371/j.cnki.chin.q.j.m.2020.02.003 Abstract （306）      PDF（pc） （495KB）（267）       Knowledge map    Save We investigate the growth of solutions of the following complex linear differential equation f’’+ A（z）f’+ B（z）f = 0,where A（z） and B（z） are analytic functions in C-{z0}, z0 ∈ C. Some estimations of lower bounded of growth of solutions of the differential equation are obtained by using the concept of lower order.  Related Articles | Metrics

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Majority Coloring of r-Regular Digraph XIA Wei-hao, SHI Mei, XIAO Ming-yue, CAI Jian-sheng, WANG Ji-hui Chinese Quarterly Journal of Mathematics    2022, 37 (2): 142-146.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.02.004 Abstract （304）      PDF（pc） （330KB）（417）       Knowledge map    Save  A majority k-coloring of a digraph D with k colors is an assignment c: V ( D ) →{ 1,2,···,k} , such that for every v∈V (D), we have c(w)=c(v) for at most half of all out-neighbors w∈N+( v ). For a natural number k≥2, a 1/k-majority coloring of a digraph is a coloring of the vertices such that each vertex receives the same color as at most a 1/k proportion of its out-neighbours. Kreutzer, Oum, Seymour, van der Zypen and Wood proved that every digraph has a majority 4-coloring and conjectured that every digraph admits a majority 3-coloring. Gir$\widetilde{a}$o, Kittipassorn and Popielarz proved that every digraph has a 1/k-majority 2k-coloring and conjectured that every digraph admits a 1\k-majority (2k−1)-coloring. We showed that every r -regular digraph D with r>36ln(2n) has a majority 3-coloring and proved that every digraph D with minimum outdegree $\delta^+>\frac{2k^2(2k-1)^2}{(k-1)^2}\ln{[(2k-1)n]}$ has a 1/k-majority (2k−1)-coloring. And we also proved that every r-regular digraph D with  $r>\frac{3k^2(2k-1)}{(k-1)^2} \ln(2n)$ has a 1/k-majority (2k−1)-coloring. Related Articles | Metrics

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A New Filled Function for Global Optimization Problems with Box Constraints QU De-qiang, WU Dan, SHANG You-lin Chinese Quarterly Journal of Mathematics    2020, 35 (4): 354-362.   DOI: 10.13371/j.cnki.chin.q.j.m.2020.04.003 Abstract （303）      PDF（pc） （286KB）（243）       Knowledge map    Save  In this paper, auxiliary function method for global optimization with box constraints is considered. First, a new non-parameter filled function which has the same local minimizers of the objective function is proposed. By the character that having same local minimizers, and these minimizers are all better than the current minimizer of the objective function, it does not need to minimize the objective function except for the first iteration in the filled function method. It changes the frame of conventional filled function methods that objective function and filled function are minimized alternately, and can effectively reduce the iterations of the algorithm and accelerate the speed of global optimization. And then the theoretical properties of the filled function are discussed and the corresponding algorithm is established. Finally, numerical experiments are made and comparisons on several test problems are shown which exhibit the feasibility and effectiveness of the algorithm. Related Articles | Metrics

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Korn’s Inequality and Divergence Equations on Generalize Orlicz Spaces WU Rui-min , WANG Song-bai Chinese Quarterly Journal of Mathematics    2020, 35 (4): 344-353.   DOI: 10.13371/j.cnki.chin.q.j.m.2020.04.002 Abstract （294）      PDF（pc） （387KB）（219）       Knowledge map    Save  Let ϕ be a generalized Orlicz function satisfying (A0), (A1), (A2), (aInc) and (aDec). We prove that the mapping  f →f #:=supB 1/\int|B||f(x)-fB|dx is continuous on Lϕ(·)(Rn) by extrapolation. Based on this result we generalize Korn’s inequality to the setting of generalized Orlicz spaces, i.e., ||\triangledown f||L^{ϕ(·)}(Ω)  \lesssim||DF|||L^{ϕ}(Ω) . Using the Calder´on–Zygmund theory on generalized Orlicz spaces, we obtain that the divergence equation divu=f has a solution u∈(W01,ϕ(·)(Ω))n such that ||\triangledown u||L^{ϕ(·)}(Ω) \lesssim ||f||L^{ϕ}(Ω). Related Articles | Metrics

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Existence and Concentration of Solutions for An Indefinite Schrodinger-Kirchhoff System CHEN Yu-song, CHANG He-jie Chinese Quarterly Journal of Mathematics    2020, 35 (1): 37-45.   DOI: 10.13371/j.cnki.chin.q.j.m.2020.01.003 Abstract （286）      PDF（pc） （615KB）（218）       Knowledge map    Save This paper is concerned with the nonlinear Schrodinger-Kirchhoff system $-(a+b \int _{R^{3}}|\nabla u|^{2} dx)  \triangle u+ \lambda V(x)u=f(x,u)$ in R3, where constants a > 0,b ≥ 0 and λ > 0 is a parameter. We require that V (χ) ∈ C(R3) and has a potential well V -1(0). Combining this with other suitable assumptions on K and ƒ, the existence of nontrivisd solutions is obtained via vaxiational methods. Furthermore, the concentration behavior of the nontrivial solution is also explored on the set V -1(0) as λ → + ∞ as well. It is worth noting that the (PS )-condition can not be directly got as done in the literature, which makes the problem more complicated. To overcome this difficulty, we adopt different method. Related Articles | Metrics

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Majorization and Fekete-Szegö Problems for Multivalent Meromorphic Functions Associated with the Mittag-Leffler Function LONG Pin-hong, GANGADHARAN Murugusundaramoorthy, HAN Hui-li, WANG Wen-shuai Chinese Quarterly Journal of Mathematics    2022, 37 (2): 111-123.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.02.001 Abstract （283）      PDF（pc） （356KB）（285）       Knowledge map    Save The main object of this present paper is to investigate the problem of majorization of certain class of multivalent meromorphic functions of complex order involving Mittag-Leffler function. Meanwhile, for this subclass the corresponding coefficient estimates and some Fekete-Szeg¨ o type inequalities are obtained. Moreover we point out some new or known consequences of our main results. Related Articles | Metrics

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General Energy Decay of Solutions for A Wave Equation with Nonlocal Nonlinear Damping and Source Terms ZHANG Hong-wei, LI Dong-hao, HU Qing-ying Chinese Quarterly Journal of Mathematics    2020, 35 (3): 302-310.   DOI: 10.13371/j.cnki.chin.q.j.m.2020.03.005 Abstract （282）      PDF（pc） （473KB）（423）       Knowledge map    Save We consider a wave equation with nonlocal nonlinear damping and source terms. We prove a general energy decay property for solutions by constructing a stable set and using the multiplier technique. The main difficult is how to handle with the nonlocal nonlinear damping term. Our result extends and improves the result in the literature such as the work by Jorge Silva and Narciso（Evolution Equation and Control Theory, 2017（6）:437-470） and Narciso（Evolution Equations and Control Theory, 2020, 9（2）: 487-508）. Related Articles | Metrics

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Volume Inequalities for Sections and Projections of Asymmetric Convex Bodies CAO Zi-xin, Li Ai-jun Chinese Quarterly Journal of Mathematics    2022, 37 (2): 178-188.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.02.007 Abstract （263）      PDF（pc） （346KB）（134）       Knowledge map    Save In this paper, we establish volume inequalities for k-dimensional sections and projections of convex bodies (not necessarily symmetric) and their polars in a more general position than John’s position. Related Articles | Metrics

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Blow-Up and Decay Estimate in a Logarithmic p-Laplace Parabolic Equation LI Ping, LI Feng-jie Chinese Quarterly Journal of Mathematics    2024, 39 (4): 331-354.   DOI: 10.13371/j.cnki.chin.q.j.m.2024.04.001 Abstract （246）      PDF（pc） （462KB）（114）       Knowledge map    Save This paper deals with an initial-boundary value problem of a fourth-order parabolic equation involving Logarithmic type p-Laplacian, which could be proposed as a model for the epitaxial growth of thin films. By using the variational method and the logarithmic type Sobolev inequality, we give some threshold results for blow-up solutions and global solutions, which could be classified by the initial energy. The asymptotic estimates about blow-up time and decay estimate of weak solutions are obtained. Related Articles | Metrics

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Induced Matching-Extendability of Halin Graphs ZHANG Qing-nan, HUI Zhi-hao, YANG Yu, WANG An, Chinese Quarterly Journal of Mathematics    2022, 37 (4): 380-385.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.04.005 Abstract （245）      PDF（pc） （344KB）（236）       Knowledge map    Save  Let G be a connected graph having a perfect matching. The graph G is said to be induced matching (IM) extendable if every induced matching M of G is contained in a perfect matching of G. In this paper, we show that Halin graph G =T ∪C is IMextendable if and only if its characteristic tree T is isomorphic to K1,3, K1,5, K1,7 or S2,2. Related Articles | Metrics

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Well-Posedness for Timoshenko System with Thermodiffusion Effects and Delay QIN Yu-ming, HAN Ning Chinese Quarterly Journal of Mathematics    2022, 37 (1): 1-9.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.01.001 Abstract （244）      PDF（pc） （293KB）（442）       Knowledge map    Save  In this paper, we study the well-posedness for the thermoelastic Timoshenko system with a constant delay and mass diffusion effects. Heat and mass exchange with the environment during thermodiffusion in Timoshenko beam. The C 0 -semigroup theory will be used to prove the well-posedness of the considered problem. Related Articles | Metrics