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Acta Mathematica Sinica,Chinese Series
2017, Vol.33 Num.6
Online: 20170615
Articles
Articles
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 colorchange 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
).
2017 Vol. 33 (6): 731747 [
Abstract
] (
3
) [
HTML
1KB] [
PDF
281KB] (
15
)
748
Han Jun YU, Jun Shan SHEN, Zhao Nan LI, Xiang Zhong FANG
Semiparametric Bayesian Inference for MeanCovariance Regression Models
In this paper, we propose a Bayesian semiparametric meancovariance regression model with known covariance structures. A mixture model is used to describe the potential nonnormal 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 nonnormal cases. We also demonstrate the implementation of our method by analyzing the data set from a sleep deprivation study.
2017 Vol. 33 (6): 748760 [
Abstract
] (
3
) [
HTML
1KB] [
PDF
215KB] (
24
)
761
Dragos Patru COVEI
A Remark on the Existence of Entire Large and Bounded Solutions to a (
k
_{1}
,
k
_{2}
)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
???2017060301???
S
_{k1}
(
λ
(
D
^{2}
u
_{1}
))+
a
_{1}
(
x
) ∇
u
_{1}

^{k1}
=
p
_{1}
(
x
)
f
_{1}
(
u
_{2}
) for
x
∈ R
^{N}
,
???2017060302???
S
_{k2}
(
λ
(
D
^{2}
u
_{2}
))+
a
_{2}
(
x
) ∇
u
_{2}

^{k2}
=
p
_{2}
(
x
)
f
_{2}
(
u
_{1}
) for
x
∈ R
^{N}
,
Here
S
_{ki}
(
λ
(
D
^{2}
u
_{i}
)) is the kiHessian operator,
a
_{1}
,
p
_{1}
,
f
_{1}
,
a
_{2}
,
p
_{2}
and
f
_{2}
are continuous functions.
2017 Vol. 33 (6): 761774 [
Abstract
] (
11
) [
HTML
1KB] [
PDF
236KB] (
23
)
775
Jing Hui QIU
A Revised PreOrder Principle and SetValued Ekeland Variational Principles with Generalized Distances
In my former paper “A preorder principle and setvalued Ekeland variational principle” (see [
J. Math. Anal. Appl.
, 419, 904–937 (2014)]), we established a general preorder principle. From the preorder principle, we deduced most of the known setvalued Ekeland variational principles (denoted by EVPs) in set containing forms and their improvements. But the preorder principle could not imply Khanh and Quy's EVP in [On generalized Ekeland's variational principle and equivalent formulations for setvalued 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 preorder principle. This revised version not only implies the original preorder principle, but also can be applied to obtain the above Khanh and Quy's EVP. In particular, we give several new setvalued EVPs, where the perturbations contain convex subsets of the ordering cone and various types of generalized distances.
2017 Vol. 33 (6): 775792 [
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,
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 Rmodule is of finite Ding projective dimension if and only if
D
_{dpsg}
^{b}
(
R
) = 0.
2017 Vol. 33 (6): 793806 [
Abstract
] (
2
) [
HTML
1KB] [
PDF
254KB] (
22
)
807
Sheng Jun FAN
Existence, Uniqueness and Approximation for
L
^{p}
Solutions of Reflected BSDEs with Generators of Onesided Osgood Type
We prove several existence and uniqueness results for
L
^{p}
(
p
> 1) solutions of reflected BSDEs with continuous barriers and generators satisfying a onesided 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): 807838 [
Abstract
] (
8
) [
HTML
1KB] [
PDF
381KB] (
10
)
839
Na Na LUAN
Chung's Law of the Iterated Logarithm for Subfractional Brownian Motion
Let
X
^{H}
= {
X
^{H}
(
t
),
t
∈ R
_{+}
} be a subfractional Brownian motion in R
^{d}
. We provide a sufficient condition for a selfsimilar 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}
.
2017 Vol. 33 (6): 839850 [
Abstract
] (
2
) [
HTML
1KB] [
PDF
219KB] (
7
)
851
Jian Xin WEI
All Good (Bad) Words Consisting of 5 Blocks
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 nonextendable 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): 851860 [
Abstract
] (
4
) [
HTML
1KB] [
PDF
205KB] (
9
)
861
Wen Peng ZHANG, Xiao Xue LI
The Fourth Power Mean of the General 2Dimensional 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 2dimensional Kloostermann sums mod
p
, and give an exact computational formula for it.
2017 Vol. 33 (6): 861867 [
Abstract
] (
9
) [
HTML
1KB] [
PDF
161KB] (
23
)
868
Yu Chao TANG, Chuan Xi ZHU, Meng WEN, Ji Gen PENG
A Splitting Primaldual Proximity Algorithm for Solving Composite Optimization Problems
Our work considers the optimization of the sum of a nonsmooth 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 primaldual 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
L
2 data error function is added to the
L
1 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): 868886 [
Abstract
] (
4
) [
HTML
1KB] [
PDF
317KB] (
30
)
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