15 June 2017, Volume 33 Issue 6

    

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  A Comparison between the Metric Dimension and Zero Forcing Number of Trees and Unicyclic Graphs

  Linda EROH, Cong X. KANG, Eunjeong YI

  Acta Mathematica Sinica. 2017, 33(6): 731-747. https://doi.org/10.1007/s10114-017-4699-4

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  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 e ∈ E(T).
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  Semiparametric Bayesian Inference for Mean-Covariance Regression Models

  Han Jun YU, Jun Shan SHEN, Zhao Nan LI, Xiang Zhong FANG

  Acta Mathematica Sinica. 2017, 33(6): 748-760. https://doi.org/10.1007/s10114-016-6357-7

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  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.
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  A Remark on the Existence of Entire Large and Bounded Solutions to a (k₁, k₂)-Hessian System with Gradient Term

  Dragos Patru COVEI

  Acta Mathematica Sinica. 2017, 33(6): 761-774. https://doi.org/10.1007/s10114-017-6291-3

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  In this paper, we study the existence of positive entire large and bounded radial positive solutions for the following nonlinear system
  ???20170603-01???S_k1 (λ(D²u₁))+ a₁ (|x|) |∇u₁|^k₁ = p₁ (|x|) f₁ (u₂) for x ∈ R^N,
  ???20170603-02???S_k2 (λ(D²u₂))+ a₂ (|x|) |∇u₂|^k₂ = p₂ (|x|) f₂ (u₁) for x ∈ R^N,
  Here S_ki (λ(D²u_i)) is the ki-Hessian operator, a₁, p₁, f₁, a₂, p₂ and f₂ are continuous functions.
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  A Revised Pre-Order Principle and Set-Valued Ekeland Variational Principles with Generalized Distances

  Jing Hui QIU

  Acta Mathematica Sinica. 2017, 33(6): 775-792. https://doi.org/10.1007/s10114-017-5062-5

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  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.
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  Singularity Categories with Respect to Ding Projective Modules

  Wen Jing CHEN, Zhong Kui LIU, Xiao Yan YANG

  Acta Mathematica Sinica. 2017, 33(6): 793-806. https://doi.org/10.1007/s10114-017-6209-0

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  We introduce the singularity category with respect to Ding projective modules, D_dpsg^b(R), as the Verdier quotient of Ding derived category D_DP^b(R) by triangulated subcategory K^b(DP), and give some triangle equivalences. Assume DP is precovering. We show that D_DP^b(R) ≌ K^-,dpb(DP) and D_dpsg^b(R) ≌ D_Ddefect^b(R). We prove that each R-module is of finite Ding projective dimension if and only if D_dpsg^b(R) = 0.
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  Existence, Uniqueness and Approximation for L^p Solutions of Reflected BSDEs with Generators of One-sided Osgood Type

  Sheng Jun FAN

  Acta Mathematica Sinica. 2017, 33(6): 807-838. https://doi.org/10.1007/s10114-016-6281-x

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  We prove several existence and uniqueness results for L^p (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.
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  Chung's Law of the Iterated Logarithm for Subfractional Brownian Motion

  Na Na LUAN

  Acta Mathematica Sinica. 2017, 33(6): 839-850. https://doi.org/10.1007/s10114-016-6090-2

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  Let X^H = {X^H(t), t ∈ R₊} be a subfractional Brownian motion in R^d. We provide a sufficient condition for a self-similar Gaussian process to be strongly locally nondeterministic and show that X^H has the property of strong local nondeterminism. Applying this property and a stochastic integral representation of X^H, we establish Chung's law of the iterated logarithm for X^H.
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  All Good (Bad) Words Consisting of 5 Blocks

  Jian Xin WEI

  Acta Mathematica Sinica. 2017, 33(6): 851-860. https://doi.org/10.1007/s10114-017-6134-2

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  Generalized Fibonacci cube Q_d(f), introduced by Ilić, Klavžar and Rho, is the graph obtained from the hypercube Q_d by removing all vertices that contain f as factor. A word f is good if Q_d(f) is an isometric subgraph of Q_d 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.
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  The Fourth Power Mean of the General 2-Dimensional Kloostermann Sums Mod p

  Wen Peng ZHANG, Xiao Xue LI

  Acta Mathematica Sinica. 2017, 33(6): 861-867. https://doi.org/10.1007/s10114-016-6347-9

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  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.
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  A Splitting Primal-dual Proximity Algorithm for Solving Composite Optimization Problems

  Yu Chao TANG, Chuan Xi ZHU, Meng WEN, Ji Gen PENG

  Acta Mathematica Sinica. 2017, 33(6): 868-886. https://doi.org/10.1007/s10114-016-5625-x

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  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.