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Acta Mathematica Sinica,Chinese Series
2017, Vol.33 Num.6
Online: 2017-06-15

731 Linda EROH, Cong X. KANG, Eunjeong YI
A Comparison between the Metric Dimension and Zero Forcing Number of Trees and Unicyclic Graphs
The metric dimension dim(G) of a graph G is the minimum number of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices. The zero forcing number Z(G) of a graph G is the minimum cardinality of a set S of black vertices (whereas vertices in V(G)\S are colored white) such that V(G) is turned black after finitely many applications of “the color-change rule”: a white vertex is converted black if it is the only white neighbor of a black vertex. We show that dim(T) ≤ Z(T) for a tree T, and that dim(G) ≤ Z(G)+1 if G is a unicyclic graph; along the way, we characterize trees T attaining dim(T) = Z(T). For a general graph G, we introduce the “cycle rank conjecture”. We conclude with a proof of dim(T) - 2 ≤ dim(T + e) ≤ dim(T) + 1 for eE(T).
2017 Vol. 33 (6): 731-747 [Abstract] ( 3 ) [HTML 1KB] [PDF 281KB] ( 15 )
748 Han Jun YU, Jun Shan SHEN, Zhao Nan LI, Xiang Zhong FANG
Semiparametric Bayesian Inference for Mean-Covariance Regression Models
In this paper, we propose a Bayesian semiparametric mean-covariance regression model with known covariance structures. A mixture model is used to describe the potential non-normal distribution of the regression errors. Moreover, an empirical likelihood adjusted mixture of Dirichlet process model is constructed to produce distributions with given mean and variance constraints. We illustrate through simulation studies that the proposed method provides better estimations in some non-normal cases. We also demonstrate the implementation of our method by analyzing the data set from a sleep deprivation study.
2017 Vol. 33 (6): 748-760 [Abstract] ( 3 ) [HTML 1KB] [PDF 215KB] ( 24 )
761 Dragos Patru COVEI
A Remark on the Existence of Entire Large and Bounded Solutions to a (k1, k2)-Hessian System with Gradient Term
In this paper, we study the existence of positive entire large and bounded radial positive solutions for the following nonlinear system
???20170603-01???Sk1 (λ(D2u1))+ a1 (|x|) |∇u1|k1 = p1 (|x|) f1 (u2) for x ∈ RN,
???20170603-02???Sk2 (λ(D2u2))+ a2 (|x|) |∇u2|k2 = p2 (|x|) f2 (u1) for x ∈ RN,
Here Ski (λ(D2ui)) is the ki-Hessian operator, a1, p1, f1, a2, p2 and f2 are continuous functions.
2017 Vol. 33 (6): 761-774 [Abstract] ( 11 ) [HTML 1KB] [PDF 236KB] ( 23 )
775 Jing Hui QIU
A Revised Pre-Order Principle and Set-Valued Ekeland Variational Principles with Generalized Distances
In my former paper “A pre-order principle and set-valued Ekeland variational principle” (see [J. Math. Anal. Appl., 419, 904–937 (2014)]), we established a general pre-order principle. From the pre-order principle, we deduced most of the known set-valued Ekeland variational principles (denoted by EVPs) in set containing forms and their improvements. But the pre-order principle could not imply Khanh and Quy's EVP in [On generalized Ekeland's variational principle and equivalent formulations for set-valued mappings, J. Glob. Optim., 49, 381–396 (2011)], where the perturbation contains a weak τ -function, a certain type of generalized distances. In this paper, we give a revised version of the pre-order principle. This revised version not only implies the original pre-order principle, but also can be applied to obtain the above Khanh and Quy's EVP. In particular, we give several new set-valued EVPs, where the perturbations contain convex subsets of the ordering cone and various types of generalized distances.
2017 Vol. 33 (6): 775-792 [Abstract] ( 2 ) [HTML 1KB] [PDF 249KB] ( 13 )
793 Wen Jing CHEN, Zhong Kui LIU, Xiao Yan YANG
Singularity Categories with Respect to Ding Projective Modules
We introduce the singularity category with respect to Ding projective modules, Ddpsgb(R), as the Verdier quotient of Ding derived category DDPb(R) by triangulated subcategory Kb(DP), and give some triangle equivalences. Assume DP is precovering. We show that DDPb(R) ≌ K-,dpb(DP) and Ddpsgb(R) ≌ DDdefectb(R). We prove that each R-module is of finite Ding projective dimension if and only if Ddpsgb(R) = 0.
2017 Vol. 33 (6): 793-806 [Abstract] ( 2 ) [HTML 1KB] [PDF 254KB] ( 22 )
807 Sheng Jun FAN
Existence, Uniqueness and Approximation for Lp Solutions of Reflected BSDEs with Generators of One-sided Osgood Type
We prove several existence and uniqueness results for Lp (p > 1) solutions of reflected BSDEs with continuous barriers and generators satisfying a one-sided Osgood condition together with a general growth condition in y and a uniform continuity condition or a linear growth condition in z. A necessary and sufficient condition with respect to the growth of barrier is also explored to ensure the existence of a solution. And, we show that the solutions may be approximated by the penalization method and by some sequences of solutions of reflected BSDEs. These results are obtained due to the development of those existing ideas and methods together with the application of new ideas and techniques, and they unify and improve some known works.
2017 Vol. 33 (6): 807-838 [Abstract] ( 8 ) [HTML 1KB] [PDF 381KB] ( 10 )
839 Na Na LUAN
Chung's Law of the Iterated Logarithm for Subfractional Brownian Motion
Let XH = {XH(t), t ∈ R+} be a subfractional Brownian motion in Rd. We provide a sufficient condition for a self-similar Gaussian process to be strongly locally nondeterministic and show that XH has the property of strong local nondeterminism. Applying this property and a stochastic integral representation of XH, we establish Chung's law of the iterated logarithm for XH.
2017 Vol. 33 (6): 839-850 [Abstract] ( 2 ) [HTML 1KB] [PDF 219KB] ( 7 )
851 Jian Xin WEI
All Good (Bad) Words Consisting of 5 Blocks
Generalized Fibonacci cube Qd(f), introduced by Ilić, Klavžar and Rho, is the graph obtained from the hypercube Qd by removing all vertices that contain f as factor. A word f is good if Qd(f) is an isometric subgraph of Qd for all d ≥ 1, and bad otherwise. A non-extendable sequence of contiguous equal digits in a word μ is called a block of μ. Ilić, Klavžar and Rho shown that all the words consisting of one block are good, and all the words consisting of three blocks are bad. So a natural problem is to study the words consisting of other odd number of blocks. In the present paper, a necessary condition for a word consisting of odd number of blocks being good is given, and all the good (bad) words consisting of 5 blocks is determined.
2017 Vol. 33 (6): 851-860 [Abstract] ( 4 ) [HTML 1KB] [PDF 205KB] ( 9 )
861 Wen Peng ZHANG, Xiao Xue LI
The Fourth Power Mean of the General 2-Dimensional Kloostermann Sums Mod p
The main purpose of this paper is using the analytic methods and the properties of Gauss sums to study the computational problem of one kind fourth power mean of the general 2-dimensional Kloostermann sums mod p, and give an exact computational formula for it.
2017 Vol. 33 (6): 861-867 [Abstract] ( 9 ) [HTML 1KB] [PDF 161KB] ( 23 )
868 Yu Chao TANG, Chuan Xi ZHU, Meng WEN, Ji Gen PENG
A Splitting Primal-dual Proximity Algorithm for Solving Composite Optimization Problems
Our work considers the optimization of the sum of a non-smooth convex function and a finite family of composite convex functions, each one of which is composed of a convex function and a bounded linear operator. This type of problem is associated with many interesting challenges encountered in the image restoration and image reconstruction fields. We developed a splitting primal-dual proximity algorithm to solve this problem. Furthermore, we propose a preconditioned method, of which the iterative parameters are obtained without the need to know some particular operator norm in advance. Theoretical convergence theorems are presented. We then apply the proposed methods to solve a total variation regularization model, in which the L2 data error function is added to the L1 data error function. The main advantageous feature of this model is its capability to combine different loss functions. The numerical results obtained for computed tomography (CT) image reconstruction demonstrated the ability of the proposed algorithm to reconstruct an image with few and sparse projection views while maintaining the image quality.
2017 Vol. 33 (6): 868-886 [Abstract] ( 4 ) [HTML 1KB] [PDF 317KB] ( 30 )
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